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2G Second Generation 3G 3${}^{\text{rd}}$ Generation 3GPP 3rd Generation Partnership Project 4G 4${}^{\text{th}}$ Generation 5G 5${}^{\text{th}}$ Generation 5GPPP 5G Infrastructure Public Private Partnership QAM quadrature amplitude modulation ADAS Advanced driver assistance system AD autonomous driving AI artificial intelligence AoA angle of arrival AoD angle of departure API application programming interface AR autoregressive ARQ automatic repeat request BER bit error rate BLER block error rate BPC Binary Power Control BPSK Binary Phase-Shift Keying BRA Balanced Random Allocation BS base station CAM cooperative awareness messages CAP Combinatorial Allocation Problem CAPEX capital expenditure CBF coordinated beamforming CBR congestion busy ratio CDD cyclic delay diversity CDF cumulative distribution function CDL clustered delay line CS Coordinated Scheduling C-ITS cooperative intelligent transportation system CSI channel state information CSIT channel state information at the transmitter D2D device-to-device DCA Dynamic Channel Allocation DCI downlink control information DE Differential Evolution DENM decentralized environmental notification messages DFO Doppler frequency offset DFT Discrete Fourier Transform DIST Distance DL downlink DMA Double Moving Average DMRS Demodulation Reference Signal D2DM D2D Mode DMS D2D Mode Selection DMRS demodulation reference symbol DPC Dirty paper coding DPS Dynamic point switching DRA Dynamic resource assignment DSA Dynamic spectrum access eMBB enhanced mobile broadband eV2X Enhanced vehicle-to-everything EIRP equivalent isotropically radiated power ERTMS European Rail Traffic Management System ETSI European Telecommunications Standards Institute FDD frequency division duplexing FR1 frequency range-1 FR2 frequency range-2 GNSS global navigation satellite system HARQ hybrid automatic repeat request HST high-speed train IAB integrated access and backhaul ITS intelligent transportation system KPI key performance indicator IEEE Institute of Electronics and Electrical Engineers IMT International Mobile Telecommunications IMU inertial measurement unit InC in-coverage IoT Internet of Things ITS intelligent transportation system LDPC low-density parity-check coding LMR land mobile radio LoS line-of-sight LTE Long Term Evolution MAC medium access control mmWave millimeter-wave MBB mobile broadband MCS modulation and coding scheme METIS Mobile Enablers for the Twenty-Twenty Information Society MIMO multiple-input multiple-output MISO multiple-input single-output ML machine learning MRC maximum ratio combining MS mode selection MSE mean square error MTC machine type communications multi-TRP multiple transmission and reception points mMTC massive machine type communications cMTC critical machine type communications NDAF Network Data Analytics Function NF network function NR New Radio NSPS national security and public safety NWC network coding OEM original equipment manufacturer OFDM orthogonal frequency division multiplexing OoC out-of-coverage PSBCH physical sidelink broadcast channel PSFCH physical sidelink feedback channel PSCCH physical sidelink control channel PSSCH physical sidelink shared channel PDCCH physical downlink control channel PDCP packet data convergence protocol PHY physical PLNC physical layer network coding PPPP proximity services per packet priority PPPR proximity services per packet reliability PSD power spectral density RLC radio link control QAM quadrature amplitude modulation QCL quasi co-location QoS quality of service QPSK quadrature phase shift keying PaC partial coverage RAISES Reallocation-based Assignment for Improved Spectral Efficiency and Satisfaction RAN radio access network RA Resource Allocation RAT Radio Access Technology RB resource block RF radio frequency RS reference signal RSRP Reference Signal Received Power SA scheduling assignment SFN Single frequency network SNR signal-to-noise ratio SINR signal-to-interference-plus-noise ratio SC-FDM single carrier frequency division modulation SFBC space-frequency block coding SCI sidelink control information SL sidelink SLAM simultaneous localization and mapping SPS semi-persistent scheduling STC space-time coding SW software TCI transmission configuration indication TBS transmission block size TDD time division duplexing TRP transmission and reception point TTI transmission time interval UAV unmanned aerial vehicle UAM urban air mobility UE user equipment UL uplink URLLC ultra-reliable and low latency communications VUE vehicular user equipment V2I vehicle-to-infrastructure V2N vehicle-to-network V2X vehicle-to-everything V2V vehicle-to-vehicle V2P vehicle-to-pedestrian ZF Zero-Forcing ZMCSCG Zero Mean Circularly Symmetric Complex Gaussian TBS transport block size SCI sidelink control information # 5G New Radio for Automotive, Rail, and Air Transport Gábor Fodor⋆‡ Julia Vinogradova† Peter Hammarberg⋆ Keerthi Kumar Nagalapur⋆ Zhiqiang (Tyler) Qi♭ Hieu Do⋆ Ricardo Blasco† Mirza Uzair Baig⋆ ⋆Ericsson Research Sweden E-mail<EMAIL_ADDRESS> ‡KTH Royal Institute of Technology Sweden. E-mail<EMAIL_ADDRESS> †Ericsson Research Finland Email: <EMAIL_ADDRESS> ♭Ericsson Research China Email<EMAIL_ADDRESS> ###### Abstract The recent and upcoming releases of the 3rd Generation Partnership Project’s 5G New Radio (NR) specifications include features that are motivated by providing connectivity services to a broad set of verticals, including the automotive, rail, and air transport industries. Currently, several radio access network features are being further enhanced or newly introduced in NR to improve 5G’s capability to provide fast, reliable, and non-limiting connectivity for transport applications. In this article, we review the most important characteristics and requirements of a wide range of services that are driven by the desire to help the transport sector to become more sustainable, economically viable, safe, and secure. These requirements will be supported by the evolving and entirely new features of 5G NR systems, including accurate positioning, reference signal design to enable multi- transmission and reception points, service-specific scheduling configuration, and service quality prediction. Keywords: 5G networks, automotive services, high-speed train, urban air mobility, positioning, QoS prediction. ## I Introduction Recent advances in wireless communications, real-time control, sensing, and battery technologies, collaborative spectrum management and sharing, and artificial intelligence are enabling the transport sector to become more cost efficient, secure, and sustainable [1]. Due to new requirements arising in road, railway, air and maritime transport, cellular connectivity, and reliable wireless communications between vehicles and road users are no longer a "nice to have", but are essential parts of cooperative intelligent transportation systems and smart cities [2]. Ericsson predicts that the number of connected cars in operation will rise to more than 500 million in 2025, and the railway sector is making the first steps to digitalize the European Rail Traffic Management System (ERTMS), which includes mission-critical control systems for train operations, including high-speed trains. The unmanned aerial vehicle (UAV) and urban air mobility (UAM) (drone) market is expected to grow from the current estimated USD $4.4$bn to $63.6$bn by 2025 [3]. Apart from smart city applications, there is a growing interest in employing connected UAVs in surface mining, seaports, oil and gas, and other large industrial facilities or in public safety situations in order to improve and optimize industrial processes, enhance operational efficiencies, and create resilience. The digitalization and increasing connectivity in the transport sector are driven by three key factors. First, there are increasing demands imposed virtually by all stakeholders – including passengers, cargo companies, vehicle (car, truck, locomotive, ship) manufacturers, public transport and rail operators, and infrastructure (road, rail, harbor) providers. This broad set of requirements includes being always connected to the Internet and enterprise networks, enjoying safe and secure journeys in urban and rural environments, and minimizing environmental impacts. At the same time, reducing capital and operational expenditures necessitates increasing digitalization, automation, and always-on connectivity, since these technologies make manufacturing, maintaining and operating transport equipment, infrastructure, and services much more efficient. Thirdly, the rapid deployment of 5G networks, and the recent advances in 6G research provide a technology push towards digitalized and connected transport services [4]. In parallel with the above trends in the transport industry, the release 15 (Rel-15) of the 3rd Generation Partnership Project (3GPP) specifications in 2016 marked the birth of the new cellular radio interface for the fifth generation (5G) systems, commonly referred to as New Radio (NR). Although mobile broadband (MBB) services continue to be the main driver for NR, this new radio technology generation inherently has much stronger support for verticals such as the transport industry, as compared to Long Term Evolution (LTE). Additionally, already in Rel-16, new technical features are introduced specifically for supporting critical machine-type communications including ultra-reliable and low latency communications (URLLC) vehicle-to-everything (V2X) services for automotive. Further enhancements targeting special connectivity requirements of the rail operations and remote control of UAVs are also being standardized in the upcoming releases. Compared with 4G systems, 5G NR adopts a new design philosophy and novel technology components, including flexible numerology and waveform design for lower and millimeter-wave frequency bands, minimizing control signaling overhead, multi-hop support by integrated access and backhaul relay, enhanced positioning, and quality of service (QoS) handling mechanisms. Also, further enhanced multiple-input multiple-output (MIMO) techniques enable 5G networks to acquire accurate channel state information (CSI) for both analog and hybrid beamforming and spatial multiplexing applications, which are important for maintaining high spectral efficiency even in high-speed road and rail transport scenarios [5]. Finally, recent 3GPP releases of 5G radio access networks pave the way for advanced radio-based positioning techniques that efficiently complement and make positioning more precise than pure satellite- based positioning techniques [6], which are highly useful for automotive and drone use cases. The present paper serves two purposes. Firstly, we summarize the technical foundations of 5G NR which can fulfill basic requirements imposed by emerging use cases in the transport sector. Secondly, based on an in-depth review of connectivity requirements of transport use cases, we highlight several important new technology enablers which will play a key role in meeting the most stringent requirements. In particular, we focus on the following, * • Positioning techniques that take advantage of combining onboard sensors and cellular network measurements; * • Reference signal design and selecting the appropriate multi- transmission and reception point (TRP) option for spectrum-efficient operations of HSTs and other high-speed user equipments; * • Service-specific scheduling techniques for V2X communications that ensure high resource utilization and service differentiation between low-latency and delay tolerant (lower than best effort) traffic types; * • Novel QoS-prediction techniques that are useful in driverless and driver- assisted road, rail, and drone transport use cases. ## II Technical Foundations of 5G NR and the Initial NR Evolution Targeting the Transport Vertical Figure 1: Key areas and relevant transport use cases in NR Rel-15, Rel-16, and Rel-17. ### II-A Major Features in Rel-15 and Rel-16 As mentioned, Rel-15 and Rel-16 of the 3GPP specifications have been largely driven by requirements of MBB services, including requirements on enhanced data rates, latency, coverage, capacity, and reliability. However, starting already in Rel-15 and continuing in the subsequent 3GPP releases, NR enables new use cases by meeting the requirements imposed by transport use cases, such as connected cars, high-speed trains, and UAVs. While Rel-15 focused on supporting MBB and URLLC applications, Rel-16/17 includes UE power savings, operation in unlicensed spectrum, industrial Internet of Things (IoT) enhancements as well as special radio access network (RAN) features such as physical layer support for unicasting sidelink (device-to-device) for advanced V2X services [7]. A key distinguishing feature of 5G NR from fourth generation (4G) systems is the substantial expansion of the frequency bands, in which NR can be deployed. For transport applications, the following NR-specific features are particularly important (see Figure 1): * • Symmetric physical layer design with orthogonal frequency division multiplexing (OFDM) waveform for all link types, including uplink, downlink, sidelink, and backhaul; * • Wide range of carrier frequencies, bandwidths, and deployment options. 3GPP aims to develop and specify components and physical layer numerology that can operate in frequencies up to 100 GHz. This implies several options for OFDM subcarrier spacing ranging from 15 kHz up to 240 kHz with a proportional change in cyclic prefix duration; * • Native support for dynamic time division duplexing (TDD) as a key technology component. In dynamic TDD, parts of a slot can be adaptively allocated to either uplink or downlink, depending on the prevailing traffic demands; * • Support for massive MIMO, that is a massive number of steerable antenna elements for both transmission and reception, utilizing channel reciprocity in TDD deployments and a flexible CSI acquisition framework. The NR channels and signals, including those used for data transmission, control signaling and synchronization are all designed for optional beamforming. In addition to flexible numerology, native support for dynamic TDD and advanced MIMO features, NR is designed using the principle of ultra-lean design, which aims at minimizing control plane and synchronization signal transmissions when data transmissions are idle. Inherent support for distributed MIMO, also referred to as multi-TRP, is introduced and fully supported in Rel-16. This feature is largely motivated by HST applications, since it allows UEs to receive multiple control and data channels per slot, which enables simultaneous data transmissions from multiple physically separated base stations. ### II-B Major Developments in Rel-17 Looking beyond Rel-16, the NR evolution will be driven by industry verticals, including a variety of transport use cases, such as V2X communications, high- speed trains, UAVs and passenger aircrafts, and maritime communications. These use cases justify new features discussed and planned for Rel-17. MIMO enhancements are expected to support multi-TRP specific tracking reference signals, single frequency network deployments, and non-coherent joint transmissions by multiple base stations, which are particularly useful for providing connectivity to high-speed trains. Furthermore, Rel-17 is studying the integration of non-terrestrial and terrestrial networks in order to support use cases for which terrestrial networks alone cannot provide the required coverage and capacity, including maritime, UAV, and UAM scenarios. ## III Overview of Intelligent Transportation Systems Services and Requirements Figure 2: Use case categories in the automotive and road transport (upper), railway (middle), and UAV and UAM (lower) segments. Figure 2 classifies the automotive, rail, and UAV/UAM use cases in use case categories, together with the key connectivity requirements per category. Regulated C-ITSs provide international or governmental regulated services for road, rail and drone traffic efficiency, sustainability, and safety. Traffic efficiency use cases have relaxed latency requirements, while safety-related data often requires URLLC. A benefit of regulation is to facilitate original equipment manufacturer (OEM) cooperation in standardized information exchange. C-ITS services may also use dedicated spectrum in certain regions; for example, for direct short-range communication using the 3GPP sidelink technology. For rail transport, C-ITS implies station dwell time control and speed/break control to optimize rail network utilization while ensuring safety. Advanced driver assistance systems and autonomous driving (AD) are increasingly employed for both road and rail transport. In Europe, for example, the next generation of the ERTMS will support well-defined levels of automation, including semi-automated (assisted) driving, driverless and unattended train operation. Similarly, advanced pilot assistance systems for UAM and passenger aircrafts are envisioned by various stake-holders of the air transport industry. For this set of applications, URLLC communication and high-accuracy positioning play crucial roles. Fleet management including remote assistance of driverless vehicles is an important application for road, rail, and UAV-based transport. This type of services aim at vehicle fleet owners such as logistics or car-sharing companies. The communication service is primarily used to monitor vehicle locations and the vehicle/driver status. With increasing level of automation in the rail industry and for UAVs, or for a fleet of driverless trucks, fleet management also includes communication support for operations monitoring and remote assistance from a control tower. Convenience and infotainment, based on MBB services for drivers, crew, and passengers are equally important in road, rail, and future UAM transport use cases. Such services deliver content such as traffic news and audio entertainment for car drivers, and gaming and video entertainment for passengers. One specific example is the concept of "Gigabit train" services, which motivate the adoption of HST scenarios in 3GPP. For this set of use cases, the most important requirement is high data rate and low latency connections, which rely heavily on the capability of tracking wireless channels at high vehicle speed. The primary focus in the logistics and connected goods category is on the tracking of transported objects (commodities, merchandise goods, cargo and so on) during the production and transport cycle of the object. Near real time tracking and status monitoring of goods are attractive for cargo companies, customers, and freight train operators. In the vehicle-as-a-sensor use case category, sensors installed in the vehicle sense the environment and can also provide anonymized data to 3rd parties. In road transport, for example, the vehicle-mounted sensors provide information for solutions such as ADAS or AD as well as for monitoring city infrastructure and road status. For rails, railway track monitoring and anomaly detection are supported by various sensors mounted on the train, effectively operating the train as a sensor. Similarly, drones can be equipped with a lot of sensors that help collect data and perform distributed or federated computation for various purposes such as forecasting cloud formation, rain, and other hazardous weather conditions. Just as with the convenience and infotaiment category, the vehicle-as-a-sensor requires high data-rate connections between vehicles or between vehicles and the cellular network at high vehicle speed and dynamic interference conditions. Telematics applications for vehicles include collecting vehicle diagnostics to monitor/adjust the vehicle, while rail telematics rail applications allow continuous status updates from trains to determine state, delay, cargo conditions, software (SW) updates, and geo-fencing. In this category, several applications (e.g. SW updates) tolerate delay, while others are more delay critical. Similarly, for air transport, telematics serve as a tool for collecting air vehicles diagnostics to monitor/adjust the vehicles. To make sense of the vast amount of data collected from vehicles in this group of use cases, the role of artificial intelligence (AI)/ machine learning (ML) is utterly important. In the reverse direction, AI/ML can also play a meaningful role in determining when to perform a certain task to the vehicle in an efficient manner. For example, ML-based spare capacity prediction, which is part of the so-called cellular network QoS prediction, can be used to predict the most economical time for SW update for a set of vehicles [8, 9]. Connected road infrastructure services are operated by cities and road authorities to monitor the state of the traffic and control its flow, such as physical traffic guidance systems, parking management and dynamic traffic signs. For railways, the communication between the rail infrastructure and the locomotive via specific transmission modules and eurobalises is used to send information from line-side electric units to the trains e.g., current speed restrictions for the coming rail segment. For UAV/UAM, an Unmanned Aircraft System Traffic Management (UTM) is used for traffic control, which requires high-accuracy 3D positioning and URLLC communication with the ground control system. ## IV Proposed New Features and Solutions to Support intelligent transportation system (ITS) Requirements Despite the recent and ongoing enhancements to 5G NR, there is still a need to further improve the technology to meet the growing demands of industry verticals. In this section, we summarize the state of the standardization of several specific features and introduce new solutions which can help fulfilling the stringent connectivity requirements of the transport sectors outlined in the preceding section. These components span both radio layers (physical, medium access control) and the service layer of the protocol stack. ### IV-A Advanced Positioning Support and Algorithms With the introduction of NR, 3GPP targets improved positioning capabilities to cater for a number of new use cases, involving indoor, industrial, and automotive applications. Cooperative manoeuvring in the C-ITS category and several ADAS applications rely on accurate positioning, which must remain operational even in global navigation satellite system (GNSS)-problematic areas. Specifically in Rel-16, a set of positioning related features are introduced, which pave the way for enhanced positioning services. These new features include new and improved uplink (UL) and downlink (DL) reference signal design, allowing larger bandwidths and beamforming, assisted by additional measurements and enhanced reporting capabilities. By supporting larger bandwidths than in LTE, higher accuracy of range estimates can be achieved, and with angle of arrival (AoA) and angle of departure (AoD) measurements new positioning schemes exploiting the spatial domain can be supported. Architecture-wise, similarly to LTE, NR positioning is based on the use of a location server. The location server collects and distributes information related to positioning (UE capabilities, assistance data, measurements, position estimates, etc.) to the other entities involved in the positioning procedures (base stations and connected vehicles). A range of positioning schemes, including DL-based and UL-based ones, are used separately or in combination to meet the accuracy requirements in vehicular scenarios. Specifically, in the millimeter-wave (mmWave)-bands, referred to as frequency range-2 (FR2) bands of NR, the AoA and AoD measurements can be enhanced by using large antenna arrays, which facilitate high resolution angular measurements. By unlocking the spatial domain, NR can significantly increase the positioning accuracies for many industrial and automotive use cases [10], [11]. Additionally, to further improve the accuracy and reliability of positioning in GNSS-problematic areas, we propose to fuse acceleration measurements provided by onboard inertial measurement units with measurements on the received NR DL reference signals from multiple NR BSs. Fusing local IMU measurements with measurements on multiple NR DL signals can reach decimeter accuracy in favourable deployments. Furthermore, high accuracy spatial and temporal measurements facilitate the use of advanced positioning schemes such as simultaneous localization and mapping (SLAM), which utilizes consecutive measurements to build a statistical model of the environment, achieving high accuracy even in extreme scenarios by utilizing measurements on DL signals of only a single base station. ### IV-B Further Enhanced MIMO to Support Multiple Transmission and Reception Points HST wireless communication is characterized by a highly time-varying channel and rapid changes of the closest TRPs to the train, resulting from the extreme high velocities. Recognizing these challenges, NR has been designed to support high mobility from day one, and includes features to enable communications with HSTs [12]. Furthermore, several multi-TRP deployment options and features developed under the general MIMO framework can be exploited to support HST communications for the "Gigabit train", while minimizing the need for handovers. We expect this technology component will also play a very important role in UAV/UAM use cases. The multi-TRP options that are the most relevant to HST communications are: * • Dynamic point switching (DPS): Data signals are transmitted from a single TRP at a given time, and the TRP used for transmission is dynamically selected based on the relative quality of channels between the train and a few closest TRPs; * • Single frequency network (SFN): All the TRPs in the SFN area transmit the same data and reference signals to the train; * • SFN with TRP-specific reference signals: The same data signal is transmitted from different TRPs, while some of the reference signals are transmitted in a TRP-specific manner. The first and the third options rely on TRP-specific reference signals, whereas the second option uses common reference signal across the TRPs in the coverage area of the SFN. In addition to supporting TRP-specific reference signals, NR supports associating different reference signals with different channel properties, such as Doppler shift and delay spread, through NR’s quasi co-location (QCL) and transmission configuration indication framework. In Rel-17, different QCL enhancements are investigated to better support advanced channel estimation schemes that can be implemented at the train and to evaluate the need for TRP-side pre-compensation algorithms. In Section V, we evaluate the necessity and performance of the multi-TRP options described above through link level evaluations. Furthermore, beam management enhancements necessary to support HST communications in the higher bands of NR are also investigated in Rel-17. ### IV-C Service-Specific Scheduling To accommodate MBB, URLLC, and machine-type communications services, NR networks employ scheduling algorithms that take into account the current service mix, prevailing channel conditions, traffic load, available carriers, and other factors. Scheduling in multiservice wireless networks has been researched for decades and a vast literature as well as practically deployed scheduling schemes exist. Interestingly, due to requirements imposed by the coexistence of URLLC and delay tolerant services, mmWave communications support in the FR2 bands and serving very high speed UE-s have stimulated renewed research interest in this topic [13]. Some recent works propose scheduling strategies to minimize end-to-end delay for time-critical services [14], or to optimize resource allocation for the coexistence of various services [13]. In NR, service-specific scheduling can also be configured to take into account the specific opportunities that are present in certain deployment scenarios. We propose to customize the scheduler in certain deployment scenarios, which can be illustrated by a scheduling configuration that is suitable for non-time-critical services. This can be applicable for background data transfer for SW updates or uploading sensor measurements in the vehicle-as-a-sensor category. This scheduling mechanism divides vehicles into a high and a low path-gain group. Vehicles belonging to the high path-gain group are eligible for medium access, while scheduling vehicles in the low path-gain group is postponed (dropped) until their path gain improves. The scheme is suitable for highly mobile users (including automotive or urban train use cases) in high-way, urban or suburban scenarios, and can be activated or de-activated based on velocity or other sensor measurements that help to configure the path gain threshold. ### IV-D QoS Prediction The NR QoS framework together with features like URLLC are successful in delivering a minimum guaranteed performance, especially in controlled scenarios. However, highly mobile UEs usually experience time-variant network performance, partly because the actual QoS often exceeds the minimum or guaranteed level, and partly because the system is occasionally not able to fulfill a QoS provision. Interestingly, in many cases, including certain C-ITS, ADAS or telematics applications, these performance fluctuations are not a problem if they can be predicted in advance. Having access to real-time QoS predictions has generated a large interest from the automotive industry [8], as it would allow service providers, mobile network users, and automotive applications to dynamically adapt their behaviors to the prevailing or imminent QoS level. This would allow for enabling services relying on continuous guaranteed performance as well as for exploiting spare capacity for large bulk data transfers in lower than best effort services. Despite the high expectations, QoS prediction is largely an open research topic. The realistically achievable performance and the applicability of this type of algorithms are still unknown. To a large extent, QoS prediction is seen as a ML application with a broad data set consisting of network measurements, device measurements, and application data [9]. In practice, different types of information may be collected with different periodicities, time horizons, resolutions, and accuracies, depending on practical and business-related constraints. Understanding the relevance of each of them is ongoing work and will be instrumental in determining the relative merits and tradeoffs of the different architecture options, which range from network-centric to application-based. As a first step towards supporting predictive QoS in mobile networks, 3GPP enhanced the NR system architecture in Rel-16 to support services providing network data analytics in the 5G core network. To this end, application programming interfaces for exposing network-based predictions were defined, the necessary procedures for collecting the relevant data for the analytics from different network functions as well as from the operations administration and management functionality. In addition, procedures providing analytics (e.g., load, network performance, data congestion, QoS sustainability and UE related analytics) to other network functions were introduced. As usual, the algorithms used to obtain the network analytics are not defined in the specification, and as said, they are a topic of ongoing research. ## V Numerical Examples ### V-A Positioning Figure 3: CDF of the obtained positioning accuracy when using NR only and fused IMU+NR positioning at 5 and 15 dB SNR. We consider a highway scenario with BSs equally spaced and placed at the side of the road. A moving vehicle following a snake-like trajectory is equipped with an onboard IMU sensor and an NR UE. A MIMO system is considered with square antenna arrays at the BS and the UE. A line-of-sight (LoS) downlink propagation is assumed with a grid of Discrete Fourier Transform beams transmitted by the BS. A sensor fusion-based positioning approach is proposed (similar to the one in [11]), for which a Kalman filter is used to fuse the measurements obtained from the IMU and the NR downlink such as the range and the angles-of-arrival. An extension to this is proposed allowing to fuse measurements from multiple BS. Positioning accuracy in terms of positioning cumulative distribution functions are compared in Figure 3 for NR-based method and for the sensor fusion-based method combining IMU with NR. It is assumed that the BSs operate at the millimeter wave frequency of 28 GHz with 256 antennas. There are 4 antennas at the UE, the UE speed is equal to 130 km/h, and signal-to-noise ratio (SNR) is equal to 5 and 15 dB. The BSs are placed at 40 m from the road with the inter- site distance equal to 200 m. The results are averaged over a total distance of 10 km. The number of BSs used to fuse the measurements from is denoted by NbFusedBS. The simulation results show that a large performance gain is obtained for the sensor fusion-based method as compared to the NR only-based method, especially at low SNR. Notice that fusion-based methods allow to achieve a decimeter level accuracy with greater than 90% probability, even at low SNR of 5dB. ### V-B Further Enhanced MIMO to Support Multiple Transmission and Reception Points Figure 4: High-speed train multi-TRP scenario with four base stations (upper) and downlink throughput as a function of train position when using SFN without/with CDD, DPS and SFN with precompensation. (The $x,y$ and $z$ axes are scaled by 10, 10 and 7 respectively for better visualization.) For evaluations, a four-TRP deployment at 2 GHz carrier frequency using 20 MHz bandwidth (50 resource blocks) with TRP hight of 35m, 30 kHz subcarrier spacing, and train speed of 500 km/h is assumed, as illustrated in Figure 4 (upper) is used. The TRP antenna orientation is set to 10 degrees downtilt with an antenna gain of 20.5 dBi. The channel between each TRP and the reception point at the train is modeled using an extended clustered delay line (CDL) channel model. The SNR at train position $D1=0$ m is 16 dB in the SFN deployment, and an hybrid automatic repeat request (HARQ) scheme with a maximum number of 3 retransmissions is employed, using a fixed modulation and coding scheme (MCS) with 64- quadrature amplitude modulation (QAM), low- density parity- check coding with code-rate = 0.428. Figure 4 (lower) shows the throughput as a function of distance in the different deployment options. In the baseline SFN transmission scheme, the throughput for UE locations, close to midpoint of two TRPs does not reach the peak throughput of the modulation and coding scheme used. This is due to the fact that the equivalent channel formed by the combination of the two dominant CDL channels with LoS components having equal and opposite Doppler shifts results in a less frequency selective channel with deep fades across some of the OFDM symbols in a slot. This channel behavior can be modified by adding a TRP-specific cyclic delay diversity which converts the effective channel close to midpoint between TRPs to a frequency selective channel without deep fades across OFDM symbols. The throughput improvement with TRP-specific cyclic delay diversity is shown in the figure. A precompensation scheme where a TRP-specific Doppler frequency offset (DFO) compensation is performed also improves the throughput close to the midpoint, as seen in the figure. However, this scheme requires TRP- specific reference signals in order for the train to estimate the Doppler shifts and additional signaling in the uplink direction to feedback the estimates. The figure also shows the performance of the DPS scheme with genie selection of the TRPs, where a single TRP closest to the train is used for data transmission. In case of DPS, the received SNR at a train position is smaller than in the case of SFN transmissions due to transmission from a single TRP. However, if sufficient SNR can be guaranteed using proper deployment, DPS achieves peak throughput of the MCS used at all train positions. The evaluation results show that a complex scheme, such as SFN with DFO precompensation, does not perform significantly better than the other alternatives. Also, a low-complexity and UE transparent scheme such as SFN with TRP-specific CDD, which can be readily employed and scaled to serve a large number of UEs, performs well by suitably altering the effective channel. ### V-C Service-Specific Scheduling Figure 5: Downlink mean user throughput as a function of the traffic density when employing different drop rates in an automotive highway scenario, in which the base station sites are deployed with 1732m inter-site distance. To illustrate the impact of customizing the scheduler to operate in specific deployment scenarios, we consider an automotive high-way scenario, in which the BS sites are deployed according to the 3GPP recommendation in [15]. Figure 5 shows that dropping low path-gain users improves the mean user throughput, especially in case of high traffic density. Specifically, when all users are scheduled simultaneously (marked as 0% Drop) the mean user throughput drops to close 0 when the traffic is around or higher than 1000 Mbps/km2. When the scheduler is configured to distinguish low and high path- gain vehicles and postpones scheduling vehicles that are momentarily have low path-gain, the average throughput significantly increases. Dropping 50% of the low path-gain users (Drop 50% line), for example, improves the mean user throughput when traffic density lies between 1000-3000 Mbps/ km2. This simple example illustrates that adjusting the drop ratio (by adjusting the threshold between low and high path gain vehicles and setting the drop ratio in the low path-gain group) – according to the deployment parameters and prevailing traffic density – can optimize the system spectral efficiency. The basic rationale for this is that for non-latency-critical services, the user data transmission can wait until the user is moving into good coverage area, while users in poor coverage area can be dropped to improve both system resource utilization and spectral efficiency. As an example, consider the highway scenario, in which vehicles drive at 140km/h. To guarantee 95% MBB-like service coverage (10 Mbps data rate for DL and 2 Mbps for UL), the DL capacity for the non-dropping case is 450 Mbps/km2 (760 Mbps/km2 for UL). It takes 50s (250s for UL) to transmit a 500M file in the DL. For the drop 50% case, for the same traffic density (450 Mbps/km2 for DL and 760 Mbps/km2 for UL), the transmission time is greatly reduced (19s for DL and 31s for UL). Although the Drop 50% case has about 50% coverage hole, this coverage hole lasts until the vehicles drive by a next BS, improve their path loss, and complete the ongoing file transmission. ### V-D QoS Prediction Figure 6: Predictive QoS: Linking network-level QoS level and the application layer (upper), Network-based prediction function utilizing ML (middle) and DL throughput prediction (lower). The most fundamental question of predictive QoS is, perhaps, the accuracy of the predictions. We have studied multiple alternatives to predict DL throughput in different time horizons. Figure 6 summarizes our findings in terms of the CDF for different prediction horizons of the difference between predicted and delivered number bits $B$ over a given time interval $\Delta t$: $\displaystyle e^{\prime}(t,\Delta t)$ $\displaystyle\triangleq\frac{\Big{\|}B_{\text{delivered}}(t)-B_{\text{predicted}}(t)\Big{\|}}{\Delta t}.$ Our results show that both short-term and long-term predictions are quite accurate in most cases. In the former case, this is due to the ability to predict short term channel quality fluctuations, while other system variables that are harder to predict (e.g., interference level, instantaneous cell load) are relatively stable. In the latter case, the longer interval averages out the instantaneous variations of channel quality and other short-term effects. In contrast, DL throughput prediction in the intermediate regime is much more challenging. In this case, rapid channel fluctuations are often not well predicted, while the averaging effect is still weak. How to bridge the gap between short- and long-term predictions is still an open question. ## VI Concluding Remarks and Outlook 5G NR was designed to enable various use cases, reach a broad range of aggressive performance targets and be deployed in both traditional and mmWave frequency bands. The initial release (Rel-15) of NR included support for flexible numerology, latency-optimized frame structure, massive MIMO, interworking between low and high frequency bands and dynamic TDD. The new features of NR in subsequent releases include enhancements for MIMO, V2X, high-speed UE, and URLLC services, more accurate positioning, and support for non-terrestrial and mission-critical communications. These new standardized features, together with proprietary and algorithmic solutions facilitate connected and intelligent transport services, including automotive, rail, air transport and public safety services. Connected and intelligent transport systems will continue to rely on ubiquitous broadband connectivity as expectations by the automotive, rail and air transport and public safety stakeholders evolve, and new business models emerge. The contours of future 6G systems are already emerging, as the standardization and research communities and end-users in the transport sector define future requirements and solutions. Based on these discussions, 6G technology candidates include both further enhancements of existing features and entirely new features. 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[ERTMS]: European Rail Traffic Management System [UAV]: unmanned aerial vehicle [UAM]: urban air mobility [UAVs]: unmanned aerial vehicle [3GPP]: 3rd Generation Partnership Project [NR]: New Radio [MBB]: mobile broadband [LTE]: Long Term Evolution [URLLC]: ultra-reliable and low latency communications [V2X]: vehicle-to-everything [QoS]: quality of service [MIMO]: multiple-input multiple-output [CSI]: channel state information [TRP]: transmission and reception point [HSTs]: high-speed train [IoT]: Internet of Things [RAN]: radio access network [OFDM]: orthogonal frequency division multiplexing [TDD]: time division duplexing [HST]: high-speed train [UEs]: user equipment [C-ITSs]: cooperative intelligent transportation system [OEM]: original equipment manufacturer [C-ITS]: cooperative intelligent transportation system [AD]: autonomous driving [ADAS]: Advanced driver assistance system [SW]: software [AI]: artificial intelligence [ML]: machine learning [ITS]: intelligent transportation system [GNSS]: global navigation satellite system [UL]: uplink [DL]: downlink [AoA]: angle of arrival [AoD]: angle of departure [mmWave]: millimeter-wave [FR2]: frequency range-2 [BSs]: base station [IMU]: inertial measurement unit [SLAM]: simultaneous localization and mapping [TRPs]: transmission and reception point [DPS]: Dynamic point switching [SFN]: Single frequency network [QCL]: quasi co-location [UE]: user equipment [BS]: base station [LoS]: line-of-sight [SNR]: signal-to-noise ratio [CDD]: cyclic delay diversity [CDL]: clustered delay line [HARQ]: hybrid automatic repeat request [MCS]: modulation and coding scheme [QAM]: quadrature amplitude modulation [DFO]: Doppler frequency offset *[CDF]: cumulative distribution function
# Efficient MPI-based Communication for GPU-Accelerated Dask Applications ††thanks: This research is supported in part by NSF grants #1818253, #1854828, #1931537, #2007991, #2018627, and XRAC grant #NCR-130002. Aamir Shafi, Jahanzeb Maqbool Hashmi, Hari Subramoni and Dhabaleswar K. (DK) Panda The Ohio State University {shafi.16, hashmi.29, subramoni.1<EMAIL_ADDRESS> ###### Abstract Dask is a popular parallel and distributed computing framework, which rivals Apache Spark to enable task-based scalable processing of big data. The Dask Distributed library forms the basis of this computing engine and provides support for adding new communication devices. It currently has two communication devices: one for TCP and the other for high-speed networks using UCX-Py—a Cython wrapper to UCX. This paper presents the design and implementation of a new communication backend for Dask—called MPI4Dask—that is targeted for modern HPC clusters built with GPUs. MPI4Dask exploits mpi4py over MVAPICH2-GDR, which is a GPU-aware implementation of the Message Passing Interface (MPI) standard. MPI4Dask provides point-to-point asynchronous I/O communication coroutines, which are non-blocking concurrent operations defined using the async/await keywords from the Python’s asyncio framework. Our latency and throughput comparisons suggest that MPI4Dask outperforms UCX by $6\times$ for 1 Byte message and $4\times$ for large messages (2 MBytes and beyond) respectively. We also conduct comparative performance evaluation of MPI4Dask with UCX using two benchmark applications: 1) sum of cuPy array with its transpose, and 2) cuDF merge. MPI4Dask speeds up the overall execution time of the two applications by an average of $3.47\times$ and $3.11\times$ respectively on an in-house cluster built with NVIDIA Tesla V100 GPUs for $1-6$ Dask workers. We also perform scalability analysis of MPI4Dask against UCX for these applications on TACC’s Frontera (GPU) system with upto $32$ Dask workers on $32$ NVIDIA Quadro RTX 5000 GPUs and $256$ CPU cores. MPI4Dask speeds up the execution time for cuPy and cuDF applications by an average of $1.71\times$ and $2.91\times$ respectively for $1-32$ Dask workers on the Frontera (GPU) system. ###### Index Terms: Python, Dask, MPI, MVAPICH2-GDR, Coroutines ## I Introduction With the end of Moore’s Law [1, 2] in sight, the performance advances in the computer industry are likely to be driven by the “Top”—1) hardware architecture, 2) software, and 3) algorithms—as noted by Leiserson recently [3]. This is in stark comparison to the vision “There’s Plenty of Room at the Bottom” laid out by Feynman [4] in 1959 referring to semiconductor physics and fabrication technology. Leiserson [3] argues that the post-Moore generation of software will focus on reducing software engineering bloat as well as exploiting parallelism, locality, and heterogeneous hardware. A representative of the “Top” is the Python programming language, which is a clear winner on the landscape of data science. Python is a classic example of a language benefiting from Moore’s law by traditionally focusing on programmer productivity and reduced development time. In the context of data science applications, Python’s popularity is due to rich set of free and open-source libraries that enable the end-to-end data processing pipeline. These libraries/packages include: core (SciPy), data preparation (NumPy, Pandas), data visualization (Matlibplot), machine learning (Scikit-learn), and deep learning (PyTorch, TensorFlow). However in the post-Moore era, it is vital that Python is able to support parallel and distributed computing as well as exploit emerging architectures especially accelerators. Two popular big data computing frameworks that enable high-performance data science in Python include Dask [5] and Apache Spark [6]. This paper focuses on Dask, which provides support for natively extending popular data processing libraries like numPy and Pandas. This allows incremental development of user applications and lesser modifications to legacy codes when executed with Dask. Traditionally Dask has mainly supported execution on hosts (CPUs) only, which means that it was not able to exploit massive parallelism offered by Graphical Processing Units (GPUs). However this has recently changed as part of the development of the NVIDIA RAPIDS library, which aims to enable parallel and distributed computation on clusters of GPUs. RAPIDS adopted a similar approach—as taken by Dask—of extending already existing Python data processing libraries for the GPU ecosystem. For instance, RAPIDS support processing of distributed data stored in cuPy (numPy-like) and cuDF (Pandas-like) formats. An overarching goal for RAPIDS project is to hide the low-level programming complexities of the CUDA compute environment from Python developers and make it easy to deploy and execute full end-to-end processing pipeline on GPUs. An example of a machine learning library provided by RAPIDS is cuML [7], which is the GPU-counterpart for Scikit-learn. In order to support execution of Dask programs on cluster of GPUs, an efficient communication layer is required. This is extended by the Dask Distributed library that provides essentials for distributed execution of Dask programs on parallel hardware. It is an asynchronous I/O application, which means that it supports non-blocking and concurrent execution of its routines/functions including communication primitives. This mandates the restriction that any communication backend—aiming to be part of Dask Distributed library—must implement coroutines that are non-blocking methods defined and awaited by using the async and await keywords respectively. Coroutines support concurrent mode of execution and hence cannot be invoked like regular Python functions. The main reason behind this non-blocking asynchronous style of programming is to avoid blocking, or delaying, networking applications for completing I/O operations. The Dask Distributed library currently provides two communication devices: one for TCP and the other for UCX-Py that is a Cython-based wrapper library to UCX [8]. The UCX-Py communication device is capable of communicating data to/from GPU device memory directly. However, it fails to deliver high-performance—as revealed by our performance evaluation detailed in Section V—since it delays progressing the communication engine by assigning it to a separate coroutine that executes periodically. We address this performance overhead by advocating an efficient design where communication coroutines also progress the communication engine. The Message Passing Interface (MPI) standard [9] is considered as the defacto programming model for writing parallel applications on modern GPU clusters. The MVAPICH2-GDR [10] library provides high-performance support for communicating data to/from GPU devices via optimized point-to-point and collective routines. As part of this paper, we design, implement, and evaluate a new communication backend, called MPI4Dask, based on the MVAPICH2-GDR library for Dask. MPI4Dask implements point-to-point communication coroutines using mpi4py [11] over MVAPICH2-GDR. To the best of our knowledge, this is the first attempt to use an GPU-enabled MPI library to handle communication requirements of Dask. This is challenging especially because the Dask Distributed library is an asynchronous I/O application. Hence any point-to- point communication operations—implemented via MPI—must be integrated as communication coroutines that typically have performance penalties over regular Python functions. An additional goal here is to avoid making unnecessary modifications to the Dask ecosystem. Another challenge that MPI4Dask addresses is to provide communication isolation and support for dynamic connectivity between Dask components including scheduler, workers, and client. Communication between Dask entities—as provided by TCP and UCX backends—is based on the abstraction of endpoints that are established when processes connect with one another. An endpoint essentially represent a direct point-to-point connection. In MPI4Dask , we devise a strategy, detailed in Section IV, that relies on sub- communicator mechanism provided by MPI—coupled with communicator duplication—to handle this. This allows us to build sub-communicators between processes to mimic a direct endpoint-based connection. In order to motivate the need for MPI4Dask, we present latency comparison between Python communication coroutines implemented using MVAPICH2-GDR and mpi4py with UCX-Py polling and event-based modes in Figure 1. The latency graphs—Figure 1(a) and Figure 1(b) depict that the performance of Python communication coroutines using MVAPICH2-GDR (with mpi4py) are roughly $5\times$ better for small messages ($1$ Byte to $16$ Bytes) and $2-3\times$ better for small-medium messages ($32$ Bytes to $4$ KBytes). For large messages ($2$ MBytes to $128$ MBytes) as shown in Figure 1(c), MVAPICH2-GDR (with mpi4py)-based coroutines are better than UCX-Py point-to-point coroutines by a factor of $3-4\times$. These performance benefits form the basis for our motivation to design and implement MPI-based communication backend for the Dask framework. (a) Latency (Small) (b) Latency (Medium) (c) Throughput (Large) Figure 1: Latency and Bandwidth comparison of MVAPICH2-GDR (using mpi4py) with UCX using Ping-pong Benchmark—based on Python Coroutines—on the RI2 Cluster (V100 GPUs). This demonstrates the performance benefits of of using MVAPICH2-GDR as compared to UCX at the Python layer and forms the major motivation of this paper. We present a detailed performance evaluation of MPI4Dask against TCP and UCX communication backends using a number of micro-benchmarks and application benchmarks—these are presented in Section V. For the micro-benchmark evaluation, MPI4Dask outperforms the UCX communication device in latency and throughput comparison by ping-pong benchmarks by $5\times$ for small messages, $2-3\times$ better for small-medium messages, and $3-4\times$ for large messages. We used these two application benchmarks: 1) sum of cuPy array and its transpose, and 2) cuDF merge on an in-house cluster and the TACC’s Frontera (GPU) cluster. On the in-house cluster, we are witnessing an average speedup of $3.47\times$ for $1-6$ Dask workers for the cuPy application. Communication time for this application has been reduced by $6.92\times$ compared to UCX communication device. For the cuDF application on the in-house cluster, there is an average speedup of $3.11\times$ for $2-6$ Dask workers and $3.22\times$ reduction in communication time. On the Frontera (GPU) cluster, MPI4Dask outperforms UCX by an average factor of $1.71\times$ for the cuPy application with $1-32$ Dask workers. For the cuDF application, MPI4Dask reduces the total execution time by an average factor of $2.91\times$ when compared to UCX for $1-32$ Dask workers. Reasons for better overall performance of MPI4Dask against its counterparts include better point-to-point performance of MVAPICH2-GDR and efficient coroutine implementation. Unlike UCX-Py that implements a separate coroutine to make progress for UCX worker, MPI4Dask ensures cooperative progression where every communication coroutine triggers the communication progression engine. ### I-A Contributions Main contributions of this paper are summarized below: 1. 1. Design and implementation of MPI4Dask that provides high-performance point-to- point communication coroutines for Python-based HPC applications. To the best of our knowledge, MPI4Dask is the first library that implements MPI-based Python common coroutines that work with the asyncio framework on cluster of GPUs. 2. 2. Integration of MPI4Dask with asynchronous Dask Distributed library. This is a pioneering effort that enables MPI-based communication for the Dask ecosystem. 3. 3. Demonstrate the performance benefits of using MPI4Dask compared to TCP and UCX devices using basic micro-benchmarks and two application benchmarks (based on cuPy and cuDF) using an in-house cluster comprising of V100/K80 GPUs. 4. 4. Perform scalability evaluation of MPI4Dask against TCP and UCX communication devices using two application benchmarks (based on cuPy and cuDF) on TACC’s Frontera (GPU) system with upto $32$ Dask workers on $32$ NVIDIA Quadro RTX 5000 GPUs and $256$ CPU cores. Rest of the paper is organized as follows. Section II presented relevant background followed by the design approach of the MPI4Dask library in Section III. Section IV presents implementation details. Later, we evaluate MPI4Dask using a communication micro-benchmark and two application-level benchmarks. Section VII concludes the paper. ## II Background This section covers the background for this paper, which includes the fundamentals of the MPI standard, the Dask framework, and the asyncio package. This section also presents related work. ### II-A Message Passing Interface (MPI) The Message Passing Interface (MPI) API is considered the defacto standard for writing parallel applications. The MPI API defines a set of point-to-point and collective communication routines that are provided as convenience functions to application developers. In the context of the Dask Distributed library, the most relevant set of functions are the non-blocking point-to-point communication functions like MPI_Isend() and MPI_Irecv(). Both of these functions return an MPI_Request object that can be used to invoke MPI_Test() function to check if the non-blocking communication operation has completed or not. In this paper, we make use of a GPU-aware MPI library called MVAPICH2-GDR [10] that provides optimized point-to-point and collective communication support for GPU devices. GPU-awareness here means that the MPI library is capable of directly communication data efficiently to/from GPU memory instead of staging it through the host system. The Dask ecosystem is implemented in the Python programming language. This implies that the MPI communication backend must also be implemented in Python. For this reason, we use the GPU- aware mpi4py library that provides Cython [12] wrappers to native MPI library—MVAPICH2-GDR in this case. For performance reasons, it is important that the Python wrapper library is capable of communicating data to/from GPU memory directly without incurring the overhead of serialization/de- serialization. mpi4py supports efficient exchange of GPU data stored in cuPy and cuDF format. ### II-B Dask Dask is a popular data science framework for Python programmers. It converts user application into a task-graph, which is later executed lazily on distributed hardware. This execution requires data exchange supported through implicit communication by the Dask runtime. The Dask ecosystem is a suite of Python packages. One such package is Dask Distributed that provides various components like scheduler, workers, and client. This library also supports point-to-point functionality between these Dask components. Figure 2 depicts the distributed execution model of a Dask program. There are three types of connections between Dask entities: 1) data connections (solid lines) for exchanging application data, 2) control connections (dotted lines) for exchanging heart-beat messages to maintain status of workers and detect any failures, and 3) dynamic connections (dashed lines) between workers to resolve dependencies, work-stealing, or achieving higher throughput. Currently Dask has two communication devices: 1) the TCP device that exploits the asynchronous Tornado library, 2) the UCX backend that uses UCX-Py [13]—a Cython wrapper library—on top of the native UCX [8] communication library. Figure 2: Dask Execution Model. The scheduler and workers form a Dask cluster. The client executes user program by connecting with the Dask cluster. ### II-C The asyncio Package Since version $3.5$, Python has introduced a new package called asyncio that allows writing concurrent non-blocking I/O applications. The main idea behind this package is to allow networking (or other I/O) applications to efficiently utilize CPU without unnecessarily getting blocked for long-running I/O operations. This is possible because as the Python program executes, it keeps defining tasks that are stored in a task queue and execute concurrently as soon as it is possible to execute them. These tasks are defined through coroutines, which are functions defined using the async keyword and invoked later using the await keyword. ## III Design Overview of MPI4Dask This section presents the design of the MPI4Dask library. We first discuss communication requirements of the Dask framework. This is followed by coverage of layered architecture of Dask with focus on communication devices. We begin with communication requirements of the Dask framework. 1. 1. Scalability: Provide scalability by exploiting low-latency and high-throughput for cluster of GPUs typically deployed in modern data centers. 2. 2. Coroutines: The communication backend for Dask Distributed library is asynchronous and executes within an event loop, as part of an asyncio application. Hence, the communication backend needs to support non-blocking point-to-point send and receive operations through asyncio coroutines defined using the async/await syntax. 3. 3. Elasticity: The Dask cluster (e.g., scheduler and a set of workers) is an elastic entity meaning that the workers dynamically join or leave the cluster. In this context, the communication backend needs to support such dynamic connectivity for clients and workers. Dask, however, also executes with static number of processes using the Dask-MPI library. In this paper we adopt this approach. 4. 4. Serialization/De-serialization: Dask applications support distributed processing of GPU-based buffers including cuPy and cuDF DataFrames. The communication engine should be able to support communication to/from GPU-based Python objects/data-structures supported by Dask. Figure 3 presents a layered architecture for Dask. The top layer represents the Dask framework, which includes support for various data structures and storage formats including Dask Bags, Arrays, and DataFrames. Also there is support for asynchronous execution through Delayed and Future objects. The user application is internally converted to a task graph by Dask. The next layer shows various packages that are part of the Dask ecosystem. These include Dask-MPI, Dask-CUDA, and Dask-Jobqueue. The next layer represents the Dask Distributed library. This package supports distributed computation through scheduler, worker, and client. This library also contains the distributed.comm module, shown here as “Comm Layer”. This layer provides the API that all Dask communication backends must implement. A subset of this API can be seen in Listing 1. As shown, the Dask Distributed library provides TCP and UCX-Py backends represented in this layer by tcp.py and ucx.py. MPI4Dask is implemented as part of the Dask Distributed “Comm Layer” as a multi-layered software sitting over mpi4py, which in turn exploits MVAPICH2-GDR for GPU- aware MPI communication. The bottom layer in Figure 3 represents cluster of GPUs as parallel hardware. Dask is also capable of running on shared memory system like a laptop or a desktop. Figure 3: Dask Layered Architecture with Communication Backends. Yellow boxes are designed and evaluated as a part of this paper. ⬇ 1class Comm(ABC): 2 @abstractmethod 3 def read(self, deserializers=None): 4 @abstractmethod 5 def write(self, msg, serializers=None, 6 on_error=None): 7 8class Listener(ABC): 9 @abstractmethod 10 async def start(self): 11 @abstractmethod 12 def stop(self): 13 14class Connector(ABC): 15 @abstractmethod 16 def connect(self, address, deserialize=True) Listing 1: Subset of the Communication Backend API as Mandated by the Dask Distributed Library ## IV Implementation Details of the MPI4Dask Library This section outlines the implementation details of the MPI4Dask communication library. ### IV-A Bootstrapping the Dask ecosystem and initializing MPI There are many ways to start execution of Dask programs. The manual way is to start the scheduler followed by multiple workers on the command line. After this, the client is ready to connect with the Dask cluster and execute the user program. In order to automate this process, Dask provides a number of utility “Cluster” classes. An example of this is the LocalCUDACluster from the Dask CUDA package. Once an instance of LocalCUDACluster object has been initiated, a client can connect to execute a user program. Other example of utility cluster classes include SLURMCluster, SGECluster, PBSCluster, and others. MPI4Dask currently requires that user initiates the execution of Dask program using the Dask-MPI package. Dask-MPI uses the bootstrapping mechanism provided by MPI libraries to start Dask scheduler, client, and one or more workers. We use the mpirun_rsh utility provided by MVAPICH2-GDR . Figure 2 illustrates this where the mpirun_rsh utility was used to start $4$ MPI processes. Here processes $0$ and $1$ assume the role of scheduler and client, while all the other processes—in this case $2-3$ become worker processes. This bootstrapping is done by the Dask-MPI package and this is the point where mpi4py and MVAPICH2-GDR are also initialized. The CUDA_VISIBLE_DEVICES environment variable is used to map Dask worker processes on a particular GPU in a node. This is particularly important when a node has multiple GPUs since we would want multiple Dask workers to utilize distinct GPUs on the system. Note that Dask-MPI does not provide any communication between Dask components and this is what we address in this paper. ### IV-B Point-to-point Communication Coroutines A challenge that we tackle as part of this work is to implement asynchronous communication coroutines using mpi4py over MVAPICH2-GDR that can be incorporated inside the Dask Distributed layer. To the best of our knowledge, this is first such effort to exploit MPI-based communication inside an asyncio application in Python. mpi4py provides two variants of point-to-point functions. The first option is to use the lowercase methods like Comm.send()/Comm.isend() to communicate data/to from Python objects. This involves picking/unpickling (serialization/de-serialization) of Python objects. The second option is to use methods like Comm.Send()/Comm.Isend() that start with uppercase letter and communicate data to/from directly from user-specified buffer. The second option is efficient and preferred for achieving high-performance in the Dask Distributed library. Since we are implementing MPI4Dask for GPU buffers, the specified buffer to mpi4py communication routines must support the __cuda_array_interface__. As previously mentioned, the Dask Distributed library is an asynchronous application due to which it executes coroutines in a non-blocking and concurrent manner. Also for this reason, using blocking MPI point-to-point methods will result in deadlock for the application. In order to tackle this, our implementation of MPI4Dask makes use of Comm.Isend()/Comm.Irecv() methods that return Request objects. Later, MPI4Dask checks for completion of pending communication using the Request.Test() method. Instead of checking the completion status in a busy-wait loop—that will result in a deadlock situation—MPI4Dask calls the asyncio.sleep() method that allows other coroutines to make progress while waiting for communication to complete. Listings 2 and 3 provide outlines of send and receive communication coroutines in MPI4Dask. ⬇ 1request = comm.Isend([buf, size], dest, tag) 2status = request.Test() 3 4while status is False: 5 await asyncio.sleep(0) 6 status = request.Test() Listing 2: The Comm.Isend()-based Send Communication Coroutine Implemented by MPI4Dask. ⬇ 1request = comm.Irecv([buf, size], src, tag) 2status = request.Test() 3 4while status is False: 5 await asyncio.sleep(0) 6 status = request.Test() Listing 3: The Comm.Irecv()-based Receive Communication Coroutine Implemented by MPI4Dask. ### IV-C Handling Large Messages Dask is a programming framework for writing data science applications and it is common for such libraries to handle and exchange large amounts of data. As a consequence it is possible that the higher layers of the Dask ecosystem attempt to communicate large messages using the underlying communication infrastructure. We have experienced this with MPI4Dask that relies on the point-to-point non-blocking primitives provided by MPI—Comm.Isend() and Comm.Irecv() methods. These functions accept an argument int count that is used to specify the size of the message being sent or received. The maximum value that this parameter can hold is $2^{31}-1$ bytes, which corresponds a message size of $2$ GB$-1$. On the other hand, the Dask Distributed library attempts to communicate messages larger than this value including upto $64$ GB. We have catered for this requirement by dividing the large message into several chunks of $1$ GB for the actual communication using the Comm.Isend() and Comm.Irecv() methods. Typically the buffers specified to these communication functions are Python objects and hence subscriptable using the slice notation array[start:end]. This approach works for numPy and cuPy arrays and hence the buffer argument can be subscripted in a loop to implement chunking. But this strategy does not work for cuDF and RAPIDS Memory Manager (RMM) buffers rmm.DeviceBuffer. For these, we implement chunking by incrementing the offset argument in the communication loop. ### IV-D Providing Communication Isolation After the initial bootstrapping, MPI4Dask has full connectivity between all processes. This is provided by the default communicator MPI_COMM_WORLD initialized by the MPI library. However using this communicator naively might lead to message interference. This is possible when a particular worker might want to use same tag for both data and control message exchanges. There are multiple ways to tackle this. The approach that we choose in this paper is to rely on MPI sub-communicators. Figure 4 outlines this approach. The $4$ MPI processes here engage with one another to form five sub-communicators represented by colored bi-directional edges between a pair of processes. The pseudocode for this is shown in Listing 4. Each process stores a handle to these sub-communicators in a table called comm_table. Creating a new MPI sub- communicator is a costly operation and for this reason all of this is done at the startup. The MPI.Group.Incl() and MPI.COMM_WORLD.Create() functions (from mpi4py) are used for creating sub-communicators that only contain two processes. Complexity of making new sub-communicators is $O(\frac{n^{2}}{2})$ where $n$ is the total number of MPI processes. Our approach has negligible overhead because these sub-communicators are initialized at the startup and re-used later during the program execution. Figure 4: Building new Sub-Communicators from MPI.COMM_WORLD at Startup. The bi-directional edges between processes represent newly built sub- communicators. ⬇ 1for i in range(size): 2 for j in range(i+1, size): 3 incls = [i, j] 4 new_group = MPI.Group.Incl(group, incls) 5 new_comm = MPI.COMM_WORLD.Create(new_group) 6 if rank == i: 7 comm_table.update({j : new_comm}) 8 else if rank == j: 9 comm_table.update({i : new_comm}) Listing 4: Nested Initialization Loop Executed at Each MPI Process during Startup to Build new Sub-communicators from MPI.COMM_WORLD. The newly built sub-communicators encapsulate only two processes and are stored in the comm_table. ### IV-E Handling Dynamic Connectivity The Dask Distributed library support connectivity between Dask components including scheduler, workers, and client. The UCX and TCP communication devices maintain information for remote endpoints since this information is required for the actual communication. In MPI4Dask, we have replaced the abstraction of endpoint with sub-communicator. This enables us to utilize the existing communication infrastructure of the Dask ecosystem. Apart from communications being setup at the startup, Dask allows dynamic connections between workers as well. In this context, new communication channels are only established in the Dask Distributed library through traditional client/server semantics of connect/accept. In MPI4Dask we handle this requirement by starting a server that listens for incoming connections—and invokes a connection handler callback function—using the asyncio.start_server() method. Dask scheduler and worker processes use this function since they act as listeners and other processes are allowed to connect with them. Later any process (scheduler, client, or worker) that attempts connecting to a listener process do so by using the asyncio.open_connection() function. When two processes connect with one another—and this is something that happens frequently in Dask at startup—each process gets the relevant sub-communicator from the comm_table by doing a lookup based on the destination process rank in the MPI.COMM_WORLD communicator. The corresponding sub-communicator from the comm_table is duplicated by using the Comm.Dup() function. This ensures that a new sub-communicator is built for every new dynamic connection. For performance reasons, we are maintaining a configurable cache of these sub- communicators. ## V Performance Evaluation This section presents performance evaluation of MPI4Dask against UCX and TCP (using IPoIB) communication devices using 1) Ping Pong micro-benchmark (Section V-A), 2) Two application benchmarks (cuPy and cuDF) on an in-house cluster (RI2) with two types of GPU nodes with NVIDIA Tesla V100s and NVIDIA Tesla K80s (Section V-B), and 3) Scalability results for the same two application benchmarks on TACC’s Frontera (GPU) cluster with NVIDIA Quadro RTX 5000 GPUs (Section V-C). The hardware specifications for the RI2 cluster and Frontera (GPU) subsystem are shown in Table I. The following versions of the software were used: UCX v1.8.0, UCX-Py v0.17.0, Dask Distributed v2.30, MVAPICH2-GDR v2.3.4, and mpi4py v3.0.3. TABLE I: Hardware specification of the in-house RI2 and TACC’s Frontera (GPU) clusters. Columns 1 and 2 titled RI2-V100 and RI2-K80 provide details for two types of nodes on the RI2 cluster. Column 3 titled RI2-V100 provide details for the nodes used on the Frontera (GPU) cluster. Specification \| RI2-V100 \| RI2-K80 \| Frontra (GPU) ---\|---\|---\|--- Number of Nodes \| 16 \| 16 \| 90 Processor Family \| Xeon Broadwell \| Xeon Broadwell \| Xeon Broadwell Processor Model \| E5-2680 v4 \| E5-2680 v4 \| E5-2620 v4 Clock Speed \| 2.4 GHz \| 2.4 GHz \| 2.1 GHz Sockets \| 2 \| 2 \| 2 Cores Per socket \| 14 \| 14 \| 16 RAM (DDR4) \| 128 GB \| 128 GB \| 128 GPU Family \| Tesla V100 \| Tesla K80 \| Quadro RTX 5000 GPUs \| 1 \| 2 \| 4 GPU Memory \| 32 GB \| 12 GB \| 16 GB Interconnect \| IB-EDR (100G) \| IB-EDR (100G) \| IB-FDR (56G) ### V-A Latency and Throughput Comparison We first perform latency and throughput comparisons between MPI4Dask and other communication backends—in particular UCX-Py—using a Ping Pong benchmark. Also we add UCX (Tag API) and MVAPICH2-GDR to our comparisons for baseline performance. UCX v1.8.0 was used alongwith ucx_perftest [14] to evaluate latency and throughput for UCX Tag API. For MVAPICH2-GDR, we used the osu_latency test that is part of the OSU Micro-Benchmark Suite (OMB) [15]. UCX-Py was also evaluated using its own Ping Pong benchmark [16]. UCX-Py has two modes of execution: 1) polling-based, and 2) event-based. The polling- based mode is latency-bound and hence more efficient than the event-based mode. Comparisons also include mpi4py running over MVAPICH2-GDR—this is labeled as MV2-GDR (mpi4py). The benchmark used for mpi4py is an in-house Python version of OMB. Lastly we plot MPI4Dask that essentially represents the MPI4Dask communication device integrated into the Dask Distributed library. We always execute the TCP device using IP over InfiniBand (IPoIB) protocol that provides best performance for this backend. For this reason, we use TCP and IPoIB interchangeably in this section. (a) Latency (Small) (b) Latency (Medium) (c) Throughput (Large) Figure 5: Latency/Bandwidth comparison of MPI4Dask with UCX-Py (Polling and Event Modes) using Ping Pong Benchmark on RI2 Cluster with V100 GPUs. UCX (Tag API) and MVAPICH2-GDR numbers are also presented for baseline performance. (a) Latency (Small) (b) Latency (Medium) (c) Throughput (Large) Figure 6: Latency/Throughput comparison of MPI4Dask with UCX-Py (Polling and Event Modes) using Ping Pong Benchmark on RI2 Cluster with K80 GPUs. UCX (Tag API) and MVAPICH2-GDR numbers are also presented for baseline performance. Figure 5(a) shows latency for message sizes $1$ Byte to $4$ KByte—here MPI4Dask outperforms UCX-Py (Polling) by $5\times$. Similarly for medium-sized messages presented in Figure 5(b)—$8$ KByte to $512$ KByte—MPI4Dask is better than UCX-Py (Polling) by $2-3\times$. In throughput comparisons for large messages, $1$ MByte and beyond, MPI4Dask outperforms UCX-Py (Polling) by $3-4\times$ in throughput as shown in Figure 5(c). It is important to note that the difference in performance between MV2-GDR (mpi4py) and MV2-GDR is minimal. This means that regular Python functions over native code do not introduce much overhead. However when executing communication coroutines—by employing MPI4Dask or UCX-Py—there is additional overhead due to the asyncio framework. We observe similar performance patterns for Tesla K80 GPUs in latency and throughput comparisons between MPI4Dask and UCX-Py (Polling) as depicted in Figure 6. As noted earlier, we are focusing on performance comparison of MPI4Dask with UCX-Py (Polling) since it is the more efficient mode of the UCX-Py library. ### V-B Application Benchmarks We also evaluated MPI4Dask against UCX-Py for two application benchmarks: 1) sum of cuPy array and its transpose, and 2) cuDF merge. The cuPy benchmark presents strong scaling results as the problem size remains the same as more Dask workers are used in the computation. On the other hand, the cuDF benchmark presents weak scaling results as the problem size increases with increase in the number of Dask workers. This evaluation was done on the in- house RI2 cluster. #### V-B1 Sum of cuPy Array and its Transpose This benchmark [17] creates a cuPy array and distributes its chunks across Dask workers. The benchmark adds these distributed chunks to their transpose, forcing the GPU data to move around over the network. The following operations are performed: y = x + x.T y = y.persist() wait(y) Performance comparison graphs for the first application benchmark—sum of cuPy array with its transpose—are shown in Figure 7. Dask follows the One Process per GPU (OPG) model of execution, which means that a single worker process with multiple threads is initiated for each GPU. On the RI2 cluster, each node has a single GPU. For this reason, we instantiate a single worker process with $28$ worker threads to fully exploit the available CPU cores. Figure 7(a) shows execution time where we are witnessing an average speedup of $3.47\times$ for $2-6$ Dask workers. Figure 7(b) shows the communication time for the benchmark run. This shows that MPI4Dask is better than UCX by $6.92\times$ on average for $2-6$ workers. Average throughput comparison for Dask workers is shown in Figure 7(c), which depicts that MPI4Dask outperform UCX by $5.17\times$ on average for $2-6$ workers. This benchmark application is presenting strong scaling results. As can be seen in Figure 7(a), the cuPy application does not exhibit impressive speedups as the number of workers are increased. This is due to the nature of this benchmark that is designed to stress and evaluate communication performance. This is suitable for this paper because we are primarily interested in comparative performance of communication devices and not demonstrating application-level performance of the Dask ecosystem. #### V-B2 cuDF Merge cuDF DataFrames are table-like data-structure that are stored in the GPU memory. As part of this application [18], a merge operation is carried out for multiple cuDF data frames. Performance comparison graphs for the second application benchmark—cuDF merge operation—are shown in Figure 8. Figure 8(a) shows execution time where we are witnessing an average speedup of $3.11\times$ for $2-6$ Dask workers. Figure 8(b) shows time for communication that is taking place. This shows that MPI4Dask is better than UCX by $3.22\times$ on average for $2-6$ workers. Average throughput comparison for Dask workers is presented in Figure 8(c), which depicts that MPI4Dask outperforms UCX by $3.82\times$ on average for $2-6$ workers. The cuDF merge benchmark is presenting weak scaling results. As can be observed in Figure 8, the execution time with MPI4Dask is increasing slightly with an increase in the number of workers. This is due to efficient communication performance provided by MPI4Dask to the Dask execution. Note that $3.2$, $6.4$, $9.6$, $12.8$, $16$, and $19.2$ GB of data is processed for $1$, $2$, $3$, $4$, $5$, and $6$ Dask workers respectively. Note that UCX-Py is executed in polling-mode that is the efficient mode. Reasons for better overall performance of MPI4Dask against other counterparts include better point-to-point performance of MVAPICH2-GDR , chunking implemented for messages greater than $1$ GB, and efficient coroutine implementation for MPI4Dask as compared to UCX-Py. Unlike UCX-Py that implements a separate coroutine to make communication progress for UCX worker, MPI4Dask ensures cooperative progression where every communication coroutine triggers the communication progression engine. (a) Execution Time Comparison (b) Communication Time Comparison (c) Aggregate Throughput Between Workers Figure 7: Sum of cuPy Array and its Transpose (cuPy Dims: 16K$\times$16K, Chunk size: 4K, Partitions: 16): Performance Comparison between IPoIB, UCX, and MPI4Dask on the RI2 Cluster. This benchmark presents strong scaling results. $28$ threads are started in a single Dask worker. (a) Execution Time Comparison (b) Communication Time Comparison (c) Aggregate Throughput Between Workers Figure 8: cuDF Merge Operation Benchmark (Chunk Size: 1E8, Shuffle: True, Fraction Match: 0.3): Performance Comparison between IPoIB, UCX, and MPI4Dask on the RI2 Cluster. This benchmark presents weak scaling results. $28$ threads are started in a single Dask worker. $3.2$ GB of data is processed per Dask worker (or GPU) ### V-C Scalability Results on the TACC Frontera (GPU) Cluster This section presents the scalability results for the two application benchmarks introduced earlier in Section V-B. This evaluation was done on the Frontera (GPU) system that is equipped with $360$ NVIDIA Quadro RTX 5000 GPUs in $90$ nodes. Following the OPG model of execution, we started $4$ processes on a single node—one process per GPU—with $8$ worker threads in each process. This evaluation is presented in Figure 9. Figure 9(a) shows the execution time comparison between TCP (using IPoIB), UCX, and MPI4Dask devices for the sum of cuPy with its transpose application with $1-32$ Dask workers—here MPI4Dask outperforms UCX by an average factor of $1.71\times$. Figures 9(b) and 9(c) shows the execution time and throughput comparison between all communication devices for the cuDF merge operation. Again, MPI4Dask outperforms UCX by an average factor of $2.91\times$ for overall execution time for $1-32$ Dask workers. The merge operation throughout depicts the rate at which workers merge the cuDF input data and it also follows a similar pattern. The performance results shown in Figure 9(a) present strong scaling results for the cuPy application, while Figures 9(b) and 9(c) present weak scaling results for the cuDF merge operation. The performance shown by $1$ Dask worker in Figures 9(b) and 9(c) is efficient due to small problem size and no communication overhead. However, the problem size becomes larger and more realistic with additional Dask workers. Note that $1.6$ GB of data is processed per GPU for the cuDF application—this means that $51.2$ GB of data is processed on $32$ GPUs. (a) Execution Time Comparison for Sum of cuPy Array and its Transpose Benchmark. (cuPy Dims: 20E3$\times$20E3, Chunk size: 5E2, Partitions: 16E2) (b) Execution Time Comparison for cuDF Merge Benchmark. (Chunk Size: 5E7, Shuffle: True, Fraction Match: 0.3) (c) Average Throughput For Workers for the cuDF Merge Benchmark Figure 9: Execution Time Comparison for Sum of cuPy Array with its Transpose Benchmark, cuDF Merge Benchmark, and Average Throughput for Dask Workers for the cuDF Merge Benchmark. This evaluation was done on the Frontera (GPU) system at TACC. This comparison is between MPI4Dask , UCX, and TCP (using IPoIB) communication devices. There are $4$ GPUs in a single node. Each Dask worker has $8$ threads. ## VI Related Work A framework similar to Dask is the Apache Spark software [6]. It supports programming data science applications using Scala, Java, Python and R. Traditionally Spark has also only supported execution on hosts (CPUs). This meant that it was not able to exploit massive parallelism offered by GPUs. However GPU support has been added by the RAPIDS project recently to the Apache Spark 3.0 through a Spark-RAPIDS plugin [19]. Vanilla versions of Apache Spark only support data communication over Ethernet or other high-speed networks using the TCP backend. There have been efforts [20, 21] to address this shortcoming for Apache Spark. The Dask Distributed library currently support two communication backends. The first device—called TCP—makes use of Python’s Tornado framework [22]. The second device—called UCX—uses a Cython wrapper called UCX-Py [13] over the UCX [8] communication library. In an earlier effort [23], we have developed a new communication device called Blink for Dask. However Blink is limited to CPU- only execution of Dask programs and hence is not relevant here. Also, recently there has been another effort [24] to develop an MPI communication layer on the Dask Distributed github repository. However, this is currently a work in progress and is not addressing GPU-based data science applications, which is the focus of this paper. At this time, it is only possible to use TCP and UCX based devices for Dask on cluster of GPUs. This paper demonstrates that MPI4Dask —developed as part of this work—delivers better performance than the current state of the art communication options for Dask on cluster of GPUs. ## VII Conclusions and Future Work Python is an emerging language on the landscape of scientific computing with support for distributed computing engines like Dask. Rivaling the Apache Spark ecosystem, Dask allows incremental parallelization of Big Data applications. Recently the Dask software has been extended, as part of the NVIDIA RAPIDS framework, to support distributed computation on cluster of GPUs. Support for GPU-based data storage and processing APIs like cuPy and cuDF—GPU-counterparts of numPy and Pandas respectively—has also been added. As part of this work, we have extended the Dask Distributed library with a new communication device called MPI4Dask based on the popular MPI standard. MPI4Dask exploits the mpi4py wrapper software on top of the MVAPICH2-GDR library to offer an efficient alternative to existing backends—TCP and UCX—from within a non- blocking asynchronous I/O framework. This is done by implementing high- performance communication coroutines using MPI that support the async/await syntax. The performance evaluation done, on an in-house cluster built with Tesla V100/K80 GPUs and the Frontera (GPU) cluster equipped with Quadro RTX 5000 GPUs—using the Ping Pong micro-benchmark and two other application benchmarks—suggest that MPI4Dask clearly outperforms other communication devices. This is due to efficient implementation of communication coroutines in MPI4Dask that use the cooperative progress approach with non-blocking point-to-point MPI calls. In the future we plan to extend MPI4Dask with support for dynamic process management, which will allow entities like workers and clients to dynamically join and leave the Dask Cluster. 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††thanks: This work was supported by the National Natural Science Foundation of China (Nos. 11805294 and 11975021), the China Postdoctoral Science Foundation (2018M631013), the Strategic Priority Research Program of Chinese Academy of Sciences (XDA10010900), the Fundamental Research Funds for the Central Universities, Sun Yat-sen University (19lgpy268), and in part by the CAS Center for Excellence in Particle Physics (CCEPP). # Event vertex and time reconstruction in large-volume liquid scintillator detectors Zi-Yuan Li School of Physics, Sun Yat-Sen University, Guangzhou 510275, China Yu-Mei Zhang<EMAIL_ADDRESS>Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-Sen University, Zhuhai 519082, China Zhen Qian School of Physics, Sun Yat-Sen University, Guangzhou 510275, China Shu Zhang School of Physics, Sun Yat-Sen University, Guangzhou 510275, China Kai-Xuan Huang School of Physics, Sun Yat-Sen University, Guangzhou 510275, China Guo-Fu Cao Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China Zi-Yan Deng Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China Gui-Hong Huang Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China Wei-Dong Li Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China Tao Lin Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China Liang-Jian Wen Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China Miao Yu School of Physics and Technology, Wuhan University, Wuhan 430072, China Jia-Heng Zou Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China Wu-Ming Luo<EMAIL_ADDRESS>Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China Zheng-Yun You<EMAIL_ADDRESS>School of Physics, Sun Yat-Sen University, Guangzhou 510275, China ###### Abstract Large-volume liquid scintillator detectors with ultra-low background levels have been widely used to study neutrino physics and search for dark matter. Event vertex and event time are not only useful for event selection but also essential for the reconstruction of event energy. In this study, four event vertex and event time reconstruction algorithms using charge and time information collected by photomultiplier tubes were analyzed comprehensively. The effects of photomultiplier tube properties were also investigated. The results indicate that the transit time spread is the main effect degrading the vertex reconstruction, while the effect of dark noise is limited. In addition, when the event is close to the detector boundary, the charge information provides better performance for vertex reconstruction than the time information. JUNO, Liquid scintillator detector, Neutrino experiment, Vertex reconstruction, Time reconstruction ## I Introduction Liquid scintillators (LSs) have widely been used as detection medium for neutrinos in experiments such as Kamioka Liquid Scintillator Antineutrino Detector (KamLAND) kamland , Borexino borexino , Double Chooz doublechooz , Daya Bay dyb , and Reactor Experiment for Neutrino Oscillation (RENO) reno . KamLAND revealed a large mixing angle (LMA) solution for solar neutrino oscillations. Borexino confirmed the Mikheyev-–Smirnov-–Wolfenstein (MSW) LMA lmamsw model in the sub-MeV region for solar neutrino oscillations. Double Chooz, Daya Bay, and RENO reported nonzero measurements for the mixing angle $\theta_{13}$. The size of such detectors varies from hundreds to thousands of cubic meters. Large-volume liquid scintillator detectors are widely used in the next generation of neutrino experiments, aiming to solve problems such as neutrinoless double-beta decay (SNO+ at the Sudbury Neutrino Observatory sno+ ) and neutrino mass ordering (Jiangmen Underground Neutrino Observatory (JUNO) junophysics ). The sensitivity of these experiments is limited by the energy resolution, detector volume, and detector background. These detectors typically contain a fiducial volume, where the signal-to-noise ratio is maximal. To distinguish between events occurring in the fiducial and non-fiducial regions, the event vertex is reconstructed using the charge and time distribution of photons collected by the photomultiplier tubes (PMTs). Most importantly, to achieve a high energy resolution, an accurate vertex is essential to correct for energy non-uniformity. In addition, the event vertex and event time information can also be used for particle identification, direction reconstruction, event classification, and other purposes. A previous study Liu_2018 investigated vertex reconstruction with time information in JUNO, without discussing the event time reconstruction, dark noise effect, and the improvement based on the charge information when the event is close to the detector boundary. The aim of this study was to analyze vertex and time reconstruction for point-like events in JUNO under more realistic conditions. The main contributions of this study are as follows: 1. $$ The event time was reconstructed to provide the start time information of the event, which was important for event alignment, event correlation, etc. 2. $$ The dark noise from PMTs was considered and its effect on the vertex reconstruction was properly controlled. 3. $$ Two types of large PMTs were considered and handled separately mainly because of the difference in transit time spread (TTS). 4. $$ An algorithm to provide a more accurate initial vertex value was developed to improve the performance, especially at the detector boundary region. 5. $$ An algorithm employing charge information to reconstruct the event vertex at the detector boundary was developed. The remainder of this paper is organized as follows. In Sec. II, a brief introduction of the JUNO detector and the configurations of the PMTs used in this study is provided. In Sec. III, the optical processes are described and a simple optical model is introduced. In Sec. IV, two simple algorithms that can quickly provide initial values are compared. In Secs. V and VI, two complex algorithms that provide relatively good performance are introduced in detail. Finally, a performance summary, the discussion, and the conclusions are provided in Secs. VII, VIII, and IX, respectively. ## II The JUNO detector Figure 1: Schematic of the JUNO detector. A schematic of the JUNO detector is shown in Fig. 1. The central detector (CD) is the main part of the JUNO detector assembly, with an acrylic sphere with a diameter of 35.4 m as inner layer. The acrylic sphere is supported by a stainless steel latticed shell with a diameter of 40.1 m and filled with approximately 20,000 tons of LS as target for neutrino detection. The composition of the LS includes linear alkylbenzene (LAB) as solvent, 2,5-diphenyloxazole (PPO) as fluor, and p-bis-(o-methylstyryl)-benzene (bis- MSB) as wavelength shifter. To collect photons, the CD is surrounded by approximately 17,600 20-inch PMTs, including 5000 Hamamatsu R12860 PMTs, 12,600 North Night Vision Technology (NNVT) GDG-6201 PMTs, and approximately 25,600 3-inch XP72B22 PMTs. Around the CD, a 2.5-m-thick water pool is used to shield external radioactivity from the surrounding rocks and is combined with approximately 2,000 20-inch GDG-6201 PMTs to serve as a water Cherenkov detector to veto cosmic ray muons. A top tracker detector, consisting of plastic scintillators, located above the water pool is used for the identification and veto of muon tracks. A more detailed description of the JUNO detector can be found in Refs. junocdr ; junophysics . The main factors affecting the reconstruction of the event vertex and time include the TTS and dark noise of the PMTs. In this study, only the 20-inch PMTs of the CD were used for reconstruction. The number and parameters of the Hamamatsu and NNVT PMTs are summarized in Table 1 pmtsys . In principle, the 3-inch PMTs with $\sigma\simeq 1.6$ ns TTS could also be included to improve the reconstruction performance; however, these PMTs 3inchpmt were not considered in this study because of their small photon detection coverage ($\sim$3%) gdml ; gdml2 . Table 1: Number and parameters of the PMTs used in the reconstruction. The TTS and dark noise rate are the mean values of the distribution measured during the mass testing. However, these are not the final values for JUNO. Company \| Number \| TTS ($\sigma$) \| Dark noise rate ---\|---\|---\|--- Hamamatsu \| 5,000 \| 1.15 ns \| 15 kHz NNVT \| 12,600 \| 7.65 ns \| 32 kHz ## III Optical processes When a charged particle deposits energy in the scintillator, the solvent enters an excited state and transfers energy to the fluor in a non-radiative manner. Scintillation photons are then emitted along the particle track through the radiative de-excitation of the excited fluor within a limited time. The emitted scintillation photons can undergo several different processes while propagating through a large LS detector. At short wavelengths ($<$ 410 nm), photons are mostly absorbed and then re-emitted at longer wavelengths, which maximizes the detection efficiency of the PMTs. At long wavelengths ($>$ 410 nm), photons mainly undergo Rayleigh scattering. A more detailed study of the wavelength-dependent absorption and re-emission can be found in Ref. lsopmdl . Additionally, the refractive indices at 420 nm are 1.50 and 1.34 for the LS and water, respectively. The difference in the refractive indices results in refraction and total reflection at the boundary of the two media, which affects the time-of-flight of the photons. When using the time information in the reconstruction, the time-of-flight is crucial, and it is calculated using the equation $\mathrm{tof}=\sum_{m}\frac{d_{m}}{v_{m}},$ (1) where $\mathrm{tof}$, $d_{m}$, and $v_{m}$ are the time-of-flight, optical path length, and effective light speed, respectively, and $m$ represents different media, in this case the LS and water, in the JUNO experiment. The acrylic sphere (thickness of 12 cm) and acrylic cover (thickness of 1 cm) in front of each PMT were ignored in this study because their refractive indices were similar to that of the LS and their thickness was small compared to that of the LS and water. ### III.1 Optical path length Figure 2: Event display of the optical path from the event vertex to the PMT in the JUNO simulation. The red circle ring is the event vertex and the gray bulbs with blue caps represent the PMTs. The optical path length can be characterized by the start and end positions in the detector, which are the vertex of the event and the position of the PMTs, respectively. The bold cyan curve in Fig. 2 shows a typical example of the optical path of photons detected by the PMT in the JUNO simulation using the event display eventdisplay1 ; eventdisplay2 , and further examples are shown by the thin green curves. There are multiple physically possible paths between these two positions, each of which has a different optical path length, as follows: 1. $$ owing to absorption and re-emission, the re-emitted photon is not in the same absorption position and the propagation direction also changes; 2. $$ owing to scattering, the photon changes the original direction of the propagation; and 3. $$ owing to refraction and total reflection, the photon does not travel in a straight line. As shown in Fig. 2, owing to the various aforementioned optical processes, it is difficult to predict the actual optical path length for each photon. In this paper, a simple optical model is proposed, which uses a straight line connecting the vertex and the PMTs to calculate the optical path length (Fig. 3), and combines with the effective light speed to correct for the time-of- flight. Using this simple optical model reasonable results can be obtained, as discussed in Sec. V. Figure 3: Optical path length from the event vertex to the $i$th PMT. $O$ denotes the center of the detector. In Fig. 3, {$\vec{r}_{0},t_{0}$} represents the event vertex and start time, {$\vec{r}_{i},t_{i}$} is the position of the $i$th PMT and the time of the earliest arriving photon detected by it. The angle between the normal direction of the $i$th PMT and the vector of the position of the $i$th PMT pointing to the event vertex is $\theta_{i}$ and $\alpha_{i}=\arccos(\hat{\vec{r}}_{0}\cdot\hat{\vec{r}}_{i})$. The optical path length of the photon arriving at the $i$th PMT is $d_{pathlength,i}=\|\vec{r}_{i}-\vec{r}_{0}\|=d_{LS,i}+d_{water,i}$ and the corresponding time-of-flight is $\mathrm{tof}_{i}$. The optical path length in the LS and water can be calculated by simply solving the trigonometric equation. ### III.2 Effective light speed According to Ref. lsopmdl , the emission spectrum of scintillation photons is in the range of approximately 300––600 nm. Typically, the group velocity of the wave packet is used to describe the photon propagation in the medium, which is given by the equation $v_{g}(\lambda)=\frac{c}{n(\lambda)-\lambda\frac{\partial{n(\lambda)}}{\partial\lambda}},$ (2) where $v_{g}$ is the group velocity, $c$ is the speed of light in vacuum, $n$ is the refractive index, and $\lambda$ is the wavelength. By fitting the Sellmeier equation optics , which describes the dispersion of the measurement in Refs. rayleighscat and h2oindex , the refractive index of the LS and water at different wavelengths is shown in the upper panel of Fig. 4. The group velocity of the LS and water can be calculated using Eq. 2 at different wavelengths, as shown in the lower panel of Fig. 4. Figure 4: Dependence of the refractive index (upper panel) and group velocity (lower panel) on the wavelength in the LS and water. The propagation speed of photons in water ($v_{water}$) was determined as the average speed weighted by the probability density function of the photon wavelength, which was obtained from a Monte Carlo (MC) simulation. As the absorption and re-emission change the initial wavelength, determining the propagation speed of photons in the LS ($v_{LS}$) is more complicated. To consider all wavelength-dependent effects that affect the propagation speed of photons, the effective light speed $v_{eff}$ is introduced. In addition, $v_{eff}$ also mitigates the effects by the simplified optical model, which, for example, ignores the refraction at the interface between the LS and water, as well as the change in the optical path length due to Rayleigh scattering. The exact value for $v_{eff}$ can be determined using a data-driven method based on the calibration data as follows: place $\gamma$ sources along the Z-axis, use $v_{LS}$ at 420 nm as the initial value of $v_{eff}$ in the reconstruction algorithm and then, calibrate $v_{eff}$ such that the source positions can be appropriately reconstructed. As no calibration data was available for JUNO, in this study, simulated calibration data were used, and the optimized values for the effective refractive index (c/$v_{eff}$) were 1.546 in the LS and 1.373 in water. In the future, the same method can be applied to the experimental calibration data. ## IV Initial value for vertex and time The TMinuit package tminuit was used for the minimization procedure in the time likelihood and in the charge likelihood algorithm introduced in Secs. V and VI. When there are multiple local minima in the parameter space, an inaccurate initial value results in local instead of global minima, resulting in a lower reconstruction efficiency. For detectors such as JUNO, the initial value needs to be treated carefully because of the total reflection, as discussed in the following subsections. ### IV.1 Charge-based algorithm The charge-based algorithm is essentially based on the charge-weighted average of the positions of the PMTs in an event, and the event vertex can be determined using the equation $\vec{r}_{0}=a\cdot\frac{\sum_{i}{q_{i}\cdot\vec{r}_{i}}}{\sum_{i}{q_{i}}},$ (3) where $q_{i}$ is the charge of the pulses detected by the $i$th PMT and $\vec{r}_{0}$ and $\vec{r}_{i}$ are defined in Fig. 3. A scale factor $a$ is introduced because the charge-based algorithm is inherently biased and an ideal point-like event in a spherical detector is covered by a uniform photocathode. Even if all propagation-related effects, such as absorption and scattering are ignored, the result of a simple integral of the intersections of all photons with the sphere surface shows that the reconstructed position of the event is 2/3 of the true position. The value of $a$ can be tuned based on the calibration data along the $Z$-axis. In this study, $a=1.3$ was used, which was sufficient to provide an initial estimate for the event vertex. Figure 5: Heatmap of $R_{rec}$ (upper panel) and $R_{rec}-R_{true}$ (lower panel) as a function of $R_{true}$ for 4-MeV $e^{+}$ uniformly distributed in space calculated by the charge-based algorithm. As can be seen in Fig. 5, even with the scale factor, owing to total reflection, the reconstructed vertex deviates up to 3 m near the detector boundary. According to Ref. optics , total reflection occurs only when the event vertex is located at an $R$ larger than $R_{LS}\cdot{n_{water}/n_{LS}}\approx 15.9$ m, where $R_{LS}$ is the radius of the acrylic sphere, $n_{LS}$ and $n_{water}$ are the refractive indices in the LS and water, respectively. The total reflection region is defined as $R>15.9$ m while $R<15.9$ m is the central region. If the result from the charge-based algorithm is used as the initial value for the time likelihood algorithm, approximately 18% of events is reconstructed at a local minimum position. In addition, it should be noted that the charge-based algorithm is not able to provide an initial value for the event generation time $t_{0}$. Therefore, a fast time-based algorithm needs to be introduced, which can provide more accurate initial values. ### IV.2 Time-based algorithm The time-based algorithm uses the distribution of the time-of-flight correction time $\Delta{t}$ (defined in Eq. 4) of an event to reconstruct its vertex and $t_{0}$. In practice, the algorithm finds the reconstructed vertex and $t_{0}$ using the following iterations: 1. 1. Apply the charge-based algorithm to obtain the initial vertex. 2. 2. Calculate time-of-flight correction time $\Delta{t}$ for the $i$th PMT as $\Delta{t}_{i}(j)=t_{i}-\mathrm{tof}_{i}(j),$ (4) where $j$ is the iteration step and $t_{i}$, $\mathrm{tof}_{i}$ are defined in Fig. 3. Plot the $\Delta{t}$ distribution for all triggered PMTs, and label the peak position as $\Delta{t}^{peak}$. 3. 3. Calculate the correction vector $\vec{\delta}[\vec{r}(j)]$ as $\vec{\delta}[\vec{r}(j)]=\frac{\sum_{i}(\frac{\Delta{t}_{i}(j)-\Delta{t}^{peak}(j)}{\mathrm{tof}_{i}(j)})\cdot(\vec{r}_{0}(j)-\vec{r}_{i})}{N^{peak}(j)},$ (5) where $\vec{r}_{0}$, and $\vec{r}_{i}$ are defined in Fig. 3. To minimize the effect of scattering, reflection, and dark noise on the bias of the reconstructed vertex, only the pulses appearing in the $(-10\leavevmode\nobreak\ \rm ns,+5\leavevmode\nobreak\ \rm ns)$ window around $\Delta{t}^{peak}$ are included. The time cut also suppresses the effect of the late scintillation photons. The number of triggered PMTs in the window is $N^{peak}$. 4. 4. If $\vec{\delta}[\vec{r}(j)]<1\leavevmode\nobreak\ \rm mm$ or $j=100$, stop the iteration; otherwise, update the vertex with $\vec{r}_{0}(j+1)=\vec{r}_{0}(j)+\vec{\delta}[\vec{r}(j)]$ and go to step 2 to start a new round of iteration. The distribution of $\Delta{t}$ at different iteration steps is shown in Fig. 6. At the beginning of the iteration, the $\Delta{t}$ distribution is wide because the initial vertex is far from the true vertex. As the number of iterations increases, the $\Delta{t}$ distribution becomes more concentrated. Finally, when the requirement in step 4 is satisfied, the iteration stops. In the final step, $\vec{r}_{0}$ is the reconstructed vertex and $\Delta{t}^{peak}$ is the reconstructed time $t_{0}$. Figure 6: $\Delta{t}$ distribution at different iteration steps $j$. After the time-of-flight correction, the $\Delta{t}$ distribution is independent of the event vertex. However, because the earliest arrival time is used, according to the first-order statistic, as discussed in Ref. fos ; fos2 ; fos3 , $t_{i}$ is related to the number of photoelectrons $N^{i}_{pe}$ detected by $i$th PMT. To reduce the bias of the vertex reconstruction, the following form of the time–$N_{pe}$ correction is applied, and in Eq. 4 $t_{i}$ is replaced by $t^{\prime}_{i}$: $t^{\prime}_{i}=t_{i}-p0/\sqrt{N^{i}_{pe}}-p1-p2/N^{i}_{pe}.$ (6) The parameters $(p0,p1,p2)$ with the corresponding values of (9.42, 0.74, $-$4.60) for Hamamatsu PMTs and (41.31, $-$12.04, $-$20.02) for NNVT PMTs were found to minimize the bias and energy dependence of the reconstruction in this study. The difference in the parameters is mainly due to the difference in the TTSs of the PMTs. Following the correction, the times of different PMTs with different values of $N_{pe}$ are aligned. Figure 7: Heatmap of $R_{rec}$ (upper panel) and $R_{rec}-R_{true}$ (lower panel) as a function of $R_{true}$ for 4-MeV $e^{+}$ uniformly distributed in space calculated by the time-based algorithm. As shown in Fig. 7, the time-based algorithm provided a more accurate reconstructed vertex than the charge-based algorithm (Fig. 5). In addition, after the time–$N_{pe}$ correction, the reconstruction shows no obvious bias within the entire detector, even in the total reflection region. The reconstructed result was used as the initial value for the time likelihood algorithm. ## V Time likelihood algorithm ### V.1 Principle of the algorithm The time likelihood algorithm uses the scintillator response function to reconstruct the event vertex. The variable residual time $t_{res}(\vec{r}_{0},t_{0})$ for the $i$th PMT can be described as $t^{i}_{res}(\vec{r}_{0},t_{0})=t_{i}-\mathrm{tof}_{i}-t_{0},$ (7) where $t^{i}_{res}$ is the residual time of the $i$th PMT and $\vec{r}_{0}$, $t_{0}$, $t_{i}$, and $\mathrm{tof}_{i}$ are defined in Fig. 3. The scintillator response function mainly consists of the emission time profile of the scintillation photons and the TTS and the dark noise of PMTs. In principle, the additional delays introduced by the absorption, re-emission, scattering, and total reflection of the photon arriving to the PMT depend on the distance between the emission position and the individual PMTs. However, the differences are only noticeable for the late arrival hits, which are largely suppressed by the requirement for the earliest arriving photons in the time likelihood algorithm. Therefore, in the first-order approximation, the scintillator response function can be considered to be the same for all positions inside the scintillator. The scintillator response function can be described as follows. As described in Sec. III, when a charged particle interacts with a scintillator molecule, the molecule is excited, then de-excites, and emits photons. Typically, the scintillator has more than one component; thus, the emission time profile of the scintillation photons, $f(t_{res})$, can be described as $f(t_{res})=\sum_{k}\frac{\rho_{k}}{\tau_{k}}e^{\frac{-t_{res}}{\tau_{k}}},\sum_{k}\rho_{k}=1,$ (8) where each ${k}$ component is characterized by its decay time $\tau_{k}$ and intensity $\rho_{k}$. The different components result from the different excited states of the scintillator molecules. To consider the spread in the arrival time of photons at the PMTs, a convolution with a Gaussian function is applied, given by $g(t_{res})=\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(t_{res}-\nu)^{2}}{2\sigma^{2}}}\cdot{f(t_{res})}.$ (9) where $\sigma$ is the TTS of PMTs and $\nu$ is the average transit time. The dark noise, which occurs without incident photons in the PMTs, is not correlated with any physical event. The fraction of the dark noise in the total number of photoelectrons $\varepsilon_{dn}$ can be calculated based on the data acquisition (DAQ) windows, dark noise rate, and light yield of the LS. The probability of dark noise $\varepsilon(t_{res})$ is constant over time, where $\int_{DAQ}{\varepsilon(t_{res})dt_{res}}=\varepsilon_{dn}$. By adding $\varepsilon(t_{res})$ to $g(t_{res})$ and renormalizing its integral to 1, the probability density function (PDF) of the scintillator response function can be written as $p(t_{res})=(1-\varepsilon_{dn})\cdot{g(t_{res})}+\varepsilon(t_{res}).$ (10) The distribution of the residual time $t_{res}$ of an event for a hypothetical vertex can be compared with $p(t_{res})$. The best-fitting vertex and $t_{0}$ are chosen by minimizing the negative log-likelihood function $\mathcal{L}{(\vec{r}_{0},t_{0})}=-\ln(\prod_{i}p(t^{i}_{res})).$ (11) The parameters in Eq. 10 can be measured experimentally tres1 ; tres2 ; tres3 ; tres4 . In this work, the PDF from the MC simulation for the methodology study was employed. ### V.2 Probability density function Figure 8: PDF of the scintillator response function for PMTs detecting different numbers of photoelectrons. The upper panel shows the response function for Hamamatsu, the lower panel for the NNVT PMTs. Figure 9: Bias of the reconstructed $R$ (left panel), $\theta$ (middle panel), and $\phi$ (right panel) for different energies calculated by the time likelihood algorithm. Figure 10: Resolution of the reconstructed $R$ (left panel), $\theta$ (middle panel), and $\phi$ (right panel) as a function of energy calculated by the time likelihood algorithm. Different colors represent different PMT configurations. The PDF of the scintillator response function for PMTs detecting a single photoelectron was obtained from the MC simulation, using a 4.4-MeV $\gamma$ source located at the center of the detector, such that the distance to all PMTs is the same. For PMTs detecting multiple photoelectrons, the time of the earliest arriving photon is biased toward an earlier time. Therefore, the PDF need to be modified according to the first-order statistic of $p(t_{res})$ or the so-called first photoelectron timing technique fos ; fos2 ; fos3 as $p_{N_{pe}}(t_{res})=N_{pe}p(t_{res})(\int_{t_{res}}^{\infty}p(x)dx)^{N_{pe}-1},$ (12) where $p_{N_{pe}}(t_{res})$ is the PDF of the scintillator response function when the PMTs detect $N_{pe}$ hits. The PDF of two kinds of PMTs is shown in Fig. 8: the upper panel is for Hamamatsu while the lower panel is for NNVT PMTs. As the PDF is affected by the time resolution of the PMTs, the PDF of the NNVT is wider because of its inferior TTS. The inset in the lower panel shows the PDF on a logarithmic scale, and the time constant contribution of the dark noise $\varepsilon(t_{res})$ is clearly visible. ### V.3 Reconstruction performance The reconstructed vertex was compared with the true vertex in spherical coordinates ($R,\theta,\phi$) for the MC $e^{+}$ samples and fitted with a Gaussian function to analyze the bias and resolution. The bias of the reconstruction is shown in Fig. 9, where different colors represent events with different energies. As can be seen in the left panel of Fig. 9, the reconstructed $R$ is consistent with the true value in the central region, while an energy-dependent bias behavior is noticeable near the detector boundary. Given its regular bias behavior, the bias can be corrected with an energy-dependent correction. Moreover, although the reconstructed $R$ is biased, there is no bias in $\theta$ and $\phi$, as shown in the middle and right panels of Fig. 9, respectively. The spatial resolution of the vertex reconstruction as a function of energy is shown in Fig. 10. The $R$ bias was corrected before the analysis of the resolution. To study the individual effect of the TTS and dark noise on the vertex reconstruction, different MC samples were produced with and without these effects. The vertex reconstruction results are shown in Fig. 10. The magenta circles represent the default PMT configuration, as described in Sec. II. The red triangles represent an ideal configuration, which assumes perfect PMTs without the effects of the TTS and dark noise. The black squares represent the configuration of PMTs including only the dark noise effect, while the blue inverted triangles represent the PMT configuration including only the TTS effect. The exact values of the vertex resolution at 1.022 MeV and 10.022 MeV are summarized in Tables 2 and 3, respectively. The energy $E_{true}$ includes the energy of the annihilation gamma rays. The light yield was approximately 1300 detected $N_{pe}$ per 1 MeV of deposited energy in JUNO, and the energy nonlinearities on the light yield were ignored in the approximation. As can be seen in Tables 2 and 3, the dark noise has no effect at high energy and its effect at low energy is also highly limited. The largest effect results from the TTS in the time likelihood algorithm. The energy-dependent vertex resolution is approximately proportional to $1/\sqrt{N_{pe}}$ fos2 . Table 2: Vertex resolution for different PMT configurations at 1.022 MeV (detection of $\sim$1328 $N_{pe}$ in total, corresponding to $\sim$370 $N_{pe}$ detected by Hamamatsu PMTs). PMT configuration \| $R$ (mm) \| $\theta$ (degrees) \| $\phi$ (degrees) ---\|---\|---\|--- Ideal \| 60 \| 0.25 \| 0.31 With dark noise only \| 62 \| 0.27 \| 0.34 With TTS only \| 89 \| 0.37 \| 0.44 With TTS and dark noise \| 103 \| 0.40 \| 0.47 With TTS and dark noise (Hamamatsu PMTs only) \| 105 \| 0.42 \| 0.49 Table 3: Vertex resolution for different PMT configurations at 10.022 MeV (detection of $\sim$13280 $N_{pe}$ in total, corresponding to $\sim$ 3700 $N_{pe}$ detected by Hamamatsu PMTs). PMT configuration \| $R$ (mm) \| $\theta$ (degrees) \| $\phi$ (degrees) ---\|---\|---\|--- Ideal \| 19 \| 0.08 \| 0.11 With dark noise only \| 19 \| 0.08 \| 0.11 With TTS only \| 31 \| 0.13 \| 0.16 With TTS and dark noise \| 31 \| 0.13 \| 0.16 With TTS and dark noise (Hamamatsu PMTs only) \| 32 \| 0.14 \| 0.17 Owing to the low time resolution of the NNVT PMTs, in Fig. 10 only the reconstruction using Hamamatsu PMTs is shown (green circles). In this study, we found that the vertex resolution with Hamamatsu PMTs was similar to that of using all PMTs. The reconstruction speed was 3.5 times faster, because the fraction of the Hamamatsu PMTs was approximately 28% of all PMTs in the CD. Figure 11: Reconstructed event time $t_{0}$ at different energies. The reconstructed event time $t_{0}$ is shown in Fig. 11. The effect of $t_{0}$ is essentially a global shift of an event to match the scintillator response function PDF; in reality, $t_{0}$ is also affected by the trigger time and the time delay from the cable. The absolute value of $t_{0}$ can be neglected; only the relative difference of different events is important for the alignment of events. The small bump near $-$1.6 ns is correlated with the $R$ bias, and the long tail on the right side results from positronium formation. The variation in the reconstructed $t_{0}$ is within a few nanoseconds. ## VI Total reflection region calculated by the charge likelihood algorithm The time likelihood method described in Sec. V introduces a bias in the $R$ direction when the reconstructing events are close to the acrylic sphere. As mentioned in Ref. Smirnov_2003 , using a charge signal with the maximum likelihood method can provide better spatial resolution than the time likelihood algorithm when an event occurs near the detector boundary. In this section, we discuss the charge likelihood algorithm to reconstruct the event vertex in the total reflection region only, while the reconstruction result in the central region is omitted. The charge likelihood algorithm is based on the distribution of the number of photoelectrons in each PMT. With the mean expected number of photoelectrons $\mu(\vec{r_{0}},E)$ detected by each PMT at a given vertex and energy, the probability of observing $N_{pe}$ on a PMT follows a Poisson distribution. Furthermore, 1. $$ Probability for the $j$th PMT with no hits: $P_{nohit}^{j}(\vec{r_{0}},E)=e^{-\mu_{j}}$, 2. $$ Probability for the $i$th PMT with $N^{i}_{pe}$ hits: $P_{hit}^{i}(\vec{r_{0}},E)=\frac{\mu_{i}^{N^{i}_{pe}}e^{-\mu_{i}}}{N^{i}_{pe}!}$. Therefore, the probability of observing a hit pattern for an event can be written as $p(\vec{r_{0}},E)=\prod_{j}{P_{nohit}^{j}(\vec{r_{0}},E)}\cdot{\prod_{i}{P_{hit}^{i}(\vec{r_{0}},E)}}.$ (13) The best-fit values of $\vec{r_{0}}$ and $E$ can be obtained by minimizing the negative log-likelihood function $\mathcal{L}{(\vec{r_{0}},E)}=-\ln(p(\vec{r_{0}},E)).$ (14) In principle, $\mu(\vec{r_{0}},E)$ can be expressed by the equation $\mu_{i}(\vec{r_{0}},E)=Y\cdot\frac{\Omega(\vec{r_{0}},r_{i})}{4\pi}\cdot\varepsilon_{i}\cdot{f(\theta_{i})}\cdot{e^{-\sum_{m}{\frac{d_{m}}{\zeta_{m}}}}}\cdot{E}+\delta_{i},$ (15) where $Y$ is the energy scale factor, $\Omega(\vec{r_{0}},r_{i})$ is the solid angle of the $i$th PMT, $\varepsilon_{i}$ is the detection efficiency of the $i$th PMT, $f(\theta_{i})$ is the angular response of the $i$th PMT, $\theta_{i}$ is defined in Fig. 3, $\zeta_{m}$ is the attenuation length attleng in materials, and $\delta_{i}$ is the expected number of dark noise. This equation is based on the assumption that the scintillation light yield is linearly proportional to the energy. Figure 12: Mean expected number of photoelectron distribution as a function of radius $R$ and angle $\alpha$. This map is obtained by placing gamma sources at 29 specific positions along the Z-axis, which can be performed using a calibration procedure Wu_2019 . However, Eq. 15 cannot describe properly the contribution of the indirect light, the effect of light shadows because of the geometric structure, and the effect of the total reflection. Another solution is to use the model- independent method described in Ref. Wu_2019 : the mean expected number of photoelectrons can be obtained by placing gamma sources at 29 specific positions along the Z-axis, which can be performed using a calibration procedure calibsys . In this study, different from Ref. Wu_2019 , we focused on the performance of the vertex reconstruction. The mean expected number of the photoelectron distributions as a function of radius $R$ and angle $\alpha$ is shown in Fig. 12, and the definition of angle $\alpha$ is shown in Fig. 3. The mean expected number of photoelectrons $\mu$ obtained from Fig. 12 was used to calculate the hit probability. Instead of reconstructing ($R,\theta,\phi$) at the same time, $\theta$ and $\phi$ were fixed at the reconstructed values provided by the time likelihood algorithm, and only the event radius $R$ was reconstructed using the charge likelihood algorithm. Therefore, the probability in Eq. 13 can be rewritten as $p(R,E)=\prod_{j}{P_{nohit}^{j}(R,E)}\cdot{\prod_{i}{P_{hit}^{i}(R,E)}}.$ (16) The reconstruction performance, focusing on the total reflection region, is shown in Figs. 13 and 14. In the total reflection region, the mean value of the reconstructed $R$ was consistent with the true $R$, and the resolutions in the $R$ direction were 81 mm at 1.022 MeV and 30 mm at 10.022 MeV. Figure 13: Bias of the reconstructed $R$ in the total reflection region at different energies calculated by the charge likelihood algorithm. Figure 14: Resolution of reconstructed $R$ as a function of energy calculated by the time likelihood and charge likelihood algorithms ($R^{3}\leavevmode\nobreak\ >\leavevmode\nobreak\ 4000\leavevmode\nobreak\ \rm{m}^{3}$) . As the charge distribution provides good radial discrimination ability, this algorithm can provide better resolution and a significantly smaller bias compared with those of the time likelihood algorithm in the total reflection region. ## VII Performance summary The execution time of the reconstruction for each event was tested on a computing cluster with Intel Xeon Gold 6238R CPUs (2.2 GHz), as shown in Fig. 15. The execution time of the charge-based algorithm was in the order of $O(10^{-4})$ s per event, which cannot be presented in the figure. The execution time of the time-based and the time likelihood algorithm was proportional to the event energy and could be reduced by using only the Hamamatsu PMTs for the reconstruction. The execution time of the charge likelihood algorithm was independent of the event energy. Figure 15: Execution time for the reconstruction for different algorithms. The resolutions of the four algorithms in the $R$ direction are shown in Fig. 16. Owing to the large bias of the charge-based algorithm, a correction to remove the position-dependent bias was applied before the analysis of the resolution. Figure 16: Resolution of the reconstructed $R$ as a function of energy for different algorithms. The charged-based algorithm is suitable for online reconstruction tasks that require high speed but do not require high resolution. The time-based algorithm does not rely on MC; it can be used as a data-driven reconstruction method. The time likelihood and charge likelihood algorithms are relatively accurate and each has its own advantages in a specific detector region. ## VIII Discussion The vertex resolutions of KamLAND and Borexino are approximately 12 cm and 10 cm at 1 MeV, respectively, and for JUNO it is approximately 10.5 cm. The diameter of JUNO (35.4 m) is several times larger than that of KamLAND (13 m) and Borexino (8.5 m). Despite its larger size, JUNO is still able to achieve a similar vertex resolution based only on the PMT time information. In this study, various effects on the vertex reconstruction for JUNO were comprehensively analyzed. As expected, the TTS of the PMT is the dominant factor. The vertex reconstruction capability of JUNO is mainly based on the Hamamatsu PMTs. Although the number of NNVT PMTs is more than twice, their time information is not useful in the vertex reconstruction because of their significantly inferior TTS. After considering the light yield of the LS and the PMT coverage, the number of photons detected by the Hamamatsu PMTs is in the same range as those of the three detectors mentioned above. This provides an explanation of their similar vertex resolutions based only on the PMT time information. To fully exploit the large PMT coverage and large number of PMTs in JUNO, the charge information of PMTs also need to be utilized in addition to the time information to constrain the event vertex, especially near the detector boundary. At the same time, the effect of the dark noise of the PMT can be mitigated with appropriate treatment. A more accurate initial value also improves the performance of the vertex reconstruction. In addition to the event vertex, the event time can also be reconstructed simultaneously, which is a useful variable for downstream analyses. In LS detectors, in addition to the scintillation photons there are also Cherenkov photons, whose effects need to be studied in the future. The $R$ bias near the detector edge in the current results also indicates that a more accurate PDF of the scintillator response function is needed to include its dependence on the position as well as the particle type. All vertex reconstruction methods based on PMT time information use the time of the first arrival photons only; in principle, later photons might be useful as well. Therefore, novel methods also need to be explored. Our preliminary studies on algorithms based on machine learning qianz_2021 showed comparable vertex reconstruction performance, and they need to be further investigated ## IX Conclusion In this study, four algorithms for the reconstruction of the event vertex and event time were investigated in detail and verified using MC samples generated by the offline software of JUNO. 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# Length functions on groups and rigidity Shengkui Ye ###### Abstract Let $G$ be a group. A function $l:G\rightarrow[0,\infty)$ is called a length function if (1) $l(g^{n})=\|n\|l(g)$ for any $g\in G$ and $n\in\mathbb{Z};$ (2) $l(hgh^{-1})=l(g)$ for any $h,g\in G;$ and (3) $l(ab)\leq l(a)+l(b)$ for commuting elements $a,b.$ Such length functions exist in many branches of mathematics, mainly as stable word lengths, stable norms, smooth measure-theoretic entropy, translation lengths on $\mathrm{CAT}(0)$ spaces and Gromov $\delta$-hyperbolic spaces, stable norms of quasi-cocycles, rotation numbers of circle homeomorphisms, dynamical degrees of birational maps and so on. We study length functions on Lie groups, Gromov hyperbolic groups, arithmetic subgroups, matrix groups over rings and Cremona groups. As applications, we prove that every group homomorphism from an arithmetic subgroup of a simple algebraic $\mathbb{Q}$-group of $\mathbb{Q}$-rank at least $2,$ or a finite-index subgroup of the elementary group $E_{n}(R)$ $(n\geq 3)$ over an associative ring, or the Cremona group $\mathrm{Bir}(P_{\mathbb{C}}^{2})$ to any group $G$ having a purely positive length function must have its image finite. Here $G$ can be outer automorphism group $\mathrm{Out}(F_{n})$ of free groups, mapping classes group $\mathrm{MCG}(\Sigma_{g})$, $\mathrm{CAT}(0)$ groups or Gromov hyperbolic groups, or the group $\mathrm{Diff}(\Sigma,\omega)$ of diffeomorphisms of a hyperbolic closed surface preserving an area form $\omega.$ ### 0.1 Introduction The rigidity phenomena have been studied for many years. The famous Margulis superrigidity implies any group homomorphism between irreducible lattices in semisimple Lie groups of real rank $\mathrm{rk}_{\mathbb{R}}(G)\geq 2$ are virtually induced by group homomorphisms between the Lie groups. Therefore, group homomorphisms from ‘higher’-rank irreducible lattices to ‘lower’-rank irreducible lattices normally have finite images. Farb, Kaimanovich and Masur [26] [39] prove that every homomorphism from an (irreducible) higher rank lattice into the mapping class group $\mathrm{MCG}(\Sigma_{g})$ has a finite image. Bridson and Wade [17] showed that the same superrigidity remains true if the target is replaced with the outer automorphism group $\mathrm{Out}(F_{n})$ of the free group. Mimura [48] proves that every homomorphism from Chevalley group over commutative rings to $\mathrm{MCG}(\Sigma_{g})$ or $\mathrm{Out}(F_{n})$ has a finite image. Many other rigidity results can be found, e.g. [49] [18] [19] [33] [54] and [53]. In this article, we study rigidity phenomena with the notion of length functions. Let $G$ be a group. We call a function $l:G\rightarrow[0,\infty)$ a length function if 1) $l(g^{n})=\|n\|l(g)$ for any $g\in G$ and $n\in\mathbb{Z};$ 2) $l(aga^{-1})=l(g)$ for any $a,g\in G;$ 3) $l(ab)\leq l(a)+l(b)$ for commuting elements $a,b.$ Such length functions exist in geometric group theory, dynamical system, algebra, algebraic geometry and many other branches of mathematics. For example, the following functions $l$ are length functions (see Section 3 for more examples with details). * • (The stable word lengths) Let $G$ be a group generated by a symmetric (not necessarily finite) set $S.$ For any $g\in G,$ the word length $\phi_{S}(w)=\min\\{n\mid g=s_{1}s_{2}\cdots s_{n},$each $s_{i}\in S\\}$ is the minimal number of elements of $S$ whose product is $g.$ The stable length is defined as $l(g)=\lim_{n\rightarrow\infty}\frac{\phi_{S}(g^{n})}{n}.$ * • (Stable norms) Let $M$ be a compact smooth manifold and $G=\mathrm{Diff}(M)$ the diffeomorphism group consisting of all self-diffeomorphisms. For any diffeomorphism $f:M\rightarrow M,$ let $\\|f\\|=\sup_{x\in M}\\|D_{x}f\\|,$ where $D_{x}f$ is the induced linear map between tangent spaces $T_{x}M\rightarrow T_{f(x)}M.$ Define $l(f)=\max\\{\lim_{n\rightarrow+\infty}\frac{\log\\|f^{n}\\|}{n},\lim_{n\rightarrow+\infty}\frac{\log\\|f^{-n}\\|}{n}\\}.$ * • (smooth measure-theoretic entropy) Let $M$ be a $C^{\infty}$ closed Riemannian manifold and $G=\mathrm{Diff}_{\mu}^{2}(M)$ consisting of diffeomorphisms of $M$ preserving a Borel probability measure $\mu.$ Let $l(f)=h_{\mu}(f)$ be the measure-theoretic entropy, for any $f\in G=\mathrm{Diff}_{\mu}^{2}(M)$. * • (Translation lengths) Let $(X,d)$ be a metric space and $G=\mathrm{Isom}(X)$ consisting of isometries $\gamma:X\rightarrow X$. Fix $x\in X,$ define $l(\gamma)=\lim_{n\rightarrow\infty}\frac{d(x,\gamma^{n}x)}{n}.$ This contains the translation lengths on $\mathrm{CAT}(0)$ spaces and Gromov $\delta$-hyperbolic spaces as special cases. * • (average norm for quasi-cocycles) Let $G$ be a group and $E$ be a Hilbert space with an $G$-action by linear isometrical action. A function $f:G\rightarrow E$ is a quasi-cocyle if there exists $C>0$ such that $\\|f(gh)-f(g)-gf(h)\\|<C$ for any $g,h\in G.$ Let $l:G\rightarrow[0,+\infty)$ be defined by $l(g)=\lim_{n\rightarrow\infty}\frac{\\|f(g^{n})\\|}{n}.$ * • (Rotation numbers of circle homeomorphisms) Let $\mathbb{R}$ be the real line and $G=\mathrm{Home}_{\mathbb{Z}}(\mathbb{R})=\\{f\mid f:R\rightarrow R$ is a monotonically increasing homeomorphism such that $f(x+n)=f(x)$ for any $n\in\mathbb{Z}\\}.$ For any $f\in\mathrm{Home}_{\mathbb{Z}}(\mathbb{R})$ and $x\in[0,1),$ the translation number is defined as $l(f)=\lim_{n\rightarrow\infty}\frac{f^{n}(x)-x}{n}.$ * • (Asymptotic distortions) Let $f$ be a $C^{1+bv}$ diffeomorphism of the closed interval $[0,1]$ or the circle $S^{1}.$ (“bv” means derivative with finite total variation.) The asymptotic distortion of $f$ is defined (by Navas [50]) as $l(f)=\lim_{n\rightarrow\infty}\mathrm{var}(\log Df^{n}).$ This gives a length function $l$ on the group $\mathrm{Diff}^{1+bv}(M)$ of $C^{1+bv}$ diffeomorphisms for $M=[0,1]$ or $S^{1}.$ * • (Dynamical degree) Let $\mathbb{C}P^{n}$ be the complex projective space and $f:\mathbb{C}P^{n}\dashrightarrow\mathbb{C}P^{n}$ be a birational map given by $(x_{0}:x_{1}:\cdots:x_{n})\dashrightarrow(f_{0}:f_{1}:\cdots:f_{n}),$ where the $f_{i}$’s are homogeneous polynomials of the same degree without common factors. The degree of $f$ is $\deg f=\deg f_{i}.$ Define $l(f)=\max\\{\lim_{n\rightarrow\infty}\log\deg(f^{n})^{\frac{1}{n}},\lim_{n\rightarrow\infty}\log\deg(f^{-n})^{\frac{1}{n}}\\}.$ This gives a length function $l:\mathrm{Bir}(\mathbb{C}P^{n})\rightarrow[0,+\infty).$ Here $\mathrm{Bir}(\mathbb{C}P^{n})$ is the group of birational maps, also called Cremona group. The terminologies of length functions are used a lot in the literature (eg. [28], [22]). However, they usually mean different things from ours (in particular, it seems that the condition 3) has not been addressed for commuting elements before). Our first observation is the following result on vanishing of length functions. ###### Theorem 0.1 Let $G_{A}=\mathbb{Z}^{2}\rtimes_{A}\mathbb{Z}$ be an abelian-by-cyclic group, where $A\in\mathrm{SL}_{2}(\mathbb{Z})$. 1. (i) When the absolute value of the trace $\|\mathrm{tr}(A)\|>2,$ any length function $l:\mathbb{Z}^{2}\rtimes_{A}\mathbb{Z}\rightarrow\mathbb{R}_{\geq 0}$ vanishes on $\mathbb{Z}^{2}.$ 2. (ii) When $\|\mathrm{tr}(A)\|=2$ and $A\neq I_{2},$ any length function $l:\mathbb{Z}^{2}\rtimes_{A}\mathbb{Z}\rightarrow\mathbb{R}_{\geq 0}$ vanishes on the direct summand of $\mathbb{Z}^{2}$ spanned by eigenvectors of $A$. ###### Corollary 0.2 Suppose that the semi-direct product $G_{A}=\mathbb{Z}^{2}\rtimes_{A}\mathbb{Z}$ acts on a compact manifold by Lipschitz homeomorphisms (or $C^{2}$-diffeomorphisms, resp.). The topological entropy $h_{top}(g)=0$ (or Lyapunov exponents of $g$ are zero, resp.) for any $g\in\mathbb{Z}^{2}$ when $\|\mathrm{tr}(A)\|>2$ or any eigenvector $g\in\mathbb{Z}^{2}$ when $\|\mathrm{tr}(A)\|=2.$ It is well-known that the central element in the integral Heisenberg group $G_{A}$ (for $A=\begin{bmatrix}1&1\\\ 0&1\end{bmatrix}$) is distorted in the word metric. When the Heisenberg group $G_{A}$ acts on a $C^{\infty}$ compact Riemannian manifold, Hu-Shi-Wang [36] proves that the topological entropy and all Lyapunov exponents of the central element are zero. These results are special cases of Theorem 0.1 and Corollary 0.2, by choosing special length functions. A length function $l:G\rightarrow[0,\infty)$ is called purely positive if $l(g)>0$ for any infinite-order element $g.$ A group $G$ is called virtually poly-positive, if there is a finite-index subgroup $H<G$ and a subnormal series $1=H_{n}\vartriangleleft H_{n-1}\vartriangleleft\cdots\vartriangleleft H_{0}=H$ such that every finitely generated subgroup of each quotient $H_{i}/H_{i+1}$ $(i=0,...,n-1)$ has a purely positive length function. Our following results are on the rigidity of group homomorphisms. ###### Theorem 0.3 Let $\Gamma$ be an arithmetic subgroup of a simple algebraic $\mathbb{Q}$-group of $\mathbb{Q}$-rank at least $2.$ Suppose that $G$ is virtually poly-positive. Then any group homomorphism $f:\Gamma\rightarrow G$ has its image finite. ###### Theorem 0.4 Let $G$ be a group having a finite-index subgroup $H<G$ and a subnormal series $1=H_{n}\vartriangleleft H_{n-1}\vartriangleleft\cdots\vartriangleleft H_{0}=H$ satisfying that 1. (i) every finitely generated subgroup of each quotient $H_{i}/H_{i+1}$ $(i=0,...,n-1)$ has a purely positive length function, i.e. $G$ is virtually poly-positive; and 2. (ii) any torsion abelian subgroup in every finitely generated subgroup of each quotient $H_{i}/H_{i+1}$ $(i=0,...,n-1)$ is finitely generated. Let $R$ be a finitely generated associative ring with identity and $E_{n}(R)$ the elementary subgroup. Suppose that $\Gamma<E_{n}(R)$ is finite-index subgroup. Then any group homomorphism $f:\Gamma\rightarrow G$ has its image finite when $n\geq 3$. ###### Corollary 0.5 Let $\Gamma$ be an arithmetic subgroup of a simple algebraic $\mathbb{Q}$-group of $\mathbb{Q}$-rank at least $2,$ or a finite-index subgroup of the elementary subgroup $E_{n}(R)$ $(n\geq 3)$ for an associative ring $R.$ Then any group homomorphism $f:E\rightarrow G$ has its image finite. Here $G$ is one of the following groups: * • a Gromov hyperbolic group, * • $\mathrm{CAT(0)}$ group, * • automorphism group $\mathrm{Aut}(F_{k})$ of a free group, * • outer automorphism group $\mathrm{Out}(F_{k})$ of a free group or * • mapping class group $\mathrm{MCG}(\Sigma_{g})$ $(g\geq 2)$. * • the group $\mathrm{Diff}(\Sigma,\omega)$ of diffeomorphisms of a closed surface preserving an area form $\omega.$ ###### Theorem 0.6 Suppose that $G$ is virtually poly-positive. Let $R$ be a finitely generated associative ring of characteristic zero such that any nonzero ideal is of a finite index (eg. the ring of algebraic integers in a number field). Suppose that $S<E_{n}(R)$ is a finite-index subgroup of the elementary group. Then any group homomorphism $f:S\rightarrow G$ has its image finite when $n\geq 3$. ###### Corollary 0.7 Let $R$ be an associative ring of characteristic zero such that any nonzero ideal is of a finite index. Any group homomorphism $f:E\rightarrow G$ has its image finite, where $E<E_{n}(R)$ is finite-index subgroup and $n\geq 3$. Here $G$ is one of the followings: * • $\mathrm{Aut}(F_{k}),\mathrm{Out}(F_{k}),\mathrm{MCG}(\Sigma_{g}),$ * • a hyperbolic group, * • a $\mathrm{CAT}(0)$ group or more generally a semi-hyperbolic group, * • a group acting properly semi-simply on a $\mathrm{CAT}(0)$ space, or * • a group acting properly semi-simply on a $\delta$-hyperbolic space, * • the group $\mathrm{Diff}(\Sigma,\omega)$ of diffeomorphisms of a hyperbolic closed surface preserving an area form $\omega.$ Some relevent cases of Theorem 0.4 and Theorem 0.6 are already established in the literature. Bridson and Wade [17] showed that any group homomorphism from an irreducible lattice in a semisimple Lie group of real rank $\geq 2$ to the mapping class group $\mathrm{MCG}(\Sigma_{g})$ must have its image finite. However, Theorem 0.3 can never hold when $\Gamma$ is a cocompact lattice, since a cocompact lattice has its stable word length purely positive. When the length functions involved in the virtually poly-positive group $G$ are required to be stable word lengths, Theorem 0.3 holds more generally for $\Gamma$ non-uniform irreducible lattices a semisimple Lie group of real rank $\geq 2$ (see Proposition 8.4). When the length functions involved in the virtually poly-positive group $G$ are given by a particular kind of quasi- cocyles, Theorem 0.3 holds more generally for $\Gamma$ with property TT (cf. Py [54], Prop. 2.2). Haettel [33] prove that any action of a high-rank a higher rank lattice on a Gromov-hyperbolic space is elementary (i.e. either elliptic or parabolic). Guirardel and Horbez [33] prove that every group homomorphism from a high-rank lattice to the outer automorphism group of torsion-free hyperbolic group has finite image. Thom [57] (Corollary 4.5) proves that any group homomorphism from a boundedly generated with property (T) to a Gromov hyperbolic group has finite image. Compared with these results, our target group $G$ and the source group $E_{n}(R)$ (can be defined over any non-commutative ring) in Theorem 0.4 are much more general. The inequalities of $n$ in Theorem 0.4, Theorem 0.6 and Corollary 0.5, Corollary 0.7 can not be improved, since $\mathrm{SL}_{2}(\mathbb{Z})$ is hyperbolic. The group $\Gamma$ in Corrolary 0.5 has Kazhdan’s property T (i.e. an arithmetic subgroup of a simple algebraic $\mathbb{Q}$-group of $\mathbb{Q}$-rank at least $2,$ or a finite-index subgroup of the elementary subgroup $E_{n}(R),$ $n\geq 3,$ for an associative ring $R$ has Kazhdan’s property T by [23]). However, there exist hyperbolic groups with Kazhdan’s property $T$ (cf. [32], Section 5.6). This implies that Corrolary 0.5 does not hold generally for groups $\Gamma$ with Kazhdan’s property T. Franks and Handel [27] prove that any group homomorphism from a quasi-simple group containing a subgroup isomorphic to the three-dimensional integer Heisenberg group, to the group $\mathrm{Diff}(\Sigma,\omega)$ of diffeomorphisms of a closed surface preserving an area form $\omega,$ has its image finite (cf. Lemma 8.3). We now study length functions on the Cremona groups. ###### Theorem 0.8 Let $\mathrm{Bir}(\mathbb{P}_{k}^{n})$ $(n\geq 2)$ be the group of birational maps on the projective space $\mathbb{P}_{k}^{n}$ over an algebraic closed field $k$. Any length function $l:\mathrm{Bir}(\mathbb{P}_{k}^{n})\rightarrow[0,+\infty)$ vanishes on the automorphism group $\mathrm{Aut}(\mathbb{P}_{k}^{n})=\mathrm{PGL}_{n+1}(k).$ When $n=2,$ a result of Blanc and Furter [9] (page 7 and Proposition 4.8.10) implies that there are three length functions $l_{1},l_{2},l_{3}$ on $\mathrm{Bir}(\mathbb{P}_{k}^{2})$ such that any element $g\in\mathrm{Bir}(\mathbb{P}_{k}^{n})$ satisfying $l_{1}(g)=l_{2}(g)=l_{3}(g)=0$ is either finite or conjugate to an element in $\mathrm{Aut}(\mathbb{P}_{k}^{2}).$ This implies that the automorphism group $\mathrm{Aut}(\mathbb{P}_{k}^{n})$ (when $k=2$) is one of the ‘largest’ subgroups of $\mathrm{Bir}(\mathbb{P}_{k}^{n})$ on which every length function vanishes. ###### Corollary 0.9 Let $G$ be a virtually poly-positive group. Any group homomorphism $f:\mathrm{Bir}(\mathbb{P}_{k}^{2})\rightarrow G$ is trivial, for an algebraic closed field $k$. In particular, Corrolary 0.9 implies that any quotient group of $\mathrm{Bir}(\mathbb{P}_{k}^{2})$ cannot act properly semisimply neither on a Gromov $\delta$-hyperbolic space nor a $\mathrm{CAT}(0)$ space. This is interesting, considering the following facts. There are (infinite-dimensional) hyperbolic space and cubical complexes, on which $\mathrm{Bir}(P_{k}^{2})$ acts isometrically (see [21], Section 3.1.2 and [45]). The Cremona group $\mathrm{Bir}(\mathbb{P}_{k}^{2})$ is sub-quotient universal: every countable group can be embedded in a quotient group of $\mathrm{Bir}(\mathbb{P}_{k}^{2})$ (see [21], Theorem 4.7). Moreover, Blanc- Lamy-Zimmermann [10] (Theorem E) proves that when $n\geq 3,$ there is a surjection from $\mathrm{Bir}(\mathbb{P}_{k}^{n})$ onto a free product of two- element groups $\mathbb{Z}/2.$ This means that Corrolary 0.9 can never hold for higher dimensional Cremona groups. As byproducts, we give characterizations of length functions on Lie groups. Our next result is that there is essentially only one length function on the special linear group $\mathrm{SL}_{2}(\mathbb{R})$: ###### Theorem 0.10 Let $G=\mathrm{SL}_{2}(\mathbb{R}).$ Any length function $l:G\rightarrow[0,+\infty)$ continuous on the subgroup $SO(2)$ and the diagonal subgroup is proposional to the translation function $\tau(g):=\inf_{x\in X}d(x,gx),$ where $X=\mathrm{SL}_{2}(\mathbb{R})/\mathrm{SO}(2)$ is the upper-half plane. More generally, we study length functions on Lie groups. Let $G$ be a connected semisimple Lie group whose center is finite with an Iwasawa decomposition $G=KAN$. Let $W$ be the Weyl group, i.e. the quotient group of the normalizers $N_{K}(A)$ modulo the centralizers $C_{K}(A)$. Our second result shows that a length function $l$ on $G$ is uniquely determined by its image on $A.$ ###### Theorem 0.11 Let $G$ be a connected semisimple Lie group whose center is finite with an Iwasawa decomposition $G=KAN$. Let $W$ be the Weyl group. 1. (i) Any length function $l$ on $G$ that is continuous on the maximal compact subgroup $K$ is determined by its image on $A.$ 2. (ii) Conversely, any length function $l$ on $A$ that is $W$-invariant (i.e. $l(w\cdot a)=l(a)$) can be extended to be a length function on $G$ that vanishes on the maximal compact subgroup $K.$ The proofs of Theorems 0.10 0.11 are based the Jordan-Chevalley decompositions of algebraic groups and Lie groups. We will prove that any length function on a Heisenberg group vanishes on the central elements (see Lemma 5.2). This is a key step for many other proofs. Based this fact, we prove Theorems 0.3, 0.6, 0.4, 0.8 by looking for Heisenberg subgroups. In Section 1, we give some elementary facts on the length functions. In Section 2, we discuss typical examples of length functions. In later sections, we study length functions on Lie groups, algebraic groups, hyperbolic groups, matrix groups and the Cremona groups. ## 1 Basic properties of length functions ### 1.1 Length functions ###### Definition 1.1 Let $G$ be a group. A function $l:G\rightarrow[0,\infty)$ is called a length function if 1) $l(g^{n})=\|n\|l(g)$ for any $g\in G$ and $n\in\mathbb{Z}.$ 2) $l(aga^{-1})=l(g)$ for any $a,g\in G.$ 3) $l(ab)\leq l(a)+l(b)$ for commuting elements $a,b.$ ###### Lemma 1.2 Any torsion element $g\in G$ has length $l(g)=0.$ Proof. Note that $l(1)=2l(1)$ and thus $l(1)=0.$ If $g^{n}=1,$ then $l(g)=\|n\|l(1)=0.$ Recall that a subset $V$ of a real vector space is a convex cone, if $av+bw\in V$ for any $v,w\in V$ and any non-negative real numbers $a,b\geq 0.$ ###### Lemma 1.3 The set $\mathrm{Func}(G)$ of all length functions on a group $G$ is a convex cone. Proof. It is obvious that for two functions $l_{1},l_{2}$ on $G,$ a non- negative linear combination $al_{1}+bl_{2}$ is a new length function. ###### Lemma 1.4 Let $f:G\rightarrow H$ be a group homomorphism between two groups $G$ and $H.$ For any length function $l:H\rightarrow[0,\infty),$ the composite $l\circ f$ is a length function on $G.$ Proof. It is enough to note that a group homomorphism preserves powers of elements, conjugacy classes and commutativity of elements. ###### Corollary 1.5 For a group $G,$ let $\mathrm{Out}(G)=\mathrm{Aut}(G)/\mathrm{Inn}(G)$ be the outer automorphism group. Then $\mathrm{Out}(G)$ acts on the set $\mathrm{Func}(G)$ of all length functions by pre-compositions $l\mapsto l\circ g,$ where $l\in\mathrm{Func}(G),$ $g\in\mathrm{Out}(G)$. This action preserves scalar multiplications and linear combinations (with non-negative coefficients). Proof. For an inner automorphism $I_{g}:G\rightarrow G$ given by $I_{g}(h)=ghg^{-1},$ the length function $l\circ I_{g}=l$ since $l$ is invariant under conjugation. Therefore, the outer automorphism group $\mathrm{Out}(G)$ has an action on $\mathrm{Func}(G).$ It is obvious that the pre-compositions preserve scalar multiplications and linear combinations with non-negative coefficients. ###### Definition 1.6 A length function $l:G\rightarrow[0,\infty)$ is primitive if it is not a composite $l^{\prime}\circ f$ for a non-trivial surjective group homomorphism $f:G\twoheadrightarrow H$ and a length function $l^{\prime}:$ $H\rightarrow[0,\infty).$ ###### Lemma 1.7 Suppose that a length function $l:G\rightarrow[0,\infty)$ vanishes on a central subgroup $H<G.$ Then $l$ factors through the quotient group $G/H.$ In other words, there exists a length function $l^{\prime}:G/H\rightarrow[0,\infty)$ such that $l=l^{\prime}\circ q,$ where $q:G\rightarrow G/H$ is the quotient group homomorphism. Proof. Write $G=\cup gH,$ the union of left cosets. For any $h\in H,$ we have $l(gh)\leq l(g)+l(h)=l(g)$ and $l(g)=l(ghh^{-1})\leq l(gh).$ Therefore, $l(gh)=l(g)$ for any $h\in H.$ Define $l^{\prime}(gH)=l(g).$ Then $l^{\prime}$ is a length function on the quotient group $G/H.$ The required property follows the definition easily. ###### Corollary 1.8 Suppose that a group $G$ has non-trivial finite central subgroup $Z(G).$ Any length function $l$ on $G$ factors through $G/Z(G).$ Proof. This follows Lemma 1.7 and Lemma 1.2. ###### Lemma 1.9 Let $G$ be a group. Suppose that any non-trivial normal subgroup $H\vartriangleleft G$ is of finite index. Then any non-vanishing length function $l:G\rightarrow[0,\infty)$ is primitive. Proof. Suppose that $l$ is a composite $l^{\prime}\circ f$ for a non-trivial surjective group homomorphism $f:G\twoheadrightarrow H$ and a length function $l^{\prime}:H\rightarrow[0,\infty).$ By the assumption of $G,$ the quotient group $H$ is finite. This implies that $l^{\prime}$ and thus $l$ vanishes, which is a contradiction. ###### Theorem 1.10 Let $\Gamma$ be an irreducible lattice in a connected irreducible semisimple Lie group of real rank $\geq 2.$ Then any non-vanishing length function $l:\Gamma\rightarrow[0,\infty)$ factors through a primitive function on $\Gamma/Z(\Gamma)$. Proof. By the Margulis-Kazhdan theorem (see [62], Theorem 8.1.2), any normal subgroup $N$ of $\Gamma$ either lies in the center of $\Gamma$ (and hence it is finite) or the quotient group $\Gamma/N$ is finite. Corollary 1.8 implies that $l$ factors through a length function $l^{\prime}$ on $\Gamma/Z(\Gamma).$ The previous lemma 1.9 implies that $l^{\prime}$ is primitive. ## 2 Examples of length functions Let’s see a general example first. Let $G$ be a goup and $f:G\rightarrow[0,+\infty)$ be a function satisfying $f(gh)\leq f(g)+f(h)$ and $f(g)=f(g^{-1})$ for any elements $g,h\in G.$ Define $l:G\rightarrow[0,+\infty)$ by $l(g)=\lim_{n\rightarrow\infty}\frac{f(g^{n})}{n}$ for any $g\in G.$ ###### Lemma 2.1 The function $l$ is a length function in the sense of Definition 1.1. Proof. For any $g\in G,$and natural numbers $n,m,$ we have $f(g^{n+m})\leq f(g^{n})+f(g^{m}).$ This means that $\\{f(g^{n})\\}_{n=1}^{\infty}$ is a subadditive sequence and thus the limit $\lim_{n\rightarrow\infty}\frac{f(g^{n})}{n}$ exists. This shows that $l$ is well-defined. From the definition of $l,$ it is clear that $l(g^{n})=\|n\|l(g)$ for any integer $n.$ Let $h\in G.$ We have $l(hgh^{-1})=\lim_{n\rightarrow\infty}\frac{f(hg^{n}h^{-1})}{n}\leq\lim_{n\rightarrow\infty}\frac{f(h)+f(g^{n})+f(h^{-1})}{n}=\lim_{n\rightarrow\infty}\frac{f(g^{n})}{n}=l(g).$ Similarly, we have $l(g)=l(h^{-1}(hgh^{-1})h)\leq l(hgh^{-1})$ and thus $l(g)=l(hgh^{-1}).$ For commuting elements $a,b,$ we have $(ab)^{n}=a^{n}b^{n}.$ Therefore, $\displaystyle l(ab)$ $\displaystyle=$ $\displaystyle\lim_{n\rightarrow\infty}\frac{f((ab)^{n})}{n}=\lim_{n\rightarrow\infty}\frac{f(a^{n}b^{n})}{n}$ $\displaystyle\leq$ $\displaystyle\lim_{n\rightarrow\infty}\frac{f(a^{n})+f(b^{n})}{n}\leq l(a)+l(b).$ Many (but not all) length functions $l$ come from subadditive functions $f.$ ### 2.1 Stable word lengths Let $G$ be a group generated by a (not necessarily finite) set $S$ satisfying $s^{-1}\in S$ for each $s\in S.$ For any $g\in G,$ the word length $\phi_{S}(w)=\min\\{n\mid g=s_{1}s_{2}\cdots s_{n},$each $s_{i}\in S\\}$ is the minimal number of elements of $S$ whose product is $g.$ The stable length $l(g)=\lim_{n\rightarrow\infty}\frac{\phi_{S}(g^{n})}{n}.$ Since $\phi_{S}(g^{n})$ is subadditive, the limit always exists. ###### Lemma 2.2 The stable length $l:G\rightarrow[0,+\infty)$ is a length function in the sense of Definition 1.1. Proof. From the definition of the word length $\phi_{S},$ it is clear that $\phi_{S}(gh)\leq\phi_{S}(g)+\phi_{S}(h)$ and $\phi_{S}(g)=\phi_{S}(g^{-1})$ for any $g,h\in G.$ The claim is proved by Lemma 2.1. When $S$ is the set of commutators, the $l(g)$ is called the stable commutator length, which is related to lots of topics in low-dimensional topology (see Calegari [20]). ### 2.2 Growth rate Let $G$ be a group generated by a finite set $S$ satisfying $s^{-1}\in S$ for each $s\in S.$ Suppose $\|\cdot\|_{S}$ is the word length of $(G,S).$ For any automorphism $\alpha:G\rightarrow G,$ define $l^{\prime}(\alpha)=\max\\{\|\alpha(s_{i})\|_{S}:s_{i}\in S\\}.$ Let $l(\alpha)=\lim_{n\rightarrow\infty}\frac{\log l^{\prime}(\alpha^{n})}{n}.$ This number $l(\alpha)$ is called the algebraic entropy of $\alpha$ (cf. [40], Definition 3.1.9, page 114). ###### Lemma 2.3 Let $\mathrm{Aut}(G)$ be the group of automorphisms of $G.$ The function $l:\mathrm{Aut}(G)\rightarrow[0,+\infty)$ is a length function in the sense of Definition 1.1. Proof. Since $\alpha(s_{i})^{-1}=\alpha^{-1}(s_{i})$ for any $s_{i}\in S,$ we know that $l^{\prime}(\alpha)=l^{\prime}(\alpha^{-1}).$ For another automorphism $\beta:G\rightarrow G,$ let $l^{\prime}(\beta)=\|\beta(s_{i})\|_{S}$ for some $s_{i}\in S.$ Suppose that $\beta(s_{i})=s_{i_{1}}s_{i_{2}}\cdots s_{i_{k}}$ with $k=l^{\prime}(\beta).$ Then $\|(\alpha\beta)(s_{i})\|_{S}=\|\alpha(s_{i_{1}})\alpha(s_{i_{2}})\cdots\alpha(s_{i_{k}})\|_{S}\leq l^{\prime}(\alpha)k.$ This proves that $l^{\prime}(\alpha\beta)\leq l^{\prime}(\alpha)l^{\prime}(\beta).$ The claim is proved by Lemma 2.1. Fix $g\in G.$ For any automorphism $\alpha:G\rightarrow G,$ define $b_{n}=\|\alpha^{n}(g)\|_{S}.$ Suppose that $g=s_{1}s_{2}\cdots s_{k}$ with $k=\|g\|_{S}.$ Note that $b_{n}=\|\alpha^{n}(g)\|_{S}=\|\alpha^{n}(s_{1})\alpha^{n}(s_{2})\cdots\alpha^{n}(s_{k})\|_{S}\leq l^{\prime}(\alpha^{n})\|g\|_{S}.$ Therefore, we have $\lim\sup_{n\rightarrow\infty}\frac{\log b_{n}}{n}\leq l(\alpha).$ This implies that $l(\alpha)$ is an upper bounded for growth rate of $\\{\|\alpha^{n}(g)\|_{S}\\}.$ The growth rate is studied a lot in geometric group theory (for example, see [43] for growth of automorphisms of free groups). ### 2.3 Matrix norms and group acting on smooth manifolds For a square matrix $A,$ the matrix norm $\\|A\\|=\sup_{\\|x\\|=1}\\|Ax\\|.$ Define the stable norm $s(A)=\lim_{n\rightarrow+\infty}\frac{\log\\|A^{n}\\|}{n}.$ Since $\\|AB\\|\leq\\|A\\|\\|B\\|$ for any two matrices $A,B,$ the sequence $\\{\log\\|A^{n}\\|\\}_{n=1}^{\infty}$ is subadditive and thus the limit exists. ###### Lemma 2.4 Let $G=\mathrm{GL}_{n}(\mathbb{R})$ be the general linear group. The function $l:G\rightarrow[0,+\infty)$ defined by $l(g)=\max\\{s(g),s(g^{-1})\\}$ is a length function in the sense of Definition 1.1. Proof. From the definition of the matrix norm, it is clear that $\log\\|gh\\|\leq\log\\|g\\|+\log\\|h\\|$ for any $g,h\in G.$ Then $l(g)=\max\\{s(g),s(g^{-1})\\}$ is a length function by Lemma 2.1. Let $M$ be a compact smooth manifold and $\mathrm{Diff}(M)$ the diffeomorphism group consisting of all self-diffeomorphisms. For any diffeomorphism $f:M\rightarrow M,$ let $\\|f\\|=\sup_{x\in M}\\|D_{x}f\\|,$ where $D_{x}f$ is the induced linear map between tangent spaces $T_{x}M\rightarrow T_{f(x)}M.$ Define $l(f)=\max\\{\lim_{n\rightarrow+\infty}\frac{\log\\|f^{n}\\|}{n},\lim_{n\rightarrow+\infty}\frac{\log\\|f^{-n}\\|}{n}\\}.$ A similar argument as the proof of the previous lemma proves the following. ###### Lemma 2.5 Let $G$ be a group acting on a smooth manifold $M$ by diffeomorphisms. The function $l:G\rightarrow[0,+\infty)$ is a length function in the sense of Definition 1.1. For an $f$-invariant Borel probability measure $\mu$ on $M,$ it is well known (see [O]) that there exists a measurable subset $\Gamma_{f}\subset M$ with $\mu(\Gamma_{f})=1$ such that for all $x\in\Gamma_{f}$ and $u\in T_{x}M,$ the limit $\chi(x,u,f)=\lim\frac{1}{n}\log\\|D_{x}f^{n}(u)\\|$ exists and is called Lyapunov exponent of $u$ at $x.$ From the definitions, we know that $\chi(x,u,f)\leq l(f)$ for any $x\in\Gamma_{f}$ and $u\in T_{x}M.$ ### 2.4 Smooth measure-theoretic entropy Let $T:X\rightarrow X$ be a measure-preserving map of the probability space $(X,\mathfrak{B},m).$ For a finite-sub-$\sigma$-algebra $A=\\{A_{1},A_{2},...,A_{k}\\}$ of $\mathfrak{B},$ denote by $\displaystyle H(A)$ $\displaystyle=$ $\displaystyle-\sum m(A_{i})\log m(A_{i}),$ $\displaystyle h(T,A)$ $\displaystyle=$ $\displaystyle\lim\frac{1}{n}H(\vee_{i=0}^{n-1}T^{-i}A),$ where $\vee_{i=0}^{n-1}T^{-i}A$ is a set consisting of sets of the form $\cap_{i=0}^{n-1}T^{-i}A_{j_{i}}.$ The entropy of $T$ is defined as $h_{m}(T)=\sup h(T,A),$ where the supremum is taken over all finite sub- algebra $A$ of $\mathfrak{B}.$ For more details, see Walters [58] (Section 4.4). ###### Lemma 2.6 Let $M$ be a $C^{\infty}$ closed Riemannian manifold and $G=\mathrm{Diff}_{\mu}^{2}(M)$ consisting of diffeomorphisms of $M$ preserving a Borel probability measure $\mu.$ The entropy $h_{\mu}$ is a length function on $\mathrm{Diff}_{\mu}^{2}(M)$ in the sense of Definition 1.1. Proof. For any $f,g\in\mathrm{Diff}_{\mu}^{2}(M)$ and integer $n,$ it is well- known that $h_{\mu}(f^{n})=\|n\|h_{\mu}(f)$ and $h_{\mu}(f)=h_{\mu}(gfg^{-1})$ (cf. [58], Theorem 4.11 and Theorem 4.13). Hu [35] proves that $h_{\mu}(fg)\leq h_{\mu}(f)+h_{\mu}(g)$ when $fg=gf.$ ### 2.5 Stable translation length on metric spaces Let $(X,d)$ be a metric space and $\gamma:X\rightarrow X$ an isometry. Fix $x\in X.$ Note that $d(x,\gamma_{1}\gamma_{2}x)\leq d(x,\gamma_{1}x)+d(\gamma_{1}x,\gamma_{1}\gamma_{2}x)=d(x,\gamma_{1}x)+d(x,\gamma_{2}x)$ and $d(x,\gamma_{1}x)=d(x,\gamma_{1}^{-1}x)$ for any isometries $\gamma_{1},\gamma_{2}.$ Define $l(\gamma)=\lim_{n\rightarrow\infty}\frac{d(x,\gamma^{n}x)}{n}.$ For any $y\in X,$ we have $\displaystyle d(x,\gamma^{n}x)$ $\displaystyle\leq$ $\displaystyle d(x,y)+d(y,\gamma^{n}y)+d(\gamma^{n}y,\gamma^{n}x)$ $\displaystyle=$ $\displaystyle 2d(x,y)+d(y,\gamma^{n}y)$ and thus $\lim_{n\rightarrow\infty}\frac{d(x,\gamma^{n}x)}{n}\leq\lim_{n\rightarrow\infty}\frac{d(y,\gamma^{n}y)}{n}.$ Similarly, we have the other direction $\lim_{n\rightarrow\infty}\frac{d(y,\gamma^{n}y)}{n}\leq\lim_{n\rightarrow\infty}\frac{d(x,\gamma^{n}x)}{n}.$ This shows that the definition of $l(\gamma)$ does not depend on the choice of $x.$ ###### Lemma 2.7 Let $G$ be a group acting isometrically on a metric space $X.$ Then the function $l:G\rightarrow[0,+\infty)$ defined by $g\longmapsto l(g)$ as above is a length function in the sense of Definition 1.1. Proof. This follows Lemma 2.1. ### 2.6 Translation lengths of isometries of CAT(0) spaces In this subsection, we will prove that the translation length on a CAT(0) space defines a length function. First, let us introduce some notations. Let $(X,d_{X})$ be a geodesic metric space, i.e. any two points $x,y\in X$ can be connected by a path $[x,y]$ of length $d_{X}(x,y)$. For three points $x,y,z\in X,$ the geodesic triangle $\Delta(x,y,z)$ consists of the three vertices $x,y,z$ and the three geodesics $[x,y],[y,z]$ and $[z,x].$ Let $\mathbb{R}^{2}$ be the Euclidean plane with the standard distance $d_{\mathbb{R}^{2}}$ and $\bar{\Delta}$ a triangle in $\mathbb{R}^{2}$ with the same edge lengths as $\Delta$. Denote by $\varphi:\Delta\rightarrow\bar{\Delta}$ the map sending each edge of $\Delta$ to the corresponding edge of $\bar{\Delta}.$ The space $X$ is called a CAT(0) space if for any triangle $\Delta$ and two elements $a,b\in\Delta,$ we have the inequality $d_{X}(a,b)\leq d_{\mathbb{R}^{2}}(\varphi(a),\varphi(b)).$ The typical examples of CAT(0) spaces include simplicial trees, hyperbolic spaces, products of CAT(0) spaces and so on. From now on, we assume that $X$ is a complete CAT(0) space. Denote by Isom$(X)$ the isometry group of $X.$ For any $g\in$ Isom$(X)$, let $\mathrm{Minset}(g)=\\{x\in X:d(x,gx)\leq d(y,gy)\text{ for any }y\in X\\}$ and let $\tau(g)=\inf\nolimits_{x\in X}d(x,gx)$ be the translation length of $g.$ When the fixed-point set $\mathrm{Fix}(g)\neq\emptyset,$ we call $g$ elliptic. When $\mathrm{Minset}(g)\neq\emptyset$ and $d_{X}(x,gx)=\tau(g)>0$ for any $x\in\mathrm{Minset}(g),$ we call $g$ hyperbolic. The group element $g$ is called semisimple if the minimal set $\mathrm{Minset}(g)$ is not empty, i.e. it is either elliptic or hyperbolic. A subset $C$ of a CAT(0) space if convex, if any two points $x,y\in C$ can connected by the geodesic segment $[x,y]\subset C.$ A group $G$ is called CAT(0) if $G$ acts properly discontinuously and cocompactly on a CAT(0) space $X$. In such a case, any infinite-order element in $G$ acts hyperbolically on $X.$ For more details on CAT(0) spaces, see the book of Bridson and Haefliger [16]. The following was proved by Ballmann-Gromov-Schroeder [2] (Lemma 6.6, page 83). The original proof was for Hardmard manifolds, which also holds for general cases. For completeness, we give details here. ###### Lemma 2.8 Let $\gamma:X\rightarrow X$ be an isometry of a complete CAT(0) space $X.$ For any $x_{0}\in X,$ we have $\tau(\gamma):=\inf_{x\in X}d(\gamma x,x)=\lim_{k\rightarrow\infty}\frac{d(\gamma^{k}x_{0},x_{0})}{k}.$ Proof. For any $p=x_{0}\in X,$ let $m$ be the middle point of $[p,\gamma p].$ We have that $d(m,\gamma m)\leq\frac{1}{2}d(p,\gamma^{2}p)$ by the convexity of length functions. Therefore, $d(p,\gamma^{2}p)\geq 2\tau(\gamma)$ and $\tau(\gamma^{2})\geq 2\tau(\gamma).$ Note that $d(p,\gamma^{2}p)\leq d(p,\gamma p)+d(\gamma p,\gamma^{2}p)=2d(p,\gamma p)$ and thus $\tau(\gamma^{2})\leq 2\tau(\gamma).$ Inductively, we have $2^{n}\tau(\gamma)\leq d(p,\gamma^{2^{n}}p)\leq 2^{n}d(p,\gamma p).$ Note that the limit $\lim_{k\rightarrow\infty}\frac{d(\gamma^{k}p,p)}{k}$ exists and is independent of $p$ (see the previous subsection). Therefore, the limit $\lim_{k\rightarrow\infty}\frac{d(\gamma^{k}p,p)}{k}$ equals to $\tau(\gamma).$ ###### Corollary 2.9 Let $X$ be a complete CAT(0) space and $G$ a group acting on $X$ by isometries. For any $g\in G,$ define $\tau(g)=\inf_{x\in X}d(x,gx)$ as the translation length. Then $\tau:G\rightarrow[0,+\infty)$ is a length function in the sense of Definition 1.1. Proof. This follows Lemma 2.8 and Lemma 2.7. ### 2.7 Translation lengths of Gromov $\delta$-hyperbolic spaces Let $\delta>0.$ A geodesic metric space $X$ is called Gromov $\delta$-hyperbolic if for any geodesic triangle $\Delta xyz$ one side $[x,y]$ is contained a $\delta$-neighborhood of the other two edges $[x,z]\cup[y,z].$ Fix $x_{0}\in X.$ Any isometry $\gamma:X\rightarrow X$ is called elliptic if $\\{\gamma^{n}x\\}_{ne\mathbb{Z}}$ is bounded. If the orbit map $\mathbb{Z}\rightarrow X$ given by $n\mapsto\gamma^{n}x_{0}$ is quasi- isometric (i.e. there exists $A\geq 1$ and $B\geq 0$ such that $\frac{1}{A}\|n-m\|-B\leq d_{X}(\gamma^{n}x_{0},\gamma^{m}x_{0})\leq A\|n-m\|+B$ for any integers $n,m$), we call $\gamma$ is hyperbolic. Otherwise, we call $\gamma$ is parabolic. Define $l(\gamma)=\lim_{n\rightarrow\infty}\frac{d(\gamma^{n}x_{0},x_{0})}{n}.$ For any group $G$ acts isometrically on a $\delta$-hyperbolic space, the function $l:G\rightarrow[0,\infty)$ is a length function by Lemma 2.7. A finitely generated group $G$ is Gromov $\delta$-hyperbolic if for some finite generating set $S,$ the Caley graph $\Gamma(G,S)$ is Gromov $\delta$-hyperbolic. Any infinite-order element $g$ in a Gromov $\delta$-hyperbolic group is hyperbolic and thus has positive length $l(g)>0$ (cf. [32], 8.1.D). For more details on hyperbolic spaces and hyperbolic groups, see the book [32] of Gromov. ### 2.8 Quasi-cocycles Let $G$ be a group and $(E,\\|\cdot\\|)$ be a normed vector space with an $G$-action by linear isometries. A function $f:G\rightarrow E$ is a quasi- cocyle if there exists $C>0$ such that $\\|f(gh)-f(g)-gf(h)\\|<C$ for any $g,h\in G.$ Let $l:G\rightarrow[0,+\infty)$ be defined by $l(g)=\lim_{n\rightarrow\infty}\frac{\\|f(g^{n})\\|}{n}.$ Note that $\\|f(g^{n+m})\\|\leq\\|f(g^{n})\\|+\\|f(g^{m})\\|+C$ for any integers $n,m\geq 0.$ This general subadditive property implies that the limit $\lim_{n\rightarrow\infty}\frac{\\|f(g^{n})\\|}{n}$ exists (see [56], Theorem 1.9.2, page 22). We call $l$ the average norm. Many applications of quasi- cocycles can be found in [49]. ###### Lemma 2.10 For any quasi-cocyle $f:G\rightarrow E,$ the average norm $l$ is a length function. Proof. For any natural number $n,$ we have $\\|f(1)-f(g^{-n})-g^{-n}f(g^{n})\\|<C$ and thus $\\|\frac{f(1)-f(g^{-n})-g^{-n}f(g^{n})}{n}\\|<\frac{C}{n}.$ Taking the limit, we have $\lim_{n\rightarrow\infty}\frac{\\|f(g^{-n})\\|}{n}=\lim_{n\rightarrow\infty}\frac{\\|f(g^{n})\\|}{n}.$ Therefore, for any $k\in\mathbb{Z},$ we have $l(g^{k})=\lim_{n\rightarrow\infty}\frac{\\|f(g^{kn})\\|}{n}=\|k\|l(g).$ For any $h\in G,$ we have $\\|f(hg^{n}h^{-1})\\|\leq\\|f(h)\\|+\\|f(h^{-1})\\|+\\|f(g^{n})\\|+2C.$ Therefore, we have $l(hgh^{-1})=\lim_{n\rightarrow\infty}\frac{\\|f(hg^{n}h^{-1})\\|}{n}\leq\lim_{n\rightarrow\infty}\frac{\\|f(g^{n})\\|}{n}=l(g).$ Similarly, we have $l(g)=l(h^{-1}(hgh^{-1})h)\leq l(hgh^{-1}).$ When $g,h$ commutes, we have $\displaystyle l(gh)$ $\displaystyle=$ $\displaystyle\lim_{n\rightarrow\infty}\frac{\\|f((gh)^{n})\\|}{n}=\lim_{n\rightarrow\infty}\frac{\\|f(g^{n}h^{n})\\|}{n}$ $\displaystyle\leq$ $\displaystyle\lim_{n\rightarrow\infty}\frac{\\|f(g^{n})\\|+\\|f(h^{n})\\|+C}{n}=l(g)+l(h).$ ### 2.9 Rotation number Let $\mathbb{R}$ be the real line and $\mathrm{Home}_{\mathbb{Z}}(\mathbb{R})=\\{f\mid f:\mathbb{R}\rightarrow\mathbb{R}$ is a monotonically increasing homeomorphism such that $f(x+n)=f(x)$ for any $n\in\mathbb{Z}\\}.$ For any $f\in\mathrm{Home}_{\mathbb{Z}}(\mathbb{R})$ and $x\in[0,1),$ the translation number is defined as $l(f)=\lim_{n\rightarrow\infty}\frac{f^{n}(x)-x}{n}.$ It is well-known that $l(f)$ exists and is independent of $x$ (see [51], Prop. 2.22, p.31). Note that every $f\in\mathrm{Home}_{\mathbb{Z}}(\mathbb{R})$ induces an orientation-preserving homeomorphism of the circle $S^{1}.$ ###### Proposition 2.11 The absolute value of the translation number $\|l\|:\mathrm{Home}_{\mathbb{Z}}(\mathbb{R})\rightarrow[0,\infty)$ is a length function in the sense of Definition 1.1. Proof. For any $f\in\mathrm{Home}_{\mathbb{Z}}(\mathbb{R})$ and $k\in\mathbb{Z}\backslash\\{0\\},$ we have that $l(f^{k})=\lim_{n\rightarrow\infty}\frac{f^{kn}(x)-x}{n}=k\lim_{n\rightarrow\infty}\frac{f^{kn}(x)-x}{nk}=kl(f).$ For any $a\in\mathrm{Home}_{\mathbb{Z}}(\mathbb{R}),$ we have that $\displaystyle\mid$ $\displaystyle l(afa^{-1})-l(f)\mid=\lim_{n\rightarrow\infty}\mid\frac{af^{n}(a^{-1}x)-x-f^{n}(x)+x}{n}\mid$ $\displaystyle=$ $\displaystyle\lim_{n\rightarrow\infty}\mid\frac{af^{n}(a^{-1}x)-f^{n}(a^{-1}x)+f^{n}(a^{-1}x)-f^{n}(x)}{n}\mid$ $\displaystyle=$ $\displaystyle 0,$ since $a$ is bounded on $[0,1]$ and $\mid f^{n}(a^{-1}x)-f^{n}(x)\mid\leq 2+\|a^{-1}x-x\|.$ For commuting elements $f,g\in\mathrm{Home}_{\mathbb{Z}}(\mathbb{R}),$ we have that $l(fg)=\lim_{n\rightarrow\infty}\frac{f^{n}(g^{n}(x))-x}{n}=\lim_{n\rightarrow\infty}\frac{f^{n}(g^{n}(x))-g^{n}(x)+g^{n}(x)-x}{n}.$ Suppose that $g^{n}(x)=k_{n}+x_{n}$ for $k_{n}\in\mathbb{Z}$ and $x_{n}\in[0,1).$ Then $\displaystyle\lim_{n\rightarrow\infty}\frac{f^{n}(g^{n}x)-g^{n}(x)+g^{n}(x)-x}{n}$ $\displaystyle=$ $\displaystyle\lim_{n\rightarrow\infty}\frac{f^{n}(0)-0+g^{n}(x)-x}{n}=l(f)+l(g).$ Therefore, we get $\|l(fg)\|\leq\|l(f)\|+\|l(g)\|.$ ###### Remark 2.12 It is actually true that the rotation number $l$ is multiplicative on any amenable group (see [51], Prop. 2.2.11 and the proof of Prop. 2.2.10, page 36). This implies that the absolute rotation number $\|l\|$ is subadditive on any amenable group. In other words, for any amenable group $G<\mathrm{Home}_{\mathbb{Z}}(\mathbb{R})$ and any $g,h\in G$ we have $\|l(gh)\|\leq\|l(g)\|+\|l(h)\|.$ ### 2.10 Asymptotic distortions Let $f$ be a $C^{1+bv}$ diffeomorphism of the closed interval $[0,1]$ or the circle $S^{1}.$ (“bv” means derivative with finite total variation.) The asymptotic distortion of $f$ is defined as $l(f)=\mathrm{dist}_{\infty}(f)=\lim_{n\rightarrow\infty}\mathrm{var}(\log Df^{n}).$ It’s proved by Eynard-Bontemps amd Navas ([24], pages 7-8) that 1. (1) $\mathrm{dist}_{\infty}(f^{n})=\|n\|\mathrm{dist}_{\infty}(f)$ for all $n\in\mathbb{Z}$; 2. (2) $\mathrm{dist}_{\infty}(hfh^{-1})=\mathrm{dist}_{\infty}(f)$ for every $C^{1+bv}$ diffeomorphism $h;$ 3. (3) $\mathrm{dist}_{\infty}(f\circ g)\leq\mathrm{dist}_{\infty}(f)+\mathrm{dist}_{\infty}(g)$ for commuting $f,g.$ Therefore, the asymptotic distortion is a length function $l$ on the group $\mathrm{Diff}^{1+bv}(M)$ of $C^{1+bv}$ diffeomorphisms for $M=[0,1]$ or $S^{1}.$ ### 2.11 Dynamical degrees of Cremona groups Let $k$ be a field and $\mathbb{P}_{k}^{n}=k^{n+1}\backslash\\{0\\}/\\{\lambda\sim\lambda x:\lambda\neq 0\\}$ be the projective space. A rational map from $\mathbb{P}_{k}^{n}$ to itself is a map of the following type $(x_{0}:x_{1}:\cdots:x_{n})\dashrightarrow(f_{0}:f_{1}:\cdots:f_{n})$ where the $f_{i}$’s are homogeneous polynomials of the same degree without common factor. The degree of $f$ is $\deg f=\deg f_{i}.$ A birational map from $\mathbb{P}_{k}^{n}$ to itself is a rational map $f:\mathbb{P}_{k}^{n}\dashrightarrow\mathbb{P}_{k}^{n}$ such that there exists a rational map $g:\mathbb{P}_{k}^{n}\dashrightarrow\mathbb{P}_{k}^{n}$ such that $f\circ g=g\circ f=\mathrm{id}.$ The group $\mathrm{Bir}(\mathbb{P}_{k}^{n})$ of birational maps is called the Cremona group (also denoted as $\mathrm{Cr}_{n}(k)$). It is well-known that $\mathrm{Bir}(\mathbb{P}_{k}^{n})$ is isomorphic to the group $\mathrm{Aut}_{k}(k(x_{1},x_{2},\cdots,x_{n}))$ of self-isomorphisms of the field $k(x_{1},x_{2},\cdots,x_{n})$ of the rational functions in $n$ indeterminates over $k.$ The (first) dynamical degree $\lambda(f)$ of $f\in\mathrm{Bir}(\mathbb{P}_{k}^{n})$ is defined as $\lambda(f)=\max\\{\lim_{n\rightarrow\infty}\deg(f^{n})^{\frac{1}{n}},\lim_{n\rightarrow\infty}\deg(f^{-n})^{\frac{1}{n}}\\}.$ Since $\deg(f^{n})^{\frac{1}{n}}$ is sub-multiplicative, the limit exists. ###### Lemma 2.13 Let $l(f)=\log\lambda(f).$ Then $l:\mathrm{Bir}(\mathbb{P}_{k}^{n})\rightarrow[0,+\infty)$ is a length function. Proof. Without loss of generality, we assume that $\lambda(f)=\lim_{n\rightarrow\infty}\deg(f^{n})^{\frac{1}{n}},$ while the other case can be considered similarly. For any $k\in\mathbb{N},$ it is easy that $l(f^{k})=\lim_{n\rightarrow\infty}\frac{\log\deg f^{nk}}{n}=kl(f).$ For any $h\in\mathrm{Bir}(\mathbb{P}_{k}^{n}),$ we have $l(hfh^{-1})=\lim_{n\rightarrow\infty}\frac{\log\deg hf^{n}h^{-1}}{n}=\lim_{n\rightarrow\infty}\frac{\log\deg f^{nk}}{n}=l(f).$ For commuting maps $f,g,$ we have $(fg)^{n}=f^{n}g^{n}.$ Therefore, $\displaystyle l(fg)$ $\displaystyle=$ $\displaystyle\lim_{n\rightarrow\infty}\frac{\log\deg f^{n}g^{n}}{n}$ $\displaystyle\leq$ $\displaystyle\lim_{n\rightarrow\infty}\frac{\log\deg f^{n}}{n}+\lim_{n\rightarrow\infty}\frac{\log\deg g^{n}}{n}=l(f)+l(g).$ This checks the three conditions of the length function. It is surprising that when $n=2$ and $k$ is an algebraically closed field, the length function $l(f)$ is given by the translation length $\tau(f)$ on an (infinite-dimensional) Gromov $\delta$-hyperbolic space (see Blanc-Cantat [8], Theorem 4.4). Some other length functions are studied by Blanc and Furter [9] for groups of birational maps, eg. dynamical number of base-points and dynamical length. ## 3 Groups with purely positive length functions ###### Definition 3.1 A length function $l$ on a group $G$ is said to be purely positive if $l(g)>0$ for any infinite-order element $g\in G.$ In this section, we show that the (Gromov) hyperbolic group, mapping class group and outer automorphism groups of free groups have purely positive length functions. First, let us recall the relevant definitions. A geodesic metric space $X$ is $\delta$-hyperbolic (for some real number $\delta>0$) if for any geodesic triangle $\Delta xyz$ in $X,$ one side is contained the $\delta$-neighborhood of the other two sides. A group $G$ is (Gromov) hyperbolic if $G$ acts properly discontinuously and cocompactly on a $\delta$-hyperbolic space $X$. ###### Definition 3.2 1. (i) An element $g$ in a group $G$ is called primitive if it cannot be writen as a proper power $\alpha^{n},$ where $\alpha\in G$ and $\|n\|\geq 2;$ 2. (ii) A group $G$ has unique-root property if every infinite-order element $g$ is a proper power of a unique (up to sign) primitive element, i.e. $g=\gamma^{n}=\gamma_{1}^{m}$ for primitive elements $\gamma,\gamma_{1}$ will imply $\gamma=\gamma^{\pm}.$ The following fact is well-known. ###### Lemma 3.3 A torsion-free hyperbolic group has unique-root property. Proof. Let $G$ be a torsion-free hyperbolic group and $1\neq g\in G.$ Suppose that $g=\gamma^{n}=\gamma_{1}^{m}$ for primitive elements $\gamma$ and $\gamma_{1}.$ The set $C_{G}(g)$ of centralizers is virtually cyclic (cf. [16], Corollary 3.10, page 462). By a result of Serre, a torsion-free virtually free group is free. Since $G$ is torsion-free, the group $C_{G}(g)$ is thus free and thus cyclic, say generated by $t$. Since $\gamma$ and $\gamma^{\prime}$ are primitive, they are $t^{\pm}.$ ###### Remark 3.4 The previous lemma does not hold for general hyperbolic groups with torsions. For example, let $G=\mathbb{Z}/2\times\mathbb{Z}.$ We have $(0,2)=(0,1)^{2}=(1,1)^{2}$ and $(0,1),$ $(1,1)$ are both primitive. For a group $G,$ let $P(G)$ be the set of all primitive elements. We call two primitive elements $\gamma,\gamma^{\prime}$ are general conjugate if there exists $g\in G$ such that $g\gamma g^{-1}=\gamma^{\prime}$ or $g\gamma^{-1}g=\gamma^{\prime}$. Let $\mathrm{CP}(G)$ be the general conjugacy classes of primitive elements. For a set $S,$ let $S_{\mathbb{R}}$ be the set of all real functions on $S$. The convex polyhedral cone spanned by $S$ is the subset $\\{\mathop{\textstyle\sum}\nolimits_{s\in S}a_{s}s\mid a_{s}\geq 0\ \\}\subset S_{\mathbb{R}}.$ ###### Lemma 3.5 Let $G$ be a torsion-free hyperbolic group. The set of all length functions on $G$ is the convex polyhedral cone spanned by the general conjugacy classes $\mathrm{CP}(G).$ Proof. Let $l$ be a length function on $G.$ Then $l$ gives an element $\mathop{\textstyle\sum}\nolimits_{s\in S}a_{s}s$ in the convex polyhedral cone by $a_{s}=l(s).$ Conversely, for any general conjugacy classes $[s]\in\mathrm{CP}(G)$ with $s$ a primitive element, let $l_{s}$ be the function defined by $l_{s}(s^{\pm})=1$ and $l_{s}(\gamma)=0$ for element $\gamma$ in any other general conjugacy classes. For any $1\neq g\in G,$ there is a unique (up to sign) primitive element $\gamma$ such that $g=\gamma^{n}.$ Define $l_{s}(g)=\|n\|l_{s}(\gamma).$ Then $l_{s}$ satisfies conditions (1) and (2) in Definition 1.1. The condition (3) is satisfied automatically, since any commuting pair of elements $a,b$ generate a cyclic group in a torsion-free hyperbolic group. Any element $\mathop{\textstyle\sum}\nolimits_{s\in S}a_{s}s$ gives a length function on $G$ as a combination of $a_{s}l_{s}$. ###### Lemma 3.6 Let $G\ $be one of the following groups: * • automorphism group $\mathrm{Aut}(F_{k})$ of a free group; * • outer automorphism group $\mathrm{Out}(F_{k})$ of a free group or * • mapping class group $\mathrm{MCG}(\Sigma_{g,m})$ (where $\Sigma_{g,m}$ is an oriented surface of genus $g$ and $m$ punctures); * • a hyperbolic group, * • a $\mathrm{CAT}(0)$ group or more generally * • a semi-hyperbolic group, * • a group acting properly semi-simply on a $\mathrm{CAT}(0)$ space, * • a group acting properly semi-simply on a $\delta$-hyperbolic space. Then $G$ has a purely positive length function. Proof. Note that hyperbolic groups and $\mathrm{CAT}(0)$ groups are semihyperbolic (see [16], Prop. 4.6 and Cor. 4.8, Chapter III.$\Gamma$). For a semihyperbolic group $G$ acting a metric space $X$ (actually $X=G$), the translation $\tau$ is a length function by Lemma 2.7. Moreover, for any infinite-order element $g\in G,$ the length $\tau(g)>0$ (cf. [16], Lemma 4.18, page 479). For group acting properly semisimply on a $\mathrm{CAT}(0)$ space (or a $\delta$-hyperbolic space), the translation $l(\gamma)=\lim_{n\rightarrow\infty}\frac{d(x,\gamma^{n}x)}{n}$ is a length function (cf. Lemma 2.7). For any hyperbolic $\gamma$, we get $l(\gamma)>0.$ For any elliptic $\gamma,$ it is finite-order since the action is proper. Alibegovic [1] proves that the stable word length of $\mathrm{Aut}(F_{n}),\mathrm{Out}(F_{n})$ are purely positive. Farb, Lubotzky and Minsky [25] prove that Dehn twists and more generally all elements of infinite order in $\mathrm{MCG}(\Sigma_{g,m})$ have positive translation length. ###### Definition 3.7 A group $G$ is called poly-positive (or has a poly-positive length), if there is a subnormal series $1=H_{n}\vartriangleleft H_{n-1}\vartriangleleft\cdots\vartriangleleft H_{0}=G$ such that every finitely generated subgroup of the quotient $H_{i}/H_{i+1}$ $(i=0,...,n-1)$ has a purely positive length function. Recall that a group $G$ is poly-free, if there is a subnormal series $1=H_{n}\vartriangleleft H_{n-1}\vartriangleleft\cdots\vartriangleleft H_{0}=G$ such that the successive quotient $H_{i}/H_{i+1}$ is free $(i=0,...,n-1).$ Since a free group is hyperbolic, it has a purely positive length function. This implies that a poly-free group is poly-positive. A group is said to have a virtual property if there is a finite-index subgroup has the property. Let $\Sigma$ be a closed oriented surfaec endowed with an area form $\omega.$ Denote by $\mathrm{Diff}(\Sigma,\omega)$ the group of diffeomorphisms preserving $\omega$ and $\mathrm{Diff}_{0}(\Sigma,\omega)$ the subgroup consisting of diffeomorphisms isotopic to the identity. ###### Lemma 3.8 When the genuse of $\Sigma$ is greater than $1,$ the group $\mathrm{Diff}_{0}(\Sigma,\omega)$ and $\mathrm{Diff}(\Sigma,\omega)$ is poly- positive. Proof. This is eseentially proved by Py [54] (Section 1). There is a group homomorphism $\alpha:\mathrm{Diff}_{0}(\Sigma,\omega)\rightarrow H_{1}(\Sigma,\mathbb{R})$ with $\ker\alpha=\mathrm{Ham}(\Sigma,\omega)$ the group of Hamiltonian diffeomorphisms of $\Sigma.$ Polterovich [53] (1.6.C.) proves that any finitely generated group of $\mathrm{Ham}(\Sigma,\omega)$ has a purely positive stable word length. Since the quotient group $\mathrm{Diff}(\Sigma,\omega)/\mathrm{Diff}_{0}(\Sigma,\omega)$ is a subgroup of the mapping class group $\mathrm{MCG}(\Sigma),$ which has a purely positive stable word length by Farb-Lubotzky-Minsky [25], the group $\mathrm{Diff}(\Sigma,\omega)$ is poly-positive. ## 4 Vanishing of length functions on abelian-by-cyclic groups We will need the following result proved in [28]. ###### Lemma 4.1 Given a group $G$, let $l:G\rightarrow[0,+\infty)$ be function such that 1) $l(e)=0;$ 2) $l(x^{n})=\|n\|l(x)$ for any $x\in G,$ any $n\in\mathbb{Z};$ 3) $l(xy)\leq l(x)+l(y)$ for any $x,y\in G.$ Then there exist a real Banach space $(\mathbb{B},\\|\\|)$ and a group homomorphism $\varphi:G\rightarrow\mathbb{B}$ such that $l(x)=\\|\varphi(x)\\|$ for all $x\in G.$ Further more, if $l(x)>0$ for any $x\neq e$, one can take $\varphi$ to be injective, i.e., an isometric embedding. Let $\mathbb{Z}^{2}\rtimes_{A}\mathbb{Z}$ be an abelian by cyclic group, where $A=\begin{bmatrix}a&b\\\ c&d\end{bmatrix}\in\mathrm{GL}_{2}(\mathbb{Z})$. We prove Theorem 0.1 by proving the following two theorems. ###### Theorem 4.2 When the absolute value of the trace $\|\mathrm{tr}(A)\|>2,$ any length function $l:\mathbb{Z}^{2}\rtimes_{A}\mathbb{Z}\rightarrow\mathbb{R}_{\geq 0}$ vanishes on $\mathbb{Z}^{2}.$ Proof. Let $A=\begin{bmatrix}a&b\\\ c&d\end{bmatrix}\in\mathrm{GL}_{2}(\mathbb{Z}).$ Suppose that $t$ is a generator of $\mathbb{Z}$ and $t\begin{bmatrix}x\\\ y\end{bmatrix}t^{-1}=A\begin{bmatrix}x\\\ y\end{bmatrix}$ for any $x,y\in\mathbb{Z}.$ Note that $(0,t^{k})(v,0)(0,t^{k})^{-1}=(A^{k}v,0)$ for any $v\in\mathbb{Z}^{2}$ and $k\in\mathbb{Z}$. Therefore, an element $v\in\mathbb{Z}^{2}$ is conjugate to $A^{k}v$ for any integer $k.$ Note that $A\begin{bmatrix}1\\\ 0\end{bmatrix}=\begin{bmatrix}a\\\ c\end{bmatrix},A^{2}\begin{bmatrix}1\\\ 0\end{bmatrix}=\begin{bmatrix}a^{2}+bc\\\ ac+dc\end{bmatrix}$ and $\begin{bmatrix}a^{2}+bc\\\ ac+dc\end{bmatrix}=(a+d)\begin{bmatrix}a\\\ c\end{bmatrix}-(ad-bc)\begin{bmatrix}1\\\ 0\end{bmatrix}.$ Therefore, we have $\displaystyle\|a+d\|l(\begin{bmatrix}1\\\ 0\end{bmatrix})$ $\displaystyle=$ $\displaystyle l((a+d)\begin{bmatrix}a\\\ c\end{bmatrix})$ $\displaystyle=$ $\displaystyle l(\begin{bmatrix}a^{2}+bc\\\ ac+dc\end{bmatrix}+(ad- bc)\begin{bmatrix}1\\\ 0\end{bmatrix})$ $\displaystyle\leq$ $\displaystyle(1+\|ad-bc\|)l(\begin{bmatrix}1\\\ 0\end{bmatrix}).$ When $ad-bc=\pm 1$ and $\|a+d\|>2,$ we must have $l(\begin{bmatrix}1\\\ 0\end{bmatrix})=0.$ Similarly, we can prove that $l(\begin{bmatrix}0\\\ 1\end{bmatrix})=0.$ Since $l$ is subadditive on $\mathbb{Z}^{2},$ we get that $l$ vanishes on $\mathbb{Z}^{2}.$ ###### Theorem 4.3 When the absolute value $\|\mathrm{tr}(A)\|=\|a+d\|=2,I_{2}\neq A\in\mathrm{SL}_{2}(\mathbb{Z}),$ any length function $l:\mathbb{Z}^{2}\rtimes_{A}\mathbb{Z}\rightarrow\mathbb{R}_{\geq 0}$ vanishes on the direct summand of $\mathbb{Z}^{2}$ spanned by eigenvectors of $A$. Proof. We may assume that $A=\begin{bmatrix}1&n\\\ 0&1\end{bmatrix},n\neq 0.$ For any integer $k\geq 0$ and $v\in\mathbb{Z}^{2},$ we have $t^{k}vt^{-k}=A^{k}v.$ Take $v=\begin{bmatrix}0\\\ 1\end{bmatrix}$ to get that $t^{k}\begin{bmatrix}0\\\ 1\end{bmatrix}t^{-k}=\begin{bmatrix}kn\\\ 0\end{bmatrix}+\begin{bmatrix}0\\\ 1\end{bmatrix}.$ Since the function $l\|_{\mathbb{Z}^{2}}$ is given by the norm of a Banach space according to Lemma 4.1, we get that $\displaystyle k\|n\|l(\begin{bmatrix}1\\\ 0\end{bmatrix})$ $\displaystyle\leq$ $\displaystyle l(t^{k}\begin{bmatrix}0\\\ 1\end{bmatrix}t^{-k})+l(\begin{bmatrix}0\\\ 1\end{bmatrix})$ $\displaystyle=$ $\displaystyle 2l(\begin{bmatrix}0\\\ 1\end{bmatrix}).$ Since $k$ is arbitrary, we get that $l(\begin{bmatrix}1\\\ 0\end{bmatrix})=0.$ ###### Remark 4.4 When $A=\begin{bmatrix}1&1\\\ 0&1\end{bmatrix},$ the semidirect product $G=\mathbb{Z}^{2}\rtimes_{A}\mathbb{Z}$ is a Heisenberg group. A length function on $G/Z(G)\cong\mathbb{Z}^{2}$ gives a length function on $G.$ In particular, a length function of $G$ may not vanish on the second component $\begin{bmatrix}0\\\ 1\end{bmatrix}\in\mathbb{Z}^{2}<G.$ ###### Remark 4.5 When $A\in\mathrm{SL}_{2}(\mathbb{Z})$ has $\|\mathrm{tr}(A)\|<2,$ the matrix $A$ is of finite order and the semi-direct product $\mathbb{Z}^{2}\rtimes_{A}\mathbb{Z}$ contains $\mathbb{Z}^{3}$ as a finite- index normal subgroup. Actually, in this case the group $\mathbb{Z}^{2}\rtimes_{A}\mathbb{Z}$ is the fundamental group of a flat $3$-manifold $M$ (see [61], Theorem 3.5.5). Therefore, the group $\mathbb{Z}^{2}\rtimes_{A}\mathbb{Z}$ acts freely properly discontinuously isometrically and cocompactly on the universal cover $\tilde{M}=\mathbb{R}^{3}.$ This means the translation length gives a purely positive length function on $\mathbb{Z}^{2}\rtimes_{A}\mathbb{Z}$. ###### Lemma 4.6 Let $A\in\mathrm{GL}_{n}(\mathbb{Z})$ be a matrix and $G=\mathbb{Z}^{n}\rtimes_{A}\mathbb{Z}$ the semi-direct product. Let $\sum_{i=0}^{n}a_{i}x^{i}$ be the characterisitic polynomial of some power $A^{k}$. Suppose that for some $k,$ there is a coefficient $a_{i}$ such that $\|a_{i}\|>\sum_{j\neq i}\|a_{j}\|.$ Any length function $l$ of $G$ vanishes on $\mathbb{Z}^{n}.$ Proof. Let $t$ be a generator of $Z$ and $tat^{-1}=Aa$ for any $a\in Z^{n}.$ Note that for any integer $m,$ we have $t^{m}at^{-m}=A^{m}a$ and $l(a)=l(A^{m}a).$ Note that $\sum_{i=0}^{n}a_{i}A^{ki}=0$ and thus $\displaystyle\sum_{i=0}^{n}a_{i}A^{ki}a$ $\displaystyle=$ $\displaystyle 0,$ $\displaystyle l(\sum_{i=0}^{n}a_{i}A^{ki}a)$ $\displaystyle=$ $\displaystyle 0$ for any $a\in\mathbb{Z}^{n}.$ Therefore, $\|a_{i}\|l(a)=\|a_{i}\|l(A^{ki}a)=l(\sum_{j\neq i}a_{j}A^{kj}a)\leq\sum_{j\neq i}\|a_{j}\|l(a).$ This implies that $l(a)=0.$ Proof of Corollary 0.2. When the group action is $C^{2},$ define $l(f)=\max\\{\lim_{n\rightarrow+\infty}\frac{\log\sup_{x\in M}\\|D_{x}f^{n}\\|}{n},\lim_{n\rightarrow+\infty}\frac{\log\sup_{x\in M}\\|D_{x}f^{-n}\\|}{n}\\}$ for any diffeomorphism $f:M\rightarrow M.$ Lemma 2.5 shows that $l$ is a length function, which is an upper bound of the Lyapunov exponents. When the group action is Lipschitz, define $L(f)=\sup_{x\neq y}\frac{d(fx,fy)}{d(x,y)}$ for a Lipschitz-homeomorphism $f:M\rightarrow M.$ Since $L(fg)\leq L(f)L(g)$ for two Lipschitz-homeomorphisms $f,g:M\rightarrow M,$ we have that $l(f):=\lim_{n\rightarrow\infty}\frac{\max\\{\log(L(f^{n})),\log(L(f^{-n}))\\}}{n}$ gives a length function by Lemma 2.1. Note that $l(f)\geq h_{top}(f)$ (see [40], Theorem 3.2.9, page 124). The vanishings of the topological entropy $h_{top}$ and the Lyapunov exponents in Corolary 0.2 are proved by Theorem 0.1 considering these length functions. ## 5 Classification of length functions on nilpotent groups The following lemma is a key step for our proof of the vanishing of length functions on Heisenberg groups. ###### Lemma 5.1 Let $G=\langle a,b,c\mid aba^{-1}b^{-1}=c,ac=ca,bc=cb\rangle$ be the Heisenberg group. Suppose that $f:G\rightarrow\mathbb{R}$ is a conjugation-invariant function, i.e. $f(xgx^{-1})=f(g)$ for any $x,g\in G.$ For any coprime integers (not-all-zero) $m,n$ and any integer $k,$ we have $f(a^{m}b^{n}c^{k})=f(a^{m}b^{n}).$ Proof. It is well-known that for any integers $n,m,$ we have $[a^{n},b^{m}]=c^{nm}.$ Actually, since $aba^{-1}b^{-1}=c,$ we have $ba^{-1}b^{-1}=a^{-1}c$ and thus $ba^{-n}b^{-1}=a^{-n}c^{n}$ for any integer $n.$ Therefore, $a^{n}ba^{-n}b^{-1}=c^{n}$ and $a^{n}ba^{-n}=c^{n}b,$ $a^{n}b^{m}a^{-n}=a^{nm}b^{m}$ for any integer $m.$ This means $[a^{n},b^{m}]=c^{nm}.$ For any coprime $m,n$, and any integer $k,$ let $s,t\in\mathbb{Z}$ such that $ms+nt=k.$ We have $a^{-m}b^{-s}a^{m}b^{s}=c^{ms},b^{-s}a^{m}b^{s}=a^{m}c^{ms},b^{-s}a^{m}b^{n}b^{s}=a^{m}b^{n}c^{ms}$ and $a^{t}b^{n}a^{-t}b^{-n}=c^{nt},a^{t}b^{n}a^{-t}=b^{n}c^{nt},a^{t}a^{m}b^{n}a^{-t}=a^{m}b^{n}c^{nt}.$ Therefore, $a^{t}(b^{-s}a^{m}b^{n}b^{s})a^{-t}=a^{t}(a^{m}b^{n}c^{ms})a^{-t}=a^{m}b^{n}c^{nt+ms}=a^{m}b^{n}c^{k}.$ When $f$ is conjugation-invariant, we get $f(a^{m}b^{n}c^{k})=f(a^{m}b^{n})$ for any coprime $m,n$, and any integer $k.$ ###### Lemma 5.2 Let $G=\langle a,b,c\mid aba^{-1}b^{-1}=c,ac=ca,bc=cb\rangle$ be the Heisenberg group. Any length function $l:G\rightarrow[0,\infty)$ (in the sense of Definition 1.1) factors through the abelization $G_{\mathrm{ab}}:=G/[G,G]\cong\mathbb{Z}^{2}.$ In other words, there is a function $l^{\prime}:G_{\mathrm{ab}}\rightarrow[0,\infty)$ such that $l^{\prime}(x^{n})=\|n\|l^{\prime}(x)$ for any $x\in G_{\mathrm{ab}},$ any integer $n$ and $l=l^{\prime}\circ q$, where $q:G\rightarrow G_{\mathrm{ab}}$ is the natural quotient group homomorphism. Proof. Let $H=\langle c\rangle\cong\mathbb{Z}$ and write $G=\cup gH$ the union of left cosets. We choose the representative $g_{ij}=a^{i}b^{j}$ with $(i,j)\in\mathbb{Z}^{2}.$ Note that the subgroup $\langle g_{ij},c\rangle$ generated by $g_{ij},c$ is isomorphic to $\mathbb{Z}^{2}$ for coprime $i,j.$ The length function $l$ is subadditive on $\langle g_{ij},c\rangle.$ By Lemma 4.1, there is a Banach space $\mathbb{B}$ and a group homomorphism $\varphi:$ $\langle g_{ij},c\rangle\rightarrow\mathbb{B}$ such that $l(g)=\\|\varphi(g)\\|$ for any $g\in$ $\langle g_{ij},c\rangle.$ Lemma 5.1 implies that $\\|\varphi(g_{ij})+\varphi(c^{k})\\|=\\|\varphi(g_{ij})\\|$ for any integer $k.$ Since $\\|\varphi(g_{ij})\\|=\\|\varphi(g_{ij})+\varphi(c^{k})\\|\geq\|k\|\\|\varphi(c)\\|-\\|\varphi(g_{ij})\\|$ for any $k,$ we have that $\\|\varphi(c)\\|=l(c)=0.$ This implies that $l(g_{ij}^{n}c^{m})=\\|\varphi(g_{ij}^{n})+\varphi(c^{m})\\|=l(g_{ij}^{n})$ for any integers $m,n.$ Moreover, for any integers $m,n$ and coprime $i,j,$ we have $a^{ni}b^{nj}c^{m}=(a^{i}b^{j})^{n}c^{k}$ for some integer $k.$ This implies that $l(a^{ni}b^{nj}c^{m})=l((a^{i}b^{j})^{n})=\|n\|l(a^{i}b^{j}).$ Therefore, the function $l$ is constant on each coset $gH.$ Define $l^{\prime}(gH)=l(g).$ Since $l(g^{k})=\|l\|l(g),$ we have that $l^{\prime}(g^{k}H)=\|k\|l^{\prime}(gH).$ The proof is finished. Denote by $S^{\prime}=\\{(m,n)\mid m,n$ are coprime integers$\\}$ be the set of coprime integer pairs and define an equivalence relation by $(m,n)\sim(m^{\prime},n^{\prime})$ if $(m,n)=\pm(m^{\prime},n^{\prime})$. Let $S=S^{\prime}/\sim$ be the equivalence classes. ###### Theorem 5.3 Let $G=\langle a,b,c\mid aba^{-1}b^{-1}=c,ac=ca,bc=cb\rangle$ be the Heisenberg group. The set of all length functions $l:G\rightarrow[0,\infty)$ (in the sense of Definition 1.1) is the convex polyhedral cone $\mathbb{R}_{\geq 0}[S]=\\{\sum_{s\in S}a_{s}s\mid a_{s}\in\mathbb{R}_{\geq 0},s\in S\\}.$ Proof. Similar to the proof of the previous lemma, we let $H=\langle c\rangle\cong\mathbb{Z}$ and write $G=\cup gH$ the union of left cosets. We choose the representative $g_{ij}=a^{i}b^{j}$ with $(i,j)\in\mathbb{Z}^{2}.$ Let $T$ be the set of length function $l:G\rightarrow[0,\infty).$ For any length functions $l,$ let $\varphi(l)=\sum_{s\in S}a_{s}s\in\mathbb{R}_{\geq 0}[S],$ where $a_{s}=l(a^{i}b^{j})$ with $(i,j)$ a representative of $s.$ Note that $l(a^{-i}b^{-j})=l(b^{-j}a^{-i}c^{ij})=l(b^{-j}a^{-i})=l(a^{i}b^{j}),$ which implies that $a_{s}$ is well-defined. We have defined a function $\varphi:T\rightarrow\mathbb{R}_{\geq 0}[S].$ If $\varphi(l_{1})=\varphi(l_{2})$ for two functions $l_{1},l_{2},$ then $l_{1}(a^{i}b^{j})=l_{2}(a^{i}b^{j})$ for coprime integers $i,j.$ Since both $l_{1},l_{2}$ are conjugation-invariant, Lemma 5.1 implies that $l_{1},l_{2}$ coincide on any coset $a^{i}b^{j}H$ and thus on the whole group $G.$ This proves the injectivity of $\varphi.$ For any $\sum_{s\in S}a_{s}s,$ we define a function $l:G=\cup a^{i}b^{j}H\rightarrow[0,\infty).$ For any coprime integers $i,j,$ define $l(a^{i}b^{j}z)=a_{s}$ for any representative $(i,j)$ of $s$ and any $z\in H.$ For any general integers $m,n$ and $z\in H,$ define $l(a^{m}b^{n}z)=l(a^{m}b^{n})=\|\gcd(m,n)\|l(a^{m/\gcd(m,n)}b^{n/\gcd(m,n)})$ and $l(z)=0.$ From the definition, it is obvious that $l$ is homogeous. Note that any element of $G$ is of the form $a^{k}b^{s}c^{t}$ for integers $k,s,t\in\mathbb{Z}.$ For any two elements $a^{k}b^{s}c^{t},a^{k^{\prime}}b^{s^{\prime}}c^{t^{\prime}}$ we have the conjugation $a^{k^{\prime}}b^{s^{\prime}}c^{t^{\prime}}a^{k}b^{s}c^{t}(a^{k^{\prime}}b^{s^{\prime}}c^{t^{\prime}})^{-1}=a^{k}b^{s}c^{t^{\prime\prime}}$ for some $t^{\prime\prime}\in\mathbb{Z}.$ Therefore, we see that $l$ is conjugation-invariant. The previous equality also shows that two elements $g,h$ are commuting if and only if they lies simutanously in $\langle a^{i}b^{j},c\rangle$ for a pair of coprime integers $i,j.$ By construction, we have $l(g)=l(h).$ This proves the surjectivity of $\varphi.$ ## 6 Length functions on matrix groups In this section, we study length functions on matrix groups $\mathrm{SL}_{n}(\mathbb{R})$. As the proofs are elementary, we present here in a separated section, without using profound results on Lie groups and algebraic groups. The following lemma is obvious. ###### Lemma 6.1 Let $G_{p,q}=\langle x,t:tx^{p}t^{-1}=x^{q}\rangle$ be a Baumslag-Solitar group. When $\|p\|\neq\|q\|,$ any length function $l$ on $G$ has $l(x)=0.$ Proof. Note that $\|p\|l(x)=l(x^{p})=l(x^{q})=\|q\|l(x),$ which implies $l(x)=0.$ Let $V^{n}$ be a finite-dimensional vector space over a field $K$ and $A:V\rightarrow V$ a unipotent linear transformation (i.e. $A^{k}=0$ for some positive integer $k).$ The following fact is from linear algebra (see the Lemma of page 313 in [4]. Since the reference is in Chinese, we repeat the proof here). ###### Lemma 6.2 $I+A$ is conjugate to a direct sum of Jordan blocks with $1s$ along the diagonal. Proof. We prove that $V$ has a basis $\\{a_{1},Aa_{1},\cdots,A^{k_{1}-1}a_{1},a_{2},Aa_{2},\cdots,A^{k_{2}-1}a_{2},\cdots,a_{s},\cdots,Aa_{s},\cdots,A^{k_{s}-1}a_{s}\\}$ satisfying $A^{k_{i}}a_{i}=0$ for each $i,$ which implies that the representation matrix of $I+A$ is similar to a direct sum of Jordan blocks with $1$ along the diagonal. The proof is based on the induction of $\dim V.$ When $\dim V=1,$ choose $0\neq v\in V.$ Suppose that $Av=\lambda v.$ Then $A^{k}v=\lambda^{k}v=0$ and thus $\lambda=0.$ Suppose that the case is proved for vector spaces of dimension $k<n.$ Note that the invariant subspace $AV$ is a proper subspace of $V$ (otherwise, $AV=V$ implies $A^{k}V=A^{k-1}V=V=0$). By induction, the subspace $AV$ has a basis $\\{a_{1},Aa_{1},\cdots,A^{k_{1}-1}a_{1},a_{2},Aa_{2},\cdots,A^{k_{2}-1}a_{2},\cdots,a_{s},\cdots,Aa_{s},\cdots,A^{k_{s}-1}a_{s}\\}.$ Choose $b_{i}\in V$ satisfying $A(b_{i})=a_{i}.$ Then $A$ maps the set $\\{b_{1},Ab_{1},\cdots,A^{k_{1}}b_{1},b_{2},Ab_{2},\cdots,A^{k_{2}}b_{2},b_{s},\cdots,Ab_{s},\cdots,A^{k_{s}}b_{s}\\}$ to the basis $\\{a_{1},Aa_{1},\cdots,A^{k_{1}-1}a_{1},a_{2},Aa_{2},\cdots,A^{k_{2}-1}a_{2},\cdots,a_{s},\cdots,Aa_{s},\cdots,A^{k_{s}-1}a_{s}\\}.$ This implies that the former set is linearly independent (noting that $A(A^{k_{i}}b_{i})=0$). Extend this set to be a $V^{\prime}s$ basis $\\{b_{1},Ab_{1},\cdots,A^{k_{1}-1}b_{1},b_{2},Ab_{2},\cdots,A^{k_{2}-1}b_{2},b_{s},\cdots,Ab_{s},\cdots,A^{k_{s}-1}b_{s},b_{s+1},\cdots,b_{s^{\prime}}\\}.$ Note that $Ab_{i}=0$ for $i\geq s+1$ and $A^{k_{i}+1}b_{i}=A^{k_{i}}a_{i}=0$ for each $i\leq s.$ This finishes the proof. ###### Corollary 6.3 Let $A_{n\times n}$ be a strictly upper triangular matrix over a field $K$ of characteristic $\mathrm{ch}(K)\neq 2$. Then $A^{2}$ is conjugate to $A.$ Proof. Suppose that $A=I+u$ for a nilpotent matrix $u.$ Lemma 6.2 implies that $A^{2}$ is conjugate to a direct sum of Jordan blocks. Without loss of generality, we assume $A$ is a Jordan block. Then $A^{2}=I+2u+u^{2}.$ By Lemma 6.2 again, $A^{2}$ is conjugate to a direct sum of Jordan blocks with $1s$ along the diagonal. The minimal polynomial of $A^{2}$ is $(x-1)^{n},$ which shows that there is only one block in the direct sum and thus $A^{2}$ is conjugate to $A.$ Recall that a matrix $A\in\mathrm{GL}_{n}(\mathbb{R})$ is called semisimple if as a complex matrix $A$ is conjugate to a diagonal matrix. A semisimple matrix $A$ is elliptic (respectively, hyperbolic) if all its (complex) eigenvalues have modulus $1$ (respectively, are $>0$). The following lemma is the complete multiplicative Jordan (or Jordan-Chevalley) decomposition (cf. [34], Lemma 7.1, page 430). ###### Lemma 6.4 Each $A\in$ $\mathrm{GL}_{n}(\mathbb{R})$ can be uniquely writen as $A=ehu,$ where $e,h,u\in\mathrm{GL}_{n}(\mathbb{R})$ are elliptic, hyperbolic and unipotent, respectively, and all three commute. The following result characterizes the continuous length functions on compact Lie groups. ###### Lemma 6.5 Let $G$ be a compact connected Lie group and $l$ a continuous length function on $G$. Then $l=0.$ Proof. For any element $g\in G,$ there is a maximal torus $T\backepsilon g.$ For finite order element $h\in T,$ we have $l(h)=0.$ Note that the set of finite-order elements is dense in $T.$ Since $l$ is continuous, $l$ vanishes on $T$ and thus $l(g)=0$ for any $g.$ ###### Theorem 6.6 Let $G=\mathrm{SL}_{n}(\mathbb{R})$ $(n\geq 2).$ Let $l:G\rightarrow[0,+\infty)$ be a length function, which is continuous on compact subgroups and the subgroup of diagonal matrices with positive diagonal entries. Then $l$ is uniquely determined by its images on the subgroup $D$ of diagonal matrices with positive diagonal entries. Proof. For any $g\in\mathrm{SL}_{n}(\mathbb{R}),$ let $g=ehu$ be the Jordan decomposition for commuting elements $e,h,u$, where $e$ is elliptic, $h$ is hyperbolic and $u$ is unipotent (see Lemma 6.4) after multiplications by suitable powers of derterminants. Then $l(g)\leq l(e)+l(h)+l(u).$ For any unipotent matrix $u,$ there is an invertible matrix $a$ such that $aua^{-1}$ is strictly upper triangular (see [34], Theorem 7.2, page 431). Lemma 6.3 implies that $u^{2}$ is conjugate to $u$. Therefore, $l(u)=0$ by Lemma 6.1. Since $l$ vainishes on a compact Lie group (cf. Lemma 6.5), we have that $l(e)=0$ for any elliptic matrix $e.$ Therefore, $l(g)\leq l(h).$ Similarly, $l(h)=l(e^{-1}gu^{-1})\leq l(g)$ which implies $l(g)=l(h).$ Note that a hyperbolic matrix is conjugate to a real diagonal matrix with positive diagonal entries. Proof of Theorem 0.10. By Theorem 6.6, the length function $l$ is determined by its image on the subgroup $D$ generated by $h_{12}(x),x\in\mathbb{R}_{>0}.$ Take $x=e^{t},t\in\mathbb{R}$. We have $l(h_{12}(e^{\frac{k}{l}}))=\frac{\|k\|}{\|l\|}l(h_{12}(e))$ for any rational number $\frac{k}{l}.$ Since $l$ satisfies the condition 2) of the definition and is continuous on $D$, we see that $l\|_{D}$ is determined by the image $l(h_{12}(e))$ (actually, any real number $t$ is a limit of a rational sequence). Note that the translation function $\tau$ vainishes on compact subgroups and is continuous on the subgroup of diagonal subgroups with positive diagonal entries (cf. [16], Cor. 10.42 and Ex. 10.43, page 320). Therefore, $l$ is proposional to $\tau.$ Actually, $\tau$ can be determined explicitly by the formula $\mathrm{tr}(A)=\pm 2$cosh$\frac{\tau(A)}{2}$ (for nonzero $\tau(A)$), where $\mathrm{tr}$ is the trace and cosh is the hyperbolic cosine function (see [5], Section 7.34, page 173). This implies that $l(A)$ is determined by the spectrum radius of $A$ (which could also be seen clearly by the matrix norm). Let $h_{1i}(x)$ ($i=2,\cdots,n$) be an $n\times n$ diagonal matrix whose $(1,1)$-entry is $x,$ $(i,i)$-entry is $x^{-1},$ while other diagonal entries are $1$s and non-diagonal entries are $0$s. The subgroup $D<\mathrm{SL}_{n}(\mathbb{R})$ of diagonal matrices with positive diagonal entries is isomorphic to $(\mathbb{R}_{>0})^{n-1}$ and $D$ is generated by the matrices $h_{1i}(x)$ ($i=2,\cdots,n$) whose $(1,1)$-entry is $x,$ $(i,i)$-entry is $x^{-1}.$ Since $h_{1i}(x)$ $(i\neq 1)$ is conjugate to $h_{12}(x),$ a length function $l:\mathrm{SL}_{n}(\mathbb{R})\rightarrow[0,+\infty)$ is completely determined by its image on the convex hull spanned by $h_{12}(e),h_{13}(e),\cdots,h_{1n}(e)$ (see Theorem 7.10 for a more general result on Lie groups). Here $e$ is the Euler’s number in the natural exponential function. ###### Corollary 6.7 Let $l:\mathrm{SL}_{2}(\mathbb{R})\rightarrow[0,+\infty)$ be a non-trivial length function that is continuous on the subgroup $SO(2)$ and the diagonal subgroup. Then $l(g)>0$ if and only if $g$ is hyperbolic. Proof. It is well-known that the elements in $\mathrm{SL}_{2}(\mathbb{R})$ are classified as elliptic, hyperbolic and parabolic elements. Moreover, the translation length $\tau$ vanishes on the compact subgroup $SO(2)$ and the parabolic elements. The corollary follows Theorem 0.10. When the length function $l$ is the asymptotic distortion function $\mathrm{dist}_{\infty}$, Corollary 6.7 is known to Navas [24] (Proposition 4). ## 7 Length functions on algebraic and Lie groups For an algebraic group $G,$ let $k[G]$ be the regular ring. For any $g\in G,$ let $\rho_{g}:k[G]\rightarrow k[G]$ be the right translation by $x.$ The following is the famous Jordan (or Jordan-Chevalley) decomposition. ###### Lemma 7.1 ([37] p.99) Let $G$ be an algebraic group and $g\in G.$ There exists unique elements $g_{s},g_{u}$ such that $g_{s}g_{u}=g_{u}g_{s},$ and $\rho_{g_{s}}$ is semisimple, $\rho_{g_{u}}$ is unipotent. ###### Lemma 7.2 [46] Let $G$ be a reductive connected algebraic group over an algebraically closed field $k.$ The conjugacy classes of unipotent elements in $G$ is finite. ###### Lemma 7.3 ([29], Theorem 3.4) If $G$ is a reductive linear algebraic group defined over a field $k$ and $g\in G(k)$ then the set of conjugacy classes in $G(k)$ which when base changed to the algebraic closed field $\bar{k}$ are equal to the conjugacy class of $g$ in $G(\bar{k})$ is in bijection with the subset of $H^{1}(\bar{k}/k,Z(g)(k)),$ the Galois cohomology group. ###### Definition 7.4 A field $k$ is of type (F) if for any integer $n$ there exist only finitely many extensions of $k$ of degree $n$ (in a fixed algebraic closure $\bar{k}$ of $k$). Examples of fields of type (F) include: the field $\mathbb{R}$ of reals, a finite field, the field of formal power series over an algebraically closed field. ###### Lemma 7.5 [Borel-Serre [12], Theorem 6.2] Let $k$ be a field of type (F) and let $H$ be a linear algebraic group defined over $k$. The set $H^{1}(\bar{k}/k,H(k))$ is finite. ###### Lemma 7.6 Let $G(k)$ be a reductive linear algebraic group over a field of type (F) and $l$ a length function on $G.$ Then $l(g)=l(g_{s}),$ where $g_{s}$ is the semisimple part of $g.$ Proof. By the Jordan decomposition $g=g_{s}g_{u},$ we have $l(g)\leq l(g_{s})+l(g_{u})$ and $l(g_{s})\leq l(g)+l(g_{u}^{-1}).$ Note that for any integer $n,$ $g_{u}^{n}$ is also unipotent. By the Lemma 7.2, Lemma 7.3 and Lemma 7.5, there are only finitely many conjugacy classes of unipotent elements. This implies that $g_{u}^{n_{1}}=g_{u}^{n_{2}}$ for distinct positive integers $n_{1},n_{2}.$ Therefore, we have $n_{1}l(g_{u})=n_{2}l(g_{u}),$ which implies that $l(g_{u})=0$ and thus $l(g)=l(g_{s}).$ A Lie group $G$ is semisimple if its maximal connected solvable normal subgroup is trivial. Let $\mathfrak{g}$ be its Lie algebra and let $\exp:\mathfrak{g}\rightarrow G$ denote the exponential map. An element $x\in\mathfrak{g}$ is real semi-simple if $Ad(x)$ is diagonalizable over $\mathbb{R}$. An element $g\in G$ is called hyperbolic (resp. unipotent) if $g$ is of the form $g=\exp(x)$ where $x$ is real semi-simple (resp. nilpotent). In either case the element $x$ is easily seen to be unique and we write $x=\log g$. The following is the Jordan decomposition in Lie groups. An element $e\in G$ is elliptic if $Ad(e)$ is diagonalizable over $\mathbb{C}$ with eigenvalues $1$. ###### Lemma 7.7 ([42], Prop. 2.1 and Remark 2.1) 1. 1. Let $g\in G$ be arbitrary. Then g may be uniquely written $g=e(g)h(g)u(g)$ where $e(g)$ is elliptic, $h(g)$ is hyperbolic and $u(g)$ is unipotent and where the three elements $e(g),h(g),u(g)$ commute. 2. 2. An element $f\in G$ commutes with $g$ if and only if $f$ commutes with the three components. Moreover, if $f,g$ commutes, then $e(fg)=e(f)e(g),h(fg)=h(f)h(g),u(fg)=u(f)u(g).$ ###### Lemma 7.8 [Eberlein, Prop. 1.14.6, page 63]Let $G$ be a connected semisimple Lie group whose center is trivial. Then there exists an integer $n\geq 2$ and an algebraic group $G^{\ast}<\mathrm{GL}_{n}(\mathbb{C})$ defined over $\mathbb{Q}$ such that $G$ is isomorphic to $G_{\mathbb{R}}^{\ast 0}$ (the connected component of $G_{\mathbb{R}}^{\ast 0}$ containing the identity) as a Lie group. Let $G=KAN$ be an Iwasawa decomposition. The Weyl group $W$ is the finite group defined as the quotient of the normalizer of $A$ in $K$ modulo the centralizer of $A$ in $K.$ For an element $x\in A,$ let $W(h)$ be the set of all elements in $A$ which are conjugate to $x$ in $G.$ ###### Lemma 7.9 ([42], Prop. 2.4) An element $h\in G$ is hyperbolic if and only if it is conjugate to an element in $A.$ In such a case, $W(h)$ is a single $W$-orbit in $A.$ ###### Theorem 7.10 Let $G$ be a connected semisimple Lie group whose center is finite with an Iwasawa decomposition $G=KAN$. Let $W$ be the Weyl group. 1. (i) Any length function $l$ on $G$ that is continuous on the maximal compact subgroup $K$ is determined by its image on $A.$ 2. (ii) Conversely, any length function $l$ on $A$ that is $W$-invariant (i.e. $l(w\cdot a)=l(a)$) can be extended to be a length function on $G$ that vanishes on the maximal compact subgroup $K.$ Proof. (i) Let $Z$ be the center of $G.$ Then $G/Z$ is connected with trivial center. For any $z\in Z,g\in G,$ we have $l(z)=0$ and $l(gz)=l(g).$ The length function $l$ factors through a length function on $G/Z.$ We may assume that $G$ has the trivial center. For any $g\in G,$ the Jordan decomposition gives $g=ehu,$ where $e$ is elliptic, $h$ is hyperbolic and $u$ is unipotent and where the three elements $e,h,u$ commute (cf. Lemma 7.7). By Lemma 7.8, the Lie group $G$ is an algebraic group. Lemma 7.6 implies that $l$ vanies on unipotent elements and $l(g)=l(eh).$ Since $l$ vanishes on $e$ (cf. Lemma 6.5), we have $l(g)=l(h).$ Therefore, the function $l$ is determined by its image on $A.$ (ii) Let $l$ be a length function $l$ on $A$ that is $W$-invariant. We first extend $l$ to the set $H$ of all the conjugates of $A.$ For any $g\in G,a\in A,$ define $l^{\prime}(gag^{-1})=l(a).$ If $g_{1}a_{1}g_{1}^{-1}=g_{2}a_{2}g_{2}^{-1}$ for $g_{1},g_{2}\in G,a_{1},a_{2}\in A,$ then $g_{1}^{-1}g_{2}a_{2}g_{2}^{-1}g_{1}=a_{1}.$ By Lemma 7.9, there exists an element $w\in W$ such that $w\cdot a_{2}=a_{1}.$ Therefore, we have $l(a_{1})=l(a_{2})$ and thus $l^{\prime}$ is well-defined on the set $H$ of conjugates of elements in $A.$ Such a set $H$ is the set consisting of hyperbolic elements by Lemma 7.9. We then extend $l^{\prime}$ on the set of all conjugates of elements in $K.$ For any $g\in G,k\in K,$ define $l^{\prime}(gkg^{-1})=0.$ If $g_{1}kg_{1}^{-1}=g_{2}ag_{2}^{-1}$ for $g_{1},g_{2}\in G,k\in K,a\in A,$ then $g_{1}^{-1}g_{2}ag_{2}^{-1}g_{1}=k.$ Then $k$ is both hyperbolic and elliptic. The only element which is both elliptic and hyperbolic is the identity element. Therefore, we have $k=a=1$ and $l^{\prime}(g_{1}kg_{1}^{-1})=l(g_{2}ag_{2}^{-1})=1.$ This shows that $l^{\prime}$ is well-defined on the set of hyperbolic elements and elliptic elements. For any unipotent element $u\in G,$ define $l^{\prime}(u)=0.$ For any element $g\in G,$ let $g=ehu$ be the Jordan decomposition. Define $l^{\prime}(g)=l^{\prime}(h).$ We check the function $l^{\prime}$ is a length function on $G.$ The definition shows that $l^{\prime}$ is conjugate invariant. For any positive integer $n$ and any $g\in G$ with Jordan decomposition $g=ehu,$ we have $g^{n}=e^{n}h^{n}u^{n}$ and thus $l^{\prime}(g^{n})=l^{\prime}(h^{n}).$ But $h^{n}$ is hyperbolic and conjugate to an element in $A$ (see Lemma 7.9). Therefore, we have $l^{\prime}(h^{n})=\|n\|l^{\prime}(h)$ and thus $l^{\prime}(g^{n})=\|n\|l^{\prime}(g).$ If $g_{1}=e_{1}h_{1}u_{1}$ commutes with $g_{2}=e_{2}h_{2}u_{2},$ then $g_{1}g_{2}=e_{1}e_{2}h_{1}h_{2}u_{1}u_{2}$ (cf. Lemma 7.7) and $l^{\prime}(g_{1}g_{2})=l^{\prime}(h_{1}h_{2}).$ Since $h_{1},h_{2}$ are commuting hyperbolic elements, they are conjuate simutaniously to elements in $A.$ Therefore, we have $l^{\prime}(h_{1}h_{2})\leq l^{\prime}(h_{1})+l^{\prime}(h_{2})$ and $l^{\prime}(g_{1}g_{2})\leq l^{\prime}(g_{1})+l^{\prime}(g_{2}).$ ###### Remark 7.11 A length function $l$ on $A$ is determined by a group homomorphism $f:A\rightarrow\mathbb{B},$ for a real Banach space $(\mathbb{B},\\|\\|),$ satisfying $l(a)=\\|f(a)\\|$ (see Lemma 4.1). The previous theorem implies that a length function $l$ on the Lie group $G$ (that is continuous on compact subgroup) is uniqely determined by such a group homomorphism $f:A\rightarrow\mathbb{B}$ such that $\\|f(a)\\|=\\|f(wa)\\|$ for any $a\in A$ and $w\in W,$ the Weyl group. Let $G$ be a connected semisimple Lie group whose center is finite with an Iwasawa decomposition $G=KAN$. Let $\exp:\mathfrak{g}\rightarrow G$ be the exponent map from the Lie algebra $\mathfrak{g}$ with subalgebra $\mathfrak{h}$ corresponding to $A$. ###### Theorem 7.12 Suppose that $l$ is a length function on $G$ that is continuous on $K$ and $A.$ Then $l$ is determined by its image on $\exp(v)$ $($unit vector $v\in\mathfrak{h})$ in a fixed closed Weyl chamber of $A.$ Proof. Let $Z$ be the center of $G.$ Then $G/Z$ is connected with trivial center. The length function $l$ factors through an length function on $G/Z$ (cf. Corrolary 1.8). For any $g\in G$ we have $g=ehu,$ where $e$ is elliptic, $h$ is hyperbolic and $u$ is unipotent and where the three elements $e,h,u$ commute (cf. Lemma 7.7). By Lemma 7.8 and Lemma 7.6, we have $l(g)=l(eh).$ Since $l$ vanishes on $e$ (cf. Lemma 6.5), we have $l(g)=l(h).$ Any element $h\in A$ is conjugate to an element in a fixed Weyl chamber $C$ (cf. [38], Theorem 8.20, page 254). For any element $\exp(x)\in C,$ with unit vector $x\in\mathfrak{h},$ the one-parameter subgroup $\exp(\mathbb{R}x)$ lies in $A.$ Since $l$ is continuous on $A,$ the function $l$ is determined by its image on $\exp(\mathbb{Q}x).$ Note that $l(e^{\frac{m}{n}x})=\frac{1}{n}l(e^{mx})=\frac{m}{n}l(e^{x})$ for any rational number $\frac{m}{n}.$ The function $l$ is determined by $l(e^{x}),$ for all unit vectors $x$ in the fixed closed Weyl chamber. ###### Corollary 7.13 Let $G$ be a connected semisimple Lie group whose center is finite of real rank $1.$ There is essentially only one length function on $G.$ In order words, any continuous length function is proportional to the translation function on the symmetric space $G/K.$ Proof. When the real rank of $G$ is $1,$ a closed Weyl chamber is of dimension $1$. Therefore, the previous theorem implies that any continuous length function is determined by its image on a unit vector in a split torus. ## 8 Rigidity of group homomorphisms on arithmetic groups Let $V$ denote a finite-dimensional vector space over $\mathbb{C}$, endowed with a $\mathbb{Q}$-structure. Recall that the arithmetic subgroup is defined as the following (cf. Borel [11], page 37). ###### Definition 8.1 Let $G$ be a $\mathbb{Q}$-subgroup of $\mathrm{GL}(V)$. A subgroup $\Gamma$ of $G_{\mathbb{Q}}$ is said to be arithmetic if there exists a lattice $L$ of $V_{\mathbb{Q}}$ such that $\Gamma$ is commensurable with $G_{L}=\\{g\in G_{\mathbb{Q}}:gL=L\\}.$ ###### Theorem 8.2 Let $\Gamma$ be an arithmetic subgroup of a simple algebraic $\mathbb{Q}$-group of $\mathbb{Q}$-rank at least $2.$ Suppose that $H$ is a group with a purely positive length function. Then any group homomorphism $f:\Gamma\rightarrow H$ has its image finite. Recall that a group $G$ is quasi-simple, if any non-trivial normal subgroup is either finite or of finite index. The Margulis-Kazhdan theorem (see [62], Theorem 8.1.2) implies that an irreducible lattice (and hence) in a semisimple Lie group of real rank $\geq 2$ is quasi-simple. ###### Lemma 8.3 Let $\Gamma$ be a finitely generated quasi-simple group with contains a Heisenberg subgroup, i.e. there are elements torsion-free elements $a,b,c\in\Gamma$ satisfying $[a,b]=c,[a,c]=[c,b]=1$. Suppose that $G$ has a virtually poly-positive length. Then any group homomorphism $f:\Gamma\rightarrow G$ has its image finite. Proof. Suppose that $G$ has a finite-index subgroup $H$ and a subnormal series $1=H_{n}\vartriangleleft H_{n-1}\vartriangleleft\cdots\vartriangleleft H_{0}=H$ such that every finitely generated subgroup of $H_{i}/H_{i+1}$ has a purely positive length function. Without loss of generality, we assume that $H$ is normal. Let $f:\Gamma\rightarrow G$ be a homomorphism. The kernel of the composite $f_{0}:\Gamma\overset{f}{\rightarrow}G\rightarrow G/H$ is finitely generated. Suppose that the image of the composite $f_{1}:\ker f_{0}\overset{f}{\rightarrow}H\rightarrow H/H_{1}$ has a purely positive length function $l.$ After passing to finite-index subgroups, we may still suppose that $\ker f_{0}$ contains a Heisenberg subgroup $\langle a,b,c\rangle.$ By Lemma 5.2, the length function $l$ vanishes on $f_{1}(c).$ Therefore $f_{1}(c^{k})=1\in H/H_{1}$ for some integer $k>0.$ The normal subgroup $\ker f_{1}$ containing $c^{k}$ is of finite index. Now we have map $\ker f_{1}\rightarrow H_{1}$ induced by $f.$ An induction argument shows that $f$ maps some power $c^{d}$ of the central element of the Heisenberg subgroup into the identity $1\in G.$ Therefore, the image of $f$ is finite. Proof of Theorem 8.2. It is well-known that that $G$ contains a $\mathbb{Q}$-split simple subgroup whose root system is the reduced subsystem of the $\mathbb{Q}$-root system of $G$ (see [13], Theorem 7.2, page 117). Replacing $G$ with this $\mathbb{Q}$-subgroup, we may assume $G$ is $\mathbb{Q}$-split and the root system of G is reduced. Because $G$ is simple and $\mathbb{Q}$-rank($G$)$\geq 2$, we know that the $\mathbb{Q}$-root system of G is irreducible and has rank at least two. Therefore, the $\mathbb{Q}$-root system of $G$ contains an irreducible subsystem of rank two, that is, a root subsystem of type $A_{2},B_{2},G_{2}$ (see [60], page 338). For $A_{2},$ choose $\\{\alpha_{1},\alpha_{3}\\}$ as a set of simple roots (see Figure 1). Then the root element $x_{\alpha_{1}+\alpha_{3}}(rs)=x_{\alpha_{2}}(rs)$ is a commutator $[x_{\alpha_{1}}(r),x_{\alpha_{3}}(s)],$ with $x_{\alpha_{2}}(rs)$ commutes with $x_{\alpha_{1}}(r),x_{\alpha_{3}}(s).$ For $G_{2}$, the long roots form a subsystem of $A_{2}.$ For $B_{2},$ choose $\\{\alpha_{1},\alpha_{4}\\}$ as a set of simple roots (see Figure 2). The long root element $x_{\alpha_{3}}(2rs)$ is a commutator $[x_{\alpha_{2}}(r),x_{\alpha_{4}}(s)]$ of the two short root elements, and $x_{\alpha_{3}}(2rs)$ commutes with $x_{\alpha_{2}}(r),x_{\alpha_{4}}(s)$ (cf. [37], Proposition of page 211). This shows that the arithmetic subgroup $\Gamma$ contains a Heisenberg subgroup. The theorem is then proved by Lemma 8.3. If we consider special length functions, general results can be proved. When we consider the stable word lengths, the following is essentially already known (cf. Polterovich [53], Corollary 1.1.D and its proof). ###### Proposition 8.4 Let $\Gamma$ be an irreducible non-uniform lattice in a semisimple connected, Lie group without compact factors and with finite center of real rank $\geq 2.$ Assume that a group $G$ has a virtually poly-positive stable word length. In other words, the group $G$ has a finite-index subgroup $H$ and a subnormal series $1=H_{n}\vartriangleleft H_{n-1}\vartriangleleft\cdots\vartriangleleft H_{0}=H$ such that every finitely generated subgroup of $H_{i}/H_{i+1}$ ($i=0,1,\cdots,n-1$) has a purely positive stable word length. Then any group homomorphism $f:\Gamma\rightarrow G$ has its image finite. Proof. Without loss of generality, we assume that $f$ takes image in $H.$ Since a lattice is finitely generated, $\Gamma$ has its image in $H_{0}/H_{1}$ finitely generated. When the image has a purely positive word length, any distorted element in $\Gamma$ must have trivial image in $H_{0}/H_{1}$ (see ). Lubotzky, Mozes and Raghunathan [44] prove that irreducible non-uniform lattices in higher rank Lie groups have non-trivial distortion elements (They actually prove the stronger result that there are elements in the group whose word length has logarithmic growth). Then a finite-index subgroup $\Gamma_{0}<\Gamma$ will have image in $H_{1}$, since high-rank irreducible lattices are quasi-simple. An induction argument finishes the proof. When we consider the length given by quasi-cocyles, the following is also essentially already known ( cf. Py [54], Prop. 2.2, following Burger-Monod [18] [19]). Recall that a locally compact group has property (TT) if any continuous rough action on a Hilbert space has bounded orbits (see [49], page 172). Burger-Monod proves that an irreducible lattice $\Gamma$ in a high-rank semisimple Lie group has property (TT). ###### Proposition 8.5 Let $\Gamma$ be an irreducible lattice in a semisimple connected, Lie group without compact factors and with finite center of real rank $\geq 2.$. Assume that a group $G$ has a virtually poly-positive average norm for quasi- cocycles. In other words, the group $G$ has a finite-index subgroup $H$ and a subnormal series $1=H_{n}\vartriangleleft H_{n-1}\vartriangleleft\cdots\vartriangleleft H_{0}=H$ such that every finitely generated subgroup of $H_{i}/H_{i+1}$ ($i=0,1,\cdots,n-1$) has a purely positive length given by a quasi-cocycle with values in Hilbert spaces. Then any group homomorphism $f:\Gamma\rightarrow G$ has its image finite. Proof. Note that a group $\Gamma$ has property (TT) if and only if $H^{1}(\Gamma;E)=0$ and $\ker(H_{b}^{2}(\Gamma;E)\rightarrow H^{2}(\Gamma;E))=0$ for any linear isometric action of $\Gamma$ on a Hilbert space $E.$ Here $H_{b}^{2}(\Gamma;E)$ is the second bounded cohomology group. Suppose that $u:\Gamma\rightarrow E$ is a quasi-cocyle. There is a bounded map $v:\Gamma\rightarrow E$ and a $1$-cocycle $w:\Gamma\rightarrow E$ such that $u=v+w,$ by Proposition 2.1 of Py [54]. Since $\Gamma$ has property T, there exists $x_{0}\in E$ such that $w(\gamma)=\gamma x_{0}-x_{0}.$ Therefore, we have $\displaystyle\frac{\\|u(\gamma^{n})\\|}{n}$ $\displaystyle=$ $\displaystyle\frac{\\|v(\gamma^{n})+w(\gamma^{n})\\|}{n}$ $\displaystyle=$ $\displaystyle\frac{\\|v(\gamma^{n})+\gamma^{n}x_{0}-x_{0}\\|}{n}$ $\displaystyle\leq$ $\displaystyle\frac{\\|v(\gamma^{n})\\|+2\\|x_{0}\\|}{n}\rightarrow 0.$ Without loss of generality, we assume that $G=H.$ Suppose that any finitely generated subgroup of $H/H_{1}$ has a purely positive average norm $l$ given by a cocycle. The composite $\Gamma\overset{f}{\rightarrow}H\rightarrow H/H_{1}$ has a finite-index kernel $\Gamma_{0}$, since $l$ vanishes on infinite-order elements of the image. This implies that $f(\Gamma_{0})$ lies in $H_{1}.$ A similar argument proves that $\ker f$ is of finite index in the general case. ## 9 Rigidity of group homomorphisms on matrix groups ### 9.1 Steinberg groups over finite rings Recall that a ring $R$ is right Artinian if any non-empty family of right ideals contains minimal elements. A ring $R$ is semi-local if $R/\mathrm{rad}(R)$ is right Artinian (see Bass’ K-theory book [3] page 79 and page 86), where $\mathrm{rad}(R)$ is the Jacobson radical. Let $n$ be a positive integer and $R^{n}$ the free $R$-module of rank $n$ with standard basis. A vector $(a_{1},\ldots,a_{n})$ in $R^{n}$ is called _right unimodular_ if there are elements $b_{1},\ldots,b_{n}\in R$ such that $a_{1}b_{1}+\cdots+a_{n}b_{n}=1$. The _stable range condition_ $\mathrm{sr}_{m}$ says that if $(a_{1},\ldots,a_{m+1})$ is a right unimodular vector then there exist elements $b_{1},\ldots,b_{m}\in R$ such that $(a_{1}+a_{m+1}b_{1},\ldots,a_{m}+a_{m+1}b_{m})$ is right unimodular. It follows easily that $\mathrm{sr}_{m}\Rightarrow\mathrm{sr}_{n}$ for any $n\geq m$. A semi-local ring has the stable range $\mathrm{sr}_{2}$ ( [3], page 267, the proof of Theorem 9.1). A finite ring $R$ is right Artinian and thus has $\mathrm{sr}_{2}.$ The stable range $\mathrm{sr}(R)=\min\\{m:R\text{ has }\mathrm{sr}_{m+1}\\}.$ Thus $\mathrm{sr}(R)=1$ for a finite ring $R$. We briefly recall the definitions of the elementary subgroups $E_{n}(R)$ of the general linear group $\mathrm{GL}_{n}(R)$, and the Steinberg groups $\mathrm{St}_{n}(R)$. Let $R$ be an associative ring with identity and $n\geq 2$ be an integer. The general linear group $\mathrm{GL}_{n}(R)$ is the group of all $n\times n$ invertible matrices with entries in $R$. For an element $r\in R$ and any integers $i,j$ such that $1\leq i\neq j\leq n,$ denote by $e_{ij}(r)$ the elementary $n\times n$ matrix with $1s$ in the diagonal positions and $r$ in the $(i,j)$-th position and zeros elsewhere. The group $E_{n}(R)$ is generated by all such $e_{ij}(r),$ i.e. $E_{n}(R)=\langle e_{ij}(r)\|1\leq i\neq j\leq n,r\in R\rangle.$ Denote by $I_{n}$ the identity matrix and by $[a,b]$ the commutator $aba^{-1}b^{-1}.$ The following lemma displays the commutator formulas for $E_{n}(R)$ (cf. Lemma 9.4 in [47]). ###### Lemma 9.1 Let $R$ be a ring and $r,s\in R.$ Then for distinct integers $i,j,k,l$ with $1\leq i,j,k,l\leq n,$ the following hold: 1. (1) $e_{ij}(r+s)=e_{ij}(r)e_{ij}(s);$ 2. (2) $[e_{ij}(r),e_{jk}(s)]=e_{ik}(rs);$ 3. (3) $[e_{ij}(r),e_{kl}(s)]=I_{n}.$ By Lemma 9.1, the group $E_{n}(R)$ $(n\geq 3)$ is finitely generated when the ring $R$ is finitely generated. Moreover, when $n\geq 3,$ the group $E_{n}(R)$ is normally generated by any elementary matrix $e_{ij}(1).$ The commutator formulas can be used to define Steinberg groups as follows. For $n\geq 3,$ the Steinberg group $\mathrm{St}_{n}(R)$ is the group generated by the symbols $\\{x_{ij}(r):1\leq i\neq j\leq n,r\in R\\}$ subject to the following relations: 1. (St$1$) $x_{ij}(r+s)=x_{ij}(r)x_{ij}(s);$ 2. (St$2$) $[x_{ij}(r),x_{jk}(s)]=x_{ik}(rs)$ for $i\neq k;$ 3. (St$3)$ $[x_{ij}(r),x_{kl}(s)]=1$ for $i\neq l,j\neq k.$ There is an obvious surjection $\mathrm{St}_{n}(R)\rightarrow E_{n}(R)$ defined by $x_{ij}(r)\longmapsto e_{ij}(r).$ For any ideal $I\vartriangleleft R,$ let $p:R\rightarrow R/I$ be the quotient map. Then the map $p$ induces a group homomorphism $p_{\ast}:\mathrm{St}_{n}(R)\rightarrow\mathrm{St}_{n}(R/I).$ Denote by $\mathrm{St}_{n}(R,I)$ (resp., $E_{n}(R,I)$) the subgroup of $\mathrm{St}_{n}(R)$ (resp., $E_{n}(R)$) normally generated by elements of the form $x_{ij}(r)$ (resp., $e_{ij}(r)$) for $r\in I$ and $1\leq i\neq j\leq n.$ In fact, $\mathrm{St}_{n}(R,I)$ is the kernel of $p_{\ast}$ (cf. Lemma 13.18 in Magurn [47] and its proof). However, $E_{n}(R,I)$ may not be the kernel of $E_{n}(R)\rightarrow E_{n}(R/I)$ induced by $p.$ ###### Lemma 9.2 When $n\geq\mathrm{sr}(R)+2,$ the natural map $\mathrm{St}_{n}(R)\rightarrow\mathrm{St}_{n+1}(R)$ is injective. In particular, when $R$ is finite, the Steinberg group $\mathrm{St}_{n}(R)$ is finite for any $n\geq 3.$ Proof. Let $W(n,R)$ be the kernel of the natural map $\mathrm{St}_{n}(R)\rightarrow\mathrm{St}_{n+1}(R).$ When $n\geq\mathrm{sr}(R)+2,$ the kernel $W(n,R)$ is trivial (cf. Kolster [41], Theorem 3.1 and Cor. 2.10). When $n$ is sufficient large, the Steinberg group $\mathrm{St}_{n}(R)$ is the universal central extension of $E_{n}(R)$ (cf. [59], Proposition 5.5.1. page 240). Therefore, the kernel $\mathrm{St}_{n}(R)\rightarrow E_{n}(R)$ is the second homology group $H_{2}(E_{n}(R);\mathbb{Z}).$ When $R$ is finite, both $E_{n}(R)$ and $H_{2}(E_{n}(R);\mathbb{Z})$ are finite. Therefore, the group $\mathrm{St}_{n}(R)$ is finite for any $n\geq 3.$ ### 9.2 Rigidity of group homomorphisms on matrix groups ###### Theorem 9.3 Suppose that $G$ is a group satisfying that 1) $G$ has a purely positive length function, i.e. there is a length function $l:G\rightarrow[0,\infty)$ such that $l(g)>0$ for any infinite-order element $g;$ and 2) any torsion abelian subgroup of $G$ is finitely generated. Let $R$ be an associative ring with identity and $\mathrm{St}_{n}(R)$ the Steinberg group. Suppose that $S<\mathrm{St}_{n}(R)$ is a finite-index subgroup. Then any group homomorphism $f:\mathrm{St}_{n}(R)\rightarrow G$ has its image finite when $n\geq 3$. Proof. Since any ring $R$ is a quotient of a free (non-commutative) ring $\mathbb{Z}\langle X\rangle$ for some set $X$ and $\mathrm{St}_{n}(R)$ is functorial with respect to the ring $R,$ we assume without loss of generality that $R=$ $\mathbb{Z}\langle X\rangle$. We prove the case $S=\mathrm{St}_{n}(R)\ $first. Let $x_{ij}=\langle x_{ij}(r):r\in R\rangle,$ which is isomorphic to the abelian group $R.$ Note that $[x_{12}(1),x_{23}(1)]=x_{13}(1)$ and $x_{13}(1)$ commutes with $x_{12}(1)$ and $x_{23}(1).$ Lemma 5.2 implies any length function vanishes on $x_{13}(1).$ By Lemma 1.4, the length $l(f(x_{13}(1)))=0.$ Note that $x_{ij}(r)$ is conjugate to $x_{13}(r)$ for any $r\in R$ and $i,j$ satisfying $1\leq i\neq j\leq n.$ Since $l$ is purely postive, we get that $f(x_{12}(1))$ is of finite order. Let $I=\ker f\|_{x_{12}}.$ Then $I\neq\varnothing,$ as $f(x_{12}(1))$ is of finite order. For any $x\in I,$ and $y\in R,$ we have $x_{12}(xy)=[x_{13}(x),x_{32}(y)].$ Therefore, $f(x_{12}(xy))=[f(x_{13}(x),f(x_{32}(y)))]=1$ and thus $xy\in I.$ Similarly, we have $f(x_{12}(yx))=f([x_{13}(y),x_{32}(x)])=1.$ This proves that $I$ is a (two- sided) ideal. Note that $f(e_{12})=R/I$ is a torsion abelian group. By the assumption 2), the quotient ring $R/I$ is finite. Let $\mathrm{St}_{n}(R,I)$ be the normal subgroup of $\mathrm{St}_{n}(R)$ generated by $x_{ij}(r),r\in I.$ There is a short exact sequence $\begin{array}[]{ccc}1\rightarrow\mathrm{St}_{n}(R,I)\rightarrow&\mathrm{St}_{n}(R)\rightarrow&\mathrm{St}_{n}(R/I)\rightarrow 1.\end{array}$ Since $R/I$ is finite, we know that $\mathrm{St}_{n}(R/I)$ is finite by Lemma 9.2. This proves that $\mathop{\mathrm{I}m}f$ is finite since $f$ factors through $\mathrm{St}_{n}(R/I)$. For general finite-index subgroup $S,$ we assume $S$ is normal in $\mathrm{St}_{n}(R)$ after passing to a finite-index subgroup of $S.$ A similar proof shows that $S$ contains $\mathrm{St}_{n}(R,I)$ for some ideal $I$ with the quotient ring $R/I$ finite. Therefore, the image $\mathop{\mathrm{I}m}f$ is finite. ###### Theorem 9.4 Suppose that $G$ is a group having a purely positive length function $l$. Let $R$ be an associative ring of characteristic zero such that any nonzero ideal is of a finite index (eg. the ring of algebraic integers in a number field). Suppose that $S<\mathrm{St}_{n}(R)$ is a finite-index subgroup of the Steinberg group. Then any group homomorphism $f:S\rightarrow G$ has its image finite when $n\geq 3$. Proof. The proof is similar to that of Theorem 9.3. Let $I=\ker f\|_{x_{12}},$ where $x_{12}=S\cap\langle x_{12}(r):r\in R\rangle.$ Since $R$ is of characteristic zero and the length $l(f(x_{12}(k)))=0$ for some integer $k,$ we have $f(x_{12}(k))$ is of finite order. Therefore, $f(x_{12}(k^{\prime}))=1$ for some integer $k^{\prime},$ which proves that the ideal $I$ is nozero. Since $I$ is of finite index in $R,$ we get that $\mathrm{St}_{n}(R,I)$ is of finite index in $S.$ This finishes the proof. Since the natural map $\mathrm{St}_{n}(R)\rightarrow E_{n}(R)$ is surjective, any group homomorphism $f:E_{n}(R)\rightarrow G$ can be lifted to be a group homomorphism $\mathrm{St}_{n}(R)\rightarrow G.$ Moreover, a finite-index subgroup $E$ of $E_{n}(R)$ is lifted to be a finite-index subgroup $S$ of $\mathrm{St}_{n}(R).$ Theorem 0.4 and Theorem 0.6 follows Theorem 9.3 and Theorem 9.4, by inductive arugments on the subnormal series as those of the proofs of Theorem 0.3. Proof of Corollary 0.5 and Corollary 0.7. For Corollary 0.5, it is enough to check the two conditions for $G$ in Theorem 0.4. Lemma 3.6 proves that $G$ has a purely positive length function. When $G$ is a $\mathrm{CAT}(0)$ group, (i.e. $G$ acts properly and cocompactly on a $\mathrm{CAT}(0)$ space), then any solvable subgroup of $G$ is finitely generated (and actually virtually abelian, see the Solvable Subgroup Theorem of [16], Theorem 7.8, page 249). When $G$ is hyperbolic, it’s well-known that $G$ contains finitely many conjugacy classes of finite subgroups and thus a torsion abelian subgroup is finite (see [16], Theorem 3.2, page 459). Birman-Lubotzky-McCarthy [7] proves that any abelian subgroup of the mapping class groups for orientable surfaces is finitely generated. Bestvina-Handel [6] proves that every solvable subgroup of $\mathrm{Out}(F_{k})$ has a finite index subgroup that is finitely generated and free abelian. When $G$ is the diffeomorphism group $\mathrm{Diff}(\Sigma,\omega),$ there is a subnormal series (see the proof of Lemma 3.8) $1\vartriangleleft\mathrm{Ham}(\Sigma,\omega)\vartriangleleft\mathrm{Diff}_{0}(\Sigma,\omega)\vartriangleleft\mathrm{Diff}(\Sigma,\omega),$ with subquoitents in $\mathrm{Ham}(\Sigma,\omega),$ $H_{1}(\Sigma,\mathbb{R})$ and the mapping class group $\mathrm{MCG}(\Sigma).$ Any abelian subgroup of a finitely generated subgroup of these groups is finitely generated. Corrolary 0.7 follows Theorem 9.4 and Lemma 3.6. ###### Remark 9.5 An infinite torsion abelian group may act properly on a simplicial tree (see [16], Example 7.11, page 250). Therefore, condition 2) in Theorem 0.4 does not hold for every group $G$ acting properly on a $\mathrm{CAT}(0)$ (or a Gromov hyperbolic) space. We don’t know whether the condition 2) can be dropped. ## 10 Length functions on Cremona groups Let $k$ be a field and $k(x_{1},x_{2},\cdots,x_{n})$ be the field of rational functions in $n$ indeterminates over $k.$ It is well-known that the Cremona group $\mathrm{Cr}_{n}(k)$ is isomorphic to the automorphism group $\mathrm{Aut}_{k}(k(x_{1},x_{2},\cdots,x_{n}))$ of the field $k(x_{1},x_{2},\cdots,x_{n})$. ###### Lemma 10.1 Let $f:k(x_{1},x_{2},\cdots,x_{n})\rightarrow k(x_{1},x_{2},\cdots,x_{n})$ be given by $f(x_{1})=\alpha x_{1},f(x_{i})=x_{i}$ for some $0\neq\alpha\in k$ and any $i=2,\cdots,n.$ Then $f$ lies in the center of a Heisenberg subgroup. In other words, there exists $g,h\in\mathrm{Cr}_{n}(k)$ such that $[g,h]=ghg^{-1}h^{-1}=f,[g,f]=1$ and $[h,f]=1.$ Proof. Let $g,h:k(x_{1},x_{2},\cdots,x_{n})\rightarrow k(x_{1},x_{2},\cdots,x_{n})$ be given by $g(x_{1})=x_{1}x_{2},g(x_{i})=x_{i}(i=2,\cdots,n)$ and $h(x_{1})=x_{1},h(x_{2})=\alpha^{-1}x_{2},h(x_{j})=x_{j}(j=3,\cdots,n).$ It can be directly checked that $[g,h]=f,[g,f]=1$ and $[h,f]=1.$ ###### Lemma 10.2 Let $l:\mathrm{Bir}(\mathbb{P}_{k}^{n})\rightarrow[0,\infty)$ be a length function $(n\geq 2)$. Then $l$ vanishes on diagonal elements and unipotent elements of $\mathrm{Aut}(\mathbb{P}_{k}^{n})=\mathrm{PGL}_{n+1}(k).$ Proof. Let $g=\mathrm{diag}(a_{0},a_{1},\cdots,a_{n})\in\mathrm{PGL}_{n+1}(k)$ be a diagonal element. Note that $l$ is subadditive on the diagonal subgroups. In order to prove $l(g)=0,$ it is enough to prove that $l(\mathrm{diag}(1,\cdots,1,a_{i},1,\cdots,1))=0,$ where $\mathrm{diag}(1,\cdots,1,a_{i},1,\cdots,1)$ is the diagonal matrix with $a_{i}$ in the $(i,i)$-th position and all other diagonal entries are $1.$ But $\mathrm{diag}(1,\cdots,1,a_{i},1,\cdots,1)$ is conjugate to $\mathrm{diag}(1,\alpha,1,\cdots,1)$ for $\alpha=a_{i}.$ Lemma 10.1 implies that $\mathrm{diag}(1,\alpha,1,\cdots,1)$ lies in the center of a Heisenberg group. Therefore, $l(\mathrm{diag}(1,\alpha,1,\cdots,1))=0$ by Lemma 5.2. This proves $l(g)=0.$ The vanishing of $l$ on unipotent elements follows Corollary 6.3 when the characteristic of $k$ is not $2$. When the characteristic of $k$ is $2,$ any unipotent element $A=I+u$ (where $u$ is nilpotent) is of finite order. This means $l(A)=0$. Proof of Theorem 0.8. When $k$ is algebraically closed, the Jordan normal form implies that any element $g\in\mathrm{PGL}_{n}(k)$ is conjugate to the form $sn$ with $s$ diagonal and $n$ the strictly upper triangular matrix. Moreover, $sn=ns.$ Therefore, $l(f)\leq l(s)+l(n).$ By Lemma 10.2, $l(s)=l(n)=0$ and thus $l(g)=0$. Proof of Corollary 0.9. Let $f:\mathrm{Bir}(\mathbb{P}_{k}^{2})\rightarrow G$ be a group homomorphism. Suppose that $G$ has a purely positive length function $l.$ By Theorem 0.8, the purely positive length function $l$ on $G$ will vanish on $f(\mathrm{PGL}_{3}(k)).$ Since $k$ is infinite and $\mathrm{PGL}_{3}(k)$ is a simple group, we get that $\mathrm{PGL}_{3}(k)$ lies in the $\ker f.$ By Noether-Castelnuovo Theorem, $\mathrm{Bir}(\mathbb{P}_{k}^{2})$ is generated by $\mathrm{PGL}_{3}(k)$ and an involution. Moreover, the $\mathrm{Bir}(\mathbb{P}_{k}^{2})$ is normally generated by $\mathrm{PGL}_{3}(k).$ Therefore, the group homomorphism $f$ is trivial. The general case is proved by an inductive argument on the subnormal series of a finite-index subgroup of $G.$ ###### Lemma 10.3 Let $\mathrm{Bir}(\mathbb{P}_{\mathbb{R}}^{n})$ $(n\geq 2)$ be the real Cremona group. Any length function $l:\mathrm{Bir}(\mathbb{P}_{k}^{n})\rightarrow[0,\infty),$ which is continuous on $\mathrm{PSO}(n+1)<\mathrm{Aut}(\mathbb{P}_{\mathbb{R}}^{n}),$ vanishes on $\mathrm{PGL}_{n+1}(\mathbb{R}).$ Proof. By Lemma 10.2, the length function $l$ vanishes on diagonal matrices of $\mathrm{PGL}_{n+1}(\mathbb{R}).$ Theorem 0.11 implies that $l$ vanishes on the whole group $\mathrm{PGL}_{n+1}(\mathbb{R}).$ Acknowledgements The author wants to thank many people for helpful discussions, including Wenyuan Yang on a discussion of hyperbolic groups, C. 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# The convergence rate of of multivariate operators on simplex in Orlicz space111This work was supported by the Ningxia Science and Technology Department [grant numbers 2019BEB04003]; and Ningxia Education Department [grant numbers NXYLXK2019A8] Wan Ma Lihong Chang Yongxia Qiang<EMAIL_ADDRESS>School of Mathematics and Computer Science, Ningxia Normal University, Guyuan City, Ningxia 756000, People’s Republic of China ###### Abstract The approximation of functions in Orlicz space by multivariate operators on simplex is considered. The convergence rate is given by using modulus of smoothness. ###### keywords: Stancu-Kantorovič operator, Meyer-König-Zeller operator, Convergence rate , Orlicz space ††journal: J. Math. Anal. Appl. ## 1 Introduction Let $\Phi(u)$ be a N-function, $\Psi$ be the complementary function of $\Phi$. We will say that $\Phi$ satisfies the $\Delta_{2}$-condition if $\Phi(2u)\leq c\Phi(u)$ for any $u\geq u_{0}\geq 0$ with some constant $c$ independent of $u.$ For $\triangle=\\{x=(x_{1},x_{2})\in\mathrm{R^{2}}:x_{1}+x_{2}\leq 1,x_{1},x_{2}\geq 0\\},$ the Orlicz space $L_{\Phi}^{\ast}(\triangle)$ corresponding to the function $\Phi$ consists of all Lebesgue-measurable functions $f(x)$ on $\triangle$ such that integral $\int_{\triangle}f(x)g(x)\mbox{d}x$ is finite for any measurable functions $g(x)$ with $\int_{\triangle}\Psi(g(x))\mbox{d}x<\infty.$ It is well-known that the space $L_{\Phi}^{\ast}(\triangle)$ becomes a complete normed space with Orlicz norm $\\|f\\|_{\Phi}=\\|f\\|_{[\Phi,\triangle]}=\sup\left\\{\left\|\int_{\triangle}f(x)g(x)\mbox{d}x\right\|:\int_{\triangle}\Psi(g(x))\mbox{d}x\leq 1\right\\}.$ It can be proved that $\\|f\\|_{\Phi}=\inf_{\alpha>0}\frac{1}{\alpha}\left\\{1+\int_{\triangle}\Phi(\alpha f(x))\mbox{d}x\right\\}.$ See [1] for the above. For $f\in L_{\Phi}^{\ast}(\triangle),$ we first extend $f(x)$ from $\triangle$ to $D=[0,1]\times[0,1]$ according to $f(x)=f(1-x)$, and then extend $f(x)$ to $\mathrm{R^{2}}$ with period 1. The nonnegative function $\Omega_{\mathrm{R^{2}}}^{2}(f,r)_{\Phi}=\sup\\{\omega_{h}^{2}(f,r)_{\Phi}:h=(h_{1},h_{2})\in\mathrm{R^{2}},\|h\|=1\\}$ of the variable $r\geq 0$ will be called the 2-th order modulus of continuity of the function $f\in L_{\Phi}^{\ast}(\triangle)$ in the Orlicz norm $\Phi.$ Here, $\|h\|=\sqrt{h_{1}^{2}+h_{2}^{2}},$ and $\omega_{h}^{2}(f,r)_{\Phi}=\sup_{\|t\|\leq r}\\|f(x+th)+f(x-th)-2f(x)\\|_{\Phi}$ is the 2-th order modulus of continuity in the direction $h$ of the function $f.$ For any Lebesgue-measurable function $f(x)$ on $\triangle,$ the functional $K_{n}(f;x)=\sum_{k_{1}=0}^{\infty}\sum_{k_{2}=0}^{\infty}{\tilde{p}_{n,k_{1},k_{2}}(x)}c_{n,k_{1},k_{2}}\int_{\triangle_{k_{1},k_{2}}}f(u)\mbox{d}u$ (1.1) is called Meyer-König-Zeller-$\mathrm{Kantorovi\check{c}}$ operator on $\triangle$ [2]; the functional $K_{n,s}(f;x)=\sum_{k+l\leq n}{b_{n,k,l,s}(x)}(n+2)^{2}\int_{I_{n,k,l}}f(u)\mbox{d}u$ (1.2) is called Stancu-$\mathrm{Kantorovi\check{c}}$ operator on $\triangle$[3], where $x\in\triangle,$ $n\in\mathrm{Z^{+}},$ $k,s,l$ are nonnegative integers, $0\leq s<\frac{n}{2},$ and ${\tilde{p}_{n,k_{1},k_{2}}(x)}=\frac{(n+k_{1}+k_{2})!}{n!k_{1}!k_{2}!}x_{1}^{k_{1}}x_{2}^{k_{2}}(1-x_{1}-x_{2})^{n+1},$ $c_{n,k_{1},k_{2}}=\frac{(n+k_{1}+k_{2})^{2}(n+k_{1}+k_{2}+1)^{2}}{(n+k_{1})(n+k_{2})},$ $\triangle_{k_{1},k_{2}}=\left[\frac{k_{1}}{n+k_{1}+k_{2}},\frac{k_{1}+1}{n+k_{1}+k_{2}+1}\right]\times\left[\frac{k_{2}}{n+k_{1}+k_{2}},\frac{k_{2}+1}{n+k_{1}+k_{2}+1}\right],$ $I_{n,k,l}=\left[\frac{k}{n+2},\frac{k+1}{n+2}\right]\times\left[\frac{l}{n+2},\frac{l+1}{n+2}\right],$ ${p_{n,k,l}(x)}=\frac{n!}{k!l!(n-k-l)!}x_{1}^{k}x_{2}^{l}(1-x_{1}-x_{2})^{n-k-l},$ ${b_{n,k,l,s}(x)}=\begin{cases}(1-x_{1}-x_{2})p_{n-s,k,l}(x),\quad k+l\leq n-s,0\leq k,l<s;\\\ (1-x_{1}-x_{2})p_{n-s,k,l}(x)+x_{1}p_{n-s,k-s,l}(x),\\\ \qquad\qquad\qquad\qquad\qquad\quad\;\;\;k+l\leq n-s,s\leq k,0\leq l<s;\\\ (1-x_{1}-x_{2})p_{n-s,k,l}(x)+x_{2}p_{n-s,k,l-s}(x),\\\ \qquad\qquad\qquad\qquad\qquad\quad\;\;\;k+l\leq n-s,s\leq l,0\leq k<s;\\\ (1-x_{1}-x_{2})p_{n-s,k,l}(x)+x_{1}p_{n-s,k-s,l}(x)+x_{2}p_{n-s,k,l-s}(x,y),\\\ \qquad\qquad\qquad\qquad\qquad\qquad k+l\leq n-s,s\leq k,s\leq l;\\\ x_{1}p_{n-s,k-s,l}(x),\quad\qquad\qquad n-s<k+l\leq n,s\leq k,0\leq l<s;\\\ x_{2}p_{n-s,k,l-s}(x),\quad\qquad\qquad n-s<k+l\leq n,0\leq k<s,s\leq l;\\\ x_{1}p_{n-s,k-s,l}(x)+x_{2}p_{n-s,k,l-s}(x),\\\ \qquad\qquad\qquad\qquad\qquad\qquad\;n-s<k+l\leq n,s\leq k,s\leq l.\end{cases}$ Denote $C$ a constant independent of $f,n,s,$ and its value can be different in different positions. The convergence rate of the operators (1.1) and (1.2) in space $L_{p}$ has been studied (see [2, 3]). This paper intends to investigate their convergence in space $L_{\Phi}^{\ast}(\triangle),$ and the main results are as follows. ###### Theorem 1.1. For $f\in L_{\Phi}^{\ast}(\triangle),$ if a N-function $\Phi(u)$ satisfies the $\Delta_{2}$-condition, then $\\|K_{n}(f)-f\\|_{\Phi}\leq C\left(\frac{1}{n}\\|f\\|_{\Phi}+\Omega_{\mathrm{R^{2}}}^{2}\left(f,\sqrt{\frac{1}{n}}\right)_{\Phi}\right).$ ###### Theorem 1.2. For $f\in L_{\Phi}^{\ast}(\triangle),$ if a N-function $\Phi(u)$ satisfies the $\Delta_{2}$-condition, then $\\|K_{n,s}(f)-f\\|_{\Phi}\leq C\left(\frac{1}{n}\\|f\\|_{\Phi}+\Omega_{\mathrm{R^{2}}}^{2}\left(f,\sqrt{\frac{1}{n}}\right)_{\Phi}\right).$ ## 2 Lemmas ###### Lemma 2.1. $K_{n}$ is a bounded linear operator, and $\\|K_{n}\\|_{\Phi}\leq 2.$ ###### Proof. The linearity of $K_{n}$ is obvious. The following proves $\\|K_{n}\\|_{\Phi}\leq 2.$ After calculation, we can get $mes\triangle_{k_{1},k_{2}}=\frac{(n+k_{1})(n+k_{2})}{(n+k_{1}+k_{2})^{2}(n+k_{1}+k_{2}+1)^{2}},$ $\int_{\triangle}\tilde{p}_{n,k_{1},k_{2}}(x)\mbox{d}x=\frac{n+1}{(n+k_{1}+k_{2}+3)(n+k_{1}+k_{2}+2)(n+k_{1}+k_{2}+1)}.$ By using the _Lemma 1_ in [2], Jensen inequality of convex function and the _Theorem 1.4_ in [1], we obtain $\displaystyle\\|K_{n}(f)\\|_{\Phi}$ $\displaystyle=\inf_{\alpha>0}\frac{1}{\alpha}\left\\{1+\int_{\triangle}\Phi\left(\alpha K_{n}(f;x)\right)\mbox{d}x\right\\}$ $\displaystyle=\inf_{\alpha>0}\frac{1}{\alpha}\left\\{1+\int_{\triangle}\Phi\left(\alpha\sum_{k_{1}=0}^{\infty}\sum_{k_{2}=0}^{\infty}\tilde{p}_{n,k_{1},k_{2}}(x)c_{n,k_{1},k_{2}}\int_{\triangle_{k_{1},k_{2}}}f(u)\mbox{d}u\right)\mbox{d}x\right\\}$ $\displaystyle\leq\inf_{\alpha>0}\frac{1}{\alpha}\left\\{1+\sum_{k_{1}=0}^{\infty}\sum_{k_{2}=0}^{\infty}\int_{\triangle}\tilde{p}_{n,k_{1},k_{2}}(x)\mbox{d}xc_{n,k_{1},k_{2}}\int_{\triangle_{k_{1},k_{2}}}\Phi\left(\alpha f(u)\right)\mbox{d}u\right\\}$ $\displaystyle\leq\inf_{\alpha>0}\frac{1}{\alpha}\\{1+2\sum_{k_{1}=0}^{\infty}\sum_{k_{2}=0}^{\infty}\int_{\triangle_{k_{1},k_{2}}}\Phi\left(\alpha f(u)\right)\mbox{d}u\\}$ $\displaystyle\leq\inf_{\alpha>0}\frac{1}{\alpha}\left\\{1+\int_{\triangle}\Phi\left(2\alpha f(u)\right)\mbox{d}u_{1}\mbox{d}u_{2}\right\\}$ $\displaystyle=\\|2f\\|_{\Phi}$ $\displaystyle=2\\|f\\|_{\Phi}.$ Namely $\\|K_{n}\\|_{\Phi}\leq 2.\qed$ ###### Lemma 2.2. $K_{n,s}$ is a bounded linear operator, and $\\|K_{n,s}\\|_{\Phi}\leq 12.$ ###### Proof. The linearity of $K_{n,s}$ is obvious. The following proves $\\|K_{n,s}\\|_{\Phi}\leq 12.$ After calculation, we can get $mesI_{n,k,l}=mesI_{n,k+s,l}=mesI_{n,k,l+s}=\frac{1}{(n+2)^{2}},$ $\int_{\triangle}p_{n-s,k,l}(x)\mbox{d}x=\frac{1}{(n-s+2)(n-s+1)}.$ By using the _Lemma 2.1_ in [3], Jensen inequality of convex function and the _Theorem 1.4_ in [1], we obtain $\displaystyle\\|K_{n,s}(f)\\|_{\Phi}$ $\displaystyle\leq\inf_{\alpha>0}\frac{1}{\alpha}\left\\{1+\sum_{k+l\leq n-s}\int_{\triangle}p_{n-s,k,l}(x)\mbox{d}x\left(\int_{I_{n,k,l}}+\int_{I_{n,k+s,l}}\int_{I_{n,k,l+s}}\right)\Phi\left(\alpha f(u)\right)\mbox{d}u\right\\}$ $\displaystyle\leq\inf_{\alpha>0}\frac{1}{\alpha}\left\\{1+\frac{3(n+2)^{2}}{(n-s+2)(n-s+1)}\int_{\triangle}\Phi(\alpha f(u))\mbox{d}u\right\\}.$ $\displaystyle\leq\inf_{\alpha>0}\frac{1}{\alpha}\left\\{1+\int_{\triangle}\Phi\left(\alpha 12f(u)\right)\mbox{d}u\right\\}$ $\displaystyle=\\|12f\\|_{\Phi}$ $\displaystyle=12\\|f\\|_{\Phi}.$ Namely $\\|K_{n,s}\\|_{\Phi}\leq 12.\qed$ ###### Lemma 2.3. The following holds for (1.1). $K_{n}(1;x)=1,\quad K_{n}\left((u_{i}-x_{i})^{i};x\right)\leq\frac{C}{n},\quad i=1,2.$ ###### Lemma 2.4. The following holds for (1.2). $K_{n,s}(1;x)=1,\quad K_{n,s}\left((u_{i}-x_{i})^{i};x\right)\leq\frac{C}{n},\quad i=1,2.$ The proof of the Lemma 2.3 and Lemma 2.4 can be obtained from the _Lemma 3_ in [2] and the _Lemma 2.1_ in [3]. ###### Lemma 2.5. If we denote $f_{r}$ the Steklov mean function for $f\in L_{\Phi}^{\ast}(\triangle),$ i.e. $f_{r}(x)=\frac{1}{r^{4}}\int_{[-r/2,r/2]^{4}}f(x+u+v)\mbox{d}s\mbox{d}t,$ then $\left\\|{f_{r}}\right\\|_{\Phi}\leq C\left\\|f\right\\|_{\Phi},$ (2.1) $\left\\|{f-f_{r}}\right\\|_{\Phi}\leq C\Omega_{R^{2}}^{2}\left({f,r}\right)_{\Phi},$ (2.2) $\left\\|{\frac{{\partial f_{r}}}{{\partial x_{1}}}}\right\\|_{\Phi}\leq C\left({\left\\|{f_{r}}\right\\|_{\Phi}+\left\\|{\frac{{\partial^{2}f_{r}}}{{\partial x_{1}^{2}}}}\right\\|_{\Phi}}\right),$ (2.3) $\left\\|{\frac{{\partial f_{r}}}{{\partial x_{2}}}}\right\\|_{\Phi}\leq C\left({\left\\|{f_{r}}\right\\|_{\Phi}+\left\\|{\frac{{\partial^{2}f_{r}}}{{\partial x_{2}^{2}}}}\right\\|_{\Phi}}\right),$ (2.4) $\left\\|{\frac{{\partial^{2}f_{r}}}{{\partial x_{1}^{2}}}}\right\\|_{\Phi}+\left\\|{\frac{{\partial^{2}f_{r}}}{{\partial x_{2}^{2}}}}\right\\|_{\Phi}+\left\\|{\frac{{\partial^{2}f_{r}}}{{\partial x_{1}\partial x_{2}}}}\right\\|_{\Phi}\leq\frac{C}{{r^{2}}}\Omega_{R^{2}}^{2}\left({f,r}\right)_{\Phi}.$ (2.5) (2.1), (2.2), (2.5) can be directly verified, and the proof of (2.3),(2.4) is similar to that of the _Lemma 1a_ in [4]. If N-function $\Phi$ satisfies the $\Delta_{2}$-condition, then $L_{\Phi}^{\ast}$ is separable. This leads to the following conclusion[5]. ###### Lemma 2.6. If N-function $\Phi$ satisfies the $\Delta_{2}$-condition, then $\left\\|\sup_{u_{1}\neq x_{1}}\frac{1}{u_{1}-x_{1}}\int_{x_{1}}^{u_{1}}\left\|\frac{\partial^{2}f_{r}(\xi,x_{2})}{\partial\xi^{2}}\right\|\mbox{d}\xi\right\\|_{\Phi}\leq C\left\\|{\frac{{\partial^{2}f_{r}}}{{\partial x_{1}^{2}}}}\right\\|_{\Phi},$ $\left\\|\sup_{u_{2}\neq x_{2}}\frac{1}{u_{2}-x_{2}}\int_{x_{2}}^{u_{2}}\left\|\frac{\partial^{2}f_{r}(x_{1},\eta)}{\partial\eta^{2}}\right\|\mbox{d}\eta\right\\|_{\Phi}\leq C\left\\|{\frac{{\partial^{2}f_{r}}}{{\partial x_{2}^{2}}}}\right\\|_{\Phi},$ $\left\\|\sup_{u_{2}\neq x_{2}}\frac{1}{u_{2}-x_{2}}\int_{x_{2}}^{u_{2}}\left(\sup_{u_{1}\neq x_{1}}\int_{x_{1}}^{u_{1}}\left\|\frac{\partial^{2}f_{r}(\xi,\eta)}{\partial\xi\partial\eta}\right\|\mbox{d}\xi\right)\mbox{d}\eta\right\\|_{\Phi}\leq C\left\\|{\frac{{\partial^{2}f_{r}}}{{\partial x_{1}\partial x_{2}}}}\right\\|_{\Phi}.$ ## 3 Proof of the main results The proof of the Theorem 1.1 and Theorem 1.2 is similar, so only the Theorem 1.1 is proved below. ###### Proof. Because $\displaystyle f_{r}(u)-f_{r}(x)=(u_{1}-x_{1})\frac{\partial f_{r}(x)}{\partial x_{1}}$ $\displaystyle+(u_{2}-x_{2})\frac{\partial f_{r}(x)}{\partial x_{2}}+\int_{x_{1}}^{u_{1}}(u_{1}-\xi)\frac{\partial^{2}f_{r}(\xi,x_{2})}{\partial\xi^{2}}\mbox{d}\xi$ $\displaystyle+\int_{x_{2}}^{u_{2}}(u_{2}-\eta)\frac{\partial^{2}f_{r}(x_{1},\eta)}{\partial\eta^{2}}\mbox{d}\eta+\int_{x_{1}}^{u_{1}}\int_{x_{2}}^{u_{2}}\frac{\partial^{2}f_{r}(\xi,\eta)}{\partial\xi\partial\eta}\mbox{d}\eta\mbox{d}\xi,$ so $\displaystyle\left\|K_{n}(f_{r};x)-f_{r}(x)\right\|\leq$ $\displaystyle\left\|K_{n}((u_{1}-x_{1});x)\right\|\left\|\frac{\partial f_{r}(x)}{\partial x_{1}}\right\|+\left\|K_{n}((u_{2}-x_{2});x)\right\|\left\|\frac{\partial f_{r}(x)}{\partial x_{2}}\right\|+$ $\displaystyle\left\|K_{n}((u_{1}-x_{1})^{2};x)\right\|\left(\sup_{u_{1}\neq x_{1}}\frac{1}{u_{1}-x_{1}}\int_{x_{1}}^{u_{1}}\left\|\frac{\partial^{2}f_{r}(\xi,x_{2})}{\partial\xi^{2}}\right\|\mbox{d}\xi\right)+$ $\displaystyle\left\|K_{n}((u_{2}-x_{2})^{2};x)\right\|\left(\sup_{u_{2}\neq x_{2}}\frac{1}{u_{2}-x_{2}}\int_{x_{2}}^{u_{2}}\left\|\frac{\partial^{2}f_{r}(x_{1},\eta)}{\partial\eta^{2}}\right\|\mbox{d}\eta\right)+$ $\displaystyle\left\|K_{n}(\|u_{1}-x_{1}\|\|u_{2}-x_{2}\|;x)\right\|\left(\sup_{u_{2}\neq x_{2}}\frac{1}{u_{2}-x_{2}}\int_{x_{2}}^{u_{2}}\left(\sup_{u_{1}\neq x_{1}}\int_{x_{1}}^{u_{1}}\left\|\frac{\partial^{2}f_{r}(\xi,\eta)}{\partial\xi\partial\eta}\right\|\mbox{d}\xi\right)\mbox{d}\eta\right).$ Noticing $\|K_{n}(\|u_{1}-x_{1}\|\|u_{2}-x_{2}\|;x)\|\leq\frac{1}{2}\|K_{n}((u_{1}-x_{1})^{2};x)\|+\|K_{n}((u_{2}-x_{2})^{2};x)\|,$ we continue the above estimation using the Lemma 2.3, Lemma 2.5 and Lemma 2.6. $\displaystyle\\|K_{n}(f_{r})-f_{r}\\|_{\Phi}$ $\displaystyle\leq\frac{C}{n}\left(\left\\|\frac{\partial f_{r}}{\partial x_{1}}\right\\|_{\Phi}+\left\\|\frac{\partial f_{r}}{\partial x_{2}}\right\\|_{\Phi}+\left\\|\frac{\partial^{2}f_{r}}{\partial x_{1}^{2}}\right\\|_{\Phi}+\left\\|\frac{\partial^{2}f_{r}}{\partial x_{2}^{2}}\right\\|_{\Phi}+\left\\|\frac{\partial^{2}f_{r}}{\partial x_{1}\partial x_{2}}\right\\|_{\Phi}\right)$ $\displaystyle\leq\frac{C}{n}\left(\left\\|f\right\\|_{\Phi}+\frac{1}{r^{2}}\Omega_{\mathrm{R}^{2}}^{2}(f,r)_{\Phi}\right).$ If $r=\sqrt{\frac{1}{n}},$ then $\\|K_{n}(f_{r})-f_{r}\\|_{\Phi}\leq\frac{C}{n}\left(\\|f\\|_{\Phi}+n\Omega_{\mathrm{R}^{2}}^{2}\left(f,\sqrt{\frac{1}{n}}\right)_{\Phi}\right).$ For $f\in L_{\Phi}^{\ast}(\triangle),$ using the Lemma 2.1 and Lemma 2.5 we get $\displaystyle\\|K_{n}(f)-f\\|_{\Phi}$ $\displaystyle\leq\\|K_{n}(f)-K_{n}(f_{r})\\|_{\Phi}+\\|K_{n}(f_{r})-f_{r}\\|_{\Phi}+\\|f_{r}-f\\|_{\Phi}$ $\displaystyle\leq 3\\|f_{r}-f\\|_{\Phi}+\\|K_{n}(f_{r})-f_{r}\\|_{\Phi}$ $\displaystyle\leq C\Omega_{\mathrm{R}^{2}}^{2}(f,r)_{\Phi}+C\left(\frac{1}{n}\\|f\\|_{\Phi}+\Omega_{\mathrm{R}^{2}}^{2}\left(f,\sqrt{\frac{1}{n}}\right)_{\Phi}\right)$ $\displaystyle\leq C\left(\frac{1}{n}\\|f\\|_{\Phi}+\Omega_{\mathrm{R}^{2}}^{2}(f,\sqrt{\frac{1}{n}})_{\Phi}\right).\hskip 99.58464pt\qed$ ## 4 Remark If N-function $\Phi(u)=u^{p}$ $(1<p<\infty),$ then $L_{\Phi}^{\ast}(\triangle)=L_{p}.$ Thus the corresponding conclusions in [2] and [3] can be obtained from the Theorem 1.1 and Theorem 1.2. ## References ## References * [1] C. 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# High efficient multipartite entanglement purification using hyperentanglement Lan Zhou,1 Pei-Shun Yan,2 Wei Zhong,2 Yu-Bo<EMAIL_ADDRESS>1 School of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210003, China 2Institute of Quantum Information and Technology, Nanjing University of Posts and Telecommunications, Nanjing, 210003, China ###### Abstract Multipartite entanglement plays an important role in controlled quantum teleportation, quantum secret sharing, quantum metrology and some other important quantum information branches. However, the maximally multipartite entangled state will degrade into the mixed state because of the noise. We present an efficient multipartite entanglement purification protocol (EPP) which can distill the high quality entangled states from low quality entangled states for $N$-photon systems in a Greenberger-Horne-Zeilinger (GHZ) state in only linear optics. After performing the protocol, the spatial-mode entanglement is used to purify the polarization entanglement and one pair of high quality polarization entangled state will be obtained. This EPP has several advantages. Firstly, with the same purification success probability, this EPP only requires one pair of multipartite GHZ state, while existing EPPs usually require two pairs of multipartite GHZ state. Secondly, if consider the practical transmission and detector efficiency, this EPP may be extremely useful for the ratio of purification efficiency is increased rapidly with both the number of photons and the transmission distance. Thirdly, this protocol requires linear optics and does not add additional measurement operations, so that it is feasible for experiment. All these advantages will make this protocol have potential application for future quantum information processing. ###### pacs: 03.67.Lx ## I Introduction Entanglement plays an important role in quantum communication and computation. Quantum teleportation teleportation , quantum key distributionQKD , dense codingdensecoding , quantum secure direct communicationQSDC1 ; QSDC2 ; QSDC3 , distributed quantum computing computation , distributed secure quantum machine learning DSQML , and other important branches all require the parties to share the entanglement. Besides the bipartite entanglement, multipartite entanglement also plays an important role in controlled quantum teleportation cteleportation1 ; cteleportation2 , quantum secret sharingQSS1 ; QSS2 ; QSS3 , quantum state sharingQSTS1 ; QSTS2 ; QSTS3 , quantum metrologymetrology1 ; metrology2 , and so on. Recently, the multipartite entangled state named Greenberger-Horne-Zeilinger (GHZ) states was been used in some important quantum communication experiment, such as long-distance measurement-device- independent multiparty quantum communication chenzb , equitable multiparty quantum communication without a trusted third party jeong1 and quantum teleportation of shared quantum secret jeong2 . The GHZ state also have been realized with superconducting system superconduct1 ; superconduct2 , trapped ions ion , and photonic system photon . The detection of the multipartite entanglement structure has also been reported detection . In a practical application, the quantum system should inevitably interact with its environment, and the environment noise will degrade the entanglement. In general, the decoherence will make the maximally entangled state become a mixed state. The degraded entanglement will decrease the efficiency of the quantum communication and it also will make the quantum communication become insecure repeater . Entanglement purification is a powerful tool to distill the high quality entangled states from the low quality entangled states purification1 ; purification2 ; purification3 ; purification4 ; purification5 ; addpurification1 ; addpurification3 ; experiment2 ; purification6 ; purification7 ; purification8 ; purification9 ; purification11 ; purification12 ; purification13 ; purification14 ; purification15 ; purification16 ; purification17 ; purification18 ; purification19 ; purification20 ; purification21 ; purification22 ; purification23 ; purification24 ; purification25 ; purification26 ; addpurification2 ; purification27 ; addpurification4 ; shengprl . Entanglement purification has been widely discussed since Bennett et al. proposed the first entanglement purification protocol (EPP) purification1 . EPPs for bipartite system were proposed in photonic system purification3 ; purification4 ; purification5 ; purification7 ; purification8 ; purification9 , electionsaddpurification1 , quantum dotspurification11 ; purification12 , atoms experiment2 ; purification13 and so on. For example, in 2001, Pan et al. proposed the polarization EPP with only feasible linear optics elements purification3 . In 2008, the EPP with spontaneous parametric down conversion (SPDC) source based on cross-Kerr nonlinearity was proposed purification7 . In this EPP, the purified high quality entangled state can be remained for further application and the remained entangled states can also be repeated to perform the purification to obtain the higher entangled states. The experiments of entanglement purification in optical system were also reported purification4 . In 2017, Chen et al. realized the nested entanglement purification for quantum repeaters. In this experiment, the entanglement purification and entanglement swapping can be realized simultaneously purification17 . On the other hand, the double-pair noise components from the SPDC source can be eliminated automatically. This work was extended to the multi-copy cases addpurification2 . The optimal entanglement purification was also investigated purification26 . Recently, the first high efficient and long-distance entanglement purification using hyperentanglement was demonstrated shengprl . The hyperentanglement was first distributed to 11 km and the spatial entanglement was used to purification polarization entanglement. The authors also demonstrated its powerful application in entanglement-based QKD. For multipartite system, Murao et al. described the first EPP with controlled- not (CNOT) gate multipurification1 . In 2003, Dur et al. described the EPP for Graph state multipurification2 . In 2007, this protocol was extended to high- dimension multipartite system with the generalized CNOT gate multipurification3 . In 2008, the multipartite EPP for polarization entangled states with cross-Kerr nonlinearity was proposed multipurification4 . In 2011, Deng proposed the multipartite EPP using entanglement link from subspace multipurification5 . In his protocol, the discussed items in conventional EPPs still have entanglement in a subspace and they can be reused with entanglement link. There are another kind of EPP for multipartite entanglement system, named deterministic EPP multipurification6 ; multipurification7 . In these EPPs, they exploit the hyperentanglement to perform the purification. Such EPPs are based on the condition that the spatial mode entanglement is robust and it does not suffer from the noise. Therefore, the spatial mode entanglement or the frequency entanglement can be completely transformed to the polarization entanglement. In conventional EPPs, they all require two copies of low quality entangled states. After performing the CNOT or similar operations, one pair of high quality entangled state is remained, if the purification is successful. On the other hand, if the purification is a failure, both pairs should be discarded. In this paper, we will describe an efficient EPP for multipartite polarization entangled systems in a GHZ state, inspired the idea of Ref.shengprl . Different from existing EPPs for multipartite system, this protocol only requires one pair of hyperentangled state. By performing the CNOT gate between two degrees of freedom, the spatial entanglement is consumed. Therefore, if the protocol is successful, one can obtain a high quality polarization entangled state. This EPP is based on linear optics and it is also feasible for current experiment condition. This protocol is organized as follows. In Sec.II, we describe this EPP for bit-flip error. In Sec.III, we describe this protocol for phase-flip error. In Sec.IV, we extend this EPP to a general case for arbitrary N-photon GHZ state. In Sec.V, we present a discussion. Finally, in Sec. VI, we will provide a conclusion. ## II Multipartite entanglement purification for bit-flip error In this section, we describe this EPP with a simple example. The three-photon GHZ states can be written as follows. $\displaystyle\|\Phi_{0}^{\pm}\rangle_{ABC}=\frac{1}{\sqrt{2}}(\|H\rangle_{A}\|H\rangle_{B}\|H\rangle_{C}\pm\|V\rangle_{A}\|V\rangle_{B}\|V\rangle_{C}),$ $\displaystyle\|\Phi_{1}^{\pm}\rangle_{ABC}=\frac{1}{\sqrt{2}}(\|H\rangle_{A}\|H\rangle_{B}\|V\rangle_{C}\pm\|V\rangle_{A}\|V\rangle_{B}\|H\rangle_{C}),$ $\displaystyle\|\Phi_{2}^{\pm}\rangle_{ABC}=\frac{1}{\sqrt{2}}(\|H\rangle_{A}\|V\rangle_{B}\|H\rangle_{C}\pm\|V\rangle_{A}\|H\rangle_{B}\|V\rangle_{C}),$ $\displaystyle\|\Phi_{3}^{\pm}\rangle_{ABC}=\frac{1}{\sqrt{2}}(\|V\rangle_{A}\|H\rangle_{B}\|H\rangle_{C}\pm\|H\rangle_{A}\|V\rangle_{B}\|V\rangle_{C}).$ (1) Here $\|H\rangle$ denotes the horizonal polarization and $\|V\rangle$ denotes the vertical polarization of the photon, respectively. Figure 1: Schematic drawing showing the principle of entanglement purification for bit-flip error. HWP45 is the half-wave plate setting as 45∘. The PBS is the polarization beam splitter which can transmit the $\|H\rangle$ polarization and reflect the $\|V\rangle$ polarization photon. The polarizing beam displacers (BD) can couple $\|H\rangle$ and $\|V\rangle$ polarization components from different spatial modes. The spatial mode GHZ states can be written as follows. $\displaystyle\|\phi_{0}^{\pm}\rangle_{ABC}=\frac{1}{\sqrt{2}}(\|a_{1}\rangle_{A}\|b_{1}\rangle_{B}\|c_{1}\rangle_{C}\pm\|a_{2}\rangle_{A}\|b_{2}\rangle_{B}\|c_{2}\rangle_{C}),$ $\displaystyle\|\phi_{1}^{\pm}\rangle_{ABC}=\frac{1}{\sqrt{2}}(\|a_{1}\rangle_{A}\|b_{1}\rangle_{B}\|c_{2}\rangle_{C}\pm\|a_{2}\rangle_{A}\|b_{2}\rangle_{B}\|c_{1}\rangle_{C}),$ $\displaystyle\|\phi_{2}^{\pm}\rangle_{ABC}=\frac{1}{\sqrt{2}}(\|a_{1}\rangle_{A}\|b_{2}\rangle_{B}\|c_{1}\rangle_{C}\pm\|a_{2}\rangle_{A}\|b_{1}\rangle_{B}\|c_{2}\rangle_{C}),$ $\displaystyle\|\phi_{3}^{\pm}\rangle_{ABC}=\frac{1}{\sqrt{2}}(\|a_{2}\rangle_{A}\|b_{1}\rangle_{B}\|c_{1}\rangle_{C}\pm\|a_{1}\rangle_{A}\|b_{2}\rangle_{B}\|c_{2}\rangle_{C}).$ (2) Here a1, a2, b1, b2, c1 and c2 are the spatial modes as shown in Fig. 1. The PBS is the polarization beam splitter which can transmit the $\|H\rangle$ polarization and reflect the $\|V\rangle$ polarization photon. The polarizing beam displacers (BD) can couple $\|H\rangle$ and $\|V\rangle$ polarization components from different spatial modes. HWP45 is the half-wave plate setting as 45∘. It can convert $\|H\rangle$ polarization to $\|V\rangle$ and $\|V\rangle$ to $\|H\rangle$ , respectively. The entanglement source $S$ emits a pair of hyperentangled GHZ state of the form $\displaystyle\|\Psi\rangle_{ABC}=\|\Phi_{0}^{+}\rangle_{ABC}\otimes\|\phi_{0}^{+}\rangle_{ABC}.$ (3) The hyperentangled GHZ state is distributed to Alice, Bob and Charlie, respectively. During distribution, if a bit-flip error occurs on both the polarization and spatial modes entangled state, the original state $\|\Psi\rangle_{ABC}$ will become a mixed state as $\displaystyle\rho_{ABC}=\rho^{P}_{ABC}\otimes\rho^{S}_{ABC}.$ (4) Here $\rho^{P}_{ABC}$ and $\rho^{S}_{ABC}$ are the mixed state in polarization and spatial modes. They can be written as $\displaystyle\rho^{P}_{ABC}=F_{1}\|\Phi_{0}^{+}\rangle_{ABC}\langle\Phi_{0}^{+}\|+(1-F_{{}_{1}})\|\Phi_{1}^{+}\rangle_{ABC}\langle\Phi_{1}^{+}\|,$ (5) and $\displaystyle\rho^{S}_{ABC}=F_{2}\|\phi_{0}^{+}\rangle_{ABC}\langle\phi_{0}^{+}\|+(1-F_{{}_{2}})\|\phi_{1}^{+}\rangle_{ABC}\langle\phi_{1}^{+}\|.$ (6) From Eq.(4), the mixed state $\rho_{ABC}$ can be described as follows. With the probability of $F_{1}\otimes F_{2}$, it is in the state $\|\Phi_{0}^{+}\rangle_{ABC}\otimes\|\phi_{0}^{+}\rangle_{ABC}$. With the probability of $(1-F_{1})(1-F_{2})$, it is in the state $\|\Phi_{1}^{+}\rangle_{ABC}\otimes\|\phi_{1}^{+}\rangle_{ABC}$. With the probability of $F_{1}(1-F_{2})$ and $(1-F_{{}_{1}})F_{2}$, they are in the states $\|\Phi_{0}^{+}\rangle_{ABC}\otimes\|\phi_{1}^{+}\rangle_{ABC}$ and $\|\Phi_{1}^{+}\rangle_{ABC}\otimes\|\phi_{0}^{+}\rangle_{ABC}$, respectively. The first case $\|\Phi_{0}^{+}\rangle_{ABC}\otimes\|\phi_{0}^{+}\rangle_{ABC}$ can be described as $\displaystyle\|\Phi_{0}^{+}\rangle_{ABC}\otimes\|\phi_{0}^{+}\rangle_{ABC}$ (7) $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}(\|H\rangle_{A}\|H\rangle_{B}\|H\rangle_{C}+\|V\rangle_{A}\|V\rangle_{B}\|V\rangle_{C})$ $\displaystyle\otimes$ $\displaystyle\frac{1}{\sqrt{2}}(\|a_{1}\rangle_{A}\|b_{1}\rangle_{B}\|c_{1}\rangle_{C}+\|a_{2}\rangle_{A}\|b_{2}\rangle_{B}\|c_{2}\rangle_{C})$ $\displaystyle=$ $\displaystyle\frac{1}{2}(\|H\rangle_{a_{1}}\|H\rangle_{b_{1}}\|H\rangle_{c_{1}}+\|H\rangle_{a_{2}}\|H\rangle_{b_{2}}\|H\rangle_{c_{2}}$ $\displaystyle+$ $\displaystyle\|V\rangle_{a_{1}}\|V\rangle_{b_{1}}\|V\rangle_{c_{1}}+\|V\rangle_{a_{2}}\|V\rangle_{b_{2}}\|V\rangle_{c_{2}})$ $\displaystyle\rightarrow$ $\displaystyle\frac{1}{2}(\|V\rangle_{a_{3}}\|V\rangle_{b_{3}}\|V\rangle_{c_{3}}+\|H\rangle_{a_{6}}\|H\rangle_{b_{6}}\|H\rangle_{c_{6}}$ $\displaystyle+$ $\displaystyle\|V\rangle_{a_{5}}\|V\rangle_{b_{5}}\|V\rangle_{c_{5}}+\|H\rangle_{a_{4}}\|H\rangle_{b_{4}}\|H\rangle_{c_{4}}).$ From Fig. 1, items $\|V\rangle_{a_{3}}\|V\rangle_{b_{3}}\|V\rangle_{c_{3}}$ and $\|H\rangle_{a_{4}}\|H\rangle_{b_{4}}\|H\rangle_{c_{4}}$ will couple in the BD1, BD2, and BD3, respectively, and finally become the polarization entangled state $\|\Phi_{0}^{+}\rangle$ in the output modes D1D2D3. On the other hand, items $\|H\rangle_{a_{6}}\|H\rangle_{b_{6}}\|H\rangle_{c_{6}}$ and $\|V\rangle_{a_{5}}\|V\rangle_{b_{5}}\|V\rangle_{c_{5}}$ will also couple in the BD4, BD5 and BD6 and become the polarization entangled state $\|\Phi_{0}^{+}\rangle$ in the output modes D4D5D6. The second case $\|\Phi_{1}^{+}\rangle_{ABC}\otimes\|\phi_{1}^{+}\rangle_{ABC}$ can evolve as $\displaystyle\|\Phi_{1}^{+}\rangle_{ABC}\otimes\|\phi_{1}^{+}\rangle_{ABC}$ (8) $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}(\|H\rangle_{A}\|H\rangle_{B}\|V\rangle_{C}+\|V\rangle_{A}\|V\rangle_{B}\|H\rangle_{C})$ $\displaystyle\otimes$ $\displaystyle\frac{1}{\sqrt{2}}(\|a_{1}\rangle_{A}\|b_{1}\rangle_{B}\|c_{2}\rangle_{C}+\|a_{2}\rangle_{A}\|b_{2}\rangle_{B}\|c_{1}\rangle_{C})$ $\displaystyle=$ $\displaystyle\frac{1}{2}(\|H\rangle_{a_{1}}\|H\rangle_{b_{1}}\|V\rangle_{c_{2}}+\|H\rangle_{a_{2}}\|H\rangle_{b_{2}}\|V\rangle_{c_{1}}$ $\displaystyle+$ $\displaystyle\|V\rangle_{a_{1}}\|V\rangle_{b_{1}}\|H\rangle_{c_{2}}+\|V\rangle_{a_{2}}\|V\rangle_{b_{2}}\|H\rangle_{c_{1}})$ $\displaystyle\rightarrow$ $\displaystyle\frac{1}{2}(\|V\rangle_{a_{3}}\|V\rangle_{b_{3}}\|H\rangle_{c_{4}}+\|H\rangle_{a_{6}}\|H\rangle_{b_{6}}\|V\rangle_{c_{5}}$ $\displaystyle+$ $\displaystyle\|V\rangle_{a_{5}}\|V\rangle_{b_{5}}\|H\rangle_{c_{6}}+\|H\rangle_{a_{4}}\|H\rangle_{b_{4}}\|V\rangle_{c_{3}}).$ Items $\|V\rangle_{a_{3}}\|V\rangle_{b_{3}}\|H\rangle_{c_{4}}$ and $\|H\rangle_{a_{4}}\|H\rangle_{b_{4}}\|V\rangle_{c_{3}}$ will become $\|\Phi_{1}^{+}\rangle$ in the output modes D1D2D3. Items $\|H\rangle_{a_{6}}\|H\rangle_{b_{6}}\|V\rangle_{c_{5}}$ and $\|V\rangle_{a_{5}}\|V\rangle_{b_{5}}\|H\rangle_{c_{6}}$ will also become $\|\Phi_{1}^{+}\rangle$ in the output modes D4D5D6. On the other hand, the cases $\|\Phi_{0}^{+}\rangle_{ABC}\otimes\|\phi_{1}^{+}\rangle_{ABC}$ and $\|\Phi_{1}^{+}\rangle_{ABC}\otimes\|\phi_{0}^{+}\rangle_{ABC}$ cannot make the three photons in the output modes D1D2D3 or D4D5D6. For example, $\|\Phi_{0}^{+}\rangle_{ABC}\otimes\|\phi_{1}^{+}\rangle_{ABC}$ will lead the three photons become $\|\Phi_{1}^{+}\rangle$ in output modes D1D2D6 or D4D5D3. Case $\|\Phi_{1}^{+}\rangle_{ABC}\otimes\|\phi_{0}^{+}\rangle_{ABC}$ will become $\|\Phi_{0}^{+}\rangle$ in the output modes D1D2D6 or D4D5D3. In this way, by selecting the output modes D1D2D3 or D4D5D6 each having a photon, they can ultimately obtain a new mixed state $\displaystyle\rho^{\prime}_{ABC}=F^{\prime}\|\Phi_{0}^{+}\rangle_{ABC}\langle\Phi_{0}^{+}\|+(1-F^{\prime})\|\Phi_{1}^{+}\rangle_{ABC}\langle\Phi_{1}^{+}\|.$ (9) Here $F^{\prime}$ is $\displaystyle F^{\prime}=\frac{F_{1}F_{2}}{F_{1}F_{2}+(1-F_{1})(1-F_{2})}.$ (10) Obviously, if $F_{1}>\frac{1}{2}$ and $F_{2}>\frac{1}{2}$, we can obtain $F^{\prime}>F_{1}$ and $F^{\prime}>F_{2}$. In this way, we complete the purification of bit-flip error and the success probability is $F_{1}F_{2}+(1-F_{1})(1-F_{2})$. ## III Multipartite entanglement purification for phase-flip error In this section, we will describe the purification of phase-flip error. If a phase-flip error occurs in polarization part and spatial mode part, the mixed state can be written as $\displaystyle\varrho_{ABC}=\varrho^{P}_{ABC}\otimes\varrho^{S}_{ABC}.$ (11) Here $\displaystyle\varrho^{P}_{ABC}=F_{3}\|\Phi_{0}^{+}\rangle_{ABC}\langle\Phi_{0}^{+}\|+(1-F_{3})\|\Phi_{0}^{-}\rangle_{ABC}\langle\Phi_{0}^{-}\|,$ (12) and $\displaystyle\varrho^{S}_{ABC}=F_{4}\|\phi_{0}^{+}\rangle_{ABC}\langle\phi_{0}^{+}\|+(1-F_{4})\|\phi_{0}^{-}\rangle_{ABC}\langle\phi_{0}^{-}\|.$ (13) Figure 2: Schematic drawing showing the principle of entanglement purification for phase-flip error. The HWP22.5 can transform the $\|H\rangle$ polarization to $\frac{1}{\sqrt{2}}(\|H\rangle+\|V\rangle)$ and $\|V\rangle$ polarization to $\frac{1}{\sqrt{2}}(\|H\rangle-\|V\rangle)$. The BS is the 50:50 beam splitter. $\|a_{1}\rangle\rightarrow\frac{1}{\sqrt{2}}(\|a_{1}\rangle+\|a_{2}\rangle)$ and $\|a_{2}\rangle\rightarrow\frac{1}{\sqrt{2}}(\|a_{1}\rangle-\|a_{2}\rangle)$. Therefore, the HWP22.5 and BS both act as the role of Hadamard operation for polarization and spatial mode qubits, respectively. The principle of phase-flip error is shown in Fig.2. Before purification, they should transform the phase-flip error to bit-flip error using the setup $T_{i},i=1,2,3$. Here HWP22.5 can perform the Hadamard operation and make $\|H\rangle\rightarrow\frac{1}{\sqrt{2}}(\|H\rangle+\|V\rangle)$ and $\|V\rangle\rightarrow\frac{1}{\sqrt{2}}(\|H\rangle-\|V\rangle)$. The beam splitter (BS) can also act as the role of Hadamard operation for spatial mode qubit. It can make $\|a_{1}\rangle\rightarrow\frac{1}{\sqrt{2}}(\|a_{1}\rangle+\|a_{2}\rangle)$ and $\|a_{2}\rangle\rightarrow\frac{1}{\sqrt{2}}(\|a_{1}\rangle-\|a_{2}\rangle)$. After performing the Hadamard operation, the GHZ states in polarization and spatial mode as shown in Eq. (1) and (2) can be rewritten as $\displaystyle\|\Psi_{0}^{+}\rangle_{ABC}$ $\displaystyle=$ $\displaystyle\frac{1}{2}(\|H\rangle_{A}\|H\rangle_{B}\|H\rangle_{C}+\|H\rangle_{A}\|V\rangle_{B}\|V\rangle_{C}$ $\displaystyle+$ $\displaystyle\|V\rangle_{A}\|H\rangle_{B}\|V\rangle_{C}+\|V\rangle_{A}\|V\rangle_{B}\|H\rangle_{C}),$ $\displaystyle\|\Psi_{0}^{-}\rangle_{ABC}$ $\displaystyle=$ $\displaystyle\frac{1}{2}(\|H\rangle_{A}\|H\rangle_{B}\|V\rangle_{C}+\|H\rangle_{A}\|V\rangle_{B}\|H\rangle_{C}$ $\displaystyle+$ $\displaystyle\|V\rangle_{A}\|H\rangle_{B}\|H\rangle_{C}+\|V\rangle_{A}\|V\rangle_{B}\|V\rangle_{C}),$ $\displaystyle\|\Psi_{1}^{+}\rangle_{ABC}$ $\displaystyle=$ $\displaystyle\frac{1}{2}(\|H\rangle_{A}\|H\rangle_{B}\|H\rangle_{C}+\|H\rangle_{A}\|V\rangle_{B}\|V\rangle_{C}$ $\displaystyle-$ $\displaystyle\|V\rangle_{A}\|H\rangle_{B}\|V\rangle_{C}-\|V\rangle_{A}\|V\rangle_{B}\|H\rangle_{C}),$ $\displaystyle\|\Psi_{1}^{-}\rangle_{ABC}$ $\displaystyle=$ $\displaystyle\frac{1}{2}(\|H\rangle_{A}\|H\rangle_{B}\|V\rangle_{C}+\|H\rangle_{A}\|V\rangle_{B}\|H\rangle_{C}$ $\displaystyle-$ $\displaystyle\|V\rangle_{A}\|H\rangle_{B}\|H\rangle_{C}-\|V\rangle_{A}\|V\rangle_{B}\|V\rangle_{C}),$ $\displaystyle\|\Psi_{2}^{+}\rangle_{ABC}$ $\displaystyle=$ $\displaystyle\frac{1}{2}(\|H\rangle_{A}\|H\rangle_{B}\|H\rangle_{C}-\|H\rangle_{A}\|V\rangle_{B}\|V\rangle_{C}$ $\displaystyle+$ $\displaystyle\|V\rangle_{A}\|H\rangle_{B}\|V\rangle_{C}-\|V\rangle_{A}\|V\rangle_{B}\|H\rangle_{C}),$ $\displaystyle\|\Psi_{2}^{-}\rangle_{ABC}$ $\displaystyle=$ $\displaystyle\frac{1}{2}(\|H\rangle_{A}\|H\rangle_{B}\|V\rangle_{C}-\|H\rangle_{A}\|V\rangle_{B}\|H\rangle_{C}$ $\displaystyle+$ $\displaystyle\|V\rangle_{A}\|H\rangle_{B}\|H\rangle_{C}-\|V\rangle_{A}\|V\rangle_{B}\|V\rangle_{C}),$ $\displaystyle\|\Psi_{3}^{+}\rangle_{ABC}$ $\displaystyle=$ $\displaystyle\frac{1}{2}(\|H\rangle_{A}\|H\rangle_{B}\|H\rangle_{C}-\|H\rangle_{A}\|V\rangle_{B}\|V\rangle_{C}$ $\displaystyle-$ $\displaystyle\|V\rangle_{A}\|H\rangle_{B}\|V\rangle_{C}+\|V\rangle_{A}\|V\rangle_{B}\|H\rangle_{C}),$ $\displaystyle\|\Psi_{3}^{-}\rangle_{ABC}$ $\displaystyle=$ $\displaystyle\frac{1}{2}(\|H\rangle_{A}\|H\rangle_{B}\|V\rangle_{C}-\|H\rangle_{A}\|V\rangle_{B}\|H\rangle_{C}$ $\displaystyle-$ $\displaystyle\|V\rangle_{A}\|H\rangle_{B}\|H\rangle_{C}+\|V\rangle_{A}\|V\rangle_{B}\|V\rangle_{C}),$ $\displaystyle\|\psi_{0}^{+}\rangle_{ABC}$ $\displaystyle=$ $\displaystyle\frac{1}{2}(\|a_{1}\rangle_{A}\|b_{1}\rangle_{B}\|c_{1}\rangle_{C}+\|a_{1}\rangle_{A}\|b_{2}\rangle_{B}\|c_{2}\rangle_{C}$ $\displaystyle+$ $\displaystyle\|a_{2}\rangle_{A}\|b_{1}\rangle_{B}\|c_{2}\rangle_{C}+\|a_{2}\rangle_{A}\|b_{2}\rangle_{B}\|c_{1}\rangle_{C}),$ $\displaystyle\|\psi_{0}^{-}\rangle_{ABC}$ $\displaystyle=$ $\displaystyle\frac{1}{2}(\|a_{1}\rangle_{A}\|b_{1}\rangle_{B}\|c_{2}\rangle_{C}+\|a_{1}\rangle_{A}\|b_{2}\rangle_{B}\|c_{1}\rangle_{C}$ $\displaystyle+$ $\displaystyle\|a_{2}\rangle_{A}\|b_{1}\rangle_{B}\|c_{1}\rangle_{C}+\|a_{2}\rangle_{A}\|b_{2}\rangle_{B}\|c_{2}\rangle_{C}),$ $\displaystyle\|\psi_{1}^{+}\rangle_{ABC}$ $\displaystyle=$ $\displaystyle\frac{1}{2}(\|a_{1}\rangle_{A}\|b_{1}\rangle_{B}\|c_{1}\rangle_{C}+\|a_{1}\rangle_{A}\|b_{2}\rangle_{B}\|c_{2}\rangle_{C}$ $\displaystyle-$ $\displaystyle\|a_{2}\rangle_{A}\|b_{1}\rangle_{B}\|c_{2}\rangle_{C}-\|a_{2}\rangle_{A}\|b_{2}\rangle_{B}\|c_{1}\rangle_{C}),$ $\displaystyle\|\psi_{1}^{-}\rangle_{ABC}$ $\displaystyle=$ $\displaystyle\frac{1}{2}(\|a_{1}\rangle_{A}\|b_{1}\rangle_{B}\|c_{2}\rangle_{C}+\|a_{1}\rangle_{A}\|b_{2}\rangle_{B}\|c_{1}\rangle_{C}$ $\displaystyle-$ $\displaystyle\|a_{2}\rangle_{A}\|b_{1}\rangle_{B}\|c_{1}\rangle_{C}-\|a_{2}\rangle_{A}\|b_{2}\rangle_{B}\|c_{2}\rangle_{C}),$ $\displaystyle\|\psi_{2}^{+}\rangle_{ABC}$ $\displaystyle=$ $\displaystyle\frac{1}{2}(\|a_{1}\rangle_{A}\|b_{1}\rangle_{B}\|c_{1}\rangle_{C}-\|a_{1}\rangle_{A}\|b_{2}\rangle_{B}\|c_{2}\rangle_{C}$ $\displaystyle+$ $\displaystyle\|a_{2}\rangle_{A}\|b_{1}\rangle_{B}\|c_{2}\rangle_{C}-\|a_{2}\rangle_{A}\|b_{2}\rangle_{B}\|c_{1}\rangle_{C}),$ $\displaystyle\|\psi_{2}^{-}\rangle_{ABC}$ $\displaystyle=$ $\displaystyle\frac{1}{2}(\|a_{1}\rangle_{A}\|b_{1}\rangle_{B}\|c_{2}\rangle_{C}-\|a_{1}\rangle_{A}\|b_{2}\rangle_{B}\|c_{1}\rangle_{C}$ $\displaystyle+$ $\displaystyle\|a_{2}\rangle_{A}\|b_{1}\rangle_{B}\|c_{1}\rangle_{C}-\|a_{2}\rangle_{A}\|b_{2}\rangle_{B}\|c_{2}\rangle_{C}),$ $\displaystyle\|\psi_{3}^{+}\rangle_{ABC}$ $\displaystyle=$ $\displaystyle\frac{1}{2}(\|a_{1}\rangle_{A}\|b_{1}\rangle_{B}\|c_{1}\rangle_{C}-\|a_{1}\rangle_{A}\|b_{2}\rangle_{B}\|c_{2}\rangle_{C}$ $\displaystyle-$ $\displaystyle\|a_{2}\rangle_{A}\|b_{1}\rangle_{B}\|c_{2}\rangle_{C}+\|a_{2}\rangle_{A}\|b_{2}\rangle_{B}\|c_{1}\rangle_{C}),$ $\displaystyle\|\psi_{3}^{-}\rangle_{ABC}$ $\displaystyle=$ $\displaystyle\frac{1}{2}(\|a_{1}\rangle_{A}\|b_{1}\rangle_{B}\|c_{2}\rangle_{C}-\|a_{1}\rangle_{A}\|b_{2}\rangle_{B}\|c_{1}\rangle_{C}$ $\displaystyle-$ $\displaystyle\|a_{2}\rangle_{A}\|b_{1}\rangle_{B}\|c_{1}\rangle_{C}+\|a_{2}\rangle_{A}\|b_{2}\rangle_{B}\|c_{2}\rangle_{C}).$ After performing the Hadamard operations, the mixed state in Eq.(11) can be rewritten as $\displaystyle\sigma_{ABC}=\sigma^{P}_{ABC}\otimes\sigma^{S}_{ABC}.$ (16) Here $\displaystyle\sigma^{P}_{ABC}=F_{3}\|\Psi_{0}^{+}\rangle_{ABC}\langle\Psi_{0}^{+}\|+(1-F_{3})\|\Psi_{0}^{-}\rangle_{ABC}\langle\Psi_{0}^{-}\|,$ (17) and $\displaystyle\sigma^{S}_{ABC}=F_{4}\|\psi_{0}^{+}\rangle_{ABC}\langle\psi_{0}^{+}\|+(1-F_{4})\|\psi_{0}^{-}\rangle_{ABC}\langle\psi_{0}^{-}\|.$ (18) Therefore, $\sigma_{ABC}$ can be described as follows. With the probability of $F_{3}F_{4}$, it is in the state $\|\Psi_{0}^{+}\rangle_{ABC}\otimes\|\psi_{0}^{+}\rangle_{ABC}$. With the probability of $F_{3}(1-F_{4})$, it is in the state $\|\Psi_{0}^{+}\rangle_{ABC}\otimes\|\psi_{0}^{-}\rangle_{ABC}$. With the probability of $(1-F_{3})F_{4}$, it is in the state $\|\Psi_{0}^{-}\rangle_{ABC}\otimes\|\psi_{0}^{+}\rangle_{ABC}$. With the probability of $(1-F_{3})(1-F_{4})$, it is in the state $\|\Psi_{0}^{-}\rangle_{ABC}\otimes\|\psi_{0}^{-}\rangle_{ABC}$. We first discuss the case $\|\Psi_{0}^{+}\rangle_{ABC}\otimes\|\psi_{0}^{+}\rangle_{ABC}$. It will evolve as $\displaystyle\|\Psi_{0}^{+}\rangle_{ABC}\otimes\|\psi_{0}^{+}\rangle_{ABC}$ (19) $\displaystyle=$ $\displaystyle\frac{1}{2}(\|H\rangle_{A}\|H\rangle_{B}\|H\rangle_{C}+\|H\rangle_{A}\|V\rangle_{B}\|V\rangle_{C}$ $\displaystyle+$ $\displaystyle\|V\rangle_{A}\|H\rangle_{B}\|V\rangle_{C}+\|V\rangle_{A}\|V\rangle_{B}\|H\rangle_{C})$ $\displaystyle\otimes$ $\displaystyle\frac{1}{2}(\|a_{1}\rangle_{A}\|b_{1}\rangle_{B}\|c_{1}\rangle_{C}+\|a_{1}\rangle_{A}\|b_{2}\rangle_{B}\|c_{2}\rangle_{C}$ $\displaystyle+$ $\displaystyle\|a_{2}\rangle_{A}\|b_{1}\rangle_{B}\|c_{2}\rangle_{C}+\|a_{2}\rangle_{A}\|b_{2}\rangle_{B}\|c_{1}\rangle_{C})$ $\displaystyle=$ $\displaystyle\frac{1}{4}(\|H\rangle_{a_{1}}\|H\rangle_{b_{1}}\|H\rangle_{c_{1}}+\|H\rangle_{a_{1}}\|H\rangle_{b_{2}}\|H\rangle_{c_{2}}$ $\displaystyle+$ $\displaystyle\|H\rangle_{a_{2}}\|H\rangle_{b_{1}}\|H\rangle_{c_{2}}+\|H\rangle_{a_{2}}\|H\rangle_{b_{2}}\|H\rangle_{c_{1}}$ $\displaystyle+$ $\displaystyle\|H\rangle_{a_{1}}\|V\rangle_{b_{1}}\|V\rangle_{c_{1}}+\|H\rangle_{a_{1}}\|V\rangle_{b_{2}}\|V\rangle_{c_{2}}$ $\displaystyle+$ $\displaystyle\|H\rangle_{a_{2}}\|V\rangle_{b_{1}}\|V\rangle_{c_{2}}+\|H\rangle_{a_{2}}\|V\rangle_{b_{2}}\|V\rangle_{c_{1}}$ $\displaystyle+$ $\displaystyle\|V\rangle_{a_{1}}\|H\rangle_{b_{1}}\|V\rangle_{c_{1}}+\|V\rangle_{a_{1}}\|H\rangle_{b_{2}}\|V\rangle_{c_{2}}$ $\displaystyle+$ $\displaystyle\|V\rangle_{a_{2}}\|H\rangle_{b_{1}}\|V\rangle_{c_{2}}+\|V\rangle_{a_{2}}\|H\rangle_{b_{2}}\|V\rangle_{c_{1}}$ $\displaystyle+$ $\displaystyle\|V\rangle_{a_{1}}\|V\rangle_{b_{1}}\|H\rangle_{c_{1}}+\|V\rangle_{a_{1}}\|V\rangle_{b_{2}}\|H\rangle_{c_{2}}$ $\displaystyle+$ $\displaystyle\|V\rangle_{a_{2}}\|V\rangle_{b_{1}}\|H\rangle_{c_{2}}+\|V\rangle_{a_{2}}\|V\rangle_{b_{2}}\|H\rangle_{c_{1}})$ $\displaystyle\rightarrow$ $\displaystyle\frac{1}{4}(\|V\rangle_{a_{3}}\|V\rangle_{b_{3}}\|V\rangle_{c_{3}}+\|V\rangle_{a_{3}}\|H\rangle_{b_{6}}\|H\rangle_{c_{6}}$ $\displaystyle+$ $\displaystyle\|H\rangle_{a_{6}}\|V\rangle_{b_{3}}\|H\rangle_{c_{6}}+\|H\rangle_{a_{6}}\|H\rangle_{b_{6}}\|V\rangle_{c_{3}}$ $\displaystyle+$ $\displaystyle\|V\rangle_{a_{3}}\|V\rangle_{b_{5}}\|V\rangle_{c_{5}}+\|V\rangle_{a_{3}}\|H\rangle_{b_{4}}\|H\rangle_{c_{4}}$ $\displaystyle+$ $\displaystyle\|H\rangle_{a_{6}}\|V\rangle_{b_{5}}\|H\rangle_{c_{4}}+\|H\rangle_{a_{6}}\|H\rangle_{b_{4}}\|V\rangle_{c_{5}}$ $\displaystyle+$ $\displaystyle\|V\rangle_{a_{5}}\|V\rangle_{b_{3}}\|V\rangle_{c_{5}}+\|V\rangle_{a_{5}}\|H\rangle_{b_{6}}\|H\rangle_{c_{4}}$ $\displaystyle+$ $\displaystyle\|H\rangle_{a_{4}}\|V\rangle_{b_{3}}\|H\rangle_{c_{4}}+\|H\rangle_{a_{4}}\|H\rangle_{b_{6}}\|V\rangle_{c_{5}}$ $\displaystyle+$ $\displaystyle\|V\rangle_{a_{5}}\|V\rangle_{b_{5}}\|V\rangle_{c_{3}}+\|V\rangle_{a_{5}}\|H\rangle_{b_{4}}\|H\rangle_{c_{6}}$ $\displaystyle+$ $\displaystyle\|H\rangle_{a_{4}}\|V\rangle_{b_{5}}\|H\rangle_{c_{6}}+\|H\rangle_{a_{4}}\|H\rangle_{b_{4}}\|V\rangle_{c_{3}}).$ From Eq. (19), item $\|V\rangle_{a_{3}}\|V\rangle_{b_{3}}\|V\rangle_{c_{3}}$, $\|V\rangle_{a_{3}}\|H\rangle_{b_{4}}\|H\rangle_{c_{4}}$, $\|H\rangle_{a_{4}}\|V\rangle_{b_{3}}\|H\rangle_{c_{4}}$ and $\|H\rangle_{a_{4}}\|H\rangle_{b_{4}}\|V\rangle_{c_{3}}$ will be in the output modes D1D2D3 and become $\displaystyle\frac{1}{2}(\|V\rangle_{D_{1}}\|V\rangle_{D_{2}}\|V\rangle_{D_{3}}+\|V\rangle_{D_{1}}\|H\rangle_{D_{2}}\|H\rangle_{D_{3}}$ (20) $\displaystyle+$ $\displaystyle\|H\rangle_{D_{1}}\|V\rangle_{D_{2}}\|H\rangle_{D_{3}}+\|H\rangle_{D_{1}}\|H\rangle_{D_{2}}\|V\rangle_{D_{3}}).$ By performing bit-flip operation on each photon, state in Eq.(20) can be changed to $\|\Psi_{0}^{+}\rangle_{ABC}$. Finally, they can change $\|\Psi_{0}^{+}\rangle_{ABC}$ to $\|\Phi_{0}^{+}\rangle_{ABC}$ by adding another Hadamard operation on each photon. On the other hand, from Eq.(19), by selecting the output modes D1D5D6, D4D5D3, or D4D2D6, they can obtain the same state in Eq.(20). In this way, they can also obtain $\|\Phi_{0}^{+}\rangle_{ABC}$. The case $\|\Psi_{0}^{-}\rangle_{ABC}\otimes\|\psi_{0}^{-}\rangle_{ABC}$ can be evolve as $\displaystyle\|\Psi_{0}^{-}\rangle_{ABC}\otimes\|\psi_{0}^{-}\rangle_{ABC}$ (21) $\displaystyle=$ $\displaystyle\frac{1}{2}(\|H\rangle_{A}\|H\rangle_{B}\|V\rangle_{C}+\|H\rangle_{A}\|V\rangle_{B}\|H\rangle_{C}$ $\displaystyle+$ $\displaystyle\|V\rangle_{A}\|H\rangle_{B}\|H\rangle_{C}+\|V\rangle_{A}\|V\rangle_{B}\|V\rangle_{C})$ $\displaystyle\otimes\frac{1}{2}(\|a_{1}\rangle_{A}\|b_{1}\rangle_{B}\|c_{2}\rangle_{C}+\|a_{1}\rangle_{A}\|b_{2}\rangle_{B}\|c_{1}\rangle_{C}$ $\displaystyle+$ $\displaystyle\|a_{2}\rangle_{A}\|b_{1}\rangle_{B}\|c_{1}\rangle_{C}+\|a_{2}\rangle_{A}\|b_{2}\rangle_{B}\|c_{2}\rangle_{C})$ $\displaystyle\rightarrow\frac{1}{4}(\|V\rangle_{a_{3}}\|V\rangle_{b_{3}}\|H\rangle_{c_{4}}+\|V\rangle_{a_{3}}\|H\rangle_{b_{6}}\|V\rangle_{c_{5}}$ $\displaystyle+$ $\displaystyle\|H\rangle_{a_{6}}\|V\rangle_{b_{3}}\|V\rangle_{c_{5}}+\|H\rangle_{a_{6}}\|H\rangle_{b_{6}}\|H\rangle_{c_{4}}$ $\displaystyle+$ $\displaystyle\|V\rangle_{a_{3}}\|V\rangle_{b_{5}}\|H\rangle_{c_{6}}+\|V\rangle_{a_{3}}\|H\rangle_{b_{4}}\|V\rangle_{c_{3}}$ $\displaystyle+$ $\displaystyle\|H\rangle_{a_{6}}\|V\rangle_{b_{5}}\|V\rangle_{c_{3}}+\|H\rangle_{a_{6}}\|H\rangle_{b_{6}}\|H\rangle_{c_{6}}$ $\displaystyle+$ $\displaystyle\|V\rangle_{a_{5}}\|V\rangle_{b_{3}}\|H\rangle_{c_{6}}+\|V\rangle_{a_{5}}\|H\rangle_{b_{6}}\|V\rangle_{c_{3}}$ $\displaystyle+$ $\displaystyle\|H\rangle_{a_{4}}\|V\rangle_{b_{3}}\|V\rangle_{c_{3}}+\|H\rangle_{a_{4}}\|H\rangle_{b_{6}}\|H\rangle_{c_{6}}$ $\displaystyle+$ $\displaystyle\|V\rangle_{a_{5}}\|V\rangle_{b_{5}}\|H\rangle_{c_{4}}+\|V\rangle_{a_{5}}\|H\rangle_{b_{4}}\|H\rangle_{c_{5}}$ $\displaystyle+$ $\displaystyle\|H\rangle_{a_{4}}\|V\rangle_{b_{5}}\|V\rangle_{c_{5}}+\|H\rangle_{a_{4}}\|H\rangle_{b_{4}}\|H\rangle_{c_{4}}).$ From Eq.(21), items $\|V\rangle_{a_{3}}\|V\rangle_{b_{3}}\|H\rangle_{c_{4}}$, $\|V\rangle_{a_{3}}\|H\rangle_{b_{4}}\|V\rangle_{c_{3}}$, $\|H\rangle_{a_{4}}\|V\rangle_{b_{3}}\|V\rangle_{c_{3}}$ and $\|H\rangle_{a_{4}}\|H\rangle_{b_{4}}\|H\rangle_{c_{4}}$ will be in the output modes D1D2D3 and become $\displaystyle\frac{1}{2}(\|V\rangle_{D_{1}}\|V\rangle_{D_{2}}\|H\rangle_{D_{3}}+\|V\rangle_{D_{1}}\|H\rangle_{D_{2}}\|V\rangle_{D_{3}}$ (22) $\displaystyle+$ $\displaystyle\|H\rangle_{D_{1}}\|V\rangle_{D_{2}}\|V\rangle_{D_{3}}+\|H\rangle_{D_{1}}\|H\rangle_{D_{2}}\|H\rangle_{D_{3}}).$ State in Eq.(22) can be changed to $\|\Psi_{0}^{-}\rangle_{ABC}$ by adding bit- flip operation on each photon. By adding another Hadamard operation on each photon, $\|\Psi_{0}^{-}\rangle_{ABC}$ can be converted to $\|\Phi_{0}^{-}\rangle_{ABC}$. On the other hand, from Eq.(21), by selecting the output modes D1D5D6, D4D5D3, or D4D2D6, they can obtain the same state in Eq.(22). In this way, they can also obtain $\|\Phi_{0}^{-}\rangle_{ABC}$. The other cases $\|\Psi_{0}^{+}\rangle_{ABC}\otimes\|\psi_{0}^{-}\rangle_{ABC}$ and $\|\Psi_{0}^{-}\rangle_{ABC}\otimes\|\psi_{0}^{-}\rangle_{ABC}$ will lead the photons in the output modes D1D2D6, D1D5D3, D4D2D3, or D4D5D6. Therefore, by selecting the cases that the output modes D1D2D3, D1D5D6, D4D5D3, or D4D2D6 contain one photon, cases $\|\Psi_{0}^{+}\rangle_{ABC}\otimes\|\psi_{0}^{-}\rangle_{ABC}$ and $\|\Psi_{0}^{-}\rangle_{ABC}\otimes\|\psi_{0}^{-}\rangle_{ABC}$ can be eliminated automatically. Finally, with the probability of $F_{3}F_{4}$, they will obtain $\|\Phi_{0}^{+}\rangle_{ABC}$. With the probability of $(1-F_{3})(1-F_{4})$, they will obtain $\|\Phi_{0}^{-}\rangle_{ABC}$. The new mixed state can be rewritten as $\displaystyle\sigma^{\prime}_{ABC}=F^{\prime\prime}\|\Psi_{0}^{+}\rangle_{ABC}\langle\Psi_{0}^{+}\|+(1-F^{\prime\prime})\|\Psi_{0}^{-}\rangle_{ABC}\langle\Psi_{0}^{-}\|.$ Here $F^{\prime\prime}$ is $\displaystyle F^{\prime\prime}=\frac{F_{3}F_{4}}{F_{3}F_{4}+(1-F_{3})(1-F_{4})}.$ (24) Similar to Eq.(10), $F^{\prime\prime}>F_{3}$ and $F^{\prime\prime}>F_{4}$ if $F_{3}>\frac{1}{2}$ and $F_{4}>\frac{1}{2}$. ## IV Arbitrary multipartite entanglement purification It is easy to extend the EPP to the arbitrary GHZ state. The $m$-photon GHZ state in polarization can be described as $\displaystyle\|\Phi_{0}^{\pm}\rangle_{m}$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}(\|H\rangle_{1}\|H\rangle_{2}\cdots\|H\rangle_{m}\pm\|V\rangle_{1}\|V\rangle_{2}\cdots\|V\rangle_{m}),$ $\displaystyle\|\Phi_{1}^{\pm}\rangle_{m}$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}(\|H\rangle_{1}\|H\rangle_{2}\cdots\|V\rangle_{m}\pm\|V\rangle_{1}\|V\rangle_{2}\cdots\|H\rangle_{m}),$ $\displaystyle\cdots$ , $\displaystyle\|\Phi_{2^{m-1}}^{\pm}\rangle_{m}$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}(\|V\rangle_{1}\|H\rangle_{2}\cdots\|H\rangle_{m}\pm\|H\rangle_{1}\|V\rangle_{2}\cdots\|V\rangle_{m}),$ Figure 3: Schematic drawing showing the principle of entanglement purification for arbitrary GHZ state. On the other hand, the $m$-photon GHZ state in spatial mode can be described as $\displaystyle\|\phi_{0}^{\pm}\rangle_{m}$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}(\|a_{1}\rangle_{1}\|b_{1}\rangle_{2}$ $\displaystyle\cdots$ $\displaystyle\|m_{1}\rangle_{m}\pm\|a_{2}\rangle_{1}\|b_{2}\rangle_{2}\cdots\|m_{2}\rangle_{m}),$ $\displaystyle\|\phi_{1}^{\pm}\rangle_{m}$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}(\|a_{1}\rangle_{1}\|b_{1}\rangle_{2}$ $\displaystyle\cdots$ $\displaystyle\|m_{2}\rangle_{m}\pm\|a_{2}\rangle_{1}\|b_{2}\rangle_{2}\cdots\|m_{1}\rangle_{m}),$ $\displaystyle\cdots$ , $\displaystyle\|\phi_{2^{m-1}}^{\pm}\rangle_{m}$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}(\|a_{2}\rangle_{1}\|b_{1}\rangle_{2}$ (26) $\displaystyle\cdots$ $\displaystyle\|m_{1}\rangle_{m}\pm\|a_{1}\rangle_{1}\|b_{1}\rangle_{2}\cdots\|m_{2}\rangle_{m}).$ As shown in Fig. 3, the entanglement source prepares the $m$-photon hyperentangled state $\|\Psi\rangle_{m}$ of the form $\displaystyle\|\Psi\rangle_{m}=\|\Phi_{0}^{+}\rangle_{m}\otimes\|\phi_{0}^{+}\rangle_{m}.$ (27) Such hyperentangled state is distributed to $m$ parties, named Bob1, Bob2, $\cdots$, and Bobm. After distribution, the initial state becomes a mixed state as $\displaystyle\rho_{m}=\rho^{P}_{m}\otimes\rho^{P}_{m}.$ (28) Here $\rho^{P}_{m}$ can be written as $\displaystyle\rho^{P}_{m}=F_{1}\|\Phi_{0}^{+}\rangle_{m}\langle\Phi_{0}^{+}\|+(1-F_{{}_{1}})\|\Phi_{1}^{+}\rangle_{m}\langle\Phi_{1}^{+}\|.$ (29) $\rho^{P}_{m}$ can be written as $\displaystyle\rho^{S}_{m}=F_{2}\|\phi_{0}^{+}\rangle_{m}\langle\phi_{0}^{+}\|+(1-F_{{}_{2}})\|\phi_{1}^{+}\rangle_{m}\langle\phi_{1}^{+}\|.$ (30) The purification is similar as described in above section. By selecting the output modes D1,D2, $\cdots$, Dm exactly contain one photon, they can ultimately obtain a high fidelity mixed state in polarization. The fidelity $F^{\prime}$ is same as it is shown in Eq. (10). On the other hand, if the phase-flip error occurs, one can also convert it to the bit-flip error, and perform the purification in a next step. In this way, one can purify the arbitrary $m$-photon GHZ state. ## V Discussion So far, we have completely described this EPP. We first described the EPP for three-photon GHZ state with a bit-flip error. Then we explained the EPP with a phase-flip error. In this way, all the errors can be purified. Finally, we extend this EPP for arbitrary GHZ state and which can be purified in the same way. In above, we suppose that the bit-flip error occurs on the first qubit. In a practical transmission, the hyperentanglement in polarization and spatial mode will suffer from different errors. For example, the bit-flip error occurs on the first qubit in polarization and which makes the polarization part become the mixed state in Eq. (5), while the bit-flip error occurs on the second qubit in spatial mode and makes the spatial part become $\displaystyle\rho^{\prime S}_{ABC}=F_{3}\|\phi_{0}^{+}\rangle_{ABC}\langle\phi_{0}^{+}\|+(1-F_{{}_{3}})\|\phi_{2}^{+}\rangle_{ABC}\langle\phi_{2}^{+}\|.$ (31) Therefore, the hyperentangled mixed state can be written as $\displaystyle\rho^{\prime}_{ABC}=\rho^{P}_{ABC}\otimes\rho^{\prime S}_{ABC}.$ (32) With the probability of F1F3, it is in the state $\|\Phi_{0}^{+}\rangle_{ABC}\otimes\|\phi_{0}^{+}\rangle_{ABC}$. Such state will make the three photons in the output modes D1D2D3 or D4D5D6. With the probability of (1-F${}_{1})$F3, it is in the state $\|\Phi_{1}^{+}\rangle_{ABC}\otimes\|\phi_{0}^{+}\rangle_{ABC}$. Such state will make the three photons in the output modes D1D2D6 or D4D5D3. With the probability of F1(1-F${}_{3})$, it is in the state $\|\Phi_{0}^{+}\rangle_{ABC}\otimes\|\phi_{2}^{+}\rangle_{ABC}$. Such state will make the three photons in the output modes D1D5D3 or D4D2D6. Finally, with the probability of (1-F1)(1-F3), it is in the state $\|\Phi_{1}^{+}\rangle_{ABC}\otimes\|\phi_{2}^{+}\rangle_{ABC}$. Such state will make the three photons in the output modes D1D5D6 or D4D2D3. In this way, by selecting the cases D1D2D3 or D4D5D6, they can ultimately obtain the polarization state $\|\Phi_{0}^{+}\rangle_{ABC}$. Interestingly, if the mixed state is described as shown in Eq.(32), the bit-flip error can be completely purified. The second case $\|\Phi_{1}^{+}\rangle_{ABC}\otimes\|\phi_{0}^{+}\rangle_{ABC}$ will lead the photons in D1D2D6 or D4D5D3 and such state will become $\|\Phi_{1}^{+}\rangle_{ABC}$. They can add a bit-flip operation on the first photon and convert it to $\|\Phi_{0}^{+}\rangle_{ABC}$ deterministically. On the other hand, the second case $\|\Phi_{0}^{+}\rangle_{ABC}\otimes\|\phi_{2}^{+}\rangle_{ABC}$ will lead the photons in the spatial modes D1D5D3 or D4D2D6 and become $\|\Phi_{2}^{+}\rangle_{ABC}$. They can also add the bit-flip operation on the second photon and convert it to $\|\Phi_{0}^{+}\rangle_{ABC}$ deterministically. The final case $\|\Phi_{1}^{+}\rangle_{ABC}\otimes\|\phi_{2}^{+}\rangle_{ABC}$ will also become $\|\Phi_{1}^{+}\rangle_{ABC}$ and can be converted to $\|\Phi_{0}^{+}\rangle_{ABC}$ deterministically. In this way, they can obtain the maximally pure entangled state $\|\Phi_{0}^{+}\rangle_{ABC}$ with the probability of 100$\%$. For a general mixed state with bit-flip error, the polarization part and spatial-mode part can be written as $\displaystyle\rho^{\prime\prime P}_{ABC}$ $\displaystyle=$ $\displaystyle F_{1}\|\Phi_{0}^{+}\rangle_{ABC}\langle\Phi_{0}^{+}\|+F_{2}\|\Phi_{1}^{+}\rangle_{ABC}\langle\Phi_{1}^{+}\|$ (33) $\displaystyle+$ $\displaystyle F_{3}\|\Phi_{2}^{+}\rangle_{ABC}\langle\Phi_{2}^{+}\|+F_{4}\|\Phi_{3}^{+}\rangle_{ABC}\langle\Phi_{3}^{+}\|,$ and $\displaystyle\rho^{\prime\prime S}_{ABC}$ $\displaystyle=$ $\displaystyle F_{4}\|\phi_{0}^{+}\rangle_{ABC}\langle\phi_{0}^{+}\|+F_{5}\|\phi_{1}^{+}\rangle_{ABC}\langle\phi_{1}^{+}\|$ (34) $\displaystyle+$ $\displaystyle F_{6}\|\phi_{2}^{+}\rangle_{ABC}\langle\phi_{2}^{+}\|+F_{7}\|\phi_{3}^{+}\rangle_{ABC}\langle\phi_{3}^{+}\|.$ Here $F_{1}+F_{2}+F_{3}+F_{4}=1$ and $F_{5}+F_{6}+F_{7}+F_{8}=1$. Similarly, by selecting the output modes D1D2D3 or D4D5D6, they can obtain a new mixed state as $\displaystyle\rho_{ABC}^{\prime\prime}$ $\displaystyle=$ $\displaystyle F^{\prime}_{1}\|\Phi_{0}^{+}\rangle_{ABC}\langle\Phi_{0}^{+}\|+F^{\prime}_{2}\|\Phi_{1}^{+}\rangle_{ABC}\langle\Phi_{1}^{+}\|$ (35) $\displaystyle+$ $\displaystyle F^{\prime}_{3}\|\Phi_{2}^{+}\rangle_{ABC}\langle\Phi_{2}^{+}\|+F^{\prime}_{4}\|\Phi_{3}^{+}\rangle_{ABC}\langle\Phi_{3}^{+}\|.$ Here $\displaystyle F^{\prime}_{1}=\frac{F_{1}F_{5}}{F_{1}F_{5}+F_{2}F_{6}+F_{3}F_{7}+F_{4}F_{8}},$ $\displaystyle F^{\prime}_{2}=\frac{F_{2}F_{6}}{F_{1}F_{5}+F_{2}F_{6}+F_{3}F_{7}+F_{4}F_{8}},$ $\displaystyle F^{\prime}_{3}=\frac{F_{3}F_{7}}{F_{1}F_{5}+F_{2}F_{6}+F_{3}F_{7}+F_{4}F_{8}},$ $\displaystyle F^{\prime}_{4}=\frac{F_{4}F_{8}}{F_{1}F_{5}+F_{2}F_{6}+F_{3}F_{7}+F_{4}F_{8}}.$ (36) If $F_{1}>\frac{1}{2}$ and $F_{5}>\frac{1}{2}$, we can also obtain $F^{\prime}_{1}>F_{1}$ and $F^{\prime}_{1}>F_{5}$. In this way, we can realize the general purification. It is interesting to calculate the purification efficiency in a practical environment. As shown in Fig. 1, the N-photon GHZ state was distributed to $N$ parties. The transmission efficiency is $\eta_{t}=e^{-\frac{L}{L_{0}}}$. The detector efficiency is $\eta_{d}$. The $\eta_{c}$ is the probability of coupling a photon to the single-photon detector. $L_{0}$ is the attenuation length of the channel (25 km for commercial fibre) munro . $L$ is transmission distance. The success probability is $p_{1}=F_{1}F_{2}+(1-F_{1})(1-F_{2})$. For N-photon purification, the total purification efficiency can be calculated as $\displaystyle P^{N}_{one}=p_{1}\eta^{N}_{t}\eta^{N}_{d}\eta^{N}_{c}.$ (37) In existing multipartite EPPs multipurification1 ; multipurification3 ; multipurification4 ; multipurification7 , they exploit two pairs of N-photon GHZ states to perform the purification. Therefore, for linear optical system, the total purification efficiency can be calculated as $\displaystyle P^{N}_{two}=\frac{1}{4}p_{1}\eta^{2N}_{t}\eta^{2N}_{d}\eta^{2N}_{c}.$ (38) The ratio of $P^{N}_{one}$ and $P^{N}_{two}$ can be calculated as $\displaystyle R=\frac{P^{N}_{one}}{P^{N}_{two}}=\frac{4}{\eta^{N}_{t}\eta^{N}_{d}\eta^{N}_{c}}=\frac{4}{(e^{-\frac{L}{L_{0}}})^{N}\eta^{N}_{d}\eta^{N}_{c}}.$ (39) If we let $\eta_{d}=0.9$, $\eta_{c}=0.95$ munro . Figure 4: The ratio $R$ of entanglement purification efficiency plotted against length of entanglement distribution. We let the photon number of GHZ state as $N=3$ and $N=6$, respectively. Figure 5: The ratio $R$ of entanglement purification efficiency plotted against photon number of GHZ state. We let $L=L_{0}=25km$. Fig. 4 shows the relationship between the coefficient $L$ and $R$. Here we let $N=3$ and $N=6$ respectively. We change the distance $L$ from 20km to 100km. The ratio $R$ increases rapidly. The $R$ can reach more than $10^{10}$ when $N=6$ and $L=100$km. In Fig. 5, we also calculated the $R$ altered with $N$. We let $L=L_{0}=25$km. We also showed that the $R$ increases rapidly with the photon number $N$. On the other hand, in existing multipartite EPPs multipurification1 ; multipurification3 ; multipurification4 ; multipurification7 , after each party performing the CNOT or similar operation, they should measure the target particles to judge that the purification is successful or not. In this EPP, the parties are not required to measure the particles and they can judge whether the purification is successful or not according to the output modes of the photons. In this way, this EPP is more economical and practical in future application. Finally, let us briefly discuss the possible realization. This protocol mainly exploits the common linear optics, such as PBS, BS, BD, HWP. Meanwhile, this protocol require the multi-partite hyperentanglement. Such hyperentanglement was also realized in experiment source , which show that this protocol is feasible in current experiment condition. ## VI Conclusion In conclusion, we have proposed the EPP for multipartite entanglement purification using hyperentanglement. After performing the EPP, the spatial entanglement can be used to purify the polarization entanglement. Different from the previous works, this EPP has several advantages. Firstly, with the same purification success probability, this EPP only requires one pair of multipartite GHZ states, while existing EPPs usually require two pairs of multipartite GHZ state. Secondly, if consider the practical transmission and detector efficiency, this EPP may be extremely useful for the ratio of purification efficiency increases rapidly with both the number of photons and the transmission distance. Thirdly, this protocol requires linear optics and does not add additional measurement operations, so that it is feasible for experiment. 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# Snapshot Hyperspectral Imaging Based on Weighted High-order Singular Value Regularization Niankai Cheng1, Hua Huang2, Lei Zhang1, and Lizhi Wang1 Corresponding author: Hua<EMAIL_ADDRESS>1School of Computer Science and Technology, Beijing Institute of Technology, Beijing 100081, China 2School of Artifical Intelligence, Beijing Normal University, Beijing 100875, China ###### Abstract Snapshot hyperspectral imaging can capture the 3D hyperspectral image (HSI) with a single 2D measurement and has attracted increasing attention recently. Recovering the underlying HSI from the compressive measurement is an ill-posed problem and exploiting the image prior is essential for solving this ill-posed problem. However, existing reconstruction methods always start from modeling image prior with the 1D vector or 2D matrix and cannot fully exploit the structurally spectral-spatial nature in 3D HSI, thus leading to a poor fidelity. In this paper, we propose an effective high-order tensor optimization based method to boost the reconstruction fidelity for snapshot hyperspectral imaging. We first build high-order tensors by exploiting the spatial-spectral correlation in HSI. Then, we propose a weight high-order singular value regularization (WHOSVR) based low-rank tensor recovery model to characterize the structure prior of HSI. By integrating the structure prior in WHOSVR with the system imaging process, we develop an optimization framework for HSI reconstruction, which is finally solved via the alternating minimization algorithm. Extensive experiments implemented on two representative systems demonstrate that our method outperforms state-of-the- art methods. ## I Introduction Hyperspectral imaging techniques, which are able to capture the 3D hyperspetral image (HSI) with multiple discrete bands at specific frequencies, have attracted increasing interests in recent years. By providing abundant spatial and spectral information simultaneously, HSI can be used for many visual tasks that traditional gray image or color image cannot accomplish. In recent years, HSI has been applied in various vision tasks, such as classification [1], segmentation [2], recognition [3] and tracking [4]. HSI consists of 2D spatial information and 1D spectral information. To obtain the 3D HSI, conventional hyperspectral imaging techniques scan the scene along 1D or 2D coordinate [5, 6, 7, 8]. This imaging is time-consuming and can not be used in dynamic scenes. To address the limitations of conventional spectral imaging techniques, many snapshot hyperspectral imaging systems based on compressive sensing theory [9, 10] has been developed. Among those numerous imaging systems, the coded aperture snapshot spectral imager (CASSI) [11, 12], stands out in the dynamic scene due to its advantage in collecting the 3D data from different wavelengths with one shot. As an advancement system of CASSI, the latest proposed design of dual-camera compressive hyperspectral imaging (DCCHI) incorporates a common panchromatic camera to collect more information simultaneously with CASSI [13, 14]. With the complementary details from the uncoded panchromatic branch, DCCHI can improve the reconstruction fidelity significantly, which obtains the potential ability to apply snapshot hyperspectral imaging into practice [15]. Since the number of snapshot hyperspectral imaging systems is far less than what is required by the Nyquist sampling frequency, the reconstruction is severely under-determined. Such a difficult issue can be addressed by solving a convex optimization problem regularized with a HSI prior. So far considerable reconstruction methods have been proposed in this domain [16, 17, 18, 19, 20, 21]. However, most of them start from expressing HSI as 1D vector or 2D matrix and cannot fully exploit the structurally spectral-spatial prior in 3D HSI. Such a compromise treatment results in poor reconstruction quality and hinder the application of snapshot hyperspectral imaging. Figure 1: Overview of the proposed method. We reconstruct HSI from the compressive measurement. Our reconstruction method, including matching, weighted high-order singular value regularization (shortened as WHOSVR in the figure) and projection, is iteratively performed. To overcome the aforementioned drawbacks, we propose a novel tensor based reconstruction method with weight high-order singular value regularization (WHOSVR). Our key observation is that, compared with the 1D vector and 2D matrix, the high-order tensor is more faithful to delivery the structure information of the 3D HSI. Such advantage of tensor motivates us to exploit the high-order representation for snapshot hyperspectral imaging reconstruction and promote the reconstruction fidelity. Specifically, we first build a high-order tensor by exploiting the high correlation in the spatial and spectral domains for each exemplar cubic patch. Then, to characterize the high-order structure prior of HSI, we propose a WHOSVR model, where the singular values of tensor are treated adaptively, to finish an accurate low- rank tensor recovery. By integrating the structure prior in WHOSVR with the system imaging process, we develop an optimization algorithm for HSI reconstruction, which is finally solved via an alternating minimization algorithm. To the best of our knowledge, it is the first time to exploiting the prior of HSI with WHOSVR for snapshot hyperspectral imaging. Extensive experiments implemented on both CASSI and DCCHI demonstrate that our method outperforms state-of-the-art methods. ## II Related Works ### II-A Hyperspectral Imaging System Conventional scanning-based hyperspectral imaging systems directly trade the temporal resolution for the spatial/ spectral resolution and thus lose the ability to record dynamic scenes [5, 6, 7, 8]. To capture the dynamic scenes, several snapshot hyperspectral imaging systems have been developed in the last decades [22, 23, 24]. However, these systems still suffer from the trade-off between the spatial and temporal resolution. Leveraging the compressive sensing theory, CASSI stands out as a promising solution for the trade-off between the spatial and temporal resolution recently. CASSI employs one disperser or two dispersers to capture a 2D encoded image of the target [11, 12]. Then the underlying HSI can be reconstructed from the compressive measurement by solving an ill-posed problem. Several hardware advancements of CASSI have been developed to improve the performance, e.g. multiple snapshot imaging system [25, 26] and spatial- spectral compressive spectral image [18]. The latest dual-camera design, i.e., DCCHI [13, 14], incorporates a co-located panchromatic camera to collect more information simultaneously with CASSI. With the complementary details from the uncoded image, DCCHI obtains a significant improvement on reconstruction quality and the potential ability to be applied into practice. In this paper, we propose a high-order tensor optimization based method to boost the reconstruction accuracy for snapshot hyperspectral imaging. Meanwhile, our method is a general approach and can be easily extends different systems. ### II-B Hyperspectral Image Reconstruction Recovering the underlying HSI from the compressive measurement plays an essential role for snapshot hyperspectral imaging. Based on compressive sensing, HSI can be reconstructed by solving optimization problems with prior knowledge based regularization. With the hypothesis that nature images owns piecewise smooth property, TV prior based methods have been widely used in snapshot hyperspectral imaging [27, 28, 29]. By conducting wavelet transform or over-completed learned dictionary as sparsity basis, various sparse reconstruction methods have been developed [12, 14, 16, 19, 17]. Matrix rank minimization regularization based on spectral-spatial correlation are also adopted [20, 21]. However, all these methods always start their modeling with vector or matrix and ignore the high-dimensionality nature of HSI, thus leading to poor reconstruction quality. In recent years, several methods on deep learning, including ADMM-Net [30] and ISTA-Net [31] have been developed for compressive sensing of nature image. However, the heterogeneity of HSI makes those methods difficult to be extended for snapshot hyperspectral imaging. A convolutional autoencoder (AE) based method was first designed for CASSI to learn a non-linear sparse representation [32]. A recent work named HSCNN [33], which treats the reconstruction task as an image enhancement task, was proposed for CASSI. HRNet utilized two separate networks to explore the spatial and spectral similarity [34]. The latest design in [35] combined the neural network and the optimization framework to learn the spectral regularization prior (SRP) and achieved suboptimal results. However, these methods all try to learn a single image prior information with network and still ignore the high-dimensionality intrinsic structure of HSI. ## III Approach ### III-A Notations and Preliminaries We first introduce notations and preliminaries as follows. Matrices are denoted as boldface capital letters, e.g., $\bm{X}$, vectors are represented with blodface lowercase letters, e.g. $\bm{x}$, and scalars are indicated as lowercase letters, e.g., $x$. A tensor of order $N$ as boldface Euler script ${\bm{\mathcal{X}}}\in{\mathbb{R}^{{I_{1}}\times{I_{2}}\times\cdot\cdot\cdot{I_{N}}}}$. By varying index $i_{n}$ while keeping the others fixed, the mode-$n$ fiber of ${\mathcal{X}}$ can be obtained. By arranging the mode-$n$ fibers of $\bm{\mathcal{X}}$ as column vectors, we get the mode-$n$ unfolding matrix $\bm{X}_{(n)}\in{\mathbb{R}^{{I_{n}}\times({I_{1}}\cdot\cdot\cdot{I_{n-1}}{I_{n+1}}\cdot\cdot\cdot{I_{N}})}}$. $\text{fold}_{n}(\cdot)$ is the operator that converts the matrix back to the tensor format along the mode-$n$. The _Frobenius_ norm of tensor is defined as the square root of the sum of the squares of all its elements, i.e., ${\left\\|{\bm{\mathcal{X}}}\right\\|_{F}}=\sqrt{\sum\nolimits_{{i_{1}}=1}^{{I_{1}}}{\sum\nolimits_{{i_{2}}=1}^{{I_{2}}}{\cdot\cdot\cdot\sum\nolimits_{{i_{N}}=1}^{{I_{N}}}{x({{i_{1}},{i_{2}},\cdot\cdot\cdot,{i_{N}}})^{2}}}}}.$ (1) The tensor $n$-mode product is defined as multiplying a tensor by a matrix in mode-$n$. For example, the $n$-mode product of ${\bm{\mathcal{X}}}\in{\mathbb{R}^{{I_{1}}\times{I_{2}}\times\cdot\cdot\cdot{I_{N}}}}$ by a matrix ${\bm{A}}\in{\mathbb{R}^{{J}\times{I_{n}}}}$, which denoted as ${\bm{\mathcal{X}}}~{}{\times}_{n}~{}{\bm{A}}$, is an $N$-order tensor ${\bm{\mathcal{B}}}\in{\mathbb{R}^{{I_{1}}\times\cdot\cdot\cdot{J}\times\cdot\cdot\cdot{I_{N}}}}$, with entries $b({i_{1}},...,{i_{n-1}},j,{i_{n-1}},...,{i_{N}})=\sum\limits_{{i_{n}}=1}^{{i_{N}}}{x\left({{i_{1}},{i_{2}},...,{i_{N}}}\right)a\left({j,{i_{n}}}\right)}.$ (2) With the Tucker decomposition [36], an $N$-order tensor ${\bm{\mathcal{X}}}$ can be decomposed into the following form: ${\bm{\mathcal{X}}}=\bm{\mathcal{G}}~{}{\times}_{1}~{}{\bm{U}_{1}}~{}{\times}_{2}~{}{\bm{U}_{2}}~{}{\times}_{3}\cdot\cdot\cdot{\times}_{N}{\bm{U}_{N}},$ (3) where ${\bm{\mathcal{G}}}\in{\mathbb{R}^{{R_{1}}\times{R_{2}}\times\cdot\cdot\cdot{R_{N}}}}$ is called the core tensor, which is similar to the singular values in matrix SVD, and ${\bm{U}_{i}}\in{\mathbb{R}^{{I_{i}}\times{R_{i}}}}$ is the orthogonal base matrix, which is also similar to the principal components matrix in matrix SVD. Therefore, the Tucker decomposition can be regarded as high-order SVD (HOSVD). Figure 2: Diagram of two representative snapshot hyperspectral imaging systems. ### III-B Observation Model We then give a brief introduction to the observation model of the two representative systems, i.e., CASSI and DCCHI. It is worth mention that our method is general in this filed and suited for other systems, such as the multiple snapshot imaging system and the spatial-spectral encoded imaging system. As shown in Fig. 2, the incident light in CASSI is first projected on the plane of a coded aperture through a objective lens. After spatial coding by the coded aperture, the modulated light goes through a relay lens and is spectrally dispersed in the vertical direction by Amici prism. Finally, the modulated and dispersed spectral information is captured by a panchromatic camera. Let $\bm{\mathcal{F}}\in{\mathbb{R}^{{I}\times{J}\times{\Lambda}}}$ denote the original HSI and $f({i,j,\lambda})$ is its element, where $1{\leq}i{\leq}I$, $1{\leq}j{\leq}J$ index the spatial coordinate and $1{\leq}\lambda{\leq}\Lambda$ indexes the spectral coordinate. The compressive measurement at position $(i,j)$ on the focal plane of CASSI can be represented as: ${y^{c}}(i,j)=\sum\nolimits_{\lambda=1}^{\Lambda}{\rho(\lambda)}\varphi(i-\phi(\lambda),j)f(i-\phi(\lambda),j,\lambda),$ (4) where $\varphi(i,j)$ denotes the modulation pattern of the coded aperture, $\phi(\lambda)$ denotes the dispersion introduced by Amici prism and $\rho(\lambda)$ is the spectral response of the detector. For brevity, let ${\bm{Y}}^{c}\in{\mathbb{R}^{{(I+\Lambda-1)}\times J}}$ denote the matrix representation of ${y^{c}}(i,j)$, and $\bm{\Phi}^{c}$ denote the forward imaging function of CASSI, which is jointly determined by $\rho(\lambda)$, $\varphi(i,j)$ and $\phi(\lambda)$. Then the matrix form of CASSI imaging can be expressed as: ${\bm{Y}}^{c}=\bm{\Phi}^{c}(\bm{\mathcal{F}}).$ (5) As shown in Fig. 2, DCCHI consists of a CASSI branch and a panchromatic camera branch. The incident light first is divided into two directions by the beam splitter equivalently. The light in one direction is captured by the CASSI system according to the above imaging principle, while the light on the other direction is captured directly by a panchromatic camera. The compressive measurement ${y^{p}}(i,j)$ on the panchromatic detector can be represented as: ${y^{p}}(i,j)=\sum\nolimits_{\lambda=1}^{\Lambda}{\rho(\lambda)}f(i,j,\lambda).$ (6) Let ${\bm{Y}}^{p}\in{\mathbb{R}^{{I\times J}}}$ denote the matrix representation of ${y^{p}}(i,j)$, and $\bm{\Phi}^{p}$ denote the forward imaging function of the panchromatic camera, which is determined by $\rho(\lambda)$. Then the matrix form of the panchromatic branch can be expressed as: ${\bm{Y}}^{p}=\bm{\Phi}^{p}(\bm{\mathcal{F}}),$ (7) A general imaging representation of snapshot hyperspectral imaging can be formulated as: ${\bm{Y}}=\bm{\Phi}(\bm{\mathcal{F}}).$ (8) Figure 3: Low-rank property analysis. We exploit the nonlocal similarity across spatial and spectral dimensions to reformulate a low-rank tensor. Then we implement HOSVD on the tensor and show the distribution of singular values in the core tensor. For CASSI, $\bm{Y}=\bm{Y}^{c}$ and $\bm{\Phi}=\bm{\Phi}^{c}$. For DCCHI, $\bm{Y}=[\bm{Y}^{c};\bm{Y}^{p}]$ and $\bm{\Phi}=[\bm{\Phi}^{c};\bm{\Phi}^{p}]$. The goal of HSI reconstruction is to estimate $\bm{\mathcal{F}}$ from the compressive measurement $\bm{Y}$. ### III-C Weighted High-order Singular Value Regularization The key of reconstruction algorithm is to fully exploit the prior information of HSI and build a suitable regularization model. So far most of existing reconstuction methods are on account of two important properties of HSI, i.e., the spatial self-similarity and the spectral correlation. The spatial self- similarity states a nature that there are many image patches around each exemplar patch with the same texture structure. While the spectral correlation indicates that HSI contains a small amount of basis materials and thus exhibits rich redundancy in spectra. In this paper, we utilize tensor low-rank regularization to take such two properties into consideration simultaneously and promote the reconstruction accuracy. TABLE I: Average reconstruction results (PSNR(dB)/SSIM/ERGAS/RMSE) of different methods on CASSI. Indexes \| TV \| AMP \| 3DSR \| NSR \| LRMA \| AE \| ISTA \| HSCNN \| HRNet \| SPR \| Ours ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- PSNR \| 23.16 \| 23.18 \| 23.636 \| 26.13 \| 25.94 \| 25.72 \| 20.60 \| 25.09 \| 22.83 \| 24.48 \| 28.05 SSIM \| 0.7130 \| 0.6600 \| 0.7311 \| 0.7610 \| 0.7930 \| 0.7720 \| 0.5499 \| 0.7334 \| 0.6648 \| 0.7395 \| 0.8302 ERGAS \| 258.32 \| 256.76 \| 245.153 \| 189.19 \| 195.63 \| 197.32 \| 344.57 \| 206.97 \| 268.65 \| 224.19 \| 153.06 RMSE \| 0.0469 \| 0.0474 \| 0.0457 \| 0.0333 \| 0.0315 \| 0.0333 \| 0.0653 \| 0.0373 \| 0.0496 \| 0.0451 \| 0.0236 For one cubic patch with the size of ${s}\times{s}\times\Lambda$ across full bands of HSI $\bm{\mathcal{F}}\in{\mathbb{R}^{{I}\times{J}\times{\Lambda}}}$, we search for its $k-1$ nearest neighbors patches in a local window. By reordering the spatial block of each band into a 1D column vector, the constructed 3-order tensor $\bm{\mathcal{S}}$ with the size of $s^{2}\times\Lambda\times k$ is formed. The constructed tensor simultaneously exhibits the spatial self-similarity (mode-1), the spectral correlation (mode-2) and the joint correlation (mode-3). Here we take the multi- dimensionality property of tensor and introduce low-rank tensor recovery model to preserve the structure information of HSI. Consequently, we build a basic regularization towards low-rank tensor recovery: ${\Gamma(\bm{\mathcal{S}})}=\tau{\left\\|{{{\bf{R}}(\bm{\mathcal{F}})}-\bm{\mathcal{S}}}\right\\|_{F}^{2}}+\text{rank}(\bm{\mathcal{S}}),$ (9) where ${\bf{R}}(\bm{\mathcal{F}})$ represents extracting the 3D tensor from $\bm{\mathcal{F}}$ and $\tau$ denotes the penalty factor. With the Tucker decomposition [36], the rank of a 3-order tensor can be defined as the sum of ranks of unfolding matrices along three modes. However, considering the rank of different modes separately still ignore the correlation between different modes. Meanwhile, estimating the rank of a matrix is still a NP-hard problem. We return to the definition of Tucker decomposition and analyze the low rank property of the constructed tensor. As shown in Fig. 3, we implement Tucker decomposition on a nonlocal tensor extracted from a clean HSI and show the distribution of singular values in the core tensor. We can see that the singular values tend to be dropping to zero fleetly, which indicates that the core tensor exhibits significant sparsity. Therefore, we can introduce a weighted $\ell_{1}$ norm to pursue the tensor rank minimization, i.e.,: $\text{rank}(\bm{\mathcal{S}})=\left\\|\bf{w}\circ\bm{\mathcal{G}}\right\\|_{1}~{}~{}s.t.\bm{\mathcal{S}}=\bm{\mathcal{G}}~{}{\times}_{1}~{}{\bm{U}_{1}}~{}{\times}_{2}~{}{\bm{U}_{2}}~{}{\times}_{3}~{}{\bm{U}_{3}},$ (10) where $\left\\|\bf{w}\circ\bm{\mathcal{G}}\right\\|_{1}=\sum\nolimits_{n}{{w_{n}}\left\|{{g_{n}}}\right\|}$ and ${g_{n}}$ is the element of $\bm{\mathcal{G}}$. The weight $w_{n}$ is set as: $w_{n}^{t+1}={c\mathord{\left/{\vphantom{1{\left({\left\|{w_{i}^{t}}\right\|+\varepsilon}\right)}}}\right.\kern-1.2pt}{\left({\left\|{w_{i}^{t}}\right\|+\varepsilon}\right)}},$ (11) where $t$ denotes the $t$-th iteration, $c$ is a positive constant number and $\varepsilon\leq 10^{-6}$. Such formulation derives a meaningful outcome that the singular values can be penalized adaptively and structure information can be better preserved. Specifically, those greater singular values in the $t$-th iteration, which deliver more important structure information, will get a smaller weight and be shrunk less at $(t+1)$-th iteration. \| \| \| \| \| ---\|---\|---\|---\|---\|--- TV \| AMP \| 3DSR \| NSR \| LRMA \| AE (20.13 / 0.6750) \| (19.61 / 0.5699) \| (20.81 / 0.7030) \| (23.46 / 0.7559) \| (22.73 / 0.8063) \| (22.84 / 0.7593) \| \| \| \| \| ISTA \| HSCNN \| HRNet \| SRP \| Ours \| GT (18.20 / 0.5173) \| 22.66 / 0.8107) \| (19.76 / 0.6314) \| (22.16 / 0.7421) \| (26.88 / 0.8756) \| (PSNR /SSIM) \| \| \| \| \| TV \| AMP \| 3DSR \| NSR \| LRMA \| AE (20.42 / 0.7696) \| (22.96 / 0.7202) \| (22.02 / 0.7796) \| (26.90 / 0.8402) \| (24.93 / 0.8633) \| (26.40 / 0.8496) \| \| \| \| \| ISTA \| HSCNN \| HRNet \| SRP \| Ours \| GT (19.66 / 0.5135) \| (25.80 / 0.8335) \| (21.51 / 0.6998) \| (23.83 / 0.7675) \| (30.39 / 0.9159) \| (PSNR / SSIM) Figure 4: Reconstructed quality comparison of CASSI. The PSNR and SSIM for the result images of _chart and stuffed toy_ and _stuffed toys_ are shown in the parenthesis. By comparing the reconstructed results and ground truth (shortened as GT in the figure), our method obtains better spatial contents and textures. By combining Equation 9 and 10, we obtain the WHOSVR model as: ${\Gamma(\bm{\mathcal{G}})}=\tau{\left\\|{{{\bf{R}}(\bm{\mathcal{F}})}-\bm{\mathcal{G}}~{}{\times}_{1}~{}{\bm{U}_{1}}~{}{\times}_{2}~{}{\bm{U}_{2}}~{}{\times}_{3}~{}{\bm{U}_{3}}}\right\\|_{F}^{2}}+\left\\|\bf{w}\circ\bm{\mathcal{G}}\right\\|_{1}.$ (12) In the following, we will illustrate the WHOSVR based reconstruction method for snapshot hyperspectral imaging. ### III-D Reconstruction Method Based on the analysis above, a general reconstruction formulation for snapshot hyperspectral imaging is proposed: $\begin{split}\mathop{\min}\limits_{\bm{\mathcal{F}},\bm{\mathcal{G}}_{l}}\;&\frac{1}{2}\left\\|{{\bm{Y}}-\bm{\Phi}(\bm{\mathcal{F}})}\right\\|_{F}^{2}+\sum\nolimits_{l=1}^{L}\big{(}\\\ &\tau\left\\|{{{\bf{R}}_{l}(\bm{\mathcal{F}})}-\bm{\mathcal{G}}_{l}~{}{\times}_{1}~{}{\bm{U}_{l,1}}~{}{\times}_{2}~{}{\bm{U}_{l,2}}~{}{\times}_{3}~{}{\bm{U}_{l,3}}}\right\\|_{F}^{2}+\left\\|{\bf{w}}_{l}\circ{\bm{\mathcal{G}}}_{l}\right\\|_{1}\big{)},\end{split}$ (13) where $L$ denotes the total number of constructed tensors. To optimize Equation 13, we adopt an alternating minimization scheme to split it into two finer subproblems: updating core tensor $\bm{\mathcal{G}}_{l}$ and updating the whole HSI $\bm{\mathcal{F}}$. #### III-D1 Updating Core Tensor $\bm{\mathcal{G}}_{l}$ By fixing HSI $\bm{\mathcal{F}}$, we can estimate each low-rank tensor $\bm{\mathcal{G}}_{l}$ independently by solving the following reformulated equation: $\mathop{\min}\limits_{\bm{\mathcal{G}}_{l}}\tau\left\\|{{{\bf{R}}_{l}(\bm{\mathcal{F}})}-\bm{\mathcal{G}}_{l}~{}{\times}_{1}~{}{\bm{U}_{l,1}}~{}{\times}_{2}~{}{\bm{U}_{l,2}}~{}{\times}_{3}~{}{\bm{U}_{l,3}}}\right\\|_{F}^{2}+\left\\|{\bf{w}}_{l}\circ\bm{\mathcal{G}}_{l}\right\\|_{1},$ (14) Given ${{\bf{R}}_{l}(\bm{\mathcal{F}})}={\hat{\bm{\mathcal{G}}}_{l}}~{}{\times}_{1}~{}{\bm{U}_{l,1}}~{}{\times}_{2}~{}{\bm{U}_{l,2}}~{}{\times}_{3}~{}{\bm{U}_{l,3}}$ be the Tucker decomposition of ${{\bf{R}}_{l}(\bm{\mathcal{F}})}$, the solution of $\bm{\mathcal{G}}_{l}$ in Equation 14 is: ${g_{n}}=\text{max}({\hat{g_{n}}}-\frac{w_{n}}{2\tau},0),$ (15) where $\hat{g_{n}}$ is the element of $\hat{\bm{\mathcal{G}}}_{l}$. TV \| AMP \| 3DSR \| NSR \| LRMA \| Ours ---\|---\|---\|---\|---\|--- \| \| \| \| \| \| \| \| \| \| Figure 5: The error maps comparison of two typical scenes on DCCHI. It shows that our method can produce relatively higher spatial fidelity. #### III-D2 Updating Whole HSI $\bm{\mathcal{F}}$ Once we obtain the core tensor $\bm{\mathcal{G}}_{l}$, the whole HSI $\bm{\mathcal{F}}$ can be updated by solving the following problem: $\begin{split}&\qquad\mathop{\min}\limits_{\bm{\mathcal{F}}}\;\frac{1}{2}\left\\|{{\bm{Y}}-\bm{\Phi}(\bm{\mathcal{F}})}\right\\|_{F}^{2}+\\\ &\sum\nolimits_{l=1}^{L}\tau\left\\|{{{\bf{R}}_{l}(\bm{\mathcal{F}})}-\bm{\mathcal{G}}_{l}{\times}_{1}{\bm{U}_{l,1}}{\times}_{2}{\bm{U}_{l,2}}{\times}_{3}{\bm{U}_{l,3}}}\right\\|_{F}^{2}.\end{split}$ (16) Equation 16 is a quadratic optimization problem, so $\bm{\mathcal{F}}$ admits a straightforward least-square solution: $\displaystyle\bm{\mathcal{F}}=\big{(}{{\bm{\Phi}^{T}}\bm{\Phi}+2\tau\sum\nolimits_{l}{{{{{{\bf{R}}_{l}^{T}}}}}{{\bf{R}}_{l}}}}\big{)}^{{\rm{-}}1}}\big{(}{{\bm{\Phi}^{T}}(\bm{Y})$ (17) $\displaystyle+2\tau\sum\nolimits_{l=1}^{L}{{{{{{\bf{R}}_{l}^{T}}}}}(\bm{\mathcal{G}}_{l}~{}{\times}_{1}~{}{\bm{U}_{l,1}}~{}{\times}_{2}~{}{\bm{U}_{l,2}}~{}{\times}_{3}~{}{\bm{U}_{l,3}})}\big{)}.$ In practice, Equation 17 can be solved by the conjugate gradient algorithm [37]. \| \| \| ---\|---\|---\|--- Figure 6: The absolute spectral error on DCCHI between the ground truth and reconstructed results of the white labels in Fig. 4. It shows that our method obtains the highest spectral accuracy. TABLE II: Average reconstruction results of different methods on DCCHI. Indexes \| TV \| AMP \| 3DSR \| NSR \| LRMA \| Ours ---\|---\|---\|---\|---\|---\|--- PSNR \| 28.51 \| 28.52 \| 28.32 \| 32.58 \| 37.45 \| 37.81 SSIM \| 0.8938 \| 0.8526 \| 0.9037 \| 0.9377 \| 0.9730 \| 0.9733 ERGAS \| 167.14 \| 140.75 \| 163.67 \| 107.71 \| 57.30 \| 51.38 RMSE \| 0.0525 \| 0.0263 \| 0.0337 \| 0.0285 \| 0.0132 \| 0.0069 ## IV Experiments In this section, we conduct experiments on two representative snapshot hyperspectral imaging systems, i.e., CASSI and DCCHI, to verify the performance of our method. ### IV-A Implementation Details We generate the mask of coded aperture in CASSI as a random Bernoulli matrix with $p=0.5$. The dispersion of the Amici prism obeys a linear distribution across the wavelength dimension. The Columbia dataset, which contains 32 various real-world objects from 400nm to 700nm (31 bands with 10nm interval), are used as synthetic data. In our experiment, the resolution of all tested images is cropped into $256\times 256$ at the center region cross full bands. Meanwhile, we use 22 HSIs for training and 10 HSIs for testing. Our algorithm is compared with 5 prior knowledge based methods, i.e., TV regularization solved by TwIST [38], 3D sparse reconstruction (3DSR) [18], approximate message passing (AMP) [17], nonlocal sparse representation (NSR) [19] and low-rank matrix approximation (LRMA) [20] and 5 deep learning based methods, i.e., AE [32], HSCNN [33], ISTA-Net (shorten as ISTA) [31], HRNet [34] and SRP[35]. The parameters for our method are set as following. The penalty factor $\tau=1$, the constant $c=0.0055$, and the cubic spatial size $s=5$ with step length of 4. We search $k=45$ nonlocal similar patches within a [-20, 20] window. We set the max iteration number of the alternating minimization scheme as 600 for the stop criterion. For the competitive methods we make great efforts to achieve the best results for all the competing methods according to their publication and released codes. We execute our experiments on a platform of the Windows 10 64-bit system with I7 6700 and 64GB RAM. For quantitative evaluation, four image quality indexes are employed, including peek signal-to-noise ratio (PSNR), structure similarity (SSIM) [39], erreur relative globale adimensionnelle de synthèse (ERGAS)[40] and root mean square error (RMSE). PSNR and SSIM measure the visual quality and the structure similarity, respectively. ERGAS and RMSE measure the spectral fidelity. Generally, a bigger PSNR and SSIM and a smaller ERGAS and RMSE suggest a better reconstruction fidelity. ### IV-B Evaluation Results The average quantitative results of all methods on CASSI are shown in Table I. The best values for each index are highlighted in bold. We can see that our method obtains remarkable promotion in PSNR, SSIM, ERGAS and RMSE compared with other methods. Specifically, our method produces noticeable quality promotion compared with TV, AMP, 3DSR and NSR. It indicates that low rank prior can recover more structure information than sparse prior. The promotion upon LRMA demonstrates that the high-dimensional tensor is more powerful than matrix to exploit the intrinsic nature of HSI. Meanwhile, promotion upon the deep learning based methods indicates that the tensor based optimization can more faithfully account for the high-dimensionality structure. For a visual comparison, we convert HSI into RGB image using the CIE color mapping function. The synthetic RGB images of two representative scenes, i.e., _chart and stuffed toy_ and _stuffed toys_ , are shown in Fig. 4. We can see that our method can produce not only clearer spatial textures but also higher spectral accuracy. Next, we evaluate the reconstruction performance on DCCHI. It should be noted that the mismatch in the two branches of DCCHI results in that block layout based learning methods can not be implemented. So here we compare our method with the prior regularization based methods. The average quantitative results are presented in Table II. It shows that our method can produce the best quantitative performance. We then show the average absolute error maps between the ground truth and restored results across spectra in Fig. 5. It can be seen that the results produced by our method are closer to the ground truth compared with other methods, which verifies that our method obtains higher spatial accuracy. Futher, we show the average absolute error curves between the ground truth and reconstructed results across spectra in Fig. 6. It demonstrates that our method obtains higher spectral accuracy. ## V Conclusion In this paper, we proposed a novel and general reconstruction method with low- rank tensor recovery based on WHOSVR for snapshot hyperspectral imaging. We introduce 3D tensors to exploit the high-order nature of HSI, including spatial self-similarity, spectral correlation and spatial-spectral joint correlation. Specifically, we first construct a 3D tensor for each exemplar cubic patch of HSI. We then proposed to utilize WHOSVR to characterize the high correlation in each mode of the formulated tensors. Finally, an iterative optimization algorithm based on WHOSVR was developed to finish high-accuracy HSI reconstruction. Through experiments implemented on two representative systems verified that our method outperforms state-of-the-art methods. ## Acknowledgment This work is supported by the National Natural Science Foundation of China under Grant No. 62072038 and No. 61922014. ## References * [1] H. 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# Numerical analysis of a deep learning formulation of elastic full waveform inversion with high order total variation regularization in different parameterization Tianze Zhang11footnotemark: 1 Jian Sun11footnotemark: 122footnotemark: 2 Kristopher A. Innanen11footnotemark: 1 Daniel Trad11footnotemark: 1 ###### Abstract We have formulated elastic seismic full waveform inversion (FWI) within a deep learning environment. Training this network with a single seismic data set is equivalent to carrying out elastic FWI. There are three main motivations for this effort. The first is an interest in developing an inversion algorithm which is more or less equivalent to standard elastic FWI but which is ready- made for a range of cloud computational architectures. The second is an interest in algorithms which can later (i.e., not in this paper) be coupled with a more involved training component, involving multiple datasets. The third is a general interest in developing the idea of theory-guiding within machine learning solutions for large geophysical problems, wherein the number of degrees of freedom within a network, and the reliance on exhaustive training data, are both reduced by constraining the network with physical rules. In our formulation, a recurrent neural network is set up with rules enforcing elastic wave propagation, with the wavefield projected onto a measurement surface acting as the synthetic data to be compared with observed seismic data. Gradients for iterative updating of an elastic model, with a variety of parameterizations and misfit functionals, can be efficiently constructed within the network through the automatic differential method. With this method, the inversion based on complex misfits can be calculated. We survey the impact of different complex misfits (based on the $l_{2}$, $l_{1}$) with high order total variation (TV) regulations on multiparameter elastic FWI recovery of models within velocity/density, modulus/density, and stiffness parameter/density parameterizations. We analyze parameter cross-talk. Inversion results on simple and complex models show that the RNN elastic FWI with high order TV regulation using $l_{1}$ norm can help mitigate cross-talk issues with gradient-based optimization methods. ## 1 Introduction It has recently been shown (Sun et al.,, 2020) that seismic wave propagation can be simulated with a specialized recurrent neural network (RNN), and that the process of training such a network with a single seismic data set is equivalent to carrying out seismic full waveform inversion (FWI). We are motivated to extend and expand on these results, because of (1) the apparent potential for wider training of such a network to combat common FWI issues such as modelling error; (2) the opportunities for efficient computation (e.g., cloud) offered by FWI realized on platforms like TensorFlow, and (3) an interest in the behavior of more complex RNN-FWI formulations than previously analyzed, e.g., multi-parameter elastic FWI. In this paper we report on progress on the third of these. The application of machine learning methods to seismic problems has been underway for decades; for example, Röth and Tarantola, (1994) presented a neural network-based inversion of time-domain seismic data for acoustic velocity depth-profiles. However, the evolution of these network approaches into deep learning methods, and the results which have subsequently become possible, make many aspects of the discipline quite new, and explain the major recent surge in development and interest. Now, novel seismic applications are being reported in fault detection, denoising, reservoir prediction and velocity inversion (e.g., Jin et al.,, 2019; Zheng et al.,, 2019; Peters et al.,, 2019; Chen et al.,, 2019; Li et al., 2019a, ; Smith et al.,, 2019; Shustak and Landa,, 2018). Specifically in seismic velocity inversion, Sun and Demanet, (2018) applied a deep learning method to the problem of bandwidth extension; Zhang et al., (2019) designed an end-to-end framework, the velocity-GAN, which generates velocity images directly from the raw seismic waveform data; Wu and Lin, (2018) trained a network with an encoder-decoder structure to model the correspondence between seismic data and subsurface velocity structures; and Yang and Ma, (2019) investigated a supervised, deep convolutional neural network for velocity-model building directly from raw seismograms. The above examples are purely data-driven, in the sense that they involve no assumed theoretical/physical relationships between the input layer (e.g., velocity model) and output layer (e.g., seismic data). We believe that the advantages of the purely data-driven methods are that once the training for the network to perform inversion is finished, take the data-driven training for a network that can perform FWI as an example, the raw seismograms can be directly mapped to the velocity models. This would become a faster inversion method compared with the conventional inversion method that requires iterations. However, seismic inversion is a sophisticated issue, so how we choose the sufficient amount of training data sets that can represent such complex wave physics features and their corresponding velocity models is a hard problem. Here we distinguish between such methods and those belonging to the theory- guided AI paradigm (e.g., Wagner and Rondinelli,, 2016; Wu et al.,, 2018; Karpatne et al.,, 2017). Theory-guided deep learning networks are, broadly, those which enforce physical rules on the relationships between variables and/or layers. This may occur through the training of a standard network with simulated or synthetic data, which are created using physical rules, or by holding some weights in a network, chosen to force the network to mimic a physical process, fixed and un-trainable. Theory-guiding was explicitly used in the former sense by Downton and Hampson, (2018), in which a network designed to predict well log properties from seismic amplitudes was trained not only with real data but with synthetics derived from the Zoeppritz equations. Bar-Sinai et al., (2019) and Raissi, (2018) built deep convolutional neural networks (or CNNs) to solve partial differential equations, i.e., explicitly using theoretical models within the design, which is an example of theory guiding in the latter sense. Sun et al., (2020), in the work we extend in this paper, similarly set up a deep recurrent neural network to simulate the propagation of a seismic wave through a general acoustic medium. The network is set up in such a way that the trainable weights correspond to the medium property unknowns (i.e., wave velocity model), and the non-trainable weights correspond to the mathematical rules (differencing, etc.) enforcing wave propagation. The output layer was the wave field projected onto a measurement surface, i.e., a simulation of measured seismic data. The training of the Sun et al., (2020) network with a single data set was shown to be an instance of full waveform inversion. Conventional (i.e., not network-based) seismic full waveform inversion, or FWI, is a complex data fitting procedure aimed at extracting important information from seismic records. Key ingredients are an efficient forward modeling method to simulate synthetic data, and a local differential approach, in which the gradient and direction are calculated to update the model. When updating several parameters simultaneously, which is a multiparameter full waveform inversion, error in one parameter tends to produce updates in others, a phenomenon referred to as inter-parameter trade-off or cross-talk (e.g., Kamei and Pratt,, 2013; Alkhalifah and Plessix,, 2014; Innanen,, 2014; Pan et al.,, 2018; Keating and Innanen,, 2019). The degree of trade-off or cross-talk between parameters can depend sensitively on the specific parameterization; even though, for instance, $\lambda$, $\mu$ and $\rho$ have the same information content as $V_{P}$, $V_{S}$, and $\rho$, updating in one can produce markedly different convergence properties and final models that doing so in the other. The coupling between different physical parameters is controlled by the physical relationships amongst these parameters, and the relationships between the parameters and the wave which interacts with them. Tarantola, (1986), by using the scattering or radiation pattern, systematically analyzed the effect of different model parameterizations on isotropic FWI. He suggested that the greater the difference in the scattering pattern between each parameter, the better the parameters can be resolved. Köhn et al., (2012) showed that across all geophysical parameterizations, within isotropic-elastic FWI, density is always the most challenging to resolve (e.g., Tarantola,, 1984; Plessix,, 2006; Kamath et al.,, 2018; Lailly,, 1983; Operto et al.,, 2013; Oh and Alkhalifah,, 2016; Pan et al.,, 2016; Keating and Innanen,, 2017). In RNN-based FWI, then, a natural step is the extension of the Sun et al., (2020) result, which involves a scalar-acoustic formulation of FWI, to a multi-parameter elastic version. Cells within a deep recurrent neural network are designed such that the propagation of information through the network is equivalent to the propagation of an elastic wavefield through a 2D heterogeneous and isotropic-elastic medium; the network is equipped to explore a range of parameterizations and misfit functionals for training based on seismic data. As with the acoustic network, the output layer is a projection of the wave field onto some defined measurement surface, simulating measured data. In addition to providing a framework for inversion methods which mixes the features of FWI with the training capacity of a deep learning network, this approach also allows for efficient calculation of the derivatives of the residual through automatic differential (AD) methods (e.g., Li et al., 2019b, ; Sambridge et al.,, 2007), an engine for which is provided by the open-source machine learning library Pytorch (Paszke et al.,, 2017). Our recurrent neural network is designed using this library. The forward simulation of wave propagation is represented by a Dynamic Computational Graph (DCG), which records how each internal parameter is calculated from all previous ones. In the inversion, the partial derivatives of the residual with respect to any parameter (within a trainable class) is computed by (1) backpropagating within the computational graph to that parameter, and (2) calculating the partial derivative along the path using the chain rule. This paper is organized as follows. First, we introduce the basic structure of the recurrent neural network and how the gradients can be calculated using the backpropagation method. Second, we demonstrate how the elastic FWI RNN cell is constructed in this paper and how the waveflieds propagate in the RNN cells. Third, we explain the $l_{2}$ and $l_{1}$ misfits with high order TV regulation and the mathematical expression for the gradients based on these misfits. Fourth, we use simple layers models and complex over-thrust models to perform inversions with various parameterizations using different misfits. Finally, we discuss the conclusions of this work. ## 2 Recurrent Neural Network A recurrent neural network (RNN) is a machine learning network suitable for dealing with data which have some sequential meaning. Within such a network, the information generated in one cell layer can be stored and used by the next layer. This design has natural applicability in the processing and interpretation of the time evolution of physical processes (Sun et al.,, 2020), and time-series data in general; examples include language modeling (Mikolov,, 2012), speech recognition Graves et al., (2013), and machine translation (Kalchbrenner and Blunsom,, 2013; Zaremba et al.,, 2014). Figure 1: Forward propagation through an example RNN. Cells are identical in form, with the output of one being the input of the next. $\textbf{O}=[O_{1},O_{2},O_{3},{\cdots}]$ are the internal variables in each cell; $\textbf{S}=[S_{1},S_{2}]$ are the inputs; $\textbf{P}=[P_{1},P_{2}]$ are the outputs; $\textbf{L}=[L_{1},L_{2}]$ are the labeled data; $\textbf{W}=[W_{1},W_{2}]$ are the trainable weights. Black dashed line indicates forward propagation. Figure 2: Residual backward propagation through an example RNN. $\textbf{R}=[R_{1},R_{2}]$ are the residuals at each RNN cell, which are calculated using absolute error($l_{1}$ norm). Gray solid line shows how gradients are calculated using back propagation from the residual to the trainable weights in RNN cell along the computational graph. Figure 1 illustrates the forward propagation of information through an example RNN, in which each RNN cell represents an instant in time: the $\textbf{O}=[O_{1},O_{2},O_{3},{\cdots}]$ are the internal variables in each RNN cell; $\textbf{S}=[S_{1},S_{2}]$ are the input at each time step; $\textbf{P}=[P_{1},P_{2}]$ are the output; $\textbf{L}=[L_{1},L_{2}]$ are the labeled data; and $\textbf{W}=[W_{1},W_{2}]$ are the trainable parameters at each time step. Mathematical operations relating the internal variables within a cell, and those relating internal variables across adjacent cells, are represented as arrows. To train this network is to select trainable weights such that labeled data L and RNN output P are as close as possible. Specifically, in training we determine the parameters W, through a gradient-based optimization involving the partial derivatives of the residual with respect to each $W_{i}$. These derivatives are determined through backpropagation, which is a repeated application the chain rule, organized to resemble flow in the reverse direction along with the arrows in the network. For the example RNN in Figure 1, this takes the form illustrated in Figure 2, in which the sequence $\textbf{R}=[R_{1},R_{2}]$ represents the residuals at each time step. Within this example, to calculate the partial derivative of $R_{1}$ with respect to $W_{1}$, we back-propagate from node $R_{1}$ to $W_{1}$: $\frac{{\partial}R_{1}}{{\partial}W_{1}}=\frac{{\partial}R_{1}}{{\partial}P_{1}}\frac{{\partial}P_{1}}{{\partial}W_{1}}=\frac{{\partial}R_{1}}{{\partial}P_{1}}\frac{{\partial}P_{1}}{{\partial}O_{3}}\frac{{\partial}O_{3}}{{\partial}W_{1}}=-2$ (1) To calculate the partial derivative of $R_{2}$ with respect to $W_{1}$, we backpropagate from $R_{2}$ to $W_{1}$: $\frac{{\partial}R_{2}}{{\partial}W_{1}}=\frac{{\partial}R_{2}}{{\partial}P_{2}}\frac{{\partial}P_{2}}{{\partial}W_{1}}=\frac{{\partial}R_{2}}{{\partial}P_{2}}\frac{{\partial}P_{2}}{{\partial}O_{5}}\frac{{\partial}O_{5}}{{\partial}O_{4}}\frac{{\partial}O_{4}}{{\partial}O_{3}}\frac{{\partial}O_{3}}{{\partial}W_{1}}=-4O_{4}$ (2) If the RNN was set up to propagate through two-time steps, the gradient for $W_{1}$ is $-4O_{4}+2$. Real RNNs are more complex and involve propagation through larger numbers of time steps, but all are optimized through a process similar to this. These derivatives are the basis for gradients in the optimization misfit function; using them the W are updated and iterations continue. ## 3 A Recurrent Neural Network formulation of EFWI Wave propagation can be simulated using suitably-designed RNNs (Sun et al.,, 2020; Richardson,, 2018; Hughes et al.,, 2019). Here we take the acoustic wave propagation approach of Sun et al., (2020) as a starting point, and formulate an RNN which simulates the propagation of an elastic wave through isotropic elastic medium. The underlying equations are the 2D velocity-stress form of the elastodynamic equations (Virieux,, 1986; Liu and Sen,, 2009): $\begin{aligned} &{\frac{\partial{{v}}_{x}}{\partial t}=\frac{1}{\rho}\left(\frac{\partial{\sigma}_{xx}}{\partial x}+\frac{\partial{\sigma}_{xz}}{\partial z}\right)}\\\ &{\frac{\partial{{v}}_{z}}{\partial t}=\frac{1}{\rho}\left(\frac{\partial{\sigma}_{xz}}{\partial x}+\frac{\partial{\sigma}_{zz}}{\partial z}\right)}\\\ &{\frac{\partial{\sigma}_{xx}}{\partial t}=({\lambda}+2{\mu})\frac{\partial{{v}}_{x}}{\partial x}+{\lambda}\frac{\partial{{v}}_{z}}{\partial z}}\\\ &{\frac{\partial{\sigma}_{zz}}{\partial t}=({\lambda}+2{\mu})\frac{\partial{{v}}_{z}}{\partial z}+{\lambda}\frac{\partial{{v}}_{x}}{\partial x}}\\\ &{\frac{\partial{\sigma}_{xz}}{\partial t}={\mu}\left(\frac{\partial{{v}}_{x}}{\partial z}+\frac{\partial{{v}}_{z}}{\partial x}\right)}\\\ \end{aligned}.,$ (3) where ${{v}}_{x}$ and ${{v}}_{z}$ are the $x$ and $z$ components of the particle velocity, ${{\sigma}_{xx}}$, ${{\sigma}_{zz}}$ and ${{\sigma}_{xz}}$ are three 2D components of the stress tensor. Discretized spatial distributions of the Lamé parameters $\lambda$ and $\mu$, and the density ${\rho}$, form the elastic model. Figure 3: The structure of each RNN. In this figure, $\partial_{x}{\sigma}_{xx}^{t}$, $\partial_{z}{\sigma}_{zz}^{t}$, $\partial_{x}{\sigma}_{xz}^{t}$, $\partial_{z}{\sigma}_{xz}^{t}$, $\partial_{x}{v}_{x}^{t+\frac{1}{2}}$, $\partial_{z}{v}_{x}^{t+\frac{1}{2}}$, $\partial_{x}{v}_{z}^{t+\frac{1}{2}}$, $\partial_{z}{v}_{z}^{t+\frac{1}{2}}$ are the internal variables. ${v}^{t-\frac{1}{2}}_{x}$, ${v}^{t-\frac{1}{2}}_{z}$, ${\sigma}_{xx}^{t}$, ${\sigma}_{zz}^{t}$, ${\sigma}_{xz}^{t}$, is communicated between the RNN cells. ${\lambda}$, ${\mu}$,$\rho$ are trainable parameters. Algorithm 1 Sequence of calculations in the RNN cell with PML boundary 1:Source: $s_{x}$, $s_{z}$; velocity and stress fields at the previous time step, parameters :$\lambda$, $\mu$, $\rho$; time step: $dt$. PML damping coefficients $d_{x}$. $d_{z}$. 2:Update velocity field at $t+\frac{1}{2}$ and stress fields at $t+1$ 3:${\sigma_{xx}^{t}}\leftarrow{\sigma_{xx}^{t}}+{s}_{x}$ 4:${\sigma_{zz}^{t}}\leftarrow{\sigma_{zz}^{t}}+{s}_{z}$ 5:$\partial_{x}{\sigma_{xx}^{t}}\leftarrow({\sigma_{xx}^{t}}{\mathbf{k}}_{x_{1}})/\rho$ 6:$\partial_{z}{\sigma_{xz}^{t}}\leftarrow({\sigma_{xz}^{t}}{\mathbf{k}}_{z_{2}})/\rho$ 7:$\partial_{x}{\sigma_{xz}^{t}}\leftarrow({\sigma_{xz}^{t}}{\mathbf{k}}_{x_{2}})/\rho$ 8:$\partial_{z}{\sigma_{zz}^{t}}\leftarrow({\sigma_{zz}^{t}}{\mathbf{k}}_{z_{1}})/\rho$ 9:${{v}_{x}^{t+\frac{1}{2}}}_{x}\leftarrow(1-dtd_{x}){{v}_{x}^{t-\frac{1}{2}}}_{x}+dt(\partial_{x}{\sigma_{xx}^{t}})$ 10:${{v}_{x}^{t+\frac{1}{2}}}_{z}\leftarrow(1-dtd_{z}){{v}_{x}^{t-\frac{1}{2}}}_{z}+dt(\partial_{z}{\sigma_{xz}^{t}})$ 11:${v}_{x}^{t+\frac{1}{2}}\leftarrow{{v}_{x}^{t+\frac{1}{2}}}_{x}+{{v}_{x}^{t+\frac{1}{2}}}_{z}$ 12:${{v}_{z}^{t+\frac{1}{2}}}_{x}\leftarrow(1-dtd_{x}){{v}_{z}^{t-\frac{1}{2}}}_{x}+dt(\partial_{x}{\sigma_{xz}^{t}})$ 13:${{v}_{z}^{t+\frac{1}{2}}}_{z}\leftarrow(1-dtd_{z}){{v}_{z}^{t-\frac{1}{2}}}_{z}+dt(\partial_{z}{\sigma_{zz}^{t}})$ 14:${v}_{z}^{t+\frac{1}{2}}\leftarrow{{v}_{x}^{t+\frac{1}{2}}}_{x}+{{v}_{x}^{t+\frac{1}{2}}}_{z}$ 15:$\partial_{x}{v}_{x}^{t+\frac{1}{2}}\leftarrow{v}_{x}^{t+\frac{1}{2}}{\mathbf{k}}_{x_{2}}$ 16:$\partial_{z}{v}_{x}^{t+\frac{1}{2}}\leftarrow{v}_{x}^{t+\frac{1}{2}}{\mathbf{k}}_{z_{1}}$ 17:$\partial_{x}{v}_{z}^{t+\frac{1}{2}}\leftarrow{v}_{z}^{t+\frac{1}{2}}{\mathbf{k}}_{x_{1}}$ 18:$\partial_{z}{v}_{z}^{t+\frac{1}{2}}\leftarrow{v}_{z}^{t+\frac{1}{2}}{\mathbf{k}}_{z_{2}}$ 19:${{\sigma_{xx}}^{t+1}}_{x}\leftarrow(1-dtdx){{\sigma_{xx}}^{t}}_{x}+dt({\lambda+2\mu})\partial_{x}{v}_{x}^{t+\frac{1}{2}}$ 20:${{\sigma_{zz}}^{t+1}}_{x}\leftarrow(1-dtdz){{\sigma_{zz}}^{t}}_{x}+dt(\lambda)\partial_{z}{v}_{z}^{t+\frac{1}{2}}$ 21:${{\sigma_{zz}}^{t+1}}\leftarrow{{\sigma_{xx}}^{t+1}}_{x}+{{\sigma_{xx}}^{t+1}}_{z}$ 22:${{\sigma_{zz}}^{t+1}}_{x}\leftarrow(1-dtdx){{\sigma_{xx}}^{t}}_{x}+dt({\lambda})\partial_{x}{v}_{x}^{t+\frac{1}{2}}$ 23:${{\sigma_{zz}}^{t+1}}_{z}\leftarrow(1-dtdz){{\sigma_{zz}}^{t}}_{z}+dt(\lambda+2\mu)\partial_{z}{v}_{z}^{t+\frac{1}{2}}$ 24:${{\sigma_{zz}}^{t+1}}\leftarrow{{\sigma_{zz}}^{t+1}}_{x}+{{\sigma_{zz}}^{t+1}}_{z}$ 25:${{\sigma_{xz}}^{t+1}}_{x}\leftarrow(1-dtdx){{\sigma_{xz}}^{t}}_{x}+dt({\mu})\partial_{z}{v}_{x}^{t+\frac{1}{2}}$ 26:${{\sigma_{xz}}^{t+1}}_{z}\leftarrow(1-dtdz){{\sigma_{xz}}^{t}}_{z}+dt(\mu)\partial_{x}{v}_{z}^{t+\frac{1}{2}}$ 27:${{\sigma_{xz}}^{t+1}}\leftarrow{{\sigma_{xz}}^{t+1}}_{x}+{{\sigma_{xz}}^{t+1}}_{z}$ In Figure 3 the structure of an RNN cell which produces a staggered-grid finite difference solution for the velocity and stress fields is illustrated. At each time step the discrete sources $s_{x}$ and $s_{z}$ act as inputs; the velocity and stress information, ${v}^{t-\frac{1}{2}}_{x}$, ${v}^{t-\frac{1}{2}}_{z}$, ${\sigma}_{xx}^{t}$, ${\sigma}_{zz}^{t}$, and ${\sigma}_{xz}^{t}$, is communicated between the RNN cells ; the partial derivative fields, $\partial_{x}{\sigma}_{xx}^{t}$, $\partial_{z}{\sigma}_{zz}^{t}$, $\partial_{x}{\sigma}_{xz}^{t}$, $\partial_{z}{\sigma}_{xz}^{t}$, $\partial_{x}{v}_{x}^{t+\frac{1}{2}}$, $\partial_{z}{v}_{x}^{t+\frac{1}{2}}$, $\partial_{x}{v}_{z}^{t+\frac{1}{2}}$, $\partial_{z}{v}_{z}^{t+\frac{1}{2}}$ are the internal variables in each RNN cell, which correspond to O in Figure 2; and, $\lambda$, $\mu$ and $\rho$ are included as trainable weights, which correspond to W in Figure 2. In Algorithm 1 pseudocode detailing these calculations within the RNN cell is provided. The $$ symbol represents the machine learning image convolution operator. This image convolution is the process of adding each element of the image to its local neighbors, weighted by the image convolution kernel. We find that this image convolution operator is also capable of calculating space partial derivatives if the convolution kernel is designed according to the finite difference coefficients. Details about the image convolution operation can be found in Podlozhnyuk, (2007). $dx$, $dz$ are the grid intervals, and the image convolution kernels are: ${\mathbf{k}}_{x_{1}}={\mathbf{a}}/dx$, ${\mathbf{k}}_{x_{2}}={\mathbf{b}}/dx$, ${\mathbf{k}}_{z_{1}}={\mathbf{a}}^{T}/dz$, and ${\mathbf{k}}_{z_{2}}={\mathbf{b}}^{T}/dz$, where ${\mathbf{a}}=[0,1/24,-9/8,9/8,-1/24]$ and ${\mathbf{b}}=[1/24,-9/8,9/8,-1/24,0]$. ${\mathbf{a}}$ and ${\mathbf{b}}$ are 1$\times$5 dimension arrays. ${\mathbf{k}}_{x_{1}}$ and ${\mathbf{k}}_{x_{2}}$ are kernels, for the image convolution process, responsible for calculating the staggered grid space partial derivative in x direction. ${\mathbf{k}}_{z_{1}}$ and ${\mathbf{k}}_{z_{2}}$ are kernels, for the image convolution process, responsible for calculating the staggered grid space partial derivative in z direction, and that is also why the arrays, ${\mathbf{a}}$ and ${\mathbf{b}}$, are transposed in ${\mathbf{k}}_{z_{1}}$ and ${\mathbf{k}}_{z_{2}}$. Space partial derivative calculated in this way is, mathematically, the same with conventional staggered grid method (e.g.Virieux, (1986)). In this study, we achieve this process in an image convolution way. In algorithm 1, in order to implement the PML boundary, all the stress fields and the velocity fields need to be split into their x and z components. In algorithm 1, $d_{x}$ and $d_{z}$ are the PML damping coefficients in x direction and z direction. $d_{x}$ can be expressed as: $d_{x}(i)=d_{0x}(\frac{i}{n_{pmlx}})^{p}$ (4) ,where represents either x or z direction. i is PML layer number starting from the effective calculation boundary. $n_{pmlx}$ is the PML layer number in $x$ direction. p is an integer and the value is from 1-4. $d_{0x}$ can be expressed as: $d_{0x}=log(\frac{1}{R})\frac{r{V_{s}}}{n_{pmlx}{\delta}_{x}}$ (5) ,where $R$ is a theoretically reflection coefficient, $r$ is a value ranging from 3-4. ${\delta}_{x}$ is the grid length in $x$ direction. $d_{z}$ can also be calculated in the same way. Figure 4: Velocity field forward and residual backward propagation under formatting of the RNN. The shot records formed at each time step correspond to the output of RNN cell P in Figure 2. Observed shot records correspond to labeled data (L in Figure 2). Residual shot record correspond to residual information (R in Figure 2). Back propagation starts from the residual shot record. The activity of the RNN network is illustrated in “unfolded” form in Figure 4. Above the unfolded network, horizontal velocity fields associated with a point source at the top left of a model are plotted at three times during propagation of the wave information through the network, the third being at $t_{max}$, the maximum receiving time. The wave field values are stored at positions selected to match multicomponent receivers; these form shot records as time evolves, which becomes the output data at each time. These shot records correspond to variables P in Figures 1 and 2, and the observed data. For FWI problem, the observed data is obtained from the true model. For real seismic data inversion, the observed data is obtained from field survey. In this RNN based elastic FWI the observed data is considered as the labeled data. The residuals are calculated at the last computational time as Figure 4 illustrated and this along with a selected norm defines the misfit function used to train the network. Algorithm 2 describes the training of the network, i.e., the process of elastic RNN FWI. Partial derivatives of the residual with respect to the trainable parameters (in this case $\lambda$, $\mu$ and $\rho$) are calculated through backpropagation using the automatic differential method, as set out in the previous section. After we have the gradients then we can use an optimization method and step length to update the trainable parameters and reduce the misfit and start another iteration. In step 4, RNN() is the network discussed above, whose output is the synthetic data; costFunc(), in step 5, is the misfit or loss function chosen to measure the difference between the synthetic data and observed data; loss.backward() begins the backpropagation within the computational graph and produces the gradients for each parameter, with which the current parameter model can be updated and another iteration started. Algorithm 2 Loop for elastic RNN FWI 1:Set trainable parameters: $\lambda$, $\mu$, ${\rho}$ in this test. 2:Set optimizers for parameters: $Optimizer_{1}$, $Optimizer_{2}$ and $Optimizer_{3}$ for ${\lambda}$, $\mu$ and ${\rho}$ respectively. 3:for iter $\in[1,maxiter]$ or not converge do 4: $D_{syn}$ = RNN($\lambda$, $\mu$, ${\rho}$): generate synthetic data 5: loss = costFunc($D_{syn}$, $D_{obs}$): calculate misfits 6: loss.backward(): Backpropagation and give gradients for the parameters 7: optimizers.step(): update parameters 8:end for The gradient calculated within the training process above essentially reproduces the adjoint-state calculations within FWI, as discussed by Sun et al., (2020). Formulated as a problem of RNN training, the gradient calculation occurs rapidly and in a manner suitable for cloud computational architectures; also, it allows the researcher to efficiently alter misfit function choices and parameterization in order to do high-level optimization. However, it involves the storage of the whole wavefield, and thus should be expected to have significant memory requirements. ## 4 Misfits with high order TV regulation Here we fist introduce the elastic RNN misfits based on $l_{2}$ norm with high order TV regularization: $\displaystyle{\mathbf{\Phi}}_{l2}^{TV}({\mathbf{m_{\lambda},{m_{\mu},m_{\rho}}}},{\alpha_{1}^{\lambda}},{\alpha_{1}^{\mu}},{\alpha_{1}^{\rho}},{\alpha_{2}^{\lambda}},{\alpha_{2}^{\mu}},{\alpha_{2}^{\rho}})=\frac{1}{2}\lVert\mathbf{D_{syn}}\mathbf{(m_{\lambda},m_{\mu},m_{\rho})}-\mathbf{D_{obs}}\rVert^{2}_{2}+$ (6) $\displaystyle{\alpha_{1}^{\lambda}}\mathbf{{\Theta}_{TV}}\mathbf{(m_{\lambda})}+{\alpha_{1}^{\mu}}\mathbf{{\Theta}_{TV}}\mathbf{(m_{\mu})}+{\alpha_{1}^{\rho}}\mathbf{{\Theta}_{{TV}}}\mathbf{(m_{\rho})}+$ $\displaystyle{\alpha_{2}^{\lambda}}\mathbf{\Upsilon_{TV}}\mathbf{(m_{\lambda})}+{\alpha_{2}^{\mu}}\mathbf{\Upsilon_{TV}}\mathbf{(m_{\mu})}+{\alpha_{2}^{\rho}}\mathbf{\Upsilon_{TV}}\mathbf{(m_{\rho})}$ ,where ${\alpha_{1}^{\lambda}}$, ${\alpha_{1}^{\mu}}$, ${\alpha_{1}^{\rho}}$, ${\alpha_{1}^{\lambda}}$, ${\alpha_{1}^{\mu}}$, ${\alpha_{1}^{\rho}}$, are vector of Lagrange multipliers, and $\mathbf{{\Theta}_{TV}}$, $\mathbf{{\Upsilon}_{TV}}$ represents first and second order TV regularization functions respectively. $\mathbf{D_{syn}}\mathbf{(m_{\lambda},m_{\mu},m_{\rho})}$ represents the synthetic data, which is the function of the model parameters, and in this equation they are $V_{P}$, $V_{S}$ and $\rho$. $\mathbf{{\Theta}_{TV}}$ and $\mathbf{{\Upsilon}_{TV}}$ represent functions for calculating the first and second order TV regulations for the models. The first order TV regulation term can be expressed as: $\displaystyle TV_{1}(\mathbf{(m)})=$ $\displaystyle\sum_{i=1}^{n-1}\sum_{j=1}^{m-1}\|M_{i+1,j}-M_{i,j}\|+\sum_{i=1}^{n-1}\sum_{j=1}^{m-1}\|M_{i,j+1}-M_{i,j}\|=$ (7) $\displaystyle\begin{pmatrix}{\nabla}_{x},{\nabla}_{z}\end{pmatrix}\begin{pmatrix}\mathbf{m},\\\ \mathbf{m}\end{pmatrix}=\begin{pmatrix}\mathbf{\mathcal{L}_{x}},\mathbf{\mathcal{L}_{z}},\end{pmatrix}\begin{pmatrix}\mathbf{m},\\\ \mathbf{m}\end{pmatrix}=\mathbf{{\Theta}_{TV}}\mathbf{(m)}$ The second order TV regulation term can be expressed as: $\displaystyle TV_{2}(\mathbf{(m)})=$ $\displaystyle\sum_{i=1}^{n-1}\sum_{j=1}^{m-1}\|M_{i+1,j}-2M_{i,j}+M_{i-1,j}\|+\sum_{i=1}^{n-1}\sum_{j=1}^{m-1}\|M_{i,j+1}-2M_{i,j}+M_{i,j-1}\|$ (8) $\displaystyle=$ $\displaystyle\begin{pmatrix}{\nabla}_{xx},{\nabla}_{zz}\end{pmatrix}\begin{pmatrix}\mathbf{m},\\\ \mathbf{m}\end{pmatrix}=\begin{pmatrix}\mathbf{\mathcal{K}_{xx}},\mathbf{\mathcal{K}_{zz}},\end{pmatrix}\begin{pmatrix}\mathbf{m},\\\ \mathbf{m}\end{pmatrix}=\mathbf{{\Upsilon}_{TV}}\mathbf{(m)}$ The derivative of ${\mathbf{\Phi}}_{l2}^{TV}$ for each parameter ,which is the gradient for $V_{P}$, $V_{S}$ and $\rho$ based on the $l_{2}^{TV}$ norm can be expressed as: $\displaystyle\begin{pmatrix}\frac{{\partial}{\mathbf{\Phi}}_{l2}^{TV}}{{\partial}{\mathbf{m_{\lambda}}}}\\\ \frac{{\partial}{\mathbf{\Phi}}_{l2}^{TV}}{{\partial}{\mathbf{m_{\mu}}}}\\\ \frac{{\partial}{\mathbf{\Phi}}_{l2}^{TV}}{{\partial}{\mathbf{m_{\rho}}}}\\\ \end{pmatrix}=\begin{pmatrix}\mathbf{G_{{l2}_{\lambda}}}\\\ \mathbf{G_{{l2}_{\mu}}}\\\ \mathbf{G_{{l2}_{\rho}}}\\\ \end{pmatrix}+\begin{pmatrix}\mathbf{R_{\lambda}}\\\ \mathbf{R_{\mu}}\\\ \mathbf{R_{\rho}}\\\ \end{pmatrix}$ (9) , where $\mathbf{G_{{l2}_{\lambda}}}$,$\mathbf{G_{{l2}_{\mu}}}$,$\mathbf{G_{{l2}_{\rho}}}$ are the gradient for $\lambda$, $\mu$, ${\rho}$. $\mathbf{R_{\lambda}}$,$\mathbf{R_{\mu}}$,$\mathbf{R_{\rho}}$ are the regulation terms and the mathematical expressions for these regulation terms are: $\displaystyle\begin{pmatrix}\mathbf{R_{\lambda}}\\\ \mathbf{R_{\mu}}\\\ \mathbf{R_{\rho}}\\\ \end{pmatrix}=\begin{pmatrix}&{\alpha_{1}^{\lambda}}\mathbf{\mathcal{L}^{T}_{x}}\mathbf{Q_{x_{\lambda}}}\hskip 7.22743pt&{\alpha_{1}^{\lambda}}\mathbf{\mathcal{L}^{T}_{z}}\mathbf{Q_{z_{\lambda}}},\hskip 7.22743pt&{\alpha_{2}^{\lambda}}\mathbf{\mathcal{K}^{T}_{xx}}\mathbf{Q_{{xx}_{\lambda}}}\hskip 7.22743pt&{\alpha_{2}^{\lambda}}\mathbf{\mathcal{K}^{T}_{zz}}\mathbf{Q_{{zz}_{\lambda}}}\\\ &{\alpha_{1}^{\mu}}\mathbf{\mathcal{L}^{T}_{x}}\mathbf{Q_{{x}_{\mu}}}\hskip 7.22743pt&{\alpha_{1}^{\mu}}\mathbf{\mathcal{L}^{T}_{z}}\mathbf{Q_{{z}_{\mu}}}\hskip 7.22743pt&{\alpha_{2}^{\mu}}\mathbf{\mathcal{K}^{T}_{xx}}\mathbf{Q_{{xx}_{\mu}}}\hskip 7.22743pt&{\alpha_{2}^{\mu}}\mathbf{\mathcal{K}^{T}_{zz}}\mathbf{Q_{{zz}_{\mu}}}\\\ &{\alpha_{1}^{\rho}}\mathbf{\mathcal{L}^{T}_{x}}\mathbf{Q_{{x}_{\rho}}}\hskip 7.22743pt&{\alpha_{1}^{\rho}}\mathbf{\mathcal{L}^{T}_{z}}\mathbf{Q_{{z}_{\rho}}}\hskip 7.22743pt&{\alpha_{2}^{\rho}}\mathbf{\mathcal{K}^{T}_{xx}}\mathbf{Q_{{xx}_{\rho}}}\hskip 7.22743pt&{\alpha_{2}^{\rho}}\mathbf{\mathcal{K}^{T}_{zz}}\mathbf{Q_{{zz}_{\rho}}}\end{pmatrix}\begin{pmatrix}\mathbf{\mathcal{L}_{x}}\\\ \mathbf{\mathcal{L}_{z}}\\\ \mathbf{\mathcal{K}_{xx}}\\\ \mathbf{\mathcal{K}_{zz}}\end{pmatrix}\cdot\begin{pmatrix}\mathbf{m_{\lambda}}\\\ \mathbf{m_{\mu}}\\\ \mathbf{m_{\rho}}\end{pmatrix}$ (10) $\begin{pmatrix}\mathbf{q_{{x}_{\lambda}}}&\mathbf{q_{{x}_{\mu}}}&\mathbf{q_{{x}_{\rho}}}\\\ \mathbf{q_{{z}_{\lambda}}}&\mathbf{q_{{z}_{\mu}}}&\mathbf{q_{{z}_{\rho}}}\\\ \mathbf{q_{{xx}_{\lambda}}}&\mathbf{q_{{xx}_{\mu}}}&\mathbf{q_{{xx}_{\rho}}}\\\ \mathbf{q_{{zz}_{\lambda}}}&\mathbf{q_{{zz}_{\mu}}}&\mathbf{q_{{zz}_{\rho}}}\\\ \end{pmatrix}=\begin{pmatrix}\mathbf{\mathcal{L}_{x}}\\\ \mathbf{\mathcal{L}_{z}}\\\ \mathbf{\mathcal{K}_{xx}}\\\ \mathbf{\mathcal{K}_{zz}}\\\ \end{pmatrix}\begin{pmatrix}\mathbf{m_{\lambda}},\mathbf{m_{\mu}},\mathbf{m_{\rho}}\end{pmatrix}$ (11) $\begin{pmatrix}\mathrm{Q_{{x}_{\lambda}}}&\mathrm{Q_{{x}_{\mu}}}&\mathrm{Q_{{x}_{\rho}}}\\\ \mathrm{Q_{{z}_{\lambda}}}&\mathrm{Q_{{z}_{\mu}}}&\mathrm{Q_{{z}_{\rho}}}\\\ \mathrm{Q_{{xx}_{\lambda}}}&\mathrm{Q_{{xx}_{\mu}}}&\mathrm{Q_{{xx}_{\rho}}}\\\ \mathrm{Q_{{zz}_{\lambda}}}&\mathrm{Q_{{zz}_{\mu}}}&\mathrm{Q_{{zz}_{\rho}}}\\\ \end{pmatrix}=\begin{pmatrix}\mathrm{\frac{1}{\|q_{{x}_{\lambda}}\|}}&\mathrm{\frac{1}{\|q_{{x}_{\mu}}\|}}&\mathrm{\frac{1}{\|q_{{x}_{\rho}}\|}}\\\ \mathrm{\frac{1}{\|q_{{z}_{\lambda}}\|}}&\mathrm{\frac{1}{\|q_{{z}_{\mu}}\|}}&\mathrm{\frac{1}{\|q_{{z}_{\rho}}\|}}\\\ \mathrm{\frac{1}{\|q_{{xx}_{\lambda}}\|}}&\mathrm{\frac{1}{\|q_{{xx}_{\mu}}\|}}&\mathrm{\frac{1}{\|q_{{xx}_{\rho}}\|}}\\\ \mathrm{\frac{1}{\|q_{{zz}_{\lambda}}\|}}&\mathrm{\frac{1}{\|q_{{zz}_{\mu}}\|}}&\mathrm{\frac{1}{\|q_{{zz}_{\rho}}\|}}\end{pmatrix}$ (12) $\mathbf{T}$ means the transpose of the matrix, $\cdot$ meas dot product. $\mathbf{q_{{x}_{\mathbf{\lambda}}}}$ represent the first order TV regularization vector in x direction for parameter ${\lambda}$. $\mathrm{q_{{x}_{{\lambda}}}}$ represent the values in vector $\mathbf{q_{{x}_{\lambda}}}$. $\mathrm{Q_{{x}_{\lambda}}}$ is the absolute inverse of $\mathrm{q_{{x}_{\lambda}}}$. $\mathrm{Q_{{x}_{\lambda}}}$ are elements in vector $\mathbf{Q_{{x}_{\lambda}}}$. $\mathbf{q_{{xx}_{\mathbf{\lambda}}}}$ represent the second order TV regularization vector in x direction for parameter ${\lambda}$. $\mathrm{q_{{xx}_{{\lambda}}}}$ represent the values in vector $\mathbf{q_{{xx}_{\lambda}}}$. $\mathrm{Q_{{xx}_{\lambda}}}$ is the absolute inverse of $\mathrm{q_{{xx}_{\lambda}}}$. $\mathrm{Q_{{xx}_{\lambda}}}$ are elements in vector $\mathbf{Q_{{xx}_{\lambda}}}$. Other values in equations (9) and (10) can be also deduced like this. $\mathbf{\mathcal{L}_{x}}$, $\mathbf{\mathcal{L}_{z}}$ are the first order differential vector to give the first order total variations in x and z directions respectively. $\mathbf{\mathcal{K}_{xx}}$, $\mathbf{\mathcal{K}_{zz}}$ are the second order differential vector to give the second order total variations in x and z directions respectively. If we were to use $l1$ norm objective function with TV regulation. The objective function can be written as: $\displaystyle{\mathbf{\Phi}}_{l1}^{TV}({\mathbf{m_{\lambda},{m_{\mu},m_{\rho}}}},{\alpha_{1}^{\lambda}},{\alpha_{1}^{\mu}},{\alpha_{1}^{\rho}},{\alpha_{2}^{\lambda}},{\alpha_{2}^{\mu}},{\alpha_{2}^{\rho}})=\lVert\mathbf{D_{syn}}\mathbf{(m_{\lambda},m_{\mu},m_{\rho})}-\mathbf{D_{obs}}\rVert+$ (13) $\displaystyle{\alpha_{1}^{\lambda}}\mathbf{{\Theta}_{TV}}\mathbf{(m_{\lambda})}+{\alpha_{1}^{\mu}}\mathbf{{\Theta}_{TV}}\mathbf{(m_{\mu})}+{\alpha_{1}^{\rho}}\mathbf{{\Theta}_{{TV}}}\mathbf{(m_{\rho})}+$ $\displaystyle{\alpha_{2}^{\lambda}}\mathbf{\Upsilon_{TV}}\mathbf{(m_{\lambda})}+{\alpha_{2}^{\mu}}\mathbf{\Upsilon_{TV}}\mathbf{(m_{\mu})}+{\alpha_{2}^{\rho}}\mathbf{\Upsilon_{TV}}\mathbf{(m_{\rho})}$ The gradient for each parameter based on l1 norm: $\displaystyle\begin{pmatrix}\frac{{\partial}{\mathbf{\Phi}}_{l1}^{TV}}{{\partial}{\mathbf{m_{\lambda}}}}\\\ \frac{{\partial}{\mathbf{\Phi}}_{l1}^{TV}}{{\partial}{\mathbf{m_{\mu}}}}\\\ \frac{{\partial}{\mathbf{\Phi}}_{l1}^{TV}}{{\partial}{\mathbf{m_{\rho}}}}\\\ \end{pmatrix}=\begin{pmatrix}\mathbf{G_{{l1}_{\lambda}}}\\\ \mathbf{G_{{l1}_{\mu}}}\\\ \mathbf{G_{{l1}_{\rho}}}\\\ \end{pmatrix}+\begin{pmatrix}\mathbf{R_{\lambda}}\\\ \mathbf{R_{\mu}}\\\ \mathbf{R_{\rho}}\\\ \end{pmatrix}$ (14) , where $\mathbf{G_{{l1}_{\lambda}}}$,$\mathbf{G_{{l1}_{\mu}}}$,$\mathbf{G_{{l1}_{\rho}}}$ are the gradient for $\lambda$, $\mu$, ${\rho}$ using $l_{1}$ norm as misfit function. The gradient calculation in $l_{1}$ norm is using a differenta adjoint source (Pyun et al., (2009) Brossier et al., (2010)). The adjoint source for the adjoint fields for $l_{1}$ norm is , In the case of real arithmetic numbers, the term $\frac{{\Delta\mathbf{d}}}{\|\Delta\mathbf{d}\|}$ corresponds to the function $sign$. In this study, we did not meet conditions when $\Delta\mathbf{d}=0$. The detail gradient expression using the adjoint state method for parameters $\lambda$, $\mu$ and $\rho$ based on $l_{2}$ and $l_{1}$ norm can be expressed as: $\begin{array}[]{l}\mathbf{G_{{l2}_{{\lambda}}}}=-\sum_{\mathbf{x_{s}}}\sum_{\mathbf{x_{g}}}\int_{0}^{T}\\\ \left(\left(\partial_{x}\tilde{u}_{x}\left(\mathbf{r},\mathbf{r}_{s},t\right)+\partial_{z}\tilde{u}_{z}\left(\mathbf{r},\mathbf{r}_{s},t\right)\right)\left(\partial_{x}\tilde{u}_{x}^{_{l2}}\left(\mathbf{r},\mathbf{r}_{g},T-t\right)+\partial_{z}\tilde{u}_{z}^{_{l2}}\left(\mathbf{r},\mathbf{r}_{g},T-t\right)\right)\right)\end{array}$ (15) $\begin{array}[]{l}\mathbf{G_{{l2}_{\mu}}}=-\sum_{\mathbf{x_{s}}}\sum_{\mathbf{x_{g}}}\int_{0}^{T}\\\ \left(\left(\partial_{z}\tilde{u}_{x}\left(\mathbf{r},\mathbf{r}_{s},t\right)+\partial_{x}\tilde{u}_{z}\left(\mathbf{r},\mathbf{r}_{s},t\right)\right)\left(\partial_{z}\tilde{u}_{x}^{_{l2}}\left(\mathbf{r},\mathbf{r}_{g},T-t\right)+\partial_{x}\tilde{u}_{z}^{_{l2}}\left(\mathbf{r},\mathbf{r}_{g},T-t\right)\right)\right)\\\ -2\left(\left(\partial_{x}\tilde{u}_{x}\left(\mathbf{r},\mathbf{r}_{s},t\right)\partial_{x}\tilde{u}_{x}^{_{l2}}\left(\mathbf{r},\mathbf{r}_{g},T-t\right)+\partial_{z}\tilde{u}_{z}\left(\mathbf{r},\mathbf{r}_{s},t\right)\partial_{z}\tilde{u}_{z}^{_{l2}}\left(\mathbf{r},\mathbf{r}_{g},T-t\right)\right)\right)\end{array}$ (16) $\mathbf{G_{{l2}_{\rho}}}=\sum_{\mathbf{x_{s}}}\sum_{\mathbf{x_{g}}}\int_{0}^{T}\\\ \left(\left(\partial_{t}\tilde{u}_{x}\left(\mathbf{r},\mathbf{r}_{s},t\right)\partial_{t}\tilde{u}_{x}^{_{l2}}\left(\mathbf{r},\mathbf{r}_{g},T-t\right)+\partial_{t}\tilde{u}_{z}\left(\mathbf{r},\mathbf{r}_{s},t\right)\partial_{t}\tilde{u}_{z}^{_{l2}}\left(\mathbf{r},\mathbf{r}_{g},T-t\right)\right)\right)$ (17) $\begin{array}[]{l}\mathbf{G_{{l1}_{{\lambda}}}}=-\sum_{\mathbf{x_{s}}}\sum_{\mathbf{x_{g}}}\int_{0}^{T}\\\ \left(\left(\partial_{x}\tilde{u}_{x}\left(\mathbf{r},\mathbf{r}_{s},t\right)+\partial_{z}\tilde{u}_{z}\left(\mathbf{r},\mathbf{r}_{s},t\right)\right)\left(\partial_{x}\tilde{u}_{x}^{_{l1}}\left(\mathbf{r},\mathbf{r}_{g},T-t\right)+\partial_{z}\tilde{u}_{z}^{_{l1}}\left(\mathbf{r},\mathbf{r}_{g},T-t\right)\right)\right)\end{array}$ (18) $\begin{array}[]{l}\mathbf{G_{{l1}_{\mu}}}=-\sum_{\mathbf{x_{s}}}\sum_{\mathbf{x_{g}}}\int_{0}^{T}\\\ \left(\left(\partial_{z}\tilde{u}_{x}\left(\mathbf{r},\mathbf{r}_{s},t\right)+\partial_{x}\tilde{u}_{z}\left(\mathbf{r},\mathbf{r}_{s},t\right)\right)\left(\partial_{z}\tilde{u}_{x}^{_{l1}}\left(\mathbf{r},\mathbf{r}_{g},T-t\right)+\partial_{x}\tilde{u}_{z}^{_{l1}}\left(\mathbf{r},\mathbf{r}_{g},T-t\right)\right)\right)\\\ -2\left(\left(\partial_{x}\tilde{u}_{x}\left(\mathbf{r},\mathbf{r}_{s},t\right)\partial_{x}\tilde{u}_{x}^{_{l1}}\left(\mathbf{r},\mathbf{r}_{g},T-t\right)+\partial_{z}\tilde{u}_{z}\left(\mathbf{r},\mathbf{r}_{s},t\right)\partial_{z}\tilde{u}_{z}^{_{l1}}\left(\mathbf{r},\mathbf{r}_{g},T-t\right)\right)\right)\end{array}$ (19) $\mathbf{G_{{l1}_{\rho}}}=\\\ \sum_{\mathbf{x_{s}}}\sum_{\mathbf{x_{g}}}\int_{0}^{T}\left(\left(\partial_{t}\tilde{u}_{x}\left(\mathbf{r},\mathbf{r}_{s},t\right)\partial_{t}\tilde{u}_{x}^{_{l1}}\left(\mathbf{r},\mathbf{r}_{g},T-t\right)+\partial_{t}\tilde{u}_{z}\left(\mathbf{r},\mathbf{r}_{s},t\right)\partial_{t}\tilde{u}_{z}^{_{l1}}\left(\mathbf{r},\mathbf{r}_{g},T-t\right)\right)\right)$ (20) $\mathbf{G_{{l2}_{\lambda}}}$, $\mathbf{G_{{l2}_{\mu}}}$, $\mathbf{G_{{l2}_{\rho}}}$, are gradients for $\lambda$, $\mu$ and $\rho$ using $l_{2}$ norm as misfit respectively. $\mathbf{G_{{l1}_{\lambda}}}$, $\mathbf{G_{{l1}_{\mu}}}$, $\mathbf{G_{{l1}_{\rho}}}$, are gradients for $\lambda$, $\mu$ and $\rho$ using $l_{1}$ norm as misfit respectively. $\tilde{u_{x}}^{_{l1}}$ and $\tilde{u_{z}}^{_{l1}}$ are the adjoint wavefields generated by the l1 norm adjoint source, $\tilde{u_{x}}^{_{l2}}$ and $\tilde{u_{z}}^{_{l2}}$ are the adjoint wavefields generated by the l2 norm adjoint source. $T$ is the total receiving time for the shot records. $\mathbf{r_{s}}$, $\mathbf{r_{g}}$ represent the source and receivers locations respectively. $\mathbf{r}$ represent the model perturbation locations for $\lambda$, $\mu$ and $\rho$ model. Figure 5 shows the gradient calculated using the adjoint state method and the Automatic Difference method. Figure 5 (a), (b), (c) are the normalized ${\lambda}$, ${\mu}$ and $\rho$ gradients calculated by using the adjont state method. Figure 5 (d), (e), (f) are the normalized ${\lambda}$, ${\mu}$ and $\rho$ gradients calculated by using the Automatic Differential method. The gradients calculated by using the Automatic Difference method, contains more information about the model, for instance the lower part of the model, indicating that they can better reconstruct the mode. Now we rewrite the misfit function as: $\displaystyle{\mathbf{\Phi}}^{TV}=\mathbf{J_{D}}+\mathbf{J_{r1}}+\mathbf{J_{r2}}$ (21) ,where $\mathbf{J_{D}}$ represents the any kind of norm misfit between observed data and synthetic data. $\mathbf{J_{r1}}={\alpha_{1}^{\lambda}}\mathbf{{\Theta}_{TV}}\mathbf{(m_{\lambda})}+{\alpha_{1}^{\mu}}\mathbf{{\Theta}_{TV}}\mathbf{(m_{\mu})}+{\alpha_{1}^{\rho}}\mathbf{{\Theta}_{{TV}}}\mathbf{(m_{\rho})}$. $\mathbf{J_{r2}}={\alpha_{2}^{\lambda}}\mathbf{\Upsilon_{TV}}\mathbf{(m_{\lambda})}+{\alpha_{2}^{\mu}}\mathbf{\Upsilon_{TV}}\mathbf{(m_{\mu})}+{\alpha_{2}^{\rho}}\mathbf{\Upsilon_{TV}}\mathbf{(m_{\rho})}$. The value $\alpha_{1}^{\lambda}$, ${\alpha_{1}^{\mu}}$, ${\alpha_{1}^{\rho}}$, $\alpha_{2}^{\lambda}$, ${\alpha_{2}^{\mu}}$, ${\alpha_{2}^{\rho}}$. are chosen according to the following formula (Guitton et al., (2012)). $T=\frac{\mathbf{J_{D}}}{\mathbf{J_{r1}}+\mathbf{J_{r2}}}$ (22) We should control the values for $\alpha_{1}^{\lambda}$, ${\alpha_{1}^{\mu}}$, ${\alpha_{1}^{\rho}}$, $\alpha_{2}^{\lambda}$, ${\alpha_{2}^{\mu}}$, ${\alpha_{2}^{\rho}}$ and keep value T between 1 and 10. T should be relatively large when, noise occurs in the data (Xiang and Zhang, (2016)). Figure 5: (a) ${\lambda}$ gradient given by the adjoint state method. (b) $\mu$ gradient based by the adjoint state method. (c) $\rho$ gradient based by the adjoint state method. (d) ${\lambda}$ gradient given by the AD method. (e) $\mu$ gradient based by the AD method. (f) $\rho$ gradient based by the AD method. ### 4.1 Parameterization testing By modifying the RNN cells, and changing the trainable parameters, we can examine the influence of parameterization on waveform inversion within the deep learning formulation. Three sets of parameter classes are considered: the velocity parameterization (D-V model), involving P-wave velocity, S-wave velocity, and density; the modulus parameterization (D-M model), involving the Lamé parameters ${\lambda}$ and ${\mu}$, and density; and, the stiffness matrix model (D-S model), involving ${C_{11}}$, $C_{44}$ and density ${\rho}$. In these tests, the size of each model is 40${\times}90$. 7 source points are evenly distributed across the surface of the model; the source is a Ricker wavelet with a dominant frequency of $30Hz$. The grid length of the model is $dx=dz=4$m. In Figure 6 (a)-(b) are true and initial $V_{P}$ model, (c)-(d) are the true and initial $V_{S}$ model, (e)-(f) are the true and initial $\rho$ model we use in this test. In Figures 7 are the inversion results using the D-V parameterization. Figure 7 (a)-(d) are the inversion results for $V_{P}$ generated by $l_{2}$, $l_{2}^{TV}$, $l_{1}$ and $l_{1}^{TV}$ norm. Figure 7 (e)-(h) are the inversion results for $V_{S}$ generated by $l_{2}$, $l_{2}^{TV}$, $l_{1}$ and $l_{1}^{TV}$ norm. Figure 7 (i)-(l) are the inversion results for $\rho$ generated by $l_{2}$, $l_{2}^{TV}$, $l_{1}$ and $l_{1}^{TV}$ norm. In Figure 7 (a), (e) and (i) we can see the cross talk between different parameters as the back arrows indicate, In Figure 7 (b), (c), (d) we can see that by changing the misfits and adding high order TV regulations, the cross talk between $V_{P}$ and $\rho$ has been reduced. In Figure 7 (h) and (l) we can see that by using $l_{1}^{TV}$ norm, the cross talk between density and $V_{S}$ has been mitigated, while in (j) and (k) we still see the cross talk between $V_{S}$ and density. From Figures 7 we can see that $l_{1}$ norm with high order TV regulation can help to mitigate the cross talk problem. Figure 6: (a) true Vp model, (b) initial Vs model, (c) true Vs model, (d) initial vs model, (e) true ${\rho}$ mode, (f) initial ${\rho}$ model. Figure 7: D-V parameterization inversion results. (a)-(d), $V_{P}$ $l_{2}$ norm, $l_{2}^{TV}$ norm, $l_{1}$ norm, $l_{1}^{TV}$ norm inversion results respectively, (e)-(h), $V_{S}$ $l_{2}$ norm, $l_{2}^{TV}$ norm, $l1$ norm $l_{1}^{TV}$ norm, inversion results respectively, (i)-(l), $\rho$ $l_{2}$ norm $l_{2}^{TV}$ norm, $l_{1}$ norm, $l_{1}^{TV}$ norm inversion results respectively. Next the modulus parameterization is examined, in which we seek to recover ${\lambda}$, ${\mu}$ and ${\rho}$ models. This occurs through a straightforward modification of the RNN cell, and again a change in the trainable parameters from velocities to moduli. Figure 8 (a)-(d) are the inversion results for $\lambda$ generated by $l_{2}$, $l_{2}^{TV}$, $l_{1}$ and $l_{1}^{TV}$ norm. Figure 8 (e)-(h) are the inversion results for $\mu$ generated by $l_{2}$, $l_{2}^{TV}$, $l_{1}$ and $l_{1}^{TV}$ norm. Figure 8 (i)-(l) are the inversion results for $\rho$ generated by $l_{2}$, $l_{2}^{TV}$, $l_{1}$ and $l_{1}^{TV}$ norm. In Figure 8 (b) and (d) we can see that by using the high order TV regulations in $l_{2}$ and $l_{1}$ norm the cross talk between density and $\lambda$ has been mitigated. Figure (j) shows that In this parameterization by using the high order TV regulation on $l_{2}$ norm can provide better inversion results for density as well. Figure 8: D-M parameterization inversion results. (a)-(d), ${\lambda}$ $l_{2}$, norm, $l_{2}^{TV}$ norm, $l_{1}$, norm $l_{1}^{TV}$ norm inversion results respectively, (e)-(h), $\mu$ $l_{2}$ norm, $l_{2}^{TV}$ norm, $l1$ norm, $l_{1}^{TV}$ norm inversion results respectively, (i)-(l), $\rho$ $l_{2}$ norm, $l_{2}^{TV}$ norm, $l_{1}$ norm, $l_{1}^{TV}$ norm inversion results respectively. Inversion results for models in the S-D parameterization are plotted in Figure 9, generated using the change of variables ${C_{11}}=V_{P}^{2}{\rho}$ and ${C_{44}}=V_{S}^{2}{\rho}$. Figure 9 (a)-(d) are the inversion results for $\lambda$ generated by $l_{2}$, $l_{2}^{TV}$, $l_{1}$ and $l_{1}^{TV}$ norm. Figure 9 (e)-(h) are the inversion results for $\mu$ generated by $l_{2}$, $l_{2}^{TV}$, $l_{1}$ and $l_{1}^{TV}$ norm. Figure 9 (i)-(l) are the inversion results for $\rho$ generated by $l_{2}$, $l_{2}^{TV}$, $l_{1}$ and $l_{1}^{TV}$ norm. In Figure 9 we can still see the cross talk between $c44$ and $c11$ as the black arrows pointing out. However this cross talk has been mitigated by in figure (d), by using the $l_{1}^{TV}$ norm misfit. The inversion results above shows that the RNN based high order TV regulation FWi based on the $l_{2}$ and $l_{1}$ norm has the ability to mitigate cross talk problem with only gradient based methods. The RNN based $l_{2}^{TV}$ RNN FWI in D-M parameterization and $l_{1}^{TV}$ RNN FWI in D-S parameterization provide better inversion results than other inversion tests. Figure 9: D-S parameterization inversion results. (a)-(d), $c11$ $l_{2}$ norm, $l_{2}^{TV}$ norm, $l_{1}$ norm, $l_{1}^{TV}$ norm inversion results respectively, (e)-(h), $c44$ $l_{2}$ norm, $l_{2}^{TV}$ norm, $l1$ norm $l_{1}^{TV}$ norm, inversion results respectively, (i)-(l), $\rho$ $l_{2}$ norm $l_{2}^{TV}$ norm, $l_{1}$ norm, $l_{1}^{TV}$ norm inversion results respectively. Figure 10: (a) true $V_{P}$ model, (b) initial $V_{P}$ model, (c) true $V_{S}$ model (d) initial Vs model, (e) ${\rho}$ true (f) initial $\rho$ Figure 11: (a) true $V_{P}$ model, (b) initial $V_{P}$ model, (c) true $V_{S}$ model (d) initial $V_{S}$ model, (e) ${\rho}$ true (f) initial $\rho$ Figure 12: M-D parameterization inversion results. (a)-(c), $\lambda$ $l_{2}$ norm, $l_{2}^{TV}$ norm, $l_{1}^{TV}$ norm inversion results respectively, (c)-(f), $\mu$ $l_{2}$ norm, $l_{2}^{TV}$ norm, $l_{1}^{TV}$ norm, inversion results respectively, (g)-(i), $\rho$ $l_{2}$ norm $l_{2}^{TV}$ norm, $l_{1}^{TV}$ norm inversion results respectively Figure 13: Profiles through the recovered elastic models. (a) Vertical $\lambda$ profiles; (b) vertical $\mu$ profiles; (c) vertical ${\rho}$ profiles Next, we will verify the proposed methods on the over-thrust model. Figure 10 (a), (c), (e) demonstrate the true models for $V_{P}$, $V_{S}$, and density $\rho$ model, and (b), (d) and (h) are the initial models for $V_{P}$, $V_{S}$, and density $\rho$ respectively. The size of the model is 121 $\times$ 240\. The grid length of the model is 10m. 12 shots are evenly distributed on the surface of the model and every grid point has a receiver. The source of the wavelet is Ricker’s wavelet with main frequency 20Hz. Figure Figure 11 shows the inversion results by using the conventional FWI. Figure Figure 11 (a) is the inversion for $V_{P}$, Figure Figure 11 (b) is the inversion for $V_{S}$, Figure Figure 11 (c) is the inversion for $\rho$. From figure 11 we can see that the overall inversion resolution by using the conventional FWI is poor. Figure 12 shows the inversion results by using D-M parameterization. Figure 12 (a)-(c) are ${\lambda}$ $l_{2}$ norm, $l_{2}^{TV}$ norm, $l_{1}^{TV}$ norm inversion results respectively, (d)-(f) are $\mu$ $l_{2}$ norm, $l_{2}^{TV}$ norm, $l_{1}^{TV}$ norm, inversion results respectively, (g)-(i) are $\rho$ $l_{2}$ norm $l_{2}^{TV}$ norm, $l_{1}^{TV}$ norm inversion results respectively. In this parameterization we get unstable inversion results for parameter $\lambda$. However, in D-M parameterization Figure 13 (f), the small half arc structure at around 1000m of the model has been recovered in , as the black arrows indicate. Figure 13 shows the profiles through the recovered elastic modules at 1000m of the models based on D-M parameterization. In Figure 13, the black lines are the true values, the yellow lines are the initial values, red lines are the inversion results for $l_{2}$ norm, green lines are the inversion results for $l_{2}^{TV}$ norm and blue lines are the inversion results for $l_{1}^{TV}$ norm. Compared with the true lines we can also see that , Figure 13 (b) and Figure 13 (c), $l_{1}^{TV}$ norm inversion results are more close to the true values. Figure 14: D-V parameterization inversion results. (a)-(d), $V_{P}$ $l_{2}$ norm, $l_{2}^{TV}$ norm, $l_{1}^{TV}$ norm inversion results respectively, (e)-(h), $V_{S}$ $l_{2}$ norm, $l_{2}^{TV}$ norm, $l_{1}^{TV}$ norm, inversion results respectively, (i)-(l), $\rho$ $l_{2}$ norm $l_{2}^{TV}$ norm, $l_{1}^{TV}$ norm inversion results respectively. Figure 15: Profiles through the recovered elastic models. (a) Vertical $V_{P}$ profiles; (b) vertical $V_{S}$ profiles; (c) vertical ${\rho}$ profiles. Figure 16: D-V parameterization inversion results . (a) Vp model misfit using $l_{2}$,$l_{2}^{TV}$ and $l_{1}^{TV}$ norm.(b) Vs model misfit using $l_{2}$,$l_{2}^{TV}$ and $l_{1}^{TV}$ norm.(c) $\rho$ model misfit using $l_{2}$,$l_{2}^{TV}$ and $l_{1}^{TV}$ norm Figure 14 shows the inversion results by using D-V parameterization. In this parameterization all the three parameters are stable. Figure 14 (a)-(c) are, $V_{P}$ $l_{2}$ norm, $l_{2}^{TV}$ norm, $l_{1}^{TV}$ norm inversion results respectively, (d)-(f) are $V_{S}$ $l_{2}$ norm, $l_{2}^{TV}$ norm, $l_{1}^{TV}$ norm, inversion results respectively, (g)-(i) are $\rho$ $l_{2}$ norm $l_{2}^{TV}$ norm, $l_{1}^{TV}$ norm inversion results respectively. Compared with the true models ,we can see that the inversion results generated by using the $l_{2}$ norm are less robust compared with other misfits. $l_{1}$ norm with high order TV regulation can provide more accurate inversion results. This can also be seen from Figure 14, which is the profiles through the recovered elastic modules at 1000m of the models. In Figure 14, the black lines are the true velocities and density, the yellow lines are the initial values, red lines are the inversion results for $l_{2}$ norm, green lines are the inversion results for $l_{2}^{TV}$ norm and blue lines are the inversion results for $l_{1}^{TV}$ norm. Figure 15 (a) shows the results for $V_{P}$. Figure 15 (b) shows the results for $V_{S}$. Figure 15 (c) shows the results for $\rho$. In all the three figures we can see that blue lines are closer to the true values compared with other lines, which means that the $l_{1}^{TV}$ norm can provide us with more accurate inversion results. Figure 16 shows the D-V parameterization inversion model misfits in each iteration. The red lines are the inversion using $l_{2}$ norm. The green lines are the inversion using $l_{2}^{TV}$ norm. The red lines are the inversion using $l_{1}^{TV}$ norm. Figure 16 shows that we can get higher accuracy inversion results by using the $l_{1}^{TV}$ norm with fewer iterations. Figure 17 shows the inversion results by using D-S parameterization. Figure 17 (a)-(c), are ${c11}$ $l_{2}$ norm, $l_{2}^{TV}$ norm, $l_{1}^{TV}$ norm inversion results respectively, (d)-(f) are $c44$ $l_{2}$ norm, $l_{2}^{TV}$ norm, $l_{1}^{TV}$ norm, inversion results respectively, (g)-(i) are $\rho$ $l_{2}$ norm $l_{2}^{TV}$ norm, $l_{1}^{TV}$ norm inversion results respectively. In this parameterization the small half arc structure at around 1000m of the model is clearly resolved in this parameterization by using the $l_{1}^{TV}$ norm. Compared with other misfits , again the $l_{1}^{TV}$ norm provide us the best resolution for the model. Figure 13 shows the profiles through the recovered elastic modules at 1000m of the models based on D-M parameterization.. Red lines, green lines and blue lines are $l_{2}$, $l_{2}^{TV}$, and $l_{1}^{TV}$ norm inversion results respectively. We can also see that in D-S parameteriation. $l_{1}^{TV}$ provide us with inversion results that is more close the to true values. Figure 18 shows how model misfits in D-M parameterization changes in each iteration. The blue line, the $l_{1}^{TV}$ norm inversion, has the fastest model misfit decline rate. We can conclude in this parameterization $l_{1}^{TV}$ can generate more accuracy inversion results with fewer iterations. Figure 17: S-D parameterization inversion results. (a)-(c), $c11$ $l_{2}$ norm, $l_{2}^{TV}$ norm, $l_{1}^{TV}$ norm inversion results respectively, (c)-(f), $c44$ $l_{2}$ norm, $l_{2}^{TV}$ norm, $l_{1}^{TV}$ norm, inversion results respectively, (g)-(i), $\rho$ $l_{2}$ norm $l_{2}^{TV}$ norm, $l_{1}^{TV}$ norm inversion results respectively. Figure 18: Profiles through the recovered elastic models based on . (a) Vertical $c11$ profiles; (b) vertical $c44$ profiles; (c) vertical ${\rho}$ profiles Figure 19: D-S parameterization inversion results . (a) $c11$ model misfit using $l_{2}$,$l_{2}^{TV}$ and $l_{1}^{TV}$ norm.(b) $c44$ model misfit using $l_{2}$,$l_{2}^{TV}$ and $l_{1}^{TV}$ norm.(c) $\rho$ model misfit using $l_{2}$,$l_{2}^{TV}$ and $l_{1}^{TV}$ norm ## 5 Random noise testing In this section, we will test the sensitivity of this deep learning method to data contaminated with random noise drawn from a Gaussian distribution. In Figure 20, we plot a trace from an example shot profile with different ratios of noise. Figure 20: Noise free and noise shotrecords. (a) Noise free record and records with Gaussian noise $std$ = $0.3$. (b) Noise free record and records with Gaussian noise $std$ = $0.5$. (c) Noise free record and records with Gaussian noise $std$ = $1$. In (a), the red signal is the record without noise, and the blue line is the record with Gaussian random noise. The mean value of the noise is zero, and the standard deviation is 0.3 ($std$ = $0.3$); in (b) the noise-free data and data with noise at $std=0.5$ are plotted; in (c) the noise-free data and the data with noise at $std=1$ are plotted. Figure 21: Random noise testing inversion results. (a),(e),(i), noise free inversion results for $V_{P}$, $V_{S}$, and density (b),(f),(j) Gaussian noise $std=0.3$ inversion results for $V_{P}$, $V_{S}$, and density, (c),(g),(k) Gaussian noise $std=0.5$ inversion results for $V_{P}$, $V_{S}$, and density, (d),(h),(l) Gaussian noise $std=1.0$ inversion results for $V_{P}$, $V_{S}$, and density. In Figure 21a, e and i the inversion results from noise-free inversion based on the RNN are plotted. In Figure 21b, f and j, the inversion results with noise at $std=0.3$ for $V_{P}$, $V_{S}$, and ${\rho}$ are plotted, respectively; in Figure 21c, g, and k, the inversion results with noise at $std=0.5$ for $V_{P}$, $V_{S}$, and ${\rho}$ are plotted; in Figure 21d, h, and l are likewise for $std=1.0$. We conclude that a moderate amount of random error in the data used for the RNN training leads to acceptable results, though some blurring is introduced and detail in the structure is lost. The $V_{S}$ recovery appears to be much more sensitive to noise than are those of $V_{P}$ or ${\rho}$. ## 6 Conclusions Elastic multi-parameter full waveform inversion can be formulated as a strongly constrained, theory-based deep-learning network. Specifically, a recurrent neural network, set up with rules enforcing elastic wave propagation, with the wavefield projected onto a measurement surface acting as the labeled data to be compared with observed seismic data, recapitulates elastic FWI but with both (1) the opportunity for data-driven learning to be incorporated, and (2) a design supported by powerful cloud computing architectures. Each cell of the recurrent neural network is designed according to the isotropic-elastic wave equation. The partial derivatives of the data residual with respect to the trainable parameters which act to represent the elastic media are calculated by using the intelligent automatic differential method. With the automatic differential method, gradients can be automatically calculated via the chain rule, guided by backpropagation along the paths within the computational graph. The automatic differential method produces a high level of computational efficiency and scalability for the calculation of gradients for different parameters in elastic media. The formulation is suitable for exploring numerical features of different misfits and different parameterizations, with an aim of improving the resolution of the recovered elastic models, and mitigate cross-talk. We compared RNN waveform inversions based on $l_{2}$, $l_{1}$ with high order total variations. We used this RNN synthetic environment to compare density- velocity (P-wave velocity, S-wave velocity, and density), modulus-density (Lamé parameters and density) and S-D ($C_{11}$, $C_{44}$ and density) parameterizations, and their exposure to cross-talk for the varying misfit functions. Our results suggest generally that this approach to full waveform inversion is consistent and stable. 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# Differentially Private SGD with Non-Smooth Losses††thanks: Corresponding author: Yiming Ying. Email<EMAIL_ADDRESS> Puyu Wang†, Yunwen Lei‡, Yiming Ying∗ and Hai Zhang† † School of Mathematics, Northwest University, Xi’an, 710127, China ‡ School of Computer Science, University of Birmingham, Birmingham B15 2TT, UK ∗Department of Mathematics and Statistics, State University of New York at Albany, Albany, NY, 12222, USA ###### Abstract In this paper, we are concerned with differentially private stochastic gradient descent (SGD) algorithms in the setting of stochastic convex optimization (SCO). Most of the existing work requires the loss to be Lipschitz continuous and strongly smooth, and the model parameter to be uniformly bounded. However, these assumptions are restrictive as many popular losses violate these conditions including the hinge loss for SVM, the absolute loss in robust regression, and even the least square loss in an unbounded domain. We significantly relax these restrictive assumptions and establish privacy and generalization (utility) guarantees for private SGD algorithms using output and gradient perturbations associated with non-smooth convex losses. Specifically, the loss function is relaxed to have an $\alpha$-Hölder continuous gradient (referred to as $\alpha$-Hölder smoothness) which instantiates the Lipschitz continuity ($\alpha=0$) and the strong smoothness ($\alpha=1$). We prove that noisy SGD with $\alpha$-Hölder smooth losses using gradient perturbation can guarantee $(\epsilon,\delta)$-differential privacy (DP) and attain optimal excess population risk $\mathcal{O}\Big{(}\frac{\sqrt{d\log(1/\delta)}}{n\epsilon}+\frac{1}{\sqrt{n}}\Big{)}$, up to logarithmic terms, with the gradient complexity $\mathcal{O}(n^{2-\alpha\over 1+\alpha}+n).$ This shows an important trade-off between $\alpha$-Hölder smoothness of the loss and the computational complexity for private SGD with statistically optimal performance. In particular, our results indicate that $\alpha$-Hölder smoothness with $\alpha\geq{1/2}$ is sufficient to guarantee $(\epsilon,\delta)$-DP of noisy SGD algorithms while achieving optimal excess risk with the linear gradient complexity $\mathcal{O}(n).$ Keywords: Stochastic Gradient Descent , Algorithmic Stability, Differential Privacy, Generalization ## 1 Introduction Stochastic gradient descent (SGD) algorithms are widely employed to train a wide range of machine learning (ML) models such as SVM, logistic regression, and deep neural networks. It is an iterative algorithm which replaces the true gradient on the entire training data by a randomized gradient estimated from a random subset (mini-batch) of the available data. As opposed to gradient descent algorithms, this reduces the computational burden at each iteration trading for a lower convergence rate [5]. There is a large amount of work considering the optimization error (convergence analysis) of SGD and its variants in the linear case [2, 19, 20, 29, 30] as well as the general setting of reproducing kernel Hilbert spaces [10, 23, 28, 36, 37, 32]. At the same time, data collected often contain sensitive information such as individual records from schools and hospitals, financial records for fraud detection, online behavior from social media and genomic data from cancer diagnosis. Modern ML algorithms can explore the fine-grained information about data in order to make a perfect prediction which, however, can lead to privacy leakage [8, 31]. To a large extent, SGD algorithms have become the workhorse behind the remarkable progress of ML and AI. Therefore, it is of pivotal importance for developing privacy-preserving SGD algorithms to protect the privacy of the data. Differential privacy (DP) [12, 14] has emerged as a well- accepted mathematical definition of privacy which ensures that an attacker gets roughly the same information from the dataset regardless of whether an individual is present or not. Its related technologies have been adopted by Google [15], Apple [25], Microsoft [11] and the US Census Bureau [1]. In this paper, we are concerned with differentially private SGD algorithms in the setting of stochastic convex optimization (SCO). Specifically, let the input space $\mathcal{X}$ be a domain in some Euclidean space, the output space $\mathcal{Y}\subseteq\mathbb{R}$, and $\mathcal{Z}=\mathcal{X}\times\mathcal{Y}.$ Denote the loss function by $\ell:\mathbb{R}^{d}\times\mathcal{Z}\mapsto[0,\infty)$ and assume, for any $z\in\mathcal{Z}$, that $\ell(\cdot,z)$ is convex with respect to (w.r.t.) the first argument. SCO aims to minimize the expected (population) risk, i.e. $\mathcal{R}(\mathbf{w}):=\mathbb{E}_{z}[\ell(\mathbf{w},z)]$, where the model parameter $\mathbf{w}$ belongs to a (not necessarily bounded) domain $\mathcal{W}\subseteq\mathbb{R}^{d}$, and the expectation is taken w.r.t. $z$ according to a population distribution $\mathcal{D}.$ While the population distribution is usually unknown, we have access to a finite set of $n$ training data points denoted by $S=\\{z_{i}\in\mathcal{Z}:i=1,2,\ldots,n\\}.$ It is assumed to be independently and identically distributed (i.i.d.) according to the distribution $\mathcal{D}$ on $\mathcal{Z}.$ In this context, one often considers SGD algorithms to solve the Empirical Risk Minimization (ERM) problem defined by $\min_{\mathbf{w}\in\mathcal{W}}\Bigl{\\{}\mathcal{R}_{S}(\mathbf{w}):=\frac{1}{n}\sum_{i=1}^{n}\ell(\mathbf{w},z_{i})\Bigr{\\}}.$ For a randomized algorithm (e.g., SGD) $\mathcal{A}$ to solve the above ERM problem, let $\mathcal{A}(S)$ be the output of algorithm $\mathcal{A}$ based on the dataset $S$. Then, its statistical generalization performance is measured by the excess (population) risk, i.e., the discrepancy between the expected risk $\mathcal{R}(\mathcal{A}(S))$ and the least possible one in $\mathcal{W}$, which is defined by $\epsilon_{\text{risk}}(\mathcal{A}(S))=\mathcal{R}(\mathcal{A}(S))-\min_{\mathbf{w}\in\mathcal{W}}\mathcal{R}(\mathbf{w}).$ Along this line, there are a considerable amount of work [35, 4, 16] on analyzing the excess risk of private SGD algorithms in the setting of SCO. However, most of such approaches often require two assumptions: 1) the loss $\ell$ is $L$-Lipschitz and $\beta$-smooth; 2) the domain $\mathcal{W}$ is uniformly bounded. These assumptions are very restrictive as many popular losses violate these conditions including the hinge loss $(1-y\mathbf{w}^{T}x)^{q}_{+}$ for $q$-norm soft margin SVM and the $q$-norm loss $\|y-\mathbf{w}^{T}x\|^{q}$ in regression with $1\leq q\leq 2.$ More specifically, the work [35] assumed the loss to be Lipschitz continuous and strongly smooth and showed that the private SGD algorithm with output perturbation can achieve $(\epsilon,\delta)$-DP and an excess risk rate $\mathcal{O}(\frac{(d\log(1/\delta))^{1/4}}{\sqrt{n\epsilon}})$ when the gradient complexity (i.e. the number of computing gradients) $T=n$. The study [4] proved, under the same assumptions, that the private SGD algorithm with gradient perturbation can achieve an optimal excess risk rate $\mathcal{O}\Big{(}\frac{\sqrt{d\log(1/\delta)}}{n\epsilon}+\frac{1}{\sqrt{n}}\Big{)}$ while guaranteeing its $(\epsilon,\delta)$-DP. To deal with the non- smoothness, it used the Moreau envelope technique to smooth the loss function and got the optimal rate. However, the algorithm is computationally inefficient with a gradient complexity $\mathcal{O}\Big{(}n^{4.5}\sqrt{\epsilon}+\frac{n^{6.5}\epsilon^{4.5}}{(d\log(1/\delta))^{2}}\Big{)}.$ The work [16] improved the gradient complexity of the algorithm to $\mathcal{O}(n^{2}\log(\frac{1}{\delta}))$ by localizing the approximate minimizer of the population loss on each phase. Recently, [3] showed that a simple variant of noisy projected SGD yields the optimal rate with gradient complexity $\mathcal{O}(n^{2})$. However, it only focused on the Lipschitz continuous losses and assumed that the parameter domain $\mathcal{W}$ is bounded. Our main contribution is to significantly relax these restrictive assumptions and to prove both privacy and generalization (utility) guarantees for private SGD algorithms with non-smooth convex losses in both bounded and unbounded domains. Specifically, the loss function $\ell(\mathbf{w},z)$ is relaxed to have an $\alpha$-Hölder continuous gradient w.r.t. the first argument, i.e., there exists $L>0$ such that, for any $\mathbf{w},\mathbf{w}^{\prime}\in\mathcal{W}$ and any $z\in\mathcal{Z}$, $\\|\partial\ell(\mathbf{w},z)-\partial\ell(\mathbf{w}^{\prime},z)\\|_{2}\leq L\\|\mathbf{w}-\mathbf{w}^{\prime}\\|_{2}^{\alpha},$ where $\\|\cdot\\|_{2}$ denotes the Euclidean norm, $\partial\ell(\mathbf{w},z)$ denotes a subgradient of $\ell$ w.r.t. the first argument. For the sake of notional simplicity, we refer to this condition as $\alpha$-Hölder smoothness with parameter $L$. The smoothness parameter $\alpha\in[0,1]$ characterizes the smoothness of the loss function $\ell(\cdot,z)$. The case of $\alpha=0$ corresponds to the Lipschitz continuity of the loss $\ell$ while $\alpha=1$ means its strong smoothness. This definition instantiates many non-smooth loss functions mentioned above. For instance, the hinge loss for $q$-norm soft- margin SVM and $q$-norm loss for regression mentioned above with $q\in[1,2]$ are $(q-1)$-Hölder smooth. In particular, we prove that noisy SGD with $\alpha$-Hölder smooth losses using gradient perturbation can guarantee $(\epsilon,\delta)$-DP and attain the optimal excess population risk $\mathcal{O}\Big{(}\frac{\sqrt{d\log(1/\delta)}}{n\epsilon}+\frac{1}{\sqrt{n}}\Big{)}$, up to logarithmic terms, with gradient complexity $\mathcal{O}(n^{2-\alpha\over 1+\alpha}+n).$ This shows an important trade-off between $\alpha$-Hölder smoothness of the loss and the computational complexity for private SGD in order to achieve statistically optimal performance. In particular, our results indicate that $\alpha$-Hölder smoothness with $\alpha\geq{1/2}$ is sufficient to guarantee $(\epsilon,\delta)$-DP of noisy SGD algorithms while achieving the optimal excess risk with linear gradient complexity $\mathcal{O}(n).$ Table 1 summarizes the upper bound of the excess population risk, gradient complexity of the aforementioned algorithms in comparison to our methods. Our key idea to handle general Hölder smooth losses is to establish the approximate non-expansiveness of the gradient mapping, and the refined boundedness of the iterates of SGD algorithms when domain $\mathcal{W}$ is unbounded. This allows us to show the uniform argument stability [24] of the iterates of SGD algorithms with high probability w.r.t. the internal randomness of the algorithm (not w.r.t. the data $S$), and consequently estimate the generalization error of differentially private SGD with non- smooth losses. Reference \| Loss \| Method \| Utility bounds \| Gradient Complexity \| Domain ---\|---\|---\|---\|---\|--- [35] \| Lipschitz \| Output \| $\mathcal{O}\Big{(}\frac{(d\log(\frac{1}{\delta}))^{\frac{1}{4}}}{\sqrt{n\epsilon}}\Big{)}$ \| $\mathcal{O}\big{(}n\big{)}$ \| bounded & smooth [4] \| Lipschitz \| Gradient \| $\mathcal{O}\Big{(}\frac{\sqrt{d\log(\frac{1}{\delta})}}{n\epsilon}+\frac{1}{\sqrt{n}}\Big{)}$ \| $\mathcal{O}\Big{(}n^{1.5}\sqrt{\epsilon}+\frac{(n\epsilon)^{2.5}}{d\log(\frac{1}{\delta})}\Big{)}$ \| bounded & smooth Lipschitz \| Gradient \| $\mathcal{O}\Big{(}\frac{\sqrt{d\log(\frac{1}{\delta})}}{n\epsilon}+\frac{1}{\sqrt{n}}\Big{)}$ \| $\mathcal{O}\Big{(}n^{4.5}\sqrt{\epsilon}+\frac{n^{6.5}\epsilon^{4.5}}{(d\log(\frac{1}{\delta}))^{2}}\Big{)}$ \| bounded [16] \| Lipschitz \| Phased Output \| $\mathcal{O}\Big{(}\frac{\sqrt{d\log(\frac{1}{\delta})}}{n\epsilon}+\frac{1}{\sqrt{n}}\Big{)}$ \| $\mathcal{O}\big{(}n\big{)}$ \| bounded & smooth Lipschitz \| Phased ERM \| $\mathcal{O}\Big{(}\frac{\sqrt{d\log(\frac{1}{\delta})}}{n\epsilon}+\frac{1}{\sqrt{n}}\Big{)}$ \| $\mathcal{O}\big{(}n^{2}\log(\frac{1}{\delta})\big{)}$ \| bounded [3] \| Lipschitz \| Gradient \| $\mathcal{O}\Big{(}\frac{\sqrt{d\log(\frac{1}{\delta})}}{n\epsilon}+\frac{1}{\sqrt{n}}\Big{)}$ \| $\mathcal{O}\big{(}n^{2}\big{)}$ \| bounded Ours \| $\alpha$-Hölder \| Output \| $\mathcal{O}\Big{(}\frac{(d\log(\frac{1}{\delta}))^{\frac{1}{4}}\sqrt{\log(\frac{n}{\delta})}}{\sqrt{n\epsilon}}\Big{)}$ \| $\mathcal{O}\big{(}n^{\frac{2-\alpha}{1+\alpha}}+n\big{)}$ \| bounded smooth $\alpha$-Hölder \| Output \| $\mathcal{O}\Big{(}\frac{\sqrt{d\log(\frac{1}{\delta})}\log(\frac{n}{\delta})}{n^{\frac{2}{3+\alpha}}\epsilon}+\frac{\log(\frac{n}{\delta})}{n^{\frac{1}{3+\alpha}}}\Big{)}$ \| $\mathcal{O}\big{(}n^{\frac{-\alpha^{2}-3\alpha+6}{(1+\alpha)(3+\alpha)}}+n\big{)}$ \| unbounded smooth $\alpha$-Hölder \| Gradient \| $\mathcal{O}\Big{(}\frac{\sqrt{d\log(\frac{1}{\delta})}}{n\epsilon}+\frac{1}{\sqrt{n}}\Big{)}$ \| $\mathcal{O}\big{(}n^{\frac{2-\alpha}{1+\alpha}}+n\big{)}$ \| bounded smooth Table 1: Comparison of different $(\epsilon,\delta)$-DP algorithms. We report the method, utility (generalization) bound, gradient complexity and parameter domain for three types of convex losses, i.e. Lipschitz, Lipschitz and smooth, and $\alpha$-Hölder smooth. Here Output, Gradient, Phased Output and Phased ERM denote output perturbation which adds Gaussian noise to the output of non- private SGD, gradient perturbation which adds Gaussian noise at each SGD update, phased output perturbation and phased ERM output perturbation [16], respectively. The gradient complexity is the total number of computing the gradient on one datum in the algorithm. Organization of the Paper. The rest of the paper is organized as follows. The formulation of SGD algorithms and the main results are given in Section 2. We provide the proofs in Section 3 and conclude the paper in Section 4. ## 2 Problem Formulation and Main Results ### 2.1 Preliminaries Throughout the paper, we assume that the loss function $\ell:\mathcal{W}\times\mathcal{Z}\rightarrow\mathbb{R}$ is convex w.r.t. the first argument, i.e., for any $z\in\mathcal{Z}$ and $\mathbf{w},\mathbf{w}^{\prime}\in\mathcal{W}$, there holds $\ell(\mathbf{w},z)\geq\ell(\mathbf{w}^{\prime},z)+\langle\partial\ell(\mathbf{w}^{\prime},z),\mathbf{w}-\mathbf{w}^{\prime}\rangle$ where $\partial\ell(\mathbf{w}^{\prime},z)$ denotes a subgradient of $\ell(\cdot,z)$ in the first argument. We restrict our attention to the (projected) stochastic gradient descent algorithm which is defined as below. ###### Definition 1 (Stochastic Gradient Descent). Let $\mathcal{W}\subseteq\mathbb{R}^{d}$ be convex, $T$ denote the number of iterations, and $\text{Proj}_{\mathcal{W}}$ denote the projection to $\mathcal{W}$. Let $\mathbf{w}_{1}=\mathbf{0}\in\mathbb{R}^{d}$ be an initial point, and $\\{\eta_{t}\\}_{t=1}^{T-1}$ be a sequence of positive step sizes. At step $t\in\\{1,\ldots,T-1\\}$, the update rule of (projected) stochastic gradient decent is given by $\mathbf{w}_{t+1}=\text{Proj}_{\mathcal{W}}\big{(}\mathbf{w}_{t}-\eta_{t}\partial\ell(\mathbf{w}_{t},z_{i_{t}})\big{)},$ (1) where $\\{i_{t}\\}$ is uniformly drawn from $[n]:=\\{1,2,\ldots,n\\}$. When $\mathcal{W}=\mathbb{R}^{d}$, then (1) is reduced to $\mathbf{w}_{t+1}=\mathbf{w}_{t}-\eta_{t}\partial\ell(\mathbf{w}_{t},z_{i_{t}}).$ For a randomized learning algorithm $\mathcal{A}:\mathcal{Z}^{n}\rightarrow\mathcal{W}$, let $\mathcal{A}(S)$ denote the model produced by running $\mathcal{A}$ over the training dataset $S$. We say two datasets $S$ and $S^{\prime}$ are neighboring datasets, denoted by $S\simeq S^{\prime}$, if they differ by a single datum. We consider the following high-probabilistic version of the uniform argument stability (UAS), which is an extension of the UAS in expectation [24]. ###### Definition 2 (Uniform argument stability). We say an algorithm $\mathcal{A}$ has $\Delta_{\mathcal{A}}$-UAS with probability at least $1-\gamma$ ($\gamma\in(0,1)$) if $\mathbb{P}_{\mathcal{A}}(\sup_{S\simeq S^{\prime}}\delta_{\mathcal{A}}(S,S^{\prime})\geq\Delta_{\mathcal{A}})\leq\gamma,$ where $\delta_{\mathcal{\mathcal{A}}}(S,S^{\prime}):=\\|\mathcal{A}(S)-\mathcal{A}(S^{\prime})\\|_{2}.$ We will use UAS to study generalization bounds with high probability. In particular, the following lemma as a straightforward extension of Corollary 8 in [7] establishes the relationship between UAS and generalization errors. The proof is given in the Appendix for completeness. ###### Lemma 1. Suppose $\ell$ is nonnegative, convex and $\alpha$-Hölder smooth with parameter $L$. Let $M_{0}=\sup_{z\in\mathcal{Z}}\ell(0,z)$ and $M=\sup_{z\in\mathcal{Z}}\\|\partial\ell(0,z)\\|_{2}$. Let $\mathcal{A}$ be a randomized algorithm with the output of $\mathcal{A}$ bounded by $G$ and $\mathbb{P}_{\mathcal{A}}(\sup_{S\simeq S^{\prime}}\delta_{\mathcal{A}}(S,S^{\prime})\geq\Delta_{\mathcal{A}})\leq\gamma_{0}.$ Then there exists a constant $c>0$ such that for any distribution $\mathcal{D}$ over $\mathcal{Z}$ and any $\gamma\in(0,1)$, there holds $\mathbb{P}_{\mathcal{S}\sim\mathcal{D}^{n},\mathcal{A}}\biggl{[}\|\mathcal{R}(\mathcal{A(S)})-\mathcal{R}_{S}(\mathcal{A(S)})\|\geq c\bigg{(}(M+LG^{\alpha})\Delta_{\mathcal{A}}\log(n)\log(1/{\gamma})+\big{(}M_{0}+(M+LG^{\alpha})G\big{)}\sqrt{n^{-1}\log(1/\gamma)}\bigg{)}\biggr{]}\leq\gamma_{0}+\gamma.$ Differential privacy [13] is a de facto standard privacy measure for a randomized algorithm $\mathcal{A}.$ ###### Definition 3 (Differential Privacy). We say a randomized algorithm $\mathcal{A}$ satisfies $(\epsilon,\delta)$-DP if, for any two neighboring datasets $S$ and $S^{\prime}$ and any event $E$ in the output space of $\mathcal{A}$, there holds $\mathbb{P}(\mathcal{A}(S)\in E)\leq e^{\epsilon}\mathbb{P}(\mathcal{A}(S^{\prime})\in E)+\delta.$ In particular, we call it satisfies $\epsilon$-DP if $\delta=0$. We also need the following concept called $\ell_{2}$-sensitivity. ###### Definition 4 ($\ell_{2}$-sensitivity). The $\ell_{2}$-sensitivity of a function (mechanism) $\mathcal{M}:\mathcal{Z}^{n}\rightarrow\mathcal{W}$ is defined as $\Delta=\sup_{S\simeq S^{\prime}}\\|\mathcal{M}(S)-\mathcal{M}(S^{\prime})\\|_{2},$ where $S$ and $S^{\prime}$ are neighboring datasets. A basic mechanism to obtain $(\epsilon,\delta)$-DP from a given function $\mathcal{M}:\mathcal{Z}^{n}\rightarrow\mathcal{W}$ is to add a random noise from a Gaussian distribution $\mathcal{N}(0,\sigma^{2}\mathbf{I}_{d})$ where $\sigma$ is proportional to its $\ell_{2}$-sensitivity. This mechanism is often referred to as Gaussian mechanism as stated in the following lemma. ###### Lemma 2 ([14]). Given a function $\mathcal{M}:\mathcal{Z}^{n}\rightarrow\mathcal{W}$ with the $\ell_{2}$-sensitivity $\Delta$ and a dataset $S\subset\mathcal{Z}^{n}$, and assume that $\sigma\geq\frac{\sqrt{2\log(1.25/\delta)}\Delta}{\epsilon}$. The following Gaussian mechanism yields $(\epsilon,\delta)$-DP: $\mathcal{G}(S,\sigma):=\mathcal{M}(S)+\mathbf{b},~{}~{}\mathbf{b}\sim\mathcal{N}(0,\sigma^{2}\mathbf{I}_{d}),$ where $\mathbf{I}_{d}$ is the identity matrix in $\mathbb{R}^{d\times d}$. Although the concept of $(\epsilon,\delta)$-DP is widely used in privacy- preserving methods, its composition and subsampling amplification results are relatively loose, which are not suitable for iterative SGD algorithms. Based on the Rényi divergence, the work [26] proposed Rényi differential privacy (RDP) as a relaxation of DP to achieve tighter analysis of composition and amplification mechanisms. ###### Definition 5 (RDP [26]). For $\lambda>1$, $\rho>0$, a randomized mechanism $\mathcal{A}$ satisfies $(\lambda,\rho)$-RDP, if, for all neighboring datasets $S$ and $S^{\prime}$, we have $D_{\lambda}\big{(}\mathcal{A}(S)\parallel\mathcal{A}(S^{\prime})\big{)}:=\frac{1}{\lambda-1}\log\int\Big{(}\frac{P_{\mathcal{A}(S)}(\theta)}{P_{\mathcal{A}(S^{\prime})}(\theta)}\Big{)}^{\lambda}dP_{\mathcal{A}(S^{\prime})}(\theta)\leq\rho,$ where $P_{\mathcal{A}(S)}(\theta)$ and $P_{\mathcal{A}(S^{\prime})}(\theta)$ are the density of $\mathcal{A}(S)$ and $\mathcal{A}(S^{\prime})$, respectively. As $\lambda\rightarrow\infty$, RDP reduces to $\epsilon$-DP, i.e., $\mathcal{A}$ satisfies $\epsilon$-DP if and only if $D_{\infty}\big{(}\mathcal{A}(S)\|\|\mathcal{A}(S^{\prime})\big{)}\leq\epsilon$ for any neighboring datasets $S$ and $S^{\prime}$. Our analysis requires the introduction of several lemmas on useful properties of RDP listed below. First, we introduce the privacy amplification of RDP by uniform subsampling, which is fundamental to establish privacy guarantees of noisy SGD algorithms. In general, a uniform subsampling scheme first draws a subset with size $pn$ uniformly at random with a subsampling rate $p\leq 1$, and then applies a known randomized mechanism to the subset. ###### Lemma 3 ([22]). Consider a function $\mathcal{M}:\mathcal{Z}^{n}\rightarrow\mathcal{W}$ with the $\ell_{2}$-sensitivity $\Delta$, and a dataset $S\subset\mathcal{Z}^{n}$. The Gaussian mechanism $\mathcal{G}(S,\sigma)=\mathcal{M}(S)+\mathbf{b}$, where $\mathbf{b}\sim\mathcal{N}(0,\sigma^{2}\mathbf{I}_{d})$, applied to a subset of samples that are drawn uniformly without replacement with subsampling rate $p$ satisfies $(\lambda,3.5p^{2}\lambda\Delta^{2}/\sigma^{2})$-RDP if $\sigma^{2}\geq 0.67\Delta^{2}$ and $\lambda-1\leq\frac{2\sigma^{2}}{3\Delta^{2}}\log\big{(}\frac{1}{\lambda p(1+\sigma^{2}/\Delta^{2})}\big{)}$. The following adaptive composition theorem of RDP establishes the privacy of a composition of several adaptive mechanisms in terms of that of individual mechanisms. We say a sequence of mechanisms $(\mathcal{A}_{1},\ldots,\mathcal{A}_{k})$ are chosen adaptively if $\mathcal{A}_{i}$ can be chosen based on the outputs of the previous mechanisms $\mathcal{A}_{1}(S),\ldots,\mathcal{A}_{i-1}(S)$ for any $i\in[k]$. ###### Lemma 4 (Adaptive Composition of RDP [26]). If a mechanism $\mathcal{A}$ consists of a sequence of adaptive mechanisms $(\mathcal{A}_{1},\ldots,\mathcal{A}_{k})$ with $\mathcal{A}_{i}$ satisfying $(\lambda,\rho_{i})$-RDP, $i\in[k]$, then $\mathcal{A}$ satisfies $(\lambda,\sum_{i=1}^{k}\rho_{i})$-RDP. Lemme 4 tells us that the derivation of the privacy guarantee for a composition mechanism is simple and direct. This is the underlying reason that we adopt RDP in our subsequent privacy analysis. The following lemma allows us to further convert RDP back to $(\epsilon,\delta)$-DP. ###### Lemma 5 (From RDP to $(\epsilon,\delta)$-DP [26]). If a randomized mechanism $\mathcal{A}$ satisfies $(\lambda,\rho)$-RDP, then $\mathcal{A}$ satisfies $(\rho+\log(1/\delta)/(\lambda-1),\delta)$-DP for all $\delta\in(0,1)$. The following lemma shows that a post-processing procedure always preserves privacy. ###### Lemma 6 (Post-processing [26]). Let $\mathcal{A}:\mathcal{Z}^{n}\rightarrow\mathcal{W}_{1}$ satisfy $(\lambda,\rho)$-RDP and $f:\mathcal{W}_{1}\rightarrow\mathcal{W}_{2}$ be an arbitrary function. Then $f\circ\mathcal{A}:\mathcal{Z}^{n}\rightarrow\mathcal{W}_{2}$ satisfies $(\lambda,\rho)$-RDP. ### 2.2 Main Results We present our main results here. First, we state a key bound of UAS for SGD when $\mathcal{W}\subseteq\mathbb{R}^{d}$ and the loss function is $\alpha$-Hölder smooth. Then, we propose two privacy-preserving SGD-type algorithms using output and gradient perturbations, and present the corresponding privacy and generalization (utility) guarantees. The utility guarantees in terms of the excess risk typically rely on two main errors: optimization errors and generalization errors, as shown soon in (3) and (4) for the algorithms with output and gradient perturbations, respectively. We will apply techniques in optimization theory to handle the optimization errors [27], and the concept of UAS [6, 17, 24], which was given in Definition 2 in Subsection 2.1, to estimate the generalization errors. #### 2.2.1 UAS bound of SGD with Non-Smooth Losses We begin by stating the key result on the distance between two iterate trajectories produced by SGD on neighboring datasets. Let $c_{\alpha,1}=\begin{cases}(1+1/\alpha)^{\frac{\alpha}{1+\alpha}}L^{\frac{1}{1+\alpha}},&\mbox{if }\alpha\in(0,1]\\\ M+L,&\mbox{if }\alpha=0.\end{cases}$ (2) and $c_{\alpha,2}=\sqrt{\frac{1-\alpha}{1+\alpha}}(2^{-\alpha}L)^{\frac{1}{1-\alpha}}$, where $M=\sup_{z\in\mathcal{Z}}\\|\partial\ell(0,z)\\|_{2}$. In addition, define $C_{\alpha}=\frac{1-\alpha}{1+\alpha}c^{\frac{2(1+\alpha)}{1-\alpha}}_{\alpha,1}\big{(}\frac{\alpha}{1+\alpha}\big{)}^{\frac{2\alpha}{1-\alpha}}+2\sup_{z\in\mathcal{Z}}\ell(0;z)$. Furthermore, let $\mathcal{B}(0,r)$ denote the Euclidean ball of radius $r>0$ centered at $0\in\mathbb{R}^{d}$. Without loss of generality, we assume $\eta>1/T$. ###### Theorem 7. Suppose that the loss function $\ell$ is convex and $\alpha$-Hölder smooth with parameter $L$. Let $\mathcal{A}$ be the SGD with $T$ iterations and $\eta_{t}=\eta<\min\\{1,1/L\\}$, and $\bar{\mathbf{w}}=\frac{1}{T}\sum_{t=1}^{T}\mathbf{w}_{t}$ be the output produced by $\mathcal{A}$. Further, let $c_{\gamma,T}=\max\Big{\\{}\big{(}3n\log(n/\gamma)/T\big{)}^{\frac{1}{2}},3n\log(n/\gamma)/T\Big{\\}}$. 1. (a) If $\ell$ is nonnegative and $\mathcal{W}=\mathbb{R}^{d}$, then, for any $\gamma\in(0,1)$, there holds $\mathbb{P}_{\mathcal{A}}\Big{(}\sup_{S\simeq S^{\prime}}\delta_{\mathcal{A}}(S,S^{\prime})\geq\Delta_{SGD}(\gamma)\Big{)}\leq\gamma,$ where $\Delta_{SGD}(\gamma)=\Big{(}e\big{(}c^{2}_{\alpha,2}T\eta^{\frac{2}{1-\alpha}}+4\big{(}M+L(C_{\alpha}T\eta)^{\frac{\alpha}{2}}\big{)}^{2}\eta^{2}\Big{(}1+\frac{T}{n}(1+c_{\gamma,T})\Big{)}\frac{T}{n}(1+c_{\gamma,T})\big{)}\Big{)}^{1/2}$. 2. (b) If $\mathcal{W}\subseteq\mathcal{B}(0,R)$ with $R>0$, then, for any $\gamma\in(0,1)$, there holds $\mathbb{P}_{\mathcal{A}}\Big{(}\sup_{S\simeq S^{\prime}}\delta_{\mathcal{A}}(S,S^{\prime})\geq\tilde{\Delta}_{\text{SGD}}(\gamma)\Big{)}\leq\gamma,$ where $\tilde{\Delta}_{\text{SGD}}(\gamma)=\Big{(}e\big{(}c^{2}_{\alpha,2}T\eta^{\frac{2}{1-\alpha}}+4\big{(}M+LR^{\alpha}\big{)}^{2}\eta^{2}\Big{(}1+\frac{T}{n}(1+c_{\gamma,T})\Big{)}\frac{T}{n}(1+c_{\gamma,T})\big{)}\Big{)}^{1/2}$. ###### Remark 1. Under the reasonable assumption of $T\geq n$, we have $c_{\gamma,T}=\mathcal{O}(\log(n/\gamma))$. Then $\Delta_{\text{SGD}}(\gamma)=\mathcal{O}\Big{(}\sqrt{T}\eta^{\frac{1}{1-\alpha}}+\frac{(T\eta)^{1+\alpha/2}\log(n/\gamma)}{n}\Big{)}$ and $\tilde{\Delta}_{\text{SGD}}(\gamma)=\mathcal{O}\Big{(}\sqrt{T}\eta^{\frac{1}{1-\alpha}}+\frac{T\eta\log(n/\gamma)}{n}\Big{)}$. In addition, if $\ell$ is strongly smooth, i.e., $\alpha=1$, the first term in the UAS bounds tends to $0$ under the typical assumption of $\eta<1$. In this case we have $\Delta_{\text{SGD}}(\gamma)=\mathcal{O}\Big{(}\frac{\big{(}T\eta\big{)}^{3/2}\log(n/\gamma)}{n}\Big{)}$ and $\tilde{\Delta}_{\text{SGD}}(\gamma)=\mathcal{O}\Big{(}\frac{T\eta\log(n/\gamma)}{n}\Big{)}$. The work [3] established the high probability upper bound of the random variable of the argument stability $\delta_{SGD}$ in the order of $\mathcal{O}(\sqrt{T}\eta+\frac{T\eta}{n})$ for Lipschitz continuous losses under an additional assumption $\gamma\geq\exp(-n/2)$. Our result gives the upper bound of $\sup_{S\simeq S^{\prime}}\delta_{SGD}(S,S^{\prime})$ in the order of $\mathcal{O}(\sqrt{T}\eta+\frac{T\eta\log(n/\gamma)}{n})$ for any $\gamma\in(0,1)$ for the case of $\alpha=0$. The work [17] gave the bound of $\mathcal{O}({T\eta}/{n})$ in expectation for Lipschitz continuous and smooth loss functions. As a comparison, our stability bounds are stated with high probability and do not require the Lipschitz condition. Under a further Lipschitz condition, our stability bounds actually recover the bound $\mathcal{O}({T\eta}/{n})$ in [17] in the smooth case. Indeed, both the term $\big{(}M+(C_{\alpha}T\eta)^{\frac{\alpha}{2}}\big{)}^{2}$ and the term $\big{(}M+LR^{\alpha}\big{)}^{2}$ are due to controlling the magnitude of gradients, and can be replaced by $L^{2}$ for $L$-Lipschitz losses. #### 2.2.2 Differentially Private SGD with Output Perturbation 1: Inputs: Data $S=\\{z_{i}\in\mathcal{Z}:i=1,\ldots,n\\}$, $\alpha$-Hölder smooth loss $\ell(\mathbf{w},z)$ with parameter $L$, the convex set $\mathcal{W}$, step size $\eta$, number of iterations $T$, and privacy parameters $\epsilon$, $\delta$ 2: Set: $\mathbf{w}_{1}=\mathbf{0}$ 3: for $t=1$ to $T$ do 4: Sample $i_{t}\sim\text{Unif}([n])$ 5: $\mathbf{w}_{t+1}=\text{Proj}_{\mathcal{W}}(\mathbf{w}_{t}-\eta\partial\ell(\mathbf{w}_{t};z_{i_{t}}))$ 6: end for 7: if $\mathcal{W}=\mathbb{R}^{d}$ then 8: let $\Delta=\Delta_{\text{SGD}}(\delta/2)$ 9: else if $\mathcal{W}\subseteq\mathcal{B}(0,R)$ then 10: let $\Delta=\tilde{\Delta}_{\text{SGD}}(\delta/2)$ 11: end if 12: Compute: $\sigma^{2}=\frac{2\log(2.5/\delta)\Delta^{2}}{\epsilon^{2}}$ 13: return: ${\mathbf{w}}_{\text{priv}}=\frac{1}{T}\sum_{t=1}^{T}\mathbf{w}_{t}+\mathbf{b}$ where $\mathbf{b}\sim\mathcal{N}(0,\sigma^{2}\mathbf{I}_{d})$ Algorithm 1 Differentially Private SGD with Output perturbation (DP-SGD- Output) Output perturbation [9, 13] is a common approach to achieve $(\epsilon,\delta)$-DP. The main idea is to add a random noise $\mathbf{b}$ to the output of the SGD algorithm, where $\mathbf{b}$ is randomly sampled from the Gaussian distribution with mean $0$ and variance proportional to the $\ell_{2}$-sensitivity of SGD. In Algorithm 1, we propose the private SGD algorithm with output perturbation for non-smooth losses in both bounded domain $\mathcal{W}\subseteq\mathcal{B}(0,R)$ and unbounded domain $\mathcal{W}=\mathbb{R}^{d}$. The difference in these two cases is that we add random noise with different variances according to the sensitivity analysis of SGD stated in Theorem 7. In the sequel, we present the privacy and utility guarantees for Algorithm 1. ###### Theorem 8 (Privacy guarantee). Suppose that the loss function $\ell$ is convex, nonnegative and $\alpha$-Hölder smooth with parameter $L$. Then Algorithm 1 (DP-SGD-Output) satisfies $(\epsilon,\delta)$-DP. According to the definitions, the $\ell_{2}$-sensitivity of SGD is identical to the UAS of SGD: $\sup_{S\simeq S^{\prime}}\delta_{SGD}(S,S^{\prime})$. In this sense, the proof of Theorem 8 directly follows from Theorem 7 and Lemma 2. For completeness, we include the detailed proof in Subsection 3.2. Recall that the empirical risk is defined by $\mathcal{R}_{S}(\mathbf{w})=\frac{1}{n}\sum_{i=1}^{n}\ell(\mathbf{w},z_{i})$, and the population risk is $\mathcal{R}(\mathbf{w})=\mathbb{E}_{z}[\ell(\mathbf{w},z)]$. Let $\mathbf{w}^{}\in\arg\min_{\mathbf{w}\in\mathcal{W}}\mathcal{R}(\mathbf{w})$ be the one with the best prediction performance over $\mathcal{W}$. We use the notation $B\asymp\tilde{B}$ if there exist constants $c_{1},c_{2}>0$ such that $c_{1}\tilde{B}<B\leq c_{2}\tilde{B}$. Without loss of generality, we always assume $\\|\mathbf{w}^{}\\|_{2}\geq 1$. ###### Theorem 9 (Utility guarantee for unbounded domain). Suppose the loss function $\ell$ is nonnegative, convex and $\alpha$-Hölder smooth with parameter $L$. Let $\mathbf{w}_{\text{priv}}$ be the output produced by Algorithm 1 with $\mathcal{W}=\mathbb{R}^{d}$ and $\eta=n^{\frac{1}{3+\alpha}}/\big{(}T(\log(\frac{1}{\gamma}))^{\frac{1}{3+\alpha}}\big{)}$. Let $T\asymp n^{\frac{-\alpha^{2}-3\alpha+6}{(1+\alpha)(3+\alpha)}}$ if $0\leq\alpha<\frac{\sqrt{73}-7}{4}$, and $T\asymp n$ else. Then, for any $\gamma\in(4\max\\{\exp(-d/8),\delta\\},1)$, with probability at least $1-\gamma$ over the randomness in both the sample and the algorithm, there holds $\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}(\mathbf{w}^{})=\\|\mathbf{w}^{}\\|_{2}^{2}\cdot\mathcal{O}\bigg{(}\frac{\sqrt{d\log(1/\delta)}{\log(n/\delta)}}{(\log(1/\gamma))^{\frac{1+\alpha}{4(3+\alpha)}}n^{\frac{2}{3+\alpha}}\epsilon}+\frac{\log(n)\big{(}\log(1/\gamma)\big{)}^{\frac{2}{3+\alpha}}{\log(n/\delta)}}{n^{\frac{1}{3+\alpha}}}\bigg{)}.$ To examine the excess population risk $\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}(\mathbf{w}^{})$, we use the following error decomposition: $\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}(\mathbf{w}^{})=[\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}(\bar{\mathbf{w}})]+[\mathcal{R}(\bar{\mathbf{w}})-\mathcal{R}_{S}(\bar{\mathbf{w}})]+[\mathcal{R}_{S}(\bar{\mathbf{w}})-\mathcal{R}_{S}(\mathbf{w}^{})]+[\mathcal{R}_{S}(\mathbf{w}^{})-\mathcal{R}(\mathbf{w}^{})],$ (3) where $\bar{\mathbf{w}}=\frac{1}{T}\sum_{t=1}^{T}\mathbf{w}_{t}$ is the output of non-private SGD. The first term is due to the added noise $\mathbf{b}$, which can be estimated by the Chernoff bound for Gaussian random vectors. The second term is the generalization error of SGD, which can be handled by the stability analysis. The third term is an optimization error and can be controlled by standard techniques in optimization theory. Finally, the last term can be bounded by $\mathcal{O}(1/\sqrt{n})$ by Hoeffding inequality. The proof of Theorem 9 is given in Subsection 3.2. Now, we turn our attention to the utility guarantee for the case with a bounded domain. ###### Theorem 10 (Utility guarantees for bounded domain). If the loss function $\ell$ is nonnegative, convex and $\alpha$-Hölder smooth with parameter $L$. Let $\mathbf{w}_{\text{priv}}$ be the output produced by Algorithm 1 with $\mathcal{W}\subseteq\mathcal{B}(0,R)$. Let $T\asymp n^{\frac{2-\alpha}{1+\alpha}}$ if $\alpha<\frac{1}{2}$, $T\asymp n$ else, and choose $\eta=1/\Big{(}T\max\Big{\\{}\frac{\sqrt{\log(n/\delta)\log(n)\log(1/\gamma)}}{\sqrt{n}},\frac{\big{(}d\log(1/\delta)\big{)}^{1/4}\sqrt{\log(n/\delta)}(\log(1/\gamma))^{1/8}}{\sqrt{n\epsilon}}\Big{\\}}\Big{)}$. Then for any $\gamma\in(4\max\\{\exp(-d/8),\delta\\},1)$, with probability at least $1-\gamma$ over the randomness in both the sample and the algorithm, there holds $\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}(\mathbf{w}^{})=\\|\mathbf{w}^{}\\|_{2}^{2}\cdot\mathcal{O}\bigg{(}\frac{\big{(}d\log(1/\delta)\big{)}^{\frac{1}{4}}(\log(1/\gamma))^{\frac{1}{8}}{\sqrt{\log(n/\delta)}}}{\sqrt{n\epsilon}}+\frac{\sqrt{\log(n)\log(1/\gamma){\log(n/\delta)}}}{\sqrt{n}}\bigg{)}.$ The definition of $\alpha$-Hölder smoothness and the convexity of $\ell$ imply the following inequalities $\\|\partial\ell(\mathbf{w};z)\\|_{2}\leq M+LR^{\alpha}\text{ and }\ell(\mathbf{w};z)\leq\ell(0;z)+MR+LR^{1+\alpha},\quad\forall z\in\mathcal{Z},\mathbf{w}\in\mathcal{W}.$ These together with Theorem 8 and Theorem 9 imply the privacy and utility guarantees in the above theorem. The detailed proof is given in Subsection 3.2. ###### Remark 2. The private SGD algorithm with output perturbation was studied in [35] under both the Lipschitz continuity and the strong smoothness assumption, where the excess population risk for one-pass private SGD (i.e. the total iteration number $T=n$) with a bounded parameter domain was bounded by $\mathcal{O}\big{(}(n\epsilon)^{-\frac{1}{2}}(d\log(1/\delta)^{\frac{1}{4}}\big{)}$. As a comparison, we show that the same rate (up to a logarithmic factor) $\mathcal{O}\big{(}(n\epsilon)^{-\frac{1}{2}}(d\log(1/\delta))^{\frac{1}{4}}\log^{\frac{1}{2}}(n/\delta)\big{)}$ can be achieved for general $\alpha$-Hölder smooth losses by taking $T=\mathcal{O}(n^{2-\alpha\over 1+\alpha}+n).$ Our results extend the output perturbation for private SGD algorithms to a more general class of non-smooth losses. #### 2.2.3 Differentially Private SGD with Gradient Perturbation 1: Inputs: Data $S=\\{z_{i}\in\mathcal{Z}:i=1,\ldots,n\\}$, loss function $\ell(\mathbf{w},z)$ with Hölder parameters $\alpha$ and $L$, the convex set $\mathcal{W}\subseteq\mathcal{B}(0,R)$, step size $\eta$, number of iterations $T$, privacy parameters $\epsilon$, $\delta$, and constant $\beta$. 2: Set: $\mathbf{w}_{1}=\mathbf{0}$ 3: Compute $\sigma^{2}=\frac{14(M+LR^{\alpha})^{2}T}{\beta n^{2}\epsilon}\Big{(}\frac{\log(1/\delta)}{(1-\beta)\epsilon}+1\Big{)}$ 4: for $t=1$ to $T$ do 5: Sample $i_{t}\sim\text{Unif}([n])$ 6: $\mathbf{w}_{t+1}=\text{Proj}_{\mathcal{W}}\big{(}\mathbf{w}_{t}-\eta(\partial\ell(\mathbf{w}_{t};z_{i_{t}})+\mathbf{b}_{t})\big{)}$, where $\mathbf{b}_{t}\sim\mathcal{N}(0,\sigma^{2}\mathbf{I}_{d})$ 7: end for 8: return: ${\mathbf{w}}_{\text{priv}}=\frac{1}{T}\sum_{t=1}^{T}\mathbf{w}_{t}$ Algorithm 2 Differentially Private SGD with Gradient perturbation (DP-SGD- Gradient) An alternative approach to achieve $(\epsilon,\delta)$-DP is gradient perturbation, i.e., adding Gaussian noise to the stochastic gradient at each update. The detailed algorithm is described in Algorithm 2, whose privacy guarantee is established in the following theorem. ###### Theorem 11 (Privacy guarantee). Suppose the loss function $\ell$ is nonnegative, convex and $\alpha$-Hölder smooth with parameter $L$. Then Algorithm 2 (DP-SGD-Gradient) satisfies $(\epsilon,\delta)$-DP if there exists $\beta\in(0,1)$ such that $\frac{\sigma^{2}}{4(M+LR^{\alpha})^{2}}\geq 0.67$ and $\lambda-1\leq\frac{\sigma^{2}}{6(M+LR^{\alpha})^{2}}\log\Big{(}\frac{n}{\lambda(1+\frac{\sigma^{2}}{4(M+LR^{\alpha})^{2}})}\Big{)}$ hold with $\lambda=\frac{\log(1/\delta)}{(1-\beta)\epsilon}+1$. Since $\mathcal{W}\subseteq\mathcal{B}(0,R)$, the Hölder smoothness of $\ell$ implies that $\\|\partial\ell(\mathbf{w}_{t},z)\\|_{2}\leq M+LR^{\alpha}$ for any $t\in[T]$ and any $z\in\mathcal{Z}$, from which we know that the $\ell_{2}$-sensitivity of the function $\mathcal{M}_{t}=\partial\ell(\mathbf{w}_{t},z)$ can be bounded by $2(M+LR^{\alpha})$. By Lemma 3 and the post-processing property of DP, it is easy to show that the update of $\mathbf{w}_{t}$ satisfies $(\frac{\log(1/\delta)}{(1-\beta)\epsilon}+1,\frac{\beta\epsilon}{T})$-RDP for any $t\in[T]$. Furthermore, by the composition theorem of RDP and the relationship between $(\epsilon,\delta)$-DP and RDP, we can show that the proposed algorithm satisfies $(\epsilon,\delta)$-DP. The detailed proof can be found in Subsection 3.3. Other than the privacy guarantees, the DP-SGD-Gradient algorithm also enjoys utility guarantees as stated in the following theorem. ###### Theorem 12 (Utility guarantee). Suppose the loss function $\ell$ is nonnegative, convex and $\alpha$-Hölder smooth with parameter $L$. Let $\mathbf{w}_{\text{priv}}$ be the output produced by Algorithm 2 with $\eta=\frac{1}{T}\max\big{\\{}\frac{\sqrt{\log(n)\log(n/\gamma)\log(1/\gamma)}}{\sqrt{n}},\frac{\sqrt{d\log(1/\delta)}(\log(1/\gamma))^{\frac{1}{4}}}{n\epsilon}\big{\\}}$. Furthermore, let $T\asymp n^{\frac{2-\alpha}{1+\alpha}}$ if $\alpha<\frac{1}{2}$, and $T\asymp n$ else. Then, for any $\gamma\in(18\exp(-Td/8),1)$, with probability at least $1-\gamma$ over the randomness in both the sample and the algorithm, there holds $\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}(\mathbf{w}^{})=\\|\mathbf{w}^{}\\|_{2}^{2}\cdot\mathcal{O}\bigg{(}\frac{\sqrt{d\log(1/\delta)\log(1/\gamma)}}{n\epsilon}+\frac{\sqrt{\log(n){\log(n/\gamma)}\log(1/\gamma)}}{\sqrt{n}}\bigg{)}.$ Our basic idea to prove Theorem 12 is to use the following error decomposition: $\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}(\mathbf{w}^{})=[\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}_{S}(\mathbf{w}_{\text{priv}})]+[\mathcal{R}_{S}(\mathbf{w}_{\text{priv}})-\mathcal{R}_{S}(\mathbf{w}^{})]+[\mathcal{R}_{S}(\mathbf{w}^{})-\mathcal{R}(\mathbf{w}^{})].$ (4) Similar to the proof of Theorem 9, the generalization error $\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}_{S}(\mathbf{w}_{\text{priv}})$ can be handled by the UAS bound, the optimization error $\mathcal{R}_{S}(\mathbf{w}_{\text{priv}})-\mathcal{R}_{S}(\mathbf{w}^{})$ can be estimated by standard techniques in optimization [[, e.g.]]Nem, and the last term $\mathcal{R}_{S}(\mathbf{w}^{})-\mathcal{R}(\mathbf{w}^{})$ can be bounded by the Hoeffding inequality. The detailed proof can be found in Subsection 3.3. ###### Remark 3. We now compare our results with the related work under a bounded domain assumption. The work [4] established the optimal rate $\mathcal{O}(\frac{1}{n\epsilon}{\sqrt{d\log(1/\delta)}}+\frac{1}{\sqrt{n}})$ for the excess population risk of private SCO algorithm in either smooth case ($\alpha=1$) or non-smooth case ($\alpha=0$). However, their algorithm has a large gradient complexity $\mathcal{O}\Big{(}n^{4.5}\sqrt{\epsilon}+\frac{n^{6.5}\epsilon^{4.5}}{(d\log(\frac{1}{\delta}))^{2}}\Big{)}$. The work [16] proposed a private phased ERM algorithm for SCO, which can achieve the optimal excess population risk for non-smooth losses with a better gradient complexity of the order $\mathcal{O}(n^{2}\log(1/{\delta}))$. The very recent work [3] improved the gradient complexity to $\mathcal{O}(n^{2})$. As a comparison, we show that SGD with gradient complexity $\mathcal{O}(n^{2-\alpha\over 1+\alpha}+n)$ is able to achieve the optimal (up to logarithmic terms) excess population risk $\mathcal{O}(\frac{1}{n\epsilon}{\sqrt{d\log(1/\delta)}}+\frac{1}{\sqrt{n}})$ for general $\alpha$-Hölder smooth losses. Our results match the existing gradient complexity for both the smooth case in [4] and the Lipschitz continuity case [3]. An interesting observation is that our algorithm can achieve the optimal utility guarantee with the linear gradient complexity $\mathcal{O}(n)$ for $\alpha\geq 1/2$, which shows that a relaxation of the strong smoothness from $\alpha=1$ to $\alpha\geq 1/2$ does not bring any harm in both the generalization and computation complexity. Now, we give a sufficient condition for the existence of $\beta$ in Theorem 11 under a specific parameter setting. ###### Lemma 13. Let $n\geq 18$, $T=n$ and $\delta=1/{n^{2}}$. If $\epsilon\geq\frac{7(n^{\frac{1}{3}}-1)+4\log(n)n+7}{2n(n^{\frac{1}{3}}-1)},$ then there exists $\beta\in(0,1)$ such that Algorithm 2 satisfies $(\epsilon,\delta)$-DP. Figure 1: The sufficient condition for the existence of $\beta$ in Lemma 13. The shaded area is the area where the sufficient condition in Lemma 13 holds true, i.e., $\epsilon\geq\big{(}7(n^{\frac{1}{3}}-1)+4\log(n)n+7\big{)}/\big{(}2n(n^{\frac{1}{3}}-1)\big{)}$. ###### Remark 4. Privacy parameters $\epsilon$ and $\delta$ together quantify the privacy risk. $\epsilon$ is often called the privacy budget controlling the degree of privacy leakage. A larger value of $\epsilon$ implies higher privacy risk. Therefore, the value of $\epsilon$ depends on how much privacy the user needs to protect. Theoretically, the value of $\epsilon$ is less than 1. However, in practice, to obtain the desired utility, a larger privacy budget, i.e., $\epsilon\geq 1$, is always acceptable [35, 33]. For instance, Apple uses a privacy budget $\epsilon=8$ for Safari Auto-play intent detection, and $\epsilon=2$ for Health types111https://www.apple.com/privacy/docs/Differential_Privacy_Overview.pdf. Parameter $\delta$ is the probability with which $e^{\epsilon}$ fails to bound the ratio between the two probabilities in the definition of differential privacy, i.e., the probability of privacy protection failure. For meaningful privacy guarantees, according to [14] the value of $\delta$ should be much smaller than $1/n$. In particular, we always choose $\delta=1/n^{2}$. For DP- SGD-Gradient algorithm, another constant we should discuss is $\beta$ which depends on the choice of the number of iterations $T$, size of training data $n$, privacy parameters $\epsilon$ and $\delta$. The appearance of this parameter is due to the use of subsampling result for RDP (see Lemma 3). The condition in Lemma 13 ensures the existence of $\beta\in(0,1)$ such that Algorithm 2 satisfies DP. In practical applications, we search in $(0,1)$ for all $\beta$ that satisfy the RDP conditions in Theorem 11. Note that the closer the $\beta$ is to $1/2$, the smaller the variance of the noise added to the algorithm in each iteration. Therefore, we choose the value that is closest to $1/2$ of all $\beta$ that meets the RDP conditions as the value of $\beta$. We end this section with a final remark on the challenges of proving DP for Algorithm 2 when $\mathcal{W}$ is unbounded. ###### Remark 5. To make Algorithm 2 satisfy DP when $\mathcal{W}=\mathbb{R}^{d}$, the variance $\sigma_{t}$ of the noise $\mathbf{b}_{t}$ added in the $t$-th iteration should be proportional to the $\ell_{2}$-sensitivity $\Delta_{t}=\\|\partial\ell(\mathbf{w}_{t},z_{i_{t}})-\partial\ell(\mathbf{w}_{t},z^{\prime}_{i_{t}})\\|_{2}$. The definition of Hölder smoothness implies that $\Delta_{t}\leq 2(M+L\\|\mathbf{w}_{t}\\|^{\alpha}_{2})$. When $\alpha=0$, we have $\Delta_{t}\leq 2(M+L)$ and the privacy guarantee can be established in a way similar to Theorem 11. When $\alpha\in(0,1]$, we have to establish an upper bound of $\\|\mathbf{w}_{t}\\|_{2}$. Since $\mathbf{w}_{t}=\mathbf{w}_{t-1}-\eta(\partial\ell(\mathbf{w}_{t-1},z_{i_{t-1}})+\mathbf{b}_{t-1})$ ($\mathbf{b}_{t-1}\sim\mathcal{N}(0,\sigma_{t-1}^{2}\mathbf{I}_{d})$), we can only give a bound of $\\|\mathbf{w}_{t}\\|_{2}$ with high probability. Thus, the sensitivity $\Delta_{t}$ can not be uniformly bounded in this case. Therefore, the first challenge is how to analyze the privacy guarantee when the sensitivity changes at each iteration and all of them can not be uniformly bounded. Furthermore, by using the property of the Gaussian vector, we can prove that $\\|\mathbf{w}_{t}\\|_{2}=\mathcal{O}(\sqrt{t\eta}+\eta\sum_{j=1}^{t-1}\sigma_{j}+\eta\sqrt{d\sum_{j=1}^{t-1}\sigma_{j}^{2}})$ with high probability. However, as mentioned above, the variance $\sigma_{t}$ should be proportional to $\Delta_{t}$ whose upper bound involves $\\|\mathbf{w}_{t}\\|^{\alpha}_{2}$. Thus, $\sigma_{t}$ is proportional to $(t\eta)^{\alpha/2}+\eta^{\alpha}(\sum_{j=1}^{t-1}\sigma_{j})^{\alpha}+\eta^{\alpha}(d\sum_{j=1}^{t-1}\sigma_{j}^{2})^{\alpha/2}.$ For this reason, it seems difficult to give a clear expression for an upper bound of $\\|\mathbf{w}_{t}\\|_{2}.$ ## 3 Proofs of Main Results Before presenting the detailed proof, we first introduce some useful lemmas on the concentration behavior of random variables. ###### Lemma 14 (Chernoff bound for Bernoulli variable [34]). Let $X_{1},\ldots,X_{k}$ be independent random variables taking values in $\\{0,1\\}$. Let $X=\sum_{i=1}^{k}X_{i}$ and $\mu=\mathbb{E}[X]$. The following statements hold. 1. (a) For any $\tilde{\gamma}\in(0,1)$, with probability at least $1-\exp\big{(}-\mu\tilde{\gamma}^{2}/3\big{)}$, there holds $X\leq(1+\tilde{\gamma})\mu$. 2. (b) For any $\tilde{\gamma}\geq 1$, with probability at least $1-\exp\big{(}-\mu\tilde{\gamma}/3\big{)}$, there holds $X\leq(1+\tilde{\gamma})\mu$. ###### Lemma 15 (Chernoff bound for the $\ell_{2}$-norm of Gaussian vector [34]). Let $X_{1},\ldots,X_{k}$ be i.i.d. standard Gaussian random variables, and $\mathbf{X}=[X_{1},\ldots,X_{k}]\in\mathbb{R}^{k}$. Then for any $t\in(0,1)$, with probability at least $1-\exp(-kt^{2}/8)$, there holds $\\|\mathbf{X}\\|_{2}^{2}\leq k(1+t).$ ###### Lemma 16 (Hoeffding inequality [18]). Let $X_{1},\ldots,X_{k}$ be independent random variables such that $a_{i}\leq X_{i}\leq b_{i}$ with probability 1 for all $i\in[k]$. Let ${X}=\frac{1}{k}\sum_{i=1}^{k}X_{i}$. Then for any $t>0$, with probability at least $1-\exp(-2t^{2}/\sum_{i}(b_{i}-a_{i})^{2})$, there holds ${X}-\mathbb{E}[{X}]\leq t.$ ###### Lemma 17 (Azuma-Hoeffding inequality [18]). Let $X_{1},\ldots,X_{k}$ be a sequence of random variables where $X_{i}$ may depend on the previous random variables $X_{1},\ldots,X_{i-1}$ for all $i=1,\ldots,k$. Consider a sequence of functionals $\xi_{i}(X_{1},\ldots,X_{i})$, $i\in[k]$. If $\|\xi_{i}-\mathbb{E}_{X_{i}}[\xi_{i}]\|\leq b_{i}$ for each $i$. Then for all $t>0$, with probability at least $1-\exp(-t^{2}/(2\sum_{i}b_{i}^{2}))$, there holds $\sum_{i=1}^{k}\xi_{i}-\sum_{i=1}^{k}\mathbb{E}_{X_{i}}[\xi_{i}]\leq t$. ###### Lemma 18 (Tail bound of sub-Gaussian variable [34]). Let $X$ be a sub-Gaussian random variable with mean $\mu$ and sub-Gaussian parameter $v^{2}$. Then, for any $t\geq 0$, we have, with probability at least $1-\exp\big{(}-t^{2}/(2v^{2})\big{)}$, that $X-\mu\leq t$. ### 3.1 Proofs on UAS bound of SGD on Non-smooth Losses Our stability analysis for unbounded domain requires the following lemma on the self-bounding property for Hölder smooth losses. ###### Lemma 19. ([21, 37]) Suppose the loss function $\ell$ is nonnegative, convex and $\alpha$-Hölder smooth with parameter $L$. Then for $c_{\alpha,1}$ defined as (2) we have $\\|\partial\ell(\mathbf{w},z)\\|_{2}\leq c_{\alpha,1}\ell^{\frac{\alpha}{1+\alpha}}(\mathbf{w},z),\quad\forall\mathbf{w}\in\mathbb{R}^{d},z\in\mathcal{Z}.$ Based on Lemma 19, we develop the following bound on the iterates produced by the SGD update (1) which is critical to analyze the privacy and utility guarantees in the case of unbounded domain. Recall that $M=\sup_{z\in\mathcal{Z}}\\|\partial\ell(0,z)\\|_{2}$. ###### Lemma 20. Suppose the loss function $\ell$ is nonnegative, convex and $\alpha$-Hölder smooth with parameter $L$. Let $\\{\mathbf{w}_{t}\\}_{t=1}^{T}$ be the sequence produced by SGD with $T$ iterations when $\mathcal{W}=\mathbb{R}^{d}$ and $\eta_{t}<\min\\{1,1/L\\}$. Then, for any $t\in[T]$, there holds $\\|\mathbf{w}_{t+1}\\|_{2}^{2}\leq C_{\alpha}\sum_{j=1}^{t}\eta_{j},$ where $C_{\alpha}=\frac{1-\alpha}{1+\alpha}c^{\frac{2(1+\alpha)}{1-\alpha}}_{\alpha,1}\big{(}\frac{\alpha}{1+\alpha}\big{)}^{\frac{2\alpha}{1-\alpha}}+2\sup_{z\in\mathcal{Z}}\ell(0;z)$. ###### Proof. The update rule $\mathbf{w}_{t+1}=\mathbf{w}_{t}-\eta_{t}\partial\ell(\mathbf{w}_{t},z_{i_{t}})$ implies that $\displaystyle\\|\mathbf{w}_{t+1}\\|_{2}^{2}$ $\displaystyle=\\|\mathbf{w}_{t}-\eta_{t}\partial\ell(\mathbf{w}_{t},z_{i_{t}})\\|_{2}^{2}=\\|\mathbf{w}_{t}\\|_{2}^{2}+\eta_{t}^{2}\\|\partial\ell(\mathbf{w}_{t},z_{i_{t}})\\|_{2}^{2}-2\eta_{t}\langle\mathbf{w}_{t},\partial\ell(\mathbf{w}_{t},z_{i_{t}})\rangle.$ (5) First, we consider the case $\alpha=0$. By the definition of Hölder smoothness, we know $\ell$ is $(M+L)$-Lipschitz continuous. Furthermore, by the convexity of $\ell$, we have $\displaystyle\eta_{t}\\|\partial\ell(\mathbf{w}_{t},z_{i_{t}})\\|_{2}^{2}-2\langle\mathbf{w}_{t},\partial\ell(\mathbf{w}_{t},z_{i_{t}})$ $\displaystyle\leq\eta_{t}\\|\partial\ell(\mathbf{w}_{t},z_{i_{t}})\\|_{2}^{2}+2\big{(}\ell(0,z_{i_{t}})-\ell(\mathbf{w}_{t},z_{i_{t}})\big{)}$ $\displaystyle\leq(M+L)^{2}+2\sup_{z\in\mathcal{Z}}\ell(0,z),$ where in the last inequality we have used $\eta_{t}<1$ and the nonnegativity of $\ell$. Now, putting the above inequality back into (5) and taking the summation gives $\\|\mathbf{w}_{t+1}\\|_{2}^{2}\leq\big{(}(M+L)^{2}+2\sup_{z\in\mathcal{Z}}\ell(0;z)\big{)}\sum_{j=1}^{t}\eta_{j}.$ (6) Then, we consider the case $\alpha=1$. In this case, Lemma 19 implies $\\|\partial\ell(\mathbf{w};z)\\|_{2}^{2}\leq 2L\ell(\mathbf{w};z)$. Therefore, $\displaystyle\eta_{t}\\|\partial\ell(\mathbf{w}_{t},z_{i_{t}})\\|_{2}^{2}-2\langle\mathbf{w}_{t},\partial\ell(\mathbf{w}_{t},z_{i_{t}})\rangle\leq 2\eta_{t}L\ell(\mathbf{w}_{t},z_{i_{t}})+2\ell(0,z_{i_{t}})-2\ell(\mathbf{w}_{t},z_{i_{t}})\leq 2\ell(0,z_{i_{t}}),$ where we have used the convexity of $\ell$ and $\eta_{t}<1/L$. Plugging the above inequality back into (5) and taking the summation yield that $\\|\mathbf{w}_{t+1}\\|_{2}^{2}\leq 2\sup_{z\in\mathcal{Z}}\ell(0,z)\sum_{j=1}^{t}\eta_{j}.$ (7) Finally, we consider the case $\alpha\in(0,1)$. According to the self-bounding property and the convexity, we know $\\|\partial\ell(\mathbf{w}_{t},z_{i_{t}})\\|_{2}\leq c_{\alpha,1}\ell^{\frac{\alpha}{1+\alpha}}(\mathbf{w}_{t},z_{i_{t}})\leq c_{\alpha,1}\big{(}\langle\mathbf{w}_{t},\partial\ell(\mathbf{w}_{t},z_{i_{t}})\rangle+\ell(0,z_{i_{t}})\big{)}^{\frac{\alpha}{1+\alpha}}.$ Therefore, for $\alpha\in(0,1)$ there holds $\displaystyle\\|\partial\ell(\mathbf{w}_{t},z_{i_{t}})\\|_{2}^{2}$ $\displaystyle\leq c^{2}_{\alpha,1}\big{(}\langle\mathbf{w}_{t},\partial\ell(\mathbf{w}_{t},z_{i_{t}})\rangle+\ell(0,z_{i_{t}})\big{)}^{\frac{2\alpha}{1+\alpha}}$ $\displaystyle=\Big{(}\frac{1+\alpha}{\alpha\eta_{t}}\big{(}\langle\mathbf{w}_{t},\partial\ell(\mathbf{w}_{t},z_{i_{t}})\rangle+\ell(0,z_{i_{t}})\big{)}\Big{)}^{\frac{2\alpha}{1+\alpha}}\cdot\Big{(}c^{2}_{\alpha,1}\big{(}\frac{1+\alpha}{\alpha\eta_{t}}\big{)}^{-\frac{2\alpha}{1+\alpha}}\Big{)}$ $\displaystyle\leq\frac{2\alpha}{1+\alpha}\Big{(}\frac{1+\alpha}{\alpha\eta_{t}}\big{(}\langle\mathbf{w}_{t},\partial\ell(\mathbf{w}_{t},z_{i_{t}})\rangle+\ell(0,z_{i_{t}})\big{)}\Big{)}+\frac{1-\alpha}{1+\alpha}\Big{(}c^{2}_{\alpha,1}\big{(}\frac{1+\alpha}{\alpha\eta_{t}}\big{)}^{-\frac{2\alpha}{1+\alpha}}\Big{)}^{\frac{1+\alpha}{1-\alpha}}$ $\displaystyle=2\eta_{t}^{-1}\big{(}\langle\mathbf{w}_{t},\partial\ell(\mathbf{w}_{t},z_{i_{t}})\rangle+\ell(0,z_{i_{t}})\big{)}+\frac{1-\alpha}{1+\alpha}c^{\frac{2(1+\alpha)}{1-\alpha}}_{\alpha,1}\big{(}\frac{\alpha}{1+\alpha}\big{)}^{\frac{2\alpha}{1-\alpha}}\eta_{t}^{\frac{2\alpha}{1-\alpha}},$ where the last inequality used Young’s inequality $ab\leq\frac{1}{p}a^{p}+\frac{1}{q}b^{q}$ with $\frac{1}{p}+\frac{1}{q}=1.$ Putting the above inequality into (5), we have $\\|\mathbf{w}_{t+1}\\|_{2}^{2}\leq\\|\mathbf{w}_{t}\\|_{2}^{2}+\frac{1-\alpha}{1+\alpha}c^{\frac{2(1+\alpha)}{1-\alpha}}_{\alpha,1}\big{(}\frac{\alpha}{1+\alpha}\big{)}^{\frac{2\alpha}{1-\alpha}}\eta_{t}^{\frac{2}{1-\alpha}}+2\ell(0,z_{i_{t}})\eta_{t},$ If the step size $\eta_{t}<1$, then $\\|\mathbf{w}_{t+1}\\|_{2}^{2}\leq\\|\mathbf{w}_{t}\\|_{2}^{2}+\bigg{(}\frac{1-\alpha}{1+\alpha}c^{\frac{2(1+\alpha)}{1-\alpha}}_{\alpha,1}\big{(}\frac{\alpha}{1+\alpha}\big{)}^{\frac{2\alpha}{1-\alpha}}+2\sup_{z\in\mathcal{Z}}\ell(0;z)\bigg{)}\eta_{t}.$ Taking a summation of the above inequality, we get $\\|\mathbf{w}_{t+1}\\|_{2}^{2}\leq\Big{(}\frac{1-\alpha}{1+\alpha}c^{\frac{2(1+\alpha)}{1-\alpha}}_{\alpha,1}\big{(}\frac{\alpha}{1+\alpha}\big{)}^{\frac{2\alpha}{1-\alpha}}+2\sup_{z\in\mathcal{Z}}\ell(0;z)\Big{)}\sum_{j=1}^{t}\eta_{j}.$ (8) The desired result follows directly from (6), (7) and (8) for different values of $\alpha.$ ∎ The following lemma shows the approximately non-expensive behavior of the gradient mapping $\mathbf{w}\mapsto\mathbf{w}-\eta\partial\ell(\mathbf{w},z)$. The case $\alpha\in[0,1)$ can be found in Lei and Ying [21], and the case $\alpha=1$ can be found in Hardt [17]. ###### Lemma 21. Suppose the loss function $\ell$ is convex and $\alpha$-Hölder smooth with parameter $L$. Then for all $\mathbf{w},\mathbf{w}^{\prime}\in\mathbb{R}^{d}$ and $\eta\leq 2/L$ there holds $\\|\mathbf{w}-\eta\partial\ell(\mathbf{w},z)-\mathbf{w}^{\prime}+\eta\partial\ell(\mathbf{w}^{\prime},z)\\|_{2}^{2}\leq\\|\mathbf{w}-\mathbf{w}^{\prime}\\|_{2}^{2}+\frac{1-\alpha}{1+\alpha}(2^{-\alpha}L)^{\frac{2}{1-\alpha}}\eta^{\frac{2}{1-\alpha}}.$ With the above preparation, we are now ready to prove Theorem 7. ###### Proof of Theorem 7. (a) Assume that $S$ and $S^{\prime}$ differ by the $i$-th datum, i.e., $z_{i}\neq z^{\prime}_{i}.$ Let $\\{\mathbf{w}_{t}\\}_{t=1}^{T}$ and $\\{\mathbf{w}^{\prime}_{t}\\}_{t=1}^{T}$ be the sequence produced by SGD update (1) based on $S$ and $S^{\prime}$, respectively. For simplicity, let $c^{2}_{\alpha,2}=\frac{1-\alpha}{1+\alpha}(2^{-\alpha}L)^{\frac{2}{1-\alpha}}$. Note that when $\mathcal{W}=\mathbb{R}^{d}$, Eq. (1) reduces to $\mathbf{w}_{t+1}=\mathbf{w}_{t}-\eta\partial\ell(\mathbf{w}_{t},z_{i_{t}})$. For any $t\in[T]$, we consider the following two cases. Case 1: If $i_{t}\neq i$, Lemma 21 implies that $\\|\mathbf{w}_{t+1}-\mathbf{w}^{\prime}_{t+1}\\|_{2}^{2}=\\|\mathbf{w}_{t}-\eta_{t}\partial\ell(\mathbf{w}_{t},z_{i_{t}})-\mathbf{w}^{\prime}_{t}+\eta_{t}\partial\ell(\mathbf{w}^{\prime}_{t},z_{i_{t}})\\|_{2}^{2}\leq\\|\mathbf{w}_{t}-\mathbf{w}^{\prime}_{t}\\|_{2}^{2}+c^{2}_{\alpha,2}\eta_{t}^{\frac{2}{1-\alpha}}.$ Case 2: If $i_{t}=i$, it follows from the elementary inequality $(a+b)^{2}\leq(1+p)a^{2}+(1+1/p)b^{2}$ that $\displaystyle\\|\mathbf{w}_{t+1}-\mathbf{w}^{\prime}_{t+1}\\|_{2}^{2}$ $\displaystyle=\\|\mathbf{w}_{t}-\eta_{t}\partial\ell(\mathbf{w}_{t},z_{i})-\mathbf{w}^{\prime}_{t}+\eta_{t}\partial\ell(\mathbf{w}^{\prime}_{t},z^{\prime}_{i})\\|_{2}^{2}$ $\displaystyle\leq(1+p)\\|\mathbf{w}_{t}-\mathbf{w}^{\prime}_{t}\\|_{2}^{2}+(1+1/p)\eta_{t}^{2}\\|\partial\ell(\mathbf{w}^{\prime}_{t},z^{\prime}_{i})-\partial\ell(\mathbf{w}_{t},z_{i})\\|_{2}^{2}.$ According to the definition of Hölder smoothness and Lemma 20, we know $\\|\partial\ell(\mathbf{w}_{t},z)\\|_{2}\leq M+L\Big{(}C_{\alpha}\sum_{j=1}^{t-1}\eta_{j}\Big{)}^{\frac{\alpha}{2}}:=c_{\alpha,t}.$ (9) Combining the above two cases and (9) together, we have $\\|\mathbf{w}_{t+1}-\mathbf{w}^{\prime}_{t+1}\\|_{2}^{2}\leq(1+p)^{\mathbb{I}_{[i_{t}=i]}}\\|\mathbf{w}_{t}-\mathbf{w}^{\prime}_{t}\\|_{2}^{2}+c^{2}_{\alpha,2}\eta_{t}^{\frac{2}{1-\alpha}}+4(1+1/p)\mathbb{I}_{[i_{t}=i]}c^{2}_{\alpha,t}\eta_{t}^{2},$ where $\mathbb{I}_{[i_{t}=i]}$ is the indicator function, i.e., $\mathbb{I}_{[i_{t}=i]}=1$ if $i_{t}=i$ and $0$ otherwise. Applying the above inequality recursively, we get $\\|\mathbf{w}_{t+1}-\mathbf{w}^{\prime}_{t+1}\\|_{2}^{2}\leq\prod_{k=1}^{t}(1+p)^{\mathbb{I}_{[i_{k}=i]}}\\|\mathbf{w}_{1}-\mathbf{w}^{\prime}_{1}\\|_{2}^{2}+\Big{(}c^{2}_{\alpha,2}\sum_{k=1}^{t}\eta_{k}^{\frac{2}{1-\alpha}}+4\sum_{k=1}^{t}c_{\alpha,k}^{2}\eta_{k}^{2}(1+1/p){\mathbb{I}_{[i_{k}=i]}}\Big{)}\prod_{j=k+1}^{t}(1+p)^{\mathbb{I}_{[i_{j}=i]}}.$ Since $\mathbf{w}_{1}=\mathbf{w}_{1}^{\prime}$ and $\eta_{t}=\eta$, we further get $\displaystyle\\|\mathbf{w}_{t+1}-\mathbf{w}^{\prime}_{t+1}\\|_{2}^{2}$ $\displaystyle\leq\prod_{j=2}^{t}(1+p)^{\mathbb{I}_{[i_{j}=i]}}\Big{(}c^{2}_{\alpha,2}t\eta^{\frac{2}{1-\alpha}}+4\eta^{2}\sum_{k=1}^{t}c^{2}_{\alpha,k}(1+1/p){\mathbb{I}_{[i_{k}=i]}}\Big{)}$ $\displaystyle\leq(1+p)^{\sum_{j=2}^{t}\mathbb{I}_{[i_{j}=i]}}\Big{(}c^{2}_{\alpha,2}t\eta^{\frac{2}{1-\alpha}}+4c^{2}_{\alpha,t}\eta^{2}(1+1/p)\sum_{k=1}^{t}\mathbb{I}_{[i_{k}=i]}\Big{)}.$ (10) Applying Lemma 14 with $X_{j}=\mathbb{I}_{[i_{j}=i]}$ and $X=\sum_{j=1}^{t}X_{j}$, for any $\exp(-t/3n)\leq\gamma\leq 1$, with probability at least $1-\frac{\gamma}{n}$, there holds $\sum_{j=1}^{t}\mathbb{I}_{[i_{j}=i]}\leq\frac{t}{n}\Big{(}1+\frac{\sqrt{3\log(1/\gamma)}}{\sqrt{t/n}}\Big{)}.$ For any $0<\gamma<\exp(-t/3n)$, with probability at least $1-\frac{\gamma}{n}$, there holds $\sum_{j=1}^{t}\mathbb{I}_{[i_{j}=i]}\leq\frac{t}{n}\Big{(}1+\frac{3\log(1/\gamma)}{t/n}\Big{)}.$ Plug the above two inequalities back into (3.1), and let $c_{\gamma,t}=\max\Big{\\{}\sqrt{\frac{3\log(n/{\gamma})}{t/n}},\frac{3\log(n/{\gamma})}{t/n}\Big{\\}}$. Then, for any $\gamma\in(0,1)$, with probability at least $1-\frac{\gamma}{n}$, we have $\\|\mathbf{w}_{t+1}-\mathbf{w}^{\prime}_{t+1}\\|_{2}^{2}\leq(1+p)^{\frac{t}{n}(1+c_{\gamma,t})}\Big{(}c^{2}_{\alpha,2}t\eta^{\frac{2}{1-\alpha}}+4c^{2}_{\alpha,t}\eta^{2}(1+1/p)\frac{t}{n}(1+c_{\gamma,t})\Big{)}.$ Let $p=\frac{1}{\frac{t}{n}(1+c_{\gamma,t})}$. Then we know $(1+p)^{\frac{t}{n}(1+c_{\gamma,t})}\leq e$ and therefore $\displaystyle\\|\mathbf{w}_{t+1}-\mathbf{w}^{\prime}_{t+1}\\|_{2}^{2}\leq e\Big{(}c^{2}_{\alpha,2}t\eta^{\frac{2}{1-\alpha}}+4c^{2}_{\alpha,t}\eta^{2}\Big{(}1+\frac{t}{n}(1+c_{\gamma,t})\Big{)}\frac{t}{n}(1+c_{\gamma,t})\Big{)}.$ (11) This together with the inequality $c^{2}_{\alpha,t}\leq\big{(}M+L(C_{\alpha}t\eta)^{\frac{\alpha}{2}}\big{)}^{2}$ due to Lemma 20, we have, with probability at least $1-\frac{\gamma}{n}$, that $\\|\mathbf{w}_{t+1}-\mathbf{w}^{\prime}_{t+1}\\|_{2}^{2}\leq e\Big{(}c^{2}_{\alpha,2}t\eta^{\frac{2}{1-\alpha}}+4\big{(}M+L(C_{\alpha}t\eta)^{\frac{\alpha}{2}}\big{)}^{2}\eta^{2}\Big{(}1+\frac{t}{n}(1+c_{\gamma,t})\Big{)}\frac{t}{n}(1+c_{\gamma,t})\Big{)}.$ By taking a union bound of probabilities over $i=1,\ldots,n$, with probability at least $1-\gamma$, there holds $\sup_{S\simeq S^{\prime}}\\|\mathbf{w}_{t+1}-\mathbf{w}^{\prime}_{t+1}\\|_{2}^{2}\leq e\Big{(}c^{2}_{\alpha,2}t\eta^{\frac{2}{1-\alpha}}+4\big{(}M+L(C_{\alpha}t\eta)^{\frac{\alpha}{2}}\big{)}^{2}\eta^{2}\Big{(}1+\frac{t}{n}(1+c_{\gamma,t})\Big{)}\frac{t}{n}(1+c_{\gamma,t})\Big{)}.$ Let $\Delta_{SGD}(\gamma)=\Big{(}e\big{(}c^{2}_{\alpha,2}T\eta^{\frac{2}{1-\alpha}}+4\big{(}M+L(C_{\alpha}T\eta)^{\frac{\alpha}{2}}\big{)}^{2}\eta^{2}\Big{(}1+\frac{T}{n}(1+c_{\gamma,T})\Big{)}\frac{T}{n}(1+c_{\gamma,T})\big{)}\Big{)}^{1/2}$. Recall that $\mathcal{A}$ is the SGD with $T$ iterations, and $\bar{\mathbf{w}}=\frac{1}{T}\sum_{t=1}^{T}\mathbf{w}_{t}$ is the output produced by $\mathcal{A}$. Hence, $\sup_{S\simeq S^{\prime}}\delta_{\mathcal{A}}(S,S^{\prime})=\sup_{S\simeq S^{\prime}}\\|\bar{\mathbf{w}}-\bar{\mathbf{w}}^{\prime}\\|_{2}$. By the convexity of the $\ell_{2}$-norm, with probability at least $1-\gamma$, we have $\sup_{S\simeq S^{\prime}}\delta_{\mathcal{A}}(S,S^{\prime})\leq\frac{1}{T}\sum_{t=1}^{T}\sup_{S\simeq S^{\prime}}\\|\mathbf{w}_{t}-\mathbf{w}_{t}^{\prime}\\|_{2}\leq\Delta_{SGD}(\gamma).$ This completes the proof of part (a). (b) For the case $\mathcal{W}\subseteq\mathcal{B}(0,R)$, the analysis is similar to the case $\mathcal{W}=\mathbb{R}^{d}$ except using a different estimate for the term $\\|\partial\ell(\mathbf{w}_{t},z)\\|_{2}$. Indeed, in this case we have $\\|\mathbf{w}_{t}\\|_{2}\leq R$, which together with the Hölder smoothness, implies $\\|\partial\ell(\mathbf{w}_{t},z)\\|_{2}\leq M+LR^{\alpha}$ for any $t\in[T]$ and $z\in\mathcal{Z}$. Now, replacing $c_{\alpha,t}=M+LR^{\alpha}$ in (9) and putting $c_{\alpha,t}$ back into (11), with probability at least $1-\frac{\gamma}{n}$, we obtain $\sup_{S\simeq S^{\prime}}\\|\mathbf{w}_{t+1}-\mathbf{w}^{\prime}_{t+1}\\|_{2}^{2}\leq e\Big{(}c^{2}_{\alpha,2}t\eta^{\frac{2}{1-\alpha}}+4\big{(}M+LR^{\alpha}\big{)}^{2}\eta^{2}\Big{(}1+\frac{t}{n}(1+c_{\gamma,t})\Big{)}\frac{t}{n}(1+c_{\gamma,t})\Big{)}.$ Now, let $\tilde{\Delta}_{SGD}(\gamma)=\Big{(}e\big{(}c^{2}_{\alpha,2}T\eta^{\frac{2}{1-\alpha}}+4\big{(}M+LR^{\alpha}\big{)}^{2}\eta^{2}\Big{(}1+\frac{T}{n}(1+c_{\gamma,T})\Big{)}\frac{T}{n}(1+c_{\gamma,T})\big{)}\Big{)}^{1/2}.$ The convexity of a norm implies, with probability at least $1-\gamma$, that $\sup_{S\simeq S^{\prime}}\delta_{\mathcal{A}}(S,S^{\prime})\leq\frac{1}{T}\sum_{t=1}^{T}\sup_{S\simeq S^{\prime}}\\|\mathbf{w}_{t}-\mathbf{w}_{t}^{\prime}\\|_{2}\leq\tilde{\Delta}_{SGD}(\gamma).$ The proof of the theorem is completed. ∎ ### 3.2 Proofs on Differentially Private SGD with Output Perturbation In this subsection, we prove the privacy and utility guarantees for output perturbation (i.e. Algorithm 1). We consider both the unbounded domain $\mathcal{W}=\mathbb{R}^{d}$ and bounded domain $\mathcal{W}\subseteq\mathcal{B}(0,R)$. We first prove Theorem 8 on the privacy guarantee of Algorithm 1. ###### Proof of Theorem 8. Let $\mathcal{A}$ be the SGD with $T$ iterations, $\bar{\mathbf{w}}=\frac{1}{T}\sum_{t=1}^{T}\mathbf{w}_{t}$ be the output of $\mathcal{A}$. First, consider the unbounded domain case, i.e., $\mathcal{W}=\mathbb{R}^{d}$. Let $I=\\{i_{1},\ldots,i_{T}\\}$ be the sequence of sampling after $T$ iterations in $\mathcal{A}$. Define $\mathcal{B}=\big{\\{}I:\sup_{S\simeq S^{\prime}}\delta_{\mathcal{A}}(S,S^{\prime})\leq\Delta_{\text{SGD}}(\delta/2)\big{\\}}.$ Part (a) in Theorem 7 implies that $\mathbb{P}(I\in\mathcal{B})\geq 1-\delta/2$. Further, according to the definitions, we know the $\ell_{2}$-sensitivity of $\mathcal{A}$ is identical to the UAS of $\mathcal{A}$. Thus, if $I\in\mathcal{B}$, then Lemma 2 with $\delta^{\prime}=\delta/2$ implies Algorithm 1 satisfies $(\epsilon,\delta/2)$-DP. For any neighboring datasets $S$ and $S^{\prime}$, let $\mathbf{w}_{\text{priv}}$ and $\mathbf{w}^{\prime}_{\text{priv}}$ be the output produced by Algorithm 1 based on $S$ and $S^{\prime}$, respectively. Hence, for any $E\subseteq\mathbb{R}^{d}$ we have $\displaystyle\mathbb{P}(\mathbf{w}_{\text{priv}}\in E)$ $\displaystyle=\mathbb{P}(\mathbf{w}_{\text{priv}}\in E\cap I\in\mathcal{B})+\mathbb{P}(\mathbf{w}_{\text{priv}}\in E\cap I\in\mathcal{B}^{c})$ $\displaystyle\leq\mathbb{P}(\mathbf{w}_{\text{priv}}\in E\|I\in\mathcal{B})\mathbb{P}(I\in\mathcal{B})+\frac{\delta}{2}\leq\Big{(}e^{\epsilon}\mathbb{P}(\mathbf{w}_{\text{priv}}^{\prime}\in E\|I\in\mathcal{B})+\frac{\delta}{2}\Big{)}\mathbb{P}(I\in\mathcal{B})+\frac{\delta}{2}$ $\displaystyle\leq e^{\epsilon}\mathbb{P}(\mathbf{w}_{\text{priv}}^{\prime}\in E\cap I\in\mathcal{B})+\delta\leq e^{\epsilon}\mathbb{P}(\mathbf{w}_{\text{priv}}^{\prime}\in E)+\delta,$ where in the second inequality we have used the definition of DP. Therefore, Algorithm 1 satisfies $(\epsilon,\delta)$-DP when $\mathcal{W}=\mathbb{R}^{d}$. The bounded domain case can be proved in a similar way by using part (b) of Theorem 7. The proof is completed. ∎ Now, we turn to the utility guarantees of Algorithm 1. Recall that the excess population risk $\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}(\mathbf{w}^{})$ can be decomposed as follows ($\bar{\mathbf{w}}=\frac{1}{T}\sum_{t=1}^{T}\mathbf{w}_{t}$) $\displaystyle\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}(\mathbf{w}^{})$ $\displaystyle=[\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}(\bar{\mathbf{w}})]+[\mathcal{R}(\bar{\mathbf{w}})-\mathcal{R}_{S}(\bar{\mathbf{w}})]+[\mathcal{R}_{S}(\bar{\mathbf{w}})-\mathcal{R}_{S}(\mathbf{w}^{})]+[\mathcal{R}_{S}(\mathbf{w}^{})-\mathcal{R}(\mathbf{w}^{})].$ (12) We now introduce three lemmas to control the first three terms on the right hand side of (12). The following lemma controls the error resulting from the added noise. ###### Lemma 22. Suppose the loss function $\ell$ is nonnegative, convex and $\alpha$-Hölder smooth with parameter $L$. Let $\mathbf{w}_{\text{priv}}$ be the output produced by Algorithm 1 based on the dataset $S=\\{z_{1},\cdots,z_{n}\\}$ with $\eta_{t}=\eta<\min\\{1,1/L\\}$. Then for any $\gamma\in(4\exp(-d/8),1)$, the following statements hold true. 1. (a) If $\mathcal{W}=\mathbb{R}^{d}$, then, with probability at least $1-\frac{\gamma}{4}$, there holds $\displaystyle\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}(\bar{\mathbf{w}})=\mathcal{O}\Big{(}(T\eta)^{\frac{\alpha}{2}}\sigma\sqrt{d}(\log(1/\gamma))^{\frac{1}{4}}+\sigma^{1+\alpha}d^{\frac{1+\alpha}{2}}(\log(1/\gamma))^{\frac{1+\alpha}{4}}\Big{)}.$ 2. (b) If $\mathcal{W}\subseteq\mathcal{B}(0,R)$ with $R>0$, then, with probability at least $1-\frac{\gamma}{4}$, we have $\displaystyle\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}(\bar{\mathbf{w}})=\mathcal{O}\Big{(}\sigma\sqrt{d}(\log(1/\gamma))^{\frac{1}{4}}+\sigma^{1+\alpha}d^{\frac{1+\alpha}{2}}(\log(1/\gamma))^{\frac{1+\alpha}{4}}\Big{)}.$ ###### Proof. (a) First, we consider the case $\mathcal{W}=\mathbb{R}^{d}$. Note that $\displaystyle\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}(\bar{\mathbf{w}})$ $\displaystyle=\mathbb{E}_{z}[\ell(\mathbf{w}_{\text{priv}},z)-\ell(\bar{\mathbf{w}},z)]\leq\mathbb{E}_{z}[\langle\partial\ell(\mathbf{w}_{\text{priv}},z),\mathbf{w}_{\text{priv}}-\bar{\mathbf{w}}\rangle]$ $\displaystyle\leq\mathbb{E}_{z}[\\|\partial\ell(\mathbf{w}_{\text{priv}},z)\\|_{2}\\|\mathbf{b}\\|_{2}]\leq(M+L\\|\mathbf{w}_{\text{priv}}\\|_{2}^{\alpha})\\|\mathbf{b}\\|_{2}$ $\displaystyle\leq(M+L\\|\bar{\mathbf{w}}\\|^{\alpha}_{2})\\|\mathbf{b}\\|_{2}+L\\|\mathbf{b}\\|_{2}^{1+\alpha},$ (13) where the first inequality is due to the convexity of $\ell$, the second inequality follows from the Cauchy-Schwartz inequality, the third inequality is due to the definition of Hölder smoothness, and the last inequality uses $\mathbf{w}_{\text{priv}}=\bar{\mathbf{w}}+\mathbf{b}$. Hence, to estimate $\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}(\bar{\mathbf{w}})$, it suffices to bound $\\|\mathbf{b}\\|_{2}$ and $\\|\bar{\mathbf{w}}\\|_{2}$. Since $\mathbf{b}\sim\mathcal{N}(0,\sigma^{2}\mathbf{I})$, then for any $\gamma\in(\exp(-d/8),1)$, Lemma 15 implies, with probability at least $1-\frac{\gamma}{4}$, that $\\|\mathbf{b}\\|_{2}\leq\sigma\sqrt{d}\Big{(}1+\Big{(}\frac{8}{d}\log\big{(}4/\gamma\big{)}\Big{)}^{\frac{1}{4}}\Big{)}.$ (14) Further, by the convexity of a norm and Lemma 20, we know $\\|\bar{\mathbf{w}}\\|_{2}\leq\frac{1}{T}\sum_{t=1}^{T}\\|\mathbf{w}_{t}\\|_{2}\leq\big{(}C_{\alpha}T\eta\big{)}^{\frac{1}{2}}.$ (15) Putting the above inequality and (14) back into (3.2) yields $\displaystyle\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}(\bar{\mathbf{w}})$ $\displaystyle\leq\big{(}M+L(C_{\alpha}T\eta)^{\frac{\alpha}{2}}\big{)}\sigma\sqrt{d}\Big{(}1+\Big{(}\frac{8}{d}\log\big{(}4/\gamma\big{)}\Big{)}^{\frac{1}{4}}\Big{)}+L\sigma^{1+\alpha}d^{\frac{1+\alpha}{2}}\Big{(}1+\Big{(}\frac{8}{d}\log\big{(}4/\gamma\big{)}\Big{)}^{\frac{1}{4}}\Big{)}^{1+\alpha}$ $\displaystyle=\mathcal{O}\Big{(}(T\eta)^{\frac{\alpha}{2}}\sigma\sqrt{d}\big{(}\log(1/\gamma)\big{)}^{\frac{1}{4}}+\sigma^{1+\alpha}d^{\frac{1+\alpha}{2}}\big{(}\log(1/\gamma)\big{)}^{\frac{1+\alpha}{4}}\Big{)}.$ This completes the proof of part (a). (b) The proof for the unbounded domain case is similar to that of the bounded domain. Since $\\|\mathbf{w}_{t}\\|_{2}\leq R$ for $t\in[T]$ in this case, then $\\|\bar{\mathbf{w}}\\|_{2}\leq\frac{1}{T}\sum_{t=1}^{T}\\|\mathbf{w}_{t}\\|_{2}\leq R.$ (16) Plugging (16) and (14) back into (3.2) yield the result in part (b). ∎ In the following lemma, we use the stability of SGD to control the generalization error $\mathcal{R}(\bar{\mathbf{w}})-\mathcal{R}_{S}(\bar{\mathbf{w}})$. ###### Lemma 23. Suppose the loss function $\ell$ is nonnegative, convex, and $\alpha$-Hölder smooth with parameter $L$. Let $\mathcal{A}$ be the SGD with $T$ iterations and $\eta_{t}=\eta<\min\\{1,1/L\\}$ based on the dataset $S=\\{z_{1},\cdots,z_{n}\\}$, and $\bar{\mathbf{w}}=\frac{1}{T}\sum_{t=1}^{T}\mathbf{w}_{t}$ be the output produced by $\mathcal{A}$. Then for any $\gamma\in(4\delta,1)$, the following statements hold true. 1. (a) If $\mathcal{W}=\mathbb{R}^{d}$, then, with probability at least $1-\frac{\gamma}{4}$, there holds $\displaystyle\mathcal{R}(\bar{\mathbf{w}})-\mathcal{R}_{S}(\bar{\mathbf{w}})=\mathcal{O}\Big{(}(T\eta)^{\frac{\alpha}{2}}\Delta_{\text{SGD}}(\delta/2)\log(n)\log(1/\gamma)+(T\eta)^{\frac{1+\alpha}{2}}\sqrt{n^{-\frac{1}{2}}\log(1/\gamma)}\Big{)}.$ 2. (b) If $\mathcal{W}\subseteq\mathcal{B}(0,R)$ with $R>0$, then, with probability at least $1-\frac{\gamma}{4}$, we have $\displaystyle\mathcal{R}(\bar{\mathbf{w}})-\mathcal{R}_{S}(\bar{\mathbf{w}})=\mathcal{O}\Big{(}\tilde{\Delta}_{\text{SGD}}(\delta/2)\log(n)\log(1/\gamma)+\sqrt{n^{-\frac{1}{2}}\log(1/\gamma)}\Big{)}.$ ###### Proof. (a) Consider the unbounded domain case. Part (a) in Theorem 7 implies, with probability at least $1-\frac{\delta}{2}$, that $\sup_{S\simeq S^{\prime}}\delta_{\mathcal{A}}(S,S^{\prime})\leq\Delta_{\text{SGD}}(\delta/2).$ (17) Since $\gamma\geq 4\delta$, then we know (17) holds with probability at least $1-\frac{\gamma}{8}$. According to the result $\\|\bar{\mathbf{w}}\\|_{2}\leq\sqrt{C_{\alpha}T\eta}$ by (15) and Lemma 1 with $G=\sqrt{C_{\alpha}T\eta}$ together, we derive the following inequality with probability at least $1-\frac{\gamma}{8}-\frac{\gamma}{8}=1-\frac{\gamma}{4}$ $\displaystyle\mathcal{R}(\bar{\mathbf{w}})-\mathcal{R}_{S}(\bar{\mathbf{w}})$ $\displaystyle\leq c\bigg{(}(M+L(C_{\alpha}T\eta)^{\frac{\alpha}{2}})\Delta_{\text{SGD}}(\delta/2)\log(n)\log({8}/{\gamma})+\big{(}\sup_{z\in\mathcal{Z}}\ell(0,z)+(M+L(T\eta)^{\frac{\alpha}{2}})\sqrt{T\eta}\big{)}\sqrt{\frac{\log({8}/{\gamma})}{n}}\bigg{)}$ $\displaystyle=\mathcal{O}\bigg{(}(T\eta)^{\frac{\alpha}{2}}\Delta_{\text{SGD}}(\delta/2)\log(n)\log(1/\gamma)+(T\eta)^{\frac{1+\alpha}{2}}\sqrt{\frac{\log(1/\gamma)}{n}}\bigg{)},$ where $c>0$ is a constant. The proof of part (a) is completed. (b) For the case $\mathcal{W}\subseteq\mathcal{B}(0,R)$, the proof follows a similar argument as part (a). Indeed, part (b) in Theorem 7 implies, with probability at least $1-\frac{\gamma}{8}$, that $\sup_{S\simeq S^{\prime}}\delta_{\mathcal{A}}(S,S^{\prime})\leq\tilde{\Delta}_{\text{SGD}}(\delta/2).$ (18) Note that $\\|\bar{\mathbf{w}}\\|_{2}\leq R$ in this case, then combining (18) and Lemma 1 with $G=R$ together, with probability at least $1-\frac{\gamma}{4}$, we have $\displaystyle\mathcal{R}(\bar{\mathbf{w}})-\mathcal{R}_{S}(\bar{\mathbf{w}})$ $\displaystyle\leq c\bigg{(}(M+LR^{\alpha})\tilde{\Delta}_{\text{SGD}}(\delta/2)\log(n)\log({8}/{\gamma})+\big{(}\sup_{z\in\mathcal{Z}}\ell(0,z)+(M+LR^{\alpha})R\big{)}\sqrt{\frac{\log({8}/{\gamma})}{n}}\bigg{)}$ $\displaystyle=\mathcal{O}\bigg{(}\tilde{\Delta}_{\text{SGD}}(\delta/2)\log(n)\log(1/\gamma)+\sqrt{\frac{\log(1/\gamma)}{n}}\bigg{)},$ where $c>0$ is a constant. This completes the proof of part (b). ∎ In the following lemma, we use techniques in optimization theory to control the optimization error $\mathcal{R}_{S}(\bar{\mathbf{w}})-\mathcal{R}_{S}(\mathbf{w}^{})$. ###### Lemma 24. Suppose the loss function $\ell$ is nonnegative, convex and $\alpha$-Hölder smooth with parameter $L$. Let $\mathcal{A}$ be the SGD with $T$ iterations and $\eta_{t}=\eta<\min\\{1,1/L\\}$ based on the dataset $S=\\{z_{1},\cdots,z_{n}\\}$, and $\bar{\mathbf{w}}=\frac{1}{T}\sum_{t=1}^{T}\mathbf{w}_{t}$ be the output produced by $\mathcal{A}$. Then, for any $\gamma\in(0,1)$, the following statements hold true. 1. (a) If $\mathcal{W}=\mathbb{R}^{d}$, then, with probability at least $1-\frac{\gamma}{4}$, there holds $\mathcal{R}_{S}(\bar{\mathbf{w}})-\mathcal{R}_{S}(\mathbf{w}^{})=\mathcal{O}\bigg{(}\eta^{\frac{1+\alpha}{2}}T^{\frac{\alpha}{2}}\sqrt{\log(1/\gamma)}+\\|\mathbf{w}^{}\\|_{2}^{1+\alpha}\sqrt{\frac{\log(1/\gamma)}{T}}+\frac{\\|\mathbf{w}^{}\\|_{2}^{2}}{\eta T}+\\|\mathbf{w}^{}\\|_{2}^{1+\alpha}\eta\bigg{)}.$ 2. (b) If $\mathcal{W}\subseteq\mathcal{B}(0,R)$ with $R>0$, then, with probability at least $1-\frac{\gamma}{4}$, we have $\mathcal{R}_{S}(\bar{\mathbf{w}})-\mathcal{R}_{S}(\mathbf{w}^{})=\mathcal{O}\bigg{(}\\|\mathbf{w}^{}\\|_{2}^{1+\alpha}\sqrt{\frac{\log(1/\gamma)}{T}}+\frac{\\|\mathbf{w}^{}\\|_{2}^{2}}{\eta T}+\\|\mathbf{w}^{}\\|_{2}^{1+\alpha}\eta\bigg{)}.$ ###### Proof. (a) We first consider the case $\mathcal{W}=\mathbb{R}^{d}$. From the convexity of $\ell$, we have $\displaystyle\mathcal{R}_{S}(\bar{\mathbf{w}})-\mathcal{R}_{S}(\mathbf{w}^{})$ $\displaystyle\leq\frac{1}{T}\sum_{t=1}^{T}\mathcal{R}_{S}(\mathbf{w}_{t})-\mathcal{R}_{S}(\mathbf{w}^{})$ $\displaystyle=\frac{1}{T}\sum_{t=1}^{T}[\mathcal{R}_{S}(\mathbf{w}_{t})-\ell(\mathbf{w}_{t},z_{i_{t}})]+\frac{1}{T}\sum_{t=1}^{T}[\ell(\mathbf{w}^{},z_{i_{t}})-\mathcal{R}_{S}(\mathbf{w}^{})]+\frac{1}{T}\sum_{t=1}^{T}[\ell(\mathbf{w}_{t},z_{i_{t}})-\ell(\mathbf{w}^{},z_{i_{t}})].$ (19) First, we consider the upper bound of $\frac{1}{T}\sum_{t=1}^{T}[\mathcal{R}_{S}(\mathbf{w}_{t})-\ell(\mathbf{w}_{t};z_{i_{t}})]$. Since $\\{z_{i_{t}}\\}$ is uniformly sampled from the dataset $S$, then for all $t=1,\ldots,T$ we obtain $\mathbb{E}_{z_{i_{t}}}[\ell(\mathbf{w}_{t},z_{i_{t}})\|\mathbf{w}_{1},...,\mathbf{w}_{t-1}]=\mathcal{R}_{S}(\mathbf{w}_{t}).$ By the convexity of $\ell$, the definition of Hölder smoothness and Lemma 20, for any $z\in\mathcal{Z}$ and all $t\in[T]$, there holds $\displaystyle\ell(\mathbf{w}_{t},z)$ $\displaystyle\leq\sup_{z}\ell(0,z)+\langle\partial\ell(\mathbf{w}_{t},z),\mathbf{w}_{t}\rangle\leq\sup_{z}\ell(0,z)+\\|\partial\ell(\mathbf{w}_{t},z)\\|_{2}\\|\mathbf{w}_{t}\\|_{2}$ $\displaystyle\leq\sup_{z}\ell(0,z)+(M+L\\|\mathbf{w}_{t}\\|_{2}^{\alpha})\\|\mathbf{w}_{t}\\|_{2}\leq\sup_{z}\ell(0,z)+M(C_{\alpha}T\eta)^{\frac{1}{2}}+L(C_{\alpha}T\eta)^{\frac{1+\alpha}{2}}.$ (20) Similarly, for any $z\in\mathcal{Z}$, we have $\displaystyle\ell(\mathbf{w}^{},z)\leq\sup_{z}\ell(0;z)+M\\|\mathbf{w}^{}\\|_{2}+L\\|\mathbf{w}^{}\\|_{2}^{1+\alpha}.$ (21) Now, combining Lemma 17 with (3.2) and noting $\eta>1/T$, we get the following inequality with probability at least $1-\frac{\gamma}{8}$ $\displaystyle\frac{1}{T}\sum_{t=1}^{T}[\mathcal{R}_{S}(\mathbf{w}_{t})-\ell(\mathbf{w}_{t},z_{i_{t}})]$ $\displaystyle\leq\big{(}\sup_{z}\ell(0,z)+M(C_{\alpha}T\eta)^{\frac{1}{2}}+L(C_{\alpha}T\eta)^{\frac{1+\alpha}{2}}\big{)}\sqrt{\frac{2\log(\frac{8}{\gamma})}{T}}=\mathcal{O}\Big{(}\eta^{\frac{1+\alpha}{2}}T^{\frac{\alpha}{2}}\sqrt{\log(1/\gamma)}\Big{)}.$ (22) According to Lemma 16, with probability at least $1-\frac{\gamma}{8}$, there holds $\displaystyle\frac{1}{T}\sum_{t=1}^{T}[\ell(\mathbf{w}^{};z_{i_{t}})-\mathcal{R}_{S}(\mathbf{w}^{})]$ $\displaystyle\leq\big{(}\sup_{z}\ell(0,z)+M\\|\mathbf{w}^{}\\|_{2}+L\\|\mathbf{w}^{}\\|_{2}^{1+\alpha}\big{)}\sqrt{\frac{\log({8}/{\gamma})}{2T}}=\mathcal{O}\Big{(}\\|\mathbf{w}^{}\\|_{2}^{1+\alpha}\sqrt{\frac{\log(1/\gamma)}{T}}\Big{)}.$ (23) Finally, we consider the term $\frac{1}{T}\sum_{t=1}^{T}[\ell(\mathbf{w}_{t},z_{i_{t}})-\ell(\mathbf{w}^{},z_{t})]$. The update rule implies $\mathbf{w}_{t+1}-\mathbf{w}^{}=\big{(}\mathbf{w}_{t}-\mathbf{w}^{}\big{)}-\eta\partial\ell(\mathbf{w}_{t},z_{i_{t}})$, from which we know $\displaystyle\\|\mathbf{w}_{t+1}-\mathbf{w}^{}\\|_{2}^{2}$ $\displaystyle=\\|\big{(}\mathbf{w}_{t}-\mathbf{w}^{}\big{)}-\eta\partial\ell(\mathbf{w}_{t},z_{i_{t}})\\|_{2}^{2}$ $\displaystyle=\\|\mathbf{w}_{t}-\mathbf{w}^{}\\|_{2}^{2}+\eta^{2}\\|\partial\ell(\mathbf{w}_{t},z_{i_{t}})\\|_{2}^{2}-2\eta\langle\partial\ell(\mathbf{w}_{t},z_{i_{t}}),\mathbf{w}_{t}-\mathbf{w}^{}\rangle.$ It then follows that $\langle\partial\ell(\mathbf{w}_{t},z_{i_{t}}),\mathbf{w}_{t}-\mathbf{w}^{}\rangle=\frac{1}{2\eta}\big{(}\\|\mathbf{w}_{t}-\mathbf{w}^{}\\|_{2}^{2}-\\|\mathbf{w}_{t+1}-\mathbf{w}^{}\\|_{2}^{2}\big{)}+\frac{\eta}{2}\\|\partial\ell(\mathbf{w}_{t},z_{i_{t}})\\|_{2}^{2}.$ Combining the above inequality and the convexity of $\ell$ together, we derive $\displaystyle\frac{1}{T}\sum_{t=1}^{T}[\ell(\mathbf{w}_{t},z_{i_{t}})-\ell(\mathbf{w}^{},z_{i_{t}})]$ $\displaystyle\leq\frac{1}{T}\sum_{t=1}^{T}\Bigl{[}\frac{1}{2\eta}\big{(}\\|\mathbf{w}_{t}-\mathbf{w}^{}\\|_{2}^{2}-\\|\mathbf{w}_{t+1}-\mathbf{w}^{}\\|_{2}^{2}\big{)}+\frac{\eta}{2}\\|\partial\ell(\mathbf{w}_{t},z_{i_{t}})\\|_{2}^{2}\Bigr{]}$ $\displaystyle\leq\frac{1}{2T\eta}\\|\mathbf{w}_{1}-\mathbf{w}^{}\\|_{2}^{2}+\frac{\eta}{2T}\sum_{t=1}^{T}\\|\partial\ell(\mathbf{w}_{t},z_{i_{t}})\\|_{2}^{2}.$ (24) Since $0\leq\frac{2\alpha}{1+\alpha}\leq 1$, Lemma 19 implies the following inequality for any $t=1,\ldots,T$ $\\|\partial\ell(\mathbf{w}_{t};z_{i_{t}})\\|_{2}^{2}\leq c_{\alpha,1}\ell^{\frac{2\alpha}{1+\alpha}}(\mathbf{w}_{t};z_{i_{t}})\leq c_{\alpha,1}\max\\{\ell(\mathbf{w}_{t};z_{i_{t}}),1\\}\leq c_{\alpha,1}\ell(\mathbf{w}_{t};z_{i_{t}})+c_{\alpha,1}.$ Putting $\\|\partial\ell(\mathbf{w}_{t};z_{i_{t}})\\|^{2}_{2}\leq c_{\alpha,1}\ell(\mathbf{w}_{t};z_{i_{t}})+c_{\alpha,1}$ back into (3.2) and noting $\\|\mathbf{w}_{1}\\|_{2}=0$, we have $\displaystyle\frac{1}{T}\sum_{t=1}^{T}[\ell(\mathbf{w}_{t},z_{i_{t}})-\ell(\mathbf{w}^{},z_{i_{t}})]$ $\displaystyle\leq\frac{\\|\mathbf{w}^{}\\|_{2}^{2}}{2\eta T}+\frac{c_{\alpha,1}\eta}{2T}\sum_{t=1}^{T}\ell(\mathbf{w}_{t},z_{i_{t}})+\frac{c_{\alpha,1}\eta}{2}.$ Rearranging the above inequality and using (21), we derive $\displaystyle\frac{1}{T}\sum_{t=1}^{T}[\ell(\mathbf{w}_{t},z_{i_{t}})-\ell(\mathbf{w}^{},z_{i_{t}})]$ $\displaystyle\leq\frac{1}{1-\frac{c_{\alpha,1}\eta}{2}}\Bigl{(}\frac{\\|\mathbf{w}^{}\\|_{2}^{2}}{2\eta T}+\frac{c_{\alpha,1}\eta}{2T}\sum_{t=1}^{T}\ell(\mathbf{w}^{},z_{i_{t}})+\frac{c_{\alpha,1}\eta}{2}\Bigr{)}$ $\displaystyle=\mathcal{O}\Big{(}\frac{\\|\mathbf{w}^{}\\|_{2}^{2}}{\eta T}+\\|\mathbf{w}^{}\\|_{2}^{1+\alpha}\eta\Big{)}.$ (25) Now, plugging (22), (23) and (3.2) back into (3.2), we derive $\mathcal{R}_{S}(\bar{\mathbf{w}})-\mathcal{R}_{S}(\mathbf{w}^{})=\mathcal{O}\Big{(}\eta^{\frac{1+\alpha}{2}}T^{\frac{\alpha}{2}}\sqrt{\log(1/\gamma)}+\\|\mathbf{w}^{}\\|_{2}^{1+\alpha}\sqrt{\frac{\log(1/\gamma)}{T}}+\frac{\\|\mathbf{w}^{}\\|_{2}^{2}}{\eta T}+\\|\mathbf{w}^{}\\|_{2}^{1+\alpha}\eta\Big{)}$ with probability at least $1-\frac{\gamma}{4}$, which completes the proof of part (a). (b) Consider the bounded domain case. Since $\\|\mathbf{w}_{t}\\|_{2}\leq R$ for any $t\in[T]$, then by the convexity of $\ell$ and the definition of Hölder smoothness, for any $z\in\mathcal{Z}$, there holds $\ell(\mathbf{w}_{t},z)\leq\sup_{z}\ell(0,z)+(M+LR^{\alpha})R.$ Combining the above inequality and Lemma 17 together, with probability at least $1-\frac{\gamma}{8}$, we obtain $\displaystyle\frac{1}{T}\sum_{t=1}^{T}[\mathcal{R}_{S}(\mathbf{w}_{t})-\ell(\mathbf{w}_{t},z_{i_{t}})]$ $\displaystyle\leq\big{(}\sup_{z}\ell(0,z)+(M+LR^{\alpha})R\big{)}\sqrt{\frac{2\log(\frac{8}{\gamma})}{T}}=\mathcal{O}\Big{(}\sqrt{\frac{\log(1/\gamma)}{T}}\Big{)}.$ (26) Since $\\|\mathbf{w}_{t+1}-\mathbf{w}^{}\\|^{2}_{2}=\\|\text{Proj}_{\mathcal{W}}\big{(}\mathbf{w}_{t}-\eta\partial\ell(\mathbf{w}_{t},z_{i_{t}})\big{)}-\mathbf{w}^{}\\|_{2}^{2}\leq\\|(\mathbf{w}_{t}-\mathbf{w}^{})-\eta\partial\ell(\mathbf{w}_{t},z_{i_{t}})\\|_{2}^{2}$, then (3.2) also holds true in this case. Putting (26), (23) and (3.2) back into (3.2), with probability at least $1-\frac{\gamma}{4}$, we have $\mathcal{R}_{S}(\bar{\mathbf{w}})-\mathcal{R}_{S}(\mathbf{w}^{})=\mathcal{O}\bigg{(}\\|\mathbf{w}^{}\\|_{2}^{1+\alpha}\sqrt{\frac{\log(1/\gamma)}{T}}+\frac{\\|\mathbf{w}^{}\\|_{2}^{2}}{\eta T}+\\|\mathbf{w}^{}\\|_{2}^{1+\alpha}\eta\bigg{)}.$ The proof is completed. ∎ Now, we are in a position to prove the utility guarantee for DP-SGD-Output algorithm. First, we give the proof for the unbounded domain case (i.e. Theorem 9). ###### Proof of Theorem 9. Note that $\mathcal{R}_{S}(\mathbf{w}^{})-\mathcal{R}(\mathbf{w}^{})=\mathcal{R}_{S}(\mathbf{w}^{})-\mathbb{E}_{S}[\mathcal{R}_{S}(\mathbf{w}^{})]$. By Hoeffding inequality and (21), with probability at least $1-\frac{\gamma}{4}$, there holds $\displaystyle\mathcal{R}_{S}(\mathbf{w}^{})-\mathcal{R}(\mathbf{w}^{})\leq\Big{(}\sup_{z\in\mathcal{Z}}\ell(0,z)+M\\|\mathbf{w}^{}\\|_{2}+L\\|\mathbf{w}^{}\\|_{2}^{1+\alpha}\Big{)}\sqrt{\frac{\log({4}/{\gamma})}{2n}}=\mathcal{O}\Big{(}\\|\mathbf{w}^{}\\|_{2}^{1+\alpha}\sqrt{\frac{\log(1/\gamma)}{n}}\Big{)}.$ (27) Combining part (a) in Lemmas 22, 23, 24 and (27) together, with probability at least $1-\gamma$, the population excess risk can be bounded as follows $\displaystyle\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}(\mathbf{w}^{})$ $\displaystyle=[\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}(\bar{\mathbf{w}})]+[\mathcal{R}(\bar{\mathbf{w}})-\mathcal{R}_{S}(\bar{\mathbf{w}})]+[\mathcal{R}_{S}(\bar{\mathbf{w}})-\mathcal{R}_{S}(\mathbf{w}^{})]+[\mathcal{R}_{S}(\mathbf{w}^{})-\mathcal{R}(\mathbf{w}^{})]$ $\displaystyle=\mathcal{O}\bigg{(}(T\eta)^{\frac{\alpha}{2}}\sigma\sqrt{d}\big{(}\log(1/\gamma)\big{)}^{\frac{1}{4}}+\sigma^{1+\alpha}d^{\frac{1+\alpha}{2}}(\log(1/\gamma))^{\frac{1+\alpha}{4}}+(T\eta)^{\frac{\alpha}{2}}\Delta_{\text{SGD}}(\delta/2)\log(n)\log(1/\gamma)+\eta^{\frac{1+\alpha}{2}}\Big{(}T^{\frac{1+\alpha}{2}}\sqrt{\frac{\log(1/\gamma)}{n}}$ $\displaystyle\qquad+T^{\frac{\alpha}{2}}\sqrt{\log(1/\gamma)}\Big{)}+\frac{\\|\mathbf{w}^{}\\|_{2}^{2}}{\eta T}+\\|\mathbf{w}^{}\\|_{2}^{1+\alpha}\eta+\\|\mathbf{w}^{}\\|_{2}^{1+\alpha}\sqrt{\frac{\log(1/\gamma)}{n}}\bigg{)}.$ (28) Plugging $\Delta_{\text{SGD}}(\delta/2)=\mathcal{O}\Big{(}\sqrt{T}\eta^{\frac{1}{1-\alpha}}+\frac{(T\eta)^{1+\frac{\alpha}{2}}\log(n/\delta)}{n}\Big{)}$ and $\sigma=\mathcal{O}(\frac{\sqrt{\log(1/\delta)}\Delta_{\text{SGD}}(\delta/2)}{\epsilon})$ back into (3.2), we have $\displaystyle\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}(\mathbf{w}^{})$ $\displaystyle=\mathcal{O}\bigg{(}T^{\frac{1+\alpha}{2}}\sqrt{\frac{\log(1/\gamma)}{n}}\eta^{\frac{1+\alpha}{2}}+\frac{T^{1+\alpha}\sqrt{d\log(1/\delta)}\big{(}\log(1/\gamma)\big{)}^{\frac{1}{4}}\log(n/\delta)}{n\epsilon}\eta^{1+\alpha}+\frac{d^{\frac{1+\alpha}{2}}(\log(1/\gamma))^{\frac{1+\alpha}{4}}\big{(}T\log(\frac{1}{\delta})\big{)}^{\frac{1+\alpha}{2}}}{\epsilon^{1+\alpha}}\eta^{\frac{1+\alpha}{1-\alpha}}$ $\displaystyle\qquad+\frac{\big{(}d\log(1/\delta)\big{)}^{\frac{1+\alpha}{2}}(\log(1/\gamma))^{\frac{1+\alpha}{4}}T^{(1+\frac{\alpha}{2})(1+\alpha)}\big{(}\log(n/\delta)\big{)}^{1+\alpha}}{(n\epsilon)^{1+\alpha}}\eta^{(1+\frac{\alpha}{2})(1+\alpha)}+\frac{T^{\frac{1+\alpha}{2}}\sqrt{d\log(\frac{1}{\delta})}\big{(}\log(1/\gamma)\big{)}^{\frac{1}{4}}}{\epsilon}\eta^{\frac{2+\alpha-\alpha^{2}}{2(1-\alpha)}}$ $\displaystyle\qquad+T^{\frac{1+\alpha}{2}}\log(n)\log(1/\gamma)\eta^{\frac{2+\alpha-\alpha^{2}}{2(1-\alpha)}}+\frac{T^{1+\alpha}\log(n/\delta)\log(n)\log(1/\gamma)}{n}\eta^{1+\alpha}+\frac{1}{\eta T}+\eta+\sqrt{\frac{\log(1/\gamma)}{n}}\bigg{)}\cdot\\|\mathbf{w}^{}\\|_{2}^{2}.$ (29) Taking the derivative of $\frac{1}{T\eta}+T^{\frac{1+\alpha}{2}}\sqrt{\frac{\log(1/\gamma)}{n}}\eta^{\frac{1+\alpha}{2}}$ w.r.t $\eta$ and setting it to $0$, then we have $\eta=n^{\frac{1}{3+\alpha}}/\big{(}T(\log(1/\gamma))^{\frac{1}{3+\alpha}}\big{)}$. Putting this $\eta$ back into (3.2), we obtain $\displaystyle\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}(\mathbf{w}^{})$ $\displaystyle=\mathcal{O}\bigg{(}\frac{n^{\frac{(2-\alpha)(1+\alpha)}{2(1-\alpha)(3+\alpha)}}\sqrt{d\log(1/\delta)}}{T^{\frac{1+\alpha}{2(1-\alpha)}}\epsilon\big{(}\log(1/\gamma)\big{)}^{\frac{1+4\alpha-\alpha^{2}}{4(1-\alpha)(3+\alpha)}}}+\frac{\sqrt{d\log(1/\delta)}\log(n/\delta)}{n^{\frac{2}{3+\alpha}}\epsilon\big{(}\log(1/\gamma)\big{)}^{\frac{1+\alpha}{4(3+\alpha)}}}+\Big{(}\frac{\sqrt{d\log(1/\delta)}\log(n/\delta)}{n^{\frac{4+\alpha}{2(3+\alpha)}}\epsilon\big{(}\log(1/\gamma)\big{)}^{\frac{1+\alpha}{4(3+\alpha)}}}\Big{)}^{1+\alpha}$ $\displaystyle\quad+\Big{(}\frac{n^{\frac{1}{(1-\alpha)(3+\alpha)}}\sqrt{d\log(1/\delta)}}{T^{\frac{1+\alpha}{2(1-\alpha)}}\epsilon\big{(}\log(1/\gamma)\big{)}^{\frac{(1+\alpha)^{2}}{4(1-\alpha)(3+\alpha)}}}\Big{)}^{1+\alpha}+\log(n)\log(n/\delta)\big{(}\log(1/\gamma)\big{)}^{\frac{2}{3+\alpha}}\Big{(}\frac{n^{\frac{2+\alpha-\alpha^{2}}{2(3+\alpha)(1-\alpha)}}}{T^{\frac{1+\alpha}{2(1-\alpha)}}}+\frac{1}{n^{\frac{2}{3+\alpha}}}+\frac{1}{n^{\frac{1}{3+\alpha}}}+\frac{n^{\frac{1}{3+\alpha}}}{T}\Big{)}\bigg{)}\cdot\\|\mathbf{w}^{}\\|_{2}^{2}.$ (30) To achieve the best rate with a minimal computational cost, we choose the smallest $T$ such that $\frac{n^{\frac{(2-\alpha)(1+\alpha)}{2(1-\alpha)(3+\alpha)}}}{T^{\frac{1+\alpha}{2(1-\alpha)}}}=\mathcal{O}(\frac{1}{n^{\frac{2}{3+\alpha}}})$, $\frac{n^{\frac{1+\alpha}{(1-\alpha)(3+\alpha)}}}{T^{\frac{(1+\alpha)^{2}}{2(1-\alpha)}}}=\mathcal{O}(\frac{1}{n^{\frac{(4+\alpha)(1+\alpha)}{2(3+\alpha)}}})$ and $\frac{n^{\frac{2+\alpha-\alpha^{2}}{2(3+\alpha)(1-\alpha)}}}{T^{\frac{1+\alpha}{2(1-\alpha)}}}+\frac{1}{n^{\frac{2}{3+\alpha}}}+\frac{n^{\frac{1}{3+\alpha}}}{T}=\mathcal{O}(\frac{1}{n^{\frac{1}{3+\alpha}}})$. Hence, we set $T\asymp n^{\frac{-\alpha^{2}-3\alpha+6}{(1+\alpha)(3+\alpha)}}$ if $0\leq\alpha\leq\frac{\sqrt{73}-7}{4}$, and $T\asymp n$ else. Now, putting the choice of $T$ back into (3.2), we derive $\displaystyle\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}(\mathbf{w}^{})=$ $\displaystyle\mathcal{O}\bigg{(}\frac{\sqrt{d\log(1/\delta)}\log(n/\delta)}{(\log(1/\gamma))^{\frac{1+\alpha}{4(3+\alpha)}}n^{\frac{2}{3+\alpha}}\epsilon}+\Bigl{(}\frac{\sqrt{d\log(1/\delta)}\log(n/\delta)}{(\log(1/\gamma))^{\frac{1+\alpha}{4(3+\alpha)}}n^{\frac{4+\alpha}{2(3+\alpha)}}\epsilon}\Bigr{)}^{1+\alpha}$ $\displaystyle\quad+\frac{\log(n)\big{(}\log(1/\gamma)\big{)}^{\frac{2}{3+\alpha}}\log(n/\delta)}{n^{\frac{1}{3+\alpha}}}\bigg{)}\cdot\\|\mathbf{w}^{}\\|_{2}^{2}.$ Without loss of generality, we assume the first term of the above utility bound is less than 1. Therefore, with probability at least $1-\gamma$, there holds $\displaystyle\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}(\mathbf{w}^{})=\\|\mathbf{w}^{}\\|_{2}^{2}\cdot\mathcal{O}\bigg{(}\frac{\sqrt{d\log(1/\delta)}\log(n/\delta)}{(\log(1/\gamma))^{\frac{1+\alpha}{4(3+\alpha)}}n^{\frac{2}{3+\alpha}}\epsilon}+\frac{\log(n)\big{(}\log(1/\gamma)\big{)}^{\frac{2}{3+\alpha}}\log(n/\delta)}{n^{\frac{1}{3+\alpha}}}\bigg{)}.$ The proof is completed. ∎ Finally, we provide the proof of utility guarantee for the DP-SGD-Output algorithm when $\mathcal{W}\subseteq\mathcal{B}(0,R)$ (i.e. Theorem 10). ###### Proof of Theorem 10. The proof is similar to that of Theorem 9. Indeed, plugging part (b) in Lemmas 22, 23, 24 and (27) back into (12), with probability at least $1-\gamma$, the population excess risk can be bounded as follows $\displaystyle\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}(\mathbf{w}^{})$ $\displaystyle=\mathcal{O}\Big{(}\sigma\sqrt{d}(\log(1/\gamma))^{\frac{1}{4}}+\sigma^{1+\alpha}d^{\frac{1+\alpha}{2}}(\log(1/\gamma))^{\frac{1+\alpha}{4}}+\tilde{\Delta}_{\text{SGD}}(\delta/2)\log(n)\log(1/\gamma)+\sqrt{\frac{\log(1/\gamma)}{n}}$ $\displaystyle\qquad+\frac{\\|\mathbf{w}^{}\\|_{2}^{2}}{T\eta}+\\|\mathbf{w}^{}\\|_{2}^{1+\alpha}\eta+\\|\mathbf{w}^{}\\|_{2}^{1+\alpha}\sqrt{\frac{\log(1/\gamma)}{n}}\Big{)}.$ Note that $\tilde{\Delta}_{\text{SGD}}(\delta/2)=\mathcal{O}(\sqrt{T}\eta^{\frac{1}{1-\alpha}}+\frac{T\eta\log(n/\delta)}{n})$ and $\sigma=\frac{\sqrt{2\log(2.5/\delta)}\tilde{\Delta}_{\text{SGD}}(\delta/2)}{\epsilon}$. Then we have $\displaystyle\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}(\mathbf{w}^{})$ $\displaystyle=\mathcal{O}\biggl{(}\big{(}\frac{T\log(n/\delta))\log(n)\log(1/\gamma)}{n}+\frac{T\sqrt{d\log(1/\delta)}\log(n/\delta)(\log(1/\gamma))^{\frac{1}{4}}}{n\epsilon}\big{)}\eta$ $\displaystyle\qquad+\big{(}\frac{\sqrt{\log(1/\delta)Td}(\log(1/\gamma))^{\frac{1}{4}}}{\epsilon}+\sqrt{T}\log(n)\log(1/\gamma)\big{)}\eta^{\frac{1}{1-\alpha}}+\frac{(Td\log(1/\delta))^{\frac{1+\alpha}{2}}(\log(1/\gamma))^{\frac{1+\alpha}{4}}}{\epsilon^{1+\alpha}}\eta^{\frac{1+\alpha}{1-\alpha}}$ $\displaystyle\qquad+\big{(}\frac{T\sqrt{d\log(1/\delta)}\log(n/\delta)(\log(1/\gamma))^{\frac{1}{4}}}{n\epsilon}\big{)}^{1+\alpha}\eta^{1+\alpha}+\frac{1}{T\eta}+\sqrt{\frac{\log(1/\gamma)}{n}}\biggr{)}\cdot\\|\mathbf{w}^{}\\|_{2}^{2}.$ (31) Consider the tradeoff between $1/\eta$ and $\eta$. Taking the derivative of $\big{(}\frac{T\log(n/\delta)\log(n)\log(1/\gamma)}{n}+\frac{T\sqrt{d\log(1/\delta)}\log(n/\delta)(\log(1/\gamma))^{1/4}}{n\epsilon}\big{)}\eta$ $+\frac{1}{T\eta}$ w.r.t $\eta$ and setting it to $0$, we have $\eta=1/\Big{(}T\max\Big{\\{}\frac{\sqrt{\log(n/\delta)\log(n)\log(1/\gamma)}}{\sqrt{n}},\frac{\big{(}d\log(1/\delta)\big{)}^{1/4}\sqrt{\log(n/\delta)}(\log(1/\gamma))^{1/8}}{\sqrt{n\epsilon}}\Big{\\}}\Big{)}$. Then putting the value of $\eta$ back into (3.2), we obtain $\displaystyle\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}(\mathbf{w}^{})$ $\displaystyle=\mathcal{O}\bigg{(}\frac{\big{(}d\log(1/\delta)\big{)}^{\frac{1}{4}}(\log(1/\gamma))^{\frac{1}{8}}\sqrt{\log(n/\delta)}}{\sqrt{n\epsilon}}+\Bigl{(}\frac{\big{(}d\log(1/\delta)\big{)}^{\frac{1}{4}}(\log(1/\gamma))^{\frac{1}{8}}\sqrt{\log(n/\delta)}}{\sqrt{n\epsilon}}\Bigr{)}^{1+\alpha}$ $\displaystyle\quad+\frac{\big{(}d\log(1/\delta)\big{)}^{\frac{1-2\alpha}{4(1-\alpha)}}(\log(1/\gamma))^{\frac{1-2\alpha}{8(1-\alpha)}}n^{\frac{1}{2(1-\alpha)}}\epsilon^{\frac{2\alpha-1}{2(1-\alpha)}}}{T^{\frac{1+\alpha}{2(1-\alpha)}}(\log(n/\delta))^{\frac{1}{2(1-\alpha)}}}+\Big{(}\frac{\big{(}d\log(1/\delta)\big{)}^{\frac{1-2\alpha}{4(1-\alpha)}}(\log(1/\gamma))^{\frac{1-2\alpha}{8(1-\alpha)}}n^{\frac{1}{2(1-\alpha)}}\epsilon^{\frac{2\alpha-1}{2(1-\alpha)}}}{T^{\frac{1+\alpha}{2(1-\alpha)}}(\log(n/\delta))^{\frac{1}{2(1-\alpha)}}}\Big{)}^{1+\alpha}$ $\displaystyle\quad+\sqrt{\log(n)\log(1/\gamma)\log(1/\delta)}\big{(}\frac{1}{\sqrt{n}}+\frac{n^{\frac{1}{2(1-\alpha)}}}{T^{\frac{1+\alpha}{2(1-\alpha)}}}\big{)}\bigg{)}\cdot\\|\mathbf{w}^{}\\|_{2}^{2}.$ Similarly, we choose the smallest $T$ such that $\frac{n^{\frac{1}{2(1-\alpha)}}}{T^{\frac{1+\alpha}{2(1-\alpha)}}}=\mathcal{O}(\frac{1}{\sqrt{n}})$. Hence, we set $T\asymp n^{\frac{2-\alpha}{1+\alpha}}$ if $\alpha<\frac{1}{2}$, and $T\asymp n$ else. Since $\frac{1}{4}\geq\frac{1-2\alpha}{2(1-\alpha)}$, we have $\displaystyle\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}(\mathbf{w}^{})=\mathcal{O}\bigg{(}$ $\displaystyle\frac{\big{(}d\log(1/\delta)\big{)}^{\frac{1}{4}}(\log(1/\gamma))^{\frac{1}{8}}\sqrt{\log(n/\delta)}}{\sqrt{n\epsilon}}+\bigl{(}\frac{\big{(}d\log(1/\delta)\big{)}^{\frac{1}{4}}(\log(1/\gamma))^{\frac{1}{8}}\sqrt{\log(n/\delta)}}{\sqrt{n\epsilon}}\bigr{)}^{1+\alpha}$ $\displaystyle+\frac{\sqrt{\log(n)\log(1/\gamma)\log(n/\delta)}}{\sqrt{n}}\bigg{)}\cdot\\|\mathbf{w}^{}\\|_{2}^{2}.$ It is reasonable to assume the first term is less than $1$ here. Therefore, with probability at least $1-\gamma$, there holds $\displaystyle\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}(\mathbf{w}^{})=\\|\mathbf{w}^{}\\|_{2}^{2}\cdot\mathcal{O}\Big{(}\frac{\big{(}d\log(1/\delta)\big{)}^{\frac{1}{4}}(\log(1/\gamma))^{\frac{1}{8}}\sqrt{\log(n/\delta)}}{\sqrt{n\epsilon}}+\frac{\sqrt{\log(n)\log(1/\gamma)\log(n/\delta)}}{\sqrt{n}}\Big{)}.$ The proof is completed. ∎ ### 3.3 Proofs on Differential Privacy of SGD with Gradient Perturbation We now turn to the analysis for DP-SGD-Gradient algorithm (i.e. Algorithm 2) and provide the proofs for Theorems 11 and 12. We start with the proof of Theorem 11 on the privacy guarantee for Algorithm 2. ###### Proof of Theorem 11. Consider the mechanism $\mathcal{G}_{t}=\mathcal{M}_{t}+\mathbf{b}_{t}$, where $\mathcal{M}_{t}=\partial\ell(\mathbf{w}_{t},z_{i_{t}})$. For any $\mathbf{w}_{t}\in\mathcal{W}$ and any $z_{i_{t}},z^{\prime}_{i_{t}}\in\mathcal{Z}$, the definition of $\alpha$-Hölder smoothness implies that $\\|\partial\ell(\mathbf{w}_{t},z_{i_{t}})-\partial\ell(\mathbf{w}_{t},z^{\prime}_{i_{t}})\\|_{2}\leq 2\big{(}M+L\\|\mathbf{w}_{t}\\|^{\alpha}_{2}\big{)}\leq 2(M+LR^{\alpha}).$ Therefore, the $\ell_{2}$-sensitivity of $\mathcal{M}_{t}$ is $2(M+LR^{\alpha})$. Let $\sigma^{2}=\frac{14(M+LR^{\alpha})^{2}T}{\beta n^{2}\epsilon}\Big{(}\frac{\log(1/\delta)}{(1-\beta)\epsilon}+1\Big{)}.$ Lemma 3 with $p=\frac{1}{n}$ implies that $\mathcal{G}_{t}$ satisfies $\Big{(}\lambda,\frac{\lambda\beta\epsilon}{T\big{(}\frac{\log(1/\delta)}{(1-\beta)\epsilon}+1\big{)}}\Big{)}$-RDP if the following conditions hold $\displaystyle\frac{\sigma^{2}}{4(M+LR^{\alpha})^{2}}\geq 0.67$ (32) and $\displaystyle\lambda-1\leq\frac{\sigma^{2}}{6(M+LR^{\alpha})^{2}}\log\Big{(}\frac{n}{\lambda(1+\frac{\sigma^{2}}{4(M+LR^{\alpha})^{2}})}\Big{)}.$ (33) Let $\lambda=\frac{\log(1/\delta)}{(1-\beta)\epsilon}+1$. We obtain that $\mathcal{G}_{t}$ satisfies $(\frac{\log(1/\delta)}{(1-\beta)\epsilon}+1,\frac{\beta\epsilon}{T})$-RDP. Then by the post-processing property of DP (see Lemma 6), we know $\mathbf{w}_{t+1}$ also satisfies $(\frac{\log(1/\delta)}{(1-\beta)\epsilon}+1,\frac{\beta\epsilon}{T})$-RDP for any $t=0,...,T-1$. Furthermore, according to the adaptive composition theorem of RDP (see Lemma 4), Algorithm 2 satisfies $(\frac{\log(1/\delta)}{(1-\beta)\epsilon}+1,\beta\epsilon)$-RDP. Finally, by Lemma 5, the output of Algorithm 2 satisfies $(\epsilon,\delta)$-DP as long as (32) and (33) hold. ∎ Now, we turn to the generalization analysis of Algorithm 2. First, we estimate the generalization error $\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}_{S}(\mathbf{w}_{\text{priv}})$ in (4). ###### Lemma 25. Suppose the loss function $\ell$ is nonnegative, convex and $\alpha$-Hölder smooth with parameter $L$. Let $\mathbf{w}_{\text{priv}}$ be the output produced by Algorithm 2 based on $S=\\{z_{1},\cdots,z_{n}\\}$ with $\eta_{t}=\eta<\min\\{1,1/L\\}$. Then for any $\gamma\in(0,1)$, with probability at least $1-\frac{\gamma}{3}$, there holds $\displaystyle\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}_{S}(\mathbf{w}_{\text{priv}})$ $\displaystyle=\mathcal{O}\Big{(}\tilde{\Delta}_{\text{SGD}}(\gamma/6)\log(n)\log(1/\gamma)+\sqrt{\frac{\log(1/\gamma)}{n}}\Big{)}.$ ###### Proof. Part (b) in Theorem 7 implies that $\tilde{\Delta}_{\text{SGD}}(\gamma/6)=\mathcal{O}\Big{(}\sqrt{T}\eta^{\frac{1}{1-\alpha}}+\frac{T\eta\log(n/\gamma)}{n}\Big{)}$ with probability at least $1-\frac{\gamma}{6}$. Since the noise added to the gradient in each iteration is the same for the neighboring datasets $S$ and $S^{\prime}$, the noise addition does not impact the stability analysis. Therefore, the UAS bound of the noisy SGD is equivalent to the SGD. According to Lemma 1 and $\\|\mathbf{w}_{\text{priv}}\\|_{2}\leq R$, we derive the following inequality with probability at least $1-(\frac{\gamma}{6}+\frac{\gamma}{6})$ $\displaystyle\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}_{S}(\mathbf{w}_{\text{priv}})$ $\displaystyle\leq c\Big{(}(M+LR^{\alpha})\tilde{\Delta}_{\text{SGD}}(\gamma/6)\log(n)\log(6/{\gamma})+\big{(}M_{0}+(M+LR^{\alpha}\big{)}R\sqrt{\frac{\log(6/{\gamma})}{n}}\Big{)}$ $\displaystyle=\mathcal{O}\Big{(}\tilde{\Delta}_{\text{SGD}}(\gamma/6)\log(n)\log(1/\gamma)+\sqrt{\frac{\log(1/\gamma)}{n}}\Big{)},$ where $c>0$ is a constant. The proof is completed. ∎ The following lemma gives an upper bound for the second term $\mathcal{R}_{S}(\mathbf{w}_{\text{priv}})-\mathcal{R}_{S}(\mathbf{w}^{})$ in (4). ###### Lemma 26. Suppose the loss function $\ell$ is nonnegative, convex and $\alpha$-Hölder smooth with parameter $L$. Let $\mathbf{w}_{\text{priv}}$ be the output produced by Algorithm 2 based on $S=\\{z_{1},\cdots,z_{n}\\}$ with $\eta_{t}=\eta<\min\\{1,1/L\\}$. Then, for any $\gamma\in(18\exp(-dT/8),1)$, with probability at least $1-\frac{\gamma}{3}$, there holds $\displaystyle\mathcal{R}_{S}(\mathbf{w}_{\text{priv}})-\mathcal{R}_{S}(\mathbf{w}^{})=\mathcal{O}\Big{(}$ $\displaystyle\\|\mathbf{w}^{}\\|_{2}^{1+\alpha}\sqrt{\frac{\log(1/\gamma)}{T}}+\frac{\\|\mathbf{w}^{}\\|_{2}^{2}}{T\eta}+\eta+\frac{\sqrt{\log(1/\delta)\log(1/\gamma)}(\\|\mathbf{w}^{}\\|_{2}+\eta)}{n\epsilon}$ $\displaystyle+\frac{\eta Td\log(\frac{1}{\delta})\sqrt{\log(\frac{1}{\gamma})}}{n^{2}\epsilon^{2}}\Big{)}.$ ###### Proof. To estimate the term $\mathcal{R}_{S}(\mathbf{w}_{\text{priv}})-\mathcal{R}_{S}(\mathbf{w}^{})$, we decompose it as $\displaystyle\mathcal{R}_{S}(\mathbf{w}_{\text{priv}})-\mathcal{R}_{S}(\mathbf{w}^{})$ $\displaystyle\leq\frac{1}{T}\sum_{t=1}^{T}[\mathcal{R}_{S}(\mathbf{w}_{t})-\ell(\mathbf{w}_{t},z_{i_{t}})]+\frac{1}{T}\sum_{t=1}^{T}[\ell(\mathbf{w}^{},z_{i_{t}})-\mathcal{R}_{S}(\mathbf{w}^{})]+\frac{1}{T}\sum_{t=1}^{T}[\ell(\mathbf{w}_{t},z_{i_{t}})-\ell(\mathbf{w}^{},z_{i_{t}})].$ (34) Similar to the analysis in (3.2) and (21), we have $\ell(\mathbf{w}^{},z)=\mathcal{O}(\\|\mathbf{w}^{}\\|_{2}^{1+\alpha})$ for all $z\in\mathcal{Z}$ and $\ell(\mathbf{w}_{t},z)=\mathcal{O}(R+R^{1+\alpha})$ for all $t=1,\ldots,T$ and $z\in\mathcal{Z}$. Therefore, Azuma-Hoeffding inequality (see Lemma 17) yields, with probability at least $1-\frac{\gamma}{9}$, that $\displaystyle\frac{1}{T}\sum_{t=1}^{T}[\mathcal{R}_{S}(\mathbf{w}_{t})-\ell(\mathbf{w}_{t},z_{t})]\leq\big{(}\sup_{z\in{\mathcal{Z}}}\ell(0,z)+\sup_{t=1,\ldots,T;z\in\mathcal{Z}}\ell(\mathbf{w}_{t},z)\big{)}\sqrt{\frac{\log({9}/{\gamma})}{2T}}=\mathcal{O}\Big{(}(R+R^{1+\alpha})\sqrt{\frac{\log(1/\gamma)}{T}}\Big{)}.$ (35) In addition, Hoeffding inequality (see Lemma 16) implies, with probability at least $1-\frac{\gamma}{9}$, that $\displaystyle\frac{1}{T}\sum_{t=1}^{T}[\ell(\mathbf{w}^{},z_{i_{t}})-\mathcal{R}_{S}(\mathbf{w}^{})]\leq(\sup_{z\in{\mathcal{Z}}}\ell(0,z)+\sup_{z\in{\mathcal{Z}}}\ell(\mathbf{w}^{},z))\sqrt{\frac{\log({9}/{\gamma})}{2T}}=\mathcal{O}\Big{(}\\|\mathbf{w}^{}\\|_{2}^{1+\alpha}\sqrt{\frac{\log(1/\gamma)}{T}}\Big{)}.$ (36) Finally, we try to bound $\frac{1}{T}\sum_{t=1}^{T}[\ell(\mathbf{w}_{t},z_{i_{t}})-\ell(\mathbf{w}^{},z_{i_{t}})]$. The SGD update rule implies that $\\|\mathbf{w}_{t+1}-\mathbf{w}^{}\\|_{2}^{2}=\\|\text{Proj}_{\mathcal{W}}\big{(}\mathbf{w}_{t}-\eta(\partial\ell(\mathbf{w}_{t},z_{i_{t}})+\mathbf{b}_{t})\big{)}-\mathbf{w}^{}\\|_{2}^{2}\leq\\|(\mathbf{w}_{t}-\mathbf{w}^{})-\eta(\partial\ell(\mathbf{w}_{t},z_{i_{t}})+\mathbf{b}_{t})\\|_{2}^{2}$, then we have $\langle\mathbf{w}_{t}-\mathbf{w}^{},\partial\ell(\mathbf{w}_{t},z_{i_{t}})\rangle\leq\frac{1}{2\eta}\big{(}\\|\mathbf{w}_{t}-\mathbf{w}^{}\\|_{2}^{2}-\\|\mathbf{w}_{t+1}-\mathbf{w}^{}\\|_{2}^{2}\big{)}+\frac{\eta}{2}\big{(}\\|\partial\ell(\mathbf{w}_{t},z_{i_{t}})\\|_{2}^{2}+\\|\mathbf{b}_{t}\\|_{2}^{2}\big{)}-\langle\mathbf{b}_{t},\mathbf{w}_{t}-\mathbf{w}^{}-\eta\partial\ell(\mathbf{w}_{t},z_{i_{t}})\rangle$. Further, noting $\\|\mathbf{w}_{1}\\|_{2}=0$, then by the convexity of $\ell$ we have $\displaystyle\frac{1}{T}\sum_{t=1}^{T}[\ell(\mathbf{w}_{t},z_{i_{t}})-\ell(\mathbf{w}^{},z_{i_{t}})]\leq\frac{\\|\mathbf{w}^{}\\|_{2}^{2}}{2T\eta}+\frac{\eta}{2T}\sum_{t=1}^{T}\\|\partial\ell(\mathbf{w}_{t},z_{i_{t}})\\|_{2}^{2}-\frac{1}{T}\sum_{t=1}^{T}\langle\mathbf{b}_{t},\mathbf{w}_{t}-\mathbf{w}^{}-\eta\partial\ell(\mathbf{w}_{t},z_{i_{t}})\rangle+\frac{\eta}{2T}\sum_{t=1}^{T}\\|\mathbf{b}_{t}\\|_{2}^{2}.$ The definition of $\alpha$-Hölder smoothness implies that $\\|\partial\ell(\mathbf{w}_{t},z_{i_{t}})\\|_{2}\leq M+L\\|\mathbf{w}_{t}\\|^{\alpha}_{2}\leq M+LR^{\alpha}$ for any $t$. Then, there hold $\displaystyle\frac{\eta}{2T}\sum_{t=1}^{T}\\|\partial\ell(\mathbf{w}_{t},z_{i_{t}})\\|_{2}^{2}$ $\displaystyle\leq\frac{\eta}{2T}\sum_{t=1}^{T}(M+L\\|\mathbf{w}_{t}\\|^{\alpha}_{2})^{2}=\mathcal{O}(\eta),$ and $\\|\mathbf{w}_{t}-\mathbf{w}^{}-\eta\partial\ell(\mathbf{w}_{t},z_{i_{t}})\\|_{2}\leq\\|\mathbf{w}^{}\\|_{2}+R+\eta(M+LR^{\alpha}).$ Since $\mathbf{b}_{t}$ is an $\sigma^{2}$-sub-Gaussian random vector, $\frac{1}{T}\langle\mathbf{b}_{t},\mathbf{w}_{t}-\mathbf{w}^{}-\eta\partial\ell(\mathbf{w}_{t},z_{i_{t}})\rangle$ is an $\frac{\sigma^{2}}{T^{2}}\big{(}\\|\mathbf{w}^{}\\|_{2}+R+\eta(M+LR^{\alpha})\big{)}^{2}$-sub- Gaussian random vector. Note that the sub-Gaussian parameter $\frac{\sigma^{2}}{T^{2}}\big{(}\\|\mathbf{w}^{}\\|_{2}+R+\eta(M+LR^{\alpha})\big{)}^{2}$ is independent of $\mathbf{w}_{t-1}$ and $\mathbf{b}_{t-1}$. Hence, $\frac{1}{T}\sum_{t=1}^{T}\langle\mathbf{b}_{t},\mathbf{w}_{t}-\mathbf{w}^{}-\eta\partial\ell(\mathbf{w}_{t},z_{i_{t}})\rangle$ is an $\frac{\sigma^{2}\sum_{t=1}^{T}(\\|\mathbf{w}^{}\\|_{2}+R+\eta(M+LR^{\alpha}))^{2}}{T^{2}}$-sub- Gaussian random vector. Since $\sigma^{2}=\mathcal{O}(\frac{T\log(1/\delta)}{n^{2}\epsilon^{2}})$, the tail bound of Sub-Gaussian variables (see Lemma 18) implies, with probability at least $1-\frac{\gamma}{18}$, that $\displaystyle\frac{1}{T}\sum_{t=1}^{T}\langle\mathbf{b}_{t},\mathbf{w}_{t}-\mathbf{w}^{}-\eta\partial\ell(\mathbf{w}_{t},z_{i_{t}})\rangle$ $\displaystyle\leq\frac{\Big{(}\sigma^{2}\big{(}\\|\mathbf{w}^{}\\|_{2}+R+\eta(M+LR^{\alpha})\big{)}^{2}\Big{)}^{\frac{1}{2}}}{\sqrt{T}}\sqrt{2\log({18}/{\gamma})}$ $\displaystyle=\mathcal{O}\Big{(}\sigma(\\|\mathbf{w}^{}\\|_{2}+\eta)\sqrt{\frac{\log(1/\gamma)}{T}}\Big{)}=\mathcal{O}\Big{(}\frac{\sqrt{\log(1/\delta)\log(1/\gamma)}(\\|\mathbf{w}^{}\\|_{2}+\eta)}{n\epsilon}\Big{)}.$ According to the Chernoff bound for the $\ell_{2}$-norm of Gaussian vector with $\mathbf{X}=[\mathbf{b}_{11},...,\mathbf{b}_{1d},\mathbf{b}_{21}...,\mathbf{b}_{Td}]\in\mathbb{R}^{Td}$(see Lemma 15), for any $\gamma\in(18\exp(-dT/8),1)$, with probability at least $1-\frac{\gamma}{18}$, there holds $\frac{\eta}{2T}\sum_{t=1}^{T}\\|\mathbf{b}_{t}\\|_{2}^{2}\leq\frac{\eta d}{2T}\Big{(}1+(\frac{1}{d}\log({18}/{\gamma}))^{\frac{1}{2}}\Big{)}T\sigma^{2}=\mathcal{O}\Big{(}\frac{\eta Td\log(\frac{1}{\delta})\sqrt{\log(\frac{1}{\gamma})}}{n^{2}\epsilon^{2}}\Big{)}.$ Therefore, with probability at least $1-\frac{\gamma}{9}$, there holds $\displaystyle\frac{1}{T}\sum_{t=1}^{T}[\ell(\mathbf{w}_{t},z_{i_{t}})-\ell(\mathbf{w}^{},z_{i_{t}})]\leq\mathcal{O}\Big{(}\frac{\\|\mathbf{w}^{}\\|_{2}^{2}}{T\eta}+\eta+\frac{\sqrt{\log(1/\delta)\log(1/\gamma)}(\\|\mathbf{w}^{}\\|_{2}+\eta)}{n\epsilon}+\frac{\eta Td\log(1/\delta)\sqrt{\log(1/\gamma)}}{n^{2}\epsilon^{2}}\Big{)}.$ (37) Putting (35), (36) and (37) back into (34), we obtain, with probability at least $1-\frac{\gamma}{3}$, that $\displaystyle\mathcal{R}_{S}(\mathbf{w}_{\text{priv}})-\mathcal{R}_{S}(\mathbf{w}^{})$ $\displaystyle=\mathcal{O}\Big{(}\\|\mathbf{w}^{}\\|_{2}^{1+\alpha}\sqrt{\frac{\log(1/\gamma)}{T}}+\frac{\\|\mathbf{w}^{}\\|_{2}^{2}}{T\eta}+\eta+\frac{\sqrt{\log({1}/{\delta})\log(1/\gamma)}(\\|\mathbf{w}^{}\\|_{2}+\eta)}{n\epsilon}+\frac{\eta Td\log({1}/{\delta})\sqrt{\log({1}/{\gamma})}}{n^{2}\epsilon^{2}}\Big{)}.$ The proof is completed. ∎ Now, we are ready to prove the utility theorem for DP-SGD-Gradient algorithm. ###### Proof of Theorem 12. The Hoeffding inequality implies, with probability at least $1-\frac{\gamma}{3}$, that $\displaystyle\mathcal{R}_{S}(\mathbf{w}^{})-\mathcal{R}(\mathbf{w}^{})\leq\big{(}\sup_{z\in\mathcal{Z}}\ell(0,z)+\sup_{z\in{\mathcal{Z}}}\ell(\mathbf{w}^{},z)\big{)}\sqrt{\frac{\log({3}/{\gamma})}{2n}}=\mathcal{O}\Big{(}\\|\mathbf{w}^{}\\|_{2}^{1+\alpha}\sqrt{\frac{\log(1/\gamma)}{n}}\Big{)}.$ Combining Lemma 25, Lemma 26 and the above inequality together, with probability at least $1-\gamma$, we obtain $\displaystyle\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}(\mathbf{w}^{})=\mathcal{O}\Big{(}$ $\displaystyle\tilde{\Delta}_{\text{SGD}}(\gamma/6)\log(n)\log(1/\gamma)+\frac{\\|\mathbf{w}^{}\\|_{2}^{2}}{T\eta}+\eta+\frac{\sqrt{\log(1/\delta)\log(1/\gamma)}(\\|\mathbf{w}^{}\\|_{2}+\eta)}{n\epsilon}$ $\displaystyle+\frac{\eta Td\log(\frac{1}{\delta})\sqrt{\log(\frac{1}{\gamma})}}{n^{2}\epsilon^{2}}+\\|\mathbf{w}^{}\\|_{2}^{1+\alpha}\sqrt{\frac{\log(1/\gamma)}{n}}\Big{)}.$ Now, putting $\tilde{\Delta}_{\text{SGD}}(\gamma/6)=\mathcal{O}(\sqrt{T}\eta^{\frac{1}{1-\alpha}}+\frac{T\eta\log(n/\gamma)}{n})$ back into the above estimate, we have $\displaystyle\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}(\mathbf{w}^{})$ $\displaystyle=\mathcal{O}\Big{(}\sqrt{T}\log(n)\log(1/\gamma)\eta^{\frac{1}{1-\alpha}}+\frac{\\|\mathbf{w}^{}\\|_{2}^{2}}{T\eta}+\eta\Big{(}\frac{Td\log(1/\delta)\sqrt{\log(1/\gamma)}}{n^{2}\epsilon^{2}}+\frac{T\log(n)\log(n/\gamma)\log(1/\gamma)}{n}\Big{)}$ $\displaystyle\qquad+\\|\mathbf{w}^{}\\|_{2}^{1+\alpha}\sqrt{\frac{\log(1/\gamma)}{n}}+\frac{\\|\mathbf{w}^{}\\|_{2}\sqrt{\log(1/\delta)\log(1/\gamma)}}{n\epsilon}\Big{)}.$ (38) To choose a suitable $\eta$ and $T$ such that the algorithm achieves the optimal rate, we consider the trade-off between $1/\eta$ and $\eta$. We take the derivative of $\frac{1}{T\eta}+\eta\big{(}\frac{Td\log(1/\delta)\sqrt{\log(1/\gamma)}}{n^{2}\epsilon^{2}}+\frac{T\log(n)\log(n/\gamma)\log(1/\gamma)}{n}\big{)}$ w.r.t $\eta$ and set it to $0$, then we have $\eta=1/T\cdot\max\big{\\{}\frac{\sqrt{\log(n)\log(n/\gamma)\log(1/\gamma)}}{\sqrt{n}},\frac{\sqrt{d\log(1/\delta)}(\log(1/\gamma))^{\frac{1}{4}}}{n\epsilon}\big{\\}}$. Putting the value of $\eta$ back into (3.3), we obtain $\displaystyle\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}(\mathbf{w}^{})$ $\displaystyle=\mathcal{O}\bigg{(}\frac{\big{(}\log(n)\log(1/\gamma)\big{)}^{\frac{1-2\alpha}{2(1-\alpha)}}n^{\frac{1}{2(1-\alpha)}}}{T^{\frac{1+\alpha}{2(1-\alpha)}}(\log(n/\gamma))^{\frac{1}{2(1-\alpha)}}}+\frac{\sqrt{d\log(1/\delta)\log(1/\gamma)}}{n\epsilon}+\frac{\sqrt{\log(n)\log(n/\gamma)\log(1/\gamma)}}{\sqrt{n}}\bigg{)}\cdot\\|\mathbf{w}^{}\\|_{2}^{2}.$ In addition, if $n=\mathcal{O}(T^{\frac{1+\alpha}{2-\alpha}})$, then there holds $\displaystyle\mathcal{R}(\mathbf{w}_{\text{priv}})-\mathcal{R}(\mathbf{w}^{})=$ $\displaystyle\\|\mathbf{w}^{}\\|_{2}^{2}\cdot\mathcal{O}\Big{(}\frac{\sqrt{d\log(1/\delta)\log(1/\gamma)}}{n\epsilon}+\frac{\sqrt{\log(n)\log(n/\gamma)\log(1/\gamma)}}{\sqrt{n}}\Big{)}.$ The above bound matches the optimal rate $\mathcal{O}\big{(}\frac{\sqrt{d\log(1/\delta)}}{n\epsilon}+\frac{1}{\sqrt{n}}\big{)}$. Furthermore, we want the algorithm to achieve the optimal rate with a low computational cost. Therefore, we set $T\asymp n^{\frac{2-\alpha}{1+\alpha}}$ if $\alpha<\frac{1}{2}$, and $T\asymp n$ else. The proof is completed. ∎ Finally, we give the proof of Lemma 13 on the existence of $\beta$ for Algorithm 2 to be $(\epsilon,\delta)$-DP. ###### Proof of Lemma 13. We give sufficient conditions for the existence of $\beta\in(0,1)$ such that RDP conditions (32) and (33) hold with $\sigma^{2}=\frac{14(M+LR^{\alpha})^{2}\lambda}{\beta n\epsilon}$ and $\lambda=\frac{2\log(n)}{(1-\beta)\epsilon}+1$ in Theorem 11. Condition (32) with $T=n$ and $\delta=\frac{1}{n^{2}}$ is equivalent to $\displaystyle f(\beta):=\beta^{2}-\Big{(}1+\frac{7}{1.34n\epsilon}\Big{)}\beta+\frac{7(2\log(n)+\epsilon)}{1.34n\epsilon^{2}}\geq 0.$ (39) If $\big{(}1+\frac{7}{1.34n\epsilon}\big{)}^{2}<\frac{28(2\log(n)+\epsilon)}{1.34n\epsilon^{2}}$, then $f(\beta)\geq 0$ for all $\beta$. Then (32) holds for any $\beta\in(0,1)$. If $\big{(}1+\frac{7}{1.34n\epsilon}\big{)}^{2}\geq\frac{28(2\log(n)+\epsilon)}{1.34n\epsilon^{2}}$, then $\beta\in(0,\beta_{1}]\cup[\beta_{2},+\infty)$ such that the above condition holds, where $\beta_{1,2}=\frac{1}{2}\Big{(}\big{(}1+\frac{7}{1.34n\epsilon}\big{)}\mp\sqrt{\big{(}1+\frac{7}{1.34n\epsilon}\big{)}^{2}-\frac{28(2\log(n)+\epsilon)}{1.34n\epsilon^{2}}}\Big{)}$ are two roots of $f(\beta)=0$. Now, we consider the second RDP condition. Plugging $\sigma^{2}=\frac{14(M+LR^{\alpha})^{2}\lambda}{\beta n\epsilon}$ back into (33), we derive $\displaystyle\frac{3\beta n\epsilon(\lambda-1)}{7\lambda}+\log(\lambda)+\log(1+\frac{7\lambda}{2\beta n\epsilon})\leq\log(n).$ (40) To guarantee (40), it suffices that the following three inequalities hold $\frac{3\beta n\epsilon(\lambda-1)}{7\lambda}\leq\frac{\log(n)}{3},$ (41) $\log(\lambda)\leq\frac{\log(n)}{3},$ (42) $\log\big{(}1+\frac{7\lambda}{2\beta n\epsilon}\big{)}\leq\frac{\log(n)}{3}.$ (43) We set $\lambda=\frac{2\log(n)}{(1-\beta)\epsilon}+1$ in the above three inequalities. Since $\lambda>1$, then (41) holds if $\beta\leq 7\log(n)/9n\epsilon$. Eq. (42) reduces to $\beta\leq 1-\frac{2\log(n)}{(n^{1/3}-1)\epsilon}$. Moreover, (43) is equivalent to the following inequality $\displaystyle g(\beta):=\beta^{2}-(1+\frac{7}{2n(n^{\frac{1}{3}}-1)\epsilon})\beta+\frac{7(2\log(n)+\epsilon)}{2n(n^{\frac{1}{3}}-1)\epsilon^{2}}\leq 0.$ (44) There exists at least one $\beta$ such that $g(\beta)\leq 0$ if $(1+\frac{7}{2n(n^{1/3}-1)\epsilon})^{2}-\frac{14(2\log(n)+\epsilon)}{n(n^{1/3}-1)\epsilon^{2}}\geq 0$, which can be ensured by the condition $\epsilon\geq\frac{7}{2n(n^{1/3}-1)}+2\sqrt{\frac{7\log(n)}{n(n^{1/3}-1)}}$. Furthermore, $g(\beta)\leq 0$ for all $\beta\in[\beta_{3},\beta_{4}]$, where $\beta_{3,4}=\frac{1}{2}\Big{(}\big{(}1+\frac{7}{2n(n^{1/3}-1)\epsilon}\big{)}\mp\sqrt{\big{(}1+\frac{7}{2n(n^{1/3}-1)\epsilon}\big{)}^{2}-\frac{14(2\log(n)+\epsilon)}{n\big{(}n^{1/3}-1\big{)}\epsilon^{2}}}\Big{)}$ are two roots of $g(\beta)=0$. Finally, note that $\max\bigg{\\{}\frac{7}{2n(n^{\frac{1}{3}}-1)}+2\sqrt{\frac{7\log(n)}{n(n^{\frac{1}{3}}-1)}},\frac{\log(n)\big{(}14\log(n)(n^{\frac{1}{3}}-1)+162n-63\big{)}}{9n\big{(}2\log(n)(n^{\frac{1}{3}}-1)-9\big{)}}\bigg{\\}}\leq\frac{7(n^{\frac{1}{3}}-1)+4\log(n)n+7}{2n(n^{\frac{1}{3}}-1)}.$ Then if $n\geq 18$ and $\epsilon\geq\frac{7(n^{\frac{1}{3}}-1)+4\log(n)n+7}{2n(n^{\frac{1}{3}}-1)},$ there hold $\displaystyle\beta_{3}\leq\min\bigg{\\{}\frac{7\log(n)}{9n\epsilon},1-\frac{2\log(n)}{(n^{\frac{1}{3}}-1)\epsilon}\bigg{\\}}$ (45) and $\displaystyle\beta_{3}\leq\beta_{1}\;\text{\ if\ }\big{(}1+\frac{7}{1.34n\epsilon^{2}}\big{)}^{2}\geq\frac{28(2\log(n)+\epsilon)}{1.34n\epsilon^{2}}.$ (46) Conditions (45) and (46) ensure the existence of at least one consistent $\beta\in(0,1)$ such that (39), (41), (42), (43) and (44) hold, which imply that (32) and (33) hold. The proof is completed. ∎ ## 4 Conclusion In this paper, we are concerned with differentially private SGD algorithms with non-smooth losses in the setting of stochastic convex optimization. In particular, we assume that the loss function is $\alpha$-Hölder smooth (i.e., the gradient is $\alpha$-Hölder continuous). We systematically studied the output and gradient perturbations for SGD and established their privacy as well as utility guarantees. For the output perturbation, we proved that our private SGD with $\alpha$-Hölder smooth losses in a bounded $\mathcal{W}$ can achieve $(\epsilon,\delta)$-DP with the excess risk rate $\mathcal{O}\Big{(}\frac{(d\log(1/\delta))^{1/4}\sqrt{\log(n/\delta)}}{\sqrt{n\epsilon}}\Big{)}$, up to some logarithmic terms, and gradient complexity $T=\mathcal{O}(n^{2-\alpha\over 1+\alpha}+n)$, which extends the results of [35] in the strongly-smooth case. We also established similar results for SGD algorithms with output perturbation in an unbounded domain $\mathcal{W}=\mathbb{R}^{d}$ with excess risk $\mathcal{O}\Big{(}\frac{\sqrt{d\log(1/\delta)}\log(n/\delta)}{n^{\frac{2}{3+\alpha}}\epsilon}+\frac{\log(n/\delta)}{n^{\frac{1}{3+\alpha}}}\Big{)}$, up to some logarithmic terms, which are the first-ever known results of this kind for unbounded domains. For the gradient perturbation, we show that private SGD with $\alpha$-Hölder smooth losses in a bounded domain $\mathcal{W}$ can achieve optimal excess risk $\mathcal{O}\Big{(}\frac{\sqrt{d\log(1/\delta)}}{n\epsilon}+\frac{1}{\sqrt{n}}\Big{)}$ with gradient complexity $T=\mathcal{O}(n^{2-\alpha\over 1+\alpha}+n).$ Whether one can derive privacy and utility guarantees for gradient perturbation in an unbounded domain still remains a challenging open question to us. Acknowledgement. This work was done while Puyu Wang was a visiting student at SUNY Albany. 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Then for any $\gamma\in(0,1)$, there holds $\mathbb{P}_{\mathcal{S}\sim\mathcal{D}^{n},\mathcal{A}}\left[\|\mathcal{R}(\mathcal{A(S)})-\mathcal{R}_{S}(\mathcal{A(S)})\|\geq c\bigg{(}(M+LG^{\alpha})\Delta_{\mathcal{A}}\log(n)\log(1/{\gamma})+\big{(}M_{0}+(M+LG^{\alpha})G\big{)}\sqrt{n^{-1}\log(1/\gamma)}\bigg{)}\right]\leq\gamma.$ ###### Proof. By the convexity of $\ell$ and the definition of $\alpha$-Hölder smoothness, we have for any $S$ and $S^{\prime}$, $\displaystyle\ell(\mathcal{A}(S),z)$ $\displaystyle\leq\sup_{z\in\mathcal{Z}}\ell(0,z)+\langle\partial\ell(\mathcal{A}(S),z),\mathcal{A}(S)\rangle\leq M_{0}+\\|\partial\ell(\mathcal{A}(S),z)\\|_{2}\\|\mathcal{A}(S)\\|_{2}$ $\displaystyle\leq M_{0}+(M+L\\|\mathcal{A}(S)\\|^{\alpha}_{2})\\|\mathcal{A}(S)\\|_{2}\leq M_{0}+(M+LG^{\alpha})G$ (47) and $\displaystyle\sup_{z\in\mathcal{Z}}\|\ell(\mathcal{A}(S),z)-\ell(\mathcal{A}(S^{\prime}),z)\|$ $\displaystyle\leq\max\big{\\{}\\|\partial\ell(\mathcal{A}(S),z)\\|_{2},\\|\partial\ell(\mathcal{A}(S^{\prime}),z)\\|_{2}\big{\\}}\\|\mathcal{A}(S)-\mathcal{A}(S^{\prime})\\|_{2}$ $\displaystyle\leq(M+LG^{\alpha})\\|\mathcal{A}(S)-\mathcal{A}(S^{\prime})\\|_{2}.$ Note $\sup_{S\simeq S^{\prime}}\delta_{\mathcal{A}}(S,S^{\prime})\leq\Delta_{\mathcal{A}}$ and $\delta_{\mathcal{A}}(S,S^{\prime})=\\|\mathcal{A}(S)-\mathcal{A}(S^{\prime})\\|_{2}$. Then for any neighboring datasets $S\simeq S^{\prime}$, we have $\displaystyle\sup_{z\in\mathcal{Z}}\|\ell(\mathcal{A}(S),z)-\ell(\mathcal{A}(S^{\prime}),z)\|$ $\displaystyle\leq(M+LG^{\alpha})\Delta_{\mathcal{A}}.$ (48) Combining Eq. (Proof.), Eq. (48) and Corollary 8 in [7] together, we derive the following probabilistic inequality $\mathbb{P}_{\mathcal{S}\sim\mathcal{D}^{n},\mathcal{A}}\left[\|\mathcal{R}(\mathcal{A(S)})-\mathcal{R}_{S}(\mathcal{A(S)})\|\geq c\bigg{(}(M+LG^{\alpha})\Delta_{\mathcal{A}}\log(n)\log(1/{\gamma})+\big{(}M_{0}+(M+LG^{\alpha})G\big{)}\sqrt{n^{-1}\log(1/\gamma)}\bigg{)}\right]\leq\gamma.$ The proof is completed. ∎ ###### Proof of Lemma 1. Let $E_{1}=\\{\mathcal{A}:\sup_{S\simeq S^{\prime}}\\|\mathcal{A}(S)-\mathcal{A}(S^{\prime})\\|_{2}\geq\Delta_{\mathcal{A}}\\}$ and $E_{2}=\Big{\\{}(S,\mathcal{A}):\|\mathcal{R}(\mathcal{A(S)})-\mathcal{R}_{S}(\mathcal{A(S)})\|\geq c\Big{(}(M+LG^{\alpha})\Delta_{\mathcal{A}}\log(n)\log(1/{\gamma})+\big{(}M_{0}+(M+LG^{\alpha})G\big{)}\sqrt{n^{-1}\log(1/\gamma)}\Big{)}\Big{\\}}$. Then by the assumption we have $\mathbb{P}_{\mathcal{A}}[\mathcal{A}\in E_{1}]\leq\gamma_{0}$. Further, according to Lemma 27, for any $\gamma\in(0,1)$, we have $\mathbb{P}_{S,\mathcal{A}}[(S,\mathcal{A})\in E_{2}\cap\mathcal{A}\notin E_{1}]\leq\gamma$. Therefore, $\displaystyle\mathbb{P}_{S,\mathcal{A}}[(S,\mathcal{A})\in E_{2}]$ $\displaystyle=\mathbb{P}_{S,\mathcal{A}}[(S,\mathcal{A})\in E_{2}\cap\mathcal{A}\in E_{1}]+\mathbb{P}_{S,\mathcal{A}}[(S,\mathcal{A})\in E_{2}\cap\mathcal{A}\notin E_{1}]$ $\displaystyle\leq\mathbb{P}[\mathcal{A}\in E_{1}]+\mathbb{P}_{S,\mathcal{A}}[(S,\mathcal{A})\in E_{2}\cap\mathcal{A}\notin E_{1}]\leq\gamma_{0}+\gamma.$ The proof is completed. ∎
# A Two-stream Neural Network for Pose-based Hand Gesture Recognition Chuankun Li, Shuai Li, Yanbo Gao, Xiang Zhang, Wanqing Li Chuankun Li is with the North University of China, Taiyuan, China (email: chuankun@nuc.edu.cn).Shuai Li and Yanbo Gao are with Shandong University. (e-mail: {shuaili<EMAIL_ADDRESS>Xiang Zhang is with University of Electronic Science and Technology of China, Chengdu, China. (e-mail: <EMAIL_ADDRESS>Wanqing Li is with the Advanced Multimedia Research Lab, University of Wollongong, Wollongong, Australia. (wanqing@uow.edu.au). ###### Abstract Pose based hand gesture recognition has been widely studied in the recent years. Compared with full body action recognition, hand gesture involves joints that are more spatially closely distributed with stronger collaboration. This nature requires a different approach from action recognition to capturing the complex spatial features. Many gesture categories, such as “Grab” and “Pinch”, have very similar motion or temporal patterns posing a challenge on temporal processing. To address these challenges, this paper proposes a two-stream neural network with one stream being a self-attention based graph convolutional network (SAGCN) extracting the short-term temporal information and hierarchical spatial information, and the other being a residual-connection enhanced bidirectional Independently Recurrent Neural Network (RBi-IndRNN) for extracting long-term temporal information. The self-attention based graph convolutional network has a dynamic self-attention mechanism to adaptively exploit the relationships of all hand joints in addition to the fixed topology and local feature extraction in the GCN. On the other hand, the residual-connection enhanced Bi-IndRNN extends an IndRNN with the capability of bidirectional processing for temporal modelling. The two streams are fused together for recognition. The Dynamic Hand Gesture dataset and First-Person Hand Action dataset are used to validate its effectiveness, and our method achieves state-of-the-art performance. ###### Index Terms: Hand Gesture Recognition, Graph Convolutional Network, Bidirectional Independently Recurrent Neural Network. ## I Introduction Hand gesture recognition (HGR) is a hot research topic in artificial intelligence and computer vision, and has a large number of applications in the human-machine systems [1, 2, 3, 4]. Nowadays, depth sensors, like Intel RealSense and Microsoft Kinect, have become easily accessible, so have the body and hand skeletons [5, 6, 7]. Accordingly, hand gesture recognition based on pose/skeleton has attracted more and more interests[8, 9, 10, 11, 12, 13]. In the past few years, hand gesture recognition based on handcrafted features has been widely reported [8, 14, 15, 16]. Generally, shape of connected joints, hand orientation, surface normal orientation are often used as spatial features, and dynamic time warping or hidden Markov model are then used to process temporal information for classification. Recently, deep learning has also been explored for hand gestures recognition. [11, 17, 12, 18, 19]. One approach is to encode joint sequences into texture images and feed into Convolutional Neural Networks (CNNs) in order to extract discriminative features for gesture recognition. Several methods [12, 19, 20, 21] have been proposed along this approach. However, these methods cannot effectively and efficiently express the dependency between joints, since hand joints are not distributed in a regular grid but in a non-Euclidean domain. To address this problem, graph convolutional networks (GCN) [22, 21, 23] expressing the dependency among joints with a graph have been proposed. For instance, the method in [22] sets four types of edges to capture relationship between non-adjacent joints. However, the topology of the graph is fixed which is ineffective to deal with varying joint relationships for different hand gestures. A typical example is that, the connection between tip of thumb and tip of forefinger in gesture “Write” is likely to be strong, but it is not the case for gestures “Prick” and “Tap”. Modelling gesture dependent collaboration among joints is especially important for robust hand gesture recognition. Furthermore, conventional ST-GCN has limited receptive field in temporal domain, hence, long-term temporal information cannot be effectively learned. Another approach is to apply Recurrent Neural Networks (RNNs) to a hand skeleton sequence for classification [11, 18, 17]. However, due to the problem of gradient exploding and vanishing in the temporal direction or gradient decay along layers in training a RNN, shallow RNNs are often adopted. This paper adopts the Independently Recurrent Neural Network (IndRNN) [24, 25] as the basic component to develop a deep residual bidirectional IndRNN (RBi- IndRNN) for effective extraction of long-term temporal features. The contributions are summarized in the following. * • A two-stream neural network is proposed. One stream is a self-attention based graph convolutional network (SAGCN) and the other is a deep residual- connection enhanced bidirectional Independently Recurrent Neural Network (RBi- IndRNN). * • The SAGCN is specifically designed to model the strong collaboration among joints in hand gestures. In particular, a global correlation map is first adaptively constructed using a self-attention mechanism to characterize pairwise relationships among all the joints and it works together with a static adjacency map of the hand skeleton to capture varying spatial patterns for different gestures. * • The deep residual bidirectional IndRNN (RBi-IndRNN) is developed, which extends the residual connection enhanced IndRNN with bidirectional processing to exploit the bidirectional temporal context and long-term temporal information for challenging gestures having similar motion patterns such as “Grab” vs “Pinch”, “Swipe Left” vs “Swipe Right”. The proposed RBi-IndRNN compensate the inability of the SAGCN in learning long-range temporal patterns. * • Extensive experiments have been performed on two datasets including Dynamic Hand Gesture dataset (DHG) [8] and First-Person Hand Action datasets (FPHA) [26] and the proposed method performs the best. Detailed analysis on the experimental results is also presented to provide insight on the proposed two- stream networks. Part of this work has been presented in ISCAS 2020 [27] where the method only consists of a plain Bi-IndRNN. This paper extends the method in [27] to a two- stream network. One stream is a self-attention based graph convolutional network for capturing spatial and short-term temporal information, and the other stream is the deep residual-connection enhanced Bi-IndRNN to exploit the long-term temporal pattern. More experiments are also conducted with thorough analysis. The rest of this article is structured as follows. Section II provides a review of the relevant literature. The proposed two-stream network is described in Section III. Extensive experiment and detailed analysis are given in section IV. Section V concludes the paper. ## II Related Work Some related and representative works on hand gesture recognition are introduced in this section. They can be categorized into two approaches: handcrafted features-based [14, 15, 28, 16, 8] and deep learning-based [11, 12, 19, 17, 20, 29]. ### II-A Handcrafted Feature based Hand Gesture Recognition Under the category of handcrafted features based methods, Ren et al. [14] modelled hand skeleton as a signature and proposed FEMD (finger-earth mover distance) metric treating each finger as a cluster and penalizing the empty finger-hole to recognize the hand gesture. Wang et al. [15] proposed a mapping score-disparity cost map as the hand representation and used a SVM classifier for recognition. Kuznetsova et al. [28] used some geometric features to characterize the hand skeleton such as point distances and angle between the two lines created by two points. The multi-layered random forest is used for classification. Marin et al. [16] constructed a feature set containing the position and direction of fingertips, and performed classification using multi-class SVM classifier for recognition of hand gestures. Smedt et al. [8] designed features based on hand shape and combined featues of wrist rotations and hand directions, and utilized temporal pyramid to model these features. ### II-B Deep Learning based Hand Gesture Recognition In recent years, deep learning based methods have been widely studied for recognizing hand gesture. Depending on the types of deep neural networks, they can be usually divided into two ways: RNNs-based and CNNs-based ones. Figure 1: Framework of the proposed two-stream network. SAGCN module focuses on hierarchical spatial information and short-term temporal information, and RBi-IndRNN focuses on long-term temporal information. For the CNNs-based approach, it takes the advantage of CNNs to process spatial hand joints or each hand joint over time as grid data and extract local features. Devineau et al. [12] employed a parallel CNN to conduct temporal convolution processing for the world coordinate sequence of the skeleton joint separately, and fused the time-varying characteristics of each joint into a full connection layer (FC) and Softmax function for hand gesture classification. Liu et al. [29] firstly used RGB and depth map based faster region proposal networks to detect hand region. Then the video was segmented by using hand positions. From segmented videos, hand-oriented spatio-temporal features were extracted using the 3D convolutional networks for recognizing the hand gestures. Narayana et al. [19] utilized a spatial focus of attention to construct 12-stream networks called a FOA-Nets. Each network extracted local features from different modality. Then a sparse network architecture was designed to fuse the 12 channels. Pavlo et al. [20] used a recurrent 3DCNN for hand gesture recognition. In the recurrent 3DCNN, connectionist temporal classification (CTC) was utilized to synchronously segment video and classify dynamic hand gestures from multi-modal data. However, these methods based on CNNs cannot utilize the graphical structure of hand joints explicitly. To exploit this, Li et al. [22] employed a hand gesture based graph convolutional network (HGCN) to capture the linkage and motion information of different joints. However, the topology of the graph is fixed which limits capability of the network. In addition, the GCN is not designed for learning long-term temporal relationship. Nguyen et al. [21] employed a neural network to exploit gesture representation from hand joints by using SPD matrix. This method focuses on relationship of local joints, but ignores two joints that are far away from each other in the matrix. Motivated by the concept of attention based graph neural network (GCN) [30, 31], we propose a self-attention based GCN (SAGCN) to effectively explore the often strong collaboration among hand joints in gesture. In the RNNs-based approach, handcrafted features or features extracted from CNN of joint sequences are used as input to RNNs to explore temporal information. Chen et al. [11] utilized angle features of joints to characterize the finger movement and global features of rotation and translation. Three LSTM networks were used to process these features, respectively. Shin et al. [18] used two local features (finger and palm features) and a global feature (Pose feature) to capture the spatial information, and used a GRU-RNN network to process each feature part. Zhang et al. [17] employed a deep framework including 3DCNN layer, ConvLSTM layer, 2DCNN layer and temporal pooling layer to capture short-term spatio-temporal information. However, due to the difficulty in constructing a deep RNN, these methods usually employ a relatively shallow RNN network. Consequently, they often fail to recognize hand gesture having complex and long-term temporal patterns. In this paper, a residual connection enhanced bidirectional Independently Recurrent Neural Network (RBi-IndRNN) is employed to address this issue. ## III Proposed Method Fig. 1 illustrates the proposed two-stream network, one stream is a self- attention based Graph Convolutional Network learning the spatial and short- term temporal features, and the other stream is a residual Bi-IndRNN learning long-term temporal features. The two streams are then fused at score level for final classification. Figure 2: Sample of hand skeleton in the DHG 14/28 dataset [8]. Referencing to the gravity center of the hand joints (purple cross), the neighbouring connected joints of each joint, called root (blue circle), are grouped into two: 1) centripetal group (yellow circle): adjacent nodes closer to the gravity center of the hand joint; 2)centrifugal group: the rest connected neighbors (green circle). ### III-A Self-attention based Graph Convolutional Network Graph Convolutional Networks (GCNs) have been proposed for structured but non- grid data and extended for recognizing human action based on skeleton. A typical example is the spatial temporal GCN (ST-GCN) [32]. However, ST-GCN mainly explores the interaction or collaboration between local (e.g. first and second order neighbouring) joints and the topology of the graph representing the human body is fixed or static. Such representation is ineffective in exploring the collaboration among non-neighbouring joints and, hence, in differentiating actions with collaboration of different body parts. This problem becomes severe in hand gesture recognition where hand joints, regardless of being connected or not, are more closely and strongly collaborated than body joints in action recognition. To tackle this problem, a self-attention based ST-GCN method is developed. As illustrated in the left image of Fig. 2, a hand skeleton graph is usually defined with each hand joint being a node and anatomical connectivity between joints being edges. The graph is expressed as an adjacency matrix whose elements representing connected neighbouring joints are one and, otherwise, zero. Convolutional operations are performed at each node, the node itself, referred to as a root node to differentiate it from its connected nodes in the following, and its neighbouring connected nodes defined by the adjacency matrix. In order to characterize the local structure, different weights are learned for the root node and its connected nodes. Since the number of its connected nodes varies from node to node, a spatial configuration partitioning strategy is developed for hand joints similarly as in [32] to generalize the operations by defining two groups of neighbouring connected nodes and assuming the same weight for nodes in the same group. Specifically, a gravity center is first generated by calculating the average coordinate of all hand joints in a frame. Using this gravity center as a reference point, the neighbouring adjacent nodes/joints of each root node are grouped into two: 1) centripetal group: adjacent nodes closer to the gravity center of the hand joint; 2) the centrifugal group: the other neighbouring nodes locating farther away from the gravity center than the root node/joint, as illustrated the right image in Fig. 2. With the above adjacency matrix and partitioning strategy, a convolution operation on each node can be expressed as follows. $\mathbf{f_{out}}=\sigma(\sum_{k}^{K_{v}}\mathbf{W_{k}f_{in}}\mathbf{A_{k}})$ (1) where $K_{v}$ is the kernel size of the spatial dimension, i.e., the number of groups of connected neighbors, and is set to 3 (the two groups plus the root). For brevity, $\mathbf{A_{k}}$ denotes the normalized adjacency matrix $\mathbf{\Lambda_{k}^{-\frac{1}{2}}(A_{k})\Lambda_{k}^{-\frac{1}{2}}}$ similarly as in[31], where $\Lambda_{k}^{ii}=\sum_{j}(A_{k}^{(ij)})$. $\mathbf{W_{k}}$ is the weight vector of the $1\times 1$ convolution operation. The nonlinear activation function ReLU is used for $\sigma$. The above adjacency matrix is predefined anatomically and static. It represents each joint and its connected neighbouring joints, hence, the convolution operation extracts local features. However, such a local process is not able to capture directly the collaboration among non-connected joints that is often dynamic within a gesture and varies from gesture to gesture. Therefore, we propose in this paper a dynamic attention matrix or map to adaptively characterize the collaboration among all nodes/joints. The attention matrix is obtained via a self-attention mechanism and is calculated in a feature space by projecting the node features with a weight $\mathbf{W_{a}}$. The attention matrix is especially useful for the spatially closely-located, connected or non-connected hand joints that have strong collaboration. The detailed process is expressed as follows: $\displaystyle\mathbf{f_{a}}$ $\displaystyle=\mathbf{W_{a}{f_{in}}}$ (2) $\displaystyle\mathbf{A_{g}}$ $\displaystyle=\frac{exp(\mathbf{f_{a}}\otimes\mathbf{f_{a}^{T}})}{\sum exp(\mathbf{f_{a}}\otimes\mathbf{f_{a}^{T}})}$ (3) where $\mathbf{f_{in}}$ is input and $\mathbf{W}$ maps the input to a feature space (done via a convolutional operation in the experiments). $\mathbf{f_{a}^{T}}$ is the transpose of $\mathbf{f_{a}}$, and $\otimes$ is matrix multiplication. $\mathbf{A_{g}}$ is an attention matrix to indicate the relationship of the pair-wise nodes/joints and is normalized to $(0,1)$ via softmax. In order to capture local structure and global collaboration among the joins together, $\mathbf{A_{g}}$ is combined with the adjacency matrix as follows. $\mathbf{f_{out}}=\sigma(\sum_{k}^{K_{v}}\mathbf{W_{k}f_{in}}\mathbf{A_{k}}+\mathbf{W_{g}f_{in}}\mathbf{A_{g}})$ (4) where $\sum_{k}^{K_{v}}\mathbf{W_{k}f_{in}}\mathbf{A_{k}}$ captures local structure of joints and their connected neighbors and $\mathbf{W_{g}f_{in}}\mathbf{A_{g}}$ captures the global collaboration among all the joints. In this way, the proposed self-attention based GCN, termed as SAGCN, can process both local and global features together. Finally, the spatial features at each time step are processed over time with convolution in the same way as TCN in [32]. Figure 3: Illustration of an SAGCN unit. Figure 4: The recognition network built upon the proposed SAGCN. Fig. 3 illustrates the proposed SAGCN. An input feature is first processed by a convolution operation to a feature space, and then multiplied and normalized with a Softmax function to obtain the attention map $A_{g}$. Together with the input adjacency matrix, the GCN and TCN process the input feature spatially and temporally with batch normalization and ReLU activation functions. $N$, $C$ and $T$ in the figure represent the total number of the vertexes, number of convolutional channels and the length of hand sequences, respectively. The structure of the proposed SAGCN used in the experiments is illustrated in Fig. 4. The network contains six layers of SAGCN units and in each SAGCN layer the numbers of kernels of the attention component, the GCN component and the TCN component are set the same. The numbers of kernels for the six SAGCN layers are 64, 64, 128, 128, 256, and 256, respectively. The stride for the convolution in the fourth TCN is set to $2$ as a pooling operation over time. After six layers of SAGCN units, global average pooling (GPA) is used to pool the spatial-temporal features, and feed into FC and Softmax function for gesture classification. Figure 5: Structure of the proposed bidirectional IndRNN. ### III-B Residual Bidirectional Independently Recurrent Neural Network (RBi- IndRNN) As seen from the architecture of an SAGCN unit, it focuses on hierarchical spatial features and relatively short-term temporal features since it has a relatively small receptive field of the temporal convolution operations. It would be difficult for an SAGCN to capture long-term temporal features, hence, to distinguish gestures that are differentiated in long-term motion patterns, such as “Swipe up” and “Swipe down”. To address this shortcoming of the SAGCN, a second stream using the recent IndRNN [24, 25] is proposed in this paper. $\mathbf{h}_{t}=\sigma(\mathbf{Wx}_{t}+\mathbf{u}\odot\mathbf{h}_{t-1}+\mathbf{b})$ (5) where $\mathbf{x}_{t}\in\mathbb{R}^{M}$ is input of IndRNN network and $\mathbf{h}_{t}\in\mathbb{R}^{N}$ represents hidden states at time-step $t$. $\mathbf{W}\in\mathbb{R}^{N\times M}$, $\mathbf{u}\in\mathbb{R}^{N}$ and $\mathbf{b}\in\mathbb{R}^{N}$ are the weights need to be learned. $\odot$ is dot product, and $N$ is the number of neurons. One of the key advantages of IndRNN, comparing with the conventional RNN, is that IndRNN can capture longer temporal information by regulating the recurrent weights to avoid gradient vanishing and exploding in training. Moreover, multiple IndRNN layers are able to be efficiently stacked to build a deeper network with low complexity. Considering the success of the Bidirectional Recurrent Neural Networks (Bi- RNN) in action recognition [33] and language modelling [34], a Bidirectional IndRNN (Bi-IndRNN) is constructed whose architecture is shown in Fig. 5. The features are extracted by two directions of temporal processing using IndRNNs, and features of two directions are concatenated and fed into next layer. The Bi-IndRNN is able to capture the temporal relationship in two directions. Considering that IndRNN can effectively work with ReLU, we further develop a residual connection enhanced Bi-IndRNN (RBi-IndRNN) as shown in Fig. 6a, by adding an identity shortcut (skip-connection) to bypass the non-linear transformation of the input feature in order to facilitate the gradient backpropagation. This skip-connection does not affect the temporal processing, but makes the deeper features a summation of the shallower features. This paper adopts a pre-activation type of residual function with batch normalization, Bi-IndRNN and then weight processing as shown in Fig. 6a. (a) (b) Figure 6: Illustration of the (a): RBi-IndRNN module and (b): proposed 6-layer RBi-IndRNN for hand gesture recognition. In the experiments, a deep RBi-IndRNN of 6-layers and the framework is shown in Fig. 6b. After six layers of RBi-IndRNN, the features of last time-step are fed into a FC layer for gesture classification. Moreover, the temporal displacement of each joint describing the movement between adjacent frames similarly as the optical flow is extracted and concatenated with original skeletal joints as input to the RBi-IndRNN. Figure 7: A sample attention matrix of gesture “Grab” obtained by the proposed SAGCN. (a) (b) Figure 8: Confusion matrices on the DHG-14 dataset using (a): RBi-IndRNN and (b): SAGCN. In order to reduce the complexity of the two-stream network training, the SAGCN and the RBi-IndRNN are trained separately. The resulted models are then used to test and the probability vectors produced by the SAGCN and RBi-IndRNN are fused by multiplication. The class with the max probability is recognized as the gesture class and can be expressed as follows. $label=argmax({v_{SAGCN}}\odot{v_{RBi-IndRNN}})$ (6) where $v$ represents a probability vector, $\odot$ is dot product, and $argmax(\cdot)$ is to find the index position with the maximum probability. ## IV Experiments ### IV-A Datasets Two widely used datasets for HGR, namely the Dynamic Hand Gesture (DHG) 14/28 dataset [8] and the First-Person Hand Action (FPHA) dataset [26], are used for experiments. The DHG 14/28 dataset [8] is captured by Intel RealSense depth cameras containing hand joint data (3-dimension world coordinate ($x$, $y$, $z$)) and depth map sequences as shown in Fig. 2. This DHG 14/28 dataset is constructed with 2800 gesture sequences containing 14 gestures performed by 20 subjects. Each sequence ranging from 20 to 50 frames is assigned with a class label. The categories of this dataset include “Swipe x (SX)”, “Swipe down (SD)”, “Rotation counter-clockwise (RCC)”, “Tap (T)”, “Rotation clockwise (RC)”, “Swipe right (SR)”, “Pinch (P)”, “Swipe up (SU)”, “Shake (SH)”, “Grab (G)”, “Swipe left (SL)”, “Swipe v (SV)”,“Expand (E)”, “Swipe + (S+)”. Depending on the number of fingers used, gestures are classified with either 14 labels or 28 labels. The evaluation protocols in [12] are adopted, namely 1960 video as training samples and other sequences for testing. And 5% of the training data is randomly selected for validation. 20 frames are sampled from each sequence and fed into the proposed networks described in Section III for training and classification. The FPHA dataset [26] contains 1175 sequences from 45 different gesture classes with high viewpoint, speed, intra-subject variability and inter- subject variability of style, viewpoint and scale. This dataset is captured in 3 different scenarios (kitchen, office and social) and performed by 6 subjects. Compare with DHG 14/28 dataset, FPHA dataset has 21 hand joints and the palm joint is missed. This is a challenging dataset due to the similar motion patterns and involvement of many different objects. The same evaluation strategy in [26] are used. ### IV-B Training Setup The experiments are conducted on the Pytorch platform using a 1070Ti GPU card. Adaptive Moment Estimation (Adam) [35] is used as optimization function of training network. The batch size is set to 64 and 32 for DHG and FPHA datasets, respectively. Dropout [36] is set to 0.2 and 0.5 and the initial learning rate is set to $210^{-4}$ and $210^{-3}$ for RBi-IndRNN and SAGCN respectively. When the validation accuracy is improved, learning rate is decayed by 10. The number of neurons in each RBi-IndRNN layer is $512$. And 1024 neurons are used in FC of the RBi-IndRNN due to the bidirectional processing. TABLE I: Comparison of w/o self-attention for the GCN on two datasets Method \| DHG-14 \| DHG-28 \| FPHA ---\|---\|---\|--- GCN \| 90.83% \| 87.74% \| 83.48% SAGCN \| 93.33% \| 91.54% \| 87.83% TABLE II: Results on the two datasets under different settings of RBi-IndRNN Method \| DHG-14 \| DHG-28 \| FPHA ---\|---\|---\|--- IndRNN(joint coordinate) \| 92.07% \| 85.82% \| 84.52% IndRNN(joint coordinate + displacement) \| 92.19% \| 88.87% \| 86.78% Bi-IndRNN(joint coordinate + displacement) \| 93.15% \| 91.13% \| 88.35% RBi-IndRNN(joint coordinate + displacement) \| 94.05% \| 91.90% \| 88.87% TABLE III: Performance of the proposed method on the DHG dataset with comparisons to the existing methods in terms of accuracy Method \| modality \| 14 gestures \| 28 gestures \| average ---\|---\|---\|---\|--- HO4D Normals [37] \| Depth \| 78.53% \| 74.03% \| 76.28% Motion Trajectories [13] \| Pose \| 79.61% \| 62.00% \| 70.80% CNN for key frames [9] \| Pose \| 82.90% \| 71.90% \| 77.40% JAS and HOG2 [10] \| Pose \| 83.85% \| 76.53% \| 80.19% RNN+Motion feature [11] \| Pose \| 84.68% \| 80.32% \| 82.50% HoHD+HoWR+SoCJ [8] \| Pose \| 88.24% \| 81.90% \| 85.07% Parallel CNN [12] \| Pose \| 91.28% \| 84.35% \| 87.82% HG-GCN [22] \| Pose \| 92.80% \| 88.30% \| 90.55% leap motion controller [38] \| Pose \| 97.62% \| 91.43% \| 94.53% PB-GRU-RNN [18] \| Pose \| 95.21% \| 93.23% \| 94.22% proposed method \| Pose \| 96.31% \| 94.05% \| 95.18% Figure 9: Confusion matrix of the proposed network on DHG-14. Figure 10: Confusion matrix of the proposed network on DHG-28. TABLE IV: Results on the FPHA dataset and comparisons to the existing methods in terms of the accuracy Method \| modality \| 14 gestures ---\|---\|--- Two stream [39] \| RGB \| 75.30% Joint angles similarities and HOG2 [10] \| Depth+Pose \| 66.78% Histogram of Oriented 4D Normals [37] \| Depth \| 70.61% JOULE [40] \| RGB+Depth+Pose \| 78.78% Novel View [41] \| Depth \| 69.21% 2-layer LSTM [42] \| Pose \| 80.14% Moving Pose [43] \| Pose \| 56.34% Lie Group [44] \| Pose \| 82.69% HBRNN [45] \| Pose \| 77.40% Gram Matrix [46] \| Pose \| 85.39% TF [47] \| Pose \| 80.69% SPD Matrix Learning [48] \| Pose \| 84.35% Grassmann Manifolds [49] \| Pose \| 77.57% proposed method \| Pose \| 90.26% ### IV-C Ablation Study on Some Key Factors #### IV-C1 Contribution of the self-attention in the SAGCN The self-attention model used in the proposed SAGCN is evaluated against the GCN. The results on the two datasets by using GCN network and SAGCN are tabulated in Table I. It shows that the self-attention model used in the proposed SAGCN significantly improves the performance, compared with original GCN network. Taking the FPHA dataset as an example, the performance is improved from $83.48\%$ to $87.83\%$ in terms of accuracy. It indicates that the collaboration among all hand joints is important and is well captured by the proposed attention model via the attention matrix. A visual illustration, an attention matrix of gesture “Grab”, is shown in Fig. 7. This matrix demonstrates and also verifies the intuition that collaboration between thumb and forefinger joints is very important for recognizing gesture “Grab”. #### IV-C2 Contribution of the bidirectional processing and residual connection in the RBi-IndRNN The bidirectional processing and residual connection in the RBi-IndRNN is evaluated against the plain IndRNN. Table II shows the recognition results under different settings. The classification accuracies of the plain IndRNN only using joint world coordinates are 92.07%, 85.82% and 84.52% on DHG14, DHG-28 and FPHA datasets, respectively, which outperforms some methods (as illustrated in the following Table IV and Table III). When using the temporal displacement and joint world coordinates, the performance is further improved to 92.19%, 88.87% and 86.78%, respectively. It improves by $3.05$ and $2.26$ percentage points for DHG-28 and FPHA datasets respectively, compared with only using joint coordinates. Meanwhile, the RBi-IndRNN with the temporal displacement performs better than Bi-IndRNN and gets the best performance as shown in Table II. #### IV-C3 Contribution of the long-term temporal feature processing using RBi-IndRNN against SAGCN As discussed in Section III, SAGCN is not capable of learning long-term temporal features, but the RBi-IndRNN is. By comparing the performance of SAGCN and RBi-IndRNN as illustrated in Table I and Table II, respectively, RBi-IndRNN outperforms the SAGCN. To further understand what types of hand gestures that SAGCN and RBi-IndRNN are better at in the proposed framework, the confusion matrices of the SAGCN and RBi-IndRNN recognition results on the DHG-14 dataset are presented in Fig. 8a and 8b, respectively. By comparing the two confusion matrices, it can be seen that SAGCN performs better for hand gestures having much spatial variation and less temporal movement such as “ Swipe left ”, while RBi-IndRNN performs better for hand gestures with complex temporal motion such as “Grab ”. This indicates that SAGCN and RBi-IndRNN are complementary to each other. Figure 11: Confusion matrix of the proposed network on the FPHA dataset. ### IV-D Results on the DHG Dataset Results of the proposed method comparing with the existing hand-crafted feature based and deep learning based method [37, 13, 10, 8, 9, 12, 21, 38, 18] are shown in Table III. It can be seen that approaches based on deep learning generally outperform handcrafted feature based methods. Compared with methods based on CNNs [9, 12], graph convolutional networks based method [22] can capture spatial relationship of hand joints better and perform better. The proposed method outperforms these approaches, and achieves the best performance. Fig. 9 and Fig. 10 show the confusion matrices on the DHG-14 and DHG-28 datasets, respectively, obtained with the proposed method. It can be seen that most hand gestures can be recognized effectively except hand gesture “Grab (G)” and “Pinch (P)”. This is mostly because some Grab gestures only using the thumb and forefinger is of similar movement with “Pinch”. Therefore, networks to explore subtle differences are still highly desired and need to be investigated in the future. However, compared with results as illustrated in 8a of the previous work [27], our proposed method improves performance on all hand gestures. ### IV-E Results on the FPHA Dataset The comparison between the existing methods and the proposed method on the FPHA dataset is shown in Table IV. It also shows that our method achieves better performance than existing methods [39, 40, 47, 46, 49]. The confusion matrix of FPHA dataset is illustrated in Fig. 11. Most of the hand gestures are able to be accurately recognized. However, it is still difficult to recognize some gestures effectively such as “open wallet”, “unfold glasses” and “take letter from envelope” as shown in the confusion matrix. This is mostly because these hand gestures involve hand-object interaction, which cannot be well captured by the skeleton/pose alone. ## V Conclusion In this paper, a two-stream neural network is presented for recognizing the pose-based hand gesture. One is an SAGCN network having a self-attention mechanism to adaptively explore the collaboration among all joints in the spatial and short-term temporal domains. The other is a RBi-IndRNN to explore the long-term temporal dependency, compensating the weakness of SAGCN in processing the temporal features. The bidirectional processing and residual connections used in the RBi-IndRNN have proven to be effective in learning temporal patterns. State-of-the-art results are achieved by our two-stream neural network on two representative hand gesture datasets. Thorough analysis with ablation studies have also been conducted, validating the effectiveness of the proposed method. ## References * [1] L. Cheng, Y. Liu, Z. Hou, M. Tan, D. Du, and M. 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Chellappa, “Human action recognition by representing 3d skeletons as points in a lie group,” in _IEEE Conference on Computer Vision and Pattern Recognition_ , 2014, pp. 588–595. * [45] Y. Du, W. Wang, and L. Wang, “Hierarchical recurrent neural network for skeleton based action recognition,” in _IEEE Conference on Computer Vision and Pattern Recognition_ , 2015, pp. 1110–1118. * [46] X. Zhang, Y. Wang, M. Gou, M. Sznaier, and O. Camps, “Efficient temporal sequence comparison and classification using gram matrix embeddings on a riemannian manifold,” in _IEEE Conference on Computer Vision and Pattern Recognition_ , 2016, pp. 4498–4507. * [47] G. Garciahernando and T. Kim, “Transition forests: Learning discriminative temporal transitions for action recognition and detection,” in _IEEE Conference on Computer Vision and Pattern Recognition_ , 2017, pp. 407–415. * [48] Z. Huang and L. Van Gool, “A riemannian network for spd matrix learning,” in _AAAI Conference on Artificial Intelligence_ , 2016, pp. 2036–2042. * [49] Z. Huang, J. Wu, and L. Van Gool, “Building deep networks on grassmann manifolds,” in _AAAI Conference on Artificial Intelligence_ , 2018, pp. 3279–3286. \| Chuankun Li received the BE degree in electronic information engineering from North University of China, Taiyuan, China, in 2012 and received the MS degree in communication and information system from North University of China, Taiyuan, China, in 2015. He received the Ph.D degree with School of electronic information engineering , Tianjin University, China in 2020. His current research interests include computer vision and machine learning. ---\|--- \| Shuai Li is currently with the School of Control Science and Engineering, ShanDong University (SDU), China, as a Professor and QiLu Young Scholar. He was with the School of Information and Communication Engineering, University of Electronic Science and Technology of China, China, as an Associate Professor from 2018-2020. He received his Ph.D. degree from the University of Wollongong, Australia, in 2018. His research interests include image/video coding, 3D video processing and computer vision. He was a co-recipient of two best paper awards at the IEEE BMSB 2014 and IIH-MSP 2013, respectively. ---\|--- \| Yanbo Gao is currently with the School of Software, Shandong University (SDU), Jinan, China, as an Associate Professor. She was with the School of Information and Communication Engineering, University of Electronic Science and Technology of China (UESTC), Chengdu, China, as a Post-doctor from 2018-2020. She received her Ph.D. degree from UESTC in 2018. Her research interests include video coding, 3D video processing and light field image coding. She was a co-recipient of the best student paper awards at the IEEE BMSB 2018. ---\|--- \| Xiang Zhang received the B.S. and M.S. degrees from University of Electronic Science and Technology of China, Chengdu, China, and the Ph.D. degree from Shanghai Jiaotong University, Shanghai, China, in 2003, 2006, and 2009, respectively. He is an Associate Professor with the School of Information and Communication Engineering, University of Electronic Science and Technology of China. His research interests include video analysis and machine learning. ---\|--- \| Wanqing Li (M’97-SM’05) received his Ph.D. in electronic engineering from The University of Western Australia. He was a Principal Researcher (98-03) at Motorola Research Lab and a visiting researcher (08, 10 and 13) at Microsoft Research US. He is currently an Associate Professor and Co-Director of Advanced Multimedia Research Lab (AMRL) of UOW, Australia. His research areas are machine learning, 3D computer vision, 3D multimedia signal processing and medical image analysis. Dr. Li currently serves as an Associate Editor for IEEE Transactions on Circuits and Systems for Video Technology and IEEE Transactions on Multimedia. He was an Associator for Journal of Visual Communication and Image Representation. ---\|---
# Machine Learning Percolation Model Shu Cheng1 Huai Zhang<EMAIL_ADDRESS>Yaolin Shi Key Laboratory of Computational Geodynamics, College of Earth and Planetary Sciences, University of Chinese Academy of Sciences, No.19(A) Yuquan Road, Shijingshan District, Beijing 100049, China Fei He1 Ka-Di Zhu<EMAIL_ADDRESS>Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education), School of Physics and Astronomy, Shanghai Jiao Tong University, 800 Dong Chuan Road, Shanghai 200240, China ###### Abstract Recent advances in machine learning have become increasingly popular in the applications of phase transitions and critical phenomena. By machine learning approaches, we try to identify the physical characteristics in the two- dimensional percolation model. To achieve this, we adopt Monte Carlo simulation to generate dataset at first, and then we employ several approaches to analyze the dataset. Four kinds of convolutional neural networks (CNNs), one variational autoencoder (VAE), one convolutional VAE (cVAE), one principal component analysis (PCA), and one $k$-means are used for identifying order parameter, the permeability, and the critical transition point. The former three kinds of CNNs can simulate the two order parameters and the permeability with high accuracy, and good extrapolating performance. The former two kinds of CNNs have high anti-noise ability. To validate the robustness of the former three kinds of CNNs, we also use the VAE and the cVAE to generate new percolating configurations to add perturbations into the raw configurations. We find that there is no difference by using the raw or the perturbed configurations to identify the physical characteristics, under the prerequisite of corresponding labels. In the case of lacking labels, we use unsupervised learning to detect the physical characteristics. The PCA, a classical unsupervised learning, performs well when identifying the permeability but fails to deduce order parameter. Hence, we apply the fourth kinds of CNNs with different preset thresholds, and identify a new order parameter and the critical transition point. Our findings indicate that the effectiveness of machine learning still needs to be evaluated in the applications of phase transitions and critical phenomena. ††preprint: APS/123-QED ## I Introduction Machine learning methods have rapidly become pervasive instruments due to better fitting quality and predictive quality in comparison with traditional models in terms of phase transitions and critical phenomena. Usually, machine learning can be divided into supervised and unsupervised learning. In the former, the machine receives a set of inputs and labels. Supervised learning models are trained with high accuracy to predict labels. The effectiveness of supervised learning has been examined by many predecessors on Ising models [1, 2, 3, 4, 5, 6, 7, 8], Kitaev chain models [3], disordered quantum spin chain models [3], Bose-Hubbard models [6], SSH models [6], SU(2) lattice gauge theory [7], topological states models [9], $q$-state Potts models [10, 11], uncorrelated configuration models [12], Hubbard models [13, 12], and $XY$ models [14], ect. On the other hand, in unsupervised learning models, there are no labels. Unsupervised learning can be used as meaningful analysis tools, such as sample generation, feature extraction, cluster analysis. Principal component analysis (PCA) is one of the unsupervised learning techniques. Recently investigators have examined the PCA’s effectiveness for exploring physical features without labels in the applications of phase transitions and critical phenomena [15, 14, 8, 13, 16, 17]. Variational autoencoder (VAE) and convolutional VAE (cVAE), another two classical unsupervised learning technique, incorporated into generative neural networks, are used for data reconstruction and dimensional reduction in respect of phase transitions and critical phenomena [8, 18]. Although machine learning approaches have been applied successfully in phase transitions and critical phenomena, there is only one study on the percolation model [17]. Motivated by predecessors, we conduct much more comprehensive studies, which combine supervised learning with unsupervised learning, to detect the physical characteristics in the percolation model. Our work is considered from the following several aspects. First, we use the former three kinds of deep convolutional neural networks (CNNs) to deduce the two order parameters and the permeabilities in the two-dimensional percolation model. our inspiration and method come from [2, 3], whose study both focus on Ising model. Nevertheless, the above CNNs are trained on the known configurations from the dataset obtained by Monte Carlo simulation. [8, 18] find that VAE and cVAE can reconstruct samples in Ising model. Hence, we use the VAE and the cVAE to generate new configurations that are out of the dataset. After generating the new configurations, we pour them into the former three kinds of CNNs, respectively. Having explored supervised learning, we now move on to unsupervised learning. Here we try to identify physical characteristics without labels in the two- dimensional percolation model. [15] takes the first principal component obtained by PCA as the order parameter in Ising model. In contrast to [15], by using preprocessing on the unpercolating clusters, [17] also successfully finds the order parameter in percolation model by PCA. In this study, we try to use the PCA to extract relevant low-dimensional representations to discover physical characteristics. In an actual situation, we may not know the labels when identifying order parameter. To overcome the difficulty associated with missing labels, [12] changes the preset threshold between the labels zero and one so as to make incorrect labels between the preset and the true thresholds. Hence, we deliberately change the preset thresholds, determined by $k$-means, between the labels zero and one. Here we use the fourth kinds of CNNs which receives the raw configurations as input and the labels determined by the preset thresholds. This paper is organized as follows. In Sec. II, we describe the two- dimensional percolation model and the dataset from Monte Carlo simulation. In Sec. III, we give a brief introduction to CNNs, VAE and cVAE, and PCA. Next, we provide dozens of machine learning models to capture the physical characteristics and discuss the results in Sec. IV. Finally, we conclude with a summary in Sec. V. ## II The two-dimensional Percolation model For percolation models, what we need to do is to capture the physical characteristics. A suitable dataset should be constructed to fulfill this objective. Various models in physical dynamics can be simulated mathematically by the Monte Carlo method, and it has been proved to be valid for using the Monte Carlo simulation to capture different physical features in phase transitions and critical phenomena [15, 14, 8]. In this study, the Monte Carlo simulation for the two-dimensional percolation model is carried out as follows. First, 40 values of permeability range from 0.41 to 0.80 with an interval of 0.01. For each permeability, the initial samples consist of 1000 percolating configurations. To train the machine learning models, the matrix $\bm{X}$ with the size of $M\times N$ (see Eq. 6) is used for storing 40,000 raw percolating configurations. $\displaystyle\bm{X}=\left(\begin{array}[]{ccccc}a_{1,1}&a_{1,2}&\ldots&a_{1,N-1}&a_{1,N}\\\ a_{2,1}&a_{2,2}&\ldots&a_{2,N-1}&a_{2,N}\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\ a_{M-1,1}&a_{M-1,2}&\ldots&a_{M-1,N-1}&a_{M-1,N}\\\ a_{M,1}&a_{M,2}&\ldots&a_{M,N-1}&a_{M,N}\\\ \end{array}\right)_{M\times N}.$ (6) In Eq. (6), $M=40,000$, $N=L\times L$, and $L=28$. $M$ and $N$ represent the number of the configurations and the lattices, respectively. Each row $\bm{R}_{i}$ ($i=1,2,\ldots,M$) in the matrix $\bm{X}$ is a configuration with one dimension, can be reshaped as the matrix $\bm{X}_{i}$ ($i=1,2,\ldots,M$) with the size of $L\times L$ (see Eq. 10). Furthermore, each column $\bm{C}_{j}$ ($j=1,2,\ldots,N$) in the matrix $\bm{X}$ represents one lattice with different configurations. Moreover, the element $a_{ij}$ ($i=1,2,\ldots,M;j=1,2,\ldots,N$) in matrix $\bm{X}$ and the element $b_{kl}$ ($k,l=1,2,\ldots,L$) in matrix $\bm{X}_{i}$ take 0 when the corresponding lattice is occupied and take 1 otherwise. $\displaystyle\bm{X}_{i}=\left(\begin{array}[]{ccccc}b_{11}&b_{12}&\ldots&b_{1L-1}&b_{1L}\\\ &&\vdots&&\\\ b_{L1}&b_{L2}&\ldots&b_{LL-1}&b_{LL}\\\ \end{array}\right)_{L\times L}.$ (10) Figure 1: (a) The relationship between the permeabilities $\\{0.41,0.42,\ldots,0.80\\}$ and the raw $\varPi(\bm{p},L)$. (b) The relationship between the permeabilities $\\{0.41,0.42,\ldots,0.80\\}$ and the raw $P(\bm{p},L)$. Except for the raw configurations, the dataset also encompasses order parameter. In the two-dimensional percolation model, order parameter includes the percolation probability $\varPi(\bm{p},L)$ and the density of the spanning cluster $P(\bm{p},L)$. $\varPi(\bm{p},L)$ refers to the probability that there is one connected path from one side to another in $\bm{X}_{i}$. That is to say, $\varPi(\bm{p},L)$ is a function of the permeability $\bm{p}$ in the system with the size of $L\times L$ (see Fig. 1(a)). With a connectivity of 4, we can identify the cluster for each lattice $b_{kl}$. The clusters are marked sequentially with an unique index. Note that the two lattices having the same index belong to the same cluster. If there are more than one cluster, the greatest cluster is chosen as the result. In this way, we can count up how many times the configurations $\\{\bm{X}_{1},\bm{X}_{2},\ldots,\bm{X}_{M}\\}$ are percolated for given $\bm{p}$ and $L$. For each permeability $p$, $\varPi(\bm{p},L)_{p}$ is expressed in Eq. (11). $\displaystyle\varPi(\bm{p},L)_{p}=\frac{1}{1000}\sum_{r=1}^{1000}{S_{r}},$ (11) Where $S_{r}$ refers that the two-dimensional configurations $\\{\bm{X}_{e},\bm{X}_{2\times e},\ldots,\bm{X}_{1000\times e}\\}$ for the permeability $p$ is percolated or not. Clearly, $S_{r}$ takes 0 if the corresponding lattice is occupied and takes 1 otherwise. $r=\\{1,2,\ldots,1000\\}$, $p\in\\{0.41,0.42,\ldots,0.80\\}$, $e=(p-0.40)\times 100$, and $\varPi(\bm{p},L)_{p}\in[0,1]$. Another representation of order parameter is the density of the spanning cluster $P(\bm{p},L)$. In contrast to the $\varPi(\bm{p},L)$, $P(\bm{p},L)$ is associated with spanning cluster. Therefore, for all configurations $\\{\bm{X}_{1},\bm{X}_{2},\ldots,\bm{X}_{M}\\}$, $P(\bm{p},L)$ is characterized by that whether or not each lattice $b_{kl}$ belongs to the total spanning cluster. Similarly, $P(\bm{p},L)$ is a function of the permeability $\bm{p}$ in the system with the size of $L\times L$ (see Fig. 1(b)). For each permeability $p$, $P(\bm{p},L)_{p}$ is expressed in Eq. (12). $\displaystyle P(\bm{p},L)_{p}=\frac{1}{1000\times L\times L}\sum_{r=1}^{1000}{S_{r}^{{}^{\prime}}}.$ (12) Where $S_{r}^{{}^{\prime}}$ counts up the total number of lattices that belong to the spanning cluster for each configuration in $\\{\bm{X}_{e},\bm{X}_{2\times e},\ldots,\bm{X}_{1000\times e}\\}$ for the permeability $p$. Obviously, $r=\\{1,2,\ldots,1000\\}$, $p\in\\{0.41,0.42,\ldots,0.80\\}$, $e=(p-0.40)\times 100$, $0\leq S_{r}^{{}^{\prime}}<L\times L$, and $P(\bm{p},L)_{p}\in[0,1)$. ## III Machine learning methods ### III.1 CNNs In this section, we will focus on the two-dimensional percolation model and the dataset obtained by the Monte Carlo simulation. This section will discuss several machine learning approaches, including CNNs, VAE and cVAE, and PCA, to deduce physical characteristics. Let us first introduce CNNs. CNNs, supervised learning methods, are particularly useful in solving realistic problem for many disciplines, such as physics[3], chemistry [19], medicine [20], economics [21], biology [22], and geophysics [23, 24], ect. In the applications of phase transitions and critical phenomena, many predecessors utilize CNNs to detect physical features, especially order parameter [1, 25, 26, 27, 10, 5, 2, 6]. In this study, the four kinds of CNNs are not only used to detect the two order parameters ($\varPi(\bm{p},L)$ and $P(\bm{p},L)$), but also to detect the permeability $\bm{p}$. Figure 2: The structure of the CNNs with four layers, including “Conv1”, “Conv2”, “FC”, and “Output”. The square with black and white lattices is percolating configurations $\\{\bm{X}_{1},\bm{X}_{2},\ldots,\bm{x}_{M}\\}$. The light yellow cuboids (“Conv1” and “Conv2”) stand for convolution layers. The bright orange cuboids stand for max-pooling layers. The light purple cuboid “FC” refer to a fully connected layer. The input layer “Input” has the size of $28\times 28$. The first layer “Conv1” with “filter1” filters has the size of “size1”$\times$“size1”. The second layer “Conv2” with “filter2” filters has the size of “size2”$\times$“size2”. The layer “Flatten” owns the size of “size3”. The third layer “FC” owns the size of “size4”. And the last layer “Output” represents for a fully connected layer with one neuron. Next, we demonstrate the architecture of the CNNs (see Fig. 2). The structure of the CNNs has four layers, including two convolution layers and two fully connected layers. The percolating configurations $\\{\bm{X}_{1},\bm{X}_{2},\ldots,\bm{X}_{M}\\}$ are taken as inputs. Consequently, the CNNs receive the corresponding order parameters ($\varPi(\bm{p},L)$ and $P(\bm{p},L)$) or the permeability $\bm{p}$ as outputs. ### III.2 VAE and cVAE In the former section (see Sec. III.1), the raw configuration $\bm{X}$ at different permeability $\bm{p}$ is generated by Monte Carlo simulation. However, what if configurations are not in the raw configurations, can we still identify the physical features as well? Here we consider to use the VAE and the cVAE to generate new configuration $\hat{\bm{X}}_{\text{VAE}}$ and $\hat{\bm{X}}_{\text{cVAE}}$, respectively. Figure 3: The structure of the VAE with an encoder and a decoder. The left large light purple cuboid refers to the encoder with two fully connected layers, i.e., “FC1”, and “$\bm{\mu}$” or “$\bm{\sigma}$”, with the size of “size1”, and “size2”. And the right large light red cuboid is the decoder with two fully connected layers, including “FC2”, and “Output” with the size of “size3”, and 784. The outputs of the encoder are the mean value “$\bm{\mu}$” and the standard deviation “$\bm{\sigma}$”. The input “$\bm{Z}$” of the decoder is sampled from the normal distribution with “$\bm{\mu}$” and “$\bm{\sigma}$”. The green cuboid consists of “$\bm{\mu}$”, “$\bm{\sigma}$”, and “$\bm{Z}$”. The rectangles with 784 black and white lattices represent percolating configuration $\bm{X}$ on the left and its reconstruction $\hat{\bm{X}}$ on the right, respectively. VAE (see Fig. 3), a generative network, bases on the variational Bayes inference proposed by [28]. Contracted with traditional AE (see Fig. S. 1), the VAE describes latent variables with probability. From that point, the VAE shows great values in data generation. Just like AE, the VAE is composed of an encoder and a decoder. The VAE uses two different CNNs as two probability density distributions. The encoder in the VAE, called the inference network $p_{\text{encoder}}(\bm{Z}\|\bm{X})$, can generate the latent variables $\bm{Z}$. And the decoder in the VAE, called the generating network $p_{\text{decoder}}(\hat{\bm{X}}\|\bm{Z})$, reconstructs the raw configuration $\bm{X}$. Unlike AE, the encoder and the decoder in VAE are constrained by the two probability density distributions. Figure 4: The structure of the cVAE with an encoder and a decoder. The left large light purple cuboid refers to the encoder with three layers, including the input layer “Input” with the size of “28$\times$28”, two convolution layers (“Conv1” and “Conv2”) with the size of “filter1$\times$size1$\times$size1” and “filter2$\times$size2$\times$size2”, a flatten layer “Flatten” with the size of “filter2$\times$size2$\times$size2”, and a fully connected layer (“$\bm{\mu}$” or “$\bm{\sigma}$”) with the size of “size3”. And the right large light red cuboid is the decoder with four layers, including the layer “$\bm{Z}$” with the size of “size3”, a fully connected layer “FC” with the size of “filter1$\times$size4$\times$size4”, a reshape layer “Reshape” with the size of “filter1$\times$size4$\times$size4”, two transposed convolution layers (“Conv3” and “Conv4”) with the size of “filter2$\times$size5$\times$size5” and “filter1$\times$28$\times$28”, and a output layer “Output” with the size of “28$\times$28”. The outputs of the encoder are the mean value “$\bm{\mu}$” and the standard deviation “$\bm{\sigma}$”. The input of the decoder “$\bm{Z}$” is sampled from the normal distribution with “$\bm{\mu}$” and “$\bm{\sigma}$”. The green cuboid is consist of “$\bm{\mu}$”, “$\bm{\sigma}$”, and “$\bm{Z}$”. The squares with 784 black and white lattices represent percolating configurations $\\{\bm{X}_{1},\bm{X}_{2},\ldots,\bm{X}_{M}\\}$ on the left and their reconstructions $\\{\hat{\bm{X}}_{1},\hat{\bm{X}}_{2},\ldots,\hat{\bm{X}}_{M}\\}$ on the right, respectively. To better reconstruct the raw two-dimensional configuration $\bm{X}$, the fully connected layer in the VAE is replaced by the convolution layer. Now we have got the cVAE. The architecture of the cVAE is shown in Fig. 3. Usually, the performance of the cVAE is better than the VAE due to the configuration $\bm{X}$ with spatial attribute. In our work, the VAE and the cVAE are both used for generating THE new configuration $\hat{\bm{X}}$. ### III.3 PCA The Sec. III.1 and Sec. III.2 focus on supervised learning, which hypothesize that the labels exist for the raw configuration $\bm{X}$ on the percolation model. However, though we can detect the permeability values $\\{0.41,0.42,\ldots,0.80\\}$ and the two order parameters ($\varPi(\bm{p},L)$ and $P(\bm{p},L)$) by supervised learning, label dearth often occurs. Thus, it is imperative to identify the labels, such as $\varPi(\bm{p},L)$, $P(\bm{p},L)$, and $\bm{p}$. Some recent studies have shown that the first principal component obtained by PCA can be regarded as typical physical quantities [15, 8, 14, 17]. Base on these studies, we explore the meaning of the first principal component on the percolation model. As it is well-known, PCA can reduce the dimension of the matrix $\bm{X}$. First, we compute the mean value $\bm{X}_{\text{mean}}=1/M\sum_{i=1}^{M}a_{ij}(i=1,2,\ldots,M;j=1,2,\ldots,N)$ for each column $\bm{C}_{j}(j=1,2,\ldots,N)$ in the matrix $\bm{X}$. Then we get the centered matrix $\bm{X}_{\text{centered}}$ that is expressed as $\bm{X}_{\text{centered}}=\bm{X}-\bm{X}_{\text{mean}}$. After obtaining $\bm{X}_{\text{centered}}$, by an orthogonal linear transformation expressed as $\bm{X}_{\text{centered}}^{T}\bm{X}_{\text{centered}}\bm{W}=\bm{\lambda}\bm{W}$, we extract the eigenvectors $\bm{W}$ and the eigenvalues $\bm{\lambda}$. The eigenvectors $\bm{W}$ are composed with $\bm{w}_{1},\bm{w}_{2},\ldots,\bm{w}_{N}$. The eigenvalues are sorted in the descending order, i.e., $\bm{\lambda}_{1}\geq\bm{\lambda}_{2}\geq\ldots\geq\bm{\lambda}_{N}\geq 0$. The normalized eigenvalues $\tilde{\bm{\lambda}}_{j}(j=1,2,\ldots,N)$ are expressed as $\bm{\lambda}_{j}/\sum_{j=1}^{N}{\bm{\lambda}_{j}}$. The row $\bm{R}_{i}$ in matrix $\bm{X}$ can be transformed into $\bm{X}^{{}^{\prime}}_{i}=\bm{R}_{i}\bm{W}$. Eq. 13 represents the statistic average every 40 intervals for each permeability $p$. This process is quite similar to the process of calculating the two order parameters, i.e., $\varPi(\bm{p},L)$ and $P(\bm{p},L)$. Table 1 shows the procedure of PCA algorithm. $\langle\bm{X}^{{}^{\prime}}\rangle=\frac{1}{1000}\sum_{i=1}^{1000}\|\bm{X}^{{}^{\prime}}_{i\times 40}\|$ (13) Table 1: The procedure of PCA Algorithm Require: the raw configuration $\bm{X}$ --- 1\. Compute the mean value $\bm{X}_{\text{mean}}$ for the column $\bm{C}_{j}$ in the matrix $\bm{X}$; 2\. Get the centered matrix $\bm{X}_{\text{centered}}$; 3\. Compute the eigenvectors $\bm{W}$ and the eigenvalues $\bm{\lambda}$ by an orthogonal linear transformation; 4\. Transform $\bm{R}_{i}$ into $\bm{X}^{{}^{\prime}}_{i}$; 5\. Get the statistic average $\langle\bm{X}^{{}^{\prime}}\rangle$ with every 40 intervals for each permeability $p$. ## IV Results and Discussion ### IV.1 Simulate the two order parameters by two CNNs In this section, we consider to use the approches in Sec. III to capture the physic features. First we make use of TensorFlow 2.2 library to perform the CNNs with four layers. To predict the two order parameters ($\varPi(\bm{p},L)$ and $P(\bm{p},L)$), two kinds of CNNs (CNNs-I and CNNs-II) are constructed. The first two layers of the former two kinds of CNNs are composed of two convolution layers (“Con1” and “Con2”), both of which possessed “filter1”=32 and “filter2”=64 filters with the size of $3\times 3$, and a stride of 1. Each convolution layer is followed by a max-pooling layer with the size of $2\times 2$. The final convolution layer “Con2” is strongly interlinked to a fully connected layer “FC” with 128 variables. The output layer “Output”, following by “FC”, is a fully connected layer. For the two convolution layers and the fully connected layer “FC”, a rectified linear unit (ReLU) $\bm{a}=\text{max}(0,\bm{x})$ [29] is chosen as activation function due to its reliability and validity. However, the output layer has no activation function. After determining the framework of CNNs-I and CNNs-II, here we mention how to train CNNs-I and CNNs-II for deducing $\varPi(\bm{p},L)$ and $P(\bm{p},L)$. First, we carry out an Adam algorithm [30] as an optimizer to update parameters, i.e., weights and biases. Then, a mini batch size of 256 and a learning rate of $10^{-4}$ are selected for its timesaving. Following this treatment, CNNs-I and CNNs-II are trained on 1000 epochs for 40,000 uncorrelated and shuffled configurations, respectively. Before training, we split $\\{\bm{X}_{1},\bm{X}_{2},\ldots,\bm{X}_{M}\\}$, $\varPi(\bm{p},L)$ and $P(\bm{p},L)$ into 32,000 training set and 8,000 testing set. While training, we monitor three indicators, including the loss function (i.e., mean squared error (MSE, see Eq. 14)), mean average error (MAE, see Eq. 15), and root mean squared error (RMSE, see Eq. 16), for training and testing set [24]. In Eq. 14-16, ${y_{i}}^{\text{raw}}$ and ${y_{i}}^{\text{pred}}$ refer to $\varPi(\bm{p},L)$/$P(\bm{p},L)$ and its predictions. If the loss function in testing set reaches the minimum, then the optimal CNNs-I and CNNs-II will be obtained. As can be seen from Fig. S. 2, these indicators gradually decrease. In Table. S. 3, the errors of the optimal CNNs-I and CNNs-II are very small. What stands out in Fig. S. 2 is that CNNs-I and CNNs-II have high stability, consistency, and faster convergence rate. $\mathrm{MSE}=\frac{1}{M}\sum_{i=1}^{M}({y_{i}}^{\text{pred}}-{y_{i}}^{\text{raw}})^{2}\ $ (14) $\mathrm{MAE}=\frac{1}{M}\sum_{i=1}^{M}\|{y_{i}}^{\text{pred}}-{y_{i}}^{\text{raw}}\|$ (15) $\mathrm{RMSE}=\sqrt{\frac{1}{M}\sum_{i=1}^{M}({y_{i}}^{\text{pred}}-{y_{i}}^{\text{raw}})^{2}}$ (16) Figure 5: (a) The relationship between the permeability values $\\{0.41,0.42,\ldots,0.80\\}$ and the raw $\varPi(\bm{p},L)$ or the statistic average from the outputs of CNNs-I. (b) The relationship between the permeability values $\\{0.41,0.42,\ldots,0.80\\}$ and the raw $P(\bm{p},L)$ or the statistic average from the outputs of CNNs-II. Before assess CNNs-I and CNNs-II, we have to explain what is meant by statistic average. Statistic average can be defined as the averages of CNNs-I’s or CNNs-II’s outputs for each permeability $p$. As shown in Fig. 5, there is a clear trend of phase transition between the permeability values $\\{0.41,0.42,\ldots,0.80\\}$ and $\varPi(\bm{p},L)$/$P(\bm{p},L)$. The two grey lines in Fig. 5, which are the same as the two blue lines in Fig. 1, represent the relationship between the permeability values $\\{0.41,0.42,\ldots,0.80\\}$ and the raw $\varPi(\bm{p},L)$ or $P(\bm{p},L)$. Likewise, the two blue lines in Fig. 5, represent the relationship between the permeability values $\\{0.41,0.42,\ldots,0.80\\}$ and the statistic average from the outputs of CNNs-I and CNNs-II. The overlapping of the two kinds of lines shows that CNNs-I and CNNs-II can be used to deduce the two order parameters and the process of phase transition. To overcome the difficulty associated with the percolation model being near the critical transition point, we truncate the dataset. Specifically, we remove the data near the critical transition point, and only retain the data far away from the critical transition point. Here we take the simulation of $\varPi(\bm{p},L)$ as an example. The retained data with the raw $\varPi(\bm{p},L)$ rangs from 0 to 0.1, and 0.9 to 1. As shown in Fig. S. 3 and the middle red points in Fig. 6, we find that CNNs-I can extrapolate $\varPi(\bm{p},L)$ to missing data by learning the retained data. Figure 6: Identification of phase transition with truncated dataset by CNNs-I. The permeability values $\\{0.41,0.42,\ldots,0.80\\}$ and $\varPi(\bm{p},L)$ connected with red points are artificially removed from the dataset. And CNNs-I makes the judgment only by learning the data associated with blue points. The grey curve show that $\varPi(\bm{p},L)$ shifts with the permeability values $\\{0.41,0.42,\ldots,0.80\\}$. This section demonstrate that CNNs-I and CNNs-II can be two effective tools for detecting $\varPi(\bm{p},L)$ and $P(\bm{p},L)$, respectively. Additional test should be made to verify that whether or not CNNs-I and CNNs-II are robust against noise. To address this issue, we deliberately invert a proportion, i.e., 5%, 10%, and 20%, of the labels for the raw $\varPi(\bm{p},L)$ and $P(\bm{p},L)$ and verify that whether or not the “artificial” noises can affect the predicted $\varPi(\bm{p},L)$ and $P(\bm{p},L)$. Fig. 7 and Fig. S. 4 demonstrate that CNNs-I and CNNs-II are robust against noise. As the labeling error rates increase, the same trend is evident in the outputs of CNNs-I and CNNs-II within a relatively small difference (see Fig. 7). Therefore, we draw the conclusion that noises have little effect on detecting $\varPi(\bm{p},L)$ and $P(\bm{p},L)$. Figure 7: Robustness of the two order parameters ($\varPi(\bm{p},L)$ and $P(\bm{p},L)$) with noises for CNNs-I and CNNs-II. (a) The relationship between the permeability values $\\{0.41,0.42,\ldots,0.80\\}$ and the raw $\varPi(\bm{p},L)$ or the statistic average from the outputs of CNNs-I under different noisy inputs. (b) The relationship between the permeability values $\\{0.41,0.42,\ldots,0.80\\}$ and the raw $P(\bm{p},L)$ or the statistic average from the outputs of CNNs-II under different noisy inputs. ### IV.2 Simulate the permeability by one CNNs Just like we use CNNs-I and CNNs-II in Sec. IV.1, here we use the same structure for CNNs-III and strategies for training the permeability $\bm{p}$. The only distinction between CNNs-III and CNNs-I/CNNs-II is that the outputs for CNNs-III are the permeability $\bm{p}$ instead of the two order parameters. Fig. S. 5 shows the performance of CNNs-III. With successive increases in epochs, the MSE, MAE, and RMSE continue to decrease until no longer dropping. Another measure of CNNs-III’s performance is concerned with the difference between the raw permeability $\bm{p}$ and its prediction $\hat{\bm{p}}$. The blue circles in Fig. 8 show that there is a strong positive correlation between the raw permeability $\bm{p}$ and its prediction $\hat{\bm{p}}$. Further statistical tests reveal that most of the gap between the raw permeability $\bm{p}$ and its prediction $\hat{\bm{p}}$ is less than 0.1. The result proves that CNNs-III has the advantage of convenient use and high precision. Figure 8: The correlation between the raw permeability $\bm{p}$ and its predictions. The blue circles refer to the relationship between the raw permeability $\bm{p}$ in dataset and its predictions $\hat{\bm{p}}$ by CNNs- III. The red circles refer to the relationship between the raw permeability $\bm{p}_{\text{ex}}$ out of dataset and its prediction $\hat{\bm{p}}_{\text{extrapolated}}$ by CNNs-III. The raw permeability $\bm{p}_{\text{ex}}$ out of dataset not only range from 0.01 to 0.4, and from 0.81 to 1.0, with an interval of 0.01. The black line refers to the correlation between the raw permeabilities $\\{\bm{p},\bm{p}_{\text{extrapolated}}\\}$ ranges from 0.01 to 1.0 with an interval of 0.01 and their average predictions. On the other hand, extrapolation ability can also reflect the performance of CNNs-III. To do this, we use Monte Carlo simulation to generate new dataset. The new permeability $\bm{p}$ not only ranges from 0.01 to 0.40, but also from 0.81 to 1.00, with an interval of 0.01. Just like Sec. II, we perform 1050 Monte Carlo steps and keep the last 1000 steps. As a consequence, 60,000 configurations are generated. The red circles in Fig. 8 exhibit the extrapolation ability of CNNs-III. Our results show that no significant correlation between the extrapolated permeability $\bm{p}_{\text{ex}}$ ranging from 0.81 to 1.00 and their prediction $\hat{\bm{p}}_{\text{extrapolated}}$. However, in Fig. 8, the results indicate that CNNs-III has good extrapolation ability for the extrapolated permeability $\bm{p}_{\text{ex}}$ from 0.01 to 0.40. ### IV.3 Generate new configurations by one VAE and one cVAE Though CNNs-I, CNNs-II, and CNNs-III are valid when detecting the two order parameters ($\varPi(\bm{p},L)$ and $P(\bm{p},L)$) and the permeability $\bm{p}$, the validity of these three CNNs is unkown for percolating configurations outside of the dataset. As is shown in Fig. 3 and Fig. 4, we use the same network structures for the VAE and the cVAE to generate new configurations ($\hat{\bm{X}}_{\text{vae}}$ and $\hat{\bm{X}}_{\text{cvae}}$). Actually, $\hat{\bm{X}}_{\text{vae}}$ and $\hat{\bm{X}}_{\text{cvae}}$ can be regarded as adding some noise into the raw configuration $\bm{X}$. Let us first consider VAE. Just like AE (see Fig. S. 1), the VAE is also composed of an encoder and a decoder. The encoder of the VAE owns two fully connected layers, both of which follows with a ReLU activation function. The first layer of the encoder possess “size1”=512 neurons. Another 512 neurons, including 256 mean “$\bm{\mu}$” and 256 variance “$\bm{\sigma}$”, are taken into account for the second layer of the encoder. By resampling from the Gaussian distribution with the mean “$\bm{\mu}$” and the variance “$\bm{\sigma}$”, we obtain 256 latent variables “$\bm{Z}$” which are the inputs of the decoder. For the decoder of the VAE, two fully connected layers follow with the outputs of the encoder. For symmetry, the first layer of the decoder also contains “size1”=512 neurons and follows with a ReLU activation function. And the output layer of the decoder contains 784 neurons which are used to reconstruct the raw configuration $\bm{X}$. Thus, the neurons in the output layer are the same as that in the input layer. Moving on now to consider cVAE. The encoder of the cVAE is composed of one input layer, two hidden convolution layers with “filter1”=32 and “filter2”=64 filters with the size of 3 and a stride of 2, and a ReLU activation function. The output layer in the encoder is a fully connected flatten layer with 800 neurons (400 mean “$\bm{\mu}$” and 400 variance “$\bm{\sigma}$”) without activation function. By resampling from the Gaussian distribution with the mean “$\bm{\mu}$” and the variance “$\bm{\sigma}$”, we obtain 400 latent variables “$\bm{Z}$”. The reason why latent variables in the cVAE is more than the VAE is that the cVAE needs to consider more complex spatial characteristic. The decoder of the cVAE is composed of an input layer with 400 latent variables “$\bm{Z}$”. A fully connected layer “FC” with 1,568 neurons is followed by “$\bm{Z}$”. After reshaping the outputs of “FC” into three dimension, we feed the data into two transposed convolution layers (“Conv3” and “Conv4”) and one output layer “Output”. The filters in these deconvolution layers are 64, 32 and 1 with the size of 3 and the stride of 2. After excluding the output layer “Output” without activation functions, there exist two ReLU activation functions followed by “Conv3” and “Conv4”, respectively. We train the VAE and the cVAE over $10^{3}$ epochs using the Adam optimizer, a learning rate of $10^{-3}$, and a mini batch size of 256. To train the VAE and the cVAE, we use the sum of binary cross-entropy (see Eq. 17) and the Kullback-Leibler (KL) divergence (see Eq. 18) as the loss function [18]. In Eq. 17-18, $\bm{x}_{i}^{\text{raw}}$ and $\bm{x}_{i}^{\text{pred}}$ represent each raw configuration with one/two dimension and its prediction. As shown in Fig. S. 6, the loss function, the binary cross-entropy, and the KL divergence vary with epochs for the VAE and the cVAE. Here we focus on the minimum value of loss function. From Fig. S. 6, the optimal cVAE performs better than the optimal VAE. $\displaystyle\text{BinaryCrossEntropy}=-\sum_{i=1}^{M}((\bm{x}_{i}^{\text{pred}}\times\text{log}(\bm{x}_{i}^{\text{raw}})$ $\displaystyle+(1-\bm{x}_{i}^{\text{pred}})\times\text{log}(1-\bm{x}_{i}^{\text{raw}})).$ (17) $\text{KL}_{\text{divergence}}=-\sum_{i=1}^{M}\Biggl{(}\bm{x}_{i}^{\text{raw}}\times\text{log}\left(\frac{\bm{x}_{i}^{\text{raw}}}{\bm{x}_{i}^{\text{pred}}}\right)\Biggr{)}.$ (18) For a more visual comparison, we show the snapshots of the raw configurations $\\{\bm{X}_{1},\bm{X}_{2},\ldots,\bm{X}_{M}\\}$, and compare them to the VAE- generated and cVAE-generated configurations in Fig. 9. As we can see from Fig. 9, the configurations from the Monte Carlo simulation, the VAE and the cVAE are very close to each other. Figure 9: Snapshots of percolating configurations for $p\in\\{0.46,0.60,0.75\\}$. The configurations in the top, middle, and bottom panels are sampled from the Monte Carlo simulation, the VAE, and the cVAE, respectively. Figure 10: The relationship between the permeability values $\\{0.41,0.42,\ldots,0.80\\}$ and the $\varPi(\bm{p},L)$ or the statistic averages from the outputs of CNNs-I that originate from the outputs of the VAE and the cVAE. (b) The relationship between the permeability values $\\{0.41,0.42,\ldots,0.80\\}$ and $P(\bm{p},L)$ or the statistic averages from the outputs of CNNs-II that originate from the outputs of the VAE and the cVAE. After reconstructing the configurations ($\hat{\bm{X}}_{\text{vae}}$ and $\hat{\bm{X}}_{\text{cvae}}$) through the VAE and the cVAE and pouring $\hat{\bm{X}}_{\text{vae}}$ and $\hat{\bm{X}}_{\text{cvae}}$ into CNNs-I, CNNs-II, and CNNs-III, we can detect $\varPi(\bm{p},L)$, $P(\bm{p},L)$, and the permeability $\bm{p}$. Fig. 10 shows the relationships between the permeability values $\\{0.41,0.42,\ldots,0.80\\}$ and the statistic average from the outputs in CNNs-I and CNNs-II, respectively. From the red and purple lines in Fig. 10, by using $\hat{\bm{X}}_{\text{vae}}$ and $\hat{\bm{X}}_{\text{cvae}}$, we can obtain the two order parameters as well. From the Fig. 11, the raw permeability $\bm{p}$ is remarkably correlated linearly with its prediction $\hat{\bm{p}}$ through the VAE/cVAE and CNNs-III. Thus, subtle change in the raw configurations does not effect the catch of physical features. Figure 11: The correlation between the raw permeability $\bm{p}$ and its predictions through VAE/cVAE and CNNs-III. The blue circles refers to the relationship between the raw permeability $\bm{p}$ and its predictions by VAE/cVAE and CNNs-III. And the black line refers to the correlation between the raw permeability $\bm{p}$ and its average predictions. ### IV.4 Identify the characteristic of the first principal component by one PCA This section will discuss how to capture the physic characteristics without labels. Various studies suggest to use PCA to identify order parameters [15, 8, 14, 17]. Therefore, we try to verify the feasibility of PCA in capturing order parameter on the percolation model. First, we perform the PCA to the raw configuration $\bm{X}$. Fig. 12 exhibit the $N$ normalized eigenvalues $\tilde{\bm{\lambda}}_{n}=\bm{\lambda}_{n}/\sum_{n=1}^{N}\bm{\lambda}_{n}$. $\tilde{\bm{\lambda}}_{n}$ is also called as the explained variance ratios. The most noteworthy information in Fig. 12 is that there is one dominant principal component $\tilde{\bm{\lambda}}_{1}$, whcih is the largest one among $\tilde{\bm{\lambda}}_{n}$ and much larger than other explained variance ratios. Thus, $\tilde{\bm{\lambda}}_{1}$ plays a key role when dealing with dimension reduction. Based on $\tilde{\bm{\lambda}}_{1}$, the raw configuration $\bm{X}$ are mapped to another matrix $\bm{Y}=\bm{X}\tilde{\bm{\lambda}}_{1}$. Figure 12: The explained variance ratios obtained from the raw configuration $\bm{X}$ by the PCA, with the horizontal axis indicating corresponding component labels. The largest value of the explained variance ratios locating at the top-left corner means that there exists one dominant principle component. In Fig 13, we construct the matrix $\bm{Y}^{{}^{\prime}}=\\{{\bm{X}\tilde{\bm{\lambda}}_{1},\bm{X}\tilde{\bm{\lambda}}_{2}}\\}$ by the first two eigenvalues and their eigenvectors. We use 40,000 blue scatter points to plot the the relationship between $\bm{X}\tilde{\bm{\lambda}}_{1}$ and $\bm{X}\tilde{\bm{\lambda}}_{2}$ on 40 permeability values ranging from 0.41 to 0.8. Just like in Fig. 12, there is only one dominant representation on the percolation model due to the first principal component is much more important than the second principal component. Figure 13: Projection of the raw configuration $\bm{X}$ onto the plane of the first two dominant principal components, i.e., $\bm{X}\tilde{\bm{\lambda}}_{1}$ and $\bm{X}\tilde{\bm{\lambda}}_{2}$. The color bar on the right indicates the permeability values $\\{0.41,0.42,\ldots,0.80\\}$. Having analysed the importance of the first principal component, we now move on to discuss the meaning of the first principal component. In Fig. 14, we focus on the quantified first principal component as a function of the permeability $\bm{p}$ and the two order parameters, i.e., $\varPi(\bm{p},L)$ and $P(\bm{p},L)$. From Fig. 14, we can see that there is a strong linear correlation between the quantified first principal component and the permeability $\bm{p}$. And the relationships between the quantified first principal component and two order parameters ($\varPi(\bm{p},L)$ and $P(\bm{p},L)$) are similar to the relationships between the permeability values $\\{0.41,0.42,\ldots,0.80\\}$ and the two order parameters. Our results are significant different from former study which demonstrates that the quantified first principal component can be taken as order parameter by data preprocessing [17]. A possible explanation may be that [17] evaluates the first principal component by removing the raw dataset with certain attributes on the percolation model. Therefore, we assume that the quantified first principal component obtained by PCA may not be taken as order parameter for various physical models. Another possible explanation is that, under certain conditions, the quantified first principal component can be regarded as order parameter. Figure 14: Taking the normalized quantified first principal component $\bm{X}\tilde{\bm{\lambda}}_{1}$ as a function of the permeability $\bm{X}$ and the two order parameters, i.e., $\varPi(\bm{p},L)$ and $P(\bm{p},L)$. ### IV.5 Identify physic characteristics by one $k$-means and one CNNs From Sec. IV.4, no significant corresponding is found between the normalized quantified first principal component and the two order parameters. Here we eagerly wonder how to identify order parameter. And another physical characteristics we desire to explore is the critical transition point. So we try to find a way to capture order parameter from the raw configurations $\\{\bm{X}_{1},\bm{X}_{2},\ldots,\bm{X}_{M}\\}$ and their permeability $\bm{p}$ on the percolation model. As it is well-known, for the two-dimensional percolating configurations, the critical transition point is equal to 0.593 in theoretical calculation. Though the critical transition point is already known, we wonder that whether or not the critical transition point can be found by machine. To do this, we first use a cluster analysis algorithm named $k$-means [31] to separate the raw configurations $\\{\bm{X}_{1},\bm{X}_{2},\ldots,\bm{X}_{M}\\}$ into two categories. The minimum and maximum value of their permeability $\bm{p}$ are 0.55 and 0.80 in the first cluster, and 0.41 and 0.65 in the second cluster. Note that there is an overlapping interval between 0.55 and 0.65 for the two categories. According to the overlapping interval, we hypothesize that the critical threshold $p_{c}$ is set to be 11 values, i.e., 0.55, 0.56, $\ldots$, and 0.65. For each raw configuration, if the permeability is smaller than $p_{c}$, there will exist no percolating cluster and its label will be marked as 0; otherwise, there will exist at least one percolating cluster and its label will be marked as 1. To detect physic characteristics, we use the fourth kinds of CNNs (CNNs-IV) with the structure in Fig. 2 for 11 preset critical thresholds from 0.55 to 0.65. Note that the output layer use a sigmoid activation function expressed as $\bm{a}=1/(1+e^{-\bm{x}})$, to make sure the outputs are between 0 and 1. Another critical thing to pay attention is that the He normal distribution initializer [32] and L2 regularization [33] are used in the layer of “Conv1”, “Conv2”, and “FC” on CNNs-IV. To avoid overfiting, in addition to L2 regularization, we also use a dropout layer with a dropout rate of 0.5 on “FC”. A Mini batch size of 512 and a learning rate of $10^{-4}$ are chosen while training CNNs-IV. The binary cross-entropy (see Eq. 17) is taken as the loss function on CNNs-IV. Another metric, used to measure the performance of CNNs-IV, is the binary accuracy (see Eq. 19). The other hyper-parameters are the same as the CNNs-I, CNNs-II, and CNNs-III. $\displaystyle\text{BinaryAccuracy}=\sum_{i=1}^{M}\frac{n_{(y_{i}^{\text{pred}}==y_{i}^{\text{raw}})}}{n_{y_{i}^{\text{pred}}}}.$ (19) Turning now to the experimental evidence on the inference ability of capturing relevant physic features. After obtaining the well-trained CNNs-IV with high accuracy (see Fig. S. 7), we obtain the outputs by pouring the raw 40,000 configurations $\\{\bm{X}_{1},\bm{X}_{2},\ldots,\bm{X}_{M}\\}$ into CNNs-IV. The statistical average of the outputs is calculated according to the 40 independent permeability values $\\{0.41,0.42,\ldots,0.80\\}$. The results of the correlational analysis are shown in Fig. 15. We set the horizontal dashed line as the threshold value 0.5. Hence, each curve is divided into two parts by the horizontal dashed line. The lower part indicates that the percolation system is not penetrated; while the upper part implies that the percolation system is penetrated. The crosspoint, where the horizontal dashed line and the red curve are intersected, has a permeability value of 0.594, which is very close to the theoretical value of 0.593 that is marked by the vertical dashed line in Fig. 15. Remarkably, the critical transition point can be calculated by CNNs-IV with the preset value of 0.60 for the sampling interval of 0.01. Therefore, CNNs-IV with the preset threshold value of 0.60 is the most effective model. In further studies, the preset threshold value may need to be enhanced by smaller sampling intervals for higher precision. Figure 15: The 11 curves show that the average outputs shifts when the preset threshold changes from 0.55 to 0.65. The average outputs and the threshold of phase transitions deduce from different preset threshold values on the percolation model by CNNs-IV. ## V Conclusions As machine learning approaches have become increasingly popular in phase transitions and critical phenomena, predecessors have pointed out that these approaches can capture physic characteristics. However, previous studies about identifying physical characteristics, especially order parameter and critical threshold, need to be further mutually validated. To highlight the possibility of effectiveness by machine learning methods, we conduct a much more comprehensive research than predecessors to reassess the machine learning approaches in phase transitions and critical phenomena. Our results show the effectiveness of machine learning approaches in phase transitions and critical phenomena than previous researchers. Precisely, we use CNNs-I, CNNs-II and CNNs-III to simulate the two order parameters, and the permeability values. To identify whether or not CNNs-I and CNNs-II are robust against noise, we add a proportion of the noises for the two order parameters. To validate the robustness of CNNs-I, CNNs-II and CNNs-III, we also use VAE and cVAE to generate new configurations that are slightly different from their raw configurations. After pouring the new configurations into the CNNs-I, CNNs-II, and CNNs-III, we achieve the results that these models are robust against noise. However, after we use PCA to reduce the dimension of the raw configurations and make a statistically significant linear correlation between the first principal component and the permeability values, no statistically significant linear correlations are found between the first principal component and the two order parameters. Clearly, the first principal component fails to be regarded as an order parameter in the two-dimensional percolation model. To identify order parameter, we use the fourth kinds of CNNs, i.e., CNNs-IV. The results show that CNNs-IV can identify new order parameter when the preset threshold value is 0.60. Surprisingly, we find that the critical transition point value is 0.594 by CNNs-IV. Although these machine learning methods are valid to explore the physical characteristics in the percolation model, the current study may still have some inevitable limitations that prevent us from making an overall judgement by these methods on the other models of phase transitions and critical phenomena. In other words, it must be acknowledged that this research is based on the two-dimensional percolation model.We are not sure of the usefulness of applying our methods to the other models. Consequently, our methods in this study may open an opportunity to other models on phase transitions and critical phenomena for further research. ###### Acknowledgements. The authors gratefully thank Yicun Guo for revising the manuscript. 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# Rationality of even-dimensional intersections of two real quadrics Brendan Hassett Department of Mathematics Brown University Box 1917 151 Thayer Street Providence, RI 02912 USA brendan<EMAIL_ADDRESS>, János Kollár Department of Mathematics Princeton University Fine Hall, Washington Road Princeton NJ 08544-1000 USA<EMAIL_ADDRESS>and Yuri Tschinkel Courant Institute New York University New York, NY 10012 USA Simons Foundation 160 Fifth Avenue New York, NY 10010 USA<EMAIL_ADDRESS> ###### Abstract. We study rationality constructions for smooth complete intersections of two quadrics over nonclosed fields. Over the real numbers, we establish a criterion for rationality in dimension four. ## 1\. Introduction Consider a geometrically rational variety $X$, smooth and projective over a field $k$. Is $X$ rational over $k$? A necessary condition is that $X(k)\neq\emptyset$, which is sufficient in dimension one, as well as for quadric hypersurfaces and Brauer-Severi varieties of arbitrary dimension. When the dimension of $X$ is at most two, rationality over $k$ was settled by work of Enriques, Manin, and Iskovksikh [Isk79]. Rationality is encoded in the Galois action on the geometric Néron-Severi group – varieties with rational points that are ‘minimal’ in the sense of birational geometry need not be rational. In dimension three, recent work [HT19a, HT19c, KP20a, BW20, BW19] has clarified the criteria for rationality: one also needs to take into account principal homogeneous spaces over the intermediate Jacobian, reflecting which curve classes are realized over the ground field. The case of complete intersections of two quadrics was an important first step in understanding the overall structure [HT19b]; rationality in dimension three is equivalent to the existence of a line over $k$ [HT19a, BW19, KP20a]. These developments stimulate investigations in higher dimensions [KP20b]; the examples considered are rational provided there are rational points. In this paper, we focus on the case of four-dimensional complete intersections of two quadrics, especially over the real numbers $\mathbb{R}$. Here we exhibit rational examples without lines and explore further rationality constructions. ###### Theorem 1.1. A smooth complete intersection of two quadrics $X\subset\mathbb{P}^{6}$ over $\mathbb{R}$ is rational if and only if $X(\mathbb{R})$ is nonempty and connected. In general, a projective variety $X$ that is rational over $\mathbb{R}$ has connected nonempty real locus $X(\mathbb{R})$. The point of Theorem 1.1 is that this necessary condition is also sufficient. ###### Corollary 1.2. A smooth complete intersection of two quadrics $X\subset\mathbb{P}^{6}$ is rational over $\mathbb{R}$ if and only if there exists a unirational parametrization $\mathbb{P}^{4}\dashrightarrow X$, defined over $\mathbb{R}$, of odd degree. Indeed, odd degree rational maps are surjective on real points, which guarantees that $X(\mathbb{R})$ is connected. Smooth complete intersections of two quadrics, of dimension at least two, are unirational provided they have a rational point; see Section 3.1 for references and discussion. We also characterize rationality in dimension six, with the exception of one isotopy class that remains open (see Section 6.2). Here is the roadmap of this paper. In Section 2 we recall basic facts about quadrics in even-dimensional projective spaces and their intersections. All interesting cohomology is spanned by the classes of projective subspaces in $X$ of half-dimension, and the Galois group acts on these classes via symmetries of the primitive part of this cohomology, a lattice for the root system $D_{2n+3}$. In Section 3 we present several rationality constructions. The isotopy classification, using Krasnov’s invariants, is presented in Section 4; we draw connections with the Weyl group actions. In Section 5 we focus attention on cases where rationality is not obvious, e.g., due to the presence of a line. In Section 6 we prove Theorem 1.1 and discuss the applicability and limitations of our constructions in dimensions four and six. We speculate on possible extensions to more general fields in Section 7. Acknowledgments: The first author was partially supported by NSF grant 1701659 and Simons Foundation Award 546235, the second author by the NSF grant DMS-1901855, and the third author by NSF grant 2000099. ## 2\. Geometric background ### 2.1. Roots and weights Let $D_{2n+3}$ be the root lattice of the corresponding Dynkin diagram, expressed in the standard Euclidean lattice $\left<L_{1},\ldots,L_{2n+3}\right>,\quad\quad L_{i}\cdot L_{j}=\delta_{ij}$ as the lattice generated by simple roots $\displaystyle R_{1}=L_{1}-L_{2},R_{2}=L_{2}-L_{3},$ $\displaystyle\ldots,R_{2n+1}=L_{2n+1}-L_{2n+2},$ $\displaystyle R_{2n+2}=L_{2n+2}-L_{2n+3},\quad$ $\displaystyle R_{2n+3}=L_{2n+2}+L_{2n+3}.$ Its discriminant group is cyclic of order four, generated by $\tfrac{1}{4}(2(R_{1}+2R_{2}+\cdots+(2n+1)R_{2n+1})+(2n+1)R_{2n+2}+(2n+3)R_{2n+3}).$ Multiplication by $-1$ acts on the discriminant via $\pm 1$. The outer automorphisms of $D_{2n+3}$ also act via automorphisms of $D_{2n+3}$ acting on the discriminant via $\pm 1$, e.g., exchanging $R_{2n+2}$ and $R_{2n+3}$ and keeping the other roots fixed. The Weyl group $W(D_{2n+3})$ acts in the basis $\\{L_{i}\\}$ via signed permutations with an even number of $-1$ entries. The outer automorphisms act via signed permutations with no constraints on the choice of signs, e.g., $L_{i}\mapsto L_{i},\quad i=1,\ldots,2n+2,\quad\quad L_{2n+3}\mapsto- L_{2n+3}.$ The odd and even half-spin representations have weights indexed by subsets $I\subset\\{1,2,\ldots,2n+3\\}$, with $\|I\|$ odd or even, written $w_{I}=\tfrac{1}{2}(\sum_{i\in I}L_{i}-\sum_{j\in I^{c}}L_{j}).$ The odd and even weights are exchanged by outer automorphisms. ### 2.2. Planes In this section, we assume that the ground field is algebraically closed of characteristic zero. Let $X\subset\mathbb{P}^{2n+2}$ be a smooth complete intersection of two quadrics. We will identify subvarieties in $X$ with their classes in the cohomology of $X$ when no confusion may arise. Let $h$ denote the hyperplane section and consider the primitive cohomology of $X$ under the intersection pairing. Reid [Rei72, 3.14] shows that $(h^{n})^{\perp}\simeq(-1)^{n}D_{2n+3}.$ In other words, the primitive sublattice of $H^{2n}(X,\mathbb{Z}(n))$ – the Tate twist of singular cohomology for the underlying complex variety – may be identified with the root lattice. This is the target of the cycle class map $\operatorname{CH}^{n}(X)\rightarrow H^{2n}(X,\mathbb{Z}(n))$ so the sign convention is natural. ###### Remark 2.1 (Caveat on signs). When $X$ is defined over $\mathbb{R}$, codimension-$n$ subvarieties $Z\subset X$ defined over $\mathbb{R}$ yield classes in $H^{2n}(X,\mathbb{Z}(n))$ that are invariant under complex conjugation. However, the corresponding classes in $H^{2n}(X,\mathbb{Z})$ are multiplied by $(-1)^{n}$. When we mention invariant classes, it is with respect to the former action. Given a plane $P\simeq\mathbb{P}^{n}\subset X$, we have $(P\cdot P)_{X}=c_{n}$ where [Rei72, 3.11] $c_{0}=1,c_{1}=-1,c_{2}=2,c_{3}=-2,\ldots,c_{n}=(-1)^{n}(\lfloor\tfrac{n}{2}\rfloor+1).$ The projection of $P$ into rational primitive cohomology takes the form $P-\tfrac{1}{4}h^{n}$ which has self-intersection $c_{n}-1/4$. The corresponding element $w_{P}\in D_{2n+3}$ has $w_{P}\cdot w_{P}=(2n+3)/4.$ By [Rei72, Cor. 3.9], we obtain bijections $\\{w_{P}\\}_{P\simeq\mathbb{P}^{n}\subset X}=\\{w_{I}\\}_{\|I\|\text{ has fixed parity }}.$ Note that the residual intersections to $\mathbb{P}^{n}\subset X$ $X\cap\mathbb{P}^{n+2}=\mathbb{P}^{n}\cup S$ give cubic scrolls $S\subset X$; these realize the weights of opposite parity. By [Rei72, Th. 3.8], two planes $P_{1}$ and $P_{2}$ are disjoint if and only if $w_{P_{1}}\cdot w_{P_{2}}=(-1)^{n+1}/4.$ For example, if $n=1$ and $w_{P_{1}}$ is identified with $(L_{1}-L_{2}-L_{3}-L_{4}-L_{5})/2$ then the relevant weights are $(L_{1}+L_{2}+L_{3}-L_{4}-L_{5})/2,\quad\ldots,\quad(L_{1}-L_{2}-L_{3}+L_{4}+L_{5})/2$ and $(-L_{1}+L_{2}+L_{3}+L_{4}-L_{5})/2,\quad\ldots,\quad(-L_{1}-L_{2}+L_{3}+L_{4}+L_{5})/2,$ a total of $10=\binom{5}{2}$ such lines. When $n=2$ and $w_{P_{1}}$ is identified with $(L_{1}-L_{2}-\cdots-L_{7})/2$ then the relevant weights are $(L_{1}+L_{2}+L_{3}+L_{4}+L_{5}-L_{6}-L_{7})/2,\ldots,(L_{1}-L_{2}-L_{3}+L_{4}+L_{5}+L_{6}+L_{7})/2$ and $(-L_{1}+L_{2}+L_{3}+L_{4}-L_{5}-L_{6}-L_{7})/2,\ldots,(-L_{1}-L_{2}-L_{3}-L_{4}+L_{5}+L_{6}+L_{7})/2,$ a total of $\binom{6}{4}+\binom{6}{3}=35=\binom{7}{3}$ planes. The planes $P_{1}$ and $P_{2}$ meet at a point if and only if $w_{P_{1}}\cdot w_{P_{2}}=(-1)^{n}\tfrac{3}{4}.$ If they meet along an $r$-plane then an excess intersection computation gives [Rei72, 3.10] $P_{1}\cdot P_{2}=(-1)^{r}(\lfloor\tfrac{r}{2}\rfloor)+1)$ and $w_{P_{1}}\cdot w_{P_{2}}=(-1)^{r+n}(\lfloor\tfrac{r}{2}\rfloor+1)-(-1)^{n}\tfrac{1}{4}.$ In particular, they meet along an $(n-1)$-plane when $w_{P_{1}}\cdot w_{P_{2}}=-(\lfloor\tfrac{n-1}{2}\rfloor+1)-(-1)^{n}\tfrac{1}{4};$ for a fixed $w_{P_{1}}$ there are $2n+3$ planes $P_{2}$ meeting $P_{1}$ in this way. For example, if $n=1$ and $w_{P_{1}}=(L_{1}-L_{2}-L_{3}-L_{4}-L_{5})/2$ then the possibilities for $w_{P_{2}}$ are $\displaystyle(L_{1}+L_{2}+L_{3}+L_{4}+L_{5})/2,\quad(-L_{1}-L_{2}+L_{3}+L_{4}+L_{5})/2,$ $\displaystyle(-L_{1}+L_{2}-L_{3}+L_{4}+L_{5})/2,\quad(-L_{1}+L_{2}+L_{3}-L_{4}+L_{5})/2,$ $\displaystyle(-L_{1}+L_{2}+L_{3}+L_{4}-L_{5})/2.$ ### 2.3. Quadrics We retain the notation of Section 2.2. Our next task is to analyze quadric $n$-folds $Q\subset X$, i.e., $Q$ a degree-two hypersurface in $\mathbb{P}^{n+1}$. Let $\\{\mathcal{Q}_{t}\\},t\in\mathbb{P}^{1}$ denote the pencil of quadric hypersurfaces cutting out $X$. The degeneracy locus $D:=\\{t\in\mathbb{P}^{1}:\mathcal{Q}_{t}\text{ singular }\\}$ consists of $2n+3$ points; since $X$ is smooth, each has rank $2n+2$. The Hilbert scheme of quadric $n$-folds $Q\subset X$ is isomorphic to the relative Fano variety of $(n+1)$-planes $\mathcal{F}(\mathcal{Q}/\mathbb{P}^{1})=\\{\Pi\simeq\mathbb{P}^{n+1}\subset\mathcal{Q}_{t}\text{ for some }t\in\mathbb{P}^{1}\\},$ which consists of $2(2n+3)$ copies of the connected isotropic Grassmannian $\operatorname{OGr}(n+1,2n+2)$. Given a quadric $Q$, its projection to rational primitive cohomology $Q-\tfrac{1}{2}h^{n}$ corresponds to an element $w_{Q}\in D_{2n+3},\quad w_{Q}\in\\{\pm L_{1},\ldots,\pm L_{2n+3}\\}.$ In particular, we have $Q\cdot Q=\begin{cases}2&\text{ if $n$ even}\\\ 0&\text{ if $n$ odd.}\end{cases}$ Residuation in a complete intersection of linear forms $Q\cup Q^{\prime}=X\cap h^{n}$ reverses signs, i.e., $w_{Q}=-w_{Q^{\prime}}$. On the other hand, if $Q_{1}$ and $Q_{2}$ are not residual then (2.1) $Q_{1}\cdot Q_{2}=1.$ We summarize this as follows: ###### Proposition 2.2. The signed permutation representation of $W(D_{2n+3})$ is realized via the action on classes $[Q]$, where $Q\subset X$ is a quadric $n$-fold. Note that there are $2^{2n+1}(2n+3)$ reducible quadrics – unions of two $n$-planes meeting in an $(n-1)$-plane – with $2^{2n}$ reducible quadrics in each copy of the isotropic Grassmannian. ## 3\. Rationality constructions We now work over an arbitrary field $k$ of characteristic zero. ### 3.1. General considerations Let $X\subset\mathbb{P}^{d+2}$ denote a smooth complete intersection of two quadrics of dimension at least two. Recall the following: * • If $X(k)\neq\emptyset$ then $X$ is unirational over $k$ and has Zariski dense rational points [CTSSD87, Rem. 3.28.3]. * • If there is a line $\ell\subset X$ defined over $k$ then projection induces a birational map $\pi_{\ell}:X\stackrel{{\scriptstyle\sim}}{{\dashrightarrow}}\mathbb{P}^{d}$. For reference, we recall Amer’s theorem [Lee07, Th. 2.2]: ###### Theorem 3.1. Let $k$ be a field of characteristic not two, $F_{1}$ and $F_{2}$ quadrics over $k$, and $\mathcal{Q}_{t}=\\{F_{1}+tF_{2}\\}$ the associated pencil of quadrics over $k(t)$. Then $X=\\{F_{1}=F_{2}=0\\}$ has an $r$-dimensional isotropic subspace over $k$ if and only if $\mathcal{Q}_{t}$ has an $r$-dimensional isotropic subspace over $k(t)$. We apply this for $k=\mathbb{R}$, where $X\subset\mathbb{P}^{d+2}$ is a smooth complete intersection of two quadrics and $\mathcal{Q}\rightarrow\mathbb{P}^{1}$ is the associated pencil. Recall Springer’s Theorem: A quadric hypersurface $\mathcal{Q}$ over a field $L$ has a rational point if it admits a rational point over some odd-degree extension of $L$. Applying this to the pencil $\mathcal{Q}\rightarrow\mathbb{P}^{1}$ associated with a complete intersection of two quadrics, with Amer’s Theorem, yields: ###### Proposition 3.2. If $d\geq 1$ and $X$ contains a subvariety of odd degree over $k$ then $X(k)\neq\emptyset$. We can prove a bit more: ###### Proposition 3.3. If $d\geq 2$ and $X$ has a curve of odd degree defined over $k$ then $X$ is rational over $k$. ###### Proof. Recall that double projection from a sufficiently general rational point $x\in X(k)$ yields a diagram $X\stackrel{{\scriptstyle\sim}}{{\dashrightarrow}}Y\rightarrow\mathbb{P}^{1}$ where $Y$ is a quadric bundle of relative dimension $d-1$. A curve $C\subset X$ of odd degree yields an multisection of this bundle of odd degree. Thus $Y\rightarrow\mathbb{P}^{1}$ has a section by Springer’s Theorem and its generic fiber $Y_{t}$ is rational over $k(\mathbb{P}^{1})$. It follows that $X$ is rational over the ground field. ∎ ###### Remark 3.4. The pencil defining $X$ gives a quadric bundle $\mathcal{Q}\rightarrow\mathbb{P}^{1}$ of relative dimension $d+1$. We apply Witt’s decomposition theorem to $[\mathcal{Q}_{t}]$ and $[Y_{t}]$, understood as quadratic forms over $k(\mathbb{P}^{1})=k(t)$, to obtain $[\mathcal{Q}_{t}]=[Y_{t}]\oplus\left(\begin{matrix}0&1\\\ 1&0\end{matrix}\right).$ Thus a section of $Y\rightarrow\mathbb{P}^{1}$ yields an isotropic line of $\mathcal{Q}\rightarrow\mathbb{P}^{1}$, and Theorem 3.1 implies that $X$ contains a line defined over $k$. ###### Corollary 3.5. (see appendix by Colliot-Thélène [HT19b, Th. A5]) Let $X\subset\mathbb{P}^{d+2}$ denote a smooth complete intersection of two quadrics of dimension at least two. Suppose there exists an irreducible positive-dimensional subvariety $W\subset X$ of odd degree, defined over $k$. Then $X$ is rational over $k$. Given these results, we focus on proving rationality in cases where $X$ does not contain lines or positive-dimensional subvarieties of odd degree. ### 3.2. Rationality using half-dimensional subvarieties We now turn to even-dimensional intersections of two quadrics $X\subset\mathbb{P}^{2n+2},\quad n\geq 1.$ Throughout, we assume that $X(k)\neq\emptyset,$ and thus $X$ is $k$-unirational and $k$-rational points are Zariski dense. Construction I: Suppose that * • $X$ has a pair of conjugate disjoint $n$-planes $P,\bar{P}$, defined over a quadratic extension $K$ of $k$. Projecting from a general $x\in X(k)$ gives a birational map $X\stackrel{{\scriptstyle\sim}}{{\dashrightarrow}}X^{\prime}\subset\mathbb{P}^{2n+1},$ where $X^{\prime}$ is a cubic hypersurface. Since $X(k)\subset X$ is Zariski dense, we may assume that the images of $P$ and $\bar{P}$ in $X^{\prime}$ remain disjoint. The ‘third point’ construction gives a birational map $\mathbf{R}_{K/k}(P)\stackrel{{\scriptstyle\sim}}{{\dashrightarrow}}X^{\prime},$ where the source variety is the restriction of scalars. We conclude that $X$ is rational over $k$. This construction appears in [CTSSD87, Th. 2.4]. Construction II: Suppose that * • $X$ has a pair of conjugate disjoint quadric $n$-folds $Q,\bar{Q}$, defined over a quadratic extension $K$ of $k$, and meeting transversally at one point. Projecting from $x\in Q\cap\bar{Q}$, which is a $k$-rational point $X$, gives a birational map $X\stackrel{{\scriptstyle\sim}}{{\dashrightarrow}}X^{\prime}\subset\mathbb{P}^{2n+1},$ where $X^{\prime}$ is a cubic hypersurface. The proper transforms $Q^{\prime},\bar{Q}^{\prime}\subset X^{\prime}$ are disjoint unless there exists a line $x\in\ell\subset\mathbb{P}^{2n+2}$ with $\\{x\\}\subsetneq\ell\cap Q,\ell\cap\bar{Q}$ as schemes. We may assume that $\ell\not\subset X$ as we already know $X$ is rational in this case. Thus the only possibility is $\ell\cap Q=\ell\cap\bar{Q}$ as length-two subschemes, which is precluded by the intersection assumption. Repeating the argument for Construction I thus gives rationality. Construction III: Suppose that * • $X$ contains a quadric $Q$ of dimension $n$, defined over $k$. Projection gives a fibration $q:\operatorname{Bl}_{Q}(X)\rightarrow\mathbb{P}^{n}$ with fibers quadrics of dimension $n$. Now suppose that $X$ contains a second $n$-fold $T$ with the property that $\deg(T)-T\cdot Q$ is odd, i.e., a multisection for $q$ of odd degree. It follows that the generic fiber of $q$ is rational and thus $Y$ is rational over $k$. When $\dim(X)=4$ a number of $T$ might work, e.g., * • a plane disjoint from $Q$, * • a second quadric meeting $Q$ in one point, * • a quartic or a sextic del Pezzo surface meeting $Q$ in one or three points, * • a degree 8 K3 surface meeting $Q$ in one, three, five, or seven points. Construction IV: Suppose that * • $\dim(X)=4$ and $X$ contains a quartic scroll $T$, defined over $k$. Geometrically, $T$ is the image of the ruled surface ${\mathbb{F}}_{0}=\mathbb{P}^{1}\times\mathbb{P}^{1}\hookrightarrow\mathbb{P}^{5}$ under the linear series of bidegree $(1,2)$. Over $\mathbb{R}$ we are interested in cases where $T=\mathbb{P}^{1}\times C$ with $C$ a nonsplit conic. We do not want to force $X$ to have lines! (Note that quartic scrolls geometrically isomorphic to ${\mathbb{F}}_{2}$ contain lines defined over the ground field and thus are not useful for our purposes.) On projecting from a point $x\in X(k)$ we get a cubic fourfold $X^{\prime}\subset\mathbb{P}^{5},$ containing a quartic scroll. The Beauville-Donagi construction [BD85] – concretely, take the image under the linear system of quadrics vanishing along $T$ – shows that $X^{\prime}$ is birational to a quadric hypersurface thus rational over $k$. Recall that an $n$-dimensional smooth variety $W\subset\mathbb{P}^{2n+1}$ is said to have one apparent double point if a generic point is contained in a unique secant to $W$. Construction V: Suppose that $X$ contains a variety $W$ defined over $k$ of dimension $n\geq 2$ such that: * • $W$ spans a $\mathbb{P}^{2n+1}$ and has one apparent double point; or * • $W$ has a rational point $w$ such that projecting from $w$ maps $W$ birationally to a variety with one apparent double point. Then $X$ is rational over $k$. As before, one projects from a rational point to a get cubic hypersurface $X^{\prime}\subset\mathbb{P}^{2n+1}$. Cubic hypersurfaces containing varieties $W$ with one apparent double point are rational [Rus00, Prop. 9]. Indeed, intersecting secant lines of $W$ with $X^{\prime}$ yields $\operatorname{Sym}^{2}(W)\dashrightarrow X^{\prime},$ which is birational when each point lies on a unique secant to $W$. Quartic scrolls in $\mathbb{P}^{5}$ have one apparent double point so Construction IV is a special case of Construction V. ## 4\. Isotopy classification We review the classification of smooth complete intersections of two quadrics $X\subset\mathbb{P}^{2n+2}$ over $\mathbb{R}$, following [Kra18]. Express $X=\\{F_{1}=F_{2}=0\\}$ where $F_{1}$ and $F_{2}$ are real quadratic forms. We continue to use $D$ for the degeneracy locus of the associated pencil $\mathcal{Q}_{t}=\\{t_{1}F_{1}+t_{2}F_{2}=0\\}$. Let $r=\|D(\mathbb{R})\|$ which is odd with $r\leq 2n+3$. Consider the signatures $(I^{+},I^{-})$ of the forms $s_{1}F_{1}+s_{2}F_{2},\quad(s_{1},s_{2})\in\mathbb{S}^{1}=\\{(s_{1},s_{2})\in\mathbb{R}^{2}:s_{1}^{2}+s_{2}^{2}=1\\}.$ Record these at the $2r$ points lying over $D$, in order as we trace the circle counterclockwise. We label each of these points with $\pm$ depending on whether the positive part $I^{+}$ of the signature increases or decreases as we cross the point. Each point of $D(\mathbb{R})$ yields a pair of antipodal points on $\mathbb{S}^{1}$ labelled with opposite signs. For example, for $n=0$ and $r=3$ admissible sequences of signatures and $\pm$’s include $(0,2)(1,1)(2,0)(2,0)(1,1)(0,2)\quad(+,+,+,-,-,-).$ and $(1,1)(1,1)(1,1)(1,1)(1,1)(1,1)\quad(+,-,+,-,+,-).$ Suppose the sequence of $\pm$’s has maximal unbroken chains of $+$’s of lengths $r_{1},r_{2},\ldots,r_{2s+1}$ where $r=r_{1}+r_{2}+\cdots+r_{2s+1}.$ The number of terms is odd because antipodal points have opposite signs. In the examples above we have $3=3$ and $3=1+1+1$. Our invariant is the sequence $(r_{1},\ldots,r_{2s+1})$ up to cyclic permutations and reversals – a complete isotopy invariant of $X$ over $\mathbb{R}$ [Kra18]. We derive a sequence of $\pm 1$’s of length $k$ from this invariant as follows: For each point of $D(\mathbb{R})$, record the sign of the discriminant of the associated rank-$(2n+2)$ quadratic form, determined by the parity of $(I^{+}-I^{-})/2$. In the examples above, we obtain $(-1,+1,-1)$ and $(+1,+1,+1)$. The number of $-1$’s is always even. The analysis in Section 2.3 shows that complex conjugation acts on $H^{2n}(X,\mathbb{Z}(n))$ in the basis $\\{L_{1},\ldots,L_{2n+3}\\}$ as a signed permutation of order two. This is a direct sum of blocks $(+1),(-1),\pm\left(\begin{matrix}0&1\\\ 1&0\end{matrix}\right).$ Actually, we may assume the sign is positive in the third case after conjugating by $\left(\begin{matrix}0&-1\\\ 1&0\end{matrix}\right).$ Suppose there are $a$ blocks $(+1)$, $2b$ blocks $(-1)$, and $c$ blocks of the third kind, with $a+2b+2c=2n+3$. These correspond to the conjugacy classes of involutions $\iota\in W(D_{2n+3})$ [Kot00, §3.2,3.3]. We have $r=a+2b$, reflecting the number of points of $D(\mathbb{R})$ with positive and negative discriminants respectively, and $2c=2n+3-r$, reflecting the number of complex- conjugate pairs in $D(\mathbb{C})\setminus D(\mathbb{R})$. The passage from isotopy classes to conjugacy classes in $W(D_{2n+3})$ results in a loss of information. We give an example for real quartic del Pezzo surfaces $X\subset\mathbb{P}^{4}$. ###### Example 4.1. The isotopy class $(5)$ has singular members with signatures $(0,4)(1,3)(2,2)(3,1)(4,0)(4,0)(3,1)(2,2)(1,3)(0,4)$ with involution given the diagonal $5\times 5$ matrix $\operatorname{diag}(1,-1,{\bf{1}},-1,1),$ where the bolded $\bf{1}$ corresponds to a degenerate fiber $\mathcal{Q}_{t}$ whose rulings sweep out quadric curves (conics) on $X$ defined over $\mathbb{R}$. (There is only one pair of such conics.) Here $X(\mathbb{R})=\emptyset$ as it is contained in an anisotropic quadric threefold. The isotopy class $(2,2,1)$ has singular members with signatures $(1,3)(2,2)(2,2)(2,2)(3,1)(3,1)(2,2)(2,2)(2,2)(1,3)$ with involution $\operatorname{diag}(-1,{\bf{1}},{\bf{1}},{\bf{1}},-1).$ This has the same Galois action but contains three pairs of conics defined over $\mathbb{R}$, represented by the bolded $\bf{1}$’s. These are distinguished cohomologically by the arrow $\mathbb{Z}^{3}\simeq H^{0}(G,\operatorname{Pic}(X_{\mathbb{C}}))\rightarrow H^{2}(G,\Gamma(\mathcal{O}^{}_{X_{\mathbb{C}}}))=\operatorname{Br}(\mathbb{R})\simeq\mathbb{Z}/2\mathbb{Z}$ of the Hochschild-Serre spectral sequence. ###### Proposition 4.2. Fix a conjugacy class $[\iota=\iota_{abc}]$ of involutions in $W(D_{2n+3})$. Consider the isotopy classes of $X(\mathbb{R})\subset\mathbb{P}^{2n+2}$ such that complex conjugation acts by $\iota$. The possible isotopy classes correspond to shuffles of $(\underbrace{1,\ldots,1}_{a\text{ times}},\underbrace{-1,\ldots,-1}_{2b\text{ times}})$ up to cyclic permutations and reversals. ###### Proof. Observe first that points in $D(\mathbb{C})\setminus D(\mathbb{R})$ are irrelevant to the Krasnov invariant, assuming the dimension is given. So it makes sense they are not relevant in the enumeration. We have already seen that sequences of the prescribed form arise from each isotopy class; we present the reverse construction. Choose such a sequence, e.g. $(1,1,1,-1,1,-1,-1,1,-1).$ The key observation is that local maxima and minima of $I^{+}-I^{-}$ – which necessarily occur at smooth points – arise precisely between points of the degeneracy locus where signs do not change. We indicate smooth fibers achieving maxima/minima with $\mid$, e.g., $1\mid 1\mid 1,-1,1,-1\mid-1,1,-1$ or equivalently $\mid 1\mid 1,-1,1,-1\mid-1,1,-1,1\mid$ after cyclic permutation. Lifting to the double cover $\mathbb{S}^{1}$ entails concatenating two such expressions: $\mid 1\mid 1,-1,1,-1\mid-1,1,-1,1\mid 1\mid 1,-1,1,-1\mid-1,1,-1,1\mid$ From this, we read off the points of $D(\mathbb{R})$ on which $I^{+}$ increases and decreases $+----++++-++++----$ which determines the Krasnov invariant – $(1,4,4)$ in this example. ∎ ## 5\. Applying quadratic forms ### 5.1. Quadric fibrations over real curves Let $C$ be a smooth projective geometrically connected curve over $\mathbb{R}$ with function field $K=\mathbb{R}(C)$. Let $Q\subset\mathbb{P}^{d+1}$ be a smooth (rank $d+2$) quadric hypersurface over $K$ and $F_{i}(Q)$ the $i$-dimensional isotropic subspaces, so that $F_{0}(Q)=Q$ and $F_{m}(Q)$ is empty when $2m>d$. If $d=2m$ then $F_{m}(Q)$ has two geometrically connected components; otherwise it is connected. Suppose that $\pi:\mathcal{Q}\rightarrow C$ is a regular projective model of $Q$, such that the fibers are all quadric hypersurfaces of rank at least $d+1$. The locus $D\subset C$ corresponding to fibers of rank $d+1$ is called the degeneracy locus. Fundamental results of Witt – see [CT96, §2] and [Sch96] for a modern formulation in terms of local-global principles – assert that: • if $d>0$ then $Q(K)\neq\emptyset$ if $\mathcal{Q}_{c}=\pi^{-1}(c)$ has a smooth real point for each $c\in C(\mathbb{R})$; * • if $d>2$ then $F_{1}(Q)(K)\neq\emptyset$ if $F_{1}(\mathcal{Q}_{c})$ has a smooth real point for each $c\in C(\mathbb{R})$. We can translate these into conditions on the signatures of the smooth fibers * • if $d>0$ then $Q(K)\neq\emptyset$ if $\mathcal{Q}_{c}$ is not definite for any $c\in(C\setminus D)(\mathbb{R})$; * • if $d>2$ then $F_{1}(Q)(K)\neq\emptyset$ if $\mathcal{Q}_{c}$ does not have signatures $(d+2,0),(d+1,1),(1,d+1)$ or $(0,d+2)$ for any $c\in(C\setminus D)(\mathbb{R})$. In other words, we have points and lines over $K$ if the fibers permit them. This reflects a general principle: Suppose $\mathcal{X}$ is regular and has a flat proper morphism $\varpi:\mathcal{X}\rightarrow C$ to a curve $C$, all defined over $\mathbb{R}$. The local-global and reciprocity obstructions to sections of $\varpi$ are reflected in the absence of continuous sections $C(\mathbb{R})\rightarrow\mathcal{X}(\mathbb{R})$ for induced map of the underlying real manifolds [Duc98]. ### 5.2. Implications of Amer’s Theorem Let $X\subset\mathbb{P}^{d+2}$ be a smooth complete intersection of two quadrics over $\mathbb{R}$ and $\mathcal{Q}\rightarrow\mathbb{P}^{1}$ the associated pencil of quadrics. The results of Section 5.1 imply that $\mathcal{Q}\rightarrow\mathbb{P}^{1}$ has a section unless the Krasnov invariant is $(d+3)$; the variety of lines $F_{1}(\mathcal{Q}/\mathbb{P}^{1})\rightarrow\mathbb{P}^{1}$ has a section unless the Krasnov invariant is $(d+3),(d+2-e,e,1)\text{ with }1\leq e\leq\tfrac{d+2}{2},\quad(d+1).$ Thus the Krasnov invariant determines which dimensional linear subspaces and quadrics appear on $X$: ###### Proposition 5.1. Let $X\subset\mathbb{P}^{d+2}$ be a smooth complete intersection of two quadrics defined over $\mathbb{R}$. The only isotopy classes of $X$ that fail to contain a line are: * • $(d+3)$ where $X(\mathbb{R})=\emptyset$; * • $(d+2-e,e,1)$ with $1\leq e\leq\tfrac{d+2}{2}$; * • $(d+1)$. The case $(d+1,1,1)$ has disconnected real locus $X(\mathbb{R})$ [Kra18, p. 117]; thus $X$ cannot be rational over $\mathbb{R}$. The cases $(d+2-e,e,1)$ with $2\leq e\leq\tfrac{d+2}{2}$ are connected. ### 5.3. Quadric $n$-folds over $\mathbb{R}$ Assume now that $X$ has even dimension $d=2n$. We may read off from the invariant $(r_{1},\ldots,r_{2s+1})$ which classes of quadric $n$-folds $Q\subset X_{\mathbb{C}}$ are realized by quadrics defined over $\mathbb{R}$. Fix a smooth real quadric hypersurface ${\bf Q}=\\{F=0\\}\subset\mathbb{P}^{2n+1}.$ The following conditions are equivalent: * • the geometric components of the variety of maximal isotropic subspaces $\operatorname{OGr}(\bf{Q})$ are defined over $\mathbb{R}$; * • the discriminant of $F$ is positive; * • the $I^{+}(F)-I^{-}(F)$ is divisible by four. This means that complex conjugation fixes the class of a maximal isotropic subspace. We also have equivalence among: * • there is a maximal isotropic subspace $\mathbb{P}^{n}\subset\bf{Q}$ defined over $\mathbb{R}$; * • the signature of $F$ is $(n+1,n+1)$. Thus quadric $n$-folds $Q\subset X\subset\mathbb{P}^{2n+2}$ defined over $\mathbb{R}$ correspond to rulings of degenerate fibers $\mathcal{Q}_{t},t\in D(\mathbb{R})$ where $\mathcal{Q}_{t}$ has signature $(n+1,n+1)$. As in Example 4.1, the corresponding $(+1)$-blocks in the complex conjugation involution $\iota\in W(D_{2n+3})$ will be designated $\bf{1}$, in boldface. ### 5.4. Analysis of the remaining even-dimensional isotopy classes We continue to assume that $X$ has even dimension $d=2n$, focusing on the isotopy classes without lines. The cases $(2n+2-e,e,1)=(e,1,2n+2-e),\quad 2\leq e\leq n+1$ have degeneracy consisting of $2n+3$ real points. The signatures of nonsingular members are $\displaystyle(1,2n+2)\ldots(e+1,2n+2-e)(e,2n+3-e)(e+1,2n+2-e)\ldots$ $\displaystyle(2n+1,2)(2n+2,1)\ldots(2n+2-e,e+1)(2n+3-e,e)$ $\displaystyle(2n+2-e,e+1)\ldots(2,2n+1)$ For $(n+1,n+1,1)$ the resulting signed permutation matrix is the diagonal matrix (5.1) $\operatorname{diag}(\underbrace{(-1)^{n},\ldots,-1,{\mathbf{1}}}_{n+1\text{ terms }},{\mathbf{1}},\underbrace{{\mathbf{1}},-1,\ldots,(-1)^{n}}_{n+1\text{ terms }}),$ with the bolded $\bf{1}$’s corresponding to singular fibers with signature $(n+1,n+1)$. The number of $+1$’s $a=\begin{cases}n+2&\text{if $n$ odd }\\\ n+3&\text{if $n$ even.}\end{cases}$ For $e\neq n+1$ we have $\operatorname{diag}(\underbrace{(-1)^{n},\ldots,(-1)^{n+e-1}}_{e\text{ terms}},(-1)^{n+e-1},\underbrace{(-1)^{n+(2n+2-e)-1},\ldots,(-1)^{n}}_{2n+2-e\text{ terms}}).$ Note that $(-1)^{n+e-1}=(-1)^{n+(2n+2-e)-1}$ so the three middle terms are equal. There is exactly one $\bf{1}$ corresponding to the singular fiber with signature $(n+1,n+1)$. The number of $+1$’s is given (5.2) $a=\begin{cases}n&\text{if $n,e$ odd }\\\ n+2&\text{if $n$ odd and $e$ even}\\\ n+3&\text{if $n$ even and $e$ odd}\\\ n+1&\text{if $n,e$ even.}\end{cases}$ For case $(2n+1)$ the signatures of nonsingular members are $\displaystyle(2,2n+1)\ldots(2n,3)(2n+1,2)(2n+2,3)\ldots(2,2n+1)$ The signed permutation matrix has one factor $\left(\begin{matrix}0&1\\\ 1&0\end{matrix}\right)$ and diagonal entries $((-1)^{n-1},\ldots,-1,1,-1,\ldots,(-1)^{n-1}).$ The number of positive terms is $a=\begin{cases}n&\text{if $n$ odd}\\\ n+1&\text{if $n$ even.}\end{cases}$ ## 6\. Application of the constructions ### 6.1. Proof of Theorem 1.1 Rationality is evident for isotopy classes of varieties that contain a line defined over $\mathbb{R}$. Propostion 5.1 enumerates the remaining cases $(5),(1,3,3),(1,2,4).$ These are covered by the following propositions. ###### Proposition 6.1. Let $X\subset\mathbb{P}^{2n+2}$ be a smooth complete intersection of two quadrics over $\mathbb{R}$ with invariant $(2n+1)$. Then $X$ is rational. ###### Proof. The analysis in Section 5.4 indicates that complex conjugation exchanges two classes of quadric $n$-folds associated to complex conjugate points in $D(\mathbb{C})\setminus D(\mathbb{R})$. Denote these by $[Q]$ and $[\bar{Q}]$ – recall from (2.1) that $[Q]\cdot[\bar{Q}]=1.$ Choosing a suitably general complex representation $Q\subset X_{\mathbb{C}}$, the intersection $Q\cap\bar{Q}$ is proper. Then $Q\cap\bar{Q}$ consists of a single rational point of $X$ with multiplicity one. In particular, the hypotheses of Construction II are satisfied. ∎ ###### Proposition 6.2. Let $X\subset\mathbb{P}^{2n+2}$ be a smooth complete intersection of two quadrics over $\mathbb{R}$ with invariant $(2n+2-e,e,1)=(e,1,2n+2-e),\quad 2\leq e\leq n+1.$ Assume that either $e$ is even or $e=n+1$. Then $X$ is rational. ###### Proof. Assume first that $e=n+1$. It follows from (5.1) in Section 5.4 that $X$ admits three classes $\begin{array}[]{c\|cccc}&h^{n}&Q_{1}&Q_{2}&Q_{3}\\\ \hline\cr h^{n}&4&2&2&2\\\ Q_{1}&2&1+(-1)^{n}&1&1\\\ Q_{2}&2&1&1+(-1)^{n}&1\\\ Q_{3}&2&1&1&1+(-1)^{n}\end{array}$ with each $Q_{i}$ realized by a quadric $n$-fold defined over $\mathbb{R}$. Construction III gives rationality in this case. Now assume that $e$ is even. The formula (5.2) shows that the numbers of $+1$’s and $-1$’s appearing in the $\iota\in W(D_{2n+3})$ associated with complex conjugation are as close as possible. If $n$ is even then we have $n+1$ of the former and $n+2$ of the latter; when $n$ is odd we have $n+2$ of the former and $n+1$ of the letter. Given an $n$-plane $P\subset X_{\mathbb{C}}$, the formulas in Section 2.2 yield $w_{P}\cdot w_{\bar{P}}=(-1)^{n+1}/4$ whence $P$ and $\bar{P}$ are disjoint in $X_{\mathbb{C}}$. Thus we may apply Construction I to conclude rationality. ∎ ###### Remark 6.3. Remark 3.4 implies that $X$ does not admit curves (or surfaces!) of odd degree defined over $\mathbb{R}$. These would force the existence of lines defined over $\mathbb{R}$, which do not exist in this isotopy class. ### 6.2. Remaining six-dimensional case To settle the rationality of six-dimensional complete intersections of two quadrics $X\subset\mathbb{P}^{6}$, there is one remaining case in the Krasnov classification: $(1,3,5)$. The sequence of signatures of nonsingular elements of $\\{\mathcal{Q}_{t}\\}$ is: $\displaystyle(1,8)(2,7)(3,6)(4,5)(5,4)(6,3)(5,4)(6,3)(7,2)$ $\displaystyle(8,1)(7,2)(6,3)(5,4)(4,5)(3,6)(4,5)(3,6)(2,7).$ The signed permutation is the diagonal matrix $\operatorname{diag}(-1,1,-1,{\bf{1}},-1,-1,-1,1,-1)$ and the invariant cycles are: $\begin{array}[]{c\|cccc}&h^{3}&Q_{1}&Q_{2}&Q_{3}\\\ \hline\cr h^{3}&4&2&2&2\\\ Q_{1}&2&0&1&1\\\ Q_{2}&2&1&0&1\\\ Q_{3}&2&1&1&0\end{array}$ Here $Q_{1}$ corresponds to the singular fiber of signature $(4,4)$ and $Q_{2}$ and $Q_{3}$ correspond to the singular fibers of signatures $(2,6)$ and $(6,2)$. If $P\subset X$ is a three-plane then $w_{P}\cdot w_{\bar{P}}=-3/4$ and $P$ and $\bar{P}$ meet in a single point. ## 7\. Extensions and more general fields We work over a field $k$ of characteristic zero. In this section, we give further examples of rationality constructions for $2n$-dimensional intersections of two quadrics over $k$, relying on special subvarieties of dimension $n$. ### 7.1. Dimension four: intersection computations Given $X\subset\mathbb{P}^{6}$, a smooth complete intersection of two quadrics, we have $\displaystyle c_{t}({\mathcal{T}}_{X})$ $\displaystyle\equiv(1+7ht+21h^{2}t^{2})/(1+2ht)^{2}\pmod{t^{3}}$ $\displaystyle\equiv(1+7ht+21h^{2}t^{2})(1-2ht+4h^{2}t^{2})^{2}\pmod{t^{3}}$ $\displaystyle\equiv 1+3ht+5h^{2}t^{2}\pmod{t^{3}}$ If $T\subset X$ is a smooth projective geometrically connected surface then $\displaystyle c_{t}({\mathcal{N}}_{T/X})$ $\displaystyle=(1+3ht+5h^{2}t^{2})/(1-K_{T}t+\chi(T)t^{2})$ $\displaystyle=1+(3h+K_{T})t+(5h^{2}+3hK_{T}+K_{T}^{2}-\chi(T))t^{2},$ where $\chi(T)$ is the topological Euler characteristic. The expected dimension of the deformation space of $T$ in $X$ is $\displaystyle\chi({\mathcal{N}}_{T/X})$ $\displaystyle=2-\tfrac{1}{2}(3h+K_{T})K_{T}+\frac{1}{2}(3h+K_{T})^{2}$ $\displaystyle\hskip 142.26378pt-(5h^{2}+3hK_{T}+K_{T}^{2}-\chi(T))$ $\displaystyle=2\chi(\mathcal{O}_{T})+\tfrac{1}{2}(-h^{2}-3hK_{T})-K_{T}^{2}+\chi(T)).$ For example, * • if $T=\mathbb{P}^{2}$ is embedded as a plane then $(T\cdot T)_{X}=2$ and $T$ is rigid; * • if $T$ is a quadric then $(T\cdot T)_{X}=2$ and $T$ moves in a three-parameter family; * • if $T$ is a quartic scroll then $(T\cdot T)_{X}=6$ and moves in a five- parameter family; * • if $T$ is a sextic del Pezzo surface then $(T\cdot T)_{X}=12$ and $T$ moves in an eight-parameter family. ### 7.2. Dimension four: surfaces with one apparent double point Recall that Construction V gives the rationality of fourfolds admitting a surface with one apparent double point. A classical result asserted by Severi – see [CMR04, Th. 4.10] for a modern proof – characterizes smooth surfaces $T\subset\mathbb{P}^{5}$ with one apparent double point, i.e., surfaces that acquire one singularity on generic projection into $\mathbb{P}^{4}$: * • $\deg(T)=4$: $T$ is a quartic scroll $T\simeq\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}(2)^{2}),\quad\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}(1)\oplus\mathcal{O}_{\mathbb{P}^{1}}(3));$ * • $\deg(T)=5$: $T$ is a quintic del Pezzo surface. Thus Construction V says that a smooth complete intersection of two quadrics $X\subset\mathbb{P}^{6}$ is rational if it contains a quartic scroll, a quintic del Pezzo surface, or a sextic del Pezzo surface with a rational point. Rationality always holds when there are positive-dimensional subvarieties of odd degree (see Section 3.1), so we focus attention to the first case. We seek criteria for the existence of a quartic scroll $T\subset X$, defined over $k$. Clearly, the class $[T]$ must be Galois-invariant; however, Galois- invariant classes need not be represented by cycles over $k$. The intersection computations above imply that $[T]=[Q_{2}]+[Q_{3}],$ where $Q_{2}$ and $Q_{3}$ represent quadric surfaces in $X$, defined over the algebraic closure. Assume that the class $[Q_{2}]+[Q_{3}]$ is Galois invariant and represents algebraic cycles defined over the ground field. We look for quartic scrolls $T\subset X$ with class $[T]=[Q_{2}]+[Q_{3}]$. ###### Remark 7.1. Over $k=\mathbb{R}$, case $(1,2,4)$ has signed permutation $\operatorname{diag}(1,-1,-1,-1,{\bf{1}},-1,1).$ The intersection form on the invariant part of $H^{4}(X_{\mathbb{C}},\mathbb{Z})$ takes the form $\begin{array}[]{c\|cccc}&h^{2}&Q_{1}&Q_{2}&Q_{3}\\\ \hline\cr h^{2}&4&2&2&2\\\ Q_{1}&2&2&1&1\\\ Q_{2}&2&1&2&1\\\ Q_{3}&2&1&1&2\end{array}$ Here we assume $Q_{1}$ (and $h^{2}-Q_{1}$) are associated with the element of $\\{\mathcal{Q}_{t}\\}$ with signature $(3,3)$ and $Q_{2}$ and $Q_{3}$ are contributed by the elements with signatures $(1,5)$ and $(5,1)$. While $Q_{1}$ is definable over $\mathbb{R}$, $Q_{2}$ and $Q_{3}$ are not definable over $\mathbb{R}$ (see Section 5.3). The requisite real cycles exist in $[T]=[Q_{2}]+[Q_{3}]$. This follows from the exact sequence in [CT15, §4], using the rationality of $X$ over $\mathbb{R}$ to ensure the vanishing of the unramified cohomology. It would be interesting to deduce this directly using cohomological machinery [Kah96]. However, we do not know whether such $X$ contain quartic scrolls, in general. Let $M$ denote the moduli space of quartic scrolls in a fixed cohomology class on $X$. There is a morphism $\begin{array}[]{rcl}M&\rightarrow&(\mathbb{P}^{6})^{\vee}\\\ T&\mapsto&\operatorname{span}(T)\end{array}$ assigning to each scroll the hyperplane it spans. Hyperplane sections $Y=H\cap X$ containing such scrolls are singular by the Lefschetz hyperplane theorem. Computations in Macaulay2 indicate that a generic such $Y$ has four ordinary singularities. If a complete intersection of two quadrics $Y\subset\mathbb{P}^{5}$ contains a quartic scroll, it contains two families of such scrolls, each parametrized by $\mathbb{P}^{3}$: These arise from residual intersections in quadrics in $I_{T}(2)/I_{Y}(2).$ Thus the residual family has class $2h^{2}-[T]=(h^{2}-[Q_{2}])+(h^{2}-[Q_{3}]).$ The hyperplane sections of $X$ with four singularities should be parametrized by a reducible surface with distinguished component $\Sigma\subset(\mathbb{P}^{6})^{\vee}$. We speculate that $\Sigma$ is a quartic del Pezzo surface, constructed as follows: Consider the pencil of quadrics $\mathcal{Q}_{t}$ defining $X$ and fix the pair of rank-six quadrics $\mathcal{Q}_{t_{2}},\mathcal{Q}_{t_{3}}$ whose maximal isotropic subspaces sweep out $Q_{2}$ and $Q_{3}$. Let $v_{i}\in\mathcal{Q}_{t_{i}}$ denote the vertices and $\ell$ the line they span, which is defined over $k$ even when $t_{2}$ and $t_{3}$ are conjugate over $k$. Projecting from $\ell$ gives a degree-four cover $X\rightarrow\mathbb{P}^{4}.$ Geometrically, the covering group is the Klein four-group and the branch locus consists of two quadric hypersurfaces $Y_{2},Y_{3}$ intersecting in a degree- four del Pezzo surface $S_{23}$. Is $\Sigma\simeq S_{23}$ over $k$? ### 7.3. Dimension six: Threefolds with one apparent double point Construction V indicates that the existence of a threefold $W\subset X$ with one apparent double point yields rationality. The following classification [CMR04] builds on constructions of Edge [Edg32]: * • $\deg(W)=5$: a scroll in planes associated with two lines and twisted cubic, or one line and two conics; * • $\deg(W)=6$: an Edge variety constructed as a residual intersection $Q\cap(\mathbb{P}^{1}\times\mathbb{P}^{3})=\Pi_{1}\cup\Pi_{2}\cup W,$ where $Q$ is a quadric hypersurface and the $\Pi_{i}\simeq\mathbb{P}^{3}$ are fibers of the Segre variety under the first projection; * • $\deg(W)=7$: an Edge variety constructed as a residual intersection $Q\cap(\mathbb{P}^{1}\times\mathbb{P}^{3})=\Pi\cup W$ with the notation as above; * • $\deg(W)=8$: a scroll in lines over $\mathbb{P}^{2}$ of the form $\mathbb{P}(\mathcal{E})$ (one-dimensional quotients of $\mathcal{E}$) where $\mathcal{E}$ is a rank-two vector bundle given as an extension $0\rightarrow\mathcal{O}_{\mathbb{P}^{2}}\rightarrow\mathcal{E}\rightarrow\mathcal{I}_{p_{1},\ldots,p_{8}}(4)\rightarrow 0$ for eight points $p_{1},\ldots,p_{8}\in\mathbb{P}^{2}$, no four collinear or seven on a conic. Given that rationality follows when there are positive-dimensional subvarieties of odd degree (see Section 3.1) we focus on the varieties of even degree. ### 7.4. Dimension six: Degree six Edge variety Let $W\subset\mathbb{P}^{7}$ denote an Edge variety arising as follows: Consider the Segre fourfold $\mathbb{P}^{1}\times\mathbb{P}^{3}\subset\mathbb{P}^{7}$ and take the residual intersection to two copies of $\mathbb{P}^{3}$ $\\{0,\infty\\}\times\mathbb{P}^{3}\subset\mathbb{P}^{1}\times\mathbb{P}^{3}$ in a quadric hypersurface. The resulting threefold $W\simeq\mathbb{P}^{1}\times\Sigma,$ where $\Sigma\subset\mathbb{P}^{3}$ is a quadric hypersurface. Note that the ideal of $W\subset\mathbb{P}^{7}$ is generated by nine quadratic forms. Complete intersections of two quadrics $W\subset Y\subset\mathbb{P}^{7}$ depend on five parameters. A Magma computation shows that a generic such $Y$ has eight ordinary singularities. Suppose we have an embedding $W\hookrightarrow X$, where $X\subset\mathbb{P}^{8}$ is a smooth complete intersection of two quadrics. For fixed $X$, the Hilbert scheme of such threefolds has dimension eight. Realizing $W\simeq\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$ we have $\displaystyle c_{1}({\mathcal{N}}_{W/X})$ $\displaystyle=$ $\displaystyle 3(h_{1}+h_{2}+h_{3}),$ $\displaystyle c_{2}({\mathcal{N}}_{W/X})$ $\displaystyle=$ $\displaystyle 8(h_{1}h_{2}+h_{1}h_{3}+h_{2}h_{3}),$ $\displaystyle c_{3}({\mathcal{N}}_{W/X})$ $\displaystyle=$ $\displaystyle 4h_{1}h_{2}h_{3}.$ The Riemann-Roch formula gives $\chi(N_{W/X})=8$. Since $(W\cdot W)_{X}=4$, the primitive class $([W]-\tfrac{3}{2}h^{3})^{2}=4-18+9=-5.$ ###### Remark 7.2. Suppose that $X$ is defined over $\mathbb{R}$, and corresponds to the $(1,2,6)$ case, with signed permutation matrix (see Section 5.4) $\operatorname{diag}(-1,1,1,1,-1,{\bf{1}},-1,1,-1).$ The invariant cycles are: $\begin{array}[]{c\|cccccc}&h^{3}&Q_{1}&Q_{2}&Q_{3}&Q_{4}&Q_{5}\\\ \hline\cr h^{3}&4&2&2&2&2&2\\\ Q_{1}&2&0&1&1&1&1\\\ Q_{2}&2&1&0&1&1&1\\\ Q_{3}&2&1&1&0&1&1\\\ Q_{4}&2&1&1&1&0&1\\\ Q_{5}&2&1&1&1&1&0\end{array}$ Here $Q_{1}$ corresponds to the singular fiber of signature $(4,4)$ and the classes $Q_{2},\ldots,Q_{5}$ correspond to the singular fibers of signatures $(2,6)$ and $(6,2)$. The class $[Q_{2}]+[Q_{3}]+[Q_{4}]+[Q_{5}]-[Q_{1}]$ has degree six and self-intersection four. Is it represented by cycles defined over $\mathbb{R}$? Does it admit an Edge variety of degree six over $\mathbb{R}$? Over more general $k$ where the requisite cycles exist? ### 7.5. Dimension six: Degree eight variety Let $\mathcal{E}$ be a stable rank-two vector bundle on $\mathbb{P}^{2}$ with invariants $c_{1}(\mathcal{E})=4L$ and $c_{2}(\mathcal{E})=8L^{2}$. Note that $\Gamma(\mathcal{E})$ has dimension eight, giving an inclusion $V:=\mathbb{P}(\mathcal{E}^{\vee})\subset\mathbb{P}^{7}.$ We have a tautological exact sequence $0\rightarrow\mathcal{O}_{V}(-\xi)\rightarrow\mathcal{E}^{\vee}_{V}\rightarrow Q\rightarrow 0,$ whence $0\rightarrow Q(\xi)\rightarrow T_{V}\rightarrow T_{\mathbb{P}^{2}}\rightarrow 0.$ Thus we have the following $\displaystyle\xi^{2}-4L\xi+8L^{2}$ $\displaystyle=0$ $\displaystyle c(Q(\xi))$ $\displaystyle=1+(2\xi-4L)+(\xi^{2}-4L\xi+8L^{2}),$ $\displaystyle c({\mathcal{T}}_{V})$ $\displaystyle=(1+3L+3L^{2})(c(Q(\xi))$ $\displaystyle=1+(2\xi-L)+(\xi^{2}-4L\xi+8L^{2}+6\xi L-12L^{2}+3L^{2})$ and we find $\displaystyle c_{1}({\mathcal{N}}_{V/X})$ $\displaystyle=$ $\displaystyle 3\xi+L,$ $\displaystyle c_{2}({\mathcal{N}}_{V/X})$ $\displaystyle=$ $\displaystyle 5\xi^{2}-\xi L+2L^{2},$ $\displaystyle c_{3}({\mathcal{N}}_{V/X})$ $\displaystyle=$ $\displaystyle 3\xi^{3}-3\xi^{2}L+2L^{2}\xi-9L^{3}.$ Note that $\deg(L^{3})=0,\quad\deg(L^{2}\xi)=1,\quad\deg(L\xi^{2})=4,\text{ and }\deg(\xi^{3})=8,$ so we conclude that $(V\cdot V)_{X}=14.$ The primitive class $[V]-2h^{3}$ has self-intersection $14-4\cdot 8+16=-2$ which means that $[V]=h^{3}+[Q_{2}]+[Q_{3}],\quad(Q_{2}\cdot Q_{3})=1,$ where $Q_{2}$ and $Q_{3}$ are classes of quadric threefolds in $X$, defined over the algebraic closure. (Up to the action of the Weyl group $W(D_{9})$ this is the only possibility.) Returning to the only remaining case in dimension six where rationality over $\mathbb{R}$ remains open (see Section 6.2): ###### Question 7.3. Let $X\subset\mathbb{P}^{8}$ be a smooth complete intersection of two quadrics over $\mathbb{R}$ in isotopy class $(1,3,5)$. Which classes of codimension- three cycles $X$ are realized over $\mathbb{R}$? Are there varieties with one apparent double point, defined over $\mathbb{R}$, representing these classes? ## References * [BD85] Arnaud Beauville and Ron Donagi. La variété des droites d’une hypersurface cubique de dimension $4$. C. R. Acad. Sci. Paris Sér. I Math., 301(14):703–706, 1985\. * [BW19] Olivier Benoist and Olivier Wittenberg. Intermediate Jacobians and rationality over arbitrary fields, 2019. arXiv:1909.12668. * [BW20] Olivier Benoist and Olivier Wittenberg. The Clemens-Griffiths method over non-closed fields. Algebr. 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# Inferring solar differential rotation through normal-mode coupling using Bayesian statistics ∗Samarth G. Kashyap Department of Astronomy and Astrophysics Tata Institute of Fundamental Research Mumbai, India ∗Srijan Bharati Das Department of Geosciences Princeton University Princeton, New Jersey, USA Shravan M. Hanasoge Department of Astronomy and Astrophysics Tata Institute of Fundamental Research Mumbai, India Martin F. Woodard NorthWest Research Associates Boulder Office, 3380 Mitchell Lane Boulder, Colorado, USA Jeroen Tromp Department of Geosciences and Program in Applied & Computational Mathematics Princeton University Princeton, New Jersey, USA (Received December 23, 2021; Revised January 18, 2021; Accepted January 21, 2021) ###### Abstract Normal-mode helioseismic data analysis uses observed solar oscillation spectra to infer perturbations in the solar interior due to global and local-scale flows and structural asphericity. Differential rotation, the dominant global- scale axisymmetric perturbation, has been tightly constrained primarily using measurements of frequency splittings via “$a$-coefficients”. However, the frequency-splitting formalism invokes the approximation that multiplets are isolated. This assumption is inaccurate for modes at high angular degrees. Analysing eigenfunction corrections, which respect cross coupling of modes across multiplets, is a more accurate approach. However, applying standard inversion techniques using these cross-spectral measurements yields $a$-coefficients with a significantly wider spread than the well-constrained results from frequency splittings. In this study, we apply Bayesian statistics to infer $a$-coefficients due to differential rotation from cross spectra for both $f$-modes and $p$-modes. We demonstrate that this technique works reasonably well for modes with angular degrees $\ell=50-291$. The inferred $a_{3}-$coefficients are found to be within $1$ nHz of the frequency splitting values for $\ell>200$. We also show that the technique fails at $\ell<50$ owing to the insensitivity of the measurement to the perturbation. These results serve to further establish mode coupling as an important helioseismic technique with which to infer internal structure and dynamics, both axisymmetric (e.g., meridional circulation) and non-axisymmetric perturbations. Sun: helioseismology — Sun: oscillations — Sun: interior — differential rotation — MCMC ††journal: ApJS††thanks: Both authors have contributed equally to this study. ## 1 Introduction The strength and variation of observed solar activity is governed by the spatio-temporal dependence of flow fields in the convective envelope (Charbonneau, 2005; Fan, 2009). Thus, understanding the physics that governs the evolution and sustenance of the activity cycle of the Sun necessitates imaging its internal layers. While differential rotation has the most significant imprint on Dopplergram images (Schou et al., 1998), signatures due to weaker effects, such as meridional circulation (Giles et al., 1997; Basu et al., 1999; Zhao & Kosovichev, 2004; Gizon et al., 2020) and magnetic fields (Gough, 1990; Dziembowski & Goode, 2004; Antia et al., 2013), are also noticeable. The ability to image these weaker effects therefore critically depends on an accurate measurement of the dominant flows. This makes inferring the strength of the dominant flows along with assigning appropriate statistical uncertainties an important area of study. Differences between normal modes of the Sun and those predicted using standard solar models may be used to constrain solar internal properties. The standard models are typically adiabatic, hydrodynamic, spherically symmetric and non- rotating, also referred to as SNRNMAIS (Lavely & Ritzwoller, 1992; Christensen-Dalsgaard et al., 1996). The usual labelling convention, using 3 quantum numbers, $(n,\ell,m)$, where $n$ denotes radial order, $\ell$ the angular degree, and $m$ the azimuthal order, are used to uniquely identify normal modes. Departures of solar structure from the SNRNMAIS are modelled as small perturbations (Christensen–Dalsgaard, 2003), which ultimately manifest themselves as observable shifts (or splittings) in the eigenfrequencies and distortions in the eigenfunctions (Woodard, 1989). The distorted eigenfunctions may be expressed as a linear combination of reference eigenfunctions and are said to be coupled with respect to the reference. Observed cross-spectra of spherical-harmonic time series corresponding to full-disk Dopplergrams are used to measure eigenfunction distortion. In the present study, we use observational data from the _Helioseismic Magnetic Imager_ (HMI) onboard the _Solar Dynamics Observatory_ (Schou et al., 2012). Different latitudes of the Sun rotate at different angular velocities, with the equator rotating faster than the poles (Howard et al., 1984; Ulrich et al., 1988). To an observer in a frame co-rotating at a specific rotation rate $\bar{\Omega}$ of the Sun, this latitudinal rotational shear is the most significant perturbation to the reference model. This large-scale toroidal flow $\Omega\,(r,\theta)$ is well approximated as being time-independent (shown to vary less than 5% over the last century in Gilman, 1974; Howard et al., 1984; Basu & Antia, 2003) and zonal, with variations only along the radius $r$ and co-latitude $\theta$. Very low $\ell\leq 5$ modes penetrate the deepest layers of the Sun and were used in earlier attempts to constrain the rotation rate in the core and radiative interior (Claverie et al., 1981; Chaplin et al., 1999; Eff-Darwich et al., 2002; Couvidat et al., 2003; Chaplin et al., 2004). However, observed solar activity is believed to be governed by the coupling of differential rotation and magnetic fields in the bulk of the convection zone (Miesch, 2005). Subsequently, studies using intermediate $\ell\leq 100$ (Duvall & Harvey, 1984; Brown & Morrow, 1987; Brown et al., 1989; Libbrecht, 1989; Duvall et al., 1996) and modes with relatively high $\ell\leq 250$ (Thompson et al., 1996; Kosovichev et al., 1997; Schou et al., 1998) yielded overall convergent results for the rotation profile. Among other features of the convection zone (Howe, 2009), these studies established the presence of shear layers at the base of the convection zone (the tachocline) and below the solar surface. Most of these studies used measurements of frequency splittings in a condensed convention known as $a$-coefficients (Ritzwoller & Lavely, 1991). The azimuthal and temporal independence make differential rotation particularly amenable to inversion via $a$-coefficients. The assumption behind this formalism is that multiplets, identified by $(n,\ell)$, are well separated in frequency from each other, known as the ‘isolated multiplet approximation’. This assumption holds true when differential rotation is the sole perturbation under consideration (Lavely & Ritzwoller, 1992), even at considerably high $\ell$. We therefore state at the outset that estimates of $a$-coefficients determined from frequency splitting serve as reliable measures of differential rotation (Chatterjee & Antia, 2009). Nevertheless, the estimation of non- axisymmetric perturbations requires a rigorous treatment honoring the cross coupling of multiplets (Hanasoge et al., 2017; Das et al., 2020). In such cases, measuring changes to the eigenfunctions is far more effective than, for instance, the $a$-coefficient formalism. As a first step, it is therefore important to explore the potential of eigenfunction corrections to infer differential rotation. The theoretical modeling of eigenfunction corrections for given axisymmetric – zonal and meridional – flow fields may be traced back to Woodard (1989), followed up by further investigations Woodard (2000); Gough & Hindman (2010); Vorontsov (2007); Schad et al. (2011). Schad et al. (2013) and Schad & Roth (2020) used observables in the form of mode-amplitude ratios to infer meridional circulation and differential rotation, respectively. In this study, we adopt the closed-form analytical expression for correction coefficients first proposed by Vorontsov (2007) and subsequently verified to be accurate up to angular degrees as high as $\ell\leq 1000$ Vorontsov (2011), henceforth V11. The method of using cross-spectral signals to fit eigenfunction corrections was first applied by Woodard et al. (2013), henceforth W13, to infer differential rotation and meridional circulation. A simple least-squares fitting, assuming a unit covariance matrix, was used for inversions in W13. Their results of odd $a$-coefficients (which encodes differential rotation), even though qualitatively similar, show a considerably larger spread than the results from frequency splittings. Moreover, the authors of W13 note that the inferred meridional flow was “less satisfactory” [than their zonal flow estimates]. Cross spectra are dominated by differential rotation, a much larger perturbation than meridional circulation. Although zonal and meridional flows are measured in different cross-spectral channels, the inference of meridional flow is affected by differential rotation through leakage. Thus, the accurate determination of odd $a$-coefficients is critical to the inference of meridional flow. The relatively large spread in inferences of differential rotation obtained by W13 may be due to (a) a poorly conditioned minimizing function with multiple local minima surrounding the expected (frequency splitting) minima, (b) a relative insensitivity of various modes to differential rotation, resulting in a flat minimizing function close to the expected minima, (c) an inaccurate estimation of the minimizing function on account of assuming a unit-data covariance matrix, and/or (d) eigenfunction corrections only yielding accurate results in the limit of large $\ell$ ($>250$), where the isolated-multiplet approximation starts worsening. In this study, we investigate the above issues and explore the potential of using eigenfunction corrections as a means to infer differential rotation using tools from Bayesian statistics. We apply the Markov Chain Monte Carlo (MCMC) algorithm (Metropolis & Ulam, 1949; Metropolis et al., 1953) using a minimizing function calculated in the L2 norm, adequately weighted by data variance. We do not bias the MCMC sampler in light of any previous measurement, effectively using an uninformed prior. The results inferred, therefore, are an independent measurement constrained only by observed cross spectra. Since Bayesian inference is a probabilistic approach to parameter estimation, we obtain joint probability-density functions in the $a$-coefficient space. This allows us to rigorously compute uncertainties associated with the measurements. We compare and qualify the results obtained with independent measurements from frequency splitting and those obtained using similar cross-spectral analysis in W13. Further, we report the inadequacy of this method for low angular-degree modes on account of poor sensitivity of spectra to rotation via $a$-coefficients. The structure of this paper is as follows. We establish mathematical notations and describe the basic physics of normal-mode helioseismology in Section 2.1. The governing equations which we use for modeling cross spectra using eigenfunction-correction coefficients are outlined in Section 2.2. Section 3 elaborates the steps for computing the observed cross spectra and building the misfit function and estimating data variance for performing the MCMC. Results are discussed in Section 4. Using the $a$-coefficients inferred from MCMC, cross-spectra are reconstructed in Section 4.1. A discussion on sensitivity of the current model to the model parameters is presented in Section 4.2. The conclusions from this work are reported in Section 5. ## 2 Theoretical Formulation ### 2.1 Basic Framework and Notation For inferring flow profiles in the solar interior, we begin by considering the system of coupled hydrodynamic equations, namely, $\displaystyle\partial_{t}\rho$ $\displaystyle=$ $\displaystyle-\bm{\nabla}\cdot(\rho\,\mathbf{v}),$ (1) $\displaystyle\rho(\partial_{t}\mathbf{v}+\mathbf{v}\cdot\bm{\nabla}\mathbf{v})$ $\displaystyle=$ $\displaystyle-\bm{\nabla}P-\rho\bm{\nabla}\phi,$ (2) $\displaystyle\partial_{t}P$ $\displaystyle=$ $\displaystyle-\mathbf{v}\cdot\bm{\nabla}P-\gamma\,P\,\bm{\nabla}\cdot\mathbf{v},$ (3) where $\rho$ is the mass density, $\mathbf{v}$ the material velocity, $P$ the pressure, $\phi$ the gravitational potential and $\gamma$ the ratio of specific heats determined by an adiabatic equation of state. The eigenstates of the Sun are modeled as linear combinations of the eigenstates of a standard solar model. Here we use model S as this reference state, which is discussed in Christensen-Dalsgaard et al. (1996). In absence of background flows, $\tilde{{\mathbf{v}}}={\bf 0}$, the zeroth-order hydrodynamic equations trivially reduce to the hydrostatic equilibrium $\bm{\nabla}\tilde{P}+\tilde{\rho}\bm{\nabla}\tilde{\phi}=0$. Hereafter, all zeroth-order static fields, unperturbed mode eigenfrequencies, eigenfunctions and amplitudes corresponding to the reference model will be indicated using tilde (to maintain consistency with notation used in W13). In response to small perturbations to the static reference model, the system exhibits oscillations $\mbox{\boldmath$\bf\xi$}(\mathbf{r},t)$. These oscillations may be decomposed into resonant “normal modes” of the system, labeled by index $k$, with characteristic frequency $\tilde{\omega}_{k}$ and spatial pattern $\tilde{}\mbox{\boldmath$\bf\xi$}_{k}$, as follows: $\bm{\xi}(\bm{r},t)=\sum_{k}\tilde{\Lambda}_{k}(t)\,\tilde{\bm{\xi}}_{k}(\bm{r})\exp(i\tilde{\omega}_{k}t),$ (4) where $\tilde{\Lambda}_{k}$ are the respective mode amplitudes and $\bm{r}=(r,\theta,\phi)$ denote spherical-polar coordinates. Linearizing eqns. (1)–(3) about the hydrostatic background model gives (for a detailed derivation refer to Christensen–Dalsgaard, 2003) $\mathcal{L}_{0}\tilde{\bm{\xi}}_{k}=-\bm{\nabla}(\tilde{\rho}c_{s}^{2}\,\bm{\nabla}\cdot\tilde{\bm{\xi}}_{k}-\tilde{\rho}g\,\tilde{\bm{\xi}}_{k}\cdot\hat{\bm{e}}_{r})-g\,\hat{\bm{e}}_{r}\bm{\nabla}\cdot(\tilde{\rho}\,\tilde{\bm{\xi}}_{k})=\tilde{\rho}\,\tilde{\omega}_{k}^{2}\,\tilde{\bm{\xi}_{k}}.$ (5) Here $\tilde{\rho}(r),c_{s}(r)$, and $g(r)$ denote density, sound speed, and gravity (directed radially inward) respectively of the reference solar model, and $\mathcal{L}\,_{0}$ is the self-adjoint unperturbed wave operator. This ensures that the eigenfrequencies $\tilde{\omega}_{k}$ are real and eigenfunctions $\tilde{\mbox{\boldmath$\bf\xi$}}_{k}$ are orthogonal. Introducing flows and other structure perturbations through the operator $\delta\mathcal{L}\,$ (e.g., magnetic fields or ellipticity) modifies the unperturbed wave equation (5) to $\tilde{\rho}\,\omega_{k}^{2}\,\bm{\xi}_{k}=\left(\mathcal{L}\,_{0}+\delta\mathcal{L}\,\right)\bm{\xi}_{k},$ (6) where $\omega_{k}=\tilde{\omega}_{k}+\delta\omega_{k}$ and $\mbox{\boldmath$\bf\xi$}_{k}=\sum_{k^{\prime}}c_{k^{\prime}}\tilde{\mbox{\boldmath$\bf\xi$}}_{k^{\prime}}$ are the eigenfrequency and eigenfunction associated with the perturbed wave operator $\mathcal{L}\,_{0}+\delta\mathcal{L}\,$. The Sun, a predominantly hydrodynamic system, is thus treated as a fluid body with vanishing shear modulus (Dahlen & Tromp, 1998). This is unfavourable for sustaining shear waves and therefore the eigenfunctions of the reference model are very well approximated as spheroidal (Chandrasekhar & Kendall, 1957), $\displaystyle\tilde{\bm{\xi}}_{k}(r,\theta,\phi)={}_{n}U{}_{\ell}(r)\,Y_{\ell m}(\theta,\phi)\,\hat{\bm{e}}_{r}+{}_{n}V{}_{\ell}(r)\,\bm{\nabla}_{1}Y_{\ell m}(\theta,\phi).$ (7) $\bm{\bm{\nabla}}_{1}=\hat{\bm{e}}_{\theta}\,\partial_{\theta}+\hat{\bm{e}}_{\phi}\,(\sin\theta)^{-1}\partial_{\phi}$ is the dimensionless lateral covariant derivative operator. Suitably normalized eigenfunctions $\tilde{\mbox{\boldmath$\bf\xi$}}_{k}$ and $\tilde{\mbox{\boldmath$\bf\xi$}}_{k^{\prime}}$, where $k^{\prime}=(n^{\prime},\ell^{\prime},m^{\prime})$, satisfy the orthonormality condition $\int_{\odot}\mathrm{d}^{3}\mathbf{r}\,\rho\,\bm{\tilde{\xi}}_{k^{\prime}}^{}\cdot\bm{\tilde{\xi}}_{k}=\delta_{n^{\prime}n}\,\delta_{\ell^{\prime}\ell}\,\delta_{m^{\prime}m}.$ (8) Since we observe only half the solar surface, orthogonality cannot be used to extract each mode separately. Windowing in the spatial domain results in spectral broadening, where contributions from neighbouring modes seep into the observed mode signal $\varphi^{\ell m}(\omega)$, as described by the leakage matrix (Schou & Brown, 1994), $\varphi^{\ell m}(\omega)=\sum_{k^{\prime}}L^{\ell m}_{k^{\prime}}\,\Lambda^{k^{\prime}}(\omega)=\sum_{k^{\prime}}\tilde{L}^{\ell m}_{k^{\prime}}\,\ \tilde{\Lambda}^{k^{\prime}}(\omega).$ (9) Here, leakage matrices $L^{\ell m}_{k^{\prime}},\tilde{L}^{\ell m}_{k^{\prime}}$ and amplitudes $\Lambda^{k^{\prime}}(\omega),\tilde{\Lambda}^{k^{\prime}}(\omega)$ of the observed surface velocity field ${\mathbf{v}}(\omega)$ correspond to the bases of perturbed ($\mbox{\boldmath$\bf\xi$}_{k^{\prime}}$) and unperturbed eigenfunctions ($\tilde{\mbox{\boldmath$\bf\xi$}}_{k^{\prime}}$), respectively, ${\mathbf{v}}=\sum_{k^{\prime}}\Lambda^{k^{\prime}}\mbox{\boldmath$\bf\xi$}_{k^{\prime}}=\sum_{k^{\prime}}\tilde{\Lambda}^{k^{\prime}}\tilde{\mbox{\boldmath$\bf\xi$}}_{k^{\prime}}.$ (10) Since leakage falls rapidly with increasing spectral distance $(\|\ell-\ell^{\prime}\|,\|m-m^{\prime}\|)$, Eqn. (9) demonstrates the entangling of modes in spectral proximity to $(\ell,m)$. The presence of a zeroth-order flow field $\tilde{{\mathbf{v}}}$ in Eqns. (1)–(3) gives rise to perturbed eigenfunctions $\mbox{\boldmath$\bf\xi$}_{k}$ and therefore introduces correction factors $c_{k}^{k^{\prime}}$ with respect to the unperturbed eigenfunctions $\tilde{\mbox{\boldmath$\bf\xi$}}_{k^{\prime}}$. $\mbox{\boldmath$\bf\xi$}_{k}=\sum_{k^{\prime}}c_{k}^{k^{\prime}}\tilde{\mbox{\boldmath$\bf\xi$}}_{k^{\prime}}.$ (11) Figure 1: Differential rotation induces 3D distortions in radial eigenfunction of unperturbed mode $(n,l)=(2,150)$ for $m=10,75,140$ at radii $r/R_{\odot}=0.95,1.0$. Each column in the upper panel correspond to 2D surfaces for the undistorted eigenfunctions $\tilde{\bm{\xi}}_{nlm}$ in the left slice and differences between distorted and undistorted eigenfunctions $\hat{r}\cdot(\bm{\xi}_{nlm}-\tilde{\bm{\xi}}_{nlm})$ in the right slice. The middle panel shows the difference in the radial variation of eigenfunctions for a chosen $(\theta_{0},\phi_{0})=(67.8^{\circ},177.6^{\circ})$. The lower panels indicate the magnitudes of the coupling coefficients that induce eigenfunction distortion, as in Eqn. (11). The self-coupling coefficients $c^{\ell m}_{\ell m}$ (i.e., $p=0$), being the most dominant, are not shown, in order to highlight the contributions of cross-coupling coefficients ($p\neq 0$.) Using this, the statistical expectation of the cross-spectral measurement is expressed as in Eqns. (14)–(17) of W13, $\langle\varphi^{\ell^{\prime}m^{\prime}}\varphi^{\ell m}\rangle=\sum_{i,j,k}\tilde{L}^{\ell^{\prime}m^{\prime}}_{j}\,\tilde{L}^{\ell m}_{k}\,c^{j}_{i}\,c^{k}_{i}\,\langle\|\Lambda^{i}(\omega)\|^{2}\rangle,$ (12) where $\langle\|\Lambda^{i}(\omega)\|^{2}\rangle$ denotes Lorentzians centered at resonant frequencies $\omega=\omega_{i}$ corresponding to the perturbed modes $\mbox{\boldmath$\bf\xi$}_{i}$. ### 2.2 Eigenfunction corrections due to axisymmetric flows This study uses the fact that eigenfunction-correction factors $c_{k}^{k^{\prime}}$ in Eqn. (11) carry information about the flow field $\tilde{{\mathbf{v}}}$. Although this problem was first addressed by Woodard (1989), a rigorous treatment using perturbative analysis of mode coupling was only presented in V11. In this section, we outline the governing equations for the eigenfunction-correction factors $c_{k}^{k^{\prime}}$ due to differential rotation and meridional circulation as shown in V11. Upon introducing flows, the model-S eigenfunctions are corrected as follows: $\mbox{\boldmath$\bf\xi$}_{\ell}=\sum_{\ell^{\prime}}c_{\ell}^{\ell^{\prime}}\,\tilde{\mbox{\boldmath$\bf\xi$}}_{\ell^{\prime}}+\delta\mbox{\boldmath$\bf\xi$}_{\ell}=\sum_{p=0,\pm 1,\pm 2,...}c_{\ell}^{\ell+p}\,\tilde{\mbox{\boldmath$\bf\xi$}}_{\ell+p}+\delta\mbox{\boldmath$\bf\xi$}_{\ell},$ (13) where $p=\ell^{\prime}-\ell$ is used to label the offset (in angular degrees) of the neighbouring mode contributing to the distortion of the erstwhile unperturbed eigenfunction $\mbox{\boldmath$\bf\xi$}_{\ell}$ — visual illustration may be found in Figure 1. Correction factors $c_{\ell}^{\ell+p}$ solely from modes with the same radial orders and azimuthal degrees are considered in Eqn. (13) and therefore labels $n$ and $m$ are suppressed. $c_{\ell,m}^{\ell+p,m^{\prime}}=0$ for $m\neq m^{\prime}$ since differential rotation and meridional circulation are axisymmetric (see selection rules imposed due to Wigner 3-$j$ symbols in Appendix A of V11). Corrections from modes belonging to a different radial order $n$ are accumulated in $\delta\mbox{\boldmath$\bf\xi$}$. Following V11 and W13, subsequent treatment ignores terms in $\delta\mbox{\boldmath$\bf\xi$}$ since it is considered to be of the order of the perturbation $\delta\mathcal{L}\,$ or smaller (rendering them at least second order in perturbed quantities). This is because the correction factor $c_{n\ell}^{n^{\prime}\ell^{\prime}}$ is non-trivial only if modes ${}_{n}\mathrm{S}_{\ell}\,$ and ${}_{n^{\prime}}\mathrm{S}_{\ell^{\prime}}\,$ are proximal in frequency space as well as the angular degree $s$ of the perturbing flow satisfies the relation $\|\ell^{\prime}-\ell\|\leq s$. For modes belonging to different dispersion branches $(n\neq n^{\prime})$, with either $\ell$ or $\ell^{\prime}$ being moderately large ($>50$) the prior conditions are not satisfied, since, for differential rotation, the largest non-negligible angular degree of perturbation is $s=5$. As shown in V11, using eigenfunction perturbations as in Eqn. (13) and eigenfrequency perturbations $\omega_{\ell}=\tilde{\omega}_{\ell}+\delta\omega_{\ell}$, the wave equation (6) reduces to an eigenvalue problem of the form $\mathbf{Z}\,\bm{\mathcal{C}}_{\ell}=\delta\omega_{\ell}\,\bm{\mathcal{C}}_{\ell},$ (14) where $\bm{\mathcal{C}}_{\ell}=\\{...,c_{\ell}^{\ell-1},c_{\ell}^{\ell},c_{\ell}^{\ell+1},...\\}$ are eigenvectors corresponding to the $(P\times P)$ self-adjoint matrix ${\mathbf{Z}}$ and $P=\mathrm{max}(\|\ell^{\prime}-\ell\|)$ denotes the largest offset of a contributing mode $\ell^{\prime}$ from $\ell$ according as Eqn. (13). From detailed considerations of first- and second-order quasi-degenerate perturbation theory, V11 showed that the following closed-form expression for correction coefficients is accurate up to angular degrees as high as $\ell=1000$: $c_{\ell}^{\ell+p}=\tfrac{1}{\pi}\int_{0}^{\pi}\cos\left[pt-\sum_{k=1,2,...}\tfrac{2}{k}\mathrm{Re}(b_{k})\sin{(kt)}\right]\times\mathrm{exp}\left[i\sum_{k=1,2,...}\tfrac{2}{k}\mathrm{Im}(b_{k})\cos{(kt)}\right]\mathrm{d}t,\qquad p=0,\pm 1,...,$ (15) where the convenient expressions for real and imaginary parts of $b_{k}$ are $\displaystyle\mathrm{Re}(b_{k})$ $\displaystyle=$ $\displaystyle\ell\left(\frac{\partial\tilde{\omega}}{\partial\ell}\right)^{-1}_{n}\sum_{s+k=\mathrm{odd}}(-1)^{\frac{s-k+1}{2}}\frac{(s-k)!!(s+k)!!}{(s+k)!}\times P_{s}^{k}\left(\frac{m}{\ell}\right)\langle\Omega_{s}\rangle_{n\ell},\quad k=1,2,...$ (16) $\displaystyle\mathrm{Im}(b_{k})$ $\displaystyle=$ $\displaystyle k\ell\left(\frac{\partial\tilde{\omega}}{\partial\ell}\right)^{-1}_{n}\sum_{s+k=\mathrm{even}}(-1)^{\frac{s-k+2}{2}}\left(\frac{2s+1}{4\pi}\right)^{1/2}\frac{(s-k-1)!!(s+k-1)!!}{(s+k)!}\times P_{s}^{k}\left(\frac{m}{\ell}\right)\langle\frac{v_{s}}{r}\rangle_{n\ell},\quad k=1,2,....$ (17) We consider only odd-$s$ dependencies of $\Omega$. The even-$s$ correspond to North-South (NS) asymmetry in differential rotation and are estimated to be weak at the surface (NS asymmetry coefficients are estimated to be an order of magnitude smaller than their symmetric counterparts; Mdzinarishvili et al., 2020). The contribution of even-$s$ components to the real part of $b_{k}$ can thus be ignored. For the asymptotic limit of high-degrees, $\mathrm{Re}(b_{k})=\ell\left(\frac{\partial\tilde{\omega}}{\partial\ell}\right)_{n}^{-1}\sum_{s+k=odd}(-1)^{\frac{k-2}{2}}\frac{s!(s-k)!!(s+k)!!}{(s+k)!s!!s!!}\times a^{n\ell}_{s}\,P_{s}^{k}\left(\frac{m}{\ell}\right),\quad k=2,4,...,$ (18) $a^{n\ell}_{s}\approx(-1)^{\frac{s-1}{2}}\frac{s!!s!!}{s!}\langle\Omega_{s}\rangle_{n\ell},\quad s=1,3,...$ (19) Figure 1 illustrates the distortion of eigenfunctions due to an equatorially symmetric differential rotation (using frequency splitting estimates of $a_{3}$ and $a_{5}$ coefficients). It can be seen that differences between distorted eigenfunctions $\bm{\xi}_{nlm}$ and their undistorted counterparts $\tilde{\bm{\xi}}_{nlm}$ are at around the $50\%$ level for some azimuthal orders. The correction coefficients, given by $c^{\ell+p,m}_{\ell,m}$, are shown in the bottom panel of Figure 1. Since the largest contribution to $\bm{\xi}_{\ell}$ comes from $\tilde{\bm{\xi}}_{\ell}$, $c^{\ell,m}_{\ell,m}(\gtrsim 0.8)$ are not plotted to highlight the corrections from neighbouring modes with $p\neq 0$. Visual inspection shows that $c^{\ell+p,m}_{\ell,m}$ have non-zero elements at $p=\pm 2,\pm 4$, as expected from selection rules due to the rotation field $\Omega_{s}(r)$ for $s=3,5$. High $\ell$ eigenfunctions are predominantly large close to the surface. Consequently, we see that their distortions are much larger at shallower than deeper depths. We choose to plot three cases — low, intermediate, and high $m$. For the extreme cases of $m=0$ and $m=\ell$, $c^{\ell+p,m}_{\ell,m}\sim 0$, since for odd $s$ and even $k$, $P_{s}^{k}(\mu)$ vanishes at $\mu=0,1$. Thus these eigenfunctions remain undistorted under an equatorially symmetric differential rotation. For sake of completeness, it may be mentioned that the finite $c^{\ell+p,m}_{\ell,m}$ for $p\neq 0$ seemingly disqualifies the frequency- splitting measurements, which assume isolated multiplets — meaning $c^{\ell+p,m}_{\ell,m}=\delta_{p,0}$. However, it does not necessarily imply that the isolated multiplet approximation is poor at these angular degrees. If the eigenfunction error $\delta\bm{\xi}_{k}$ incurred on neglecting cross- coupling is of order $\mathcal{O}(\epsilon)$ then it can be shown (see Chapter 8 of Freidberg, 2014; Cutler, 2017) that the error in estimating eigenfrequency $\delta\omega_{k}$ is at most of order $\mathcal{O}(\epsilon^{2})$, where $\epsilon$ is small. To illustrate this further, if the error in estimating eigenfunction distortion on neglecting cross coupling $(p\neq 0)$ is written as $\epsilon\,\bm{\xi}_{\ell+p}$, then from inspecting Eqn. (13), we see that $\epsilon\sim\|c_{\ell}^{\ell+p}\|$. Upon investigating the $(n,\ell)=(2,150)$ case presented in Figure 1 for $p\neq 0$, we find $c_{\ell}^{\ell+p}\lesssim\mathcal{O}(10^{-1})$. The equivalent error incurred in eigenfrequency estimation may be computed according to the discussion in Section 4.4. This yields $\delta\omega/\omega\lesssim\mathcal{O}(10^{-2})$ in the range $150\leq\ell\leq 250$ thereby confirming the above argument for $\epsilon\sim 10^{-1}$. Given the leakage matrices and Lorentzians, the forward problem of modeling $\langle\varphi^{\ell^{\prime}m^{\prime}}\varphi^{\ell m}\rangle$ requires constructing eigenfunction corrections $c^{\ell+p}_{\ell}$ using the $a$-coefficients in Eqn. (18) and the poloidal flow in Eqn. (17). Thus, for axisymmetric flows, the cross spectra for moderately large $\ell_{1}$ and $\ell_{2}$ from Eqn. (12) may be written more explicitly as $\langle\varphi^{\ell_{1},m_{1}}\,\varphi^{\ell_{2},m_{1}}\rangle=\sum_{p,p^{\prime},\ell,m}\,\tilde{L}_{\ell+p,m}^{\ell_{1},m_{1}}\,\tilde{L}_{\ell+p^{\prime},m}^{\ell_{2},m_{2}}\,c_{\ell,m}^{\ell+p,m}\,c_{\ell,m}^{\ell+p^{\prime},m}\,\langle\|\Lambda^{\ell,m}(\omega)\|^{2}\rangle.$ (20) The leakage matrices $\tilde{L}_{\ell+p,m}^{\ell_{1},m_{1}}$ impose bounds on the farthest modes that leak into mode amplitude $\varphi^{\ell m}$. This is because $\tilde{L}_{\ell+p,m}^{\ell_{1},m_{1}}$ is non-zero only when $\ell+p\in[\ell_{1}-\delta\ell,\ell_{1}+\delta\ell]$ and $m\in[m_{1}-\delta m,m_{1}+\delta m]$, where $\delta\ell$ and $\delta m$ are the farthest spectral offsets. Thus, for a given $\ell$, we must determine the correction coefficients $c_{\ell,m}^{\ell+p,m}$ such that $p\in[\ell_{1}-\delta\ell-\ell,\ell_{1}+\delta\ell-\ell]$. Similar bounds on $p^{\prime}$ in $c_{\ell,m}^{\ell+p^{\prime},m}$ are imposed by the second leakage matrix $\tilde{L}_{\ell+p^{\prime},m}^{\ell_{2},m_{2}}$, namely, $\langle\varphi^{\ell_{1},m_{1}}\,\varphi^{\ell_{1}+\Delta\ell,m_{1}}\rangle=\sum_{p,p^{\prime},\ell,m}\,\tilde{L}_{\ell+p,m}^{\ell_{1},m_{1}}\,\tilde{L}_{\ell+p^{\prime},m}^{\ell_{1}+\Delta\ell,m_{1}}\,c_{\ell,m}^{\ell+p,m}\,c_{\ell,m}^{\ell+p^{\prime},m}\,\langle\|\Lambda^{\ell,m}(\omega)\|^{2}\rangle.$ (21) Being significantly weaker than differential rotation, we neglect the contribution of meridional circulation (Imada & Fujiyama, 2018; Gizon et al., 2020) to the eigenfunction corrections. ## 3 Data Analysis Figure 2: Cross-spectral signal for $\ell=200$, $\Delta\ell=2$ and $n=0$. Panel (a, b): Observed cross-spectrum corresponding to $m^{+}$ and $m^{-}$. Panel (c, d): Derotated cross spectrum corresponding to $m^{+}$ and $m^{-}$. Panel (e, f): $D^{\ell,\Delta\ell,\pm}_{n}$. The baseline is indicated by the dashed blue line. The blue dots represent observations from the five 72-day time series and the red curve corresponds to the expectation value of the cross-spectrum. We use the full-disk 72-day gap-filled spherical-harmonic time series $\varphi^{\ell m}(t)$, which are recorded at a cadence of 45 seconds by HMI (Larson & Schou, 2015). The data are available for harmonic degrees in the range $\ell\leq 300$. The time series is transformed to the frequency domain to obtain $\varphi^{\ell m}(\omega)$. The negative-frequency components are associated with the negative $m$ components using the symmetry relation (Appendix A) $\varphi^{\ell,-\|m\|}(\omega)=(-1)^{\|m\|}\varphi^{\ell,\|m\|}(-\omega).$ (22) The ensemble average of the cross spectrum is computed by averaging five continuous 72-day time series, which corresponds to 360 days of helioseismic data. The eigenfrequencies of the unperturbed model $\tilde{\omega}_{n\ell m}$ are degenerate in $m$, i.e., $\tilde{\omega}_{n\ell m}=\tilde{\omega}_{n\ell 0}$. Rotation breaks spherical symmetry and lifts the degeneracy in $m$. As in W13, we show the cross-spectrum for $n=0$ and $\ell=200$, $\Delta\ell=2$ in Figure 2. The effect of rotation is visible through the inclination of the ridges in the $m-\nu$ spectrum, as seen in Panels (a, b) of the figure. The multiple vertical ridges are due to leakage of power. The cross spectra are derotated and stacked about the central frequency, corresponding to $m=0$, which is shown in Panels (c, d) of Figure 2. In order to improve the signal-to-noise ratio, the stacked cross spectrum is summed over azimuthal order $m$. This quantity is used to determine the extent of coupling, denoted by $D^{\ell,\Delta\ell,\pm}_{n}$. The $-$ ($+$) signs indicates summation over negative (positive) $m$. For notational convenience, we define $m^{+}$ when referring to $m\geq 0$ and $m^{-}$ to denote $m\leq 0$. The operation of stacking (derotating) the original spectra is denoted by $\mathcal{S}_{m}$. Since differential rotation affects only the real part of the cross-spectrum (see Eqn. 15), $D^{\ell,\Delta\ell,\pm}_{n}$ refers to the real part of the cross-spectrum. $D_{n}^{\ell,\Delta\ell,\pm}(\omega)=\left\langle\sum_{m^{\pm}}\mathcal{S}_{m}\left(\text{Re}\left[\varphi^{\ell m}(\omega)\varphi^{\ell+\Delta\ell,m}(\omega)\right]\right)\right\rangle.$ (23) The cross-spectral model is a combination of Lorentzians and is based on Eqn. (21). The HMI-pipeline analysis provides us with mode amplitudes and linewidths for multiplets $(n,\ell)$. The $m$ dependence of frequency, $\omega_{n\ell m}-\omega_{nl0}$, is encoded in 36 frequency-splitting coefficients $(a^{nl}_{1},a^{nl}_{2},...,a^{nl}_{36})$. These values are used to construct the Lorentzians for the model, which is denoted by $M^{\ell,\Delta\ell,\pm}$ and expressed as $M^{\ell,\Delta\ell,\pm}_{n}(\omega)=\sum_{m\pm}\mathcal{S}_{m}\left(\sum_{p,p^{\prime},\ell,m^{\prime}}\,\tilde{L}_{\ell+p,m^{\prime}}^{\ell_{1},m}\,\tilde{L}_{\ell+p^{\prime},m^{\prime}}^{\ell_{1}+\Delta\ell,m}\,c_{\ell,m^{\prime}}^{\ell+p,m^{\prime}}\,c_{\ell,m^{\prime}}^{\ell+p^{\prime},m^{\prime}}\,\langle\|\Lambda^{\ell,m^{\prime}}_{n}(\omega)\|^{2}\rangle\right).$ (24) As seen in Panels (e, f) of Figure 2, the cross spectra sit on a non-zero baseline. This is a non-seismic background and hence is explicitly fitted for before further analysis of the data. The complete model of the cross spectrum involves leakage from the power spectrum, eigenfunction coupling, as well as the non-seismic background, i.e., the data $D^{\ell,\Delta\ell,\pm}_{n}(\omega)$ is modelled as $M^{\ell,\Delta\ell,\pm}_{n}(\omega)+b^{\ell,\Delta\ell,\pm}_{n}(\omega)$. The baseline $b_{n}^{\ell,\Delta\ell,\pm}(\omega)$ is computed by considering 50 frequency bins on either side, far from resonance, and fitting a straight line through them, in a least-squares sense. The model $M^{\ell,\Delta\ell,\pm}_{n}(\omega)$ depends on the $a^{nl}_{3}$ and $a^{nl}_{5}$ splitting coefficients via the eigenfunction-correction coefficients $c^{\ell+p}_{\ell}$. A Bayesian-analysis approach is used to estimate the values $(a^{nl}_{3},a^{nl}_{5})$, using MCMC, described in Section 3.1. The misfit function that quantifies the goodness of a chosen model is given by $\Xi_{n}=\sum_{l,\omega,\pm}\left(\frac{D^{\ell,\Delta\ell,\pm}_{n}(\omega)-(M^{\ell,\Delta\ell,\pm}_{n}(\omega)+b^{\ell,\Delta\ell,\pm}_{n}(\omega))}{\sigma^{\ell,\Delta\ell,\pm}_{n}(\omega)}\right)^{2},$ (25) where $[\sigma^{\ell,\Delta\ell,\pm}_{n}(\omega)]^{2}$ denotes the variance of the data $D^{\ell,\Delta\ell,\pm}_{n}(\omega)$ and is given by $[\sigma^{\ell,\Delta\ell,\pm}_{n}(\omega)]^{2}=\left\langle\left(\sum_{m\pm}\mathcal{S}_{m}\left[\phi^{\ell,m}(\omega)\,\phi^{\ell+\Delta\ell,m}(\omega)\right]-D^{\ell,\Delta\ell,m\pm}\right)^{2}\right\rangle.$ (26) ### 3.1 Bayesian Inference: MCMC Bayesian inference is a statistical method to determine the probability distribution functions (PDF) of the inferred model parameters. For data $D$ and model parameters $a$, the posterior PDF $p(a\|D)$, which is the conditional probability of the model given data, may be constructed using the likelihood function $p(D\|a)$ and a given prior PDF of the model parameters $p(a)$. The prior encapsulates information about what is already known about the model parameters $a$. $p(a\|D)\propto p(D\|a)p(a).$ (27) The constant of proportionality is the normalization factor for the posterior probability distribution, which may be difficult to compute. The sampling of these PDFs is performed using MCMC, which involves performing a biased random walk in parameter space. Starting from an initial guess of parameters, a random change is performed. The move is accepted or rejected based on the ratio of the posterior probability at the two locations. Hence, the normalization factor is superfluous to the MCMC method. Bayesian MCMC analysis has been used quite extensively in astrophysical problems (Saha & Williams, 1994; Christensen & Meyer, 1998; Sharma, 2017, and references therein) and terrestrial seismology (Sambridge & Mosegaard, 2002, and references therein). However, the use of MCMC in global helioseismology has been limited as compared to terrestrial seismology (Jackiewicz, 2020). The aim of the current calculation is the estimation of $(a_{3}^{n\ell},a_{5}^{n\ell})$ that best reproduce the observed cross-spectra from the model, given by Eqn. (24), where it is seen that the coupling coefficients $c^{\ell+p}_{\ell}$ depend on $(a_{3}^{n\ell},a_{5}^{n\ell})$. However, because of leakage, neighbouring $\ell$ corresponding to the spectrum in question also contribute to the cross-spectrum. Hence, the spectrum of $(\ell,\Delta\ell)$ depends on $(a_{3}^{n\ell^{\prime}},a_{5}^{n\ell^{\prime}})$ for $\ell^{\prime}\in[\ell-\delta\ell,\ell+\Delta\ell+\delta\ell]$. Since we only consider mode leakage at the same radial order $n$, we are forced to simultaneously estimate all the $(a_{3}^{n\ell},a_{5}^{n\ell})$ for a given $n$. For instance, at $n=0$, we have 52 modes with $\ell<250$, and 94 spectra corresponding to $\Delta\ell=2,4$, for both $m^{+}$ and $m^{-}$ branches. In this case, there are 52 $(a_{3}^{0\ell},a_{5}^{0\ell})$ pairs that need to be estimated and 188 spectra which need to be modeled. Performing inversions on a high dimensional, jagged landscape is a challenge as the fine tuning of regularization is tedious. However, since we have a model which encodes the dependence of the $a$ coefficients on the cross-spectrum, we could “brute- force” the estimation of parameters. The utility of MCMC is that it enables us to sample the entire parameter space. Since the inference of the posterior PDF depends strongly on the prior, it is instructive to use an uninformed or flat prior. For the MCMC simulations, we use the Python package emcee by Foreman-Mackey et al. (2013). The package is based on the affine invariant ensemble sampler by Goodman & Weare (2010). Multiple random walkers are used to sample high- dimensional parameter spaces efficiently. We use a flat prior for all $a_{3}^{n\ell}$ and $a_{5}^{n\ell}$ given by $\displaystyle p(a_{3})=\frac{1}{20}\qquad 15\leq a_{3}\leq 35\qquad\text{and}\qquad p(a_{5})=\frac{1}{16}\qquad-16\leq a_{5}\leq 0,$ (28) and zero everywhere else for all $(\ell,n)$. This is motivated by the results of frequency splittings. For modes near the surface, i.e., for low values of $\nu_{n\ell}/\ell$, $a_{3}$ has been measured to be nearly $22$ nHz and $a_{5}$ is $-4$ nHz. The likelihood function is defined as $p(D\|a)=\exp(-\Xi_{n}),$ (29) where $\Xi_{n}$ is the misfit given by Eqn. (25). Flat priors enable us to sample the likelihood function in the given region in parameter space. We perform MCMC inversions for $n=0,1,..8$ and find that the likelihood function is unimodal in all model parameters. For the sake of illustration, a smaller computation is presented in Appendix B. Radial order $n$ \| Range of $\ell$ for $(a_{3},a_{5})$ ---\|--- 0 \| 192–241, 241–281, 271–289 1 \| 80–120, 110–150, 140–183 2 \| 60–100, 90–130, 120–161 3 \| 43–73, 73–113, 103–145 4 \| 40–80, 70–110, 100–140 5 \| 46–86, 76–116, 106–146 6 \| 58–98, 88–128, 118–138 7 \| 64–104, 94–114 8 \| 73–103 Table 1: List of modes $(n,\ell)$ used in MCMC. These are marked as black dot in Figure 1. Figure 3: Classification of modes. ## 4 Results and Discussion The MCMC analysis is performed for each radial order separately. The current model only considers leakage between modes of the same radial order and hence the ideal way of estimating the parameters would be to estimate all $(a_{3},a_{5})$ at a given radial order by modelling all the cross-spectra at the same radial order. However, this makes the problem computationally very demanding as the MCMC method used requires at least $2k+1$ random walkers for $k$ different parameters to be fit. To work around this, we break the entire set of parameters into chunks of 40 pairs, while ensuring an overlap of 10 pairs between the chunks. In Table 1, we list the set of $\ell$’s for which MCMC sampling is performed and parameters are estimated. Figure 1 marks the multiplets $(n,\ell)$ available from the HMI pipeline. The multiplets whose modes are used for this study are labelled as black dots. The red dots, which are located at lower $\ell$, correspond to those modes which have contributions from neighbouring radial orders within the temporal- frequency window. This gets worse for $\ell<20$, where contributions from neighbouring radial orders may be seen even near central peaks. Modelling these spectra would require including coupling across radial orders, which is not the case in the present analysis. Thus we only use modes corresponding to $\nu_{n\ell}/\ell<45$. Figure 1 also marks unused HMI-resolved modes as blue dots on either side of the black dots (used modes). This is because we consider only modes that may be fully modelled with parameters available from the HMI pipeline. Modelling a the degree $\ell$ requires mode parameters corresponding to modes from $(\ell-\delta\ell)$ to $(\ell+\delta\ell)$. The existence of unresolved modes (with no mode-parameter information from the HMI pipeline) in this region means that modelling is incomplete, i.e., there would be peaks in the observed spectrum that are missed by the model. Hence, such modes are not considered for the present work. For any given radial order, the first $\delta\ell$ and the last $\delta\ell$ modes cannot be modelled and thus we see blue points on either side of the set of black dots in Figure 1. The results of the MCMC analysis at all the radial orders are combined and presented in Figure 6. We note that that the confidence intervals become larger for higher $\nu/\ell$. The reasons for this are discussed in Section 4.2. Estimates of $a$-coefficients are largely in agreement with the splitting coefficients — although the most probable values of the coupling-derived parameters are different from their splitting counterparts, they predominantly lie within the 1-$\sigma$ confidence interval. The confidence intervals of $a_{3}$ and $a_{5}$ are nearly the same size. We obtain better results, in terms of the spread in the inferred $a_{3}$-coefficients, than W13. This may be attributed to the consideration of data variance as well as simultaneous fitting for model parameters using a Bayesian approach. For instance, the spread of $a_{3}$ in the range $0<\nu/\ell\lesssim 40$ is seen to be in the range 7.5–30 nHz in W13, whereas our estimates are in the range 15–26 nHz. The present method allows us to quantify the 1-$\sigma$ confidence interval around the most probable values for estimated $a$-coefficients, whereas W13 have shown only inversion values of $a$-coefficients without their respective uncertainties. However, we also note that the estimates of $a_{5}$ from Bayesian analysis are comparable to the least-squares inversions of W13. ### 4.1 Reconstructed power and cross spectra Figure 4: Cross spectrum for $\ell=222$ and $\Delta\ell=0,2,4$. The upper panels correspond to $m^{+}$ and lower panels to $m^{-}$. The black curve shows observed data. The blue curve is the model before considering eigenfunction coupling and the red curve corresponds to model constructed using parameters estimated from MCMC. Figure 5: Cross spectrum for $\ell=70$ and $\Delta\ell=0,2,4$. The upper panels correspond to $m^{+}$ and lower panels to $m^{-}$. The black curve shows observed data. The blue curve is the model before considering eigenfunction coupling and the red curve corresponds to the model constructed using parameters estimated from MCMC. The $a$-coefficients obtained from the MCMC analysis are used to reconstruct cross-spectra, e.g., Figure 4 shows the cross spectrum for $(n=0,\ell=222)$. It may be seen that, before considering eigenfunction corrections (in the absence of differential rotation), the spectrum shown in blue is considerably different — in both magnitude and sign — from the observed data. After including eigenfunction corrections, which have been estimated from MCMC, we see that the model is in close agreement with the data. In the intermediate-$\ell$ range, we show cross-spectra for $(n=4,\ell=70)$ in Figure 5. The corrections due to eigenfunction distortion are markedly less significant when compared to $(\ell=222,n=0)$, demonstrating loss of sensitivity of the model to the coupling coefficients. Figure 6: Inferred $a_{3}$ and $a_{5}$ coefficients from MCMC are shown as black dots with 1-$\sigma$ confidence intervals. The values from frequency splitting are shown in red. ### 4.2 Sensitivity of $a$-coefficients to differential rotation Mode coupling has diminished sensitivity in estimating $a$-coefficients for low-$\ell$ modes. The coupling coefficients $c_{\ell}^{\ell+p}$ depend on the real and imaginary parts of $b_{k}$. Differential rotation contributes to only the real part of $b_{k}$ (Eqn. [16]) and the dependence on $\ell$ appears through the factor $\ell(\partial\omega_{nl}/\partial\ell)^{-1}$. The plot of eigenfrequencies $\omega_{n\ell}$ against $\ell$ is known to flatten for higher $\ell$. Hence, $\partial\omega_{n\ell}/\partial\ell$ is large for small $\ell$ and small for large $\ell$ (see Figure 1 in Rhodes et al., 1997). This results in $b_{k}$ being small for low $\ell$ and its magnitude increases with $\ell$, causing this decreased sensitivity to low $\ell$. The lower sensitivity implies that the misfit function $S$ is flatter at lower $\ell$. To demonstrate this, we compute $S$ over $\ell=80$–245 for a range of values of $a-$coefficients and determine how wide or flat $S$ is in the neighbourhood of the optimal solution. Figure 7 shows that the misfit is wide for $\ell=80$ and it becomes sharper with increasing $\ell$. As the highest-resolved mode for $n=1$ corresponds to $\ell=179$, we consider the radial order $n=0$ in order to study this in an extended region of $\ell$. The first two panels show the colour map of the misfit function. Near the optimal value $a^{n\ell}_{s}/a^{n\ell}_{FS}=1$, the synthetic misfit falls to $0$. This is possible as the synthetic data is noise free and it can be completely modeled. The misfit increases on either side of the optimum value. The second panel shows the scaled misfit for HMI data, which is close to $1$ at the optimum, increasing on either side of the optimal value. We see the dark patch become wider at lower $\ell$, indicating the flatness of the misfit function for low $\ell$. The likelihood function, which is defined to be $\exp(-S)$, is approximated as a Gaussian in the vicinity of the optimum. The width of this Gaussian is treated as a measure of the width of the misfit function $S$, with wider misfit implying lower sensitivity to $a-$coefficients. This is shown in the third panel of Figure 7, where we see a decreasing trend in misfit width, indicating that the sensitivity of mode coupling increases with $\ell$. Figure 7: Sensitivity of spectral fitting to $a$-coefficients as a function of angular degree $\ell$. Top panel shows the variation of misfit between synthetic data calculated using frequency-splitting $a$-coefficients $a_{s,\mathrm{FS}}^{n\ell}$ and synthetic spectra computed from a scaled set of $a$ coefficients $a_{s}^{n\ell}$. A well-defined minimum along $a_{s}^{n\ell}/a_{s,\mathrm{FS}}^{n\ell}=1.0$, which broadens towards smaller $\ell$, shows a drop in sensitivity of the spectra to variations in $a$ coefficients, as predicted by theory. Middle panel shows the sensitivity of $a$ coefficients, but now computed using the misfit between HMI and synthetic spectra computed from a scaled set of $a$ coefficients $a_{s}^{n\ell}$. While it has the same qualitative drop in $a$-coefficient sensitivity for decreasing $\ell$, the ridge of the minimum (darkest patch) is seen to deviate from $a_{s,\mathrm{FS}}^{n\ell}$. Bottom panel shows in black the effective variance of misfit for each $\ell$. The narrowing confinement of the data misfit towards higher $\ell$ is seen as a decreasing effective variance with increasing $\ell$. The red line shows the increase in the factor $\ell/(\partial\omega_{n\ell}/\partial\ell)$ that enhances sensitivity at higher $\ell$, as predicted by Eqn. (18). The areas corresponding to radial orders $n=0,1$ are indicated on top of each plot. ### 4.3 Scaling factor for synthetic spectra The model constructed using mode parameters obtained from the HMI pipeline needs to be scaled to match the observations. This scaling factor has to be empirically determined. Since there is no well-accepted convention to estimate this factor, it is worthwhile to explore different methods of its estimating. We employ three different methods to infer the scale factor and show that the results are nearly identical. * • Consider all the power spectra for a given radial order and perform a least- squares fitting for the scale factor $N_{0}$. * • Fit for the scale factor $N_{\ell}$ as a function of the spherical harmonic degree $\ell$ by considering all power spectra at a given radial order. * • Include the scale factor as an independent parameter to be estimated in the MCMC analysis. Figure 8 shows that all the independent ways of estimating the scale factor are within 5% of each other, indicating robustness. Figure 8: The red line corresponds to $N_{0}$. The gray region corresponds to 5% error from $N_{0}$. The gray points correspond to $N_{\ell}$ and the solid black lines are from each MCMC simulation. The right-most panel shows the histogram of all the gray points, taken from all radial orders. ### 4.4 How good is the isolated multiplet approximation for $\ell\leq 300$? Figure 9: The relative offset of $L_{2}^{\text{QDPT}}$ as compared to that of $L_{2}^{\text{DPT}}$ (see Eqn [30,31]) under the perturbation of an axisymmetric differential rotation $\Omega(r,\theta)$ as observed in the Sun. An increase in intensity of the color scale indicates worsening of the isolated multiplet approximation. The measures of offset are plotted for the HMI-resolved multiplets shown in Figure 1. Estimation of $a$ coefficients through frequency splitting measurements assumes validity of the isolated multiplet approximation using degenerate perturbation theory (DPT). However, an inspection of the distribution of the multiplets in $\nu-\ell$ space (as shown in Fig. [1]) shows that it is natural to expect this approximation to worsen with increasing $\ell$. This necessitates carrying out frequency estimation respecting cross-coupling of modes across multiplets, also known as quasi-degenerate perturbation theory (QDPT). A detailed discussion on DPT and QDPT in the context of differential rotation can be found in Ritzwoller & Lavely (1991) and Lavely & Ritzwoller (1992). In this section we discuss the goodness of the isolated multiplet approximation in estimating $a_{n\ell}$ due to $\Omega(r,\theta)$ for all the HMI-resolved modes shown in Figure 1. A similar result but for $\ell\leq 30$ was presented in Appendix G of Das et al. (2020). In Figure 9 we color code multiplets to indicate the departure of frequency shifts obtained from QDPT $\delta{}_{n}\omega{}_{\ell m}^{Q}$ as compared to shifts obtained from DPT $\delta{}_{n}\omega{}_{\ell m}^{D}$. Strictly speaking, carrying out the eigenvalue problem in the QDPT formalism causes garbling of the quantum numbers —$n$, $\ell$, and $m$ are no longer good quantum numbers— and prevents a one-to-one mapping of unperturbed to perturbed modes. This prohibits an explicit comparison of frequency shifts on a singlet-by-singlet basis. However, modes belonging to the same multiplet can still be identified visually and grouped together. So, to quantify the departure of $\delta{}_{n}\omega{}_{\ell m}^{Q}$ from $\delta{}_{n}\omega{}_{\ell m}^{D}$ we calculate the Frobenius norm of these frequency shifts corresponding to each multiplet: $\displaystyle L_{2}^{\text{QDPT}}$ $\displaystyle=$ $\displaystyle\sqrt{\sum_{m}(\delta{}_{n}\omega{}_{\ell m}^{\text{Q}})^{2}}\qquad\text{for cross-coupling,}$ (30) $\displaystyle L_{2}^{\text{DPT}}$ $\displaystyle=$ $\displaystyle\sqrt{\sum_{m}(\delta{}_{n}\omega{}_{\ell m}^{\text{D}})^{2}}\qquad\text{for self-coupling.}$ (31) The color scale intensity in Figure 9 indicates the relative offset of $L_{2}^{\text{QDPT}}$ as compared to $L_{2}^{\text{DPT}}$ for a multiplet $(n,\ell)$ marked as an ‘o’. Larger offset indicates the degree of worsening of the isolated multiplet approximation. We find that the largest error incurred using DPT instead of QPDT is 0.27% this is found to be at $\ell=300$. This clearly shows that even for the $f$ mode (which is the most susceptible to errors) the frequency splitting $a$-coefficients are exceptionally accurate. ## 5 Conclusion Most of what is currently known about solar differential rotation is derived from from $a$-coefficients using frequency splitting measurements. Inferring these $a$-coefficients involves invoking the isolated multiplet approximation based on degenerate perturbation theory. Although this approximation works well even for high $\ell\leq 300$ modes, reasons motivating the need to investigate the possibility of erroneous $a$-coefficients from frequency splitting measurements at even higher $\ell$ stem from a combination of two effects, namely, the increasing proximity of modes (in frequency) along the same radial branch, and spectral-leakage from neighbouring modes. Partial visibility of the Sun causes broadening of peaks in the spectral domain, referred to as mode leakage (Schou & Brown, 1994; Hanasoge, 2018). This causes proximal modes at high $\ell$ to widen and resemble continuous ridges in observed spectra. As a result, spectral-peak identification for frequency- splitting measurements are harder and increasingly inaccurate. Moreover, since the $a$-coefficient formalism breaks down for non-axisymmetric perturbations, considering techniques which respect cross-coupling becomes indispensable. Thus, mode coupling becomes more relevant in these regimes, and it is important to investigate the potential of mode-coupling techniques as compared to frequency splittings. Hence, this study was directed towards answering the following broad questions. (i) Can mode-coupling via MCMC use information stored in eigenfunction distortions to constrain differential rotation as accurately as frequency splittings? This would also serve to compare the potential of a Bayesian approach with the least square inversion performed in W13. (ii) Can this technique further increase the accuracy of $a_{n\ell}$ at $\ell\geq 150$? We already know that higher $\ell$ estimates are increasingly precise and accurate from W13. (iii) What are the uncertainties in estimating $a_{n\ell}$ using mode-coupling theory and do they fall within 1-$\sigma$ of frequency splitting estimates? (iv) Why are mode-coupling results poorer in the low $\ell$ regime? This is seen in earlier studies, which aimed to go deeper into the convection zone and obtained significantly imprecise and inaccurate results (Woodard et al., 2013; Schad & Roth, 2020). The approach in this study is broadly based on the theoretical formulations from V11 and modelling from W13. However, the novelty of the current work lies in 3 main aspects. (a) The MCMC analysis enabled exploration of the complete parameter space, and it was found that the chosen misfit function is unimodal in nature, for all degrees $\ell$ and radial orders $n$. This establishes that the method of normal-mode coupling does return a unique value of $(a_{3},a_{5})$. (b) Leakage of power occurs for modes in the same radial order $n$ and hence the determination of $(a_{3},a_{5})$ in a consistent manner would involve simultaneous estimation of splitting coefficients for all $\ell$ and the same radial order. However, the number of parameters is large and hence we break it into chunks of 40 pairs of $(a_{3},a_{5})$ per MCMC, with an overlap of $N_{o}$ pairs of the parameters between two different chunks. To settle on a reasonable value of $N_{o}$, we perform a simple experiment. From MCMC simulations with different overlap numbers $N_{o}=\\{0,2,4,6,8,10\\}$, we find that for $N_{o}>6$, the inferred $a$-coefficients vary less than 1-$\sigma$ and therefore reasonably stable for larger $N_{o}$. Hence, we choose the modal overlap number $N_{o}=10$ for computation at all radial orders. (c) Since a large number of splitting coefficients are determined simultaneously, a corresponding number of spectra is used. Hence, estimation of the data variance becomes critical in order to appropriately weight different data points according to their noise levels. These improvements lead to a better estimate of differential rotation using mode coupling. The inference of rotation at lower $\ell$ ($<50$) suffers for two reasons. (a) Low sensitivity of the model to the $a$-coefficients. (b) Proximity of modes of radial orders $(n+1)$ and $(n-1)$ to modes at radial order $n$. Since the current model only accounts for leakage of power within the same radial order, a chosen frequency window in data would contain peaks from neighboring radial orders, which are not modelled. Hence, an improvement might be achieved at lower $\ell$ by modelling the interaction of modes of different radial orders. Finally, in this study we also show that even though frequency splitting is much more precise for low $\ell\leq 150$, mode coupling estimates of differential rotation improves at high $\ell\geq 200$. Therefore, it is expected that mode-coupling would be comparable to (or possibly more accurate than) frequency splitting for very high $\ell\geq 300$. This would then allow one to compare mode-coupling estimates of shallow, small-scale structures with results from methods in local helioseismology. Going this high in angular degree for mode-coupling, however, introduces some challenges: (a) The computation of leakage matrices for high $\ell$ is very expensive. (b) $\partial\omega/\partial\ell$ decreases as $\ell$ grows and the spectrum becomes a continuous ridge in frequency space making it harder to resolve the modes completely. In conclusion, there remains scope for improvement and related lines of study. In this study, we have ignored the even-$s$ components of $\Omega_{s}$, which are the NS-asymmetric components of differential rotation. These components have been estimated to be small at the surface and are anticipated to be small in the interior. However, this assumption may be premature given that prior estimates of interior rotation-asymmetries are based on non-seismic surface measurements. Since the V11 formalism is capable of accommodating the estimation of even-$s$ components as well, this could be the focus of a future investigation. Additionally, the current analysis was performed after summing up the stacked cross-spectrum. Although this was done to improve the signal- to-noise ratio, the spectrum at different azimuthal orders $m$ are not identical. Hence a more complete computation would involve the misfit computed using the full spectrum as a function of $m$. This may possibly lead to better results of the $a$-coefficients, as there exists structure in the azimuthal order (see Fig. [2]), which is lost after summation. The authors of this study are grateful to Jesper Schou (Max Planck Institute for Solar System Research) for numerous insightful discussions as well as detailed comments that helped us improve the quality of the manuscript. The authors thank the anonymous referee for valuable suggestions that helped improve the text and figures in this manuscript. ## Appendix A Spherical harmonics symmetry relations Consider a time-varying, real-valued scalar field on a sphere $\phi(\theta,\phi,t)$. The spherical harmonic components are given by $\phi^{l,\|m\|}(t)=\int_{\Omega}{\mathrm{d}}\Omega Y^{l,\|m\|}(\theta,\phi)\phi(\theta,\phi,t)=(-1)^{\|m\|}\int_{\Omega}{\mathrm{d}}\Omega Y^{l,-\|m\|}\phi(\theta,\phi,t)=(-1)^{\|m\|}\phi^{l,-\|m\|}(t)$ (32) where $d\Omega$ is the area element, the integration being performed over the entire surface of the sphere. After performing a temporal Fourier transform, we have $\phi^{l,\|m\|}(\omega)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}{\mathrm{d}}te^{-i\omega t}\phi^{l,\|m\|}(t)=(-1)^{\|m\|}\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}{\mathrm{d}}te^{-i\omega t}\phi^{l,-\|m\|}(t)$ (33) $\phi^{l,-\|m\|}(\omega)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}{\mathrm{d}}te^{i\omega t}\phi^{l,-\|m\|}(t)=(-1)^{\|m\|}\phi^{l,\|m\|}(-\omega)\implies\phi^{l,-\|m\|}(\omega)=(-1)^{\|m\|}\phi^{l,\|m\|}(-\omega)$ (34) ## Appendix B MCMC: An illustrative Case We present an MCMC estimation of $a$-coefficients using a smaller set of modes (and hence model parameters). The smaller number of model parameters lets us present all the marginal probabilities in a single plot. The MCMC walkers are shown in Figure 10. In spite of using a flat prior, the likelihood function is sharp enough to bias the walkers to move towards the region of optimal solution within $\sim 500$ iterations. It can be seen that different walkers start off randomly at different locations in parameter space and ultimately converge to the same region around the optimal solution. After removing the iterations from the “burn-in” period, where the walkers are still exploring a larger parameter space, histograms are plotted and marginal probability distributions are obtained. Figure 11 shows one such estimation of $(a_{3},a_{5})$ for $n=0$ and $\ell$ in the range $200$ to $202$. It can be seen that the marginal posterior probability distributions for each of the parameters are unimodal. This tells us that the currently defined misfit function has a unique minimum. Note that this distribution was obtained using a flat prior and hence the resulting posterior distributions are essentially sampling the likelihood function. It is also worth noting that for the range of $\ell$’s chosen, the confidence intervals are $<1$ nHz. 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# Spin Hall effect of light under arbitrarily polarized and unpolarized light Minkyung Kim Department of Mechanical Engineering, Pohang University of Science and Technology (POSTECH), Pohang 37673, Republic of Korea Dasol Lee Department of Mechanical Engineering, Pohang University of Science and Technology (POSTECH), Pohang 37673, Republic of Korea Junsuk Rho Department of Mechanical Engineering, Pohang University of Science and Technology (POSTECH), Pohang 37673, Republic of Korea Department of Chemical Engineering, Pohang University of Science and Technology (POSTECH), Pohang 37673, Republic of Korea<EMAIL_ADDRESS> ###### Abstract The spin Hall effect of light (SHEL), which refers to a spin-dependent and transverse splitting at refraction and reflection phenomena, inherently depends on the polarization states of the incidence. Most of the previous research have focused on a horizontally or vertically polarized incidence, in which the analytic formula of the shift is well-formulated and SHEL appears symmetrically in both shift and intensity. However, the SHEL under an arbitrarily polarized or unpolarized incidence has remained largely unexplored. Whereas the SHEL under other polarization is sensitive to incident polarization and is asymmetrical, here we demonstrate that the SHEL is independent of the incident polarization and is symmetrical in shift if Fresnel coefficients of the two linear polarization are the same. The independence of the shift with respect to the incident polarization is proved both analytically and numerically. Moreover, we prove that under an unpolarized incidence composed of a large number of completely random polarization states, the reflected beam is split in half into two circularly polarized components that undergo the same amount of splitting but in opposite directions. This result means that under unpolarized incidence, the SHEL occurs exactly the same as under horizontally or vertically polarized incidence. We believe that the incident-polarization-independent SHEL can broaden the applicability of the SHEL to cover optical systems in which polarization is ill-defined. ## 1 Introduction Spin-orbit coupling is a universal phenomenon that can be observed in a variety of fields in physics including classical mechanics 1, quantum physics 2, and photonics 3. In particular, there exist diverse kinds of spin-orbit interactions in photonics because of rich physics enabled by the two electric and magnetic vector fields 4. Among them, one interesting spin-orbit related feature is the spin Hall effect of light (SHEL) 5, 6, 7, also known as an Imbert-Fedorov shift 8, 9, which reveals itself as a transverse and spin- dependent splitting of a finitely-thick beam at an optical interface. A physical mechanism that underpins the spin-dependent shift is the transverse nature of light, $\mathbf{k}\cdot\mathbf{E}=0$, that makes the incidence contain a bundle of slightly differently defined polarization bases 10. Despite its long history tracing back to the mid-19th century, the SHEL has regained a booming interest recently especially in photonics and metamaterials communities. A variety of nanophotonic devices and metamaterials have been proposed to enlarge the spin-dependent shift 11, 12, 13, 14, 15, 16, 17, 18, to increase the efficiency 19, and to exploit the SHEL to identify geometric parameters 20, 21, electric and magnetic properties 22, 23, and chemical reactions 24, 25 with high precision. Except for the studies of asymmetric SHEL, in which the magnitude of the shift is spin-dependent 26, 27, 28, 29, 30, most previous studies have focused only on horizontally or vertically polarized incidence. By symmetry, a horizontally or vertically polarized light is split at the optical interface into equal amplitudes of left circularly polarized (LCP) and right circularly polarized (RCP) beams, which shift by the same magnitude but with opposite signs 6. However, under an arbitrary incident polarization, neither the magnitude of the shift nor the intensity of the two circularly polarized components is symmetrical in general; in other words, the arbitrarily polarized incidence is split into LCP and RCP unevenly in both shift and intensity. Furthermore, the shift is sensitive to the polarization states of the incidence, resulting in a different amount of splitting as the incident polarization varies. Until very recently, a well-defined polarization state has been regarded as a prerequisite of spin-orbit related phenomena. However, it has been reported recently that even unpolarized light can induce a transverse spin 31. Here, we demonstrate that if reflection coefficients for $s$ and $p$ polarizations are equal to each other, the shift is degenerate for any arbitrarily polarized incidence and that the SHEL appears symmetrically in both shift and intensity under unpolarized incidence, as it does under a horizontally or vertically polarized incidence. First, we prove both analytically and numerically that the SHEL is independent of the polarization state of the incidence when the two linear polarizations have the same reflection coefficients. In such a case, LCP and RCP components of the reflected beam are shifted evenly, by the same amount but in opposite directions. The intensities of the LCP and RCP components are generally asymmetrical under an arbitrarily given incident polarization, but they also become symmetrical when the incidence is unpolarized, i.e., when the incidence is a superposition state of a vast number of completely random polarizations. We show that when the reflection coefficients are degenerate, the SHEL appears symmetrically in both shift and intensity, thereby having the whole symmetries of the SHEL under horizontally or vertically polarized incidence even under unpolarized incidence. In stark contrast to the previous studies of the SHEL where the incidence has a well- defined single polarization state, our work will extend the field of the SHEL to include unpolarized sources. ## 2 Results and Discussion ### 2.1 The analytical proof of incident-polarization-independent shift This section is devoted to proving the incident-polarization-independent shift theoretically by using a wave packet model. To do so, we examine how an incidence characterized by a single arbitrary polarization transforms at an interface through the reflection. An incident Gaussian beam propagating in the $x_{I}$-$z_{I}$ plane along the $z_{I}$-axis can be expressed in momentum space as $\bm{\psi}_{I}=\begin{pmatrix}\psi_{I}^{H}\\\ \psi_{I}^{V}\end{pmatrix}\psi_{0},$ (1) where the superscripts $H$ and $V$ correspond to horizontal and vertical polarization respectively, $\begin{pmatrix}\psi^{H}_{I}&\psi^{V}_{I}\end{pmatrix}^{T}$ is the Jones vector of the incidence, and $\psi_{0}$ is a Gaussian term given as $\psi_{0}=\frac{\omega_{0}}{\sqrt{2\pi}}\exp(-\frac{\omega_{0}^{2}}{4}(k_{x}^{2}+k_{y}^{2})),$ (2) where $\omega_{0}$ is a beam waist, $k_{0}$ is the incident wave vector, and $k_{x}\equiv k_{Ix}=-k_{Rx}$ and $k_{y}\equiv k_{Iy}=k_{Ry}$ are $x$\- and $y$-components of the wave vector. The Jones vectors of the incident and reflected beams represented in a linear basis are related to each other 32 $\begin{pmatrix}\psi^{H}_{R}\\\ \psi^{V}_{R}\end{pmatrix}=\begin{pmatrix}r_{p}&\frac{k_{y}}{k_{0}}(r_{p}+r_{s})\cot{\theta_{i}}\\\ -\frac{k_{y}}{k_{0}}(r_{p}+r_{s})\cot{\theta_{i}}&r_{s}\end{pmatrix}\begin{pmatrix}\psi^{H}_{I}\\\ \psi^{V}_{I}\end{pmatrix},$ (3) where the subscripts $I$ and $R$ correspond to the incident and reflected beam respectively, $r_{s}$ and $r_{p}$ are Fresnel reflection coefficients for $s$ and $p$ polarization, and $\theta_{i}$ is an incident angle. The reflected beam is composed of the Jones vector and the Gaussian part similarly to Eq. 1, $\bm{\psi}_{R}=\begin{pmatrix}\psi_{R}^{H}\\\ \psi_{R}^{V}\end{pmatrix}\psi_{0}.$ (4) The reflected beam can be represented in a circular basis by using a basis transformation $\begin{pmatrix}\psi^{+}_{R}\\\ \psi^{-}_{R}\end{pmatrix}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&i\\\ 1&-i\end{pmatrix}\begin{pmatrix}\psi^{H}_{R}\\\ \psi^{V}_{R}\end{pmatrix},$ (5) where the superscripts $+$ and $-$ denote LCP and RCP respectively. Since the wave vector component along the transverse axis is much smaller than the wave number ($k_{y}/k_{0}\ll 1$), the first-order Taylor expansion of $1\pm x\approx\exp(\pm x)$ can be applied. Then, Eq. 3 can be expressed as $\begin{pmatrix}\psi^{+}_{R}\\\ \psi^{-}_{R}\end{pmatrix}=\frac{1}{\sqrt{2}}\begin{pmatrix}r_{p}\exp(+ik_{y}\triangle_{H})&ir_{s}\exp(+ik_{y}\triangle_{V})\\\ r_{p}\exp(-ik_{y}\triangle_{H})&-ir_{s}\exp(-ik_{y}\triangle_{V})\end{pmatrix}\begin{pmatrix}\psi^{H}_{I}\\\ \psi^{V}_{I}\end{pmatrix},$ (6) where $\displaystyle\triangle_{H}$ $\displaystyle=\frac{\cot{\theta_{i}}}{k_{0}}(1+\frac{r_{s}}{r_{p}}),$ $\displaystyle\triangle_{V}$ $\displaystyle=\frac{\cot{\theta_{i}}}{k_{0}}(1+\frac{r_{p}}{r_{s}}).$ (7) Then the shift can be obtained by using a position operator $\mathbf{r}=i\hbar\partial_{\mathbf{p}}=i\partial_{\mathbf{k}}$ to calculate an average value of $y$ position of the reflected beam as $\delta^{\pm}=\text{Re}\frac{\expectationvalue{i\partial_{k_{y}}}{\psi_{R}^{\pm}\psi_{0}}}{\bra{\psi_{R}^{\pm}\psi_{0}}\ket{\psi_{R}^{\pm}\psi_{0}}}.$ (8) If the polarization state of the incidence is either horizontal ($\psi_{I}^{H}=1,\psi_{I}^{V}=0$) or vertical ($\psi_{I}^{H}=0,\psi_{I}^{V}=1$), then $\psi_{R}^{\pm}$ are eigenstates of the position operator $i\partial_{k_{y}}$ with eigenvalues of $\mp\triangle_{H}$ and $\mp\triangle_{V}$ respectively. On the other hand, $i\partial_{k_{y}}\psi_{0}=-\psi_{0}k_{y}\omega_{0}^{2}/2$, and this odd function has no contribution in the indefinite integral. Therefore, the shift can be obtained by taking the eigenvalues of $\psi_{R}^{\pm}$, so Eq. 8 gives the well-known formulas 33, 6 $\displaystyle\delta_{H}^{\pm}$ $\displaystyle=\mp\frac{\cot{\theta_{i}}}{k_{0}}\text{Re}(1+\frac{r_{s}}{r_{p}}),$ $\displaystyle\delta_{V}^{\pm}$ $\displaystyle=\mp\frac{\cot{\theta_{i}}}{k_{0}}\text{Re}(1+\frac{r_{p}}{r_{s}}).$ (9) Under an incidence that is polarized neither horizontally nor vertically, the shift should be calculated by substituting $\psi_{R}^{\pm}$ (Eq. 6) into Eq. 8 and by performing integration in momentum space. Interestingly, when the Fresnel coefficients of the two linear polarizations are equal ($r_{s}=r_{p}\equiv r$), the two equations in Eq. 7 are degenerate ($\triangle_{H}=\triangle_{V}\equiv\triangle$), then $\psi_{R}^{\pm}$ become eigenstates of the operator $i\partial_{k_{y}}$ with eigenvalues of $\mp\triangle$ for any incident polarization. Consequently, similarly to the instance under a horizontally or vertically polarized incidence, Eq. 8 reduces to $\delta^{\pm}=\mp\text{Re}(\triangle)$, which is equivalent to Eq. 9 but under an arbitrarily polarized incidence. This formula of the shift contains neither $\psi^{H}_{I}$ nor $\psi^{V}_{I}$ but only depends on $\theta_{i}$. The independence of $\delta^{\pm}$ with respect to the incident polarization clearly shows that an arbitrarily polarized incidence is shifted by the same distance, regardless of the polarization states of the incidence. Another important attribute of the SHEL that originates from $r_{s}=r_{p}$ is that the splitting occurs symmetrically in shift ($\lvert\delta^{+}\rvert=\lvert\delta^{-}\rvert$) under an arbitrarily polarized incidence. This result is opposed to the conventional wisdom that a circularly or elliptically polarized incidence is split asymmetrically into LCP and RCP 26, 27, 28, 29, 30. Although the condition of $r_{s}=r_{p}$ seems to imply the polarization- independent reflection, it is not true. Instead, $r_{s}=r_{p}$ is associated with the $\pi$ phase shift between the electric fields of $s$\- and $p$-polarized reflected beam because of the sign convention 4. Polarization- independent reflection ($r_{s}=-r_{p}$) is another possible solution of $\triangle_{H}=\triangle_{V}$, but is excluded because it leads to zero shift. Here we consider only the reflection type of the SHEL for demonstration but this scheme is also applicable to the transmission type by matching $t_{s}=-t_{p}$. ### 2.2 The intensities of circularly polarized components of the reflected beam under arbitrarily polarized incidence To understand how the SHEL occurs systematically, not only the shifts but also the intensities of the LCP and RCP components of the reflected beam should be considered. For a given incident polarization, the intensities can be calculated as $I_{R}^{\pm}=\frac{\bra{\psi_{R}^{\pm}\psi_{0}}\ket{\psi_{R}^{\pm}\psi_{0}}}{\bra{\psi_{I}^{H}\psi_{0}}\ket{\psi_{I}^{H}\psi_{0}}+\bra{\psi_{I}^{V}\psi_{0}}\ket{\psi_{I}^{V}\psi_{0}}}.$ (10) Considering that both $\psi_{I}^{H}$ and $\psi_{I}^{V}$ are complex, the intensities of the two circularly polarized beams are generally not symmetrical ($I_{R}^{+}\neq I_{R}^{-}$). Especially, when $r_{s}=r_{p}$, Eq. 10 can be simplified to $I_{R}^{\pm}=\lvert r\rvert^{2}\Big{[}\frac{1}{2}\pm\lvert\psi_{I}^{H}\rvert\lvert\psi_{I}^{V}\rvert\sin\Big(\text{arg}(\psi_{I}^{H})-\text{arg}(\psi_{I}^{V})\Big{missing})\Big{]}.$ (11) This result shows that even when $r_{s}=r_{p}$ is satisfied, $I_{R}^{+}=I_{R}^{-}$ if the incidence is linearly polarized, and $I_{R}^{+}\neq I_{R}^{-}$ otherwise. It also conforms to our intuition in that only linearly polarized light is a superposition of equal amounts of LCP and RCP whereas the others such as elliptically and circularly polarized light are not. ### 2.3 The SHEL under arbitrarily polarized incidence and unpolarized incidence Figure 1: Schematics of the SHEL under various polarization states of incidence in a general case ($r_{s}\neq r_{p}$). (a) The SHEL in real space and (b) corresponding intensity profiles of the reflected beam along the transverse axis when the incidence is horizontally or vertically polarized. The two circularly polarized reflected beams are symmetrical in shift ($\lvert\delta^{+}\rvert=\lvert\delta^{-}\rvert$) and intensity ($I_{R}^{+}=I_{R}^{-}$). (c) The SHEL in real space and (d) corresponding intensity profiles of the reflected beam when the incidence has an arbitrary polarization. Both symmetries are preserved under linear polarization but are broken ($\lvert\delta^{+}\rvert\neq\lvert\delta^{-}\rvert$ and $I_{R}^{+}\neq I_{R}^{-}$) under circularly or elliptically polarized incidence. For clear visualization, only the central wave vector component of the incident and reflected beams are plotted, instead of the wave packets with finite beam waist. The width of the central wave vectors of the reflected beams shown in Fig. 1a and c represent their intensities. Using Eq. 8 and Eq. 11, which provide the shift and intensities of the circularly polarized components of the reflected beam respectively, we now illustrate how the SHEL occurs under a given incident polarization. Firstly, a general case of $r_{s}\neq r_{p}$ is considered. Under a horizontally or vertically polarized incidence, as is well-known from many previous publications 6, 33, 34, the reflected beam is split in half into LCP and RCP ($I_{R}^{+}=I_{R}^{-}$), which undergoes the equal amount of the shift along the opposite direction ($\lvert\delta^{+}\rvert=\lvert\delta^{-}\rvert$) (Fig. 1a and b). This symmetrical splitting originates from the high symmetries of the incident polarization state and maintains under other linear polarization. In contrast, under a circularly or elliptically polarized incidence, the polarization breaks the symmetries and thus both the shift and intensity are asymmetrical ($\lvert\delta^{+}\rvert\neq\lvert\delta^{-}\rvert$ and $I_{R}^{+}\neq I_{R}^{-}$) (Fig. 1c and d). Figure 2: Schematics of the SHEL for various polarization states of incidence when $r_{s}=r_{p}$. (a) The SHEL in real space and (b) corresponding intensity profiles of the reflected beam along the transverse axis under arbitrarily polarized incidence. The shift is incident-polarization-independent and is symmetrical ($\lvert\delta^{+}\rvert=\lvert\delta^{-}\rvert$). Intensity is symmetrical ($I_{R}^{+}=I_{R}^{-}$) only under a linear polarization, and is asymmetrical ($I_{R}^{+}\neq I_{R}^{-}$) under a circularly or elliptically polarized incidence. (c) The SHEL in real space and (d) corresponding intensity profiles of the reflected beam under unpolarized incidence. Both the shift and intensity are symmetrical ($\lvert\delta^{+}\rvert=\lvert\delta^{-}\rvert$ and $I_{R}^{+}=I_{R}^{-}$). For clear visualization, only the central wave vector component of the incident and reflected beams are plotted, instead of the wave packets with finite beam waist. The width of the central wave vectors of the reflected beams shown in Fig. 2a and c represent their intensities. In contrast, if $r_{s}=r_{p}$, the shift becomes symmetrical ($\lvert\delta^{+}\rvert=\lvert\delta^{-}\rvert$) under arbitrarily polarized incidence (Fig. 2a and b) as proved in the previous section. More importantly, any incidence is split by the same amount regardless of its polarization state. However, despite the degeneracy in the shift, the intensities of the LCP and RCP components are symmetrical ($I_{R}^{+}=I_{R}^{-}$) only under a linear polarization but are polarization-dependent and are asymmetrical ($I_{R}^{+}\neq I_{R}^{-}$) in a general circularly or elliptically polarized incidence. The intensities can be made symmetrical by using unpolarized incidence. Here, the unpolarized state refers to a superposition of a vast number of randomly polarized light. Then the total incidence has no phase difference between the two components of the Jones vector due to the randomness and hence the sine term in Eq. 11 averages to zero. Therefore, the unpolarized incidence is split in half into LCP and RCP ($I_{R}^{+}=I_{R}^{-}$), the shifts of which have the same magnitude ($\lvert\delta^{+}\rvert=\lvert\delta^{-}\rvert$) just as a horizontally or vertically polarized incidence does (Fig. 2c and d). On the other hand, when $r_{s}$ and $r_{p}$ have nonzero phase difference, the SHEL has asymmetrical intensity ($I_{R}^{+}\neq I_{R}^{-}$) under unpolarized incidence. This phenomenon occurs because when $r_{s}\neq r_{p}$, one can still obtain the formula of the intensities of the LCP and RCP components of the reflected beam akin to Eq. 11, but the sine term of that formula includes not only the phase difference between $\psi_{I}^{H}$ and $\psi_{I}^{V}$ but also that between $r_{s}$ and $r_{p}$. It prevents $I_{R}^{\pm}$ from being degenerate, resulting in asymmetrical intensities under unpolarized incidence. ### 2.4 Numerical demonstration of the incident-polarization-independent SHEL To confirm the incident-polarization-independent SHEL numerically, a Monte Carlo simulation is performed by generating $N$ numbers of randomly oriented polarization states, then using each of them as an incidence. We consider a paraxial regime in which the polarization is defined in a two-dimensional sense. The $n$-th incidence ($n\in\\{1,2,...,N\\}$) has a well-defined polarization that corresponds to the Stokes parameters ($S_{1}^{(n)},S_{2}^{(n)},S_{3}^{(n)}$) distributed randomly on a Poincaré sphere (Fig. 3a, left), and have a degree of paraxial two-dimensional polarization equal to unity ($P_{2D}^{(n)}=\sqrt{\sum_{i=1}^{3}\big{(}S_{i}^{(n)}\big{)}^{2}}/S_{0}^{(n)}=1$). The averaged Stokes parameters ($\sum_{n=1}^{N}{S_{i}^{(n)}/N}$ for $i=1,2,3$) converge to zero as $N$ increases and are all less than $5\times 10^{-3}$ when $N=1000$ (Fig. 3a, right). The total incidence satisfies the condition of two- dimensional unpolarized light, that is 31, $(S_{0},S_{1},S_{2},S_{3})\propto(1,0,0,0)$ and $P_{2D}=0$. Figure 3: The SHEL under arbitrarily polarized incidence. (a) Poincaré sphere representing $N$ randomly distributed polarization states of incidence (left) and the averaged Stokes parameters (right). (b) Amplitude and (c) phase of $r_{s}$ and $r_{p}$ at an interface between an isotropic ($\varepsilon_{1}=2$) and anisotropic ($\varepsilon_{2x}=\varepsilon_{2z}=4.1,\varepsilon_{2y}=1$) media. Yellow dashed lines indicate $\theta_{0}$, an incident angle at which $r_{s}=r_{p}$ is satisfied. (d, e) Scatter plots of (d) $\delta^{+(n)}/\lambda$ and (e) $\delta^{-(n)}/\lambda$, where $\lambda$ is the wavelength, under $N$ independent incidences at the interface between the isotropic and anisotropic media. Solid curves represent $\delta_{H,V}^{\pm}/\lambda$ obtained by Eq. 9. (f) Standard deviation of the $N$ values of $\delta^{\pm(n)}/\lambda$ at each $\theta_{i}$. For simulation, $N=1000,w_{0}=10^{3}\lambda,z_{R}=10\lambda$ are used. To investigate the polarization dependency of the SHEL under arbitrarily polarized incidence, an interface between an isotropic ($\varepsilon_{1}$) and anisotropic ($\varepsilon_{2x}=\varepsilon_{2z}\neq\varepsilon_{2y}$) media is considered. Whereas $r_{s}$ and $r_{p}$ are coupled to each other by Fresnel equations 4 at an interface between two isotropic media, $s$\- and $p$-polarized light experience the anisotropic medium, in which the optic axis lies perpendicular to the incident plane, independently. When $\varepsilon_{2x}=\varepsilon_{2z}>\varepsilon_{1}>\varepsilon_{2y}$, light propagating from the isotropic medium to the anisotropic one has degenerate Fresnel coefficients for $s$ and $p$ polarization ($r_{s}=r_{p}$) at a certain angle. This attribute originates from the occurrence of total internal reflection under the $s$-polarized light and the resultant $\pi$ phase shift in the Fresnel coefficients. We use the material parameters given as $\varepsilon_{1}=2,\varepsilon_{2x}=\varepsilon_{2z}=4.1,\varepsilon_{2y}=1$. In such a case, $r_{s}=r_{p}$ is satisfied at a certain angle defined as $\theta_{0}$ (Fig. 3b and c). A feasible design of such a medium and details of the reflection at the boundary of an anisotropic medium can be found in Supporting Information. Because the analytic formula of the shift is known explicitly only for a horizontally or vertically polarized incidence, the shift under arbitrarily polarized incidence is obtained by taking the average $y$ position of the reflected beam profile numerically. Thus, we first obtain the spatial distribution of the reflected Gaussian beam. By combining Eq. 1-5 and then applying a Taylor expansion to reflection coefficients $r_{p,s}\approx r_{p,s}(\theta_{i})+\frac{k_{Ix}}{k_{0}}\frac{\partial r_{p,s}}{\partial\theta_{i}},$ (12) the field distribution of the reflected beam can be found in momentum space. Then the spatial distribution of the reflected beam can be obtained by applying a Fourier transform $\tilde{\psi}^{\pm}_{R}(x_{R},y_{R},z_{R})=\iint{dk_{Rx}dk_{Ry}\psi^{\pm}_{R}(k_{Rx},k_{Ry})\psi_{0}\exp\big(i(k_{Rx}x_{R}+k_{Ry}y_{R}+k_{Rz}z_{R})\big{missing})}.$ (13) Each circularly polarized component of a reflected beam under a horizontally and vertically polarized incidence has a spatial profile of $\displaystyle\tilde{\psi}^{\pm}_{R,H}=$ $\displaystyle\frac{1}{\sqrt{2\pi}\omega_{0}}\frac{z_{0}}{z_{0}+iz_{R}}\exp(-\frac{k_{0}}{2}\frac{x_{R}^{2}+y_{R}^{2}}{z_{0}+iz_{R}})$ $\displaystyle\times\Big{[}r_{p}-i\frac{x_{R}}{z_{0}+iz_{R}}\dot{r_{p}}\mp\frac{y_{R}\cot{\theta_{i}}}{z_{0}+iz_{R}}(r_{p}+r_{s})\mp i\frac{x_{R}y_{R}\cot{\theta_{i}}}{(z_{0}+iz_{R})^{2}}(\dot{r_{p}}+\dot{r_{s}})\Big{]}\exp(ik_{r}z_{R}),$ $\displaystyle\tilde{\psi}^{\pm}_{R,V}=$ $\displaystyle\frac{\pm i}{\sqrt{2\pi}\omega_{0}}\frac{z_{0}}{z_{0}+iz_{R}}\exp(-\frac{k_{0}}{2}\frac{x_{R}^{2}+y_{R}^{2}}{z_{0}+iz_{R}})$ $\displaystyle\times\Big{[}r_{s}-i\frac{x_{R}}{z_{0}+iz_{R}}\dot{r_{s}}\mp\frac{y_{R}\cot{\theta_{i}}}{z_{0}+iz_{R}}(r_{p}+r_{s})\mp i\frac{x_{R}y_{R}\cot{\theta_{i}}}{(z_{0}+iz_{R})^{2}}(\dot{r_{p}}+\dot{r_{s}})\Big{]}\exp(ik_{r}z_{R}),$ (14) where the second subscript corresponds to incident polarization, the dot notation indicates the first derivative with respect to $\theta_{i}$, and $z_{0}=k_{0}\omega_{0}^{2}/2$ is the Rayleigh length. These four field distributions correspond to the elements of a matrix that transforms the Jones vector of the $n$-th incident polarization to the $n$-th reflected beam profile, i.e., $\tilde{\psi}^{\pm(n)}_{R}=\tilde{\psi}^{\pm}_{R,H}\psi_{I}^{H(n)}+\tilde{\psi}^{\pm}_{R,V}\psi_{I}^{V(n)}.$ (15) Eq. 15 provides the spatial profiles of the two circularly polarized components of the reflected beam for each of the given incident polarization. We conduct $N$ independent calculations with parameters given as: $N=1000,w_{0}=10^{3}\lambda,z_{R}=10\lambda$ where $\lambda$ is the wavelength. In the simulation, $\lambda$ is set as 600 nm but has no influence on the results. The shifts that each of the given incidence undergoes are obtained by calculating beam centroids of the reflected beam as $\delta^{\pm(n)}=\frac{\expectationvalue{y_{R}}{\tilde{\psi}_{R}^{\pm(n)}}}{\bra{\tilde{\psi}_{R}^{\pm(n)}}\ket{\tilde{\psi}_{R}^{\pm(n)}}},$ (16) and are plotted as black dots in Fig. 3d and e, after normalized by $\lambda$. Because the shift depends on the incident polarization, the scatter plots exhibit many different $\delta^{\pm(n)}/\lambda$ at a fixed $\theta_{i}$. However, at $\theta_{0}$ at which $r_{s}=r_{p}$ is satisfied and thus $\delta_{H}^{\pm}=\delta_{V}^{\pm}$, the values of $\delta^{\pm(n)}/\lambda$ all appear at the intersection point regardless of the incident polarization (Fig. 3d and e, yellow dashed lines). The convergence of the shift for all incident polarization is also confirmed by the zero standard deviation of $\delta^{\pm(n)}/\lambda$ at $\theta_{0}$ (Fig. 3f). The standard deviation is nonzero at all $\theta_{i}$ other than $\theta_{0}$. The large standard deviation near the Brewster angle results from the diverging shift under horizontally polarized incidence. In several consecutive Monte Carlo simulations in which the $N$ polarization states are randomly redistributed, the fine details of the scatter plot in Fig. 3d and e change, but the overall tendency remains unaltered. Besides $\theta_{0}$, there exists another intersection of $\delta^{\pm}_{H,V}/\lambda$ near 60∘, but the shift is incident-polarization-dependent because $\triangle_{H}\neq\triangle_{V}$ at this angle and hence $\psi_{R}^{\pm}$ are not eigenstates of $i\partial_{k_{y}}$. Figure 4: Shift distribution represented in a Poincaré sphere when (a) $\theta_{i}=\theta_{0}$, (b) $\theta_{i}=20^{\circ}\neq\theta_{0}$, and (c) $\theta_{i}=40^{\circ}\neq\theta_{0}$. The location of a point on the sphere manifests the polarization states of the $n$-th incidence; the color represents $\delta^{\pm(n)}/\lambda$. Top and bottom corresponds to the shift of LCP and RCP components respectively. The boundary of the sphere is omitted for clarity. Fig. 3d and e show the distributions of the shift for a given $\theta_{i}$, but do not provide the correspondence between the polarization state and the shift. Thus, we examine the distribution of $\delta^{\pm(n)}/\lambda$ in Poincaré sphere for three different $\theta_{i}$ (Fig. 4). The location of a point on the sphere manifests the polarization states of the $n$-th incidence, and the color represents $\delta^{\pm(n)}/\lambda$. As confirmed by Fig. 3d-f, the shift under an arbitrarily polarized incidence are all degenerate and are also symmetrical for LCP and RCP components when $\theta_{i}=\theta_{0}$ (Fig. 4a). However, at other $\theta_{i}=20^{\circ}\neq\theta_{0}$, the shift depends on the incident polarization, showing variations of the shift over the sphere (Fig. 4b). This tendency is more noticeable under circular and elliptical polarization than under linear polarization. In contrast, at $\theta_{i}=40^{\circ}$, incidence that has Stokes parameters near $(1,0,0)$ produce shifts with large deviation (Fig. 4c). This trend occurs because of the diverging shift under the horizontally polarized light near the Brewster angle. Furthermore, the shift distribution at $\theta_{i}\neq\theta_{0}$ shows that the splitting is not symmetrical ($\lvert\delta^{+(n)}\rvert\neq\lvert\delta^{-(n)}\rvert$) under arbitrarily polarized incidence if $r_{s}\neq r_{p}$ (Fig. 4b and c). ### 2.5 Numerical demonstration of the SHEL under unpolarized incidence Figure 5: Field profiles of the reflected Gaussian beam. Spatial distributions of (a) $\lvert\tilde{\psi}_{R}^{+}\rvert^{2}$, (b) $\lvert\tilde{\psi}_{R}^{-}\rvert^{2}$, and (c) their difference $\lvert\tilde{\psi}_{R}^{-}\rvert^{2}-\lvert\tilde{\psi}_{R}^{+}\rvert^{2}$ at $\theta_{0}$. (d) Spatial distribution of $\lvert\tilde{\psi}_{R}^{-}\rvert^{2}-\lvert\tilde{\psi}_{R}^{+}\rvert^{2}$ at $\theta_{i}=50^{\circ}\neq\theta_{0}$. (e, f) Intensity profiles in an arbitrary unit (A.U.) of the two circularly polarized components of the reflected beam (blue: $\lvert\tilde{\psi}_{R}^{+}\rvert^{2}$, red: $\lvert\tilde{\psi}_{R}^{-}\rvert^{2}$) and their difference (yellow) along the transverse axis at $x_{R}=0$ at (e) $\theta_{0}$ and (f) $\theta_{i}=50^{\circ}\neq\theta_{0}$. For better visualization, $\lvert\tilde{\psi}_{R}^{-}\rvert^{2}-\lvert\tilde{\psi}_{R}^{+}\rvert^{2}$ are exaggerated in Fig. 5e and f. Because of the linearity of Eq. 15, the total reflected beam under unpolarized incidence can be obtained by taking a superposition of all $\tilde{\psi}^{\pm(n)}_{R}$ for a set of given incident polarizations: $\tilde{\psi}^{\pm}_{R}=\sum_{n=1}^{N}\tilde{\psi}^{\pm(n)}_{R}/N$. For completeness, the intensity distributions of the two circularly polarized components of the total reflected beam and their difference are presented in Fig. 5. The splitting of the LCP and RCP is not readily apparent (Fig. 5a and b), but their intensity difference, which is proportional to $S_{3}$, shows the symmetric and spin-dependent shift along the transverse direction at $\theta_{0}$ (Fig. 5c). This SHEL under unpolarized incidence can be understood as a result of the superposition of the spin-dependent splittings under $N$ differently polarized incidences, in which the magnitude of the splittings are all degenerate. This field profile is obtained under unpolarized incidence but resembles that under a horizontally or vertically polarized incidence 35, 13. In contrast, at $\theta_{i}=50^{\circ}\neq\theta_{0}$, this splitting exhibits significantly distinct features (Fig. 5d) for the following reasons. Firstly, the shift is incident-polarization-dependent; the $N$ differently polarized incidences are split by different amounts of displacement, then superposed. Secondly, the $\lvert\tilde{\psi}_{R}^{-}\rvert^{2}-\lvert\tilde{\psi}_{R}^{+}\rvert^{2}$ distribution of a single sign is attributed to the asymmetrical splitting of intensity ($I_{R}^{+}\neq I_{R}^{-}$) as a result of the phase difference between $r_{s}$ and $r_{p}$. If the shift is not large enough, the asymmetry in intensity leads to a single sign of $S_{3}$ distribution (Fig. 5d). For better understanding, the intensities along $x_{R}=0$ are shown in Fig. 5e and f. Whereas the unpolarized incidence is split symmetrically into two circularly polarized components at $\theta_{0}$ (Fig. 5e), the SHEL at $\theta_{i}\neq\theta_{0}$ is asymmetrical in both shift and intensity (Fig. 5f). ## 3 Conclusion The SHEL is known as a symmetrical splitting of a refracted or reflected beam into two circularly polarized beams along the transverse axis at an optical interface, but such symmetries only appear when the incidence is horizontally or vertically polarized. Here we show that the two circularly polarized components of the reflected beam undergo incident-polarization-independent and symmetrical splitting under an arbitrary incident polarization if $r_{s}=r_{p}$. The polarization independence of the shift is proved theoretically, then numerically using a Monte Carlo simulation. The intensities of the circularly polarized components of the reflected beams are symmetrical under unpolarized incidence, thereby appearing exactly the same as under horizontally or vertically polarized incidence. Lastly, an interface at which $r_{s}=r_{p}$ is suggested by adjoining isotropic and anisotropic dielectric media. In contrast to the previous research on the SHEL in which a well-defined polarization state is preliminary, the incident-polarization- independent SHEL can widen the boundaries and possible applications of the SHEL to cover unpolarized or ill-defined polarized sources. ## 4 Method ### 4.1 Monte Carlo simulation For the random polarization states, the elements of the Jones vector of the $n$-th incidence are defined as $\psi_{I}^{H,V(n)}=\lvert\psi_{I}^{H,V(n)}\rvert\exp(i\phi^{H,V(n)})$ where the magnitude $\lvert\psi_{I}^{H,V(n)}\rvert$ and the phase $\phi^{H,V(n)}$ are real scalars that are randomly chosen in $(0,1)$ and $(-\pi,\pi)$ respectively for $n\in\\{1,...,N\\}$. Then the Jones vector $\begin{pmatrix}\psi_{I}^{H(n)}&\psi_{I}^{V(n)}\end{pmatrix}^{T}$ is normalized by its magnitude $\sqrt{\lvert\psi_{I}^{H(n)}\rvert^{2}+\lvert\psi_{I}^{V(n)}\rvert^{2}}$. ### 4.2 Reflection coefficients at an interface between isotropic and anisotropic media At an interface between the isotropic (medium 1, permittivity $\varepsilon_{1}$) and anisotropic (medium 2, permittivity $\varepsilon_{2}=\text{diag}(\varepsilon_{2x},\varepsilon_{2y},\varepsilon_{2z})$) media, the Fresnel reflection coefficients can be obtained by solving Maxwell’s equations: $\displaystyle r_{s}=$ $\displaystyle\frac{\sqrt{\varepsilon_{1}-\beta^{2}}-\sqrt{\varepsilon_{2y}-\beta^{2}}}{\sqrt{\varepsilon_{1}+\beta^{2}}+\sqrt{\varepsilon_{2y}-\beta^{2}}},$ $\displaystyle r_{p}=$ $\displaystyle\frac{\sqrt{\varepsilon_{1}-\beta^{2}}/\varepsilon_{1}-\sqrt{\varepsilon_{2x}-\beta^{2}\varepsilon_{2x}/\varepsilon_{2z}}/\varepsilon_{2x}}{\sqrt{\varepsilon_{1}-\beta^{2}}/\varepsilon_{1}+\sqrt{\varepsilon_{2x}-\beta^{2}\varepsilon_{2x}/\varepsilon_{2z}}/\varepsilon_{2x}}$ (17) where $\beta=\sqrt{\varepsilon_{1}}\sin\theta_{i}$ is the propagation constant. ## References * 1 A. 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# Strongly coupled Yukawa plasma layer in a harmonic trap Hong Pan Department of Physics, Boston College, Chestnut Hill, Massachusetts, 02467, USA Gabor J. Kalman Department of Physics, Boston College, Chestnut Hill, Massachusetts, 02467, USA Peter Hartmann Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O.Box. 49, H-1525 Budapest, Hungary ###### Abstract Observations made in dusty plasma experiments suggest that an ensemble of electrically charged solid particles, confined in an elongated trap, develops structural inhomogeneities. With narrowing the trap the particles tend to form layers oriented parallel with the trap walls. In this work we present theoretical and numerical results on the structure of three-dimensional many- particle systems with screened Coulomb (Yukawa) inter-particle interaction in the strongly coupled liquid phase, confined in one-dimensional harmonic trap, forming quasi-2D configurations. Particle density profiles are calculated by means of the hypernetted chain approximation (HNC), showing clear signs of layer formation. The mechanism behind the formation of layer structure is discussed and a method to predict the number of layers is presented. Molecular dynamics (MD) simulations provide validation of the theoretical results and detailed microscopic insights. ###### pacs: 52.27.Gr, 52.27.Lw Strongly coupled charged particle systems (e.g. plasmas), where the inter- particle interaction energy exceeds the thermal kinetic energy of the constituent particles, can often be approximated by the one component plasma (OCP) model [1]. In cases when the dynamics of one dominant particle species decouples from that of the other components, it might be sufficient to provide a detailed description only for the former, while approximating the contribution of rest with a continuous neutralizing background. In the case of warm dense matter, for instance, the ions can follow classical particle trajectories, while the electrons experience quantum degeneracy and realize a homogeneous background [2, 3]. In the case of dusty plasmas it is the very different time scales and charge states that decouple the dust dynamics (with characteristic times in the order of 10 ms and $10^{4}$ elementary charges per dust) from the microscopic interactions with the electrons and ions present in the gas discharge (with typical times in the ns to $\mu$s range and unit charges). In both cases, only one of the plasma components (the ions in warm dens matter, and the dust particles in dusty plasmas) needs to be traced explicitly, the contribution of the background is reflected in the particular shape of the inter-particle interaction. If the polarizability of the isotropic neutralizing background is taken into account, the system can be approximated by the Yukawa one component plasma (YOCP) model. In this case the electrostatic pairwise interaction between the particles becomes screened, which screening can be approximated by the Dedye-Hückel mechanism resulting in an exponential decay superimposed to the bare Coulomb potential. Since the pioneering experiments published in 1994 [4, 5, 6] laboratory dusty plasmas have been extensively used to gain insight into the microscopic details of macroscopic and collective phenomena. In large area radio frequency (RF) discharges, using micrometer sized monodisperse spherical dust particles it is easy to form a single layer of highly charged dust grains. This ensemble can form large (consisting of tens of thousands of dust grains) structures with the particles ordered in triangular lattices. Simply by changing discharge conditions (e.g. the RF power) or by other forms of external energy coupling (e.g. laser heating [7] or DC pulsing [8]) the system can undergo a solid to liquid transition and stabilize in the strongly coupled liquid state. However, the apparent single layer structure does not mean that the system is strictly two-dimensional. The vertical confinement is defined by the interplay of gravity and the electric field in the RF sheath, ultimately forming a potential well experienced by each dust particle. Both the vertical equilibrium position and the effective trap frequency depend on the charge-to- mass ($q/m$) ratio of the dust grains and the discharge conditions. At finite temperature the dust particles experience small amplitude vertical oscillations, forming quasi-2D configurations. Early experimental investigations on quasi-2D system have been published by Lin I [9, 10] observing anomalous diffusion, as well as layering and slow dynamics. Theoretical work on quasi-2D systems include studies of in-plane and out-of-plane polarized wave propagation and structural transitions into multi- layered configurations [11, 12, 13, 14, 15] building heavily on numerical simulations. An analytical approach targeting the particle density distribution in the trap would provide deeper insight into the double or triple layer formation, as observed in the experiments. A similar problem has been studied by Wrighton [16, 17, 18] for spherical confinement comparing mean field and hypernetted chain calculations to numerical results. In Klumov’s work [19], crystallization of quasi-2D system under both harmonic and hard wall confinement was studied, in which the wetting phenomenon was observed in the hard wall case. Further theoretical studies on radially, as well as linearly confined liquids [20, 21, 22, 23, 24, 25] include the prediction of the radial density distribution, mechanisms of oscillatory density profiles and ensuing solvation forces, in both the weak and strong coupling regimes. In this work, we study the evolution of the density profile of quasi-2D Yukawa systems in a one-dimensional harmonic trap in the strongly coupled liquid state. The system is infinite in the $x$ and $y$ directions, where the particles can move freely. A harmonic trapping potential is applied in the $z$ direction. Let $n_{\rm s}$ denote the surface density of the quasi-2D layer projected onto the $x$-$y$ plane. In this case we define $a$ as the Wigner- Seitz radius in the projected plane as $a^{2}=1/(\pi n_{\rm s})$. The Yukawa interaction potential energy and the harmonic trap potential are, respectively $\varphi(r)=q^{2}\frac{e^{-\kappa\,r/a}}{r},~{}~{}~{}~{}~{}{\rm and}~{}~{}~{}V(z)=\frac{m\omega_{\rm t}^{2}z^{2}}{2},$ (1) where $r$ is the three-dimensional inter-particle distance, $\kappa$ is the dimensionless Yukawa screening parameter, $\omega_{\rm t}$ is the trap frequency, $q$ and $m$ are the electric charge and mass, equal for all particles. The strength of the electrostatic coupling can be characterized with the Coulomb coupling parameter, nominally expressing the ratio of the potential to kinetic energies per particle, and is defined as $\Gamma=\beta q^{2}/a,$ (2) where the thermodynamic $\beta=1/(k_{\rm B}T)$. For Yukawa systems an effective coupling $\Gamma_{\rm eff}(\kappa)<\Gamma$ [12, 26] can be introduced, that depends on the Yukawa screening parameter a does more accurately characterize the state of a particular system. The strongly coupled domain is defined by $\Gamma\gg 1$. Using the nominal 2D plasma frequency $\omega_{\rm p}^{2}=2\pi q^{2}n_{s}/(ma)$ to define the frequency and time units we introduce the dimensionless parameter $t$ characterizing the strength of the trapping potential, as $t^{2}=\frac{\omega_{\rm t}^{2}}{\omega_{\rm p}^{2}}=\frac{m\omega_{\rm t}^{2}a}{2\pi q^{2}n_{\rm s}}=\frac{m\omega_{\rm t}^{2}a^{3}}{2q^{2}}.$ (3) Within the frame of the quasi-2D YOCP model the behavior of a system is fully determined by the three dimensionless parameters $\Gamma$, $\kappa$ and $t$. Both the equilibrium properties and the dynamics of the system have been studied by molecular dynamics (MD) simulation and by theoretical analysis. Here we report on the equilibrium studies. Our MD simulations trance the trajectories of 10 000 particles in an external trapping potential defined by $V(z)$ and periodic boundary conditions in $x$ and $y$ in a cubic simulation domain. Initial positions are assigned to the particles based on a simple initial barometric estimate. Inter-particle forces are summed for all particles within a radius of $R\approx 44a$ for each particle. Before conducting any measurements the system is given enough time (approximately of 4 000 plasma oscillation cycles) to reach equilibrium during the initial thermalization phase using the velocity back-scaling thermostat. This is verified by observing the temperature stability after the thermaization is turned off. During the measurements data is collected and averaged over approx. 2 000 plasma oscillation cycles (100 000 time-steps). In order to obtain a theoretical description of the density distribution within the trap we invoke the density functional theory (DFT) [27, 28]. By minimizing the grand potential, a general, but quite complex expression for $n(r)$, the 3D number density has been reported by Evans [28] (eq. 26 in chapter 3), see also the Appendix of [16] in terms of $c\left(r,r^{\prime};[n(r)]\right)$, the direct correlation function (DCF) of the system. The notation emphasizes that the DCF is a unique, albeit unknown functional of the density profile. $c(r,r^{\prime})$ is also connected with $h(r,r^{\prime})$, the pair correlation function through the Ornstein-Zernike equation (OZ). $h(r,r^{\prime})=c(r,r^{\prime})+\int c(r,r^{\prime\prime})n(r^{\prime\prime})h(r^{\prime\prime},r^{\prime})~{}{\rm d}r^{\prime\prime}$ (4) The latter is related to the pair distribution function through $g(r,r^{\prime})=h(r,r^{\prime})+1$. The density profile is determined by the self-consistency relation [27] $n(r)\propto\exp\left[-\beta V(r)+\int n(r^{\prime})c_{0}(r-r^{\prime};n)~{}{\rm d}r^{\prime}\right],$ (5) where $c_{0}$ is the DCF of a reference system with uniform density. The structure of eq. (5) tells us that $c(r,r^{\prime})$ plays the role of the effective interaction potential in the system (note that $\varphi(r)$ does not appear explicitly in eq. (5)), $-\beta\varphi_{\rm eff}(r-r^{\prime})=c_{0}(r-r^{\prime})$ (6) In order to solve $c$ and $h$ simultaneously, the OZ relation provides the first equation, the second relation is derived by applying the hypernetted chain (HNC) approximation [29, 27]. HNC has been successfully used in various problems relating to strongly coupled Coulomb and Yukawa systems [30]. The HNC approximation is based on the neglect of the so-called “bridge” (or irreducible, i.e. not derivable from the combined operations of “parallel connection” and “series connection” of Mayer diagrams) diagrams: their contribution is assumed to be negligible in the case of long range potentials. As a result, one obtains the general basic relationship $g(r,r^{\prime})=\exp\left[-\beta\varphi(r-r^{\prime})+h(r,r^{\prime})-c(r,r^{\prime})\right],$ (7) which in combination with the OZ equation (4), provides a solution for $h(r,r^{\prime})$ and $c(r,r^{\prime})$. Restricting the solutions to homogeneous and isotropic functions, DCF satisfies the simpler variant of the OZ equation $h(r)=c(r)+{\bar{n}}\int c(\|r-r^{\prime}\|)h(r^{\prime})~{}{\rm d}r^{\prime}.$ (8) In equation (7), when $r$ is large enough, $g(r)\rightarrow 1$, $h(r)\rightarrow 0$, the expression reduces to $-\beta\varphi_{(}r-r^{\prime})=c(r-r^{\prime})$. Using this asymptotic formula for the whole range of $r$, one arrives to the mean field (MF) approximation. Another self-consistent approach to calculating DCF for homogeneous fluids was described in [31]. Since the system is uniform in the $x$-$y$ plane, the density is non-uniform only along the $z$ direction, the parametrization can be simplified to $n(r)=n(z)$, with the normalization condition $n_{s}=\int n(z)~{}{\rm d}z$. The integral in equation (5) can be split into$z$ part and radial part separately. We can write eq. (5) as $n(z)\propto\exp\left[-\beta V(z)+2\pi\iint n(z^{\prime})c_{0}(z,z^{\prime},\rho^{\prime};n)\rho^{\prime}~{}{\rm d}\rho^{\prime}~{}{\rm d}z^{\prime}\right].$ (9) We can rewrite eq. (9) in terms of the dimensionless quantities $\tilde{n}=na^{2}$, $\tilde{z}=z/a$, $\tilde{U}(z)=\frac{\beta}{\Gamma}U(z)$, etc. as $\displaystyle\tilde{n}(\tilde{z})$ $\displaystyle=$ $\displaystyle\tilde{n_{s}}\frac{\exp[-\Gamma\tilde{U}(\tilde{z})]}{\int_{-\infty}^{\infty}\exp[-\Gamma\tilde{U}(\tilde{z})]~{}{\rm d}\tilde{z}}$ (10) $\displaystyle\tilde{U}(\tilde{z})$ $\displaystyle=$ $\displaystyle t^{2}\tilde{z}^{2}-\tilde{W}(\tilde{z})$ $\displaystyle\tilde{W}(\tilde{z})$ $\displaystyle=$ $\displaystyle\frac{2\pi}{\Gamma}\iint\tilde{n}(\tilde{z}^{\prime})c_{0}(\tilde{z},\tilde{z}^{\prime},\tilde{\rho}^{\prime};n)\tilde{\rho}^{\prime}~{}{\rm d}\tilde{\rho}^{\prime}~{}{\rm d}\tilde{z}^{\prime}$ $\rho$ being the 2D distance between two projected particle positions in the $x$-$y$ plane. Then, provided that $c_{0}(r-r^{\prime};n)$ is known one can obtain the density profile by the iterative solution of eq. (10). In view of eq. (6) the simplest approximation for $c_{0}(z-z^{\prime},\rho)$ is to ignore correlations and set $\varphi_{\rm eff}(r,r^{\prime})=\varphi(r-r^{\prime})$, i.e. $c_{0}(r-r^{\prime})=-\beta\varphi(r-r^{\prime})$ (11) This is tantamount to a mean field approximation. Substituting (11) into (10), (10) reduces to $\displaystyle U(z)=t^{2}z^{2}+2\iint n(z^{\prime})\frac{\exp[-\kappa d(z,z^{\prime},\rho^{\prime})]}{d(z,z^{\prime},\rho^{\prime})}\rho^{\prime}~{}{\rm d}\rho^{\prime}~{}{\rm d}z^{\prime}$ $\displaystyle d(z,z^{\prime},\rho^{\prime})=\sqrt{(z-z^{\prime})^{2}+\rho^{\prime 2}}.$ (12) In the sequel we drop the symbol for the dimensionless quantities. It should be kept in mind that all the length variables are in the unit of the Wigner- Seitz radius $a$. In the following, for simplicity, we only discuss the results for $\kappa=0.4$. The results of the MF calculation and their comparison with the results of the MD simulation are given in Fig. 1. If there is no particle-particle interaction, then $\Gamma\rightarrow 0$, the profile is Gaussian $n(z)\propto\exp[-\Gamma t^{2}z^{2}]$, mapping the Maxwell distribution of non-interacting particles. Proceeding to higher $\Gamma$ values, we expect the MF method to gradually fail to reproduce the numerical data as it is only valid for weak coupling (low $\Gamma$, i.e. low density or high temperature). Inspecting the MF profiles in Fig. 1, covering moderate and strong $\Gamma$ values, we observe that those reasonably match the MD results at low $\Gamma$ values, while fail spectacularly at high $\Gamma$-s. In particular MF does not provide the non-monotonic behavior related to layer formation. In fact, it has been proved analytically that the MF density profiles have to be monotonic on both sides of the maximum [32], and thus MF is unable to predict the formation of multiple layers. Figure 1: Density profile comparison between the MD simulation and the MF approximation (continuous line) for different $\Gamma$ and t values. (a) $\Gamma=8$, $t=0.2$, (b) $\Gamma=16$, $t=0.1$, (c) $\Gamma=64$, $t=0.2$, (d) $\Gamma=64$, $t=0.4$ Next we calculate the DCF from the HNC approximation, then derive the density profile. The $\bar{n}$ in eq. (8) is chosen as the average number density of the plasma between the two layer boundaries. The boundary is defined arbitrarily as the points where the density value is one percent of the maximum density in the layer. This, of course requires the knowledge of the density, whose determination is the purpose of the calculation. All this lends itself to an iteration scheme. The resulting protocol for the numerical calculation is portrayed in Fig. 2. The algorithm is based on work published in [33]. Figure 2: Numerical iteration loop for calculating density profile. Figure 3: Density profile comparison between MD simulation (continuous line) and the HNC approximation, for different trapping strengths. $\Gamma=32$ Figure 4: Density profile comparison between MD simulation (continuous line) and the HNC approximation, for different trapping strengths. $\Gamma=64$ The major improvement of the HNC over the MF calculation is that it correctly reproduces the splitting of the system into multiple layers. Figs. 3 and 4 show comparisons with MD data at moderately high coupling values for cases when multi-peak profiles form. The similar multi-peak profile was reported in previous works [34, 20, 23, 24, 35, 36, 37]. Generally, both $t$ and $\Gamma$ can affects the density profile formation, but the acting mechanisms are different. From Figs. 3 and 4 we can see that when $t$ decreases, the layer becomes wider, developing more peaks (layers) in the density profile. The coupling parameter $\Gamma$ only affect the density modulation amplitude. The relation between the trapping strength and the density profile was also studied in [38, 39]. MD results in Figs. 5 and 6 show density profiles for a set of $\Gamma$ values. With increasing $\Gamma$, the system develops very sharp density peaks with deep minima between the peaks, a sign of the formation of crystal-like ordering (no exchange of particle between layers). As $\Gamma$ decreases, the particles experience larger vertical oscillation amplitudes, and the sides of neighboring peaks in the density profile fuse. The position and the number of the density peaks is mostly independent of the coupling, the distribution in the liquid state resembles that of the solid with lower amplitude of the density modulation. \| 3 \| 3 \| 3 \| 4 \| 4 \| 5 \| 5 \| 6 \| 6 \| 7 \| peak number ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- 0.34 \| 0.3 \| 0.22 \| 0.2 \| 0.14 \| 0.12 \| 0.09 \| 0.08 \| 0.07 \| 0.06 \| t N \| $\sqrt{N\pi}$ \| 3.5 \| 4.2 \| 6 \| 7 \| 10 \| 12 \| 15 \| 17.5 \| 19 \| 22 \| layer width 2 \| 2.5066 \| 3.5 \| 4.2 \| 6 \| 7 \| 10 \| 12 \| 15 \| 17.5 \| 19 \| 22 \| $\frac{\mbox{layer width}}{N-1}$ 3 \| 3.0700 \| 1.75 \| 2.1 \| 3 \| 3.5 \| 5 \| 6 \| 7.5 \| 8.75 \| 9.5 \| 11 4 \| 3.5449 \| 1.1667 \| 1.4 \| 2 \| 2.3333 \| 3.3333 \| 4 \| 5 \| 5.8333 \| 6.3333 \| 7.3333 5 \| 3.9633 \| 0.875 \| 1.05 \| 1.5 \| 1.75 \| 2.5 \| 3 \| 3.75 \| 4.375 \| 4.75 \| 5.5 6 \| 4.3416 \| 0.7 \| 0.84 \| 1.2 \| 1.4 \| 2 \| 2.4 \| 3 \| 3.5 \| 3.8 \| 4.4 7 \| 4.6895 \| 0.5833 \| 0.7 \| 1 \| 1.1667 \| 1.6667 \| 2 \| 2.5 \| 2.9167 \| 3.1667 \| 3.6667 8 \| 5.0133 \| 0.5 \| 0.6 \| 0.8571 \| 1 \| 1.4286 \| 1.7143 \| 2.1429 \| 2.5 \| 2.7143 \| 3.1429 Table 1: Number of layers with different trapping strength, $\Gamma=512$, $\kappa=0.4$, the row with green color corresponds to the predicted $N$ value. Based on the idea that a multiple layers may be regarded as a slab extracted from a 3D lattice [40, 41, 42], one may determine the number of layers by a simple algorithm. Assuming there are $N$ layers, for simplicity a planar square lattice is associated to each layer. In this case the unit square side length is $x_{N}=\sqrt{\pi N}$. The inter-layer distance is $d_{N}=\frac{w}{N-1}$, where $w$ is the distance between the two outermost peaks. Comparing $x_{N}$ and $d_{N}$, the lowest value of $N$ at which $x_{N}>d_{N}$ is the prediction for the number of layers (see Table 1). In [40] the total energy of the system with different number of layers, including the particle interaction and the trapping energy is calculated. Despite operating with different physical quantities, the principles of the two methods are equivalent. In [43], the shell structure of the density profile in a confined colloidal particle system was studied by Monte Carlo simulation. Figure 5: Evolution of the density profiles with increasing $\Gamma$ values as determined by MD simulations. $t=0.1$ Figure 6: Evolution of the density profiles with increasing $\Gamma$ values as determined by MD simulations. $t=0.2$ Figure 7: HNC (lines) and MD (symbols) density profile comparison at high $\Gamma$. $\Gamma=300$. (a) $t=0.05$, (b) $t=0.07$. A similar configuration was studied in [22] applying both the MF and HNC formalism, however, as those results are restricted to the weak coupling regime no layer formation was reported. In that work the correlation energy, based on local density approximation (LDA), and with this the excess chemical potential was calculated. In anther study [44] the density profile of a polyelectrolyte system in a harmonic trap was calculated. A similar shell structure formation was observed. In [45] a similar DFT method was applied on hard-core Yukawa dusty plasma in a spherically harmonic trap. In [46] a study on van der Waals fluids with different confining potentials was presented including the modulation of the density profiles. In our calculations, relying on the HNC approximation, the stability of the solution scheme appears to be limited to $\Gamma\leq 300$ (for $\kappa=0.4$), at higher couplings no converged solution could be found. For this reason comparing the HNC and MD result at the strongest coupling is performed at $\Gamma=300$. Fig. 7 shows that even though the HNC result underestimates the density modulation amplitude, it predicts correctly the number and position of the peaks. In conclusion, in this paper, we have analyzed the density profile of quasi-2D Yukawa plasma in harmonic trap in the strongly coupled liquid regime. We utilize the HNC approximation to calculate the density profile, which matches MD simulation result very well at low to moderate couplings. At stronger coupling, near the solidification the HNC calculations underestimate the density modulation amplitudes. We have presented a method to predict the number of layers in the solid phase based on the comparison between the in- plane inter-particle distance and the inter-layer distance. We confirm the validity of this prediction in the liquid state as well. An outlook into future research directions include: (1) theoretical models beyond the standard HNC approximation could be applied to derive the direct correlation function $c(r)$; (2) a method for the more accurate incorporation of correlation effect could be worked out to extend applicability to higher $\Gamma$-s, where the system is in an intermediate state between liquid and solid phases; (3) application of the current model to anharmonic confinement potentials, like hard-wall traps. ###### Acknowledgements. The authors thank Jeff Wrighton’s help on numerical calculation in this paper. 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# Increasing Cluster Size Asymptotics for Nested Error Regression Models Ziyang Lyu and A.H. Welsh Mathematical Sciences Institute and Research School of Finance Actuarial Studies and Statistics Australian National University (today) ###### Abstract This paper establishes asymptotic results for the maximum likelihood and restricted maximum likelihood (REML) estimators of the parameters in the nested error regression model for clustered data when both of the number of independent clusters and the cluster sizes (the number of observations in each cluster) go to infinity. Under very mild conditions, the estimators are shown to be asymptotically normal with an elegantly structured covariance matrix. There are no restrictions on the rate at which the cluster size tends to infinity but it turns out that we need to treat within cluster parameters (i.e. coefficients of unit-level covariates that vary within clusters and the within cluster variance) differently from between cluster parameters (i.e. coefficients of cluster-level covariates that are constant within clusters and the between cluster variance) because they require different normalisations and are asymptotically independent. Key words: asymptotic independence; maximum likelihood estimator; mixed model; REML estimator; variance components. ## 1 Introduction Regression models with nested errors (also called random intercept or homogeneous correlation models) are widely used in applied statistics to model relationships in clustered data; they were introduced for survey data, by Scott and Holt (1982) and Battese et al. (1988), and, for longitudinal data, by Laird and Ware (1982). The models are usually fitted (see Harville (1977)) by assuming normality and computing maximum likelihood or restricted maximum likelihood (REML) estimators. As these estimators are nonlinear, asymptotic results provide an important way to understand their properties and then to construct approximate inferences about the unknown parameters. The usual asymptotic results applied to these estimators from Hartley and Rao (1967), Anderson (1969), Miller (1977), Das (1979), Cressie and Lahiri (1993), and Richardson and Welsh (1994) increase the number of clusters while keeping the size of each cluster fixed or bounded. However, there are many applications, particularly with survey data, with large cluster sizes; for example, Arora and Lahiri (1997) give an example with $43$ clusters and cluster sizes ranging from $95$ to $633$ and such examples are common in analysing poverty data (Pratesi, 2016). In addition, there are theoretical problems (e.g. in prediction, see Jiang (1998)) for which both the number of clusters and the cluster sizes need to increase. Therefore, in this paper, we study the asymptotic properties of normal-theory maximum likelihood and REML estimators of the parameters in the nested error regression model as both the number of clusters and the cluster sizes tend to infinity. Suppose that we observe on the $j$th unit in the $i$th cluster the vector $[y_{ij},\mathbf{x}_{ij}^{T}]^{T}$, where $y_{ij}$ is a scalar response variable and $\mathbf{x}_{ij}$ is a vector of explanatory variables or covariates, $j=1,\ldots,m_{i}$, $i=1,\ldots,g$. The nested error regression model specifies that $y_{ij}=\beta_{0}+\mathbf{x}_{ij}^{T}\boldsymbol{\beta}_{s}+\alpha_{i}+e_{ij},\qquad j=1,\ldots,m_{i},\,i=1,\ldots,g,$ (1) where $\beta_{0}$ is the intercept, $\boldsymbol{\beta}_{s}$ is the slope parameter, $\alpha_{i}$ is a random effect representing a random cluster effect and $e_{ij}$ is an error term. We assume that the $\\{\alpha_{i}\\}$ and $\\{e_{ij}\\}$ are all mutually independent with mean zero and variances (called the variance components) $\sigma_{\alpha}^{2}$ and $\sigma_{e}^{2}$, respectively; we do not assume normality. This regression model treats clusters as independent with constant (i.e. homogeneous) correlation within clusters. It is a particular, simple linear mixed model that is widely used in fields such as small area estimation (see Rao and Molina (2015)) to model and make predictions from clustered data, so our results are immediately useful. In addition, its simplicity allows us to use elementary methods to gain insight into exactly what is going on and obtain explicit, highly interpretable results as the cluster sizes increase. These arguments and results form the basis for how to proceed to more complicated cases, with multiple variance components. When the random effects and errors are normally distributed, the likelihood for the parameters and the REML criterion can be obtained analytically. Irrespective of whether normality holds or not, we refer to these functions as the likelihood and the REML criterion for the model (1) and the values of the parameters that maximise them as maximum likelihood and REML estimators, respectively. For our results, we make very simple assumptions: essentially finite “$4+\delta$” moments for the random effects and errors (instead of normality) and, allowing the explanatory variables to be fixed or random, conditions analogous to finite “$2+\delta$” moments for the explanatory variables. We allow $g\to\infty$ and $\min_{1\leq i\leq g}m_{i}\to\infty$ without any restriction on the rates. We obtain asymptotic representations for both the maximum likelihood and REML estimators that give the influence functions of these estimators, are very useful for deriving results when we combine these estimators with other estimators, and lead to central limit theorems for these estimators and asymptotic inferences for the unknown parameters. The normalisation is by a diagonal matrix which is easy to interpret. These results provide new and striking insights. First, we need to separate and treat within cluster parameters (i.e. coefficients of unit-level covariates that vary within clusters and the within cluster variance $\sigma_{e}^{2}$) differently from between cluster parameters (i.e. coefficients of cluster-level covariates that are constant within clusters and the between cluster variance $\sigma_{\alpha}^{2}$). We make explicit the fact that the information for within cluster parameters grows with $n=\sum_{i=1}^{g}m_{i}$ and the information for between cluster parameters grows with $g$ so they require different normalisations. The asymptotic variance matrix which we obtain explicitly has a very tidy and easy to interpret block diagonal structure. Second, there are good reasons for centering the within cluster covariates about their cluster means and then including the cluster means as contextual effect variables in the between cluster covariates (see for example Yoon and Welsh (2020) for references) but our asymptotic results (which include both cases) show that increasing cluster size has asymptotically the same effect as the centering (although without increasing the number of between cluster parameters) and also asymptotically orthogonalises the variance components. These apparently simple insights are new and not available from the existing literature. The few results in the literature that allow both the number of clusters and the cluster size to go to infinity do not give the same insights as our results. Jiang (1996) proved consistency and asymptotic normality of the maximum likelihood and REML estimators for a wide class of linear mixed models allowing increasing cluster sizes. He later showed this condition is required for studying the empirical distribution of the empirical best linear unbiased predictors (EBLUPs) of the random effects (Jiang, 1998). Xie and Yang (2003) obtained results for generalized estimating equation regression parameter estimators with increasing cluster size which potentially relate to our estimators, but their estimators do not include the variance components so the results do not apply to our estimators. The difficulties with trying to apply general results to particular models like (1) are that it can be difficult to understand the conditions and interpret the main result. To illustrate, increasing cluster size in Jiang (1996) is a part of other complicated assumptions and, for particular examples, he needed further conditions on the way the cluster size increases, making it difficult to see whether there is any restriction on the relationship between the cluster size and the number of clusters and leaving open questions of whether the conditions are minimal or not. Also, although Jiang did give some nested model examples which satisfy his main invariant class $AI^{4}$ condition, this condition is quite complicated. In terms of their main results, both Jiang (1996) and Xie and Yang (2003) normalise the estimators by the product of general (nondiagonal) matrices, producing results which are difficult to interpret and do not provide the insights our results provide. We introduce notation to describe the maximum likelihood and REML estimators for the parameters in (1)-(2), specify the conditions and state our main results in Section 2. We discuss the results in Section 3 and give the proofs in Section 4. ## 2 Results We gain important insights by partitioning the vector of covariates $\mathbf{x}_{ij}$ into the $p_{w}$-vector $\mathbf{x}_{ij}^{(w)}$ of within cluster covariates and the $p_{b}$-vector $\mathbf{x}_{i}^{(b)}$ of between cluster covariates. As noted in the Introduction, it is also often useful to center the within cluster covariates about their cluster means and then expand the between cluster covariate vector to include the cluster means of the within cluster covariates. Specifically, for a single within cluster covariate $x_{ij}$, we can make the regression function either $\beta_{0}+x_{ij}\beta_{2}$ or the centered form $\beta_{0}+\bar{x}_{i}\beta_{1}+(x_{ij}-\bar{x}_{i})\beta_{2}$. This centering ensures that $\sum_{j=1}^{m_{i}}\mathbf{x}_{ij}^{(w)}=\boldsymbol{0}_{[p_{w}:1]}$ for all $i=1,\ldots,g$, where $\boldsymbol{0}_{[p:q]}$ denotes the $p\times q$ matrix of zeros, and as it orthogonalises the between and within covariates, has advantages for interpreting and fitting the model (Yoon and Welsh, 2020) as well as increasing flexibility. We leave this as choice for the modeller; our analysis handles both cases as well as the cases in which there are no within cluster or no between cluster covariates because they are all special cases of the model (1) which we re-express as $y_{ij}=\beta_{0}+\mathbf{x}_{i}^{(b)T}\boldsymbol{\beta}_{1}+\mathbf{x}_{ij}^{(w)T}\boldsymbol{\beta}_{2}+\alpha_{i}+e_{ij},\qquad j=1,\ldots,m_{i},\,i=1,\ldots,g,$ (2) where $\beta_{0}$ is the unknown intercept, $\boldsymbol{\beta}_{1}$ is the unknown between cluster slope parameter and $\boldsymbol{\beta}_{2}$ is the unknown within cluster slope parameter. We treat the covariates as fixed; when they are random, we condition on them, though we omit this from the notation. We assume throughout that the true model that describes the data generating mechanism is (2) with general parameter $\boldsymbol{\omega}=[\boldsymbol{\beta}_{0},\boldsymbol{\beta}_{1}^{T},\sigma_{\alpha}^{2},\boldsymbol{\beta}_{2}^{T},\sigma_{e}^{2}]^{T}$, true parameter $\dot{\boldsymbol{\omega}}=[\dot{\beta}_{0},\dot{\boldsymbol{\beta}}_{1}^{T},\dot{\sigma}_{\alpha}^{2},\dot{\boldsymbol{\beta}}_{2}^{T},\dot{\sigma}_{e}^{2}]^{T}$ and take all expectations under the true model. The order of the parameters in $\boldsymbol{\omega}$ and $\dot{\boldsymbol{\omega}}$ groups the between parameters and the within parameters together and simplifies the presentation of our results. To simplify notation, let $\tau_{i}=m_{i}/(\sigma_{e}^{2}+m_{i}\sigma_{a}^{2})$ with true value $\dot{\tau}_{i}$, $m_{L}=\min_{1\leq i\leq g}m_{i}$, $\begin{split}&\bar{y}_{i}=\frac{1}{{m_{i}}}\sum_{j=1}^{m_{i}}y_{ij},\quad\bar{\mathbf{x}}_{i}^{(w)}=\frac{1}{m_{i}}\sum_{j=1}^{m_{i}}\mathbf{x}_{ij}^{(w)},\quad S_{w}^{y}=\sum_{i=1}^{g}\sum_{j=1}^{m_{i}}(y_{ij}-\bar{y}_{i})^{2},\\\ &\mathbf{S}_{w}^{xy}=\sum_{i=1}^{g}\sum_{j=1}^{m_{i}}(\mathbf{x}_{ij}^{(w)}-\bar{\mathbf{x}}_{i}^{(w)})(y_{ij}-\bar{y}_{i}),\quad\text{and}\quad\\\ &\mathbf{S}_{w}^{x}=\sum_{i=1}^{g}\sum_{j=1}^{m_{i}}(\mathbf{x}_{ij}^{(w)}-\bar{\mathbf{x}}_{i}^{(w)})(\mathbf{x}_{ij}^{(w)}-\bar{\mathbf{x}}_{i}^{(w)})^{T}.\end{split}$ The log-likelihood for the parameters in the model (after discarding constant terms) is $\begin{split}l(\boldsymbol{\omega})&=\frac{1}{2}\sum_{i=1}^{g}\log(\tau_{i})-\frac{n-g}{2}\log\sigma_{e}^{2}-\frac{1}{2\sigma_{e}^{2}}(S_{w}^{y}-2\mathbf{S}_{w}^{xyT}\boldsymbol{\beta}_{2}+\boldsymbol{\beta}_{2}^{T}\mathbf{S}_{w}^{x}\boldsymbol{\beta}_{2})\\\ &\qquad-\frac{1}{2}\sum_{i=1}^{g}\tau_{i}(\bar{y}_{i}-\beta_{0}-\mathbf{x}_{i}^{(b)T}\boldsymbol{\beta}_{1}-\bar{\mathbf{x}}_{i}^{(w)T}\boldsymbol{\beta}_{2})^{2}.\end{split}$ (3) To maximize $l(\boldsymbol{\omega})$ and find the maximum likelihood estimator $\hat{\boldsymbol{\omega}}$ of $\boldsymbol{\omega}$, we differentiate (3) with respect to $\boldsymbol{\omega}$ to obtain the estimating function $\boldsymbol{\psi}(\boldsymbol{\omega})$ and then solve the estimating equation $\boldsymbol{0}_{[p_{b}+p_{w}+3:1]}=\boldsymbol{\psi}(\boldsymbol{\omega})$. The components of $\boldsymbol{\psi}(\boldsymbol{\omega})$ are $\begin{split}&l_{\beta_{0}}(\boldsymbol{\omega})=\sum_{i=1}^{g}\tau_{i}(\bar{y}_{i}-\beta_{0}-\mathbf{x}_{i}^{(b)T}\boldsymbol{\beta}_{1}-\bar{\mathbf{x}}_{i}^{(w)T}\boldsymbol{\beta}_{2}),\\\ &\mathbf{l}_{\boldsymbol{\beta}_{1}}(\boldsymbol{\omega})=\sum_{i=1}^{g}\tau_{i}\mathbf{x}_{i}^{(b)}(\bar{y}_{i}-\beta_{0}-\mathbf{x}_{i}^{(b)T}\boldsymbol{\beta}_{1}-\bar{\mathbf{x}}_{i}^{(w)T}\boldsymbol{\beta}_{2}),\\\ &l_{\sigma_{\alpha}^{2}}(\boldsymbol{\omega})=-\frac{1}{2}\sum_{i=1}^{g}\tau_{i}+\frac{1}{2}\sum_{i=1}^{g}\tau_{i}^{2}(\bar{y}_{i}-\beta_{0}-\mathbf{x}_{i}^{(b)T}\boldsymbol{\beta}_{1}-\bar{\mathbf{x}}_{i}^{(w)T}\boldsymbol{\beta}_{2})^{2},\\\ &\mathbf{l}_{\boldsymbol{\beta}_{2}}(\boldsymbol{\omega})=\frac{1}{\sigma_{e}^{2}}\mathbf{S}_{w}^{xy}-\frac{1}{\sigma_{e}^{2}}\mathbf{S}_{w}^{x}\boldsymbol{\beta}_{2}+\sum_{i=1}^{g}\tau_{i}\bar{\mathbf{x}}_{i}^{(w)}(\bar{y}_{i}-\beta_{0}-\mathbf{x}_{i}^{(b)T}\boldsymbol{\beta}_{1}-\bar{\mathbf{x}}_{i}^{(w)T}\boldsymbol{\beta}_{2}),\\\ &l_{\sigma_{e}^{2}}(\boldsymbol{\omega})=-\frac{1}{2}\sum_{i=1}^{g}m_{i}^{-1}\tau_{i}-\frac{n-g}{2\sigma_{e}^{2}}+\frac{1}{2\sigma_{e}^{4}}(S_{w}^{y}-2\mathbf{S}_{w}^{xyT}\boldsymbol{\beta}_{2}+\boldsymbol{\beta}_{2}^{T}\mathbf{S}_{w}^{x}\boldsymbol{\beta}_{2})\\\ &\qquad\qquad+\frac{1}{2}\sum_{i=1}^{g}m_{i}^{-1}\tau_{i}^{2}(\bar{y}_{i}-\beta_{0}-\mathbf{x}_{i}^{(b)T}\boldsymbol{\beta}_{1}-\bar{\mathbf{x}}_{i}^{(w)T}\boldsymbol{\beta}_{2})^{2}.\end{split}$ (4) Let $\boldsymbol{\psi}(\boldsymbol{\omega})^{T}=[\boldsymbol{\psi}^{(b)}(\boldsymbol{\omega})^{T},\boldsymbol{\psi}^{(w)}(\boldsymbol{\omega})^{T}]$, where $\boldsymbol{\psi}^{(b)}(\boldsymbol{\omega})^{T}=[l_{\beta_{0}}(\boldsymbol{\omega}),\mathbf{l}_{\boldsymbol{\beta}_{1}}(\boldsymbol{\omega})^{T},l_{\sigma_{\alpha}^{2}}(\boldsymbol{\omega})]$ are the estimating functions for the between cluster parameters and $\boldsymbol{\psi}^{(w)}(\boldsymbol{\omega})^{T}=[\mathbf{l}_{\boldsymbol{\beta}_{2}}(\boldsymbol{\omega})^{T},l_{\sigma_{e}^{2}}(\boldsymbol{\omega})]$ are the estimating functions for the within cluster parameters. The derivatives of the estimating functions which we write as $\nabla\boldsymbol{\psi}(\boldsymbol{\omega})$ and their expected values under the model are given in the Appendix. To control the estimating function and derive the asymptotic properties of $\hat{\boldsymbol{\omega}}$ from the estimating equation, we impose the following condition. Condition A * 1. The model (2) holds with true parameters $\dot{\boldsymbol{\omega}}$ inside the parameter space $\Omega$. * 2. The number of clusters $g\to\infty$ and the minimum number of observations per cluster $m_{L}\to\infty$. * 3. The random variables $\\{\alpha_{i}\\}$ and $\\{e_{ij}\\}$ are independent and identically distributed and there is a $\delta>0$ such that $\operatorname{E}\|\alpha_{i}\|^{4+\delta}<\infty$ and $\operatorname{E}\|e_{ij}\|^{4+\delta}<\infty$ for all $i=1,\ldots,g$ and $j\in\mathcal{S}_{i}$. * 4. Suppose that the limits $\mathbf{c}_{1}=\lim_{g\rightarrow\infty}g^{-1}\sum_{i=1}^{g}\mathbf{x}_{i}^{(b)}$, $\mathbf{C}_{2}=\lim_{g\rightarrow\infty}g^{-1}\sum_{i=1}^{g}\mathbf{x}_{i}^{(b)}\mathbf{x}_{i}^{(b)^{T}}$ and $\mathbf{C}_{3}=\lim_{g\to\infty}\lim_{m_{L}\to\infty}n^{-1}\mathbf{S}_{w}^{x}$ exist and the matrices $\mathbf{C}_{2}$ and $\mathbf{C}_{3}$ are positive definite. Suppose further that $\lim_{g\to\infty}\lim_{m_{L}\to\infty}\frac{1}{g}\sum_{i=1}^{g}\|\bar{\mathbf{x}}_{i}^{(w)}\|^{2}<\infty$, and there is a $\delta>0$ such that $\lim_{g\to\infty}g^{-1}\sum_{i=1}^{g}\|\mathbf{x}_{i}^{(b)}\|^{2+\delta}<\infty$ and $\lim_{g\to\infty}\lim_{m_{L}\to\infty}n^{-1}\sum_{i=1}^{g}\sum_{j=1}^{m_{i}}\|\mathbf{x}_{ij}^{(w)}-\bar{\mathbf{x}}_{i}^{(w)}\|^{2+\delta}<\infty$. These are very mild conditions which are often satisfied in practice. Conditions A3 and A4 ensure that limits needed to ensure the existence of the asymptotic variance of the estimating function exist and that we can establish a Lyapounov condition and hence a central limit theorem for the estimating function. They also ensure that minus the appropriately normalised second derivative of the estimating function converges to $\mathbf{B}$ given in (11) below. Unlike in the case of fixed $m_{i}$, A4 does not involve unknown parameters through the weights $\dot{\tau}_{i}$. Our main result is the following theorem which we prove in Section 4. ###### Theorem 1. Suppose Condition A holds. Then, as $g,m_{L}\to\infty$, there is a solution $\hat{\boldsymbol{\omega}}$ to the estimating equations $\boldsymbol{0}_{[p_{b}+p_{w}+3:1]}=\boldsymbol{\psi}(\boldsymbol{\omega})$, satisfying $\|\mathbf{K}^{1/2}(\hat{\boldsymbol{\omega}}-\dot{\boldsymbol{\omega}})\|=O_{p}(1)$, where $\mathbf{K}=\operatorname{diag}(g,g\boldsymbol{1}_{p_{b}}^{T},g,n\boldsymbol{1}_{p_{w}}^{T},n)$ with $\boldsymbol{1}_{p}$ the $p$ vector of ones. Moreover, $\hat{\boldsymbol{\omega}}$ has the asymptotic representation $\mathbf{K}^{1/2}(\hat{\boldsymbol{\omega}}-\dot{\boldsymbol{\omega}})=\mathbf{B}^{-1}\mathbf{K}^{-1/2}\boldsymbol{\xi}+o_{p}(1),$ (5) where $\mathbf{B}$ is given by (11) below and $\boldsymbol{\xi}=[\xi_{\beta_{0}},\boldsymbol{\xi}_{\boldsymbol{\beta}_{1}}^{T},\xi_{\sigma_{\alpha}^{2}},\boldsymbol{\xi}_{\boldsymbol{\beta}_{2}}^{T},\xi_{\sigma_{e}^{2}}]^{T}$ has components $\begin{split}&\xi_{\beta_{0}}=\frac{1}{\dot{\sigma}_{\alpha}^{2}}\sum_{i=1}^{g}\alpha_{i},\qquad\boldsymbol{\xi}_{\boldsymbol{\beta}_{1}}=\frac{1}{\dot{\sigma}_{\alpha}^{2}}\sum_{i=1}^{g}\mathbf{x}_{i}^{(b)}\alpha_{i},\qquad\xi_{\sigma_{\alpha}^{2}}=\frac{1}{2\dot{\sigma}_{\alpha}^{4}}\sum_{i=1}^{g}(\alpha_{i}^{2}-\dot{\sigma}_{\alpha}^{2}),\\\ &\boldsymbol{\xi}_{\boldsymbol{\beta}_{2}}=\frac{1}{\dot{\sigma}_{e}^{2}}\sum_{i=1}^{g}\sum_{j=1}^{m_{i}}(\mathbf{x}_{ij}^{(w)}-\bar{\mathbf{x}}_{i}^{(w)})e_{ij}\qquad\mbox{and}\qquad\xi_{\sigma_{e}^{2}}=\frac{1}{2\dot{\sigma}_{e}^{4}}\sum_{i=1}^{g}\sum_{j=1}^{m_{i}}(e_{ij}^{2}-\dot{\sigma}_{e}^{2}).\end{split}$ It follows that $\mathbf{K}^{1/2}(\hat{\boldsymbol{\omega}}-\dot{\boldsymbol{\omega}})\xrightarrow{D}N(\boldsymbol{0},\mathbf{C}),$ where $\begin{split}\mathbf{C}&=\left[\begin{matrix}\dot{\sigma}_{\alpha}^{2}d&\dot{\sigma}_{\alpha}^{2}\mathbf{d}_{1}^{T}&\operatorname{E}\alpha_{1}^{3}&\boldsymbol{0}_{[1:p_{w}]}&0\\\ \dot{\sigma}_{\alpha}^{2}\mathbf{d}_{1}&\dot{\sigma}_{\alpha}^{2}\mathbf{D}_{2}&\boldsymbol{0}_{[p_{b}:1]}&\boldsymbol{0}_{[p_{b}:p_{w}]}&\boldsymbol{0}_{[p_{b}:1]}\\\ \operatorname{E}\alpha_{1}^{3}&\boldsymbol{0}_{[1:p_{b}]}&\operatorname{E}\alpha_{1}^{4}-\dot{\sigma}_{\alpha}^{4}&\boldsymbol{0}_{[1:p_{w}]}&0\\\ \boldsymbol{0}_{[p_{w}:1]}&\boldsymbol{0}_{[p_{w}:p_{b}]}&\boldsymbol{0}_{[p_{w}:1]}&\dot{\sigma}_{e}^{2}\mathbf{C}_{3}^{-1}&\boldsymbol{0}_{[p_{w}:1]}\\\ 0&\boldsymbol{0}_{[1:p_{b}]}&0&0&\operatorname{E}e_{ij}^{4}-\dot{\sigma}_{e}^{4}\end{matrix}\right]\end{split}$ with $d=1//(1-\mathbf{c}_{1}^{T}\mathbf{C}_{2}^{-1}\mathbf{c}_{1})$, $\mathbf{d}_{1}=-\mathbf{c}_{1}^{T}\mathbf{C}_{2}^{-1}/(1-\mathbf{c}_{1}^{T}\mathbf{C}_{2}^{-1}\mathbf{c}_{1})$ and $\mathbf{D}_{2}=\mathbf{C}_{2}^{-1}+\mathbf{C}_{2}^{-1}\mathbf{c}_{1}\mathbf{c}_{1}^{T}\mathbf{C}_{2}^{-1}/(1-\mathbf{c}_{1}^{T}\mathbf{C}_{2}^{-1}\mathbf{c}_{1})$. We now consider REML estimation. To describe REML, we group the parameters into the regression parameters $\boldsymbol{\beta}=[\beta_{0},\boldsymbol{\beta}_{1}^{T},\boldsymbol{\beta}_{2}^{T}]$ and variance components $\boldsymbol{\theta}=[\sigma_{\alpha}^{2},\sigma_{e}^{2}]^{T}$.The REML criterion function is obtained by replacing the regression parameters $\boldsymbol{\beta}$ in the log-likelihood (3) by their maximum likelihood estimators for each fixed $\boldsymbol{\theta}$ to produce a profile log- likelihood for $\boldsymbol{\theta}$ and then adding an adjustment term. Let $\mathbf{z}_{i}=[1,\mathbf{x}_{i}^{(b)T},\bar{\mathbf{x}}_{i}^{(w)T}]^{T}$, $\mathbf{w}=[0,\boldsymbol{0}_{[1:p_{b}]},\mathbf{S}_{w}^{xyT}]^{T}$ and $\mathbf{W}=\mbox{block diag}(0,\boldsymbol{0}_{[p_{b}:p_{b}]},\mathbf{S}_{w}^{x})$. Then, for each fixed $\boldsymbol{\theta}$, we solve the estimating equations in (4) for $\boldsymbol{\beta}$ to obtain $\begin{split}\hat{\boldsymbol{\beta}}(\boldsymbol{\theta})&=\boldsymbol{\Delta}(\boldsymbol{\theta})^{-1}\Big{(}\sum_{i=1}^{g}\tau_{i}\mathbf{z}_{i}\bar{y}_{i}+\sigma_{e}^{-2}\mathbf{w}\Big{)},\quad\mbox{ with }\quad\boldsymbol{\Delta}(\boldsymbol{\theta})=\sum_{i=1}^{g}\tau_{i}\mathbf{z}_{i}\mathbf{z}_{i}^{T}+\sigma_{e}^{-2}\mathbf{W},\end{split}$ and the REML criterion function is given by $l_{R}(\boldsymbol{\theta};\mathbf{y})=l(\hat{\boldsymbol{\beta}}(\boldsymbol{\theta}),\boldsymbol{\theta};\mathbf{y})-\frac{1}{2}\log\left\\{\|\boldsymbol{\Delta}(\boldsymbol{\theta})\|\right\\}.$ The REML estimator $\hat{\boldsymbol{\theta}}_{R}$ of $\boldsymbol{\theta}$ is the maximiser of the REML criterion function $l_{R}(\boldsymbol{\theta};\mathbf{y})$; we call $\hat{\boldsymbol{\beta}}_{R}=\hat{\boldsymbol{\beta}}(\hat{\boldsymbol{\theta}}_{R})$ the REML estimator of $\boldsymbol{\beta}$ and write $\hat{\boldsymbol{\omega}}_{R}=(\hat{\beta}_{R0},\hat{\boldsymbol{\beta}}_{R1}^{T},\hat{\sigma}_{R\alpha}^{2},\hat{\boldsymbol{\beta}}_{R2}^{T},\hat{\sigma}_{Re}^{2})^{T}$. Since $\boldsymbol{\Delta}(\boldsymbol{\theta})$ does not depend on $\boldsymbol{\beta}$, the REML estimator is also the maximiser of the adjusted log-likelihood $l_{A}(\boldsymbol{\beta},\boldsymbol{\theta};\mathbf{y})=l(\boldsymbol{\omega};\mathbf{y})-\frac{1}{2}\log\left\\{\|\Delta(\boldsymbol{\theta})\|\right\\}.$ That is, we can find the REML estimator in one step instead of two (Patefield, 1977) by maximising $l_{A}(\boldsymbol{\beta},\boldsymbol{\theta};\mathbf{y})$. In either case, the estimating function is $\boldsymbol{\psi}_{A}(\boldsymbol{\omega})=[l_{A\beta_{0}}(\boldsymbol{\omega}),\,\mathbf{l}_{A\beta_{1}}(\boldsymbol{\omega})^{T},\,l_{A\sigma_{\alpha}^{2}}(\boldsymbol{\omega}),\,\mathbf{l}_{A\beta_{2}}(\boldsymbol{\omega})^{T},\,l_{A\sigma_{e}^{2}}(\boldsymbol{\omega})]^{T}$. The derivatives $l_{A\beta_{0}}(\boldsymbol{\omega})=l_{\beta_{0}}(\boldsymbol{\omega})$, $\mathbf{l}_{A\beta_{1}}(\boldsymbol{\omega})=\mathbf{l}_{\beta_{1}}(\boldsymbol{\omega})$ and $\mathbf{l}_{A\beta_{2}}(\boldsymbol{\omega})=\mathbf{l}_{\beta_{2}}(\boldsymbol{\omega})$, while $\begin{split}&l_{A\sigma_{\alpha}^{2}}(\boldsymbol{\omega})=l_{\sigma_{\alpha}^{2}}(\boldsymbol{\omega})-\frac{1}{2}\mbox{trace}\Big{\\{}\boldsymbol{\Delta}(\boldsymbol{\theta})^{-1}\frac{\partial\boldsymbol{\Delta}(\boldsymbol{\theta})}{\partial\sigma_{\alpha}^{2}}\Big{\\}}\quad\mbox{ with }\quad\frac{\partial\boldsymbol{\Delta}(\boldsymbol{\theta})}{\partial\sigma_{\alpha}^{2}}=-\sum_{i=1}^{g}\tau_{i}^{2}\mathbf{z}_{i}\mathbf{z}_{i}^{T}\\\ &l_{A\sigma_{e}^{2}}(\boldsymbol{\omega})=l_{\sigma_{e}^{2}}(\boldsymbol{\omega})-\frac{1}{2}\mbox{trace}\Big{\\{}\boldsymbol{\Delta}(\boldsymbol{\theta})^{-1}\frac{\partial\boldsymbol{\Delta}(\boldsymbol{\theta})}{\partial\sigma_{e}^{2}}\Big{\\}}\quad\mbox{ with }\\\ &\qquad\frac{\partial\boldsymbol{\Delta}(\boldsymbol{\theta})}{\partial\sigma_{e}^{2}}=-\sum_{i=1}^{g}m_{i}^{-1}\tau_{i}^{2}\mathbf{z}_{i}\mathbf{z}_{i}^{T}-\sigma_{e}^{-4}\mathbf{W}.\end{split}$ We show that the REML estimator is asymptotically equivalent to the maximum likelihood estimator by showing that the contribution from the adjustment terms to the estimating function is asymptotically negligible. This yields the following theorem which we prove in Section 4. ###### Theorem 2. Suppose Condition A holds. Then, as $g,m_{L}\to\infty$, there is a solution $\hat{\boldsymbol{\omega}}_{R}$ to the adjusted likelihood estimating equations satisfying $\|\mathbf{K}^{1/2}(\hat{\boldsymbol{\omega}}_{R}-\dot{\boldsymbol{\omega}})\|=O_{p}(1)$ and $\mathbf{K}^{\frac{1}{2}}(\hat{\boldsymbol{\omega}}_{R}-\hat{\boldsymbol{\omega}})=\ o_{p}(1),$ so Theorem 1 applies to the REML estimator. ## 3 Discussion Theorems 1 and 2 establish the asymptotic equivalence, asymptotic representations and asymptotic normality for the maximum likelihood and REML estimators of the parameters in the nested error regression model under very mild conditions when both the number of clusters and the cluster sizes increase to infinity. In this section we interpret and discuss these results before pointing out possible directions for future work. We can estimate $\dot{\boldsymbol{\omega}}$ consistently when $g\to\infty$ with bounded cluster sizes but we need to let $m_{L}\to\infty$ to estimate the random effects $\\{\alpha_{i}\\}$ consistently (Jiang, 1998). If $m_{L}\to\infty$ but $g$ is held fixed, we can estimate the within cluster variance $\dot{\sigma}_{e}^{2}$ consistently but not the between cluster variance $\dot{\sigma}_{\alpha}^{2}$. These considerations motivate allowing both $g\to\infty$ and $m_{L}\to\infty$. The asymptotic representation shows that the influence function of the maximum likelihood and REML estimators under the model is given by the summands of $\mathbf{B}^{-1}\boldsymbol{\xi}$. Explicitly, at a point $[\alpha_{i},e_{ij},\mathbf{x}^{(b)T}_{i},\mathbf{x}_{ij}^{(w)T}]^{T}$ (which we suppress in the notation), the influence function is the $(p_{b}+p_{w}+3)$-vector function $\boldsymbol{\lambda}=[\lambda_{\beta_{0}},\boldsymbol{\lambda}_{\boldsymbol{\beta}_{1}}^{T},\lambda_{\sigma_{\alpha}^{2}},\boldsymbol{\lambda}_{\boldsymbol{\beta}_{2}}^{T},\lambda_{\sigma_{e}^{2}}]^{T}$, where $\begin{split}&\lambda_{\beta_{0}}=\\{(1-\mathbf{c}_{1}^{T}\mathbf{C}_{2}^{-1}\mathbf{x}_{i}^{(b)})/(1-\mathbf{c}_{1}^{T}\mathbf{C}_{2}^{-1}\mathbf{c}_{1})\\}\alpha_{i},\\\ &\boldsymbol{\lambda}_{\boldsymbol{\beta}_{1}}=\\{\mathbf{C}_{2}^{-1}\mathbf{x}_{i}^{(b)}+(1-\mathbf{c}_{1}^{T}\mathbf{C}_{2}^{-1}\mathbf{x}_{i}^{(b)})/(1-\mathbf{c}_{1}^{T}\mathbf{C}_{2}^{-1}\mathbf{c}_{1})\\}\mathbf{x}_{i}^{(b)}\alpha_{i},\\\ &\lambda_{\sigma_{\alpha}^{2}}=\alpha_{i}^{2}-\dot{\sigma}_{\alpha}^{2},\qquad\boldsymbol{\lambda}_{\boldsymbol{\beta}_{2}}=\mathbf{C}_{3}^{-1}(\mathbf{x}_{ij}^{(w)}-\bar{\mathbf{x}}_{i}^{(w)})e_{ij}\qquad\mbox{and}\qquad\lambda_{\sigma_{e}^{2}}=e_{ij}^{2}-\dot{\sigma}_{e}^{2}.\end{split}$ These expressions are not easy to obtain directly because the between and within parameters are estimated at different rates. As is well-known, the estimators are not robust because the influence function is unbounded in the covariates, random effect and error. The central limit theorem allows us to construct asymptotic confidence intervals for the parameters in the model. An asymptotic $100(1-\gamma)\%$ confidence interval for $\dot{\beta}_{1k}$ is $[\hat{\beta}_{1k}-\Phi^{-1}(1-\gamma/2)\hat{\sigma}_{\alpha}d_{kk}^{(b)1/2}/g^{1/2},\,\,\hat{\beta}_{1k}+\Phi^{-1}(1-\gamma/2)\hat{\sigma}_{\alpha}d_{kk(b)}^{1/2}/g^{1/2}],$ where $d_{kk}^{(b)}$ is the $k$th diagonal element of $\hat{\mathbf{C}}_{2}^{-1}+\hat{\mathbf{C}}_{2}^{-1}\hat{\mathbf{c}}_{1}\hat{\mathbf{c}}_{1}^{T}\hat{\mathbf{C}}_{2}^{-1}/(1-\hat{\mathbf{c}}_{1}^{T}\hat{\mathbf{C}}_{2}^{-1}\hat{\mathbf{c}}_{1})$ with $\hat{\mathbf{c}}_{1}=g^{-1}\sum_{i=1}^{g}\mathbf{x}_{i}^{(b)}$ and $\hat{\mathbf{C}}_{2}=g^{-1}\sum_{i=1}^{g}\mathbf{x}_{i}^{(b)}\mathbf{x}_{i}^{(b)^{T}}$, and an asymptotic $100(1-\gamma)\%$ confidence interval for $\dot{\beta}_{2r}$ is $[\hat{\beta}_{2r}-\Phi^{-1}(1-\gamma/2)\hat{\sigma}_{e}d_{rr}^{(w)1/2}/n^{1/2},\,\,\hat{\beta}_{2r}+\Phi^{-1}(1-\gamma/2)\hat{\sigma}_{e}d_{rr}^{(w)1/2}/n^{1/2}],$ where $d_{rr}^{(w)}$ is the $r$th diagonal element of $(\mathbf{S}_{w}^{x}/n)^{-1}$. Setting the confidence interval on the $\log$ scale and then backtransforming, an asymptotic $100(1-\gamma)\%$ confidence interval for $\dot{\sigma}_{\alpha}$ is $\begin{split}[\hat{\sigma}_{\alpha}\exp\\{-\Phi^{-1}(1-\gamma/2)&(\hat{\mu}_{4\alpha}-\hat{\sigma}_{\alpha}^{4})^{1/2}/2g^{1/2}\hat{\sigma}_{\alpha}^{2}\\},\,\\\ &\hat{\sigma}_{\alpha}\exp\\{\Phi^{-1}(1-\gamma/2)(\hat{\mu}_{4\alpha}-\hat{\sigma}_{\alpha}^{4})^{1/2}/2g^{1/2}\hat{\sigma}_{\alpha}^{2}\\}],\end{split}$ where $\hat{\mu}_{4\alpha}=g^{-1}\sum_{i=1}^{g}(\bar{y}-\hat{\beta}_{0}-\mathbf{x}_{i}^{(b)^{T}}\hat{\boldsymbol{\beta}}_{1}-\bar{\mathbf{x}}_{i}^{(w)^{T}}\hat{\boldsymbol{\beta}}_{2})^{4}$ estimates $\operatorname{E}\alpha_{1}^{4}$. Squaring the endpoints gives an asymptotic $100(1-\gamma)\%$ confidence interval for $\dot{\sigma}_{\alpha}^{2}$. The results show explicitly that the between and within parameters are estimated at different rates and the form of $\mathbf{C}$ shows that, even without assuming normality, the maximum likelihood and REML estimators of the within parameters are asymptotically independent of the estimators of the between parameters. That is, the two sets of parameters are asymptotically orthogonal. The within cluster regression parameter is asymptotically orthogonal to the within cluster variance and the between cluster slope parameter is asymptotically orthogonal to the between cluster variance, but the intercept is only asymptotically orthogonal to the between cluster variance when the random effect distribution is symmetric. When the cluster sizes are fixed, the maximum likelihood and REML estimators all converge to the true parameters at the same rate ($g^{-1/2}$) and the expression for their asymptotic variance is much more complicated. Appending a subscript $m$ to emphasise that the cluster sizes are fixed at their upper bounds, the asymptotic variance of the estimators is $g^{-1}\mathbf{C}_{m}=g^{-1}\mathbf{B}_{m}^{-1}\mathbf{A}_{m}\mathbf{B}_{m}^{-1}$, where $\mathbf{B}_{m}=-\lim_{g\to\infty}g^{-1}\operatorname{E}\nabla\psi(\dot{\boldsymbol{\omega}})$ (which we can obtain from (12)) and $\mathbf{A}_{m}=\lim_{g\to\infty}g^{-1}\operatorname{Var}\\{\psi(\dot{\boldsymbol{\omega}})\\}$. We require assumptions on the convergence of weighted means and weighted products of covariates to ensure the existence of $\mathbf{B}_{m}$ and $\mathbf{A}_{m}$. Under these assumptions, in general, $\mathbf{B}_{m}$ is block diagonal for $[\boldsymbol{\beta}^{T},\boldsymbol{\theta}^{T}]^{T}$, although it is not block diagonal for $\boldsymbol{\omega}$ because the $(\sigma_{\alpha}^{2},\sigma_{e}^{2})$ term is nonzero. When $\bar{\mathbf{x}}_{i}^{(w)}=\boldsymbol{0}_{[p_{w}:1]}$ for all $i=1,\ldots,g$, the $(\boldsymbol{\beta}_{2},\beta_{0})$ and $(\boldsymbol{\beta}_{2},\boldsymbol{\beta}_{1})$ terms are zero, but this does not affect the $(\sigma_{\alpha}^{2},\sigma_{e}^{2})$. The matrix $\mathbf{A}_{m}$ involves third and fourth moments and is rarely evaluated in the non-normal case; general expressions are given in Field et al. (2008) and expressions specific to the model (2) are available from the authors on request. It is in general not block diagonal for $[\boldsymbol{\beta}^{T},\boldsymbol{\theta}^{T}]^{T}$ unless both $\operatorname{E}(\alpha_{1}^{3})=0$ and $\operatorname{E}(e_{11}^{3})=0$. The centering condition makes the covariance of $\mathbf{l}_{\boldsymbol{\beta}_{2}}(\dot{\boldsymbol{\omega}})$ with all the other components of $\boldsymbol{\psi}(\dot{\boldsymbol{\omega}})$ equal zero, but not the covariance between $l_{\beta_{0}}(\dot{\boldsymbol{\omega}})$ or $\mathbf{l}_{\boldsymbol{\beta}_{2}}(\dot{\boldsymbol{\omega}})$ with $l_{\sigma_{\alpha}^{2}}(\dot{\boldsymbol{\omega}})$ or $l_{\sigma_{e}^{2}}(\dot{\boldsymbol{\omega}})$. These limits are not block diagonal for $\boldsymbol{\omega}$ even when both $\operatorname{E}(\alpha_{1}^{3})=0$ and $\operatorname{E}(e_{11}^{3})=0$ (because the covariance between $l_{\sigma_{\alpha}^{2}}(\dot{\boldsymbol{\omega}})$ and $l_{\sigma_{e}^{2}}(\dot{\boldsymbol{\omega}})$ is nonzero). Of course, the nonzero terms in $\mathbf{C}_{m}$ involve limits of weighted averages which differ from those in $\mathbf{C}$. In working with the model (2), we have both between cluster and within cluster regression parameters to estimate. If we have no between cluster covariates,we discard $\boldsymbol{\beta}_{1}$, while if we have no within cluster covariates, we discard $\boldsymbol{\beta}_{2}$. The results for these cases can be obtained as special cases of the general results by deleting the components of vectors and the rows and columns of matrices corresponding to the discarded parameter. If there are no between cluster covariates in the model (there is no $\boldsymbol{\beta}_{1}$ in the model), we drop rows and columns $2$ to $p_{b}+1$ from $\mathbf{C}$. If there are no within cluster covariates (there is no $\boldsymbol{\beta}_{2}$ in the model), we drop rows and columns $p_{b}+3$ to $p_{b}+p_{w}+2$ from $\mathbf{C}$. There is a corresponding simplification to Condition A4. We have treated the covariates in the model as fixed, conditioning on them when they are random. As noted by Yoon and Welsh (2020), when the covariates are random, it makes sense to treat them as having a similar covariance structure to the response. That is, $\mathbf{x}_{i}^{(b)}$ are independent with mean $\boldsymbol{\mu}_{x}^{(b)}$ and variance $\boldsymbol{\Sigma}_{x}^{(b)}$, and the $\mathbf{x}_{ij}^{(w)}$ are independent in different clusters but correlated within clusters with mean $\boldsymbol{\mu}_{x}^{(w)}$, variance $\boldsymbol{\Upsilon}_{x}^{(w)}+\boldsymbol{\Sigma}_{x}^{(w)}$ and within cluster covariance $\boldsymbol{\Upsilon}_{x}^{(w)}$. The two types of covariates can be correlated. Condition A holds if both covariates have finite $2+\delta$ moments. We have $\mathbf{c}_{1}=\boldsymbol{\mu}_{x}^{(b)}$,$\mathbf{C}_{2}=\boldsymbol{\Sigma}_{x}^{(b)}+\boldsymbol{\mu}_{x}^{(b)}\boldsymbol{\mu}_{x}^{(b)T}$ and $\mathbf{C}_{3}=\boldsymbol{\Sigma}_{x}^{(w)}$. If $\mathbf{x}_{i}^{(b)}$ contains $\bar{\mathbf{x}}_{i}^{(w)}$, the terms $\boldsymbol{\mu}_{x}^{(b)}$ and variance $\boldsymbol{\Sigma}_{x}^{(b)}$ contain $\operatorname{E}\bar{\mathbf{x}}_{i}^{(w)}=\boldsymbol{\mu}_{x}^{(w)}$, $\operatorname{Var}(\bar{\mathbf{x}}_{i}^{(w)})=\boldsymbol{\Upsilon}_{x}^{(w)}+m_{i}^{-1}\boldsymbol{\Sigma}_{x}^{(w)}\to\boldsymbol{\Upsilon}_{x}^{(w)}$, as $m_{L}\to\infty$, and the covariance between $\mathbf{x}_{i}^{(b)}$ and $\bar{\mathbf{x}}_{i}^{(w)}$. One motivation for allowing the cluster size to increase with the number of clusters is that, as we have noted, this is required for consistent prediction of the random effects (Jiang, 1998). We have not considered prediction of the random effects explicitly in this paper but will do so in follow up work. The present paper makes an important step towards tackling prediction for sample survey applications because our results allow subsampling within clusters. In particular if the model (2) holds for the finite population, then noninformative subsampling of units within clusters ensures that the sample data satisfy the same model and hence that we can apply Theorems 1 and 2. The model we have considered is a simple linear mixed model. It is of interest to extend our results to more general linear mixed models and indeed to generalized linear mixed models. It is clear that we can extend the hierarchical structure of the model and allow for more variance components. The effect is to increase the sets of parameters so that there is a set for each level in the hierarchy. The estimators in each level converge at different rates and the limit distribution has a diagonal block for each level in the hierarchy. The maximum likelihood and REML estimators are not the only estimators of interest for the parameters of linear mixed models. Other estimators (including robust estimators) are available and it is also of interest to derive their asymptotic properties. We expect that the form of the asymptotic covariance matrices for these estimators will be block diagonal with a separate block for the parameters at each level in the hierarchy, just as we found for the maximum likelihood and REML estimators. Finally, Jiang (1996) also allowed the number of covariates to increase asymptotically and showed that the maximum likelihood and REML estimators have different asymptotic properties in this case. This is also an interesting problem to consider in the framework of this paper. ## 4 Proofs The proofs of Theorems 1 and 2 are presented in Subsection 4.1. The supporting lemmas used in these proofs are then proved in Subsections 4.2 and 4.3. ### 4.1 Proofs of Theorems 1 and 2 Proof. Write $\begin{split}\mathbf{K}^{-1/2}\psi(\boldsymbol{\omega})&=\mathbf{K}^{-1/2}\boldsymbol{\xi}-\mathbf{B}\mathbf{K}^{1/2}(\boldsymbol{\omega}-\dot{\boldsymbol{\omega}})+T_{1}+T_{2}(\boldsymbol{\omega})+T_{3}(\boldsymbol{\omega}),\end{split}$ where ${\mathbf{B}}=\lim_{g\to\infty}\lim_{m_{L}\to\infty}-\mathbf{K}^{-1/2}\operatorname{E}\nabla\psi(\boldsymbol{\omega})\mathbf{K}^{-1/2}$, $T_{1}=\mathbf{K}^{-1/2}\\{\psi(\dot{\boldsymbol{\omega}})-\xi\\}$, $T_{2}(\boldsymbol{\omega})=\mathbf{K}^{-1/2}\operatorname{E}\\{\psi(\boldsymbol{\omega})-\psi(\dot{\boldsymbol{\omega}})\\}+\mathbf{B}\mathbf{K}^{1/2}(\boldsymbol{\omega}-\dot{\boldsymbol{\omega}})$ and $T_{3}(\boldsymbol{\omega})=\mathbf{K}^{-1/2}[\psi(\boldsymbol{\omega})-\psi(\dot{\boldsymbol{\omega}})-\operatorname{E}\\{\psi(\boldsymbol{\omega})-\psi(\dot{\boldsymbol{\omega}})\\}]$. If we can show that $\|T_{1}\|=o_{p}(1)$, $\underset{\boldsymbol{\omega}\in\mathcal{N}}{\sup}\|T_{2}(\boldsymbol{\omega})\|=o(1)$ and $\underset{\boldsymbol{\omega}\in\mathcal{N}}{\sup}\|T_{3}(\boldsymbol{\omega})\|=o_{p}(1)$, respectively, then uniformly on $\mathcal{N}$, we have $\mathbf{K}^{-1/2}\psi(\boldsymbol{\omega})=\mathbf{K}^{-1/2}\boldsymbol{\xi}-\mathbf{B}\mathbf{K}^{1/2}(\boldsymbol{\omega}-\dot{\boldsymbol{\omega}})+o_{p}(1).$ (6) Multiplying by $(\boldsymbol{\omega}-\dot{\boldsymbol{\omega}})^{T}\mathbf{K}^{1/2}$, $(\boldsymbol{\omega}-\dot{\boldsymbol{\omega}})^{T}\psi(\boldsymbol{\omega})=(\boldsymbol{\omega}-\dot{\boldsymbol{\omega}})^{T}\boldsymbol{\xi}-(\boldsymbol{\omega}-\dot{\boldsymbol{\omega}})^{T}\mathbf{K}^{1/2}\mathbf{B}\mathbf{K}^{1/2}(\boldsymbol{\omega}-\dot{\boldsymbol{\omega}})+o_{p}(1).$ Since $B$ is positive definite, the right-hand side of is negative for $M$ sufficiently large. Therefore, according to Result 6.3.4 of Ortega and Rheinboldt (1973), a solution to the estimating equations exists in probability and satisfies $\|\mathbf{K}^{1/2}(\hat{\boldsymbol{\omega}}-\dot{\boldsymbol{\omega}})\|=O_{p}(1)$, so $\hat{\boldsymbol{\omega}}\in\mathcal{N}$. This allows us to substitute $\hat{\boldsymbol{\omega}}$ for $\boldsymbol{\omega}$ in (6) and rearrange the terms to obtain the asymptotic representation for $\hat{\boldsymbol{\omega}}$; the central limit theorem follows from the asymptotic representation and the central limit theorem for $\boldsymbol{\xi}$ that we establish in Lemma 1. It remains to show that that remainder terms in (4.1) are of smaller order and can be ignored. In Lemma 2, we establish $\|T_{1}\|=o_{p}(1)$ by showing that the result holds for each component of $T_{1}={\mathbf{K}}^{-1/2}\\{{\boldsymbol{\psi}}(\dot{\boldsymbol{\omega}})-\boldsymbol{\xi}\\}$ by applying Chebychev’s inequality and calculating the variances of the components. Our approach to handling $T_{2}(\boldsymbol{\omega})$ and $T_{3}(\boldsymbol{\omega})$ is inspired by Bickel (1975) who applied similar arguments to one-step regression estimators. The approach was extended to maximum likelihood and REML estimators in linear mixed models by Richardson and Welsh (1994); the bounds we use require more care with increasing cluster size. For $T_{2}(\boldsymbol{\omega})$, we have $\begin{split}\underset{\boldsymbol{\omega}\in\mathcal{N}}{\sup}\|T_{2}(\boldsymbol{\omega})\|&\leq\underset{\boldsymbol{\omega}\in\mathcal{N}}{\sup}\left\|\mathbf{K}^{-1/2}\left[\operatorname{E}\left\\{\boldsymbol{\psi}(\boldsymbol{\omega})-\boldsymbol{\psi}(\dot{\boldsymbol{\omega}})\right\\}-\operatorname{E}\nabla\boldsymbol{\psi}(\dot{\boldsymbol{\omega}})(\boldsymbol{\omega}-\dot{\boldsymbol{\omega}})\right]\right\|\\\ &\quad+\underset{\boldsymbol{\omega}\in\mathcal{N}}{\sup}\|\mathbf{K}^{-1/2}\operatorname{E}\nabla\boldsymbol{\psi}(\dot{\boldsymbol{\omega}})(\boldsymbol{\omega}-\dot{\boldsymbol{\omega}})+\mathbf{B}\mathbf{K}^{1/2}(\boldsymbol{\omega}-\dot{\boldsymbol{\omega}})\|\\\ &\leq M\underset{\boldsymbol{\omega}\in\mathcal{N}}{\sup}\left\\|\mathbf{K}^{-1/2}\left\\{\operatorname{E}\nabla\boldsymbol{\psi}(\boldsymbol{\Omega})-\operatorname{E}\nabla\boldsymbol{\psi}(\dot{\boldsymbol{\omega}})\right\\}\mathbf{K}^{-1/2}\right\\|\\\ &\quad+M\\|-\mathbf{K}^{-1/2}\operatorname{E}\nabla\boldsymbol{\psi}(\dot{\boldsymbol{\omega}})\mathbf{K}^{-1/2}-\mathbf{B}\\|\\\ &\leq M\underset{\boldsymbol{\omega}\in\mathcal{N}}{\sup}\left\\|\mathbf{K}^{-1/2}\left\\{\operatorname{E}\nabla\boldsymbol{\psi}(\boldsymbol{\Omega})-\operatorname{E}\nabla\boldsymbol{\psi}(\dot{\boldsymbol{\omega}})\right\\}\mathbf{K}^{-1/2}\right\\|+M\\|\mathbf{B}_{n}-\mathbf{B}\\|,\end{split}$ where the rows of $\boldsymbol{\Omega}$ are possibly different but lie between $\boldsymbol{\omega}$ and $\dot{\boldsymbol{\omega}}$ and $\mathbf{B}_{n}=-\mathbf{K}^{-1/2}\operatorname{E}\nabla\psi(\dot{\boldsymbol{\omega}})\mathbf{K}^{-1/2}$. In Lemma 4 we show that $\\|\mathbf{B}_{n}-\mathbf{B}\\|=o(1)$ and in Lemma 5, $\underset{\boldsymbol{\omega}\in\mathcal{N}}{\sup}\left\\|\mathbf{K}^{-1/2}\left\\{\operatorname{E}\nabla\boldsymbol{\psi}(\boldsymbol{\Omega})-\operatorname{E}\nabla\boldsymbol{\psi}(\dot{\boldsymbol{\omega}})\right\\}\mathbf{K}^{-1/2}\right\\|=o(1).$ Finally, to handle $T_{3}(\boldsymbol{\omega})$, decompose $\mathcal{N}=\\{\boldsymbol{\omega}:\|\mathbf{K}^{1/2}(\boldsymbol{\omega}-\dot{\boldsymbol{\omega}})\|\leq M\\}$ into the set of $N=O(g^{\frac{1}{4}})$ smaller cubes $\mathcal{C}=\\{\mathcal{C}(\mathbf{t}_{k})\\}$, where $\mathcal{C}(\mathbf{t})=\\{\boldsymbol{\omega}:\|\mathbf{K}^{1/2}(\mathbf{t}-\dot{\boldsymbol{\omega}})\|\leq Mg^{-1/4}\\}$. We first show that $\|T_{2}(\boldsymbol{\omega})\|=o(1)$ holds over the set of indices $\mathbf{t}_{k}=(t_{k1},t_{k2},t_{k3},t_{k4},t_{k5})^{T}$ for the cubes in $\mathcal{C}$ and then that the difference between taking the supremum over a fine grid of points and over $\mathcal{N}$ is small. Using Chebychev’s inequality, for any $\eta>0$, we have $\begin{split}\operatorname{Pr}&\left(\max_{1\leq k\leq N}\|\mathbf{K}^{-1/2}[\boldsymbol{\psi}(\mathbf{t}_{k})-\boldsymbol{\psi}(\dot{\boldsymbol{\omega}})-\operatorname{E}\\{\boldsymbol{\psi}(\mathbf{t}_{k})-\boldsymbol{\psi}(\dot{\boldsymbol{\omega}})\\}]\|>\eta\right)\\\ &\leq\sum_{k=1}^{N}\operatorname{Pr}\left(\|\mathbf{K}^{-1/2}[\boldsymbol{\psi}(\mathbf{t}_{k})-\boldsymbol{\psi}(\dot{\boldsymbol{\omega}})-\operatorname{E}\\{\boldsymbol{\psi}(\mathbf{t}_{k})-\boldsymbol{\psi}(\dot{\boldsymbol{\omega}})\\}]\|>\eta\right)\\\ &\leq\eta^{-2}\sum_{k=1}^{N}\operatorname{E}\|\mathbf{K}^{-1/2}[\boldsymbol{\psi}(\mathbf{t}_{k})-\boldsymbol{\psi}(\dot{\boldsymbol{\omega}})-\operatorname{E}\\{\boldsymbol{\psi}(\mathbf{t}_{k})-\boldsymbol{\psi}(\dot{\boldsymbol{\omega}})\\}]\|^{2}\\\ &=\eta^{-2}g^{-1}\sum_{k=1}^{N}\operatorname{E}\|\boldsymbol{\psi}^{(b)}(\mathbf{t}_{k})-\boldsymbol{\psi}^{(b)}(\dot{\boldsymbol{\omega}})-\operatorname{E}\\{\boldsymbol{\psi}^{(b)}(\mathbf{t}_{k})-\boldsymbol{\psi}^{(b)}(\dot{\boldsymbol{\omega}})\\}\|^{2}\\\ &\qquad+\eta^{-2}n^{-1}\sum_{k=1}^{N}\operatorname{E}\|\boldsymbol{\psi}^{(w)}(\mathbf{t}_{k})-\boldsymbol{\psi}^{(w)}(\dot{\boldsymbol{\omega}})-\operatorname{E}\\{\boldsymbol{\psi}^{(w)}(\mathbf{t}_{k})-\boldsymbol{\psi}^{(w)}(\dot{\boldsymbol{\omega}})\\}\|^{2}\\\ &=\eta^{-2}g^{-1}\sum_{k=1}^{N}\mbox{trace}[\operatorname{Var}\\{\boldsymbol{\psi}^{(b)}(\mathbf{t}_{k})-\boldsymbol{\psi}^{(b)}(\dot{\boldsymbol{\omega}})\\}]\\\ &\qquad+\eta^{-2}n^{-1}\sum_{k=1}^{N}\mbox{trace}[\operatorname{Var}\\{\boldsymbol{\psi}^{(w)}(\mathbf{t}_{k})-\boldsymbol{\psi}^{(w)}(\dot{\boldsymbol{\omega}})\\}].\end{split}$ We show in Lemma 3 that the variances $\operatorname{Var}\\{\psi^{(b)}(\mathbf{t}_{k})-\psi^{(b)}(\dot{\boldsymbol{\omega}})\\}$ and $\operatorname{Var}\\{\psi^{(w)}(\mathbf{t}_{k})-\psi^{(w)}(\dot{\boldsymbol{\omega}})\\}$ are uniformly bounded by $L$, say, so $\begin{split}\operatorname{Pr}\Big{(}\max_{1\leq k\leq N}\|\mathbf{K}^{-1/2}[\boldsymbol{\psi}(\mathbf{t}_{k})-&\boldsymbol{\psi}(\dot{\boldsymbol{\omega}})-\operatorname{E}\\{\boldsymbol{\psi}(\mathbf{t}_{k})-\boldsymbol{\psi}(\dot{\boldsymbol{\omega}})\\}]\|>\eta\Big{)}\\\ &\leq\eta^{-2}LN\\{g^{-1}(2+p_{b})+n^{-1}(1+p_{w})\\}=o(1),\end{split}$ using the fact that $N=O(g^{1/4})$. Using Taylor expansion, we get $\begin{split}\underset{1\leq k\leq N}{\max}\underset{\boldsymbol{\omega}\in\mathcal{C}(\mathbf{t}_{k})}{\sup}&\|\mathbf{K}^{-1/2}[\boldsymbol{\psi}(\boldsymbol{\omega})-\boldsymbol{\psi}(\mathbf{t}_{k})-\operatorname{E}\\{\boldsymbol{\psi}(\boldsymbol{\omega})-\boldsymbol{\psi}(\mathbf{t}_{k})\\}]\|\\\ &=\underset{1\leq k\leq N}{\max}\underset{\boldsymbol{\omega}\in\mathcal{C}(\mathbf{t}_{k})}{\sup}\|\mathbf{K}^{-1/2}[\nabla\boldsymbol{\psi}(\boldsymbol{\Omega}_{k})(\boldsymbol{\omega}-\mathbf{t}_{k})-\operatorname{E}\nabla\boldsymbol{\psi}(\boldsymbol{\Omega}_{k})(\boldsymbol{\omega}-\mathbf{t}_{k})]\|\\\ &\leq M\underset{\boldsymbol{\omega}\in\mathcal{N}}{\sup}g^{-1/4}\\|\mathbf{K}^{-1/2}\\{\nabla\boldsymbol{\psi}(\boldsymbol{\Omega})-\operatorname{E}\nabla\boldsymbol{\psi}(\boldsymbol{\Omega})\\}\mathbf{K}^{-1/2}\\|,\end{split}$ where the rows of $\boldsymbol{\Omega}_{k}$ are between $\mathbf{t}_{k}$ and $\boldsymbol{\omega}$. The result follows from Lemma 6. $\Box$ Proof. Let $\mathbf{K}_{\boldsymbol{\beta}}=\mbox{diag}(g,g\boldsymbol{1}_{p_{b}},n\boldsymbol{1}_{p_{w}})$ and write $\begin{split}\|g^{-1/2}\\{l_{A\sigma_{\alpha}^{2}}(\boldsymbol{\omega})-l_{\sigma_{\alpha}^{2}}(\boldsymbol{\omega})\\}\|&\leq\frac{1}{2g^{1/2}}\|\mbox{trace}\Big{\\{}\mathbf{K}_{\boldsymbol{\beta}}^{1/2}\boldsymbol{\Delta}(\boldsymbol{\theta})^{-1}\mathbf{K}_{\boldsymbol{\beta}}^{1/2}\mathbf{K}_{\boldsymbol{\beta}}^{-1/2}\frac{\partial\boldsymbol{\Delta}(\boldsymbol{\theta})}{\partial\sigma_{\alpha}^{2}}\mathbf{K}_{\boldsymbol{\beta}}^{-1/2}\Big{\\}}\|\\\ \|n^{-1/2}\\{l_{A\sigma_{e}^{2}}(\boldsymbol{\omega})-l_{\sigma_{e}^{2}}(\boldsymbol{\omega})\\}\|&\leq\frac{1}{2n^{1/2}}\|\mbox{trace}\Big{\\{}\mathbf{K}_{\boldsymbol{\beta}}^{1/2}\boldsymbol{\Delta}(\boldsymbol{\theta})^{-1}\mathbf{K}_{\boldsymbol{\beta}}^{1/2}\mathbf{K}_{\boldsymbol{\beta}}^{-1/2}\frac{\partial\boldsymbol{\Delta}(\boldsymbol{\theta})}{\partial\sigma_{e}^{2}}\mathbf{K}_{\boldsymbol{\beta}}^{-1/2}\Big{\\}}\|.\end{split}$ Then from Lemma 4 and the arguments establishing the convergence of $\mathbf{B}_{n}$ to $\mathbf{B}$, we can show that uniformly in $\boldsymbol{\omega}\in\mathcal{N}$ as $g,m_{L}\rightarrow\infty$, the matrices $\mathbf{K}_{\boldsymbol{\beta}}^{-1/2}\boldsymbol{\Delta}(\boldsymbol{\theta})\mathbf{K}_{\boldsymbol{\beta}}^{-1/2}$, $\mathbf{K}_{\boldsymbol{\beta}}^{-1/2}\frac{\partial\boldsymbol{\Delta}(\boldsymbol{\theta})}{\partial\sigma_{\alpha}^{2}}\mathbf{K}_{\boldsymbol{\beta}}^{-1/2}$ and $\mathbf{K}_{\boldsymbol{\beta}}^{-1/2}\frac{\partial\boldsymbol{\Delta}(\boldsymbol{\theta})}{\partial\sigma_{e}^{2}}\mathbf{K}_{\boldsymbol{\beta}}^{-1/2}$ all converge to $(p_{b}+p_{w}+1)\times(p_{b}+p_{w}+1)$ matrices with finite elements. Consequently, both $\|g^{-1/2}\\{l_{A\sigma_{\alpha}^{2}}(\boldsymbol{\omega})-l_{\sigma_{\alpha}^{2}}(\boldsymbol{\omega})\\}\|=o_{p}(1)$ and $\|n^{-1/2}\\{l_{A\sigma_{e}^{2}}(\boldsymbol{\omega})-l_{\sigma_{e}^{2}}(\boldsymbol{\omega})\\}\|=o_{p}(1)$ uniformly in $\boldsymbol{\omega}\in\mathcal{N}$, and the result follows from Theorem 1. $\Box$ ### 4.2 Lemmas for the estimating function $\boldsymbol{\psi}$ We prove a central limit theorem for $\boldsymbol{\xi}$ and that $T_{1}=o_{p}(1)$ (i.e $\boldsymbol{\psi}(\dot{\boldsymbol{\omega}})$ can be approximated by $\boldsymbol{\xi}$). We also prove that the variances of the components of $\boldsymbol{\psi}(\boldsymbol{\omega})-\boldsymbol{\psi}(\dot{\boldsymbol{\omega}})$ are uniformly bounded ###### Lemma 1. Suppose Condition A holds. Then, as $g,m_{L}\to\infty$, $\mathbf{K}^{-1/2}\boldsymbol{\xi}\xrightarrow{D}N(\boldsymbol{0},\mathbf{A})$, where $\begin{split}&\mathbf{A}=\\\ &\left[\begin{matrix}1/\dot{\sigma}_{\alpha}^{2}&\mathbf{c}_{1}^{T}/\dot{\sigma}_{\alpha}^{2}&\operatorname{E}\alpha_{1}^{3}/(2\dot{\sigma}_{\alpha}^{6})&\boldsymbol{0}_{[1:p_{w}]}&0\\\ \mathbf{c}_{1}/\dot{\sigma}_{\alpha}^{2}&\mathbf{C}_{2}/\dot{\sigma}_{\alpha}^{2}&\mathbf{c}_{1}\operatorname{E}\alpha_{1}^{3}/(2\dot{\sigma}_{\alpha}^{6})&\boldsymbol{0}_{[p_{b}:p_{w}]}&\boldsymbol{0}_{[p_{b}:1]}\\\ \operatorname{E}\alpha_{1}^{3}/(2\dot{\sigma}_{\alpha}^{6})&\mathbf{c}_{1}^{T}\operatorname{E}\alpha_{1}^{3}/(2\dot{\sigma}_{\alpha}^{6})&(\operatorname{E}\alpha_{1}^{4}-\dot{\sigma}_{\alpha}^{4})/(4\dot{\sigma}_{\alpha}^{8})&\boldsymbol{0}_{[1:p_{w}]}&0\\\ \boldsymbol{0}_{[p_{w}:1]}&\boldsymbol{0}_{[p_{w}:p_{b}]}&\boldsymbol{0}_{[p_{w}:1]}&\mathbf{C}_{3}/\dot{\sigma}_{e}^{2}&\boldsymbol{0}_{[p_{w}:1]}\\\ 0&\boldsymbol{0}_{[1:p_{b}]}&0&\boldsymbol{0}_{[1:p_{w}]}&(\operatorname{E}e_{11}^{4}-\dot{\sigma}_{e}^{4})/(4\dot{\sigma}_{e}^{8})\end{matrix}\right].\end{split}$ (7) ###### Proof. The components of $\boldsymbol{\xi}$ are sums of independent random variables with zero means and finite variances. Since $\\{\alpha_{i}\\}$ and $\\{e_{ij}\\}$ are independent, it is straightforward to compute $\operatorname{Var}\\{\mathbf{K}^{-1/2}\boldsymbol{\xi}\\}=\mathbf{A}_{n}$, and then from Condition A4, as $g,\,m_{L}\rightarrow\infty$, to show that $\mathbf{A}_{n}\rightarrow\mathbf{A}$. Partition $\boldsymbol{\xi}$ into $\boldsymbol{\xi}^{(b)}$ containing the first $p_{b}+2$ elements (corresponding to the between parameters) and $\boldsymbol{\xi}^{(w)}$ containing the remaining $p_{w}+1$ elements (corresponding to the within parameters). Partition $\mathbf{A}$ conformably into the block diagonal matrix with diagonal blocks $\mathbf{A}_{11}$ and $\mathbf{A}_{22}$, where $\mathbf{A}_{11}$ is $(p_{b}+2)\times(p_{b}+2)$ and $\mathbf{A}_{22}$ is $(p_{w}+1)\times(p_{w}+1)$. We prove that $g^{-1/2}\boldsymbol{\xi}^{(b)}\xrightarrow{D}N(\boldsymbol{0},\mathbf{A}_{11})$ and $n^{-1/2}\boldsymbol{\xi}^{(w)}\xrightarrow{D}N(\boldsymbol{0},\mathbf{A}_{22})$, and the result then follows from the fact that $\boldsymbol{\xi}^{(b)}$ and $\boldsymbol{\xi}^{(w)}$ are independent. Write $\boldsymbol{\xi}^{(b)}=\sum_{i=1}^{g}\boldsymbol{\xi}_{i}^{(b)}$, where the summands $\boldsymbol{\xi}_{i}^{(b)}=[\xi_{i\beta_{0}},\boldsymbol{\xi}_{i\boldsymbol{\beta}_{1}}^{T},\xi_{i\sigma_{\alpha}^{2}}]^{T}$, and let $\mathbf{a}$ be a fixed $(p_{b}+2)$-vector satisfying $\mathbf{a}^{T}\mathbf{a}=1$. Then $g^{-1/2}\mathbf{a}^{T}\boldsymbol{\xi}^{(b)}$ is a sum of independent scalar random variables with mean zero and finite variance. It follows from the $c_{r}$-inequality and Conditions A3 - A4 that Lyapunov’s condition holds. Consequently $g^{-1/2}\mathbf{a}^{T}\boldsymbol{\xi}^{(b)}$ converges in distribution to $N(0,\mathbf{a}^{T}\mathbf{A}_{11}\mathbf{a})$, as $g\to\infty$ and the result follows from the Cramer-Wold device (Billingsley, 1999, p 49). The proof that $n^{-\frac{1}{2}}\boldsymbol{\xi}^{(w)}$ converges to $N(0,\mathbf{A}_{22})$, as $g,m_{L}\to\infty$, is similar. ∎ In the proofs of Lemmas 2-6, we use the following simple bounds which we gather here for convenience. Uniformly on $\boldsymbol{\omega}\in\mathcal{N}$, there exist fixed constants $0<L_{1}<L_{2}<\infty$ such that both $m_{i}L_{2}^{-1}\leq m_{i}\dot{\sigma}_{\alpha}^{2}\leq\dot{\sigma}_{e}^{2}+m_{i}\dot{\sigma}_{\alpha}^{2}=m_{i}\dot{\sigma}_{\alpha}^{2}\\{\dot{\sigma}_{e}^{2}/m_{i}\dot{\sigma}_{\alpha}^{2}+1\\}\leq m_{i}L_{1}^{-1}$ and for $g$ sufficiently large, $m_{i}L_{2}^{-1}\leq\sigma_{e}^{2}+m_{i}\sigma_{\alpha}^{2}\leq m_{i}L_{1}^{-1}$ hold. It follows that uniformly both in $\boldsymbol{\omega}\in\mathcal{N}$ and $1\leq i\leq g$, $\begin{split}&L_{1}\leq\tau_{i},\dot{\tau}_{i}\leq L_{2},\qquad\|\tau_{i}-\dot{\tau}_{i}\|\leq L_{2}^{2}M(m_{L}^{-1}n^{-1/2}+g^{-1/2})\leq O(g^{-1/2})\\\ &\|\tau_{i}^{2}(1-\dot{\tau}_{i}^{-1}\tau_{i})\|=O(g^{-1/2}),\quad\|\tau_{i}^{2}-\dot{\tau}_{i}^{2}\|\leq\|\tau_{i}-\dot{\tau}_{i}\|\|\tau_{i}+\dot{\tau}_{i}\|=O(g^{-1/2}).\end{split}$ (8) We also require the moments of $\bar{e}_{i}=m_{i}^{-1}\sum_{j=1}^{m_{i}}e_{ij}$ which are $\begin{split}&\operatorname{E}\bar{e}_{i}=0,\qquad\operatorname{Var}\bar{e}_{i}=m_{i}^{-1}\dot{\sigma}_{e}^{2},\qquad\operatorname{E}(\bar{e}_{i}^{3})=m_{i}^{-2}\operatorname{E}e_{11}^{3},\\\ &\operatorname{E}(\bar{e}_{i}^{4})=m_{i}^{-2}3\dot{\sigma}_{e}^{4}+m_{i}^{-3}\\{\operatorname{E}e_{11}^{4}-3\dot{\sigma}_{e}^{4}\\}\leq m_{i}^{-2}3\dot{\sigma}_{e}^{4}+m_{i}^{-3}\operatorname{E}e_{11}^{4};\end{split}$ (9) see for example (Cramér, 1946, p 345). These imply that $\operatorname{Var}(\alpha_{i}+\bar{e}_{i})=(\dot{\sigma}_{\alpha}^{2}+m_{i}^{-1}\dot{\sigma}_{e}^{2})=\dot{\tau}_{i}^{-1}$. ###### Lemma 2. Suppose Condition A holds. Then $\|T_{1}\|=o_{p}(1)$. ###### Proof. We establish the result for each component of $T_{1}={\mathbf{K}}^{-1/2}\\{{\boldsymbol{\psi}}(\dot{\boldsymbol{\omega}})-\boldsymbol{\xi}\\}$. Write $\bar{y}_{i}=\dot{\beta}_{0}+\mathbf{x}_{i}^{(b)T}\dot{\boldsymbol{\beta}}_{1}+\bar{\mathbf{x}}_{i}^{(w)T}\dot{\boldsymbol{\beta}}_{2}+\alpha_{i}+\bar{e}_{i}=\mathbf{z}_{i}^{T}\dot{\boldsymbol{\beta}}+\alpha_{i}+\bar{e}_{i}$, where $\bar{e}_{i}=m_{i}^{-1}\sum_{j=1}^{m_{i}}e_{ij}$, and $y_{ij}-\bar{y}_{i}=(\mathbf{x}_{ij}^{(w)}-\bar{\mathbf{x}}_{i}^{(w)})^{T}\dot{\boldsymbol{\beta}}_{2}+e_{ij}-\bar{e}_{i}$. Then $\begin{split}\mathbf{l}_{\boldsymbol{\beta}_{1}}(\dot{\boldsymbol{\omega}})-\boldsymbol{\xi}_{\boldsymbol{\beta}_{1}}&=\sum_{i=1}^{g}(\dot{\tau}_{i}-1/\dot{\sigma}_{\alpha}^{2})\mathbf{x}_{i}^{(b)}\alpha_{i}+\sum_{i=1}^{g}\dot{\tau}_{i}\mathbf{x}_{i}^{(b)}\bar{e}_{i}.\end{split}$ The $k$th components of the two sums in the last line have mean zero and variances $\operatorname{Var}\\{\sum_{i=1}^{g}(\dot{\tau}_{i}-1/\dot{\sigma}_{\alpha}^{2})x_{ik}^{(b)}\alpha_{i}\\}=\sum_{i=1}^{g}m_{i}^{-2}\dot{\tau}_{i}^{2}\dot{\sigma}_{e}^{4}x_{ik}^{(b)2}\dot{\sigma}_{\alpha}^{2}=O(m_{L}^{-2}g)$ and $\operatorname{Var}\\{\sum_{i=1}^{g}\dot{\tau}_{i}x_{ik}^{(b)}\bar{e}_{i(s)}\\}=\sum_{i=1}^{g}m_{i}^{-1}\dot{\tau}_{i}^{2}\dot{\sigma}_{e}^{2}x_{ik}^{(b)2}=O(m_{L}^{-1}g),$ respectively, using (8) and (9). It follows that $\mathbf{l}_{\boldsymbol{\beta}_{1}}(\dot{\boldsymbol{\omega}})-\boldsymbol{\xi}_{\boldsymbol{\beta}_{1}}=o_{p}(g^{1/2})$ and, by essentially the same argument, $l_{\beta_{0}}(\dot{\boldsymbol{\omega}})-\xi_{\beta_{0}}=o_{p}(g^{1/2})$. For the estimating equation for the between variance component, write $\begin{split}l_{\sigma_{\alpha}^{2}}(\dot{\boldsymbol{\omega}})-\xi_{\sigma_{\alpha}^{2}}=\frac{1}{2}\sum_{i=1}^{g}(\dot{\tau}_{i}^{2}-1/\dot{\sigma}_{\alpha}^{4})(\alpha_{i}^{2}-\dot{\sigma}_{\alpha}^{2})+\frac{1}{2}\sum_{i=1}^{g}\dot{\tau}_{i}^{2}(2\bar{e}_{i}\alpha_{i}+\bar{e}_{i}^{2}-\dot{\sigma}_{e}^{2}/m_{i}).\end{split}$ From (8) and (9), the variances of the sums are $O(m_{L}^{-2}g)$ and $O(m_{L}^{-1}g)$ so $l_{\sigma_{\alpha}^{2}}(\dot{\boldsymbol{\omega}})-\xi_{\sigma_{\alpha}^{2}}=o_{p}(g^{1/2})$. Next, we can write $\mathbf{S}_{w}^{xy}=\mathbf{S}_{w}^{x}\dot{\boldsymbol{\beta}}_{2}+\sum_{i=1}^{g}\sum_{j=1}^{m_{i}}(\mathbf{x}_{ij}^{(w)}-\bar{\mathbf{x}}_{i}^{(w)})e_{ij}$ so, for the within slope parameter, $\begin{split}\mathbf{l}_{\boldsymbol{\beta}_{2}}(\dot{\boldsymbol{\omega}})-\boldsymbol{\xi}_{\boldsymbol{\beta}_{2}}&=\sum_{i=1}^{g}\dot{\tau}_{i}\bar{\mathbf{x}}_{i}^{(w)}(\alpha_{i}+\bar{e}_{i})=o_{p}(n^{-1/2}),\end{split}$ because $g=o(n)$. For the within variance component, expanding $S_{w}^{y}$, we show that $\begin{split}&S_{w}^{y}-2\dot{\boldsymbol{\beta}}_{2}^{T}\mathbf{S}_{w}^{xy}+\dot{\boldsymbol{\beta}}_{2}^{T}\mathbf{S}_{w}^{x}\dot{\boldsymbol{\beta}}_{2}=(n-g)\dot{\sigma}_{e}^{2}+\sum_{i=1}^{g}\sum_{j=1}^{m_{i}}(e_{ij}^{2}-\dot{\sigma}_{e}^{2})+o_{p}(n^{1/2}).\end{split}$ (10) It then follows that $\begin{split}l_{\sigma_{e}^{2}}(\dot{\boldsymbol{\omega}})-\xi_{\sigma_{e}^{2}}&=\frac{1}{2}\sum_{i=1}^{g}m_{i}^{-1}\dot{\tau}_{i}^{2}(\alpha_{i}^{2}-\dot{\sigma}_{\alpha}^{2}+2\alpha_{i}\bar{e}_{i}+\bar{e}_{i}^{2}-\dot{\sigma}_{e}^{2}/m_{i})+o_{p}(n^{1/2}).\end{split}$ Since $\operatorname{E}(\alpha_{i}^{2}-\dot{\sigma}_{\alpha}^{2}+2\alpha_{i}\bar{e}_{i}+\bar{e}_{i}^{2}-\dot{\sigma}_{e}^{2}/m_{i})^{2}=\operatorname{E}(\alpha_{1}^{2}-\dot{\sigma}_{\alpha}^{2})^{2}+m_{i}^{-1}4\dot{\sigma}_{\alpha}^{2}\dot{\sigma}_{e}^{2}+\operatorname{Var}(\bar{e}_{i}^{2})$, we have $l_{\sigma_{e}^{2}}(\dot{\boldsymbol{\omega}})-\xi_{\sigma_{e}^{2}}=o_{p}(n^{1/2})$, which completes the proof. ∎ ###### Lemma 3. Suppose Condition A holds. Then, there exists a finite constant $L$ such that $\begin{split}&\underset{\boldsymbol{\omega}\in\mathcal{N}}{\sup}\operatorname{Var}\\{l_{\beta_{0}}(\boldsymbol{\omega})-l_{\beta_{0}}(\dot{\boldsymbol{\omega}})\\}\leq L,\quad\underset{\boldsymbol{\omega}\in\mathcal{N}}{\sup}\operatorname{Var}\\{l_{\beta_{1k}}(\boldsymbol{\omega})-l_{\beta_{1k}}(\dot{\boldsymbol{\omega}})\\}\leq L,\\\ &\underset{\boldsymbol{\omega}\in\mathcal{N}}{\sup}\operatorname{Var}\\{l_{\sigma_{\alpha}^{2}}(\boldsymbol{\omega})-l_{\sigma_{\alpha}^{2}}(\dot{\boldsymbol{\omega}})\\}\leq L,\quad\underset{\boldsymbol{\omega}\in\mathcal{N}}{\sup}\operatorname{Var}\\{l_{\beta_{2r}}(\boldsymbol{\omega})-l_{\beta_{2r}}(\dot{\boldsymbol{\omega}})\\}\leq L,\\\ &\underset{\boldsymbol{\omega}\in\mathcal{N}}{\sup}\operatorname{Var}\\{l_{\sigma_{e}^{2}}(\boldsymbol{\omega})-l_{\sigma_{e}^{2}}(\dot{\boldsymbol{\omega}})\\}\leq L,\,\,\,\,k=1,\ldots,p_{b},r=1,\ldots,p_{w}.\end{split}$ ###### Proof. We write out $\boldsymbol{\psi}(\boldsymbol{\omega})-\boldsymbol{\psi}(\dot{\boldsymbol{\omega}})$, take the variance (which eliminates all no-stochastic terms) and then bound the variance uniformly on $\boldsymbol{\omega}\in\mathcal{N}$ using (8) and (9). For example,we have $\begin{split}l_{\beta_{0}}(\boldsymbol{\omega})-l_{\beta_{0}}(\dot{\boldsymbol{\omega}})&=\sum_{i=1}^{g}\tau_{i}\mathbf{z}_{i}^{T}(\dot{\boldsymbol{\beta}}-\boldsymbol{\beta})+\sum_{i=1}^{g}(\tau_{i}-\dot{\tau}_{i})(\alpha_{i}+\bar{e}_{i})\end{split}$ so $\begin{split}\operatorname{Var}\\{l_{\beta_{0}}(\boldsymbol{\omega})-l_{\beta_{0}}(\dot{\boldsymbol{\omega}})\\}&=\sum_{i=1}^{g}(\tau_{i}-\dot{\tau}_{i})^{2}/\dot{\tau}_{i}\\\ &\leq 2L_{1}^{-1}L_{2}^{2}M^{2}\sum_{i=1}^{g}(m_{i}^{-2}n^{-1}+g^{-1})\leq 4L_{1}^{-3}M^{2}.\end{split}$ The argument for each of the remaining terms is similar ∎ ### 4.3 Lemmas for the derivative of $\boldsymbol{\psi}$ A key part of the proof of Theorem 1 is using the mean value theorem to obtain a linear approximation for $\boldsymbol{\psi}(\boldsymbol{\omega})$. We apply the mean value theorem to each (real) element of ${\boldsymbol{\psi}}(\boldsymbol{\omega})$ so we need to allow different arguments (i.e. values of $\boldsymbol{\omega}$) in each row of the derivative matrix. Let $\boldsymbol{\Omega}$ be a $(p_{b}+p_{w}+3)\times(p_{b}+p_{w}+3)$ matrix and write $\nabla{\boldsymbol{\psi}}(\boldsymbol{\Omega})$ to mean that each row of the derivative $\nabla{\boldsymbol{\psi}}$ is evaluated at the corresponding row of $\boldsymbol{\Omega}$. We also partition $\nabla{\boldsymbol{\psi}}(\boldsymbol{\Omega})$ into submatrices conformably with the between cluster and within cluster parameters. Leting $\boldsymbol{\Omega}^{(b)}$ contain the first $p_{b}+2$ rows and $\boldsymbol{\Omega}^{(w)}$ the remaining $p_{w}+1$ rows of $\boldsymbol{\Omega}$, we can write $\begin{split}\nabla{\boldsymbol{\psi}}(\boldsymbol{\Omega})=\left[\begin{matrix}\nabla{\boldsymbol{\psi}^{(bb)}}({\boldsymbol{\Omega}}^{(b)})&\nabla{\boldsymbol{\psi}}^{(bw)}({\boldsymbol{\Omega}}^{(b)})\\\ \nabla{\boldsymbol{\psi}}^{(wb)}({\boldsymbol{\Omega}}^{(w)})&\nabla{\boldsymbol{\psi}}^{(ww)}({\boldsymbol{\Omega}}^{(w)})\end{matrix}\right].\end{split}$ The arguments to $\nabla\boldsymbol{\psi}^{(wb)}$ and $\nabla\boldsymbol{\psi}^{(bw)}$ are potentially different but, when they are the same, these matrices are the transposes of each other. When the rows of $\boldsymbol{\Omega}$ all equal $\boldsymbol{\omega}^{T}$, we simplify the notation by replacing $\boldsymbol{\Omega}$ and its submatrices by $\boldsymbol{\omega}$. (We discard the transpose because there is no ambiguity in doing so and the notation looks unnecessarily complicated when it is retained.) Again discarding the transpose, we also use $\boldsymbol{\omega}$ as a generic symbol to represent any of the rows of $\boldsymbol{\Omega}$ when the specific choice of row is not important. ###### Lemma 4. Suppose Condition A holds. Then, as $g,m_{L}\to\infty$, $\left\\|\mathbf{B}_{n}-\mathbf{B}\right\\|=o(1)$, where $\mathbf{B}_{n}=-\mathbf{K}^{-1/2}\operatorname{E}\nabla\boldsymbol{\psi}(\dot{\boldsymbol{\omega}})\mathbf{K}^{-1/2}$ and $\mathbf{B}=\left[\begin{matrix}1/\dot{\sigma}_{\alpha}^{2}&\mathbf{c}_{1}^{T}/\dot{\sigma}_{\alpha}^{2}&0&\boldsymbol{0}_{[1:p_{w}]}&0\\\ \mathbf{c}_{1}/\dot{\sigma}_{\alpha}^{2}&\mathbf{C}_{2}/\dot{\sigma}_{\alpha}^{2}&\boldsymbol{0}_{[p_{b}:1]}&\boldsymbol{0}_{[p_{b}:p_{w}]}&\boldsymbol{0}_{[p_{b}:1]}\\\ 0&\boldsymbol{0}_{[1:p_{b}]}&1/(2\dot{\sigma}_{\alpha}^{4})&\boldsymbol{0}_{[1:p_{w}]}&0\\\ \boldsymbol{0}_{[p_{w}:1]}&\boldsymbol{0}_{[p_{w}:p_{b}]}&\boldsymbol{0}_{[p_{w}:1]}&\mathbf{C}_{3}/\dot{\sigma}_{e}^{2}&\boldsymbol{0}_{[p_{w}:1]}\\\ 0&\boldsymbol{0}_{[1:p_{b}]}&0&\boldsymbol{0}_{[1:p_{w}]}&1/(2\dot{\sigma}_{e}^{4})\end{matrix}\right].$ (11) We have $\mathbf{A}=\mathbf{B}$ under normality, but not otherwise. ###### Proof. From the expressions for the elements of $\operatorname{E}\nabla{\boldsymbol{\psi}}(\boldsymbol{\Omega})$ given in the Appendix, we have $\begin{split}&\mathbf{B}_{n}=\\\ &\left[\begin{matrix}g^{-1}\sum_{i=1}^{g}\dot{\tau}_{i}&g^{-1}\sum_{i=1}^{g}\dot{\tau}_{i}\mathbf{x}_{i}^{(b)T}&0&\mathbf{f}^{T}&0\\\ g^{-1}\sum_{i=1}^{g}\dot{\tau}_{i}\mathbf{x}_{i}^{(b)}&g^{-1}\sum_{i=1}^{g}\dot{\tau}_{i}\mathbf{x}_{i}^{(b)}\mathbf{x}_{i}^{(b)T}&\boldsymbol{0}_{[p_{b}:1]}&\mathbf{H}&\boldsymbol{0}_{[p_{b}:1]}\\\ 0&\boldsymbol{0}_{[1:p_{b}]}&(2g)^{-1}\sum_{i=1}^{g}\dot{\tau}_{i}^{2}&\boldsymbol{0}_{[1:p_{w}]}&r\\\ \mathbf{f}&\mathbf{H}^{T}&\boldsymbol{0}_{[p_{w}:1]}&\mathbf{S}_{w}^{x}/n\dot{\sigma}_{e}^{2}+\mathbf{P}&\boldsymbol{0}_{[p_{w}:1]}\\\ 0&\boldsymbol{0}_{[1:p_{b}]}&r&\boldsymbol{0}_{[1:p_{w}]}&q\end{matrix}\right],\end{split}$ (12) where $\mathbf{f}=(gn)^{-1/2}\sum_{i=1}^{g}\dot{\tau}_{i}\bar{\mathbf{x}}_{i}^{(w)T}$, $\mathbf{H}=(gn)^{-1/2}\sum_{i=1}^{g}\dot{\tau}_{i}\mathbf{x}_{i}^{(b)}\bar{\mathbf{x}}_{i}^{(w)T}$, $\mathbf{P}=n^{-1}\sum_{i=1}^{g}\dot{\tau}_{i}\bar{\mathbf{x}}_{i}^{(w)}\bar{\mathbf{x}}_{i}^{(w)T}$ and $q=(2n)^{-1}\sum_{i=1}^{g}m_{i}^{-2}\dot{\tau}_{i}^{2}+(n-g)/(2n\dot{\sigma}_{e}^{4})$ and $r=(4gn)^{-1/2}\sum_{i=1}^{g}m_{i}^{-1}\dot{\tau}_{i}^{2}$. It is straightforward to show from Conditions A3-A4 and the fact that $\dot{\tau}_{i}\rightarrow 1/\dot{\sigma}_{\alpha}^{2}$ uniformly in $1\leq i\leq g$ as $m_{L}\to\infty$ that $\begin{split}\|g^{-1}\sum_{i=1}^{g}\dot{\tau}_{i}\bar{x}_{ik}^{(b)}-c_{1k}/\dot{\sigma}_{\alpha}^{2}\|&\leq\max_{1\leq i\leq g}\|\dot{\tau}_{i}-1/\dot{\sigma}_{\alpha}^{2}\|g^{-1}\sum_{i=1}^{g}\|\bar{x}_{ik}^{(b)}\|\\\ \quad&+\|g^{-1}\sum_{i=1}^{g}\bar{x}_{ik}^{(w)}-c_{1k}\|/\dot{\sigma}_{\alpha}^{2}=o(1).\end{split}$ Similar arguments can be applied to establish the convergence of the terms the remaining terms in $-g^{-1}\operatorname{E}\nabla{\boldsymbol{\psi}^{(bb)}}({\boldsymbol{\Omega}}^{(b)})$ and $-n^{-1}\operatorname{E}\nabla{\boldsymbol{\psi}}^{(ww)}({\boldsymbol{\Omega}}^{(w)})$. Finally, similar arguments can be used to show that $(n/g)^{1/2}$ times the entries in the off-diagonal blocks $(ng)^{-1/2}\operatorname{E}\nabla{\boldsymbol{\psi}}^{(bw)}({\boldsymbol{\Omega}}^{(b)})$ and $(ng)^{-1/2}\operatorname{E}\nabla{\boldsymbol{\psi}}^{(wb)}({\boldsymbol{\Omega}}^{(w)})$ converge and then using the fact that $g=o(n)$ to show that the entries in the off-diagonal blocks converge to zero. ∎ The convergence result for the expected derivative of the estimating equation that we require in order to handle $T_{2}(\boldsymbol{\omega})$ is established in Lemma 5. ###### Lemma 5. Suppose Condition A holds. Then, as $g,m_{L}\to\infty$, $\underset{\boldsymbol{\omega}\in\mathcal{N}}{\sup}\left\\|\mathbf{K}^{-1/2}\left\\{\operatorname{E}\nabla\boldsymbol{\psi}(\boldsymbol{\Omega})-\operatorname{E}\nabla\boldsymbol{\psi}(\dot{\boldsymbol{\omega}})\right\\}\mathbf{K}^{-1/2}\right\\|=o(1).$ ###### Proof. It is enough to show the uniform convergence to zero of the elements of $g^{-1}\\{\operatorname{E}\nabla\boldsymbol{\psi}^{(bb)}(\boldsymbol{\Omega}^{(b)})-\operatorname{E}\nabla\boldsymbol{\psi}^{(bb)}(\dot{\boldsymbol{\omega}})\\}$, $g^{-1/2}n^{-1/2}\\{\operatorname{E}\nabla\boldsymbol{\psi}^{(bw)}(\boldsymbol{\Omega}^{(b)})-\operatorname{E}\nabla\boldsymbol{\psi}^{(bw)}(\dot{\boldsymbol{\omega}})\\}$ and $n^{-1}\\{\operatorname{E}\nabla\boldsymbol{\psi}^{(ww)}(\boldsymbol{\Omega}^{(w)})-\operatorname{E}\nabla\boldsymbol{\psi}^{(ww)}(\dot{\boldsymbol{\omega}})\\}$. These are all deterministic matrices so the result is obtained by directly bounding the components of these matrices. In addition to the bounds (8), we also use the fact that, uniformly in $\boldsymbol{\omega}\in\mathcal{N}$, $\|\mathbf{z}_{i}^{T}(\dot{\boldsymbol{\beta}}-\boldsymbol{\beta})\|\leq Mg^{-1/2}\\{(1+\|\mathbf{x}_{i}^{(b)}\|)+(g/n)^{1/2}\|\bar{\mathbf{x}}^{(w)}_{i}\|\\}$ to obtain bounds of the form $\|g^{-1}\sum_{i=1}^{g}\tau_{i}^{2}\mathbf{z}_{i}^{T}(\dot{\boldsymbol{\beta}}-\boldsymbol{\beta})\|\leq L_{2}^{2}Mg^{-1}\sum_{i=1}^{g}g^{-1/2}\\{(1+\|\mathbf{x}_{i}^{(b)}\|)+(g/n)^{1/2}\|\bar{\mathbf{x}}^{(w)}_{i}\|\\}=O(g^{-1/2}).$ Combining these bounds, we can show that, uniformly in $\boldsymbol{\omega}\in\mathcal{N}$, $g^{-1}\\{\operatorname{E}\nabla\boldsymbol{\psi}^{(bb)}(\boldsymbol{\Omega}^{(b)})-\operatorname{E}\nabla\boldsymbol{\psi}^{(bb)}(\dot{\boldsymbol{\omega}})\\}=O(g^{-1/2})$, $g^{-1/2}n^{-1/2}\\{\operatorname{E}\nabla\boldsymbol{\psi}^{(bw)}(\boldsymbol{\Omega}^{(b)})-\operatorname{E}\nabla\boldsymbol{\psi}^{(bw)}(\dot{\boldsymbol{\omega}})\\}=O(n^{-1/2})$ and $n^{-1}\\{\operatorname{E}\nabla\boldsymbol{\psi}^{(ww)}(\boldsymbol{\Omega}^{(w)})-\operatorname{E}\nabla\boldsymbol{\psi}^{(ww)}(\dot{\boldsymbol{\omega}})\\}=O(n^{-1/2})$ and the result follows. ∎ The final result we require in order to handle $T_{3}(\boldsymbol{\omega})$ and complete the proof of Theorem 1 is given in Lemma 6. ###### Lemma 6. Suppose Condition A holds. As $g,m_{L}\to\infty$, $\begin{split}&\underset{\boldsymbol{\omega}\in\mathcal{N}}{\sup}g^{-1/4}\\|\mathbf{K}^{-1/2}\\{\nabla\boldsymbol{\psi}(\boldsymbol{\Omega})-\operatorname{E}\nabla\boldsymbol{\psi}(\boldsymbol{\Omega})\\}\mathbf{K}^{-1/2}\\|=o_{p}(1).\end{split}$ ###### Proof. Arguing as in the proof of Lemma 5, it is enough to show the uniform convergence to zero of the elements of $g^{-5/4}\\{\nabla\boldsymbol{\psi}^{(bb)}(\boldsymbol{\Omega}^{(b)})-\operatorname{E}\nabla\boldsymbol{\psi}^{(bb)}(\boldsymbol{\Omega}^{(b)})\\}$, $g^{-3/4}n^{-1/2}\\{\nabla\boldsymbol{\psi}^{(bw)}(\boldsymbol{\Omega}^{(b)})-\operatorname{E}\nabla\boldsymbol{\psi}^{(bw)}(\boldsymbol{\Omega}^{(b)})\\}$ and $g^{-1/4}n^{-1}\\{\nabla\boldsymbol{\psi}^{(ww)}(\boldsymbol{\Omega}^{(w)})-\operatorname{E}\nabla\boldsymbol{\psi}^{(ww)}(\boldsymbol{\Omega}^{(w)})\\}$. We use the bounds (8) and the fact that, by direct calculation of means and variances, we have $\begin{split}&\sum_{i=1}^{g}(1+\|\mathbf{x}_{i}^{(b)}\|)\|\alpha_{i}+\bar{e}_{i}\|=O_{p}(g),\quad\sum_{i=1}^{g}\|\bar{\mathbf{x}}_{i}^{(w)}\|\|\alpha_{i}+\bar{e}_{i}\|=O_{p}(g)\quad\mbox{and}\\\ &\sum_{i=1}^{g}\|(\alpha_{i}+\bar{e}_{i})^{2}-\dot{\sigma}_{\alpha}^{2}-m_{i}^{-1}\dot{\sigma}_{e}^{2}\|=O_{p}(g).\end{split}$ For the derivatives with respect to the variance components, we have $\begin{split}\|l_{\sigma_{\alpha}^{2}\sigma_{\alpha}^{2}}(\boldsymbol{\omega})-&\operatorname{E}l_{\sigma_{\alpha}^{2}\sigma_{\alpha}^{2}}(\boldsymbol{\omega})\|\leq L_{2}^{3}\sum_{i=1}^{g}\|(\alpha_{i}+\bar{e}_{i})^{2}-\dot{\sigma}_{\alpha}^{2}-m_{i}^{-1}\dot{\sigma}_{e}^{2}\|\\\ &+2L_{2}^{3}M\sum_{i=1}^{g}\\{g^{-1/2}(1+\|\mathbf{x}_{i}^{(b)}\|)+n^{-1/2}\|\bar{\mathbf{x}}_{i}^{(w)}\|\\}\|\alpha_{i}+\bar{e}_{i}\|=O_{p}(g),\end{split}$ $\begin{split}\|l_{\sigma_{\alpha}^{2}\sigma_{e}^{2}}(\boldsymbol{\omega})-&\operatorname{E}l_{\sigma_{\alpha}^{2}\sigma_{e}^{2}}(\boldsymbol{\omega})\|\leq m_{L}^{-1}L_{2}^{3}\sum_{i=1}^{g}\|(\alpha_{i}+\bar{e}_{i})^{2}-\dot{\sigma}_{\alpha}^{2}-m_{i}^{-1}\dot{\sigma}_{e}^{2}\|\\\ &+m_{L}^{-1}2L_{2}^{3}M\sum_{i=1}^{g}\\{g^{-1/2}(1+\|\mathbf{x}_{i}^{(b)}\|)+n^{-1/2}\|\bar{\mathbf{x}}_{i}^{(w)}\|\\}\|\alpha_{i}+\bar{e}_{i}\|=O_{p}(m_{L}^{-1}g).\end{split}$ and $\begin{split}\|l_{\sigma_{e}^{2}\sigma_{e}^{2}}(\boldsymbol{\omega})&-\operatorname{E}l_{\sigma_{e}^{2}\sigma_{e}^{2}}(\boldsymbol{\omega})\|\leq\sigma_{e}^{-6}\|2(\dot{\boldsymbol{\beta}}_{2}-\boldsymbol{\beta}_{2})^{T}\sum_{i=1}^{g}\sum_{j=1}^{m_{i}}(\mathbf{x}_{ij}^{(w)}-\bar{\mathbf{x}}_{i}^{(w)})e_{ij}+\sum_{i=1}^{g}\sum_{j=1}^{m_{i}}(e_{ij}^{2}-\dot{\sigma}_{e}^{2})\\\ &-\sum_{i=1}^{g}({m_{i}}\bar{e}_{i}^{2}-\dot{\sigma}_{e}^{2})\|+m_{L}^{-2}L_{2}^{3}\sum_{i=1}^{g}\|(\alpha_{i}+\bar{e}_{i})^{2}-\dot{\sigma}_{\alpha}^{2}-m_{i}^{-1}\dot{\sigma}_{e}^{2}\|\\\ &+2m_{L}^{-2}L_{2}^{3}\sum_{i=1}^{g}\\{g^{-1/2}(1+\|\mathbf{x}_{i}^{(b)}\|)+n^{-1/2}\|\bar{x}_{ik}^{(w)}\|\\}\|\alpha_{i}+\bar{e}_{i}\|=O_{p}(n)\end{split}$ because $g<n$ implies $m_{L}^{-2}g^{1/2}<m_{L}^{-2}g<n$. ∎ ## Appendix A Appendix: The derivative and expected derivative of $\boldsymbol{\psi}$ For the first row in $\nabla{\boldsymbol{\psi}}^{(bb)}({\boldsymbol{\Omega}}^{(b)})$, we have $\begin{split}&l_{\beta_{0}\beta_{0}}(\boldsymbol{\omega})=-\sum_{i=1}^{g}\tau_{i},\quad\mathbf{l}_{\beta_{0}\boldsymbol{\beta}_{1}}(\boldsymbol{\omega})^{T}=-\sum_{i=1}^{g}\tau_{i}\mathbf{x}_{i}^{(b)T},\quad l_{\beta_{0}\sigma_{\alpha}^{2}}(\boldsymbol{\omega})=-\sum_{i=1}^{g}\tau_{i}^{2}(\bar{y}_{i}-\mathbf{z}_{i}^{T}\boldsymbol{\beta});\end{split}$ for rows $k=2,\ldots,p_{b}+1$ in $\nabla{\boldsymbol{\psi}}^{(bb)}({\boldsymbol{\Omega}}^{(b)})$, we have $\begin{split}&l_{\beta_{1k}\beta_{0}}(\boldsymbol{\omega})=-\sum_{i=1}^{g}\tau_{i}\mathbf{x}_{ik}^{(b)},\quad\mathbf{l}_{\beta_{1k}\boldsymbol{\beta}_{1}}(\boldsymbol{\omega})^{T}=-\sum_{i=1}^{g}\tau_{i}\mathbf{x}_{ik}^{(b)}\mathbf{x}_{i}^{(b)T},\\\ &l_{\beta_{1k}\sigma_{\alpha}^{2}}(\boldsymbol{\omega})=-\sum_{i=1}^{g}\tau_{i}^{2}\mathbf{x}_{ik}^{(b)}(\bar{y}_{i}-\mathbf{z}_{i}^{T}\boldsymbol{\beta});\end{split}$ and for the $(p_{b}+2)$th row in $\nabla{\boldsymbol{\psi}}^{(bb)}({\boldsymbol{\Omega}}^{(b)})$, we have $\begin{split}&l_{\sigma_{\alpha}^{2}\beta_{0}}(\boldsymbol{\omega})=-\sum_{i=1}^{g}\tau_{i}^{2}(\bar{y}_{i}-\mathbf{z}_{i}^{T}\boldsymbol{\beta}),\quad\mathbf{l}_{\sigma_{\alpha}^{2}\boldsymbol{\beta}_{1}}(\boldsymbol{\omega})^{T}=-\sum_{i=1}^{g}\tau_{i}^{2}\mathbf{x}_{i}^{(b)T}(\bar{y}_{i}-\mathbf{z}_{i}^{T}\boldsymbol{\beta}),\\\ &l_{\sigma_{\alpha}^{2}\sigma_{\alpha}^{2}}(\boldsymbol{\omega})=\frac{1}{2}\sum_{i=1}^{g}\tau_{i}^{2}-\sum_{i=1}^{g}\tau_{i}^{3}(\bar{y}_{i}-\mathbf{z}_{i}^{T}\boldsymbol{\beta})^{2}.\end{split}$ The rows of $\nabla{\boldsymbol{\psi}}^{(bw)}({\boldsymbol{\Omega}}^{(b)})$ are $\begin{split}&\mathbf{l}_{\beta_{0}\boldsymbol{\beta}_{2}}(\boldsymbol{\omega})^{T}=-\sum_{i=1}^{g}\tau_{i}\bar{\mathbf{x}}_{i}^{(w)T},\quad l_{\beta_{0}\sigma_{e}^{2}}(\boldsymbol{\omega})=-\sum_{i=1}^{g}m_{i}^{-1}\tau_{i}^{2}(\bar{y}_{i}-\mathbf{z}_{i}^{T}\boldsymbol{\beta});\\\ &\mathbf{l}_{\boldsymbol{\beta}_{1k}\boldsymbol{\beta}_{2}}(\boldsymbol{\omega})^{T}=-\sum_{i=1}^{g}\tau_{i}x_{ik}^{(b)}\bar{\mathbf{x}}_{i}^{(w)T},\quad l_{\beta_{1k}\sigma_{e}^{2}}(\boldsymbol{\omega})=-\sum_{i=1}^{g}m_{i}^{-1}\tau_{i}^{2}\mathbf{x}_{ik}^{(b)}(\bar{y}_{i}-\mathbf{z}_{i}^{T}\boldsymbol{\beta});\quad\\\ &\mathbf{l}_{\sigma_{\alpha}^{2}\boldsymbol{\beta}_{2}}(\boldsymbol{\omega})^{T}=-\sum_{i=1}^{g}\tau_{i}^{2}\bar{\mathbf{x}}_{i}^{(w)T}(\bar{y}_{i}-\mathbf{z}_{i}^{T}\boldsymbol{\beta}),\quad\\\ &l_{\sigma_{\alpha}^{2}\sigma_{e}^{2}}(\boldsymbol{\omega})=\frac{1}{2}\sum_{i=1}^{g}m_{i}^{-1}\tau_{i}^{2}-\sum_{i=1}^{g}m_{i}^{-1}\tau_{i}^{3}(\bar{y}_{i}-\mathbf{z}_{i}^{T}\boldsymbol{\beta})^{2};\end{split}$ $k=1,\ldots,p_{b}$, and, finally, the rows of $\nabla{\boldsymbol{\psi}}^{(ww)}({\boldsymbol{\Omega}}^{(w)})$ are $\begin{split}&\mathbf{l}_{\beta_{2k}\boldsymbol{\beta}_{2}}(\boldsymbol{\omega})^{T}=-\frac{1}{\sigma_{e}^{2}}\mathbf{S}_{wk}^{xT}-\sum_{i=1}^{g}\tau_{i}\bar{x}_{ik}^{(w)}\bar{\mathbf{x}}_{i}^{(w)T};\\\ &l_{\beta_{2k}\sigma_{e}^{2}}(\boldsymbol{\omega})=-\frac{1}{\sigma_{e}^{4}}\\{S_{wk}^{xy}-\mathbf{S}_{wk}^{xT}\boldsymbol{\beta}_{2}\\}-\sum_{i=1}^{g}m_{i}^{-1}\tau_{i}^{2}\bar{x}_{ik}^{(w)}(\bar{y}_{i}-\mathbf{z}_{i}^{T}\boldsymbol{\beta});\\\ &\mathbf{l}_{\sigma_{e}^{2}\boldsymbol{\beta}_{2}}(\boldsymbol{\omega})^{T}=-\frac{1}{\sigma_{e}^{4}}\\{\mathbf{S}_{w}^{xyT}-\boldsymbol{\beta}_{2}^{T}\mathbf{S}_{w}^{x}\\}-\sum_{i=1}^{g}m_{i}^{-1}\tau_{i}^{2}\bar{\mathbf{x}}_{i}^{(w)T}(\bar{y}_{i}-\mathbf{z}_{i}^{T}\boldsymbol{\beta}),\\\ \ &l_{\sigma_{e}^{2}\sigma_{e}^{2}}(\boldsymbol{\omega})=\frac{1}{2}\sum_{i=1}^{g}m_{i}^{-2}\tau_{i}^{2}+\frac{n-g}{2\sigma_{e}^{4}}-\frac{1}{\sigma_{e}^{6}}(S_{w}^{y}-2\boldsymbol{\beta}_{2}^{T}\mathbf{S}_{w}^{xy}+\boldsymbol{\beta}_{2}^{T}\mathbf{S}_{w}^{x}\boldsymbol{\beta}_{2})\\\ &\qquad\qquad-\sum_{i=1}^{g}m_{i}^{-2}\tau_{i}^{3}(\bar{y}_{i}-\mathbf{z}_{i}^{T}\boldsymbol{\beta})^{2},\end{split}$ $k=1,\ldots,p_{w}$. Here we have written $\mathbf{S}_{wk}^{xT}$ for the $k$th row of $\mathbf{S}_{w}^{x}$ so $\mathbf{S}_{w}^{x}=[\mathbf{S}_{w1}^{x},\ldots,\mathbf{S}_{wp_{w}}^{x}]^{T}$ and $x_{ik}^{(b)}$, $\bar{x}_{ik}^{(w)}$and $S_{wk}^{xy}$ for the $k$th element of $\mathbf{x}_{i}^{(b)}$, $\bar{\mathbf{x}}_{i}^{(w)}$ and $\mathbf{S}_{wk}^{xy}$, respectively, so $\mathbf{x}_{i}^{(b)}=[x_{ik}^{(b)}]$, $\bar{\mathbf{x}}_{ik}^{(w)}=[\bar{x}_{ik}^{(w)}]$ and $\mathbf{S}_{wk}^{xy}=[S_{wk}^{xy}]$. When we need to address the elements of $\mathbf{S}_{w}^{x}$, we write $\mathbf{S}_{w}^{x}=[S_{wkr}^{x}]$. We calculate the expected derivative matrix using $\operatorname{E}(\bar{y}_{i}-\mathbf{z}_{i}^{T}\boldsymbol{\beta})^{2}=\left\\{\mathbf{z}_{i}^{T}(\dot{\boldsymbol{\beta}}-\boldsymbol{\beta})\right\\}^{2}+\dot{\tau}_{i}^{-1}$, $\operatorname{E}(\mathbf{S}_{w}^{xy})=\mathbf{S}_{w}^{x}\boldsymbol{\beta}_{2}$ and $\operatorname{E}(S_{w}^{y}-2\boldsymbol{\beta}_{2}^{T}\mathbf{S}_{w}^{xy}+\boldsymbol{\beta}_{2}^{T}\mathbf{S}_{w}^{x}\boldsymbol{\beta}_{2})=(\dot{\boldsymbol{\beta}}_{2}-\boldsymbol{\beta}_{2})^{T}\mathbf{S}_{w}^{x}(\dot{\boldsymbol{\beta}}_{2}-\boldsymbol{\beta}_{2})+(n-g)\dot{\sigma}_{e}^{2}$. The first row of $\operatorname{E}\nabla{\boldsymbol{\psi}}^{(bb)}({\boldsymbol{\Omega}}^{(b)})$ is $\begin{split}&\operatorname{E}l_{\beta_{0}\beta_{0}}(\boldsymbol{\omega})=-\sum_{i=1}^{g}\tau_{i},\quad\operatorname{E}\mathbf{l}_{\beta_{0}\boldsymbol{\beta}_{1}}(\boldsymbol{\omega})^{T}=-\sum_{i=1}^{g}\tau_{i}\mathbf{x}_{i}^{(b)T},\\\ &\operatorname{E}l_{\beta_{0}\sigma_{\alpha}^{2}}(\boldsymbol{\omega})=-\sum_{i=1}^{g}\tau_{i}^{2}\mathbf{z}_{i}^{T}(\dot{\boldsymbol{\beta}}-\boldsymbol{\beta});\end{split}$ rows $k=2,\ldots,p_{b}+1$ of $\operatorname{E}\nabla{\boldsymbol{\psi}}^{(bb)}({\boldsymbol{\Omega}}^{(b)})$ are $\begin{split}&\operatorname{E}l_{\beta_{1k}\beta_{0}}(\boldsymbol{\omega})=-\sum_{i=1}^{g}\tau_{i}\mathbf{x}_{ik}^{(b)},\quad\operatorname{E}\mathbf{l}_{\beta_{1k}\boldsymbol{\beta}_{1}}(\boldsymbol{\omega})^{T}=-\sum_{i=1}^{g}\tau_{i}\mathbf{x}_{ik}^{(b)}\mathbf{x}_{i}^{(b)T},\\\ &\operatorname{E}l_{\beta_{1k}\sigma_{\alpha}^{2}}(\boldsymbol{\omega})=-\sum_{i=1}^{g}\tau_{i}^{2}\mathbf{x}_{ik}^{(b)}\mathbf{z}_{i}^{T}(\dot{\boldsymbol{\beta}}-\boldsymbol{\beta});\end{split}$ and the $(p_{b}+2)$th row of $\operatorname{E}\nabla{\boldsymbol{\psi}}^{(bb)}({\boldsymbol{\Omega}}^{(b)})$ is $\begin{split}&\operatorname{E}l_{\sigma_{\alpha}^{2}\beta_{0}}(\boldsymbol{\omega})=-\sum_{i=1}^{g}\tau_{i}^{2}\mathbf{z}_{i}^{T}(\dot{\boldsymbol{\beta}}-\boldsymbol{\beta}),\quad\operatorname{E}\mathbf{l}_{\sigma_{\alpha}^{2}\boldsymbol{\beta}_{1}}(\boldsymbol{\omega})^{T}=-\sum_{i=1}^{g}\tau_{i}^{2}\mathbf{x}_{i}^{(b)T}\mathbf{z}_{i}^{T}(\dot{\boldsymbol{\beta}}-\boldsymbol{\beta}),\\\ &\operatorname{E}l_{\sigma_{\alpha}^{2}\sigma_{\alpha}^{2}}(\boldsymbol{\omega})=\frac{1}{2}\sum_{i=1}^{g}\tau_{i}^{2}(1-2\dot{\tau}_{i}^{-1}\tau_{i})-\sum_{i=1}^{g}\tau_{i}^{3}\\{\mathbf{z}_{i}^{T}(\dot{\boldsymbol{\beta}}-\boldsymbol{\beta})\\}^{2}.\end{split}$ The first $p_{w}$ columns of $\operatorname{E}\nabla{\boldsymbol{\psi}}^{(bw)}({\boldsymbol{\Omega}}^{(b)})$ are $\begin{split}&\operatorname{E}\mathbf{l}_{\beta_{0}\boldsymbol{\beta}_{2}}(\boldsymbol{\omega})^{T}=-\sum_{i=1}^{g}\tau_{i}\bar{\mathbf{x}}_{i}^{(w)T},\quad\operatorname{E}\mathbf{l}_{\boldsymbol{\beta}_{1k}\boldsymbol{\beta}_{2}}(\boldsymbol{\omega})^{T}=-\sum_{i=1}^{g}\tau_{i}x_{ik}^{(b)}\bar{\mathbf{x}}_{i}^{(w)T};\\\ &\operatorname{E}\mathbf{l}_{\sigma_{\alpha}^{2}\boldsymbol{\beta}_{2}}(\boldsymbol{\omega})^{T}=-\sum_{i=1}^{g}\tau_{i}^{2}\bar{\mathbf{x}}_{i}^{(w)T}\mathbf{z}_{i}^{T}(\dot{\boldsymbol{\beta}}-\boldsymbol{\beta}),\end{split}$ $k=1,\ldots,p_{b}$. The last column of $\operatorname{E}\nabla{\boldsymbol{\psi}}^{(bw)}({\boldsymbol{\Omega}})$ is $\begin{split}&\operatorname{E}l_{\beta_{0}\sigma_{e}^{2}}(\boldsymbol{\omega})=-\sum_{i=1}^{g}m_{i}^{-1}\tau_{i}^{2}\mathbf{z}_{i}^{T}(\dot{\boldsymbol{\beta}}-\boldsymbol{\beta});\\\ &\operatorname{E}l_{\boldsymbol{\beta}_{1k}\sigma_{e}^{2}}(\boldsymbol{\omega})=-\sum_{i=1}^{g}m_{i}^{-1}\tau_{i}^{2}x_{ik}^{(b)}\mathbf{z}_{i}^{T}(\dot{\boldsymbol{\beta}}-\boldsymbol{\beta});\\\ &\operatorname{E}l_{\sigma_{\alpha}^{2}\sigma_{e}^{2}}(\boldsymbol{\omega})=\frac{1}{2}\sum_{i=1}^{g}m_{i}^{-1}\tau_{i}^{2}(1-2\dot{\tau}_{i}^{-1}\tau_{i})-\sum_{i=1}^{g}m_{i}^{-1}\tau_{i}^{3}\left\\{\mathbf{z}_{i}^{T}(\dot{\boldsymbol{\beta}}-\boldsymbol{\beta})\right\\}^{2},\end{split}$ $k=1,\ldots,p_{b}$, and, finally, the rows of $\operatorname{E}\nabla{\boldsymbol{\psi}}^{(ww)}({\boldsymbol{\Omega}})$ are $\begin{split}&\operatorname{E}\mathbf{l}_{\beta_{2k}\boldsymbol{\beta}_{2}}(\boldsymbol{\omega})^{T}=-\frac{1}{\sigma_{e}^{2}}\mathbf{S}_{wk}^{xT}-\sum_{i=1}^{g}\tau_{i}\bar{x}_{ik}^{(w)}\bar{\mathbf{x}}_{i}^{(w)T},\,\,\,\,k=1,\ldots,p_{w};\\\ &\operatorname{E}l_{\beta_{2k}\sigma_{e}^{2}}(\boldsymbol{\omega})=-\frac{1}{\sigma_{e}^{4}}\mathbf{S}_{wk}^{xT}(\dot{\boldsymbol{\beta}}_{2}-\boldsymbol{\beta}_{2})-\sum_{i=1}^{g}m_{i}^{-1}\tau_{i}^{2}\bar{x}_{ik}^{(w)}\mathbf{z}_{i}^{T}(\dot{\boldsymbol{\beta}}-\boldsymbol{\beta}),\,\,\,\,k=1,\ldots,p_{w};\\\ &\operatorname{E}\mathbf{l}_{\sigma_{e}^{2}\boldsymbol{\beta}_{2}}(\boldsymbol{\omega})^{T}=-\frac{1}{\sigma_{e}^{4}}(\dot{\boldsymbol{\beta}}_{2}-\boldsymbol{\beta}_{2})^{T}\mathbf{S}_{w}^{x}-\sum_{i=1}^{g}m_{i}^{-1}\tau_{i}^{2}\bar{\mathbf{x}}_{ik}^{(w)T}\mathbf{z}_{i}^{T}(\dot{\boldsymbol{\beta}}-\boldsymbol{\beta})\\\ &\operatorname{E}l_{\sigma_{e}^{2}\sigma_{e}^{2}}(\boldsymbol{\omega})=\frac{1}{2}\sum_{i=1}^{g}m_{i}^{-1}\tau_{i}^{2}(1-2\dot{\tau}_{i}^{-1}\tau_{i})+\frac{n-g}{2\sigma_{e}^{4}}\big{(}1-2\frac{\dot{\sigma}_{e}^{2}}{\sigma_{e}^{2}}\big{)}\\\ &\qquad\qquad\qquad\qquad-\frac{1}{\sigma_{e}^{6}}(\dot{\boldsymbol{\beta}}_{2}-\boldsymbol{\beta}_{2})^{T}\mathbf{S}_{w}^{x}(\dot{\boldsymbol{\beta}}_{2}-\boldsymbol{\beta}_{2})-\sum_{i=1}^{g}m_{i}^{-2}\tau_{i}^{3}\\{\mathbf{z}_{i}^{T}(\dot{\boldsymbol{\beta}}-\boldsymbol{\beta})\\}^{2}.\end{split}$ ## References * Anderson [1969] T.W. 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# Study to improve the performance of interferometer with ultra-cold atoms Xiangyu Dong1, Shengjie Jin1, Hongmian Shui1, Peng Peng1, Xiaoji Zhou1,⋆ 1State Key Laboratory of Advanced Optical Communication System and Network, Department of Electronics, Peking University, Beijing 100871, China > Ultra-cold atoms provide ideal platforms for interferometry. The macroscopic > matter-wave property of ultra-cold atoms leads to large coherent length and > long coherent time, which enable high accuracy and sensitivity to > measurement. Here, we review our efforts to improve the performance of the > interferometer. We demonstrate a shortcut method for manipulating ultra-cold > atoms in an optical lattice. Compared with traditional ones, this shortcut > method can reduce manipulation time by up to three orders of magnitude. We > construct a matter-wave Ramsey interferometer for trapped motional quantum > states and significantly increase its coherence time by one order of > magnitude with an echo technique based on this method. Efforts have also > been made to enhance the resolution by multimode scheme. Application of a > noise-resilient multi-component interferometer shows that increasing the > number of paths could sharpen the peaks in the time-domain interference > fringes, which leads to a resolution nearly twice compared with that of a > conventional double-path two-mode interferometer. With the shortcut method > mentioned above, improvement of the momentum resolution could also be > fulfilled, which leads to atomic momentum patterns less than 0.6 $\hbar > k_{L}$. To identify and remove systematic noises, we introduce the methods > based on the principal component analysis (PCA) that reduce the noise in > detection close to the $1/\sqrt{2}$ of the photon-shot noise and separate > and identify or even eliminate noises. Furthermore, we give a proposal to > measure precisely the local gravity acceleration within a few centimeters > based on our study of ultracold atoms in precision measurements. Keywords: Precision Measurement, Ultra-cold atoms, Atomic interferometer, Gravity measurements PACS: 42.50.Dv, 67.10.Ba, 07.60.Ly, 91.10.Pp ## 1 Introduction Precision measurement is the cornerstone of the development of modern physics. Atom-based precision measurement is an important part. In recent years, ultra- cold atoms have attracted extensive interest in different fields (?, ?, ?, ?), because of their macroscopic matter-wave property (?, ?). This property will lead to large coherent length and long coherent time, which enable high fringe contrast (?, ?, ?). Hence, with these advantages, ultra-cold atoms provide ideal stages for precision measurement (?, ?) and have numerous applications (?, ?, ?, ?, ?, ?, ?, ?), ranging from inertia measurements (?, ?, ?) to precision time keeping (?, ?). For example, a Bose-Einstein condensate is used as an atomic source for a high precision sensor, which is released into free fall for up to 750 ms and probed with a $T=130$ ms Mach-Zehnder atom interferometer based on Bragg transitions (?). A trapped geometry is realized to probe gravity by holding ultra-cold cesium atoms for 20 seconds (?), which suppresses the phase variance due to vibrations by three to four orders of magnitude, overcoming the dominant noise source in atom-interferometric gravimeters. With ultra-cold atoms in an optical cavity, the detection of weak force can achieve a sensitivity of 42 $\rm{yN}/\sqrt{\rm{Hz}}$, which is a factor of 4 above the standard quantum limit (?). With the superiority mentioned above, it is obvious to consider an interferometer by ultra-cold atoms to further precision measurements. Interferometers with atoms propagating in free fall are ideally suited for inertia measurements (?, ?, ?, ?, ?, ?, ?). Meanwhile, with atoms held in tight traps or guides, they are better to measure weak localized interactions. For example, a direct measurement to the Casimir-Polder force is performed by I. Carusotto et al. in 2005, which is as large as $10^{-4}$ gravity (?). However, ultra-cold atoms still get some imperfections needed to be surmounted when combining with interferometry. The macroscopic matter-wave property is generated simultaneously with non-linear atom-atom interactions. Phase diffusion caused by interactions limits the coherence time, and ultimately restricts the sensitivity of interferometers. Besides the interrogation time, the momentum splitting as well as the path number also has an impact on the sensitivity. It has been demonstrated by experiments of multipath interferometers that, interferometric fringes can be sharpened due to the higher-harmonic phase contributions of the multiple energetically equidistant Zeeman states (?, ?), whereas a decrease in the average number of atoms per path causes a greater susceptibility to shot noise. Equilibrium between these parameters could lead to an optimal resolution. In addition, we should also pay attention to the signal analysis procedure as the interferometric information is mainly extracted from the signal detected. The resulting resolution severely relies on the probing system. In this review, we mainly introduce our experimental developments that study these fundamental and important issues to improve the performance of interferometer with ultra-cold atoms. The main developments are concentrated in three aspects: increasing coherent time, using multimode scheme and reducing systematic noises. A. Enhanced resolution by increasing coherent time. We introduce an effective and fast (few microseconds) methods, for manipulating ultra-cold atoms in an optical lattice (OL), which can be used to construct the atomic interferometer and increase the coherent time to finally get a higher resolution. This shortcut loading method is a designed pulse sequence, which can be used for preparing and manipulating arbitrary pure states and superposition states. Another advantage of this method is that the manipulation time is much shorter than traditional methods (100 ms$\to$100 $\rm{\mu s}$). Based on this shortcut method, we constructed an echo-Ramsey interferometer (RI) with motional Bloch states (at zero quasi-momentum on S- and D-bands of an OL) (?). Thanks to the rapidity of shortcut methods, more time could be used for the RI process. We identified the mechanisms that reduced the RI contrast, and greatly increased the coherent time (1.3 ms $\to$14.5 ms) by a quantum echo process, which eliminated the influence of contrast attenuation mechanisms mostly. B. Enhanced resolution by multimode scheme. Several efforts have been made to avoiding the decays of interferometric resolution because of the experimental noises. We demonstrated that the improvement of the phase resolution could be accomplished by a noise-resilient multi-component interferometric scheme. With the relative phase of different components remaining stable, increasing the number of paths could sharpen the peaks in the interference fringes, which led to a resolution nearly twice compared with that of a conventional double-path two-mode interferometer. Moreover, improvement of the momentum resolution was fulfilled with optical lattice pulses. We got results of atomic momentum patterns with intervals less than the double recoil momentum. The momentum pattern exhibited 10 main peaks. C. Enhanced resolution by removing the systematic noise. The method to identify and remove systematic noises for ultra-cold atoms is also introduced in this paper. For improving the quality of absorption image, which is the basic detection result in ultra-cold atoms experiments, we developed an optimized fringe removal algorithm (OFRA), making the noise close to the theoretical limit as $1/\sqrt{2}$ of the photon-shot noise. Besides, for the absorption images after preprocessing by OFRA, we applied the principal component analysis to successfully separate and identify noises from different origins of leading contribution, which helped to reduce or even eliminate noises via corresponding data processing procedures. Furthermore, based on our study of ultracold atoms in precision measurements, we demonstrated a scheme for potential compact gravimeter with ultra-cold atoms in a small displacement. The text structure is as follows. In Sec.2, a shortcut method manipulating ultra-cold atoms in an optical lattice and an Echo-Ramsey interferometry with motional quantum states are introduced, which can increase the coherent time. In Secs.3, we prove that the resolution can be increased using a double-path multimode interferometer with spinor Bose-Einstein condensates (BECs) or an optical pulse, both of which can be classified into multimode scheme. In Secs.4, methods for identifying and reducing the systematic noises for ultra- cold atoms are demonstrated. Finally, we give a proposal on gravimeter with ultra-cold atoms in Secs.5. ## 2 Enhanced resolution by increasing coherent time Figure 1: Schematic diagram of the shortcut method (take ground state preparation as an example). (a) At the beginning, the BECs are formed in a weak harmonic trap. (b) Time sequence of shortcut method. (c) Mapping the shortcut process onto the Bloch sphere. The track $A\to C\to\|S\rangle$ and track $A\to B\to E\to M\to\|S\rangle$ represent one pulse and two pulses shortcut process, respectively. (d) After this shortcut process, the desired states of an 3D optical lattice are prepared. (e) Band structure of 1D OL with different quasi-momentum $q$ when $V_{0}=10\;E_{r}$. Reproduced with permission from Ref. (?). The macroscopic coherent properties of ultra-cold atoms (?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?) are conducive to precise measurement. To make full use of the advantages of ultra-cold atomic coherence properties, one method is to reduce the manipulation time and another is to suppress the attenuation of coherence. Firstly, we demonstrated a shortcut process for manipulating BECs trapped in an OL (?, ?, ?). By optimizing the parameters of the pulses, which constitute the sequence of the shortcut process, We can get extremely high fidelity and robustness for manipulating BECs into the desired states, including the ground state, excited states, and superposition states of a one, two or three- dimensional OLs. Another advantage of this method is that the manipulation time is much shorter than that in traditional methods (100 ms$\to$100 $\rm{\mu s}$). This shortcut is composed of optical lattice pulses and intervals that are imposed on the system before the lattice is switched on. The time durations and intervals in the sequence are optimized to transfer the initial state to the target state with high fidelity. This shortcut procedure can be completed in several tens of microseconds, which is shorter than the traditional method (usually hundreds of milliseconds). It can be applied to the fast manipulation of the superposition of Bloch states. Then, based on this method, we constructed an echo-Ramsey interferometer (RI) with motional Bloch states (at zero quasi-momentum on S- and D-bands of an OL) (?). The key to realizing a RI is to design effective $\pi$\- and $\pi/2$ pulses, which can be obtained by the shortcut method (?, ?, ?). Thanks to the rapidity of shortcut methods, more time can be used for the RI process. We identified the mechanisms that reduced the RI contrast, and greatly increased the coherent time (1.3 ms $\to$14.5 ms) by a quantum echo process, which eliminated the influence of most contrast attenuation mechanisms. ### 2.1 Shortcut manipulating ultra-cold atoms Efficient and fast manipulation of BECs in OLs can be used for precise measurements, such as constructing atom-based interferometers and increasing the coherent time of these interferometers. Here we demonstrate an effective and fast (around 100 $\rm{\mu s}$) method for manipulating BECs from an arbitrary initial state to a desired OL state. This shortcut method is a designed pulse sequence, in which the parameters, such as duration and interval of each step, are optimized to maximize fidelity and robustness of the final state. With this shortcut method, the pure Bloch states with even or odd parity and superposition states of OLs can be prepared and manipulated. In addition, the idea can be extended to the case of two- or three-dimensional OLs. This method has been verified by experiments many times and is very consistent with the theoretical analysis (?, ?, ?, ?, ?, ?, ?, ?). We used the simplest one-dimensional standing wave OL to demonstrate the design principle of this method. The OL potential is $V_{OL}(x)=V_{0}\cos^{2}{kx}$, where $V_{0}$ is used to characterize the depth of the OL. Supposing that the target state $\left\|{\psi_{a}}\right\rangle$ is in the OL with depth $V_{0}$, $m$-step preloading sequence has been applied on the initial state $\left\|\psi_{i}\right\rangle$. The final states $\left\|{\psi_{f}}\right\rangle$ is given by $\centering\|\psi_{f}\rangle=\prod_{j=1}^{m}\hat{Q}_{j}\|\psi_{i}\rangle,\@add@centering$ (1) where $\hat{Q}_{j}=e^{-i\hat{H}_{j}t_{j}}$ is the evolution operator of the $j$th step. By maximizing the fidelity $Fidelity=\|\langle\psi_{a}\|\psi_{f}\rangle\|^{2},$ (2) we can get the optimal parameters $\hat{H}_{j}$ and $t_{j}$. This preprocess is called a shortcut method, which can be used for loading atoms into different bands of an optical lattice. For example, the shortcut loading ultra-cold atoms into S-band in a one-dimensional optical lattice is shown in Fig. 1. By setting different initial state and target state, different time sequences can be designed to manipulate atoms, to build different interferometers, which greatly saves the coherent time. Based on this shortcut method, we can prepare exotic quantum states (?, ?, ?) and construct interferometer with motional quantum states of ultra-cold atoms (?). ### 2.2 Increasing coherent time in a Ramsey interferometry with motional Bloch states of ultra-cold atoms Suppressing the decoherence mechanism in the atomic interferometer is beneficial for increasing coherence time and improving the measurement accuracy. Here we demonstrated an echo method can increase the coherent time for Ramsey interferometry with motional Bloch states (at zero quasi-momentum on S- and D-bands of an OL) of ultra-cold atoms (?). The RI can be applied to the measurement of quantum many-body effects. The key challenge for the construction of this RI is to achieve $\pi$\- and $\pi/2$-pulses, because there is no selection rule for Bloch states of OLs. The $\pi$\- or $\pi/2$-pulse sequences can be obtained by the shortcut method (?, ?, ?, ?, ?), which precisely and rapidly manipulates the superposition of BECs at the zero quasi-momentum on the 1st and 3rd Bloch bands. Retaining the OL, we observed the interference between states and measured the decay of coherent oscillations. We identified the mechanisms that reduced the RI contrast: thermal fluctuations, laser intensity fluctuation, transverse expansion induced by atomic interaction, and the nonuniform OL depth. Then, we greatly increased the coherent time (1.3 ms $\to$14.5 ms) by a quantum echo process, which eliminated the influence of most contrast attenuation mechanisms. #### 2.2.1 Ramsey interferometer in an optical lattice This RI starts from BECs of ${}^{87}\rm{Rb}$ at the temperature 50 nK similar to our previous work (?, ?, ?, ?, ?, ?, ?, ?, ?). Then a 1D standing wave OL is formed (Fig. 2). After a shortcut sequence, the BEC is transferred into the ground band of the OL, denoted as $\phi_{S,0}$. Figure 2: Experimental configuration for a Ramsey interferometer in a $V_{0}=10\;E\mathrm{r}$ lattice: (a) The BEC is divided into discrete pancakes in $yz$ plane by an 1-dimensional optical lattice along $x$ axis with a lattice constant $d=426$ nm. (b) Band energies for the S-band and the D-band. (c) Time sequences for the Ramsey interferometry. The atoms are first loaded into the S band of OL, followed by the RI sequence: $\pi/2$ pulse, holding time $t_{OL}$, and the second $\pi/2$ pulse. Finally band mapping is used to detect the atom number in the different bands. (d) The used pulse sequences designed by an optimised shortcut method. Reproduced with permission from Ref. (?). Figure 3: (a) Change of $p_{\mathrm{D}}$, the population of atoms in the D-band, over time $t_{OL}$ with temperature $T=50$ nK. (b) Influence of different mechanisms on the RI. (c) Characteristic time $\tau$ for the different number of $\pi$ pulse $n$ and different temperatures. The circles, squares, and diamonds represent the experimental results and lines are fitting curves. Reproduced with permission from Ref. (?). Fig. 2(b) illustrates that the RI is constructed with Bloch states $\phi_{i,q}$, which includes the ground band $\|S\rangle$, the third band $\|D\rangle$, and their superposition state $\psi=a_{S}\|S\rangle+a_{D}\|D\rangle$ (denoted as $\binom{a_{S}}{a_{D}}$). It is difficult to realize the interferometer with this pseudo-spin system, because there is no selection rule for Bloch states of OLs. However, thanks to the existence of a coherent, macroscopic matter-wave, a $\pi/2$-pulse for BECs in an OL can be obtained, where the Bloch states $\|S\rangle$ and $\|D\rangle$ are to be manipulated to $\|\psi_{1}\rangle$=$(\|S\rangle+\|D\rangle)/\sqrt{2}$ and $\|\psi_{2}\rangle$=$(-\|S\rangle+\|D\rangle)/\sqrt{2}$, respectively. Fig. 2(d) shows the $\pi/2$ pulse we used for the RI (?, ?, ?) with fidelities of $98.5\%$ and $98.0\%$ respectively (?, ?, ?, ?). Fig. 2(c) illustrates the whole process of RI. First, the BEC is transferred to $\|S\rangle$ of an OL. Then, a $\pi/2$-pulse $\hat{L}(\pi/2)$ is applied to atoms, where $\hat{L}(\pi/2)\binom{1}{0}=\frac{1}{\sqrt{2}}\binom{1}{1}$. After holding time $t_{OL}$ and another $\pi/2$-pulse, the state is: $\psi_{f}=\hat{L}(\pi/2)\hat{Q}(t_{OL})\hat{L}(\pi/2)\psi_{i},$ (3) where $\hat{L}(\alpha)=(\cos\frac{\alpha}{2}-i\sin\frac{\alpha}{2})\hat{\sigma}_{y}$. The operator $\hat{Q}(t_{OL})=(\cos\omega t+i\sin\omega t)\hat{\sigma}_{z}$ ($\omega$ corresponds to the energy gap between S and D bands at zeros quasi- momentum). Fig. 3(a) depicts the results $p_{D}(t_{OL})=N_{D}/(N_{S}+N_{D})$ at different $t_{OL}$ for the RI process ($\hat{R}(\pi/2)-\hat{U}(t_{OL})-\hat{R}(\pi/2)$). $N_{\mathrm{S}}$ ($N_{\mathrm{D}}$) represents the number of atoms in S-band (D-band). The period of the oscillation of $p_{D}$ is $41.1\pm 1.0\;\mu s$. This period related to the band gap and the theoretical value is $40.8\;\mu s$. From Fig. 3(a), we can see that the amplitude, or the contrast $C(t_{OL})$, decreases with the increase of $t_{OL}$, where $p_{D}(t_{OL})=[1+C(t_{OL})\cos(\omega t_{OL}+\phi)]/2.$ (4) We defined a characteristic time $\tau$, which corresponds to the time when the $C(t_{OL})$ decreases to $1/e$. Temperature can affect the length of $\tau$. #### 2.2.2 Contrast decay mechanisms To improve RI’s coherent time and performance, we should analyze the mechanisms that cause RI signal attenuation. By solving the Gross-Pitaevskii equation(GPE), which considers the mechanism that may lead to decay, we can get the process of contrast decay in theory. In Fig. 3(b), The following mechanisms are introduced in turn: the effect of the imperfection of the $\pi/2$ pulse (brown dashed line), inhomogeneity of laser wavefront (blue dotted line), the transverse expansion caused by the many-body interaction (blue dashed line), laser intensity fluctuation (the dash-dotted line), and the thermal fluctuations (the orange solid line). Fig. 3(b) illustrates that the theoretical (the orange solid line) and experimental (black dots) curves of the final result are very consistent. #### 2.2.3 An echo-Ramsey interferometer with motional Bloch states of BECs In order to extend the coherence time $\tau$, we proposed a quantum echo method. The echo process refers to a designed $\pi$ pulse ($\hat{L}(\pi)$) that flips the atomic populations of the two bands. So the evolution operator of Echo-RI is $\hat{L}(\pi/2)[\hat{Q}(t_{OL}/2n)\hat{L}(\pi)\hat{Q}(t_{OL}/2n)]^{n}\hat{L}(\pi/2)$, where $n$ is the number of the $\pi$ pulse inserted between the two $\pi/2$ pulses. Table 1: The effects for the contrast decay. Decay factor \| Beam inhomogeneity \| Echo recovery ---\|---\|--- $Dephasing$ \| Momentum dispersion \| Yes $Collision$ \| Unbalance of population \| Yes $Decoherence$ \| fluctuation \| No Fig. 3(c) illustrates the characteristic time $\tau$ for different $n$ and temperatures. And the effects for the contrast decay are listed in Table. 1. It can be seen from Fig. 3(c) that the interferometer with the longest characteristic time (14.5 ms) was obtained when $n\geq 6$ and $T=50$ nK. ## 3 Enhanced resolution by multimode scheme As an essential indicator, the resolution evaluates the performance of interferometers. The resolution is theoretically restricted to shot-noise limit, or sub-shot noise limit (?, ?), however, it will decay easily due to other experimental noises, with those upper limits beyond reach. Therefore, we have made several efforts to increase the resolution in practice. Improvement of the phase resolution was accomplished by a noise-resilient multi-component interferometric scheme. With the relative phase of different components remaining stable, increasing the number of paths could sharpen the peaks in the interference fringes, which leads to a resolution nearly twice compared with that of a conventional double-path two-mode interferometer with hardly any attenuation in visibility. Moreover, improvement of the momentum resolution is fulfilled with optical lattice pulses. Under the condition of 10 $\rm{E_{R}}$ OL depth, atomic momentum patterns with interval less than the double recoil momentum can be achieved, exhibiting 10 main peaks, respectively, where the minimum one we have given was 0.6 $\hbar k_{L}$. The demonstration of these techniques is shown in the next four subsections. ### 3.1 Time evolution of two-component Bose-Einstein condensates with a coupling drive For the multicomponent interferometer, it is necessary to study the interference characteristics of multi-component ultra-cold atoms. Here we introduced a basic method to deal with this problem, which simulates the time evolution of the relative phase in two-component Bose-Einstein condensates with a coupling drive (?). We considered a two-component Bose-Einstein condensate system with weak nonlinear interatomic interactions and coupling drive. In the formalism of the second quantization, the Hamiltonian of such a system can be written as $\hat{H}=\hat{H}_{1}+\hat{H}_{2}+\hat{H}_{int}+\hat{H}_{driv},$ (5) $\hat{H}_{i}=\int dx\Psi_{i}^{\dagger}(x)[-\frac{\hbar^{2}}{2m}\nabla^{2}+V_{i}(x)+U_{i}(x)\Psi_{i}^{\dagger}(x)\Psi_{i}(x)]\Psi_{i}(x),$ (6) $\hat{H}_{int}=U_{12}\int dx\Psi_{1}^{\dagger}(x)\Psi_{2}^{\dagger}(x)\Psi_{1}(x)\Psi_{2}(x),$ (7) $\hat{H}_{driv}=\int dx[\Psi_{1}^{\dagger}(x)\Psi_{2}(x)e^{i\omega_{rf}t}+\Psi_{1}(x)\Psi_{2}^{\dagger}(x)e^{-i\omega_{rf}t}],$ (8) where $i=1$ and $2$. Then the interference between two BEC’s is $I(t)=\frac{1}{2}N+\frac{1}{2}(N_{1}-N_{2})\cos{\omega_{rf}t}+\frac{1}{2}e^{-A(t)}\sin{\omega_{rf}t}\mathcal{R}(t).$ (9) Previous analysis can be used to simulate the time evolution of the relative phase in two-component Bose-Einstein condensates with a coupling drive, as well as to study the interference of multi-component ultra-cold atoms. This simulation would help to construct a multimode interferometer of a spinor BEC (see Subsecs. 3.3). ### 3.2 Parallel multicomponent interferometer with a spinor Bose-Einstein condensate Figure 4: (a) One typical interference picture. These spatial interference fringes come from the five sub-magnetic states of $\left\|F=2\right\rangle$ hyperfine level. (b1-5) Density distributions corresponding to different sub- magnetic components respectively, where the points are the experimental data and the curves are fitting results according to the empirical expression (?, ?, ?). (c) Average of 15 consecutive experimental shots with a visibility reduction to zero for the chosen state $\left\|m_{F}=-1\right\rangle$. Reproduced with permission from Ref. (?). Figure 5: (a1)-(a4) Histograms of relative phases distributions $(\phi_{2}-\phi_{1},\phi_{-2}-\phi_{-1},\phi_{2}-\phi_{-2},and\;\phi_{1}-\phi_{-1})$ respectively. These relative phases show good reproducibility, for the first two are concentrated at about $0^{o}$, while the latter two are concentrated at about $180^{o}$ in 61 consecutive experimental shots; (b) Relative phase distributions of 41 consecutive experimental shots with $t_{0}=3.6$ ms. Distributions of relative phases $\phi_{2}-\phi_{1}$ and $\phi_{1}-\phi_{-1}$ are shown in (b1) and (b2); (c)When $t_{0}=3.5$ ms, distributions of relative phases $\phi_{2}-\phi_{1}$ and $\phi_{1}-\phi_{-1}$ are shown in (c1) and (c2). The polar plots of relative phase vs visibility (shown as angle vs radius) are shown as these insets, respectively, where the value of visibility is an average of the visibility involved in calculation. Reproduced with permission from Ref. (?). Revealing the wave-particle duality, Young’s double-slit interference experiment plays a critical role in the foundation of modern physics. Other than quantum mechanical particles such as photons or electrons which had been proved in this stunning achievement, ultra-cold atoms with long coherent time have got the potential of precision measurements when utilizing this interferometric structure. Here we have demonstrated a parallel multi-state interferometer structure (?) in a higher spin atom system (?, ?, ?), which was achieved by using our spin-2 BEC of ${}^{87}\rm{Rb}$ atoms. The experimental scheme is described as following. After the manufacture of Bose-Einstein condensates in an optical-magnetic dipole trap, we switched off the optical harmonic trap and populate the condensates from $\left\|F=2,m_{F}=2\right\rangle$ state to $\left\|m_{F}=2\right\rangle$ and $\left\|m_{F}=1\right\rangle$ sub-magnetic level equally. After the evolution in a gradient magnetic field for time $t_{1}$, these two wave packets were converted again into multiple $m_{F}$ states ($m_{F}=\pm 2,\pm 1,0$) as our spin states, leading to the so-called parallel path. All these states were allowed to evolve for another period time $t_{2}$, then the time-of-flight (TOF) stage $t_{3}$ for absorption imaging. Spatial interference fringes had been observed in all the spin channels. Here, we used the technique of spin projection with Majorana transition (?, ?, ?) by switching off the magnetic field pulses nonadiabatically to translate the atoms into different Zeeman sublevels. The spatial separation of atom cloud in different Zeeman states was reached by Stern-Gerlach momentum splitting in the gradient magnetic field. A typical picture after 26 ms TOF is shown in Fig. 4(a). Fig. 4(b1-b5) are the density distributions for each interference fringes. To reach the maximal visibility, we studied the correlation between the interference fringes’ visibility and the time interval applying Stern-Gerlach process. Though separated partially, the interfering wave pockets must overlap in a sort of way. The optimal visibility was about 0.6, corresponding to $t_{1}=210\;\rm{\mu s}$ and $t_{2}=1300\;\rm{\mu s}$. We also measured the fringe frequencies of different components, which exhibited a weak dependence on $m_{F}$. Special attention is required in Fig. 4(c). After an average of 15 consecutive CCD shots in repeated experiments, the interference fringe almost disappeared for the chosen state $\left\|m_{F}=-1\right\rangle$. This result manifested the phase difference between the two copies of each component in every experimental run is evenly distributed. The poor phase repeatability could be attributed to uncontrollable phase accumulation in Majorana transitions. However, the relative phase across the spin components remained the same after more than 60 continuous experiments, just as Fig. 5(a) illustrates. Furthermore, evidence has been spotted that the relative phase can be controlled by changing the time $t_{0}$ before the first Majorana transition, as shown in Fig. 5(b)(c), paving a way towards noise-resilient multicomponent parallel interferometer or multi-pointer interferometric clocks (?). ### 3.3 Implementation of a double-path multimode interferometer using a spinor Bose-Einstein condensate Figure 6: (a1-a3) Single-shot spatial interference pattern with five interference modes after TOF = 26 ms. Fringes of each mode are (a1)in phase (a2)partially in phase (a3)complementary in space. (b1-b3) Black points are the experimental data by integrating the image in panels (a1-a3) along the z direction. Red solid lines are fitted by Thomas-Fermi Distribution (?). Visibilities are 0.55, 0.24, and 0.05, respectively. (c) Schematic of the spatial interference image. $\Delta\phi(T_{d},T_{N})$ is the relative phase between adjacent mode fringes. The fringe in each color represents the interference between the two wave packets of a single mode. Reproduced with permission from Ref. (?). Figure 7: Dependence of the visibility on the number of modes N and initial relative phase $\phi_{m_{F}}$ of the same mode in two paths. (a) Dependence on N in a situation that $\phi_{m_{F}}$ are all zero. FWHM of the N-mode fringe is 2/N times that of the two-mode fringe. (b) Dependence on $\phi_{m_{F}}$ using $N=4$ as an example. The green dashed line, red solid line, and purple dotted line show the fringes with $(\phi_{1},\phi_{2},\phi_{3},\phi_{4})=(0,0,0,0)$, $(0,0,\pi,\pi)$, and $(0.7\pi,0.2\pi,0.5\pi,\pi)$, respectively. Reproduced with permission from Ref. (?). The experiment described above was achieved by Stern-Gerlach momentum splitting, separating the wave pockets in different spin states or Zeeman sub- magnetic states in space. The conclusion that relative phases across the spin components remain stable gives us an inspiration to carry on the double-path multimode matter wave interferometer scheme. With the number of paths increased, it will suppress the noise and improve the resolution (?, ?, ?, ?, ?, ?) compared with the conventional double-path single-mode structure. The results show that resolution of the phase measurements is increased nearly twice in time domain interferometric fringes (?). The experimental procedure is similar to the previous one. The major difference lies in the splitting stage, during the optical harmonic trap participating in the preparation of the condensates is not going to switch off until the TOF stage, thus the Stern-Gerlach process in the gradient magnetic field mentioned above cannot significantly split the wave packets. With different momentum atomic clouds are spatially separated only for tens of nanometers, approximately 1% of the BEC size, thus well overlapped (?). As a result, multi-modes from two paths will interfere in one region instead of five. Another difference lies in the second spin projection with non-adiabatic Majorana transition. Here we replace it with a radio frequency pulse for its higher efficiency as a 1 to 5 beam splitter, although that we still use it to transfer the initial condensates into $\left\|m_{F}=2\right\rangle$ and $\left\|m_{F}=1\right\rangle$ sub-magnetic levels. The performance of Majorana transition is better than RF pulse as a 1 to 2 beam splitter. There are also some changes with experimental parameters that count a little and we would not discuss them here. Hence the global view of our interferometer is as follows: The magnetic sublevels are considered as modes in the interferometer, each has its own different phase evolution rates in gradient magnetic field. The double path configuration is made up of Majorana transition as well as the evolution of the first two $m_{F}$ superposed states during time $T_{d}$, makes up (path I, path II). RF pulse leading to the multiple $m_{F}$ superposed states together with their evolution in time $T_{N}$ forms the multi-modes configuration. During the TOF stage, atomic clouds expand and interfere with each other. Owing to the different state-dependent phase evolution rate $\omega_{m_{F}}^{(I,II)}$, the absorption image shows something more than spatial interference fringes, which is a periodic dependence of the visibility on phase evolution time as the function $V_{N}(T_{d},T_{N})$. We refer to it as the time domain interference. Fig. 6(a) shows a group of absorption images with various combinations of $T_{d}$, $T_{N}$. The observed fringe is a superposition of the interference fringes of different modes. Consequently, the visibility depends on the relative phase $\Delta\phi(T_{d},T_{N})$ between the interference fringes of each mode [Fig. 6(c)] and can also be modulated. By carefully analyzing with expression $V_{N}(T_{d},T_{N})=\langle\Psi^{(I)}\|\Psi^{(II)}\rangle$[34], we can acquire the expression of the relative phase between two adjacent components: $\displaystyle\begin{split}\Delta\phi(T_{d},T_{N})&=(\Delta\phi_{m_{F}}-\Delta\theta_{m_{F}-1})\\\ &=\Delta\omega T_{N}+\Delta\theta\end{split}$ (10) where $\Delta\omega$ is the relative phase evolution rate between the two paths, $\Delta\theta$ is the relative initial phase introduced through the double path stage $T_{d}$. Yet we have already demonstrated that the visibility $V_{N}$ is modulated with the period $2\pi/\Delta\omega$ along with how the time domain fringe emerges theoretically. A remarkable feature of the multi-modes interferometer is the enhancement of resolution, which is defined as (fringe period)/(full width at half maximum). We have investigated the resolution of the time domain fringe experimentally and theoretically. It can be influenced by parameters like modes number and initial phase, which is $R(N,\phi_{m_{F}})$. $\phi_{m_{F}}$ refers to the initial relative phase of $m_{F}$ states accumulated in double paths $T_{d}$. Fig. 7 is the numerical results considering an arbitrary number of modes. Fig. 7(a) is under the condition that the phases $\phi_{m_{F}}$ are all the same for any modes. In that case, if we denote $\Delta\omega T_{N}=2n\pi/N$, then the visibility achieves $V_{N}=1$ when n is the multiple of N and a major peak is observed in this case. A remarkable feature of our interferometer is the enhancement of resolution by $N/2$ times without any changes in visibility nor periodical time. It is the harmonics that cause the peak width to decrease with the number of modes increasing in this case (?). Fig. 7(b) indicates $\phi_{m_{F}}$ varies from mode to mode for comparison. Neither the maximum visibility $V_{N}=1$ nor the minimum could be reached. Meanwhile, the time domain fringe shows more than one main peak in one period. Therefore, the initial phase $\phi_{m_{F}}$ needs to be well controlled to achieve the highest possible visibility and clear interference fringe in the time domain. We also experimentally study the time domain fringes. The experimental data (not depicted here) coincides with the numerical results of Fig. 7(b) red line, testifying its superiority to the resolution of the phase measurement. Moreover, the relative phase evolution rate $\Delta\omega$ can be controlled by adjusting the difference between the two paths accumulated in $T_{d}$ stage (?, ?, ?). With enhanced resolution, the sensitivity of interferometric measurements of physical observables can also be improved by properly assigning measurable quantities to the relative phase between two paths, as long as the modes do not interact with each other (?, ?, ?). ### 3.4 Atomic momentum patterns with narrower interval Figure 8: (a) Shortcut method for loading atoms: (a1) after the first two pulses and the $30$ ms holding time in the OL and the harmonic trap, the state becomes the superposition of the Bloch states in S-band with quasi-momenta taking the values throughout the FBZ, and is denoted by $\left\|{\psi\left(0\right)}\right\rangle$. Then $1$ms band mapping is added. (a2) The single pulse acted on the superposed state $\left\|{\psi\left(0\right)}\right\rangle$. (b) the superposed Bloch states of S-band spreading in the FBZ (black circles). The top Patterns in (c) and (d) are the TOF images in experiments. The lower part of (c) and (d) depicts the atomic distributions in experiments (red circles) and theoretical simulations (blue solid lines). There are seven peaks in (c) and ten peaks in (d) with $q=\pm 3\;\hbar k_{L}$. Reproduced with permission from Ref. (?). For ultra-cold atoms used in precise measurement, improving the precision of momentum manipulation is also conducive to improving the measurement resolution. The method to get atomic momentum patterns with narrower interval has been proposed and verified by experiments (?). Here we applied the shortcut pulse to realize the atomic momentum distribution with high resolutions for a superposed Bloch states spreading in the ground band of an OL. While difficult to prepare this superposition of Bloch states, it can be overcome by the shortcut method. First, the atoms are loaded in the superposition of S- and D-bands $(\|S,q=0\rangle+\|D,q=0\rangle)/\sqrt{2}$, where $q$ is the quasi-momentum. Fig. 8(a1) depicts the loading sequence. The atoms in S and D bands Collision between atoms in S and D bands will cause the atoms to gradually transfer to S band with non-zero quasi-momentum. After $30$ ms, as shown in Fig. 8(b), atoms cover the entire ground band from $q=-\hbar k_{L}$ to $\hbar k_{L}$. The momentum distribution of the initial state is a Gaussian-like shape. After an OL standing-wave pulse, which is similar to that in the shortcut process, the different patterns with the narrower interval can be obtained. The standing-wave pulse sequence is shown in Fig. 8(a2). Fig. 8(c) and (d) show the different designs for patterns of multi modes with various numbers of peaks under OL depth 10 $E_{r}$, where the top figures are the absorption images after the pulse and a $25$ ms TOF. The red circles are the experimental results of the atomic distribution along the x-axis from the TOF images. These results are very close to the numerical simulation result (blue lines). For the numerical simulation, we can get the initial superposition of states by fitting the experimental distributions (Fig. 8(b)). Fig. 8(c) and (d) depict the atomic momentum distribution with resolutions of $0.87\;\hbar k_{L}$ interval (seven peaks within $q=\pm 3\;\hbar k_{L}$) and $0.6\;\hbar k_{L}$ (ten peaks within $q=\pm 3\;\hbar k_{L}$) interval, respectively. The superposed states with different quasi-momenta in the ground band cause the narrow interval (far less than double recoil momentum) between peaks, which is useful to improve the resolution of atom interferometer (?). ## 4 Enhanced resolution by removing the systematic noise Noise identification as well as removal is crucial when extracting useful information in ultra-cold atoms absorption imaging. In general concept, systematic noises of cold atom experiments originate from two sources, one is the process of detection, such as optical absorption imaging; the other is the procedure of experiments, such as the instability of experimental parameters. Here we provided an OFRA scheme, reducing the noise to a level near the theoretical limit as $1/\sqrt{2}$ of the photo-shot noise. When applying the PCA, we found that the noise origins, which mainly come from the fluctuations of atom number and spatial positions, much fewer than the data dimensions of TOF absorption images. These images belong to BECs in one dimensional optical lattice, where the data dimension is actually the number of image pixels. If the raw TOF data can be preprocessed with normalization and adaptive region extraction methods, these noises can be remarkably attenuated or even wiped out. PCA of the preprocessed data exhibits a more subtle noises structure. When we compare the practical results with the numerical simulations, the few dominant noise components reveal a strong correlation with the experimental parameters. These encouraging results prove that the OFRA as well as PCA can be a promising tool for analysis in interferometry with higher precision (?, ?). ### 4.1 Optimized fringe removal algorithm for absorption images Optical absorption imaging is an important detection technique to obtain information from matter waves experiments. By comparing the recorded detection light field with the light field in the presence of absorption, we can easily attain the atoms’ spatial distribution. However, due to the inevitable differences between two recorded light field distributions, detection noises are unavoidable. Figure 9: Comparison between ordinary method and OFRA method. The integral of the atomic distribution in the red box in (a1) and (b1) correspond to (a2) and (b2). The atomic distribution (blue dots) is fitted by a bi-modal function to extract the temperature of atoms, which is shown in (c). Reproduced with permission from Ref. (?). Therefore, we have demonstrated an OFRA scheme to generate an ideal reference light field. With the algorithm, noise generated by the light field difference could be eliminated, leading to a noise close to the theoretical limit (?). The OFRA scheme is based on the PCA, we confirmed its validity by experiments of triangular optical lattices. The experimental configuration has been described in our prior work (?, ?). When the experiment was in process, the depth of the lattice was adiabatically raised to a final value, followed by a hold time of 20 ms to keep the atoms in the lattice potential before the optical absorption imaging. There are several parameters to characterize the triangle lattice system (?), among which the visibility, the condensate fraction, and the temperature matter. Fig. 9 (a2) and Fig. 9 (b2) are bimodal fitting to the scattering peaks by summing up the atomic distribution within the red box in the direction perpendicular to the center. Here Fig. 9 (a) stands for the common way of calculation and 9(b) for the OFRA. The bi-modal curve is composed of two parts: a Gaussian distribution for the thermal component and an inverse parabola curve for the condensed atoms. For each part, the column densities along the imaging axis can be written as $\displaystyle n_{th}(x)$ $\displaystyle=$ $\displaystyle\frac{n_{th}(0)}{g_{2}(1)}g_{2}[\exp(-(x-x_{0})^{2}/\sigma_{T}^{2})],$ (11) $\displaystyle n_{c}(x)$ $\displaystyle=$ $\displaystyle n_{c}(0)\max[1-\frac{(x-x_{0})^{2}}{\chi^{2}}].$ In the formula there are 5 parameters accounting for the bi-mode fitting, the amplitude of two components $n_{th}(0)$ and $n_{c}(0)$, the width of two components $\sigma_{T}$, $\chi$ and the center position $x_{0}$ of the atomic cloud. The Bose function is defined as $g_{j}(z)=\sum_{i}z^{i}/i^{j}$. In practice, we performed the least-squares fitting of $n_{th}(x)+n_{c}(x)$ to the real distribution obtained from the imaging. From the fit, we can get the atom number and width of the two components separately. Note that the measurement in Fig. 9 (9c) is performed at different lattice depths. For each lattice depth, 30 experiments have been performed to acquire the statistical results. The temperature is given as $T=1/2M\sigma_{T}^{2}/t_{TOF}^{2}/k_{B}$, where $M$ described the atom mass and $t_{TOF}$ width of the thermal part (?). For the number of condensed atoms, the fitting outcome is less affected by the fringe shown in Fig. 9 (a). Whereas the influence of the fringe on the fitting of temperature is much more evident. The temperature is proportional to the width of the Gaussian distribution $\sigma_{T}$ as mentioned above. Fig. 9(c) shows the temperature extracted from the TOF absorption images with and without the OFRA separately, namely Fig. 9(a1) and 9(b1). Fig. 9(a2) and 9(b2) are the corresponding integrated one-dimensional atomic distributions for each method. Fig. 9 depicts that the temperature we get with the common way of calculation has a large error of 400 nK, extraordinary higher than the initial BEC temperature of 90 nK. The turning on the procedure of lattice potential would indeed lead to a limited heating effect, nevertheless the proportion of condensed atom should be reduced significantly considering our system has been heated up by 4 times. This is still not consistent with the observation. However, the temperature is measured with much small variance at a much reasonable value if we dive into the OFRA scheme. For example, the measured temperature is 123.5 nK for a lattice depth of $V=4\;E_{r}$, with 183.9 nK for $V=9\;E_{r}$. Comparison between these two results illustrates that only by using the fringe removal algorithm we can get a reliable result, especially in the case of small atom numbers when fitting physical quantities such as the temperature. In conclusion, with this algorithm, we can measure parameters with higher contradiction to the conventional methods. The OFRA scheme is easy to implement in absorption imaging-based matter-wave experiments as well. There is no need to do any changes to the experimental system, only some algorithmic modifications matter. ### 4.2 Extraction and identification of noise patterns for ultracold atoms in an optical lattice Furthermore, on the basis of the absorption images after preprocessing by OFRA, the PCA method is used to identify the external noise fluctuation of the system caused by the imperfection of the experimental system. The noise can be reduced or even eliminated by the corresponding data processing program. It makes the task more difficult that these external systematic noises are often coupled, covered by nonlinear effects and a large number of pixels. PCA provides a good method to solve this problem (?, ?, ?, ?, ?). PCA can decompose the fluctuations in the experimental data into eigenmodes and provide an opportunity to separate the noises from different sources. For BEC in a one-dimensional OL, it was proved that PCA could be applied to the TOF images, where it successfully separated and recognizing noises from different main contribution sources, and reduced or even eliminated noises by data processing programs (?). The purpose of PCA is to use the smallest set of orthogonal vectors, called principal components (PCs) to approximate the variations of data while preserving the information of datasets as much as possible. The PCs correspond to the fluctuations of the experimental system, which can help to distinguish the main features of fluctuations. In the experimental system of BECs, the data are usually TOF images. A specific TOF image $A_{i}$ can be represented by the sum of the average value of the images and its fluctuation: $\displaystyle A_{i}=\bar{A}+\sum\varepsilon_{ij}P_{j},$ (12) Here $P_{j}$ is the different eigenmodes of fluctuation, $\varepsilon_{ij}$ is the weight of the eigenmodes $P_{j}$. Taking the BEC experiment (?, ?, ?, ?) in an OL as an example, we demonstrated the protocol of PCA for extraction noise. A TOF image for ultra-cold atoms in experiments can be represented as a $h\times w$ matrix. The PCA progress, shown in Fig. 10, will be applied to the images: Figure 10: Process diagram of PCA method. (a) Transform the $h\times w$ matrix (raw images) into a $1\times hw$ vector, denoted by $A_{i}$. And stack these vectors together. (b) Calculate the mean vector $\bar{A}=\frac{1}{n}\sum_{i=1}^{n}A_{i}$ and the fluctuations $\delta_{i}=A_{i}-\bar{A}$. Leaving only the fluctuation term in the matrix. (c) Stack $\delta_{i}$ together to form a matrix $X=\left[\delta_{1},\delta_{2},\cdots,\delta_{n}\right]$. Then the covariance matrix $S$ is obtained by $S=\frac{1}{n-1}X\cdot X^{T}$. (d) Decompose covariance matrix so that ${{V}^{-1}{S}{V}={D}}$ where D is a diagonal matrix. (e) Transform eigenvectors of interest back to a new TOF image. Reproduced with permission from Ref. (?). (1) Transform the $h\times w$ matrix into a $1\times hw$ vector, denoted by $A_{i}$. (2) Express $A_{i}$ as the sum of the average value $\bar{A}$ and the fluctuation $\delta_{i}$, where $\bar{A}=\frac{1}{n}\sum_{i=1}^{n}A_{i}$ and $\delta_{i}=A_{i}-\bar{A}$. (3) Stack $\delta_{i}$ together to form a matrix $X=\left[\delta_{1},\delta_{2},\cdots,\delta_{n}\right]$. Then the covariance matrix $S$ is obtained by $S=\frac{1}{n-1}X\cdot X^{T}$. (4) Decompose covariance matrix with ${{V}^{-1}{S}{V}={D}}$, where $V$ is the matrix of eigenvectors. Figure 11: PCA results of the TOF images. (a) Example of a raw TOF image. (b)-(f) correspond to the fluctuation in atom number (b), atom position (c)(d), peak width (e) and normal phase fraction (f), respectively. (a2), (b2), (d2), (e2), and (f2) are the integrated results of atom distributions along x direction. (c2) is for the atom distribution along z direction. The blue lines are the experimental results, and the orange lines are the simulation results. Reproduced with permission from Ref. (?). Fig. 11 shows the PCA results of the TOF images. Fig. 11(b)-(f) correspond to the first to fifth PCs, respectively. The first PC is from the fluctuation in the atom number. The normalization process can be applied to reducing the fluctuation in the atomic number. Because for a macroscopic wave function $\Psi{\left(\textbf{r}\right)}=\sqrt{N}\phi{\left(\textbf{r}\right)}$ (?), we usually concentrate on the relative density distribution, instead of the $\sqrt{N}$. After the normalization, the impact of this PC becomes very small. The second and third PCs correspond to the position fluctuations along the $z$\- and $x$-directions, respectively. The fluctuation in spatial position of the TOF images originates from the vibration of the system structure, such as the OL potential, trapping potential, and imaging system. We used a dynamic extraction method to eliminate the fluctuation in spatial position. We chose a region whose center is also the center of the density distribution. We first set a criterion to determine the center of the density distribution in the extraction area, and then used this center as the center of the new area to extract the new one. We repeated this process until the region to be extracted becomes stable. The fourth PC is from the fluctuation in the width of the Bragg peaks in the TOF images. The final PC shown in Fig. 11(f1) comes from the normal phase fraction fluctuation. By studying the first five feature images, we have identified the physical origins of several PCs leading to the main contributions. We numerically simulated this understanding using the GPE with external fluctuation terms, and got very consistent results (?). It is helpful to understand the physical origins of PCs in designing a pretreatment to reduce or even eliminate fluctuations in atom number, spatial position and other sources. Even in the absence of any knowledge of the system, the PCA method is very effective to analyze the noise, so that it can be applied to interferometers with higher precision (?, ?). ## 5 Proposal on gravity measurements Figure 12: Schematic of the experimental protocol. ${}^{87}\rm{Rb}$ atoms are evaporation cooled as a Bose-Einstein Condensate in the $\left\|F=2,m_{F}=0\right\rangle$ initial state. However, they are transferred to $\left\|F=1,m_{F}=0\right\rangle$ as the two arms of interferometry, followed by $P$ sequence of accelerate optical lattice pulse to maintain the atoms against gravity: When atoms fall to a velocity of $q\times v_{Recoil}$, they acquire a velocity of $2q\times v_{Recoil}$ upwards. The delay $T_{Bloch}$ is chosen as $T_{Bloch}=2qv_{Recoil}/g$ to eliminate the fall caused by gravity. Still, the probe beam should consist a laser light resonant with the $F=1$ ground state to upper levels, thus we can take absorption photos of $F=1$ population for analysis. Inertia measurements (?, ?, ?, ?, ?, ?, ?), especially those for gravity acceleration $g$, have always drawn lots of attention. Until now, the performance of atom interferometry has reached a sensitivity of $8\times 10^{-9}$ at 1 second (?, ?), pushing forward the determination of the Newtonian gravitational constant G (?, ?) or the verification of equivalence principle (?, ?). Yet the bulky size of these quantum sensors strictly restricts their application for on-site measurements. Therefore, based on our previous study of ultra-cold atoms in precision measurements, we intend to precisely measure the local gravity acceleration with our ${}^{87}\rm{Rb}$ Bose-Einstein Condensates in a small displacement. Note that this conception bases on previous research of Perrier Cladé (?). Fig. 12 illustrates the protocol of this BEC gravimeter. Instead of the Mach- Zender method $(\pi/2-\pi-\pi/2)$ widely used in free-falling or atomic fountain gravimeters, we utilize the Ramsey-Bordé approach by two pairs of $\pi/2$ Raman pulses to get a small volume. Application of the Doppler- sensitive Raman beam rather than the 6.8 GHz microwave field provides a far more efficient way to realize larger momentum splitting, which will significantly boost the interference resolution. Raman light pulse can also attain the effects of velocity distribution. Consequently, the first pair of $\pi/2$ pulses selects the initial velocity while the second pair can measure the final distribution. It should be noticed that right after the velocity selection step, a cleaning light pulse resonant with the $D_{2}$ line will shine on the condensate to clear away atoms remaining in $F=2$ state, leading to the two arms in interferometry. The critical feature lies in the evolving stage between the two pairs of $\pi/2$ pulses. By periodically inverting the velocity, the two arms shall replicate parabolic trajectory in a confined volume, without any decreasing of interrogation time. Choosing appropriate parameters, displacement of ultra- cold atoms can be limited in a few centimeters, at least 1 order less than that of a Mach-Zender gravimeter. This assumption is accomplished by a succession of Bloch oscillation (BO) in a pulsed accelerated optical lattice which transfers many photon recoils to the condensates (?, ?, ?, ?). Here the application of ultra-cold atoms instead of optical molasses (?) makes it more efficient when loading the atoms in the first Brillouin zone adiabatically, owing to their wave function consistency and narrow velocity distribution. This pulsed accelerated optical lattice should be manufactured along the direction of gravity with higher lattice depth, to minimize the effect of Landau-Zener Tunneling loss. To deduce the value of g, we may scan the evolving time between the two pairs of $\pi/2$ pulses, keeping the Raman frequency of each pair of $\pi/2$ pulses fixed. When the time interval equals $2Pqv_{Recoil}/g$, where $P$ is the number of pulses and $q$ is the number of recoil velocities($v_{Recoil}$) obtained by a single pulse (shown in Fig. 12), the two arms are in phase and the value of g can be extrapolated. Here the absorption image is used to extract the interference information, due to the number of atoms being one order less than that obtained by the conventional method. We also give a qualitative analysis of this compact BEC gravimeter. Besides its enormous potential in transportable instruments, prospective sensitivity maybe even better. This encouraging outlook can be attributed to a longer coherent time of ultra-cold atoms where the phase shift scales quadratically. The smaller range of movement possesses other superiorities. Systematic errors stemming from the gradients of residual magnetic fields and light fields become negligible, especially for Gouy phase and wave-front aberrations (?, ?, ?). Furthermore, the value of the gravity acceleration is averaged over a smaller height compared with the Mach-Zender ones. Finally, this vertical Bloch oscillation technique offers a remarkable ability to coherently and efficiently transfer photon momenta (?), though decoherence induced by the inhomogeneity of the optical lattice must be taken into consideration. In conclusion, we believe that this compact BEC gravimeter will have a sensitivity of a few tens of ${\mu}$Gal at least. The falling distance will be no more than 2 centimeters. Further improvement should be possible by performing atom chip-assisted BEC preparation as well as the interaction- suppressing mechanism (?). Gravity measurements with sub-${\mu}$Gal accuracies in miniaturized, robust devices are sure to come in the future. ## 6 Conclusion In summary, we review our recent experimental developments on the performance of interferometer with ultra-cold atoms. First, we demonstrated a method for effective preparation of a BEC in different bands of an optical lattice within a few tens of microseconds, reducing the loading time by up to three orders of magnitude as compared to adiabatic loading. Along with this shortcut method, a Ramsey interferometer with band echo technique is employed to atoms within an OL, enormously extend the coherence time by one order of magnitude. Efforts to boost the resolution with multimode scheme is made as well. Application of a noise-resilient multi-component interferometric scheme shows that increasing the number of paths could sharpen the peaks in the time-domain interference fringes, which leads to a resolution nearly twice compared with that of a conventional double-path two-mode interferometer. We can somehow boost the momentum resolution meanwhile. The patterns in the momentum space have got an interval far less than the double recoil momentum, where the narrowest one is given as $0.6\;\hbar k_{L}$. However, these advancements are inseparable with our endeavor to optimize data analysis based on the PCA. Extrinsic systematic noise for absorption imaging can be reduced efficiently. A scheme for potential compact gravimeter with ultra-cold atoms has been proposed. 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# Re-evaluation of Spin-Orbit Dynamics of Polarized $e^{+}e^{-}$ Beams in High Energy Circular Accelerators and Storage Rings: an approach based on a Bloch equation††thanks: Based on a talk at IAS, Hong Kong, January 17, 2019. Also available as article: _Int. J. Mod. Phys._ , vol. A35, Nos. 15 & 16, 2041003, 2020. Moreover available as DESY Report 20-137. ###### Abstract We give an overview of our current/future analytical and numerical work on the spin polarization in high-energy electron storage rings. Our goal is to study the possibility of polarization for the CEPC and FCC-ee. Our work is based on the so-called Bloch equation for the polarization density introduced by Derbenev and Kondratenko in 1975. We also give an outline of the standard approach, the latter being based on the Derbenev-Kondratenko formulas. Keywords: electron storage rings, spin-polarized beams, polarization density, FCC, CEPC, stochastic . differential equations, method of averaging. Klaus Heinemann 111Corresponding author. Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, USA <EMAIL_ADDRESS> Daniel Appelö 222Now at Michigan State University, USA. (email: <EMAIL_ADDRESS> Department of Applied Mathematics, University of Colorado Boulder, Boulder, CO 80309-0526, USA <EMAIL_ADDRESS> Desmond P. Barber Deutsches Elektronen-Synchrotron (DESY) Hamburg, 22607, Germany and: Department of Mathematics and Statistics, University of New Mexico Albuquerque, NM 87131, USA <EMAIL_ADDRESS> Oleksii Beznosov Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, USA <EMAIL_ADDRESS> James A. Ellison Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, USA <EMAIL_ADDRESS> PACS numbers:29.20.db,29.27.Hj,05.10.Gg ###### Contents 1. 1 Introduction 2. 2 Sketching the standard approach based on the Derbenev-Kondratenko formulas 3. 3 The Bloch equation and the Reduced Bloch equation in the laboratory frame 4. 4 The Reduced Bloch equation in the beam frame 5. 5 The Effective Reduced Bloch equation in the beam frame 6. 6 Next steps 7. 7 Acknowledgement ## 1 Introduction This paper is an update on a talk by K. Heinemann at the IAS Mini-Workshop on Beam Polarization in Future Colliders on January 17, 2019, in Hong Kong [1]. Our ultimate goal is to examine the possibility of high polarization for CEPC and FCC-ee. We will first briefly review the “standard” approach which is based on the Derbenev-Kondratenko formulas [2]. These formulas rely, in part, on plausible assumptions grounded in deep physical intuition. So the following question arises: do the Derbenev-Kondratenko formulas tell full story? In fact there is an alternative approach based on a Bloch-type equation for the polarization density [3] which we call 333Note that in previous work we sometimes called it the full Bloch equation. the Bloch equation (BE) and which we believe can deliver more information than the standard approach even if the latter includes potential correction terms [4]. So we aim to determine the domain of applicability of the Derbenev-Kondratenko formulas and the possibility in theory of polarization at the CEPC and FCC-ee energies. Of course both approaches focus on the equilibrium polarization and the polarization time. We use the name “Bloch” to reflect the analogy with equations for magnetization in condensed matter [5]. This paper concentrates on the Bloch approach. The cost of the numerical computations in the Bloch approach is considerable since the polarization density depends on six phase-space variables plus the time variable so that the numerical solution of the BE, the BE being a system of three PDEs in seven independent variables, is a nontrivial task which cannot be pursued with traditional approaches like the finite difference method. However we see at least five viable methods: 1. 1. Approximating the BE by an effective BE via the Method of Averaging and solving the effective BE via spectral phase-space discretization, e.g., a collocation method, plus an implicit-explicit time discretization. 2. 2. Solving the system of stochastic differential equations (SDEs), which underlies the BE, via Monte-Carlo spin tracking. See Ref. 6 for the system of SDEs underlying the BE. 3. 3. Solving the Fokker-Planck equation, which underlies the BE, via the Gram- Charlier method. 4. 4. Solving the BE via a deep learning method. 5. 5. Solving the system of SDEs in a way that allows connections with the Derbenev- Kondratenko formulas to be established. We will dwell on Method 1 in this paper. We plan to validate this method by one of the other four methods. More details on Method 1 can be found in Ref. 6. The method of averaging we use is discussed in Refs. 7-12. One hope tied to Method 1 is that the effective BE gives analytical insights into the spin- resonance structure of the bunch. Note that Methods 1-4 are independent of the standard approach. In particular they do not rely on the invariant spin field. Note also that Methods 1-3 and 5 are based on knowing the system of SDEs, which underlies the BE. For details of this system of SDEs, see the invited ICAP18 paper of Ref. 6. Regarding Method 2 there is a large literature on the numerical solution of SDEs, see Refs. 13, 14 and references in Ref. 15. By neglecting the spin-flip terms and the kinetic-polarization term in the BE one obtains an equation that we call the Reduced Bloch equation (RBE). The RBE approximation is sufficient for computing the radiative depolarization rate due to stochastic orbital effects and it shares the terms with the BE that are challenging to discretize. For details on our phase-space discretization and time discretization of the RBE, see Refs. 6,16,17 and 18. We proceed as follows. In Section 2 we sketch the standard approach. In Section 3 we present, for the laboratory frame, the BE and its restriction, the RBE. In Section 4 we discuss the RBE in the beam frame and in Section 5 we show how, in the beam frame, the effective RBE is obtained via the method of averaging. In Section 6 we describe ongoing and future work. ## 2 Sketching the standard approach based on the Derbenev-Kondratenko formulas We define the “time” $\theta=2\pi s/C$ where $s$ is the distance around the ring and $C$ is the circumference. We denote by $y$ a position in six- dimensional phase space of accelerator coordinates which we call beam-frame coordinates. In particular, following Ref. 19, $y_{6}$ is the relative deviation of the energy from the reference energy. Then if, $f=f(\theta,y)$ denotes the normalized $2\pi$-periodic equilibrium phase-space density at $\theta$ and $y$ and $\vec{P}_{loc}=\vec{P}_{loc}(\theta,y)$ denotes the local polarization vector of the bunch we have $\displaystyle\int dy\;f(\theta,y)=1\;,\quad\int dy\;f(\theta,y)\vec{P}_{loc}(\theta,y)=\vec{P}(\theta)\;,$ (1) where $\vec{P}(\theta)$ is the polarization vector of the bunch at $\theta$. For a detailed discussion about $\vec{P}_{loc}$, see, e.g., Ref. 20. Here and in the following we use arrows on three-component column vectors. Central to the standard approach is the invariant spin field (ISF) $\hat{n}=\hat{n}(\theta,y)$ defined as a normalized periodic solution of the Thomas-BMT-equation in phase space, i.e., $\displaystyle\partial_{\theta}\hat{n}=L_{\rm Liou}(\theta,y)\hat{n}+\Omega(\theta,y)\hat{n}\;,$ (2) such that 1. 1. $\Big{\|}\hat{n}(\theta,y)\Big{\|}=1$, 2. 2. $\hat{n}(\theta+2\pi,y)=\hat{n}(\theta,y)$, and where $L_{\rm Liou}$ denotes the Hamiltonian part of the Fokker-Planck operator $L_{\rm FP}^{y}$, the latter being introduced in Section 3 below. For some of our work on the ISF see Refs. 21 and 22. The unit vector of the ISF on the closed orbit is denoted by $\hat{n}_{0}(\theta)$ and it is easily obtained as an eigenvector of the one-turn spin-transport map on the closed orbit [19]. There are many methods for computing the ISF but none are trivial (for a recent technique see Ref. 23). In fact the existence, in general, of the invariant spin field is a mathematical issue which is only partially resolved, see, e.g., Ref. 21. The standard approach assumes that a function $P_{\rm DK}=P_{\rm DK}(\theta)$ exists such that $\displaystyle\vec{P}_{loc}(\theta,y)\approx P_{\rm DK}(\theta)\hat{n}(\theta,y)\;.$ (3) Thus, by (1) and (3), $\displaystyle\vec{P}(\theta)=\int dy\;f(\theta,y)\vec{P}_{loc}(\theta,y)\approx P_{\rm DK}(\theta)\int dy\;f(\theta,y)\hat{n}(\theta,y)\;.$ (4) The approximation (3) leads to [2] $\displaystyle P_{\rm DK}(\theta)=P_{\rm DK}(\infty)(1-e^{-\theta/\tau_{\rm DK}})+P_{\rm DK}(0)e^{-\theta/\tau_{\rm DK}}\;,$ (5) where $\tau_{\rm DK}$ and $P_{\rm DK}(\infty)$ are given by the Derbenev- Kondratenko formulas $\displaystyle P_{\rm DK}(\infty):=\frac{\tau_{0}^{-1}}{\tau_{\rm DK}^{-1}}\;,$ (6) $\displaystyle\tau_{\rm DK}^{-1}:=\frac{5\sqrt{3}}{8}\frac{r_{e}\gamma_{0}^{5}\hbar}{m}\frac{C}{4\pi^{2}}\int_{0}^{2\pi}\;d\theta\Big{\langle}\frac{1}{\|R\|^{3}}[1-\frac{2}{9}(\hat{n}\cdot{\hat{\beta}})^{2}+\frac{11}{18}\Big{\|}\partial_{y_{6}}\hat{n}\Big{\|}^{2}]\Big{\rangle}_{\theta}\;,$ (7) $\displaystyle\tau_{0}^{-1}:=\frac{r_{e}\gamma_{0}^{5}\hbar}{m}\frac{C}{4\pi^{2}}\int_{0}^{2\pi}\;d\theta\Big{\langle}\frac{1}{\|R\|^{3}}\hat{b}\cdot[\hat{n}-\partial_{y_{6}}\hat{n}]\Big{\rangle}_{\theta}\;,$ (8) with * • $\Big{\langle}\cdots\Big{\rangle}_{\theta}\equiv\int dy\;f(\theta,y)\cdots$ * • $\hat{b}=\hat{b}(\theta,y)\equiv$ normalized magnetic field, $\hat{\beta}=\hat{\beta}(\theta,y)\equiv$ normalized velocity vector, $\gamma_{0}\equiv$ Lorentz factor of the reference particle, $R(\theta,y)\equiv$ radius of curvature in the external magnetic field, $r_{e}\equiv$ classical electron radius, $m\equiv$ rest mass of electrons or positrons. By (4) and for large $\theta$ $\displaystyle\vec{P}(\theta)\approx P_{\rm DK}(\infty)\int dy\;f(\theta,y)\hat{n}(\theta,y)\;,$ (9) where $P_{\rm DK}(\infty)$ is given by (6) and where the rhs of (9) is the approximate equilibrium polarization vector. Note that the latter is $2\pi$-periodic in $\theta$ since $f(\theta,y)$ and $\hat{n}(\theta,y)$ are $2\pi$-periodic in $\theta$. Defining $\displaystyle\tau_{dep}^{-1}:=\frac{5\sqrt{3}}{8}\frac{r_{e}\gamma_{0}^{5}\hbar}{m}\frac{C}{4\pi^{2}}\int_{0}^{2\pi}\;d\theta\Big{\langle}\frac{1}{\|R\|^{3}}\frac{11}{18}\Big{\|}\partial_{y_{6}}\hat{n}\Big{\|}^{2}\Big{\rangle}_{\theta}\;,$ (10) we can write (7) as $\displaystyle\tau_{\rm DK}^{-1}=\tau_{dep}^{-1}+\frac{5\sqrt{3}}{8}\frac{r_{e}\gamma_{0}^{5}\hbar}{m}\frac{C}{4\pi^{2}}\int_{0}^{2\pi}\;d\theta\Big{\langle}\frac{1}{\|R\|^{3}}[1-\frac{2}{9}(\hat{n}\cdot{\hat{\beta}})^{2}]\Big{\rangle}_{\theta}\;.$ (11) For details on (6), (7), (8), (10) and (11) see, e.g., Refs. 24 and 19. We now briefly characterize the various terms in the Derbenev-Kondratenko formulas. First, $\tau_{dep}^{-1}$ is the radiative depolarization rate. Secondly, the term $\frac{r_{e}\gamma_{0}^{5}\hbar}{m}\frac{C}{4\pi^{2}}\int_{0}^{2\pi}\;d\theta\Big{\langle}\frac{1}{\|R\|^{3}}\hat{b}\cdot\hat{n}\Big{\rangle}_{\theta}$ in $\tau_{0}^{-1}$ and the term $\frac{5\sqrt{3}}{8}\frac{r_{e}\gamma_{0}^{5}\hbar}{m}\frac{C}{4\pi^{2}}\int_{0}^{2\pi}\;d\theta\Big{\langle}\frac{1}{\|R\|^{3}}\Big{\rangle}_{\theta}$ in $\tau_{\rm DK}^{-1}$ cover the Sokolov-Ternov effect. Lastly, the term $-\frac{r_{e}\gamma_{0}^{5}\hbar}{m}\frac{C}{4\pi^{2}}\int_{0}^{2\pi}\;d\theta\Big{\langle}\frac{1}{\|R\|^{3}}\hat{b}\cdot[\partial_{y_{6}}\hat{n}]\Big{\rangle}_{\theta}$ in $\tau_{0}^{-1}$ covers the kinetic polarization effect and the term in $\tau_{\rm DK}^{-1}$ which is proportional to $2/9$ covers the Baier-Katkov correction. We now sketch three approaches for computing $P_{\rm DK}(\infty)$ via the Derbenev-Kondratenko formulas. All three approaches use (6) but they differ in how $\tau_{0}^{-1}$ and $\tau_{\rm DK}^{-1}$ are computed. * (i) Compute $\tau_{0}^{-1}$ via (8) and $\tau_{\rm DK}^{-1}$ via (7) by computing $f$ and $\hat{n}$ as accurately as needed. * (ii) Approximate $\tau_{0}^{-1}$ by neglecting the usually-small kinetic polarization term in (8) and by approximating the remaining term in (8) by replacing $\hat{n}$ by $\hat{n}_{0}$. Compute $\tau_{\rm DK}^{-1}$ via (11) where $\tau_{dep}^{-1}$ is not computed via (10) but via Monte-Carlo spin tracking and where the remaining terms in (11) are approximated by using the $\hat{n}_{0}$-axis. 444Prominent Monte-Carlo spin tracking codes are SLICKTRACK by D.P. Barber [19], SITROS by J. Kewisch [19], Zgoubi by F. Meot [25], PTC/FPP by E. Forest [26], and Bmad by D. Sagan [27]. This approach provides a useful first impression avoiding the computation of $f$ and $\hat{n}$. For more details on this approach see Ref. 19. Monte-Carlo tracking can also be extended beyond integrable orbital motion to include, as just one example, beam-beam forces. Note that Monte-Carlo tracking just gives an estimate of $\tau_{dep}^{-1}$ but it does not provide an explanation. Nevertheless, insights into sources of depolarization can be obtained by switching off terms in the Thomas-BMT equation. In principal such diagnoses can also be applied in approach (i). Such investigations can the systematized under the heading of “spin matching” [19]. * (iii) Compute $\tau_{0}^{-1}$ via (8) and $\tau_{\rm DK}^{-1}$ via (7) by linear approximation in orbit and spin variables via the so-called SLIM formalism [19]. Approach (ii) is the most practiced while approach (i) is only feasible if one can compute $f$ and $\hat{n}$ as accurately as needed (which is not easy!). Approach (iii), which was historically the first, is very simple and is often used for ballparking $P_{\rm DK}(\infty)$. Since the inception of the Derbenev-Kondratenko formulas correction terms to the rhs of (10) have been suspected. See Refs. 4, 28 as well as Z. Duan’s contribution to this workshop. These correction terms, associated with so-called resonance crossing, in turn associated with large energy spread, are not as well understood as the rhs of (10), partly because of their peculiar form. Nevertheless, careful observation of spin motion during the Monte-Carlo tracking in approach (ii), might provide a way to investigate their existence and form. ## 3 The Bloch equation and the Reduced Bloch equation in the laboratory frame In the previous section we used the beam frame and we will do so later. However the BE was first presented in Ref. 3 for the laboratory frame and in that frame it also has its simplest form. In this section we focus on the laboratory frame. In a semiclassical probabilistic description of an electron or positron bunch the spin-orbit dynamics is described by the spin-$1/2$ Wigner function $\rho$ (also called the Stratonovich function) written as $\displaystyle\rho(t,z)=\frac{1}{2}[f_{lab}(t,z)I_{2\times 2}+\vec{\sigma}\cdot\vec{\eta}_{lab}(t,z)]\;,$ (12) with $z=(\vec{r},\vec{p})$ where $\vec{r}$ and $\vec{p}$ are the position and momentum vectors of the phase space and $t$ is the time, and where $f_{lab}$ is the phase-space density of particles normalized by $\int dzf_{lab}(t,z)=1$, $\vec{\eta}_{lab}$ is the polarization density of the bunch and $\vec{\sigma}$ is the vector of the three Pauli matrices. As explained in Ref. 20, $\vec{\eta}_{lab}$ is proportional to the spin angular momentum density. In fact it is given by $\vec{\eta}_{lab}(t,z)=f_{lab}(t,z)\vec{P}_{loc,lab}(t,z)$ where $\vec{P}_{loc,lab}$ is the local polarization vector. Thus $f_{lab}=Tr[\rho]$ and $\vec{\eta}_{lab}=Tr[\rho\vec{\sigma}]$. The polarization vector $\vec{P}_{lab}(t)$ of the bunch is $\vec{P}_{lab}(t)=\int dz\vec{\eta}_{lab}(t,z)=\int dzf_{lab}(t,z)\vec{P}_{loc,lab}(t,z)$. Then, by neglecting collective effects and after several other approximations, the phase-space density evolves according to Ref. 3 via $\displaystyle\partial_{t}f_{lab}=L_{FP}^{lab}(t,z)f_{lab}\;.$ (13) Using the units as in Ref. 3 the Fokker-Planck operator $L_{FP}^{lab}$ is defined by $\displaystyle L_{FP}^{lab}(t,z):=L_{Liou}^{lab}(t,z)+\vec{F}_{rad}(t,z)+\vec{Q}_{rad}(t,z)+\frac{1}{2}\sum_{i,j=1}^{3}\partial_{p_{i}}\partial_{p_{j}}{\cal E}_{ij}(t,z)\;,$ (14) where $\displaystyle L_{Liou}^{lab}(t,z):=-\partial_{\vec{r}}\cdot\frac{1}{m\gamma(\vec{p})}\vec{p}-\partial_{\vec{p}}\cdot[e\vec{E}(t,\vec{r})+\frac{e}{m\gamma(\vec{p})}(\vec{p}\times\vec{B}(t,\vec{r}))]\;,$ (15) $\displaystyle\vec{F}_{rad}(t,z):=-\frac{2}{3}\frac{e^{4}}{m^{5}\gamma(\vec{p})}\|\vec{p}\times\vec{B}(t,\vec{r})\|^{2}\vec{p}\;,$ (16) $\displaystyle\vec{Q}_{rad}(t,z):=\frac{55}{48\sqrt{3}}\sum_{j=1}^{3}\;\frac{\partial[\lambda(t,z)\vec{p}p_{j}]}{\partial p_{j}}\;,$ (17) $\displaystyle{\cal E}_{ij}(t,z):=\frac{55}{24\sqrt{3}}\lambda(t,z)p_{i}p_{j}\;,\quad\lambda(t,z):=\hbar\frac{\|e\|^{5}}{m^{8}\gamma(\vec{p})}\|\vec{p}\times\vec{B}(t,\vec{r})\|^{3}\;,$ (18) $\displaystyle\gamma(\vec{p}):=\frac{1}{m}\sqrt{\|\vec{p}\|^{2}+m^{2}}\;,$ (19) and with $e$ being the electric charge of the electron or positron and $\vec{E}$ and $\vec{B}$ being the external electric and magnetic fields. The Fokker-Planck operator $L_{FP}^{lab}$ whose explicit form is taken from Ref. 3 is a linear second-order partial differential operator and, with some additional approximations, is commonly used for electron synchrotrons and storage rings, see Ref. 29 and Section 2.5.4 in Ref. 19. As usual, since it is minuscule compared to all other forces, the Stern-Gerlach effect from the spin onto the orbit is neglected in (13). The polarization density $\vec{\eta}_{lab}$ evolves via eq. 2 in Ref. 3, i.e., via that which we call the Bloch equation, namely $\displaystyle\partial_{t}\vec{\eta}_{lab}=L_{\rm FP}^{lab}(t,z)\vec{\eta}_{lab}+M(t,z)\vec{\eta}_{lab}$ $\displaystyle\quad-[1+\partial_{\vec{p}}\cdot\vec{p}]\lambda(t,z)\frac{1}{m\gamma(\vec{p})}\frac{\vec{p}\times\vec{a}(t,z)}{\|\vec{a}(t,z)\|}f_{lab}(t,z)\;,$ (20) where $\displaystyle M(t,z):=\Omega^{lab}(t,z)-\lambda(t,z)\frac{5\sqrt{3}}{8}[I_{3\times 3}-\frac{2}{9m^{2}\gamma^{2}(\vec{p})}\vec{p}\vec{p}^{T}]\;,$ (21) $\displaystyle\vec{a}(t,z):=\frac{e}{m^{2}\gamma^{2}(\vec{p})}(\vec{p}\times\vec{B}(t,\vec{r}))\;.$ (22) The BE was derived in Ref. 3 from the semiclassical approximation of quantum electrodynamics and it is a generalization, to the whole phase space, of the Baier-Katkov-Strakhovenko equation which just describes the evolution of polarization along a single deterministic trajectory [30]. Note also that, while the BE was new in 1975, the orbital Fokker-Planck equation (13) was already known thanks to research of the 1950s, e.g., Schwinger’s paper on quantum corrections to synchrotron radiation [31]. The skew-symmetric matrix $\Omega^{lab}(t,z)$ takes into account the Thomas-BMT spin-precession effect. Thus in the laboratory frame the Thomas-BMT-equation (2) reads as $\displaystyle\partial_{t}\hat{n}_{lab}=L_{Liou}^{lab}(t,z)\hat{n}_{lab}+\Omega^{lab}(t,z)\hat{n}_{lab}\;.$ (23) The quantum aspect of (13) and (20) is embodied in the factor $\hbar$ in $\lambda(t,z)$. For example $\vec{Q}_{rad}$ is a quantum correction to the classical radiation reaction force $\vec{F}_{rad}$. The terms $-\lambda(t,z)\frac{5\sqrt{3}}{8}\vec{\eta}_{lab}$ and $-\lambda(t,z)\frac{1}{m\gamma(\vec{p})}\frac{\vec{p}\times\vec{a}(t,z)}{\|\vec{a}(t,z)\|}f_{lab}(t,z)$ take into account spin flips due to synchrotron radiation and encapsulate the Sokolov-Ternov effect. The term $\lambda(t,z)\frac{5\sqrt{3}}{8}\frac{2}{9m^{2}\gamma^{2}(\vec{p})}\vec{p}\vec{p}^{T}\vec{\eta}_{lab}$ encapsulates the Baier-Katkov correction, and the term $\partial_{\vec{p}}\cdot\vec{p}\;\lambda(t,z)\frac{1}{m\gamma(\vec{p})}\frac{\vec{p}\times\vec{a}(t,z)}{\|\vec{a}(t,z)\|}f_{lab}(t,z)$ encapsulates the kinetic-polarization effect. The only terms in (20) which couple the three components of $\vec{\eta}_{lab}$ are the Thomas-BMT term and the Baier-Katkov correction term. As mentioned above, there exists a system of SDEs underlying (20) (for details, see Ref. 6). In particular, $f_{lab}$ and $\vec{\eta}_{lab}$ are related to a spin-orbit density ${\cal P}_{lab}={\cal P}_{lab}(t,z,\vec{s})$ via $\displaystyle f_{lab}(t,z)=\int_{{\mathbb{R}}^{3}}\;d\vec{s}\;{\cal P}_{lab}(t,z,\vec{s})\;,$ (24) $\displaystyle\vec{\eta}_{lab}(t,z)=\int_{{\mathbb{R}}^{3}}\;d\vec{s}\;\vec{s}\;{\cal P}_{lab}(t,z,\vec{s})\;,$ (25) where ${\cal P}_{lab}$ satisfies the Fokker-Planck equation corresponding to the system of SDEs in Ref. 6. These SDEs can be used as the basis for a Monte- Carlo spin tracking algorithm, i.e., for Method 2 mentioned in Section 1 above. This would extend the standard Monte-Carlo spin tracking algorithms, which we mentioned in Section 2 above, by taking into account all physical effects described by (20), i.e., the Sokolov-Ternov effect, the Baier-Katkov correction, the kinetic-polarization effect and, of course, spin diffusion. If we ignore the spin-flip terms and the kinetic-polarization term in the BE then (20) simplifies to the RBE $\displaystyle\partial_{t}\vec{\eta}_{lab}=L_{FP}^{lab}(t,z)\vec{\eta}_{lab}+\Omega^{lab}(t,z)\vec{\eta}_{lab}\;.$ (26) The RBE models spin diffusion due to the effect of the stochastic orbital motion on the spin and thus contains those terms of the BE which are related to the radiative depolarization rate $\tau_{dep}^{-1}$. This effect is clearly seen in the SDEs (see, e.g., (28) and (29)). ## 4 The Reduced Bloch equation in the beam frame In the beam frame, i.e., in the accelerator coordinates $y$ of Section 2, the RBE (26) becomes $\displaystyle\partial_{\theta}\vec{\eta}=L_{\rm FP}^{y}(\theta,y)\vec{\eta}+\Omega(\theta,y)\vec{\eta}\;.$ (27) Because the coefficients of $L_{\rm FP}^{y}$ are $\theta$-dependent, the RBE (27) is numerically and analytically quite complex. So we first approximate it by treating the synchrotron radiation as a perturbation. Then, in order to solve it numerically to determine the long-time behavior that we need, we address the system of SDEs underlying (27) and apply the refined averaging technique presented in Ref. 32 (see also 7), for the orbital dynamics, and extend it to include spin. The averaged SDEs are then used to construct an approximate RBE which we call the effective RBE. The system of SDEs underlying (27) reads as 555We denote the random dependent variables like $Y$ in (28) by capital letters to distinguish them from independent variables like $y$ in (27). $\displaystyle\frac{dY}{d\theta}=(A(\theta)+\epsilon_{R}\delta A(\theta))Y+\sqrt{\epsilon_{R}}\sqrt{\omega(\theta)}e_{6}\xi(\theta)\;,$ (28) $\displaystyle\frac{d\vec{S}}{d\theta}=[\Omega_{0}(\theta)+\epsilon_{S}C(\theta,Y)]\vec{S}\;,$ (29) where the orbital dynamics has been linearized in $Y$ and $\Omega=\Omega_{0}+\epsilon_{S}C$ has been linearized in $Y$ so that $\displaystyle C(\theta,Y)=\sum_{j=1}^{6}\;C_{j}(\theta)Y_{j}\;.$ (30) Also, $A(\theta)$ is a Hamiltonian matrix representing the nonradiative part of the orbital dynamics and $Y$ has been scaled so that $\epsilon_{R}$ is the size of the orbital effect of the synchrotron radiation. Thus $\epsilon_{R}\delta A(\theta)$ represents the orbital damping effects due to synchrotron radiation and the cavities, $\sqrt{\epsilon_{R}}\xi(\theta)$ represents the associated quantum fluctuations, $\xi$ is the white noise process and $e_{6}:=(0,0,0,0,0,1)^{T}$. In the spin equation (29), $\Omega_{0}$ is the closed-orbit contribution to $\Omega$ so that $\epsilon_{S}C(\theta,Y)$ is what remains and $C(\theta,Y)$ is chosen $O(1)$. Hence $\epsilon_{S}$ estimates the size of $\Omega-\Omega_{0}$. Both $\Omega_{0}(\theta)$ and $C(\theta,Y)$ are, of course, skew-symmetric $3\times 3$ matrices. We are interested in the situation where $\epsilon_{R}$ and $\epsilon_{S}$ are small in some appropriate sense. Eqs. (28) and (29) can be obtained by transforming the system of SDEs in Ref. 6 from the laboratory frame to the beam frame [33]. However, since in this section we only deal with the RBE (not with the BE), (28) and (29) can also be found in older expositions on spin in high-energy electron storage rings, e.g., Ref. 34. Note that these expositions make approximations as for example with the linearity of (28) in $Y$ and the linearity of $C(\theta,Y)$ in $Y$. With (28) and (29) the evolution equation for the spin-orbit joint probability density ${\cal P}={\cal P}(\theta,y,\vec{s})$ is the following spin-orbit Fokker-Planck equation $\displaystyle\partial_{\theta}{\cal P}=L_{\rm FP}^{y}(\theta,y){\cal P}-\partial_{\vec{s}}\cdot\Biggl{(}\biggl{(}\Omega(\theta,y)\vec{s}\biggr{)}{\cal P}\Biggr{)}\;,$ (31) where $L_{\rm FP}^{y}$ is the orbital Fokker-Planck operator. The phase-space density $f$ and the polarization density $\vec{\eta}$ corresponding to ${\cal P}$ are defined by $\displaystyle f(\theta,y)=\int_{{\mathbb{R}}^{3}}\;d\vec{s}\;{\cal P}(\theta,y,\vec{s})\;,\quad\vec{\eta}(\theta,y)=\int_{{\mathbb{R}}^{3}}\;d\vec{s}\;\vec{s}\;{\cal P}(\theta,y,\vec{s})\;,$ (32) which are the beam-frame analogs of (24) and (25). The local polarization vector $\vec{P}_{loc}$ from Section 2 above is related to $f$ and $\vec{\eta}$ by $\displaystyle\vec{\eta}(\theta,y)=f(\theta,y)\vec{P}_{loc}(\theta,y)\;.$ (33) The RBE (27) follows from (31) by differentiating (32) w.r.t. $\theta$ and by using the Fokker-Planck equation for ${\cal P}$. This proves that (28) and (29) is the system of SDEs which underlie the RBE (27). For (27), see also Ref. 20. ## 5 The Effective Reduced Bloch equation in the beam frame The effective RBE is, by definition, an approximation of the RBE (27) obtained by approximating the system of SDEs (28) and (29) using the method of averaging, see Refs. 7-12. We call the system of SDEs underlying the effective RBE the effective system of SDEs. We now discuss first-order averaging in the case where $\epsilon:=\epsilon_{S}=\epsilon_{R}$ is small. To apply the method of averaging to (28) and (29) we must transform them to a standard form for averaging, i.e., we must transform the variables $Y,\vec{S}$ to slowly varying variables. We do this by using a fundamental solution matrix $X$ of the unperturbed $\epsilon=0$ part of (28), i.e., $\displaystyle X^{\prime}=A(\theta)X\;,$ (34) and a fundamental solution matrix $\Phi$ of the unperturbed $\epsilon=0$ part of (29), i.e., $\displaystyle\Phi^{\prime}=\Omega_{0}(\theta)\Phi\;.$ (35) We thus transform $Y$ and $\vec{S}$ into the slowly varying $U$ and $\vec{T}$ via $\displaystyle Y(\theta)=X(\theta)U(\theta)\;,\quad\vec{S}(\theta)=\Phi(\theta)\vec{T}(\theta)\;.$ (36) Hence (28) and (29) are transformed to $\displaystyle U^{\prime}=\epsilon{\cal D}(\theta)U+\sqrt{\epsilon}\sqrt{\omega(\theta)}X^{-1}(\theta)e_{6}\xi(\theta)\;,$ (37) $\displaystyle\vec{T}^{\prime}=\epsilon{\mathfrak{D}}(\theta,U)\vec{T}\;,$ (38) where ${\cal D}$ and ${\mathfrak{D}}$ are defined by $\displaystyle{\cal D}(\theta):=X^{-1}(\theta)\delta A(\theta)X(\theta)\;,$ (39) $\displaystyle{\mathfrak{D}}(\theta,U):=\Phi^{-1}(\theta)C(\theta,X(\theta)U)\Phi(\theta)\;.$ (40) Of course, (37) and (38) carry the same information as (28) and (29). Now, applying the method of averaging to (37) and (38), we obtain the following effective system of SDEs $\displaystyle V^{\prime}=\epsilon\bar{\cal D}V+\sqrt{\epsilon}{\cal B}(\xi_{1},...,\xi_{k})^{T}\;,$ (41) $\displaystyle\vec{T}_{a}^{\prime}=\epsilon\bar{\mathfrak{D}}(V)\vec{T}_{a}\;,$ (42) where the bar denotes $\theta$-averaging, i.e., the operation $\lim_{T\rightarrow\infty}(1/T)\int_{0}^{T}d\theta\cdots$. Moreover $\xi_{1},...,\xi_{k}$ are statistically independent versions of the white noise process and ${\cal B}$ is a $6\times k$ matrix which satisfies ${\cal B}{\cal B}^{T}=\bar{\cal E}$ with $k=rank(\bar{\cal E})$ and where $\bar{\cal E}$ is the $\theta$-average of $\displaystyle{\cal E}(\theta)=\omega(\theta)X^{-1}(\theta)e_{6}e_{6}^{T}X^{-T}(\theta)\;.$ (43) For physically reasonable $A$ and $\Omega$ the fundamental matrices $X$ and $\Phi$ are quasiperiodic functions whence ${\cal D},{\mathfrak{D}}(\cdot,U)$ and ${\cal E}$ are quasiperiodic functions so that their $\theta$ averages $\bar{\cal D},\bar{\mathfrak{D}}(V)$ and $\bar{\cal E}$ exist. Our derivation of (41) from (37) is discussed in some detail in Ref. 6. We are close to showing that $U=V+O(\epsilon)$ on $\theta$-intervals of length $O(1/\epsilon)$ and it seems likely that this error is valid for $0\leq\theta<\infty$, because of the radiation damping. This is a refinement of Ref. 32 and assumes a non-resonance condition. Since the sample paths of $U$ are continuous and $U$ is slowly varying it seems likely that $\vec{T}_{a}$ is a good approximation to $\vec{T}$ and we are working on the error analysis. Spin-orbit resonances will be an important focus in the construction of $\bar{\mathfrak{D}}(V)$ from (40) which contains both the orbital frequencies in $X$ and the spin precession frequency in $\Phi$. Since, by definition, the effective system of SDEs underly the effective RBE, the latter can be obtained from the former in the same way as we obtained (27) from (28) and (29) (recall the discussion after (32)). Thus the evolution equation for the spin-orbit probability density ${\cal P}_{V}={\cal P}_{V}(\theta,{\rm v},\vec{t})$ is the following Fokker-Planck equation: $\displaystyle\partial_{\theta}{\cal P}_{V}=L^{V}_{\rm FP}({\rm v}){\cal P}_{V}-\epsilon\partial_{\vec{t}}\cdot\Biggl{(}\biggl{(}\bar{\mathfrak{D}}({\rm v})\vec{t}\biggr{)}{\cal P}_{V}\Biggr{)}\;,$ (44) where $\displaystyle L^{V}_{\rm FP}({\rm v})=-\epsilon\sum_{j=1}^{6}\partial_{{\rm v}_{j}}(\bar{\cal D}{\rm v})_{j}+\frac{\epsilon}{2}\sum_{i,j=1}^{6}\bar{\cal E}_{ij}\partial_{{\rm v}_{i}}\partial_{{\rm v}_{j}}\;.$ (45) The polarization density $\vec{\eta}_{V}$ corresponding to ${\cal P}_{V}$ is defined by $\displaystyle\vec{\eta}_{V}(\theta,{\rm{\rm v}})=\int_{{\mathbb{R}}^{3}}\;d\vec{t}\;\vec{t}\;{\cal P}_{V}(\theta,{\rm v},\vec{t})\;,$ (46) so that, by (44), the effective RBE is $\displaystyle\partial_{\theta}\vec{\eta}_{V}=L^{V}_{\rm FP}({\rm v})\vec{\eta}_{V}+\epsilon\bar{\mathfrak{D}}({\rm v})\vec{\eta}_{V}\;.$ (47) This then is the focus of our approach in Method 1. For more details on this section, see Refs. 6, 17 and 18. ## 6 Next steps * • Further development of Bloch-equation approach (numerical and theoretical), i.e., of Method 1 and with a realistic lattice. * • Development of validation methods, i.e., Methods 2-4. Note that Method 2 is an extension of the standard Monte-Carlo spin tracking algorithms and for that matter we will study Refs. 13, 14 and 15. * • Investigating the connection between the Bloch-equation approach and the standard approach based on the Derbenev-Kondratenko formulas, and studying the potential for correction terms [4] to $\tau_{\rm DK}^{-1}$ by using the RBE. ## 7 Acknowledgement This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award Number DE- SC0018008. ## References * [1] K. Heinemann, Re-evaluation of Spin-Orbit Dynamics of Polarized $e^{+}e^{-}$ Beams in High Energy Circular Accelerators and Storage Rings: Bloch equation approach, Invited talk at IAS Mini-Workshop on Beam Polarization in Future Colliders, Hong Kong, Jan. 17, 2019. See also: http://iasprogram.ust.hk/hep/2019 and K. Heinemann et al., _Int. J. Mod. Phys._ , vol. A35, Nos. 15 & 16, 2041003, 2020. * [2] Ya.S. Derbenev, A.M. Kondratenko, Polarization kinetics of particles in storage rings, _Sov. Phys. JETP_ , vol. 37, p. 968, 1973. * [3] Ya.S. Derbenev, A.M. Kondratenko, Relaxation and equilibrium state of electrons in storage rings, _Sov. Phys. Dokl._ , vol. 19, p. 438, 1975. * [4] Z. Duan, M. Bai, D.P. Barber, Q. Qin, A Monte-Carlo simulation of the equilibrium beam polarization in ultra-high energy electron (positron) storage rings, _Nucl. Instr. Meth._ , vol. A793, pp. 81-91, 2015. See also arXiv:physics/1505.02392v2. * [5] F. Bloch, Nuclear Induction, _Phys. 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[figure]style=plain,subcapbesideposition=top # Kinetic modelling of three-dimensional shock/laminar separation bubble instabilities in hypersonic flows over a double wedge Saurabh S. Sawant1<EMAIL_ADDRESS>V. Theofilis2,3 D. A. Levin1 1Department of Aerospace, University of Illinois at Urbana-Champaign, 104 S. Wright St, Champaign, Illinois, USA 2 School of Engineering, University of Liverpool, The Quadrangle, Brownlow Hill, L69 3GH, UK 3 Escola Politecnica, Universidade São Paulo, Av. Prof. Mello Moraes 2231, CEP 5508-900, São Paulo- SP, Brasil ###### Abstract Linear global instability of the three-dimensional (3-D), spanwise-homogeneous laminar separation bubble (LSB) induced by shock-wave/boundary-layer interaction (SBLI) in a Mach 7 flow of nitrogen over a $30^{\circ}-55^{\circ}$ double wedge is studied. At these conditions corresponding to a freestream unit Reynolds number, $Re_{1}=$5.2\text{\times}{10}^{4}$$ m-1, the flow exhibits rarefaction effects and comparable shock-thicknesses to the size of the boundary-layer at separation. This, in turn, requires the use of the high- fidelity Direct Simulation Monte Carlo (DSMC) method to accurately resolve unsteady flow features. We show for the first time that the LSB sustains self-excited, small- amplitude, 3-D perturbations that lead to spanwise-periodic flow structures not only in and downstream of the separated region, as seen in a multitude of experiments and numerical simulations, but also in the internal structure of the separation and detached shock layers. The spanwise-periodicity length and growth rate of the structures in the two zones are found to be identical. It is shown that the linear global instability leads to low-frequency unsteadiness of the triple point formed by the intersection of separation and detached shocks, corresponding to a Strouhal number of $St\sim 0.02$. Linear superposition of the spanwise-homogeneous base flow and the leading 3-D flow eigenmode provides further evidence of the strong coupling between linear instability in the LSB and the shock layer. ###### keywords: Authors should not enter keywords on the manuscript, as these must be chosen by the author during the online submission process and will then be added during the typesetting process (see Keyword PDF for the full list). Other classifications will be added at the same time. ## 1 Introduction Laminar SBLI has been a topic of extensive study since the best part of last century. The early experimental and theoretical work primarily focused on the upstream influence of disturbances in boundary layers, as can be found in seminal contributions such as Czarnecki & Mueller (1950); Liepmann et al. (1951); Lighthill & Newman (1953); Lighthill (1953, 2000); Chapman et al. (1958); Stewartson (1964). In subsequent research, triple deck theory (Stewartson & Williams, 1969; Smith, 1986; Neiland, 2008) was developed and used to understand boundary layer instability mechanisms that lead to separation in supersonic and hypersonic flows over compression ramps at moderate to high Reynolds numbers (Rizzetta et al., 1978; Cowley & Hall, 1990; Smith & Khorrami, 1991; Cassel et al., 1995; Korolev et al., 2002; Fletcher et al., 2004). More recent topics of study on shock-induced LSB include 3-D effects (Lusher & Sandham, 2020), unsteadiness and underlying instability mechanisms (Sansica et al., 2016), and the coupling between LSB and shock structure (Tumuklu et al., 2018b; Sawant et al., 2019). Experimental investigations of hypersonic SBLI primarily exist on compression ramps in flow regimes from laminar to turbulent and are at large $Re_{1}\sim O(10^{6}-10^{7})$ m-1 (Holden, 1963, 1978; Needham, 1965b, a; Elfstrom, 1971, 1972; Hankey Jr & Holden, 1975; Simeonides & Haase, 1995; Schneider, 2004; Roghelia et al., 2017; Chuvakhov et al., 2017). Experiments on flows over double wedges can be found at moderate Reynolds number, $Re_{1}\sim O(10^{5}-10^{6})$ m-1, but are limited, exhibit far more complicated SBLI than compression ramps, and suffer from test times significantly lower than the characteristic times involved in the development of instabilities and unsteadiness. Schrijer et al. (2006, 2009) performed experiments at $Re_{1}\sim O(10^{7})$ m-1 and $M=7$ in a Ludwieg tube facility and observed an unsteady flow exhibiting Edney type-$VI$ and $V$ interactions on 15∘-30∘ and 15∘-45∘ double wedge configurations, respectively. In a moderate Reynolds number regime, $Re_{1}\sim O(10^{5})$ m-1, Hashimoto (2009) performed experiments in a free piston shock tunnel, where the flow of air on a double wedge was tested for 300 $\mu$s at $Re_{1}=0.18-$3.5\text{\times}{10}^{6}$$ m-1 and $M=7$, and was found to exhibit Edney type-$V$ interactions. Recently, Swantek & Austin (2015) have performed experiments on a 30∘-25∘ double wedge in Hypervelocity Expansion Tube (HET) facility to test air and nitrogen at $Re_{1}=0.44-$4.6\text{\times}{10}^{6}$$ m-1 and $M=4.01-7.14$, where the test times ranged from 361-562 $\mu$s. Experiments of Knisely & Austin (2016) include nitrogen flow over the 30∘-55∘ double wedge geometry considered in this work at higher $Re_{1}=0.435-$1.1\text{\times}{10}^{6}$$ m-1 and $M=6.64-7.14$, where, in addition to HET facility, the T5 reflected shock tunnel was used that allows for a longer test time of 1 ms. On the numerical side, existing studies in laminar and transitional regimes primarily use compressible Navier-Stokes equations to understand different aspects of SBLI in hypersonic flows over compression ramps such as three- dimensionality of an LSB (Rudy et al., 1991), stability of hypersonic boundary layers (Balakumar et al., 2005), development of 3-D instability in the form of spanwise periodic striations of the LSB (Egorov et al., 2011; Dwivedi et al., 2019), and formation of secondary vortices and their fragmentation inside an LSB (Gai & Khraibut, 2019). Efforts such as these have been extended to simulate laminar SBLI over double wedge geometries (Knight et al., 2017). Durna et al. (2016) simulated a 2-D Mach 7 flow of nitrogen over a double wedge at the 2 MJ low enthalpy conditions of Swantek and Austin ($Re_{1}=$1.1\text{\times}{10}^{6}$$) to study the effect of the aft wedge angle on the flow characteristics with additional, recently included 3-D effects (Durna & Celik, 2020). Sidharth et al. (2018) carried out global stability analysis and Direct Numerical Simulation (DNS) of a Mach 5 perfect gas flow at $Re_{1}=$1.36\text{\times}{10}^{6}$$ over double ramps with forward and aft angles of 12∘ and 12∘-22∘, respectively. For aft angle of 20∘ and greater, they observed a linear instability of the 2-D separation bubble in the absence of upstream perturbations and associated that with streamwise streaks in wall temperature near the reattachment region. Recently, Reinert et al. (2020) simulated 3-D flows at Mach 7 over a 30∘-55∘ double wedge at the experimental conditions of Swantek & Austin (2015) and Knisely & Austin (2016) for much longer flow times than the duration of the experiments and reported unsteady asymmetric 3-D separation bubble. However, despite extensive numerical and experimental work, the physics of complicated SBLI formed in a hypersonic flow over double-wedge configurations is not well-understood. In this work, we investigate questions about the instability mechanism of a 3-D LSB, the coupling between the shock structure and LSB, and the low-frequency unsteadiness of the shock structure. We focus on the linear instability of a shock-induced, 3-D LSB formed in a Mach 7 nitrogen flow over a 30∘-55∘ double wedge configuration at a freestream unit Reynolds number of $Re_{1}=$5.2\text{\times}{10}^{4}$$ corresponding to an altitude of about 60 km. We will show that even for this lower density free stream condition, that is typically not studied, our fully-resolved, kinetic DSMC simulations of this complex flow allow us to study the strong coupling between the separation shock and the LSB. In our previous work, a 2-D (spanwise independent) flow over the same configuration and freestream conditions was simulated by Tumuklu et al. (2019), who demonstrated that the flow reaches a steady-state in $\sim 0.9~{}ms$ after the leading damped global modes, recovered by the residuals algorithm (Theofilis et al., 2000) and proper orthogonal decomposition (POD), have decayed. Yet we will show in our 3-D treatment using this 2-D, steady base flow of Tumuklu et al. (2019), that indeed the flow is linearly unstable to self-excited, small-amplitude, spanwise-homogeneous perturbations and will ultimately transition to turbulence. An important goal of the research discussed in this paper is to understand the effect of three-dimensionality on the coupling between the LSB and shock structure. The accurate modelling of the internal shock structure is an essential feature of relevance to the study of linear instability in high- speed boundary layer flow, where coupling between the separation bubble and the shock has been demonstrated through the amplitude function of the underlying global modes in a number of studies (e.g. Crouch et al., 2007). In the hypersonic regime, Tumuklu et al. (2018a, b) used DSMC to study the effect of unit Reynolds number, $Re_{1}=0.935-$3.74\text{\times}{10}^{5}$~{}m^{-1}$, on laminar SBLI in a Mach 16 nitrogen flow over an axisymmetric double cone configuration. The authors demonstrated a strong coupling between oscillations of the shock structure and instability of the laminar separated flow region through the spatial structure of the amplitude functions as well as Kelvin- Helmholtz waves formed at the contact surface downstream of the triple point. In this work, we focus on the coupling mechanism between a fully 3-D LSB and shock, and show that the instability in the LSB as well as inside the strong gradient region of shocks are intimately related. To capture the complex physics of SBLI with highest fidelity, we use the DSMC method. Continuous developments spanning the past fifty decades have resulted in this method being well-suited for the study of the physics of unsteady laminar SBLI to deliver accurate results in five critical aspects: (a) computations of molecular thermal fluctuations (Garcia, 1986; Bruno, 2019; Sawant et al., 2020), (b) calculation of anisotropic stresses and heat fluxes in strong shock layers (M $>>$ 1.6) (Bird, 1970; Cercignani et al., 1999), (c) prediction of rarefaction effects such as velocity slip and temperature jump (Moss, 2001; Moss & Bird, 2005; Tumuklu et al., 2018a), (d) quantification of translational, rotational, and vibrational nonequilibrium (Sawant et al., 2018), and (e) time-accurate evolution of self-excited perturbations (Tumuklu et al., 2018a, b, 2019). As a result, the method is gaining momentum in the study of hydrodynamic instabilities (Bird, 1998; Stefanov et al., 2002a, b, 2007; Kadau et al., 2004, 2010; Gallis et al., 2015, 2016). In our flow, even though the freestream Knudsen number of $3.0\text{\times}{10}^{-3}$ is continuum, the local Knudsen number in the shock-LSB region is much higher due to the steep gradients in macroscopic flow parameters. These non-continuum features, also known as local rarefaction zones, are crucial to understanding the coupling between the shock and LSB, as this work will demonstrate. Furthermore, the high-fidelity kinetic modelling of these regions is crucial because the thicknesses of shocks and the boundary-layer in the separation region are comparable. The time-accuracy of the DSMC method in modelling unsteady evolution of 3-D perturbations allows for quantification of the low-frequency unsteadiness of the shock structure. This phenomenon has been extensively investigated in turbulent SBLI at $Re_{1}\sim O(10^{6}-10^{8})$ using DNS and large eddy simulation (LES) (see Pirozzoli & Grasso, 2006; Touber & Sandham, 2009; Piponniau et al., 2009; Grilli et al., 2012; Priebe & Martín, 2012; Clemens & Narayanaswamy, 2014; Gaitonde, 2015; Priebe et al., 2016; Pasquariello et al., 2017), where numerical studies report a Strouhal number associated with unsteadiness within a range of 0.01 to 0.05, consistent with findings of many experiments (Dussauge et al., 2006). However, in hypersonic, 3-D laminar SBLI, such investigations are sparse. Tumuklu et al. (2018b) observed a similar Strouhal number of $\sim$0.08, corresponding to the bow shock oscillation in their axisymmetric flow over a double cone simulated using DSMC. In the 3-D, Mach 7, finite-span double-wedge flow simulation of Reinert et al. (2020), however, such unsteadiness was not observed for conditions at a freestream flow enthalpy of 8 MJ, although, the Reynolds number of their case was a factor of 8 higher. In this work, we show that our 8 MJ enthalpy, Mach 7, spanwise-periodic flow over the same configuration (at eight-times lower density) exhibits low-frequency unsteadiness after the onset of linear instability. Finally, topology analysis of separated flows is an important way to characterize and compare complex 3-D flows for different input conditions and shapes (see, e.g. Lighthill, 1963; Tobak & Peake, 1982; Perry & Hornung, 1984a, b; Perry & Chong, 1987; Dallmann, 1983, 1985). Topologies of 3-D flow constructed from the linear superposition of the leading stationary eigenmode due to a linear instability of an LSB and 2-D base flow were analyzed by Rodríguez & Theofilis (2010) in the incompressible regime and Robinet (2007); Boin et al. (2006) in the compressible regime involving oblique SBLI. In the analysis presented here, we estimate the changes in the 3-D wall-streamline topology for an increasing amplitude of the 3-D perturbations based on a linear combination of the 2-D base flow and 3-D perturbations. In this simplified approach, we will demonstrate that the wall-streamline signature is very different depending on whether the coupling is considered. The paper is organized as follows: section 2 describes the methodology, which includes a brief description of the DSMC method in section 2.1 and details about numerical models, the DSMC solver, the input conditions, and flow initialization in section 2.2. The features of 2-D base flow are described in section 3. Section 4 is devoted to the key findings of this paper. Section 4.1 describes the linear instability mechanism and its spatial origin through a detailed discussion of boundary layer profiles. The correlation between the shock and the separation bubble is explained in section 4.2. The surface rarefaction effects are described in section 4.3, whereas the isocontours of spanwise periodic flow structures are discussed in section 4.4. The topology of the LSB is discussed in section 5, first without taking into account the coupling between the bubble and the shock in section 5.1 and then the effect of their coupling in section 5.2. The important findings are summarized in section 6. ## 2 Methodology ### 2.1 The DSMC algorithm The equation for the evolution of velocity distribution function of molecules, $f(t,\vec{r},\vec{v})$ with respect to time $t$, position vector $\vec{r}$, and instantaneous velocity vector $\vec{v}$, is written as, $\begin{split}\frac{\partial f}{\partial t}+(\vec{v}\cdot\nabla)f+\left(\frac{\vec{F}}{m}\cdot\nabla_{v}\right)f=\left[\frac{df}{dt}\right]_{coll}\\\ \end{split}$ (1) where $\nabla$ and $\nabla_{v}$ are gradient operators in physical and velocity spaces, respectively. The first, second, and third terms on the left- hand side describe the change of $f$ with time, change due to convection of molecules in physical space, and that in the velocity space, respectively. The latter can happen due to the action of external conservative force per unit mass $\vec{F}/m$, such as gravity or electric field, which are ignored in this work. The right-hand side (RHS) term accounts for changes in $f$ in an element of space-velocity phase-space due to intermolecular collisions. For a thorough description, see Vincenti & Kruger (1965). The DSMC method (see Bird, 1994) decouples the advection of molecules and their intermolecular collisions. Each simulated particle represents $F_{n}$ amount of actual gas molecules and is advected for a discrete timestep. Based on the choice of boundary conditions, particles are introduced, removed, or reflected from the domain boundaries and interacted with the embedded surfaces using gas-surface collision models for the duration of the timestep. They are then mapped to an adaptively refined collision mesh ($C$-mesh) that encompasses the flow domain and ensures the spatial proximity of particles that are potential candidates for binary collisions. Next follows a collision scheme, which selects particle pairs that are collided based on the appropriate (elastic/inelastic) collision cross-section and are assigned with post-collisional instantaneous velocities and internal energies. Macroscopic flow parameters of interest such as pressure, velocities, etc., can be derived from the microscopic properties of simulated particles using statistical relations of kinetic theory. These parameters are represented on the sampling mesh ($S$-mesh), which has coarser cells than the $C$-mesh. Note that the unique characteristic of the DSMC method, the advection-collision decoupling, is justified if the local cell size of $C$-Mesh, $\Delta x$, is smaller than the local mean-free-path of molecules, $\lambda$, and the timestep, $\Delta t$, is lower than the mean-collision-time, $\tau$. A sufficient number of instantaneous particles in the smallest collision cells must also be ensured for unbiased collisions. Conveniently, the satisfaction of only these three numerical criteria leads to accurate modelling of internal structure of shocks, their mutual interaction, and surface rarefaction effects. This warrants the use of DSMC for detailed modelling of SBLIs at high altitudes, compared to ad-hoc techniques of modelling shocks in computational fluid dynamics (CFD) simulations that fall short of accurately capturing the internal structure of shocks. ### 2.2 The numerical implementation and flow initialization The fulfillment of the numerical criteria demands a large number of computational particles and collision cells. To overcome this challenge, we have previously developed an octree-based, 3-D DSMC solver known as Scalable Unstructured Gas-dynamic Adaptive mesh-Refinement (SUGAR-3D). See Sawant et al. (2018) for a comprehensive account of the implementation strategies, validation, and performance studies of the solver. In summary, the code takes advantage of message-passing-interface (MPI) for parallel communication between processors, adaptive mesh refinement (AMR) of coarser Cartesian octree cells to achieve spatial fidelity at regions of strong gradients, a cutcell algorithm to correctly capture physics in the vicinity of embedded surfaces, a domain decomposition strategy based on Morton-Z space-filling-curves, capability of parallel input/output, inclusion of thermal nonequilibrium models, and numerous run-time memory reduction strategies. In the octree-based AMR framework, the $C$-mesh is formed from a user-defined, uniform Cartesian grid. The cells of this grid are known as ‘root’ cells, which are recursively subdivided into eight parts until the local cell-size is smaller than the local mean-free-path. Note that a subdivision based on the above criterion is performed only if there are at least 32 particles in a collision cell. The satisfaction of both of these criteria in the presented flow over a double wedge requires $\sim$60 billion computational particles and $\sim$4.5 billion collision cells of an adaptively refined octree grid. See the appendix of Sawant et al. (2019) for the details of convergence study. Table 1: Freestream and numerical parameters for the Mach 7 nitrogen flow. Parameters \| Values ---\|--- Unit Reynolds number, $Re_{1}$ \| $5.22\text{\times}{10}^{4}$ Knudsen numbera \| $3.2\text{\times}{10}^{-3}$ Number density, $n_{1}$/(m3) \| 1022 Streamwise velocity, $u_{x,1}$/(m.s-1) \| 3812 Equilibrium translational temperature, $T_{tr,1}$/(K) \| 710 Surfaceb temperature, $T_{s}$/(K) \| 298.5 Species mass, $m$/(kg) \| $4.65\text{\times}{10}^{-26}$ Species diameter, $d$/(m) \| $4.17\text{\times}{10}^{-10}$ Viscosity index, $\omega$ \| 0.745 Reference temperature, Tr/(K) \| 273 Parker model parameters, Zr,∞ and T∗/(K) \| 18.5 and 91 Vibrational characteristic temperature, $\theta$/(K) \| 3371 Domain size, ($L_{x}$,$L_{y}$,$L_{z}$)/(mm) \| (80, 28.8, 80) Number of octree and sampling cells along ($X$,$Y$,$Z$) \| (400, 144, 400) Number of gas-surface interaction cells along ($X$,$Y$,$Z$) \| (25, 10, 25) Fnc \| $6.1\text{\times}{10}^{7}$ Timestep, $\Delta t$/(ns) \| 5 Adaptive mesh refinement interval /($\mu$s) \| 5 Relaxation probability computation interval /($\mu$s) \| 1 * • a Based on the length of the lower wedge, 50.8 mm. * • b Surface is fully accommodated (Bird, 1994), i.e., isothermal. * • c Number of actual molecules represented by a computational particle. The DSMC specific input and numerical parameters used in this work are listed in table 1. Note that the Cartesian coordinates are used with streamwise, spanwise, streamwise-normal directions as $X$, $Y$, and $Z$, respectively. The code uses the majorant frequency scheme (MFS) of Ivanov & Rogasinsky (1988) derived using the Kac stochastic model for the selection of collision pair and the variable hard sphere (VHS) model for elastic collisions. Appendix A describes an essential modification to the computation of maximum collision cross-section used in the MFS scheme for accurate spectral analysis of unsteady flows simulated on adaptively refined grids. For rotational relaxation, the Borgnakke & Larsen (1975) model is employed with rates by Parker (1959) and DSMC correction factors (Lumpkin III et al., 1991; Gimelshein et al., 2002) to account for the temperature dependence of the rotational probability. For vibrational relaxation, the semi-empirical expression of Millikan & White (1963) is used with the high-temperature correction of Park (1984). For this work, the SUGAR solver is also employed with spanwise-periodic boundary conditions as follows. Suppose a particle, during its discrete movement, intersects the spanwise domain boundary, $Y=0$ or $Y=L_{y}$, within a period, $\delta t$, smaller than the timestep. In that case, its spanwise position index is changed to the periodically opposite $Y$ boundary index, i.e., $Y=L_{y}$ or $Y=0$, respectively. After this translation, the particle continues its movement for the remaining period, $\Delta t-\delta t$. This simple algorithm is implemented in SUGAR’s parallel framework by ensuring that the processors containing a portion of the flow domain adjacent to any $Y$-boundary must also store the information of processors containing the periodically opposite portion of the domain. Such information includes the ‘location code array’ and the triangulated panels of the embedded surface. The location code arrays are special arrays used in the efficient particle mapping strategy based on the Morton-based space-filling-curve approach. See (Sawant et al., 2018) for details of these arrays and optimized gas-surface interaction strategies employed in the SUGAR code. The 2-D, steady-state solution of the flow over a double wedge, previously simulated by Tumuklu et al. (2019), is extruded in the spanwise direction ($Y$) with as many replicas as the number of spanwise octrees. See figure 1 for understanding the simulation domain setup in the $X-Z$ plane, where $X$ and $Z$ are streamwise and streamwise-normal directions. The spanwise $Y$ boundaries are periodic. From the inlet boundary, $X=0$, inward-directed ($X>0$) local Maxwellian flow is introduced at an average number density, bulk velocity, and temperature of $n_{1}$, $u_{x,1}$, and $T_{tr,1}$, respectively. Particles with the same properties are also introduced within one mean-free- path distance from the $Z$ boundaries, such that the streamlines of the flow are parallel to the $Z$-boundaries. If particles move out of the domain from either $X$ or $Z$ boundaries, they are deleted. The chosen spanwise extent of $L_{y}$=28.8 mm was estimated from a preliminary simulation with a span length of 72 mm for 30 flow times and was expected to contain four spanwise periodic structures. However, this turned out to be an underestimate, because when linear instability was detected after 50 flow times, the flow was found to exhibit a much larger spanwise wavelength. The spanwise extent of the current simulation is long enough to capture one linearly growing periodic structure. Contours and isocontours detailing spanwise periodic structures are shown with two periodic wavelengths for clarity. Note that a flow time, $T$, is defined as the time it takes for the flow to traverse a length of the separation bubble in the base (or mean) flow, $L_{s}=40$ mm, at a freestream velocity of $u_{x,1}$ where $L_{s}$ is defined as a straight-line distance from the separation point, $P_{s}$, to the reattachment point, $P_{R}$. Note that the spanwise-periodic simulation takes $\sim$5 hours per flow time using 19.2k Intel Xeon Platinum 8280 (“Cascade Lake”) processors of the Frontera supercomputer (2019). ## 3 Features of Two-dimensional Base flow Figure 1 shows the typical features of an Edney-IV type SBLI (Edney, 1968) in the base (or mean) flow, which is similar to that observed on double cones (Druguet et al., 2005; Babinsky & Harvey, 2011). The base flow macroscopic parameters are denoted by the subscript ‘$b$’. For details of the time evolution of 2-D SBLI interaction over the double wedge, see the work of Tumuklu et al. (2019). In summary, these features are formed by the interaction of a leading-edge attached (oblique) and detached (bow) shocks generated by the lower and upper wedge surfaces, respectively. This interaction generates a transmitted shock that impinges on the upper wedge surface and increases the pressure and heat flux at the reattachment (or impingement) location, $P_{R}$. The induced adverse pressure gradient results in the separation of the supersonic boundary layer on the lower wedge surface at $P_{S}$ and the formation of flow recirculation zone in the vicinity of the intersection of two surfaces, also known as the hinge. Inside the separation bubble, a shear layer is represented by the line contour of $u_{x}=0$ from $P_{S}$ to $P_{R}$. The separation zone significantly alters the SBLI system, such that the compression waves generated at the separation coalesce into a separation shock that interacts with the attached and the detached shocks at triple points. Two contact surfaces, $C_{1}$ and $C_{2}$, are formed downstream of triple points $T_{1}$ and $T_{2}$, respectively. The former is between two supersonic streams formed downstream of the separation shock, and the latter is between the lower supersonic and upper, hotter subsonic flow formed downstream of the detached shock. The transmitted shock is also affected by the contact surface $C_{1}$ and causes the reattachment point to move downstream, and the separation bubble to increase in size. A reflected shock is formed downstream of the transmitted shock to guide the supersonic stream along the upper wedge surface. If the upper wedge surface were longer, such interaction would have resulted in a $\lambda$-shock pattern, which was observed on the double cone by Tumuklu et al. (2018a, b). Instead, the flow encounters the corner of the upper wedge and goes through the Prandtl-Meyer expansion. [] [] Figure 1: (a) SBLI features shown in the magnitude of mass density gradient of the base flow, $\|\nabla\rho_{b}\|$, normalized by $\rho_{1}L_{s}^{-1}$, where $\rho_{1}$ is freestream mass density. Contour levels are shown in (b). (b) On same flowfield, overlay of wall-normal directions $S$ and $R$, and numerical probes $b$ inside separation bubble ($X$=48.496 mm, $Z$=24.270 mm), $r$ at reattachment (64.396, 44.358), $s$ in the separation shock (44.165, 32.597), $c$ near contact surface (65.191, 56.593), $t$ at the triple point $T_{2}$ (48.347, 41.624), $f$ in the freestream (39.212, 49.722). $S$ and $R$ directions intersect the $X$-axis at 62 and 127 mm, respectively. The initial 2-D SBLI system moves slightly downstream within the first 30 flow times because of low spanwise relaxation that leads to a decrease in pressure downstream of the primary shocks. This spanwise relaxation is induced by the thermal fluctuations of spanwise velocity about zero in the spanwise periodic simulation. This is consistent with the fact that all macroscopic quantities fluctuate about their mean (Landau & Lifshitz, 1980, chapter XII). A strictly imposed zero bulk velocity in the purely 2-D solution is unrealistic in that it does not account for such thermal fluctuations. The new 2-D flow state is defined by spanwise and temporally averaging the solution between 48 to 60 flow times. This is referred to as the base state, which fosters the growth of linear instability, detectable after 50 flow times. Note that the DSMC-derived instantaneous data at 90.5 flow times, shown in this work, _i.e._ , the boundary-layer profiles shown in section 4.1, the perturbation flow field contours shown in section 4.2, isocontours shown in section 4.4, and the perturbation field used for superposition in section 5, are noise-filtered using the POD method (see Appendix B). In spite of molecular fluctuations, DSMC allows for the detection of the onset of instability. Statistical mechanics predicts the standard deviation in the fluctuations of the directed bulk velocity such as $u_{x}$ in a gas at local equilibrium as, $\sqrt{R\langle T_{tr}\rangle/\langle N\rangle}$, where $R$, $\langle T_{tr}\rangle$, $\langle N\rangle$ are the gas constant, average translational temperature and average number of particles (Hadjiconstantinou et al., 2003; Landau & Lifshitz, 1980, chapter XII). Similarly, we can estimate the level of spanwise fluctuations about the spanwise average in a 2-D flow at local equilibrium conditions exhibiting small-amplitude, self- excited fluctuations by calculating $\sqrt{R\langle T_{tr}\rangle_{s}/\langle N\rangle_{s}}$. Subscript ‘$s$’ attached to the averaged quantities denote a spanwise average. If the DSMC-computed standard deviation is greater than the equilibrium estimate, then the fluctuations are not entirely thermal but are due to self-excited linear instability. The only exception is the finite thick region of shock layers, where additional fluctuations are present due to strong translational nonequilibrium (Sawant et al., 2020). This test was used as a first confirmation of the onset of linear instability at approximately 50 flow times, when the self-excited fluctuations in the separation bubble became slightly but noticeably larger than the thermal fluctuations. ## 4 Three-dimensional Instability Mechanisms ### 4.1 Linear instability: growth rate and spatial origin A linear instability responsible for making the 2-D base flow unstable to self-excited spanwise-homogeneous perturbations is verified in figure 2. Figure 2 shows the good comparison of the temporal evolution of perturbation rotational temperature $\tilde{T}_{rot}$, obtained from DSMC and a 2-D linear function that fits the DSMC solution. Note that the perturbation part of a macroscopic flow variable $Q\in(n,u_{x},u_{y},u_{z},T_{tr},T_{rot},T_{vib})$ is given by subtracting the 2-D base flow state $Q_{b}(x,z)$ as, $\epsilon\tilde{Q}(x,y,z,t)=Q(x,y,z,t)-Q_{b}(x,z)$ (2) Note that $\epsilon<<1$, which indicates the the perturbation is small. $\tilde{n}$ is the perturbation number density, $\tilde{u}_{x},\tilde{u}_{y},\tilde{u}_{z}$ are perturbation velocities in the $X$, $Y$, and $Z$ directions, and $\tilde{T}_{tr},\tilde{T}_{rot},\tilde{T}_{vib}$ are perturbation translational, rotational, and vibrational temperatures, respectively. A DSMC- computed perturbation flow parameter $\tilde{Q}\in(\tilde{n},\tilde{u}_{x},\tilde{u}_{y},\tilde{u}_{z},\tilde{T}_{tr},\tilde{T}_{rot},\tilde{T}_{vib})$ is fitted by a linear function written as, $\centering\begin{split}\tilde{Q}(x,y,z,t)&=\hat{Q}(x,z)\exp{(i\Theta)}+c.c.\\\ \end{split}\@add@centering$ (3) where $\hat{Q}(x,z)$ is a spanwise homogeneous amplitude function, and $\Theta$ is a phase function of the linear perturbation that has the form, $\centering\Theta=\beta y-\Omega t\@add@centering$ (4) $\beta=2\pi/L_{y}$ is a real spatial wavenumber indicating spanwise wavelength of the mode, $\Omega=\omega_{r}+i\omega_{i}$ is a complex parameter, whose real part indicates frequency and the imaginary part is the growth rate in time $t$, and $c.c.$ indicates complex conjugation so that $\tilde{Q}$ is real. A 2-D linear fit is performed using the generalized least-squares method using Python’s LMFIT (Version 1.0.1) module, which gives the mean value of unknown fit parameters, $\omega_{i}$, $\hat{Q}$, $\omega_{r}$, and $1\sigma$-uncertainty (standard error) in these parameters. These are listed in table 2. Note that by keeping $\omega_{r}$ as an unknown resulted in a small number for $\omega_{r}$ and imposing it as $\omega_{r}=0$ did not change the value of other three fit parameters, indicating that the linearly growing mode is stationary. [] [] [] Figure 2: (a) At probe $b$ inside the separation bubble, (left) temporal evolution of DSMC-derived perturbation rotational temperature, $\tilde{T}_{rot}$, normalized by freestream temperature, $T_{tr,1}$, and (right) 2-D linear fit. (b) Comparison of linear fits of all residuals at a spanwise location that corresponds to the peak. For $\tilde{T}_{rot}$, it is indicated at $Y/L_{y}$=0.88. Same holds true for $\tilde{T}_{tr}$ and $\tilde{T}_{vib}$. For $\tilde{n}$, $\tilde{u}_{x}$, and $\tilde{u}_{z}$, it is at 1.38, whereas for $\tilde{u}_{y}$, it is at 1.13. (c) Comparison of the linear fit of $\tilde{u}_{y}$ through the peak at probes $b$, $s$, $r$, $c$. For $b$ and $s$, the peak location is at $Y/L_{y}$=1.13, whereas for $r$ and $c$, it is at 0.63. Similar linear fits are performed on other DSMC-computed macroscopic flow parameters, and a 1-D extracted curve passing through the peak spanwise structure, such as that marked in figure 2 by a dashed line, is compared in figure 2. All curves are parallel to each other, indicating similar growth rates. Also, figure 2 shows the comparison of curve-fitting functions through the peak structure of $\tilde{u}_{y}$ of probe $b$ with probes at other important locations, $s$, $r$, and $c$. Nearly parallel curves are observed, which indicates that linear growth is global. By comparing the absolute values of the amplitude of $\tilde{u}_{y}$, it is seen that probes $r$, $b$, $s$, $c$ have largest to lowest amplitude, indicating decreasing magnitude of perturbation. The average of the mean growth rate for each parameter listed in table 2 is $\omega_{i}=5.0$ kHz, with bounds of +0.16% and -0.16%. A maximum deviation of 11.4% is observed at probe $c$. Table 2: 2-D linear curve fit parameters in equations 8 and 4 corresponding to figures 2 and 2. Perturbation parametera \| Growth rate $\omega_{i}$/(kHz) \| Amplitude $\hat{Q}$ ---\|---\|--- $\tilde{n}$ \| 4.91 $\pm$ 0.06% \| -5.013e+19 $\pm$ 0.24% $\tilde{u}_{x}$ \| 4.90 $\pm$ 0.07% \| -0.1613 $\pm$ 0.30% $\tilde{u}_{z}$ \| 4.95 $\pm$ 0.08% \| -0.1108 $\pm$ 0.33% $\tilde{T}_{tr}$ \| 4.88 $\pm$ 0.04% \| 0.5111 $\pm$ 0.17% $\tilde{T}_{rot}$ \| 4.88 $\pm$ 0.05% \| 0.5128 $\pm$ 0.19% $\tilde{T}_{vib}$ \| 5.15 $\pm$ 0.11% \| 0.1560 $\pm$ 0.51% $\tilde{u}_{y}$ \| 4.89 $\pm$ 0.10% \| 0.0762 $\pm$ 0.43% $\tilde{u}_{y}$ (at $s$) \| 5.12 $\pm$ 0.26% \| 0.03648 $\pm$ 1.14% $\tilde{u}_{y}$ (at $r$) \| 4.77 $\pm$ 0.11% \| -0.0914 $\pm$ 0.46% $\tilde{u}_{y}$ (at $c$) \| 5.55 $\pm$ 0.66% \| -0.0092 $\pm$ 3.20% * • a Probe locations other than $b$ are explicitly denoted. In comparison, Tumuklu et al. (2019), using the POD analysis, had found a least damped eigenmode of $-5.88$ kHz that leads the 2-D (spanwise independent) solution to reach steady state, unlike we find here. Also, our growth rate is larger than that obtained by Sidharth et al. (2018), which is consistent with their finding that a larger growth rate is expected for a larger angle difference between the upper and lower wedges. They performed a Mach 5 hypersonic flow of calorically perfect gas and obtained a nondimensional growth rate of approximately $7.5\text{\times}{10}^{-4}$ for a 12∘-20∘ double wedge (angle difference of 8∘). Following their nondimensionalization, where the growth rate is multiplied by the $\delta_{99}$ boundary-layer thickness at separation equal to 3.35 mm, and divided by the freestream velocity downstream of the leading-edge shock derived from the inviscid shock theory (Anderson, 2003) for observed shock angle of 41∘, $u_{x,2}=2930.8$, we obtain a value of $0.0057$. Now we turn to the question of the spatial origin of the linear instability and answer whether these spanwise structures seen in figure 2 start upstream, at or inside the separation bubble by comparing the boundary layer profiles at wall-normal directions $d_{1}$ to $d_{10}$ shown in figure 3. These are denoted in figure 3 on top of the contours of pressure gradient magnitude, $\|\nabla p_{b}\|$, in the base flow, which identifies the location of shock structure and the recirculation zone. The shear layer ($u_{x}=0$) and the separation and reattachment points are also overlaid. Along each wall-normal direction, three boundary layer profiles are shown–one in the base flow and two at $T$=90.5 on spanwise locations $Y/L_{y}$=0.88 (A) and 1.38 (B). These spanwise locations correspond to a spanwise peak and a trough of the local- streamwise (or wall-tangential) velocity so that the maximum spanwise deviation at $T=90.5$ from the base flow state can be assessed. For profiles corresponding to the lower wedge, $d_{1}$ to $d_{6}$, the local-streamwise velocity, denoted as $u_{t,l}$, is plotted as a function of wall-normal height $H_{l}$. Subscript $`t^{\prime}$ stands for the wall-tangential (or local- streamwise) component and $`l^{\prime}$ is associated with the lower wedge surface. For profiles corresponding to upper wedge, $d_{7}$ to $d_{10}$, the local-streamwise velocity, denoted as $u_{t,u}$, is plotted as a function of wall-normal height $H_{u}$. Similarly, subscript $`l^{\prime}$ is associated with the lower wedge surface. Note that $H_{l}$ and $H_{u}$ are zero at the respective surfaces. The boundary layer profiles just upstream of separation shock ($d_{1}$), at the separation ($d_{2}$), and just downstream of separation ($d_{3}$) are shown in figure 3. At $d_{1}$, all profiles overlap, indicating that the flow is 2-D upstream of the separation. Along $d_{2}$, at the separation, the absolute maximum difference of 0.72% of the freestream velocity, $u_{x,1}$, is seen between $A$ and $B$ profiles at $H_{l}/(0.1L_{y})=0.29$, which indicates spanwise modulation. The difference decreases above this height but remains nonzero even inside the shock layer, indicating the origin of linear instability inside the interaction region of the separation shock layer with the LSB. Profiles $A$ and $B$ also differ from the base state profile, indicating deviation from the base flow. Along $d_{3}$, just inside the separation bubble, $A$ and $B$ profiles deviate from each other by a maximum of 1% at $H_{l}/(0.1L_{y})=0.7$. Further inside the separation bubble, along directions $d_{4}$, $d_{5}$, and $d_{6}$, similar profiles are shown in figures 3 and 3, where the latter figure is a zoom of the rectanular boxed region denoted in the former. The absolute maximum deviation between $A$ and $B$ profiles increases along the local streamwise direction. At $d_{4}$, $d_{5}$ and $d_{6}$, it is 1.34, 1.92, 2.52% at locations $H_{l}/(0.1L_{y})=0.88,1.11,1.57$, respectively, For $d_{5}$ and $d_{6}$ directions, these profiles are on either side of their respective base profiles, indicating spanwise modulation about the base flow. On the upper wedge surface, the boundary layer profiles are shown along $d_{7}$ to $d_{10}$ in figures 3 and 3, where the latter figure is a zoom of the rectangular boxed region denoted in the former. The difference between $A$ and $B$ is even larger on the upper wedge, indicating larger amplitude of spanwise perturbations. At $d_{7}$, $d_{8}$, $d_{9}$, it is 2.8, 3.33, 3.34% at locations $H_{u}/(0.1L_{y})=0.6,0.59,0.62$, respectively, At $d_{10}$ at the reattachment location, the maximum difference decreases to 2.46% at $H_{u}/(0.1L_{y})=0.44$. The generalized inflection point (GIP) is also denoted on each boundary layer profile (open circle). Profiles $d_{1}$, $d_{2}$ and $d_{10}$ have only one GIP, whereas profiles $d_{3}$ to $d_{9}$, inside the separation bubble, have two GIPs. The GIP closest to the wall is induced in the recirculating flow between the shear layer and the surface. The GIP located farthest from the wall is induced between the shear layer and the supersonic flow outside the separation bubble. From profiles $d_{3}$ to $d_{6}$, the upper inflection point moves further away from the wall as the distance between the top enclosure of the bubble and the wall increases. The lower inflection point, more clearly seen in the respective zooms, also moves away from the surface as the distance between the shear layer and the wall increases. From profiles $d_{7}$ to $d_{9}$, both inflection points move closer to the wall. Additionally, notice that each profile exhibits a non-zero local streamwise velocity at the wall, the magnitude of which is maximum before the separation, lowest inside the separation zone on the lower wedge, and relatively larger on the upper wedge. This variation is explained by the rarefaction effects at the wall, more details of which are provided in section 4.3. [] [] [] [] [] [] Figure 3: (a) Base flow pressure gradient magnitude, $\|\nabla p_{b}\|$, normalized by $p_{1}L_{s}^{-1}$, where $p_{1}$ is freestream pressure. Overlaid wall-normal directions $d_{1}$ to $d_{10}$ are shown at a local streamwise distance from the hinge normalized by the length of separation $L_{s}$ as -0.625, -0.5225, -0.375, -0.25, -0.125, 0, 0.125, 0.25, 0.375, 0.51, respectively. (b) Local streamwise velocity tangential to lower wedge surface $u_{t,l}$ normalized by freestream velocity, $u_{x,1}$, versus wall- normal height at locations $d_{1}$, $d_{2}$, $d_{3}$. Insert shows zoom of the marked rectangular box. (c) Similar profiles at locations $d_{4}$, $d_{5}$, $d_{6}$. (d) Zoom of the rectanular region marked in (c). (e) Local streamwise velocity tangential to upper wedge surface, $u_{t,u}$, normalized by $u_{x,1}$ versus wall-normal height at locations $d_{7}$, $d_{8}$, $d_{9}$, $d_{10}$. (f) Zoom of the rectanular region marked in (e). Legends for (b) to (f):( ) base state profile, ( ) profile on an $X-Z$ slice passing through location $A$ ($Y/L_{y}$=0.88) at $T=90.5$, ( ) profile on an $X-Z$ slice passing through $B$ ($Y/L_{y}$=1.38) at $T=90.5$. ### 4.2 Correlation between the shock and separation bubble [] [] [] Figure 4: (a) Contours of $\tilde{u}_{y}$ normalized by $u_{x,1}$ at $T$=90.5 on a plane defined along wall-normal direction $S$, marked in figure 1. $X$ and $Y$ axes are the normalized span and the wall-normal height, respectively. (b) Perturbation pressure gradient magnitude, $\|\nabla\tilde{p}\|$, normalized by $p_{1}L_{s}^{-1}$ along direction $S$ as a function of wall-normal height at two spanwise locations $A$ and $B$. (c) Contours of $\tilde{u}_{y}$ normalized by $u_{x,1}$ at $T$=90.5 on plane defined along wall-normal direction $R$, marked in figure 1. Overlaid line contours: ( ) $\|\nabla\tilde{p}\|=$6.121\text{\times}{10}^{-3}$$ and ( ) $\tilde{\omega}_{y}$=0. The self-excited linear instability leads to the presence of spanwise periodic flow structures in perturbation flow parameters with a spanwise wavelength of $L_{y}$. Figure 4 shows the contours of spanwise perturbation velocity, $\tilde{u}_{y}$, at $T$=90.5 in the wall-normal planes $S$ and $R$ denoted in figure 1. On the $S$-plane, the spanwise periodic flow structures inside the separation bubble are seen between the surface ($H_{l}$=0) and the upper envelope of the separation bubble at $H_{l}=0.15L_{y}$ where the spanwise vorticity, $\tilde{\omega}_{y}$, is zero, as shown in figure 4. These structures have elliptical cross-sections with major and minor axes of lengths roughly equal to 0.4$L_{y}$ and 0.2$L_{y}$, respectively. Note that the upper envelope of the bubble also has a spanwise sinusoidal shape. The overlaid line contours of zero spanwise vorticity between $0<H_{l}<0.1L_{y}$ that are elliptical in shape shows a 90∘ phase shift in its spanwise mode and that of the spanwise velocity, _i.e._ , the center of the circular structure of $\tilde{\omega}_{y}$ is at $Y$=0.88$L_{y}$, inbetween a peak and a trough of $\tilde{u}_{y}$. The spanwise vorticity of the flow is negative inside these elliplical shaped contour lines of $\tilde{\omega}_{y}=0$, i.e. the flow rolls down the surface, and it is positive outside this zone and below the $\tilde{\omega}_{y}$=0 contour line at $H_{l}=0.15L_{y}$, i.e. the flow rolls up the surface. This shows that the flow moves in the spanwise direction while swirling about the spanwise axis ($Y$). Further away from the wall, figure 4 shows, for the first time, the spanwise periodic flow structures inside the strong gradient region of the separation shock ($0.36<H_{l}/L_{y}<0.44$). These structures are in phase with structures inside the separation bubble and they have the same periodicity length. This is consistent with the boundary-layer profiles shown in the previous section that showed the origin of linear instability inside the separation shock layer and the linear stability analysis that showed identical growth rate inside the LSB (probe $b$) and the separation shock (probe $s$). Note that the approximate boundary of the finite shock is marked by dashed horizontal lines corresponding to the isocontour line of normalized perturbation pressure gradient magnitude, $\|\nabla\tilde{p}\|=$0.612\text{\times}{10}^{-2}$$. To justify the choice of this value, figure 4 shows the variation of $\|\nabla\tilde{p}\|$ as a function of wall-normal height, $H_{l}$, along the $S$-plane at two spanwise locations, $A$ ($Y/L_{y}$=0.88) and $B$ ($Y/L_{y}$=1.38). The rapid increase of $\|\nabla\tilde{p}\|$ at $H_{l}=0.36L_{y}$ is indicative of the separation shock, inside of which the value of $\|\nabla\tilde{p}\|$ far exceeds that in the vicinity of the surface. Note that the thickness of the shock layer, $0.083L_{y}=2.39$ mm, is comparable the boundary-layer thickness at separation, $\delta_{99}=3.35$ mm. The locations $A$ and $B$ correspond to the peak and trough of the sinusoidal modulation of $\|\nabla\tilde{p}\|$ inside the separation shock. The difference between the two profiles also highlights the spanwise changes inside the shock layer. In the $R$-plane at the reattachment, a similar contour plot of $\tilde{u}_{y}$ is shown in figure 4, which exhibits spanwise periodic structures inside the reattached boundary layer. Such structures also exist in the vicinity of contour line $\tilde{\omega}_{y}$=0 at $H_{u}=0.36L_{y}$, which indicates the presence of a contact surface $C_{2}$ downstream of the triple point $T_{2}$ at the intersection of separation and detached shocks. Further away from the wall, the contour lines of $\|\nabla\tilde{p}\|$ at $H_{u}=0.61L_{y}$ and $0.677L_{y}$ indicate the approximate layer of detached shock, which is slightly smaller in thickness than the separation shock because the detached shock strength is higher. The spanwise structures inside this shock are not as noticeable as the separation shock. Additionally, figure 5 shows that the spanwise structures inside the separation bubble are present in the contours of all other perturbation flow parameters. Interestingly, inside the separation shock, all flow parameters exhibit spanwise modulations, as shown in the inserts of respective figures. The minimum (negative) and maximum (positive) values of spanwise structures in $\tilde{u}_{t,l}$, $\tilde{u}_{n,l}$, and $\tilde{n}$ are at spanwise location $Y/L_{y}$=0.88 ($A$) and 1.38 ($B$), respectively. All three perturbation temperatures have primary spanwise structures adjacent to the wall having minimum and maximum values at spanwise locations $Y/L_{y}$=1.38 and 0.88, respectively, i.e., 180∘ out of phase with that of velocities and number density. $\tilde{T}_{tr}$ and $\tilde{T}_{rot}$ also exhibit secondary structures right above the primary structures within $0.1<H_{l}<0.15$. Such secondary structures are also seen in $\tilde{u}_{n,l}$ and $\tilde{n}$, but are farther along the height within $0.2<H_{l}<0.35$. Furthermore, the onset of global linear instability ($T$=50) in the separation bubble is followed by the low-frequency unsteadiness of the shock structure. Figure 6 shows the spatio-temporal variation of normalized perturbation number density, $\tilde{n}$, at the triple point $T_{2}$ formed by the intersection of the detached and separation shocks. To capture one cycle of unsteadiness, the simulation had to be continued much longer up to $T$=165. Figure 6 shows that the triple point starts to oscillate at $T$=70 and its motion remains 2-D up to approximately $T$=85, as there is no variation in $\tilde{n}$ along the spanwise direction within this period. Afterword, however, linear instability begins at the triple point, which results in spanwise modulation of $\tilde{n}$. After $T$=100, we can see the presence of both the linear instability and the low-frequency unsteadiness at the triple point, where we see spanwise structures changing in time. These features are more clearly seen in figure 6 at spanwise locations $A$ and $B$. The period of oscillation is 54 $T$, which corresponds to the Strouhal number $St$ of 0.0185, defined based on the length of the separation bubble in the base flow, $L_{s}=$40 mm, and the freestream velocity, $u_{x,1}=$3812 m.s-1 as, $St=\frac{fL_{s}}{u_{x,1}}$ (5) This number is within the low-frequency range, $0.01\leq St\leq 0.05$, reported in the literature (see section 1). [] [] [] [] [] [] Figure 5: Contours of perturbation macroscopic flow parameters at $T$=90.5 on a plane defined along $S$, same as figure 4. (a) number density $\tilde{n}$ (b) local streamwise velocity (i.e., direction perpendicular to $S$), $u_{t,l}$, (c) wall-normal velocity (in the direction of $S$), $u_{n,l}$, (d) translational temperature, $\tilde{T}_{tr}$, (e) rotational temperture,$\tilde{T}_{rot}$, (f) vibrational temperature, $\tilde{T}_{vib}$. All quantities are normalized by freestream values, i.e., number density by $n_{1}$, velocities by $u_{x,1}$, and temperatures by $T_{tr,1}$. [] [] Figure 6: (a) At probe $t$ in the vicinity of the triple point $T_{2}$, denoted in figure 1, the temporal evolution of DSMC-derived perturbation number density, $\tilde{n}$, normalized by $n_{1}$, indicating low-frequency unsteadiness. (b) Normalized $\tilde{n}$ at spanwise locations $A$ and $B$, also marked in (a), indicating the period of unsteadiness. ### 4.3 Rarefaction effects in the surface parameters To understand the flow behaviour near the wall, figure 7 shows surface parameters at two spanwise locations $A$ ($Y/L_{y}=0.88$) and $B$ ($Y/L_{y}=1.38$) at the latest timestep $T$=90.5 and in the base state. Figure 7 shows local-streamwise (tangential) and spanwise velocity slips, $V_{t}$ and $V_{l}$, respectively, and figure 7 shows the local mean-free-path adjacent to the wall, $\lambda$, and the translational temperature jump at the surface, $T_{s}$. Velocity slip and temperature jump are rarefaction effects that are proportional to the Knudsen layer in the vicinity of the wall (Kogan, 1969; Chambre & Schaaf, 1961). Within this layer, two classes of molecules coexist–those reflected from the wall (in our case, diffusely), and those impinging on the wall which enters this layer from the outside region. As a result, the average velocity and temperature of the gas are different from the respective velocity and temperature of the wall. The Knudsen layer is approximately on the order of $\lambda$, the profile of which is noisy because it is obtained on the adaptively refined $C$-mesh. Note that $\lambda$ is inversely proportional to number density, $n$, and proportional to the translational temperature, $T_{tr}^{\omega-0.5}$, where $\omega=0.745$ is the viscosity index of the gas. Figure 7 shows a maximum tangential velocity slip of 2.16% of the freestream velocity at the leading edge ($X$=10 mm), which decreases along the local streamwise direction to 0.6% at $X$=32 mm. Tumuklu et al. (2019) had obtained a maximum velocity slip of 2.45% at the leading edge in their 2-D flow simulation of nitrogen over a double wedge. A large slip at the leading edge is due to the increased rarefaction of gas induced by steep gradients of the leading edge shock. It can be seen from figure 7 that $\lambda$ adjacent to the wall also follows the same behavior as $V_{t}$ in the local streamwise direction, although they are not exactly proportional to each other by a constant factor. Just upstream of the separation, $P_{S}$, within a region from $X$=32 to 36 mm, the local streamwise velocity, $u_{t,l}$, as well as $V_{t}$ decrease rapidly and become zero at the separation point, $P_{S}$ ($X$=36 mm). $\lambda$ also decreases within this region as there is a rapid increase in number density, $n$, and a decrease in translational temperature, $T_{tr}$, near the wall (not shown). Inside the recirculation zone, from $P_{S}$ to $P_{R}$, the point of reattachment, $V_{t}$ is negative because the flow impinging on the wall is opposite to the local streamwise direction. $V_{t}$ and $\lambda$ remain constant on the lower wedge, where the latter is about 3.69% of the freestream mean-free-path, $\lambda_{1}$. On the upper wedge, $V_{t}$ increases in magnitude and so does $\lambda$, as $n$ decreases and $T_{tr}$ increases in the local streamwise direction. From $P_{R}$ to the upper corner of the wedge, $V_{t}$ continues to increase similar to $\lambda$ as the rates of decrease of $n$ and increase of $T_{tr}$ are larger. At the location of expansion on the shoulder, $V_{t}$ decreases a bit before it plateaus. The profiles of $V_{t}$ at $A$, $B$, and the base state, are similar to each other, indicating no significant change so far due to linearly growing mode. The lateral slip, $V_{l}$, also remains within 0.078% on the entire surface of the wedge. [] [] [] [] Figure 7: Surface macroscopic flow parameters in the base state and at the latest time at two spanwise locations $A$ ($Y/L_{y}=0.88$) and $B$ ($Y/L_{y}=1.38$). Note that the base state profiles are time-averaged from 48 to 53 flow times and those at $A$ and $B$ from 85 to 90 flow times. (a) Surface velocity slips $V_{t}$ and $V_{y}$, normalized by $u_{x,1}$. (b) $\lambda$ adjacent to the wall normalized by freestream mean-free-path $\lambda_{1}$, and temperature jump $T_{s}$, normalized by $T_{tr,1}$. (c) The heat transfer and pressure coefficients, $C_{h}$ and $C_{p}$, respectively. (d) A zoom of the boxed regions marked in (c). The translational temperature jump, $T_{s}$ follows a similar behavior as $V_{t}$, where it is maximum at the leading-edge of the wedge and decreases up to the recirculation region, in which it remains constant on the lower wedge and increases on the upper wedge. From $P_{R}$ to the upper corner of the wedge, the rate of increase of temperature jump is larger, whereas on the shoulder, it plateaus. Also, no difference is seen in the profiles of $T_{s}$ at $A$, $B$, and the base state. Figure 7 shows the surface heat flux and pressure coefficients, $C_{h}$ and $C_{p}$, respectively. Similar to local streamwise velocity and temperature slips, $C_{h}$ is maximum at the leading edge of the wedge, decreases along the local streamwise direction, and remains at a nearly constant minimum value from the separation to the hinge. On the upper wedge surface, it increases rapidly up to the upper corner of the wedge, while the rate of increase is larger beyond $X$=61 mm. The pressure coefficient, $C_{p}$, is constant on the lower wedge, which increases sharply between $X$=32 to 38 mm, which is the local streamwise region in the vicinity of the separation point. Inside the recirculation zone on the lower wedge, $C_{p}$ is nearly constant but increases rapidly on the upper wedge up to the top corner of the wedge, where it is maximum. On the shoulder of the wedge, both coefficients decrease significantly. These coefficients are similar in value for profiles $A$, $B$, and the base state, yet figure 7 shows a zoom of the boxed region marked in figure 7, to highlight small differences in these profiles on the lower wedge surface inside the recirculation zone. $C_{h}$ is at most 11.8% higher for $A$ and 10.43% lower for $B$ than the base state, indicating spanwise modulation about the base state. $C_{p}$ is at most 0.852% higher for $A$ than $B$, while both profiles are higher than the base state, indicating a small overall increase in pressure. ### 4.4 Spanwise periodic flow structures [] [] [] [] [] [] Figure 8: At $T$=90.5, isocontour surfaces of $\tilde{u}_{y}$ normalized by $u_{x,1}$ and vorticity components $\tilde{\omega}_{x}$, $\tilde{\omega}_{y}$, $\tilde{\omega}_{z}$ normalized by the local vorticity mangitude. (a) $\tilde{u}_{y}$ in side view along with overlaid cut-boundaries $P$ and $Q$ with arrows attached to them that denote normal vectors [-0.7193 $\hat{i}$ \+ 0.6946 $\hat{k}$] and [0.7193 $\hat{i}$ \- 0.6946 $\hat{k}$], respectively. If extended, the cut-boundaries would intersect the $X$ axis at 21.3 and 14 mm, respectively. (b) $\tilde{u}_{y}$ on the normal side of cut-boundary $P$, (c) $\tilde{u}_{y}$ on the normal side of cut-boundary $Q$, (d) $\tilde{\omega}_{x}$, (e) $\tilde{\omega}_{y}$, (f) $\tilde{\omega}_{z}$. Isocontours of all vorticity components are shown on the normal side of cut- boundary $P$. In summary, the 2-D base flow is unstable to self-excited, small-amplitude, spanwise-homogeneous perturbations, and a linearly growing stationary global mode is observed, which is characterized by spanwise periodic structures in the perturbation flow fields. The spanwise perturbation velocity, $\tilde{u}_{y}=u_{y}$, which was zero at the beginning of the simulation, attains a sinusoidally varying amplitude not only inside the separation bubble but also inside shock layers and downstream of triple points. This section shows the spanwise periodic sinusoidal flow structures in $\tilde{u}_{y}$ and vorticity components. Between the wedge surface and the cut-boundary $P$, marked in figure 8, the spanwise periodic structures are shown in figure 8. The cut-boundary $P$ cuts through the outer isosurface of $\tilde{u}_{y}=0.07\%$ of $u_{x,1}$ to reveal core structures having a larger magnitude of $\tilde{u}_{y}=0.14\%$. The spanwise structures are seen to extend downstream of the reattachment and on the shoulder of the wedge. The global mode is also present in the subsonic and supersonic regions downstream of separation and detached shocks, respectively, as seen from figure 8 upstream of the cut-boundary $Q$ marked in figure 8. Such global behavior is expected due to the strong coupling of shocks and the separation bubble. Finally, the spanwise mode in the isocontours of $X$, $Y$, and $Z$ perturbation vorticity components are also shown in figures 8, 8, and 8, respectively. The $X$ and $Z$ components are in phase with each other and 90∘ out of phase with the $Y$ component. ## 5 Topology of Three-dimensional Laminar Separation Bubble This section investigates the changes in wall-streamlines and three- dimensionality inside the separation bubble by linearly superposing to the 2-D normalized base flow field, $Q_{b}$, a 3-D normalized perturbation field, $\tilde{Q}$, with a small amplitude, $\epsilon$, ranging from 0.005 to 0.1, using equation 2. Note that the velocity field of the base flow is normalized by the $X$-directional freestream velocity component, $u_{x,1}$. The perturbation velocity field at $T=90.5$ is normalized in two ways–by the absolute maximum component of velocity inside the separation bubble, i.e., inside the zone marked in figure 9 (section 5.1) and by the absolute maximum component of velocity in the entire flow field, which is located in the detached shock near the triple point $T_{2}$ (section 5.2). This distinction will highlight why one cannot draw conclusions about flow topology by decoupling the shock and a separation bubble. In the former case, the absolute maximum values of normalized $X$, $Y$, and $Z$ perturbation velocities inside the zone marked in figure 9 are 0.954, 0.455, and 1, respectively. In the latter case, these are 1, 0.0565, and 0.518, respectively. Figure 9: At $Y/L_{y}=0.38$, the $X$-perturbation velocity, $\tilde{u}_{x}$, normalized by the absolute maximum component of velocity inside the zone marked by a dashed line. Note that the zone is extended in the entire span. ### 5.1 Analysis without the coupling of shock and separation bubble [] [] [] [] Figure 10: Wall streamlines in the flow constructed by superposition of scaled 2-D base flow with scaled linear perturbations having amplitude $\epsilon$. (a) $\epsilon=0.005$, (b) $\epsilon=0.01$, (c) $\epsilon=0.05$, (d) $\epsilon=0.1$. Figure 10 shows profiles of wall-streamlines in the superposed flow field for different amplitudes, $\epsilon$. The signature observed in figure 10 typically results from small-amplitude spanwise homogeneous perturbations to the 2-D separation bubble, as was shown by Rodríguez & Theofilis (2010) in an incompressible flow. A series of critical points are formed on the separation and reattachment lines between which the wall-streamlines are slightly bent in the spanwise direction, indicating three-dimensionality of the separated flow. At a saddle point of separation, $S_{s}$, on the line of separation, the flow is attracted in the local streamwise direction and is diverted in the spanwise direction. In the middle of two $S_{s}$ points on the line of separation, a node point of separation, $N_{s}$, is formed, where the flow coming from both saddle points meets and leaves in the wall-normal direction. On the reattachment line, a node point of attachment, $N_{a}$, is formed, where the flow coming from the wall-normal direction is diverted in the spanwise and local streamwise directions. Between two $N_{a}$ points, a saddle point of attachment, $S_{a}$, is formed where the flow coming from the spanwise direction is diverted in the local streamwise direction. Figure 10 shows a similar pattern for a larger amplitude of $\epsilon=0.01$; however, now the node and saddle points on the separation and reattachment lines are not colinear in the local streamwise direction. As a result, the flow exhibits two new saddle points near the hinge, as seen in figure 10 for $\epsilon=0.05$. At the larger amplitude of $\epsilon=0.1$, the two saddle points are aligned with the node points of separation and reattachment; however, in the vicinity of the line connecting the saddle point of separation and reattachment, two counter-rotating foci, $F_{1}$ and $F_{2}$, are formed. Further increase in amplitude may lead to the merging of points $S_{a}$ and $N_{s}$ on the lower wedge and $S_{s}$ and $N_{a}$ on the upper wedge such that the node points on the separation and reattachment lines will disappear. Such a signature would resemble a simple $U$-shaped separation, first classified by Perry & Hornung (1984b). However, these speculations are beyond the purview of linear analysis. We will also see in section 5.2 that such topology cannot be studied without accounting for the perturbations in the shock. [] [] [] [] Figure 11: Three-dimensionality of the flow inside the separation bubble shown using volume streamlines. A volume line is a streamline traveling through 3-D volume data rather than being confined to a surface (Tecplot-360, 2020 R1). As in the earlier figure, the flow is constructed by superposition of scaled 2-D base flow with scaled linear perturbations having amplitude $\epsilon$. (a) $\epsilon=0.005$, (b) $\epsilon=0.01$, (c) $\epsilon=0.05$, (d) $\epsilon=0.1$. The increasing three-dimensionality of the separation bubble is seen in figure 11 for superpositions with the above amplitudes. Comparison of figures 11 and 11 shows increasing spanwise modulation of recirculating streamlines from $\epsilon=0.005$ to $\epsilon=0.01$, while the axis of rotation remains parallel to the spanwise-direction ($Y$). For $\epsilon=0.05$ in figure 11, the streamlines become 3-D, where the axes of rotation are seen to deviate from $Y$, and spanwise modulation is increased. For $\epsilon=0.1$ in figure 11, the streamlines are fully 3-D, where the axes of rotation diverge significantly from $Y$, so much that at some locations it is perpendicular to $Y$. ### 5.2 Analysis with the coupling of shock and separation bubble [] [] [] [] Figure 12: Streamwise velocity, $u_{x}$, obtained from the superposition of normalized base and perturbed flow field showing corrugations of (a) the separation shock at $H_{y}/L_{y}$=0.41 on the $S$-plane and (b) the detached shock at $H_{u}/L_{y}$=0.65 on the $R$-plane. These locations are inside the shock layer as seen from figures 4 and 4. Corresponding legends: ( ) $\epsilon=0.005$, ( ) $\epsilon=0.01$, ( ) $\epsilon=0.05$, ( ) $\epsilon=0.1$. (c) Wall streamlines and (d) volume lines inside the separation bubble for $\epsilon=0.1$. When the perturbation velocity field is normalized by maximum perturbation velocity component inside the shock, the linear coupling of shock and the separation bubble is taken into account. Figure 12 shows the features of the superposed flow field. The spanwise corrugations of the separation and detached shocks in flow fields composed with four increasing amplitudes of linear perturbations are seen in figures 12 and 12, respectively. The three- dimensionality of the separation shock is more than the detached shock and becomes prominent for the largest amplitude of $\epsilon=0.1$. The wall- streamlines in figure 12 for $\epsilon=0.1$ reveal alternate node and saddle points on the separation and reattachment lines, where the node and saddle points on the two lines are not aligned. This topology is similar to that in figure 10 for amplitude $\epsilon=0.005$, when the effect of shock was not taken into account. Similarly, the recirculation streamline inside the separation bubble shows a low degree of spanwise modulation, where the axis of rotation is the $Y$-axis for the largest amplitude of $\epsilon=0.1$ in figure 12, similar to figure 11 for $\epsilon=0.005$ (without coupling). Therefore, the coupled analysis indicates that the deviation from two-dimensionality in the shock structure dominates the deviation in the separation bubble. As a result, the study of three-dimensionality in the topology of an LSB cannot be done by ignoring their coupling with shock structures. ## 6 Conclusion The 3-D LSB induced by a laminar SBLI on a spanwise-periodic, Mach 7 hypersonic flow of nitrogen over a $30^{\circ}-55^{\circ}$ double wedge was simulated using the massively parallel SUGAR DSMC solver using billions of computational particles and collision cells on an adaptively refined octree grid. The fully resolved kinetic solution resulted in accurate modelling of the internal structure of shocks, surface rarefaction effects, thermal nonequilibrium, and time-accurate evolution of 3-D, self-excited perturbations. This is the first simulation that analyzes the linear instability of a 2-D base flow to self-excited, small-amplitude, spanwise- homogeneous perturbations in the low Reynolds number regime. In line with the findings of Tumuklu et al. (2018b) of Mach 16 flows over axisymmetric double cone and Tumuklu et al. (2019) of a 2-D, Mach 7 flow over the double wedge, the 3-D LSB was found to be strongly coupled with the separation and detached shocks. The presence of linear instability led to the formation of spanwise periodic flow structures in 3-D perturbations of macroscopic flow parameters not only inside the LSB, but also in the internal structure of the separation shock. The spanwise periodicity length of the structures at these two zones was found to be the same and their amplitude was found to grow with an average, linear temporal growth rate of 5.0 kHz $\pm$ 0.16%. We obtained a larger value of $0.0057$ for the nondimensional growth rate compared to that of Sidharth et al. (2018) for double wedges with lower angles, which is qualitatively consistent. The boundary-layer profiles in the 2-D base flow were compared with those obtained from the 3-D flow with perturbations at $T$=90.5 at two spanwise locations corresponding to the peak and trough of the spanwise sinusoidal mode. The comparison these profiles upstream and downstream as well as at the point of separation revealed that the linear instability originates in the interaction region of the separation shock with the LSB. The difference between the peak and trough of wall-tangential velocities revealed that the amplitude of perturbations increases inside the recirculation zone from the separation to the reattachment point. All boundary-layer profiles exhibited nonzero wall-tangential velocities at the wall in the Knudsen layer region. The profiles inside the separation zone also showed the presence of two GIPs, one between the wall and shear layer and the other between the shear layer and supersonic flow outside the bubble. The onset of linear instability at $T=50$ was followed by the low-frequency unsteadiness of the triple point $T_{2}$ at $T=70$. The oscillation frequency corresponds to a Strouhal number of $St\sim 0.02$, consistent with the existing literature on turbulent SBLI, but in contrast with the 3-D, finite- span double wedge simulation of Reinert et al. (2020) at a factor of eight times higher density which did not reveal such unsteadiness. To resolve these predictions, the slow linear growth and long time-scale of low-frequency unsteadiness ($\sim 0.57$ ms) suggests that experimental test times must be significantly long to capture these effects. In addition, the long-time ($T>100$) spatio-temporal evolution of the flow at the triple point $T_{2}$ revealed for the first time the presence of spanwise corrugation as well as sinusoidal oscillations in time. Finally, the topology signature in the wall-streamlines of the 3-D flow constructed by superposition of the 2-D base flow and 3-D linear perturbations was analyzed with and without accounting for the coupling between the shocks and the LSB. For a given amplitude of perturbations, significant differences were observed in the topology with versus without coupling. The analysis with coupling also revealed an increase in the corrugation of the separation and detached shocks with increase in amplitude of 3D perturbations. These findings further emphasize that, at these conditions, the 3-D changes to the topology of an LSB cannot be studied without taking into account the coupling with the shock structure. Acknowledgements. The authors acknowledge the Texas Advanced Computing Center (TACC) at the University of Texas at Austin for providing high performance computing resources on Frontera supercomputer under the Leadership Resource Allocation (LRAC) award CTS20001 of 200k SUs that have contributed to the research results reported within this paper. This work also used the Stampede2 supercomputing resources of 400k SUs provided by the Extreme Science and Engineering Discovery Environment (XSEDE) TACC through allocation TG- PHY160006. A part of the simulation was also carried out on Blue Waters supercomputer under projects ILL-BAWV and ILL-BBBK. The Blue Waters sustained- petascale computing project is supported by the National Science Foundation (awards OCI-0725070 and ACI-1238993) the State of Illinois, and as of December, 2019, the National Geospatial-Intelligence Agency. Blue Waters is a joint effort of the University of Illinois at Urbana-Champaign and its National Center for Supercomputing Applications. In addition, the authors thank Dr. Ozgur Tumuklu for providing the 2-D steady flow solution. Funding. The research conducted in this paper is supported by the Office of Naval Research under the grant No. N000141202195 titled, “Multi-scale modelling of unsteady shock-boundary layer hypersonic flow instabilities” with Dr. Eric Marineau as the program officer. Declaration of Interests. The authors report no conflict of interest. Author ORCID. Authors may include the ORCID identifers as follows. S. Sawant, https://orcid.org/0000-0002-2931-9299; D. Levin, https://orcid.org/0000-0002-6109-283X; V. Theofilis, https://orcid.org/0000-0002-7720-3434. ## Appendix A In a typical DSMC simulation, the collision pairs selected using the MFS or the no time counter (NTC) scheme are allowed to collide with probability, $P_{c}=\frac{\sigma_{T}c_{r}}{(\sigma_{T}c_{r})_{max}}$ (6) where $\sigma_{T}=\pi d^{2}$ is the total cross-section, $d$ is the molecular diameter, and $c_{r}$ is the relative speed. The maximum collision cross- section, $(\sigma_{T}c_{r})_{max}$, is stored for each collision cell and is estimated at the beginning of the simulation to a reasonably large value. Bird estimates this number as [Sec. 11.1 Bird 1994], $(\sigma_{T}c_{r})_{max}=(\pi d_{r}^{2})300\sqrt{T_{tr}/300}$ (7) where $d_{r}$ is the reference molecular diameter. As the simulation progresses, the parameter is updated if a larger value is encountered in a collision cell. However, a problem occurs at an AMR step, where the old $C$-mesh is deleted, and a new one is constructed. For the newly created collision cells, an estimate of $(\sigma_{T}c_{r})_{max}$ is required. If the parameter value is arbitrarily guessed based on equation 7, then the instantaneous temporal signals of macroscopic parameters exhibit kinks at the timesteps when the AMR step is performed. Although these kinks decay in approximately 3 to 4 $\mu$s, they can spuriously reveal a dominant frequency equal to the inverse of the time period between two AMR steps. To avoid the corruption of instantaneous signals with such artificial perturbations, at an AMR step, each root cell stores the smallest value of $(\sigma_{T}c_{r})_{max}$ among all of its collision cells before deleting the $C$-mesh. After a new $C$-mesh is formed, the value stored in the root is assigned as the lowest estimated guess to all collision cells in a given root. Those newly formed collision cells, for which the actual value of $(\sigma_{T}c_{r})_{max}$ must be larger than that assigned as an estimate, quickly update to this value within the next 0.2 $\mu$s. This strategy avoids the kinks in the instantaneous residual. ## Appendix B This appendix shows the use of the POD method (Luchtenburg et al., 2009) to remove the statistical noise in instantaneous perturbation macroscopic flow parameter fields obtained from DSMC. The use of the POD method to reduce statistical noise in a stochastic simulation can be found in a number of resources (Grinberg, 2012; Tumuklu et al., 2019). This method performs the singular value decomposition (SVD) of the input data matrix $\mathcal{D}$ formed from the solution of any given macroscopic flow parameter such that the number of rows and columns are equal to the number of total sampling cells $N_{c}$ in the DSMC domain and the instantaneous time snapshots $N_{s}$, respectively. The SVD procedure results in the decomposition, $\centering\begin{split}\mathcal{D}&=\phi\mathcal{S}\mathcal{T}\\\ \end{split}\@add@centering$ (8) where $\phi$ is the matrix of spatial modes having dimensions $N_{c}\times N_{r}$, $N_{r}$ is the user-specified rank of the reduced SVD approximation to $\mathcal{D}$, $\mathcal{S}$ is the square diagonal matrix of singular values having dimensions $N_{r}\times N_{r}$, and $\mathcal{T}$ is the matrix of temporal modes of dimensions $N_{r}\times N_{s}$. The $i^{th}$ spatial and temporal modes are stored in the $i^{th}$ column of $\phi$ and row of $\mathcal{T}$, respectively. The singular values in $\mathcal{S}$ are arranged in decreasing order, and their square corresponds to the amount of energy in the mode. After the decomposition, a reduced-order, noise-filtered representation of $\mathcal{D}$ can be constructed by forming a new data matrix $\mathcal{D}_{2}$ from a user-specified number of ranks $N_{r2}$, which is smaller than $N_{r}$. $N_{r2}$ is chosen such that the difference between any time snapshot of $\mathcal{D}_{2}$ and that of $\mathcal{D}$ is within statistical noise. [] [] [] Figure 13: (a) Modal energy in perturbation macroscopic flow parameters based on singular values obtained from proper orthogonal decomposition. (b) Contours of unfiltered (raw DSMC data) $\tilde{u}_{y}$ normalized by $u_{x,1}$ at $T$=90.5 on a plane defined along wall-normal direction $S$ as in figure 4. Overlaid on it the contour lines of noise-filtered reconstruction of $\tilde{u}_{y}$ from the first two proper orthogonal modes. (c) Comparison of unfiltered (DSMC) and filtered (POD) $\tilde{u}_{y}$ along lines $L_{1}$ and $L_{2}$ denoted in (b). For the double wedge solution, the data matrix for each macroscopic flow parameter was formed by the number of sampling cells, $N_{c}=$23.04\text{\times}{10}^{6}$$ and number of time snapshots, $N_{s}=450$. The instantaneous snapshots were collected from $T$=48.0312 to 90.9162, at an interval of 0.0953 flow time, which corresponds to the frequency of 1 MHz. Initially, $N_{r}$=10 was chosen; however, $N_{r2}=2$ was sufficient as the modal energy of higher modes is less than 10%, as shown in figure 13. The modal energy, $E_{i}$, of the $i^{th}$ mode is defined as, $\centering\begin{split}E_{i}&=\frac{S_{i}^{2}}{\sum_{j=1}^{N_{r}}S_{j}^{2}}\end{split}\@add@centering$ (9) where $S_{i}$ is the $i^{th}$ singular value. The total modal energy of the first two modes of perturbation parameters other than $\tilde{u}_{y}$ is almost 70%. For $\tilde{u}_{y}$, this number is lower because the shock structure has little influence on its flowfield, and it is composed only of a slowly growing linear mode and statistical noise. Note that the data matrix itself requires 77.24 GBs of run time memory, larger than the typical compute nodes of supercomputing clusters. Therefore, the method was parallelized based on the Tall and Skinny QR factorization (TSQR) algorithm (Sayadi & Schmid, 2016) to overcome storage requirements and speed up the SVD procedure. Figure 13 shows the original noise-contained DSMC solution of perturbation spanwise velocity at $T$=90.5 on the $S$-plane wall-normal to the lower wedge along with the noise-filtered contour lines of the solution reconstructed using POD. The figure also shows two horizontal dashed lines $L_{1}$ and $L_{2}$ along which the DSMC data is extracted and compared in figure 13. 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# Generalized Adler-Moser Polynomials and Multiple vortex rings for the Gross- Pitaevskii equation Weiwei Ao and Yehui Huang and Yong Liu and Juncheng Wei ###### Abstract. New finite energy traveling wave solutions with small speed are constructed for the three dimensional Gross-Pitaevskii equation $i\Psi_{t}=\Delta\Psi+(1-\|\Psi\|^{2})\Psi,$ where $\Psi$ is a complex valued function defined on ${\mathbb{R}}^{3}\times{\mathbb{R}}$. These solutions have the shape of $2n+1$ vortex rings, far away from each other. Among these vortex rings, $n+1$ of them have positive orientation and the other $n$ of them have negative orientation. The location of these rings are described by the roots of a sequence of polynomials with rational coefficients. The polynomials found here can be regarded as a generalization of the classical Adler-Moser polynomials and can be expressed as the Wronskian of certain very special functions. The techniques used in the derivation of these polynomials should have independent interest. ## 1\. Introduction In this paper, we are interested in the existence of solutions with the shape of multiple vortex rings, to the nonlinear Schrödinger type problem (1.1) $\displaystyle i\Psi_{t}\,=\triangle\Psi+\Big{(}1-\|\Psi\|^{2}\Big{)}\Psi,$ where $\triangle=\partial^{2}_{y_{1}}+\partial^{2}_{y_{2}}+\partial^{2}_{y_{3}}$ is the Laplacian operator in ${\mathbb{R}}^{3}$. Equation (1.1), usually called Gross-Pitaevskii equation (GP), is a well-known mathematical model arising in various physical contexts such as nonlinear optics and Bose-Einstein condensates, see for instance [30]. Traveling wave solutions of the GP equation play important role in its long time dynamics. If $\Psi$ is a traveling wave type solution of the form $\Psi(y,t)\,=\,{\tilde{u}}\big{(}y_{1},\,y_{2},\,y_{3}-ct\big{)},$ then ${\tilde{u}({\tilde{y}}_{1},{\tilde{y}}_{2},{\tilde{y}}_{3})}$ will be a solution of the nonlinear elliptic problem (1.2) $\displaystyle\,-\,i\,c\,\frac{\partial\tilde{u}}{\partial{\tilde{y}}_{3}}\,=\,\triangle{\tilde{u}}\,+\,\Big{(}1-\|{\tilde{u}}\|^{2}\Big{)}{\tilde{u}}.$ The existence or nonexistence of traveling wave solutions to (1.2) with $\tilde{u}\to 1$ as $\|{\tilde{y}}\|\to\infty$ has attracted much attention in the literature, initiated from the work of Jones, Putterman, Roberts [22, 23], where they studied the equation from the physical point of view and obtained solutions with formal and numerical calculation. They carried out their computation in dimension two and three, and find that the solution branches in these two cases have different properties. In particular, in the energy- momentum diagram, the branch in 2D is smooth, while the branch in 3D has a cusp singularity. In any case, the solutions they found have traveling wave speed $c$ less than $\sqrt{2}$ (the sound speed in this context, appears after taking the Madelung transform for the GP equation). A natural question is whether there exist solutions whose traveling speed is larger than the sound speed. In this respect, the nonexistence of finite energy solutions with $c>\sqrt{2}$ is rigorously proved by Gravejat in [18, 19]. This result is also true for $c=\sqrt{2}$ in $\mathbb{R}^{2}$, but the higher dimensional case is still open. The first rigorous mathematical proof of the existence is carried out in [10], where solutions in 2D with small traveling speed are obtained using mountain pass theorem. Later on, the existence of small speed solutions in dimension larger than two are proved in [11], also based on the mountain pass theorem. In [10], a different approach, minimizing the action functional with fixed momentum, is applied to get the existence of solutions with large momentum in dimension $N\geq 3$. This method is further developed in [8] to all dimensions, yielding existence or nonexistence of solutions for any fixed momentum. The asymptotic profile of these solutions are also studied in the above mentioned papers. In particular, for $c$ close to $0,$ in 2D, these solutions have two vortice and around them, the solution is close to the degree one vortex solution of the the Ginzburg-Landau equation; while in 3D, the solutions have the shape of a single vortex ring, see also [12]. We also refer to the paper [7] by F. Bethuel, P. Gravejat and J. Saut and the references therein for more details and discussions. The question of existence for all traveling speed $c\in(0,\sqrt{2})$ is quite delicate. It is proved by Maris in [27] that in dimension $N>2$, one can minimize the action under a Pohozaev constraint, obtaining solutions in the full speed interval $(0,\sqrt{2})$. Unfortunately, this argument breaks down in 2D, thus leaving the problem open in this dimension. Recently, Bellazzini- Ruiz [2] proved that the existence of almost all subsonic speed in 2D, using mountain pass argument. They also recovered the results of Maris in 3D. Note that when the parameter $c=0,$ equation (1.2) reduces to the Ginzburg- Landau equation: (1.3) $\Delta u+u\left(1-\left\|u\right\|^{2}\right)=0.$ In $\mathbb{R}^{2}$, for each $\tau\in\mathbb{Z}\backslash\left\\{0\right\\},$ it is known that the Ginzburg-Landau equation $\left(\ref{Landau}\right)$ has a degree $\tau$ vortex solution. In the polar coordinate, it has the form $S_{\tau}\left(r\right)e^{i\tau\theta}$. The function $S_{\tau}$ is real valued and vanishes exactly at $r=0.$ It satisfies $-S_{\tau}^{\prime\prime}-\frac{1}{r}S_{\tau}^{\prime}+\frac{\tau^{2}}{r^{2}}S_{\tau}=S_{\tau}\left(1-S_{\tau}^{2}\right)\text{ in }\left(0,+\infty\right).$ This equation indeed has a unique solution $S_{\tau}$ with $S_{\tau}\left(0\right)=0$ and $S_{\tau}\left(+\infty\right)=1$ and $S_{\tau}^{\prime}\left(r\right)>0.$ See [17, 31] for a proof. Recently, based on the vortex solutions of the Ginzburg-Landau equation, multi-vortex traveling wave solutions to (1.2) were constructed in [25] using Lyapunov-Schmidt reduction method. These solutions have $\frac{n(n+1)}{2}$ pairs of vortex-anti vortex configuration, where the location of the vortex points are determined by the roots of the Adler-Moser Polynomials. It is worth pointing out that the Adler-Moser polynomials arise naturally from the rational solutions of the KdV equation. We also mention that as $c$ tends to $\sqrt{2},$ a suitable rescaled traveling waves will converge to solutions of the KP-I equation, which is an important integrable system, see [6, 13]. Interestingly, the KP-I equation is actually a two dimensional generalization of the classical KdV equation. Hence in the context of GP equation, we see the KP-I equation in the transonic limit and KdV in the small speed limit. The inherent reason behind this phenomena is still to be explored. As a related result, we would like to mention that numerical simulation has been performed in [14] to illustrate the higher energy solutions of the GP equation. Denote the degree $\pm 1$ vortex solutions of the Ginzburg-Landau equation $\left(\ref{Landau}\right)$ as $v_{+}=e^{i\theta}S_{1}\left(r\right),v_{-}=e^{-i\theta}S_{1}\left(r\right).$ To better explain our main result in this paper, let us recall the following result proved in [25], which provides a family of multi-vortex solutions in dimension $2$. ###### Theorem 1.1 ([25]). In $\mathbb{R}^{2}$, for each $n\leq 34,$ there exists $c_{0}>0,$ such that for all $c\in\left(0,c_{0}\right),$ the equation $\left(\ref{TWK0}\right)$ has a solution $u_{c}$, with $u_{c}=\prod\limits_{k=1}^{n\left(n+1\right)/2}\left[v_{+}\left(z-c^{-1}p_{k}\right)v_{-}\left(z+c^{-1}p_{k}\right)\right]+o\left(1\right),$ where $p_{k}$, $k=1,...,n\left(n+1\right)/2$ are roots of the Adler-Moser polynomials. In this paper, we construct new traveling waves for $c$ close to $0$ in 3D. The solutions will have multiple vortex rings. By our construction below, it turns out that the location of the vortex points are closely related to the following system (Balancing condition): (1.4) $\left\\{\begin{array}[c]{l}{\displaystyle\sum\limits_{j=1,j\neq k}^{m}}\frac{1}{\mathbf{a}_{k}-\mathbf{a}_{j}}-{\displaystyle\sum\limits_{j=1}^{n}}\frac{1}{\mathbf{a}_{k}-\mathbf{b}_{j}}=-n,\text{ for }k=1,...,m,\\\ -{\displaystyle\sum\limits_{j=1,j\neq k}^{n}}\frac{1}{\mathbf{b}_{k}-\mathbf{b}_{j}}+{\displaystyle\sum\limits_{j=1}^{m}}\frac{1}{\mathbf{b}_{k}-\mathbf{a}_{j}}=-m,\text{ for }k=1,...,n.\end{array}\right.$ Here $\mathbf{a}_{j},j=1,...,m,$ $\mathbf{b}_{\ell},\ell=1,...,n$, are complex numbers in the $z=x_{1}+ix_{2}$ plane. The integer $m$ actually denotes the number of positively oriented vortex rings and $n$ denotes the number of negatively oriented ones. Moreover, the solvability of our original problem is related to the nondegeneracy of the linearized operator $dF$ of the map $F$ defined by (5.2). The make our construction possible, the solution $\mathbf{a}_{j},\mathbf{b}_{\ell}$ to the system (1.4) has to satisfy some symmetric properties. We therefore introduce the following condition: ($\mathcal{M}$). $m>n$. The points $\mathbf{a}_{j},\mathbf{b}_{\ell}$, $j=1,...,m$, $\ell=1,...,n$ are all distinct. The set of points of $\\{\mathbf{a}_{1},...,\mathbf{a}_{m}\\}$ and $\\{\mathbf{b}_{1},...,\mathbf{b}_{n}\\}$ are both symmetric with respect to the $x_{1}$ axis. We use Lyapunov-Schmidt reduction method to construct multi-vortex ring solutions. Our main result is the following: ###### Theorem 1.2. Suppose $\mathbf{a}_{j},\mathbf{b}_{\ell}$, $j=1,...,m$, $\ell=1,...,n$ is a solution of (1.4) satisfying condition $\mathcal{M}$ and the linearized operator $dF$ of (5.2) is non-degenerate at this solution in the sense defined in Section 5. Then for all $\varepsilon>0$ sufficiently small, there exists an axially symmetric solution $u=u(\sqrt{y_{1}^{2}+y_{2}^{2}},y_{3})$ to equation (2.2), and $u$ has $m$ positively oriented vortex rings and $n$ negatively oriented vortex rings. The distance of the vortex rings to the axis is of the order $O(\varepsilon^{-1})$, while the mutual distance of two vortex rings are of the order $O(\varepsilon^{-1}/\|\ln\varepsilon)\|$. After scaling back by the factor $\varepsilon\|\ln\varepsilon\|$, the position of the vortex rings in the $(x_{1},x_{2})$ plane is close to suitable $x_{1}$-translation of those points $\\{\mathbf{a}_{1},...,\mathbf{a}_{m},\mathbf{b}_{1},...,\mathbf{b}_{n}\\}$. More precise description of the solutions can be found in the course of the proof. From the proof in Section 5, we can see that in the case of two positively oriented vortex rings and one negatively oriented vortex ring, there exists solutions to (1.4) and the corresponding linearized operator of (5.2) is non-degenerate. Hence one can construct traveling wave solutions with three vortex rings. We also show in Section 5.2 and Section 6 that (1.4) has solutions satisfying $\mathcal{M}$, provided that $m=n+1$. (Surprisingly, if $m-n>1$, we have not found any solutions satisfying $\mathcal{M}$.) When $m=n+1$ the location of the vortex points are determined by the roots of generating polynomials which have recurrence relations and can be explicitly written down using certain Wronskians. These generating polynomials are natural generalizations of the classical Adler-Moser polynomials. We refer to Section 6 for more details. Let us point out that traveling wave solutions of the Schrodinger map equation with single vortex ring has been constructed in [24]. In principle, our method in this paper can also be applied to this equation and other related equation such as the Euler equation. The dimension three case (with obvious extension to higher dimensions) studied in the present paper actually has some new properties compared to the 2D case. Roughly speaking, the main difference of the 2D and 3D case is the following. In 2D, the vortex location of our solutions is determined by the Adler-Moser polynomials. These polynomials can be obtained by method of integrable systems and are well studied. However, in 3D, due to the presence of additional terms in the equation, the vortex location is not determined by Adler-Moser polynomials. Indeed they are determined by a sequence of polynomials, which can be regarded as a generalization of Adler-Moser polynomials, and up to our knowledge, are new. We have to find these new generating polynomials using some techniques from the theory of integrable systems. This step is nontrivial and may have independent interest. If we rescale the Gross-Pitaevskii equation $x=\epsilon^{-1}\bar{x}$, then the distance between the locations of the vortex rings obtained in Theorem 1.2 is of the order ${\mathcal{O}}(\frac{1}{\|\log\epsilon\|})$. Note that this distance is much smaller than the leapfrogging region in which the distance between the vortex rings is of the order ${\mathcal{O}}(\frac{1}{\sqrt{\|\log\epsilon\|}})$. For the dynamics of vortex rings in the leapfrogging region for the Gross-Pitaevskii equation we refer to Jerrard-Smets [21] and the references therein. The paper is organized as follows. In Section 2, we formulate the 3D problem as a two dimensional one. In Section 3, we introduce the approximate multi- vortex ring solutions and estimate their error. Section 4 is devoted to the study of a nonlinear projected problem. This is more or less standard. The main part of the paper is Section 5 and Section 6, where we get the reduced problem for the position of the vortex points and study some generating polynomials whose roots determine the location of the vortex rings. Acknowledgement W. Ao is supported by NSFC no. 11631011, no. 11801421, and no. 12071357. Y. Liu is partially supported by NSFC no. 11971026 and “The Fundamental Research Funds for the Central Universities WK3470000014”. J. Wei is partially supported by NSERC of Canada. ## 2\. Formulation of the problem We are looking for a solution to problem (1.1) in the form $\Psi(y,t)\,=\,{\tilde{u}}\Big{(}y_{1},y_{2},y_{3}-ct\Big{)}.$ Then $\tilde{u}$ must satisfy (2.1) $-ic\frac{\partial\tilde{u}}{\partial y_{3}}=\Delta\tilde{u}+(1-\|\tilde{u}\|^{2})\tilde{u}.$ Let $\varepsilon>0$ be a small parameter. We would like to seek solutions with traveling speed $c=\varepsilon\|\ln\varepsilon\|$. Equation (1.2) then becomes (2.2) $-i\varepsilon\|\ln\varepsilon\|\frac{\partial\tilde{u}}{\partial y_{3}}=\Delta\tilde{u}+(1-\|\tilde{u}\|^{2})\tilde{u}.$ We require the solution $\tilde{u}$ satisfies $\tilde{u}(y)\to 1\mbox{ as }\|y\|\to\infty.$ We are interested in the solutions axially symmetric with respect to the $y_{3}$ axis. Let us introduce $x_{1}=\sqrt{y^{2}_{1}+y^{2}_{2}},x_{2}=y_{3},$ and $z=x_{1}+ix_{2},\,u(x_{1},x_{2})=\tilde{u}(y_{1},y_{2},y_{3}).$ Then we get the following equation satisfied by $u$: (2.3) $-i\varepsilon\|\ln\varepsilon\|\frac{\partial u}{\partial x_{2}}=\Delta_{(x_{1},x_{2})}u+\frac{1}{x_{1}}\frac{\partial u}{\partial x_{1}}+(1-\|u\|^{2})u,$ with boundary conditions $\frac{\partial}{\partial x_{1}}u(0,x_{2})=0,\,u\to 1\mbox{ as }\|z\|\to\infty.$ Observe that the problem (6.2) is invariant under the following two transformations: $u(z)\to\overline{u(\bar{z})},\,u(z)\to u(-\bar{z}).$ Thus we impose the following symmetry on the solutions $u$: $\Sigma=\\{u(z)=\overline{u(\bar{z})},\,u(z)=u(-\bar{z}).\\}$ This symmetry will play an important role in our analysis. As a conclusion, if we write $u(x_{1},x_{2})=u_{1}(x_{1},x_{2})+iu_{2}(x_{1},x_{2}),$ then $u_{1}$ and $u_{2}$ enjoy the following conditions: (2.4) $\displaystyle\begin{aligned} u_{1}(x_{1},x_{2})=u_{1}(-x_{1},x_{2}),&\qquad u_{1}(x_{1},x_{2})=u_{1}(x_{1},-x_{2}),\\\ u_{2}(x_{1},x_{2})=u_{2}(-x_{1},x_{2}),&\qquad u_{2}(x_{1},x_{2})=-u_{2}(x_{1},-x_{2}),\\\ \frac{\partial u_{1}}{\partial x_{1}}(0,x_{2})=0,&\qquad\frac{\partial u_{2}}{\partial x_{1}}(0,x_{2})=0.\end{aligned}$ We now have a two dimensional elliptic system with Neumann boundary condition $\frac{\partial u}{\partial x_{1}}(0,x_{2})=0$. Compared with the two dimensional problem studied in [25], there are two differences: Firstly, there is an extra term $\frac{1}{x_{1}}\frac{\partial u}{\partial x_{1}}$; Secondly, the coefficient in front of $\frac{\partial u}{\partial x_{2}}$ becomes $\varepsilon\|\ln\varepsilon\|$, instead of $\varepsilon$. Some remarks are in order. We aim to construct multi-vortex ring solutions to (2.2). For single vortex ring, one can use $v^{+}(x-p)v^{-}(x+p)$ as a good approximate solution for the equation (6.2). But for multi-vortex rings, the vortex-anti vortex pairs are not good enough because of the extra term $\frac{1}{x_{1}}\frac{\partial u}{\partial x_{1}}$ and the Neumann boundary condition. So we need to use more accurate approximate solution which we will explain in the next section. ## 3\. The approximate solution In this section, we would like to define a family of approximate solutions for the equation (6.2). ### 3.1. The first approximate solution We consider $\mathcal{K}$ distinct points $p_{j}=(p_{j,1},p_{j,2})$, $j=1,...,\mathcal{K}$, lying in the right half of the $z$ plane. Let us define $p_{j}^{}=-\bar{p}_{j}$ for $j=1,\cdots,\mathcal{K}$. We also denote $p_{\mathcal{K}+j}=p_{j}^{}$ for $j=1,\cdots,\mathcal{K}$. Intuitively, these points represent the location of the vortex rings. We also suppose that the set of points $\\{p_{1},...,p_{\mathcal{K}}\\}$ is symmetric with respect to the $x_{1}$ axis. Moreover, we will assume: (A1.) (3.1) $\displaystyle\rho$ $\displaystyle:=min_{\ell\neq j,1\leq\ell,j\leq\mathcal{K}}\\{\|p_{\ell}-p_{j}\|\\}\sim\frac{1}{\varepsilon\|\ln\varepsilon\|},$ and $\|p_{j,1}\|\sim\frac{1}{\varepsilon}$ for $j=1,\cdots,\mathcal{K}.$ n order to understand more clearly the difference between the 2D and 3D case, let us now following the strategy of [25] to define an approximate solution. Let $S=S_{1}$ be the function associated to the degree one vortex solution of the Ginzburg-Landau equation, defined in the first section. Define $u_{j}:=S(\|z-p_{j}\|)e^{i\tau_{j}\theta_{j}},\,\ j=1,\cdots,\mathcal{K},$ where $\theta_{j}$ is the angle around $p_{j}$, $\tau_{j}=+1\mbox{ or}-1$, corresponding to the degree $\pm 1$ vortex. We then set $u_{j}=S(\|z-p_{j}\|)e^{-i\tau_{j-\mathcal{K}}\theta_{j}},\,\ j=\mathcal{K}+1,\cdots,2\mathcal{K}$ where $\theta_{j}$ is the angle around $p_{j}$. The reason of defining these functions is the following: Projecting a vortex ring onto the $z$ plane, we get two circles in the right and left plane with different orientation. Here $u_{j}$ and $u_{j+\mathcal{K}}$ can be viewed as a vortex-antivortex pair. We now define the first approximate solution as (3.2) $U=\Pi_{j=1}^{2\mathcal{K}}u_{j}.$ We will see that this approximate solution is not good enough to handle the 3D case and later on we will introduce a refined approximate solution. Note that at this moment, we still haven’t decided the sign of the degree of the vortex. This will also be done later on. Since each vortex in the right half plane has a vortex in the left plane with opposite sign, we can check directly that $U\to 1$ as $\|z\|\to\infty$. We will see that the approximate solution satisfies the boundary and symmetry condition (2.4). In fact, by the choice of the vortex points, one has ###### Lemma 3.1. The approximate solution $U$ has the following symmetry: $U(\bar{z})=\bar{U}(z),\,\,U(z^{})=U(z)$ where $z^{}=-\bar{z}$. ###### Proof. This is the result by the definition of the vortex points. Since $v_{-}(z)=\overline{v_{+}(z)}$, one has $\begin{split}U(z^{})&=\Pi_{j=1}^{\mathcal{K}}[u_{j}(z^{}-p_{j})u_{\mathcal{K}+j}(z^{}-p_{j}^{})]\\\ &=\Pi_{j=1}^{\mathcal{K}}[u_{j}((z-p_{j}^{})^{})u_{\mathcal{K}+j}((z-p_{j})^{})]\\\ &=\Pi_{j=1}^{\mathcal{K}}[u_{\mathcal{K}+j}(z-p_{j}^{})u_{j}(z-p_{j})]=U(z),\end{split}$ and since the points $\\{p_{j}\\}$ are invariant with respect to the reflection across the $x_{1}$ axis, we have $\begin{split}U(\bar{z})&=\Pi_{j=1}^{\mathcal{K}}[u_{j}(\bar{z}-p_{j})u_{\mathcal{K}+j}(\bar{z}-p_{j}^{})]\\\ &=\Pi_{j=1}^{\mathcal{K}}[\bar{u}_{j}(z-\bar{p}_{j})\bar{u}_{\mathcal{K}+j}(z-\bar{p}_{j}^{})]=\bar{U}(z).\end{split}$ This finishes the proof. ∎ ### 3.2. The error of the first approximate solution Firstly, we estimate the error of the first approximate solution $U$. Since it satisfies the symmetry and boundary condition (2.4), one only need to consider in the domain $\\{x_{1}>0\\}$. Recall that the degree $\pm 1$ vortex satisfies $S^{\prime\prime}(r)+\frac{1}{r}S^{\prime}(r)-\frac{1}{r^{2}}S(r)+(1-S^{2})S=0.$ It has the following properties([29]): ###### Lemma 3.2. The vortex solution satisfies the following properties: * (i). $S(0)=0,\,S^{\prime}(r)>0,\,S(r)\in(0,1)$; * (ii.) $S(r)=1-\frac{1}{2r^{2}}+O(\frac{1}{r^{4}})$ as $r\to\infty$; * (iii). $S(r)=a_{0}r-\frac{a_{0}}{8}r^{3}+O(r^{5})$ as $r\to 0$ where $a_{0}$ is a positive constant. In this subsection, we are going to estimate the error caused by the first approximation $U$. Use $E_{1}$ to denote the error : $\displaystyle E_{1}=i\varepsilon\|\ln\varepsilon\|\frac{\partial U}{\partial x_{2}}+\Delta U+(1-\|U\|^{2})U+\frac{1}{x_{1}}\frac{\partial U}{\partial x_{1}}.$ We have $\begin{split}\Delta U&=\Delta(u_{1}\cdots u_{2\mathcal{K}})\\\ &=\sum_{\ell=1}^{\mathcal{K}}(\Delta u_{\ell}\Pi_{j\neq\ell}u_{j})+\sum_{j\neq\ell}\nabla u_{\ell}\cdot\nabla u_{j}\Pi_{t\neq\ell,j}u_{t}.\end{split}$ On the other hand, writing $\rho_{j}=\|u_{j}\|^{2}-1$, one has $\begin{split}\|U\|^{2}-1&=\Pi_{j=1}^{2\mathcal{K}}(1+\rho_{j})-1\\\ &=\sum_{j}\rho_{j}+\sum_{j=1}^{2\mathcal{K}}\mathcal{Q}_{j},\end{split}$ where $\mathcal{Q}_{j}=\sum_{i_{1}<i_{2}<\cdots<i_{i}}(\rho_{i_{1}}\cdots\rho_{i_{i}})$. Using the fact that $\Delta u_{j}-\rho_{j}u_{j}=0$, we get $\begin{split}\Delta U+(1-\|U\|^{2})U&=\sum_{\ell\neq j}(\nabla u_{\ell}\cdot\nabla u_{j}\Pi_{t\neq\ell,j}u_{t})-U\sum_{j=2}^{2\mathcal{K}}\mathcal{Q}_{j}.\end{split}$ Let $\varphi_{0}=\sum_{j=1}^{\mathcal{K}}{\tau_{j}}(\theta_{j}-\theta_{\mathcal{K}+j}),$ and $\,r_{j}=\|z-p_{j}\|,\,r_{\mathcal{K}+j}=\|z-q_{j}\|,r_{j,1}=x_{1}-p_{j,1},r_{j,2}=x_{2}-p_{j,2}.$ We have $\begin{split}\frac{1}{x_{1}}\frac{\partial U}{\partial x_{1}}&=\frac{1}{x_{1}}\Big{[}e^{i\varphi_{0}}\frac{\partial}{\partial x_{1}}(\Pi_{j=1}^{\mathcal{K}}S(r_{j})S(r_{\mathcal{K}+j}))+ie^{i\varphi_{0}}\frac{\partial\varphi_{0}}{\partial x_{1}}\Pi_{j=1}^{\mathcal{K}}S(r_{j})S(r_{\mathcal{K}+j})\Big{]}\\\ &=\Big{(}\sum_{j=1}^{2\mathcal{K}}\frac{1}{x_{1}}\frac{S^{\prime}(r_{j})}{S(r_{j})}\frac{r_{j,1}}{r_{j}}+\frac{i}{x_{1}}\frac{\partial\varphi_{0}}{\partial x_{1}}\Big{)}U.\end{split}$ Similarly, there holds $\begin{split}\frac{\partial U}{\partial x_{2}}&=\sum_{j=1}^{2\mathcal{K}}\partial_{x_{2}}u_{j}\Pi_{\ell\neq j}u_{\ell}\\\ &=\Big{(}\sum_{j=1}^{2\mathcal{K}}\frac{S^{\prime}(r_{j})}{S(r_{j})}\frac{r_{j,2}}{r_{j}}+i\frac{\partial\varphi_{0}}{\partial x_{2}}\Big{)}U.\end{split}$ Combining the above computations, we obtain $\begin{split}E_{1}&=\sum_{\ell\neq j}\frac{(\nabla u_{\ell}\cdot\nabla u_{j})}{u_{\ell}\,u_{j}}U-U\sum_{j=2}^{2\mathcal{K}}\mathcal{Q}_{j}\\\ &+\Big{(}\sum_{j=1}^{2\mathcal{K}}\frac{1}{x_{1}}\frac{S^{\prime}(r_{j})}{S(r_{j})}\frac{r_{j,1}}{r_{j}}+\frac{i}{x_{1}}\frac{\partial\varphi_{0}}{\partial x_{1}}\Big{)}U\\\ &+i\varepsilon\|\ln\varepsilon\|\Big{(}\sum_{j=1}^{2\mathcal{K}}\frac{S^{\prime}(r_{j})}{S(r_{j})}\frac{r_{j,2}}{r_{j}}+i\frac{\partial\varphi_{0}}{\partial x_{2}}\Big{)}U.\end{split}$ In the sequel, we denote $\|p_{j,1}\|$ by $d_{j}$. Direct computation yields $\begin{split}\frac{\partial\varphi_{0}}{\partial x_{2}}&=\sum_{j=1}^{\mathcal{K}}\tau_{j}\big{[}\frac{\partial\theta_{j}}{\partial x_{2}}-\frac{\partial\theta_{\mathcal{K}+j}}{\partial x_{2}}\big{]}\\\ &=\sum_{j=1}^{\mathcal{K}}\tau_{j}\Big{[}\frac{x_{1}-p_{j,1}}{r^{2}_{j}}-\frac{x_{1}-p^{}_{j,1}}{r^{2}_{\mathcal{K}+j}}\Big{]}\\\ &=\sum_{j=1}^{\mathcal{K}}\tau_{j}\frac{2d_{j}(x_{1}^{2}-p_{j,1}^{2}-(x_{2}-p_{j,2})^{2})}{r_{j}^{2}r_{\mathcal{K}+j}^{2}}.\end{split}$ We also have $\begin{split}\frac{1}{x_{1}}\frac{\partial\varphi_{0}}{\partial x_{1}}&=\frac{1}{x_{1}}\sum_{j=1}^{\mathcal{K}}\tau_{j}\Big{[}\frac{\partial\theta_{j}}{\partial x_{1}}-\frac{\partial\theta_{\mathcal{K}+j}}{\partial x_{1}}\Big{]}\\\ &=-\frac{1}{x_{1}}\sum_{j=1}^{\mathcal{K}}\tau_{j}\Big{[}\frac{x_{2}-p_{j,2}}{r^{2}_{j}}-\frac{x_{2}-p^{}_{j,2}}{r^{2}_{\mathcal{K}+j}}\Big{]}.\end{split}$ Observe that $\frac{1}{x_{1}}\frac{\partial\varphi_{0}}{\partial x_{1}}$ contributes to the imaginary part of the error $E_{1}$. Note that away from the vortex point $p_{j}$, this decays only at the rate $O(r_{j}^{2})$, which is not sufficient for our construction. Hence the vortex-antivortex pair is not enough to be a good approximate solution. ### 3.3. The reference vortex ring In order the get rid of these singularities, one need more accurate approximations for the vortex ring. In [21], leap frogging behavior of the vortex rings to the GP equation has been analyzed. Indeed, our construction in this paper is partly inspired by these leap frogging behavior. Following the analysis performed in [21], we introduce the potential function $A_{a}$, which satisfies the following equation: (3.3) $\left\\{\begin{array}[]{l}-div\Big{(}\frac{1}{x_{1}}\nabla(x_{1}A_{a})\Big{)}=2\pi\delta_{a}\mbox{ in }H,\\\ A_{a}=0\mbox{ on }\partial H,\end{array}\right.$ where $H=\\{(x_{1},x_{2})\in\mathbb{R}^{2},\,x_{1}>0\\}$ and $a\in H$. For the region $\\{x_{1}\leq 0\\}$, we consider the odd extension of $A_{a}$. The expression of $A_{a}$ can be integrated explicitly in terms of complete elliptic integrals (see [20, 21]). We emphasize that in the literature, there are different notations concerning the definition of complete elliptic integrals, mainly about its arguments. Let $r:=r_{a}=\|z-a\|$. When $r_{a}=o(\|a_{1}\|)$, one has the following asymptotic behavior (3.4) $A_{a}(z)=\Big{(}\ln\frac{a_{1}}{r_{a}}+3\ln 2-2\Big{)}+O\Big{(}\frac{r_{a}}{a_{1}}\|\ln\frac{r_{a}}{a_{1}}\|\Big{)}$ and (3.5) $\partial_{r}A_{a}=-\frac{1}{r}+O(\frac{1}{a_{1}}),$ and for $x_{1}\to 0$ (3.6) $A_{a}(x_{1},x_{2})=\frac{x_{1}a_{1}^{2}}{a_{1}^{3}+x_{2}^{2}}\mbox{ as }\frac{x_{1}}{a_{1}}\to 0.$ Up to a constant phase factor, there exists a unique unimodular map $u_{a}^{}\in C^{\infty}(H\setminus\\{a\\},S^{1})$ such that (3.7) $x_{1}(iu_{a}^{},\nabla u_{a}^{})=x_{1}j(u_{a}^{})=-\nabla^{\perp}(x_{1}A_{a}),$ where $j(u)=u\times\nabla u=(iu,\nabla u)=Re(iu\nabla\bar{u}).$ In the sense of distribution, we have $\left\\{\begin{array}[]{l}div(x_{1}j(u_{a}^{}))=0,\\\ curl(j(u_{a}^{}))=2\pi\delta_{a},\end{array}\right.$ and the function $u_{a}^{}$ corresponds to a singular vortex ring centered at $a$. If we denote by $u_{a}^{}=e^{i\varphi_{a}}$, then by (3.7), one has (3.8) $\partial_{1}\varphi_{a}=\partial_{2}A_{a},\,\partial_{2}\varphi_{a}=-\frac{1}{x_{1}}\frac{\partial(x_{1}A_{a})}{\partial x_{1}}.$ So from the definition of $\varphi_{a}$ and the boundary condition of $A_{a}$, one has $\left\\{\begin{array}[]{l}\Delta\varphi_{a}+\frac{1}{x_{1}}\frac{\partial\varphi_{a}}{\partial x_{1}}=0\mbox{ in }H,\\\ \partial_{1}\varphi_{a}(0,x_{2})=0\mbox{ on }\partial H.\end{array}\right.$ Moreover, using the relation of $\varphi_{a}$ and $A_{a}$ in (3.8), one has (3.9) $\nabla\varphi_{a}=\frac{1}{r_{a}}\nabla^{\perp}r_{a}+O(\frac{1}{x_{1}}\log\frac{a_{1}}{r_{a}})\mbox{ for }\frac{r_{a}}{a_{1}}\to 0$ and (3.10) $\|\nabla\varphi_{a}\|\leq\frac{C}{1+r_{a}}\mbox{ for }r_{a}\geq 1.$ So near the vortex point $a$, $\nabla\varphi_{a}$ can be viewed as a perturbation of $\nabla\theta_{a}$. ### 3.4. Improvement of the first approximate solution We will use $u_{a}^{}$ instead of the vortex-anti vortex pair $e^{i(\theta_{j}-\theta_{\mathcal{K}+j})}$ to define a more accurate approximate vortex ring. In view of the symmetry condition (2.4), the vortex ring associated to a point $a\in H$ will defined to be $S(\|z-a\|)S(\|z-a^{}\|)u_{a}^{}(x)=S(\|z-a\|)S(\|z-a^{}\|)e^{i\varphi_{a}(z)}.$ We can also decompose $\varphi_{a}$ as $\varphi_{a}(z)=\theta_{a}(z)-\theta_{a^{}}(z)+\tilde{\varphi}_{a}.$ Note that the difference $\tilde{\varphi}_{a}$ can be analyzed around the vortex point $a$ using the asymptotic behavior of $A$. Define $\varphi_{d}=\sum_{j=1}^{\mathcal{K}}\tau_{j}\varphi_{p_{j}}=\varphi_{0}+\sum_{j=1}^{\mathcal{K}}\tau_{j}\tilde{\varphi}_{p_{j}}:=\varphi_{0}+\tilde{\varphi}_{\bf p}.$ Then our final approximation will be defined as $\mathcal{U}(x)=U(x)e^{i\tilde{\varphi}_{\bf p}}=\Pi_{j=1}^{2\mathcal{K}}S_{p_{j}}(x)e^{i\varphi_{d}}.$ Namely, we replace the function $\varphi_{0}$ by $\varphi_{d}$ in the first approximate solution. Since $\\{p_{j}\\}$ satisfies (A1), one can see that the new approximate solution will satisfy the symmetry condition (2.4). ### 3.5. Error of the final approximation Now the new error becomes $\begin{split}E_{2}&=i\varepsilon\|\ln\varepsilon\|\frac{\partial\mathcal{U}}{\partial x_{2}}+\Delta\mathcal{U}+(1-\|\mathcal{U}\|^{2})\mathcal{U}+\frac{1}{x_{1}}\frac{\partial\mathcal{U}}{\partial x_{1}}\\\ &:=E_{21}+E_{22}.\end{split}$ Here $E_{21}$ is the first term in the left hand side. We use $B_{l}(p)$ to denote the ball of radius $l$ centered at the point $p$. We have the following error estimate: ###### Lemma 3.3. There exists a constant $C$ such that for all small $\varepsilon$ and all points $p_{j}$ satisfying (A1), we have $\sum_{j=1}^{2\mathcal{K}}\\|E_{2}\\|_{L^{9}(B_{3}(p_{j}))}\leq C\varepsilon\|\ln\varepsilon\|.$ Moreover, we have $E_{2}=i\mathcal{U}[R_{1}+iR_{2}]$, with $R_{1},\,R_{2}$ real valued and $\begin{split}\|R_{1}\|&\leq C\sum_{j=1}^{2\mathcal{K}}\frac{O(\varepsilon^{1-\delta})}{(1+r_{j})^{3}},\\\ \|R_{2}\|&\leq C\sum_{j=1}^{2\mathcal{K}}\frac{O(\varepsilon^{1-\delta})}{1+r_{j}},\end{split}$ for any $\delta\in(0,1)$, if $\|z-p_{j}\|>1$ for all $j$. ###### Proof. We compute, in $B_{\frac{\rho}{5}}(p_{j})$, (3.11) $\begin{split}\frac{\partial\mathcal{U}}{\partial x_{2}}&=\frac{\partial\Pi_{j=1}^{2\mathcal{K}}S_{j}}{\partial x_{2}}e^{i\varphi_{d}}+i\mathcal{U}\nabla\varphi_{d}\\\ &=\big{[}\sum_{j=1}^{2\mathcal{K}}\frac{S^{\prime}(r_{j})}{S(r_{j})}\frac{r_{j,2}}{r_{j}}+i\nabla\varphi_{d}\big{]}\mathcal{U}.\end{split}$ Hence in $(\cup_{j=1}^{2\mathcal{K}}B_{3}(p_{j}))^{c}$, by (3.10), we have $\begin{split}Re\Big{[}\frac{E_{21}}{i\mathcal{U}}\Big{]}&=\varepsilon\|\ln\varepsilon\|\big{[}\sum_{j=1}^{2\mathcal{K}}\frac{S^{\prime}(r_{j})}{S(r_{j})}\frac{r_{j,2}}{r_{j}}\big{]}\\\ &\leq C\sum_{j=1}^{2\mathcal{K}}\frac{O(\varepsilon^{1-\delta})}{(1+r_{j})^{3}},\\\ Im\Big{[}\frac{E_{21}}{i\mathcal{U}}\Big{]}&=\varepsilon\|\ln\varepsilon\|[\nabla\varphi_{d}]\\\ &\leq C\sum_{j=1}^{2\mathcal{K}}\frac{O(\varepsilon^{1-\delta})}{(1+r_{j})}.\end{split}$ We also have $\\|i\varepsilon\|\ln\varepsilon\|\,\partial_{x_{2}}\mathcal{U}\\|_{L^{9}(\cup_{j}\\{r_{j}\leq 3\\})}\leq C\varepsilon\|\ln\varepsilon\|.$ Note that the $L^{\infty}$ norm is not bounded near $p_{j}$, due to the presence of $\ln r_{j}$ term. Next, letting $S_{j}=S(r_{j})$ and using the fact that $\Delta S_{j}-\frac{S_{j}}{r_{j}^{2}}+(1-S_{j}^{2})S_{j}=0,$ one has (3.12) $\begin{split}E_{22}&=\Delta\mathcal{U}+(1-\|\mathcal{U}\|^{2})\mathcal{U}+\frac{1}{x_{1}}\frac{\partial\mathcal{U}}{\partial x_{1}}\\\ &=\mathcal{U}\Big{[}\sum_{j=1}^{2\mathcal{K}}\frac{1}{r_{j}^{2}}-\|\nabla\varphi_{d}\|^{2}+\frac{1}{x_{1}}\sum_{j=1}^{2\mathcal{K}}\frac{S^{\prime}(r_{j})}{S(r_{j})}\partial_{x_{1}}r_{j}-\sum_{j=1}^{2\mathcal{K}}\mathcal{Q}_{j}\\\ &+2i\sum_{j=1}^{2\mathcal{K}}\frac{S^{\prime}(r_{j})}{S(r_{j})}\nabla r_{j}\cdot\nabla\varphi_{d}\Big{]}\end{split}$ where we have used the fact that $\Delta\varphi_{d}+\frac{1}{x_{1}}\frac{\partial}{\partial x_{1}}\varphi_{d}=0.$ By carefully checking the terms, using (3.4)-(3.10), away from the vortex points, one has $\begin{split}Re\Big{[}\frac{E_{22}}{i\mathcal{U}}\Big{]}&\leq C\sum_{j=1}^{2\mathcal{K}}\frac{O(\varepsilon^{1-\delta})}{(1+r_{j})^{3}},\\\ Im\Big{[}\frac{E_{22}}{i\mathcal{U}}\Big{]}&\leq C\sum_{j=1}^{2\mathcal{K}}\frac{O(\varepsilon^{1-\delta})}{(1+r_{j})}.\end{split}$ Moreover, $\\|E_{22}\\|_{L^{9}(\cup_{j}\\{r_{j}\leq 3\\})}\leq C\varepsilon\|\ln\varepsilon\|.$ Combining the estimates for $E_{21}$ and $E_{22}$, we obtain the desired estimates. ∎ ## 4\. Linear theory Now we set up the reduction procedure. The linear theory we use here will be the same one as that of [25]. We recall the framework developed there in the sequel. As usual, we shall look for a solution of (6.2) in the form: (4.1) $u:=\left(\mathcal{U}+\mathcal{U}\eta\right)\chi+\left(1-\chi\right)\mathcal{U}e^{\eta},$ where $\chi$ is a cutoff function such that $\chi(x)=\sum_{j=1}^{2\mathcal{K}}\tilde{\chi}(x-p_{j})$ and $\tilde{\chi}(s)=1$ for $s\leq 1$ and $\tilde{\chi}(s)=0$ for $s\geq 2$ and $\eta=\eta_{1}+\eta_{2}i$ is complex valued function close to $0$ in suitable norm which will be introduced below. We also assume that $\eta$ has the same symmetry as $\mathcal{U}.$ Note that near the vortice, $u$ is obtained from $\mathcal{U}$ by an additive perturbation; while away from the vortice, $u$ is of the form $\mathcal{U}e^{\eta}$. The reason of choosing the perturbation $\eta$ in the form (4.1) is explained in Section 3 of [16] and also in [25]. Essentially, the form of the perturbation far away from the origin makes it easier to handle the decay rates of the error away from the origin. The conditions imposed on $u$ in (2.4) can be transmitted to $\eta$: (4.2) $\displaystyle\begin{aligned} \eta_{1}(x_{1},x_{2})=\eta_{1}(-x_{1},x_{2}),&\qquad\eta_{1}(x_{1},x_{2})=-\eta_{1}(x_{1},-x_{2}),\\\ \eta_{2}(x_{1},x_{2})=\eta_{2}(-x_{1},x_{2}),&\qquad\eta_{2}(x_{1},x_{2})=\eta_{2}(x_{1},-x_{2}),\\\ \frac{\partial\eta_{1}}{\partial x_{1}}(0,x_{2})=0,&\qquad\frac{\partial\eta_{2}}{\partial x_{1}}(0,x_{2})=0.\end{aligned}$ In view of (4.1), we can write $u=\mathcal{U}e^{\eta}+\gamma,$ where $\gamma:=\chi\mathcal{U}\left(1+\eta-e^{\eta}\right).$ Note that $\gamma$ is localized near the vortex points and of the order $o(\eta),$ for $\eta$ small. Set $\mathcal{A}:=\left(\chi+\left(1-\chi\right)e^{\eta}\right)\mathcal{U}.$ Then $u$ can be written as $u=\mathcal{U}\eta\chi+\mathcal{A}.$ Following the computation in [25], we get $\left(1-\left\|u\right\|^{2}\right)u=\left(\mathcal{U}\eta\chi+\mathcal{A}\right)\left(1-\left\|\mathcal{U}e^{\eta}+\gamma\right\|^{2}\right).$ The equation for $\eta$ becomes (4.3) $-\mathcal{A}\mathbb{L}\left(\eta\right)=\left(1+\eta\right)\chi E_{2}\left(\mathcal{U}\right)+\left(1-\chi\right)e^{\eta}E_{2}\left(\ U\right)+N_{0}\left(\eta\right),$ where $E_{2}\left(\mathcal{U}\right)$ represents the error of the approximate solution $\mathcal{U}$, and (4.4) $\mathbb{L}\eta:=i\varepsilon\frac{\partial\eta}{\partial x_{2}}+\Delta\eta+2u^{-1}\nabla u\cdot\nabla\eta-2\left\|u\right\|^{2}\eta_{1}+\frac{1}{x_{1}}\frac{\partial\eta}{\partial x_{1}},$ while $N_{0}$ is $o(\eta),$ and explicitly given by $\displaystyle N_{0}\left(\eta\right)$ $\displaystyle:=\left(1-\chi\right)\mathcal{U}e^{\eta}\left\|\nabla\eta\right\|^{2}+i\varepsilon\|\ln\varepsilon\|\left(\mathcal{U}\left(1+\eta-e^{\eta}\right)\right)\partial_{x_{2}}\chi$ $\displaystyle+\frac{1}{x_{1}}\left(\mathcal{U}\left(1+\eta-e^{\eta}\right)\right)\partial_{x_{1}}\chi+2\nabla\left(\mathcal{U}\left(1+\eta-e^{\eta}\right)\right)\cdot\nabla\chi+\mathcal{U}\left(1+\eta-e^{\eta}\right)\Delta\chi$ $\displaystyle-2\mathcal{U}\left\|\mathcal{U}\right\|^{2}\eta\eta_{1}\chi-\left(\mathcal{A}+\mathcal{U}\eta\chi\right)\left[\left\|\mathcal{U}\right\|^{2}\left(e^{2\eta_{1}}-1-2\eta_{1}\right)+\left\|\gamma\right\|^{2}+2\operatorname{Re}\left(\mathcal{U}e^{\eta}\bar{\gamma}\right)\right].$ Let us write this equation as (4.5) $\mathbb{L}\left(\eta\right)=-{\mathcal{U}}^{-1}E_{2}\left(\mathcal{U}\right)+N\left(\eta\right),$ where $\begin{split}N(\eta)&=-\left\|\mathcal{U}\right\|^{2}\left(e^{2\eta_{1}}-1-2\eta_{1}\right)+\left\|\nabla\eta\right\|^{2}\\\ &+i\varepsilon\|\ln\varepsilon\|\mathcal{A}^{-1}\left(\mathcal{U}\left(1+\eta-e^{\eta}\right)\right)\partial_{x_{2}}\chi+\frac{1}{\mathcal{A}x_{1}}\left(\mathcal{U}\left(1+\eta-e^{\eta}\right)\right)\partial_{x_{1}}\chi\\\ &+2\mathcal{A}^{-1}\nabla\left(\mathcal{U}\left(1+\eta-e^{\eta}\right)\right)\cdot\nabla\chi\\\ &+\mathcal{A}^{-1}\mathcal{U}\left(1+\eta-e^{\eta}\right)\Delta\chi-\mathcal{A}^{-1}\mathcal{U}\chi\left\|\nabla\eta\right\|^{2}-\left\|\gamma\right\|^{2}-2\operatorname{Re}\left(\mathcal{U}e^{\eta}\bar{\gamma}\right)\\\ &+\mathcal{A}^{-1}\mathcal{U}\eta\chi\left[{\mathcal{U}}^{-1}E_{2}\left(\mathcal{U}\right)-2\left\|\mathcal{U}\right\|^{2}\eta_{1}-\left\|\mathcal{U}\right\|^{2}\left(e^{2\eta_{1}}-1-2\eta_{1}\right)-\left\|\gamma\right\|^{2}-2\operatorname{Re}\left(\mathcal{U}e^{\eta}\bar{\gamma}\right)\right].\end{split}$ This nonlinear equation, equivalent to the original GP equation, is the one we eventually want to solve. Observe that in $N\left(\eta\right)$, except $\left\|\mathcal{U}\right\|^{2}\left(e^{2\eta_{1}}-1-2\eta_{1}\right)-\left\|\nabla\eta\right\|^{2},$ other terms are all localized near the vortex points. ### 4.1. A Linear problem By the definition of our vortex configuration, one can see that the terms contain $\varepsilon\|\ln\varepsilon\|$ and $\frac{1}{x_{1}}$ can be viewed as small perturbation near the vortex points. Let us first consider the following linear problem: (4.6) $\mathbb{L}(\eta)=h,\,Re\int_{\mathbb{R}^{2}}\overline{\mathcal{U}\eta}Z_{\ell j}=0,\,\eta\,\mbox{ satisfies }\,(\ref{bdyofpsi}),$ where $Z_{\ell j}=\alpha_{\ell}\nabla u_{\ell}\,\tilde{\rho}_{\ell}(x),\alpha_{\ell}=\frac{\mathcal{U}}{u_{\ell}},$ and $\tilde{\rho}_{\ell}$ is a cutoff function centered at $p_{\ell}$ with support in $B_{\frac{\rho}{5}}(p_{\ell})$. We shall establish a priori estimates for this problem. The following weighted norms and linear theory has been studied in [25]. Recall that $r_{j},j=1,\cdot\cdot\cdot,\mathcal{K},$ represent the distance to the $j$-th vortex point. Let $w$ be a weight function defined by $w(z):=\left(\sum_{j=1}^{2\mathcal{K}}\left(1+r_{j}\right)^{-1}\right)^{-1}.$ This function measures the minimal distance from the point $z$ to those vortex points. We use $B_{a}\left(z\right)$ to denote the ball of radius $a$ centered at $z.$ Let $\alpha,\sigma\in\left(0,1\right)$ be small positive numbers. For complex valued function $\eta=\eta_{1}+\eta_{2}i,$ we define the following weighted norm: $\displaystyle\left\\|\eta\right\\|_{\ast}$ $\displaystyle=\left\\|u\eta\right\\|_{W^{2,9}\left(w<3\right)}+\left\\|w^{1+\sigma}\eta_{1}\right\\|_{L^{\infty}\left(w>2\right)}+\left\\|w^{2+\sigma}(\|\nabla\eta_{1}\|+\|\nabla^{2}\eta_{1}\|)\right\\|_{L^{\infty}\left(w>2\right)}$ $\displaystyle+\sup_{z\in\left\\{w>2\right\\}}\sup_{z_{1},z_{2}\in B_{w/3}\left(z\right)}\left(\frac{\left\|\nabla\eta_{1}\left(z_{1}\right)-\nabla\eta_{1}\left(z_{2}\right)\right\|+\left\|\nabla^{2}\eta_{1}\left(z_{1}\right)-\nabla^{2}\eta_{1}\left(z_{2}\right)\right\|}{w\left(z\right)^{-2-\sigma-\alpha}\left\|z_{1}-z_{2}\right\|^{\alpha}}\right)$ $\displaystyle+\left\\|w^{\sigma}\eta_{2}\right\\|_{L^{\infty}\left(w>2\right)}+\left\\|w^{1+\sigma}\nabla\eta_{2}\right\\|_{L^{\infty}\left(w>2\right)}+\left\\|w^{2+\sigma}\nabla^{2}\eta_{2}\right\\|_{L^{\infty}\left(w>2\right)}$ $\displaystyle+\sup_{z\in\left\\{w>2\right\\}}\sup_{z_{1},z_{2}\in B_{w/3}\left(z\right)}\left(w\left(z\right)^{1+\sigma+\alpha}\frac{\left\|\nabla\eta_{2}\left(z_{1}\right)-\nabla\eta_{2}\left(z_{2}\right)\right\|}{\left\|z_{1}-z_{2}\right\|^{\alpha}}\right)$ $\displaystyle+\sup_{z\in\left\\{w>2\right\\}}\sup_{z_{1},z_{2}\in B_{w/3}\left(z\right)}\left(w\left(z\right)^{2+\sigma+\alpha}\frac{\left\|\nabla^{2}\eta_{2}\left(z_{1}\right)-\nabla^{2}\eta_{2}\left(z_{2}\right)\right\|}{\left\|z_{1}-z_{2}\right\|^{\alpha}}\right).$ Basically, the norm means that the real part of $\eta$ decays like $w^{-1-\sigma}$ and its first and second derivatives decay like $w^{-2-\sigma}$. Moreover, the imaginary part of $\eta$ only decays as $w^{-\sigma}$, but its first and second derivative decay as $w^{-1-\sigma}$ and $w^{-2-\sigma}$ respectively. It is worth mentioning that the Hölder norms are taken into account in the definition because eventually we shall use the Schauder estimates. Moreover, near the vortex points, we use the $L^{p}$ norm, because the $L^{\infty}$ norm is not bounded there. On the other hand, for complex valued function $h=h_{1}+ih_{2},$ we define the following weighted Hölder norm $\displaystyle\left\\|h\right\\|_{\ast\ast}$ $\displaystyle:=\left\\|uh\right\\|_{L^{9}\left(w<3\right)}+\left\\|w^{1+\sigma}h_{1}\right\\|_{L^{\infty}\left(w>2\right)}$ $\displaystyle+\left\\|w^{2+\sigma}\nabla h_{1}\right\\|_{L^{\infty}\left(w>2\right)}+\left\\|w^{2+\sigma}h_{2}\right\\|_{L^{\infty}\left(w>2\right)}$ $\displaystyle+\sup_{z\in\left\\{w>2\right\\}}\sup_{z_{1},z_{2}\in B_{w/3}\left(z\right)}\left(w\left(z\right)^{2+\sigma+\alpha}\frac{\left\|\nabla h_{1}\left(z_{1}\right)-\nabla h_{1}\left(z_{2}\right)\right\|}{\left\|z_{1}-z_{2}\right\|^{\alpha}}\right)$ $\displaystyle+\sup_{z\in\left\\{w>2\right\\}}\sup_{z_{1},z_{2}\in B_{w/3}\left(z\right)}\left(w\left(z\right)^{2+\sigma+\alpha}\frac{\left\|h_{2}\left(z_{1}\right)-h_{2}\left(z_{2}\right)\right\|}{\left\|z_{1}-z_{2}\right\|^{\alpha}}\right).$ This definition tells us that the real and imaginary parts of $h$ have different decay rates. Moreover, intuitively we require $h_{1}$ to gain one more power of decay at infinity after taking one derivative. The choice of this norm is partly decided by the decay and smooth properties of $E_{2}(\mathcal{U})$. We have the following a priori estimate for solutions of the equation (4.6). ###### Lemma 4.1 (Proposition 4.5 in [25]). Let $\varepsilon>0$ be small. Suppose $\eta$ is a solution of (4.6) with $\left\\|h\right\\|_{\ast\ast}<\infty$. Then $\left\\|\eta\right\\|_{\ast}\leq C\varepsilon^{-\sigma}\left\|\ln\varepsilon\right\|\left\\|h\right\\|_{\ast\ast}$ where $C$ is a constant independent of $\varepsilon$ and $h$. We now consider the following linear projected problem: (4.7) $\left\\{\begin{array}[]{l}\mathbb{L}(\eta)=h+\sum_{j=1}^{2\mathcal{K}}\sum_{j=1}^{2}c_{\ell j}Z_{\ell j},\\\ Re\int_{\mathbb{R}^{2}}\overline{{\mathcal{U}}\eta}Z_{\ell j}\,dx=0,\\\ \eta\mbox{ satisfies }(\ref{bdyofpsi}).\end{array}\right.$ We state the following existence result: ###### Proposition 4.2. There exists constant $C$, depending only on $\alpha,\,\sigma$ such that for all $\varepsilon$ small, the following holds: if $\\|h\\|_{}<\infty$, there exists a unique solution $(\eta,\\{c_{\ell j}\\})=T_{\varepsilon}(h)$ to (4.7). Furthermore, there holds $\\|\eta\\|_{}\leq C\varepsilon^{-\sigma}\|\ln\varepsilon\|\\|h\\|_{*}.$ ###### Proof. The proof is similar to that of Proposition 4.1 in [16]. Instead of solving (4.7) in $\mathbb{R}^{2}$, we solve it in a bounded domain first: $\left\\{\begin{array}[]{l}\mathbb{L}(\eta)=h+\sum_{\ell=1}^{2\mathcal{K}}\sum_{j=1}^{2}c_{\ell j}Z_{\ell j},\,Re\int_{B_{M}}\overline{\mathcal{U}\eta}Z_{\ell j}\,dx=0\mbox{ in }B_{M}(0)\\\ \eta=0\mbox{ on }\partial B_{M}(0),\\\ \eta\mbox{ satisfies \, the \, condition \, (\ref{bdyofpsi})},\end{array}\right.$ where $M$ large enough. By the standard proof of a priori estimates, we also obtain the following estimates for any solution $\eta_{M}$ of the above problem with $\\|\eta_{M}\\|_{}\leq C\varepsilon^{-\sigma}\|\ln\varepsilon\|\\|h\\|_{}.$ By working in the Sobolev space $H_{0}^{1}(B_{M})$, the existence will follow by Fredholm alternatives. Now letting $M\to\infty$, we obtain a solution of the required properties. ∎ ### 4.2. Projected nonlinear problem From now on, we will denote by $T_{\varepsilon}(h)$ the solution of (4.7). We consider the following nonlinear projected problem : (4.8) $\left\\{\begin{array}[]{l}\mathbb{L}(\eta)+\frac{E_{2}(\mathcal{U})}{\mathcal{U}}+N(\eta)=\sum_{\ell=1}^{2\mathcal{K}}\sum_{j=1}^{2}c_{\ell j}Z_{\ell j},\\\ Re\int_{\mathbb{R}^{2}}\overline{\mathcal{U}\eta}Z_{\ell j}\,dx=0,\\\ \eta\mbox{ satisfies \, the \, condition }\,(\ref{bdyofpsi}).\end{array}\right.$ Using the operator $T_{\varepsilon}$ defined in Proposition 4.2, we can write the above problem as $\eta=T_{\varepsilon}(\frac{E_{2}(\mathcal{U})}{\mathcal{U}}-N(\eta)):=G_{\varepsilon}(\eta).$ Using the error estimates in Lemma 3.3, we have for $r_{j}\sim\varepsilon^{-1}$, $Re(\frac{E_{2}}{\mathcal{U}})\sim\frac{\varepsilon^{1-\delta}}{r_{j}},\,Im(\frac{E_{2}}{\mathcal{U}})\sim\frac{\varepsilon^{1-\delta}}{r_{j}^{3}}.$ More precisely, if one check the express of the error, and using the explicate expression for $A_{a}$ in Section 2, one can check by direct calculation that for $r_{j}>>\varepsilon^{-1}$, $Re(\frac{E_{2}}{\mathcal{U}})\sim\frac{\varepsilon^{1-\delta}}{r_{j}^{2}},\,Im(\frac{E_{2}}{\mathcal{U}})\sim\frac{\varepsilon^{1-\delta}}{r_{j}^{3}}.$ Taking this into account, one has $\\|{\mathcal{U}}^{-1}E_{2}(\mathcal{U})\\|_{}\leq C\varepsilon^{1-\delta}$ for any $\delta>0$. Let $\eta\in B:=\\{\\|\eta\\|_{}\leq C\varepsilon^{1-\beta}\\}$ for $\beta\in(\delta+\sigma,1)$. Then using the explicit form of $N(\eta)$, we have $\\|G_{\varepsilon}(\eta)\\|_{}\leq C(\\|N(\eta)\\|_{}+\\|{\mathcal{U}}^{-1}E_{2}(\mathcal{U})\\|_{})\leq C\varepsilon^{1-\beta}$ and $\\|G_{\varepsilon}(\eta)-G_{\varepsilon}(\tilde{\eta})\\|_{}\leq o(1)\\|\eta-\tilde{\eta}\\|_{}$ for all $\eta,\,\tilde{\eta}\in B$. By contraction mapping theorem, we obtain the following: ###### Proposition 4.3. There exists constant $C$ and $\beta$ small, depending only on $\alpha,\,\sigma$ such that for all $\varepsilon$ small, the following holds: there exists a unique solution $(\eta_{\varepsilon,\\{p_{i}\\}},\\{c_{ij}\\})=T_{\varepsilon}(h)$ to (4.8). Furthermore, there holds $\\|\eta\\|_{}\leq C\varepsilon^{1-\beta},$ and $\eta_{\varepsilon,\\{p_{i}\\}}$ is continuous in $\\{p_{i}\\}$. ## 5\. The reduced problem and the multiple vortex rings solutions ### 5.1. The reduced problem To find a real solution to problem (4.5), we solve the reduced problem by finding the positions of the vortex points $\\{p_{i}\\}$ such that the coefficients $c_{\ell j}$ in (4.8) are zero for small $\varepsilon$. In the previous section, we have deduced the existence of $\eta$ to the projected nonlinear problem: $\mathbb{L}\eta+\frac{E_{2}}{\mathcal{U}}+N(\eta)=\sum_{j}c_{j}\frac{\nabla u_{j}}{u_{j}}\tilde{\rho}_{j}(x).$ So $c_{j}=0$ is equivalent to (5.1) $Re\int_{\mathbb{R}^{2}}u_{j}[\mathcal{L}\eta+\frac{E_{2}}{\mathcal{U}}+N(\eta)]\nabla\bar{u}_{j}dx=0.$ By the relation of $u_{j}\mathbb{L}$ and $L_{0}(\phi_{j})$ where $\phi_{j}=u_{j}\eta$, $u_{j}\mathbb{L}(\eta)=L_{0}(\phi_{j})+o(\frac{1}{\rho^{2}})\phi_{j},$ where $L_{0}(\phi)=\Delta\phi+(1-S^{2})\phi-2Re(\bar{u}_{0}\phi)u_{0}.$ One has, by integration by parts, $\begin{split}Re\int_{\mathbb{R}^{2}}u_{j}\mathbb{L}\eta\,\nabla\bar{u}_{j}dx&=Re\int_{\mathbb{R}^{2}}(L_{0}+O(\frac{1}{\rho^{2}})\phi_{j}\,\nabla\bar{u}_{j}dx\\\ &=Re\int_{\mathbb{R}^{2}}\phi_{j}L_{0}(\nabla\bar{u}_{j})dx+o(\varepsilon)=o(\varepsilon),\end{split}$ and using the expression of $N(\psi)$, $Re\int_{\mathbb{R}^{2}}u_{j}N(\eta)\,\nabla\bar{u}_{j}dx=o(\varepsilon).$ We now compute $Re\int_{\mathbb{R}^{2}}u_{j}\frac{E_{2}}{\mathcal{U}}\nabla\bar{u}_{j}dx.$ Recall that one can write $\mathcal{U}$ as $u_{j}\alpha_{j}$, where $\alpha_{j}=\pi_{\ell\neq j}u_{\ell}e^{i\tilde{\varphi}_{\bf p}}$, near each vortex point $p_{j}$. By (3.11) in Section 3, we have $\begin{split}&Re\int_{\mathbb{R}^{2}}\frac{E_{21}}{\alpha_{j}}{\nabla\bar{u}_{j}}dx\\\ &=Re(i\varepsilon\|\ln\varepsilon\|)\int_{\mathbb{R}^{2}}\big{[}\frac{S^{\prime}(r_{j})}{S(r_{j})}\frac{r_{j,2}}{r_{j}}+i\partial_{x_{2}}\varphi_{d}\big{]}\Big{(}\frac{S^{\prime}(r_{j})}{S(r_{j})}\nabla r_{j}-i{\tau_{j}}\nabla\theta_{j}\Big{)}S^{2}(r_{j})dx\\\ &+o(\varepsilon)\\\ &=-\varepsilon\|\ln\varepsilon\|{\tau_{j}}\int_{\mathbb{R}^{2}}SS^{\prime}(r)\Big{(}\partial_{x_{2}}\varphi_{d}-{\tau_{j}}\frac{x_{2}}{r}\nabla\theta\Big{)}dx+o(\varepsilon)\\\ &=-\varepsilon\|\ln\varepsilon\|{\tau_{j}}\int_{\mathbb{R}^{2}}SS^{\prime}(r)\Big{(}\frac{x_{1}}{r^{2}}\nabla r-\frac{x_{2}}{r}\nabla\theta\Big{)}dx+o(\varepsilon)\\\ &=(-\pi\varepsilon\|\ln\varepsilon\|{\tau_{j}},\,0)+o(\varepsilon),\end{split}$ where we have used the estimate (3.6). On the other hand, by (3.12), $\begin{split}&Re\int_{\mathbb{R}^{2}}\frac{E_{22}}{\alpha_{j}}{\nabla\bar{u}}_{j}dx\\\ &=\int_{\mathbb{R}^{2}}\Big{(}\sum_{\ell}\frac{1}{r_{\ell}^{2}}-\|\nabla\varphi_{d}\|^{2}+\frac{1}{x_{1}}\sum_{\ell}\frac{S^{\prime}(r_{\ell})}{S(r_{\ell})}\partial_{x_{1}}r_{\ell}\Big{)}S(r_{j})S^{\prime}(r_{j})\nabla r_{j}dx\\\ &+2{\tau_{j}}\int\sum_{\ell}\frac{S^{\prime}(r_{\ell})}{S(r_{\ell})}\nabla r_{\ell}\cdot\nabla\varphi_{d}\nabla\theta_{j}S^{2}(r_{j})dx+o(\varepsilon)\\\ &=-\int_{\mathbb{R}^{2}}\|\nabla\varphi_{d}\|^{2}\nabla r_{j}S(r_{j})S^{\prime}(r_{j})dx\\\ &+\frac{1}{p_{j,1}}\int_{\mathbb{R}^{2}}(S^{\prime}(r))^{2}\partial_{x_{1}}r\nabla rdx\\\ &+2{\tau_{j}}\int_{\mathbb{R}^{2}}\nabla r_{j}\cdot\nabla\varphi_{d}\nabla\theta_{j}S(r_{j})S^{\prime}(r_{j})\,dx+o(\varepsilon)\\\ &=I_{1}+o(\varepsilon).\end{split}$ Recall the relation of $\varphi_{d}$ and $\psi$ in (3.8), one has $\nabla\varphi_{d}=\Big{(}\sum_{j=1}^{\mathcal{K}}{\tau_{j}}\partial_{2}A_{p_{j}},\,\,-\sum_{j=1}^{\mathcal{K}}{\tau_{j}}(\frac{A_{p_{j}}}{x_{1}}+\partial_{1}A_{p_{j}})\Big{)}.$ It has been shown in [20] that $A_{a}(x_{1},x_{2})=\sqrt{\frac{a_{1}}{x_{1}}}\frac{1}{\kappa}\Big{[}(2-\kappa^{2})K(\kappa^{2})-2E(\kappa^{2})\Big{]},$ where $\kappa^{2}(x)=\frac{4a_{1}x_{1}}{x_{1}^{2}+a_{1}^{2}+(x_{2}-a_{2})^{2}+2a_{1}x_{1}}$ and $K,\,E$ are the complete elliptic integrals of first and second kind, i.e., $\begin{split}K(s)&=\int_{0}^{\frac{\pi}{2}}(1-s\sin^{2}\theta)^{-\frac{1}{2}}d\theta,\\\ E(s)&=\int_{0}^{\frac{\pi}{2}}(1-s\sin^{2}\theta)^{\frac{1}{2}}d\theta.\end{split}$ They satisfy $\begin{split}K^{\prime}(s)=K(1-s),\,\,E^{\prime}(s)=E(1-s)\mbox{ for }1<s<1.\end{split}$ Note that $A_{\lambda a}(\lambda x)=A_{a}(x)$, and for $s\to 1$, $\begin{split}K(s)&=-\frac{1}{2}\ln(1-s)(1+\frac{1-s}{4})+\ln 4+O(1-s),\\\ E(s)&=1-\ln(1-s)\frac{1-s}{4}+O(1-s).\end{split}$ Moreover, as we mentioned before, when $r=\|z-a\|=o(\|a_{1}\|)$, $A_{a}(z)=\Big{(}\ln\frac{a_{1}}{r}+3\log(2)-2\Big{)}+O\Big{(}\frac{r}{a_{1}}\|\ln\frac{r}{a_{1}}\|\Big{)}$ and $\partial_{r}A_{a}=-\frac{1}{r}+O(\frac{1}{a_{1}}).$ Combining all these, one has $\begin{split}&-\int_{\mathbb{R}^{2}}\|\nabla\varphi_{d}\|^{2}\nabla r_{j}S(r_{j})S^{\prime}(r_{j})dx\\\ &+2{\tau_{j}}\int_{\mathbb{R}^{2}}\nabla r_{j}\cdot\nabla\varphi_{d}\nabla\theta_{j}S(r_{j})S^{\prime}(r_{j})\,dx\\\ &=-2{\tau_{j}}\int_{\mathbb{R}^{2}}S(r_{j})S^{\prime}(r_{j})\big{(}\nabla\theta_{j}\cdot\nabla\varphi_{d}\,\nabla r_{j}-\nabla r_{j}\cdot\nabla\varphi_{d}\,\nabla\theta_{j}\big{)}+o(\varepsilon)\\\ &=2{\tau_{j}}\Big{(}\frac{{\tau_{j}}}{p_{j,1}}\int\frac{S(r_{j})S^{\prime}(r_{j})}{r_{j}}A_{p_{j}}(x)dx+\frac{\pi}{p_{j,1}}\sum_{\ell\neq j}{\tau_{\ell}}A_{p_{\ell}}(p_{j}),\,\,0\Big{)}\\\ &+2{\tau_{j}}\pi\sum_{\ell\neq j}{\tau_{\ell}}\nabla A_{p_{\ell}}(p_{j})+o(\varepsilon)\\\ &=\frac{2\pi}{p_{j,1}}\Big{(}\ln p_{j,1}+c_{0}+\sum_{\ell\neq j}{\tau_{j}\tau_{\ell}}A_{p_{\ell}}(p_{j}),\,0\Big{)}+2\pi\sum_{\ell\neq j}{\tau_{j}\tau_{\ell}}\nabla A_{p_{\ell}}(p_{j})+o(\varepsilon)\end{split}$ where (5.2) $c_{0}=3\ln 2-2-\frac{1}{\pi}\int\frac{S(r)S^{\prime}(r)\ln r}{r}dx.$ So one has $I_{1}=\frac{2\pi}{p_{j,1}}\Big{(}\ln p_{j,1}+c_{1}+\sum_{\ell\neq j}{\tau_{j}\tau_{\ell}}A_{p_{\ell}}(p_{j}),\,0\Big{)}+2\pi\sum_{\ell\neq j}\tau_{j}\tau_{\ell}\nabla A_{p_{\ell}}(p_{j})+o(\varepsilon)$ where $c_{1}=c_{0}+\frac{1}{2}\int_{0}^{\infty}S^{\prime}(r)rdr.$ By the above estimates, $\begin{split}&Re\int_{\mathbb{R}^{2}}\frac{E_{2}}{\alpha_{j}}\partial_{x_{1}}\bar{u}_{j}dx\\\ &=-\pi\Big{[}{\tau_{j}}\varepsilon\|\ln\varepsilon\|-\frac{2\ln p_{j,1}}{p_{j,1}}-\frac{2c_{1}}{p_{j,1}}-2\sum_{\ell\neq j}\tau_{j}\tau_{\ell}\frac{A_{p_{\ell}}(p_{j})}{p_{j,1}}-2\sum_{\ell\neq j}\tau_{j}\tau_{\ell}\partial_{1}A_{p_{\ell}}(p_{j})\Big{]}\\\ &+o(\varepsilon)\end{split}$ and $Re\int_{\mathbb{R}^{2}}\frac{E_{2}}{\alpha_{j}}\partial_{x_{2}}\bar{u}_{j}dx=2\pi\sum_{\ell\neq j}\tau_{j}\tau_{\ell}\partial_{2}A_{p_{\ell}}(p_{j})+o(\varepsilon).$ We now have the following reduced problem: ###### Lemma 5.1. The reduced problem (5.1) is equivalent to the following system of the vortex points $\\{p_{j}\\}$: (5.3) $\begin{split}&{\tau_{j}}\varepsilon\|\ln\varepsilon\|-\frac{2\ln p_{j,1}}{p_{j,1}}-\frac{2c_{1}}{p_{j,1}}-2\sum_{\ell\neq j}\tau_{j}\tau_{\ell}\frac{A_{p_{\ell}}(p_{j})}{p_{j,1}}-2\sum_{\ell\neq j}\tau_{j}\tau_{\ell}\partial_{1}A_{p_{\ell}}(p_{j})=o(\varepsilon),\end{split}$ and (5.4) $2\sum_{\ell\neq j}\tau_{j}\tau_{\ell}\partial_{2}A_{p_{\ell}}(p_{j})=o(\varepsilon).$ Using the scaling invariance $A_{\lambda a}(\lambda x)=A_{a}(x),$ if we denote by $p_{j}=\frac{\tilde{p}_{j}}{\varepsilon},$ where $\|\tilde{p}_{j,1}\|=O(1)$, we can get the reduced problem for $\tilde{p}_{j}$: (5.5) $\begin{split}&{\tau_{j}}\|\ln\varepsilon\|+\frac{2\ln\varepsilon}{\tilde{p}_{j,1}}-\frac{2\ln\tilde{p}_{j,1}}{\tilde{p}_{j,1}}-\frac{2c_{1}}{\tilde{p}_{j,1}}-2\sum_{\ell\neq j}\tau_{j}\tau_{\ell}\frac{A_{\tilde{p}_{\ell}}(\tilde{p}_{j})}{\tilde{p}_{j,1}}\\\ &-2\sum_{\ell\neq j}\tau_{j}\tau_{\ell}\partial_{1}A_{\tilde{p}_{\ell}}(\tilde{p}_{j})=o(1),\end{split}$ and (5.6) $2\sum_{\ell\neq j}\tau_{j}\tau_{\ell}\partial_{2}A_{\tilde{p}_{\ell}}(\tilde{p}_{j})=o(1).$ Using the asymptotic behavior of $A_{a}(x)$ and $A^{\prime}_{a}(r)$, and recall that $\|\tilde{p}_{\ell}-\tilde{p}_{j}\|\sim\frac{1}{\ln\varepsilon},\,\|\tilde{p}_{j,1}\|\sim O(1),$ we obtain the following equivalent reduced problem: (5.7) $\left\\{\begin{array}[]{l}{\tau_{j}}\|\ln\varepsilon\|+\frac{2\ln\varepsilon}{\tilde{p}_{j,1}}+2\sum_{\ell\neq j}\tau_{j}\tau_{\ell}\frac{\tilde{p}_{j,1}-\tilde{p}_{\ell,1}}{\|\tilde{p}_{j}-\tilde{p}_{\ell}\|^{2}}=o(\ln\varepsilon),\\\ 2\sum_{\ell\neq j}\tau_{j}\tau_{\ell}\frac{\tilde{p}_{j,2}-\tilde{p}_{\ell,2}}{\|\tilde{p}_{j}-\tilde{p}_{\ell}\|^{2}}=o(1).\end{array}\right.$ ### 5.2. Vortex locations and their generating polynomials In this section, we construct a family of polynomials whose roots will correspond to the locations of the vortex rings. For each rescaled vortex point $\tilde{p}_{j},j=1,...,\mathcal{K},$ we have associated a degree $\tau_{j}=\pm 1.$ To analyze the reduced problem in a more precise way, let us relabel those points with $\tau_{j}=1$ by $\tilde{p}_{1}^{+},...,\tilde{p}_{m}^{+}$ and those with $\tau=-1$ will be denoted by $\tilde{p}_{1}^{-},...,\tilde{p}_{n}^{-}.$ We then write $\displaystyle\tilde{p}_{j}^{+}$ $\displaystyle=\alpha_{0}+\alpha+\frac{1}{\left\|\ln\varepsilon\right\|}\mathbf{a}_{j},\text{ for }j=1,...,m,$ $\displaystyle\tilde{p}_{j}^{-}$ $\displaystyle=\alpha_{0}+\alpha+\frac{1}{\left\|\ln\varepsilon\right\|}\mathbf{b}_{j},\text{ for }j=1,...,n.$ Here $\alpha_{0}$ is a fixed constant only depends on $m,n,$ and $\alpha=o\left(1\right)$ depends on $\varepsilon.$ Inserting these into the reduced problem (5.7), we find that, at the main order, $\left(\mathbf{a}_{1},...,\mathbf{a}_{m},\mathbf{b}_{1},...,\mathbf{b}_{n}\right)$ should satisfy the following system: $\left\\{\begin{array}[c]{l}{\displaystyle\sum\limits_{j=1,j\neq k}^{m}}\frac{1}{\mathbf{a}_{k}-\mathbf{a}_{j}}-{\displaystyle\sum\limits_{j=1}^{n}}\frac{1}{\mathbf{a}_{k}-\mathbf{b}_{j}}=\frac{1}{2}-{\alpha_{0}}^{-1},\text{ for }k=1,...,m,\\\ -{\displaystyle\sum\limits_{j=1,j\neq k}^{n}}\frac{1}{\mathbf{b}_{k}-\mathbf{b}_{j}}+{\displaystyle\sum\limits_{j=1}^{m}}\frac{1}{\mathbf{b}_{k}-\mathbf{a}_{j}}=\frac{1}{2}+{\alpha_{0}}^{-1},\text{ for }k=1,...,n.\end{array}\right.$ This can be regarded as a balancing condition between the multiple vortex rings. Adding together the $m+n$ equations in the balancing condition, we find that a necessary condition for the existence of a balancing configuration is $\left({\alpha_{0}}^{-1}-\frac{1}{2}\right)m+\left({\alpha_{0}}^{-1}+\frac{1}{2}\right)n=0.$ It follows that $\alpha_{0}=2\frac{m+n}{m-n}.$ Therefore, we are lead to consider the system (5.8) $\left\\{\begin{array}[c]{l}{\displaystyle\sum\limits_{j=1,j\neq k}^{m}}\frac{1}{\mathbf{a}_{k}-\mathbf{a}_{j}}-{\displaystyle\sum\limits_{j=1}^{n}}\frac{1}{\mathbf{a}_{k}-\mathbf{b}_{j}}=-n,\text{ for }k=1,...,m,\\\ -{\displaystyle\sum\limits_{j=1,j\neq k}^{n}}\frac{1}{\mathbf{b}_{k}-\mathbf{b}_{j}}+{\displaystyle\sum\limits_{j=1}^{m}}\frac{1}{\mathbf{b}_{k}-\mathbf{a}_{j}}=-m,\text{ for }k=1,...,n.\end{array}\right.$ To find solutions to this system, we define the generating polynomial as $P\left(x\right):={\displaystyle\prod\limits_{j=1}^{m}}\left(x-\mathbf{a}_{j}\right),\text{ \ }Q\left(x\right):={\displaystyle\prod\limits_{j=1}^{n}}\left(x-\mathbf{b}_{j}\right).$ If $\mathbf{a}_{j},\mathbf{b}_{j}$ satisfy $\left(\ref{Balance}\right),$ then (5.9) $P^{\prime\prime}Q-2P^{\prime}Q^{\prime}+PQ^{\prime\prime}+nP^{\prime}Q-mPQ^{\prime}=0.$ The case of $m=n$ has been studied in [25] . In this case, the system $\left(\ref{Balance}\right)$ is equivalent to $\left\\{\begin{array}[c]{l}{\displaystyle\sum\limits_{j=1,j\neq k}^{m}}\frac{1}{\mathbf{a}_{k}-\mathbf{a}_{j}}-{\displaystyle\sum\limits_{j=1}^{n}}\frac{1}{\mathbf{a}_{k}-\mathbf{b}_{j}}=-1,\text{ for }k=1,...,m,\\\ {\displaystyle\sum\limits_{j=1,j\neq k}^{n}}\frac{1}{\mathbf{b}_{k}-\mathbf{b}_{j}}-{\displaystyle\sum\limits_{j=1}^{m}}\frac{1}{\mathbf{b}_{k}-\mathbf{a}_{j}}=1,\text{ for }k=1,...,n.\end{array}\right.$ The polynomial solutions of this system are connected with theory of integrable system. Indeed, letting $\phi=\frac{Q}{P}\exp\left(x\right)$ and $u=2\left(\ln P\right)^{\prime\prime}.$ The equation $\left(\ref{PQ}\right)$ can be rewritten as $\phi^{\prime\prime}+u\phi=\phi.$ This equation appears as the first equation in the Lax pair of the KdV equation and has the Darboux invariance property. The polynomial solutions of $\left(\ref{PQ}\right)$ in this case are given by the Adler-Moser polynomials. From the view point of numerical computation, the equation $\left(\ref{PQ}\right)$ is indeed easier than $\left(\ref{Balance}\right).$ Note that our construction of multiple vortex ring solutions requires that all the points $\mathbf{a}_{j},j=1,...,m$ and $\mathbf{b}_{j},j=1,...,n$ are distinct from each other. Therefore we require that the polynomials $P$, $Q$ satisfy the following condition: (H1) $P,Q$ have no repeated roots. Our construction also requires the following condition: (H2) The set of points $\left\\{\mathbf{a}_{1},\cdots,\mathbf{a}_{m},\text{ }\mathbf{b}_{1},\cdots,\mathbf{b}_{n}\right\\}$ are symmetric with respect to the $x_{1}$ axis. Observe that equation $\left(\ref{PQ}\right)$ implies that if $X_{0}$ is a common root of $P$ and $Q,$ then necessarily $X_{0}$ is a repeated root of $P$ or $Q.$ We observe that due to the translation invariance of the equation in the balancing condition, we can normalize the polynomials $P,$ $Q$ as $\displaystyle P\left(x\right)$ $\displaystyle=s_{1}+s_{2}x+...+s_{m-1}x^{m-1}+x^{m},$ $\displaystyle Q\left(x\right)$ $\displaystyle=t_{1}+t_{2}x+...+t_{n-2}x^{n-2}+x^{n}.$ That is, the $x^{n-1}$ term in $Q$ can be chosen to be zero. In this section, we would like to find some solution pair $(P,Q)$ using software such as Maple. Then in the next section, we shall use techniques of integrable system to find a sequence of solution pairs, with explicit Wronskian representation. Let us consider the case of $m+n\leq 12.$ With this constraints, we find, using Maple, that there exist polynomial solutions to $\left(\ref{PQ}\right)$ satisfying (H1) and whose roots satisfy (H2), if further $\left(m,n\right)$ are one of the cases in the set $S:=\\{\left(2,1\right),\left(3,2\right),\left(4,3\right),\left(5,4\right),\left(6,5\right)\\}.$ Indeed, if $\left(m,n\right)=\left(2,1\right),$ then $\left(\ref{PQ}\right)$ has a solution of the form $P\left(x\right)=x^{2}-2x+2,\text{ }Q\left(x\right)=x.$ If $\left(m,n\right)=\left(3,2\right),$ then $\left(\ref{PQ}\right)$ has solution: $P\left(x\right)=x^{3}-2x^{2}+\frac{7}{2}x-\frac{3}{2},\text{ }Q\left(x\right)=x^{2}+1.$ If $\left(m,n\right)=\left(4,3\right),$ then $\left(\ref{PQ}\right)$ has solution: $\displaystyle P\left(x\right)$ $\displaystyle=x^{4}-2x^{3}+\frac{44}{9}x^{2}-\frac{89}{27}x+\frac{533}{324},$ $\displaystyle\text{ }Q\left(x\right)$ $\displaystyle=x^{3}+\frac{13}{6}x+\frac{13}{54}.$ If $\left(m,n\right)=\left(5,4\right),$ then $\left(\ref{PQ}\right)$ has solution: $\displaystyle P\left(x\right)$ $\displaystyle=x^{5}-2x^{4}+\frac{449}{72}x^{3}-\frac{749}{144}x^{2}+\frac{12919}{2592}x-\frac{16015}{15552},$ $\displaystyle\text{ }Q\left(x\right)$ $\displaystyle=x^{4}+\frac{61}{18}x^{2}+\frac{16}{27}x+\frac{1337}{1296}.$ When $\left(m,n\right)=\left(6,5\right),$ we have $\displaystyle P\left(x\right)$ $\displaystyle=x^{6}-2x^{5}+\frac{2269}{300}x^{4}-\frac{193279}{27000}x^{3}+\frac{10810499}{1080000}x^{2}-\frac{57115601}{16200000}x+\frac{3980046413}{2916000000},$ $\displaystyle Q\left(x\right)$ $\displaystyle=x^{5}+\frac{1669}{360}x^{3}+\frac{3607}{3600}x^{2}+\frac{1112099}{324000}x+\frac{23805769}{48600000}.$ The roots of $P,Q$ listed above are solutions of the balancing system. Here we list them in the order $\mathbf{a}_{1},...,\mathbf{a}_{m},\mathbf{b}_{1},...,\mathbf{b}_{n}$ and denote it by $\mathcal{P}_{\left(m,n\right)}.$ The numerical value can be listed as below: $\displaystyle\mathcal{P}_{\left(2,1\right)}$ $\displaystyle:\left(1+i,1-i,0\right),$ $\displaystyle\mathcal{P}_{\left(3,2\right)}$ $\displaystyle:\left(0.56,0.72+1.48i,0.72-1.48i,i,-i\right),$ $\displaystyle\mathcal{P}_{\left(4,3\right)}$ $\displaystyle:(0.393-0.57i,0.393+0.57i,0.607-1.76i,0.607+1.76i,$ $\displaystyle-0.11,0.055-1.48i,0.055+1.48i),$ $\displaystyle\mathcal{P}_{\left(5,4\right)}$ $\displaystyle:(0.255,0.322-0.938i,0.322+0.938i,0.55-1.948i,0.55+1.948i,$ $\displaystyle-0.107-0.567i,-0.107+0.567i,0.107-1.758i,0.107+1.758i),$ $\displaystyle\mathcal{P}_{\left(6,5\right)}$ $\displaystyle:(0.191-0.395i,0.191+0.395i,0.29-1.2i,0.29+1.2i,0.52-2.09i,$ $\displaystyle 0.52+2.09i,-0.145,-0.078-0.94i,-0.078+0.94i,0.15-1.95i,0.15+1.95i).$ Figure 1. $(m,n)=(2,1)$ Figure 2. $(m,n)=(4,3)$ Figure 3. $(m,n)=(6,5)$ Let us denote the pair $(P,Q)$ for $(m,n)=(j,j-1)$ as $(P_{j},Q_{j})$. Then for the above examples, we can see that $P_{j}$ is simply a translation in the $x$ variable of $Q_{j+1}$. We will see in the next section that this is true for all $m=n+1$. Next let us consider the linearized operator around the solution. Let us denote the left hand side of the $j$-th equation of (5.8) by $F_{j}.$ Then we can compute the linearization $dF$ of the map $F:\left(\mathbf{a}_{1},...,\mathbf{a}_{m},\mathbf{b}_{1},...,\mathbf{b}_{n}\right)\rightarrow\left(F_{1},...,F_{m+n}\right).$ $dF$ evaluated at the point $\mathcal{P}_{\left(m,n\right)}$ is a matrix, which can be explicitly computed. The solvability of our original reduced problem is closely related to the nondegeneracy of $dF.$ Since any translation of the $\left(\mathbf{a},\mathbf{b}\right)$ is still a solution to the balancing system, necessarily the determinant of this matrix is zero. That is, $0$ is an eigenvalue of $dF$. Observe that $\left(1,1,....,1\right)$ is an eigenvector. We call $\left(\mathbf{a}_{1},...,\mathbf{a}_{m},\mathbf{b}_{1},...,\mathbf{b}_{n}\right)$ nondegenerated, if the kernel of $dF$ is one dimensional. One can check by explicit computations that for the solutions $\mathcal{P}_{\left(m,n\right)}$ listed above, they are all nondegenerated. It is worth pointing out that if $\left(m,n\right)$ is not in $S,$ there may still have polynomials $P,Q$ satisfying $\left(\ref{PQ}\right),$ but with repeated roots. For instance, when $\left(m,n\right)=\left(4,1\right),$ it has a solution with $P\left(x\right)=x^{4}+4x^{3},\text{ \ }Q\left(x\right)=x.$ When $\left(m,n\right)=\left(5,3\right),$ it has a solution with $P\left(x\right)=x^{5}-\frac{4}{3}x^{4}+\frac{4}{3}x^{3}-\frac{8}{9}x^{2}+\frac{8}{27}x,\text{ }Q\left(x\right)=x^{3}.$ A given pair $\left(m,n\right)$ can be used in the construction of multiple vortex rings, if there exist polynomial solutions to $\left(\ref{PQ}\right)$ satisfying (H1) and (H2). In this respect, there are many questions remain to be answered. For instances, are there infinitely many such pairs? If $\left(m,n\right)$ is such a pair, is it necessarily that $m=n+1?$ Is the balancing configuration unique up to translation? These questions will be partially answered in the next section. Now let us come back to our original reduced problem (5.7) of the GP equation. For each $\left(m,n\right)\in S.$ We have a special solution $\left(\mathbf{a}_{1}^{0},...,\mathbf{a}_{m}^{0},\mathbf{b}_{1}^{0},...,\mathbf{b}_{n}^{0}\right)$ given by $\mathcal{P}_{\left(m,n\right)}.$ If we define vector $\beta$ by $\displaystyle\mathbf{a}_{j}$ $\displaystyle=\mathbf{a}_{j}^{0}+\beta_{j},j=1,...,m,$ $\displaystyle\mathbf{b}_{j}$ $\displaystyle=\mathbf{b}_{j}^{0}+\beta_{j+m},j=1,...,n,$ then the reduced problem (5.7) takes the form (5.10) $dF\left(\beta\right)=G\left(\alpha,\beta\right)+\alpha\alpha_{0}^{-2}\mathbf{e}_{1}\mathbf{,}$ where $G\left(\alpha,\beta\right)=o\left(1\right)$ as $\varepsilon\rightarrow 0,$ with higher order dependence on $\alpha,\beta,$ and $\mathbf{e}_{1}$ is a $m+n$ dimensional column vector whose first $m$ entries are equal to $-1$ and the last $n$ entries are all equal to $1.$ Note that $dF$ is in general not a symmetric matrix. However, since $dF$ is nondegenerated, the kernel of $\left(dF\right)^{T}$ is spanned by $\mathbf{e}_{2}:=\left(1,...,1\right).$ Using the fact that $m-n=1,$ we find that the projection of the right hand side of $\left(\ref{redu}\right)$ onto $\mathbf{e}_{2}$ is equal to $G\cdot\mathbf{e}_{2}-\alpha\alpha_{0}^{-2}.$ Now let us consider the projected problem (5.11) $dF\left(\beta\right)=G\left(\alpha,\beta\right)+\alpha\alpha_{0}^{-2}\mathbf{e}_{1}-\frac{G\cdot\mathbf{e}_{2}-\alpha\alpha_{0}^{-2}}{m+n}\mathbf{e}_{2}.$ Note that for each fixed small $\alpha,$ using the nondegeneracy of the solution $\left(\mathbf{a}_{1}^{0},...,\mathbf{a}_{m}^{0},\mathbf{b}_{1}^{0},...,\mathbf{b}_{n}^{0}\right),$ the projected system $\left(\ref{pro}\right)$ can be solved and a solution $\beta$ depending on $\alpha.$ With this $\beta,$ we then can solve the equation $G\cdot\mathbf{e}_{2}-\alpha\alpha_{0}^{-2}=0$ by a contraction mapping argument. Hence the reduced problem $\left(\ref{redu}\right)$ can be finally solved. Once this is done, with the help of linear theory of Section 4, arguments similar as that of [25] yield a solution to the GP equation, satisfying the conclusion of Theorem 1.2 . ## 6\. Recurrence relations and Wronskian representation of the generating polynomials In this section, we show that the generating polynomials of the balancing system discussed in the previous section have recurrence relations in the case of $m=n+1$, and can be explicitly written down using certain Wronskians. The main result of this section is the following ###### Theorem 6.1. There exists a sequence of polynomials $\mathcal{P}_{n},n=1,...,$ such that (6.1) $\mathcal{P}_{n+1}^{\prime\prime}\mathcal{P}_{n}-2\mathcal{P}_{n+1}^{\prime}\mathcal{P}_{n}^{\prime}+\mathcal{P}_{n+1}^{\prime\prime}\mathcal{P}_{n}+n\mathcal{P}_{n+1}^{\prime}\mathcal{P}_{n}-\left(n+1\right)\mathcal{P}_{n+1}\mathcal{P}_{n}^{\prime}=0,$ where $\mathcal{P}_{n}$ is of degree $n$, $\mathcal{P}_{1}=x$ and $\mathcal{P}_{2}=x^{2}-2x+2.$ Moreover, up to a constant factor(see $\left(\ref{cons}\right)$), these polynomials can be written as $\exp\left(-\frac{n\left(n-1\right)x}{2}\right)W\left(\omega_{1},...,\omega_{n}\right),$ where $W$ represents the Wronskian, $\omega_{j}=\left(x-a_{j}\right)\exp\left(\left(j-1\right)x\right),$ and $a_{1}=0,$ $a_{j+1}=a_{j}+\frac{2}{j}.$ These polynomials can be regarded as a generalization of the Adler-Moser polynomials. There are other types of generalization of the Adler-Moser polynomials, see, for instance [26]. We also refer to [3, 4, 15, 25] and the references cited therein for more discussion in this direction. Recall that in the previous section, we derived the equation (6.2) $P^{\prime\prime}Q-2P^{\prime}Q^{\prime}+PQ^{\prime\prime}+nP^{\prime}Q-\left(n+1\right)PQ^{\prime}=0.$ For $n=1,$ we have found that $P\left(x\right)=x^{2}-2x+2,$ $Q\left(x\right)=x$ is a solution. To solve this equation for general $n,$ we define $\phi=\frac{Q}{P}.$ Direction computation shows that the equation $\left(\ref{e1}\right)$ can be written as (6.3) $\phi^{\prime\prime}+\left(2\left(\ln P\right)^{\prime\prime}-\left(\ln P\right)^{\prime}\right)\phi-\left(n+1\right)\phi^{\prime}=0.$ Note that the equation in this form is different from the one considered by Adler-Moser, in the sense that we have two additional terms corresponding to $\left(\ln P\right)^{\prime}$ and $\phi^{\prime}.$ Moreover, equation (6.2) is not of the standard Hirota bilinear form. This significantly complicates the analysis. For $n=1,$ we already know that equation $\left(\ref{e3}\right)$ has the solution $\bar{\phi}\left(x\right):=\frac{Q}{P}=\frac{x}{x^{2}-2x+2}.$ It is worth pointing out, although not necessarily relevant to our later analysis, $\bar{\phi}$ is smooth in the whole line. Equation $\left(\ref{e3}\right)$ is a second order ODE, it has another solution linearly independent with $\bar{\phi}.$ One can check that $\phi^{\ast}$ defined below is such a solution. Explicitly, $\phi^{\ast}\left(x\right):=\frac{2x^{3}-10x^{2}+21x-16}{x^{2}-2x+2}\exp\left(2x\right).$ Note that $\phi^{\ast}$ can also be written as $\phi^{\ast}=\left(\int_{-\infty}^{x}\frac{\exp\left[\left(n+1\right)s\right]}{\bar{\phi}^{2}}ds\right)\bar{\phi}\left(x\right).$ Next we discuss the generalized Darboux transformation adapted to equation (6.3). The following result can be found in the last section of [28]. ###### Lemma 6.2. Suppose $\phi=\phi_{1}$ and $\phi=\phi_{2}$ are two solutions of the equation $u_{2}\phi^{\prime\prime}+u_{1}\phi^{\prime}+u_{0}\phi=0.$ Then the functions $\tilde{\phi}:=\phi_{2}^{\prime}-\frac{\phi_{1}^{\prime}\phi_{2}}{\phi_{1}}$ satisfies $\tilde{u}_{2}\tilde{\phi}^{\prime\prime}+\tilde{u}_{1}\tilde{\phi}^{\prime}+\tilde{u}_{0}\tilde{\phi}=0,$ where $\tilde{u}_{2}=u_{2},\tilde{u}_{1}=u_{1}+u_{2}^{\prime},\tilde{u}_{0}=u_{0}+u_{1}^{\prime}+2u_{2}\left(\ln\phi_{1}\right)^{\prime\prime}+u_{2}^{\prime}\left(\ln\phi_{1}\right)^{\prime}.$ To apply this lemma, we write equation $\left(\ref{e3}\right)$ as $e^{-x}\phi^{\prime\prime}+e^{-x}\left(2\left(\ln P\right)^{\prime\prime}-\left(\ln P\right)^{\prime}\right)\phi-e^{-x}\left(n+1\right)\phi^{\prime}=0.$ Let us define the new potential $\displaystyle\tilde{u}_{0}$ $\displaystyle:=e^{-x}\left(2\left(\ln P\right)^{\prime\prime}-\left(\ln P\right)^{\prime}\right)+\left(n+1\right)e^{-x}+2e^{-x}\left(\ln\phi^{\ast}\right)^{\prime\prime}-e^{-x}\left(\ln\phi^{\ast}\right)^{\prime}.$ $\displaystyle\tilde{u}_{1}$ $\displaystyle=-e^{-x}\left(n+1\right)-e^{-x},$ and the new function $\Phi_{1}:=\bar{\phi}^{\prime}-\frac{\phi^{\ast\prime}\bar{\phi}}{\phi^{\ast}}=\frac{W\left(\phi^{\ast},\bar{\phi}\right)}{\phi^{\ast}}=\frac{e^{2x}}{\phi^{\ast}}.$ Then using the generalized Darboux transformation described in the previous lemma, we have $e^{-x}\Phi_{1}^{\prime\prime}+\tilde{u}_{0}\Phi_{1}+\tilde{u}_{1}\Phi_{1}^{\prime}=0.$ That is, (6.4) $e^{-x}\Phi_{1}^{\prime\prime}+e^{-x}\left(2\left(\ln P_{3}\right)^{\prime\prime}-\left(\ln P_{3}\right)^{\prime}\right)\Phi_{1}-\left(n+2\right)e^{-x}\Phi_{1}^{\prime}=0,$ where the polynomial $P_{3}$ is defined by $P_{3}:=P\phi^{\ast}e^{-2x}=2x^{3}-10x^{2}+21x-16.$ Equation $\left(\ref{eqp3}\right)$ precisely has the form $\left(\ref{e3}\right).$ An important property is that the equation $\left(\ref{eqp3}\right)$ has another solution $\Phi_{1}^{\ast}:=\frac{36x^{4}-312x^{3}+1136x^{2}-1972x+1357}{2x^{3}-10x^{2}+21x-16}e^{3x}.$ The computations tell us that if $\mathcal{P}_{n}$ is a sequence of polynomials satisfies the conclusion of Theorem 6.1, then we expect the equation $\phi^{\prime\prime}+\left(2\left(\ln\mathcal{P}_{n+1}\right)^{\prime\prime}-\left(\ln\mathcal{P}_{n+1}\right)^{\prime}\right)\phi-\left(n+1\right)\phi^{\prime}=0,$ has two linearly independent solutions, of the form $\phi_{1}=\frac{\mathcal{P}_{n}}{\mathcal{P}_{n+1}},\phi_{2}=\frac{\mathcal{P}_{n+2}}{\mathcal{P}_{n+1}}e^{\left(n+1\right)x}.$ The Wronskian $W\left(\phi_{1},\phi_{2}\right)$ should be equal to $ce^{\left(n+1\right)x}$ for some constant $c.$ Hence we get the following recursive relations between $\mathcal{P}_{n},\mathcal{P}_{n+1},\mathcal{P}_{n+2}:$ $\left(\frac{\mathcal{P}_{n}}{\mathcal{P}_{n+1}}\right)^{\prime}\frac{\mathcal{P}_{n+2}}{\mathcal{P}_{n+1}}e^{\left(n+1\right)x}-\left(\frac{\mathcal{P}_{n}}{\mathcal{P}_{n+1}}\right)\left(\frac{\mathcal{P}_{n+2}}{\mathcal{P}_{n+1}}e^{\left(n+1\right)x}\right)^{\prime}=ce^{\left(n+1\right)x}.$ That is, $\left(\mathcal{P}_{n}^{\prime}\mathcal{P}_{n+1}-\mathcal{P}_{n}\mathcal{P}_{n+1}^{\prime}\right)\mathcal{P}_{n+2}-\mathcal{P}_{n}\left(\mathcal{P}_{n+2}^{\prime}\mathcal{P}_{n+1}-\mathcal{P}_{n+2}\mathcal{P}_{n+1}^{\prime}+\left(n+1\right)\mathcal{P}_{n+2}\mathcal{P}_{n+1}\right)=c\mathcal{P}_{n+1}^{3}.$ This can be written as $\mathcal{P}_{n}^{\prime}\mathcal{P}_{n+2}-\mathcal{P}_{n}P_{n+2}^{\prime}-\left(n+1\right)\mathcal{P}_{n}\mathcal{P}_{n+2}=c\mathcal{P}_{n+1}^{2}.$ If we normalize the polynomials $\mathcal{P}_{n}$ such that the highest order term is $x^{n}.$ Then the constant $c$ satisfies $-\left(n+1\right)=c$ We get the following recurrence relations (6.5) $\mathcal{P}_{n}^{\prime}\mathcal{P}_{n+2}-\mathcal{P}_{n}P_{n+2}^{\prime}-\left(n+1\right)\mathcal{P}_{n}\mathcal{P}_{n+2}+\left(n+1\right)\mathcal{P}_{n+1}^{2}=0.$ When we are given $\mathcal{P}_{n},\mathcal{P}_{n+1},$ the recurrence equation $\left(\ref{re}\right)$ can be integrated, and we expect that the resulted function $\mathcal{P}_{n+2}$ is a polynomial. However, in this step, we will not get a free parameter in this polynomial, because solution of the homogeneous equation has an exponential factor. To show that integrating $\left(\ref{re}\right)$ indeed yields a polynomial, we proceed to find the explicit formula of the sequence $\mathcal{P}_{n}$ which satisfies $\left(\ref{re}\right).$ Let us consider the sequence $a_{j}$ defined through the recurrence $a_{1}=0,$ and $a_{j+1}=a_{j}+\frac{2}{j}.$ Let us define functions $\omega_{j}$ by (6.6) $\omega_{j}=\left(x-a_{j}\right)\exp\left(\left(j-1\right)x\right).$ Then we define functions $\mathcal{P}_{n}$ through the Wronskian (6.7) $\mathcal{P}_{n}:=c_{n}\exp\left(-\frac{n\left(n-1\right)}{2}x\right)W\left(\omega_{1},...,\omega_{n}\right),$ where (6.8) $c_{n}=\left[\left(n-1\right)!{\displaystyle\prod\limits_{1\leq i<j\leq n-1}}\left(j-i\right)\right]^{-1}.$ The normalizing constant $c_{n}$ is used to ensure that the highest order term of $\mathcal{P}_{n}$ is $x^{n}.$ Note that $\mathcal{P}_{n}$ defined by $\left(\ref{Wron}\right)$ are indeed polynomials of degree $n,$ and its leading coefficient is a determinant of Vandermont type. ###### Lemma 6.3. $\mathcal{P}_{n}$ defined by $\left(\ref{Wron}\right)$ satisfies the three- term recurrence relation $\left(\ref{re}\right).$ ###### Proof. For national simplicity, we write $W\left(\omega_{1},...,\omega_{k}\right)$ as $W_{k}.$ Using $\left(\ref{Wron}\right)$ and the fact that $\frac{c_{n+1}^{2}}{c_{n}c_{n+2}}=n+1,$ we see that to prove $\left(\ref{re}\right),$ it suffices to prove (6.9) $W_{n}^{\prime}W_{n+2}-W_{n}W_{n+2}^{\prime}+nW_{n}W_{n+2}+\left(n+1\right)^{2}e^{x}W_{n+1}^{2}=0.$ Following Adler-Moser [1], for any function $\xi,$ we define $W_{k}\left(\xi\right):=W\left(\omega_{1},...,\omega_{k},\xi\right).$ Then we have the Jacobi identity(see [1], Lemma 1) (6.10) $\left(W_{k}\left(\xi\right)\right)^{\prime}W_{k+1}-W_{k}\left(\xi\right)W_{k+1}^{\prime}-W_{k+1}\left(\xi\right)W_{k}=0$ Direct computation tells us that $\omega_{j+1}^{\prime\prime}=j^{2}\omega_{j}\exp\left(x\right).$ Using this relation and its differentiation and the fact that $\omega_{1}=x$, we obtain $W_{k}\left(1\right)=\left(-1\right)^{k}\left(\left(k-1\right)!\right)^{2}\exp\left[\left(k-1\right)x\right]W_{k-1}.$ We then compute $\displaystyle\left(-1\right)^{k}\left(\left(W_{k}\left(1\right)\right)^{\prime}W_{k+1}-W_{k}\left(1\right)W_{k+1}^{\prime}-W_{k+1}\left(1\right)W_{k}\right)$ $\displaystyle=\left(\left[\left(k-1\right)!\right]^{2}\exp\left[\left(k-1\right)x\right]W_{k-1}\right)^{\prime}W_{k+1}$ $\displaystyle-\left(\left(k-1\right)!\right)^{2}\exp\left[\left(k-1\right)x\right]W_{k-1}W_{k+1}^{\prime}+\left(k!\right)^{2}\exp\left[kx\right]W_{k}^{2}.$ Dividing the right hand side by $\left[\left(k-1\right)!\right]^{2}\exp\left(k-1\right)x,$ we get $W_{k-1}^{\prime}W_{k+1}-W_{k-1}W_{k+1}^{\prime}+\left(k-1\right)W_{k-1}W_{k+1}+k^{2}\exp\left(x\right)W_{k}^{2}.$ By the Jacobi identity, this has to be zero. Letting $k=n+1,$ we get $\left(\ref{refi}\right).$ This finishes the proof. ∎ The conclusion of Theorem 6.1 follows immediately from Lemma 6.3 and the Darboux invariance property discussed above. Hence we have abundant candidates of balancing configurations of multiple vortex rings. In principle, the nondegeneracy of these configuration could be proved using similar idea as that of [25]. We leave this to a further study. Finally, let us comment on the reason why we restrict to the case $m=n+1.$ Indeed, our original equation in Section 5 to be solved is (6.11) $P^{\prime\prime}Q-2P^{\prime}Q^{\prime}+PQ^{\prime\prime}+nP^{\prime}Q-mPQ^{\prime}=0$ Let $Q=x,$ $n=1$ and $m\geq 3.$ Then the degree $m$ polynomial $P$ satisfying $\left(\ref{mn}\right)$ necessarily has the factor $x^{3}.$ Hence $P$ and $Q$ has a common root and can’t be used in our construction. 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Non Linéaire, 11 (1994), no. 4,427-440. Weiwei Ao School of Mathematics and Statistics Wuhan University, Wuhan, Hubei, China Email<EMAIL_ADDRESS> Yehui Huang School of Mathematics and Physics, North China Electric Power University, Beijing, China, Email<EMAIL_ADDRESS> Yong Liu Department of Mathematics, University of Science and Technology of China, Hefei, China, Email<EMAIL_ADDRESS> Juncheng Wei Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada, V6T 1Z2 Email<EMAIL_ADDRESS>
# Knowledge Graph Completion with Text-aided Regularization Tong Chen Sirou Zhu Yiming Wen Zhaomin Zheng Language Technology Institute Carnegie Mellon University Pittsburgh, PA 15213 {tongc2, sirouz, yimingwe<EMAIL_ADDRESS> ###### Abstract Knowledge Graph Completion is a task of expanding the knowledge graph/base through estimating possible entities, or proper nouns, that can be connected using a set of predefined relations, or verb/predicates describing interconnections of two things. Generally, we describe this problem as adding new edges to a current network of vertices and edges. Traditional approaches mainly focus on using the existing graphical information that is intrinsic of the graph and train the corresponding embeddings to describe the information; however, we think that the corpus that are related to the entities should also contain information that can positively influence the embeddings to better make predictions. In our project, we try numerous ways of using extracted or raw textual information to help existing KG embedding frameworks reach better prediction results, in the means of adding a similarity function to the regularization part in the loss function. Results have shown that we have made decent improvements over baseline KG embedding methods. ## 1 Introduction The Knowledge Graph/Knowledge Base is a collection of structured tuples of information connecting entities via relations, where entities are generalized concepts or names to be appeared in head/tail positions, forming nodes/vertices in the graph and relations are verbs or predicates describing the way two entities are logically connected, forming edges in the graph. Formally, Knowledge Graph can be represented as a directed multigraph that may have multiple edges with the same vertices. Knowledge Graphs have a lot of applications, such as Structured Search, Question-Answering Systems, Recommendation Systems etc. However, most of the Knowledge Graphs are large but far from complete. In this project, we aim to provide a modern approach to create or predict missing links in the graph, formally done by link prediction: given two elements of a triple, predict missing items. Let’s formally define our task. Let $\mathcal{E}$ be be the set of entities and $\mathcal{R}$ be the set of relations. Then a KG is defined as a collection of triple facts ($\mathit{e_{s},r,e_{d}}$), and the task of knowledge graph reasoning is to find the missing entry in the triple facts. Previously, the focus of knwoledge graph embedding and completion mainly focuses on using various methods to catch the intrinsic, latent semantics of the knowledge graph, which more or less rely on the existing items or structures of the graph itself; but it is another level of concern that the information would be limited to the existing relations or semantics of the knowledge graph, so to complete and expand a knowledge graph, it would be very helpful to introduce external information from text corpora that one knowledge graph or even commonsense graph would rely on, be it general (like FreeBase or YAGO) or industry-specific (such asUMLS). Many recent joint graph and text embedding methods have been focusing on learning better knowledge graph embeddings for reasoning (Han et al. (2018)), but we consider reaching for better graph embeddings in a more language- oriented sense. In this research, we would propose a general framework of using regularization techniques to train a set of entity embeddings that can capture the nuances of relation and connection between the entities that may not lie in the knowledge graph structure itself, but more on the text corpus side; through becoming a constituent part of regularization, we make our framework more compatible to more of the commonly available knowledge graph embedding models, and have the potential to migrate to state-of-the-art models of the time. ## 2 Related Work ### 2.1 Knowledge Graph Reasoning Given a knowledge graph, the task of knowledge graph reasoning includes predicting missing links between entities, predicting missing entities, and predicting whether a graph triple is true or false. A variety of methods have been applied to the task of knowledge graph reasoning, and recently embedding- based methods are gaining popularity and yielding promising results, such as linear models in Bordes et al. (2011), matrix factorization models in Trouillon et al. (2016), convolutional neural networks in Yang et al. (2014). In spite of promising results, these models have limited interpretability. Reinforcement learning models have better interpretability, such MINERVA in Das et al. (2017), DeepPath in Xiong et al. (2017) and Multi-hop in Lin et al. (2018), which exploit policy network. ### 2.2 Open-world Knowledge Graph Completion. Using only information inside the static and structured knowledge graph limits our ability to learn representations for embeddings and relations. Therefore, (Shi & Weninger (2018)) raised the concept of Open-world Knowledge Graph Completion problem, which is to utilize external information so as to connect unseen entities to the knowledge graph. There are several attempts to utilize text corpus to improve Knowledge Graph embeddings, such as the mutual attention model by (Han et al. (2018)) and Latent Relation Language Models in Hayashi et al. (2019). However, these approaches have the following limitations: First of all, the models focuses on local information in the text corpus. Each entity is trained with just its description without considering corpus-level statistics. Secondly, the model need to align two disjoint latent spaces, the Knowledge Graph space and the text corpus space. Thirdly, the models tend to closely couple between the Knowledge Graph and Text method. Last but not least, the models typically provide no way of learning relation- specific representations of entities. ### 2.3 Text Embedding Models and Vector Sets. Word embeddings have been proved to be useful in NLP tasks as standalone features (Turian et al. (2010)). The key idea of word embedding is to use multi-dimentional vectors to represent the meanings of words. Influential models can be divided into two categories, count-based models and prediction- based models. A good example of count-based models is GloVe, introduced by Pennington et al. in 2014, which is a log-linear model trained on window based local co-occurrence information about word pairs in Pennington et al. (2014a). Popular prediction-based models include Continuous bag of words (CBOW) models, skip-gram models and transformer-based models. Based on these state-of-the-art models, pre-trained vector sets such as Word2Vec in Mikolov et al. (2013), FastText in Mikolov et al. (2018), and GloVe, are released and are ready to be used in NLP tasks. In our model, word embedding can be used both statically and dynamically. The static way is to initialize the entity weight by pretrained embedding from results in Pennington et al. (2014a), Mikolov et al. (2013), and train the entity embedding with models in Dettmers et al. (2018) and Bordes et al. (2011). The dynamic way is to combine the loss functions of word embedding model and knowledge graph model together to train the entity and relation embedding. ## 3 Text-regularized KG Completion In this research, we would propose a general framework of using regularization techniques to train a set of entity embeddings that can capture the nuances of relation and connection between the entities that may not lie in the knowledge graph structure itself, but more on the text corpus side. Many recent joint graph and text embedding methods have been focusing on learning better knowledge graph embeddings for reasoning in Han et al. (2018), but we consider reaching for better graph embeddings in a more language-oriented sense. Existing jointly training methods, like Toutanova et al. (2015), Han et al. (2018), and Hayashi et al. (2019), share the following shortcomings: * • These methods typically only focus on a neighboring/localized space that is labeled relevant to some given entity-relation-entity triples; their embeddings are only trained with the related sentence description that are labeled in the corpus. * • They do not take corpus level statistics and language model relativities into account, due to the same reason that they rely on labeled partitioned datasets. * • They need to align two disjoint latent/embedding spaces (the KG embeddings and text embeddings), which are usually not quite overlapping considering the word tokens available from both sides. * • Entities representations are typically generic, not accounting for relation variations, so no way of learning relation-specific embeddings are offered. Here we would propose a method of text-enhanced knowledge graph embedding, which uses similarity functions as regularizers towards the training loss of the knowledge graph. The underlied motivation is that, we would typically find similar descriptions and especially, overlapping words for two entities/concepts that falls into similar domains and same categories of name types; in the meantime, these entities would also share similar characteristics in their connectivity, topology, and types of relations linking to them in the knowledge graph. Thus from the description or context regarding the concept, we would be able to train a similar set of embeddings for these two concepts. For example, both Microsoft and Amazon are concept names that describes a company, and their descriptions would definitely cover the fact that they both have headquarters in Seattle, and they both have a founder (who are Bill Gates and Jeff Bezos, respectively). To learn a corpus-regularized representations for the relations and entities, we would need to build a loss function that properly trains the embedding towards better predictability. we could define a loss function $L_{(}text)=Sim(x,y)=f(x,y\|text)$ on the domain $x,y\in\mathbf{R}^{\|E\|\|E\|}$ that captures the similarities between relevant descriptions or context of entities $x$ and $y$, as shown in the figure 1. There are lots of room of decision over which type of similarity function $f$ to be used, such as word overlap on context, TF-IDF based on word pairs, dot-product of existing trained embeddings, and so on. Figure 1: Illustration of a similarity function over the corpus space. Given two entities (e.g., Carnegie Mellon University and Stanford, or Pittsburgh), we calculate the similarities between them using any defined function that is defined across the text corpus. Formally, we would define a new loss function that serves the purpose of minimizing losses from both the side of knowledge base and the one from the text: $\mathit{L}=\sum_{(\mathbf{e_{1},e_{2},r})\in KG}\lambda_{1}\times\mathit{L}_{KG}(\mathbf{e_{1},e_{2},r})+\lambda_{2}\times\mathit{L}_{Text}(\mathbf{e_{1},e_{2}})$ To exemplify one possible similarity function, we could propose a more GLoVePennington et al. (2014a)-like approach, which could make entities that have similar descriptions being closer to each other in the vector embedding space. $\mathit{L}=\sum_{(\mathbf{e_{1},e_{2},r})\in KG}\lambda_{1}\times\mathit{L}_{KG}(\mathbf{e_{1},e_{2},r})+\lambda_{2}\times(\mathbf{e_{1}^{T}e_{2}+b_{e1}+b_{e2}-X_{e_{1},e_{2}}})$ ## 4 Variants of Similarity functions Here we show some types of similarity functions that we have tested with. They include converging to generic word embeddings, similarity calculation using entity-recognized newspaper text, and ones using associated wikipedia article text. ### 4.1 Variant A: Converging to existing embeddings A Quick way to use existing fruits of extracted textual information to begin with is converging to existing word embeddings. Given the premises that we are trying to incorporate textual structures into entity embeddings, we can well assume that word embeddings should also contain information about connections between words. Considering the apparent fact that the latent structures underlying in the knowledge graphs should be drastically different than the ones extracted from text (which could be more focusing on predicting the next words that would follow the current word), we would prefer to converge the differences between the embeddings of the entities, and the ones when they are converted to words. Also considering the fact that the latent structures’ distribution and the number of dimensions should be different between the two types of embeddings, we incorporate a fully connection layer to extract and redistribute the features, and make the differences comparable. The easiest and the most prominent example would be using cosine similarity to converge the two: $\displaystyle\mathbf{Sim}(\mathbf{e_{1}},\mathbf{e_{2}})=\cos(E_{\mathbf{e_{1}}}-E_{\mathbf{e_{2}}},f(w_{\mathbf{e_{1}}}-w_{\mathbf{e_{2}}}))$ $\displaystyle f(w_{\mathbf{e_{1}}}-w_{\mathbf{e_{2}}})=NN(w_{\mathbf{e_{1}}}-w_{\mathbf{e_{2}}})$ Another would be using rank distance defined by (Santus et al. (2018)) that would extract the most prominent ranks of the embeddings and compare the differences between the two: $\displaystyle\mathbf{Sim}(\mathbf{e_{1}},\mathbf{e_{2}})=\sum_{r\in\text{intersect}}\frac{2}{(E_{\mathbf{e_{1}}}-E_{\mathbf{e_{2}}}+f(w_{\mathbf{e_{1}}}-w_{\mathbf{e_{2}}})}$ $\displaystyle f(w_{\mathbf{e_{1}}}-w_{\mathbf{e_{2}}})=NN(w_{\mathbf{e_{1}}}-w_{\mathbf{e_{2}}})$ ### 4.2 Variant B: Using Entity-tagged Text In (Pennington et al. (2014a)), word similarities can be extracted from a corpus by constructing a co-occurrence matrix. Similarly, entity similarities can be calculated in the same manner. Given a corpus, we can construct matrix $\mathbf{X}$ where $\mathbf{X_{i,j}}$ is the number of times entity $\mathbf{j}$ occurs in the context of entity $\mathbf{i}$. We select New York Time corpus as the test dataset for extracting co-occurrence matrix $\mathbf{X}$. Summing over the entities in the knowledge graph, we have our Entity-tagged text regularization term: $\displaystyle\mathit{L}_{Entity-tagged}\sum_{(\mathbf{e_{1},e_{2},r})\in KG}\mathbf{w_{i}^{T}w_{j}-logX_{i,j}}$ ### 4.3 Variant C: Using Associated Wikipedia Text As described above, similar entities should have similar text descriptions on Wikipedia and similar text structures. For example, Wikipedia documents of counties in the US should all talk about its climate, economy, population, etc, and actors should all talk about their career works and life experiences and so on. To exploit such similarity to regularize our entity embedding, we measure two correlations between the Wikipedia documents of entities, which are, first, a modified Relaxed Word Mover Distance (Kusner et al. (2015)) between two entity documents, and second, TF-IDF similarity between two entity documents. #### 4.3.1 Relaxed Word Mover Distance In (Kusner et al. (2015)), the Relaxed Word Mover Distance is calculated as $\text{RWMD}(\mathbf{e_{1},\mathbf{e_{2}}})=\sum_{i\in\mathbf{e_{1}}}c_{i}\min_{j\in\mathbf{e_{2}}}dist(i,j)$ , where $i,j$ are index of words in entity documents, $c_{i}$ is the normalized frequency of word $i$, and $dist(i,j)$ is the distance between the pre-trained word embedding of word $i$ and $j$. In our proposed method, to boost the calculation, instead of using the distance between word embedding, we use the the dot product of two word embedding to measure their similarity, and accordingly, we introduce the concept of RWMD-Gain, which is the maximum gain of transformation from the document of an entity to another and is as follows: $\displaystyle\text{RWMD- Gain}(\mathbf{e_{1},e_{2}})=\sum_{i\in\mathbf{e_{1}}}c_{i}\max_{j\in\mathbf{e_{2}}}sim(i,j)$ We employ the bidirectional gain as the semantic similarity of two entity document, which is $\displaystyle\text{Gain}(e_{1},e_{2})=\text{RWMD- Gain}(e_{1},e_{2})+\text{RWMD-Gain}(e_{2},e_{1})$ To sum over all instances in the knowledge graph, we have our RMWD regularization term: $\displaystyle\mathit{L}_{Text-RWMD}=\sum_{(\mathbf{e_{1},e_{2},r})\in KG}log(\text{Gain}(\mathbf{e_{1}},\mathbf{e_{2}}))\|\mathbf{e_{1}}-\mathbf{e_{2}}\|^{2}_{2}$ #### 4.3.2 TF-IDF Similarity To compute the TF-IDF similarity between two entity documents, we first convert each document $\mathit{d}$ to a TF-IDF vector $\mathbf{w_{d}}$, in which each element can be calculated by the following formula: $\displaystyle w_{i,d}=tf_{i,d}\times log(\frac{N}{df_{i}})$ where $\mathit{i}$ represents the index of the word; $tf_{i,d}$ represents the occurrence of word $\mathit{i}$ in document $\mathit{d}$; $\mathit{N}$ represents the total number of documents and $df_{i}$ represents the number of documents containing word $\mathit{i}$. The similarity of document $d_{x}$ and document $d_{y}$ is the dot product of such $\mathbf{w_{x}}$, $\mathbf{w_{y}}$. $\displaystyle Sim(x,y)=dot(\mathbf{w_{x}},\mathbf{w_{y}})$ ## 5 Experiments ### 5.1 Data Sources and formulations From (Fu et al. (2019)) we select the FB60K dataset as the base knowledge graph 111https://github.com/thunlp/OpenNRE. In the aforementioned paper, the authors studied the datasets and found that the relation distributions of the two datasets are very imbalanced; but still, It is the KG dataset that contains the most entities in the “FreeBase”-related KG variants, like FB17K, FB15K-237, etc. In the original dataset, all entity names are its Freebase ID; to properly link the partially anonymized graph to the truth raw text, we employed an openly available dictionary to convert thme from ID to its corresponding names. By linking the ID to the names, we can see that the FB60K dataset mainly contains famous or obscure locations, celebrity names, schools, sports unions, etc. On the external sources of the three variants, we would list them here: • Variant A (using existing embeddings): we employ the GLoVe trained embeddings (Pennington et al. (2014b)) as the embeddings we would want to converge to. Specifically, since word embeddings are normally single words, while entities are mostly proper noun phrases, we generate the final embedding of the entity by calculating the average over all the constituent words that is related to the entity (an entity can have multiple words and multiple names to refer to) according to the open dictionaries. * • Variant B (using entity labeled text): we employ the NYT10 dataset that is supplied along with the FB60K dataset. Entities are labeled using common named entity recognition toolkits, and the source corpus text is excerpted from the New York Times corpus. They are also labeled what relations the sentence would contain, but currently we haven’t employed these. An analysis to the information overlap (i.e., alignment) between the corpus and the KG in Table 1. Higher CT/CE (triple-entity ratio) indicates adding corpus-edges to the KG increases the average degree more significantly, leading to more reduction in sparsity. * • Variant C (using linked Wikipedia articles): we employ the articles from the Wikipedia dump in December 2019. By linking the entity FreeBase ID to its name, and by finding these names’ English Wikipedia article, we managed to extract over 54,000 articles linking to existing entities (around 80%). We use Stanford CoreNLP Tokenizer (Manning et al. (2014)) to tokenize and lower- letter all the words in the text after we remove structured text and tables, which are text that cannot be easily consumed. We choose TransE (Bordes et al. (2011)) as the base KG embedding method as the base of comparison between all the variants. TransE is an old methods of calculating KG embeddings, but it is the easiest to implement and has still been holding decent performances despite several newer and more state-of-the- art models being introduced since. Dataset \| #triples(C) \| #triple(G) \| #entities(C) \| #entities(G) \| #rel(C) \| #rel(G) \| S(train) \| S(test) \| CT/CE \| CR/KR ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- FB60K-NYT10 \| 172,448 \| 268,280 \| 63,696 \| 69,514 \| 57 \| 1,327 \| 570k \| 172k \| 2.71 \| 0.04 Table 1: The dataset information. #triples(C) & #triples(G) denote the number of triples in the corpus and the KG respectively, and so on. S(train) denotes the number of sentences in the training corpus, while S(test) denotes the number of sentences in the testing corpus. CT/CE denotes triple-entity ratio. Lower triple-entity ratio indicates less triples per entity in average can be extracted from the corpus. CR/KR denotes corpus-relation-quantity/KG-relation- quantity ratio. Lower CR/KR indicates less information overlap between the corpus and the KG. ### 5.2 Experimental Result We run all of our proposed variants on FB60K-NYT10 dataset. Our baseline method is TransE (Bordes et al. (2011)). We add different proposed regularization terms to TransE individually to compare their performance. The experimental results are shown in Table 2. Method \| Hits@1 \| Hits@3 \| Hits@10 \| MRR \| Hits@1 Filtered \| Hits@3 Filtered \| Hits@10 Filtered \| MRR Filtered ---\|---\|---\|---\|---\|---\|---\|---\|--- TransE \| 0.2144 \| 0.4414 \| 0.5431 \| 0.3379 \| 0.3392 \| 0.6245 \| 0.7016 \| 0.4902 TransE + Cosine \| 0.2303 \| 0.4455 \| 0.5436 \| 0.3487 \| 0.3957 \| 0.6310 \| 0.7035 \| 0.5216 TransE + GloVe-NYT \| 0.2112 \| 0.4366 \| 0.5436 \| 0.3363 \| 0.3378 \| 0.6086 \| 0.6997 \| 0.4841 TransE + GloVe-Wiki \| 0.2305 \| 0.4474 \| 0.5493 \| 0.3500 \| 0.3552 \| 0.6170 \| 0.6951 \| 0.4957 TransE + GloVe-RWMD \| 0.2417 \| 0.4000 \| 0.5147 \| 0.3396 \| 0.4292 \| 0.5816 \| 0.6787 \| 0.5194 Table 2: Experimental results running on FB60K-NYT10 dataset. The best of all metrics are highlighted with bold. ### 5.3 Experimental Analysis From table 2, we can see that generally TransE + Cosine and TransE + GloVe- wiki outperform the other methods and baseline. For HR@1, TranE + GloVe-RWMD gives the best result and outperforms baseline significantly. For HR@3, HR@10, and MRR, TransE + GloVe is the best one. GloVe-RWMD considers the semantic distances, which helps finding the best suitable entity, but negatively impacts the score of the potential entities. TransE + Cosine and GloVe-wiki are more simple and intuitive, which consider only the word frequencies and proves to be more useful when considering a list of entities. The amount of injected information directly influences similarity matrix quality. Wiki is comparably larger then New York Times, and larger amount of information gives more accurate entity similarity score, so GloVe-Wiki outperforms Glove-NYT. ## 6 Conclusion From the experiments, our methods showed decent improvements compared to the baseline model. Although TransE is a prudent model, we believe the proposed regularization methods can fit to later state-of-the-art models with modest adaptations. However, due to time limitation, the hyper-parameters were not fine-tuned in the experiments so the results did not show a very significant improvement. Therefore, future work can be done on further improving the implementation of regularization, or on choosing a better set of parameters. Moreover, the design of decoders can be further improved to transform textual latent features to entity or knowledge graph latent features. After all, we believe that more information should always be better than less, but we need to find a good way to utilize it. In this project, we diversely explored text-assisted KG embedding or reasoning methods. However, given the fact that these methods all require extensive frameworks to extract external information, we think there might be unexplored while simpler ways to migrate textual latent features to enrich KG embeddings. 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# Non-unitary triplet superconductivity tuned by field-controlled magnetization –URhGe, UCoGe and UTe2– Kazushige Machida Department of Physics, Ritsumeikan University, Kusatsu 525-8577, Japan ###### Abstract We report on a theoretical study on ferromagnetic superconductors, URhGe, and UCoGe and identify the pairing state as a non-unitary spin-triplet one with time reversal symmetry broken, analogous to superfluid 3He-A phase. A recently found superconductor UTe2 with almost ferromagnet is analyzed by the same manner. Through investigating their peculiar upper critical field $H_{\rm c2}$ shapes, it is shown that the pairing symmetry realized in all three compounds can be tuned by their magnetization curves under applied fields. This leads to the reentrant $H_{\rm c2}$ in URhGe, an S-shaped in UCoGe and an L-shaped $H_{\rm c2}$ in UTe2 observed under the field direction parallel to the magnetic hard $b$-axis in orthorhombic crystals in common. The identification with double chiral form: ${\bf d}(k)=(\hat{b}+i\hat{c})(k_{b}+ik_{c})$ in UTe2 naturally enables us to understand (1) multiple phases with A1, A2, and A0 phases observed under pressure, (2) the enhanced reentrant $H_{\rm c2}$ for the off-axis direction is associated with first order meta-magnetic transition, and (3) Weyl point nodes oriented along the magnetic easy $a$-axis. All three compounds are found to be topologically rich solid-state materials worth further investigation. ## I Introduction The competing orders are at the heart of the strongly correlated systems in general where multiple long-range or short range orderings, such as superconductivity (SC), ferromagnetism (FM), spin and charge density waves are emerging out of the strong interactions in condensed matter systems. This is particularly true in the case of unconventional superconductivity, which is often associated with other orderings mentioned kato1 ; kato2 . A good example is high temperature cuprate superconductors in which various coexisting or mutually repulsive orderings are found kivelson . There has been much attention focused on ferromagnetic superconductors aokireview , such as UGe2 UGe2 , URhGe URhGe , and UCoGe UCoGe in recent years. A new member of such a superconductor, UTe2 with $T_{\rm c}$=1.6K ran ; aoki2 , which is almost ferromagnetic is discovered quite recently and attracts much excitement. Those systems are contrasted with the coexisting materials of magnetism and superconductivity in (RE)Rh4B4 (RE: 4f rare earth elements) and Chevrel compounds (RE)Mo6S8 in that the 4f electrons responsible for magnetism are localized spatially and distinctive from the conduction electrons machidareview . Here the 5f electrons responsible for magnetism are more subtle in that they participate both magnetism and superconductivity. UTe2 has been investigated experimentally knebel ; daniel ; miyake ; ran2 ; metz ; mad ; tokunaga ; sonier ; nakamine ; hayes ; 1 ; 2 ; 3 ; 4 ; rosa ; aokiP ; andreyUTe2 ; kittaka and theoretically xu ; ishizuka ; shick ; nevi ; fidrysiak ; yarzhemsky ; fujimoto ; lebed ; kmiyake . Simultaneously renewed interest on the former three compounds are developing. These heavy Fermion materials belong to a strongly correlated system that is heavily governed by the 5f electrons, which form a coherent narrow band with a large mass enhancement below the Kondo temperature. Because the upper critical field $H_{\rm c2}$ in those compounds exceeds the Pauli paramagnetic limitation, a triplet or odd parity pairing state is expected to be realized aokireview . However, detailed studies of the pairing symmetry remain lacking despite of the fact that previous knowledge of the first three compounds is accumulated for over two decades. Thus now it is a good chance to understand those “old” materials URhGe and UCoGe together with the new UTe2 by seeking some common features. The prominent SC properties observed commonly in these superconductors are as follows: When $H$ is applied parallel to the magnetic hard $b$-axis in orthorhombic crystals, $H_{\rm c2}$ exhibits the reentrant behavior in URhGe, where the SC state that disappeared reappears at higher fields levy , or an S-shape in UCoGe aokiS and an L-shape in UTe2 knebel in the $H$-$T$ plane. Above the superconducting transition temperature $T_{\rm c}$, ferromagnetic transition occurs in URhGe and UCoGe. Thus, the SC state survives under a strong internal field, resulting from an exchange interaction between the conduction and the 5f electrons. However, in UTe2 “static” FM has not been detected although FM fluctuations are probed ran ; miyake ; tokunaga ; sonier above $T_{\rm c}$, i.e. there is a diverging static susceptibility along the magnetic easy $a$-axis ran ; miyake and the nuclear relaxation time $1/T_{2}$ in NMR tokunaga . The gap structure is unconventional, characterized by either a point in UTe2 aoki2 ; ran or line nodes in the others aokireview . There is clear experimental evidence for double transitions: the two successive second order SC phase transitions seen in specific heat experiments as distinctive jumps systematically change under pressure ($P$) in UTe2 daniel . A similar indication for double SC transitions in ambient pressure is observed in UCoGe at $T_{\rm c2}\sim$0.2K manago1 ; manago2 where the nuclear relaxation time $1/T_{1}T$ in NMR experiments exhibits a plateau corresponding to the “half residual density of states (DOS)” value at the intermediate $T$ below $T_{\rm c}=$0.5K. Upon further lowering of $T$, it stars decreasing again at 0.2 K. Recent specific heat $C/T$ data for several high quality samples of UTe2 aoki2 ; ran ; kittaka commonly show the residual DOS amounting to $0.5N(0)$, which is half of the normal DOS $N(0)$, while some exhibit zero residual DOS metz . Thus, this “residual” half DOS issue is currently controversial. We propose a method to resolve this issue, discussed later in this paper. To understand these three spin-polarized superconductors URhGe, UCoGe, and UTe2 in a unified way, we develop a phenomenological theory based on the assumption that the three compounds are coherently described in terms of the triplet pairing symmetry analogous to the superfluid 3He-A-phase leggett . It is instructive to remember that the A1-A2 phase transition is induced by an applied field, which is observed as the clear double specific heat jumps halperin . The originally degenerate transition temperatures for the A phase are split into the A1 and A2 phases under applied fields mermin . Therefore, to address the experimental facts mentioned above, we postulate the A-phase-like triplet pair symmetry, which responds to the spontaneous FM and/or induced moment under perpendicular external fields, to yield the A1-A2 double transitions. This scenario coherently explains the observed reentrant $H_{\rm c2}$ in URhGe, the S-shape in UCoGe, and the L-shape in UTe2 for the field direction along the magnetic hard $b$-axis in a unified way. As mentioned above, the A1-A2 phase transition in 3He A-phase halperin is controlled by the linear Zeeman effect due to the applied field, which acts to split $T_{\rm c}$ mermin . In the spin polarized superconductors, $T_{\rm c}$ is controlled by the spontaneous and/or field-induced magnetic moment, which is linearly coupled to the non-unitary order parameter. We employ the Ginzburg-Landau (GL) theory to describe these characteristic $H_{\rm c2}$ curves. We also identify the pairing symmetry by group theoretic classification machida based on our previous method ozaki1 ; ozaki2 . The pairing symmetry is a non-unitary triplet machida ; ohmi ; machida2 , where the d-vector is a complex function that points perpendicular to the magnetic easy axis in zero-field. The gap function possesses either a point or line node with a possibly chiral $p$-wave orbital form. This is maximally consistent with the SC characteristics obtained so far in UTe2, such as the STM observation mad of chiral edge states, the polar Kerr experiment hayes , which shows time reversal symmetry breaking, and other various thermodynamic measurements. The arrangement of this paper is following. We set up the theoretical framework to explain those experimental facts in the three compounds, URhGe, UCoGe and UTe2 in Section II. The theory is based on the Ginzburg-Landau theory for the order parameter with three components. The quasi-particle spectra in the triplet states are examined to understand thermodynamic behaviors for the materials. In Section III we investigate the generic phase transitions of the present pairing state under fields applied to various field directions relative to the orthorhombic crystalline axes. In order to prepare analyzing the experimental data for URhGe, UCoGe and UTe2 which exhibit a variety of the $H_{\rm c2}$ such as reentrant SC (RSC), S-shaped, and L-shaped one, the magnetization curves for three compounds are studied in detail and evaluated the curves when the experimental data are absent in Section IV. We apply the present theory to the three compounds and explain the peculiar $H_{\rm c2}$ curves observed in Section V, including the multiple phase diagrams in UTe2 under pressure. Section VI devotes to detailed discussions on the gap structure, and pairing symmetries for each material. Summary and conclusion are given in the final section VII. The present paper is a full paper version of the two short papers by the author short1 ; short2 . ## II Theoretical Framework ### II.1 Ginzburg-Landau theory In order to understand a variety of experimental phenomena exhibited by the three compounds in a common theoretical framework, we start with the most generic Ginzburg-Landau (GL) theory for a spin triplet state. This is general enough to allow us to describe the diversity of those systems. Among abundant spin triplet, or odd-parity paring states we assume an A-phase like pairing state described by the complex $\bf d$-vector with three components: $\displaystyle{\bf d}(k)=\phi(k){\vec{\eta}}=\phi(k)({\vec{\eta}}^{\prime}+i{\vec{\eta}}^{\prime\prime}).$ (1) ${\vec{\eta}}^{\prime}$ and ${\vec{\eta}}^{\prime\prime}$ are real three dimensional vectors in the spin space for Cooper pairs, and $\phi(k)$ is the orbital part of the pairing function. This is classified group-theoretically under the overall symmetry: $\displaystyle\rm{SO}(3)_{\rm spin}\times{\rm D}^{\rm orbital}_{\rm 2h}\times{\rm U}(1)$ (2) with the spin, orbital and gauge symmetry respectively machida ; annett . In this study, we adopt the weak spin-orbit coupling scheme ozaki1 ; ozaki2 which covers the strong spin-orbit (SO) case as a limit. The strength of the SO coupling depends on materials and is to be appropriately tuned relative to the experimental situations. It will turn out to be crucial to choose the weak SO coupling case in understanding the $H_{\rm c2}$ phase diagrams with peculiar shapes: This allows the $\bf d$-vector rotation under an applied field whose strength is determined by the SO coupling. Note that in the strong SO coupling scheme the $\bf d$-vector rotation field is infinite because the Cooper pair spin is locked to crystal lattices. There exists U(1)$\times$Z2 symmetry in this pairing, i.e., invariance under ${\bf d}\rightarrow-{\bf d}$ and gauge transformations. We emphasize here that this SO(3) triple spin symmetry of the pairing function is expressed by a complex three component vectorial order parameter $\vec{\eta}=(\eta_{a},\eta_{b},\eta_{c})$ in the most general. It will turn out later to be important also to describe complex multiple phase diagram, consisting of five distinctive phases, but this is a minimal framework which is necessary and sufficient. Under the overall symmetry expressed by Eq. (2) the most general Ginzburg- Landau free energy functional up to the quadratic order is written down as $\displaystyle F^{(2)}=\alpha_{0}(T-T_{\rm c0}){\vec{\eta}}\cdot{\vec{\eta}}^{\star}+b\|{\vec{M}}\cdot{\vec{\eta}}\|^{2}+cM^{2}{\vec{\eta}}\cdot{\vec{\eta}}^{\star}$ $\displaystyle+i\kappa{\vec{M}}\cdot{\vec{\eta}}\times{\vec{\eta}}^{\star}$ (3) with $b$ and $c$ positive constants. The last invariant with the coefficient $\kappa$ comes from the non-unitarity of the pairing function in the presence of the spontaneous moment and field induced $\vec{M}(H)$, which are to break the SO(3) spin space symmetry in Eq. (2). We take $\kappa>0$ without loss of generality. This term responds to external field directions differently through their magnetization curves. The fourth order term in the GL functional is given by machida ; annett $\displaystyle F^{(4)}={\beta_{1}\over 2}({\vec{\eta}}\cdot{\vec{\eta}}^{\star})^{2}+{\beta_{2}\over 2}\|{\vec{\eta}}^{2}\|^{2}.$ (4) Because the fourth order terms are written as $\displaystyle F^{(4)}={\beta_{1}\over 2}({\vec{\eta}}^{\prime}\cdot{\vec{\eta}}^{\prime}+{\vec{\eta}}^{\prime\prime}\cdot{\vec{\eta}}^{\prime\prime})^{2}+{\beta_{2}\over 2}[({\vec{\eta}}^{\prime}\cdot{\vec{\eta}}^{\prime}-{\vec{\eta}}^{\prime\prime}\cdot{\vec{\eta}}^{\prime\prime})^{2}$ $\displaystyle+4({\vec{\eta}}^{\prime}\cdot{\vec{\eta}}^{\prime\prime})^{2}]$ (5) for $\beta_{1},\beta_{2}>0$, we can find a minimum when $\|{\vec{\eta}}^{\prime}\|=\|{\vec{\eta}}^{\prime\prime}\|$ and ${\vec{\eta}}^{\prime}\perp{\vec{\eta}}^{\prime\prime}$. Notably, the weak coupling estimate machida leads to ${\beta_{1}/\beta_{2}}=-2$. Thus we have to resort to the strong coupling effects in the following arguments in order to stabilize an A1 phase. It is convenient to introduce $\displaystyle\eta_{\pm}={1\over\sqrt{2}}(\eta_{b}\pm i\eta_{c})$ (6) for ${\bf M}=(M_{a},0,0)$ where we denote the $a$-axis as the magnetic easy axis in this and next sections. From Eq. (3) the quadratic term $F^{(2)}$ is rewritten in terms of $\eta_{\pm}$ and $\eta_{a}$ as $\displaystyle F^{(2)}=\alpha_{0}\\{(T-T_{\rm c1})\|\eta_{+}\|^{2}+(T-T_{\rm c2})\|\eta_{-}\|^{2}$ $\displaystyle+(T-T_{\rm c3})\|\eta_{a}\|^{2}\\}$ (7) with $\displaystyle T_{\rm c1,2}=T_{\rm c0}\pm{\kappa\over\alpha_{0}}M_{a},$ $\displaystyle T_{\rm c3}=T_{\rm c0}-{b\over\alpha_{0}}M^{2}_{a}.$ (8) The actual second transition temperature is modified to $\displaystyle T^{\prime}_{\rm c2}=T_{\rm c0}-{\kappa M_{a}\over\alpha_{0}}\cdot{{\beta_{1}-\beta_{2}}\over{2\beta_{2}}}$ (9) because of the fourth order GL terms in Eq. (4). And also $T_{\rm c3}$ starts decreasing in the linear $\|M_{a}\|$ in stead of $M^{2}_{a}$ mentioned above just near $\|M_{a}\|\ll 1$. This comes from the renormalization of $T_{c3}$ in the presence of $\|\eta_{+}\|^{2}\propto(T_{c1}-T)$ and $\|\eta_{-}\|^{2}\propto(T_{c2}-T)$. Those terms give rise to the $\|M_{a}\|$-linear suppression of $T_{c3}$ through fourth order terms. Here we note that among the GL fourth order terms, $Re(\eta_{a}^{2}\eta_{+}\eta_{-})$ in Eq. (4) becomes important in interpreting the $H_{\rm c2}$ data later because it is independent of the signs of the GL parameters $\beta_{1}$ and $\beta_{2}$. For $1\leq{\beta_{1}/\beta_{2}}\leq 3$, $\displaystyle T^{\prime}_{c2}>T_{c2}=T_{\rm c0}\pm{\kappa\over\alpha_{0}}M_{a}.$ (10) This could lead to the modification of the otherwise symmetric phase diagram: $\displaystyle T_{c1}-T_{c0}=T_{c0}-T_{c2}.$ (11) The fourth order contribution of Eq. (9) to $T_{c2}$ may become important to quantitatively reproduce the $H$-$T$ phase diagram, such as the asymmetric L-shape $H_{\rm c2}^{b}$ observed in UTe2 knebel . Note that the ratio of the specific heat jumps to $\displaystyle{\Delta C(T_{c1})\over\Delta C(T_{c2})}={T_{c1}\over T_{c2}}\cdot{\beta_{1}\over{\beta_{1}+\beta_{2}}}.$ (12) The jump at $T_{c2}$ can be quite small for $T_{c1}\gg T_{c2}$. The FM moment $M_{a}$ acts to shift the original transition temperature $T_{\rm c0}$ and split it into $T_{c1}$, $T_{c2}$, and $T_{c3}$ as shown in Fig. 1. Here, the A1 and A2 phases correspond to $\|\uparrow\uparrow>$ pair and $\|\downarrow\downarrow>$ pair, respectively and the A0 phase is $\|\uparrow\downarrow>+\|\downarrow\uparrow>$ for the spin quantization axis parallel to the magnetization direction $M_{a}$. According to Eq. (8), $T_{c1}$ ($T_{c2}$) increases (decrease) linearly as a function of $M_{a}$ while $T_{c3}$ decreases quadratically as $M^{2}_{a}$ far away from the degeneracy point shown there (the red dot). The three transition lines meet at $M_{a}$=0 where the three components $\eta_{i}$ ($i=+,-,a$) are all degenerate, restoring SO(3) spin space symmetry. Thus away from the degenerate point at $M_{a}$=0, the A0 phase starts at $T_{\rm c3}$ quickly disappears from the phase diagram. Below $T_{\rm c2}$ ($T_{\rm c3}$) the two components $\eta_{+}$ and $\eta_{-}$ coexist, symbolically denoted by A1+A2. Note that because their transition temperatures are different, A1+A2 is not the so-called A-phase which is unitary, but generically non-unitary except at the degenerate point $M_{a}$=0 where the totally symmetric phase is realized with time reversal symmetry preserved. Likewise below $T_{\rm c3}$ all the components coexist; A1+A2+A0 realizes. Figure 1: (color online) Generic phase diagram in the $T$ and $M_{a}$ plane. $T_{\rm c1}$ ($T_{\rm c2}$) for the A1 (A2) phase increases (decreases) linearly in $M_{a}$. The third phase A0 decreases quadratically in $M_{a}$ away from the degenerate point at $M_{a}$=0. Under an applied field with the vector potential $\bf A$, the gradient GL energy is given $\displaystyle F_{grad}=\sum_{\nu=a,b,c}\\{K_{a}\|D_{x}\eta_{\nu}\|^{2}+K_{b}\|D_{y}\eta_{\nu}\|^{2}+K_{c}\|D_{z}\eta_{\nu}\|^{2}\\}$ (13) where $D_{j}=-i\hbar\partial_{j}-2eA_{j}/c$ and the mass terms are characterized by the coefficients $K_{j}$ ($j=a,b,c$) in D2h. We emphasize here as seen from this form of Eq. (13) that $H_{\rm c2}$ for the three components each starting at $T_{\rm cj}$ ($j=1,2,3$) intersects each other, never avoiding or leading to a level repulsion. The level repulsion may occur for the pairing states belonging to multi-dimensional representations (see for example [repulsion1, ; repulsion2, ; repulsion3, ; repulsion4, ] in UPt3). The external field $H$ implicitly comes into $T_{\rm cj}$ ($j=1,2,3$) through $M_{a}(H)$ in addition to the vector potential $A$. This gives rise to the orbital depairing mentioned above. The magnetic coupling $\kappa$, which is a key parameter to characterize materials of interest in the following, is estimated mermin by $\displaystyle\kappa=T_{\rm c}{N^{\prime}(0)\over N(0)}ln(1.14\omega/T_{\rm c})$ (14) where $N^{\prime}(0)$ is the energy derivative of the normal density of states $N(0)$ at the Fermi level and $\omega$ is the energy cut-off. This term arises from the electron-hole asymmetry near the Fermi level. $\kappa$ indicates the degree of this asymmetry, which can be substantial for a narrow band. Thus the Kondo coherent band in heavy Fermion materials, such as in our case, is expected to be important. We can estimate $N^{\prime}(0)/N(0)\sim 1/E_{\rm F}$ with the Fermi energy $E_{\rm F}$. Because $T_{\rm c}$=2mK and $E_{\rm F}$=1K in superfluid 3He, $\kappa\sim 10^{-3}$. In the present compounds $T_{\rm c}\sim$1K and $E_{\rm F}\sim T_{\rm K}$ with the $T_{\rm K}$ Kondo temperature being typically aokireview 10$\sim$50K. Thus $\kappa$ is much larger than that of superfluid 3He and is an order of $1\sim 10^{-1}$. We will assign the $\kappa$ value for each compound to reproduce the phase diagram in the following as tabulated in Table I. ### II.2 Quasi-particle spectrum for general triplet state If we choose ${\vec{\eta}}^{\prime}=\eta_{b}{\hat{b}}$ and ${\vec{\eta}}^{\prime\prime}=\eta_{c}{\hat{c}}$ with $\eta_{a}=0$ for the magnetic easy $a$-axis, the quasi-particle spectra are calculated by $\displaystyle E_{k,\sigma}=\sqrt{\epsilon(k)^{2}+(\|{\vec{\eta}}\|^{2}\pm\|{\vec{\eta}}\times{\vec{\eta}}^{\star}\|)\phi(k)^{2}}$ (15) or $\displaystyle E_{k,\sigma}=\sqrt{\epsilon(k)^{2}+\Delta_{\sigma}(k)^{2}},$ (16) where the gap functions for two branches are $\displaystyle\Delta_{\uparrow}(k)$ $\displaystyle=$ $\displaystyle\|\eta_{b}+\eta_{c}\|\phi(k)$ $\displaystyle\Delta_{\downarrow}(k)$ $\displaystyle=$ $\displaystyle\|\eta_{b}-\eta_{c}\|\phi(k)$ $\displaystyle\Delta_{0}(k)$ $\displaystyle=$ $\displaystyle\|\eta_{a}\|\phi(k).$ (17) Note that if $\|\eta_{c}\|=0$, $\Delta_{\uparrow}(k)=\Delta_{\downarrow}(k)$, which is nothing but the A-phase leggett . When $\|\eta_{b}\|=\|\eta_{c}\|$, $\Delta_{\uparrow}(k)\neq 0$ and $\Delta_{\downarrow}(k)=0$, which is the non- unitary A1 phase for $\eta_{a}=0$. The gap in one of the two branches vanishes and the other remains ungapped. Therefore, if we assume that in the normal state $N_{\uparrow}(0)=N_{\downarrow}(0)$, the A1 phase is characterized by having the ungapped DOS $N_{\downarrow}(0)=N(0)/2$ with $N(0)=N_{\uparrow}(0)+N_{\downarrow}(0)$. Generically, however, since $N_{\uparrow}(0)\neq N_{\downarrow}(0)$, that is, $N_{\uparrow}(0)>N_{\downarrow}(0)$ in the A1 phase, which is energetically advantageous than the A2 phase, the “residual DOS” is equal to $N_{\downarrow}(0)$, which is likely less-than-half rather then more-than-half physically. In the non-unitary state with the complex $\bf d$-vector, the time reversal symmetry is broken. In the most general case where all components $\eta_{a}$, $\eta_{b}$, and $\eta_{c}$ are no-vanishing, the quasi-particle spectra are calculated by diagonalizing the $4\times 4$ eigenvalue matrix. Namely in terms of Eq. (17) the spectrum is given by $\displaystyle E^{2}_{k}=\epsilon(k)^{2}+{1\over 2}{\bigl{\\{}}\Delta^{2}_{\uparrow}(k)+\Delta^{2}_{\downarrow}(k)+2\Delta^{2}_{0}(k)$ $\displaystyle\pm\sqrt{(\Delta^{2}_{\uparrow}(k)-\Delta^{2}_{\downarrow}(k))^{2}+4\Delta^{2}_{0}(k)(\Delta^{2}_{\uparrow}(k)+\Delta^{2}_{\downarrow}(k))}{\bigr{\\}}}.$ (18) It is easy to see that this spectrum is reduced to Eq. (16) when $\Delta_{0}(k)=0$. This spectrum characterizes the phase $A_{1}+A_{2}+A_{0}$ realized in UTe2 under pressure as we will see shortly. ## III Prototypes of phase transitions Let us now consider the action of the external field $H_{b}$ applied to the magnetic hard $b$-axis on the FM moment $M_{a}$, pointing parallel to the $a$-axis. The $a$-axis component of the moment $M_{a}(H_{b})$ generally decreases as it rotates toward the $b$-axis as shown in Fig. 2(b). As discussed in the next section in more details based on experimental data, it is observed in URhGe through the neutron experiment levy . Here we display the generic and typical magnetization curves of $M_{a}$ and $M_{b}$ in Fig. 2(c) where $H_{R}$ denotes a characteristic field for $M_{b}(H_{b})=M_{a}(H_{b}=0)$. The induced moment $M_{b}$ reaches the spontaneous FM moment $M_{a}$ at zero field by rotating the FM moment, implying that the FM moment points to the $b$-axis above $H_{R}$. Experimentally, it is realized by the so-called meta-magnetic transition via a first order transition in URhGe levy and UTe2 miyake or gradual change in UCoGe knafoCo . As displayed in Fig. 2(a), by increasing $H_{b}$, $T_{c1}$ ($T_{c2}$) decreases (increases) according to Eq. (8). The two transition lines $T_{c1}(H_{b})$ =$T_{c2}(H_{b})$ meet at $H_{b}=H_{R}$. As $H_{b}$ is further increased, $T_{c1}$ also increases by rotating the $\bf d$-vector direction such that the $\bf d$-vector becomes perpendicular to ${\bf M}_{b}$, which maximally gains the magnetic coupling energy $i\kappa{\vec{M}}\cdot{\vec{\eta}}\times{\vec{\eta}}^{\star}$ in Eq. (3). This process occurs gradually or suddenly, depending on the situations of the magnetic subsystem and the spin-orbit coupling that locks the $\bf d$-vector to the underlying lattices. Therefore $H_{R}$ may indicate simultaneously the $\bf d$-vector rotation. It should be noted, however, that if the spin-orbit coupling is strong, the $\bf d$-vector rotation is prevented. In this case $H^{b}_{\rm c2}$ exhibits a Pauli limited behavior as observed in UTe2 under pressure aokiP . Figure 2: (color online)(a) Prototype phase diagram in the $T$ and $H_{b}$ plane where $H_{b}$ is parallel to the magnetic hard $b$-axis and the moment $M_{a}$ points to the easy $a$-axis. The two transition lines of $T_{c1}$ and $T_{c2}$ (red curves) initially decreases and increases respectively as $H_{b}$ increases toward the degeneracy point at $T_{R}$. There the projection of the FM moment $M_{a}$ vanishes. During this process, by rotating the $\bf d$-vector to catch the magnetization $M_{b}(H_{b})$ (the green lines) instead of $M_{a}(H_{b})$, $H^{(1)}_{\rm c2}$ and $H^{(2)}_{\rm c2}$ turn around their directions. (b) Under the perpendicular field $H_{b}$ the spontaneous moment $M_{a}$ rotates toward the $b$-direction. The projection $M_{b}(H_{b})$ of $M_{a}$ on the $b$-axis increases. (c) The rotation field $H_{R}$ is indicated as the red dot where $M_{b}(H_{b})=M_{a}(H=0)$. In Fig. 3 we show prototype phase diagrams for different situations. In addition to that displayed in Fig. 3(a), which is the same as in Fig. 2(a), there is the case in which $T_{c1}$ is bent before reaching $H_{R}$ as shown in Fig. 3(b). The magnetization curve $M_{b}(H_{b})$ starting at $T_{c0}$ exceeds the decreasing $M_{a}$ at a lower field $H_{\rm CR}$ defined by $M_{b}(H_{b})=M_{a}(H_{b})$. $H^{(1)}_{\rm c2}$ turns around there by rotating the $\bf d$-vector. We will see this case in the following analysis. In the $H_{c}$ case for the field direction parallel to the another hard $c$-axis the phase diagram is shown in Fig. 3(c). Since $H_{c}$ does not much influence on $M_{a}(H_{c})$, both $H^{(1)}_{\rm c2}$ and $H^{(2)}_{\rm c2}$ are suppressed by the orbital depression of $H_{c}$. When magnetic field is applied to the magnetic easy $a$-axis, the spontaneous moment $M_{a}(H_{a})$ increases monotonically, as shown in Fig. 3(d). According to Eq. (8), $T_{c1}$ ($T_{c2}$) increases (decreases) as $H_{a}$ increases. Thus, theoretically $H_{c2}^{(1)}$ can have a positive slope at $T_{c1}$. However, the existing data on UCoGe wu indicate that it is negative as seen shortly. This is because the strong orbital depairing $H^{\prime 0}_{\rm c2}$ overcomes the positive rise of $T_{c1}$. Moreover, $H_{c2}^{(2)}$ is strongly suppressed by both $T_{c2}$ and the orbital effect $H^{\prime 0}_{\rm c2}$, resulting in a low $H^{a}_{c2}$, compared with $H^{b}_{c2}$. This $H_{c2}$ anisotropy is common in these compounds aokireview . From the above considerations, the enhanced $H^{b}_{c2}$ is observed because the higher part of the field in $H_{c2}$ belongs to $H_{c2}^{(2)}$, which has a positive slope. Figure 3: (color online) Two types (a) and (b) of the phase diagram for $H\parallel b$ with the $b$-axis (hard axis). (a) is the same as in Fig. 2(a). (b) At $H_{\rm CR}$ defined by $M_{b}(H_{b})=M_{a}(H_{b})$, $H^{(1)}_{\rm c2}$ turns round by rotating the $\bf d$-vector to catch $M_{b}$ starting from $T_{c0}$. (c) $H\parallel c$ with the $c$-axis (another hard axis). (d) $H\parallel a$ with the $a$-axis (easy axis). The green lines are the respective magnetization curves and the red curves are $H^{(1)}_{\rm c2}$ and $H^{(2)}_{\rm c2}$. Within the GL scheme it is easy to estimate $H_{\rm c2}$ as follows. We start with the $H_{\rm c2}$ expression: $H_{\rm c2}(T)=A_{0}\\{T_{c}(H_{\rm c2})-T\\}$ (19) with $A_{0}={\Phi_{0}\over 2\pi\hbar^{2}}4m\alpha_{0}$, $m$ effective mass, and $\Phi_{0}$ quantum unit flux. Here $T_{c}$ depends on $H$ though $M_{a}(H)$ as described above. Thus, the initial slope of $H^{\prime}_{\rm c2}$ at $T_{c}$ is simply given by $H^{\prime}_{\rm c2}(T)=A_{0}{dT_{c}\over dH_{\rm c2}}H^{\prime}_{\rm c2}-A_{0}.$ (20) It is seen that if ${dT_{c}/dH}=0$ for the ordinary superconductors, $H^{\prime 0}_{\rm c2}(T)=-A_{0}<0$. The slope $H^{\prime}_{\rm c2}(T)$ is always negative. However, Eq. (20) is expressed as $H^{\prime}_{\rm c2}(T)={-A_{0}\over{1-A_{0}({dT_{c}\over dH_{\rm c2}}})},$ (21) or $\displaystyle{1\over\|H^{\prime}_{\rm c2}\|}$ $\displaystyle=$ $\displaystyle{1\over\|H^{\prime 0}_{\rm c2}\|}+\|{dT_{\rm c}\over dH_{\rm c2}}\|$ (22) $\displaystyle=$ $\displaystyle{1\over\|H^{\prime 0}_{\rm c2}\|}+{1\over\|{dH_{\rm c2}\over dT_{\rm c}(H)}\|}.$ The condition for attaining the positive slope, $H^{\prime}_{\rm c2}(T)>0$ implies $\|H^{\prime 0}_{\rm c2}\|>({dH\over dT_{\rm c}})$ at $H_{\rm c2}$. This is a necessary condition to achieve S-shaped or L-shaped $H_{\rm c2}$ curves experimentally observed. This is fulfilled when $\|H^{\prime 0}_{\rm c2}\|$ is large enough, that is, the orbital depairing is small, $\|{dT_{c}\over{dH}}\|$ at $H_{\rm c2}$ is large, or the $T_{c}$ rise is strong enough. It is noted that when $1-A_{0}({dT_{c}\over dH_{\rm c2}})=0$, the $H_{\rm c2}(T)$ curve has a divergent part in its slope, which is observed in UCoGe as a part of the S-shape. It is clear from the above that when $dT_{c}/dH<0$, $\|H^{\prime}_{\rm c2}(T)\|<\|H^{\prime 0}_{\rm c2}\|$ because the two terms in Eq. (22) are added up to further depress $H_{\rm c2}(T)$. In this case the slope $\|H^{\prime}_{\rm c2}\|$ is always smaller than the original $\|H^{\prime 0}_{\rm c2}\|$ as expected. In Fig. 4 we show the changes of $H_{\rm c2}$ when the competition between the orbital suppression and $T_{c}(M)$ varies. We start from the orbital limited $H^{\rm WHH}_{\rm c2}$ curve with $T_{c}$ unchanged as a standard one. When $T_{c}(M)$ decreases with increasing $H$, the resulting $H_{\rm c2}$ is further suppressed compared with $H^{\rm WHH}_{\rm c2}$ as shown in Fig. 4(a). $T_{c}(M)$ as a function of $H$ through $M(H)$ becomes increasing as shown in Fig. 4(b), $H_{\rm c2}$ is enhanced compared to $H^{\rm WHH}_{\rm c2}$, exceeding the $H^{\rm WHH}_{\rm c2}$ value. Figure. 4(c) displays the case where $T_{c}(M)$ increases stronger than that in Fig. 4(b), $H_{\rm c2}$ has a positive slope and keeps increasing until it hits the upper limit $H^{\rm AUL}_{\rm c2}$. There exists the absolute upper limit (AUL) for $H_{\rm c2}$. Even though $T_{c}(M)$ keeps increasing with increasing $M(H)$, $H_{\rm c2}$ terminates at a certain field because a material has its own coherent length $\xi$ which absolutely limits $H^{\rm AUL}_{\rm c2}=\Phi_{0}/2\pi\xi^{2}$. Beyond $H^{\rm AUL}_{\rm c2}$ there exists no superconducting state. Figure 4: (color online) $H_{\rm c2}$ changes due to the competition between the orbital depairing and $T_{c}(M)$. (a) $T_{c}(M)$ decreases as a function of the applied field $H$. The orbital depairing is added up to further depress $H_{\rm c2}$ than $H^{\rm WHH}_{\rm c2}$. (b) $T_{c}(M)$ increases as a function of the applied field $H$, competing with the orbital depairing. The resulting $H_{\rm c2}$ is enhanced compared with $H^{\rm WHH}_{\rm c2}$. (c) $T_{c}(M)$ increases strongly as a function of the applied field $H$. $H_{\rm c2}$ has a positive initial slope and keeps growing until hitting the absolute upper limit $H^{\rm AUL}_{\rm c2}$. Then $H_{\rm c2}$ follows this boundary. There could be several types of $H^{b}_{\rm c2}$ curves for $H$ applied to the $b$-axis (hard axis), depending on several factors: (A) the magnitude of the spontaneous moment $M_{a}$, (B) its growth rate against $H_{b}$, (C) the coupling constant $\kappa$, (D) the relative position of $T_{c0}$ on the temperature axis. Possible representative $H^{b}_{\rm c2}$ curves are displayed in Figs. 5(a), 5(b) and 5(c). When the hypothetical $T_{c0}$ is situated in the negative temperature side, the realized phase is only the A1-phase at $H_{b}=0$. In high field regions SC reappears as the reentrant SC (RSC) by increasing $M_{b}(H_{b})$, which is shown in Fig. 5(a). The reentrant SC is separated from the lower SC. As shown in Fig. 5(b) the two transition temperatures, $T_{c1}$ and $T_{c2}$ are realized at $H_{b}=0$, that is, it shows double transitions at zero field, giving rise to the $A_{1}$ and $A_{2}$ phases. The three phases $A_{1}$, $A_{2}$ and $A$ appear in a finite $H_{b}$ region. $H_{\rm c2}$ could have an S-shape. This corresponds to either Figs. 3(a) or (b). When the separation between $T_{c1}$ and $T_{c2}$ becomes wider because of increasing the spontaneous moment $M_{a}$ and/or the larger magnetic coupling $\kappa$, $H_{\rm c2}$ has an L-shape as displayed in Fig. 5(c). This could happen also when the moment rotation field $T_{R}$ is situated at relatively lower field than the overall $H_{\rm c2}$. In the following we discuss those typical $H_{\rm c2}$ behaviors based on the realistic magnetization curves for each compound, reproduce the observed $H_{\rm c2}$ curves and predict the existence of the multiple phase diagram. Figure 5: (color online) Schematic typical phase diagrams for $H$ parallel to the $b$-axis with $A_{1}$, $A_{2}$ and $A$ phases, whose structure depends on the position of $T_{c0}$ and the separation of $T_{c1}$ and $T_{c2}$. The absolute upper limit $H_{\rm c2}^{\rm AUL}$ is indicated as the grey region. (a) The reentrant SC situated at high fields such as in URhGe. (b) S-shape $H_{\rm c2}$ with the double transitions from the $A_{1}$ to the $A$ phase such as in UCoGe. (c) L-shape $H_{\rm c2}$ where the high field phase is the $A_{2}$ phase such as in UTe2. ## IV Magnetization curves In order to understand their peculiar $H_{\rm c2}$ shapes and resulting pairing symmetry in three compounds, it is essential to know their magnetic responses to applied magnetic fields as mentioned above. Here we analyze their magnetism and estimate the magnetization curves of the spontaneous moment under the transverse field, which is not probed by conventional magnetization measurements. In the following we consider the cases of URhGe and UCoGe, and UTe2 with the $c$-axis and $a$-axis are the easy axes respectively as tabulated in Table I. We mainly discuss URhGe as an typical example. The concepts introduced here are applied to the other systems with appropriately changing the notation for the magnetic easy axis. Table 1: Magnetic properties and $\kappa$ values materials \| Curie temp.[K] \| easy axis \| moment[$\mu_{B}$] \| $\kappa$[K/$\mu_{B}$] ---\|---\|---\|---\|--- URhGe \| 9.5 \| $c$-axis \| Mc=0.4 \| 2.0 UCoGe \| 2.5 \| $c$-axis \| Mc=0.06 \| 1.8 UTe2 \| – \| $a$-axis \| $\sqrt{\langle{\rm M}_{a}^{2}\rangle}$=0.48 \| 6.9 ### IV.1 Rigid rotation picture: Spontaneous moment rotation When the applied field $H_{b}$ is directed to the hard axis, or the $b$-axis, the spontaneous moment $M_{c}(H_{b})$ pointing to the $c$-axis in URhGe rotates gradually toward the applied field direction. At around $H_{R}=12$T, $M_{c}(H_{b})$ quickly turns to the $b$-direction by rotating the moment as shown in Fig. 6. We define the crossing field $H_{\rm\rm CR}$ at which $M_{c}(H_{b})=M_{b}(H_{b})$. Note that $H_{R}$ and $H_{\rm\rm CR}$ are different concepts as is clear from Fig. 6 and also in UCoGe where $H_{\rm\rm CR}\sim$ a few T and $H_{R}$=45T knafoCo . Simultaneously and correspondingly, the $M_{b}(H_{b})$ moment jumps via a first order transition. Above $H_{b}>H_{R}$ the spontaneous moment is completely aligned along the $b$-axis as seen from Fig. 6. This phenomenon is often called as the meta-magnetic transition. But this is just the moment rotation since it is demonstrated that the total magnetization $\sqrt{M^{2}_{c}(H_{b})+M_{b}^{2}(H_{b})}$ hardly changes and remains a constant during this first order transition process levy . This implies that $M_{c}(H_{b})=M_{c}\cos(\alpha(H_{b}))$, and $M_{b}(H_{b})=M_{c}\sin(\alpha(H_{b}))$ with $\alpha(H_{b})$ being the rotation angle of $M_{c}(H_{b})$ from the $c$-axis. The rotation angle $\alpha(H_{b})$ is accurately measured by the neutron scattering experiment by Lévy, et allevy who construct the detailed map of the rotation angle in the $H_{b}$ and $H_{c}$ plane. This rotation process is mirrored by the magnetization curve of $M_{b}(H_{b})$ so that the projection of $M_{c}(H_{b})$ onto the $b$-axis manifests itself on $M_{b}(H_{b})$ as shown in Fig. 6. The crossing of $M_{b}(H_{b})$ and $M_{c}(H_{b})$ occurs around at $H_{\rm CR}=9\sim 10$T, corresponding to roughly $M_{c}(H_{b})/\sqrt{2}\sim M_{b}(H_{b})$. That is, $M_{c}(H_{b})$ rotates by the angle $\alpha=45^{\circ}$ from the $c$-axis at $H_{\rm\rm CR}$. This first order phase transition phenomenon in URhGe under the transverse field is neatly described by Mineev mineev using the GL theory. This is within more general framework of the so-called meta-magnetic transition theory based on the GL phenomenology wohlfarth ; shimizu ; yamada for itinerant ferromagnets. Figure 6: (color online) The ferromagnetic spontaneous moment $M_{c}(H_{b})$ rotation indicated by the green arrow under the field $H\parallel b$ in URhGe. At $H_{b}=H_{R}$, it completely orients along the $b$-axis direction via a first order transition where $M_{b}(H_{b})$ shows a jump of the magnetization. $H_{CR}$ is defined by the field $M_{c}(H_{b})=M_{b}(H_{b})$. The rotation angle $\alpha$ from the $c$-axis is measured by neutron experiment [levy, ]. The magnetization curves $M_{b}(H_{b})$ and $M_{c}(H_{c})$ are from [hardy, ]. Those considerations based on the experimental facts demonstrate to hold “a rigid moment rotation picture”. We assume this picture applicable to the other compounds too. ### IV.2 Extraction of the $M_{b}$ moment for the tilted fields from the $b$-axis data When the applied field direction is rotated from the $b$-axis toward the easy axis $c$ by the angle $\theta$, the magnetization curves are measured by Nakamura, et al nakamura . It is obvious that the measured magnetization $M(\theta)$ contains the contribution from the spontaneous moment ${\bf M}_{c}$ projected onto the applied field direction, that is, $M_{c}\sin(\theta)$. This is confirmed experimentally at least lower fields up to $H<5$T and $T$=2K aokiprivate . Thus in this situation, we can extract the $M_{b}(H)$ curves by simply subtracting the contribution $M_{c}\sin(\theta)$ from the measured data nakamura . The result is shown in Fig. 7(a). It is seen that by increasing the angle $\theta$, the first order transition field $H_{R}$ shifts to higher fields and the jump gets smaller compared to the $b$-axis case, reflecting that the moment projection onto the applied field direction decreases. This method is valid only for the small angle $\theta$ and relatively small field regions because here the $M_{c}$ moment is assumed to be fixed under the action of small field component along the $c$-axis. It may be difficult to extract reliably the $M_{b}(H)$ information for further high fields even though the tilting angle is small, and also for larger angles $\theta$. There are two factors to be taken into account, which are internally related: One is that the $c$-component magnetic field acts to prevent the moment from further rotating it toward the $b$-axis upon increasing tilting field $H$ by $\theta$ from the $b$-axis. This “rotation angle locking effect” becomes important for the field just before $H_{R}(\theta)$ where the moment ultimately rotates completely along the $b$-axis in the higher fields. The other factor to be considered is the modification of the free energy landscape of the $M_{b}$ versus $M_{c}$ space. As mentioned, the first order transition of the moment rotation is described by Mineev mineev who considers the competition between the ferromagnetic state at $M_{c}$ and the paramagnetic state with $M_{b}$ stabilized by the Zeeman effect due to the external field $H_{b}$ within a GL free energy theory. The transverse field $H_{b}$ necessarily destabilizes the second order FM phase transition at $H_{R}$ because $H_{b}$ contributes negatively to the quartic term coefficient of $M_{c}^{4}$, giving rise to a first order transition. The extra term coming from the tilting field helps to stabilize the ferromagnetic state, preventing the first order transition, thus making $H_{R}$ to higher field and the magnetization jump smaller. Thus it is not easy to extract reliably the $M_{b}(H)$ under this free energy landscape modification. In the followings, we confine our arguments for small $\theta$ and use approximate $M_{b}(H)$ forms, which are enough for our purposes to understand the peculiar $H_{c2}$. Figure 7: (color online) (a) The magnetization component of $M_{b}(H)$ in URhGe under the field direction tilted from the $b$-axis toward the $c$-axis by $\theta$, estimated from the experimental data of $M(H)$ [nakamura, ]. (b) The magnetization $M(H)$ in URhGe under the field direction tilted from the $b$-axis toward the $a$-axis by $\phi$ estimated from the experimental data (dots) of $M(H)$ [hardy, ], including magnetization curves for three $a$, $b$ and $c$-directions for reference. Figure 8: (color online) (a) The magnetization curves for three $a$, $b$, and $c$-axes in UCoGe. Here the crossing points $H^{b}_{\rm\rm CR}$ and $H^{a}_{\rm\rm CR}$ at which each curve surpasses the spontaneous moment $M_{c}(H=0)=0.06\mu_{B}$. (b) The magnetization curves of $M_{b}(H)$ for the field directions tilted from the $b$-axis toward the $c$-axis by the angle $\theta$(degrees) in UTe2. $\theta=23.7^{\circ}$ corresponds to $H\parallel(011)$ direction measured by [miyakeprivate, ]. Those are estimated by the method explained in the main text. The inset shows the magnetization curves for three $a$, $b$, and $c$-axes in UTe2. $H_{R}$ is the first order transition for the moment rotation from the $a$-axis to the $b$-axis. ### IV.3 Applied field rotation from the $b$-axis to the hard axis In the case for the tilting angle $\phi$ from the $b$-axis toward the other hard axis $a$ of URhGe, it is known levy2 that $H_{R}(\phi)$ is scaled to $H_{R}(\phi)\propto 1/\cos(\phi)$, which is also the case in UTe2 ran2 . This means that only the $M_{b}$ projection onto the $a$-axis matters to understand the magnetization process. Therefore, we can easily reconstruct the $M(\phi)$ by using the experimental data of $M_{b}(H_{b})$ except for the fact that the induced $M_{a}(\phi)$ also contributes to $M(\phi)$. This can be accomplished by an “elliptic formula” derived as follows: We start with $M_{b}(H_{b})$ and $M_{a}(H_{a})$ measured by usual magnetization experiments shown in Fig. 7(b). Assuming the linearity assumption: $M_{b}(\phi)=\chi_{b}H\cos(\phi)$ and $M_{a}(\phi)=\chi_{a}H\sin(\phi)$ with $\chi_{i}$ $(i=a,b)$ being the magnetic susceptibility, we add up the two components, $\displaystyle M(\phi)$ $\displaystyle=$ $\displaystyle M_{b}\cos\phi+M_{a}\sin\phi$ (23) $\displaystyle=$ $\displaystyle(\chi_{b}\cos^{2}(\phi)+\chi_{a}\sin^{2}(\phi))H$ $\displaystyle=$ $\displaystyle M_{b}(H_{b})\cos^{2}\phi+M_{a}(H_{a})\sin^{2}\phi.$ We call it an “elliptic formula”. Since the rotation field is given by $H_{R}(\phi)={H_{R}^{b}\over\cos(\phi)}$ (24) with $H_{R}^{b}$ the rotation field for the $b$-axis, we obtain at $H=H_{R}$ $M(\phi)=M_{b}(H_{R}){\bigl{(}}\cos\phi+{\chi_{a}\over\chi_{b}}\cdot{\sin^{2}\phi\over\cos^{2}\phi}\bigr{)}.$ (25) This formula gives the magnetization curve consisting of a straight line from $H=0$ up to $H_{R}$. The magnetization jump at $H_{R}$ is calculated by projecting the jump $\delta M_{b}$ in $M_{b}(H_{b})$, namely $\delta M_{b}\cos(\phi)$. The resulting reconstructions of $M(\phi)$ for various tilting angles are shown in Fig. 7(b). By construction, when $\phi\rightarrow 90^{\circ}$, $M(\phi)\rightarrow M_{a}(H_{a})$. We notice that the resulting $M(\phi)$ includes the contribution from $M_{a}$. Those results should be checked experimentally and will be used to reproduce the RSC in URhGe. As shown in Fig. 8(b) this idea is also applied to UTe2 where the RSC appears centered around $\theta=$35∘ from the $b$-axis toward the another hard axis $c$. As a final comment on the magnetization of UCoGe shown in Fig. 8(a), it should be mentioned that since $H_{R}\sim 45$T knafoCo , for the following discussions on this system the characteristic magnetic fields $H^{b}_{\rm\rm CR}\sim 6$T and $H^{a}_{\rm\rm CR}\sim 7$T are relevant to notice from this figure. We also note that two magnetization curves $M_{b}$ and $M_{a}$ behave similarly. It is anticipated that $H_{\rm c2}$ for the two directions should be resemble. This is indeed the case as will be seen next. ## V Application to experiments on three compounds Let us now examine the present theory to understand a variety of experiments on the three compounds, URhGe, UCoGe and UTe2. In order to clarify the essential points of the problem and for the discussions followed to be transparent, and to minimize the free adjustable parameters, we take a simplified minimal version of the present theory. It is quite easy to finely tune our theory by introducing additional parameters such as $\beta_{1}$ and $\beta_{2}$ in the GL theory Eq. (3) for each compound if necessary. We assume that $\displaystyle T_{\rm c1}=T_{\rm c0}+{\kappa}M_{a},$ $\displaystyle T_{\rm c2}=T_{\rm c0}-{\kappa}M_{a},$ $\displaystyle T_{\rm c3}=T_{\rm c0}-bM^{2}_{a}$ (26) for the spontaneous FM moment $M_{a}$ with the easy $a$-axis. We have redefined $\kappa/\alpha_{0}$ as $\kappa$ and $b/\alpha_{0}$ as $b$, ignoring the correction in Eq. (9) from the higher order GL terms. Since $\kappa$ is a converter of the units from $\mu_{B}$ to K, we further simplify the notation in that $\kappa M$ having the dimension of temperature in [K] is denoted as $M$ in [K] in the following phase diagrams as mentioned before. We use the $\kappa$ values for three compounds throughout this paper as shown in Table I where the magnetic properties are also summarized. In the following, we intend to produce the observed $H_{\rm c2}$ curves only qualitatively, not quantitatively. This is because the experimental $H_{\rm c2}$ shapes somewhat depend on the experimental methods. For example, see Fig. 1 in Ref. [wu1, ] where $H_{\rm c2}$ shapes slightly differ each other, depending on the criteria adopted either by the mid-point of the resistivity drop, the zero-resistivity, or by thermalconductivity. We here consider the sharpest curve among them when several choices are available. #### V.0.1 $H\parallel b$: Reentrant SC URhGe exhibits the ferromagnetic transition at $T_{\rm Cuire}=9.5$K where the magnetic easy axis is the $c$-axis and the FM moment $M_{c}=0.4\mu_{B}$. The superconducting transition is at $T_{c}=0.4$K under the ferromagnetic state which is persisting to the lowest $T$. When the field $H$ is applied parallel to the $b$-axis, the superconducting state reappears in a higher field region while the low field SC phase disappears at $H_{\rm c2}\sim 2$T. This reentrant superconducting state (RSC) is explained in Fig. 9, using the knowledge shown in Fig. 7. First we plot the magnetization curves for $M_{c}(H_{b})$ and $M_{b}(H_{b})$ in the $H$-$T$ plane by choosing the $\kappa=2.0{{\rm K}/\mu_{\rm B}}$ in Eq. (26) with $M_{a}$ replaced by $M_{c}$. $M_{c}(H_{b})$ starts from $T_{c1}$ and $T_{c2}$ and decreases by increasing $H_{b}$ which acts to rotate the spontaneous ferromagnetic moment toward the $b$-axis as mentioned above. Thus $T_{c1}(H_{b})=T_{c0}+{\kappa}M_{c}(H_{b})$ and $T_{c2}(H_{b})=T_{c0}-{\kappa}M_{c}(H_{b})$ decreases and increases respectively with increasing $H_{b}$ according to Eq. (26). The splitting $2{\kappa}M_{c}(H_{b})$ between $T_{c1}(H_{b})$ and $T_{c2}(H_{b})$ diminishes and meets at the rotation field $H_{\rm R}$=12T where the two transition temperatures are going to be degenerate. $M_{b}(H_{b})$ starting at $T_{c0}$ quickly increases there. Thus as shown in Fig. 9, $H_{\rm c2}$ starting at $T_{c1}$ disappears at a low field because the orbital depairing dominates over the magnetization effect as explained above. Namely, since the decrease of $T_{c1}(H_{b})$ is slow as a function of $H_{b}$, $H_{\rm c2}$ obeys the usual WHH curve, a situation similar to that shown in Fig. 4(a). Here $\|H^{\prime}_{\rm c2}(M)\|\gg\|H^{\prime orb}_{\rm c2}\|$. However, in the higher fields the upper transition temperature $T_{c1}(H_{b})$ becomes $\displaystyle T_{c1}(H_{b})=T_{c0}+{\kappa}M_{b}(H_{b})$ (27) by rotating the $\bf d$-vector so that now it is perpendicular to the $b$-axis in order to grasp the magnetization $M_{b}(H_{b})$. This $\bf d$-vector rotation field corresponds to the field where $\displaystyle T_{c1}(H_{b})=T_{c0}+{\kappa}M_{c}(H_{b})\simeq T_{c0}+{\kappa}M_{b}(H_{b}),$ (28) namely, the $M_{c}(H_{b})$ vector projection onto the $b$-axis $M_{c}/\sqrt{2}\sim M_{b}(H_{b})$ as understood from Fig. 6. Since $M_{b}(H_{b})$ is strongly enhanced at and above $H_{\rm R}$, the A1 phase reappears by following the magnetization curve $T_{c0}+{\kappa}M_{b}(H_{b})$. It ultimately hits the $H^{\rm AUL}_{\rm c2}$ boundary. The RSC finally ceases to exist beyond this boundary. This corresponds to that in Fig. 4(c). The existence of the $H^{\rm AUL}_{\rm c2}$ will be demonstrated later in Fig. 14 where we compile various $H_{\rm c2}$ data for URhGe, including those under hydrostatic pressure miyake2 and uni-axial pressure aoki-uni along the $b$-axis. Figure 9: (color online) The phase diagram for the $H_{b}$(T) versus $T$(K) plane. $M_{c}(H_{b})$ is estimated from the neutron scattering data in Ref. [levy, ] and $M_{b}(H_{b})$ comes from the magnetization curve measured in Ref. [hardy, ]. The red dots for $H_{\rm c2}$ are the experimental data points in Ref. [aokireview, ]. The red continuous line indicates $H_{\rm c2}$ which starts at $T_{c1}$ and is suppressed by the orbital depairing effect. It reappears again by following the formula $T_{c1}(H_{b})=T_{c0}+\kappa M_{b}(H_{b})$ near $H_{R}=11$T. $H^{\prime\rm orb}_{\rm c2}$ ($H^{\prime}_{\rm c2}(M)$) is the slope due to the orbital depairing ($T_{c1}(M_{b})$). #### V.0.2 $\theta$-rotation from $b$ to $c$-axis When the direction of the magnetic field turns from the $b$-axis to the easy $c$-axis, $T_{R}$ moves up to higher fields and disappears quickly around $\theta\sim 5^{\circ}$ as shown in Fig. 7(a). According to those magnetization behaviors, we construct the $H_{\rm c2}$ phase diagram in Fig. 10. It is seen that the field-direction tilting away from the $b$-axis to the $c$-axis results in the decrease of the magnetization $M_{b}(H)$, corresponding to the counter-clock wise changes of the magnetization curves in Fig. 10. Thus the RSC region shifts to higher fields with shrinking their areas and eventually disappears by entering the $H_{\rm c2}^{\rm AUL}$ region. Figure 10: (color online) Reentrant SC (Ref. [levy2, ]) for various $\theta$ values measured from the $b$-axis ($\theta=0$) toward the $c$-axis in URhGe. As $\theta$ increases (0.79∘, 1.65∘, 3.64∘, and 5.64∘), the magnetization curves (far left scale) starting at $T_{c0}$ grows slowly, pushing up the RSC regions to higher fields. The magnetization data are from Fig. 7(a) for $\theta\neq 0$ and Ref. [hardy, ] for $\theta=0$. Figure 11: (color online) Detailed RSC structures (Ref. [levy2, ]) in $T$-$H$ plane (left scale) are displayed. The triangle areas in each $\theta$ are RSC. RSC moves right as $\theta$ increases. The magnetization curve data (right scale) corrected as explained in Fig. 7(a) are originally from Ref. [nakamura, ]. The detailed phase diagram in the reentrant region is depicted in Fig. 11 where the magnetization curves of $M_{b}(H)$ in Fig. 7(a) are overlaid. According to the present theory, $H_{\rm c2}$ follows faithfully $M_{b}(H)$ in the high fields because the strong increase tendency of the magnetization $M_{b}(H)$ overcomes the orbital depression. The characteristics of those phase diagrams are; As $\theta$ increases, (1) The RSC moves up to further higher fields. (2) As $H$ further increases, within the small angles of $\theta$ up to $6^{\circ}\sim 7^{\circ}$ the RSC fades out upon entering $H_{\rm c2}^{\rm AUL}$ region. Those characteristics (1) and (2) nicely match with the experimental observations. The triangle-like shapes for RSC will be seen later in UTe2 (see Fig. 23). #### V.0.3 $\phi$-rotation from $b$ to $a$-axis When the magnetic field direction turns to the other hard $a$-axis from the $b$-axis by the angle $\phi$, the expected magnetization curves are evaluated in Fig. 7(b). Using those magnetization curves, we construct the $H_{\rm c2}$ phase diagrams for various $\phi$ values in Fig. 12. As the angle $\phi$ increases, the magnetization $M(H)$ decreases, corresponding to the clock-wise changes in Fig. 12 and the first order rotation field $H_{R}$ is pushed to higher fields simply because of the projection effect onto the $b$-axis as mentioned in section IV-C. As a consequence, the RSC moves to higher fields persisting up to higher angle $\phi$ until finally entering $H_{\rm c2}^{\rm AUL}$ region. It is confirmed experimentally that it persists at least up to $H_{\rm c2}\sim 25$T levy . According to the present results, the RSC can exist still to higher fields. This can be checked by experiments. Here we notice an important fact that in order to explain the persistence of RSC as a function of $\phi$ up to higher fields, it is essential to use the magnetization curves in Fig. 7(b) where the magnetization contains the component $M_{a}$ in addition to $M_{b}$. It is clear that only $M_{b}$ fails to reproduce the RSC phase diagram. This means that the $\bf d$-vector rotates so as to catch both components $M_{a}$ and $M_{b}$, thus the $\bf d$-vector is always perpendicular to the vectorial sum ${\bf M}_{a}+{\bf M}_{b}$. This is contrasted with the $\theta$ rotation case mentioned above where the $\bf d$-vector is perpendicular to ${\bf M}_{b}$. This intriguing anisotropy in the $\bf d$-vector rotation relative to the magnetic easy axis might be related to the underlying magnetism in URhGe and/or the spin structure of the Cooper pair symmetry assumed as $SO(3)$ originally. This spin space anisotropy should be investigated in future. In Fig. 13 we summarize the phase boundary of the RSC determined above. The band of the RSC region is tightly associated with the $H_{R}(\phi)$ curves, which are proportional to $H_{R}(\phi)\propto 1/\cos(\phi)$. This is contrasted with the lower field $H_{\rm c2}$ which is nearly independent of the angle $\phi$. The intrinsic $H_{\rm c2}$ anisotropy is quite small in URhGe. This means the importance of the magnetization rotation field $H_{R}(\phi)$, ensuring the appearance of the RSC, and pointing to the simple mechanism for the origin of RSC. It grossly follows the ${\bf M}_{b}$ projection onto the $b$-axis. This is also true for the RSC in UTe2, which will be explained shortly. The physics is common. Figure 12: (color online) RSC phase diagram in the $T$-$H$ plane for various fields rotated from the $b$-axis toward the $a$-axis by the angle $\phi$. This is constructed by using the magnetization data (right scale) shown in Fig. 7(b). When the magnetization hits the real axis $T>0$, RSC appears in high field regions. The lower field $H_{\rm c2}$ is common for all $\phi$. Figure 13: (color online) Phase boundary of the reentrance SC (RSC) as a function of the angle $\phi$ measured from the $b$-axis to the $a$-axis constructed from Fig. 12. The blue (green) line indicates the upper (lower) boundary of the RSC. The brown line is the magnetization rotation field $H_{R}(\phi)$. The dots are experimental data points by Ref. [levy2, ]. The triangles denote the lower field $H_{\rm c2}$ which is almost independent of $\phi$. #### V.0.4 Pressure effects Before starting out to analyze the experimental data taken under hydrostatic miyake2 and uni-axial pressure aoki-uni on URhGe, we summarize the relevant data for the $H_{\rm c2}$ phase diagram with the field applied to the $b$-axis in Fig. 14. Here we list up the data under hydrostatic pressure and uni-axial pressure along the $b$-axis. (1) It is clear to see that all the $H_{\rm c2}$ are limited by the common boundary $H^{\rm AUL}_{\rm c2}$. Beyond $H^{\rm AUL}_{\rm c2}$ there exists no $H_{\rm c2}$ data. (2) It is also evident to see that the $H_{R}$ data points under pressure remarkably line up along the bottom of the boundary, forming $H^{\rm AUL}_{\rm c2}$ as an envelop. In the following we utilize those experimental facts and take into account those in investigating and reconstructing the $H_{\rm c2}$ phase diagrams. In Fig. 15 we show the $H_{\rm c2}$ data points taken when $H$ is applied along the $b$-axis under uni-axial pressure $\sigma=1.0GPa$, which is listed in Fig. 14. Those data are explained in a similar way shown above. Here $H_{\rm c2}$ starting at $T_{c1}$ is strongly bent due to the sharp $M_{b}(H_{b})$ rise concomitant with the $\bf d$-vector rotation to catch $M_{b}(H_{b})$ shown by the green line in Fig. 15. Since $M_{b}(H_{b})$ starts at the temperature $T_{c0}$ midway between $T_{c1}$ and $T_{c2}$ separated by $2M_{c}$, the second transition temperature $T_{c2}$ is found to locate there where the $A_{2}$ phase begins developing while the remaining large region is occupied by the $A_{1}$ phase. Now we see the multiple phases in this situation, which is absent under the ambient pressure in URhGe. We can estimate the spontaneous moment $M_{c}$ under $\sigma$=1.0GPa as $M_{c}=0.06\mu_{B}$ on the simple assumption that $\kappa$ is unchanged under the uni-axial pressure. Figure 14: (color online) Phase diagram for $H\parallel b$ taken under hydrostatic pressure (Ref. [miyake2, ]) and uni-axial pressure along the $b$-axis (Ref. [aoki-uni, ]) on URhGe. All data at the rotation field $H_{R}$ line up along the $H^{\rm AUL}_{\rm c2}$ boundary, evidencing the existence of $H^{\rm AUL}_{\rm c2}$. Figure 15: (color online) Multiple phase diagram consisting of the $A_{1}$ and $A_{2}$ phases under uni-axial pressure $\sigma=1.0$GPa in URhGe. The data points of $H_{\rm c2}\parallel b$ are taken from Ref. [aoki-uni, ]. Two transitions at $T_{c1}$ and $T_{c2}$ separated by $2M_{c}$ are identified. $H_{R}$ is the moment rotation field found experimentally aoki-uni . The green line indicates the magnetization curve of $M_{b}$ starting at $T_{c0}$. Figure 16: (color online) Phase diagrams ($H\parallel b$) under uni-axial pressure, including the ambient pressure (a) in Fig. 9 and $\sigma=1.0$GPa (e) in Fig. 15. The data are from Ref. [aoki- uni, ]. Continuous and systematic evolution of the multiple phase diagrams with guide lines are seen. (a) $\sigma=0$GPa, (b) $\sigma=0.2$GPa, (c) $\sigma=0.5$GPa. (d) $\sigma=0.8$GPa, (e) $\sigma=1.0$GPa, and (f) $\sigma=1.2$GPa. We analyze the experimental data available under uni-axial pressure aoki-uni displayed in Fig. 16. It is seen that the continuous and systematic evolution of the multiple phase diagrams under uni-axial pressure. Namely, as uni-axial pressure $\sigma$ increases, three characteristic temperatures $T_{c1}$, $T_{c0}$ and $T_{c2}$ shifts together to higher temperatures. $T_{c2}$ appears at a finite temperature ($T>0$) around $\sigma\sim 0.8$GPa, keeping to move up with increasing further $\sigma$. The separation of $T_{c1}$ and $T_{c2}$ becomes narrow because the spontaneous moment $M_{c}$ gets diminished, corresponding to the observed Curie temperature decrease under uni-axial pressure aoki-uni (see Fig. 17(b)). We show the changes of three temperatures $T_{c1}$, $T_{c0}$ and $T_{c2}$ assigned thus in Fig. 17(a). The separation between $T_{c1}$ and $T_{c2}$ determined by $M_{c}$ diminishes simply because $M_{c}$ decreases as $\sigma$ increases. This results in $T_{c2}>0$ appearing above $\sigma>0.8$GPa, where the double transitions at $H$=0 should be observed. It is remarkable to see that upon approaching $\sigma=1.2$GPa from below, all the transition temperatures are converging toward $\sigma_{\rm cr}=1.2$GPa. This means that above this pressure, the genuine symmetric $A$ phase is realized because the symmetry breaking parameter $M_{c}$ vanishes where the spin symmetry of the pair function restores $SO(3)$ full symmetry, a situation similar to that shown in Fig. 1 (also see Fig. 25 later). At the critical pressure $\sigma_{cr}$=1.2GPa the pairing state is analogous to superfluid 3He-$A$ phase. The resulting analysis of the spontaneous moment $M_{c}$ is shown in Fig. 17(b), revealing a monotonous decrease as $\sigma$ increases. This tendency is matched with the lowering of the Curie temperature, which is observed experimentally aoki-uni . It is interesting to see the linear changes of $T_{c1}$, $T_{c0}$, $T_{c2}$, and $M_{c}$ near the critical uni-axial pressure $\sigma_{\rm cr}=1.2$GPa. This linear relationship is similar to those in UTe2 under hydrostatic pressure around the critical pressure $P_{cr}$=0.2GPa (see Fig. 25 later). Figure 17: (color online) (a) The resulting $T_{c1}$, $T_{c0}$ and $T_{c2}$ obtained from the analysis in Fig. 16 are displayed. The linear changes of those characteristic temperatures $T_{c1}$, $T_{c0}$ and $T_{c2}$ are found, corresponding to the linear decrease in $M_{c}$. The second transition $T_{c2}$ begins appearing above $\sigma>0.8$GPa where the double transitions are expected at $H$=0. (b) The resulting $M_{c}$ change as a function of uni- axial pressure $\sigma$. The observed Curie temperatures (Ref. [aoki-uni, ]) are also shown. It is consistent with the obtained decreasing tendency of $M_{c}$ as $\sigma$ increases. ### V.1 UCoGe UCoGe is another ferromagnetic superconductor worth checking our theory in the same framework for URhGe. Major differences from URhGe in the previous section lie in the fact that (1) The small spontaneous moment $M_{c}=0.06\mu_{B}$. (2) The field induced moments of $M_{b}$ and $M_{a}$ in the hard axes are comparable in magnitude as shown in Fig. 8(a). (3) The magnetization rotation field $H_{R}\sim 45$T is far above $H_{\rm c2}$. Those are contrasted with URhGe with the distinctive induced moment for $M_{b}$ that ultimately leads to the RSC. However, $H_{\rm CR}$ is situated at low fields $6\sim 8$T in UCoGe. #### V.1.1 $H\parallel b$: S-shaped $H_{\rm c2}$ and multiple phases In Fig. 18 we show the result for the phase diagram in $H\parallel b$, assuming that $\kappa=1.8{K\over\mu_{B}}$. The two transition temperatures $T_{c1}$ and $T_{c2}$ are split by $M_{c}=0.06\mu_{B}$. Under the applied field $H_{b}$, the spontaneous moment $M_{c}(H_{b})$ decreases. $T_{c1}$ and $T_{c2}$ approach each other to meet at $H^{b}_{\rm CR}\sim 6$T. Before meeting there, the upper $T_{c1}(H_{b})$ increases and catches the magnetization $M_{b}(H_{b})$ by rotating the $\bf d$-vector direction from the $c$-perpendicular direction to the $b$-perpendicular direction. This results in an S-shaped $H_{\rm c2}$ curve which eventually reaches $H^{\rm AUL}_{\rm c2}$, giving the extrapolated $H^{b}_{\rm c2}\sim 25$T. We notice here that the initial slope of $H^{b}_{\rm c2}$ is small, extrapolated to $H^{b}_{\rm c2}$ less than a few T, which is comparable to $H^{c}_{\rm c2}\sim 0.5$T. This means that the intrinsic $H_{\rm c2}$ anisotropy is within the range of the usual effective mass anisotropy. The same nearly isotropic $H_{\rm c2}$ behavior was just emphasized in URhGe (see Fig. 13). The superficial $H_{\rm c2}$ anisotropy with the order of $H^{b}_{\rm c2}/H^{c}_{\rm c2}$=25T/0.5T$\sim 50$ is an artifact due to ignoring the origin of the S-shaped $H^{b}_{\rm c2}$. This is often pointed out as one of the major mysteries in UCoGe aokireview . It is important to notice that because we identify $T_{c2}=0.2$K there must exist the phase boundary of $A_{1}$ and $A_{2}$ phases. According to thermal- conductivity measurement in Ref. [wu, ] as a function of $H_{b}$, there indeed exists an anomalous thermal-conductivity jump at 10T and low $T$ indicated as the red dot on the $H$-axis in Fig. 18. This nicely matches our identification of the $A_{2}$ phase boundary line, a situation similar to the characteristics in Fig. 4(c) and Fig. 5(b). This assignment is consistent with the $H^{c}_{\rm c2}$ phase diagram as shown shortly. Figure 18: (color online) The S-shaped phase diagram for UCoGe in $H\parallel b$. $H^{b}_{\rm c2}$ starts at $T_{c1}$ is initially depressed by the orbital depairing. At around the crossing field $H_{\rm CR}$ it turns toward higher $T$ due to the $\bf d$-vector rotation to catch $M_{b}(H_{b})$ denoted by the green line, forming the S-shape. At further high fields after hitting $H^{\rm AUL}_{\rm c2}$, $H^{b}_{\rm c2}$ follows it. The experimental data points come from [aokiS, ] and the point at $T$=0 and 10T from [wu, ]. Figure 19: (color online) (a) Weaken S-shaped $H^{a}_{\rm c2}$ for $H\parallel a$ in UCoGe because $H_{\rm\rm CR}$ moves up compared to $H^{b}_{\rm c2}$ case shown in Fig. 18. The data are from Ref. [aokiS, ]. (b) $H^{c}_{\rm c2}$ for $H\parallel c$ in UCoGe. The data [wu, ] clearly show the anomaly around 0.3T, indicating the multiple phases identified as $A_{1}$, $A_{2}$, and $A_{3}$. The magnetization curve of $M_{c}(H_{c})$ is displayed as the green dots, showing the weak rise in this scale. Both $H^{c}_{\rm c2}$ starting at $T_{c1}$ and $T_{c1}$ are thus dominated by the orbital depairing without help of the magnetization. The four points denoted by the red triangles are read off from the thermal-conductivity anomalies [taupin, ]. #### V.1.2 $H\parallel a$ As already shown in Fig. 8(a), the magnetization curves of $M_{b}$ and $M_{a}$ is quite similar. The crossing field $H^{i}_{\rm\rm CR}$ ($i=a$ and $b$) at which $M_{b}(H_{b})$ and $M_{a}(H_{a})$ reach $M_{c}=0.06\mu_{B}$ is seen to be $H^{a}_{\rm\rm CR}\sim 8$T and $H^{b}_{\rm\rm CR}\sim 6$T. Thus $H^{c}_{\rm c2}$ curve is anticipated to be similar too. Indeed the result is shown in Fig. 19(a). Even though the S-shaped $H^{b}_{\rm c2}$ is weaken, it is still seen a weak anomaly at around $H^{a}_{\rm\rm CR}\sim 8$T which is a signature that $T_{c1}(M)$ in Eq. (26) catches $M_{a}(H_{a})$ by rotating the $\bf d$-vector whose direction is perpendicular to the $c$-axis. It is now perpendicular to the $a$-axis. We also point out that the $A_{1}$ and $A_{2}$ phase diagram is essentially the same as in $H\parallel b$ and the extrapolated $H^{a}_{\rm c2}\sim$22T is comparable to $H^{b}_{\rm c2}\sim$25T. #### V.1.3 $H\parallel c$ and multiple phases We display the analysis for the phase diagram in $H\parallel c$ in Fig. 19(b). The existing experimental data clearly indicate that $H^{c}_{\rm c2}$ consists of the two parts where the $H^{c}_{\rm c2}$ enhancement is visible at low $T$ and high $H$. Thus the phase diagram is divided into the three phases, $A_{1}$, $A_{2}$ and $A_{3}$ where $A_{3}$ is genuine spin down-down pair while $A_{2}$ is a mixture of up-up and down-down pairs, or a distorted $A$ phase with different population of the two spin pairs. As indicated in Fig. 19(b) as the green dots, the magnetization curve of $M_{c}(H_{c})$ is weakly increasing in this scale. Thus the slope at $T_{c1}$ is exclusively governed by the orbital depairing, implying that this comes from the effective mass along the $c$-axis. As mentioned above, the anisotropy of the initial slopes in $H_{\rm c2}$ at $T_{c1}$ is determined by their effective mass anisotropy. #### V.1.4 Rotation $\phi$ from the $b$-axis toward the $a$-axis Finally we touch upon the case of the field rotation from the $b$-axis toward the $a$-axis by $\phi$ as shown in Fig. 20. As $H$ is turned from the $b$-axis toward the other hard $a$-axis, the crossing field $H_{\rm CR}$ increases as shown in Fig. 8(a). Since $M_{c}(H)$ becomes slowly increasing as $H$ increases, the orbital depression gets stronger and flattens the initial slopes of $H_{\rm c2}(\phi)$ at $T_{c1}$, eventually approaching $H^{a}_{\rm c2}$ as shown in Fig. 19(a). This is already realized in $\phi=11.4^{\circ}$ case seen from it. It should be pointed out again that those initial slopes at $T_{c1}$ for those $\phi$ values only slightly change, implying that the initial slope is determined by the effective masses, namely the orientational dependent Fermi velocities. So far we assumed that $\kappa=1.8{K/\mu_{B}}$ under the condition of the existence of the second transition $T_{c2}=0.2$K. But we are warned that if those suggestive signatures of the second phase $A_{2}$ coming from thermal- conductivity measurements wu ; taupin may be an artifact, then the forgoing arguments go through and are almost unchanged by taking $\kappa=3.6{K/\mu_{B}}$ without the A2 phase. Namely we are still in ambiguous situations to finally pin down the system parameters. Therefore, it is urgent to confirm or refute the existence of the second transition in order to go further from here. Figure 20: (color online) $H_{\rm c2}(\phi)$ for $\phi=0$∘, 3.2∘, 6.8∘ and 11.4∘ from the $b$-axis toward the $a$-axis in UCoGe. The data are from Ref. [aokiS, ]. As $\phi$ increases, $M_{c}$ grows slowly as a function of $H$ (the counter-clock wise rotation of the $M_{c}$ curves), pushing up $H_{\rm CR}$ to higher fields. This results in the decreases of $H_{\rm c2}(\phi)$ because the orbital suppression becomes dominant. The enhanced $H_{\rm c2}$ becomes diminished as $\phi$ increases. ### V.2 UTe2 To coherently explain a variety of physical properties of superconducting state in UTe2 accumulated experimentally in the same context of the other compounds, URhGe and UCoGe, we need a basic assumption that the ferromagnetic fluctuations are slow enough compared to the electron motion of the conduction electrons, which condense at $T_{c}$. The slow FM fluctuation moments characterized by the non-vanishing square root-mean averaged value $\sqrt{\langle(\delta M_{a})^{2}\rangle}$ over time and space $\langle\cdots\rangle$ are assumed to be able to break the spin symmetry SO(3) of the Copper pairs. In the following we denote this spontaneous and instantaneous FM moment simply $M_{a}$=$\sqrt{\langle(\delta M_{a})^{2}\rangle}$, whose magnitude is adjusted in order to best reproduce the $H_{\rm c2}$ phase diagram as we will see next. #### V.2.1 $H\parallel b$-axis We follow the same method mentioned above for URhGe and UCoGe to understand the observed L-shaped $H^{b}_{\rm c2}$ applied to the magnetic hard $b$-axis. Here we assume that $M_{a}=0.48\mu_{B}$ and $\kappa=6.9{K/\mu_{B}}$. As seen from Fig. 21, $H^{b}_{\rm c2}$ starts from $T_{c1}$=1.6K, following $M_{a}(H_{b})$ which decreases with increasing $H_{b}$ toward $H_{\rm\rm CR}$. $H_{\rm\rm CR}$ is roughly estimated from the magnetization curves shown in the inset of Fig. 8(b) as around 20T. Above $H_{b}>H_{\rm CR}$ the $\bf d$-vector rotates in order to catch the magnetization $M_{b}(H_{b})$, which strongly increases from $T_{c0}$. $H^{b}_{\rm c2}$ begins following it to grow and forms the upper part of the L-shape. It eventually reaches $H_{R}$=32T where the first order transition occurs. As shown in the inset of Fig. 8(b) the magnetization jump at $H_{R}$ amounts to 0.6$\mu_{B}$ known experimentally miyake . The reached magnetization (the horizontal green line) is deep outside $H^{\rm AUL}_{\rm c2}$ shown by dark black colored region in Fig. 21. Therefore $H^{b}_{\rm c2}$ simply stops when it hits the $H_{R}$ line. Those features nicely reproduce the experimental characteristics shown in Fig. 21. Figure 21: (color online) The L-shaped $H^{b}_{\rm c2}$ observed in [knebel, ] is shown (red dots). $H^{b}_{\rm c2}$ starting at $T_{c1}$ follows the orbital suppression plus the $M_{a}$ depression by $H_{b}$ toward $H_{\rm\rm CR}$. When it approaches the strong increasing $M_{b}(H_{b})$, the $\bf d$-vector rotates and catches $M_{b}(H_{b})$ to grow. This forms the upper part of the L-shape. In further high fields $H^{b}_{\rm c2}$ reaches $H_{R}$=32T and disappears there by hitting $H^{\rm AUL}_{\rm c2}$. The green curve denotes the magnetization curve $M_{b}(H_{b})$ shown in the inset of Fig. 8(b)[miyake, ]. #### V.2.2 $\phi$ rotation from the $b$-axis toward the $a$-axis When the field tilts from the $b$-axis toward the magnetic easy $a$-axis by the angle $\phi$, the magnetization $M_{b}(H_{b})$ growth becomes slow compared to that for the $b$-axis as shown in Fig. 22. Those counter-clock wise changes of $M_{b}(H_{b})$ for various angle $\phi$ in Fig. 22 are estimated by the method explained in section IV. Therefore, $H_{\rm c2}$ is bent upward in the upper part of their L-shaped ones while the lower parts are hardly changed because this is mainly limited by the orbital suppression. Those $H_{\rm c2}(\phi)$ curves for various $\phi$ values eventually reach their own $H^{\rm AUL}_{\rm c2}$ which depends on $\phi$, followed by the orbital suppression. Then $H_{\rm c2}(\phi)$ finally disappears abruptly by hitting $H_{R}(\phi)$. If those $H_{\rm c2}(\phi)$ curves extrapolate naively to higher fields beyond $H_{R}(\phi)$, we find $H^{\rm AUL}_{\rm c2}(\phi)$ as shown in the inset of Fig. 22, indicating that $H^{\rm AUL}_{\rm c2}(\phi)$ changes strongly within a few degrees, peaking at $H\parallel b$ sharply. We are not able to explain this peaking phenomenon at this moment. A similar peaking phenomenon is also observed in the $\theta$ side too, where the $H_{\rm c2}(\theta)$ peak occurs at $\theta\sim 35^{\circ}$. Thus the SC region in the $\phi$-$H$ plane is quite limited to small angles up to $\phi\sim 6.3^{\circ}$. As will show next, this is similar to the $\theta$ case where the high field SC $H_{\rm c2}(\theta\sim 35^{\circ})$=60T is observed in a narrow angle $\theta$ region above $T_{R}(\theta)$. Figure 22: (color online) $H_{\rm c2}(\phi)$ for $\phi$=0∘, 2∘, 4∘, 5.2∘, and 6.3∘ from the $b$-axis toward the $a$-axis. $M_{b}(H)$ grows slowly with increasing $\phi$. $H_{\rm c2}(\phi)$ curves bent over. Before hitting $H_{R}(\phi)$ which ultimately limits it, $H_{\rm c2}(\phi)$ turns around with the negative slope because they reach their own $H^{\rm AUL}_{\rm c2}(\phi)$. $M_{b}(H)$ for each $\phi$ is estimated by Eq. (25). The data (dots) are from [knebel, ]. The inset shows $H^{\rm AUL}_{\rm c2}(\phi)$ estimated by extrapolating the straight lines toward higher fields beyond $H_{R}(\phi)$. #### V.2.3 $\theta$ rotation from the $b$-axis toward the $c$-axis It is remarkable to see the extremely high $H_{\rm c2}(\theta=35^{\circ})\sim 60$T when the field is tilted from the $b$-axis toward the other magnetic hard $c$-axis ran2 . This is detached from the low field $H_{\rm c2}(\theta)\sim 8$T. This low field SC part is nearly independent of $\theta$. This $H_{\rm c2}$ isotropy was seen also in URhGe (see Fig. 13) and UCoGe. This extremely high $H_{\rm c2}(\theta=35^{\circ})$ can be understood by the present framework as follows. We begin with the $H\parallel b$ case discussed in Fig. 21. Upon increasing $\theta$, the magnetization $M_{b}(H)$ becomes slow to grow. Around $\theta=12^{\circ}$ the upper part of the L-shaped $H_{\rm c2}$ separates into two parts as shown in Fig. 23 which is observed georg . And eventually this RSC part disappears above $\theta>12^{\circ}$, leaving only the lower $H_{\rm c2}$ part at around 10T. Further increasing $\theta$, the magnetization $M_{b}(H)$ starting from $T_{c2}$ becomes relevant because as explained in Fig. 8(b), $M_{b}(H)$ becomes small and the magnetization jump also diminishes. Around $\theta=35^{\circ}$ the magnetization curves are just available for the reentrant SC to appear at higher fields above the respective $H_{R}(\theta)$. This RSC is shown in Fig. 23. This is because the state reached after the first order jump is now within the $H^{\rm AUL}_{\rm c2}$ allowed region. Thus RSC only appears within the narrow angle region centered at $\theta=35^{\circ}$. Those RSC regions are characterized by a triangle like shape as observed in [ran2, ]. This RSC shape resembles those in Figs. 9 and 10 for URhGe. Figure 23: (color online) $H_{\rm c2}(\theta)$ for various $\theta$, which is measured from the $b$-axis toward the $c$-axis. The magnetization curves of $M_{b}(H)$ starting at $T_{c2}$ and $T_{c0}$ evaluated before (see Fig. 8(b)) lead to the reentrant SC for $\theta=35^{\circ}$ in addition to the low $H_{\rm c2}$. For the lower angle of $\theta=12^{\circ}$ the two separate SC are formed. Here the $\theta=0^{\circ}$ case ($H\parallel b$) is shown for reference. It is seen that the magnetization curves only around $\theta\sim 35^{\circ}$ allow RSC to appear. #### V.2.4 Phase diagrams under pressure and multiple phases Let us examine the pressure effects on the $H_{\rm c2}$ phase diagram, which give us another testing ground to check the present scenario. In Fig. 24 (a) we show the data (dots) of $H^{b}_{\rm c2}$ for $H\parallel b$ under $P=0.4$GPa aokiP together with our analysis. It is seen that since the magnetization curve $M_{b}(H_{b})$ denoted by the green line strongly increases, $H^{b}_{\rm c2}$ started at $T_{c1}$ exhibits a bent toward higher temperatures at around $H_{\rm\rm CR}$. The two magnetization curves started from $T_{c1}$ and $T_{c2}$ meet at $H_{\rm\rm CR}$. After passing the field $H_{\rm\rm CR}$, $H^{b}_{\rm c2}$ with a positive slope heads toward $H_{R}=30$T, which is observed as the first order transition aokiP . The same feature is observed so far several times in URhGe under uni-axial pressure such as in Fig. 15 and UCoGe in Fig. 18. The second transition at $T_{c2}$ with the A2 phase is clearly found experimentally shown there detected by AC calorimetry by Aoki, et al aokiP . Moreover, the lower $H^{b}_{\rm c2}$ started from $T_{c2}$ shows an anomaly at around 5T in Fig. 24(a), suggesting the third transition $T_{c3}$. This identification is quite reasonable when we see Fig. 24(b) where the $H\parallel a$ case is displayed for the same $P=0.4$GPa. Indeed we can consistently identify $T_{c3}$ in this field orientation too. According to our theory three phases $A_{1}$, $A_{2}$, and $A_{0}$ correspond to $T_{c1}$, $T_{c2}$, and $T_{c3}$ respectively as shown there. In the high fields, we enumerate further phases $A_{4}$ and $A_{5}$. Those lower $T$ and high $H$ phases are the mixtures of the fundamental three phases $A_{1}$, $A_{2}$, and $A_{0}$ except for $A_{5}$, which is genuine $A_{0}$. For example, the $A_{4}$ phase consists of the $A_{1}$ and $A_{0}$ phases. In Fig. 24(c) we show the data of $H^{b}_{\rm c2}$ for $H\parallel b$ under $P=1.0$GPa aokiP together with our analysis. As $P$ increases, the first order transition field $H_{R}$ becomes lower, here it is $H_{R}=$20T at $P=1.0$GPa from 30T at $P=0.4$GPa. $H^{b}_{\rm c2}$ just follows a straight line due to the orbital depairing all the way up to $H_{R}$ where the magnetization $M_{b}(H_{b})$ denoted by the green line exhibits the magnetization jump. This jump is large enough to wipe out the SC state there. Thus $H^{b}_{\rm c2}$ now follows a horizontal line at $H_{R}=20$T. This is the same case as in $H^{b}_{\rm c2}$ seen in the ambient pressure (see Fig. 21). The main difference from the ambient case is that the second transition at $T_{c2}$ is now visible and observable because the FM moment $M_{a}$ diminishes under pressure and the pressure $P=0.4$GPa is situated near the critical pressure at $P=0.2$GPa (see Fig. 25). This proves the consistency of our scenario. As shown in Fig. 24(d) where at $P$=0.7GPa for $H\parallel a$ the $H_{\rm c2}$ data points are quoted from Ref. [aokiP, ], we draw the three continuous lines to connect those points. We find the missing third transition along the $T$-axis at $T_{c3}$=0.5K. Note that the tricritial point with three second order lines is thermodynamically forbidden yip . The multiple phases are enumerated, such as $A_{1}$, $A_{2}$, and $A_{0}$ at the zero-field and $A_{4}$, and $A_{5}$ at finite fields. Those phases are consisting of the coexistence of the plural fundamental three components $A_{1}$, $A_{2}$, and $A_{0}$. Namely, those are characterized by A1 at $T_{\rm c1}$, A${}_{2}\rightarrow$A1+A2 at $T_{\rm c2}$, A${}_{0}\rightarrow$ A1+A2+A0 at $T_{\rm c3}$, A${}_{4}\rightarrow$A1+A0, and A${}_{5}\rightarrow$$A_{0}$. It is understood that this phase diagram is quite exhaustive, no further state is expected in our framework. At the intersection points in Fig. 21(d) the four transition lines should always meet together according to the above general rule and thermodynamic considerations yip . The lines indicate how those three phases interact each other, by enhancing or suppressing. $T_{\rm c3}$ could be raised by the presence of the A1 and A2 phases due to the fourth order term $Re(\eta^{2}_{a}\eta_{+}\eta_{-})$ mentioned in section II. Figure 24: (color online) $H_{\rm c2}$ and the associated internal phase transition lines under hydrostatic pressure $P$ in UTe2. (a) $P$=0.4GPa and $H//b$. (b) $P$=0.4GPa and $H//a$. (c) $P$=1.0GPa and $H//b$. (d) $P$=0.7GPa and $H//a$. The data denoted by the red dots are from Ref. [aokiP, ]. $T_{c1}$ and $T_{c2}$ at $H=0$ are split by the magnetization $M_{a}$ which decreases under the applied field $H_{b}$ as shown in (a) and (c). This decrease of $M_{a}(H_{b})$ is compensated by growing of the magnetization $M_{b}(H_{b})$ as denoted by the green lines there. In Fig. 25 we compile all the data daniel ; aokiP of the phase transitions in the $T$-$P$ plane at $H=0$. As $P$ increases from $P=0$, $T_{c1}$ ($T_{c2}$) decreases (increases) to meet at the critical pressure $P_{cr}=0.2$GPa where $T_{c3}$ is also merging to converge all three transition lines. This critical pressure corresponds to the degenerate point where the symmetry breaking parameter $M_{a}$ vanishes and the three phases $A_{1}$, $A_{2}$, and $A_{0}$ becomes degenerate, restoring the full SO(3) spin symmetry at this critical point. Upon further increasing $P$, the three phases are departing from there. The three data points for $T_{c3}$ (the three red triangles on the $T_{c3}$ line in Fig. 25 are inferred from Fig. 24). The fact that $T_{c1}$ and $T_{c2}$ behave linearly in $P$ is understood as the linear relationship between $P$ and $M_{a}(P)$, leading to the linear changes of $T_{c1}$ and $T_{c2}$. This linear relationship is also seen in Fig. 17. Simultaneously a strong departure of $T_{c3}$ from the critical pressure. This is because $T_{c3}$ changes in proportion of $M_{a}^{2}$ as mentioned before (see Eq. (26)). This $T$-$P$ phase diagram is similar to that shown in Fig. 1 globally and topologically, proving that the present scenario is valid for this compound too. Figure 25: (color online) $T$-$P$ phase diagram in UTe2 with three transition temperatures $T_{c1}$, $T_{c2}$ and $T_{c3}$ corresponding to the $A_{1}$, $A_{2}$, and $A_{0}$ phases respectively. At the degenerate point of $P_{\rm\rm cr}=0.2$GPa all three phases converges. The lines for $T_{c1}$ and $T_{c2}$ as a function of $P$ indicate that the underlying symmetry breaking field $M_{a}$ changes linearly with $P$, leading to the globally quadratic variation of $T_{c3}$ from the degenerate point. The red (dark blue) round dots are from the experiment [aokiP, ] ([daniel, ]) except for the three red triangle points at $P$=0.40, 0.54 and 0.70GPa for $T_{c3}$, which are inferred from Fig. 24. ## VI Pairing symmetry ### VI.1 Gap symmetries and nodal structures The classification of the gap or orbital symmetries allowed in the present orthorhombic crystal has been done before ohmi ; annett . Among those classified pairing states, the appropriate gap function $\phi(k)$ is selected as follows: $\phi(k)=k_{a}k_{b}k_{c}$ (A1u), $\phi(k)=k_{b}$ (B1u), $\phi(k)=k_{c}$ (B2u), and $\phi(k)=k_{a}$ (B3u). The gap structure is characterized by the line nodes for those states. They are all candidates for URhGe and UCoGe as tabulated in Table II. This leads to the overall pairing function: ${\bf d}(k)=(\vec{a}\pm i\vec{b})\phi(k)$, which breaks the time reversal symmetry. This gap structure with the line nodes is consistent with the NMR experiment manago1 , reporting that 1/T1 is proportional to $T^{3}$ at low temperatures. The line nodes also suggested by other experiments on UCoGe taupin ; wu . As for UTe2, the specific heat experiments ran ; aoki2 ; metz ; kittaka exhibit $C/T\sim T^{2}$, suggesting that the gap structure is characterized by point nodes. This is also consistent with the microwave measurements 1 . Then we have to resort, an ad hoc orbital function, namely $\phi(k)=k_{b}+ik_{c}$ beyond the group-theoretical classification scheme ozaki1 ; ozaki2 , thus the resulting overall pairing function is given by ${\bf d}(k)=(\vec{b}\pm i\vec{c})(k_{b}+ik_{c})$. This pairing state is also the time reversal broken state both in spin and orbital parts. The point nodes are oriented along the $a$-axis determined by angle resolved specific heat experiment kittaka . This is characterized by the Weyl nodes analogous to superfluid 3He-A phase mizushima1 ; mizushima2 . This double chiral state both in the spin space and orbital space might be energetically advantageous because the spin and orbital moments for Cooper pairs are parallel, namely the orbital angular moment ${\bf L}$ that is spontaneously induced by this chiral state can gain the extra energy through the coupling ${\bf M}_{s}\cdot{\bf L}$ with the spontaneous magnetic moment ${\bf M}_{s}\propto{\bf d}\times{\bf d}^{\ast}$. This is consistent with the experiments by angle-resolved specific heat measurement kittaka , the STM observation mad , and the polar Kerr experiment hayes among other thermodynamic experiments nakamine . ### VI.2 Residual density of states All the compounds exhibit more or less the residual density of states at the lowest $T$ limit in the specific heat measurements aokireview ; kittaka . This is not a dirt effect of the samples used, but it is intrinsic deeply rooted to the pairing state identified as the A1 phase. In the A1 phase the superconducting DOS has intrinsically the “residual density of states”. Since $T_{c1}$ with the A1 phase is higher than $T_{c2}$ with the A2 phase, it is reasonable to expect the the DOS $N_{A_{1}}(0)$ in the A1 phase is larger than that in the A2 phase, that is, $N_{A_{1}}(0)>N_{A_{2}}(0)$ because in the Zeeman split bands, the major spin component band with larger DOS preferentially forms the higher $T_{c}$ superconducting state rather than the minority band. It is quite reasonable physically that in UTe2 at the ambient pressure the observed “residual density of states” corresponding to $N_{A_{2}}(0)$ is less than 50$\%$. Table 2: Possible Pairing Functions Compound \| spin part \| orbital part $\phi(k)$ ---\|---\|--- URhGe \| $\vec{a}\pm i\vec{b}$ \| $k_{a}k_{b}k_{c}$(A1u), $k_{b}$(B1u), $k_{c}$(B2u), $k_{a}$(B3u) UCoGe \| $\vec{a}\pm i\vec{b}$ \| $k_{a}k_{b}k_{c}$(A1u), $k_{b}$(B1u), $k_{c}$(B2u), $k_{a}$(B3u) UTe2 \| $\vec{b}\pm i\vec{c}$ \| $k_{b}+ik_{c}$ ### VI.3 Multiple phase diagram Our three component spin-triplet state leads intrinsically and naturally to a multiple phase diagram consisting of the A0 phase at $T_{c3}$, A1 at $T_{c1}$, and A2 at $T_{c2}$ as shown in Fig. 1 under non-vanishing symmetry breaking field due to the spontaneous moment. Depending on external conditions, such as $T$, $H$, and its direction, or pressure, etc, the structure of the multiple phase diagram is varied as explained. In fact under $P$, the successive double transitions are clearly observed in UTe2 daniel and they vary systematically in their $P$-$T$ phase diagram of Fig. 25. We see even the third transition centered around the critical pressure $P_{cr}=0.2$GPa. At the ambient pressure on UTe2 the occurrence of the second transition is debated hayes ; rosa , including the detailed internal phase lines. But they agree upon the existence of the multiple phases. As for UCoGe, the thermalconductivity experiment taupin indicates an anomaly at $T=0.2K$, which coincides roughly with our prediction shown in Figs. 18 and 19. As a function of $H(\parallel b)$, the thermalconductivity anomaly is detected as a sudden increase at $H\sim 0.6H_{\rm c2}$ (see Fig. 5 in Ref. [wu1, ]). Moreover, under $H$ parallel to the easy $c$-axis, the $H_{\rm c2}$ curve in Fig. 19(b) shows an enhancement at low $T$ indicative of the underlaying phase transition (see Fig. 2(b) in Ref. [wu, ]). According to the NMR by Manago et al manago1 ; manago2 , $1/TT_{1}$ presents a similar $T$ behavior, such as a plateau at $\sim N(0)/2$ and then sudden drop upon lowering $T$, as mentioned above. We propose to conduct further careful experiments to detect the A1-A2 transitions in this compound. In URhGe at the ambient pressure shown in Fig. 9 both low field phase and the RSC phase belong to the A1 phase. However, under the uni-axial pressure along the $b$-axis, there is a good chance to observe the second transition as explained in Figs. 15 and 16. Therefore to confirm the generic multiple phase diagram for all three compound shown in Fig. 1 is essential to establish the present scenario and also detect characteristics of each pairing state associated with those multiple phases. ### VI.4 Symmetry breaking mechanism For URhGe and UCoGe the “static” FM transitions are firmly established, there is no doubt for the spontaneous FM moment to be a symmetry breaking field. Slow FM fluctuations are found in UTe2 ran ; miyake ; tokunaga ; sonier which could be the origin of the symmetry breaking of $T_{\rm c1}\neq T_{\rm c2}$ under the assumption that FM fluctuations are slow compared to the conduction electron motion. A similar observation is made in UPt3: The fluctuating antiferromagnetism (AF) at $T_{N}=5$K is detected only through the fast probe: “nominally elastic” neutron diffraction aeppli ; trappmann and undetected through other “static” probes, such as specific heat, $\mu$SR, and NMR. Thus the AF fluctuating time scale is an order of MHz or faster. This is believed to be the origin of the double transition in UPt3 UPt3 ; sauls . In UTe2 it is essential and urgent to characterize the observed ferromagnetic fluctuations in more detail, such as fluctuation time scale, or spatial correlation. Elastic and inelastic neutron scattering experiments are ideal tools for it, which was the case in UPt3. It may be too early to discuss the pairing mechanism before confirming the non-unitary spin triplet state. There already exists an opinion appel which advocates longitudinal ferromagnetic fluctuations to help stabilizing a spin triplet state before the discoveries of those compounds. A problem of this sort is how to prove or refute it, otherwise it is not direct evidence and remains only circumstantial one. We need firm objective “evidence” for a pairing mechanism. Theory must be verifiable. ### VI.5 Common and different features As already seen, URhGe, UCoGe, and UTe2 are viewed coherently from the unified point: the non-unitary triplet state. They share the common features: (1) The unusual $H_{\rm c2}$ curves occur for the field direction parallel to the magnetic hard $b$-axis, where the magnetization curve $M_{b}(H_{b})$ exhibits the first order transition at $H_{R}$ for URhGe and UTe2, corresponding to the FM moment rotation. (2) Under pressure they show the critical point behaviors $P_{cr}=0.2$GPa for UTe2 and $\sigma_{cr}=1.2$GPa for URhGe at which the split $T_{c1}$ and $T_{c2}$ converges, leading to the $SO(3)$ spin symmetry for Cooper pairs. (3) The multiple phases, including the reentrant SC, are observed and explained in URhGe and UTe2 and expected to be confirmed for UCoGe. (4) The GL parameter $\kappa$ characterizing the strength of the symmetry breaking are tabulated in Table I, showing the similar values for three compounds. As a general tendency $\kappa$ is likely larger when the FM moment is larger because it is originated from the particle-hole asymmetry of the density of states $N(0)$ at the Fermi level. There are different features: (1) The nodal structures are points oriented along the magnetic easy $a$-axis in UTe2 while lines in URhGe and UCoGe. (2) Under the ambient pressure, $H_{\rm c2}$ curves are seemingly different as in Fig. 9 for URhGe, Fig. 18 for UCoGe, and Fig. 21 for UTe2. But it is now understood as mere differences in $T_{c0}$ or the FM moments as the symmetry breaker. From this comparison, the superconductivity in UTe2, URhGe and UCoGe should be understood by the unified view point, which is more resourceful and productive than considered differently and individually. ### VI.6 Double chiral non-unitary state in UTe2 Since UTe2 attracts much attention currently, it is worth summing up our thoughts on this system to challenge novel experiments. When combining the experimental observations of the chiral current along the wall by STM mad and the angle-resolved specific heat experiment kittaka , the double chiral non- unitary symmetry described by ${\bf d}(k)=({\hat{b}}+i{\hat{c}})(k_{b}+ik_{c})$ is quite possible: This pairing state produces the chiral current at the edges of domain walls, consistent with the former observation. And it is consistent with the polar Kerr experiment hayes which shows the broken time reversal symmetry. In this pairing state the point nodes orient along the magnetic easy $a$-axis, which is supported by the angle-resolved specific heat experiment kittaka . This experiment further indicates the unusual Sommerfeld coefficient $\gamma(H)$ in the superconducting state for $H$ along the $a$-axis. The low energy quasi- particle excitations naively expected for the point nodes miranovic is absent. This lack of the nodal excitations is understood by taking into account that $T_{c}$ depends on $H$ through the magnetization. This is indeed consistent with the notion of the field-tuned SC developed throughout the present paper. ## VII Summary and Conclusion We have discussed the superconducting properties of URhGe, UCoGe, and UTe2 in detail in terms of a non-unitary spin triplet pairing state in a unified way. The spontaneous static ferromagnetic moment in URhGe and UCoGe, and the slowly fluctuating instantaneous ferromagnetic moment in UTe2 break the spin $SO(3)$ symmetry in the degenerate triplet pairing function with three components. Those produce the various types of the $H_{\rm c2}$ curves that are observed. The possible pairing function is described by the complex ${\bf d}$-vector, whose direction is perpendicular to the magnetic easy axis at zero-field. Its direction changes under applied field parallel to the magnetic hard $b$-axis common in three compounds. This $\bf d$-vector rotation is driven by the induced magnetic moment under applied fields. Thus the SC order parameter is tunable by the magnetic field in this sense, ultimately leading to the reentrant SC in URhGe, S-shape in UCoGe, and L-shape $H_{\rm c2}$ in UTe2. As for UTe2, we can study a variety of topological properties, such as Weyl nodes associated with the point nodes, known in 3He A-phase mizushima1 ; mizushima2 , which was difficult to access experimentally and remains unexplored in the superfluid 3He. We can hope to see in UTe2 similar exotic vortices and Majorana zero modes predicted in 3He phase mizushima1 ; mizushima2 ; tsutsumi1 ; tsutsumi2 . There are several outstanding problems to be investigated in future, such as the pairing mechanism leading to the present non-unitary state where longitudinal spin fluctuations are plausible, but how to prove or to refute it. That is a question. As a next step, microscopic theory and detailed calculations are definitely needed beyond the present GL framework where the most simplified version is adopted in order to just illustrate the essential points. 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# A note on Minkowski formula of conformal Killing-Yano 2-form Xiaoxiang Chai Korea Institute for Advanced Study, Seoul 02455, South Korea <EMAIL_ADDRESS> ###### Abstract. We study the Minkowski formula of conformal Killing-Yano two-forms in a spacetime of constant curvature. We establish the spacetime Alexandrov theorem with a free boundary. ## 1\. Introduction The Minkowski formula states that for a smooth closed hypersurface $X:\Sigma\to\mathbb{R}^{n}$, (1) $(n-k)\int_{\Sigma}\sigma_{k-1}\mathrm{d}\mu=k\int_{\Sigma}\sigma_{k}\langle X,\nu\rangle.$ Here $\sigma_{k}$ is the $k$-th elementary symmetric functions of principal curvatures of $\Sigma$. It has found itself many applications in Riemannian geometry for example a proof of the celebrated Alexandrov theorem which says that an closed embedded hypersurface of constant mean curvature must be an sphere. The same ideas of proof lead to a free boundary generalization due to Wang-Xia [WX19] establishing the rigidity of spherical caps in balls of space forms. Both closed and the free boundary settings made use of a specially chosen conformal Killing vector field. Tachibana introduced the conformal Killing-Yano two-form as a generalization of conformal Killing vector field. ###### Definition 1 (Tachibana [Tac69]). A two-form $Q$ on an $(n+1)$-dimensional spacetime is called a conformal Killing-Yano 2-form if for every vector field $X,Y$ and $Z$ the following identity holds (2) $(\nabla_{X}Q)(Y,Z)+(\nabla_{Y}Q)(X,Z)=[2\langle X,Y\rangle\langle\xi,Z\rangle-\langle X,Z\rangle\langle\xi,Y\rangle-\langle Y,Z\rangle\langle\xi,X\rangle]$ where $\xi=\frac{1}{n}\operatorname{div}Q$. We call $\xi$ the associated 1-form of $Q$. In physics literature, these two forms are usually termed as hidden symmetry and can give information about the spacetime. See for example [JŁ06] and the references therein. Besides its physical significance, mathematically the conformal Killing-Yano two-forms are also interesting. In particular, they also allow a Minkowski type formula. Chen, Wang, Yau [CWY19] expressed quasilocal masses using this Minkowski formula. The authors of [WWZ17] established a spacetime version of the Alexandrov theorem for codimension two spacelike hypersurfaces via the Minkowski formula. In this work, we are going to extend results in [WWZ17] where they used only conformal Killing-Yano two-forms $r\mathrm{d}r\wedge\mathrm{d}t$. First we state the spacetime CMC condition with free boundary. ###### Definition 2. We say that $\Sigma^{2}$ in a spacetime $\mathbb{R}^{3,1}$ is CMC with free boundary if $\Sigma$ admits a null normal vector field $L$ with $\langle\vec{H},L\rangle$ is constant, $(DL)^{\bot}=0$ and $\Sigma$ meets the de Sitter sphere $\mathbf{S}^{2,1}$ orthogonally. Of course, one can allow arbitrary spacetime and boundary in the above definition. One interesting problem related to such surfaces is the uniqueness problem of a topological disk (cf. [FS15]). Without the free boundary condition, similar questions can be asked for two-spheres in $3+1$ dimensional de Sitter sphere (cf. [Che69]) . One can also ask whether a spacelike graph over $\mathbb{R}^{2}$ in $\mathbb{R}^{3,1}$ with $\langle\vec{H},L\rangle=0$ and $(DL)^{\bot}=0$ is linear which is analogous to the Bernstein problem for minimal graphs. We generalize the spacetime Alexandrov theorem to the free boundary settings via establishing a spacetime Heintz-Karcher inequality Theorem 4. We state here the theorem in the Minkowski spacetime. ###### Theorem 1. Let $\Sigma$ be a codimension two, future incoming null embedded submanifold in the $(3+1)$-dimensional Minkowski spacetime with free boundary on the de Sitter sphere $\mathbf{S}^{2,1}$. If $\Sigma$ lies in a half spacetime, and there exists a null vector field $\underline{L}$ such that along $\Sigma$ that $\langle\vec{H},\underline{L}\rangle$ is a positive constant and $(D\underline{L})^{\bot}=0$. Then $\Sigma$ lies in a shear free null hypersurface. The theorem is a direct corrollary from Theorem 4 and similar proofs as in [WWZ17, Theorem 3.14]. The article is organized as follows: In Section 2, we collect basics of spacetime of constant curvature and the conformal Killing-Yano two-forms they admit. In Section 3, we prove a spacetime Heintz-Karcher inequality with a free boundary leading to a free boundary, spacetime Alexandrov theorem. We mention briefly the generalization to higher order curvatures. Acknowledgements I would like to thank Xia Chao, Wang Ye-kai for their interest and advice in an earlier version of this work. I would also like to acknowledge the support of Korea Institute for Advanced Study under the research number MG074401. ## 2\. Conformal Killing-Yano 2-form on spacetime of constant curvature A spacetime of dimension $3+1$ can only admit 20 conformal Killing-Yano two- forms. Actually, if a spacetime admits all twenty of them, then the spacetime has to be a spacetime of constant curvature. Note that for similar statements are also true for conformal Killing vector fields. In Minkowski, de Sitter and anti-de Sitter spacetime, these two forms are found explicitly. See the works by Jezierski and Lukasik [JŁ06, Jez08]. Now we collect some basics of these spacetimes and the conformal Killing-Yano two forms that live on them. ### 2.1. Minkowski spacetime Let $(x^{0},x^{1},x^{2},x^{3})$ be the standard coordinates of the Minkowski space $\mathbb{R}^{3,1}$, define (3) $\displaystyle\mathcal{D}$ $\displaystyle=-x^{0}\mathrm{d}x^{0}+x^{1}\mathrm{d}x^{1}+x^{2}\mathrm{d}x^{2}+x^{3}\mathrm{d}x^{3},$ (4) $\displaystyle\mathcal{T}_{0}$ $\displaystyle=-\mathrm{d}x^{0},$ (5) $\displaystyle\mathcal{T}_{i}$ $\displaystyle=\mathrm{d}x^{i},$ (6) $\displaystyle\mathcal{L}_{0i}$ $\displaystyle=-x^{0}\mathrm{d}x^{i}+x^{i}\mathrm{d}x^{0}.$ The conformal Killing-Yano 2-forms on Minkowski spacetime $\mathbb{R}^{3,1}$ are (7) $\mathcal{T}_{\mu}\wedge\mathcal{T}_{\nu},\mathcal{D}\wedge\mathcal{T}_{\mu},\ast(\mathcal{D}\wedge\mathcal{T}_{\mu})\text{ and }\mathcal{D}\wedge\mathcal{L}_{\mu\nu}+\frac{1}{2}\langle\mathcal{D},\mathcal{D}\rangle\mathcal{T}_{\mu}\wedge\mathcal{T}_{\nu},$ where $\ast$ is the Hodge star operator and $\mu,\nu$ range from 0 to 3. See [JŁ06] for a calculation. Note that all are still conformal Killing-Yano 2-forms on $\mathbb{R}^{n,1}$ except $\ast(\mathcal{D}\wedge\mathcal{T}_{\mu})$. We remark that the last one in (7) can be used to prove formulas relating the center of mass (See [MT16]) and a Brown-York type quasi-local quantity by following similar procedures in [CWY19]. ### 2.2. Anti-de Sitter spacetime We recall some basics of four-dimensional anti-de Sitter spacetime. The anti- de Sitter spacetime ad$\mathbf{S}^{3,1}$ is defined to be the set in $\mathbb{R}^{3,2}$ (8) $-(y^{0})^{2}+(y^{1})^{2}+(y^{2})^{2}+(y^{3})^{2}-(y^{4})^{2}=-1$ with metric induced from $\eta=-(\mathrm{d}y^{0})^{2}+(\mathrm{d}y^{1})^{2}+(\mathrm{d}y^{2})^{2}+(\mathrm{d}y^{3})^{2}-(\mathrm{d}y^{4})^{2}$. We will use coordinates of the Poincaré ball model by setting $r=\sqrt{(x^{1})^{2}+(x^{2})^{2}+(x^{3})^{2}}$, $y^{0}=\tfrac{1+r^{2}}{1-r^{2}}\cos t$, $y^{4}=\tfrac{1+r^{2}}{1-r^{2}}\sin t$ and $y^{i}=\tfrac{2x^{i}}{1-r^{2}}$. The metric of ad$\mathbf{S}^{3,1}$ is then $-(\tfrac{1+r^{2}}{1-r^{2}})^{2}\mathrm{d}t^{2}+\tfrac{4\sum_{i}(\mathrm{d}x^{i})^{2}}{(1-r^{2})^{2}}$. It is shown in [Jez08] that the conformal Killing-Yano 2-forms in four- dimensional anti-de Sitter spacetime are (9) $\mathrm{d}y^{0}\wedge\mathrm{d}y^{i},\mathrm{d}y^{0}\wedge\mathrm{d}y^{4},\mathrm{d}y^{i}\wedge\mathrm{d}y^{4},\mathrm{d}y^{i}\wedge\mathrm{d}y^{j}$ and their Hodge duals with respect to the anti-de Sitter metric. We fix the frame $\theta^{i}=\tfrac{2}{1-r^{2}}\mathrm{d}x^{i}$ and $\theta^{0}=\tfrac{1+r^{2}}{1-r^{2}}\mathrm{d}t$. Let $\omega=\tfrac{2}{1-r^{2}}\mathrm{d}r$, then the length of $\omega$ is one. We have (10) $\mathrm{d}y^{i}=\theta^{i}+y^{i}r\omega,\mathrm{d}y^{4}=\cos t\theta^{0}+\tfrac{2r}{1-r^{2}}\omega\sin t.$ Note that $y^{4}$ and $y^{i}$ are static potentials, that is $\nabla_{i}\mathrm{d}y^{\mu}=y^{\mu}\theta^{i}$ and $\nabla_{0}\mathrm{d}y^{\mu}=-y^{\mu}\theta^{0}$ for each $\mu=0,1,\ldots,4$. Here $\nabla_{\mu}$ denotes covariant derivative with respect to the vector field $(\theta^{\mu})^{\sharp}$. Then it is easy to obtain that (11) $\operatorname{div}(\mathrm{d}y^{i}\wedge\mathrm{d}y^{4})=3(y^{i}\mathrm{d}y^{4}-y^{4}\mathrm{d}y^{i}).$ Note that $y^{i}\mathrm{d}y^{4}-y^{4}\mathrm{d}y^{i}$ is a Killing 1-form. Using the properties of Hodge operators, we find that $\operatorname{div}(\ast(\mathrm{d}y^{2}\wedge\mathrm{d}y^{3}))$ vanishes. We remark that the 2-form $\mathrm{d}y^{i}\wedge\mathrm{d}y^{4}$ can be used similarly as in [CWY15] to recover a formula relating the integrals of Ricci tensor and Brown-York type mass vector of an asymptotically hyperbolic manifold. These formulas are overlooked by the authors of [CWY15]. The original proof is due to [MTX17]. ### 2.3. de Sitter spacetime The case with de Sitter spacetime is similar to the anti-de Sitter case (See [Jez08]). We consider here the $3+1$ dimensional case i.e. $\mathbf{S}^{3,1}$. The de Sitter spacetime is the subset (12) $y_{0}^{2}+y_{1}^{2}+y_{2}^{2}+y_{3}^{2}-y_{4}^{2}=1$ in $\mathbb{R}^{4,1}$ with the metric inherited from the standard Lorentz metric of $\mathbb{R}^{4,1}$. We use the coordinate change (13) $\displaystyle y^{0}$ $\displaystyle=\tfrac{1-r^{2}}{1+r^{2}}\cosh t,$ (14) $\displaystyle y^{i}$ $\displaystyle=\tfrac{2x^{i}}{1+r^{2}}$ (15) $\displaystyle y^{4}$ $\displaystyle=\tfrac{1-r^{2}}{1+r^{2}}\sinh t,$ where $r=\sqrt{\sum_{i=1}^{3}(x^{i})^{2}}<1$. Now the metric of the de Sitter spacetime $\mathbf{S}^{3,1}$ takes the form (16) $\eta=-(\tfrac{1-r^{2}}{1+r^{2}})^{2}\mathrm{d}t^{2}+\tfrac{4}{(1+r^{2})^{2}}[(\mathrm{d}x^{1})^{2}+(\mathrm{d}x^{2})^{2}+(\mathrm{d}x^{3})^{2}].$ It is shown in [Jez08] that the conformal Killing-Yano 2-forms in four- dimensional de Sitter spacetime are (17) $\mathrm{d}y^{0}\wedge\mathrm{d}y^{i},\mathrm{d}y^{0}\wedge\mathrm{d}y^{4},\mathrm{d}y^{i}\wedge\mathrm{d}y^{4},\mathrm{d}y^{i}\wedge\mathrm{d}y^{j}$ and their Hodge duals with respect to the de Sitter metric. We fix the frame $\theta^{i}=\tfrac{2}{1+r^{2}}\mathrm{d}x^{i}$ and $\theta^{0}=\tfrac{1-r^{2}}{1+r^{2}}\mathrm{d}t$. Let $\omega=\tfrac{2}{1+r^{2}}\mathrm{d}r$, then the length of $\omega$ is one. We have (18) $\mathrm{d}y^{i}=\theta^{i}-y^{i}r\omega,\mathrm{d}y^{4}=\cosh t\theta^{0}-\tfrac{2r}{1+r^{2}}\omega\sinh t.$ Note that $y^{4}$ and $y^{i}$ are static potentials, that is $\nabla_{i}\mathrm{d}y^{\mu}=-y^{\mu}\theta^{i}$ and $\nabla_{0}\mathrm{d}y^{\mu}=y^{\mu}\theta^{0}$ for each $\mu=0,1,\ldots,4$. Here $\nabla_{\mu}$ denotes covariant derivative with respect to the vector field $(\theta^{\mu})^{\sharp}$. Then it is easy to obtain that (19) $\operatorname{div}(\mathrm{d}y^{i}\wedge\mathrm{d}y^{4})=-3(y^{i}\mathrm{d}y^{4}-y^{4}\mathrm{d}y^{i}).$ Note that $y^{i}\mathrm{d}y^{4}-y^{4}\mathrm{d}y^{i}$ is a Killing 1-form. We found also easily that $\operatorname{div}(\ast(\mathrm{d}y^{2}\wedge\mathrm{d}y^{3}))$ vanishes. ## 3\. Spacetime Alexandrov theorem with free boundary We start by proving a Minkowski formula for a codimension two spacelike hypersurface in $\mathbb{R}^{3,1}$ with boundary meeting orthogonally with the de Sitter sphere. The result is related to mean curvature only, the generalization to higher order curvatures is quite straightforward. The Minkowski spacetime is used as a prototype. First, we fix a conformal Killing-Yano 2-form in Minkowski spacetime $\mathbb{R}^{3,1}$ (20) $Q=\mathcal{D}\wedge\mathcal{L}_{0i}+\frac{1}{2}[1+\langle\mathcal{D},\mathcal{D}\rangle]e^{0}\wedge e^{i}.$ The associated 1-form is $\xi:=\tfrac{1}{n}\operatorname{div}Q=\mathcal{L}_{0i}$ since ###### Lemma 1. The divergence of the 2-form $Q=\mathcal{D}\wedge\mathcal{L}_{0i}+\tfrac{1}{2}\langle\mathcal{D},\mathcal{D}\rangle\mathrm{d}x^{0}\wedge\mathrm{d}x^{i}$ is given by $\operatorname{div}Q=3\mathcal{L}_{0i}$. Define the Minkowski unit ball (21) $\mathcal{B}^{3,1}=\\{x\in\mathbb{R}^{3,1}:\langle x,x\rangle\leqslant 1\\}.$ The boundary of $\mathcal{B}^{3,1}$ is the de Sitter sphere $\mathbf{S}^{2,1}$. It is easy to check that $\mathcal{D}^{\sharp}\lrcorner Q$ is zero along $\partial\mathcal{B}^{3,1}$, so $Q$ has no components normal to $\mathbf{S}^{2,1}$. ###### Theorem 2. Let $\Sigma$ be an immersed oriented spacelike codimension two submanifolds of the Minkowski spacetime $\mathbb{R}^{3,1}$, $\partial\Sigma$ lies in the de Sitter sphere $\mathbf{S}^{2,1}$ and $\Sigma$ meets $\mathbf{S}^{2,1}$ orthogonally. For any null vector field $\underline{L}$ of $\Sigma$, we have (22) $\int_{\Sigma}[(n-1)\langle\xi,\underline{L}\rangle+Q(\vec{H},\underline{L})+Q(\partial_{a},(D^{a}\underline{L})^{\bot})]\mathrm{d}\mu=0.$ ###### Proof. Define $\mathcal{Q}=Q(\partial_{a},\underline{L})\mathrm{d}u^{a}$ on $\Sigma$ and the proof is almost the same as Theorem 2.2 of [WWZ17]. We include their proof for convenience. Let $\underline{\chi}=\langle D_{a}\underline{L},\partial_{b}\rangle$. Consider the 1-form $\mathcal{Q}=Q(\partial_{a},\underline{L})\mathrm{d}u^{a}$, we have (23) $\displaystyle\operatorname{div}\mathcal{Q}$ $\displaystyle=\nabla_{a}\mathcal{Q}^{a}-Q(\nabla_{a}\partial_{a},\underline{L})$ (24) $\displaystyle=(D^{a}Q)(\partial_{a},\underline{L})+Q(\vec{H},\underline{L})+Q(\partial_{a},D^{a}\underline{L})$ (25) $\displaystyle=(n-1)\langle\xi,\underline{L}\rangle+Q(\vec{H},\underline{L})+\underline{\chi}_{ab}Q^{ab}+Q(\partial_{a},(D^{a}\underline{L})^{\bot})$ (26) $\displaystyle=(n-1)\langle\xi,\underline{L}\rangle+Q(\vec{H},\underline{L})+Q(\partial_{a},(D^{a}\underline{L})^{\bot}).$ Integration by parts and noting that $Q$ has no components normal to the de Sitter sphere. ∎ ### 3.1. A monotonicity formula Let $\Sigma$ be a spacelike submanifold of codimension two in a spacetime $(\mathcal{S}^{3,1},g)$ which admits a Killing-Yano two form $Q$. Here, $\mathcal{S}$ is either one of the four dimensional Minkowski, de Sitter and anti de Sitter spacetime. We require that $Q$ has no normal component normal to a support hypersurface $S$. Suppose that $\langle\vec{H},\underline{L}\rangle\neq 0$, we define the following functional (27) $\mathcal{F}(\Sigma,[\underline{L}])=(n-1)\int_{\Sigma}\frac{\langle\xi,\underline{L}\rangle}{\langle\vec{H},\underline{L}\rangle}\mathrm{d}\mu-\frac{1}{2}\int_{\Sigma}Q(L,\underline{L})\mathrm{d}\mu.$ Note $\mathcal{F}$ is invariant under the change $L\to aL$ and $\underline{L}\to\frac{1}{a}\underline{L}$. Let $\chi$ and $\underline{\chi}$ be respectively the second fundamental form with respect to $L$, $\underline{L}$; let $\underline{C}_{0}$ denote the future incoming null hypersurface of $\Sigma$. $\underline{C}_{0}$ is obtained by taking the collection of all null geodesics emanating from $\Sigma$ with initial velocity $\underline{L}$. We then extend it to a future directed null vector field along $\underline{C}_{0}$. Consider the evolution of $\Sigma$ along $\underline{C}_{0}$ by a family of immersions $F:\Sigma\times[0,T)\to\underline{C}_{0}$ satisfying (28) $\left\\{\begin{array}[]{l}\frac{\partial F}{\partial s}(x,s)=\varphi(x,s)\underline{L},\\\ F(x,0)=F_{0}(x),\\\ \Sigma\bot S\end{array}\right.$ for some positive function $\varphi(x,s)$. We have the following monotonicity property of the flow $\varphi$. ###### Theorem 3. Suppose that $\langle\vec{H},\underline{L}\rangle>0$ for some null vector field $\underline{L}$. Then $\mathcal{F}(F(\Sigma,s),[\underline{L}])$ is monotone decreasing under the flow. ###### Proof. See Theorem 3.2 of [WWZ17]. We only have to use the extra fact that $Q$ has no components normal to the de Sitter space as in the proof of Theorem 2. ∎ The monotonicity property leads to a spacetime Heintz-Karcher inequality. More, specifically, if under certain flow $\varphi$, the surface $\Sigma$ with $\langle\vec{H},\underline{L}\rangle>0$ flows into a submanifold of the time slice $\\{x^{0}=0\\}$ at $s=T$ and for $\Sigma$ (29) $\mathcal{F}(\Sigma,[L])\geqslant 0$ holds provided $\varphi(\Sigma,T)\subset\\{x^{0}=0\\}$ and $\mathcal{F}(\varphi(\Sigma,T),[L])\geqslant 0$. ###### Lemma 2. For any $\Sigma\subset\\{x^{0}=0\\}$, $\mathcal{F}(\Sigma,[L])\geqslant 0$ reduces to (30) $(n-1)\int_{\Sigma}\tfrac{x^{i}}{H}\mathrm{d}\mu\geqslant\int_{\Sigma}\langle X_{\partial_{i}},\nu\rangle\mathrm{d}\mu.$ ###### Proof. We have $\underline{L}=\partial_{t}-e_{n}$ where $e_{n}$ is a unit normal. So $\langle\vec{H},\underline{L}\rangle=H$ where $H$ is the mean curvature of $\Sigma$ in $\mathbf{B}^{n}$. We have that $\xi=\mathcal{L}_{0i}=x^{i}\mathrm{d}x^{0}$, so $\langle\xi,\underline{L}\rangle=x^{i}$. Also, (31) $Q(L,\underline{L})=Q(\partial_{t}+\nu,\partial_{t}-\nu)=2\langle X_{\partial_{i}},\nu\rangle,$ where $X_{a}=\langle X,a\rangle X+\tfrac{1}{2}(\|X\|^{2}+1)a$ where $a=a^{i}\partial_{i}$ is a constant vector in $\mathbb{R}^{n}$. It easily leads to (30). ∎ Note that this is precisely an inequality proven already by Wang-Xia [WX19, (5.5)] with the assumption that $\Sigma$ has positive mean curvature and lies in a half ball. Combining with their result, we have ###### Theorem 4 (spacetime Heintz-Karcher inequality). If there exists a flow $\varphi$ of a hypersurface $\Sigma$ with $\langle\vec{H},\underline{L}\rangle>0$ for some null vector field $\underline{L}$ and a free boundary on $\mathbf{S}^{2,1}$ which flows $\Sigma$ into the half unit ball of the slice $\\{x^{0}=0\\}$, then we have the inequality (32) $\int_{\Sigma}\frac{\langle\xi,\underline{L}\rangle}{\langle\vec{H},\underline{L}\rangle}\mathrm{d}\mu\geqslant\frac{1}{2(n-1)}\int_{\Sigma}Q(L,\underline{L})\mathrm{d}\mu.$ Equality occurs if and only if $\Sigma$ lies in a shear free null hypersurface with free boundary on $\mathbf{S}^{n-1,1}$. ###### Proof. Let $\Sigma_{t}=\varphi_{t}(\Sigma)$, then for each $t>0$, the equality holds. Suppose that $\Sigma_{T}\subset\\{x^{0}=0\\}$ for some $T>0$. So $\Sigma_{T}$ has to be a spherical cap orthogonal to the unit sphere in $\mathbb{R}^{n}$ according to [WX19]. In particular, under the flow $\varphi$, $\Sigma_{t}$ foliates a shear free null hypersurface $S$ with free boundary. ∎ ### 3.2. Anti-de Sitter case Theorems 2, 4 and 1 work as well in the case with $\partial\Sigma=\emptyset$. The same proof also adapts in the anti-de Sitter and de Sitter settings. We use the notations in Section 2.2. For simplicity, we set $i$ to be 1, we use the 2-forms $\mathrm{d}y^{1}\wedge\mathrm{d}y^{4}$ and $\ast(\mathrm{d}y^{2}\wedge\mathrm{d}y^{3})$ only. Note that the Hodge star operator commutes with the covariant derivative. Using this, we see easily that $\operatorname{div}(\ast(\mathrm{d}y^{2}\wedge\mathrm{d}y^{3}))$ vanishes. We use the 2-form $Q=\mathrm{d}y^{1}\wedge\mathrm{d}y^{4}+l\ast(\mathrm{d}y^{2}\wedge\mathrm{d}y^{3})$ where $l>0$ is a positive constant. We define the surface $\mathcal{B}^{3,1}$ to be the surface with distance less than $d$ from the point $t=0$, $r=0$ where $\cosh d=l$. If $Y_{1},Y_{2}\in\operatorname{ad}\mathbf{S}^{3,1}$ (using the embedding into $\mathbb{R}^{3,2}$) are two points which can be connected via a spacelike geodesic, then the distance from $Y_{1}$ to $Y_{2}$ is $\cosh d=-\eta(Y_{1},Y_{2})>0$. The boundary $S=\partial\mathcal{B}^{3,1}$ is a timelike hypersurface of dimension three of constant distance from the point $t=0$, $r=0$ and it is umbilical hence null geodesics intrinsic to $S$ are also null geodesic in ad$\mathbf{S}^{3,1}$. It is the analog of de Sitter sphere which is of constant distance to the origin in Minkowski spacetime. It is a tedious task to check that $Q$ has no component normal to $S$. We state here the spacetime Heintz-Karcher inequality and leave the spacetime Alexandrov theorem to the reader. ###### Theorem 5. (spacetime Heintz-Karcher inequality in $\mathcal{B}^{3,1}$) If there exists a flow $\varphi$ of a hypersurface $\Sigma$ with $\langle\vec{H},\underline{L}\rangle>0$ for some null vector field $\underline{L}$ and a free boundary on $S$ which flows $\Sigma$ into the half geodesic ball of the slice $\\{t=0\\}$, then we have the inequality (33) $\int_{\Sigma}\frac{\langle\xi,\underline{L}\rangle}{\langle\vec{H},\underline{L}\rangle}\mathrm{d}\mu\geqslant\frac{n}{2(n-1)}\int_{\Sigma}Q(L,\underline{L})\mathrm{d}\mu.$ Equality occurs if and only if $\Sigma$ lies in a shear free null hypersurface. ###### Proof. The proof is the same with Theorem 4. We only have to verify when $t=0$ the inequality holds. Let $\nu$ be the unit normal of $\Sigma$ in the $\\{t=0\\}$ slice. Indeed, when $t=0$, $\xi=y^{1}\mathrm{d}y^{4}$ and $\underline{L}=e_{0}-\nu=\tfrac{1-r^{2}}{1+r^{2}}\partial_{t}-\nu,$ so $\langle\xi,\underline{L}\rangle=y^{1}=\tfrac{2x^{1}}{1-r^{2}}.$ We turn to $Q(L,\underline{L})$. We have $(\mathrm{d}y^{1}\wedge\mathrm{d}y^{4})(L,\underline{L})=2\mathrm{d}y^{1}(\nu)$ and $(\mathrm{d}y^{1})^{\sharp}=\tfrac{1}{2}\partial_{1}+(x^{1}x^{j}\partial_{j}-\tfrac{1}{2}r^{2}\partial_{1}).$ And $\ast(\mathrm{d}y^{2}\wedge\mathrm{d}y^{3})=-\theta^{1}\wedge\theta^{0}+y^{2}(x^{1}\theta^{2}-x^{2}\theta^{1})\wedge\theta^{0}+y^{3}(x^{1}\theta^{3}-x^{3}\theta^{1})\wedge\theta^{0},$ so the 1-form $(\ast(\mathrm{d}y^{2}\wedge\mathrm{d}y^{3}))(\cdot,e_{0})$ is dual to $-\tfrac{1}{2}\partial_{1}+(x^{1}x^{j}\partial_{j}-\tfrac{1}{2}r^{2}\partial_{1})$. As usual, $Q(L,\underline{L})=2Q(\nu,e_{0})$. Thus, (34) $Q(L,\underline{L})=2\langle X_{\partial_{1}},\nu\rangle,$ where $X_{a}=(1+l)\left[x^{k}a_{k}x^{j}\partial_{j}-\tfrac{1}{2}(r^{2}+\tfrac{l-1}{l+1})a\right]$ with $a=a^{j}\partial_{j}$ being a constant vector in $\mathbb{R}^{n}$. Letting $l=\tfrac{1+R_{\mathbb{R}}^{2}}{1-R_{\mathbb{R}}^{2}}$, (33) reduces to also [WX19]. ∎ ###### Remark 1. It is easy to check that the higher dimensional analog of $\ast(\mathrm{d}y^{2}\wedge\mathrm{d}y^{3})$ in the $n$-dimensional anti-de Sitter spacetime $\operatorname{ad}\mathbf{S}^{n}=\\{-(y^{0})^{2}+(y^{1})^{2}+\cdots+(y^{n})^{2}-(y^{n+1})^{2}=1\\}$ is (35) $-e^{1}\wedge e^{0}+\sum_{i\neq 1}^{n}y^{i}(x^{1}e^{i}-x^{i}e^{1})\wedge e^{0}.$ ### 3.3. de Sitter case We calculate below the quantities needed for a theorem parallel to Theorem 33. We follow similar notations and omit the the statements or details. Generalizing to higher dimension is also straightforward. The conformal Killing-Yano 2-form is $Q=\mathrm{d}y^{4}\wedge\mathrm{d}y^{1}+l\ast(\mathrm{d}y^{3}\wedge\mathrm{d}y^{2})$ and its associated 1-form $\xi=\operatorname{div}Q=3y^{1}\mathrm{d}y^{4}-y^{4}\mathrm{d}y^{1}.$ Notice the order of the superscripts. Within the slice $\\{t=0\\}$, we have that $\xi=y^{1}\mathrm{d}y^{4}$ and $\underline{L}=e_{0}-\nu=\tfrac{1+r^{2}}{1-r^{2}}\partial_{t}-\nu$ and $\langle\xi,\underline{L}\rangle=\tfrac{2x^{1}}{1+r^{2}}.$ We turn to $Q(L,\underline{L})$. We have $(\mathrm{d}y^{4}\wedge\mathrm{d}y^{1})(L,\underline{L})=-2\mathrm{d}y^{1}(\nu).$ Note that $A:=-(\mathrm{d}y^{1})^{\sharp}=-\tfrac{1}{2}\partial_{1}-(\tfrac{1}{2}r^{2}\partial_{1}-x^{1}x^{j}\partial_{j})$ and $\ast(\mathrm{d}y^{3}\wedge\mathrm{d}y^{2})=\theta^{1}\wedge\theta^{0}+y^{2}(x^{1}\theta^{2}-x^{2}\theta^{1})\wedge\theta^{0}+y^{3}(x^{1}\theta^{3}-x^{3}\theta^{1})\wedge\theta^{0},$ so the 1-form $\ast(\mathrm{d}y^{3}\wedge\mathrm{d}y^{2})(\cdot,e_{0})$ is dual to $B:=\tfrac{1}{2}\partial_{1}-\tfrac{1}{2}r^{2}\partial_{1}+x^{1}x^{j}\partial_{j}.$ $A+lB$ is then (36) $X_{\partial_{1}}:=(1+l)\left[x^{1}x^{j}\partial_{j}+\tfrac{1}{2}(\tfrac{1-l}{l+1}-r^{2})\partial_{1}\right].$ Therefore $Q(L,\underline{L})=2\langle X_{\partial_{1}},\nu\rangle$. Setting $\tfrac{1-l}{1+l}=\|x\|^{2}$ with $0<l<1$ recovers the form of [WX19]. We have not given the support hypersurface of the boundary yet. To this end, we fix a point $O=\\{t=0,r=0\\}$, let $S$ be the hypersurface in $\mathbf{S}^{3,1}$ be a hypersurface of constant distance $d$ from the point $O$ where $\cos d=l$. It is fairly easy to check that $Q$ has no components to the hypersurface $S$. ## References * [Alm66] F. J. Almgren, Jr. Some interior regularity theorems for minimal surfaces and an extension of Bernstein’s theorem. Ann. of Math. (2), 84:277–292, 1966\. * [Cal67] Eugenio Calabi. Minimal immersions of surfaces in Euclidean spheres. J. Differential Geometry, 1:111–125, 1967. * [Che69] Shiing-shen Chern. Simple proofs of two theorems on minimal surfaces. Enseign. Math. (2), 15:53–61, 1969. * [CWY15] Po-Ning Chen, Mu-Tao Wang, and Shing-Tung Yau. Conserved Quantities in General Relativity: From the Quasi-Local Level to Spatial Infinity. Communications in Mathematical Physics, 338(1):31–80, 2015. * [CWY19] Po-Ning Chen, Mu-Tao Wang, and Shing-Tung Yau. The Minkowski formula and the quasi-local mass. Ann. 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# Martingale convergence Theorems for Tensor Splines Markus Passenbrunner Institute of Analysis, Johannes Kepler University Linz, Austria, 4040 Linz, Altenberger Strasse 69<EMAIL_ADDRESS> ###### Abstract. In this article we prove martingale type pointwise convergence theorems pertaining to tensor product splines defined on $d$-dimensional Euclidean space ($d$ is a positive integer), where conditional expectations are replaced by their corresponding tensor spline orthoprojectors. Versions of Doob’s maximal inequality, the martingale convergence theorem and the characterization of the Radon-Nikodým property of Banach spaces $X$ in terms of pointwise $X$-valued martingale convergence are obtained in this setting. Those assertions are in full analogy to their martingale counterparts and hold independently of filtration, spline degree, and dimension $d$. ###### Key words and phrases: Tensor product spline orthoprojectors, Almost everywhere convergence, Maximal functions, Radon-Nikodým property, Martingale methods ###### 2010 Mathematics Subject Classification: 41A15, 42B25, 46B22, 42C10, 60G48 ## 1\. Introduction In this article we prove pointwise convergence theorems pertaining to tensor product splines defined on $d$-dimensional Euclidean space in the spirit of the known results for martingales. We begin by discussing the situation for martingales and, subsequently, for one-dimensional splines. For martingales, we use [6] and [2] as references. Let $(\Omega,(\mathscr{F}_{n}),\mathbb{P})$ be a filtered probability space. A sequence of integrable functions $(f_{n})_{n\geq 1}$ is a _martingale_ if $\mathbb{E}(f_{n+1}\|\mathscr{F}_{n})=f_{n}$ for any $n$, where we denote by $\mathbb{E}(\cdot\|\mathscr{F}_{n})$ the conditional expectation operator with respect to the $\sigma$-algebra $\mathscr{F}_{n}$. This operator is the orthoprojector onto the space of $\mathscr{F}_{n}$-measurable $L^{2}$-functions and it can be extended to act on the Lebesgue-Bochner space $L^{1}_{X}$ for any Banach space $(X,\\|\cdot\\|)$. Observe that if $f\in L^{1}_{X}$, the sequence $(\mathbb{E}(f\|\mathscr{F}_{n}))$ is a martingale. In this case, we have that $\mathbb{E}(f\|\mathscr{F}_{n})$ converges almost surely to $\mathbb{E}(f\|\mathscr{F})$ with $\mathscr{F}=\sigma(\cup_{n}\mathscr{F}_{n})$. A crucial step in the proof of this convergence theorem is _Doob’s maximal inequality_ $\mathbb{P}\\{\sup_{n}\\|f_{n}\\|>t\\}\leq\frac{\sup_{n}\\|f_{n}\\|_{L^{1}_{X}}}{t},\qquad t>0,$ which states that the martingale maximal function $\sup\\|f_{n}\\|$ is of weak type $(1,1)$. For general scalar-valued martingales, we have the following convergence theorem: any martingale $(f_{n})$ that is bounded in $L^{1}$ has an almost sure limit function contained in $L^{1}$. This limit can be identified as the Radon-Nikodým derivative of the $\mathbb{P}$-absolutely continuous part of the measure $\nu$ defined by (1.1) $\nu(A)=\lim_{m}\int_{A}f_{m}\,\mathrm{d}\mathbb{P},\qquad A\in\cup_{n}\mathscr{F}_{n}.$ This limit exists because of the martingale property of $(f_{n})$. The same convergence theorem as above holds true for $L^{1}_{X}$-bounded $X$-valued martingales $(f_{n})$, provided there exists a Radon-Nikodým derivative of the $\mathbb{P}$-absolutely continuous part of the now $X$-valued measure $\nu$ in (1.1). Banach spaces $X$ where this is always possible are said to have the _Radon-Nikodým property_ (RNP) (see Definition 2.3). The RNP of a Banach space is even characterized by martingale convergence meaning that in any Banach space $X$ without RNP, we can find a non-convergent and $L^{1}_{X}$-bounded martingale. Consider now the special case where each $\sigma$-algebra $\mathscr{F}_{n}$ is generated by a partition of a bounded interval $I\subset\mathbb{R}$ into finitely many intervals $(I_{n,i})_{i}$ of positive length as atoms of $\mathscr{F}_{n}$. In this case, $(\mathscr{F}_{n})$ is called an _interval filtration_ on $I$. Then, the characteristic functions $(\mathbbm{1}_{I_{n,i}})$ of those atoms are a sharply localized orthogonal basis of $L^{2}(\mathscr{F}_{n})$ w.r.t. Lebesgue measure $\lambda=\|\cdot\|$. If we want to preserve the localization property of the basis functions, but at the same time consider spaces of functions with higher smoothness, a natural candidate are spaces of piecewise polynomial functions of order $k$, given by $\displaystyle S^{k}(\mathscr{F}_{n})=\\{f:I\to\mathbb{R}\ \|\ $ $f$ is $k-2$ times continuously differentiable and $\displaystyle\qquad\text{a polynomial of order $k$ on each atom of $\mathscr{F}_{n}$}\\},$ where $k$ is an arbitrary positive integer. One reason for this is that $S^{k}(\mathscr{F}_{n})$ admits a special basis, the so called B-spline basis $(N_{n,i})_{i}$, that consists of non-negative and localized functions $N_{n,i}$. Here, the term “localized” means that the support of each function $N_{n,i}$ consists of at most $k$ neighbouring atoms of $\mathscr{F}_{n}$. A second reason is that if $(\mathscr{F}_{n})$ is an increasing sequence of interval $\sigma$-algebras, then the sequence of corresponding spline spaces $S^{k}(\mathscr{F}_{n})$ is increasing as well. Note that the properties of the B-spline functions $(N_{n,i})$ imply that they do not form an orthogonal basis of $S^{k}(\mathscr{F}_{n})$ for $k\geq 2$. For more information on spline functions, see e.g. [11]. Let $P_{n}^{k}$ be the orthogonal projection operator onto $S^{k}(\mathscr{F}_{n})$ with respect to the $L^{2}$ inner product on $I$ equipped with the Lebesgue measure. Since the space $S^{1}(\mathscr{F}_{n})$ consists of piecewise constant functions, $P_{n}^{1}$ is the conditional expectation operator with respect to the $\sigma$-algebra $\mathscr{F}_{n}$ and the Lebesgue measure. In general, the operator $P_{n}^{k}$ can be written in terms of the B-spline basis $(N_{n,i})$ as (1.2) $P_{n}^{k}f=\sum_{i}\int_{I}fN_{n,i}\,\mathrm{d}\lambda\cdot N_{n,i}^{},$ where the functions $(N_{n,i}^{})$, contained in the spline space $S^{k}(\mathscr{F}_{n})$, are the biorthogonal (or dual) system to the B-spline basis $(N_{n,i})$. Due to the uniform boundedness of the B-spline functions $N_{n,i}$, we are able to insert functions $f$ in formula (1.2) that are contained not only in $L^{2}$, but in the Lebesgue-Bochner space $L^{1}_{X}$, thereby extending the operator $P_{n}^{k}$ to $L^{1}_{X}$. Similarly to the definition of martingales, we adopt the following notion introduced in [7]: let $(f_{n})_{n\geq 1}$ be a sequence of functions in the space $L^{1}_{X}$. We call this sequence a _$k$ -martingale spline sequence_ (adapted to $(\mathscr{F}_{n}))$ if $P_{n}^{k}f_{n+1}=f_{n},\qquad n\geq 1.$ The local nature of the B-splines and the nestedness of the spaces $(S^{k}(\mathscr{F}_{n}))_{n}$ ultimately allow us to transfer the classical martingale theorems discussed above to $k$-martingale spline sequences adapted to _arbitrary_ interval filtrations ($\mathscr{F}_{n}$) and for any positive integer $k$, just by replacing conditional expectation operators with the spline projection operators $P_{n}^{k}$. Indeed, for any positive integer $k$, we have the following results. 1. (i) (Shadrin’s theorem) There exists a constant $C$ (depending only on $k$ and not on $(\mathscr{F}_{n})$) such that $\sup_{n}\\|P_{n}^{k}:L^{1}_{X}\to L^{1}_{X}\\|\leq C.$ 2. (ii) (Doob’s inequality for splines) There exists a constant $C$ such that for any $k$-martingale spline sequence $(f_{n})$, $\|\\{\sup_{n}\\|f_{n}\\|>t\\}\|\leq C\frac{\sup_{n}\\|f_{n}\\|_{L^{1}_{X}}}{t},\qquad t>0.$ 3. (iii) (Pointwise convergence of spline projections) For any Banach space $X$ and any $f\in L^{1}_{X}$, the sequence $P_{n}^{k}f$ converges almost everywhere to some $L^{1}_{X}$-function. 4. (iv) (RNP characterization by pointwise spline convergence) For any Banach space $X$, the following statements are equivalent: 1. (a) $X$ has RNP, 2. (b) every $k$-martingale spline sequence that is bounded in $L^{1}_{X}$ converges almost everywhere to an $L^{1}_{X}$-function. We give a few comments regarding the proofs of the statements (i)–(iv) above. Property (i), for arbitrary $k$, was proved by A. Shadrin in the groundbreaking paper [12]. We also refer to the article [3] by M. v. Golitschek, who gives a substantially shorter proof of (i). It should be noted that in the case $k=1$, due to Jensen’s inequality for conditional expectations, we can choose $C=1$ in (i). Property (ii) is proved in [9]. By a standard argument for passing from a weak type (1,1) inequality of a maximal function to a.e. convergence for $L^{1}_{X}$-functions, item (iii) is proved in [9] in the case that $\cup_{n}\mathscr{F}_{n}$ generates the Borel-$\sigma$-algebra on $I$ and in [5] in general. We also identify the limit of $P_{n}^{k}f$ as $P_{\infty}f$, where $P_{\infty}$ is (the $L^{1}_{X}$-extension of) the orthogonal projector onto the $L^{2}$-closure of $\cup_{n}S^{k}(\mathscr{F}_{n})$. The implication (a)$\implies$(b) in item (iv) is also proved in [5], whereas the reverse implication (b)$\implies$(a) is shown in [7] by constructing a non-convergent $k$-martingale spline sequence with values in Banach spaces $X$ without RNP for any positive integer $k$. In this article we are concerned with similar results pertaining to tensor product spline projections. Let $d$ be a positive integer and, for $j=1,\ldots,d$, let $(\mathscr{F}_{n}^{j})$ be an interval filtration on the interval $I\subset\mathbb{R}$. Filtrations $(\mathscr{F}_{n})$ of the form $\mathscr{F}_{n}=\mathscr{F}_{n}^{1}\otimes\cdots\otimes\mathscr{F}_{n}^{d}$ will be called an _interval filtration_ on the cube $I^{d}$. Then, the atoms of $\mathscr{F}_{n}$ are of the form $A_{1}\times\cdots\times A_{d}$ with atoms $A_{j}$ in $\mathscr{F}_{n}^{j}$. For a tuple $k=(k_{1},\ldots,k_{d})$ consisting of $d$ positive integers, denote by $P_{n}^{k}$ the orthogonal projector with respect to $d$-dimensional Lebesgue measure $\|\cdot\|=\lambda^{d}$ onto the tensor product spline space $S^{k_{1}}(\mathscr{F}_{n}^{1})\otimes\cdots\otimes S^{k_{d}}(\mathscr{F}_{n}^{d})$. The tensor product structure of $P_{n}^{k}$ immediately allows us to conclude (i) in this case, i.e., $P_{n}^{k}$ is bounded on $L^{1}_{X}(I^{d})$ by a constant depending only on $k$ (cf. also [8, Corollary 3.1]). Similarly to the one-dimensional case above, we then introduce the following notion: ###### Definition 1.1. Let $(\mathscr{F}_{n})$ be an interval filtration on a $d$-dimensional cube $I^{d}$. A sequence of functions $(f_{n})_{n\geq 1}$ in the space $L^{1}_{X}(I^{d})$ is a _$k$ -martingale spline sequence_ (adapted to ($\mathscr{F}_{n}$)) if $P_{n}^{k}f_{n+1}=f_{n},\qquad n\geq 1.$ The implication (b)$\implies$(a) in item (iv) for martingale spline sequences on $I^{d}$ can easily be deduced from its one-dimensional version as well. Indeed, for Banach spaces $X$ without RNP we get, for any positive integer $k_{1}$, a non-convergent $X$-valued $k_{1}$-martingale spline sequence $(f_{n}^{1})$ on $I$. Then, $f_{n}(x_{1},\ldots,x_{d})=f_{n}^{1}(x_{1})$ is a non-convergent $X$-valued $(k_{1},\ldots,k_{d})$-martingale spline sequence on $I^{d}$ for any choice of positive integers $k_{2},\ldots,k_{d}$. The main objective of this article is to prove the remaining assertions (ii), (iii) and the implication (a)$\implies$(b) in item (iv) for martingale spline sequences on $I^{d}$. The basic idea in the proof of (ii) for $d=1$ (see [9, Proposition 2.3]) is the pointwise bound (1.3) $\\|P_{n}^{k}f(x)\\|\leq C_{k}\mathscr{M}_{\rm HL}f(x)$ of $P_{n}^{k}$ by the _Hardy-Littlewood maximal function_ (1.4) $\mathscr{M}_{\rm HL}f(x)=\sup_{J\ni x}\frac{1}{\|J\|}\int_{J}\\|f(y)\\|\,\mathrm{d}y,$ where $\sup$ is taken over all intervals $J$ that contain the point $x$. This is enough to imply (ii) for $d=1$ as it is a well known fact that $\mathscr{M}_{\rm HL}$ itself satisfies the weak type (1,1) bound $\|\\{\mathscr{M}_{\rm HL}f>t\\}\|\leq\frac{3}{t}\\|f\\|_{L^{1}_{X}},\qquad t>0.$ In dimensions $d>1$, by using this ad-hoc approach (see [8, Proposition 3.3]) one would need the _strong maximal function_ $\mathscr{M}_{\rm S}f(x)$ on the right hand side of (1.3), where $\mathscr{M}_{\rm S}f(x)$ is defined by the same formula (1.4) as $\mathscr{M}_{\rm HL}f(x)$, but where $\sup$ is taken over all $d$-dimensional axis-parallel rectangles $J\subset I^{d}$ containing the point $x$. As a matter of fact, this is not enough to derive (ii), since the best possible weak type inequality for $\mathscr{M}_{\rm S}$ is true only in the Orlicz space $L(\log L)^{d-1}$ (see [1, 4, 10]), which is a strict subset of $L^{1}$. Here we show how to employ the martingale spline structure, especially nestedness of atoms, to avoid the usage of the strong maximal function $\mathscr{M}_{\rm S}$ altogether and replace it by an intrinsic maximal function that is (as we will show) of weak type $(1,1)$. This is crucial in the proof of the statements (ii), (iii), (iv) for any dimension $d$. Those statements are in full analogy to the martingale and one-dimensional spline results. The validity of (ii) and (iii) for martingale spline sequences on $I^{d}$ solves a problem stated in [8]. The organization of this article is as follows. In Section 2 we collect a few basic facts about vector measures needed in the sequel. In Section 3, we prove items (ii) and (iii) for martingale spline sequences on $I^{d}$ (Proposition 3.1 and Theorem 3.3 respectively). In Section 4, the implication (a)$\implies$(b) of item (iv) is proved in this case (Theorem 4.1) under the restriction that $\cup_{n}\mathscr{F}_{n}$ generates the Borel-$\sigma$-algebra on $I^{d}$. In Section 5, we show this assertion for general interval filtrations on $I^{d}$ and give an explicit formula for the pointwise limit of martingale spline sequences. ## 2\. Preliminaries We refer to the book [2] by J. Diestel and J.J. Uhl for basic facts on vector valued integration, martingales, vector measures and the results that follow. Let $\Omega$ be a set, $\mathscr{A}$ an algebra of subsets of $\Omega$ and $(X,\\|\cdot\\|)$ a Banach space. A function $\nu:\mathscr{A}\to X$ is a _(finitely additive) vector measure_ if, whenever $E_{1},E_{2}\in\mathscr{A}$ are disjoint, we have $\nu(E_{1}\cup E_{2})=\nu(E_{1})+\nu(E_{2})$. If, in addition, $\nu(\cup_{n=1}^{\infty}E_{n})=\sum_{n=1}^{\infty}\nu(E_{n})$ in the norm topology of $X$ for all sequences $(E_{n})$ of mutually disjoint members of $\mathscr{A}$ such that $\cup_{n=1}^{\infty}E_{n}\in\mathscr{A}$, then $\nu$ is a _countably additive vector measure_. The _variation_ $\|\nu\|$ of a finitely additive vector measure $\nu$ is the set function $\|\nu\|(E)=\sup_{\pi}\sum_{A\in\pi}\\|\nu(A)\\|,$ where the supremum is taken over all partitions $\pi$ of $E$ into a finite number of mutually disjoint members of $\mathscr{A}$. If $\nu$ is a finitely additive vector measure, then the variation $\|\nu\|$ is monotone and finitely additive. The measure $\nu$ is of _bounded variation_ if $\|\nu\|(\Omega)<\infty$. If $\mu:\mathscr{A}\to[0,\infty)$ is a finitely additive measure and $\nu:\mathscr{A}\to X$ is a finitely additive vector measure, $\nu$ is _$\mu$ -continuous_, if $\lim_{\mu(E)\to 0}\nu(E)=0$. If $\mu_{1},\mu_{2}:\mathscr{A}\to[0,\infty)$ are two finitely additive measures on $\mathscr{A}$, $\mu_{1}$ and $\mu_{2}$ are mutually _singular_ if for each $\varepsilon>0$ there exists a set $A\in\mathscr{A}$ so that $\mu_{1}(A^{c})+\mu_{2}(A)\leq\varepsilon.$ ###### Theorem 2.1 (Lebesgue decomposition of vector measures). Let $\mathscr{A}$ be an algebra of subsets of the set $\Omega$. Let $\nu:\mathscr{A}\to X$ be a finitely additive vector measure of bounded variation. Let $\mu:\mathscr{A}\to[0,\infty)$ be a finitely additive measure. Then there exist unique finitely additive vector measures of bounded variation $\nu_{c},\nu_{s}$ so that 1. (1) $\nu=\nu_{c}+\nu_{s}$, $\|\nu\|=\|\nu_{c}\|+\|\nu_{s}\|$, 2. (2) $\nu_{c}$ is $\mu$-continuous, 3. (3) $\|\nu_{s}\|$ and $\mu$ are mutually singular. This theorem can be found in [2, Theorem 9 on p. 31]. The following theorem is part of [2, Theorem 2 on p. 27] after using [2, Proposition 15 on p. 7]. ###### Theorem 2.2 (Extension theorem). Let $\mathscr{A}$ be an algebra of subsets of a set $\Omega$ and let $\mathscr{F}$ be the $\sigma$-algebra generated by $\mathscr{A}$. Let $\nu:\mathscr{A}\to X$ be a countably additive vector measure of bounded variation. Then, $\nu$ has a unique countably additive extension $\overline{\nu}:\mathscr{F}\to X$. ###### Definition 2.3 ([2, Definition 3, p. 61]). A Banach space $X$ admits the _Radon-Nikodým property (RNP)_ if for every measure space $(\Omega,\mathscr{F})$, for every positive, finite, countably additive measure $\mu$ on $(\Omega,\mathscr{F})$ and for every $\mu$-continuous, countably additive vector measure $\nu$ of bounded variation, there exists a function $f\in L^{1}_{X}(\Omega,\mathscr{F},\mu)$ such that $\nu(A)=\int_{A}f\,\mathrm{d}\mu,\qquad A\in\mathscr{F}.$ ## 3\. Maximal functions of Tensor spline projectors Let $d$ be a positive integer and let $(\mathscr{F}_{n})=(\mathscr{F}_{n}^{1}\otimes\cdots\otimes\mathscr{F}_{n}^{d})$ be an interval filtration on $I^{d}$ for some interval $I=(a,b]$ with $a<b$ and $a,b\in\mathbb{R}$. Each $\sigma$-algebra $\mathscr{F}_{n}$ is then generated by a finite, mutually disjoint family $\\{I_{n,i}:i\in\Lambda\\}$, $\Lambda\subset\mathbb{Z}^{d}$, of $d$-dimensional rectangles of the form $I_{n,i}=\prod_{\ell=1}^{d}(a_{\ell},b_{\ell}]$ for some $a\leq a_{\ell}<b_{\ell}\leq b$. We assume that $\Lambda$ is of the form $\Lambda^{1}\times\cdots\times\Lambda^{d}$ where for each $\ell=1,\ldots,d$, $\Lambda^{\ell}$ is a finite set of consecutive integers and the rectangles $I_{n,i}$ have the property that they are ordered in the same way as $\mathbb{R}^{d}$, i.e., if $i,j\in\Lambda$ with $i_{\ell}<j_{\ell}$ then the projection of $I_{n,i}$ onto the $\ell$th coordinate axis lies to the left of the projection of $I_{n,j}$ onto the $\ell$th coordinate axis. For $x\in I^{d}$, let $A_{n}(x)$ be the uniquely determined atom (rectangle) $A\in\mathscr{F}_{n}$ so that $x\in A$. For two atoms $A,B\in\mathscr{F}_{n}$, define $d_{n}(A,B):=\|i-j\|_{1}$ if $A=I_{n,i}$ and $B=I_{n,j}$ and where $\|w\|_{1}=\sum_{\ell=1}^{d}\|w_{\ell}\|$ denotes the $\ell^{1}$ norm of the vector $w$. If $U=\cup_{\ell}A_{\ell}$ and $V=\cup_{\ell}B_{\ell}$ are (finite) unions of atoms in $\mathscr{F}_{n}$, we set $d_{n}(U,V)=\min_{\ell,m}d_{n}(A_{\ell},B_{m})$. Additionally, for a non- negative integer $s$, define $A_{n,s}(x)$ to be the union of all atoms $A$ in $\mathscr{F}_{n}$ with $d_{n}(A,A_{n}(x))\leq s$. Moreover, for a Borel set $B\subset I^{d}$, let $A_{n,s}(B)=\cup_{x\in B}A_{n,s}(x)$. For each $\ell=1,\ldots,d$, let $k_{\ell}$ be a positive integer. Define the tensor product spline space of order $k=(k_{1},\ldots,k_{d})$ associated to $\mathscr{F}_{n}$ as $S_{n}:=S^{k_{1}}(\mathscr{F}_{n}^{1})\otimes\cdots\otimes S^{k_{d}}(\mathscr{F}_{n}^{d}).$ The space $S_{n}$ admits the tensor product B-spline basis $(N_{n,i})_{i}$ defined by $N_{n,i}=N_{n,i_{1}}^{1}\otimes\cdots\otimes N_{n,i_{d}}^{d},$ where $(N_{n,i_{\ell}}^{\ell})_{i_{\ell}}$ denotes the B-spline basis of $S^{k_{\ell}}(\mathscr{F}_{n}^{\ell})$ that forms a partition of unity. The support $E_{n,i}=\operatorname{supp}N_{n,i}$ of $N_{n,i}$ is composed of at most $k_{1}\cdots k_{d}$ neighouring atoms of $\mathscr{F}_{n}$. Consider the orthogonal projection operator $P_{n}=P_{n}^{k}$ onto $S_{n}$ with respect to $d$-dimensional Lebesgue measure $\|\cdot\|=\lambda^{d}$. Using the B-spline basis and its biorthogonal system $(N_{n,i}^{})$, the orthogonal projector $P_{n}$ is given by (3.1) $P_{n}f=\sum_{i}\int_{I^{d}}fN_{n,i}\,\mathrm{d}\lambda^{d}\cdot N_{n,i}^{},\qquad f\in L^{1}_{X}(I^{d}).$ The dual B-spline functions $N_{n,i}^{}$ admit the following crucial geometric decay estimate (3.2) $\|N_{n,i}^{}(x)\|\leq C\frac{q^{d_{n}(E_{n,i},A_{n}(x))}}{\|\operatorname{re}(E_{n,i}\cup A_{n}(x))\|},\qquad x\in I^{d},$ for some constants $C$ and $q\in[0,1)$ that depend only on $k$, where $\operatorname{re}(S)$ denotes the smallest, axis-parallel rectangle containing the set $S$. This inequality was shown in [9, Theorem 1.2] for $d=1$ and if $d>1$, (3.2) is a consequence of the fact that $N_{n,i}^{}$ is the tensor product of one-dimensional dual B-spline functions. Inserting this estimate in formula (3.1) for $P_{n}f$ and as $E_{n,i}$ consists of at most $k_{1}\cdots k_{d}$ neighbouring atoms of $\mathscr{F}_{n}$, setting $C_{k}:=C(k_{1}\cdots k_{d})q^{-\|k\|_{1}}$, we get the pointwise estimate (3.3) $\\|P_{n}f(x)\\|\leq C_{k}\sum_{A\text{ atom of }\mathscr{F}_{n}}b_{n}(q,\\|f\\|\,\mathrm{d}\lambda^{d},A,x),\qquad f\in L^{1}_{X}(I^{d})$ introducing the expression (3.4) $b_{n}(q,\theta,A,x)=\frac{q^{d_{n}(A,A_{n}(x))}}{\|\operatorname{re}(A\cup A_{n}(x))\|}\theta(A),\qquad A\text{ atom in }\mathscr{F}_{n},x\in I^{d}$ for a positive, finitely additive measure $\theta$ on the algebra $\mathscr{A}=\cup_{m}\mathscr{F}_{m}$. In view of inequality (3.3), it suffices to consider, instead of the maximal function of the projection operators $P_{n}$, the maximal functions given by (3.5) $\mathscr{M}_{K}\theta(x)=\sup_{n\geq K}\sum_{A\text{ atom of }\mathscr{F}_{n}}b_{n}(q,\theta,A,x)\qquad x\in I^{d},f\in L^{1}(I^{d})$ for any positive integer $K$ and some fixed parameter $q\in[0,1)$. If we abbreviate by $\mathscr{M}f$ the maximal function $\mathscr{M}_{1}(\|f\|\,\mathrm{d}\lambda^{d})$, we have the following weak type (1,1) result. ###### Proposition 3.1. The maximal function $\mathscr{M}$ is of weak type (1,1), i.e. there exists a constant $C$ depending only on the dimension $d$ and on the parameter $q<1$, so that we have the inequality $\|\\{\mathscr{M}f>t\\}\|\leq\frac{C}{t}\\|f\\|_{L^{1}},\qquad t>0,\ f\in L^{1}(I^{d}).$ ###### Proof. Set $B=I^{d}$, $K=1$ and $\theta=\|f\|\,\mathrm{d}\lambda^{d}$ in Theorem 3.2 below and observe that the geometric series in equation (3.6) converges. ∎ The following result about the maximal operators $\mathscr{M}_{K}$ is the focal point in our investigations. ###### Theorem 3.2. Let $(\mathscr{F}_{n})$ be an interval filtration on $I^{d}$ and let $\theta$ be a non-negative, finitely additive measure on the algebra $\mathscr{A}=\cup_{n}\mathscr{F}_{n}$. Then, for any Borel set $B\subset I^{d}$ and any positive integer $K$, (3.6) $\|B\cap\\{\mathscr{M}_{K}\theta>t\\}\|\leq\frac{C}{t}\cdot\sum_{s=0}^{\infty}q^{s/2}(s+1)^{d-1}\theta\big{(}A_{K,s}(B)\big{)},\qquad t>0$ for some constant $C$ depending only on $q$ and $d$. ###### Proof. Set $G_{t}=B\cap\\{\mathscr{M}_{K}\theta>t\\}$ and let $x\in G_{t}$. Then, there exists an index $n\geq K$ so that $\sum_{A\text{ atom of }\mathscr{F}_{n}}b_{n}(q,\theta,A,x)>t.$ Letting $c=(2\sum_{\ell=0}^{\infty}\rho^{\ell})^{d}=\big{(}2/(1-\rho)\big{)}^{d}<\infty$ with $\rho=q^{1/2}$, we obtain that there exists at least one atom $F$ of the $\sigma$-algebra $\mathscr{F}_{n}$ so that (3.7) $b_{n}(\rho,\theta,F,x)>t/c.$ Therefore, for $x\in G_{t}$, we choose $n_{x}<\infty$ to be the minimal index $n\geq K$ so that there exists an atom $F$ of $\mathscr{F}_{n_{x}}$ satisfying inequality (3.7). We choose a particular atom $F$ of $\mathscr{F}_{n_{x}}$ with this property which will be denoted by $F_{x}$. The collection of atoms $\\{A_{n_{x}}(x):x\in G_{t}\\}$ is nested and covers the set $G_{t}$. Thus, there exists a maximal countable subcollection $\\{A_{n_{x}}(x):x\in\Gamma\\}$ consisting of mutually disjoint sets that still covers $G_{t}$. Perform the following estimate using inequality (3.7): (3.8) $\displaystyle\|G_{t}\|$ $\displaystyle\leq\sum_{x\in\Gamma}\|A_{n_{x}}(x)\|\leq\frac{c}{t}\cdot\sum_{x\in\Gamma}\rho^{d_{n_{x}}(F_{x},A_{n_{x}}(x))}\theta(F_{x})$ $\displaystyle=\frac{c}{t}\cdot\sum_{s\geq 0}\rho^{s}\sum_{m\in\mathbb{Z}^{d}:\|m\|_{1}=s}\Big{(}\sum_{x\in\Gamma_{m}}\theta(F_{x})\Big{)},$ where for $m\in\mathbb{Z}^{d}$, $\Gamma_{m}$ is the set of all $x\in\Gamma$ so that, if $A_{n_{x}}(x)=I_{n_{x},i}$ and $F_{x}=I_{n_{x},j}$ for some $i,j\in\mathbb{Z}^{d}$, we have $i-j=m$. Next, we show that for each $m\in\mathbb{Z}^{d}$, the collection $\\{F_{x}:x\in\Gamma_{m}\\}$ consists of mutually disjoint sets. Assume the contrary, i.e. for some $m\in\mathbb{Z}^{d}$ there exist two points $x,y\in\Gamma_{m}$ that are different from each other with $F_{x}\cap F_{y}\neq\emptyset$. For definiteness, assume that $n_{x}\geq n_{y}$, and thus the nestedness of the $\sigma$-algebras $(\mathscr{F}_{n})$ implies $F_{x}\subseteq F_{y}$. Assume that $i,i^{\prime},j,j^{\prime}\in\mathbb{Z}^{d}$ are such that $I_{n_{x},i}=A_{n_{x}}(x),\quad I_{n_{y},i^{\prime}}=A_{n_{y}}(y),\quad I_{n_{x},j}=F_{x},\quad I_{n_{y},j^{\prime}}=F_{y}.$ Since $x,y\in\Gamma_{m}$, we know that $i-j=m=i^{\prime}-j^{\prime}$. Therefore, since $\mathscr{F}_{n_{x}}$ is finer than $\mathscr{F}_{n_{y}}$ and by the inclusion $F_{x}\subseteq F_{y}$, we have (3.9) $\operatorname{re}(F_{x}\cup A_{n_{x}}(x))\subseteq\operatorname{re}(F_{y}\cup A_{n_{y}}(x))\subseteq\operatorname{re}(F_{y}\cup A_{n_{y}}(y)).$ Moreover, this and the definition of the distance $d_{n_{y}}$ implies (3.10) $d_{n_{y}}(F_{y},A_{n_{y}}(y))\geq d_{n_{y}}(F_{y},A_{n_{y}}(x)).$ Combining (3.9) and (3.10) yields $b_{n_{y}}(\rho,\theta,F_{y},x)\geq b_{n_{y}}(\rho,\theta,F_{y},y)$; additionally, by definition of $n_{y},F_{y}$ we have the inequality $b_{n_{y}}(\rho,\theta,F_{y},y)>t/c$. Together, this implies $b_{n_{y}}(\rho,\theta,F_{y},x)>t/c.$ As $n_{x}\geq K$ is the minimal index so that such an inequality at the point $x$ is possible and $n_{x}\geq n_{y}$ we get that $n_{x}=n_{y}=:n$. Since $A_{n}(x)\cap A_{n}(y)=\emptyset$ we know that in this case $i\neq i^{\prime}$ and $x,y\in\Gamma_{m}$ implies $i-j=m=i^{\prime}-j^{\prime}$. Together, this yields $j\neq j^{\prime}$ which means $F_{x}\cap F_{y}=\emptyset$, contradicting the assumption $F_{x}\subseteq F_{y}$. Therefore, $F_{x}$ and $F_{y}$ are disjoint, concluding the proof of the fact that $\\{F_{x}:x\in\Gamma_{m}\\}$ consists of mutually disjoint sets for each $m\in\mathbb{Z}^{d}$. If $(U_{j})$ is a countable collection of disjoint members of $\mathscr{A}$ and if $U\in\mathscr{A}$ with $\cup_{j=1}^{\infty}U_{j}\subset U$, then $\sum_{j=1}^{\infty}\theta(U_{j})\leq\theta(U)$, since for finite sums this is clear by finite additivity and positivity of $\theta$ and the general case follows by passing to infinity. We apply this simple fact to the sum $\sum_{x\in\Gamma_{m}}\theta(F_{x})$ with $U=A_{K,\|m\|_{1}}(B)$ to obtain from (3.8) $\displaystyle\|G_{t}\|$ $\displaystyle\leq\frac{c}{t}\cdot\sum_{s=0}^{\infty}\rho^{s}\theta\big{(}A_{K,s}(B)\big{)}\Big{(}\sum_{\|m\|_{1}=s}1\Big{)}\leq\frac{2^{d}c}{t}\sum_{s=0}^{\infty}\rho^{s}(s+1)^{d-1}\theta\big{(}A_{K,s}(B)\big{)},$ which is the conclusion of the theorem. ∎ Combining Proposition 3.1 with the bound (3.3) on the operators $P_{n}$, we obtain that the maximal function of the spline projectors $P_{n}$ also satisfies a weak type (1,1) inequality (3.11) $\|\\{\sup_{n}\\|P_{n}f\\|>t\\}\|\leq\frac{C\\|f\\|_{L^{1}_{X}}}{t},\qquad t>0,\ f\in L^{1}_{X}(I^{d}),$ for some constant $C$ depending only on $k$. This proves Doob’s inequality (ii) on page ii for martingale spline sequences on $I^{d}$. Indeed, given a martingale spline sequence $(f_{n})$ on $I^{d}$, apply (3.11) to the function $f=f_{m}$ for a fixed positive integer $m$ and pass $m\to\infty$ to get (ii) for martingale spline sequences on $I^{d}$. As a corollary, we have the following result about almost everywhere convergence of $P_{n}f$ for $f\in L^{1}_{X}(I^{d})$, proving (iii) for tensor spline projections. ###### Theorem 3.3. Let $X$ be any Banach space and let $f\in L^{1}_{X}(I^{d})$. Then, there exists $g\in L^{1}_{X}(I^{d})$ such that $P_{n}f\to g\qquad\text{$\lambda^{d}$-almost everywhere}.$ ###### Remark. (i) The proof of Theorem 3.3 follows along the same lines as the proof of the one-dimensional case [5, Theorem 3.2] and uses standard arguments for passing from a weak type maximal inequality of the form (3.11) to almost everywhere convergence of $P_{n}f$ for $L^{1}$-functions $f$. For this argument, a dense subset of $L^{1}$ is needed, for which it is “clear” that pointwise convergence takes place. In [5, Lemma 3.1], for one-dimensional splines, this dense set is chosen to be the space of continuous functions $C(\bar{I})$ on the closure of the interval $I$. For arbitrary dimensions $d$, we can use $C(\bar{I})\otimes\cdots\otimes C(\bar{I})$ as dense subset of $L^{1}$, for which it is a consequence of the one-dimensional convergence result [5, Lemma 3.1] and its tensor product structure that $P_{n}f$ converges pointwise for $f\in C(\bar{I})\otimes\cdots\otimes C(\bar{I})$. (ii) As in the one-dimensional case, the limit function $g$ in Theorem 3.3 can be identified explicitly as the ($L^{1}_{X}$-extension of the) orthogonal projection of the function $f$ onto the closure of $\cup_{n}S_{n}$, which, in the particular case that $\cup_{n}\mathscr{F}_{n}$ generates the Borel-$\sigma$-algebra on $I^{d}$, coincides with the function $f$. We also note another immediate corollary of Theorem 3.2 that will be used later. ###### Corollary 3.4. Let $(\mathscr{F}_{n})$ be an interval filtration on $I^{d}$ and let $\theta$ be a non-negative, finitely additive measure on the algebra $\mathscr{A}=\cup_{n}\mathscr{F}_{n}$. Let $D\in\mathscr{A}$ be arbitrary and set $L_{t}:=\Big{\\{}x\in I^{d}:\limsup_{n}\sum_{A\text{ atom of }\mathscr{F}_{n}}b_{n}(q,\theta,A,x)>t\Big{\\}}.$ Let $R$ be a non-negative integer. If $B\subset D$ is a Borel set such that $A_{K,R}(B)\subset D$ for some $K$, we have (3.12) $\|B\cap L_{t}\|\leq\frac{C}{t}\Big{(}\theta(D)+\sum_{s>R}q^{s/2}(s+1)^{d-1}\theta(I^{d})\Big{)},\qquad t>0$ for some constant $C$ depending only on $d$ and $q$. ###### Proof. This just follows from Theorem 3.2 by noting that $L_{t}\subset\\{\mathscr{M}_{K}\theta>t\\}$ for any positive integer $K$. ∎ ###### Remark. Assume that in Corollary 3.4, the measure $\theta$ is a $\sigma$-additive Borel measure on $\bar{I}^{d}$ and replace the term $\theta(A)$ in the definition (3.4) of $b_{n}$ by the term $\theta(\overline{A})$ with the closure $\overline{A}$ of $A$ in $\bar{I}^{d}$. Then, the assertion of Corollary 3.4 still holds if we replace $\theta(D)$ and $\theta(I^{d})$ on the right hand side of (3.12) by $\theta(\overline{D})$ and $\theta(\bar{I}^{d})$ respectively. Indeed, the only modification in the proof of Theorem 3.2 is that we have to replace $F_{x}$ by $\overline{F_{x}}$ in (3.8), but this only gives an additional factor of $2^{d}$ on the right hand side of (3.6) and (3.12) since each point of $\bar{I}^{d}$ is contained in at most $2^{d}$ closures of disjoint rectangles. ## 4\. The Convergence Theorem for dense filtrations In this section, we show the remaining implication (a)$\implies$(b) of item (iv) on page iv for martingale spline sequences on $I^{d}$ in the case where $\mathscr{A}:=\cup_{n}\mathscr{F}_{n}$ generates the Borel-$\sigma$-algebra on $I^{d}$. We restrict ourselves to this special setting in this section to present the crucial arguments in a concise form. In order to lift the subsequent result from this hypothesis, we use technical arguments in the spirit of those in the proof of the one-dimensional result [5, Sections 4 and 6]. This will be presented in detail in Section 5. ###### Theorem 4.1. Let $(\mathscr{F}_{n})$ be an interval filtration on $I^{d}$ so that $\mathscr{A}=\cup_{n}\mathscr{F}_{n}$ generates the Borel-$\sigma$-algebra and let $X$ be a Banach space with RNP. Let $(g_{n})$ be an $X$-valued martingale spline sequence adapted to $(\mathscr{F}_{n})$ with $\sup_{n}\\|g_{n}\\|_{L^{1}_{X}}<\infty$. Then, there exists $g\in L^{1}_{X}(I^{d})$ so that $g_{n}\to g$ almost everywhere with respect to Lebesgue measure $\lambda^{d}$. ###### Remark. As for martingales (see [2]), the basic proof idea of this result is to define a vector measure $\nu$ based upon the martingale spline sequence $(g_{n})$, whose absolutely continuous part with respect to Lebesgue measure $\lambda^{d}$ has a density $g\in L^{1}_{X}$ by the RNP of $X$, which is then the a.e. limit of $g_{n}$. ###### Proof. Part I: The limit operator $T$. For $f\in S_{m}$ and $n\geq m$, since the operator $P_{n}$ is selfadjoint and using the martingale spline property of the sequence $(g_{n})$, $\displaystyle\int_{I^{d}}g_{n}\cdot f\,\mathrm{d}\lambda^{d}$ $\displaystyle=\int_{I^{d}}g_{n}\cdot P_{m}f\,\mathrm{d}\lambda^{d}=\int_{I^{d}}P_{m}g_{n}\cdot f\,\mathrm{d}\lambda^{d}=\int_{I^{d}}g_{m}\cdot f\,\mathrm{d}\lambda^{d}.$ This means in particular that for all $f\in\cup_{m}S_{m}$, the limit of $\int_{I^{d}}g_{n}\cdot f\,\mathrm{d}\lambda^{d}$ exists, so we define the linear operator $T:\cup_{m}S_{m}\to X,\qquad f\mapsto\lim_{n}\int_{I^{d}}g_{n}\cdot f\,\mathrm{d}\lambda^{d}.$ We can write $g_{n}$ in terms of this operator $T$. Indeed, by the martingale spline property of $(g_{n})$ again, (4.1) $\displaystyle g_{n}=P_{n}g_{n}$ $\displaystyle=\sum_{i}\int_{I^{d}}g_{n}N_{n,i}\,\mathrm{d}\lambda^{d}\cdot N_{n,i}^{}$ $\displaystyle=\sum_{i}\lim_{m}\int_{I^{d}}g_{m}N_{n,i}\,\mathrm{d}\lambda^{d}\cdot N_{n,i}^{}=\sum_{i}(TN_{n,i})N_{n,i}^{}.$ By Alaoglu’s theorem, we may choose a subsequence $\ell_{n}$ such that the bounded sequence of measures $\\|g_{\ell_{n}}\\|_{X}\,\mathrm{d}\lambda^{d}$ converges in the weak-topology on the space of Radon measures on the closure $\bar{I}^{d}$ of $I^{d}$ to some finite scalar measure $\mu$ on $\bar{I}^{d}$, i.e. $\lim_{n\to\infty}\int_{\bar{I}^{d}}f\\|g_{\ell_{n}}\\|\,\mathrm{d}\lambda^{d}=\int_{\bar{I}^{d}}f\,\mathrm{d}\mu,\qquad f\in C(\bar{I}^{d}).$ For a fixed positive integer $m$, we then get another subsequence of $(\ell_{n})$, again denoted by $(\ell_{n})$, so that for each atom $A$ of $\mathscr{F}_{m}$, the sequence $\\|g_{\ell_{n}}\\|\,\mathrm{d}\lambda^{d}$ converges to some Radon measure $\mu_{A}$ on the closure $\overline{A}$ of $A$ satisfying $\mu=\sum_{A\text{ atom of }\mathscr{F}_{m}}\mu_{A}$. Each function $f\in S_{m}$ is continuous and a polynomial in the interior $A^{\circ}$ of each atom $A\in\mathscr{F}_{m}$. Denote by $f_{A}$ the continuous function on the closure $\overline{A}$ of $A$ that coincides with $f$ on $A^{\circ}$. Then, for $\ell_{n}\geq m$ and $f\in S_{m}$ $\displaystyle\\|Tf\\|$ $\displaystyle=\Big{\\|}\int_{I^{d}}fg_{\ell_{n}}\,\mathrm{d}\lambda^{d}\Big{\\|}\leq\int_{I^{d}}\|f\|\\|g_{\ell_{n}}\\|\,\mathrm{d}\lambda^{d}$ $\displaystyle=\sum_{A\text{ atom of }\mathscr{F}_{m}}\int_{\overline{A}}\|f_{A}\|\\|g_{\ell_{n}}\\|\,\mathrm{d}\lambda^{d}\rightarrow\sum_{A\text{ atom of }\mathscr{F}_{m}}\int_{\bar{I}^{d}}\|f_{A}\|\,\mathrm{d}\mu_{A}$ $\displaystyle\leq\sum_{A\text{ atom of }\mathscr{F}_{m}}\int_{\bar{I}^{d}}\limsup_{s\to y}\|f(s)\|\,\mathrm{d}\mu_{A}(y)=\int_{\bar{I}^{d}}\limsup_{s\to y}\|f(s)\|\,\mathrm{d}\mu(y).$ For $f\in\cup_{n}S_{n}$ define (4.2) $\\|f\\|:=\int_{\bar{I}^{d}}\limsup_{s\to y}\|f(s)\|\,\mathrm{d}\mu(y),$ which is a seminorm on $\cup_{n}S_{n}$. As for $L^{p}$-spaces, we factor out the functions $f\in\cup_{n}S_{n}$ with $\\|f\\|=0$ in order to get a norm. Then, denote by $W$ the completion of $\cup_{n}S_{n}$ in this norm and extend the operator $T$ to $W$ continuously. Part II: Representing $T$ in terms of a vector measure $\nu$. Let $Q=\prod_{\ell=1}^{d}(a_{\ell},b_{\ell}]$ be an arbitrary atom of the $\sigma$-algebra $\mathscr{F}_{n}$ for some positive integer $n$. Let $\ell\in\\{1,\ldots,d\\}$ be an arbitrary coordinate direction. If the order of the polynomials $k_{\ell}$ in direction $\ell$ equals $1$ (piecewise constant case), we set $f_{m}^{\ell}=\mathbbm{1}_{(a_{\ell},b_{\ell}]}$ for $m\geq n$, which satisfies $f_{m}^{\ell}\in S^{k_{\ell}}(\mathscr{F}_{m}^{\ell})$. If $k_{\ell}>1$, we first choose an open interval $O$ and a closed interval $C$ (both in $I$) so that $C\subseteq(a_{\ell},b_{\ell}]\subseteq O$ and $\|O\setminus C\|\leq 1/m$. The sets $C$ and $O$ are chosen so that as many endpoints of $C$ and $O$ coincide with the corresponding endpoints of $(a_{\ell},b_{\ell}]$ as possible. Then, let $f_{m}^{\ell}\in\cup_{j}S^{k_{\ell}}(\mathscr{F}_{j}^{\ell})$ be a non- negative function that is bounded by $1$ and satisfies $\operatorname{supp}f_{m}^{\ell}\subset O\qquad\text{and}\qquad f_{m}^{\ell}\equiv 1\text{ on }C\cap I.$ Such a function exists since $\mathscr{A}$ generates the Borel-$\sigma$-algebra if one additionally notices the facts that B-splines form a partition of unity and have localized support. If we define $f_{m}=f_{m}^{1}\otimes\cdots\otimes f_{m}^{d}$, the sequence $(f_{m})$ is Cauchy in $\cup_{j}S_{j}$ with respect to the norm in (4.2) and we let $I_{Q}$ be the limit in $W$ of the sequence $(f_{m})$ satisfying $\\|TI_{Q}\\|=\lim_{m\to\infty}\\|Tf_{m}\\|\leq\mu(\overline{Q})$ (here, the closure of $Q$ is taken in $\bar{I}^{d}$). This definition of $I_{Q}$ also has the property that if $Q$ is an atom in $\mathscr{F}_{n}$ and $(Q_{j})_{j=1}^{\ell}$ is a finite sequence of disjoint atoms $Q_{j}$ in $\mathscr{F}_{n_{j}}$ with $n_{j}\geq n$ and $Q=\cup_{j=1}^{\ell}Q_{j}$, we have $I_{Q}=\sum_{j=1}^{\ell}I_{Q_{j}}$. Therefore, if $\mathscr{F}_{n}\ni A=\cup_{j=1}^{\ell}Q_{j}$ for some disjoint atoms $(Q_{j})_{j=1}^{\ell}$ in $\mathscr{F}_{n}$, it is well defined to set $I_{A}=\sum_{j=1}^{\ell}I_{Q_{j}}\in W.$ Based upon that, we define the finitely additive vector measure $\nu$ on $(I^{d},\mathscr{A})$ with values in $X$ by (4.3) $\nu(A):=T(I_{A}),\qquad A\in\mathscr{A}.$ This vector measure $\nu$ is of bounded variation, since if $\pi$ is a finite partition of $I^{d}$ into sets of $\mathscr{A}$ and if $m<\infty$ is the minimal index so that $A\in\mathscr{F}_{m}$ for all $A\in\pi$, we have $\sum_{A\in\pi}\\|T(I_{A})\\|\leq\sum_{Q\text{ atom in }\mathscr{F}_{m}}\\|T(I_{Q})\\|\leq\sum_{Q\text{ atom in }\mathscr{F}_{m}}\mu(\overline{Q})\leq 2^{d}\mu(\bar{I}^{d}),$ as each point in $\bar{I}^{d}$ is contained in at most $2^{d}$ closures of atoms of $\mathscr{F}_{m}$. Observe that for all $f\in\cup_{n}S_{n}$, we have (4.4) $\int_{I^{d}}f\,\mathrm{d}\nu=T(f).$ Indeed, each $f\in\cup_{n}S_{n}$ can be approximated uniformly by linear combinations of characteristic functions of atoms of the form $\chi_{m}:=\sum_{Q\text{ atom in }\mathscr{F}_{m}}\alpha_{Q}\mathbbm{1}_{Q}$ as $m\to\infty$, which then also has the property that $f_{m}:=\sum_{Q\text{ atom in }\mathscr{F}_{m}}\alpha_{Q}I_{Q}\to f$ in $W$ as $m\to\infty$ and thus also $Tf_{m}\to Tf$ in $X$ by the continuity of the operator $T$. As, by definition (4.3) of $\nu$, we have $\int\chi_{m}\,\mathrm{d}\nu=Tf_{m}$, equation (4.4) follows by letting $m\to\infty$. Part III: Conclusion. Continuing the calculation in equation (4.1), using the measure $\nu$ and (4.4), (4.5) $g_{n}=\sum_{i}\int_{I^{d}}N_{n,i}\,\mathrm{d}\nu\cdot N_{n,i}^{}.$ Apply Lebesgue’s decomposition Theorem 2.1 to the measure $\nu$ with respect to $\lambda^{d}$ to get two finitely additive measures $\nu_{c},\nu_{s}$ of bounded variation with (4.6) $\nu=\nu_{c}+\nu_{s},$ where $\nu_{c}$ is $\lambda^{d}$-continuous and $\|\nu_{s}\|$ is singular to $\lambda^{d}$. As $\lambda^{d}$ is countably additive, so is the $\lambda^{d}$-continuous measure $\nu_{c}$ and by the extension theorem (Theorem 2.2) extends uniquely to a countably additive vector measure $\overline{\nu_{c}}$ on the Borel-$\sigma$-algebra on $I^{d}$, which, by the RNP of $X$ can be written as $\,\mathrm{d}\overline{\nu_{c}}=g\,\mathrm{d}\lambda^{d}$ for some $g\in L^{1}_{X}$. Therefore, $g_{n}=\sum_{i}\int_{I^{d}}N_{n,i}g\,\mathrm{d}\lambda^{d}\cdot N_{n,i}^{}+\sum_{i}\int_{I^{d}}N_{n,i}\,\mathrm{d}\nu_{s}\cdot N_{n,i}^{}.$ The first part on the right hand side of this equation equals $P_{n}g$ for the $L^{1}_{X}$ function $g$ and this converges a.e. to $g$ by Theorem 3.3 and the remark following it. The second part, denoted by $P_{n}\nu_{s}$, converges to $0$ almost everywhere, which we will now show. Let $t>0$ be arbitrary and let $G_{t}:=\\{y\in I^{d}:\limsup_{n}\\|P_{n}\nu_{s}(y)\\|>t\\}$. Then, let $\varepsilon>0$ be arbitrary and choose $D\in\mathscr{A}$ with the property $\lambda^{d}(D^{c})+\|\nu_{s}\|(D)\leq\varepsilon$, which is possible since $\|\nu_{s}\|$ is singular to $\lambda^{d}$. By (3.3), replacing $\\|f\\|\,\mathrm{d}\lambda^{d}$ with $\|\nu_{s}\|$, $\displaystyle\\|P_{n}\nu_{s}(y)\\|$ $\displaystyle\leq C_{k}\sum_{A\text{ atom of }\mathscr{F}_{n}}b_{n}(q,\|\nu_{s}\|,A,y)$ for some constants $C_{k}$ and $0<q<1$ depending only on $k$ with $b_{n}$ as in (3.4). Therefore, $G_{t}\subset L_{t/C_{k}}$ with $L_{u}=\Big{\\{}y\in I^{d}:\limsup_{n}\sum_{A\text{ atom of }\mathscr{F}_{n}}b_{n}(q,\|\nu_{s}\|,A,y)>u\Big{\\}}.$ We apply Lemma 5.1 below (with $Y=I^{d}$) to the measure $\theta=\|\nu_{s}\|$ on $\mathscr{A}$ and the set $D$ to get, for any $u>0$, the estimate $\|L_{u}\|=\|D^{c}\cap L_{u}\|+\|D\cap L_{u}\|\leq\varepsilon+C\varepsilon/u$. Since this is true for any $\varepsilon>0$, we obtain $\|L_{u}\|=0$ for any $u>0$. Thus, $\|\\{y\in I^{d}:\limsup_{n}\\|P_{n}\nu_{s}(y)\\|>0\\}\|=\Big{\|}\bigcup_{r=1}^{\infty}G_{1/r}\Big{\|}\leq\Big{\|}\bigcup_{r=1}^{\infty}L_{1/(C_{k}r)}\Big{\|}=\lim_{r\to\infty}\|L_{1/(C_{k}r)}\|=0,$ which completes the proof of the theorem. ∎ ## 5\. The convergence theorem for arbitrary filtrations Now we discuss the necessary modifications in the proof of Theorem 4.1 when the interval filtration $(\mathscr{F}_{n})$ on $I^{d}$ is allowed to be arbitrary. Assume for some Banach space $X$ with RNP, $(g_{n})$ is an $X$-valued martingale spline sequence adapted to $(\mathscr{F}_{n})$ with $\sup_{n}\\|g_{n}\\|_{L^{1}_{X}}<\infty$. Part I of the proof of Theorem 4.1 does not use the density of the filtration $(\mathscr{F}_{n})$ in $I^{d}$, which means that we get an operator $T:\cup_{n}S_{n}\to X$ and a finite measure $\mu$ on $\bar{I}^{d}$ satisfying (5.1) $\\|Tf\\|\leq\int_{\bar{I}^{d}}\limsup_{s\to y}\|f(s)\|\,\mathrm{d}\mu(y),\qquad f\in\cup_{n}S_{n}.$ The operator $T$ is then extended continuously to the completion $W$ of $\cup_{n}S_{n}$ w.r.t. the norm on the right hand side of (5.1). With the aid of this operator, the martingale spline sequence $(g_{n})$ is written as (5.2) $g_{n}=\sum_{i}(TN_{n,i})N_{n,i}^{}.$ We distinguish the analysis of the convergence of $g_{n}(y)$ depending on in which coordinate direction the filtration $(\mathscr{F}_{n})$ is dense at the point $y$. To this end, for $\ell=1,\ldots,d$, we define $\Delta_{n}^{\ell}\subset\bar{I}$ to be the set of all endpoints of atoms in the $\sigma$-algebra $\mathscr{F}_{n}^{\ell}$. Next, let $U^{\ell}$ be the complement (in $\bar{I}$) of the set of all accumulation points of $\cup_{n}\Delta_{n}^{\ell}$. Note that $U^{\ell}$ is open (in $\bar{I}$), thus it can be written as a countable union of disjoint open intervals $(U^{\ell}_{j})_{j}$. Let $\displaystyle B_{j}^{\ell}=\\{a\in\partial U_{j}^{\ell}:\text{ there is no sequence of points in $U_{j}^{\ell}\cap(\cup_{n}\Delta_{n}^{\ell})$ }\text{that converges to $a$}\\}$ and define $V_{j}^{\ell}=U_{j}^{\ell}\cup B_{j}^{\ell}$ and $V^{\ell}:=\cup_{j}V_{j}^{\ell}$. ###### Lemma 5.1. Let $(\mathscr{F}_{n})$ be an interval filtration on $I^{d}$ and let $\theta$ be a non-negative, finitely additive and finite measure on $\mathscr{A}$. For $\varepsilon>0$, let $D\in\mathscr{A}$ with $\theta(D)\leq\varepsilon$ and $L_{t}:=\Big{\\{}x\in I^{d}:\limsup_{n}\sum_{A\text{ atom of }\mathscr{F}_{n}}b_{n}(q,\theta,A,x)>t\Big{\\}}.$ Then, there exists a finite constant $C$, depending only on $q$ and on $d$ so that $\|D\cap L_{t}\cap Y\|\leq\frac{C\varepsilon}{t},\qquad t>0,$ with $Y=(V^{1})^{c}\times\cdots\times(V^{d})^{c}$. ###### Proof. We shrink the set $D$ properly to then apply Corollary 3.4. This is done as follows. Since $D\in\mathscr{A}$, we can write it as $D=\cup_{j=1}^{L}Q_{j}$ for disjoint atoms $(Q_{j})$ of some $\sigma$-algebra $\mathscr{F}_{n}$. For each $j$, we have $Q_{j}=Q_{j}^{1}\times\cdots\times Q_{j}^{d}$ for some intervals $Q_{j}^{\ell}$, $\ell=1,\ldots,d$. Assume without restriction that for all $\ell\in\\{1,\ldots,d\\}$, the interior of the interval $Q_{j}^{\ell}$ contains at least two points from $(V^{\ell})^{c}$, since otherwise we would have $\|Q_{j}\cap L_{t}\cap Y\|\leq\|Q_{j}\cap Y\|=0$. Fix $\ell\in\\{1,\ldots,d\\}$, set $\eta=\varepsilon/(tL\|I\|^{d-1}d)$ and define the interval $J^{\ell}\subset Q_{j}^{\ell}$ such that 1. (1) $Q_{j}^{\ell}\setminus J^{\ell}$ has two connected components and in each one there exists a point of $(V^{\ell})^{c}$ that has positive distance to $J^{\ell}$ and to the boundary of $Q_{j}^{\ell}$, 2. (2) $\|(Q_{j}^{\ell}\setminus J^{\ell})\cap(V^{\ell})^{c}\|\leq\eta$. This is possible since $Q_{j}^{\ell}\cap(V^{\ell})^{c}$ contains at least two points. Then, set $Q_{j}^{\prime}=J^{1}\times\cdots\times J^{d}$ and $B=\cup_{j=1}^{L}Q_{j}^{\prime}$ and we get, by the choice of $\eta$, (5.3) $\|D\cap L_{t}\cap Y\|\leq\|(D\setminus B)\cap Y\|+\|B\cap L_{t}\|\leq\varepsilon/t+\|B\cap L_{t}\|.$ Choose the positive integer $R$ sufficiently large so that $\sum_{s>R}(s+1)^{d-1}q^{s/2}\theta(I^{d})\leq\varepsilon$. Then, there exists an integer $K$ so that $A_{K,R}(B)\subset D$, which is true by construction of $B$. Apply now Corollary 3.4 to get $\|B\cap L_{t}\|\leq C\varepsilon/t$, which together with (5.3) implies the assertion of the lemma. ∎ ###### Remark. Assume that in Lemma 5.1, the measure $\theta$ is a $\sigma$-additive Borel measure on $\bar{I}^{d}$ and replace the term $\theta(A)$ in the definition (3.4) of $b_{n}$ by the term $\theta(\overline{A})$ with the closure $\overline{A}$ of $A$ in $\bar{I}^{d}$. Then, the assertion of Lemma 5.1 still holds with an additional factor of $2^{d}$ on the constant $C$, since the same is true for Corollary 3.4. For a point $y=(y^{1},\ldots,y^{d})\in I^{d}$, each coordinate $y^{\ell}$ is either contained in some set $V^{\ell}_{j_{\ell}}$ or in $(V^{\ell})^{c}$. After rearranging the coordinates, we assume that $y\in F$, where $F=F^{1}\times\cdots\times F^{d}$ with $F^{\ell}=V_{j_{\ell}}^{\ell}$ if $\ell\leq s$ and $F^{\ell}=(V^{\ell})^{c}$ if $\ell>s$ for some $s\in\\{0,\ldots,d\\}$. We want to split $g_{n}(y)=\sum_{i}(TN_{n,i})N_{n,i}^{}(y)$ into the parts where $T$ acts on functions restricted to the set $F_{\delta}$ for $\delta\in\\{0,1\\}^{d}$ with $F_{\delta}=E^{1}\times\cdots\times E^{d}$ where $E^{\ell}=F^{\ell}$ if $\delta_{\ell}=0$ and $E^{\ell}=(F^{\ell})^{c}$ if $\delta_{\ell}=1$. In order to construct elements in $W$ that correspond to the functions $N_{n,i}\mathbbm{1}_{F_{\delta}}$, we need the following lemma. ###### Lemma 5.2. For any $\ell\in\\{1,\ldots,d\\}$, let $f\in S^{k_{\ell}}(\mathscr{F}_{n}^{\ell})$ for some $n$. For any interval $V_{j}^{\ell}$, there exists a sequence $(h_{m})$ of functions $h_{m}\in S^{k_{\ell}}(\mathscr{F}_{m}^{\ell})$, open intervals $O_{m}$ and closed intervals $C_{m}$ (both in $\bar{I}$) satisfying 1. (1) $O_{m}\to V_{j}^{\ell}$ as $m\to\infty$, 2. (2) $\operatorname{supp}h_{m}\subset O_{m}$, 3. (3) $h_{m}\equiv f$ on $C_{m}\cap I$, 4. (4) The closure of $O_{m}\setminus C_{m}$ converges to the empty set as $m\to\infty$. ###### Proof. Without loss of generality, assume that $f=N_{n,i}^{\ell}$ for some integer $i$. For $m\geq n$, we can write $N_{n,i}^{\ell}=\sum_{r}\lambda_{m,r}N_{m,r}^{\ell},$ where the absolute value of each coefficient $\lambda_{m,r}$ is $\leq 1$. Set $h_{m}=\sum_{r\in\Lambda_{m}}\lambda_{m,r}N_{m,r}^{\ell},$ where the set $\Lambda_{m}$ is defined to contain precisely those indices $r$ so that the support of $N_{m,r}^{\ell}$ intersects $V_{j}^{\ell}$ but the (Euclidean) distance between the support of $N_{m,r}^{\ell}$ and $\partial U_{j}^{\ell}\setminus B_{j}^{\ell}$ is positive. The function $h_{m}$ is then contained in $S^{k_{\ell}}(\mathscr{F}_{m}^{\ell})$ and satisfies $\|h_{m}\|\leq 1$. With this setting, the support of $h_{m}$ is contained in $O_{m}$ for some open interval $O_{m}$ and $h_{m}\equiv N_{n,i}^{\ell}$ on some closed interval $C_{m}\subset O_{m}$. Since the endpoints of $V_{j}^{\ell}$ are accumulation points of $\cup_{n}\Delta_{n}^{\ell}$ or endpoints of $I$, the intervals $O_{m}$ and $C_{m}$ can be chosen to satisfy items (1) and (4). ∎ Let now $(h_{j,m}^{\ell})_{m}$ be the sequence of functions from Lemma 5.2 corresponding to a function $f^{\ell}\in S^{k_{\ell}}(\mathscr{F}_{n_{\ell}}^{\ell})$ for some positive integer $n_{\ell}$ and the set $V_{j}^{\ell}$. 1. (1) If $E^{\ell}=V_{j_{\ell}}^{\ell}$, set $h_{m}^{\ell}=h_{j_{\ell},m}^{\ell}$. 2. (2) If $E^{\ell}=(V_{j_{\ell}}^{\ell})^{c}$, set $h_{m}^{\ell}=f^{\ell}-h_{j_{\ell},m}^{\ell}$. 3. (3) If $E^{\ell}=V^{\ell}$, set $h_{m}^{\ell}=\sum_{j=1}^{m}h_{j,K_{m}}^{\ell}$. 4. (4) If $E^{\ell}=(V^{\ell})^{c}$, set $h_{m}^{\ell}=f^{\ell}-\sum_{j=1}^{m}h_{j,K_{m}}^{\ell}$. Then, define $h_{m}=h_{m}^{1}\otimes\cdots\otimes h_{m}^{d}$. Since $\cup_{j\geq m}\overline{V_{j}^{\ell}}$ tends to the empty set as $m\to\infty$ for each $\ell$, and due to the properties guaranteed by Lemma 5.2 of the functions $(h_{m}^{\ell})$, if $K_{m}$ is chosen sufficiently large, $h_{m}\in S_{K_{m}}$ is Cauchy in the Banach space $W$ and its limit will be denoted by $(f^{1}I_{E^{1}})\otimes\cdots\otimes(f^{d}I_{E^{d}})$. If $f^{\ell}=N_{n,i_{\ell}}^{\ell}$ is some B-spline function for all $\ell$ and some positive integer $n$, we will also write $N_{n,i}I_{F_{\delta}}$ for this limit in $W$, which (by (5.1)) satisfies (5.4) $\\|T(N_{n,i}I_{F_{\delta}})\\|=\lim_{m}\\|Th_{m}\\|\leq\liminf_{m}\mu(\overline{\operatorname{supp}h_{m}})\leq\mu\big{(}F_{\delta}\cap\overline{\operatorname{supp}N_{n,i}}\big{)}.$ This construction allows us to decompose the martingale spline sequence $g_{n}$ into $g_{n}=\sum_{\delta\in\\{0,1\\}^{d}}g_{n,\delta},\qquad\text{with }g_{n,\delta}=\sum_{i}T(N_{n,i}I_{F_{\delta}})N_{n,i}^{}\text{ for }\delta\in\\{0,1\\}^{d}.$ We treat the sequence $(g_{n,\delta})_{n}$ for each fixed $\delta\in\\{0,1\\}^{d}$ separately. Case 1: We begin by considering the case where one of the first $s$ coordinates of $\delta$ equals one. Without restriction assume that the first coordinate of $\delta$ equals one. Write $N_{n,i}^{}=N_{n,i_{1}}^{1}\otimes N_{n,i_{2}}^{>1}$, with $i=(i_{1},i_{2})$ for an integer $i_{1}$ and a $(d-1)$-tuple of integers $i_{2}$, thus $g_{n,\delta}$ can be written as (5.5) $g_{n,\delta}(y_{1},y_{2})=\sum_{i_{2}}\Big{(}\sum_{i_{1}}T(N_{n,i}I_{F_{\delta}})N_{n,i_{1}}^{1}(y_{1})\Big{)}N_{n,i_{2}}^{>1}(y_{2}),\qquad(y_{1},y_{2})\in F.$ Fix $y_{1}\in U_{j_{1}}^{1}$ and $t>0$. Let $\varepsilon>0$ and denote by $A_{n}^{1}(y_{1})$ the atom in $\mathscr{F}_{n}^{1}$ that contains the point $y_{1}$. Then, $\beta:=\inf_{n}\|A_{n}^{1}(y_{1})\|>0$ since $U_{j_{1}}^{1}$ does not contain accumulation points of $\cup_{n}\Delta_{n}^{1}$. Choose an open interval $O\supseteq V_{j_{1}}^{1}$ so that $\mu\big{(}(O\setminus V_{j_{1}}^{1})\times\bar{I}^{d-1}\big{)}\leq\varepsilon\mu(\bar{I}^{d})$. Then, choose $M$ sufficiently large so that for all $n\geq M$, we have $q^{d_{n}(A_{n}^{1}(y_{1}),B_{n})}\leq\varepsilon$ for all atoms $B_{n}$ in $\mathscr{F}_{n}^{1}$ with $B_{n}\cap O^{c}\neq\emptyset$. This is possible since the endpoints of $V_{j_{1}}^{1}$ are accumulation points of $\cup_{n}\Delta_{n}^{1}$. Split the sum over $i_{1}$ in (5.5) into indices $i_{1}$ so that $\overline{\operatorname{supp}N_{n,i_{1}}^{1}}\subseteq O$ and its complement and use the geometric decay estimate (3.2) for the dual B-splines $N_{n,i_{1}}^{1}$ and $N_{n,i_{2}}^{>1}$ and estimate (5.4). With the measures $\theta_{1}(A)=\frac{1}{\beta}\mu\big{(}(O\setminus V_{j_{1}}^{1})\times A\big{)},\qquad\theta_{2}(A)=\frac{\varepsilon}{\beta}\mu\big{(}\bar{I}\times A\big{)}$ satisfying $\max\\{\theta_{1}(\bar{I}^{d-1}),\theta_{2}(\bar{I}^{d-1})\\}\leq\varepsilon\mu(\bar{I}^{d})/\beta$ and the notation $\mathscr{F}_{n}^{>1}=\mathscr{F}_{n}^{2}\otimes\cdots\otimes\mathscr{F}_{n}^{d}$, we then obtain for $n\geq M$ $\\|g_{n,\delta}(y_{1},y_{2})\\|\leq C\sum_{\text{$A$ atom of $\mathscr{F}_{n}^{>1}$}}\big{(}b_{n}(q,\theta_{1},A,y_{2})+b_{n}(q,\theta_{2},A,y_{2})\big{)},$ where the expressions $b_{n}(q,\theta,A,y_{2})$ are defined as in (3.4), but with $\theta(A)$ replaced by $\theta(\overline{A})$. Here and in the following, the letter $C$ denotes a constant that depends only on $k,d,q$ and that may change from line to line. Then, applying Corollary 3.4 (using also the remark succeeding it) in dimension $d-1$ with $B=D=\bar{I}^{d-1}$, we estimate $\displaystyle\|\\{y_{2}:\limsup_{n}\\|g_{n,\delta}(y_{1},y_{2})\\|>t\\}\|$ $\displaystyle\leq C\frac{\varepsilon\mu(\bar{I}^{d})}{t\beta}.$ We have this inequality for any $\varepsilon>0$, which implies $\|\\{y_{2}:\limsup_{n}\\|g_{n,\delta}(y_{1},y_{2})\\|>t\\}\|=0$. As this is true for any $t>0$ and any $y_{1}\in U_{j_{1}}^{1}$, we get $g_{n,\delta}\to 0$ almost everywhere on $F$. Case 2: Next, consider the case where $\delta\neq 0$ but the first $s$ coordinates of $\delta$ equal $0$. Write $N_{n,i}^{}=N_{n,i_{1}}^{\leq s}\otimes N_{n,i_{2}}^{>s}$ where $i=(i_{1},i_{2})$ for an $s$-tuple of integers $i_{1}$ and a $(d-s)$-tuple of integers $i_{2}$, thus, $g_{n,\delta}$ can be written as $\displaystyle g_{n,\delta}(y_{1},y_{2})$ $\displaystyle=\sum_{i_{2}}\Big{(}\sum_{i_{1}}T(N_{n,i}I_{F_{\delta}})N_{n,i_{1}}^{\leq s}(y_{1})\Big{)}N_{n,i_{2}}^{>s}(y_{2}),\qquad(y_{1},y_{2})\in F.$ Denote by $A_{m}^{\leq s}(y_{1})$ the atom $A$ in $\mathscr{F}_{m}^{1}\otimes\cdots\otimes\mathscr{F}_{m}^{s}$ with $y_{1}\in A$. If we fix $y_{1}\in U_{j_{1}}^{1}\times\cdots\times U_{j_{s}}^{s}$, we know that $\beta:=\inf_{m}\|A_{m}^{\leq s}(y_{1})\|>0$. Next, define $Y=F^{s+1}\times\cdots\times F^{d}$ and $Z=E^{s+1}\times\cdots\times E^{d}$. Moreover, let $\mathscr{F}_{m}^{>s}=\mathscr{F}_{m}^{s+1}\otimes\cdots\otimes\mathscr{F}_{m}^{d}$ and define the measure $\theta(A)=\mu(\bar{I}^{s}\times(A\cap Z))$. Observe that $\theta(Y)=0$, since $E^{\ell}\cap F^{\ell}=\emptyset$ for some $\ell>s$ by the form of $\delta$. Using estimate (3.2) for the dual B-spline functions and estimate (5.4) bounding the operator $T$ in terms of $\mu$, $\displaystyle\\|g_{n,\delta}(y_{1},y_{2})\\|$ $\displaystyle\leq\frac{C}{\beta}\sum_{\text{$A$ atom of $\mathscr{F}_{n}^{>s}$}}b_{n}(q,\theta,A,y_{2})$ where the expression $b_{n}(q,\theta,A,y_{2})$ is defined as in (3.4), but with $\theta(A)$ replaced with $\theta(\overline{A})$. Approximate $Y$ by a sequence of sets $Y_{m}\in\mathscr{F}_{m}^{>s}$ with $Y_{m}\to Y$. Then, for each $\varepsilon>0$, there exists a positive integer $m(\varepsilon)$ with $\|Y\setminus Y_{m(\varepsilon)}\|\leq\varepsilon$ and $\theta(Y_{m(\varepsilon)})\leq\varepsilon$. For $t>0$, apply Lemma 5.1 (and the remark succeeding it) in dimension $d-s$ with $D=Y_{m(\varepsilon)}$ to deduce (5.6) $\|L_{t}\cap Y\|\leq\|Y_{m(\varepsilon)}\cap L_{t}\cap Y\|+\|Y\setminus Y_{m(\varepsilon)}\|\leq\frac{C\varepsilon}{t}+\varepsilon$ with $L_{t}=\Big{\\{}y_{2}\in I^{d-s}:\limsup_{n}\sum_{\text{ $A$ atom of $\mathscr{F}_{n}^{>s}$}}b_{n}(q,\theta,A,y_{2})>t\Big{\\}}.$ Since (5.6) holds for any $\varepsilon>0$, we obtain $\|L_{t}\cap Y\|=0$ for any $t>0$, which gives that for any fixed $y_{1}$, $g_{n,\delta}(y_{1},y_{2})$ converges to $0$ a.e. in $y_{2}\in Y$. Summarizing and combining this with Case 1 for $\delta$, we have $g_{n,\delta}\to 0$ a.e. on $F$ as $n\to\infty$ if one of the coordinates of $\delta$ equals $1$. Case 3: It remains to consider the case $g_{n,0}$, i.e. the choice $\delta=0$. For each $\ell\leq s$, the B-splines $(N_{n,r}^{\ell})_{r}$ whose supports intersect $V_{j_{\ell}}^{\ell}$ can be indexed in such a way that for each fixed $r$, the function $N_{n,r}^{\ell}\mathbbm{1}_{V_{j_{\ell}}^{\ell}}$ converges uniformly to a function $\bar{N}_{r}^{\ell}$ as $n\to\infty$ (cf. [5, Section 4]). This is the case since the interior of $V_{j_{\ell}}^{\ell}$ does not contain any accumulation points of $\cup_{n}\Delta_{n}^{\ell}$. Depending on whether the endpoints of $V_{j_{\ell}}^{\ell}$ can be approximated from inside of $V_{j_{\ell}}^{\ell}$ by points in $\cup_{n}\Delta_{n}^{\ell}$, there are different possibilities for the index set $\Lambda^{\ell}$ of the functions $(\bar{N}_{r}^{\ell})_{r\in\Lambda^{\ell}}$. It can either be finite, infinite on one side or bi-infinite. We have the following biorthogonal functions to $(\bar{N}_{r}^{\ell})_{r}$ that admit the same geometric decay estimate (3.2) than the dual B-spline functions $N_{n,r}^{\ell}$. This result is similar to [5, Lemma 4.2]. ###### Lemma 5.3. Let $\ell\in\\{1,\ldots,d\\}$. For each $r\in\Lambda^{\ell}$, the sequence $N_{n,r}^{\ell}$ converges uniformly on each atom of $\mathscr{A}^{\ell}=\cup_{n}\mathscr{F}_{n}^{\ell}$ contained in $V_{j_{\ell}}^{\ell}$ to some function $\bar{N}_{r}^{\ell}$ satisfying the estimate (5.7) $\|\bar{N}_{r}^{\ell}(y)\|\leq C\frac{q^{d(A(y),E_{r})}}{\|\operatorname{re}(A(y)\cup E_{r})\|},\qquad y\in U_{j_{\ell}}^{\ell},$ denoting by $A(y)$ the atom of $\mathscr{A}^{\ell}$ that contains the point $y$, by $E_{r}$ the support of $\bar{N}_{r}^{\ell}$ and by $d(A(y),E_{r})$ the number of atoms in $\mathscr{A}^{\ell}$ between $A(y)$ and $E_{r}$. ###### Proof. Fix the index $r\in\Lambda^{\ell}$, the point $y\in U_{j_{\ell}}^{\ell}$ and $\varepsilon>0$. Since $r\in\Lambda^{\ell}$ is fixed, the support $E_{n,r}$ of $N_{n,r}^{\ell}$ intersects $U_{j_{\ell}}^{\ell}$ for some index $n$ and we know that $\beta=\inf_{m}\|E_{m,r}\|>0$. Additionally, set $\gamma=\|A(y)\|$. Without restriction, we assume that $\beta,\gamma\leq 1$. Next, we choose $L$ sufficiently large so that $Lq^{L}\leq\varepsilon\beta\gamma$ and, for any positive integer $n$, $d_{n}(A_{n}(y),E_{n,r})\leq L$. Moreover, choose an open interval $O\supseteq V_{j_{\ell}}^{\ell}$ satisfying $\|O\setminus U_{j_{\ell}}^{\ell}\|\leq\varepsilon\beta\gamma/L$. Based on that, choose $M$ sufficiently large so that each of the intervals $(\inf O,y)$ and $(y,\sup O)$ contains at least $L$ points of $\Delta_{M}^{\ell}$ and so that, for indices $\nu$ with $\|\nu-r\|\leq 2L$ we have (5.8) $\\|N_{n,\nu}^{\ell}-N_{m,\nu}^{\ell}\\|_{L^{\infty}(U_{j_{\ell}}^{\ell})}\leq\varepsilon\beta\gamma/L,\qquad m,n\geq M.$ For $n\geq m\geq M$, expand the function $N_{m,r}^{\ell}$ in the basis $(N_{n,\nu}^{\ell})_{\nu}$ as (5.9) $N_{m,r}^{\ell}=\sum_{\nu}\alpha_{r\nu}N_{n,\nu}^{\ell}.$ The coefficients $\alpha_{r\nu}$ are bounded by a constant independently of $r,\nu$ and $m,n$ as we will now see. To this end, we use the geometric decay inequality (3.2) for the dual B-spline functions $N_{m,r}^{\ell}$ to obtain $\displaystyle\|\alpha_{r\nu}\|$ $\displaystyle=\Big{\|}\int_{I}N_{m,r}^{\ell}N_{n,\nu}^{\ell}\,\mathrm{d}\lambda\Big{\|}\leq C\sum_{\text{ $A$ atom of $\mathscr{F}_{m}^{\ell}$}}\frac{q^{d_{m}(A,E_{m,r})}}{\|\operatorname{re}(A\cup E_{m,r})\|}\int_{A}N_{n,\nu}^{\ell}\,\mathrm{d}\lambda$ $\displaystyle\leq C\sum_{\text{ $A$ atom of $\mathscr{F}_{m}^{\ell}$}}\frac{q^{d_{m}(A,E_{m,r})}}{\|\operatorname{re}(A\cup E_{m,r})\|}\|A\|\leq C\sum_{\text{ $A$ atom of $\mathscr{F}_{m}^{\ell}$}}q^{d_{m}(A,E_{m,r})}\leq C.$ Denoting $f_{\nu}=N_{m,\nu}^{\ell}-N_{n,\nu}^{\ell}$, whose absolute value is bounded by $1$, $\displaystyle\delta_{r\nu}$ $\displaystyle=\int_{I}N_{m,r}^{\ell}N_{m,\nu}^{\ell}\,\mathrm{d}\lambda=\int_{I}N_{m,r}^{\ell}N_{n,\nu}^{\ell}\,\mathrm{d}\lambda+\int_{I}N_{m,r}^{\ell}f_{\nu}\,\mathrm{d}\lambda=\alpha_{r\nu}+\int_{I}N_{n,r}^{\ell}f_{\nu}\,\mathrm{d}\lambda.$ For indices $\nu$ with $\|\nu-r\|\leq 2L$, we now estimate this last integral, by decomposing it into the integrals $I_{1},I_{2},I_{3}$ over $U_{j_{\ell}}^{\ell}$, $O\setminus U_{j_{\ell}}^{\ell}$ and $O^{c}$, respectively. By estimate (5.8) and the fact that the integral of $N_{n,r}^{\ell}$ is smaller than a constant $C$ by (3.2), the integral $\|I_{1}\|$ can be bounded by $C\varepsilon\beta\gamma/L$. For the second integral, we use the fact that the integrand is bounded by $C/\beta$ and the measure estimate for $O\setminus U_{j_{\ell}}^{\ell}$ to deduce $\|I_{2}\|\leq C\varepsilon\gamma/L$. For the remaining integral $I_{3}$, we note that on $O^{c}$, the function $N_{n,r}^{\ell}$ is bounded by $Cq^{L}/\beta$, which, together with estimate (3.2) and the choice of $L$ implies $\|I_{3}\|\leq C\varepsilon\gamma/L$. Summarizing, this implies $\|\alpha_{r\nu}-\delta_{r\nu}\|\leq C\varepsilon\gamma/L,\qquad\|\nu-r\|\leq 2L.$ This can be used to estimate the difference between $N_{m,r}^{\ell}$ and $N_{n,r}^{\ell}$ for $n\geq m\geq M$ pointwise as follows $\displaystyle\|N_{m,r}^{\ell}(y)-N_{n,r}^{\ell}(y)\|$ $\displaystyle=\Big{\|}\sum_{\nu}(\alpha_{r\nu}-\delta_{r\nu})N_{n,\nu}^{\ell}(y)\Big{\|}\leq C\varepsilon+\sum_{\nu:\|\nu-r\|>2L}\|\alpha_{r\nu}N_{n,\nu}^{\ell}(y)\|,$ by using the bound $\|N_{n,\nu}^{\ell}(y)\|\leq C/\gamma$. By the choice of $L$, the inequality $\|\nu-r\|>2L$ implies $d_{n}(A_{n}(y),E_{n,\nu})>L$ and thus, the geometric decay estimate for $N_{n,\nu}^{\ell}$, the boundedness of $\alpha_{r\nu}$ and the choice of $L$ implies that the latter sum is bounded by $C\varepsilon$. This, in turn, implies the estimate $\|N_{m,r}^{\ell}(y)-N_{n,r}^{\ell}(y)\|\leq C\varepsilon$ and thus the convergence of $N_{n,r}^{\ell}(y)$, which is uniform in $A(y)$ since all the estimates above only depend on $A(y)$ and not on the particular point $y$. Now, estimate (5.7) follows from the corresponding estimate of $N_{n,r}^{\ell}$ by letting $n\to\infty$. ∎ Write $F=Z\times Y$ with $Z=F^{1}\times\cdots\times F^{s}=V_{j_{1}}^{1}\times\cdots\times V_{j_{s}}^{s}$ and $Y=F^{s+1}\times\cdots\times F^{d}$. For an $s$-tuple of integers $i_{1}=(r_{1},\ldots,r_{s})$ and a $(d-s)$-tuple of integers $i_{2}=(r_{s+1},\ldots,r_{d})$, set $N_{m,i_{1}}^{\leq s}I_{Z}=N_{m,r_{1}}^{1}I_{F^{1}}\otimes\cdots\otimes N_{m,r_{s}}^{s}I_{F^{s}}$ and $N_{n,i_{2}}^{>s}I_{Y}=N_{n,r_{s+1}}^{s+1}I_{F^{s+1}}\otimes\cdots\otimes N_{n,r_{d}}^{d}I_{F^{d}}$. The uniform convergence of $N_{m,r_{\ell}}^{\ell}\mathbbm{1}_{V_{j_{\ell}}^{\ell}}$ to $\bar{N}_{r_{\ell}}^{\ell}$ for $\ell\leq s$ as $m\to\infty$ implies that for fixed $n$ and $i_{1}$, the sequence $(N_{m,i_{1}}^{\leq s}I_{Z}\otimes N_{n,i_{2}}^{>s}I_{Y})$ converges in $W$ to some element as $m\to\infty$, which we denote by $\bar{N}_{i_{1}}^{\leq s}\otimes N_{n,i_{2}}^{>s}I_{Y}$. By the continuity of $T$, we also have $T(N_{m,i_{1}}^{\leq s}I_{Z}\otimes N_{n,i_{2}}^{>s}I_{Y})\to T(\bar{N}_{i_{1}}^{\leq s}\otimes N_{n,i_{2}}^{>s}I_{Y})$ in $X$ as $m\to\infty$. Using the expressions $T(\bar{N}_{i_{1}}^{\leq s}\otimes N_{n,i_{2}}^{>s}I_{Y})$ and the dual functions $\bar{N}_{i_{1}}^{\leq s}=\bar{N}_{r_{1}}^{1}\otimes\cdots\otimes\bar{N}_{r_{s}}^{s}$ to $\bar{N}_{i_{1}}^{\leq s}$ given by Lemma 5.3, define (5.10) $u_{n}=\sum_{i_{1},i_{2}}T(\bar{N}_{i_{1}}^{\leq s}\otimes N_{n,i_{2}}^{>s}I_{Y})(\bar{N}_{i_{1}}^{\leq s}\otimes N_{n,i_{2}}^{>s}).$ Next, we show that the sequence $(g_{n,0})$ and the sequence $(u_{n})$ have the same a.e. limit. Indeed, for fixed $y_{1}\in U_{j_{1}}^{1}\times\cdots\times U_{j_{s}}^{s}$, the difference of those two functions has the form $\displaystyle g_{n,0}(y_{1},\cdot)-u_{n}(y_{1},\cdot)$ $\displaystyle=\sum_{i_{2}}N_{n,i_{2}}^{>s}\Big{[}\sum_{i_{1}}T\big{(}(N_{n,i_{1}}^{\leq s}I_{Z}-\bar{N}_{i_{1}}^{\leq s})\otimes N_{n,i_{2}}^{>s}I_{Y}\big{)}N_{n,i_{1}}^{\leq s}(y_{1})$ $\displaystyle\qquad+\sum_{i_{1}}T(\bar{N}_{i_{1}}^{\leq s}\otimes N_{n,i_{2}}^{>s}I_{Y})\big{(}N_{n,i_{1}}^{\leq s}(y_{1})-\bar{N}_{i_{1}}^{\leq s}(y_{1})\big{)}\Big{]}.$ Denote $\mathscr{F}_{n}^{>s}=\mathscr{F}_{n}^{s+1}\otimes\cdots\otimes\mathscr{F}_{n}^{d}$. Using (3.2) for $N_{n,i_{2}}^{>s}$ and $N_{n,i_{1}}^{\leq s}$, Lemma 5.3, the uniform boundedness and the localized support of $\bar{N}_{i_{1}}^{\leq s}$, and the bound (5.4) of the operator $T$ in terms of the measure $\mu$, we obtain for all $\varepsilon>0$ an index $M$ so that for $n\geq M$ (5.11) $\\|g_{n,0}(y_{1},y_{2})-u_{n}(y_{1},y_{2})\\|\leq\sum_{\text{$A$ atom of $\mathscr{F}_{n}^{>s}$}}b_{n}(q,\theta,A,y_{2}),$ where $\theta$ is the measure given by $\theta(A)=\varepsilon\mu\big{(}\bar{I}^{s}\times(A\cap Y)\big{)}$ and the expression $b_{n}(q,\theta,A,y_{2})$ is defined as in (3.4), but with $\theta(A)$ replaced with $\theta(\overline{A})$. By Corollary 3.4 (and the remark succeeding it) with $B=D=\bar{I}^{d-s}$ we obtain $\|L_{t}\|\leq C\theta(\bar{I}^{d-s})/t\leq C\varepsilon\mu(\bar{I}^{d})/t$ with $L_{t}=\\{y_{2}\in I^{d-s}:\limsup_{n}\sum_{\text{$A$ atom of $\mathscr{F}_{n}^{>s}$}}b_{n}(q,\theta,A,y_{2})>t\\}.$ This implies, using also (5.11), $\|\\{y_{2}\in I^{d-s}:\limsup_{n}\\|g_{n,0}(y_{1},y_{2})-u_{n}(y_{1},y_{2})\\|>t\\}\|=0$ for any $t>0$, i.e., $g_{n,0}$ and $u_{n}$ have the same a.e. limit on $F$. Therefore, in order to identify the a.e. limit of $(g_{n,0})$ on $F$ (which, by Cases $1$ and $2$, is also the a.e. limit of $(g_{n})$), we identify the a.e. limit of $(u_{n})$. Similar to Part II in the proof of Theorem 4.1, we want to construct, for each $i_{1}$, a vector measure $\nu_{i_{1}}$ on $\mathscr{A}^{>s}=\cup_{n}\mathscr{F}_{n}^{>s}$ based on the expressions $T(\bar{N}_{i_{1}}^{\leq s}\otimes N_{n,i_{2}}^{>s}I_{Y})$. The aim is to have, for each B-spline function $N_{n,i_{2}}^{>s}$, the representation (5.12) $\int N_{n,i_{2}}^{>s}\,\mathrm{d}\nu_{i_{1}}=T(\bar{N}_{i_{1}}^{\leq s}\otimes N_{n,i_{2}}^{>s}I_{Y}).$ To this end, let $Q=\prod_{\ell=s+1}^{d}(a_{\ell},b_{\ell}]$ be an atom of the $\sigma$-algebra $\mathscr{F}_{n}^{>s}$ for some positive integer $n$. For the definition of the measure $\nu_{i_{1}}(Q)$, we approximate the characteristic function $\mathbbm{1}_{Q}$ of $Q$ by spline functions $f_{m}^{s+1}\otimes\cdots\otimes f_{m}^{d}$ contained in $\cup_{j}\big{(}S^{k_{s+1}}(\mathscr{F}_{j}^{s+1})\otimes\cdots\otimes S^{k_{d}}(\mathscr{F}_{j}^{d})\big{)}$, which will be done as follows. Let $\ell\in\\{s+1,\ldots,d\\}$. If the order of the polynomials $k_{\ell}$ in direction $\ell$ equals $1$ (piecewise constant case), we set $f_{m}^{\ell}=\mathbbm{1}_{(a_{\ell},b_{\ell}]}$ for $m\geq n$, which satisfies $f_{m}^{\ell}\in S^{k_{\ell}}(\mathscr{F}_{m}^{\ell})$. If $k_{\ell}>1$, we apply the following construction of the approximation $f_{m}^{\ell}$ of the characteristic function of the interval $(a_{\ell},b_{\ell}]$. If $a_{\ell}$ is contained in the countable set $\cup_{j}\partial U_{j}^{\ell}$ and if $a_{\ell}$ is not an endpoint of $I$, we choose $c\in(a_{\ell},a_{\ell}+1/m)$ that is not contained in $\cup_{j}\partial U_{j}^{\ell}$. Otherwise, set $c=a_{\ell}$. Similarly, if $b_{\ell}$ is contained in $\cup_{j}\partial U_{j}^{\ell}$ and if $b_{\ell}$ is not an endpoint of $I$, we choose $d\in(b_{\ell},b_{\ell}+1/m)$ that is not contained in $\cup_{j}\partial U_{j}^{\ell}$. Otherwise, set $d=b_{\ell}$. Put $J(x)=\begin{cases}V_{j}^{\ell},&\text{if }x\in U_{j}^{\ell},\\\ \emptyset,&\text{otherwise,}\end{cases}$ and define the interval $J=\Big{(}(c,d]\setminus J(c)\Big{)}\cup J(d),$ which has the property that $J\cap(V^{\ell})^{c}=(c,d]\cap(V^{\ell})^{c}$. We then choose a closed interval $C$ and an open interval $O$ (both in $I$) with $C\subseteq J\subseteq O$ and the property $\|O\setminus C\|\leq 1/m$. The sets $C$ and $O$ are chosen so that as many endpoints of $C$ and $O$ coincide with the corresponding endpoints of $(c,d]$ as possible. Then, let $f_{m}^{\ell}\in\cup_{j}S^{k_{\ell}}(\mathscr{F}_{j}^{\ell})$ be a non- negative function that is bounded by $1$ and satisfies $\operatorname{supp}f_{m}^{\ell}\subseteq O\qquad\text{and}\qquad f_{m}^{\ell}\equiv 1\text{ on }C\cap I.$ This is possible since if $c$ or $d$ are endpoints of $J$, they are contained in $\big{(}\cup_{j}\overline{U_{j}^{\ell}})^{c}$ and thus can be approximated from both sides with grid points $\cup_{j}\Delta_{j}^{\ell}$. Otherwise, the endpoints of $J$ are also endpoints of some set $V_{j}^{\ell}$, which are accumulation points of $\cup_{j}\Delta_{j}^{\ell}$ as well. Then, define $f_{m}=f_{m}^{s+1}\otimes\cdots\otimes f_{m}^{d}$ which gives, for each index $i_{1}$, a Cauchy sequence $\bar{N}_{i_{1}}^{\leq s}\otimes f_{m}I_{Y}$ in $W$. Its limit will be written as $\bar{N}_{i_{1}}^{\leq s}\otimes(I_{Q}\cdot I_{Y})$. Then, continuing in a similar fashion as in Part II of the proof of Theorem 4.1, we make sense of the expression $T\big{(}\bar{N}_{i_{1}}^{\leq s}\otimes(I_{A}\cdot I_{Y})\big{)}$ for any $A\in\mathscr{A}^{>s}$ and define the measure $\nu_{i_{1}}(A)=T\big{(}\bar{N}_{i_{1}}^{\leq s}\otimes(I_{A}\cdot I_{Y})\big{)}$ for $A\in\mathscr{A}^{>s}$ whose total variation satisfies $\|\nu_{i_{1}}\|(I^{d-s})\leq 2^{d-s}\mu(\overline{\operatorname{supp}\bar{N}_{i_{1}}^{\leq s}}\times Y)$. Additionally, for any B-spline function $N_{n,i_{2}}^{>s}$, we have equation (5.12). Now, as in Part III of the proof of Theorem 4.1, denoting by $w_{i_{1}}$ the Radon-Nikodým density of the absolutely continuous part of $\nu_{i_{1}}$ with respect to Lebesgue measure $\lambda^{d-s}$, $u_{n}(y_{1},y_{2})=\sum_{i_{1}}\bar{N}_{i_{1}}^{\leq s}(y_{1})(P_{n}^{>s}\nu_{i_{1}})(y_{2})\to g(y_{1},y_{2}):=\sum_{i_{1}}\bar{N}_{i_{1}}^{\leq s}(y_{1})w_{i_{1}}(y_{2})$ as $n\to\infty$ for almost every $(y_{1},y_{2})\in F$. Using the estimate from Lemma 5.3 for $\bar{N}_{i_{1}}^{\leq s}$ and the above estimate for the total variation of the measures $\nu_{i_{1}}$, we obtain that $\\|g\\|_{L^{1}_{X}(F)}\leq C\cdot\mu(F)$. Thus, we have proven the following theorem: ###### Theorem 5.4. Let $(\mathscr{F}_{n})$ be an interval filtration on $I^{d}$ and let $X$ be a Banach space with RNP. Let $(g_{n})$ be an $X$-valued martingale spline sequence adapted to $(\mathscr{F}_{n})$ with $\sup_{n}\\|g_{n}\\|_{L^{1}_{X}}<\infty$. Then, there exists $g\in L^{1}_{X}(I^{d})$ so that $g_{n}\to g$ almost everywhere with respect to Lebesgue measure $\lambda^{d}$. ###### Remark. Employing the notation developed in this section, we emphasize that the pointwise limit $g$ has the explicit representation $g(y_{1},y_{2}):=\sum_{i_{1}}\bar{N}_{i_{1}}^{\leq s}(y_{1})w_{i_{1}}(y_{2}),\qquad(y_{1},y_{2})\in F,$ where $\bar{N}_{i_{1}}^{\leq s}$ are the functions given by Lemma 5.3 corresponding to $F^{1}\times\cdots\times F^{s}=V_{j_{1}}^{1}\times\cdots\times V_{j_{s}}^{s}$ and the function $w_{i_{1}}$ is the Radon-Nikodým density of the absolutely continuous part (w.r.t Lebesgue measure $\lambda^{d-s}$) of the measure $A\mapsto T\big{(}\bar{N}_{i_{1}}^{\leq s}\otimes(I_{A}\cdot I_{Y})\big{)}$ with $Y=(V^{s+1})^{c}\times\cdots\times(V^{d})^{c}$. ### Acknowledgements The author is supported by the Austrian Science Fund FWF, project P32342. ## References * [1] M. de Guzmán. An inequality for the Hardy-Littlewood maximal operator with respect to a product of differentiation bases. Studia Math., 49:185–194, 1973/74. * [2] J. Diestel and J. J. Uhl, Jr. Vector measures. American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis, Mathematical Surveys, No. 15. * [3] M. v. Golitschek. On the $L_{\infty}$-norm of the orthogonal projector onto splines. A short proof of A. Shadrin’s theorem. J. Approx. Theory, 181:30–42, 2014. * [4] B. Jessen, J. Marcinkiewicz, and A. Zygmund. Note on the differentiability of multiple integrals. Fundamenta Mathematicae, 25(1):217–234, 1935. * [5] P. F. X. Müller and M. Passenbrunner. Almost everywhere convergence of spline sequences. Israel J. Math., 240(1):149–177, 2020. * [6] J. Neveu. Discrete-parameter martingales. North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, revised edition, 1975. Translated from the French by T. P. Speed, North-Holland Mathematical Library, Vol. 10. * [7] M. Passenbrunner. Spline characterizations of the Radon-Nikodým property. Proc. Amer. Math. Soc., 148(2):811–824, 2020. * [8] M. Passenbrunner and J. Prochno. On almost everywhere convergence of tensor product spline projections. Michigan Math. J., 68(1):3–17, 2019. * [9] M. Passenbrunner and A. Shadrin. On almost everywhere convergence of orthogonal spline projections with arbitrary knots. J. Approx. Theory, 180:77–89, 2014. * [10] S. Saks. On the strong derivatives of functions of intervals. Fundamenta Mathematicae, 25(1):235–252, 1935. * [11] L. L. Schumaker. Spline functions: basic theory. Cambridge Mathematical Library. Cambridge University Press, Cambridge, third edition, 2007. * [12] A. Shadrin. The $L_{\infty}$-norm of the $L_{2}$-spline projector is bounded independently of the knot sequence: a proof of de Boor’s conjecture. Acta Math., 187(1):59–137, 2001.
# Nonstationary Stochastic Multiarmed Bandits: UCB Policies and Minimax Regret ††thanks: This work was supported by NSF Award IIS-1734272 Lai Wei Vaibhav Srivastava L. Wei and V. Srivastava are with the Department of Electrical and Computer Engineering. Michigan State University, East Lansing, MI 48823 USA. e-mail<EMAIL_ADDRESS>e-mail<EMAIL_ADDRESS> ###### Abstract We study the nonstationary stochastic Multi-Armed Bandit (MAB) problem in which the distribution of rewards associated with each arm are assumed to be time-varying and the total variation in the expected rewards is subject to a variation budget. The regret of a policy is defined by the difference in the expected cumulative rewards obtained using the policy and using an oracle that selects the arm with the maximum mean reward at each time. We characterize the performance of the proposed policies in terms of the worst-case regret, which is the supremum of the regret over the set of reward distribution sequences satisfying the variation budget. We extend Upper-Confidence Bound (UCB)-based policies with three different approaches, namely, periodic resetting, sliding observation window and discount factor and show that they are order-optimal with respect to the minimax regret, i.e., the minimum worst-case regret achieved by any policy. We also relax the sub-Gaussian assumption on reward distributions and develop robust versions the proposed polices that can handle heavy-tailed reward distributions and maintain their performance guarantees. ###### Index Terms: Nonstationary multiarmed bandit, variation budget, minimax regret, upper- confidence bound, heavy-tailed distributions. ## I Introduction Uncertainty and nonstationarity of the environment are two of the major barriers in decision-making problems across scientific disciplines, including engineering, economics, social science, neuroscience, and ecology. An efficient strategy in such environments requires balancing several tradeoffs, including _exploration-versus-exploitation_ , i.e., choosing between the most informative and the empirically most rewarding alternatives, and _remembering- versus-forgetting_ , i.e., using more but possibly outdated information or using less but recent information. The stochastic MAB problem is a canonical formulation of the exploration- versus-exploitation tradeoff. In an MAB problem, an agent selects one from $K$ options at each time and receives a reward associated with it. The reward sequence at each option is assumed to be an unknown i.i.d random process. The MAB formulation has been applied in many scientific and technological areas. For example, it is used for opportunistic spectrum access in communication networks, wherein the arm models the availability of a channel [1, 2]. In MAB formulation of online learning for demand response[3, 4], an aggregator calls upon a subset of users (arms) who have an unknown response to the request to reduce their loads. MAB formulation has also been used in robotic foraging and surveillance [5, 6, 7, 8] and acoustic relay positioning for underwater communication [9], wherein the information gain at different sites is modeled as rewards from arms. Besides, contextual bandits are widely used in recommender systems [10, 11], wherein the acceptation of a recommendation corresponds to the rewards from an arm. The stationarity assumption in classic MAB problems limits their utility in these applications since channel usage, robot working environment and people’s preference are inherently uncertain and evolving. In this paper, we relax this assumption and study non-stationary stochastic MAB problems. Robbins [12] formulated the objective of the stochastic MAB problem as minimizing the _regret_ , that is, the loss in expected cumulative rewards caused by failing to select the best arm every time. In their seminal work, Lai and Robbins [13], followed by Burnetas and Katehakis [14], established a logarithm _problem-dependent_ asymptotic lower bound on the regret achieved by any policy, which has a leading constant determined by the underlying reward distributions. A general method of constructing UCB rules for parametric families of reward distributions is also presented in [13], and the associated policy is shown to attain the logarithm lower bound. Several subsequent UCB- based algorithms [15, 16] with efficient finite time performance have been proposed. The adversarial MAB [17] is a paradigmatic nonstationary problem. In this model, the bounded reward sequence at each arm is arbitrary. The performance of an policy is evaluated using the _weak regret_ , which is the difference in the cumulated reward of a policy compared against the best single action policy. A $\Omega(\sqrt{KT})$ lower bound on the weak regret and a near- optimal policy Exp$3$ is also presented in [17]. While being able to capture nonstationarity, the generality of the reward model in adversarial MAB makes the investigation of globally optimal policies very challenging. The nonstationary stochastic MAB can be viewed as a compromise between stationary stochastic MAB and adversarial MAB. It maintains the stochastic nature of the reward sequence while allowing some degree of nonstationarity in reward distributions. Instead of the weak regret analyzed in adversarial MAB, a strong notion of regret defined with respect to the best arm at each time step is studied in these problems. A broadly studied nonstationary problem is _piecewise stationary_ MAB, wherein the reward distributions are piecewise stationary. To deal with the remembering-versus-forgetting tradeoff, the idea of using discount factor to compute the UCB index is proposed in [18]. Garivier and Moulines [19] present and analyze Discounted UCB (D-UCB) and Sliding-Window UCB (SW-UCB), in which they compute the UCB using discounted sampling history and recent sampling history, respectively. They pointed out that if the number of change points $N_{T}$ is available, both algorithms can be tuned to achieve regret close to the $\Omega(\sqrt{KN_{T}T})$ regret lower bound. In our earlier work [20], the near optimal regret is achieved using deterministic sequencing of explore and exploit with limited memory. Other works handle the change of reward distributions in an adaptive manner by adopting change point detection techniques [21, 22, 23, 24, 25]. A more general nonstationary problem is studied in [26], wherein the cumulative maximum variation in mean rewards is subject to a variation budget $V_{T}$. Additionally, the authors in [26] establish a $\Omega((KV_{T})^{\frac{1}{3}}T^{\frac{2}{3}})$ minimax regret lower bound and propose the Rexp$3$ policy. In their subsequent work [27], they tune Exp$3$.S policy from [17] to achieve near optimal worst-case regret. Discounted Thomson Sampling (DTS) [28] has also been shown to have good experimental performance within this general framework. However, we are not aware of any analytic regret bounds for the DTS algorithm. In this paper, we follow the more general nonstationary stochastic MAB formulation in [26] and design UCB-based policies that achieve efficient performance in environments with sub-Gaussian as well as heavy-tailed rewards. We focus on UCB-based policies instead of EXP$3$-type policies because EXP$3$-type policies require bounded rewards and have large variance in cumulative rewards [17]. Additionally, by using robust mean estimator, UCB- based policies for light-tailed rewards can be extended to handle heavy-tailed reward distributions, which exist in many domains such as social networks [29] and financial markets [30]. The major contributions of this work are: * • Assuming the variation density $V_{T}/T$ is known, we extend MOSS [31] to design Resetting MOSS (R-MOSS) and Sliding-Window MOSS (SW-MOSS). Also, we show D-UCB can be tuned to solve the problem. * • With rigorous analysis, we show that R-MOSS and SW-MOSS achieve the exact order-optimal minimax regret and D-UCB achieves near-optimal worst-case regret. * • We relax the bounded or sub-Gaussian assumption on the rewards required by Rexp$3$ and SW-UCB and design policies robust to heavy-tailed rewards. We show the theoretical guarantees on the worst-case regret can be maintained by the robust policies. The remainder of the paper is organized as follows. We formulate nonstationary stochastic MAB with variation budget in Section II and review some preliminaries in Section III. In Section IV, we present and analyze three UCB policies: R-MOSS, SW-MOSS and D-UCB. We present and analyze algorithms for nonstationary heavy-tailed bandit in Section V. We complement the theoretical results with numerical illustrations in Section VI and conclude this work in Section VII. ## II Problem Formulation We consider a nonstationary stochastic MAB problem with $K$ arms and a horizon length $T$. Let $\mathcal{K}\mathrel{\mathop{\mathchar 58\relax}}=\\{1,\dots,K\\}$ be the set of arms and $\mathcal{T}\mathrel{\mathop{\mathchar 58\relax}}=\\{1,\dots,T\\}$ be the sequence of time slots. The reward sequence $\left\\{X_{t}^{k}\right\\}_{t\in\mathcal{T}}$ for each arm $k\in\mathcal{K}$ is composed of independent samples from potentially time-varying probability distribution function sequence $f_{\mathcal{T}}^{k}\mathrel{\mathop{\mathchar 58\relax}}=\left\\{f_{t}^{k}(x)\right\\}_{t\in\mathcal{T}}$. We refer to the set $\mathcal{F}_{T}^{\mathcal{K}}=\left\\{f_{\mathcal{T}}^{k}\;\|\;k\in\mathcal{K}\right\\}$ containing reward distribution sequences at all arms as the _environment_. Let $\mu_{t}^{k}=\mathbb{E}[X_{t}^{k}]$. Then, the _total variation_ of $\mathcal{F}_{T}^{\mathcal{K}}$ is defined by $v\big{(}\mathcal{F}_{T}^{\mathcal{K}}\big{)}\mathrel{\mathop{\mathchar 58\relax}}=\sum_{t=1}^{T-1}\sup_{k\in\mathcal{K}}\>\mathinner{\\!\left\lvert\mu_{t+1}^{k}-\mu_{t}^{k}\right\rvert},$ (1) which captures the non-stationarity of the environment. We focus on the class of non-stationary environments that have the total variation within a _variation budget_ $V_{T}\geq 0$ which is defined by $\mathcal{E}(V_{T},T,K)\mathrel{\mathop{\mathchar 58\relax}}=\big{\\{}\mathcal{F}_{T}^{\mathcal{K}}\;\|\;v\big{(}\mathcal{F}_{T}^{\mathcal{K}}\big{)}\leq V_{T}\big{\\}}.$ At each time slot $t\in\mathcal{T}$, a decision-making agent selects an arm $\varphi_{t}\in\mathcal{K}$ and receives an associated random reward $X_{t}^{\varphi_{t}}$. The objective is to maximize the expected value of the _cumulative reward_ $S_{T}\mathrel{\mathop{\mathchar 58\relax}}=\sum_{t=1}^{T}X_{t}^{\varphi_{t}}$. We assume that $\varphi_{t}$ is selected based upon past observations $\\{X_{s}^{\varphi_{s}},\varphi_{s}\\}_{s=1}^{t-1}$ following some policy $\rho$. Specifically, $\rho$ determines the conditional distribution $\mathbb{P}^{\rho}\left(\varphi_{t}=k\;\|\;\\{X_{s}^{\varphi_{s}},\varphi_{s}\\}_{s=1}^{t-1}\right)$ at each time $t\in\\{1,\dots,T-1\\}$. If $\mathbb{P}^{\rho}\left(\cdot\right)$ takes binary values, we call $\rho$ deterministic; otherwise, it is called stochastic. Let the expected reward from the best arm at time $t$ be $\mu_{t}^{}=\max_{k\in\mathcal{K}}\mu_{t}^{k}.$ Then, maximizing the expected cumulative reward is equivalent to minimizing the _regret_ defined by $R^{\rho}_{T}\mathrel{\mathop{\mathchar 58\relax}}=\sum_{t=1}^{T}\mu_{t}^{}-\mathbb{E}^{\rho}[S_{T}]=\mathbb{E}^{\rho}\Bigg{[}\sum_{t=1}^{T}\mu_{t}^{}-\mu_{t}^{\varphi_{t}}\Bigg{]},$ where the expectation is with respect to different realization of $\varphi_{t}$ that depends on obtained rewards through policy $\rho$. Note that the performance of a policy $\rho$ differs with different $\mathcal{F}_{T}^{\mathcal{K}}\in\mathcal{E}(V_{T},T,K)$. For a fixed variation budget $V_{T}$ and a policy $\rho$, the _worst-case regret_ is the regret with respect to the worst possible choice of environment, i.e., $R_{\textup{worst}}^{\rho}(V_{T},T,K)=\sup_{\mathcal{F}_{T}^{\mathcal{K}}\in\mathcal{E}(V_{T},T,K)}\>R_{T}^{\rho}.$ In this paper, we aim at designing policies to minimize the worst-case regret. The optimal worst-case regret achieved by any policy is called the _minimax regret_ , and is defined by $\inf_{\rho}\sup_{\mathcal{F}_{T}^{\mathcal{K}}\in\mathcal{E}(V_{T},T,K)}\>R_{T}^{\rho}.$ We will study the nonstationary MAB problem under the following two classes of reward distributions: ###### Assumption 1 (Sub-Gaussian reward). For any $k\in\mathcal{K}$ and any $t\in\mathcal{T}$, distribution $f_{t}^{k}(x)$ is $1/2$ sub-Gaussian, i.e., $\forall\lambda\in\mathbb{R}\mathrel{\mathop{\mathchar 58\relax}}\mathbb{E}\left[\exp(\lambda(X_{t}^{k}-\mu))\right]\leq\exp\left(\frac{\lambda^{2}}{8}\right).$ Moreover, for any arm $k\in\mathcal{K}$ and any time $t\in\mathcal{T}$, $\mathbb{E}\left[X_{t}^{k}\right]\in[a,a+b]$, where $a\in\mathbb{R}$ and $b>0$. ###### Assumption 2 (Heavy-tailed reward). For any arm $k\in\mathcal{K}$ and any time $t\in\mathcal{T}$, $\mathbb{E}\left[(X_{t}^{k})^{2}\right]\leq 1$. ## III Preliminaries In this section, we review existing minimax regret lower bounds and minimax policies from literature. These results apply to both sub-Gaussian and heavy- tailed rewards. The discussion is made first for $V_{T}=0$. Then, we show how the minimax regret lower bound for $V_{T}=0$ can be extended to establish the minimax regret lower bound for $V_{T}>0$. To this end, we review two UCB algorithms for the stationary stochastic MAB problem: UCB1 and MOSS. In the later sections, they are extended to design a variety of policies to match with the minimax regret lower bound for $V_{T}>0$. ### III-A Lower Bound for Minimax Regret when $V_{T}=0$ In the setting of $V_{T}=0$, for each arm $k\in\mathcal{K}$, $\mu_{t}^{k}$ is identical for all $t\in\mathcal{T}$. In stationary stochastic MAB problems, the rewards from each arm $k\in\mathcal{K}$ are independent and identically distributed, so they belong to the environment set $\mathcal{E}(0,T,K)$. According to [32], if $V_{T}=0$, the minimax regret is no smaller than $1/20\sqrt{KT}$. This result is closely related to the standard logarithmic lower bound on regret for stationary stochastic MAB problems as discussed below. Consider a scenario in which there is a unique best arm and all other arms have identical mean rewards such that the gap between optimal and suboptimal mean rewards is $\Delta$. From [33], for such a stationary stochastic MAB problem $R_{T}^{\rho}\geq C_{1}\frac{K}{\Delta}\ln\Big{(}\frac{T\Delta^{2}}{K}\Big{)}+C_{2}\frac{K}{\Delta},$ (2) for any policy $\rho$, where $C_{1}$ and $C_{2}$ are some positive constants. It needs to be noted that for $\Delta=\sqrt{K/T}$, the above lower bound becomes $C_{2}\sqrt{KT}$, which matches with the lower bound $1/20\sqrt{KT}$. ### III-B Lower Bound for Minimax Regret when $V_{T}>0$ In the setting of $V_{T}>0$, we recall here the minimax regret lower bound for nonstationary stochastic MAB problems. ###### Lemma 1 (Minimax Lower Bound: $V_{T}>0$ [26]). For the non-stationary MAB problem with $K$ arms, time horizon $T$ and variation budget $V_{T}\in[1/K,T/K]$, $\inf_{\rho}\sup_{\mathcal{F}_{T}^{\mathcal{K}}\in\mathcal{E}(V_{T},T,K)}R_{T}^{\rho}\geq C(KV_{T})^{\frac{1}{3}}T^{\frac{2}{3}},$ where $C\in\mathbb{R}_{>0}$ is some constant. To understand this lower bound, consider the following non-stationary environment. The horizon $\mathcal{T}$ is partitioned into epochs of length $\tau=\big{\lceil}{K^{\frac{1}{3}}({T/V_{T}})^{\frac{2}{3}}}\big{\rceil}$. In each epoch, the reward distribution sequences are stationary and all the arms have identical mean rewards except for the unique best arm. Let the gap in the mean be $\Delta=\sqrt{K/\tau}$. The index of the best arm switches at the end of each epoch following some unknown rule. So, the total variation is no greater than $\Delta T/\tau$, which satisfies the variation budget $V_{T}$. Besides, for any policy $\rho$, we know from (2) that worst case regret in each epoch is no less than $C_{2}\sqrt{K\tau}$. Summing up the regret over all the epochs, minimax regret is lower bounded by $T/\tau\times C_{2}\sqrt{K\tau}$, which is consistent with Lemma 1. ### III-C UCB Algorithms in Stationary Environments The family of UCB algorithms uses the principle called optimism in the face of uncertainty. In these policies, at each time slot, a UCB index which is a statistical index composed of both mean reward estimate and the associated uncertainty measure is computed at each arm, and the arm with the maximum UCB is picked. Within the family of UCB algorithms, two state-of-the-art algorithms for the stationary stochastic MAB problems are UCB$1$ [15] and MOSS [31]. Let $n_{k}(t)$ be the number of times arm $k$ is sampled until time $t-1$, and $\hat{\mu}_{k,n_{k}(t)}$ be the associated empirical mean. Then, UCB$1$ computes the UCB index for each arm $k$ at time $t$ as $g_{k,t}^{\textup{UCB1}}=\hat{\mu}_{k,n_{k}(t)}+\sqrt{\frac{2\ln t}{n_{k}(t)}}.$ It has been proved in [15] that, for the stationary stochastic MAB problem, UCB1 satisfies $R_{T}^{\textup{UCB1}}\leq 8\sum_{k\mathrel{\mathop{\mathchar 58\relax}}\Delta_{k}>0}\frac{\ln T}{\Delta_{k}}+\left(1+\frac{\pi^{2}}{3}\right)\sum_{k=1}^{K}\Delta_{k},$ where $\Delta_{k}$ is the difference in the mean rewards from arm $k$ and the best arm. In [31], a simple variant of this result is given by selecting values for $\Delta_{k}$ to maximize the upper bound, resulting in $\sup_{\mathcal{F}_{T}^{\mathcal{K}}\in\mathcal{E}(0,T,K)}R_{T}^{\textup{UCB1}}\leq 10\sqrt{(K-1)T(\ln T)}.$ Comparing this result with the lower bound on the minimax regret discussed in Section III-A, there exists an extra factor $\sqrt{\ln T}$. This issue has been resolved by the MOSS algorithm. With prior knowledge of horizon length $T$, and the UCB index for MOSS is expressed as $g_{k,t}^{\textup{MOSS}}=\hat{\mu}_{k,n_{k}(t)}+\sqrt{\frac{\max\Big{(}\ln\Big{(}\frac{T}{Kn_{k}(t)}\Big{)},0\Big{)}}{n_{k}(t)}}.$ We now recall the worst-case regret upper bound for MOSS. ###### Lemma 2 (Worst-case regret upper bound for MOSS [31]). For the stationary stochastic MAB problem ($V_{T}=0$), the worst-case regret of the MOSS algorithm satisfies $\sup_{\mathcal{F}_{T}^{\mathcal{K}}\in\mathcal{E}(0,T,K)}R_{T}^{\text{MOSS}}\leq 49\sqrt{KT}.$ ## IV UCB Algorithms for Sub-Gaussian Nonstationary Stochastic MAB Problems In this section, we extend UCB$1$ and MOSS to design nonstationary UCB policies for scenarios with $V_{T}>0$. Three different techniques are employed, namely periodic resetting, sliding observation window and discount factor, to deal with the remembering-forgetting tradeoff. The proposed algorithms are analyzed to provide guarantees on the worst-case regret. We show their performances match closely with the lower bound in Lemma 1. The following notations are used in later discussions. Let $N=\left\lceil T/\tau\right\rceil$, for some $\tau\in\\{1,\dots,T\\}$, and let $\\{\mathcal{T}_{1},\ldots,\mathcal{T}_{N}\\}$ be a partition of time slots $\mathcal{T}$, where each epoch $\mathcal{T}_{i}$ has length $\tau$ except possibly $\mathcal{T}_{N}$. In particular, $\mathcal{T}_{i}=\Big{\\{}1+(i-1)\tau\>,\ldots,\>\min\left(i\tau,T\right)\Big{\\}},\;i\in\\{1,\dots,N\\}.$ Let the maximum mean reward within $\mathcal{T}_{i}$ be achieved at time $\tau_{i}\in\mathcal{T}_{i}$ and arm $\kappa_{i}$, i.e., $\mu^{\kappa_{i}}_{\tau_{i}}=\max_{t\in\mathcal{T}_{i}}\;\mu_{t}^{}$. We define the variation within $\mathcal{T}_{i}$ as $v_{i}\mathrel{\mathop{\mathchar 58\relax}}=\sum_{t\in\mathcal{T}_{i}}\>\sup_{k\in\mathcal{K}}\>\mathinner{\\!\left\lvert\mu_{t+1}^{k}-\mu_{t}^{k}\right\rvert},$ where we trivially assign $\mu_{T+1}^{k}=\mu_{T}^{k}$ for all $k\in\mathcal{K}$. Let $\mathbf{1}\left\\{\cdot\right\\}$ denote the indicator function and $\mathinner{\\!\left\lvert\cdot\right\rvert}$ denote the cardinality of the set, if its argument is a set, and the absolute value if its argument is a real number. ### IV-A Resetting MOSS Algorithm Periodic resetting is an effective technique to preserve the freshness and authenticity of the information history. It has been employed in [26] to modify Exp$3$ to design Rexp$3$ policy for nonstationary stochastic MAB problems. We extend this approach to MOSS and propose nonstationary policy Resetting MOSS (R-MOSS). In R-MOSS, after every $\tau$ time slots, the sampling history is erased and MOSS is restarted. The pseudo-code is provided in Algorithm 1 and the performance in terms of the worst-case regret for is established below. Input : $V_{T}\in\mathbb{R}_{\geq 0}$ and $T\in\mathbb{N}$ Set : $\tau=\Big{\lceil}{K^{\frac{1}{3}}\left(T/V_{T}\right)^{\frac{2}{3}}}\Big{\rceil}$ Output : sequence of arm selection 1while _$t\leq T$_ do 2if _$\mod(t,\tau)=0$_ then 3Restart the MOSS policy; Algorithm 1 R-MOSS ###### Theorem 3. For the sub-Gaussian nonstationary MAB problem with $K$ arms, time horizon $T$, variation budget $V_{T}>0$, and $\tau=\Big{\lceil}{K^{\frac{1}{3}}\left(T/V_{T}\right)^{\frac{2}{3}}}\Big{\rceil}$, the worst case regret of R-MOSS satisfies $\sup_{\mathcal{F}_{T}^{\mathcal{K}}\in\mathcal{E}(V_{T},T,K)}R_{T}^{\textup{R-MOSS}}\in\mathcal{O}((KV_{T})^{\frac{1}{3}}T^{\frac{2}{3}}).$ ###### Sketch of the proof. Note that one run of MOSS takes place in each epoch. For epoch $\mathcal{T}_{i}$, define the set of _bad arms_ for R-MOSS by $\mathcal{B}_{i}^{\textup{R}}\mathrel{\mathop{\mathchar 58\relax}}=\left\\{k\in\mathcal{K}\;\|\;\mu_{\tau_{i}}^{\kappa_{i}}-\mu_{\tau_{i}}^{k}\geq 2v_{i}\right\\}.$ (3) Notice that for any $t_{1},t_{2}\in\mathcal{T}_{i}$, $\mathinner{\\!\left\lvert\mu_{t_{1}}^{k}-\mu_{t_{2}}^{k}\right\rvert}\leq v_{i},\quad\forall k\in\mathcal{K}.$ (4) Therefore, for any $t\in\mathcal{T}_{i}$, we have $\displaystyle\mu_{t}^{}-\mu_{t}^{\varphi_{t}}$ $\displaystyle\leq\mu_{\tau_{i}}^{\kappa_{i}}-\mu_{t}^{\varphi_{t}}\leq\mu_{\tau_{i}}^{\kappa_{i}}-\mu_{\tau_{i}}^{\varphi_{t}}+v_{i}.$ Then, the regret from $\mathcal{T}_{i}$ can be bounded as the following, $\displaystyle\mathbb{E}\bigg{[}\sum_{t\in\mathcal{T}_{i}}\mu_{t}^{}-\mu_{t}^{\varphi_{t}}\bigg{]}$ $\displaystyle\leq\mathinner{\\!\left\lvert\mathcal{T}_{i}\right\rvert}v_{i}+\mathbb{E}\bigg{[}\sum_{t\in\mathcal{T}_{i}}\mu_{\tau_{i}}^{\kappa_{i}}-\mu_{\tau_{i}}^{\varphi_{t}}\bigg{]}$ $\displaystyle\leq 3\mathinner{\\!\left\lvert\mathcal{T}_{i}\right\rvert}v_{i}+S_{i},$ (5) where $\displaystyle S_{i}=\mathbb{E}\bigg{[}\sum_{t\in\mathcal{T}_{i}}\sum_{k\in\mathcal{B}_{i}^{\textup{R}}}\mathbf{1}\left\\{\varphi_{t}=k\right\\}\left(\mu_{\tau_{i}}^{\kappa_{i}}-\mu_{\tau_{i}}^{\varphi_{t}}-2v_{i}\right)\bigg{]}$. Now, we have decoupled the problem, enabling us to the generalize the analysis of MOSS in stationary environment [31] to bound $S_{i}$. We will only specify the generalization steps and skip the details for brevity. First notice inequality (4) indicates that for any $k\in\mathcal{B}_{i}^{\textup{R}}$ and any $t\in\mathcal{T}_{i}$, $\mu_{t}^{\kappa_{i}}\geq\mu_{\tau_{i}}^{\kappa_{i}}-v_{i}\text{ and }\mu_{t}^{k}\leq\mu_{\tau_{i}}^{k}+v_{i}.$ So, at any $t\in\mathcal{T}_{i}$, $\hat{\mu}_{{\kappa_{i}},n_{\kappa_{i}}(t)}$ concentrate around a value no smaller than $\mu_{\tau_{i}}^{\kappa_{i}}-v_{i}$, and $\hat{\mu}_{k,n_{k}(t)}$ concentrate around a value no greater than $\mu_{\tau_{i}}^{k}+v_{i}$ for any $k\in B_{i}^{\textup{R}}$. Also $\mu_{\tau_{i}}^{\kappa_{i}}-v_{i}\geq\mu_{\tau_{i}}^{k}+v_{i}$ due to the definition in (3). In the analysis of MOSS in stationary environment [31], the UCB of each suboptimal arm is compared with the best arm and each selection of suboptimal arm $k$ contribute $\Delta_{k}$ in regret. Here, we can apply a similar analysis by comparing the UCB of each arm $k\in B_{i}^{\textup{R}}$ with $\kappa_{i}$ and each selection of arm $k\in B_{i}^{\textup{R}}$ contributes $(\mu_{\tau_{i}}^{\kappa_{i}}-v_{i})-(\mu_{\tau_{i}}^{k}+v_{i})$ in $S_{i}$. Accordingly, we borrow the upper bound in Lemma 2 to get $S_{i}\leq 49\sqrt{K\mathinner{\\!\left\lvert\mathcal{T}_{i}\right\rvert}}$. Substituting the upper bound on $S_{i}$ into (5) and summarizing over all the epochs, we conclude that $\sup_{\mathcal{F}_{T}^{\mathcal{K}}\in\mathcal{E}(V_{T},T,K)}R_{T}^{\textup{R-MOSS}}\leq 3\tau V_{T}+\sum_{i=1}^{N}49\sqrt{K\tau},$ which implies the theorem. ∎ The upper bound in Theorem 3 is in the same order as the lower bound in Lemma 1. So, the worst-case regret for R-MOSS is order optimal. ### IV-B Sliding-Window MOSS Algorithm We have shown that periodic resetting coarsely adapts the stationary policy to a nonstationary setting. However, it is inefficient to entirely remove the sampling history at the restarting points and the regret accumulates quickly close to these points. In [19], a sliding observation window is used to erase the outdated information smoothly and more efficiently utilize the information history. The authors proposed the SW-UCB algorithm that intends to solve the MAB problem with piece-wise stationary mean rewards. We show that a similar approach can also deal with the general nonstationary environment with a variation budget. In contrast to SW-UCB, we integrate the sliding window technique with MOSS instead of UCB1 and achieve the order optimal worst-case regret. Let the sliding observation window at time $t$ be $\mathcal{W}_{t}\mathrel{\mathop{\mathchar 58\relax}}=\left\\{\min(1,t-\tau),\ldots,t-1\right\\}$. Then, the associated mean estimator is given by $\hat{\mu}_{n_{k}(t)}^{k}\\!=\\!\frac{1}{n_{k}(t)}\\!\sum_{s\in\mathcal{W}_{t}}\\!\\!X_{s}\mathbf{1}\\{\varphi_{s}=k\\},\,\,n_{k}(t)=\\!\sum_{s\in\mathcal{W}_{t}}\\!\\!\mathbf{1}{\\{\varphi_{s}=k\\}}.$ For each arm $k\in\mathcal{K}$, define the UCB index for SW-MOSS by $g_{t}^{k}=\hat{\mu}_{n_{k}(t)}^{k}+c_{n_{k}(k)},\,\,c_{n_{k}(t)}=\sqrt{\eta\frac{\max\Big{(}\ln\Big{(}\frac{\tau}{Kn_{k}(t)}\Big{)},0\Big{)}}{n_{k}(t)}},$ where $\eta>1/2$ is a tunable parameter. With these notations, SW-MOSS is defined in Algorithm 2. To analyze it, we will use the following concentration bound for sub-Gaussian random variables. Input : $V_{T}\in\mathbb{R}_{>0}$, $T\in\mathbb{N}$ and $\eta>1/2$ Set : $\tau=\Big{\lceil}{K^{\frac{1}{3}}\left(T/V_{T}\right)^{\frac{2}{3}}}\Big{\rceil}$ Output : sequence of arm selection 1Pick each arm once. 2while _$t\leq T$_ do Compute statistics within $\mathcal{W}_{t}=\left\\{\min(1,t-\tau),\ldots,t-1\right\\}$: $\hat{\mu}_{n_{k}(t)}^{k}\\!=\\!\frac{1}{n_{k}(t)}\\!\sum_{s\in\mathcal{W}_{t}}\\!\\!X_{s}\mathbf{1}\\{\varphi_{s}=k\\},\,\,n_{k}(t)=\\!\sum_{s\in\mathcal{W}_{t}}\\!\\!\mathbf{1}{\\{\varphi_{s}=k\\}}$ Pick arm $\displaystyle\varphi_{t}=\arg\max_{k\in\mathcal{K}}\,\hat{\mu}_{n_{k}(t)}^{k}+\sqrt{\eta\frac{\max\Big{(}\ln\Big{(}\frac{\tau}{Kn_{k}(t)}\Big{)},0\Big{)}}{n_{k}(t)}}$; Algorithm 2 SW-MOSS ###### Fact 1 (Maximal Hoeffding inequality[34]). Let $X_{1},\ldots,X_{n}$ be a sequence of independent $1/2$ sub-Gaussian random variables. Define $d_{i}\mathrel{\mathop{\mathchar 58\relax}}=X_{i}-\mu_{i}$, then for any $\delta>0$, $\displaystyle\mathbb{P}\bigg{(}\exists m\in\\{1,\dots,n\\}\mathrel{\mathop{\mathchar 58\relax}}\sum_{i=1}^{m}d_{i}\geq\delta\bigg{)}\leq\exp\left(-{2\delta^{2}}/{n}\right)$ and $\displaystyle\mathbb{P}\bigg{(}\exists m\in\\{1,\dots,n\\}\mathrel{\mathop{\mathchar 58\relax}}\sum_{i=1}^{m}d_{i}\leq-\delta\bigg{)}\leq\exp\left(-{2\delta^{2}}/{n}\right).$ At time $t$, for each arm $k\in\mathcal{K}$ define $M_{t}^{k}\mathrel{\mathop{\mathchar 58\relax}}=\frac{1}{n_{k}(t)}\sum_{s\in\mathcal{W}_{t}}\mu_{s}^{k}\mathbf{1}_{\\{\varphi_{s}=k\\}}.$ Now, we are ready to present concentration bounds for the sliding window empirical mean $\hat{\mu}_{n_{k}(t)}^{k}$. ###### Lemma 4. For any arm $k\in\mathcal{K}$ and any time $t\in\mathcal{T}$, if $\eta>1/2$, for any $x>0$ and $l\geq 1$, the probability of event $A\mathrel{\mathop{\mathchar 58\relax}}=\big{\\{}\hat{\mu}_{n_{k}(t)}^{k}+c_{n_{k}(t)}\leq M_{t}^{k}-x,n_{k}(t)\geq l\big{\\}}$ is no greater than $\frac{(2\eta)^{\frac{3}{2}}}{\ln(2\eta)}\frac{K}{\tau x^{2}}\exp\left(-{x^{2}l}/{\eta}\right).$ (6) The probability of event $B\mathrel{\mathop{\mathchar 58\relax}}=\big{\\{}\hat{\mu}_{n_{k}(t)}^{k}-c_{n_{k}(t)}\geq M_{t}^{k}+x,n_{k}(t)\geq l\big{\\}}$ is also upper bounded by (6). ###### Proof. For any $t\in\mathcal{T}$, let $u_{i}^{kt}$ be the $i$-th time slot when arm $k$ is selected within $\mathcal{W}_{t}$ and let $d_{i}^{kt}=X_{u_{i}^{kt}}^{k}-\mu_{u_{i}^{kt}}^{k}$. Note that $\mathbb{P}\left(A\right)\leq\mathbb{P}\bigg{(}\exists m\in\left\\{l,\ldots,\tau\right\\}\mathrel{\mathop{\mathchar 58\relax}}\frac{1}{m}\sum_{i=1}^{m}d_{i}^{kt}\leq-x-c_{m}\bigg{)},$ Let $a=\sqrt{2\eta}$ such that $a>1$. We now apply a peeling argument [35, Sec 2.2] with geometric grid $a^{s}l<m\leq a^{s+1}l$ over $\left\\{l,\ldots,\tau\right\\}$. Since $c_{m}$ is monotonically decreasing in $m$, $\displaystyle\mathbb{P}\bigg{(}\exists m\in\\{l,\ldots,\tau\\}\mathrel{\mathop{\mathchar 58\relax}}\frac{1}{m}\sum_{i=1}^{m}d_{i}^{kt}\leq-x-c_{m}\bigg{)}$ $\displaystyle\leq$ $\displaystyle\sum_{s\geq 0}\mathbb{P}\bigg{(}\exists m\in[a^{s}l,a^{s+1}l)\mathrel{\mathop{\mathchar 58\relax}}\sum_{i=1}^{m}d_{i}^{kt}\leq-a^{s}l\left(x+c_{a^{s+1}l}\right)\bigg{)}.$ According to Fact 1, the above summand is no greater than $\displaystyle\sum_{s\geq 0}\mathbb{P}\bigg{(}\exists m\in[1,a^{s+1}l)\mathrel{\mathop{\mathchar 58\relax}}\sum_{i=1}^{m}d_{i}^{kt}\leq-a^{s}l\left(x+c_{a^{s+1}l}\right)\bigg{)}$ $\displaystyle\leq$ $\displaystyle\sum_{s\geq 0}\exp\left(-2\frac{a^{2s}l^{2}}{\left\lfloor a^{s+1}l\right\rfloor}\left(x^{2}+c_{a^{s+1}l}^{2}\right)\right)$ $\displaystyle\leq$ $\displaystyle\sum_{s\geq 0}\exp\left(-2a^{s-1}lx^{2}-\frac{2\eta}{a^{2}}\ln\left(\frac{\tau}{Ka^{s+1}l}\right)\right)$ $\displaystyle=$ $\displaystyle\sum_{s\geq 1}\frac{Kla^{s}}{\tau}\exp\left(-2a^{s-2}lx^{2}\right).$ Let $b=2x^{2}l/a^{2}$. It follows that $\displaystyle\sum_{s\geq 1}\frac{Kla^{s}}{\tau}\exp\left(-ba^{s}\right)\leq\frac{Kl}{\tau}\int_{0}^{+\infty}a^{y+1}\exp\big{(}-ba^{y}\big{)}dy$ $\displaystyle=$ $\displaystyle\frac{Kla}{\tau\ln(a)}\int_{1}^{+\infty}\exp(-bz)dz\quad\left(\text{where we set }z=a^{y}\right)$ $\displaystyle=$ $\displaystyle\frac{Klae^{-b}}{\tau b\ln(a)},$ which concludes the bound for the probability of event $A$. By using upper tail bound, similar result exists for event $B$. ∎ We now leverage Lemma 4 to get an upper bound on the worst-case regret for SW- MOSS. ###### Theorem 5. For the nonstationary MAB problem with $K$ arms, time horizon $T$, variation budget $V_{T}>0$ and $\tau=\Big{\lceil}{K^{\frac{1}{3}}\left(T/V_{T}\right)^{\frac{2}{3}}}\Big{\rceil}$, the worst-case regret of SW-MOSS satisfies $\sup_{\mathcal{F}_{T}^{\mathcal{K}}\in\mathcal{E}(V_{T},T,K)}R_{T}^{\text{SW- MOSS}}\in\mathcal{O}((KV_{T})^{\frac{1}{3}}T^{\frac{2}{3}}).$ ###### Proof. The proof consists of the following five steps. Step 1: Recall that $v_{i}$ is the variation within $\mathcal{T}_{i}$. Here, we trivially assign $\mathcal{T}_{0}=\emptyset$ and $v_{0}=0$. Then, for each $i\in\\{1,\dots,N\\}$, let $\Delta_{i}^{k}\mathrel{\mathop{\mathchar 58\relax}}=\mu_{\tau_{i}}^{\kappa_{i}}-\mu_{\tau_{i}}^{k}-2v_{i-1}-2v_{i},\quad\forall k\in\mathcal{K}.$ Define the set of bad arms for SW-MOSS in $\mathcal{T}_{i}$ as $\mathcal{B}_{i}^{\textup{SW}}\mathrel{\mathop{\mathchar 58\relax}}=\left\\{k\in\mathcal{K}\;\|\;\Delta_{i}^{k}\geq\epsilon\right\\},$ where we assign $\epsilon=4\sqrt{e\eta K/\tau}$. Step 2: We decouple the regret in this step. For any $t\in\mathcal{T}_{i}$, since $\mathinner{\\!\left\lvert\mu_{t}^{k}-\mu_{\tau_{i}}^{k}\right\rvert}\leq v_{i}$ for any $k\in\mathcal{K}$, it satisfies that $\displaystyle\mu_{t}^{}-\mu_{t}^{\varphi_{t}}$ $\displaystyle\leq\mu_{\tau_{i}}^{\kappa_{i}}-\mu_{t}^{\varphi_{t}}$ $\displaystyle\leq\mu_{\tau_{i}}^{\kappa_{i}}-\mu_{\tau_{i}}^{\varphi_{t}}+v_{i}$ $\displaystyle\leq\mathbf{1}\left\\{\varphi_{t}\in\mathcal{B}_{i}^{\textup{SW}}\right\\}(\Delta_{i}^{\varphi_{t}}-\epsilon)+2v_{i-1}+3v_{i}+\epsilon.$ Then we get the following inequalities, $\displaystyle\sum_{t\in\mathcal{T}}\mu_{t}^{}-\mu_{t}^{\varphi_{t}}$ $\displaystyle\leq$ $\displaystyle\sum_{i=1}^{N}\sum_{t\in\mathcal{T}_{i}}\mathbf{1}\left\\{\varphi_{t}\in\mathcal{B}_{i}^{\textup{SW}}\right\\}(\Delta_{i}^{\varphi_{t}}-\epsilon)+2v_{i-1}+3v_{i}+\epsilon$ $\displaystyle\leq$ $\displaystyle 5\tau V_{T}+T\epsilon+\sum_{i=1}^{N}\sum_{t\in\mathcal{T}_{i}}\mathbf{1}\left\\{\varphi_{t}\in\mathcal{B}_{i}^{\textup{SW}}\right\\}(\Delta_{i}^{\varphi_{t}}-\epsilon).$ (7) To continue, we take a decomposition inspired by the analysis of MOSS in [31] below, $\displaystyle\sum_{t\in\mathcal{T}_{i}}\mathbf{1}\left\\{\varphi_{t}\in\mathcal{B}_{i}^{\textup{SW}}\right\\}\left(\Delta_{i}^{\varphi_{t}}-\epsilon\right)$ $\displaystyle\leq$ $\displaystyle\sum_{t\in\mathcal{T}_{i}}\mathbf{1}\bigg{\\{}\varphi_{t}\in\mathcal{B}_{i}^{\textup{SW}},g_{t}^{\kappa_{i}}>M_{t}^{\kappa_{i}}-\frac{\Delta_{i}^{\varphi_{t}}}{4}\bigg{\\}}\Delta_{i}^{\varphi_{t}}$ (8) $\displaystyle+$ $\displaystyle\sum_{t\in\mathcal{T}_{i}}\mathbf{1}\bigg{\\{}\varphi_{t}\in\mathcal{B}_{i}^{\textup{SW}},g_{t}^{\kappa_{i}}\leq M_{t}^{\kappa_{i}}-\frac{\Delta_{i}^{\varphi_{t}}}{4}\bigg{\\}}\left(\Delta_{i}^{\varphi_{t}}-\epsilon\right),$ (9) where summands (8) describes the regret when arm $\kappa_{i}$ is fairly estimated and summand (9) quantifies the regret incurred by underestimating arm $\kappa_{i}$. Step 3: In this step, we bound $\mathbb{E}\left[\eqref{overestimate}\right]$. Since $g_{t}^{\varphi_{t}}\geq g_{t}^{\kappa_{i}}$, $\displaystyle\eqref{overestimate}\leq$ $\displaystyle\sum_{t\in\mathcal{T}_{i}}\mathbf{1}\bigg{\\{}\varphi_{t}\in\mathcal{B}_{i}^{\textup{SW}},g_{t}^{\varphi_{t}}>M_{t}^{\kappa_{i}}-\frac{\Delta_{i}^{\varphi_{t}}}{4}\bigg{\\}}\Delta_{i}^{\varphi_{t}}$ $\displaystyle=$ $\displaystyle\sum_{k\in\mathcal{B}_{i}^{\textup{SW}}}\sum_{t\in\mathcal{T}_{i}}\mathbf{1}\bigg{\\{}\varphi_{t}=k,g_{t}^{k}>M_{t}^{\kappa_{i}}-\frac{\Delta_{i}^{k}}{4}\bigg{\\}}\Delta_{i}^{k}.$ (10) Notice that for any $t\in\mathcal{T}_{i-1}\cup\mathcal{T}_{i}$, $\mathinner{\\!\left\lvert\mu_{t}^{k}-\mu_{\tau_{i}}^{k}\right\rvert}\leq v_{i-1}+v_{i},\quad\forall k\in\mathcal{K}.$ It indicates that an arm $k\in\mathcal{B}_{i}^{\textup{SW}}$ is at least $\Delta_{i}^{k}$ worse in mean reward than arm $\kappa_{i}$ at any time slot $t\in\mathcal{T}_{i-1}\cup\mathcal{T}_{i}$. Since $\mathcal{W}_{t}\subset\mathcal{T}_{i-1}\operatorname{\cup}\mathcal{T}_{i}$, for any $t\in\mathcal{T}_{i}$ $M_{t}^{\kappa_{i}}-M_{t}^{k}\geq\Delta_{i}^{k}\geq\epsilon,\quad\forall k\in\mathcal{B}_{i}^{\textup{SW}}.$ It follows that $\eqref{overestimate2}\leq\sum_{k\in\mathcal{B}_{i}^{\textup{SW}}}\sum_{t\in\mathcal{T}_{i}}\mathbf{1}\bigg{\\{}\varphi_{t}=k,g_{t}^{k}>M_{t}^{k}+\frac{3\Delta_{i}^{k}}{4}\bigg{\\}}\Delta_{i}^{k}.$ (11) Let $t_{s}^{ik}$ be the $s$-th time slot when arm $k$ is selected within $\mathcal{T}_{i}$. Then, for any $k\in\mathcal{B}_{i}^{\textup{SW}}$, $\displaystyle\sum_{t\in\mathcal{T}_{i}}\mathbf{1}{\bigg{\\{}\varphi_{t}=k,g_{t}^{k}>M_{t}^{k}+\frac{3\Delta_{i}^{k}}{4}\bigg{\\}}}$ $\displaystyle=$ $\displaystyle\sum_{s\geq 1}\mathbf{1}{\bigg{\\{}g_{t_{s}^{ik}}^{k}>M_{t_{s}^{ik}}^{k}+\frac{3\Delta_{i}^{k}}{4}\bigg{\\}}}$ $\displaystyle\leq$ $\displaystyle l_{i}^{k}+\sum_{s\geq l_{i}^{k}+1}\mathbf{1}{\bigg{\\{}g_{t_{s}^{ik}}^{k}>M_{t_{s}^{ik}}^{k}+\frac{3\Delta_{i}^{k}}{4}\bigg{\\}}},$ (12) where we set $l_{i}^{k}=\bigg{\lceil}{\eta\Big{(}\frac{4}{\Delta_{i}^{k}}\Big{)}^{2}\ln\left(\frac{\tau}{\eta K}\Big{(}\frac{\Delta_{i}^{k}}{4}\Big{)}^{2}\right)}\bigg{\rceil}$. Since $\Delta_{i}^{k}\geq\epsilon$, for $k\in\mathcal{B}_{i}^{\textup{SW}}$, we have $l_{i}^{k}\geq\Big{\lceil}{\eta\left({4}/{\Delta_{i}^{k}}\right)^{2}\ln\left(\frac{\tau}{\eta K}\left({\epsilon}/{4}\right)^{2}\right)}\Big{\rceil}\geq\eta\left({4}/{\Delta_{i}^{k}}\right)^{2},$ where the second inequality follows by substituting $\epsilon=4\sqrt{e\eta K/\tau}$. Additionally, since $t_{1}^{ik},\ldots,t_{s-1}^{ik}\in\mathcal{W}_{t_{s}^{ik}}$, we get $n_{k}(t_{s}^{ik})\geq s-1$. Furthermore, since $c_{m}$ is monotonically decreasing with $m$, $c_{n_{k}(t_{s}^{k})}\leq c_{l_{i}^{k}}\leq\sqrt{\frac{\eta}{l_{i}^{k}}\ln\left(\frac{\tau}{\eta K}\bigg{(}\frac{\Delta_{i}^{k}}{4}\bigg{)}^{2}\right)}\leq\frac{\Delta_{i}^{k}}{4},$ for $s\geq l_{i}^{k}+1$. Therefore, $\eqref{overestimate_k}\leq l_{i}^{k}+\sum_{s\geq l_{i}^{k}+1}\mathbf{1}{\bigg{\\{}g_{t_{s}^{ik}}^{k}-2c_{n_{k}(t_{s}^{ik})}>M_{t_{s}^{ik}}^{k}+\frac{\Delta_{i}^{k}}{4}\bigg{\\}}}.$ By applying Lemma 4, considering $n_{k}(t_{s}^{ik})\geq s-1$, $\displaystyle\sum_{s\geq l_{i}^{k}+1}\mathbb{P}{\bigg{\\{}g_{t_{s}^{ik}}^{k}-2c_{n_{k}(t_{s}^{ik})}>M_{t_{s}^{ik}}^{k}+\frac{\Delta_{i}^{k}}{4}\bigg{\\}}}$ $\displaystyle\leq$ $\displaystyle\sum_{s\geq l_{i}^{k}}\frac{(2\eta)^{\frac{3}{2}}}{\ln(2\eta)}\frac{K}{\tau}\bigg{(}\frac{4}{\Delta_{i}^{k}}\bigg{)}^{2}\exp\left(-\frac{s}{\eta}\bigg{(}\frac{\Delta_{i}^{k}}{4}\bigg{)}^{2}\right)$ $\displaystyle\leq$ $\displaystyle\int_{l_{i}^{k}-1}^{+\infty}\frac{(2\eta)^{\frac{3}{2}}}{\ln(2\eta)}\frac{K}{\tau}\bigg{(}\frac{4}{\Delta_{i}^{k}}\bigg{)}^{2}\exp\left(-\frac{y}{\eta}\bigg{(}\frac{\Delta_{i}^{k}}{4}\bigg{)}^{2}\right)\,dy$ $\displaystyle\leq$ $\displaystyle\frac{(2\eta)^{\frac{3}{2}}}{\ln(2\eta)}\frac{\eta K}{\tau}\bigg{(}\frac{4}{\Delta_{i}^{k}}\bigg{)}^{4}.$ (13) Let $h(x)=16\eta/x\ln\left({\tau x^{2}}/{16\eta K}\right)$ which achieves maximum at $4e\sqrt{\eta K/\tau}$. Combining (13), (12), (11), and (10), we obtain $\displaystyle\mathbb{E}[\eqref{overestimate}]\leq$ $\displaystyle\sum_{k\in\mathcal{B}_{i}}\frac{(2\eta)^{\frac{3}{2}}}{\ln(2\eta)}\frac{\eta K}{\tau}\frac{256}{\left(\Delta_{i}^{k}\right)^{3}}+l_{i}^{k}\Delta_{i}^{k}$ $\displaystyle\leq$ $\displaystyle\sum_{k\in\mathcal{B}_{i}}\frac{(2\eta)^{\frac{3}{2}}}{\ln(2\eta)}\frac{\eta K}{\tau}\frac{256}{\left(\Delta_{i}^{k}\right)^{3}}+h(\Delta_{i}^{k})+\Delta_{i}^{k}$ $\displaystyle\leq$ $\displaystyle\sum_{k\in\mathcal{B}_{i}}\frac{(2\eta)^{\frac{3}{2}}}{\ln(2\eta)}\frac{\eta K}{\tau}\frac{256}{\epsilon^{3}}+h\left(4e\sqrt{\eta K/\tau}\right)+b$ $\displaystyle\leq$ $\displaystyle\left(\frac{2.6\eta}{\ln(2\eta)}+3\sqrt{\eta}\right)\sqrt{K\tau}+Kb.$ Step 4: In this step, we bound $\mathbb{E}[\eqref{underestimate}]$. When event $\left\\{\varphi_{t}\in\mathcal{B}_{i}^{\textup{SW}},g_{t}^{\kappa_{i}}\leq M_{t}^{\kappa_{i}}-{\Delta_{i}^{\varphi_{t}}}/{4}\right\\}$ happens, we know $\Delta_{i}^{\varphi_{t}}\leq 4M_{t}^{\kappa_{i}}-4g_{t}^{\kappa_{i}}\text{ and }g_{t}^{\kappa_{i}}\leq M_{t}^{\kappa_{i}}-\frac{\epsilon}{4}.$ Thus, we have $\displaystyle\mathbf{1}\bigg{\\{}\varphi_{t}\in\mathcal{B}_{i}^{\textup{SW}},g_{t}^{\kappa_{i}}\leq M_{t}^{\kappa_{i}}-\frac{\Delta_{i}^{\varphi_{t}}}{4}\bigg{\\}}\left(\Delta_{i}^{\varphi_{t}}-\epsilon\right)$ $\displaystyle\leq$ $\displaystyle\mathbf{1}{\left\\{g_{t}^{\kappa_{i}}\leq M_{t}^{\kappa_{i}}-\frac{\epsilon}{4}\right\\}}\times\big{(}4M_{t}^{\kappa_{i}}-4g_{t}^{\kappa_{i}}-\epsilon\big{)}\mathrel{\mathop{\mathchar 58\relax}}=Y$ Since $Y$ is a nonnegative random variable, its expectation can be computed involving only its cumulative density function: $\displaystyle\mathbb{E}\left[Y\right]$ $\displaystyle=\int_{0}^{+\infty}\mathbb{P}\left(Y>x\right)dx$ $\displaystyle\leq\int_{0}^{+\infty}\mathbb{P}\Big{(}4M_{t}^{\kappa_{i}}-4g_{t}^{\kappa_{i}}-\epsilon\geq x\Big{)}dx$ $\displaystyle=\int_{\epsilon}^{+\infty}\mathbb{P}\Big{(}4M_{t}^{\kappa_{i}}-4g_{t}^{\kappa_{i}}>x\Big{)}dx$ $\displaystyle\leq\int_{\epsilon}^{+\infty}\frac{16(2\eta)^{\frac{3}{2}}}{\ln(2\eta)}\frac{K}{\tau x^{2}}dx=\frac{16(2\eta)^{\frac{3}{2}}}{\ln(2\eta)}\frac{K}{\tau\epsilon}.$ Hence, $\mathbb{E}[\eqref{underestimate}]\leq{16(2\eta)^{\frac{3}{2}}}K\mathinner{\\!\left\lvert\mathcal{T}_{i}\right\rvert}/\left(\ln(2\eta)\tau\epsilon\right).$ Step 5: With bounds on $\mathbb{E}\left[\eqref{overestimate}\right]$ and $\mathbb{E}[\eqref{underestimate}]$ from previous steps, $\displaystyle\mathbb{E}[\eqref{regret_sw}]\leq$ $\displaystyle 5\tau V_{T}+T\epsilon+N\left(\frac{2.6\eta}{\ln(2\eta)}+3\sqrt{\eta}\right)\sqrt{K\tau}$ $\displaystyle+NKb+\frac{16(2\eta)^{\frac{3}{2}}}{\ln(2\eta)}\frac{KT}{\tau\epsilon}\leq C(KV_{T})^{\frac{1}{3}}T^{\frac{2}{3}}$ for some constant $C$, which concludes the proof. ∎ We have shown that SW-MOSS also enjoys order optimal worst-case regret. One drawback of the sliding window method is that all sampling history within the observation window needs to be stored. Since window size is selected to be $\tau=\big{\lceil}{K^{\frac{1}{3}}({T}/{V_{T}})^{\frac{2}{3}}}\big{\rceil}$, large memory is needed for large horizon length $T$. The next policy resolves this problem. ### IV-C Discounted UCB Algorithm The discount factor is widely used in estimators to forget old information and put more attention on the recent information. In [19], such an estimation is used together with UCB$1$ to solve the piecewise stationary MAB problem, and the policy designed is called Discounted UCB (D-UCB). Here, we tune D-UCB to work in the nonsationary environment with variation budget $V_{T}$. Specifically, the mean estimator used is discounted empirical average given by $\displaystyle\hat{\mu}_{\gamma,t}^{k}$ $\displaystyle=\frac{1}{n_{\gamma,t}^{k}}\sum_{s=1}^{t-1}\gamma^{t-s}\mathbf{1}\\{\varphi_{s}=k\\}X_{s},$ $\displaystyle n_{\gamma,t}^{k}$ $\displaystyle=\sum_{s=1}^{t-1}\gamma^{t-s}\mathbf{1}\\{\varphi_{s}=k\\},$ where $\gamma=1-{K^{-\frac{1}{3}}({T}/{V_{T}})^{-\frac{2}{3}}}$ is the discount factor. Besides, the UCB is designed as $g_{t}^{k}=\hat{\mu}_{t}^{k}+2c_{t}^{k}$, where $c_{\gamma,t}^{k}=\sqrt{\xi\ln(\tau)/n_{\gamma,t}^{k}}$ for some constant $\xi>1/2$. The pseudo code for D-UCB is reproduced in Algorithm 3. It can be noticed that the memory size is only related to the number of arms, so D-UCB requires small memory. Input : $V_{T}\in\mathbb{R}_{>0}$, $T\in\mathbb{N}$ and $\xi>\frac{1}{2}$ Set : $\gamma=1-{K^{-\frac{1}{3}}({T}/{V_{T}})^{-\frac{2}{3}}}$ Output : sequence of arm selection 1for _$t\in\\{1,\dots,K\\}$ _ do Pick arm $\varphi_{t}=t$ and set $n^{t}\leftarrow\gamma^{K-t}$ and $\hat{\mu}^{t}\leftarrow X_{t}^{t}$; 2while _$t\leq T$_ do Pick arm $\displaystyle\varphi_{t}=\arg\max_{k\in\mathcal{K}}\hat{\mu}^{k}+2\sqrt{\frac{\xi\ln(\tau)}{n^{k}}}$; For each arm $k\in\mathcal{K}$, set $n^{k}\leftarrow\gamma n^{k}$; Set $n^{\varphi_{t}}\leftarrow n^{\varphi_{t}}+1\>\&\>\hat{\mu}^{\varphi_{t}}\leftarrow\hat{\mu}^{\varphi_{t}}+\frac{1}{n^{\varphi_{t}}}(X_{t}^{\varphi_{t}}-\bar{X}^{\varphi_{t}});$ Algorithm 3 D-UCB To proceed the analysis, we review the concentration inequality for discounted empirical average, which is an extension of Chernoff-Hoeffding bound. Let $M_{\gamma,t}^{k}\mathrel{\mathop{\mathchar 58\relax}}=\frac{1}{n_{\gamma,t}^{k}}\sum_{s=1}^{t-1}\gamma^{t-s}\mathbf{1}\\{\varphi_{s}=k\\}\mu_{s}^{k}.$ Then, the following fact is a corollary of [19, Theorem 18]. ###### Fact 2 (A Hoeffding-type inequality for discounted empirical average with a random number of summands). For any $t\in\mathcal{T}$ and for any $k\in\mathcal{K}$, the probability of event $A=\left\\{{\hat{\mu}_{\gamma,t}^{k}-M_{\gamma,t}^{k}}\geq\delta/\sqrt{n_{\gamma,t}^{k}}\right\\}$ is no greater than $\left\lceil\log_{1+\lambda}(\tau)\right\rceil\exp\left(-2\delta^{2}\big{(}1-{\lambda^{2}}/{16}\big{)}\right)$ (14) for any $\delta>0$ and $\lambda>0$. The probability of event $B=\left\\{\hat{\mu}_{\gamma,t}^{k}-M_{\gamma,t}^{k}\leq-\delta/\sqrt{n_{\gamma,t}^{k}}\right\\}$ is also upper bounded by (14). ###### Theorem 6. For the nonstationary MAB problem with $K$ arms, time horizon $T$, variation budget $V_{T}>0$, and $\gamma=1-{K^{-\frac{1}{3}}({T}/{V_{T}})^{-\frac{2}{3}}}$, if $\xi>1/2$, the worst case regret of D-UCB satisfies $\sup_{\mathcal{F}_{T}^{\mathcal{K}}\in\mathcal{E}(V_{T},T,K)}R_{T}^{\textup{D-UCB}}\leq C\ln(T)(KV_{T})^{\frac{1}{3}}T^{\frac{2}{3}}.$ ###### Proof. We establish the theorem in four steps. Step 1: In this step, we analyze $\big{\|}{\mu_{\gamma,t}^{k}-M_{\gamma,t}^{k}}\big{\|}$ at some time slot $t\in\mathcal{T}_{i}$. Let $\tau^{\prime}={\log_{\gamma}\big{(}(1-\gamma)\xi\ln(\tau)/b^{2}\big{)}}$ and take $t-\tau^{\prime}$ as a dividing point, then we obtain $\displaystyle\mathinner{\\!\left\lvert\mu_{\tau_{i}}^{k}-M_{\gamma,t}^{k}\right\rvert}\leq$ $\displaystyle\frac{1}{n_{\gamma,t}^{k}}\sum_{s=1}^{t-1}\gamma^{t-s}\mathbf{1}\\{\varphi_{s}=k\\}\mathinner{\\!\left\lvert\mu_{\tau_{i}}^{k}-\mu_{s}^{k}\right\rvert}$ $\displaystyle\leq$ $\displaystyle\frac{1}{n_{\gamma,t}^{k}}\sum_{s\leq t-\tau^{\prime}}\gamma^{t-s}\mathbf{1}\\{\varphi_{s}=k\\}\mathinner{\\!\left\lvert\mu_{\tau_{i}}^{k}-\mu_{s}^{k}\right\rvert}$ (15) $\displaystyle+$ $\displaystyle\frac{1}{n_{\gamma,t}^{k}}\sum_{s\geq t-\tau^{\prime}}^{t-1}\gamma^{t-s}\mathbf{1}\\{\varphi_{s}=k\\}\mathinner{\\!\left\lvert\mu_{\tau_{i}}^{k}-\mu_{s}^{k}\right\rvert}.$ (16) Since $\mu_{t}^{k}\in[a,a+b]$ for all $t\in\mathcal{T}$, we have $\eqref{bias: dis}\leq b$. Also, ${\eqref{bias: dis}}\leq\frac{1}{n_{\gamma,t}^{k}}\sum_{s\leq t-\tau^{\prime}}b\gamma^{t-s}\leq\frac{b\gamma^{\tau^{\prime}}}{(1-\gamma)n_{\gamma,t}^{k}}=\frac{\xi\ln(\tau)}{bn_{\gamma,t}^{k}}.$ Accordingly, we get ${\eqref{bias: dis}}\leq\min\left(b,\frac{\xi\ln(\tau)}{bn_{\gamma,t}^{k}}\right)\leq\sqrt{\frac{\xi\ln(\tau)}{n_{\gamma,t}^{k}}}.$ Furthermore, for any $t\in\mathcal{T}_{i}$, $\eqref{bias: dis2}\leq\max_{s\in[t-\tau^{\prime},t-1]}\mathinner{\\!\left\lvert\mu_{\tau_{i}}^{k}-\mu_{s}^{k}\right\rvert}\leq\sum_{j=i-n^{\prime}}^{i}v_{j},$ where $n^{\prime}=\lceil{\tau^{\prime}/\tau}\rceil$ and $v_{j}$ is the variation within $\mathcal{T}_{j}$. So we conclude that for any $t\in\mathcal{T}_{i}$, $\mathinner{\\!\left\lvert\mu_{\kappa_{i}}^{k}-M_{\gamma,t}^{k}\right\rvert}\leq c_{\gamma,t}^{k}+\sum_{j=i-n^{\prime}}^{i}v_{j},\quad\forall k\in\mathcal{K}.$ (17) Step 2: Within partition $\mathcal{T}_{i}$, let $\hat{\Delta}_{i}^{k}=\mu_{\tau_{i}}^{\kappa_{i}}-\mu_{\tau_{i}}^{k}-2\sum_{j=i-n^{\prime}}^{i}v_{j},$ and define a subset of bad arms as $\mathcal{B}_{i}^{\textup{D}}=\bigg{\\{}k\in\mathcal{K}\>\|\>\hat{\Delta}_{i}^{k}\geq\epsilon^{\prime}\bigg{\\}},$ where we select $\epsilon^{\prime}=4\sqrt{\xi\gamma^{1-\tau}K\ln(\tau)/\tau}$. Since $\mathinner{\\!\left\lvert\mu_{t}^{k}-\mu_{\tau_{i}}^{k}\right\rvert}\leq v_{i}$ for any $t\in\mathcal{T}_{i}$ and for any $k\in\mathcal{K}$ $\displaystyle\sum_{t\in\mathcal{T}}\mu_{t}^{}-\mu_{t}^{\varphi_{t}}\leq\sum_{i=1}^{N}\sum_{t\in\mathcal{T}_{i}}\mu_{\tau_{i}}^{\kappa_{i}}-\mu_{\tau_{i}}^{\varphi_{t}}+v_{i}$ $\displaystyle\leq$ $\displaystyle\tau V_{T}+\sum_{i=1}^{N}\sum_{t\in\mathcal{T}_{i}}\bigg{[}\mathbf{1}\left\\{\varphi_{t}\in\mathcal{B}_{i}^{\textup{D}}\right\\}\hat{\Delta}_{i}^{\varphi_{t}}+2\sum_{j=i-n^{\prime}}^{i}v_{j}+\epsilon^{\prime}\bigg{]}$ $\displaystyle\leq$ $\displaystyle(2n^{\prime}+3)\tau V_{T}+N\epsilon^{\prime}\tau\\!+\\!\sum_{i=1}^{N}\sum_{k\in\mathcal{B}_{i}^{\textup{D}}}\\!\\!\hat{\Delta}_{i}^{k}\sum_{t\in\mathcal{T}_{i}}\mathbf{1}\left\\{\varphi_{t}=k\right\\}.$ (18) Step 3: In this step, we bound $\mathbb{E}\big{[}\hat{\Delta}_{i}^{k}\sum_{t\in\mathcal{T}_{i}}\mathbf{1}\left\\{\varphi_{t}=k\right\\}\big{]}$ for an arm $k\in\mathcal{B}_{i}^{\textup{D}}$. Let $t_{i}^{k}(l)$ be the $l$-th time slot arm $k$ is selected within $\mathcal{T}_{i}$. From arm selection policy, we get $g_{t}^{\varphi_{t}}\geq g_{t}^{\kappa_{i}}$, which result in $\sum_{t\in\mathcal{T}_{i}}\mathbf{1}\left\\{\varphi_{t}=k\right\\}\leq l_{i}^{k}+\sum_{t\in\mathcal{T}_{i}}\mathbf{1}{\Big{\\{}g_{t}^{k}\geq g_{t}^{\kappa_{i}},t>t_{i}^{k}(l_{i}^{k})\Big{\\}}},$ (19) where we pick $l_{i}^{k}=\left\lceil{16\xi\gamma^{1-\tau}\ln(\tau)/{\big{(}\hat{\Delta}_{i}^{k}\big{)}^{2}}}\right\rceil$. Note that $g_{t}^{k}\geq g_{t}^{\kappa_{i}}$ is true means at least one of the followings holds, $\displaystyle\hat{\mu}_{\gamma,t}^{k}$ $\displaystyle\geq M_{\gamma,t}^{k}+c_{\gamma,t}^{k},$ (20) $\displaystyle\hat{\mu}_{\gamma,t}^{\kappa_{i}}$ $\displaystyle\leq M_{\gamma,t}^{\kappa_{i}}-c_{\gamma,t}^{\kappa_{i}},$ (21) $\displaystyle M_{\gamma,t}^{\kappa_{i}}+c_{\gamma,t}^{\kappa_{i}}$ $\displaystyle<M_{\gamma,t}^{k}+3c_{\gamma,t}^{k}.$ (22) For any $t\in\mathcal{T}_{i}$, since every sample before $t$ within $\mathcal{T}_{i}$ has a weight greater than $\gamma^{\tau-1}$, if $t>t_{i}^{k}(l_{i}^{k})$, $\displaystyle c_{\gamma,t}^{k}=\sqrt{\frac{\xi\ln(\tau)}{n_{\gamma,t}^{k}}}\leq\sqrt{\frac{\xi\ln(\tau)}{\gamma^{\tau-1}l_{i}^{k}}}\leq\frac{\hat{\Delta}_{i}^{k}}{4}.$ Combining it with (17) yields $\displaystyle M_{\gamma,t}^{\kappa_{i}}-M_{\gamma,t}^{k}$ $\displaystyle\geq\mu_{\tau_{i}}^{\kappa_{i}}-\mu_{\tau_{i}}^{k}-c_{\gamma,t}^{\kappa_{i}}-c_{\gamma,t}^{k}-2\sum_{j=i-n^{\prime}}^{i}v_{j}$ $\displaystyle\geq\hat{\Delta}_{i}^{k}-c_{\gamma,t}^{\kappa_{i}}-c_{\gamma,t}^{k}\geq 3c_{\gamma,t}^{k}-c_{\gamma,t}^{\kappa_{i}},$ which indicates (22) is false. As $\xi>1/2$, we select $\lambda=4\sqrt{1-1/(2\xi)}$ and apply Fact 2 to get $\mathbb{P}(\text{\eqref{h1} is true})\leq\left\lceil\log_{1+\lambda}(\tau)\right\rceil\tau^{-2\xi(1-{\lambda^{2}}/{16})}\leq\frac{\left\lceil\log_{1+\lambda}(\tau)\right\rceil}{\tau}.$ The probability of (21) to be true shares the same bound. Then, it follows from (19) that $\mathbb{E}\big{[}\hat{\Delta}_{i}^{k}\sum_{t\in\mathcal{T}_{i}}\mathbf{1}\left\\{\varphi_{t}=k\right\\}\big{]}$ is upper bounded by $\displaystyle\hat{\Delta}_{i}^{k}l_{i}^{k}+\hat{\Delta}_{i}^{k}\sum_{t\in\mathcal{T}_{i}}\mathbb{P}\left(\text{\eqref{h1} or~{}\eqref{h2} is true}\right)$ $\displaystyle\leq$ $\displaystyle\frac{16\xi\gamma^{1-\tau}\ln(\tau)}{\hat{\Delta}_{i}^{k}}+\hat{\Delta}_{i}^{k}+2\hat{\Delta}_{i}^{k}\left\lceil\log_{1+\lambda}\left(\tau\right)\right\rceil$ $\displaystyle\leq$ $\displaystyle\frac{16\xi\gamma^{1-\tau}\ln(\tau)}{\epsilon^{\prime}}+b+2b\left\lceil\log_{1+\lambda}\left(\tau\right)\right\rceil,$ (23) where we use $\epsilon^{\prime}\leq\hat{\Delta}_{i}^{k}\leq b$ in the last step. Step 4: From (18) and (23), and plugging in the value of $\epsilon^{\prime}$, an easy computation results in $\displaystyle R_{T}^{\textup{D-UCB}}\leq$ $\displaystyle(2n^{\prime}+3)\tau V_{T}+8N\sqrt{\xi\gamma^{1-\tau}K\tau\ln(\tau)}$ $\displaystyle+2Nb+2Nb\log_{1+\lambda}\left(\tau\right),$ where the dominating term is $(2n^{\prime}+3)\tau V_{T}$. Considering $\tau^{\prime}=\frac{\ln\big{(}(1-\gamma)\xi\ln(\tau)/b^{2}\big{)}}{\ln{\gamma}}\leq\frac{-\ln\big{(}(1-\gamma)\xi\ln(\tau)/b^{2}\big{)}}{1-\gamma},$ we get $n^{\prime}\leq C^{\prime}\ln(T)$ for some constant $C^{\prime}$. Hence there exists some absolute constant $C$ such that $R_{T}^{\textup{D-UCB}}\leq C\ln(T)(KV_{T})^{\frac{1}{3}}T^{\frac{2}{3}}.$ ∎ Although discount factor method requires less memory, there exists an extra factor $\ln(T)$ in the upper bound on the worst-case regret for D-UCB comparing with the minimax regret. This is due to the fact that discount factor method does not entirely cut off outdated sampling history like periodic resetting or sliding window techniques. ## V UCB Policies for Heavy-tailed Nonstationary Stochastic MAB Problems In this section, we propose and analyze UCB algorithms for non-stationary stochastic MAB problem with heavy-tailed rewards defined in Assumption 2. We first recall a minimax policy for the stationary heavy-tailed MAB problem called Robust MOSS [36]. We then extend it to nonstationary setting and design resetting robust MOSS algorithm and sliding-window robust MOSS algorithm. ### V-A Background on Robust MOSS algorithm for the stationary heavy-tailed MAB problem Robust MOSS algorithm handles stationary heavy-tailed MAB problems in which the rewards have finite moments of order $1+\epsilon$, for $\epsilon\in(0,1]$. For simplicity, as stated in Assumption 2, we restrict our discussion to $\epsilon=1$. Robust MOSS uses the saturated empirical mean instead of the empirical mean. Let $n_{k}(t)$ be the number of times that arm $k$ has been selected until time $t-1$. Pick $a>1$ and let $h(m)=a^{\left\lfloor\log_{a}\left(m\right)\right\rfloor+1}$. Let the saturation limit at time $t$ be defined by $B_{n_{k}(t)}\mathrel{\mathop{\mathchar 58\relax}}=\sqrt{\frac{h(n_{k}(t))}{\ln_{+}\left(\frac{T}{Kh(n_{k}(t))}\right)}},$ where $\ln_{+}(x)\mathrel{\mathop{\mathchar 58\relax}}=\max(\ln x,1)$. Then, the saturated empirical mean estimator is defined by $\bar{\mu}_{n_{k}(t)}\mathrel{\mathop{\mathchar 58\relax}}=\frac{1}{n_{k}(t)}\sum_{s=1}^{t-1}\mathbf{1}\\{\varphi_{s}=k\\}\operatorname{sat}(X_{s},B_{n_{k}(t)}),$ (24) where $\operatorname{sat}(X_{s},B_{m})\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{sign}(X_{s})\min\big{\\{}\mathinner{\\!\left\lvert X_{s}\right\rvert},B_{m}\big{\\}}.$ The Robust MOSS algorithm initializes by selecting each arm once and subsequently, at each time $t$, selects the arm that maximizes the following upper confidence bound $g^{k}_{n_{k}(t)}=\bar{\mu}^{k}_{n_{k}(t)}+(1+\zeta)c_{n_{k}(t)},$ where $c_{n_{k}(t)}=\sqrt{{\ln_{+}\big{(}\frac{T}{Kn_{k}(t)}\big{)}}/{n_{k}(t)}}$, $\zeta$ is an positive constant such that $\psi(2\zeta/a)\geq 2a/\zeta$ and $\psi(x)=(1+1/x)\ln(1+x)-1$. Note that for $x\in(0,\infty)$, function $\psi(x)$ is monotonically increasing in $x$. ### V-B Resetting robust MOSS for the non-stationary heavy-tailed MAB problem Similarly to R-MOSS, Resetting Robust MOSS (R-RMOSS) restarts Robust MOSS after every $\tau$ time slots. For a stationary heavy-tailed MAB problem, it has been shown in [36] that the worst-case regret of Robust MOSS belongs to $\mathcal{O}(\sqrt{KT})$. This result along with an analysis similar to the analysis for R-MOSS in Theorem 3 yield the following theorem for R-RMOSS. For brevity, we skip the proof. ###### Theorem 7. For the nonstationary heavy-tailed MAB problem with $K$ arms, horizon $T$, variation budget $V_{T}>0$ and $\tau=\Big{\lceil}{K^{\frac{1}{3}}\left(T/V_{T}\right)^{\frac{2}{3}}}\Big{\rceil}$, if $\psi(2\zeta/a)\geq 2a/\zeta$, the worst-case regret of R-RMOSS satisfies $\sup_{\mathcal{F}_{T}^{\mathcal{K}}\in\mathcal{E}(V_{T},T,K)}R_{T}^{\textup{R-RMOSS}}\in\mathcal{O}((KV_{T})^{\frac{1}{3}}T^{\frac{2}{3}}).$ ### V-C Sliding-window robust MOSS for the non-stationary heavy-tailed MAB problem In Sliding-Window Robust MOSS (SW-RMOSS), $n_{k}(t)$ and $\bar{\mu}_{n_{k}(t)}$ are computed from the sampling history within $\mathcal{W}_{t}$, and $c_{n_{k}(t)}=\sqrt{{\ln_{+}\big{(}\frac{\tau}{Kn_{k}(t)}\big{)}}/{n_{k}(t)}}$. To analyze SW-RMOSS, we want to establish a similar property as Lemma 4 to bound the probability about an arm being under or over estimated. Toward this end, we need the following properties for truncated random variable. ###### Lemma 8. Let $X$ be a random variable with expected value $\mu$ and $\mathbb{E}[X^{2}]\leq 1$. Let $d\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{sat}(X,B)-\mathbb{E}[\operatorname{sat}(X,B)]$. Then for any $B>0$, it satisfies (i) $\mathinner{\\!\left\lvert d\right\rvert}\leq 2B$ (ii) $\mathbb{E}[d^{2}]\leq 1$ (iii) $\mathinner{\\!\left\lvert\mathbb{E}[\operatorname{sat}(X,B)]-\mu\right\rvert}\leq 1/B$. ###### Proof. Property (i) follows immediately from definition of $d$ and property (ii) follows from $\mathbb{E}[d^{2}]\leq\mathbb{E}\big{[}\operatorname{sat}^{2}(X,B)\big{]}\leq\mathbb{E}\big{[}X^{2}\big{]}.$ To see property (iii), since $\mu=\mathbb{E}\big{[}X\big{(}\mathbf{1}{\left\\{\mathinner{\\!\left\lvert X\right\rvert}\leq B\right\\}}+\mathbf{1}{\left\\{\mathinner{\\!\left\lvert X\right\rvert}>B\right\\}}\big{)}\big{]},$ one have $\displaystyle\mathinner{\\!\left\lvert\mathbb{E}[\operatorname{sat}(X,B)]-\mu\right\rvert}$ $\displaystyle\leq\mathbb{E}\left[\left(\mathinner{\\!\left\lvert X\right\rvert}-B\right)\mathbf{1}{\left\\{\mathinner{\\!\left\lvert X\right\rvert}>B\right\\}}\right]$ $\displaystyle\leq\mathbb{E}\left[\mathinner{\\!\left\lvert X\right\rvert}\mathbf{1}{\left\\{\mathinner{\\!\left\lvert X\right\rvert}>B\right\\}}\right]\leq\mathbb{E}\left[{X^{2}}/{B}\right].$ ∎ Moreover, we will also use a maximal Bennett type inequality as shown in the following. ###### Lemma 9 (Maximal Bennett’s inequality [37]). Let $\left\\{X_{i}\right\\}_{i\in\\{1,\dots,n\\}}$ be a sequence of bounded random variables with support $[-B,B]$, where $B\geq 0$. Suppose that $\mathbb{E}[X_{i}\|X_{1},\ldots,X_{i-1}]=\mu_{i}$ and $\operatorname{Var}[X_{i}\|X_{1},\ldots,X_{i-1}]\leq v$. Let $S_{m}=\sum_{i=1}^{m}(X_{i}-\mu_{i})$ for any $m\in\\{1,\dots,n\\}$. Then, for any $\delta\geq 0$ $\displaystyle\mathbb{P}\left(\exists{m\in\\{1,\dots,n\\}}\mathrel{\mathop{\mathchar 58\relax}}S_{m}\geq\delta\right)\leq\exp\left(-\frac{\delta}{B}\psi\left(\frac{B\delta}{nv}\right)\right),$ $\displaystyle\mathbb{P}\left(\exists{m\in\\{1,\dots,n\\}}\mathrel{\mathop{\mathchar 58\relax}}S_{m}\leq-\delta\right)\leq\exp\left(-\frac{\delta}{B}\psi\left(\frac{B\delta}{nv}\right)\right).$ Now, we are ready to establish a concentration property for saturated sliding window empirical mean. ###### Lemma 10. For any arm $k\in\\{1,\dots,K\\}$ and any $t\in\left\\{K+1,\ldots,T\right\\}$, if $\psi(2\zeta/a)\geq 2a/\zeta$, the probability of either event $A=\big{\\{}g^{k}_{t}\leq M_{t}^{k}-x,n_{k}(t)\geq l\big{\\}}$ or event $B=\big{\\{}g^{k}_{t}-2c_{n_{k}(t)}\geq M_{t}^{k}+x,n_{k}(t)\geq l\big{\\}}$, for any $x>0$ and any $l\geq 1$, is no greater than $\frac{2a}{\beta^{2}\ln(a)}\frac{K}{\tau x^{2}}(\beta x\sqrt{h(l)/a}+1)\exp\left(-\beta x\sqrt{h(l)/a}\right),$ where $\beta=\psi\left(2\zeta/a\right)/(2a)$. ###### Proof. Recall that $u_{i}^{kt}$ is the $i$-th time slot when arm $k$ is selected within $\mathcal{W}_{t}$. Since $c_{m}$ is a monotonically decreasing in $m$, $1/B_{m}=c_{h(m)}\leq c_{m}$ due to $h(m)\geq m$. Then, it follows from property (iii) in Lemma 8 that $\displaystyle\mathbb{P}(A)\\!$ $\displaystyle\leq\mathbb{P}\bigg{(}\\!\exists m\\!\in\\!\\{l,\ldots,\tau\\}\\!\mathrel{\mathop{\mathchar 58\relax}}\\!\bar{\mu}^{k}_{m}\leq\sum_{i=1}^{m}\\!\frac{\mu_{u_{i}^{kt}}^{k}}{m}\\!\\!-(1+\zeta)c_{m}\\!-x\\!\bigg{)}$ $\displaystyle\leq\mathbb{P}\bigg{(}\\!\exists m\\!\in\\!\\{l,\ldots,\tau\\}\\!\mathrel{\mathop{\mathchar 58\relax}}\\!\sum_{i=1}^{m}\\!\frac{\bar{d}_{im}^{kt}}{m}\\!\leq\\!\frac{1}{B_{m}}\\!-(1+\zeta)c_{m}\\!-x\\!\bigg{)}$ $\displaystyle\leq\mathbb{P}\bigg{(}\\!\exists m\\!\in\\!\\{l,\ldots,\tau\\}\\!\mathrel{\mathop{\mathchar 58\relax}}\\!\frac{1}{m}\sum_{i=1}^{m}\bar{d}_{im}^{kt}\leq-x-\zeta c_{m}\bigg{)},\,$ (25) where $\bar{d}_{im}^{kt}=\operatorname{sat}\big{(}X_{u_{i}^{kt}}^{k},B_{m}\big{)}-\mathbb{E}\big{[}\operatorname{sat}\big{(}X_{u_{i}^{kt}}^{k},B_{m}\big{)}\big{]}$. Recall we select $a>1$. Again, we apply a peeling argument with geometric grid $a^{s}\leq m<a^{s+1}$ over time interval $\\{l,\ldots,\tau\\}$. Let $s_{0}=\left\lfloor\log_{a}(l)\right\rfloor$. Since $c_{m}$ is monotonically decreasing with $m$, $\eqref{prob:A}\leq\\!\\!\sum_{s\geq s_{0}}\\!\mathbb{P}\Bigg{(}\\!\exists m\in[a^{s},a^{s+1})\\!\mathrel{\mathop{\mathchar 58\relax}}\\!\sum_{i=1}^{m}\bar{d}_{im}^{kt}\leq\\!-a^{s}\left(x+\zeta c_{a^{s+1}}\right)\\!\\!\bigg{)}.$ For all $m\in[a^{s},a^{s+1})$, since $B_{m}=B_{a^{s}}$, from Lemma 8 we know $\mathinner{\\!\left\lvert\bar{d}_{im}^{kt}\right\rvert}\leq 2B_{a^{s}}$ and $\mathbf{Var}\left[\bar{d}_{im}^{kt}\right]\leq 1$. Continuing from previous step, we apply Lemma 9 to get $\displaystyle\eqref{prob:A}\leq$ $\displaystyle\sum_{s\geq s_{0}}\exp\left(-\frac{a^{s}\left(x+\zeta c_{a^{s+1}}\right)}{2B_{a^{s}}}\psi\left(\frac{2B_{a^{s}}}{a}\left(x+\zeta c_{a^{s+1}}\right)\right)\right)$ $\displaystyle\left(\text{since }\psi(x)\text{ is monotonically increasing}\right)$ $\displaystyle\leq$ $\displaystyle\sum_{s\geq s_{0}}\exp\left(-\frac{a^{s}\left(x+\zeta c_{a^{s+1}}\right)}{2B_{a^{s}}}\psi\left(\frac{2\zeta}{a}B_{a^{s}}c_{a^{s+1}}\right)\right)$ (substituting $c_{a^{s+1}}$, $B_{a^{s}}$ and using $h(a^{s})=a^{s+1}$) $\displaystyle=$ $\displaystyle\sum_{s\geq s_{0}+1}\exp\left(-a^{s}\left(\frac{x}{B_{a^{s-1}}}+\zeta c_{a^{s}}^{2}\right)\frac{\psi\left(2\zeta/a\right)}{2a}\right)$ $\displaystyle\left(\text{since }\zeta\psi(2\zeta/a)\geq 2a\right)$ $\displaystyle\leq$ $\displaystyle\frac{K}{\tau}\sum_{s\geq s_{0}+1}a^{s}\exp\left(-a^{s}\frac{x}{B_{a^{s-1}}}\frac{\psi\left(2\zeta/a\right)}{2a}\right).$ (26) Let $b={x\psi\left(2\zeta/a\right)}/(2a)$. Since $\ln_{+}(x)\geq 1$ for all $x>0$, $\displaystyle\eqref{sum:1}\leq$ $\displaystyle\frac{K}{\tau}\sum_{s\geq s_{0}+1}a^{s}\exp\left(-b\sqrt{a^{s}}\right)$ $\displaystyle\leq$ $\displaystyle\frac{K}{\tau}\int_{s_{0}+1}^{+\infty}a^{y}\exp\big{(}-b\sqrt{a^{y-1}}\big{)}dy$ $\displaystyle=$ $\displaystyle\frac{K}{\tau}a\int_{s_{0}}^{+\infty}a^{y}\exp\big{(}-b\sqrt{a^{y}}\big{)}dy$ $\displaystyle=$ $\displaystyle\frac{K}{\tau}\frac{2a}{\ln(a)b^{2}}\int_{b\sqrt{a^{s_{0}}}}^{+\infty}z\exp\big{(}-z\big{)}dz\,(\text{where }z=b\sqrt{a^{y}})$ $\displaystyle\leq$ $\displaystyle\frac{K}{\tau}\frac{2a}{\ln(a)b^{2}}(b\sqrt{a^{s_{0}}}+1)\exp(-b\sqrt{a^{s_{0}}}),$ which concludes the proof. ∎ With Lemma 10, the upper bound on the worst-case regret for SW-RMOSS in the nonstationary heavy-tailed MAB problem can be analyzed similarly as Theorem 5. ###### Theorem 11. For the nonstationary heavy-tailed MAB problem with $K$ arms, time horizon $T$, variation budget $V_{T}>0$ and $\tau=\Big{\lceil}{K^{\frac{1}{3}}\left(T/V_{T}\right)^{\frac{2}{3}}}\Big{\rceil}$, if $\psi(2\zeta/a)\geq 2a/\zeta$, the worst-case regret of SW-RMOSS satisfies $\sup_{\mathcal{F}_{T}^{\mathcal{K}}\in\mathcal{E}(V_{T},T,K)}R_{T}^{\textup{SW- RMOSS}}\leq C(KV_{T})^{\frac{1}{3}}T^{\frac{2}{3}}.$ ###### Sketch of the proof. The procedure is similar as the proof of Theorem 5. The key difference is due to the nuance between the concentration properties on mean estimator. Neglecting the leading constants, the probability upper bound in Lemma 4 has a factor $\exp(-x^{2}l/\eta)$ comparing with $(\beta x\sqrt{h(l)/a}+1)\exp\left(-\beta x\sqrt{h(l)/a}\right)$ in Lemma 10. Since both factors are no greater than $1$, by simply replacing $\eta$ with $(1+\zeta)^{2}$ and taking similar calculation in every step except inequality (13), comparable bounds that only differs in leading constants can be obtained. Applying Lemma 10, we revise the computation of (13) as the following, $\displaystyle\sum_{s\geq l_{i}^{k}+1}\mathbb{P}{\bigg{\\{}g_{t_{s}}^{k}-2c_{n_{k}(t_{s})}>M_{t_{s}}^{k}+\frac{\Delta_{i}^{k}}{4}\bigg{\\}}}$ $\displaystyle\leq$ $\displaystyle\sum_{s\geq l_{i}^{k}}C^{\prime}\left(\frac{\beta\Delta_{i}^{k}}{4}\sqrt{\frac{h(l)}{a}}+1\right)\exp\left(-\frac{\beta\Delta_{i}^{k}}{4}\sqrt{\frac{h(l)}{a}}\right)$ $\displaystyle\leq$ $\displaystyle\int_{l_{i}^{k}-1}^{+\infty}C^{\prime}\left(\frac{\beta\Delta_{i}^{k}}{4}\sqrt{\frac{y}{a}}+1\right)\exp\left(-\frac{\beta\Delta_{i}^{k}}{4}\sqrt{\frac{y}{a}}\right)\,dy$ $\displaystyle\leq$ $\displaystyle\frac{6a}{\beta^{2}}\frac{2a}{\beta^{2}\ln(a)}\frac{K}{\tau}\bigg{(}\frac{4}{\Delta_{i}^{k}}\bigg{)}^{4}.$ (27) where $C^{\prime}={2aK}\big{(}{4}/{\Delta_{i}^{k}}\big{)}^{2}/{\big{(}\beta^{2}\ln(a)\tau\big{)}}$.The second inequality is due to the fact that $(x+1)\exp(-x)$ is monotonically decreasing in $x$ for $x\in[0,\infty)$ and $h(l)>l$. In the last inequality, we change the lower limits of the integration from $l_{i}^{k}-1$ to $0$ since $l_{i}^{k}\geq 1$ and plug in the value of $C^{\prime}$. Comparing with (13), this upper bound only varies in constant multiplier. So is the worst-regret upper bound. ∎ ###### Remark 1. The benefit of discount factor method is that it is memory friendly. This advantage is lost if truncated empirical mean is used. As $n_{k}(t)$ could both increase and decrease with time, the truncated point could both grow and decline, so all sampling history needs to be recorded. It remains an open problem how to effectively using discount factor in a nonstationary heavy- tailed MAB problem. ## VI Numerical Experiments We complement the theoretical results in previous section with two Monte-Carlo experiments. For the light-tailed setting, we compare R-MOSS, SW-MOSS and D-UCB in this paper with other state-of-art policies. For the heavy-tailed setting, we test the robustness of R-RMOSS and SW-RMOSS against both heavy- tailed rewards and nonstationarity. Each result in this section is derived by running designated policies $500$ times. And parameter selections for compared policies are strictly coherent with referred literature. ### VI-A Bernoulli Nonstationay Stochastic MAB Experiment To evaluated the performance of different policies, we consider two nonstationary environment as shown in Figs. 1(a) and 1(b), which both have $3$ arms with nonstationary Bernoulli reward. The success probability sequence at each arm is a Brownian motion in environment $1$ and a sinusoidal function of time $t$ in environment $2$. And the variation budget $V_{T}$ is $8.09$ and $3$ respectively. (a) Environment $1$ (b) Environment $2$ (c) Regrets for environment $1$ (d) Regrets for environment $2$ Figure 1: Comparison of different policies. The growths of regret in Figs. 1(c) and 1(d) show that UCB based policies (R-MOSS, SW-MOSS, and D-UCB) maintain their superior performance against adversarial bandit based policies (Rexp$3$ and Exp$3$.S) for stochastic bandits even in nonstationary settings, especially for R-MOSS and SW-MOSS. Besides, DTS outperforms other polices when the best arm does not switch. While each switch of the best arm seems to incur larger regret accumulation for DTS, which results in a lager regret compared with SW-MOSS and R-MOSS. ### VI-B Heavy-tailed Nonstationay Stochastic MAB Experiment Again we consider the $3$-armed bandit problem with sinusoidal mean rewards. In particular, for each arm $k\in\\{1,2,3\\}$, $\mu_{t}^{k}=0.3\sin\left(0.001\pi t+2k\pi/3\right),\quad t\in\\{1,\dots,5000\\}.$ Thus, the variation budget is $3$. Besides, mean reward is contaminated by additive sampling noise $\nu$, where $\mathinner{\\!\left\lvert\nu\right\rvert}$ is a generalized Pareto random variable and the sign of $\nu$ has equal probability to be “$+$” and “$-$”. So the probability distribution for $X_{t}^{k}$ is $f_{t}^{k}(x)=\frac{1}{2\sigma}\left(1+\frac{\xi\mathinner{\\!\left\lvert x-\mu_{t}^{k}\right\rvert}}{\sigma}\right)^{-\frac{1}{\xi}-1}\,\text{for }x\in(-\infty,+\infty).$ We select $\xi=0.4$ and $\sigma=0.23$ such that Assumption 2 is satisfied. We select $a=1.1$ and $\zeta=2.2$ for both R-RMOSS and SW-RMOSS such that condition $\psi(2\zeta/a)\geq 2a/\zeta$ is met. (a) Regret (b) Histogram of $R_{T}$ Figure 2: Performances with heavy-tailed rewards. Fig. 2(a) show RMOSS based polices and slightly outperform MOSS based polices in heavy-tailed settings. While by comparing the estimated histogram of $R_{T}$ for different policies in Fig. 2(b), R-RMOSS and SW-RMOSS have a better consistency and a smaller possibility of a particular realization of the regret deviating significantly from the mean value. ## VII Conclusion We studied the general nonstationary stochastic MAB problem with variation budget and provided three UCB based policies for the problem. Our analysis showed that the proposed policies enjoy the worst-case regret that is within a constant factor of the minimax regret lower bound. Besides, the sub-Gaussian assumption on reward distributions is relaxed to define the nonstationary heavy-tailed MAB problem. We show the order optimal worst-case regret can be maintained by extending the previous policies to robust versions. There are several possible avenues for future research. In this paper, we relied on passive methods to balance the remembering-versus-forgetting tradeoff. 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# CSI-Based Localization with CNNs Exploiting Phase Information Anastasios Foliadis12, Mario H. Castañeda Garcia1, Richard A. Stirling- Gallacher1, Reiner S. Thomä2 1Munich Research Center, Huawei Technologies Duesseldorf GmbH, Munich, Germany 2Electronic Measurements and Signal Processing, Technische Universität Ilmenau, Ilmenau, Germany {anastasios.foliadis, mario.castaneda<EMAIL_ADDRESS><EMAIL_ADDRESS> ###### Abstract In this paper we study the use of the Channel State Information (CSI) as fingerprint inputs of a Convolutional Neural Network (CNN) for localization. We examine whether the CSI can be used as a distinct fingerprint corresponding to a single position by considering the inconsistencies with its raw phase that cause the CSI to be unreliable. We propose two methods to produce reliable fingerprints including the phase information. Furthermore, we examine the structure of the CNN and more specifically the impact of pooling on the positioning performance, and show that pooling over the subcarriers can be more beneficial than over the antennas. ###### Index Terms: Localization, Positioning, Deep Learning, CSI, Fingerprint, Neural Network ## I Introduction Advances in mobile communications and the development of Internet of Things (IoT) has introduced a large variety of new applications in a number of different areas of modern life. One important requirement in several of these applications is the estimation of the user’s position. Although the ubiquity of Global Positioning System (GPS) provides a great solution for outdoor localization, other alternatives are needed indoors. Many different solutions have been proposed in the literature for indoor positioning, ranging from classical approaches, like angle of arrival (AoA) and time of arrival (ToA) based, to pattern recognition approaches. More specifically, the ability to store and transmit large amounts of data has directed the focus on using deep learning. Additionally, with the 5th Generation (5G) network being deployed, providing high data rates and bandwidth, the number of antennas on devices is increasing, enabled by the mm- Wave operation frequency. For coherent communication, the multi-antenna channel between a user and the base station (BS) is estimated using pilot symbols. The estimated channel referred as channel state information (CSI) can serve as a fingerprint for localization. For localization based on fingerprint inputs, a database for a given environment is created offline and during the online phase the UE’s position is estimated, by matching its signal to the fingerprint map. There exist a number of different approaches to implement the mapping. These range from conventional, like maximum likelihood and least squares, to machine learning, like k-nearest neighbors and neural networks. What is considered a fingerprint also differs depending on the application. In [1] the mapping is done using convolutional neural networks (CNNs), achieving a sub-meter accuracy with simulated and real measurements by utilizing the real and imaginary parts of the CSI. In [2], again a CNN was used with real measurements, but with its inputs consisting of a combination of raw features (real and imaginary), polar features and time-domain features. More complex neural network configurations were used in [3] using as input the time-domain channel impulse response. The authors of [4] were able to achieve sub-centimeter accuracy by employing a denoising technique and an ensemble of neural networks. They considered only the magnitude of the channel, since they identified that phase measurements at the same position can change over time. Phase spatial inconsistency arising because of implementation aspects is a common issue affecting the CSI. For this reason several prior works either ignore or completely reject the phase and focus primarily on the magnitude. However, the phase could embed important information of the underlying channel for localization purposes. In [5] and [6], a transformation per antenna is proposed to calibrate the phase of multi-antenna measurements used as inputs to a CNN. In this paper, we propose two techniques to obtain a processed phase which is consistent and reliable for localization. In contrast to [5] and [6], we propose the same transformation across all antennas to preserve valuable AoA information. By employing CNNs, we show that the use of the processed phase improves the localization accuracy compared to when using the raw phase. To optimize the CNN based on the structure of the CSI, we also examine the impact of the pooling layer on the localization performance. In the remainder of this paper we describe the system model in Section II. In Section III, we present the proposed methods to address the phase issues. In Section IV we introduce the machine leaning approach that we utilize and in Section V we present the results of our simulations. We present our conclusions in Section VI. ## II System Model and Database Description ### II-A System Model Due to its ubiquity in wireless communications and ease of deployment, we consider the use of orthogonal frequency division multiplexing (OFDM) waveform for localization. In addition to the degrees of freedom that the subcarriers in OFDM provide, we can exploit the multiple antennas at the transmitter and receiver. For simplicity we consider a static uplink setup with a single transmit antenna at a UE and multiple receive antennas at the BS, i.e. a single-input multiple-output (SIMO) system. The uplink SIMO channel estimated at the BS is given by: $\boldsymbol{H}=\left[\boldsymbol{h}_{0},\boldsymbol{h}_{1},...,\boldsymbol{h}_{N_{\text{C}}-1}\right]\in\mathbb{C}^{N_{\text{R}}\times N_{\text{C}}}$ (1) where $N_{\text{C}}$ is the number of subcarriers and $N_{\text{R}}$ the number of antennas at the receiver. $\boldsymbol{h}_{n}\in\mathbb{C}^{N_{\text{R}}}$ is the vector describing the CSI for the receive antenna array at the $n$-th subcarrier. Since $\boldsymbol{H}$ is based on the underlying transfer function between receiver and transmitter, it can be used to obtain a distinct fingerprint for each measured position of a UE. There exist many techniques to estimate the complex channels based on transmitted pilots. In reality $\boldsymbol{H}$ is the effective channel, i.e. it includes timing offsets and hardware imperfections. Such disturbances may hinder the ability of the raw CSI to provide a distinct fingerprint for each position. According to the analysis in[7] and [8], these timing offsets between the oscillators of the transmitter and receiver, influence the estimated channel $\boldsymbol{H}$ and thus its phase at the $k$-th antenna and $n$-th subcarrier, where $k\in[0,N_{\text{R}}-1]$ and $n\in[0,N_{\text{C}}-1]$, based on the actual channel $\tilde{\boldsymbol{H}}$ can be expressed as follows: $\angle{\boldsymbol{H}_{n,k}}=\angle{\tilde{\boldsymbol{H}}_{n,k}}+(\tau_{\text{p}}+\tau_{\text{s}})n+\tau_{\text{c}}+\beta+\epsilon_{n,k}$ (2) where the phase of the actual channel at the $k$-th antenna and $n$-th subcarrier is $\angle{\tilde{\boldsymbol{H}}_{n,k}}$. $\tau_{\text{p}}$ is the symbol time offset (STO), $\tau_{\text{s}}$ the sampling time offset, $\tau_{\text{c}}$ the carrier frequency offset, $\beta$ is the phase locked loop (PLL) phase offset and $\epsilon_{n,k}$ is random noise. Due to the continuous timing drift of transmitter and receiver, the estimated channel would not be constant even if the underlying channel does not change. This makes the raw phase practically unusable as a distinct fingerprint for positioning, as pointed out in [5] and [6]. ### II-B Database Description To evaluate our proposals we use channel measurements which were described in [2] using 3 different antenna configurations for the BS in a $2.5\times 2.5$ m indoor area shown in Fig. 1. The different antenna configurations for the BS consist of a Uniform Rectangular Array (URA) of $8\times 8$ antennas, a Uniform Linear Array (ULA) of 64 antennas and a distributed (DIS) configuration of 8 ULA arrays with 8 antennas each. For each configuration, the BS has $N_{\text{R}}=64$ receive antennas. The spacing between adjacent antenna elements in the ULAs and URA is 70 mm. The UE was equipped with a single antenna. Uplink SIMO channel measurements were performed for equidistantly spaced UE locations (5 mm apart), within the green area inf Fig. 1. In [2], the carrier frequency was 2.61 GHz with a bandwidth of 20 MHz and $N_{\text{C}}=100$ subcarriers. Figure 1: Database measurement scenarios (figure taken from [2]). Distances are indicated in millimeters. (a) Channel magnitude across subcarriers (b) Channel phase across subcarriers Figure 2: Channel Fingerprints based on raw CSI (a) Phase difference method across subcarriers (b) Phase alignment method across subcarriers Figure 3: Phase fingerprints based on proposed methods ## III Fingerprint based on Phase Information An important aspect for localization based on fingerprint inputs is that the fingerprint that corresponds to each position is desired to be unique and consistent, during both online and offline phases. Otherwise the correct mapping of a new measurement to the database may not be possible. In particular, the magnitude of the estimated channel can be expected to be spatially consistent, e.g. does not vary significantly (besides some additive noise) over measurements at the same position. The spatial consistency of the magnitude can also be expected for measurements corresponding to very nearby positions. This can be observed in Fig. 2(a), where the magnitude over the subcarriers for the first antenna in the ULA is depicted for three neighboring sampled positions of the database from [2]. On the other hand, the raw phase of the estimated channel is usually not spatially consistent as described before. This is shown in Fig. 2(b), where the phase over the subcarriers for the first antenna in the ULA is depicted for the same three neighboring positions considered in Fig. 2(a). However, for our purposes we are not interested in estimating the phase of the actual channel $\tilde{\boldsymbol{H}}$, as our goal is merely to acquire a distinct and consistent fingerprint for each position, which is to say for each channel between UE and BS. In the following, we present two methods which produce reliable fingerprints by considering both phase and magnitude, while also preserving the Angle of Arrival (AoA) information embedded in the relationship between the phases of adjacent antennas. The AoA information can be exploited for localization. ### III-A Phase Difference When the same oscillator is used for all antennas, the phase offsets in (2) are the same for each antenna. Thus, by taking the difference between phases of two adjacent antennas, we eliminate the phase offsets. The phase difference between adjacent antennas for each subcarrier $n$ can be used to produce a distinct fingerprint: $\phi_{n,k}=(\angle\boldsymbol{H}_{n,k}-\angle\boldsymbol{H}_{n,(k+1)\text{mod}N_{\text{R}}})+\epsilon$ (3) where we used the modulo operator to include the phase difference between the last and the first antennas. In this way, we do not lose any information when creating the fingerprints. Based on phase difference, the fingerprint associated with the $k$-th antenna and $n$-th subcarrier for each position would be: $\boldsymbol{H}_{D}[n,k]=\|\boldsymbol{H}_{n,k}\|e^{\text{j}\left(\angle\boldsymbol{H}_{n,k}-\angle\boldsymbol{H}_{n,(k+1)modN_{\text{R}}}\right)}.$ (4) Here we have managed to create a distinct fingerprint $\boldsymbol{H}_{D}$, for each position, whose phase at each element $[n,k]$ is the phase difference of antennas $k$ and $k+1$ of the original matrix $\boldsymbol{H}$ at subcarrier $n$ while its magnitude is the same as $\boldsymbol{H}$. Also, the difference of phases, takes into account the relationship between antennas. The fingerprint quality offered by the phase difference can be seen in Fig. 3(a), where the phase difference of the first and second antennas in the ULA over the subcarriers is shown for the same positions as in Fig. 2(b). ### III-B Phase Alignment (a) Phase wrapping of $\angle H_{D}[n,k]$ (b) Phase Sine (c) Phase Cosine Figure 4: Phase fingerprint considering phase wrapping By considering (3), we see that $\tau_{\text{p}}$ and $\tau_{\text{s}}$ influence the slope of the phases across subcarriers, while $\tau_{\text{c}}$ and $\beta$ simply add a constant offset. These offsets can be mitigated by rotating and shifting the channel across the subcarriers for each antenna as proposed in [6]. However, by applying such a transformation for each antenna separately as suggested in [5], the AoA information that is embedded in the relationship between the phases of adjacent antennas is lost, i.e., it can not be exploited for localization. To preserve this information, we propose to apply the same transformation across the subcarriers for all antennas. Thus, with this method we obtain the fingerprint $\boldsymbol{H}_{A}$, where the value associated with the $k$-th antenna and $n$-th subcarrier is calculated as: $\boldsymbol{H}_{A}[n,k]=\boldsymbol{H}[n,k]e^{-j(\lambda n+b)}$ (5) where $\lambda$ is the reference slope of the subcarrier phases, and $b$ is the reference offset, which are determined as follows. Firstly, we fit a linear regression model for the phase over the subcarriers of each antenna, resulting in $\angle\boldsymbol{H}[n,k]=\lambda_{k}n+b_{k}+\zeta_{n,k}$ (6) where $\zeta_{n,k}$ is the statistical error of the regression model which is minimized. Thus, in contrast to [5] and [6] where the slope $\lambda$ is calculated from the difference of the phases of first and last subcarriers and the offset $b$ as the mean value of the phases across subcarrier, the parameters in (5) are calculated as $\lambda=\frac{1}{N_{\text{R}}}\sum_{k=1}^{N_{\text{R}}}\lambda_{k},\quad\quad b=b_{1}$ (7) By calculating the slope $\lambda$ this way we avoid the possibility that it will be affected by outliers. As our first method, the proposed phase alignment results in more reliable fingerprints as compared to the raw phase. This can be seen in Fig. 3(b), where the phase of $\boldsymbol{H}_{A}$ over the subcarriers associated with the first antenna in the ULA is shown for the same positions considered in Fig. 2b. ### III-C Phase Wrapping Phase wrapping is another problem that can impair the fingerprint. As can be seen in Fig. 4(a) for a given antenna and two neighboring positions of the database in [2], phase measurements close to $-\pi$ may fluctuate across the subcarriers, and in some cases the phase wraps around to $\pi$ due to noise. Fig. 4(a) shows an example of the phase wrapping issue, where the phases across the subcarriers, associated with the first antenna of the ULA of [2], for two similar positions is shown. While the phase measurements of both positions fluctuate around $-\pi$, they do not wrap around to $\pi$ in the same way, creating two different fingerprints. The most common method to address this issue is to simply unwrap the phase [5]. This approach, however, is unreliable under noisy conditions as it could lead to large phase values, since the errors accumulate with unwrapping. Such large values then dominate the fingerprint and small variations in the range $[-\pi,\pi)$ will have less influence. We propose to leverage the fact that for any angle $\theta$ we have $\exp(\text{j}\theta)=\cos(\theta)+\text{j}\sin(\theta)$, such that the information provided by $\theta$ is encoded in $\sin(\theta)$ and $\cos(\theta)$, which are continuous everywhere from $\pi$ to $-\pi$. In Fig. 4(b) and 4(c) we plot the sine and cosine of the phase $\angle\boldsymbol{H}_{D}[n,k]$, in contrast to 4(a), we see that the fingerprint quality is preserved when using $\sin{(\angle\boldsymbol{H}_{D}[n,k])}$ and $\cos{(\angle\boldsymbol{H}_{D}[n,k])}$. This indicates that the use of real and imaginary parts of the complex valued matrix can solve the wrapping problem since they can be expressed by the magnitude and the cosine and sine of the phase respectively. Additionally, this enables the phase and magnitude to be processed separately, using different and suitable techniques for each one, by using the sine and cosine to represent only the phase information. The downside is that to fully represent the phase fingerprint one must use both those functions, increasing the amount of data to be processed. These techniques can also be used on channel estimates based on ray-tracing simulations, even though there are no timing offsets. In this way, the fingerprints from simulations match the fingerprint from measurements, and can be used as an extra layer of information as in [9]. ## IV Localization using CNNs Although there are several localization schemes based on fingerprint inputs, we focus on using CNNs. For a UE at position $\boldsymbol{r}\in\mathbb{R}^{2}$, we describe the channel with the function $f$, meaning $f(\boldsymbol{r})=\boldsymbol{H}_{\star}$, where $\boldsymbol{H}_{\star}\in\\{\boldsymbol{H},\boldsymbol{H}_{D},\boldsymbol{H}_{A},\boldsymbol{H}_{A}^{\prime}\\}$ as described in Section II, with $\boldsymbol{H}_{A}^{\prime}$ being the fingerprint based on the method proposed in [5]. We will attempt to approximate the inverse function, $f^{-1}(\boldsymbol{H}_{\star})=\boldsymbol{r}$, by using CNNs, which have shown promising results for positioning [1],[3]. The reason is that CNNs have some features that could be beneficial for the considered type of inputs. ### IV-A Convolutional Neural Networks In a CNN, the input is convoluted with a matrix of smaller dimension, called the kernel. It is almost certainly followed by the pooling operation which is used to reduce the data at the output. Usually, the term CNN is used to describe a NN that uses the convolution operation at some layer. In addition to the two dimensions (antennas and subcarriers) that our input matrix has, the CNN may also use a third dimension, meaning multiple matrices can be stacked at the input. In machine learning terminology, the input matrices are called channels (not to be confused with the wireless channel). This convolutional layer leverages the idea of sparse interactions [10]. A conventional fully connected layer is learning parameters that describe the interactions between each and every one element of the input, while the CNN makes use of the smaller kernel to learn only the interactions between neighboring elements of the input. As the wireless channel between neighboring antennas and subcarriers is usually more correlated than the channels from antennas or subcarriers which are farther apart, the use of CNNs is appealing. Figure 5: Neural Network Model ### IV-B Pooling As previously described, after every convolutional layer there is a pooling layer which downsamples the output of the previous layer. The pooling function replaces that output with a summary statistic of the nearby outputs [10]. For example, the max pooling (which we consider in this work) reports the maximum output within a rectangular region of size $p_{ant}\times p_{sc}$, where $p_{ant}$ and $p_{sc}$ are the configurable sizes of the pooling in the antenna and subcarrier dimension, respectively. Besides reducing the dimension of the data, the pooling layer’s purpose is to make the output invariant to small translations of the input [10]. In our case, the two dimensions of the input matrix are antennas and subcarriers. We expect that small translations in the antenna dimension are important to be detected, since that provides the AoA information. On the other hand adjacent subcarriers within the coherence bandwidth may not provide additional information, as these subcarriers can be correlated. Thus, it may be more beneficial to pool over subcarriers, as pooling over the antennas may lead to a reduction of the angular resolution. ## V Simulation Results ### V-A Neural Network Setup We consider the CNN depicted in Fig. 5 with the input being convoluted with 32 different kernels of dimensions $4\times 4$. The resulting matrices are pooled, which is followed again by a convolution and pooling layer. The outputs of that layer are vectorized and inputted into four dense layers. The last layer has only 2 neurons expressing the position estimate. The training set was 80% of the database of [2] and test set was 20%. The training procedure starts with a batch size of 32 samples and is increased to 128, 256 and 1024 with each transition set after 30 epochs. The loss function is defined as the Euclidean mean distance of the estimated position and the real position. All the activation functions are set as the Rectified linear unit (ReLU) [10], except the last one which is linear, and the input data is normalized from 0 to 1, with respect to all the data in the training set. Lastly, we consider as a metric the mean error (ME) given by the Euclidean distance between the estimated and actual position in the test set. ### V-B Different Fingerprint Inputs TABLE I: One-Channel Input Input \| ME (m) ---\|--- $\|\boldsymbol{H}\|$ \| 0.03805 $\angle\boldsymbol{H}$ \| 0.04251 $\angle\boldsymbol{H}_{D}$ \| 0.04088 $\angle\boldsymbol{H}_{A}$ \| 0.03246 $\angle\boldsymbol{H}_{A}^{\prime}$ \| 0.08142 We first show the results for one, two and three number of input channels considering magnitude and phase information. For all the following configurations the pooling layers had a dimension of $4\times 4$ ($p_{sc}=p_{ant}=4$). We define $\|\boldsymbol{H}_{\star}\|\in\mathbb{R}^{N_{\text{R}}\times N_{\text{C}}}$ and $\angle\boldsymbol{H}_{\star}\in\mathbb{R}^{N_{\text{R}}\times N_{\text{C}}}$ as the element-wise absolute value and angle operator, respectively, of the fingerprint matrix $\boldsymbol{H}_{\star}\in\\{\boldsymbol{H},\boldsymbol{H}_{D},\boldsymbol{H}_{A},\boldsymbol{H}_{A}^{\prime}\\}$. For the following results, we consider only the ULA antenna configuration (see Fig. 1). For one input channel of the CNN, Table I lists the ME when using the magnitude of the channel, the raw phase and the two different processed phase inputs resulting from the phase difference and phase alignment methods, proposed in Section II. We see that the phase alignment method, not only outperforms using the raw phase, but actually achieves the best performance, even better than using only the magnitude. We also observe that performing a different phase alignment for each antenna as in [5], deteriorates the performance since the relationship of the phases between antennas (i.e., AoA information) is lost. Table II presents the results with two input channels, including magnitude and phase as well as the real and imaginary part of the different considered fingerprint matrices, to address the phase wrapping. In addition, we also considered the sine and cosine of $\angle\boldsymbol{H}_{D}$, thereby using only phase information. Similar to Table I, we see that properly processing the phase largely improves the results. We also see that using Re($\boldsymbol{H}_{D}$) and Im($\boldsymbol{H}_{D}$) outperforms using the $\sin(\angle\boldsymbol{H}_{D})$ and $\cos(\angle\boldsymbol{H}_{D})$, as the former includes also magnitude information. TABLE II: Two-Channel Input Input I \| Input II \| ME (m) ---\|---\|--- $\|\boldsymbol{H}\|$ \| $\angle\boldsymbol{H}$ \| 0.03126 $\|\boldsymbol{H}_{D}\|$ \| $\angle\boldsymbol{H}_{D}$ \| 0.03810 $\|\boldsymbol{H}_{A}\|$ \| $\angle\boldsymbol{H}_{A}$ \| 0.02792 $\|\boldsymbol{H}_{A}^{\prime}\|$ \| $\angle\boldsymbol{H}_{A}^{\prime}$ \| 0.03719 Re($\boldsymbol{H}$) \| Im($\boldsymbol{H}$) \| 0.01809 Re($\boldsymbol{H}_{A}$) \| Im($\boldsymbol{H}_{A}$) \| 0.01614 Re($\boldsymbol{H}_{D}$) \| Im($\boldsymbol{H}_{D}$) \| 0.01316 Re($\boldsymbol{H}_{A}^{\prime}$) \| Im($\boldsymbol{H}_{A}^{\prime}$) \| 0.03478 sin($\angle\boldsymbol{H}_{D}$) \| cos($\angle\boldsymbol{H}_{D}$) \| 0.01425 Lastly, Table III provides results with three channel inputs, where we can use the magnitude of the channel with the sine and cosine of the phase of the considered fingerprints. As in previous results, the use of properly processed phase information provides the best performance. From both Table II and III we see that using the fingerprints based on matrix $\boldsymbol{H}_{D}$ while also addressing phase wrapping achieved the best performance. The small improvement when using three channels can be attributed to the fact the the CNN is able to employ different processing for phase and magnitude, and extract the relevant information in each case. TABLE III: Three-Channel Input Input I \| Input II \| Input III \| ME (m) ---\|---\|---\|--- $\|\boldsymbol{H}\|$ \| sin($\angle\boldsymbol{H}$) \| cos($\angle\boldsymbol{H}$) \| 0.01981 $\|\boldsymbol{H}_{A}\|$ \| sin($\angle\boldsymbol{H}_{A}$) \| cos($\angle\boldsymbol{H}_{A}$) \| 0.01734 $\|\boldsymbol{H}_{D}\|$ \| sin($\angle\boldsymbol{H}_{D}$) \| cos($\angle\boldsymbol{H}_{D}$) \| 0.01290 ### V-C Pooling In this subsection, we analyze the impact of different pooling options $[p_{ant},p_{sc}]$ on the positioning performance, considering the ULA, URA and DIS antenna configurations from [2]. For the evaluation, we use $\|\boldsymbol{H}_{D}\|$, $\sin(\angle\boldsymbol{H}_{D})$ and $\cos(\angle\boldsymbol{H}_{D})$ as the input channels of the CNN shown in Fig. 5. In Table IV, we show the ME for different pooling options with $p_{ant}\times p_{sc}=4$, such that that resulting CNNs have the same complexity. For each antenna configuration, pooling over the subcarriers, i.e. [1,4], leads to the smallest ME, while the largest ME is obtained when pooling over the antennas, i.e. [4,1]. For a given pooling option, the best performance is achieved with the distributed antenna configuration, as it collects CSI at distinct locations around a UE’s position. On the other hand, the worst performance results by using the URA, since its resolution on the horizontal plane where the UE lies, is smaller compared to the other antenna configurations. The results in Table IV suggest that it is more beneficial to pool over the subcarriers than over the antennas. Thus, in Fig. 6 we examine the ME with pooling option $[1,p_{sc}]$ for different values of $p_{sc}$. For each antenna configuration, we see there is an optimum pooling $p_{sc}$ over the subcarriers, which we posit that it depends on the coherence bandwidth of the channel. The lowest ME attained for each of the ULA, DIST and URA antenna configurations are 6.11 mm, 5.20 mm and 9.81 mm respectively, which is lower than the ones reported in [2]. This was achieved by using the optimal size of pooling $p_{sc}$, which is different for each antenna configuration, and the three channel input: $\|\boldsymbol{H}_{D}\|,\sin(\angle\boldsymbol{H}_{D}),\cos(\angle\boldsymbol{H}_{D})$. TABLE IV: Different Pooling Options Pooling [$p_{ant}$, $p_{sc}$] \| Antenna Configuration \| ME (m) ---\|---\|--- [1,4] \| ULA \| 0.00612 [2,2] \| ULA \| 0.01010 [4,1] \| ULA \| 0.01633 [1,4] \| distributed \| 0.00521 [2,2] \| distributed \| 0.00732 [4,1] \| distributed \| 0.00982 [1,4] \| URA \| 0.01096 [2,2] \| URA \| 0.01183 [4,1] \| URA \| 0.01908 Figure 6: ME for different pooling dimensions ## VI Conclusion We have examined the use of CSI over multiple antennas and subcarriers, as fingerprint inputs of a CNN for UE localization. As the raw phase of the estimated channel cannot be used as a consistent fingerprint, we have presented different methods for producing reliable fingerprints based on phase information. Although the proposed methods have been evaluated with CNNs, they can also be used for other localization schemes based on fingerprints. For different number of inputs of a CNN, simulation results have shown that UE localization can be improved with properly processed phase information. We have also investigated the impact of different pooling options on the positioning performance with CNNs, showing that it is more beneficial to pool over the subcarriers than over the antennas. Simulation results have shown there is an optimum pooling size over the subcarriers, whose dependency on the coherence bandwidth is part of future work. ## References * [1] M. Widmaier, M. Arnold, S. Dörner, S. Cammerer, and S. Brink, “Towards practical indoor positioning based on massive MIMO systems,” _2019 IEEE 90th Vehicular Technology Conference (VTC2019-Fall)_ , 2019. * [2] S. D. Bast, A. P. Guevara, and S. Pollin, “CSI-based positioning in massive MIMO systems using convolutional neural networks,” _2020 IEEE 91st Vehicular Technology Conference (VTC2020-Spring)_ , 2020. * [3] A. Niitsoo, T. Edelhäußer, E. Eberlein, N. Hadaschik, and C. 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# Artificial Intelligence Prediction of Stock Prices using Social Media Kavyashree Ranawat Durham University School of Engineering and Computing Sciences Durham University Lower Mountjoy, South Rd, Durham DH1 3LE, United Kingdom <EMAIL_ADDRESS> &Stefano Giani Durham University School of Engineering and Computing Sciences Durham University Lower Mountjoy, South Rd, Durham DH1 3LE, United Kingdom <EMAIL_ADDRESS> ###### Abstract The primary objective of this work is to develop a Neural Network based on LSTM to predict stock market movements using tweets. Word embeddings, used in the LSTM network, are initialised using Stanford’s GloVe embeddings, pretrained specifically on 2 billion tweets. To overcome the limited size of the dataset, an augmentation strategy is proposed to split each input sequence into 150 subsets. To achieve further improvements in the original configuration, hyperparameter optimisation is performed. The effects of variation in hyperparameters such as dropout rate, batch size, and LSTM hidden state output size are assessed individually. Furthermore, an exhaustive set of parameter combinations is examined to determine the optimal model configuration. The best performance on the validation dataset is achieved by hyperparameter combination 0.4,8,100 for the dropout, batch size, and hidden units respectively. The final testing accuracy of the model is 76.14%. _K_ eywords LSTM $\cdot$ Twitter $\cdot$ Stock Prediction $\cdot$ APPLE $\cdot$ Neural Networks $\cdot$ VADER ## Introduction Twitter is a microblogging and social media platform that allows users to communicate via short messages (280 characters) known as tweets [1, 2, 3]. It enables millions of users to express their opinions on a daily basis on a variety of different topics ranging from reviews on products and services to users’ political and religious views, making Twitter a potent tool for gauging public sentiment [4]. Thus, it manifestly follows that twitter data can be regarded as a corpus, forming the basis on which predictions can be made, and researchers have indeed exploited this fact to seek trends by performing numerous and varied analyses. A characteristic feature of the stock market is volatility and there is no general equation describing the prediction of stock prices, which is a complex function of a range of different factors. The methods of stock market prediction can be broadly classified into Technical Analysis and Fundamental Analysis [5]. The latter involves the consideration of macroeconomic factors as well as industry specific news and events to guide investment strategies [5]. The analysis of public sentiment via tweets performed in this project can be regarded as an aspect of Fundamental Analysis. Although the prediction of stock prices is highly nuanced, the Efficient Market Hypothesis (EMH), propounded by Eugene Farma in the 1960’s, suggested a relation between public opinion and stock prices [6]. The semi-strong form of the EMH implies that current events and new public information have a significant bearing on market trends [1, 6]. This view is supported by Nosfinger who draws upon evidence from several studies in the field of Behavioural Finance to reinforce that changes in aggregate stock price as well as the high degree of market volatility can be, in part, attributed to public emotion [7]. Numerous studies have been successful in unveiling and proving the perceived existence of a relationship between public mood gathered from social media and stock market trends [7, 8, 9, 10, 11, 12, 13, 14, 15]. The vast majority of ML (Machine Learning) techniques applied in this sector have integrated the characteristics of NLP (Natural Language Processing) to extract and quantify the sentiment of public opinion expressed via social media. SVM (Support Vector Machines) [8], Random Forest [9], and KNN (K-Nearest Neighbour) [11] classifiers have yielded impressive classification accuracies in this application. The only caveat is that most studies consider the compound effect of historic prices and public sentiment, thereby discounting the exclusive impact of sentiment. Some studies have gone beyond the classic ML approach, employing deep learning methods such as ANNs (Artificial Neural Networks) in one form or another for the purposes of making predictions [12, 13, 14, 15, 16, 17]. Bollen et al have established that collective public mood is predictive of DJIA (Dow Jones Industrial Average) closing values by making use of Granger Causality Analysis and SOFNN (Self-Organized Fuzzy Neural Networks). Many researchers have built on this work and others have explored alternate deep learning models such as MLP (Multi Layer Perceptron), CNN (Convolutional Neural Network) + LSTM (Long Short Term Memory) for market prediction. The primary focus of this work was on the development of a variant of RNN (Recursive Neural Network), known as LSTM, capable of predicting short-term price movements. Owing to the volatile and unpredictable nature of the stock market, it is plausible that the relationship between the societal mood and economic indicators perhaps is more complex and nuanced than linear. Deep learning methods are felicitous for this application in that hidden layers can exploit the inherent relational complexity and can potentially extract these implicit relationships. It is for this reason that an LSTM structure was selected as the principal model in this work. The popularity of RNNs in NLP and stock prediction tasks is attributed to the fact that they consider the temporal effect of events which is a significant advantage over other NNs (Neural Networks). With the aid of a popular sentiment analysis tool, known as VADER, the degree of correlation between the sentiment expressed via tweets and stock price direction was also investigated for the purposes of comparison with the results from the LSTM architecture. ## 1 VADER Implementation VADER is a state of the art technique employed by researchers in sentiment analysis tasks. One aspect of this work involves using VADER to explore the degree of correlation between public opinion and sentiment expressed via twitter and stock market direction. As aforementioned, VADER is a gold standard lexicon and rule-based tool for sentiment analysis [18, 19]. Developed and empirically validated by Hutto and Gilbert, the VADER lexicon is characteristically attuned to text segments in the social media domain [20, 21]. Unlike other lexicon approaches, VADER takes into account that microblog text often contains slang, emoticons, and abbreviated text [21]. It not only provides the semantic orientation of words but also quantifies sentiment intensity by considering generalisable heuristics such as word order, capitalisation, degree modifiers etc [21]. In this application, VADER was used to generate the polarity scores of tweets, including a compound score (normalised between 1 and -1) which reflects the combined effect of the degree of positivity, negativity, and neutrality expressed in a tweet. ### 1.1 Experimental Procedure The aim is to investigate the correlation between two variables; VADER scores and stock market trajectory. Firstly, tweets containing the APPLE stock ticker symbol were cleaned, using the algorithm described in the next section. Stock data associated with APPLE, among the Big Four technology companies, is deemed as a suitable choice upon which to perform analysis for several reasons; a detailed rationale is provided in a subsequent section. The compound score for each tweet was generated using VADER. Subsequently, the average of the scores for all tweets in a single day was taken. To obtain values for the second variable i.e. stock market movement, the direction of stock price movement was quantified. If the next-day close price of the security is greater than that of the current day, the value for that day is defined as 1, else it is defined as 0. After generating the values for both variables, a special case of the Pearson coefficient, known as Point biserial correlation coefficient, was applied on the data to determine the correlation. This metric is commonly used when one variable is continuous and the other is categorical, as is the case in this application, where the VADER scores are continuous whereas the stock price change data is dichotomous (binary) [22]. The biserial correlation method, however, requires the continuous variable to be normally distributed [22]. Therefore, the distribution of the VADER scores was plotted and a roughly normal distribution was obtained (as shown in figure 1), allowing the use of the biserial correlation method. Figure 1: Distribution of VADER scores (continuous variable). ### 1.2 Further Modifications Modifications were made to the stock price change data by redefining the classification strategy. Previously, it was based on a delay of 1 day. It is plausible, however, that the effect of information and opinions presented in tweets may take longer to manifest and reflect in the asset price. To explore this theory, a delay size of days in the range [1,7] was taken. For example, a delay size of 2 indicates that if the asset price at the close of the trading day 2 days hence is higher, it is classed as 1, else it is classed as 0. Figure 2: Correlation coefficient values for different delay periods. Another configuration was assessed after making modifications to the VADER scores. If a particular tweet has more retweets than others i.e. it has been frequently shared by users, it could indicate that the information contained in that tweet has a notable influence on other users. These tweets could potentially play a greater role in impacting an incremental change within the stock market. By taking a simple average of the compound scores of tweets to arrive at a singular value for a specific day, it is not possible to gauge the contribution of important tweets. Thus, based on the number of retweets per tweet, a weighted average of the compound scores was taken to ensure accurate representation of all tweets. The correlation was then determined using the modified scores. ### 1.3 Results The correlation obtained for variable delay configurations is shown graphically in figure 2. It is evident that the correlation for the original configuration using a delay of 1 day is 0.0812 (or 8.12%), suggesting a notably poor linear relationship between the sentiment expressed in tweets and market movement. Using delays higher than 1 generally tend to improve the degree of correlation, with a delay of 4 days yielding the highest correlation (30.37%). This result supports the assumption that there exists a lag between the release of public opinion and its consequent reflection in stock price. A decreasing trend is observed as the delay size is increased beyond 4 days. A potential implication of this could be that the information contained on a particular day is irrelevant after large delay periods and no longer has any bearing on stock market values. The retweets-based weighted average configuration results in a small negative correlation value of -8.87%, which is in complete contrast to the original configuration. There is ambiguity in what the underlying cause of the generated results could be. There is a possibility that the inclusion of retweets has no impact in moulding future stock values. Alternatively, it could also be that the lack in the number of data points is masking the true contribution of the retweets. It can be concluded from the results that VADER is not able to present any strong association between public sentiment and market trajectory. The low correlation values could be attributed to the use of an insufficient number of samples for both variables. The final VADER scores are derived by taking an average of all compound scores, which is a simplistic and naive approach. The inability of the lexicon-based tool to identify any notable relationships between the variables could also be due to the inadequacies inherent in the development of VADER sentiment. However, as mentioned earlier, there is a high possibility that the relationship between the variables is not linear and is perhaps more complex and nuanced. This could be one of the reasons why using a metric such as correlation, used to assess strength of a linear relationship, is not able to detect any significant associations. ## 2 Neural Network Model ANNs, inspired by the behaviour of neurons in biological systems, are a dense interconnection of nodes or pre-processing units connected in layers, which have the ability to discover complex relationships between inputs and outputs [23]. There are many varying implementations of ANNs, which differ in terms of network architecture, properties and complexity. Considering the sequential nature of tweets, it is essential to use a network which considers the temporal effect of an input sequence. RNNs satisfy this criterion and are indeed suitable for application in tasks of this nature. However, the main drawback of RNNs is their inability in capturing long term dependencies in an input sequence i.e. the Vanishing Gradient problem [24, 25]. For example, when an input sentence is fed into the network, the error must be backpropagated through the network in order to update the weights. If the input is long, the gradients diminish exponentially during backpropagation, resulting in virtually no contribution from the state in earlier time steps. It is particularly problematic when using the sigmoid activation function as its derivative lies in the range [0, 0.25], resulting in highly diminished gradient values after repeated multiplication. A variant of the vanilla RNN network, the LSTM, can overcome this limitation [24, 25], albeit not entirely, and is used in this application. Figure 3 shows a high level abstraction of the LSTM model architecture, consisting of one stacked LSTM layer. The input side shows the vectors, concatenated in the Embedding Layer, being input to the LSTM cells in the stacked layer. The output side depicts the hidden state outputs of the LSTM layer being taken as inputs by a sigmoid-activated node to make output predictions. Figure 3: General schematic of the LSTM Network. The input side shows the vectors, concatenated in the Embedding Layer, being input to the LSTM cells in the stacked layer. The output side depicts the hidden state outputs of the LSTM layer being taken as inputs by a sigmoid-activated node to make output predictions. ### 2.1 Preprocessing Raw tweets contain a great deal of noise which needs to be eliminated in order to extract relevant information from tweets and improve the predictive performance of the applied algorithm. Tweets contain twitter handles, URLs, numeric characters, and punctuation which do not contribute meaningfully to the analysis. A preprocessing algorithm was applied to remove these elements, convert words to lower case, and tokenize the words present in a tweet. The cleaned tweets were concatenated so as to form one input sequence for a given day. Ordinal or integer encoding was used to map all the words in the vocabulary to an integer value, resulting in an input vector $\\{w_{1},w_{2},w_{3},…,w_{n}\\}$, where $w_{t}$ corresponds to a unique integer index representing a feature (or word) in the vocabulary. Although LSTMs can take inputs of variable length [26], post-padding was applied as only vectors with homogeneous dimensionality can be used with the Keras Embedding Layer. For each input vector of length $t\in[1,n]$, $n-t$ zeros or dummy features must be appended, where $n$ is the length of the longest encoded vector. This produces vectors in an $m$-dimensional feature space, where $m$ is the total number of unique samples i.e. number of unique phrases or tweets fed to the network. ### 2.2 Embedding NLP tasks for textual representation and feature extraction commonly use BoW (Bag of Words) models owing to their flexibility and simplicity. The traditional BoW has two variants: N-gram BoW and TF-IDF (Term Frequency- Inverse Document Frequency). The former reduces dimensionality of the feature set by extracting phrases comprising $N$ words while the latter considers the frequency of words whilst considering the effect of rare words. The main drawback is that BoW fails to take into account word order and context and results in sparse representations. This work explores the use of GloVe (Global Vectors for Word Representation) which has gained momentum in text classification problems [27, 28]. GloVe overcomes the sparsity problems associated with the BoW model by generating dense vector representations and projecting the vectors to a markedly lower dimensional space. It has the ability of capturing the semantic and syntactic relationships that are present between words, where words with a similar meaning are locally clustered in the vector space. Stanford’s GloVe embeddings, trained specifically on 2 billion tweets, were used to project each feature as a 200-dimensional vector [27, 28]. The weights of the feature vectors were initialised using the pre-trained embeddings but adjusted with the progression of training to improve classification performance. Each integer encoded feature, $w_{t}$, corresponds to an embedding vector, $\textbf{x}_{t}$, where $\textbf{x}_{t}\in\mathbb{R}^{p}$. Owing to the fact that each feature is represented as a 200-dimensional vector, $p=200$. A padded input feature vector is thus represented in the embedding layer as $\textbf{X}\in\mathbb{R}^{np}$, formed as a result of concatenation of $m$ vector embeddings. In figure 3, although the embedding layer is not explicitly shown, the vector embeddings which constitute it can be seen as inputs to the LSTM layer at the corresponding time steps. ### 2.3 LSTM Layer The main distinction between a neuron in the vanilla RNN and an LSTM cell lies in the presence of a cell state vector, whose contents at each time state are maintained and modified via an LSTM gating mechanism [29, 30, 31]. The information flow in an LSTM memory cell is regulated by three primary gates viz. forget gate, input gate, and output gate [29, 30]. Figure 4 shows the schematic of an LSTM cell, including the gating mechanisms used to achieve its functionality. Figure 4: Structure of an LSTM cell, showing the role of the primary gates and flow of information within the cell to form the current memory state and hidden output state. At a certain time step, $k$, the vector embedding, $\textbf{x}_{k}$, along with the previous hidden output, $\textbf{h}_{k}$, will be used by the input and forget gates to update the internal state of the cell [29, 32]. The output gate combines the inputs and the current cell state, $\textbf{c}_{k}$, to determine the information to be carried over to the next cell in the repeating structure [29]. In this fashion, LSTMs can control the contribution of those words and word relationships that have a higher impact on prediction, whilst penalising those that are less significant. Equation 1 is a matrix representation [32] of the outputs of the gates. Equations 2 and 3 show the outputs of the cell, where the output vector is obtained by an element wise multiplication process [29, 32]. $\begin{pmatrix}i\\\ f\\\ o\\\ g\\\ \end{pmatrix}=\begin{pmatrix}\sigma\\\ \sigma\\\ \sigma\\\ \tanh\\\ \end{pmatrix}W\begin{pmatrix}x_{k}\\\ h_{k-1}\par\end{pmatrix}$ (1) $\displaystyle c_{k}$ $\displaystyle=f\odot c_{k-1}+i\odot g$ (2) $\displaystyle h_{k}$ $\displaystyle=o\odot\tanh c_{k}$ (3) where $\textbf{i},\textbf{f},$ and o are the outputs of the input gate, forget gate, and output gate respectively and g is the output of an additional gate which aids in updating cell memory. W is the weights matrix and $\sigma$ and $\tanh$ represent the sigmoid and $\tanh$ non linearities. Note that the system of equations in (1) also contains a bias term for each gate output. Dropout, a regularization method, is utilized to prevent the model from overfitting [25]. Overfitting occurs when the model learns the statistical noise present in the dataset, capturing unnecessary complex relationships and thus, resulting in decreased generalisability. During training, it is possible for neighbouring neurons to become co-dependent, inhibiting the effectiveness of individual neurons. Dropout causes a proportion of the nodes or outputs in the layer to become inactive, thereby forcing the model to become more robust. This results in an increase in the network weights, which must be scaled by the dropout rate after completion of training. ### 2.4 Output Layer A singular output node with a sigmoid activation function, presented in equation 4, was used for the purpose of classifying trend. $\sigma(z)=\frac{1}{1+e^{-z}}$ (4) where $z$ is the activation of the output node. The estimated probability returned by the node was compared against a threshold probability in order to perform binary classification. If the output probability for a given input sequence $\sigma(z)\geq 0.5$, the input was labelled as 1, predicting an increase in asset price for the following trading day. If the output probability did not exceed this threshold, the input was labelled as 0, indicating either no change or a decrease in next-day price. A binary cross entropy cost function, $J_{bce}$, was used as given by equation 5 [33]. $\displaystyle J_{bce}$ $\displaystyle=-\frac{1}{m}\sum_{j=1}^{m}[y_{j}\times log(\sigma(z_{j}))$ (5) $\displaystyle+(1-y_{j})\times log(1-\sigma(z_{j}))]$ where $y_{j}$ is the $j^{th}$ target variable or the actual class label from a set of $m$ training samples. The cost or error function is representative of how accurately the model predicts target values, using a given set of network parameters. The main aim is to optimise or minimise the cost function, updating the weights and biases of connections in the network as a result. A mini-batch Stochastic Gradient was used during backpropagation to allow the model to converge to a global cost minimum [25]. This optimum state represents a model configuration where the error between the actual and predicted values is at a minimum and the network can successfully detect patterns between word embeddings essential for classification. The learning rate determines the rate at which the tunable weights approach the global minimum and must be chosen judiciously. A very large value would risk overshooting the minimum and a learning rate that is too small will significantly delay convergence. ## 3 Experimental Procedure Twitter data was obtained from followthehashtag [34], an online resource containing a readily available corpus of tweets. Approximately 167,000 tweets mentioning or associated with APPLE stocks were used for analysis. APPLE is among the companies currently dominating the technological sector and is regarded as a suitable choice upon which to base analysis. Owing to its popularity and the fact that it has the largest market capitalisation out of all NASDAQ 100 companies, it is fair to assume that twitter contains sufficient information relating to its stocks. The stock price data was sourced from Yahoo Finance [35]. The granularity of stock data considered is 1 day i.e. daily changes in stock price were computed to capture the essence of short term price fluctuation. The input tweets were labelled according to the scheme described previously. Concatenation of tweets leads to an aggregate of 48 input samples, corresponding to 48 unique days. Due to the limited number of samples, each input was divided into 100 subsets, whilst keeping the labelling of the subsets consistent with that of the original day. Subsequently, 4800 input samples for the network were obtained. For the initial experimentation, a naive model configuration was used. This model forms the basis on which further improvements in performance can be achieved. The next section discusses the effects of using different network types, hyperparameter optimisation, and varying split values (for tuning the number of input samples) on model performance. For the initial configuration, consisting of 4800 samples, the training/testing/validation split was 70/20/10 i.e. training was performed on 3360 samples, testing was performed on 960 samples, and validation was performed on the remaining 480 samples. The validation set is used to configure the model so as to obtain the hyperparameters which give the best performance. The testing data is only used once after the network has been configured to give an unbiased evaluation of model performance. A single LSTM stacked layer was utilised with a dropout value of 0.2 and 100 hidden units (used for determining the dimension of the LSTM outputs). The gradients and weights are updated according to a batch size of 32. The performance of the initial configuration is reported in the next section. The primary metric used to assess performance is accuracy. Another commonly used metric is the F1 score, which is the harmonic mean of the precision and recall [36]. Precision refers to the percentage of instances correctly predicted as positive with respect to all instances classified as positive by the model, whereas recall refers to the percentage correctly classified as positive out of all positive classes [36]. The F1 score is also calculated for varying implementations discussed in the next section . However, it is only needed when there is a greater cost associated with either the false positives or false negatives. As this is not applicable for this task and class distribution is even (as shown in figure 5), it is only computed to ensure consistency in results and thus not reported for all configurations. Figure 5: Bar graphs representing an even class distribution ($\approx$ 58/42) of stock up or down in the original dataset. ## 4 Results The initial model configuration, described in the previous section, gives an impressive classification (testing) accuracy of 74.58%. A confusion matrix, displayed in figure 6, summarizes the classification performance of the model. Figure 6: Confusion Matrix showing classifier performance at the deep functional level. It indicates that the model is able to correctly identify the 0 class and 1 class with an accuracy of 76% and 73% respectively. The F1 score for this configuration is 71.76%, reinforcing model performance as reflected in the testing accuracy result. The achieved accuracy is far superior to the random guessing threshold of 50%. This result indicates the effectiveness of NNs in this task, which is in contrast with the results of the correlation analysis performed previously. This output reinforces the claim that NNs have the ability to detect nuanced patterns and produce complex mappings between the input and output. ### 4.1 Effect of Splitting Dataset In the original model, the concatenated tweets, resulting in 48 samples, were split into 100 subsets per input sample to augment the dataset. To investigate the effect of modifying the number of subsets per sample on overall performance, values for the input splits were selected in the range [25, 450]. Figure 7 shows the dependence of classification accuracy as a function of split size. In general, selecting large values of split size leads to performance degradation. Splitting up all tweets on a particular day into higher subsets can result in insufficient information contained within a unique sample, deteriorating prediction capability. It is reasonable to assume that on any given day, some tweets cause the price to go in the opposite direction to that observed in the stock market, however, the aggregate impact of other tweets outweigh this effect. As a result, higher subdivisions do not accurately capture the true nature of the task. To determine if this trend continues, an extreme case was considered i.e. the maximum logical split value was considered. Using all 60,233 filtered tweets as individual inputs to the network, the observed accuracy was 62.37%, validating the observed graphical results. On the other end of the spectrum, using the concatenated tweets in their unaltered form will reflect all the necessary information on a given day for the model to make predictions. However, in this work, this will entail using a considerably limited number of training instances, hindering the network’s ability to learn effectively and leading to erroneous outputs. The training results of this configuration further confirmed this intuition as it gave the worst performance in comparison with using other subset values. The training time for this configuration i.e. using no splits was also significantly higher than any other value tested. As the input sequence length is maximum in this case, a significant number of LSTM cells is required. This discernibly increases processing time and degrades performance due to the emergence of vanishing gradients. Therefore, there exists a trade-off between loss of information and creating a reasonably sized dataset. In light of this fact, a split size of 150 per day was selected for subsequent analyses as it is able to achieve a satisfactory balance of the aforementioned performance variables. Figure 7: Variation of classification accuracy with the split size per input sample. ### 4.2 Hyperparameter Optimisation To improve the results of the original model configuration, the hyperparameters employed in the NN were optimised. In particular, the dropout rate, batch size, and number of LSTM output hidden units were varied to investigate their impact on classification results. Each hyperparamter was considered in isolation, with all other model variables remaining unchanged, to determine its exclusive impact. The same experimental procedure was also applied to a different network configuration known as bidirectional LSTMs. In strict terms, the standard LSTM structure used in this work is called a unidirectional LSTM network which differs from the bidirectional LSTM structure. The bidirectional network is a variant of the classical LSTM, where information flows in both directions between LSTM cells such that the cell, at every time step, is able to maintain previous and future input information [31]. This is in contrast to the unidirectional structure, where each cell contains only past information. Table 1 shows the classification accuracies achieved by varying the network dropout percentage. It is apparent that the unidirectional structure tends to perform well on dropout values higher than 0.2. However, the bidirectional structure shows no notable trends and as such, no conclusive inference can be drawn. The best accuracy (78.12%) is achieved by the unidirectional architecture, using a dropout value of 0.3. The implications of utilising higher values of dropout are that a higher proportion of neurons become inactive, increasing the robustness of the model and resulting in better testing accuracies. Table 1: Effect of varying dropout rates Value \| Unidirectional \| Bidirectional ---\|---\|--- \| Accuracy \| Epoch \| Accuracy \| Epoch 0.2 \| 74.58 \| 6,9 \| 75.63 \| 7 0.3 \| 78.12 \| 4 \| 74.79 \| 3 0.4 \| 77.29 \| 7 \| 72.71 \| 8 0.5 \| 76.25 \| 6 \| 75.63 \| 3 0.6 \| 75.00 \| 5,10 \| 76.46 \| 7 Table 2: Effect of varying batch size Value \| Unidirectional \| Bidirectional ---\|---\|--- \| Accuracy \| Epoch \| Accuracy \| Epoch 8 \| 78.33 \| 2 \| 77.92 \| 8 16 \| 78.12 \| 2 \| 75.00 \| 1 32 \| 74.58 \| 6,9 \| 75.63 \| 7 64 \| 74.79 \| 6 \| 72.29 \| 6 128 \| 73.54 \| 8 \| 73.12 \| 3 Table 3: Effect of varying lstm output hidden units Value \| Unidirectional \| Bidirectional ---\|---\|--- \| Accuracy \| Epoch \| Accuracy \| Epoch 100 \| 74.58 \| 6,9 \| 75.63 \| 7 128 \| 74.38 \| 6 \| 77.08 \| 4 256 \| 75.42 \| 3 \| 75.21 \| 10 512 \| 76.46 \| 4 \| 74.58 \| 2 Table 2, which highlights the effect of variable batch sizes, indicates that there is a decline in the accuracy level with increasing batch size for the unidirectional structure. Mini-batch sizes of 8 and 16 yield comparable results, with batch size 8 achieving a remarkable accuracy of 78.33%. Similarly to the dropout case, there are no identifiable trends for the bidirectional structure, however the lowest batch size (8) performs the best for this variant of LSTM as well. The batch size determines the number of training instances after which the gradients and weights of the network are updated. The impressive performance of the lower batch sizes can be attributed to a more robust convergence as well as the network’s ability to circumvent local minima. The final parameter used for optimisation is the number of hidden state units of the LSTM cells. The hidden state units determine the dimensionality of the output space of the LSTM layer i.e. the dimensionality of the LSTM output vectors $\textbf{H}=\\{h_{1},h_{2},h_{3},...,h_{n}\\}$. As presented in Table 3, the accuracy shows an overall increase with an increase in the output dimensionality for the unidirectional framework. Although performance gains are observed upon using larger values of hidden state size, it is at the cost of an exponential increase in the number of parameters and processing time. Values greater than 512 are not used in this study as the resulting models will be prone to overfitting owing to their marked complexity [37]. The bidirectional variant performs better when 128 hidden state units are used however results in a performance degradation for higher values. Due to the inherent characteristics of the bidirectional model, the dimensionality of the LSTM output is double that of its unidirectional counterpart. This ultimately leads to an inordinately complex model that is more prone to overfitting. Some neural network structures are known to achieve satisfactory results by employing two hidden layers. Therefore, two identical LSTM stacked layers with the same hyperparameter values as the original configuration were integrated in the network. The classification accuracy achieved by this configuration was 75.10%, which is lower than the value (76.67%) achieved using a single hidden layer implementation. Hence, it is deemed apposite to forego such an architecture. A notable observation, based on the hyperparameter tuning results, is the performance of the bidirectional structure. It has the ability to preserve past and future values in each cell, thereby allowing the network to gain a fuller context of the information present in the input tweets. In theory, this should lead to improvements in the overall predictions made by the model. However, not only does the bidirectional implementation produce comparable results overall but it also occasionally generates lower accuracies than the unidirectional model. ### 4.3 Optimal Model In order to discover the optimal model configuration, experiments were conducted using an exhaustive set of combinations of the hyperparameters. Different combinations of the dropout rate, mini-batch size, and hidden state size were deployed, with the range of hyperparameter values limited to the those outlined in Table 1, Table 2, and Table 3. The combination 0.4 (dropout), 8 (batch size), and 100 (hidden units) produces the best results, obtaining a validation accuracy of 81.04%. Improvements in accuracy cannot be attained by merely using those hyperparameter values which give the highest accuracy when altered independently i.e. using the combination 0.3,8,512. Therefore, it can be argued that there exists some degree of interaction between the variables when varied simultaneously. The optimal model configuration is thus given by the combination 0.4,8,100. To perform an unbiased evaluation of the model, the testing data (960 samples) was used. The model produces a testing accuracy of 76.14% when presented with the unseen testing data. Although the model exhibits a remarkable performance in absolute terms, its results are specific to APPLE and are not generalisable to other technology companies. There could be different patterns and inherent complexities within the twitter datasets of other companies which could lead to similar or contrasting results to that observed in this analysis. ## 5 Conclusion The objective of this work is to develop a model capable of predicting the direction of next-day stock market fluctuations using twitter messages. Tweets associated with APPLE, regarded among the Big Four technology companies, is used as the basis for this analysis. The primary focus of this work is in the development, configuration, and deployment of an LSTM structure. A correlation analysis is briefly explored to determine the relationship between VADER scores, quantifying the sentiment of tweets, and stock market movement. Using Point biserial correlation coefficient as the measurement metric, a low correlation value of 0.0812 is obtained. Alternate configurations are considered based on the time taken for information contained in tweets to manifest in market movement and a retweet-weighted average configuration. A delay size of 4 days results in the highest correlation value (0.3037). However, it is apparent from these results that VADER is not able to extract any strong relations between societal sentiment and market direction. In order to initialise the weights of the LSTM network, GloVe embeddings, pre- trained on a sizeable twitter corpus, are utilised. Due to the limited number of training samples, the effect of splitting the dataset for augmentation is analysed. A value of 150 is selected for splitting the input sequence for each day into subsets as it provides a satisfactory trade-off between information loss and a suitable representation of dataset size. Hyperparameter tuning is performed using the validation set and independently varying the dropout rate, batch size, and hidden unit size to further optimise model performance. To determine the optimum configuration, an exhaustive set of varying combinations of selected parameters is tested. A combination of 0.4,8,100 performs the best on the validation set, achieving a testing accuracy of 76.14%. Despite the level of accuracy being an impressive standalone result, twitter datasets from other technological companies need to be analysed and contrasted with results from this study. This relative comparison will allow the LSTM network performance to be gauged more accurately and in a broader context, enabling the formation of generalisable results. The application of technical indicators such as historical price data can also be explored in conjunction with the components of Fundamental Analysis used in this task to provide more input information vital to classification. Word embeddings are effective in projecting words/features occurring in similar contexts within a neighbouring vector space. However, it is common for tweets to contain words expressing opposite sentiments that are collocated. This leads to erroneous representations of these fundamentally different words as similar vectors, mitigating their discriminative ability as required for classification [38]. Further work should be undertaken to incorporate linguistic lexicons such as SentiNet to capture the effect of word similarity and sentiment [27]. 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# PROGRESSIVE IMAGE SUPER-RESOLUTION VIA NEURAL DIFFERENTIAL EQUATION ###### Abstract We propose a new approach for the image super-resolution (SR) task that progressively restores a high-resolution (HR) image from an input low- resolution (LR) image on the basis of a neural ordinary differential equation. In particular, we newly formulate the SR problem as an initial value problem, where the initial value is the input LR image. Unlike conventional progressive SR methods that perform gradual updates using straightforward iterative mechanisms, our SR process is formulated in a concrete manner based on explicit modeling with a much clearer understanding. Our method can be easily implemented using conventional neural networks for image restoration. Moreover, the proposed method can super-resolve an image with arbitrary scale factors on continuous domain, and achieves superior SR performance over state- of-the-art SR methods. ## 1 Introduction Image super-resolution (SR) is a classic low-level vision task that aims to recover a high-resolution (HR) image from a given low-resolution (LR) input image. For several decades, a large volume of literature documents the high demand of SR technique in various vision applications. However, SR problem still remains a challenge and is difficult to solve because it is a highly ill-posed inverse problem. With the recent development of deep learning technology, numerous deep- learning-based SR methods [1, 2, 3] have been presented, and they have shown plausible results. To further improve the SR performance, many researchers have attempted to restore the high-quality image by recovering the fine details of the LR input image progressively [4, 5]. Many previous works hinged on this progressive SR procedure are based on a variant of feedback network in the human visual system [6], and they show satisfactory SR results. However, owing to lack of theoretical clarity on the progressive system, these approaches need to develop a well-engineered method. For example, the number of iterations for the gradual refinements [7] and complicated learning strategies [5] as well as the network architectures [4, 8] are considered to improve the SR performance. Several researchers have conducted studies on differential equations to solve the image restoration problems [9, 10]. They also have developed progressive approaches, but these approaches are limited to modeling the prior and/or likelihood models. In this study, we introduce a neural ordinary differential equation (NODE [11]). formulation that describes an explicitly defined progressive SR procedure from the LR to HR images via a neural network. In particular, we reconstruct the HR image by numerically solving the initial value problem originated from the proposed ODE formulation, given the LR image as an initial condition. With the aid of the proposed ODE, our method eases implementation using conventional restoration networks and ODE solvers without any exertion to improve the performance. Furthermore, by simply changing the initial condition of our formulation at the test-time, ours can naturally handle a continuous-valued scale factor. Extensive experiments demonstrate the superiority of the proposed method over state-of-the-art SR approaches. ## 2 Proposed Method Fig. 1: (a) Overview of the proposed SR approach (NODE-SR). $\\{t_{i}\\}_{0\leq i\leq m}$ is a strictly decreasing sequence and $t_{m}=1$. Solid orange line represents our SR process that starts with the initial condition $\mathcal{I}(t_{0})$ until we reconstruct the final HR image $\mathcal{I}(1)$. (b) The neural network $f$ takes an input image $\mathcal{I}(t)$ with the scale factor $t$ and outputs the desired high- frequency detail. ### 2.1 Progressive Super-Resolution Formulation Existing SR methods utilizing progressive SR process [12, 4, 5] are based on iterative multi-stage approaches and can be viewed as variants of the following: $I_{n}=g_{n-1}(I_{n-1})\quad(n\leq N),$ (1) where $n$ denotes the iteration step, $I_{0}$ denotes the given initial input LR image, and $I_{n}$ is the iteratively refined image from its previous state $I_{n-1}$. These approaches typically produce multiple intermediate HR images during the refinement, and the rendered image at the last $N$-th iteration [7, 5] or a combined version of the multiple intermediate images ($\\{I_{n}\\}_{1\leq n\leq N}$) [13, 4] becomes the final SR result. Although these previous progressive methods show promising SR results, they still have some limitations. First, these methods need plenty of time and effort in determining the network configurations including the number of progressive updates $N$ and hyper-parameter settings, and designing cost functions to train the SR networks $g$. In addition, well-engineered and dedicated learning strategy, such as curriculum learning [5] and recursive supervision [13], is required for each method. This complication comes from the lack of clear understanding on their intermediate image states $\\{I_{n}\\}$. To alleviate these problems, we formulate the progressive SR process with a differential equation. This allows us to implement and train the SR networks in an established way while outperforming the performance of conventional progressive SR process. Assume that $(I_{HR})\downarrow_{t}$ is a downscaled version of a ground-truth clean image $I_{HR}$ using a traditional SR kernel (e.g., bicubic) with a scaling factor $\frac{1}{t}$. We then define $\mathcal{I}(t)$ by upscaling $(I_{HR})\downarrow_{t}$ using that SR kernel with a scaling factor ${t}$ so that $I_{HR}$ and $\mathcal{I}(t)$ have the same spatial resolution (see the illustration of Generating LR image in Figure 1(a)). Note that $t\geq 1$, and $\mathcal{I}(1)$ denotes the ground-truth clean image $I_{HR}$. To model a progressive SR process,we first estimate the high-frequency image residual with a neural network. Specifically, when $t$ is a conventional discrete- scaling factor (e.g., x2, x3, and x4), image residual between $\mathcal{I}(t)$ and $\mathcal{I}(t-1)$ can be modeled using a neural network $f_{\text{discrete}}$ as: $\mathcal{I}(t-1)-\mathcal{I}(t)=f_{\text{discrete}}(\mathcal{I}(t),t).$ (2) Notably, $\mathcal{I}(t-1)$ includes more high-frequency details than $\mathcal{I}(t)$ without loss of generality. In our method, we model the slightest image difference to formulate a continuously progressive SR process. Therefore, we take the scale factor $t$ to continuous domain, and reformulate (2) as an ODE with a neural network $f$ as: $\frac{d\mathcal{I}(t)}{dt}=f(\mathcal{I}(t),t,\theta),$ (3) where $\theta$ denotes the trainable parameter of the network $f$. Using this formulation, we can predict the high-frequency image detail required to slightly enhance $\mathcal{I}(t)$ with the network $f$. (Note that we can obtain $\mathcal{I}(t)$ with any rational number $t$ by adding padding to the border of image before resizing and then center cropping the image.) As existing SR neural networks have been proven to be successful at predicting the high-frequency residual image [2], we can use conventional SR architectures as our network $f$ in (3) without major changes. ### 2.2 Single Image Super-Resolution with Neural Ordinary Differential Equation In this section, we explain how to super-resolve a given LR image with a continuous scaling factor using our ODE-based SR formulation in (3). First, we obtain $\mathcal{I}(t_{0})$ by upscaling the given LR input image (Test time LR image in Figure 1(a)) using the bicubic SR kernel to a desired output resolution with a scaling factor $t_{0}$ . Next, we solve the ODE initial value problem in (3) with the initial condition $\mathcal{I}(t_{0})$ by integrating the neural network $f$ from $t_{0}$ to $1$ to acquire the high- quality image $\mathcal{I}(1)$ as follows: $\mathcal{I}(1)=\mathcal{I}(t_{0})+\int_{t_{0}}^{1}f(\mathcal{I}(t),t,\theta)dt.$ (4) Specifically, we approximate the high-quality image $\mathcal{I}(1)$ given a fully trained neural network $f$, network parameter $\theta$, initial condition $\mathcal{I}(t_{0})$, and integral interval $[t_{0},1]$ using an ODE solver ($ODESolve()$) as: $\mathcal{I}(1)\approx ODESolve(\mathcal{I}(t_{0}),f,\theta,[t_{0},1]).$ (5) Thus, our method does not need to consider the stop condition (i.e., the number of feedback iterations) of the progressive SR process unlike conventional approaches [7, 5]. Notably, during the training phase, we need to employ an ODE solver which allows end-to-end training using backpropagation with other components such as the neural network $f$. Unlike other progressive SR methods [13, 5], we do not require any other learning strategies like curriculum learning during the training phase. Fig. 2: Visual comparisons with conventional progressive SR methods (DRRN, SRFBN). For different scale factors (x2, and x4) intermediate HR images are visualized, and #it indicates the number of updates used to render results by DRRN and SRFBN. $\hat{I}()$ denotes the predicted results by our NODE-RDN. Fig. 3: Visual comparison of NODE-RDN (ours) with Meta-RDN on scale x2.5 and x4. In addition, our formulation is made upon a continuous context, allows a continuous scale factor $t_{0}$ where $t_{0}\geq 1$. This makes our method able to handle the arbitrary-scale SR problem. To train the deep neural network $f$, and learn the parameter $\theta$ in (5), we minimize the loss summed over scale factors $t$ using the L1 loss function as: $\mathcal{L}(\theta)=\sum_{t}\\|I_{HR}-ODESolve(\mathcal{I}(t),f,\theta,[t,1])\\|_{1}.$ (6) By minimizing the proposed loss function, our network parameter $\theta$ is trained to estimate the image detail to be added into the network input as in (3). ## 3 Experimental Results In this section, we carry out extensive experiments to demonstrate the superiority of the proposed method, and add various quantitative and qualitative comparison results. We also provide detailed analysis of our experimental results. ### 3.1 Implementation details We use VDSR [2] and RDN [3] as backbone CNN architectures for our network $f$ with slight modifications. For each CNN architecture, we change the first layer to feed the scale factor $t$ as an additional input. To be specific, we extend the input channel from 3 to 4, and the pixel locations of the newly concatenated channel (4-th channel) are filled with a scalar value $t$ as shown in Figure 1(b). In addition, for RDN, we remove the last upsampling layer so that input and output resolutions are the same in our work. Note that, no extra parameters are added except for the first layers of the networks. To train and infer the proposed SR process, we use Runge–Kutta (RK4) method as our ODE solver in (6). For simplicity, our approaches with VDSR and RDN backbones are called NODE-VDSR and NODE-RDN in the remaining parts of the experiments, respectively. We use the DIV2K [14] dataset to train our NODE- VDSR and NODE-RDN. We train the network by minimizing the L1 loss in (6) with the Adam optimizer ($\beta_{1}=0.9$, $\beta_{2}=0.999$, $\epsilon=10^{-8}$) [15]. The initial learning rate is set as $10^{-4}$, which is then decreased by half every 100k gradient update steps, and trained for 600k iterations in total. The mini-batch size of NODE-VDSR is 16 (200x200 patches), but our NODE- RDN takes 8 patches as a mini-batch (130x130 patches) owing to the memory limit of our graphic units. Similar to the training settings in Meta-SR [16], we train the network $f$ by randomly changing the scale factor $t$ in (6) from 1 to 4 with a stride of 0.1 (i.e., $t\in\\{1.1,1.2,1.3,...,4\\}$). Dataset \| Scale \| Bicubic \| DRCN \| LapSRN \| DRRN \| PRLSR \| SRFBN \| NODE-RDN (ours) \| NODE-RDN+ (ours) ---\|---\|---\|---\|---\|---\|---\|---\|---\|--- Set14 \| x2 \| 30.24/0.8688 \| 33.04/0.9118 \| 33.08/0.913 \| 33.23/0.9136 \| 33.69/0.9191 \| 33.82/0.9196 \| 33.90/0.9209 \| 33.95/0.9214 x3 \| 27.55/0.7742 \| 29.76/0.8311 \| 29.87/0.833 \| 29.96/0.8349 \| 30.43/0.8436 \| 30.51/0.8461 \| 30.53/0.8465 \| 30.59/0.8473 x4 \| 26.00/0.7027 \| 28.02/0.7670 \| 28.19/0.772 \| 28.21/0.7721 \| 28.71/0.7838 \| 28.81/0.7868 \| 28.76/0.7866 \| 28.83/0.7877 B100 \| x2 \| 29.56/0.8431 \| 31.85/0.8942 \| 31.80/0.895 \| 32.05/0.8973 \| 32.25/0.9005 \| 32.29/0.9010 \| 32.34/0.9025 \| 32.38/0.9028 x3 \| 27.21/0.7385 \| 28.80/0.7963 \| 28.81/0.797 \| 28.95/0.8004 \| 29.14/0.8060 \| 29.24/0.8084 \| 29.25/0.8094 \| 29.28/0.8100 x4 \| 25.96/0.6675 \| 27.23/0.7233 \| 27.32/0.728 \| 27.38/0.7284 \| 27.64/0.7378 \| 27.72/0.7409 \| 27.72/0.7410 \| 27.75/0.7417 Urban100 \| x2 \| 26.88/0.8403 \| 30.75/0.9133 \| 30.41/0.910 \| 31.23/0.9188 \| 32.35/0.9308 \| 32.62/0.9328 \| 32.81/0.9345 \| 32.97/0.9355 x3 \| 24.46/0.7349 \| 27.15/0.8276 \| 27.06/0.827 \| 27.53/0.8378 \| 28.27/0.8541 \| 28.73/0.8641 \| 28.81/0.8644 \| 28.94/0.8662 x4 \| 23.14/0.6577 \| 25.14/0.7510 \| 25.21/0.756 \| 25.44/0.7638 \| 26.22/0.7892 \| 26.60/0.8015 \| 26.56/0.7985 \| 26.68/0.8010 Table 1: Comparison with progressive SR methods on the benchmark datsets (Set14 [17], B100 [18], and Urban100 [19]). We provide average PSNR/SSIM values for scaling factors x2, x3, and x4. Our NODE-RDN and NODE-RDN+ show the best performance. Red and blue colors denote the best and second best results, respectively. Methods Scale \| x1.1 \| x1.2 \| x1.3 \| x1.4 \| x1.5 \| x1.6 \| x1.7 \| x1.8 \| x1.9 \| x2.0 \| x2.1 \| x2.2 \| x2.3 \| x2.4 \| x2.5 ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- bicubic \| 36.56 \| 35.01 \| 33.84 \| 32.93 \| 32.14 \| 31.49 \| 30.90 \| 30.38 \| 29.97 \| 29.55 \| 29.18 \| 28.87 \| 28.57 \| 28.31 \| 28.13 VDSR \| - \| - \| - \| - \| - \| - \| - \| - \| - \| 31.90 \| - \| - \| - \| - \| - VDSR+t \| 39.51 \| 38.44 \| 37.15 \| 36.04 \| 34.98 \| 34.15 \| 33.39 \| 32.78 \| 32.22 \| 31.70 \| 31.27 \| 30.86 \| 30.53 \| 30.2 \| 29.91 NODE-VDSR (ours) \| 41.46 \| 39.36 \| 37.75 \| 36.51 \| 35.38 \| 34.49 \| 33.70 \| 33.07 \| 32.50 \| 31.95 \| 31.52 \| 31.09 \| 30.76 \| 30.42 \| 30.12 RDN \| - \| - \| - \| - \| - \| - \| - \| - \| - \| 32.34 \| - \| - \| - \| - \| - RDN+t \| 42.83 \| 39.92 \| 38.18 \| 36.87 \| 35.71 \| 34.80 \| 33.99 \| 33.34 \| 32.77 \| 32.22 \| 31.76 \| 31.33 \| 30.99 \| 30.64 \| 30.34 Meta-RDN \| 42.82 \| 40.04 \| 38.28 \| 36.95 \| 35.86 \| 34.90 \| 34.13 \| 33.45 \| 32.86 \| 32.35 \| 31.82 \| 31.41 \| 31.06 \| 30.62 \| 30.45 NODE-RDN (ours) \| 43.22 \| 40.06 \| 38.35 \| 37.02 \| 35.86 \| 34.95 \| 34.14 \| 33.47 \| 32.89 \| 32.34 \| 31.89 \| 31.46 \| 31.12 \| 30.76 \| 30.46 NODE-RDN+ (ours) \| 43.33 \| 40.13 \| 38.40 \| 37.07 \| 35.90 \| 34.99 \| 34.17 \| 33.50 \| 32.93 \| 32.38 \| 31.93 \| 31.50 \| 31.16 \| 30.80 \| 30.50 Methods Scale \| x2.6 \| x2.7 \| x2.8 \| x2.9 \| x3.0 \| x3.1 \| x3.2 \| x3.3 \| x3.4 \| x3.5 \| x3.6 \| x3.7 \| x3.8 \| x3.9 \| x4.0 ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- bicubic \| 27.89 \| 27.66 \| 27.51 \| 27.31 \| 27.19 \| 26.98 \| 26.89 \| 26.59 \| 26.60 \| 26.42 \| 26.35 \| 26.15 \| 26.07 \| 26.01 \| 25.96 VDSR \| - \| - \| - \| - \| 28.83 \| - \| - \| - \| - \| - \| - \| - \| - \| - \| 27.29 VDSR+t \| 29.64 \| 29.39 \| 29.15 \| 28.93 \| 28.74 \| 28.55 \| 28.38 \| 28.22 \| 28.05 \| 27.89 \| 27.76 \| 27.58 \| 27.47 \| 27.34 \| 27.20 NODE-VDSR (ours) \| 29.85 \| 29.61 \| 29.36 \| 29.14 \| 28.94 \| 28.75 \| 28.58 \| 28.41 \| 28.25 \| 28.08 \| 27.96 \| 27.79 \| 27.66 \| 27.54 \| 27.40 RDN \| - \| - \| - \| - \| 29.26 \| - \| - \| - \| - \| - \| - \| - \| - \| - \| 27.72 RDN+t \| 30.06 \| 29.80 \| 29.55 \| 29.33 \| 29.12 \| 28.92 \| 28.76 \| 28.59 \| 28.43 \| 28.26 \| 28.13 \| 27.95 \| 27.84 \| 27.71 \| 27.58 Meta-RDN \| 30.13 \| 29.82 \| 29.67 \| 29.40 \| 29.30 \| 28.87 \| 28.79 \| 28.68 \| 28.54 \| 28.32 \| 28.27 \| 28.04 \| 27.92 \| 27.82 \| 27.75 NODE-RDN (ours) \| 30.18 \| 29.93 \| 29.67 \| 29.45 \| 29.25 \| 29.05 \| 28.88 \| 28.71 \| 28.54 \| 28.37 \| 28.24 \| 28.07 \| 27.96 \| 27.81 \| 27.72 NODE-RDN+ (ours) \| 30.22 \| 29.97 \| 29.71 \| 29.49 \| 29.28 \| 29.05 \| 28.92 \| 28.74 \| 28.58 \| 28.41 \| 28.28 \| 28.12 \| 28.00 \| 27.87 \| 27.75 Table 2: Average PSNR values on the B100 [18] evaluated with different scale factors. The best performance is shown in bold number. ### 3.2 Comparison with Progressive SR Methods First, we compare our NODE-RDN with several state-of-the-art progressive SR methods: DRCN [13], LapSRN [12], DRRN [7], PRLSR [8], and SRFBN [5]. As in [20], self-ensemble method is used to further improve NODE-RDN (denoted as NODE-RDN+). Note that, our NODE-RDN and NODE-RDN+ can handle multiple scale factors $t$ including non-integer scale factors (e.g., x1.5) using the same network parameter. In contrast, other approaches are required to be trained for certain discrete integer scale factors (x2, x3, and x4) separately, resulting in a distinct parameter set for each scale factor. Nevertheless, quantitative restoration results in Table 1 show that our NODE-RDN, NODE-RDN+ consistently outperforms conventional progressive SR methods for the discrete integer scaling factors (x2, x3, and x4) in terms of PSNR. In Figure 2, we investigate intermediate images produced during the progressive SR process with the scale factors x2 and x4. Final results by DRRN are obtained after 25 iterations, and the final results by SRFBN are obtained with 4 iterations as in their original settings. We provide 4 intermediate HR images during the updates for visual comparisons. For our NODE-RDN, intermediate image states are represented as $\hat{\mathcal{I}}(t_{i})$ where ${1\leq t_{i}\leq t_{0}}$ and $\hat{\mathcal{I}}(t_{i})=ODESolve(\mathcal{I}(t_{0}),f,\theta,[t_{0},t_{i}])$. We observe that DRRN and SRFBN fail to progressively refine patches with high- frequency details, while our NODE-RDN can gradually improve the intermediate images and render promising results at the final states. ### 3.3 Comparison with Multi-scale SR Methods Our approach can handle a continuous scale factor for the SR task, thus we compare ours with existing multi-scale SR methods that can handle continuous scale factors: VDSR [2] and Meta-SR [16]. Notably, Meta-SR implemented using RDN (i.e., Meta-RDN) is the current state-of-the-art SR approach. In Table 2, we show quantitative results compared to existing SR methods (VDSR, RDN, and Meta-RDN). Note that, VDSR+t and RDN+t are modified versions of VDSR and RDN to take the scale factor $t$ as an additional input of the networks and have the same input and output resolutions as in our network $f$. We also compare our method with these new baselines (VDSR+t and RDN+t) for fair comparisons. We evaluate the SR performance on the B100 benchmark dataset by increasing the scaling factor from 1.1 to 4. Interestingly, we observe that NODE-VDSR outperforms VDSR and VDSR+t at every scale by a large margin although VDSR and VDSR+t have similar network architecture to our NODE-VDSR. Similarly, NODE-RDN shows better performance than Meta-RDN and RDN+t. We also provide qualitative comparison results with Meta-SR in Figure 3, and we see that our NODE-RDN recovers much clearer edges than Meta-RDN. ## 4 Conclusion In this work, we proposed a novel differential equation for the SR task to progressively enhance a given input LR image, and allow continuous-valued scale factor. Image difference between images over different scale factors is physically modeled with a neural network, and formulated as a NODE. To restore a high-quality image, we solve the ODE initial value problem with the initial condition given as an input LR image. The main difference with existing progressive SR methods is that our formulation is based on the physical modeling of the intermediate images, and adds fine high-frequency details gradually. 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# Photoproduction $\gamma p\to K^{+}\Lambda(1520)$ in an effective Lagrangian approach Neng-Chang Wei School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing 101408, China Yu Zhang School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing 101408, China Fei Huang<EMAIL_ADDRESS>School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing 101408, China De-Min Li<EMAIL_ADDRESS>School of Physics and Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001, China ###### Abstract The data on differential cross sections and photon-beam asymmetries for the $\gamma p\to K^{+}\Lambda(1520)$ reaction have been analyzed within a tree- level effective Lagrangian approach. In addition to the $t$-channel $K$ and $K^{\ast}$ exchanges, the $u$-channel $\Lambda$ exchange, the $s$-channel nucleon exchange, and the interaction current, a minimal number of nucleon resonances in the $s$ channel are introduced in constructing the reaction amplitudes to describe the data. The results show that the experimental data can be well reproduced by including either the $N(2060)5/2^{-}$ or the $N(2120)3/2^{-}$ resonance. In both cases, the contact term and the $K$ exchange are found to make significant contributions, while the contributions from the $K^{\ast}$ and $\Lambda$ exchanges are negligible in the former case and considerable in the latter case. Measurements of the data on target asymmetries are called on to further pin down the resonance contents and to clarify the roles of the $K^{\ast}$ and $\Lambda$ exchanges in this reaction. $K^{+}\Lambda(1520)$ photoproduction, effective Lagrangian approach, photon- beam asymmetries ###### pacs: 25.20.Lj, 13.60.Le, 14.20.Gk, 13.75.Jz ## I Introduction The traditional $\pi N$ elastic and inelastic scattering experiments have provided us with abundant knowledge of the mass spectrum and decay properties of the nucleon resonances ($N^{\ast}$’s). Nevertheless, both the quark model Isgur:1977ef ; Koniuk:1979vy and lattice QCD Edwards:2011jj ; Edwards:2012fx calculations predict more resonances than have been observed in the $\pi N$ scattering experiments. The resonances predicated by quark model or lattice QCD but not observed in experiments are called “missing resonances”, which are supposed to have small couplings to the $\pi N$ channel and, thus, escape from experimental detection. In the past few decades, intense efforts have been dedicated to search for the missing resonances in meson production reaction channels other than $\pi N$. In particular, the $\rho N$, $\phi N$, and $\omega N$ production reactions in the nonstrangeness sector and the $KY$, $K^{\ast}Y$ ($Y=\Lambda,\Sigma$) production reactions in the strangeness sector have been widely investigated both experimentally and theoretically. In the present paper, we focus on the $\gamma p\to K^{+}\Lambda(1520)$ reaction process. The threshold of the $K^{+}\Lambda(1520)$ photoproduction is about $2.01$ GeV, and, thus, this reaction provides a chance to study the $N^{\ast}$ resonances in the $W\sim 2.0$ GeV mass region in which we have infancy information as shown in the latest version of the Review of Particle Physics (RPP) Tanabashi:2018oca . Besides, the isoscalar nature of $\Lambda(1520)$ allows only the $I=1/2$ $N^{\ast}$ resonances exchanges in the $s$ channel, which simplifies the reaction mechanisms of the $K^{+}\Lambda(1520)$ photoproduction. Experimentally, the cross sections for the reaction $\gamma p\to K^{+}\Lambda(1520)$ have been measured at SLAC by Boyarski et al. in 1971 for photon energy $E_{\gamma}=11$ GeV Boyarski:1970yc , and by the LAMP2 group in 1980 at $E_{\gamma}=2.8$$-$$4.8$ GeV Barber:1980zv . In 2010, the LEPS Collaboration measured the differential cross sections and photon-beam asymmetries ($\Sigma$) at Spring-8 for $\gamma p\to K^{+}\Lambda(1520)$ at energies from threshold up to $E_{\gamma}=2.6$ GeV at forward $K^{+}$ angles Kohri:2009xe . In 2011, the SAPHIR Collaboration measured the cross sections at the Electron Stretcher Accelerator (ELSA) for the $K^{+}\Lambda(1520)$ photoproduction in the energy range from threshold up to $E_{\gamma}=2.65$ GeV Wieland:2010cq . Recently, the differential and total cross sections for the $K^{+}\Lambda(1520)$ photoproduction were reported by the CLAS Collaboration at energies from threshold up to the center-of-mass energy $W=2.86$ GeV over a large range of the $K^{+}$ production angle Moriya:2013hwg . Theoretically, the $K^{+}\Lambda(1520)$ photoproduction reaction has been extensively investigated based on effective Lagrangian approaches by four theory groups in $11$ publications Nam:2005uq ; Nam:2006cx ; Nam:2009cv ; Nam:2010au ; Toki:2007ab ; Xie:2010yk ; Xie:2013mua ; Wang:2014jxb ; He:2012ud ; He:2014gga ; Yu:2017kng . In Refs. Nam:2005uq ; Nam:2006cx ; Nam:2009cv ; Nam:2010au , Nam et al. found that the contact term and the $t$-channel $K$ exchange are important to the cross sections of $\gamma p\to K^{+}\Lambda(1520)$, while the contributions from the $t$-channel $K^{\ast}$ exchange and the $s$-channel nucleon resonance exchange are rather small. In Refs. Toki:2007ab ; Xie:2010yk ; Xie:2013mua ; Wang:2014jxb , Xie, Wang, and Nieves et al. found that apart from the contact term and the $t$-channel $K$ exchange, the $u$-channel $\Lambda$ exchange and the $s$-channel $N(2120)3/2^{-}$ [previously called $D_{13}(2080)$] exchange are also important in describing the cross-section data for $\gamma p\to K^{+}\Lambda(1520)$, while the contribution from the $t$-channel $K^{\ast}$ exchange is negligible in this reaction. In Refs. He:2012ud ; He:2014gga , He and Chen found that the contribution from the $t$-channel $K^{\ast}$ exchange in $\gamma p\to K^{+}\Lambda(1520)$ is also considerable besides the important contributions from the contact term, the $t$-channel $K$ exchange, the $u$-channel $\Lambda$ exchange, and the $s$-channel $N(2120)3/2^{-}$ exchange. In Ref. Yu:2017kng , Yu and Kong studied the $\gamma p\to K^{+}\Lambda(1520)$ reaction within a Reggeized model, and they claimed that the important contributions to this reaction are coming from the contact term, the $t$-channel $K$ exchange, and the $t$-channel $K^{\ast}_{2}$ exchange, while the contribution from the $t$-channel $K^{\ast}$ exchange is minor. One observes that the common feature reported in all the above-mentioned publications of Refs. Nam:2005uq ; Nam:2006cx ; Nam:2009cv ; Nam:2010au ; Toki:2007ab ; Xie:2010yk ; Xie:2013mua ; Wang:2014jxb ; He:2012ud ; He:2014gga ; Yu:2017kng is that the contributions from the contact term and the $t$-channel $K$ exchange are important to the $\gamma p\to K^{+}\Lambda(1520)$ reaction. Even so, the reaction mechanisms of $\gamma p\to K^{+}\Lambda(1520)$ claimed by those four theory groups are quite different. In particular, there are no conclusive answers which can be derived from Refs. Nam:2005uq ; Nam:2006cx ; Nam:2009cv ; Nam:2010au ; Toki:2007ab ; Xie:2010yk ; Xie:2013mua ; Wang:2014jxb ; He:2012ud ; He:2014gga ; Yu:2017kng for the following questions: Are the contributions from the $t$-channel $K^{\ast}$ exchange and $u$-channel $\Lambda$ exchange significant or not in this reaction, does one inevitably need to introduce nucleon resonances in the $s$ channel to describe the data, and if yes, is the $N(2120)3/2^{-}$ resonance the only candidate needed in this reaction and what are the parameters of it? Figure 1: Predictions of photon-beam asymmetries at $\cos\theta=0.8$ as a function of the photon laboratory energy for $\gamma p\to K^{+}\Lambda(1520)$ from Ref. Xie:2010yk (blue dashed line), the fit II of Ref. Xie:2013mua (red solid line), Ref. He:2012ud (green dotted line), and Ref. He:2014gga (black dot-dashed line). The data are located in $0.6<\cos\theta<1$ and taken from the LEPS Collaboration Kohri:2009xe (blue square). On the other hand, the data on photon-beam asymmetries for $\gamma p\to K^{+}\Lambda(1520)$ reported by the LEPS Collaboration in 2010 Kohri:2009xe have never been well reproduced in previous publications of Refs. Nam:2005uq ; Nam:2006cx ; Nam:2009cv ; Nam:2010au ; Toki:2007ab ; Xie:2010yk ; Xie:2013mua ; Wang:2014jxb ; He:2012ud ; He:2014gga . As an illustration, we show in Fig. 1 the theoretical results on photon-beam asymmetries from Refs. Xie:2010yk ; He:2012ud ; Xie:2013mua ; He:2014gga calculated at $\cos\theta=0.8$ and compared with the data located at $0.6<\cos\theta<1$. It is true that the data bins in scattering angles are wide; nevertheless, it has been checked that the averaged values of theoretical beam-asymmetry results in $0.6<\cos\theta<1$ are comparable with those calculated at $\cos\theta=0.8$. One sees that, in the energy region $E_{\gamma}>2$ GeV, even the signs of the photon-beam asymmetries predicated by these theoretical works are opposite to the data. In the Regge model analysis of Ref. Yu:2017kng , the photon-beam asymmetries have indeed been analyzed, but there the differential cross-section data have been only qualitatively described, and the structures of the angular distributions exhibited by the data were missing due to the lack of nucleon resonances in the $s$-channel interactions. The purpose of the present work is to perform a combined analysis of the available data on both the differential cross sections and the photon-beam asymmetries for $\gamma p\to K^{+}\Lambda(1520)$ within an effective Lagrangian approach, and, based on that, we try to get a clear understanding of the reaction mechanism of $\gamma p\to K^{+}\Lambda(1520)$. In particular, we aim to clarify whether the $t$-channel $K^{\ast}$ exchange and the $u$-channel $\Lambda$ exchange are important or not and what the resonance contents and their associated parameters are in this reaction. As discussed above, previous publications of Refs. Nam:2005uq ; Nam:2006cx ; Nam:2009cv ; Nam:2010au ; Toki:2007ab ; Xie:2010yk ; Xie:2013mua ; Wang:2014jxb ; He:2012ud ; He:2014gga can describe only the differential cross-section data, and they gave diverse answers to these questions. It is expected that more reliable results on the resonance contents and the roles of $K^{\ast}$ and $\Lambda$ exchanges in this reaction can be obtained from the theoretical analysis which can result in a satisfactory description of the data on both the differential cross sections and the photon-beam asymmetries. The present paper is organized as follows. In Sec. II, we briefly introduce the framework of our theoretical model, including the effective interaction Lagrangians, the resonance propagators, and the phenomenological form factors employed in this work. The results of our model calculations are shown and discussed in Sec. III. Finally, a brief summary and conclusions are given in Sec. IV. ## II Formalism (a) $s$ channel (b) $t$ channel (c) $u$ channel (d) Interaction current Figure 2: Generic structure of the amplitude for $\gamma p\to K^{+}\Lambda(1520)$. Time proceeds from left to right. The outgoing $\Lambda^{\ast}$ denotes $\Lambda(1520)$. The full amputated photoproduction amplitude for $\gamma N\to K\Lambda(1520)$ in our tree-level effective Lagrangian approach can be expressed as Wang:2017tpe ; Wang:2018vlv ; Wang:2020mdn ; Wei:2019imo $M^{\nu\mu}\equiv M^{\nu\mu}_{s}+M^{\nu\mu}_{t}+M^{\nu\mu}_{u}+M^{\nu\mu}_{\rm int},$ (1) with $\nu$ and $\mu$ being the Lorentz indices for outgoing $\Lambda(1520)$ and incoming photon, respectively. The first three terms $M^{\nu\mu}_{s}$, $M^{\nu\mu}_{t}$, and $M^{\nu\mu}_{u}$ stand for the amplitudes resulted from the $s$-channel $N$ and $N^{\ast}$ exchanges, the $t$-channel $K$ and $K^{\ast}$ exchanges, and the $u$-channel $\Lambda$ exchange, respectively, as diagrammatically depicted in Fig. 2. They can be calculated straightforward by using the effective Lagrangians, propagators, and form factors provided in the following part of this section. The last term in Eq. (1) represents the interaction current arising from the photon attaching to the internal structure of the $\Lambda(1520)NK$ vertex. In practical calculation, the interaction current $M^{\nu\mu}_{\rm int}$ is modeled by a generalized contact current Haberzettl:1997 ; Haberzettl:2006 ; Haberzettl:2011zr ; Huang:2012 ; Huang:2013 ; Wang:2017tpe ; Wang:2018vlv ; Wei:2019imo ; Wang:2020mdn ; Wei:2020fmh : $M^{\nu\mu}_{\rm int}=\Gamma_{\Lambda^{\ast}NK}^{\nu}(q)C^{\mu}+M^{\nu\mu}_{\rm KR}f_{t}.$ (2) Here $\Gamma_{\Lambda^{\ast}NK}^{\nu}(q)$ is the vertex function of $\Lambda(1520)NK$ coupling governed by the Lagrangian of Eq. (16): $\Gamma_{\Lambda^{\ast}NK}^{\nu}(q)=-\frac{g_{\Lambda^{\ast}NK}}{M_{K}}\gamma_{5}q^{\nu},$ (3) with $q$ being the four-momentum of the outgoing $K$ meson; $M^{\nu\mu}_{\rm KR}$ is the Kroll-Ruderman term governed by the Lagrangian of Eq. (15): $M^{\nu\mu}_{\rm KR}=\frac{g_{\Lambda^{\ast}NK}}{M_{K}}g^{\nu\mu}\gamma_{5}Q_{K}\tau,$ (4) with $Q_{K}$ being the electric charge of outgoing $K$ meson and $\tau$ being the isospin factor of the Kroll-Ruderman term; $f_{t}$ is the phenomenological form factor attached to the amplitude of $t$-channel $K$ exchange, which is given by Eq. (39); $C^{\mu}$ is an auxiliary current introduced to ensure the gauge invariance of the full photoproduction amplitude of Eq. (1). Note that the photoproduction amplitudes will automatically be gauge invariant in the cases that there are no form factors and the electromagnetic couplings are obtained by replacing the partial derivative by its covariant form in the corresponding hadronic vertices. In practical calculation, one has to introduce the form factors in hadronic vertices (cf. Sec. II.3) which violate the gauge invariance. The auxiliary current $C^{\mu}$ is then introduced to compensate the gauge violation caused by the form factors. Following Refs. Haberzettl:2006 ; Haberzettl:2011zr ; Huang:2012 , for the $\gamma N\to K\Lambda(1520)$ reaction, the auxiliary current $C^{\mu}$ is chosen to be $C^{\mu}=-Q_{K}\tau\frac{f_{t}-\hat{F}}{t-q^{2}}(2q-k)^{\mu}-\tau Q_{N}\frac{f_{s}-\hat{F}}{s-p^{2}}(2p+k)^{\mu},$ (5) with $\hat{F}=1-\hat{h}\left(1-f_{s}\right)\left(1-f_{t}\right).$ (6) Here $p$, $q$, and $k$ denote the four-momenta for incoming $N$, outgoing $K$, and incoming photon, respectively; $Q_{K}$ and $Q_{N}$ are electric charges of $K$ and $N$, respectively; $f_{s}$ and $f_{t}$ are phenomenological form factors for $s$-channel $N$ exchange and $t$-channel $K$ exchange, respectively; $\hat{h}$ is an arbitrary function going to unity in the high- energy limit and set to be $1$ in the present work for simplicity; $\tau$ depicts the isospin factor for the corresponding hadronic vertex. Alternatively, one can rewrite the auxiliary current $C^{\mu}$ in Eq. (5) as $\displaystyle C^{\mu}=$ $\displaystyle- Q_{K}\tau(2q-k)^{\mu}\frac{f_{t}-1}{t-q^{2}}\left[1-\hat{h}\left(1-f_{s}\right)\right]$ $\displaystyle-\tau Q_{N}(2p+k)^{\mu}\frac{f_{s}-1}{s-p^{2}}\left[1-\hat{h}\left(1-f_{t}\right)\right].$ (7) One sees clearly that if there are no form factors, i.e., $f_{t}=f_{s}=1$, one has $C^{\mu}\to 0$ and, consequently, $M^{\nu\mu}_{\rm int}\to M^{\nu\mu}_{KR}$. We mention that the auxiliary current $C^{\mu}$ in Eq. (5) works for both real and virtual photons; i.e., the amplitudes we constructed in Eq. (1) are gauge invariant for both photo- and electroproduction of $K^{+}\Lambda(1520)$. In Ref. Nam2013 , another prescription for keeping gauge invariance of the $K^{+}\Lambda(1520)$ electroproduction amplitudes was introduced, where additional terms are considered besides those for photoproduction reactions. In the rest of this section, we present the effective Lagrangians, the resonance propagators, the form factors, and the interpolated $t$-channel Regge amplitudes employed in the present work. ### II.1 Effective Lagrangians In this subsection, we list all the Lagrangians used in the present work. For further simplicity, we define the operators $\Gamma^{(+)}=\gamma_{5}\qquad{\rm and}\qquad\Gamma^{(-)}=1,$ (8) the field $\Lambda^{\ast}=\Lambda(1520),$ (9) and the field-strength tensor $F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu},$ (10) with $A^{\mu}$ denoting the electromagnetic field. The Lagrangians needed to calculate the amplitudes for nonresonant interacting diagrams are $\displaystyle{\cal L}_{\gamma NN}$ $\displaystyle=$ $\displaystyle-\,e\bar{N}\\!\left[\\!\left(\hat{e}\gamma^{\mu}-\frac{\hat{\kappa}_{N}}{2M_{N}}\sigma^{\mu\nu}\partial_{\nu}\\!\right)\\!A_{\mu}\right]\\!N,$ (11) $\displaystyle{\cal L}_{\gamma KK}$ $\displaystyle=$ $\displaystyle ie\\!\left[K^{+}\left(\partial_{\mu}K^{-}\right)-K^{-}\left(\partial_{\mu}K^{+}\right)\right]\\!A^{\mu},$ (12) $\displaystyle{\cal L}_{\gamma K{K^{\ast}}}$ $\displaystyle=$ $\displaystyle e\frac{g_{\gamma K{K^{\ast}}}}{M_{K}}\varepsilon^{\alpha\mu\lambda\nu}\left(\partial_{\alpha}A_{\mu}\right)\left(\partial_{\lambda}K\right)K^{\ast}_{\nu},$ (13) $\displaystyle{\cal L}_{\gamma\Lambda\Lambda^{\ast}}$ $\displaystyle=$ $\displaystyle-\,ie\frac{g^{(1)}_{\Lambda^{\ast}\Lambda\gamma}}{2M_{\Lambda}}{\bar{\Lambda}}^{\ast\mu}\gamma^{\nu}F_{\mu\nu}\Lambda$ (14) $\displaystyle+\,e\frac{g^{(2)}_{\Lambda^{\ast}\Lambda\gamma}}{(2M_{\Lambda})^{2}}{\bar{\Lambda}}^{\ast\mu}F_{\mu\nu}\partial^{\nu}\Lambda+\text{H.\,c.},$ $\displaystyle{\cal L}_{\gamma\Lambda^{\ast}NK}$ $\displaystyle=$ $\displaystyle- iQ_{K}\frac{g_{\Lambda^{\ast}NK}}{M_{K}}\bar{\Lambda}^{{}^{\ast}\mu}A_{\mu}K\gamma_{5}N+\text{H.\,c.},$ (15) $\displaystyle{\cal L}_{\Lambda^{\ast}NK}$ $\displaystyle=$ $\displaystyle\frac{g_{\Lambda^{\ast}NK}}{M_{K}}{\bar{\Lambda}}^{\ast\mu}\left(\partial_{\mu}K\right)\gamma_{5}N+\text{H.\,c.},$ (16) $\displaystyle{\cal L}_{\Lambda^{\ast}NK^{\ast}}$ $\displaystyle=$ $\displaystyle-\frac{ig_{\Lambda^{\ast}NK^{\ast}}}{M_{K^{\ast}}}\bar{\Lambda}^{{}^{\ast}\mu}\gamma^{\nu}\left(\partial_{\mu}K^{\ast}_{\nu}-\partial_{\nu}K^{\ast}_{\mu}\right)N$ (17) $\displaystyle+\,\text{H.\,c.},$ $\displaystyle{\cal L}_{\Lambda NK}$ $\displaystyle=$ $\displaystyle-ig_{\Lambda NK}\bar{\Lambda}\gamma_{5}KN+\text{H.\,c.},$ (18) where $M_{K^{\ast}}$, $M_{K}$, $M_{N}$, and $M_{\Lambda}$ denote the masses of $K^{\ast}$, $K$, $N$, and $\Lambda$, respectively; $\hat{e}$ stands for the charge operator and $\hat{\kappa}_{N}=\kappa_{p}\left(1+\tau_{3}\right)/2+\kappa_{n}\left(1-\tau_{3}\right)/2$ with the anomalous magnetic moments $\kappa_{p}=1.793$ and $\kappa_{n}=-1.913$. The coupling constant $g_{\gamma KK^{\ast}}=0.413$ is calculated by the radiative decay width of $K^{\ast}\to K\gamma$ given by RPP Tanabashi:2018oca with the sign inferred from $g_{\gamma\pi\rho}$ Garcilazo:1993av via the flavor SU(3) symmetry considerations in conjunction with the vector-meson dominance assumption. The coupling constants $g^{(1)}_{\Lambda^{\ast}\Lambda\gamma}$ and $g^{(2)}_{\Lambda^{\ast}\Lambda\gamma}$ are fit parameters, but only one of them is free since they are constrained by the $\Lambda(1520)$ radiative decay width $\Gamma_{\Lambda(1520)\to\Lambda\gamma}=0.133$ MeV as given by RPP Tanabashi:2018oca . The value of $g_{\Lambda^{\ast}NK}=10.5$ is determined by the decay width of $\Lambda(1520)\to NK$, $\Gamma_{\Lambda(1520)\to NK}=7.079$ MeV, as advocated by RPP Tanabashi:2018oca . The coupling constant $g_{\Lambda^{\ast}NK^{\ast}}$ is a parameter to be determined by fitting the data. The coupling constant $g_{\Lambda NK}\approx-14$ is determined by the flavor SU(3) symmetry, $g_{\Lambda NK}=\left(-3\sqrt{3}/5\right)g_{NN\pi}$ with $g_{NN\pi}=13.46$. For nucleon resonances in the $s$ channel, the Lagrangians for electromagnetic couplings read Wang:2017tpe ; Wang:2018vlv ; Wei:2019imo ; Wang:2020mdn $\displaystyle{\cal L}_{RN\gamma}^{1/2\pm}$ $\displaystyle=$ $\displaystyle e\frac{g_{RN\gamma}^{(1)}}{2M_{N}}\bar{R}\Gamma^{(\mp)}\sigma_{\mu\nu}\left(\partial^{\nu}A^{\mu}\right)N+\text{H.\,c.},$ (19) $\displaystyle{\cal L}_{RN\gamma}^{3/2\pm}$ $\displaystyle=$ $\displaystyle-\,ie\frac{g_{RN\gamma}^{(1)}}{2M_{N}}\bar{R}_{\mu}\gamma_{\nu}\Gamma^{(\pm)}F^{\mu\nu}N$ (20) $\displaystyle+\,e\frac{g_{RN\gamma}^{(2)}}{\left(2M_{N}\right)^{2}}\bar{R}_{\mu}\Gamma^{(\pm)}F^{\mu\nu}\partial_{\nu}N+\text{H.\,c.},$ $\displaystyle{\cal L}_{RN\gamma}^{5/2\pm}$ $\displaystyle=$ $\displaystyle e\frac{g_{RN\gamma}^{(1)}}{\left(2M_{N}\right)^{2}}\bar{R}_{\mu\alpha}\gamma_{\nu}\Gamma^{(\mp)}\left(\partial^{\alpha}F^{\mu\nu}\right)N$ (21) $\displaystyle\pm\,ie\frac{g_{RN\gamma}^{(2)}}{\left(2M_{N}\right)^{3}}\bar{R}_{\mu\alpha}\Gamma^{(\mp)}\left(\partial^{\alpha}F^{\mu\nu}\right)\partial_{\nu}N$ $\displaystyle+\,\text{H.\,c.},$ $\displaystyle{\cal L}_{RN\gamma}^{7/2\pm}$ $\displaystyle=$ $\displaystyle ie\frac{g_{RN\gamma}^{(1)}}{\left(2M_{N}\right)^{3}}\bar{R}_{\mu\alpha\beta}\gamma_{\nu}\Gamma^{(\pm)}\left(\partial^{\alpha}\partial^{\beta}F^{\mu\nu}\right)N$ (22) $\displaystyle-\,e\frac{g_{RN\gamma}^{(2)}}{\left(2M_{N}\right)^{4}}\bar{R}_{\mu\alpha\beta}\Gamma^{(\pm)}\left(\partial^{\alpha}\partial^{\beta}F^{\mu\nu}\right)\partial_{\nu}N$ $\displaystyle+\,\text{H.\,c.},$ and the Lagrangians for hadronic couplings to $\Lambda(1520)K$ read $\displaystyle{\cal L}_{R\Lambda^{\ast}K}^{1/2\pm}$ $\displaystyle=$ $\displaystyle\frac{g^{(1)}_{R\Lambda^{\ast}K}}{M_{K}}\bar{\Lambda}^{\ast\mu}\Gamma^{(\pm)}\left(\partial_{\mu}K\right)R+\text{H.\,c.},$ (23) $\displaystyle{\cal L}_{R\Lambda^{\ast}K}^{3/2\pm}$ $\displaystyle=$ $\displaystyle\frac{g^{(1)}_{R\Lambda^{\ast}K}}{M_{K}}\bar{\Lambda}^{\ast\mu}\gamma_{\nu}\Gamma^{(\mp)}\left(\partial^{\nu}K\right)R_{\mu}$ (24) $\displaystyle+\,i\frac{g^{(2)}_{R\Lambda^{\ast}K}}{M_{K}^{2}}\bar{\Lambda}^{\ast}_{\alpha}\Gamma^{(\mp)}\left(\partial^{\mu}\partial^{\alpha}K\right)R_{\mu}+\text{H.\,c.},$ $\displaystyle{\cal L}_{R\Lambda^{\ast}K}^{5/2\pm}$ $\displaystyle=$ $\displaystyle i\frac{g^{(1)}_{R\Lambda^{\ast}K}}{M_{K}^{2}}\bar{\Lambda}^{\ast\alpha}\gamma_{\mu}\Gamma^{(\pm)}\left(\partial^{\mu}\partial^{\beta}K\right)R_{\alpha\beta}$ (25) $\displaystyle-\,\frac{g^{(2)}_{R\Lambda^{\ast}K}}{M_{K}^{3}}\bar{\Lambda}^{\ast}_{\mu}\Gamma^{(\pm)}\left(\partial^{\mu}\partial^{\alpha}\partial^{\beta}K\right)R_{\alpha\beta}$ $\displaystyle+\,\text{H.\,c.},$ $\displaystyle{\cal L}_{R\Lambda^{\ast}K}^{7/2\pm}$ $\displaystyle=$ $\displaystyle-\frac{g^{(1)}_{R\Lambda^{\ast}K}}{M_{K}^{3}}\bar{\Lambda}^{\ast\alpha}\gamma_{\mu}\Gamma^{(\mp)}\left(\partial^{\mu}\partial^{\beta}\partial^{\lambda}K\right)R_{\alpha\beta\lambda}$ (26) $\displaystyle-\,i\frac{g^{(2)}_{R\Lambda^{\ast}K}}{M_{K}^{4}}\bar{\Lambda}^{\ast}_{\mu}\Gamma^{(\mp)}\left(\partial^{\mu}\partial^{\alpha}\partial^{\beta}\partial^{\lambda}K\right)R_{\alpha\beta\lambda}$ $\displaystyle+\,\text{H.\,c.},$ where $R$ designates the $N^{\ast}$ resonance and the superscript of ${\cal L}_{RN\gamma}$ and ${\cal L}_{R\Lambda^{\ast}K}$ denotes the spin and parity of the resonance $R$. The coupling constants $g_{RN\gamma}^{(i)}$ and $g^{(i)}_{R\Lambda^{\ast}K}$ $(i=1,2)$ are fit parameters. Actually, only the products of $g_{RN\gamma}^{(i)}g^{(j)}_{R\Lambda^{\ast}K}$ $(i,j=1,2)$ are relevant to the reaction amplitudes, and they are what we really fit in practice. In Ref. Nam:2005uq , the off-shell effects for spin-$3/2$ resonances in $\gamma p\to K^{+}\Lambda(1520)$ have been tested. It was found that the off- shell effects are small and the off-shell parameter $X$ can be set to zero. In the present work, we simply ignore the off-shell terms in the interaction Lagrangians for high spin resonances and leave this issue for future work. ### II.2 Resonance propagators We follow Ref. Wang:2017tpe to use the following prescriptions for the propagators of resonances with spin $1/2$, $3/2$, $5/2$, and $7/2$: $\displaystyle S_{1/2}(p)$ $\displaystyle=$ $\displaystyle\frac{i}{{p\\!\\!\\!/}-M_{R}+i\Gamma_{R}/2},$ (27) $\displaystyle S_{3/2}(p)$ $\displaystyle=$ $\displaystyle\frac{i}{{p\\!\\!\\!/}-M_{R}+i\Gamma_{R}/2}\left(\tilde{g}_{\mu\nu}+\frac{1}{3}\tilde{\gamma}_{\mu}\tilde{\gamma}_{\nu}\right),$ (28) $\displaystyle S_{5/2}(p)$ $\displaystyle=$ $\displaystyle\frac{i}{{p\\!\\!\\!/}-M_{R}+i\Gamma_{R}/2}\,\bigg{[}\,\frac{1}{2}\big{(}\tilde{g}_{\mu\alpha}\tilde{g}_{\nu\beta}+\tilde{g}_{\mu\beta}\tilde{g}_{\nu\alpha}\big{)}$ (29) $\displaystyle-\,\frac{1}{5}\tilde{g}_{\mu\nu}\tilde{g}_{\alpha\beta}+\frac{1}{10}\big{(}\tilde{g}_{\mu\alpha}\tilde{\gamma}_{\nu}\tilde{\gamma}_{\beta}+\tilde{g}_{\mu\beta}\tilde{\gamma}_{\nu}\tilde{\gamma}_{\alpha}$ $\displaystyle+\,\tilde{g}_{\nu\alpha}\tilde{\gamma}_{\mu}\tilde{\gamma}_{\beta}+\tilde{g}_{\nu\beta}\tilde{\gamma}_{\mu}\tilde{\gamma}_{\alpha}\big{)}\bigg{]},$ $\displaystyle S_{7/2}(p)$ $\displaystyle=$ $\displaystyle\frac{i}{{p\\!\\!\\!/}-M_{R}+i\Gamma_{R}/2}\,\frac{1}{36}\sum_{P_{\mu}P_{\nu}}\bigg{(}\tilde{g}_{\mu_{1}\nu_{1}}\tilde{g}_{\mu_{2}\nu_{2}}\tilde{g}_{\mu_{3}\nu_{3}}$ (30) $\displaystyle-\,\frac{3}{7}\tilde{g}_{\mu_{1}\mu_{2}}\tilde{g}_{\nu_{1}\nu_{2}}\tilde{g}_{\mu_{3}\nu_{3}}+\frac{3}{7}\tilde{\gamma}_{\mu_{1}}\tilde{\gamma}_{\nu_{1}}\tilde{g}_{\mu_{2}\nu_{2}}\tilde{g}_{\mu_{3}\nu_{3}}$ $\displaystyle-\,\frac{3}{35}\tilde{\gamma}_{\mu_{1}}\tilde{\gamma}_{\nu_{1}}\tilde{g}_{\mu_{2}\mu_{3}}\tilde{g}_{\nu_{2}\nu_{3}}\bigg{)},$ where $\displaystyle\tilde{g}_{\mu\nu}$ $\displaystyle=$ $\displaystyle-\,g_{\mu\nu}+\frac{p_{\mu}p_{\nu}}{M_{R}^{2}},$ (31) $\displaystyle\tilde{\gamma}_{\mu}$ $\displaystyle=$ $\displaystyle\gamma^{\nu}\tilde{g}_{\nu\mu}=-\gamma_{\mu}+\frac{p_{\mu}{p\\!\\!\\!/}}{M_{R}^{2}},$ (32) and the summation over $P_{\mu}$ $\left(P_{\nu}\right)$ in Eq. (30) goes over the $3!=6$ possible permutations of the indices $\mu_{1}\mu_{2}\mu_{3}$ $\left(\nu_{1}\nu_{2}\nu_{3}\right)$. In Eqs. (27)$-$(32), $M_{R}$ and $\Gamma_{R}$ are the mass and width of resonance $R$ with four-momentum $p$, respectively. ### II.3 Form factors In practical calculation of the reaction amplitudes, a phenomenological form factor is introduced in each hadronic vertex. For the $t$-channel meson exchanges, we adopt the following form factor Wang:2017tpe ; Wang:2020mdn ; Wang:2018vlv ; Wei:2019imo : $\displaystyle f_{M}(q^{2}_{M})=\left(\frac{\Lambda_{M}^{2}-M_{M}^{2}}{\Lambda_{M}^{2}-q^{2}_{M}}\right)^{2},$ (33) and for the $s$-channel and $u$-channel baryon exchanges, we use Wang:2017tpe ; Wang:2020mdn ; Wang:2018vlv ; Wei:2019imo $\displaystyle f_{B}(p^{2}_{x})=\left(\frac{\Lambda_{B}^{4}}{\Lambda_{B}^{4}+\left(p_{x}^{2}-M_{B}^{2}\right)^{2}}\right)^{2}.$ (34) Here, $q_{M}$ denotes the four-momentum of the intermediate meson in the $t$ channel, and $p_{x}$ stands for the four-momentum of the intermediate baryon in $s$ and $u$ channels with $x=$ $s$ and $u$, respectively. $\Lambda_{M(B)}$ is the corresponding cutoff parameter. In the present work, in order to reduce the number of adjustable parameters, we use the same cutoff parameter $\Lambda_{B}$ for all the nonresonant diagrams, i.e., $\Lambda_{B}\equiv\Lambda_{K}=\Lambda_{K^{\ast}}=\Lambda_{\Lambda}=\Lambda_{N}$. The parameter $\Lambda_{B}$ and the cutoff parameter $\Lambda_{R}$ for $N^{\ast}$ resonances are determined by fitting the experimental data. ### II.4 Interpolated $t$-channel Regge amplitudes A Reggeized treatment of the $t$-channel $K$ and $K^{\ast}$ exchanges is usually employed to economically describe the high-energy data, which corresponds to the following replacement of the form factors in Feynman amplitudes: $\displaystyle f_{K}(q_{K}^{2})\to{\cal F}_{K}(q_{K}^{2})=$ $\displaystyle\left(\frac{s}{s_{0}}\right)^{\alpha_{K}(t)}\frac{\pi\alpha^{\prime}_{K}}{\sin[\pi\alpha_{K}(t)]}$ $\displaystyle\times\frac{t-M^{2}_{K}}{\Gamma[1+\alpha_{K}(t)]},$ (35) $\displaystyle f_{K^{\ast}}(q_{K^{\ast}}^{2})\to{\cal F}_{K^{\ast}}(q_{K^{\ast}}^{2})=$ $\displaystyle\left(\frac{s}{s_{0}}\right)^{\alpha_{K^{\ast}}(t)-1}\frac{\pi\alpha^{\prime}_{K^{\ast}}}{\sin[\pi\alpha_{K^{\ast}}(t)]}$ $\displaystyle\times\frac{t-M^{2}_{K^{\ast}}}{\Gamma[\alpha_{K^{\ast}}(t)]}.$ (36) Here $s_{0}$ is a mass scale which is conventionally taken as $s_{0}=1$ GeV2, and $\alpha^{\prime}_{M}$ is the slope of the Regge trajectory $\alpha_{M}(t)$. For $M=K$ and $K^{\ast}$, the trajectories are parameterized as Wang:2019mid $\displaystyle\alpha_{K}(t)$ $\displaystyle=0.7~{}{\rm GeV}^{-2}\left(t-m_{K}^{2}\right),$ (37) $\displaystyle\alpha_{K^{\ast}}(t)$ $\displaystyle=1+0.85~{}{\rm GeV}^{-2}\left(t-m_{K^{\ast}}^{2}\right).$ (38) Note that, in Eqs. (35) and (36), degenerate trajectories are employed for $K$ and $K^{\ast}$ exchanges; thus, the signature factors reduce to $1$. In the present work, we use the so-called interpolated Regge amplitudes for the $t$-channel $K$ and $K^{\ast}$ exchanges. The idea of this prescription is that at high energies and small angles one uses Regge amplitudes, and at low energies one uses Feynman amplitudes, while in the intermediate energy region an interpolating form factor is introduced to ensure a smooth transition from the low-energy Feynman amplitudes to the high-energy Regge amplitudes. This hybrid Regge approach has been applied to study the $\gamma p\to K^{+}\Lambda(1520)$ reaction in Refs. Nam:2010au ; Wang:2014jxb ; He:2014gga ; Yu:2017kng and the other reactions in Refs. Wang:2015hfm ; Wang:2017plf ; Wang:2017qcw ; Wang:2019mid . Instead of making the replacements of Eqs. (35) and (36) in a pure Reggeized treatment, in this hybrid Regge model the amplitudes for $t$-channel $K$ and $K^{\ast}$ exchanges are constructed by making the following replacements of the form factors in the corresponding Feynman amplitudes: $f_{M}(q_{M}^{2})\to{\cal F}_{R,M}={\cal F}_{M}(q_{M}^{2})R+f_{M}(q_{M}^{2})\left(1-R\right),$ (39) where ${\cal F}_{M}(q_{M}^{2})$ $(M=K,K^{\ast})$ is defined in Eqs. (35) and (36) and $R=R_{s}R_{t}$ with $\displaystyle R_{s}=$ $\displaystyle\frac{1}{1+e^{-(s-s_{R})/s_{0}}},$ $\displaystyle R_{t}=$ $\displaystyle\frac{1}{1+e^{-(t+t_{R})/t_{0}}}.$ (40) Here $s_{R}$, $t_{R}$, $s_{0}$, and $t_{0}$ are parameters to be determined by fitting the experimental data. The auxiliary current $C^{\mu}$ introduced in Eq. (5) and the interaction current $M^{\nu\mu}_{\rm int}$ given in Eq. (2) ensures that the full photoproduction amplitude of Eq. (1) satisfies the generalized Ward-Takahashi identity and, thus, is fully gauge invariant Haberzettl:2006 ; Haberzettl:2011zr ; Huang:2012 . Note that our prescription for $C^{\mu}$ and $M^{\nu\mu}_{\rm int}$ is independent of any particular form of the $t$-channel form factor $f_{K}(q_{K}^{2})$, provided that it is normalized as $f_{K}(q_{K}^{2}=M_{K}^{2})=1$. One sees that, when the interpolated Regge amplitude is employed for $t$-channel $K$ exchange, the replacement of Eq. (39) still keeps the the normalization condition of the form factor: $\lim_{q^{2}_{K}\to M^{2}_{K}}{\cal F}_{R,K}=1.$ (41) Therefore, as soon as we do the same replacement of Eq. (39) for the form factor of $t$-channel $K$ exchange everywhere in $C^{\mu}$ and $M^{\nu\mu}_{\rm int}$, the full photoproduction amplitude still satisfies the generalized Ward-Takahashi identity and, thus, is fully gauge invariant. ## III Results and discussion As discussed in the introduction section of this paper, the reaction $\gamma p\to K^{+}\Lambda(1520)$ has been theoretically investigated based on effective Lagrangian approaches by four theory groups in $11$ publications Nam:2005uq ; Nam:2006cx ; Nam:2009cv ; Nam:2010au ; Toki:2007ab ; Xie:2010yk ; Xie:2013mua ; Wang:2014jxb ; He:2012ud ; He:2014gga ; Yu:2017kng . The common feature of the results from these theoretical works is that the contributions from the contact term and the $t$-channel $K$ exchange are important for the $\gamma p\to K^{+}\Lambda(1520)$ reaction. Apart from that, no common ground has been found by these theoretical works for the reaction mechanisms of $\gamma p\to K^{+}\Lambda(1520)$. In particular, different groups gave quite different answers for the following questions: Are the contributions from the $t$-channel $K^{\ast}$ exchange and $u$-channel $\Lambda$ exchange significant or not in this reaction, are the nucleon resonances introduced in the $s$ channel indispensable or not to describe the available data, and, if yes, what are the resonance contents and their associated parameters in this reaction? On the other hand, we notice that even though the data on photon-beam asymmetries for $\gamma p\to K^{+}\Lambda(1520)$ have been reported by the LEPS Collaboration in 2010, they have never been well reproduced in previous theoretical publications of Refs. Nam:2005uq ; Nam:2006cx ; Nam:2009cv ; Nam:2010au ; Toki:2007ab ; Xie:2010yk ; Xie:2013mua ; Wang:2014jxb ; He:2012ud ; He:2014gga . One believes that these photon-beam-asymmetry data will definitely put further constraints on the reaction amplitudes. In Ref. Yu:2017kng , the photon-beam-asymmetry data have indeed been analyzed, but there, the structures of the angular distributions exhibited by the data are missed due to the lack of nucleon resonances. In a word, all previous theoretical publications in regards to $\gamma p\to K^{+}\Lambda(1520)$ are divided over the reaction mechanism and the resonance contents and parameters of this reaction. A simultaneous description of the differential cross-section data and the photon-beam-asymmetry data still remains to be accomplished. The purpose of the present work is to get a clear understanding of the reaction mechanism of $\gamma p\to K^{+}\Lambda(1520)$ based on a combined analysis of the available data on both the differential cross sections and the photon-beam asymmetries within an effective Lagrangian approach. As the differential cross-section data exhibit clear bump structures in the near- threshold region, apart from the $N$, $K$, $K^{\ast}$, and $\Lambda$ exchanges and the interaction current in the nonresonant background, we introduce as few as possible near-threshold nucleon resonances in the $s$ channel in constructing the $\gamma p\to K^{+}\Lambda(1520)$ reaction amplitudes to reproduce the data. Figure 3: Differential cross sections for $\gamma p\to K^{+}\Lambda(1520)$ at a few selected scattering angles as a function of the photon incident energy. The black solid lines, red dot-double-dashed lines, blue dashed lines, and green dot-dashed lines denote the results obtained by including the $N(2000)5/2^{+}$, $N(2040)3/2^{+}$, $N(2100)1/2^{+}$, and $N(2190)7/2^{-}$ resonances in the $s$ channel, respectively. Data are taken from the CLAS Collaboration Moriya:2013hwg (red circles) and the LEPS Collaboration Kohri:2009xe (blue squares). For $\cos\theta=0.85$, the CLAS data at $\cos\theta=0.84$ ($E_{\gamma}<3.25$ GeV) and $\cos\theta=0.83$ ($E_{\gamma}>3.25$ GeV) are shown. In the most recent version of RPP Tanabashi:2018oca , there are six nucleon resonances near the $K^{+}\Lambda(1520)$ threshold, namely, the $N(2000)5/2^{+}$, $N(2040)3/2^{+}$, $N(2060)5/2^{-}$, $N(2100)1/2^{+}$, $N(2120)3/2^{-}$, and $N(2190)7/2^{-}$ resonances. If none of these nucleon resonances are introduced in the construction of the $s$-channel reaction amplitudes, we find that it is not possible to achieve a simultaneous description of both the differential cross-section data and the photon-beam- asymmetry data in our model. We then try to reproduce the data by including one of these six near-threshold resonances. If we include one of the $N(2000)5/2^{+}$, $N(2040)3/2^{+}$, $N(2100)1/2^{+}$, and $N(2190)7/2^{-}$ resonances, we find that the obtained theoretical results for differential cross sections and photon-beam asymmetries have rather poor fitting qualities. As an illustration, we show in Fig. 3 the differential cross sections at a few selected scattering angles as a function of the incident photon energy which are obtained by including one of the $N(2000)5/2^{+}$ (black solid lines), $N(2040)3/2^{+}$ (red dot-double-dashed lines), $N(2100)1/2^{+}$ (blue dashed lines), and $N(2190)7/2^{-}$ (green dot-dashed lines) resonances and compared with the corresponding data Kohri:2009xe ; Moriya:2013hwg . One sees clearly from Fig. 3 that the fits with one of the $N(2040)3/2^{+}$, $N(2100)1/2^{+}$, and $N(2190)7/2^{-}$ resonances fail to describe the differential cross sections at $\cos\theta=0.95$, and the fit with the $N(2000)5/2^{+}$ resonance fails to reproduce the differential cross-section data at the other three selected scattering angles. In a word, none of these four fits that includes one of the $N(2000)5/2^{+}$, $N(2040)3/2^{+}$, $N(2100)1/2^{+}$, and $N(2190)7/2^{-}$ resonances can well describe the differential cross-section data. Thus, they are excluded to be acceptable fits. On the other hand, if either the resonance $N(2060)5/2^{-}$ or the resonance $N(2120)3/2^{-}$ is considered, a simultaneous description of both the differential cross-section data and the photon-beam-asymmetry data can be satisfactorily obtained, which will be discussed below in detail. Consequently, these two fits, i.e., the ones including the $N(2060)5/2^{-}$ or the $N(2120)3/2^{-}$ resonance, are treated as acceptable. When an additional resonance is further included, the fit quality will be improved a little bit, since one has more adjustable model parameters. But, in this case, one would obtain too many solutions with similar fitting qualities, and meanwhile the fitted error bars of adjustable parameters are also relatively large. As a consequence, no conclusive conclusion can be drawn about the resonance contents and parameters extracted from the available data for the considered reaction. We thus conclude that the available differential cross-section data and the photon-beam-asymmetry data for $\gamma p\to K^{+}\Lambda(1520)$ can be described by including one of the $N(2060)5/2^{-}$ and $N(2120)3/2^{-}$ resonances and postpone the analysis of these available data with two or more nucleon resonances until more data for this reaction become available in the future. Table 1: Fitted values of model parameters. The asterisks below resonance names represent the overall status of these resonances evaluated by RPP Tanabashi:2018oca . The numbers in the brackets below the resonance masses and widths denote the corresponding values advocated by RPP Tanabashi:2018oca . $\sqrt{\beta_{\Lambda^{\ast}K}}A_{j}$ represents the reduced helicity amplitude for resonance with $\beta_{\Lambda^{\ast}K}$ denoting the branching ratio of resonance decay to $\Lambda(1520)K$ and $A_{j}$ standing for the helicity amplitude with spin $j$ for resonance radiative decay to $\gamma p$. \| Fit A \| Fit B ---\|---\|--- $s_{R}$ $[{\rm GeV}^{2}]$ \| $5.17\pm 0.02$ \| $3.80\pm 0.12$ $s_{0}$ $[{\rm GeV}^{2}]$ \| $0.81\pm 0.02$ \| $8.00\pm 0.05$ $t_{R}$ $[{\rm GeV}^{2}]$ \| $0.80\pm 0.03$ \| $1.16\pm 0.07$ $t_{0}$ $[{\rm GeV}^{2}]$ \| $1.60\pm 0.07$ \| $0.96\pm 0.07$ $\Lambda_{B}$ $[{\rm MeV}]$ \| $748\pm 2$ \| $770\pm 5$ $g^{(1)}_{\Lambda^{\ast}\Lambda\gamma}$ \| $0.00\pm 0.01$ \| $8.99\pm 0.51$ $g_{\Lambda^{\ast}NK^{\ast}}$ \| $-22.48\pm 0.91$ \| $-54.22\pm 3.72$ \| $N(2060){5/2}^{-}$ \| $N(2120){3/2}^{-}$ \| $\ast$$\ast$$\ast$ \| $\ast$$\ast$$\ast$ $M_{R}$ $[{\rm MeV}]$ \| $2020\pm 1$ \| $2184\pm 2$ \| $[2030$$-$$2200]$ \| $[2060$$-$$2160]$ $\Gamma_{R}$ $[{\rm MeV}]$ \| $200\pm 30$ \| $83\pm 4$ \| $[300$$-$$450]$ \| $[260$$-$$360]$ $\Lambda_{R}$ $[{\rm MeV}]$ \| $1086\pm 3$ \| $2000\pm 52$ $\sqrt{\beta_{\Lambda^{\ast}K}}A_{1/2}$ $[10^{-3}\,{\rm GeV}^{-1/2}]$ \| $3.07\pm 0.02$ \| $3.04\pm 0.11$ $\sqrt{\beta_{\Lambda^{\ast}K}}A_{3/2}$ $[10^{-3}\,{\rm GeV}^{-1/2}]$ \| $0.54\pm 0.02$ \| $5.27\pm 0.19$ $g^{(2)}_{R\Lambda^{\ast}K}/g^{(1)}_{R\Lambda^{\ast}K}$ \| $-1.26\pm 0.01$ \| $-4.06\pm 0.28$ As discussed above, we introduce nucleon resonances as few as possible in constructing the reaction amplitudes to describe the available data for $\gamma p\to K^{+}\Lambda(1520)$. It is found that a simultaneous description of both the differential cross-section data and the photon-beam-asymmetry data can be achieved by including either the $N(2060)5/2^{-}$ resonance or the $N(2120)3/2^{-}$ resonance. We thus get two acceptable fits named as “fit A,” which includes the $N(2060)5/2^{-}$ resonance, and “fit B,” which includes the $N(2120)3/2^{-}$ resonance. The fitted values of the adjustable model parameters in these two fits are listed in Table 1, and the corresponding results on differential cross sections and photon-beam asymmetries are shown in Figs. 4$-$6. In Table 1, for $u$-channel $\Lambda$ exchange, only the value of the coupling constant $g^{(1)}_{\Lambda^{\ast}\Lambda\gamma}$ is listed. The other coupling constant $g^{(2)}_{\Lambda^{\ast}\Lambda\gamma}$ is not treated as a free parameter, since it is constrained by the $\Lambda(1520)$ radiative decay width $\Gamma_{\Lambda(1520)\to\Lambda\gamma}=0.133$ MeV as given by RPP Tanabashi:2018oca , which results in $g^{(2)}_{\Lambda^{\ast}\Lambda\gamma}=2.13$ in fit A and $-13.01$ in fit B, respectively. The asterisks below the resonance names represent the overall status of these resonances evaluated in the most recent RPP Tanabashi:2018oca . One sees that both the $N(2060)5/2^{-}$ and the $N(2120)3/2^{-}$ resonances are evaluated as three-star resonances. The symbols $M_{R}$, $\Gamma_{R}$, and $\Lambda_{R}$ denote the resonance mass, width, and cutoff parameter, respectively. The numbers in brackets below the resonance mass and width are the corresponding values estimated by RPP. It is seen that the fitted masses of the $N(2060)5/2^{-}$ and $N(2120)3/2^{-}$ resonances are comparable with their values quoted by RPP, while the fitted widths for these two resonances are smaller than the corresponding RPP values. For resonance couplings, since in the tree-level calculation only the products of the resonance hadronic and electromagnetic coupling constants are relevant to the reaction amplitudes, we list the reduced helicity amplitudes $\sqrt{\beta_{\Lambda^{\ast}K}}A_{j}$ for each resonance instead of showing their hadronic and electromagnetic coupling constants separately Huang:2013 ; Wang:2017tpe ; Wang:2018vlv ; Wang:2019mid . Here $\beta_{\Lambda^{\ast}K}$ is the branching ratio for resonance decay to $\Lambda(1520)K$, and $A_{j}$ is the helicity amplitude with spin $j$ ($j=1/2,3/2$) for resonance radiative decay to $\gamma p$. We have, in total, as shown in Figs. 4$-$6, $220$ data points in the fits. Fit A has a global $\chi^{2}/N=2.10$, and fit B has a global $\chi^{2}/N=2.63$. Note that, in the fitting procedure, $11.6\%$ and $5.92\%$ systematic errors for the data from the CLAS Collaboration and the LEPS Collaboration, respectively, have been added in quadrature to the statistical errors Moriya:2013hwg ; Kohri:2009xe . Overall, one sees that both the differential cross-section data and the photon-beam-asymmetry data have been well described simultaneously in both fit A and fit B. Figure 4: Differential cross sections for $\gamma p\to K^{+}\Lambda(1520)$ as a function of $\cos\theta$ from fit A (left panel) and fit B (right panel). The symbols $W$ and $E_{\gamma}$ denote the center-of-mass energy of the whole system and the photon laboratory energy, respectively, both in MeV. The black solid lines represent the results calculated from the full amplitudes. The red dotted lines, blue dashed lines, green dot-dashed lines, cyan double-dot- dashed lines, and magenta dot-double-dashed lines denote the individual contributions from the interaction current, the $t$-channel $K$ exchange, the $t$-channel $K^{\ast}$ exchange, the $s$-channel $N^{\ast}$ resonance exchange, and the $u$-channel $\Lambda$ exchange, respectively. The scattered symbols are data from the CLAS Collaboration Moriya:2013hwg . Figure 5: Differential cross sections for $\gamma p\to K^{+}\Lambda(1520)$ at a few selected scattering angles as a function of the photon incident energy from fit A (left panel) and fit B (right panel). The notations for the lines are the same as in Fig. 4. Data are taken from the CLAS Collaboration Moriya:2013hwg (red circles) and the LEPS Collaboration Kohri:2009xe (blue squares). For $\cos\theta=0.85$, the CLAS data at $\cos\theta=0.84$ ($E_{\gamma}<3.25$ GeV) and $\cos\theta=0.83$ ($E_{\gamma}>3.25$ GeV) are shown. Figures 4 and 5 show the differential cross sections for $\gamma p\to K^{+}\Lambda(1520)$ resulted from fit A (left panels), which includes the $N(2060)5/2^{-}$ resonance, and fit B (right panels), which includes the $N(2120)3/2^{-}$ resonance. There, the black solid lines represent the results calculated from the full reaction amplitudes. The red dotted lines, blue dashed lines, green dot-dashed lines, cyan double-dot-dashed lines, and magenta dot-double-dashed lines denote the individual contributions from the interaction current, the $t$-channel $K$ exchange, the $t$-channel $K^{\ast}$ exchange, the $s$-channel $N^{\ast}$ resonance exchange, and the $u$-channel $\Lambda$ exchange, respectively. The individual contributions from the $s$-channel nucleon exchange are too small to be clearly shown in these figures. One sees from Figs. 4 and 5 that the differential cross-section data are well reproduced in both fit A (left panels) and fit B (right panels). Note that in Fig. 5, for $\cos\theta=0.85$, the CLAS data at $\cos\theta=0.84$ ($E_{\gamma}<3.25$ GeV) and $\cos\theta=0.83$ ($E_{\gamma}>3.25$ GeV) are shown. That explains why in Fig. 4 the theoretical results agree with the CLAS data at high-energy forward angles but in Fig. 5 the theoretical differential cross sections at $\cos\theta=0.85$ overestimate the CLAS data at the last two energy points. From Figs. 4 and 5, one sees that, in fit A, the contribution from the interaction current [cf. Eq. (2)] plays a rather important role in the whole energy region. In the near-threshold region, the differential cross sections are dominated by the interaction current and the $N(2060)5/2^{-}$ resonance exchange. Actually, the contributions from these two terms are responsible for the sharp rise of differential cross sections near the $K^{+}\Lambda(1520)$ threshold, in particular, the bump structure near $E_{\gamma}\approx 2$ GeV at forward angles as exhibited by the LEPS data in Fig. 5. The $t$-channel $K$ exchange is seen to contribute significantly at higher energies and forward angles. The $t$-channel $K^{\ast}$ exchange has tiny contributions at high- energy forward angles, while the contributions from the $u$-channel $\Lambda$ exchange are negligible. In fit B, the interaction current plays a dominant role in the whole energy region and is also responsible for the sharp rise of the differential cross sections at forward angles near the $K^{+}\Lambda(1520)$ threshold. The bump structure near $E_{\gamma}\approx 2$ GeV at forward angles as exhibited by the LEPS data in Fig. 5 is caused by the $N(2120)3/2^{-}$ resonance on the base of the background dominated by the interaction current. The $t$-channel $K$ exchange and the $u$-channel $\Lambda$ exchange have significant contributions at forward and backward angles, respectively, mostly at higher energies. Considerable contributions are also seen from the $t$-channel $K^{\ast}$ exchange at high-energy forward angles. Figure 6: Photon-beam asymmetries for $\gamma p\to K^{+}\Lambda(1520)$ at $\cos\theta=0.8$ as a function of the photon incident energy from fit A (left panel) and fit B (right panel). The black solid lines represent the results calculated from the full amplitudes. The red dotted lines, blue dashed lines, green dot-dashed lines, cyan double-dot-dashed lines, and magenta dot-double- dashed lines denote the results obtained by switching off the contributions of the interaction current, the $t$-channel $K$ exchange, the $t$-channel $K^{\ast}$ exchange, the $s$-channel $N^{\ast}$ resonance exchange, and the $u$-channel $\Lambda$ exchange, respectively, from the full model. Data are in the bin $0.6<\cos\theta<1$ and taken from the LEPS Collaboration Kohri:2009xe . The results of photon-beam asymmetries for $\gamma p\to K^{+}\Lambda(1520)$ from fit A and fit B are shown, respectively, in the left and right panels in Fig. 6. There, the black solid lines represent the results calculated from the full amplitudes. The red dotted lines, blue dashed lines, green dot-dashed lines, cyan double-dot-dashed lines, and magenta dot-double-dashed lines denote the results obtained by switching off the contributions of the interaction current, the $t$-channel $K$ exchange, the $t$-channel $K^{\ast}$ exchange, the $s$-channel $N^{\ast}$ resonance exchange, and the $u$-channel $\Lambda$ exchange, respectively, from the full model. One sees that the photon-beam-asymmetry data are well reproduced in both fits. In fit A, when the contributions of the $N(2060){5/2}^{-}$ resonance exchange are switched off from the full model, one gets almost zero beam asymmetries. We have checked and found that the $N(2060){5/2}^{-}$ resonance exchange alone results in negligible beam asymmetries. This means that it is the interference between the $N(2060){5/2}^{-}$ resonance exchange and the other interaction terms that is crucial for reproducing the experimental values of the beam asymmetries. A similar observation also holds for the interaction current [cf. Eq. (2)]. The interaction current alone results in almost zero beam asymmetries, but one gets rather negative beam asymmetries when the contributions from the interaction current are switched off from the full model. This means that the interference between the interaction current and the other interaction terms is very important for reproducing the beam asymmetries. Switching off the contributions of the individual terms other than the $N(2060){5/2}^{-}$ resonance exchange and the interaction current from the full model does not affect too much the theoretical beam asymmetries. In fit B, the interaction current alone is found to result in almost zero beam asymmetries, the same as in fit A. Nevertheless, it is seen from Fig. 6 that one gets rather negative beam asymmetries when the contributions of the interaction current are switched off from the full model, showing the importance of the interference of the interaction current and the other interacting terms in photon-beam asymmetries for $\gamma p\to K^{+}\Lambda(1520)$. Switching off the contributions of the individual terms other than the interaction current from the full model would not affect the theoretical beam asymmetries too much. In Ref. Kohri:2009xe , it is expected that the positive values of the $K^{+}\Lambda(1520)$ asymmetries indicate a much larger contribution from the $K^{\ast}$ exchange. In both fit A and fit B of the present work, we have checked and found that the $K^{\ast}$ exchange alone does result in positive beam asymmetries, but, when the contributions of the $K^{\ast}$ exchange are switched off from the full model, the calculated beam asymmetries do not change significantly. In particular, the theoretical beam asymmetries are still positive and close to the experimental values when the contributions of the $K^{\ast}$ exchange are switched off from the full model. Figure 7: Total cross sections for $\gamma p\to K^{+}\Lambda(1520)$ predicated by fit A (left panel) and fit B (right panel). Notations for the lines are the same as in Fig. 4. Data are taken from the CLAS Collaboration Moriya:2013hwg but not included in the fits. Figure 7 shows the total cross sections for $\gamma p\to K^{+}\Lambda(1520)$ predicated from fit A (left panel) and fit B (right panel), which are obtained by integrating the corresponding differential cross sections calculated in these two fits. In Fig. 7, the black solid lines represent the results calculated from the full reaction amplitudes. The red dotted lines, blue dashed lines, green dot-dashed lines, cyan double-dot-dashed lines, and magenta dot-double-dashed lines denote the individual contributions from the interaction current, the $t$-channel $K$ exchange, the $t$-channel $K^{\ast}$ exchange, the $s$-channel $N^{\ast}$ resonance exchange, and the $u$-channel $\Lambda$ exchange, respectively. The individual contributions from the $s$-channel nucleon exchange are too small to be clearly shown in these figures. Note that the data for the total cross sections of $\gamma p\to K^{+}\Lambda(1520)$ are not included in the fits. Even so, one sees that, in both fit A and fit B, the theoretical total cross sections are in good agreement with the data. In fit A, the $s$-channel $N(2060){5/2}^{-}$ exchange, the interaction current, and the $t$-channel $K$ exchange provide the most important contributions to the total cross sections, while the contributions from the $u$-channel $\Lambda$ exchange, the $s$-channel $N$ exchange, and the $t$-channel $K^{\ast}$ exchange are negligible. The bump structure near $E_{\gamma}\approx 2$ GeV is caused mainly by the $N(2060){5/2}^{-}$ resonance exchange and the interaction current. The sharp rise of the total cross sections near the $K^{+}\Lambda(1520)$ threshold is dominated by the $s$-channel $N(2060){5/2}^{-}$ exchange. In fit B, the dominant contributions to the total cross sections come from the interaction current, which is also responsible for the sharp rise of the total cross sections near the $K^{+}\Lambda(1520)$ threshold. The individual contributions from the $s$-channel $N(2120){3/2}^{-}$ exchange, the $t$-channel $K$ and $K^{\ast}$ exchanges, and the $u$-channel $\Lambda$ exchange are considerable, while those from the $s$-channel $N$ exchange are negligible to the total cross sections. Comparing the individual contributions in fit A and fit B, one sees that the contributions from the resonance exchange are rather important in fit A, but they are much smaller in fit B. The contributions from the $t$-channel $K^{\ast}$ exchange and the $u$-channel $\Lambda$ exchange are negligible in fit A, but they are considerable in fit B. In both fits, the interaction current provides dominant contributions, and the $t$-channel $K$ exchange results in considerable contributions to the cross sections. As mentioned in the introduction section, the $K^{+}\Lambda(1520)$ photoproduction reaction has been theoretically investigated based on effective Lagrangian approaches by four theory groups in $11$ publications Nam:2005uq ; Nam:2006cx ; Nam:2009cv ; Nam:2010au ; Toki:2007ab ; Xie:2010yk ; Xie:2013mua ; Wang:2014jxb ; He:2012ud ; He:2014gga ; Yu:2017kng . In these previous publications, the photon-beam-asymmetry data reported by the LEPS Collaboration in 2010 Kohri:2009xe have never been well reproduced except in Ref. Yu:2017kng . But in Ref. Yu:2017kng , the structures of the angular distributions exhibited by the data are missed due to the lack of nucleon resonances in the employed Reggeized model. As shown in Figs. 4$-$6, the present work for the first time presents a simultaneous description of the data on both differential cross sections and photon-beam asymmetries within an effective Lagrangian approach. The common feature of the results from the previous works Nam:2005uq ; Nam:2006cx ; Nam:2009cv ; Nam:2010au ; Toki:2007ab ; Xie:2010yk ; Xie:2013mua ; Wang:2014jxb ; He:2012ud ; He:2014gga ; Yu:2017kng is that the contributions from the contact term and the $t$-channel $K$ exchange are important for the $\gamma p\to K^{+}\Lambda(1520)$ reaction. This feature has also been observed in the present work, as illustrated in Fig. 7. The contributions of nucleon resonance exchanges are reported to be small in Refs. Nam:2005uq ; Nam:2006cx ; Nam:2009cv ; Nam:2010au , while the $N(2120)3/2^{-}$ exchange is found to be important to the cross sections of $\gamma p\to K^{+}\Lambda(1520)$ in Refs. Toki:2007ab ; Xie:2010yk ; Xie:2013mua ; Wang:2014jxb ; He:2012ud ; He:2014gga . In the present work, we found that, to get a satisfactory description of the data on both differential cross sections and photon-beam asymmetries of $\gamma p\to K^{+}\Lambda(1520)$, the exchange of at least one nucleon resonance in the $s$ channel needs to be introduced in constructing the reaction amplitudes. The required nucleon resonance could be either the $N(2060){5/2}^{-}$ or the $N(2120){3/2}^{-}$, both evaluated as three-star resonances in the most recent version of RPP Tanabashi:2018oca . In the fit with the $N(2060){5/2}^{-}$ resonance, the contributions of the resonance exchange are found to be rather important to the cross sections, and, in particular, they are responsible for the sharp rise of the cross sections near the $K^{+}\Lambda(1520)$ threshold, as can be seen in Fig. 7. In the fit with the $N(2120){3/2}^{-}$ resonance, although much smaller than those of the interaction current, the contributions of the resonance exchange are still considerable to the cross sections. In Refs. Nam:2005uq ; Nam:2006cx ; Nam:2009cv ; Nam:2010au ; Toki:2007ab ; Xie:2010yk ; Xie:2013mua ; Wang:2014jxb ; Yu:2017kng , the $t$-channel $K^{\ast}$ exchange is found to provide negligible contributions. In Refs. He:2012ud ; He:2014gga , it is reported that the contributions of the $t$-channel $K^{\ast}$ exchange are considerable to the cross sections. In our present work, the contributions of the $t$-channel $K^{\ast}$ exchange are negligible in the fit with the $N(2060){5/2}^{-}$ resonance, and are considerable in the fit with the $N(2120){3/2}^{-}$ resonance. As for the $u$-channel $\Lambda$ exchange, important contributions are reported in Refs. Toki:2007ab ; Xie:2010yk ; Xie:2013mua ; Wang:2014jxb ; He:2012ud ; He:2014gga , while in the present work, considerable contributions of this term are seen only in the fit with the $N(2120){3/2}^{-}$ resonance. Figure 8: Predictions of target nucleon asymmetries for $\gamma p\to K^{+}\Lambda(1520)$ from fit A (black solid lines) and fit B (blue dashed lines) at two selected center-of-mass energies. From Figs. 4$-$7 one sees that the fit with the $N(2060){5/2}^{-}$ resonance (fit A) and the fit with the $N(2120){3/2}^{-}$ resonance (fit B) describe the data on differential cross sections and photon-beam asymmetries for $\gamma p\to K^{+}\Lambda(1520)$ almost equally well. In Fig. 8, we show the predictions of the target nucleon asymmetries ($T$) from fit A (black solid lines) and fit B (blue dashed lines) at two selected center-of-mass energies. One sees that unlike the differential cross sections and the photon-beam asymmetries, the target nucleon asymmetries predicted by fit A and fit B are quite different. Future experimental data on target nucleon asymmetries are expected to be able to distinguish the fit A and fit B of the present work and to further clarify the resonance content, the resonance parameters, and the reaction mechanism for the $\gamma p\to K^{+}\Lambda(1520)$ reaction. ## IV Summary and conclusion The photoproduction reaction $\gamma p\to K^{+}\Lambda(1520)$ is of interest since the $K^{+}\Lambda(1520)$ has isospin $1/2$, excluding the contributions of the $\Delta$ resonances from the reaction mechanisms, and the threshold of $K^{+}\Lambda(1520)$ is at $2.01$ GeV, making this reaction more suitable than $\pi$ production reactions to study the nucleon resonances in a less-explored higher resonance mass region. Experimentally, the data for $\gamma p\to K^{+}\Lambda(1520)$ on differential cross sections, total cross sections, and photon-beam asymmetries are available from several experimental groups Boyarski:1970yc ; Barber:1980zv ; Kohri:2009xe ; Wieland:2010cq ; Moriya:2013hwg , with the photon-beam- asymmetry data coming from the LEPS Collaboration Kohri:2009xe and the most recent differential and total cross-section data coming from the CLAS Collaboration Moriya:2013hwg . Theoretically, the cross-section data for $\gamma p\to K^{+}\Lambda(1520)$ have been analyzed by several theoretical groups Nam:2005uq ; Nam:2006cx ; Nam:2009cv ; Nam:2010au ; Toki:2007ab ; Xie:2010yk ; Xie:2013mua ; Wang:2014jxb ; He:2012ud ; He:2014gga within effective Lagrangian approaches, and the photon-beam-asymmetry data Kohri:2009xe have been reproduced only in Ref. Yu:2017kng within a Reggeized framework. In the latter, the apparent structures of the angular distributions exhibited by the data are missing due to the lack of nucleon resonances in $s$-channel interactions in the Regge model. In these publications, the reported common feature for the $\gamma p\to K^{+}\Lambda(1520)$ reaction is that the contributions from the contact term and the $t$-channel $K$ exchange are important to the cross sections of this reaction. Nevertheless, the reaction mechanisms of $\gamma p\to K^{+}\Lambda(1520)$ claimed by different theoretical groups are quite different. In particular, there are no conclusive answers for the questions of whether the contributions from the $t$-channel $K^{\ast}$ exchange and $u$-channel $\Lambda$ exchange are significant or not, whether the introduction of nucleon resonances in the $s$ channel is inevitable or not for describing the data, and if yes, what resonance contents and parameters are needed in this reaction. In the present work, we performed a combined analysis of the data on both the differential cross sections and photon-beam asymmetries for $\gamma p\to K^{+}\Lambda(1520)$ within an effective Lagrangian approach. We considered the $t$-channel $K$ and $K^{\ast}$ exchange, the $u$-channel $\Lambda$ exchange, the $s$-channel nucleon and nucleon resonance exchanges, and the interaction current, with the last one being constructed in such a way that the full photoproduction amplitudes satisfy the generalized Ward-Takahashi identity and, thus, are fully gauge invariant. The strategy for introducing the nucleon resonances in the $s$ channel used in the present work was that we introduce nucleon resonances as few as possible to describe the data. For the first time, we achieved a satisfactory description of the data on both the differential cross sections and the photon-beam asymmetries for $\gamma p\to K^{+}\Lambda(1520)$. We found that either the $N(2060){5/2}^{-}$ or the $N(2120){3/2}^{-}$ resonance needs to be introduced in constructing the $s$-channel reaction amplitudes in order to get a simultaneous description of the data on differential cross sections and photon-beam asymmetries for $\gamma p\to K^{+}\Lambda(1520)$. In both cases, the contributions of the interaction current and the $t$-channel $K$ exchange are found to dominate the background contributions. The $s$-channel resonance exchange is found to be rather important in the fit with the $N(2060){5/2}^{-}$ resonance and to be much smaller but still considerable in the fit with the $N(2120){3/2}^{-}$ resonance. The contributions of the $t$-channel $K^{\ast}$ exchange and the $u$-channel $\Lambda$ exchange are negligible in the fit with the $N(2060){5/2}^{-}$ resonance and are significant in the fit with the $N(2120){3/2}^{-}$ resonance. 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# Self-similar analysis of the time-dependent compressible and incompressible boundary layers including heat conduction Imre Ferenc Barna1, Krisztián Hriczó2, Gabriella Bognár2, and László Mátyás3 1 Wigner Research Center for Physics, Konkoly-Thege Miklós út 29 - 33, 1121 Budapest, Hungary 2 University of Miskolc, Miskolc-Egyetemváros 3515, Hungary, 3Department of Bioengineering, Faculty of Economics, Socio-Human Sciences and Engineering, Sapientia Hungarian University of Transylvania, Libertătii sq. 1, 530104 Miercurea Ciuc, Romania ###### Abstract We investigate the incompressible and compressible heat conducting boundary layer with applying the two-dimensional self-similar Ansatz. Analytic solutions can be found for the incompressible case which can be expressed with special functions. The parameter dependencies are studied and discussed in details. In the last part of our study we present the ordinary differential equation (ODE) system which is obtained for compressible boundary layers. ###### pacs: 47.10.-g,47.10.ab,47.10.ad ## I Introduction It is evident that the study of hydrodynamical equations has a crucial role in engineering and science as well. It is also clear that numerous classifications exist for various flow systems. One class of fluid flows is the field of boundary layer. The development of this scientific field started with the pioneering work of Prandtl prandt who used scaling arguments and derived that half of the terms of the Naiver-Stokes equations are negligible in boundary layer flows. In 1908 Blasius blasius gave the solutions of the steady-state incompressible two-dimensional laminar boundary layer equation forms on a semi-infinite plate which is held parallel to a constant unidirectional flow. Later Falkner and Skan falkner ; falkner1 generalized the solutions for steady two-dimensional laminar boundary layer that forms on a wedge, i.e. flows in which the plate is not parallel to the flow. An exhaustive description of the hydrodynamics of boundary layers can be found in the classical textbook of Schlichting sch recent applications in engineering is discussed by Hori hori . The mathematical properties of the corresponding partial differential equations (PDEs) attracted remarkable interest as well. Without completeness we mention some of the available mathematical results. Libby and Fox libby derived some solutions using perturbation method. Ma and Hui ma gave similarity solution to the boundary layer problems. Burde burde1 ; burde2 ; burde3 gave additional numerous explicit analytic solutions in the nineties. Weidman weid presented solutions for boundary layers with additional cross flows. Ludlow and coworkers lud evaluated and analyzed solutions with similarity methods as well. Vereshchagina ver investigated the spatial unsteady boundary layer equations with group fibering. Polyanin in his papers poy1 ; poy2 presents numerous independent solutions derived with various methods like general variable separation. Makinde mak investigated the laminar falling liquid film with variable viscosity along an inclined heated plate problem using perturbation technique together with a special type of Hermite – Padé approximation. In nanofluids the importance of buoyancy AnSa2016 , aspects on bioconvection MaAn2016 , and possible modified viscosity SaKoAn2016 are also discussed. One may find exact solutions for the oscillatory shear flow in SaGiKo2017 ; SaGi2018 . Bognár bogn applied the steady-state boundary layer flow equations for non- Newtonian fluids and presented self-similar results. Later it was generalized bognhri , and the steady-state heat conduction mechanism was included in the calculations as well. Certain parameters of the nanofluid can be tuned by varying the amount of nanoparticles in the fluid MaEbTe1993 ; Ch1995 ; Ng2007 . In our former studies we investigated three different kind of Rayleigh-Bénard heat conduction problems imre1 ; imre2 ; imre3 which are full two-dimensional viscous flows coupled to the heat conduction equation. We might say that the heated boundary layer equations - from the mathematical point of view - show some similarities to the Rayleigh-Bénard problem. These last five publications of us bogn ; bognhri ; imre1 ; imre2 ; imre3 led us to the decision that it would be worst examining heated boundary layers with the self-similar Ansatz. In the following we apply the Sedov type self-simiar Ansatz sedov ; zeldovich to the original partial differential equation (PDE) systems of incompressible and compressible boundary layers with heat conduction and reduce them to coupled non-linear ordinary differential equation (ODE) system. For the incompressible case the ODE system can be solved with quadrature giving analytic solutions for the velocity, pressure and temperature fields. Due, to our knowledge there are no self-similar solutions known and analyzed for any type of time-dependent boundary layer equations including heat conduction. ## II Theory ### II.1 The incompressible case We start with the PDE system of $\displaystyle\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}$ $\displaystyle=$ $\displaystyle 0,$ (1) $\displaystyle\frac{\partial p}{\partial y}$ $\displaystyle=$ $\displaystyle 0,$ (2) $\displaystyle\rho_{\infty}\frac{\partial u}{\partial t}+\rho_{\infty}\left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\right)$ $\displaystyle=$ $\displaystyle\mu\frac{\partial^{2}u}{\partial y^{2}}-\frac{\partial p}{\partial x},$ (3) $\displaystyle\rho_{\infty}c_{p}\frac{\partial T}{\partial t}+\rho_{\infty}c_{p}\left(u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}\right)$ $\displaystyle=$ $\displaystyle\kappa\frac{\partial^{2}T}{\partial y^{2}},$ (4) where the dynamical variables are the two velocities components $u(x,y,t),v(x,y,t)$ of the fluid the pressure $p(x,y,t)$ and the temperature $T(x,y,t)$. The additional physical parameters are $\rho_{\infty},c_{p},\mu,\kappa,$ the fluid density at asymptotic distances and times, the heat capacity at fixed pressure, the kinematic viscosity and the thermal diffusivity, respectively. It is important to emphasize at this point, that this description for the heated boundary layer is valid for small velocities in laminar flow, only. More information can be found in the classical book of Schlichting sch ($8^{th}$ addition page 211). Outside the laminar flow regime a viscous heating term should be added to the final temperature equation with the form of $\mu(u_{y})^{2}$. (A similar analysis for that system is already in progress and will be the topic of our next distinct study.) There is no general fundamental theory for nonlinear PDEs, but over time, some intuitive methods have evolved, most of them can be derived from symmetry considerations. Numerous (almost arbitrary) functions can be constructed which couple the temporal and spatial variables to a new reduced variable from intuitive reasons. Our long term experience shows that two of them are superior to all others and have direct physical meanings. These are the traveling wave and the self-similar Ansätze. The first is more or less well known from the community of physicists and engineers and, has the form of $G(x,t)=f(x\mp ct)$ and we may call $\eta=x\mp ct$ as the new reduced variable, where c is the propagation speed of the corresponding wave. Here $G(x,t)$ is the investigated dynamical variable in the PDE. $G(x,t)$ could be any physically relevant property, like temperature, electric field or the like. This Ansatz can be applied to any kind of PDE and will mimic the general wave property of the investigated physical system. The second (and not so well known) is the self-similar Ansatz with the from of $G(\eta)=t^{-\alpha}f(x/t^{\beta})$. There $\alpha$ and $\beta$ are two free real parameters, it can be shown that this Ansatz automatically gives the Gaussian or fundamental solution of the diffusion (or heat conduction) equation. In general, and this is the key point here, this trial functions helps us to get a deeper insight into the dispersive and decaying behavior of the investigated physical system. This is the main reason why we use it in this form. Viscous fluid dynamic equations automatically fulfill this condition, therefore it is highly probable, that this Ansatz leads to physically rational solutions. It is easy to modify the original form of the Ansatz to two (or even three) spacial dimensions and generalize it to multiple dynamical variables, hereupon we apply the following form of: $\displaystyle u(x,y,t)$ $\displaystyle=$ $\displaystyle t^{-\alpha}f(\eta),\hskip 28.45274ptv(x,y,t)=t^{-\delta}g(\eta),$ $\displaystyle T(x,y,t)$ $\displaystyle=$ $\displaystyle t^{-\gamma}h(\eta),\hskip 28.45274ptp(x,y,t)=t^{-\epsilon}i(\eta),$ (5) with the new argument $\eta=\frac{x+y}{t^{\beta}}$ of the shape functions. (To avoid later physical interpretation problems of negative values we define temperature as a temperature difference relative to the average $T=\tilde{T}-T_{av}$.) All the exponents $\alpha,\beta,\gamma,\delta$ are real numbers. (Solutions with integer exponents are called self-similar solutions of the first kind, non-integer exponents generate self-similar solutions of the second kind.) It is important to emphasize that the obtained results fulfill well-defined initial and boundary problems of the original PDE system via fixing their integration constants of the derived ODE system. The shape functions $f,g,h$ and $i$ could be any continuous functions with existing first and second continuous derivatives and will be evaluated later on. The logic, the physical and geometrical interpretation of the Ansatz were exhaustively analyzed in all our former publications imre1 ; imre2 ; imre3 therefore we skip it here. The general scheme of the calculation, how the self-similar exponents can derived is given in imre4 in details. The main idea is the following: after having done the spatial and temporal derivatives of the Ansatz the obtained terms should be replaced into the original PDE system. Due to the derivations all terms pick up an extra time dependent factor like $t^{-\alpha-1}$ or $t^{-2\beta}$ because of the reduction mechanism the new variable of the shape functions is now $\eta$ therefore all kind of extra time dependences have to be canceled. Therefore all the exponents of the time dependences eg. $\alpha+1$ or $2\beta$ should cancel each other which dictates a relation among the self-similar variables. In our very first paper we gave all the details of this kind of a calculation for the non-compressible newtonian three dimensional Navier-Stokes equation imre4 . The main points are, that $\alpha,\delta,\gamma,\epsilon$ are responsible for the rate of decay and $\beta$ is for the rate of spreading of the corresponding dynamical variable for positive exponents. Negative self-similar exponents (except for some extreme cases) mean unphysical, exploding and contracting solutions. The numerical values of the exponents are now the following: $\alpha=\beta=\delta=1/2,\hskip 28.45274pt\epsilon=1,\hskip 28.45274pt\gamma=\textrm{arbitrary real number}.$ (6) Exponents with numerical values of one half mean the regular Fourier heat conduction (or Fick’s diffusion) process. One half values for the exponent of the velocity components and unit value exponent for the pressure decay are usual for the incompressible Navier-Stokes equation imre4 . The obtained ODE system reads $\displaystyle f^{\prime}+g^{\prime}$ $\displaystyle=$ $\displaystyle 0,$ (7) $\displaystyle i^{\prime}$ $\displaystyle=$ $\displaystyle 0,$ (8) $\displaystyle\rho_{\infty}\left(-\frac{f}{2}-\frac{f^{\prime}\eta}{2}\right)+\rho_{\infty}(ff^{\prime}+gf^{\prime})$ $\displaystyle=$ $\displaystyle\mu f^{\prime\prime}-i^{\prime},$ (9) $\displaystyle\rho_{\infty}c_{p}\left(-\gamma h-\frac{h^{\prime}\eta}{2}\right)+\rho_{\infty}c_{p}(fh^{\prime}+gh^{\prime})$ $\displaystyle=$ $\displaystyle\kappa h^{\prime\prime},$ (10) where prime means derivation in respect to the variable $\eta$. The first two equations are total derivatives and can be integrated directly yielding: $f+g=c_{1}$ and $i=c_{2}$. Having total derivatives in a dynamical systems automatically mean conserved quantities, (the first of them is now mass conservation). After some straightforward algebraic manipulation we arrive to a separate second order ODE for the velocity shape which is also a total derivative and can be integrated leading to: $\mu f^{\prime}+\rho_{\infty}f\left(\frac{\eta}{2}-c_{1}\right)-c_{2}=0,$ (11) with the analytic solution of $\displaystyle f=$ $\displaystyle\left(\frac{c_{2}\sqrt{\pi}e^{-\frac{\rho_{\infty}c_{1}^{2}}{\mu}}\cdot\emph{erf}\left[\frac{1}{2}\sqrt{-\frac{\rho_{\infty}}{\mu}}\eta+\frac{\rho_{\infty c_{1}}}{\sqrt{-\mu\rho_{\infty}}}\right]}{\sqrt{-\mu\rho_{\infty}}}+c3\right)\cdot\emph{e}^{\frac{\eta(-\eta+4c_{1})\rho_{\infty}}{4\mu}}$ (12) where erf means the usual error function NIST . Note, that for the positive real constants $\rho_{\infty},\mu$ the complex quantity $\sqrt{-\rho_{\infty}\mu}$ appears in the argument of the error functions and as a complex multiplicative prefactor simultaneously making the final result a pure real function. The second important thing is to note, that for the $c_{1}=c_{2}=0$ trivial integration constants the solution is simplified to the Gaussian function of $f=c_{4}e^{-\frac{\rho_{\infty\eta^{2}}}{4\mu}}.$ (13) This means that the velocity flow process shows similarity to the regular diffusion of heat conduction phenomena. Similar solutions (containing exponential and error functions) were found for the stationary velocity field by Weyburne in 2006 with probability distribution function methodology wey . Figure (1) shows the general velocity shape function (12) for various parameter sets. The choice of these parameters are arbitrary, we are not limited to real fluid parameters, however we try to create the most general and most informative figures, which mimic the general features of the solution function. The functions are the modification of the error function. The crucial parameter is the ratio $\rho_{\infty}/\mu$, if this is larger than unity then the function tends to a sharp Gaussian. Figure 1: The graphs of the velocity shape function $f(\eta)$ in Eq. (12) for three different parameter sets ($c_{1},c_{2},c_{3},\mu,\rho_{\infty}$). The solid, dashed and dotted lines are for $(1,0,1,4.1,0.9)$, $(2,-1,0.5,2.5,1)$ and $(2,2,0.3,10,1)$, respectively. Figure 2: The velocity distribution function $u(x,y=0,t)=\frac{1}{t^{1/2}}f(\eta)$ for the third parameter set presented on the previous figure. Figure (1) presents the velocity distribution function. Note, the very sharp peak in the origin and the extreme quick time decay along the time axis. There is a separate ODE for the temperature distribution as well $\frac{\kappa}{\rho_{\infty}c_{p}}h^{\prime\prime}-h^{\prime}\left(c_{1}-\frac{\eta}{2}\right)+\gamma h=0.$ (14) For the most general case (when $\gamma$ is an arbitrary real number,) and $c_{1}\neq 0$ the solutions of Eq. (14) can be expressed with the Kummer M and Kummer U functions NIST $h=c_{2}M\left(\gamma,\frac{1}{2};-\frac{c_{p}\rho_{\infty}[\eta-2c_{1}]^{2}}{4\kappa}\right)+c_{3}U\left(\gamma,\frac{1}{2};-\frac{c_{p}\rho_{\infty}[\eta-2c_{1}]^{2}}{4\kappa}\right).$ (15) M is regular in the origin and U is irregular, therefore we investigate only the properties of M which means $(c_{3}=0)$. The M and U functions form a complete orthogonal function system if the argument is linear. Now, the argument is quadratic, - in our former studies we found numerous such solutions, for incompressible imre4 or for compressible imre5 multidimensional Navier-Stokes or Euler equations – however, we still do not know the physical message of this property. It can be easily proven with the definition of the Kummer functions using the Pochhammer symbols NIST , that for negative integer $\gamma$ values our results can be expanded into finite order polynomials, which are divergent for large arguments $\eta$. For non-integer $\gamma<0$ values, we get infinite divergent polynomials as well. The most relevant parameter of the solutions is evidently $\gamma$. The integral constant $c_{1}$ just shifts the solutions parallel to the $x$ axis, $c_{2}$ scales the solutions, and $c_{p}\rho_{\infty}/\kappa$ parameter just scales the width of the solution. Figure (3) presents three different solutions for various positive $\gamma$ values. (All negative $\gamma$ values mean divergent shape functions for large $\eta$s which are unphysical and outside of our scope.) Note, larger $\gamma$s mean more oscillations. For a better understanding we present the projection of the total solution of the temperature field $T(t,x,y)$ on Figure (4) for the $y=0$ coordinates. Figure 3: The graphs of the temperature shape function Eq. (15) for three different parameter sets ($\gamma,c_{2},c_{3},c_{p},\rho_{\infty},\kappa$). The solid, dashed and dotted lines are for $(0.8,4,0,1,0.9,0.3)$, $(3.4,4,0,1,1,0.6)$ and $(6.3,4,0,1,3,10)$, respectively. Figure 4: The temperature distribution function $T(x,y=0,t)=\frac{1}{t^{1}}h(\eta)$ for the first parameter set presented on the previous figure. For some special values of $\gamma$ the temperature shape function can be expressed with other simpler special functions. For values of $\gamma=\pm\frac{1}{2}$ and $0$ the shape functions all contain the error function. Negative integer $\gamma$s result even order polynomials. (E.g. $\gamma=-1$ defines the shape function of $f=(c_{2}+c_{3})\cdot(2\kappa+c_{p}\rho_{\infty}[\eta-2c_{1}]^{2})$. ) Polynomials are divergent in infinity therefore are out of our physical interest. For the sake of completeness we present the solutions for the pressure as well. The ODE of the shape function is trivial with the solution of: $i^{\prime}=0,\hskip 28.45274pti=c_{4}.$ (16) Therefore, the final pressure distribution reads: $p(x,y,t)=t^{-\epsilon}\cdot i(x,y,t)=\frac{c_{4}}{t},$ (17) which means that the pressure is constant in the entire space at a given time point, but has a quicker time decay than the velocity field. ### II.2 The compressible case In the last part of our study we investigate the compressible boundary layer equations. The starting PDE system is now changed to the following: $\displaystyle\frac{\partial\rho}{\partial t}+\frac{\partial\rho}{\partial x}u+\rho\frac{\partial u}{\partial x}+\frac{\partial\rho}{\partial y}v+\rho\frac{\partial v}{\partial y}$ $\displaystyle=$ $\displaystyle 0,$ (18) $\displaystyle\frac{\partial p}{\partial y}$ $\displaystyle=$ $\displaystyle 0,$ (19) $\displaystyle\rho\frac{\partial u}{\partial t}+\rho\left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\right)$ $\displaystyle=$ $\displaystyle\mu\frac{\partial^{2}u}{\partial y^{2}}-\frac{\partial p}{\partial x},$ (20) $\displaystyle c_{p}\rho\frac{\partial T}{\partial t}+c_{p}\rho\left(u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}\right)$ $\displaystyle=$ $\displaystyle k\frac{\partial^{2}T}{\partial y^{2}},$ (21) the notation of all the variables are the same as for the incompressible case. For closing constitutive equation (or with other name ”equation of state” (EOS)) we apply the ideal gas $p=R\rho T$ where $R$ is the universal gas constant. (Of course, there are numerous EOS available for physically relevant materials, and each gives us an additional new system to investigate, but that lies outside the scope of our present study.) For the dynamical variables we apply the next self-similar Ansatz of: $\displaystyle\rho(x,y,t)$ $\displaystyle=$ $\displaystyle t^{-\alpha}f(\eta),\hskip 28.45274ptu(x,y,t)=t^{-\gamma}g(\eta),$ (22) $\displaystyle v(x,y,t)$ $\displaystyle=$ $\displaystyle t^{-\delta}h(\eta),\hskip 28.45274ptT(x,y,t)=t^{-\epsilon}i(\eta),$ (23) with the usual new variable of $\eta=\frac{x+y}{t^{\beta}}$. To obtain a closed ODE system the following relations must held for the similarity exponents $\alpha=0,\hskip 28.45274pt\beta=\delta=\gamma=\epsilon=1/2.$ (24) Note, that now all the exponents have fixed numerical values. The $\alpha=0$ means two things, first the density as dynamical variable has no spreading property (just decay $\beta>0$), second, the first continuity ODE is not a total derivative and cannot be integrated directly. This system has an interesting peculiarity, our experience showed, that the incompressible Navier-Stokes (NS) equation imre4 has all fixed self-similar exponents and the compressible one imre5 has one free exponent. It is obvious that an extra free exponent makes the mathematical structure richer leaving more room to additional solutions. (As we mentioned above, self-similar exponents with the value of one half has a close connection to regular Fourier type heat conduction mechanism.) Parallel, the obtained ODE system reads $\displaystyle-\frac{1}{2}\eta f^{\prime}+fg^{\prime}+f^{\prime}g+f^{\prime}h+fh^{\prime}$ $\displaystyle=$ $\displaystyle 0,$ (25) $\displaystyle R(f^{\prime}i+fi^{\prime})$ $\displaystyle=$ $\displaystyle 0,$ (26) $\displaystyle f\left(-\frac{g}{2}-\frac{g^{\prime}\eta}{2}\right)+f(gg^{\prime}+gh^{\prime})$ $\displaystyle=$ $\displaystyle\mu g^{\prime\prime}-R(f^{\prime}i+fi^{\prime}),$ (27) $\displaystyle c_{p}f\left(-\frac{i}{2}-\frac{i^{\prime}\eta}{2}\right)+c_{p}f(gi^{\prime}+hi^{\prime})$ $\displaystyle=$ $\displaystyle\kappa i^{\prime\prime},$ (28) where prime means derivation in respect to $\eta$. Having done some non-trivial algebraic steps a decoupled ODE can be derived for the density field. First, the pressure equation (26) can be integrated, then $i(\eta)$ can be expressed, after the derivatives $i^{\prime}$ and $i^{\prime\prime}$ can be evaluated, then plugging it into (28) the $(g+h)$ quantity can be expressed with $f,f^{\prime}$ and $f^{\prime\prime}$. Finally, calculating the derivatives of $(f+g)$ and substituting them into (25) an independent ODE can be deduced for the density shape function. These algebraic manipulations are more compound and contain many more steps what we had in the past for various flow systems like imre3 ; imre4 . With the conditions $f(\eta)\neq 0$ and $f^{\prime}(\eta)\neq 0$, the next highly non-linear ODE can be derived $-\kappa f^{\prime}f^{2}f^{\prime\prime\prime}+f^{\prime\prime}\left(\kappa f^{2}f^{\prime\prime}+2\kappa ff^{\prime 2}+\frac{1}{2}c_{p}f^{4}\right)+f^{\prime 2}\left(-2\kappa f^{\prime 2}-c_{p}f^{\prime}f^{2}\cdot\eta-\frac{3}{2}c_{p}f^{3}\right)=0.$ (29) Such ODEs have no analytic solutions for any kind of parameter set (of course $\kappa\neq 0$ and $c_{p}\neq 0$). Therefore, pure numerical integration processes have to be applied. We have to mention, that an analogous fourth- order non-linear ODE was derived in the viscous heated Bénard system imre3 and was analyzed with numerical means. The shape function of the temperature field can be easily derived from (26) without any additional derivation $i=\frac{c_{1}}{Rf}.$ (30) We have to note two things here. First, the condition of $f\neq 0$ should hold. Second, the numerical value $c_{1}$ of the integration constant fixes the absolute magnitude of the temperature. The final physical field quantity which has to be determined is the velocity shape function and distribution. Note, that due to our original Ansatz the two velocity components cannot be determined separately from each other, only the $g+h$ is possible to evaluate. This can be easily done from (25) if we introduce the variable $L:=g+h$. Now the ODE is $L^{\prime}f+Lf^{\prime}-\frac{\eta f^{\prime}}{2}=0.$ (31) The formal solution now became trivial, namely $L=g+h=\frac{\int_{0}^{\eta}\omega f(\omega)d\omega+c_{2}}{2f(\eta)}.$ (32) This means that our Ansatz is not unique for the velocity field because the $x$ and $y$ coordinates are handled on the same footing. The in-depth numerical analysis of the density (29) and the velocity (32) shape functions lies outside the scope of the present study. Here, we just wanted to present that incompressible and compressible flow systems having initially comparable PDE systems, which describe similar processes, but behave completely differently during a self-similar analysis. Such derivations always give a glimpse into the deep mathematical layers of non-linear PDE systems. ## III Summary and Outlook We analyzed the incompressible and compressible time-dependent boundary flow equations with additional heat conduction mechanism with the self-similar Ansatz. Analytic solutions were derived for the incompressible flow. The velocity fields can be expressed with the error functions (in some special cases with Gaussian functions) and the temperature with the Kummer functions. The last one has the most complex mathematical structure including some oscillations. It is often asked what are analytic results are good for, we may say that our analytic solution could help to test complex numerical fluid dynamics program packages, new numerical routines endre or PDE solvers. For a $t=t_{0}$ starting time point the time propagation is exactly given by the analitic formula and can be compared to the results of any numerical scheme. In the second part of our treatise we investigated the compressible time- dependent boundary flow equations with additional heat conduction again with the self-similar Ansatz. For closing constitutive equation, the ideal gas EOS was used. It is impossible to derive analytic solutions for the dynamical variables from the coupled ODE system. However, highly non-linear independent ODEs exist for each dynamical variables which can be integrated numerically. An in-depth analysis could be the subject of a next publication. Work is in progress to apply our self-similar method to more realistic complex boundary layer flows containing viscous heating or other mechanisms. ## IV Authors Contributions The corresponding author (Imre Ferenc Barna) had the original idea of the study, performed all the calculations, created the figures and wrote large part of the manuscript. The second and third authors (Krisztián Hriczó and Gabriella Bognár) checked the written manuscript, improved the language of the final text and gave some general instructions. The third author (Gabriella Bognár) organized the financial support and the general founding. The last author (László Mátyás) checked the literature of the investigated scientific field, corrected the manuscript and had an everyday contact with the first author. ## V Acknowledgments One of us (I.F. Barna) was supported by the NKFIH, the Hungarian National Research Development and Innovation Office. This study was supported by project no. 129257 implemented with the support provided from the National Research, Development and Innovation Fund of Hungary, financed under the $K\\_18$ funding scheme. ## VI Conflicts of Interest The authors declare no conflict of interest. ## VII Data Availability The data that supports the findings of this study are all available within the article. ## References * (1) L. Prandtl, Verhandlungen 3. Int. Math. Kongr. Heidelberg 3, 484 (1904). * (2) H. Blasius, Z. Angew. Math. Phys. 56, 1 (1908). * (3) V. M. Falkner and S. W. Skan, Aero. Res. Coun. Rep. and Mem. No. 1314 (1930). * (4) V.M. Falkner and S.W. Skan, Phil. Mag., 12, 865 (1931). * (5) H. Schlichting and K. Gersten Boundary-Layer Theory, Springer, 2017. * (6) Y. Hori, Hydrodynamic Lubrication, Springer, 2006. * (7) P.A. Libby and Fox, J. Fluid Mech. 17, 3 (1963). * (8) P.K.H Ma and W.H. Hui, J. Fluid Mech. 216, 537 (1990). * (9) G.I. Burde, Quart. J. Mech.Appl. 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# Cross Chest Graph for Disease Diagnosis with Structural Relational Reasoning Gangming Zhao Baolian Qi Jinpeng Li Gangming Zhao and Baolian Qi contributed equally to this work.Gangming Zhao is with the Department of Computer Science, The University of Hong Kong, Hong Kong.Qibao Lian and Jinpeng Li are with University of Chinese Academy of Sciences, Beijing, China ###### Abstract Locating lesions is important in the computer-aided diagnosis of X-ray images. However, box-level annotation is time-consuming and laborious. How to locate lesions accurately with few, or even without careful annotations is an urgent problem. Although several works have approached this problem with weakly- supervised methods, the performance needs to be improved. One obstacle is that general weakly-supervised methods have failed to consider the characteristics of X-ray images, such as the highly-structural attribute. We therefore propose the Cross-chest Graph (CCG), which improves the performance of automatic lesion detection by imitating doctor’s training and decision-making process. CCG models the intra-image relationship between different anatomical areas by leveraging the structural information to simulate the doctor’s habit of observing different areas. Meanwhile, the relationship between any pair of images is modeled by a knowledge-reasoning module to simulate the doctor’s habit of comparing multiple images. We integrate intra-image and inter-image information into a unified end-to-end framework. Experimental results on the NIH Chest-14 database (112,120 frontal-view X-ray images with 14 diseases) demonstrate that the proposed method achieves state-of-the-art performance in weakly-supervised localization of lesions by absorbing professional knowledge in the medical field. ## 1 Introduction Chest radiographs are a type of medical images that can be conveniently acquired for disease diagnosis. With the rapid development of deep learning, automatic disease detection in chest X-ray images has become an important task in the computer-aided diagnosis. Figure 1: CCG network models the intra-image relationship between different anatomical areas by leveraging the structural information to simulate the doctor’s habit of observing different areas. Meanwhile, the relationship between any pair of images is modeled by a knowledge-reasoning module to simulate the doctor’s habit of comparing multiple images. Deep convolutional neural networks (DCNN) have been widely applied in many computer vision tasks, such as image classification [6, 18] , object detection [3, 5, 16, 15, 10] and semantic segmentation [12, 17]. To achieve good performance in these tasks, substantial images with careful annotations are needed. Encouraged by the success of DCNN in computer vision, some researches have directly applied DCNN models to analyze the medical images but cannot achieve the same performance as in the natural images. The reasons lie in two folds: 1. it is expensive to acquire accurate localization or classification labels in chest X-ray images. 2. there exists much professional knowledge in medical images that DCNN cannot exploit well. Therefore, how to exploit the professional knowledge into DCNN models for solving these two questions still opens a fully challenging problem. Our work transfers the knowledge into DCNN models to reduce the problem of shortage of carefully annotated images. Recent work paid much attention to utilize professional knowledge of chest X-ray images into DCNN frameworks. However, they just proposed a simple fused strategy to embed low-level information of chest X-ray into models, such as Liu et al. [9] utilized contrastive learning to provide more localization information with the help of healthy images. Zhao et al. [22] proposed to exploit the contralateral information of chest X-ray via a simple fusion module. These methods only exploit the apparent information of chest-Xray images. They all overlooked the inner structure information of chest X-rays. Therefore, they cannot apply their methods into real applications. In this paper, we propose a Cross Chest Graph Network (CCG-Net) as shown in Fig 1, which firstly utilizes deep expert knowledge to automatical detect disease in chest X-ray images. We have known that medical experts have much experience in finding out disease and how to treat patients. In fact, the actions of medical experts consist of two phases: training and decision-making processes. They pay much time to learn distinguish disease and embed their experience into the decision process. During the training process, experts would like to observe different areas and find out the relationship between any pair of images. Our CCG-Net aims to model the observation way by a knowledge-reasoning module to simulate the doctor’s habit of comparing multiple images. Then we integrate intra-image and inter-image information into a unified end-to-end framework. Inspired from the experience of medical experts, our proposed CCG-Net consists of four modules, 1. an end-to-end framework for deciding where and what is a disease, 2. a inter-image relation module, which formulates the training process of medical experts, to compare multiple images, 3. a intra-image knowledge learning module, which builds the local relation graph for different patches of chest X-ray images. Due to their highly structured property, every chest X-ray image can be divided into several patches, we build a patch-wise relation graph on them, 4. a knowledge reasoning module, which excavates the inner knowledge from cross-image structural features. The last three operations (2, 3, and 4) are similar to medical experts’ training process, which learn intra-image and inter-image information to gain professional knowledge. The first operation embeds the learned knowledge into DCNN frameworks leading to better disease diagnosis models. Above all, our contribution consists of three folds: * • We propose CCG-Net, which is the first to formulate the medical experts’ training process by building relation graphs in the intra-image and inter- image information of chest X-ray images. More generally, it provides inspiration to address medical vision tasks with much professional knowledge like in chest X-ray images. * • We divide the experts’ professional actions into two stages including training and decision-making processes. In addition, we utilize intra-image and inter- image relation to learn much professional knowledge that would be embedded in an end-to-end detection framework. * • We achieve state-of-the-art results on the localization of NIH ChestX-ray14. ## 2 Related Work ### 2.1 Disease Detection Object detection is one of the most important computer vision tasks, aiming to localize and classify. Due to their strong feature representation ability, DCNN achieved much progress in object detection tasks. For detection tasks, DCNN methods consist of two style framework: 1. two-stage models, such as RCNN series [16], 2. one-stage models, such as YOLO [15] and SSD [10]. However, for disease detection, because of the shortage in careful annotations, traditional detection framework cannot directly be applied in chest X-ray images. Besides, since there is much distortion caused by other chest X-ray tissues, such low contrast also causes the difficulty of disease finding. Weakly supervised object detection (WSOD) can be considered as an effective method to solve these problems. Based on CAM [23], researchers proposed many techniques to use only image-level labels to detect objects. Although there is no enough detection supervision, WSOD still achieved much progress. However, researchers still face a big challenge when it comes to disease detection in medical images. the existence of much professional knowledge greatly limits the development of the applications of DCNN in medical fields. Therefore, in this paper, we are inspired by the experts’ learning and decision processes to propose CCG-Net, which not only exploits a larger amount of knowledge in chest X-ray images but also builds a unified framework to detect disease in an end- to-end style. ### 2.2 Knowledge-based Disease Diagnosis Automatical disease diagnosis is a key problem in medical fields. However, due to the shortage of careful annotations and the existence of much professional knowledge, DCNN methods cannot achieve a good performance in medical tasks, especially such a tough problem: disease detection in chest X-ray images. To exploit medical knowledge and embed it into DCNN frameworks, researchers paid much effort to utilize medical experts’ experience for disease diagnosis. Wang et al. [20] firstly proposed a carefully annotated chest X-ray dataset and led to a series of work that focuses on using image-level labels to localize the disease. Li et al. [8] integrated classification and localization in a whole framework with two multi instance-level losses and performed better. Liu et al. [9] improved their work to propose contrastive learning of paired samples, which utilizes healthy images to provide more localization information for disease detection. Zhao et al. [22] proposed to utilize the symmetry information in a chest X-ray to improve the disease localization performance. Figure 2: The network consists of four modules: 1. an end-to-end framework for disease detection under weakly-supervised settings, 2. the inter-image relation module among different samples, 3. the intra-image knowledge learning module based on the thoracic spatial structure, 4. the knowledge reasoning module mining cross-image structural features. Our four modules are tightly related and can be easily integrated into an end-to-end framework. Besides, many works applied relation knowledge models to chest X-ray diagnosis. Ypsilantis et al. [21], Pesce et al. [14], and Guan et al. [4] proposed to build a relation attention model fusing DCNN models achieved much progress. Li et al. [7] proposed a knowledge-graph based on medical reports and images to determine the dependencies among chest X-ray images. Cheng et al. [11] also proposed a new total strongly supervised dataset for tuberculosis detection. However, they all overlooked the structural relation among chest X-ray images. In this paper, we propose to build a structural, relational graph for disease detection under weakly supervised scenarios in chest X-ray images. Specifically, we build the global and local graph in chest X-ray via three modules: 1. a inter-image relation module, 2. a intra-image knowledge learning module, 3. knowledge reasoning module. Furthermore, we integrate three modules into an end-to-end framework to jointly train our network. Our proposed three relational modules provide better supervision since we exploit the local structural knowledge and global relation among different samples. ## 3 Method ### 3.1 Overview Given the images $X=\\{x_{1},x_{2},...,x_{n}\\}$. Our proposed framework consists of four modules: * • The End to End Framework is to localize and classify the disease in chest X-ray images. In our paper, we utilize the same multi-instance level losses used in [9] and [8]. * • Inter-image Relation Module, which includes a learnable matrix $G\in R^{n\times n}$. We also use a contrast-constrained loss to share similar information of $X$ and exploit their contrasted structural knowledge. We build a cross-sample graph for them to exploit the dependencies among different samples. The graph $G\in R^{n\times n}$ is to build the inter-image relation among sampled samples, which is a learnable matrix, and every element is initialized by $\frac{1}{n}$. The element $g_{ij}$ of $G$, $i,j\in\\{1,2,...,n\\}$, represents the similarity wight of images $x_{i}$ and $x_{j}$. * • Intra-image Knowledge Learning, which firstly acquires patch-wise features of different images. Then the network can achieve a new image graph via building a structural knowledge-based module. We denote this graph as $G_{k}\in R^{n\times n}$. Assumed that the number of patches are $\|p_{i}\|$ and $\|p_{j}\|$ of images $x_{i}$ and $x_{j}$. The graph $G_{k}$ would be calculated on using the graph $G_{l}\in R^{\|p_{i}\|\times\|p_{j}\|}$, which learns the relationship between different paired patches of images. * • Knowledge Reasoning Module, which is based on cross image structural knowledge. When we get the whole structural information of different images, we will utilize it to reason the inner structural dependencies among different patches in different images. ### 3.2 End to End Framework The end to end framework is to localize and classify the disease in chest X-ray images in a coarse-grained style. More specifically, the input images $X=\\{x_{1},x_{2},...,x_{n}\\}$ of the module are resized to $512\times 512$. ResNet-50 pre-trained from the ImageNet dataset is adopted as the backbone for this module. We use the feature map $F$ after C5 (last convolutional output of 5th-stage), which is 32 times down-sampled from the input image, and of size $2048\times 16\times 16$. Each grid in the feature map denotes the existent probability of disease. We pass $F$ through two $1\times 1$ convolutional layers and a sigmoid layer to obtain the class-aware feature map P of size $C\times H\times W$, where $C$ is the number of classes. Then we follow the paradigm used in [9], computing losses and making predictions in each channel for the corresponding class. For images with box-level annotations, if the grid in the feature map overlaps with the projected ground truth box, we assign label 1 to the grid. Otherwise, we assign 0 to it. Therefore, we use the binary cross-entropy loss as used in [9] for each grid: $L^{k}_{i}(\emph{P})=\sum_{j}-y_{ij}^{k}\log(p_{ij}^{k})-\sum_{j}(1-y_{ij}^{k})\log(1-p_{ij}^{k})$ (1) where $k$, $i$, and $j$ are the index of classes, samples, and grids respectively. $y^{k}_{ij}$ denotes the target label of the grid and $p^{k}_{ij}$ denotes the predicted probability of the grid. For images with only image-level annotations, we use the MIL loss used in [8]. $\begin{split}L^{k}_{i}(\emph{P})=-&y^{k}_{i}\log(1-\prod_{j}(1-p^{k}_{ij}))\\\ -&(1-y^{k}_{i})\log(\prod_{j}(1-p^{k}_{ij}))\end{split}$ (2) where $y^{k}_{i}$ denotes the target label of the image. For this end to end framework, the whole loss $L_{base}$ as shown in Fig. 2, is formulated as follows. $\begin{split}L_{base}=\sum_{i}\sum_{k}\lambda^{k}_{i}\beta_{B}L^{k}_{i}(\emph{P})+(1-\lambda^{k}_{i})L^{k}_{i}(\emph{P})\end{split}$ (3) where $\lambda^{k}_{i}\in{0,1}$ denotes if the $k_{th}$ class in the $i_{th}$ sample has box annotation, and $\beta_{B}$ is the balance weight of the two losses and is set to 4. ### 3.3 Inter-image Relation Module Inter-image relation is formulated as a learnable matrix $G\in R^{n\times n}$. A contrast-constrained loss is used to share similar information of $X$ and exploit their contrasted structural knowledge, as following equation. $\begin{split}L_{IR}=\frac{\sum_{(u,v)\in G}G(u,v)D(F_{u},F_{v})}{n\times n}\end{split}$ (4) $D(\cdot)$ is the distance metric function, where it is a Euclidean distance. $F_{u}$ and $F_{v}$ means the feature map after C5 of the image $x_{u}$ and $x_{v}$. We build a cross-sample graph for them to exploit the dependencies among different samples. The graph $G\in R^{n\times n}$ is to build the inter- image relation among sampled samples, which is a learnable matrix, and every element is initialized by $\frac{1}{n}$. The element $g_{ij}$ of $G$, $i,j\in\\{1,2,...,n\\}$, represents the similarity wight of images $x_{i}$ and $x_{j}$. G is adaptively adjusted during training processes and changes with diverse inputs to exploit the relationship fully. ### 3.4 Intra-image Knowledge Learning Intra-image Knowledge Learning, which firstly utilizes Simple linear iterative clustering (SLIC) [1], a super-pixel method to generate the patches for different images. Assumed that the patches of the image $x_{i}$ is $p_{i}={p^{i}_{1},p^{i}_{2},...,p^{i}_{m}}$. Then the network can achieve a new image graph via building a structural knowledge-based module with the help of $p_{i},i\in{1,2,...,n}$. We denote the graph as $G_{k}\in R^{n\times n}$, which is the intra-image graph between paired images $x_{i}$ and $x_{j}$. The graph $G_{k}$ is calculated on using the graph $G_{l}\in R^{\|p_{i}\|\times\|p_{j}\|}$, which learns the dependencies among different paired patches of images. Then the same contrast-constrained loss using this graph to provide more structural knowledge for the whole framework. $\begin{split}L_{IK}=\frac{\sum_{(u,v)\in G_{k}}G_{k}(u,v)D(F_{u},F_{v})}{n\times n}\end{split}$ (5) $\begin{split}G_{k}=W_{l}(G_{l})\end{split}$ (6) Where, $W_{l}$ is a fully connected layer and $\begin{split}G_{l}(l,p)=D^{{}^{\prime}}(H_{l},H^{{}^{\prime}}_{p}),l\in{1,2,...,\|p_{i}\|},p\in{1,2,...,\|p_{j}\|}\end{split}$ (7) $H_{l}$ is the hash code [19] of the patch $p^{i}_{l}$ in the image $x_{i}$ and $H^{{}^{\prime}}_{p}$ is the hash code of the patch $p^{j}_{p}$ in the image $x_{j}$. $D^{{}^{\prime}}(\cdot)$ is the Hamming distance. ### 3.5 Knowledge Reasoning Module In addition to previous efforts to focus on information in a whole image, we also explored the value of cross-image semantic relations in the medical object. The correlations between patches across images are emphasized, especially, the correlations between corresponding patches in two images. Knowledge Reasoning Module focuses on the correlations of two images. After getting the feature map $F_{u}$ and $F_{v}$ of the images, the affinity matrix $P$ is firstly calculated between $F_{u}$ and $F_{v}$. $P=F^{\mathrm{T}}_{u}W_{P}F_{v}\in\mathbb{R}^{HW\times HW}$ where the feature map $F_{u}\in\mathbb{R}^{C\times HW}$ and $F_{v}\in\mathbb{R}^{C\times HW}$ are flattened into matrix formats, and $W_{P}\in\mathbb{R}^{C\times C}$ is a learnable matrix. The affinity matrix $P$ represents the similarity of all pairs of patches in $F_{u}$ and $F_{v}$. Then $P$ is normalized column-wise to get the attention map of $F_{u}$ for each patch in $F_{v}$ and row-wise to get the attention map of $F_{v}$ for each patch in $F_{u}$. $F^{{}^{\prime}}_{u}=F_{u}softmax(P)\in\mathbb{R}^{C\times HW}$ $F^{{}^{\prime}}_{v}=F_{v}softmax(P^{\mathrm{T}})\in\mathbb{R}^{C\times HW}$ where $softmax(P)$ and $softmax(P^{\mathrm{T}})$ pay attention to the similar patches of the feature map $F_{u}$ and $F_{v}$ respectively. Therefore, they can be used to enhance $F_{u}$ and $F_{v}$ respectively, so that similar patches in $F_{u}$ and $F_{v}$ are highlighted. The cross-image method can extract more contextual information between images than using a single image. This module exploits the context of other related images to improve the reasoning ability of the feature map, which is beneficial to the localization and classification of disease in chest X-ray images. Furthermore, we exploit the enhanced feature map to calculate the new similarity between the paired images to gain a more strong supervisor. $\begin{split}L_{KR}=\frac{\sum_{(u,v)\in G_{k}^{{}^{\prime}}}G_{k}^{{}^{\prime}}(u,v)D(F^{{}^{\prime}}_{u},F^{{}^{\prime}}_{v})}{n\times n}\end{split}$ (8) T (IoU) \| Models \| Atelectasis \| Cardiomegaly \| Effusion \| Infiltration \| Mass \| Nodule \| Pneumonia \| Pneumothorax \| Mean ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- 0.3 \| X, Wang [20] \| 0.24 \| 0.46 \| 0.30 \| 0.28 \| 0.15 \| 0.04 \| 0.17 \| 0.13 \| 0.22 Z, Li [8] \| 0.36 \| 0.94 \| 0.56 \| 0.66 \| 0.45 \| 0.17 \| 0.39 \| 0.44 \| 0.49 J, Liu [9] \| 0.53 \| 0.88 \| 0.57 \| 0.73 \| 0.48 \| 0.10 \| 0.49 \| 0.40 \| 0.53 \| Ours \| 0.44 \| 0.86 \| 0.68 \| 0.84 \| 0.47 \| 0.29 \| 0.67 \| 0.40 \| 0.60 0.5 \| X, Wang [20] \| 0.05 \| 0.18 \| 0.11 \| 0.07 \| 0.01 \| 0.01 \| 0.03 \| 0.03 \| 0.06 Z, Li [8] \| 0.14 \| 0.84 \| 0.22 \| 0.30 \| 0.22 \| 0.07 \| 0.17 \| 0.19 \| 0.27 J, Liu [9] \| 0.32 \| 0.78 \| 0.40 \| 0.61 \| 0.33 \| 0.05 \| 0.37 \| 0.23 \| 0.39 \| Ours \| 0.27 \| 0.86 \| 0.48 \| 0.72 \| 0.53 \| 0.14 \| 0.58 \| 0.35 \| 0.49 0.7 \| X, Wang [20] \| 0.01 \| 0.03 \| 0.02 \| 0.00 \| 0.00 \| 0.00 \| 0.01 \| 0.02 \| 0.01 Z, Li [8] \| 0.04 \| 0.52 \| 0.07 \| 0.09 \| 0.11 \| 0.01 \| 0.05 \| 0.05 \| 0.12 J, Liu [9] \| 0.18 \| 0.70 \| 0.28 \| 0.41 \| 0.27 \| 0.04 \| 0.25 \| 0.18 \| 0.29 \| Ours \| 0.20 \| 0.86 \| 0.48 \| 0.68 \| 0.32 \| 0.14 \| 0.54 \| 0.30 \| 0.44 Table 1: The comparison results of disease localization among the models using 50% unannotated images and 80% annotated images. For each disease, the best results are bolded. T (IoU) \| Models \| Atelectasis \| Cardiomegaly \| Effusion \| Infiltration \| Mass \| Nodule \| Pneumonia \| Pneumothorax \| Mean ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- 0.1 \| Z, Li [8] \| 0.59 \| 0.81 \| 0.72 \| 0.84 \| 0.68 \| 0.28 \| 0.22 \| 0.37 \| 0.57 J, Liu [9] \| 0.39 \| 0.90 \| 0.65 \| 0.85 \| 0.69 \| 0.38 \| 0.30 \| 0.39 \| 0.60 Ours \| 0.66 \| 0.88 \| 0.79 \| 0.85 \| 0.69 \| 0.28 \| 0.40 \| 0.47 \| 0.63 0.3 \| J, Liu [9] \| 0.34 \| 0.71 \| 0.39 \| 0.65 \| 0.48 \| 0.09 \| 0.16 \| 0.20 \| 0.38 Baseline \| 0.36 \| 0.69 \| 0.35 \| 0.64 \| 0.44 \| 0.08 \| 0.02 \| 0.23 \| 0.35 Ours \| 0.31 \| 0.79 \| 0.37 \| 0.75 \| 0.40 \| 0.06 \| 0.24 \| 0.27 \| 0.40 0.5 \| J, Liu [9] \| 0.19 \| 0.53 \| 0.19 \| 0.47 \| 0.33 \| 0.03 \| 0.08 \| 0.11 \| 0.24 Baseline \| 0.18 \| 0.51 \| 0.14 \| 0.47 \| 0.27 \| 0.03 \| 0.01 \| 0.12 \| 0.22 Ours \| 0.19 \| 0.71 \| 0.14 \| 0.52 \| 0.31 \| 0.08 \| 0.05 \| 0.13 \| 0.27 0.7 \| J, Liu [9] \| 0.08 \| 0.30 \| 0.09 \| 0.25 \| 0.19 \| 0.01 \| 0.04 \| 0.07 \| 0.13 Baseline \| 0.11 \| 0.34 \| 0.06 \| 0.32 \| 0.20 \| 0.01 \| 0.00 \| 0.06 \| 0.14 Ours \| 0.06 \| 0.64 \| 0.08 \| 0.38 \| 0.19 \| 0.01 \| 0.08 \| 0.09 \| 0.19 Table 2: The comparison results of disease localization among the models using 100% unannotated images and no any annotated images. For each disease, the best results are bolded. T (IoU) \| Models \| Atelectasis \| Cardiomegaly \| Effusion \| Infiltration \| Mass \| Nodule \| Pneumonia \| Pneumothorax \| Mean ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- 0.3 \| J, Liu [9] \| 0.55 \| 0.73 \| 0.55 \| 0.76 \| 0.48 \| 0.22 \| 0.39 \| 0.30 \| 0.50 Baseline \| 0.47 \| 0.84 \| 0.65 \| 0.82 \| 0.33 \| 0.04 \| 0.57 \| 0.29 \| 0.50 Ours \| 0.49 \| 0.87 \| 0.66 \| 0.88 \| 0.48 \| 0.10 \| 0.51 \| 0.20 \| 0.52 0.5 \| J, Liu [9] \| 0.36 \| 0.57 \| 0.37 \| 0.62 \| 0.34 \| 0.13 \| 0.23 \| 0.17 \| 0.35 Baseline \| 0.27 \| 0.76 \| 0.39 \| 0.58 \| 0.24 \| 0.02 \| 0.39 \| 0.21 \| 0.36 Ours \| 0.26 \| 0.80 \| 0.41 \| 0.67 \| 0.15 \| 0.06 \| 0.42 \| 0.18 \| 0.37 0.7 \| J, Liu [9] \| 0.19 \| 0.47 \| 0.20 \| 0.41 \| 0.22 \| 0.06 \| 0.12 \| 0.11 \| 0.22 Baseline \| 0.14 \| 0.62 \| 0.20 \| 0.42 \| 0.07 \| 0.00 \| 0.23 \| 0.08 \| 0.22 Ours \| 0.18 \| 0.71 \| 0.20 \| 0.50 \| 0.20 \| 0.02 \| 0.29 \| 0.06 \| 0.27 Table 3: The comparison results of disease localization among the models using 100% unannotated images and 40% annotated images. For each disease, the best results are bolded. Figure 3: Visualization of the predicted results on both the baseline model and our method. The first column shows the original images, the second and third columns show baseline and our method. The green bounding box and red area mean the the ground truth and prediction. The graph $G_{k}^{{}^{\prime}}$ is calculated on using the graph $G^{{}^{\prime}}_{l}\in R^{\|p_{i}\|\times\|p_{j}\|}$. $\begin{split}G_{k}^{{}^{\prime}}=W^{{}^{\prime}}_{l}(G^{{}^{\prime}}_{l})\end{split}$ (9) where $W^{{}^{\prime}}_{l}$ is a fully connected layer and $\begin{split}G_{l}^{{}^{\prime}}(l,p)=&D^{{}^{\prime}}(P_{l},P_{p}),\\\ &l\in\\{1,2,...,\|p_{i}\|\\},p\in\\{1,2,...,\|p_{j}\|\\}\end{split}$ (10) $P_{l}$ is the $l$-th feature patch of $F^{{}^{\prime}}_{u}$ and $P_{p}$ is the $p$-th feature patch of $F^{{}^{\prime}}_{v}$, respectively. ### 3.6 Training Loss The overall loss function during the training is a weighted combination of four loss functions, $\displaystyle L_{all}=w_{1}L_{base}+w_{2}L_{IR}+w_{3}L_{IK}+w_{4}L_{KR}$ (11) where $\sum^{4}_{i=1}w_{i}=1$. In our experiments, we always set $w_{i}=0.25,i\in{1,2,..,4}$. ### 3.7 Training and Test Figure 4: Visualization of the generated heatmap and ground truth of our method, where the green bounding box means the ground truth. Data \| Models \| Atelectasis \| Cardiomegaly \| Effusion \| Infiltration \| Mass \| Nodule \| Pneumonia \| Pneumothorax \| Mean ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- 0.5_0.8 \| X, Wang [20] \| 0.01 \| 0.03 \| 0.02 \| 0.00 \| 0.00 \| 0.00 \| 0.01 \| 0.02 \| 0.01 Z, Li [8] \| 0.04 \| 0.52 \| 0.07 \| 0.09 \| 0.11 \| 0.01 \| 0.05 \| 0.05 \| 0.12 J, Liu [9] \| 0.18 \| 0.70 \| 0.28 \| 0.41 \| 0.27 \| 0.04 \| 0.25 \| 0.18 \| 0.29 Baseline \| 0.34 \| 1.00 \| 0.40 \| 0.68 \| 0.11 \| 0.14 \| 0.65 \| 0.00 \| 0.41 IK \| 0.22 \| 0.82 \| 0.36 \| 0.56 \| 0.32 \| 0.14 \| 0.25 \| 0.35 \| 0.38 IR \| 0.24 \| 0.82 \| 0.40 \| 0.56 \| 0.32 \| 0.07 \| 0.38 \| 0.30 \| 0.39 KR \| 0.24 \| 0.89 \| 0.32 \| 0.68 \| 0.26 \| 0.14 \| 0.21 \| 0.30 \| 0.38 (IR+IK) \| 0.20 \| 0.86 \| 0.48 \| 0.68 \| 0.32 \| 0.14 \| 0.54 \| 0.30 \| 0.44 IR+IK+KR \| 0.27 \| 0.86 \| 0.40 \| 0.56 \| 0.37 \| 0.14 \| 0.13 \| 0.30 \| 0.38 1.0_0.0 \| J, Liu [9] \| 0.08 \| 0.30 \| 0.09 \| 0.25 \| 0.19 \| 0.01 \| 0.04 \| 0.07 \| 0.13 Baseline \| 0.11 \| 0.34 \| 0.06 \| 0.32 \| 0.20 \| 0.01 \| 0.00 \| 0.06 \| 0.14 IK \| 0.10 \| 0.59 \| 0.07 \| 0.37 \| 0.20 \| 0.00 \| 0.13 \| 0.06 \| 0.19 GR \| 0.06 \| 0.61 \| 0.07 \| 0.28 \| 0.14 \| 0.00 \| 0.05 \| 0.08 \| 0.16 IK \| 0.09 \| 0.63 \| 0.06 \| 0.36 \| 0.22 \| 0.00 \| 0.09 \| 0.07 \| 0.19 IR+IK \| 0.06 \| 0.64 \| 0.08 \| 0.38 \| 0.19 \| 0.01 \| 0.08 \| 0.09 \| 0.19 IR+IK+KR \| 0.12 \| 0.51 \| 0.07 \| 0.36 \| 0.22 \| 0.03 \| 0.02 \| 0.07 \| 0.17 1.0_0.4 \| J, Liu [9] \| 0.19 \| 0.47 \| 0.20 \| 0.41 \| 0.22 \| 0.06 \| 0.12 \| 0.11 \| 0.22 Baseline \| 0.14 \| 0.62 \| 0.20 \| 0.42 \| 0.07 \| 0.00 \| 0.23 \| 0.08 \| 0.22 IK \| 0.14 \| 0.66 \| 0.09 \| 0.47 \| 0.15 \| 0.00 \| 0.30 \| 0.06 \| 0.23 GR \| 0.14 \| 0.75 \| 0.24 \| 0.42 \| 0.11 \| 0.00 \| 0.26 \| 0.12 \| 0.25 KR \| 0.13 \| 0.68 \| 0.20 \| 0.47 \| 0.19 \| 0.06 \| 0.17 \| 0.08 \| 0.25 IR+IK \| 0.13 \| 0.72 \| 0.13 \| 0.43 \| 0.20 \| 0.00 \| 0.23 \| 0.06 \| 0.24 IR+IK+KR \| 0.18 \| 0.71 \| 0.20 \| 0.50 \| 0.20 \| 0.02 \| 0.29 \| 0.06 \| 0.27 Table 4: The comparison results of disease localization among the models using three sets of data at T(IoU)=0.7, including 50% unannotated and 80% annotated images (0.5_0.8), 100% unannotated and no any annotated images (1.0_0.0), and 100% unannotated and 40% unannotated images (1.0_0.4). For each disease, the best results are bolded. Training All the models are trained on NIH chest X-ray dataset using the SGD algorithm with the Nesterov momentum. With a total of 9 epochs, the learning rate starts from 0.001 and decreases by 10 times after every 4 epochs. Additionally, the weight decay and the momentum is 0.0001 and 0.9, respectively. All the weights are initialized by pre-trained ResNet [6] models on ImageNet [2]. The mini batch size is set to 2 with the NVIDIA 1080Ti GPU. All models proposed in this paper are implemented based on PyTorch [13]. Testing We also use the threshold of 0.5 to distinguish positive grids from negative grids in the class-wise feature map as described in [8] and [9]. All test setting is same as [9], we also up-sampled the feature map before two last fully convolutional layers to gain a more accurate localization result. ## 4 Experiments ### 4.1 Dataset and Evaluation Metrics Dataset. NIH chest X-ray dataset [20] include 112,120 frontal-view X-ray images of 14 classes of diseases. There are different diseases in each image. Furthermore, the dataset contains 880 images with 984 labeled bounding boxes. We follow the terms in [8] and [9] to call 880 images as “annotated” and the remaining 111,240 images as “unannotated”. Following the setting in [9], we also resize the original 3-channel images from resolution of $1024\times 1024$ to $512\times 512$ without any data augmentation techniques. Evaluation Metrics. We follow the metrics used in [8]. The localization accuracy is calculated by the IoU (Intersection over Union) between predictions and ground truths. Since it is a coarse-grained task, our localization predictions are discrete small rectangles. The eight diseases with ground truth boxes is reported in our paper. The localization result is regarded as correct when $IoU>T(IoU)$, where T() is the threshold. ### 4.2 Comparison with the State-of-the-art In order to evaluate the effectiveness of our models for weakly supervised disease detection, we design the experiments on three sets of data and conduct a 5-fold cross-validation. In the first experiment, we use the 50% unannotated images and 80% annotated images for training, and test the models with the remaining 20% annotated images. In the second experiment, we use the 100% unannotated images and no any annotated images for training, and test the models with all annotated images. In the third experiment, we use the 100% unannotated images and 40% annotated images for training, and test the models with remaining 60% annotated images. Additionally, our experimental results are mainly compared with four methods. The first method is X, Wang [20], which proposes a carefully annotated chest X-ray dataset and a unified weakly supervised multi-label image classification and disease localization framework. The second method is Z, Li [8], which uses fully convolutional neural network to localize and classify the disease in chest X-ray images. The third method is J, Liu [9], which proposes contrastive learning of paired samples to provide more localization information for disease detection. The last method is our baseline model, which is a unified end-to-end framework that doesn’t use our approach to locate and classify the disease. In the first experiment, we compare the localization results of our model with [20], [8] and [9]. We can observe that our model outperforms existing methods in most cases, as shown in Table 1. Particularly, with the increase of T(IoU), our model has greater advantages over the reference models. For example, when T(IoU) is 0.3, the mean accuracy of our model is 0.60, and outperforms [20], [8] and [9] by 0.38, 0.11 and 0.07 respectively. However, when T(IoU) is 0.7, the mean accuracy of our model is 0.44, and outperforms [20], [8] and [9] by 0.43, 0.32 and 0.15 respectively. Overall, the experimental results shown in Table 1 demonstrate that our method is more accurate for disease localization and classification, which provides a great role for clinical practices. In the second experiment, we train our model without any annotated images comparing the first experiment. Since [8] only provides the results when T(IoU) = 0.1, in order to better show the performance of our model, we add an evaluation method of T(IoU) = 0.1. It can be seen that our model outperforms [8] and [9] in most cases, as shown in Table 2. For example, when T(IoU) is 0.1, the mean accuracy of our models is 0.63, which is 0.06 higher than [8], and 0.03 higher than [9]. Furthermore, when T(IoU) is 0.7, the mean localization result of our model is 0.19, which is 0.06 higher than [8] and 0.05 higher than [9]. Compared with the baseline model, our approach performs better in most classes except for “Atelectasis” and “Nodule”. The trend stays the same that at higher T(IoU), our approach demonstrates more advantages over baseline methods. The added unannotated training samples contribute more than the removed annotated ones in those classes, which implies that our approach can better utilize the unannotated samples. The overall results show that even without annotated data used for training, our approach can achieve decent localization results. In the third experiment, we use more annotated images comparing the second experiment. We compare the localization results of our model with [9] in same data setting. It can be seen that our model outperforms [9] in most cases, as shown in Table 3. With T(IoU) = 0.3 and 0.7, our model outperforms [9] by 0.02 and 0.05 respectively. Similar improvements are achieved comparing the second experiment. Overall, the experimental results demonstrate that our method can improve the performance of models with limited annotated images. To better demonstrate the final effect of our approach on disease localization and classification, we visualize some of typical predictions of both the baseline model and our method, as shown in Figure 3. The first column shows the original images, the second and third columns show baseline model and our method. The green bounding box and red area mean the ground truth and prediction. It can be seen that our models can predict more accurate in most cases comparing the baseline model. For example, the class “Atelectasis” and “Nodule”, the localization reslut of the baseline model is completely inconsistent with the ground truth, but the localization reslut of our method is consistent with the ground truth. It shows that using the structural information of intra-image and inter-image can improve the performance of automatic lesion detection. Additionally, we also visualize the generated heatmap and ground truth of our model, as shown in Figure 4. It can be seen that the proposed method can effectively locate and classify medical images. ### 4.3 Ablation Studies In this section, we explore the influence of different modules on our method for ablation studies. To evaluate our method more comprehensively, we build 6 models, including the model of the end to end framework (Baseline), the model with the intra-image knowledge learning (IK), the model with the inter-image relation module (IR), the model with the knowledge reasoning (KR), the model combining the inter-image relation module and the intra-image knowledge learning (IR+IK), the model combining the inter-image relation module, the intra-image knowledge learning and the knowledge reasoning module (IR+IK+KR). Table 4 shows the results of the three experiments mentioned in section 4.2 at T(IOU)=0.7. It can be seen that our method performs better in most classes except for “Atelectasis”, “Effusion” and “Mass” comparing [20], [8] and [9]. Furthermore, comparing the baseline model, it can be observed that the performance of our other models are improved in most cases, which shows that our method is effective for improving model performance. However, a model does not always maintain the advantage in the three experiments, for example, the model (IR+IK) achieves the best performance in the data (0.5_0.8), the model (IK), the model (KR) and the model (IR+IK) achieve the best performance in the data (1.0_0.0), and the model (IR+IK+KR) achieves the best performance in the data (1.0_0.4). Overall, the experimental results demonstrate that using structural relational information can improve the performance of models. For different experimental data, our models can achieve different results. It is difficult for us to determine which model is the best, but we can be sure that our method is effective, because no matter what kind of data we use, our models achieve great improvement. Particularly, the method can achieve good localization results even without any annotation images for training. ## 5 Conclusion By imitating doctor’s training and decision-making process, we propose the Cross-chest Graph (CCG) to improve the performance of automatic lesion detection under limited supervision. CCG models the intra-image relationship between different anatomical areas by leveraging the structural information to simulate the doctor’s habit of observing different areas. Meanwhile, the relationship between any pair of images is modeled by a knowledge-reasoning module to simulate the doctor’s habit of comparing multiple images. We integrate intra-image and inter-image information into a unified end-to-end framework. 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# Everything You Wanted to Know About Noninvasive Glucose Measurement and Control Prateek Jain Amit M. Joshi Saraju P. Mohanty Dept. of ECE Dept. of ECE Computer Science and Engineering MNIT, Jaipur, India MNIT, Jaipur, India University of North Texas, USA<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS> ###### Abstract Diabetes is a chronicle disease where the body of a human is irregular to dissolve the blood glucose properly. The diabetes is due to lack of insulin in human body. The continuous monitoring of blood glucose is main important aspect for health care. Most of the successful glucose monitoring devices is based on methodology of pricking of blood. However, such kind of approach may not be advisable for frequent measurement. The paper presents the extensive review of glucose measurement techniques. The paper covers various non- invasive glucose methods and its control with smart healthcare technology. To fulfill the imperatives for non-invasive blood glucose monitoring system, there is a need to configure an accurate measurement device. Noninvasive glucose-level monitoring device for clinical test overcomes the problem of frequent pricking for blood samples. There is requirement to develop the Internet-Medical-Things (IoMT) integrated Healthcare Cyber-Physical System (H-CPS) based Smart Healthcare framework for glucose measurement with purpose of continuous health monitoring. The paper also covers selective consumer products along with selected state of art glucose measurement approaches. The paper has also listed several challenges and open problems for glucose measurement. ###### Index Terms: Smart Healthcare, Internet-of-Medical-Things (IoMT), Healthcare Cyber-Physical System (H-CPS), Diabetes, Glucose measurement, Non invasive measurement, Spectroscopy and calibration ## I Introduction The glucose is considered as important source of energy for the human body. The body requires blood glucose of normal range (80 to 150 mg/dl) in order to perform the daily activities [1]. However, the higher or lower value of glucose would lead to various complication inside the body. At the same time, insulin is also crucial hormone generated inside the body from the food intake. The glucose is produced from the food digestion which enters the blood cell to supply the energy and also helps in the growth. In case, the insulin is not properly generated then blood would accumulate the high glucose concentration. Fig. 1 illustrates the closed-loop of glucose generation and consumption in human body [2]. A consistently high blood glucose concentration is possible if the generation of $\alpha$ cells is larger as compared to that of the $\beta$ cells. Because of this condition, enough insulin is not secreted in the body for glucose consumption. This condition refers to as the Diabetes Mellitus. Diabetes is termed as chronic disease which defines high blood glucose levels inside the human body. The unbalanced glycemic profile is main reason for the cause of diabetic condition. The rate of prevalence for Non Communicable Diseases (NCD)/Chronic Disease has increased with many fold from last several years. There are around 20 million death reported yearly through cardiovascular disease, for which high blood glucose is significant predisposing factors. Moreover, people with diabetes are more affected during the viral pandemic outbreaks [3, 4, 5]. Figure 1: Illustration of the closed loop form of glucose generation and consumption [2]. There has been exponential growth of diabetes patients over past few years because of obesity, unhealthy diet plan, old-age population, and inactive lifestyle. Diabetes is considered as one of the fastest growing health challenges, with the number of adults living with diabetes having more than tripled over the past 2 decades (Refer Fig. 2) [6]. The prevalence of diabetes around the world was 9.3% during 2019 with approximate 463 million people. It is expected to rise to 578 million by 2030 with 10.2% prevalence rate and the same would be 10.9% with 700 million population by 2045. It has been observed that prevalence is quite higher in urban to 10.8% whereas 7.2% in rural region. Almost half of the diabetes patients unaware about their situation due to lack of knowledge. The diabetes has indeed global outbreak which has affected presently almost 1 in 10 people around the world. It is projected that more than 0.5 billion adults would suffer from the diabetes in the next decade [7]. As per the report from International Diabetes Federation (IDF), the death from diabetes has large number than combined death from Malaria (0.6mio), HIV/AIDS (1.5mio) and tuberculosis (1.5mio) [8]. There are around 8 million new patients are being added to diabetic community every year. This has grown the demands immensely for the effective diabetic management. It is important to monitor the blood glucose over time to time for avoiding late- stage complication from diabetes. This has necessitate the design of various reliable and robust solutions for efficient diabetes management. The market of diabetes devices has also grown rapidly with significant requisite for frequent glucose measurement for better glycemic profile control. Figure 2: Global trend of Diabetes, Adopted from [6]. Diabetes is one of the major chronic disease which has long-term impact of the well-being life of a person. Diabetes Mellitus (DM) is considered as physiological dysfunctions with high blood glucose because of insufficient insulin, insulin resistance, or excess generation of glucagon [9]. It is the critical health issue of $21^{st}$ century. Type 2 Diabetes (T2DM) has shown rapid growth around the world from past few years. Any form of diabetes may lead to complications in various body parts which increase the possibility of premature death. The higher value of blood glucose known as hyperglycemia, would lead to thickening of blood vessels which could resulted in kidneys damage and loss of sight and some times even to these organs failure. Diabetes is also associated with limb amputation, peripheral vascular diseases and myocardial. Contrary, the low blood glucose defined as hypoglycemia may occur in Type 1 Diabetes Patients (T1DM) for excessive insulin dosage [10]. The most common symptoms for hypoglycemia pateints are dizziness, sweating and fatigue and in the worst case it can lead to coma and death. The diabetic patients would have several common symptoms such as thirsty, tiredness, changes in vision, consistently hungriness, unexpected weight loss and the excretion of urine within short durations [11]. If the diabetes remain untreated over the period of time, it may cause blindness, heart stroke, kidney disease, lower limb amputation and blindness. It would lead to increase the probability of death almost 50% higher in comparison of the patients without diabetes. The diabetes also brings the additional financial burden for the treatment and point of care. The diabetic patients could also result in loss of productivity at workplace and may lead to disability. There are several health issues which may also arise from diabetes like depression, digestive problem, anxiety disorders, mood disorder and eating habits change. The diabetes could be controlled with proper diet plan, through some physical exercise, insulin dosage and medicines. The early stage of diabetes is possible to control with oral medicines. The diabetes control also helps to reduce the associated risk of high blood pressure, cardiovascular and amputation. The rest of the article is organized in the following manner: Section II briefly presents different types of diabetes while making case for the need of glucose level monitoring. Section III presents overview of various types of glucose-level measurement mechanisms. Section IV provides details of available approaches for noninvasive glucose-level monitoring. Section V has discussions on various post-processing and calibration techniques for noninvasive glucose- level monitoring. Section VI briefly discusses various consumer products for noninvasive glucose level measurement. Section VII presents the approaches for glucose-level control and corresponding consumer products. Section VIII provides the Internet-of-Medical-Things (IoMT) perspectives of glucose level measurements and control in healthcare Cyber-Physical Systems (H-CPS) that makes smart healthcare possible. Section IX outlines the shortcomings and open problems of glucose-level measurements and control. Section X summarizes the learning of this comprehensive review work. ## II The Health Issue of Diabetes and Need for Glucose-Level Measurement This Section presents details of different types of diabetes, the health issues arise due to diabetes, while making case for the need of glucose level monitoring. ### II-A Types of Diabetes The diabetes occurs because of insufficient insulin with respect to glucose generated inside the body. The insulin from body is either insufficient or not any which is generated from beta cells of the pancreas. In case of diabetes, the cells of liver, muscles and fat unable to balance glucose insulin effectively. The diabetes are classified mainly in three categories: Type 1 diabetes, Type 2 diabetes and gestational diabetes (Refer Fig. 3) [12]. Figure 3: Different types of diabetes and their symptoms. For diabetes of type-1, the pancreas does not produce insulin inside the body which is resulted in a weak immune system. This results in a person who is unable to generate insulin naturally [2, 13]. In case of type 2 diabetes, the amount of insulin from pancreas is not sufficient to maintain glycemic profile of the body. Gestational diabetes usually occurs in a pregnant woman at later stage of the delivery. in the year 2020, total 2 billion adults around the globe suffers from overweight, and 300 million of them are obese. In addition, a minimum of 155 million children in the world is overweight or obese. It is projected that the prevalence of hyperglycemia is 8.0% and expected to increase to 10% by 2025 [7]. There has been concern for diabetic people specially in developing countries due to increase in Type 2 Diabetes cases rapidly at earlier age which have overweight children even before puberty. Whereas for developed countries, most of people have high blood glucose at age around 60 years. Most frequently affected are at middle aged between 35 and 64 in developed countries [6]. In 2019, 69.2 millions population in India had Type-2 diabetes. Approximately 2.35 million adults have Type-1 diabetes. In general, there are around 5% adults have been considered for Type-1 diabetic patients while the others 90-95% are of Type-2 diabetic patients. Type-1 diabetic patient must have insulin to control the blood glucose level. Type-2 diabetic patients can control their glucose level by following an optimized diet with medication and a regular physical exercise schedule. ### II-B The Health Crisis due to Diabetes The diabetes mainly occurs due to unbalanced glucose insulin level of the body where insulin is demolished and muscles and cells are not able to generate insulin properly [14, 13]. The probability of death would also increase upto 50% in comparison to non-diabetes case. The control action of the diabetes would be possible using proper precautionary measure after frequent glucose measurements. Therefore, there is a real need for smart healthcare solution which would provide instant self measurement of blood glucose with high accuracy. Figure 4: Diseases in human body due to diabetes Hyperglycemia is the major issue which has been considered by several health organizations at worldwide level [15, 16]. There are several attempts which have been used for glucose measurement [17]. There have been substitutional work using various techniques to make the device more familiar with clinicians and patients [18]. Diabetes is possible in the age group 18 to 80 years usually [19, 20]. The normal range of glucose is in the range of 70-150 mg/dL and pathophysiological would be from 40 mg/dL to 550 mg/dL [21]. One of the emerging issues is to design the glucose measurement device for continuous health care monitoring [22]. The devices for monitoring the glucose level are available for last two decades [23]. ### II-C Glucose Measurement: A Brief History The glucose meter (aka glucometer) is a portable medical device for predicting the glucose level concentration in the blood [24, 25]. It may also be a strip based dipped into any substance and determined the glucose profile. It is a prime device for blood glucose measurement by people with diabetes mellitus or hypoglycemia. With the objective of glucose monitoring device advancement, the concept of the biosensor has been proposed earlier in 1962 by Lyons and Clark from Cincinnati. Clark is known as the “father of biosensors”, and modern-day glucose sensor which is used daily by millions of diabetics. This glucose biosensor had been composed with an inner oxygen semipermeable membrane, a thin layer of GOx, an outer dialysis membrane and an oxygen electrode. Enzymes could be gravitated at an electrochemical detector to form an enzyme electrode [26]. However, the main disadvantage of first-generation glucose biosensors was that there was the requirement of high operation potential of hydrogen peroxide amperometric measurement for high selectivity. The first-generation glucose biosensors were replaced by mediated glucose biosensors (second- generation glucose sensors). The proposed biosensors till present scenario represent the advancements in terms of portability of device and precision in measurement. But, due to some environmental and measurement limitations; these biosensors were not taken for real-time diagnosis. The history of glucose measurement is shown in Fig. 5 [27]. Figure 5: History of Glucose Measurement. Figure 6: Invasive versus Noninavsive Glucose Measurement. ### II-D Glucose Measurement Technique Presently, the glucose monitoring is carried out either laboratory based technique or home based monitoring. These both approaches are invasive in nature which provides discomfort by blood pricking and it only helps to measure the glucose measurement at that point of time. It is also not very convenient for the user to take out blood samples multiple times in a day and many patients are reluctant to opt such type of solution. Therefore, significant changes of glycemic profile may go unnoticed because of unanticipated side effects and low compliance from the patients. This could impact on improper insulin dosage and unknown food ingredient. However, they are reliable solution due to their good sensitivity and higher accuracy for glucose measurement [28, 29]. The novel approach for glucose measurement has been explored from past several years which is based on the principle of physical detection than conventional chemical based principle. Such non-invasive based method does not require the blood sample but uses the interstitial fluid (ISF) for glucose molecule detection. There are several attempts in the same direction for glucose measurement through sweat, saliva, tears and skin surface [30]. However, the main challenge is to have precise measurement, good sensitivity and reliability from such measurement. Such approach could be suitable for Continuous Glucose Measurement (CGM) and self monitoring purpose. Such CGM techniques would provide the frequent measurement in a day which would helpful for better glucose control and also for the necessary preventive actions for hyperglycemia and hypoglycemia patients. Such kind of techniques would also support for the dietician and healthcare provider to prepare proper diet plan according to glucose fluctuation for the patient. ### II-E The Need for Continuous Glucose Measurement (CGM) The measurement of glucose could be done through non-invasive, semi (or minimal) invasive and invasive approach. The frequent measurement may not be possible using invasive method which can cause trauma. The semi-invasive and non-invasive could be useful for Continuous Glucose Measurement (CGM) without any pricking of the blood. However, the non-invasive glucose measurement is most suitable technique which helps to measure the blood glucose painlessly [31]. CGM assist to have proper blood glucose level analysis at each prandial mode. It helps to measure glucose insulin level after insulin secretion,hysical exercise or subsequent to medication. The frequent glucose reading also helpful to endocrinologist for providing the proper prescription. It mainly helps for type 1 diabetic patients to take care of their insulin dosage over the period of time. The proper diet management could be possible with help of recurrent glucose monitoring and flow diagram of CGM is shown as Fig. 7 [28, 29]. The CGM is useful for the patients for frequent glucose measurement over the period of time. This would helpful to identify the average blood glucose value for the last 90 days, by which glycated haemoglobin (HbA1c) can be determined. Figure 7: The objectives of continuous glucose monitoring. ## III Approaches for Glucose-Level Measurement: A Broad Overview This Section discusses an overview of various types of glucose-level measurement mechanisms. In the past, many works has done for the glucose measurement. They can be invasive, non-invasive, or minimally invasive. A lot of works has been completed based on the non-invasive technique. They are technically based on optical and non-optical methods. Some of the optical techniques used methods based on Raman Spectroscopy, NIR spectroscopy, and PPG method. A taxonomy of the different methods is provided in Fig. 8 [25, 28, 29, 32]. Figure 8: An overview of the Glucose Measurement Options [32, 28, 29, 25]. ### III-A Invasive Methods Many commercial continuous blood glucose measurement devices use cost- effective electrochemical sensors [33]. They are available to respond quickly for glucose detection in blood [34]. Lancets (for pricking the blood) is used at the primary stage for blood glucose monitoring for various commercial devices available in the market [35]. The frequent measurement through the process is so much panic due to picking the blood sample from the fingertip more than 3-4 times in a day for frequent monitoring[36]. The low invasive biosensor for glucose monitoring has been developed with glucose oxidase that require around 1mm penetration inside the skin for measurement [37]. The technique of photometric was attempted to detect glucose with help of small blood volumes [38]. ### III-B Minimally Invasive Methods The minimally invasive method using prototype sensor was developed to have frequent monitoring of glucose tissue [39]. The sensor is wearable and is implanted on membrane which contains the immobilized glucose oxidase. The glucose monitoring through implantable devices were developed [40]. The semi or minimal invasive method using biosensors designed for diabetes patient [41]. The wearable micro system explored for frequent measurement of glucose [42]. Similarly, there was an attempt of continuous glucose monitoring with help of microfabricated biosensor through transponder chip [43]. The signal coming out of transponder chip was used for the calibration for semi invasive approach of Dexcom sensor [44]. The diabetes control explored by glucose sensor with artificial pancreas system [45]. The minimal invasive approaches have limitations mainly accuracy and may have shorter life span for monitoring. This is a wearable microsystem for the continuous monitoring of the blood glucose. It’s a minimally invasive method for the glucose monitoring. The main idea behind this is that it uses micro-actuator which consists the shape memory alloy (SMA) for the extraction of the blood sample from human skin [46]. An upgraded version of SMA is used for the implementation of PCB. Because of it’s feasibility and performance, it can be considered as the first wearable device for the glucose monitoring but it is large in size which makes it inconvenient. ### III-C Non-invasive Methods Non-invasive measurement would mitigate all the previous issues and would provide painless and accurate solutions [47, 48]. The non-invasive glucose measurement solution for smart healthcare had developed through portable measurement [31]. A lot of approaches have been introduced for glucose measurement [49]. The non-invasive measurement are more convenient for continuous glucose measurement in comparison to invasive method and semi invasive [47], [48]. The glucose measurement with help of optical method has observed more reliable and precise in the literature [50]. The popular optical methods include non-invasive measurement such as Raman spectroscopy, near infer-red spectroscopy,polarimetric,scattering spectroscopy [51], photoacoustic spectroscopy [52] etc. For the development of a non-invasive measurement device, it is considered by the researcher that the device would be much convenient for the user’s perspective [53, 54]. Calibration of the blood glucose to interstitial glucose dynamics have been considered for the accuracy of continuous glucose monitoring system [55, 56]. Several calibration algorithms have been developed and implemented for portable setup [57]. There has been several concious efforts towards the development of the self- monitoring system [58]. Figure 9: NIR Spectroscopy Mechanism of Serum Glucose Measurement. ### III-D Invasive Versus Non-invasive Glucose Measurements: The Trade-Offs Recent glucose measurement methods for the ever-increasing the diabetic patients over the world are invasive, time-consuming, painful and a bunch of the disposable items which constantly burden for the household budget. The non-invasive glucose measurement technique overcomes such limitations, for which this has become significantly researched era. Although, there is tradeoff between these two methods which is represented in Fig. 10. Figure 10: Representation of Tradeoffs between Invasive and Non-invasive Glucose Measurement. ### III-E Capillary Glucose versus Serum Glucose for Noninvasive Measurement The serum glucose value is precise which is always close to actual blood glucose measurement with compare to capillary glucose level. Traditional approaches able to measure capillary glucose instantly but the serum glucose measurement identification is difficult. It is observed that the glucose level of capillary is always higher than serum glucose. The accurate measurement of blood glucose would help for appropriate control actions. Therefore, it is really important to measure the serum glucose than the capillary glucose which is more reliable for medication. Capillary blood glucose measurement has been used widely than serum glucose estimation for medication purpose. The serum glucose is not possible for continuous glucose measurement or frequent measurement for diabetes. The blood glucose is controlled in much better way if one can measure serum glucose at regular interval. Laboratory analysis of glycosylated haemoglobin (HbA1c) which provides 6-8 weeks blood glucose measurement is also being done through the serum blood only. For the non- invasive measurement point of view, serum and capillary glucose are being measured through the optical spectroscopy. The mechanism of blood glucose measurement is based on received IR light after absorptions and scattering from glucose molecules which flow in blood vessels. The methodology is quite similar for both types of glucose measurement except the post-processing computation models which are necessary for blood glucose estimation. ### III-F Non-invasive Method for Glucose Level Estimation by Saliva As the most convenient method to estimate glucose level is via saliva [59] and is used for children and adults. This saliva has specific type of parts which can be defined as: (1) gland-specific saliva and (2) whole saliva. The collection of the Gland-specific saliva is done by individual glands like parotid, Sub mandibular, sublingual, and minor salivary glands. This diagnosis is done by the history of the patient in terms of associated risk factors, family history, age, sex, duration of diabetes,and any associated illness. Other Glucose measuring methods consist of measurement using photo-metric glucometers requiring very small sample volumes [60]. The basic approach is based on the reaction of the chemical test strip that reacts with sample. Measurement is done by capturing the reflections of the test area and then glucose level is estimated. It requires validation in large number of patients. ## IV Approaches for Noninvasive Glucose-Level Measurement This Section presents detailed discussions of various available approaches for noninvasive glucose-level monitoring. There have been several efforts for noninvasive glucose measurement using optical techniques [27, 29, 61, 62]. These techniques are mainly based on various spectroscopy based methods. For the development of a non-invasive measurement device, it is considered by the researcher that the device would be much convenient for the users perspective. Fig. 11 presents summary of various types of noninvasive glucose measurement techniques, whereas their comparative perspectives are presented in Fig. 12. A qualitative comparative perspective of various noninvasive methods is summarized in Table IV. Figure 11: Various spectroscopy techniques for noninvasive glucose measurement. Figure 12: Comparative Perspective of Various popular spectroscopy techniques for noninvasive glucose measurement. Table I: Qualitative comparison of various noninvasive glucose-level monitoring methods. Technique \| Advantages \| Disadvantages ---\|---\|--- Near Infra-Red (NIR) \| • The signal intensity is directly proportional to glucose molecule • The glucose detection concept would work with other interfacing substance such as plastic or glass \| • The glucose signal weak comparatively so complex machine learning model is required for interpretation • High scattering level Mid Infra-Red (MIR) \| • The glucose molecule absorption stronger • Low scattering \| • The light has limited penetration with tissue • Noise is present in the signal so water and other non-glucose metabolites would be detected. Far Infra-Red (FIR)/Thermal emission spectroscopy \| • Frequent Calibration is not required • Least sensitive towards scattering \| • The radiation intensity depends on temprature and substance thickness • Strong absorption with water so it is difficult to have precise glucose measurement Raman Spectroscopy \| • Less sensitivity towards temperature and water • High specificity \| • Requirement of the laser radiation source hence it can dangerous cell for CGM • Susceptible towards noise interference so low SNR Photo acoustic \| • Simple and compact sensor design • Optical radiation will not harmful for the tissue \| • Signal is vulnerable towards acoustic noise, temperature,motion etc. • It carries some noise from some non-glucose blood components Polarimetry \| • The laser intensity variation will not change much the glucose prediction \| • Requirement external laser source and requires proper alignment with eye • sensitive for the change in PH and temperature Reverse Iontophoresis \| • Based on simple enzyme based electrode system • Highly accurate as it measure glucose from interstitial fluid \| • Difficult to have proper calibration • Not so user-friendly approach due to passing of the current through skin Fluorescence \| • Highly sensitive for glucose molecule detection due to immune for light scattering • Good sensitivity because of distinctive optical properties \| • Very much sensitive for local pH and/or oxygen, • Suffers from foreign body reaction Bio impedance spectroscopy \| • Comparatively less extensive • Easy for CGM \| • Prone towards sweating, motion and temperature • Require large calibration period Millimetre and Microwave sensing \| • Deep penetration depth for precise glucose measurement • No risk for ionization \| • Poor selectivity • Very much sensitive for physiological parameters such as sweating, breathing and cardiac activity Optical Coherence Tomography \| • High resolution and good SNR • Not vulnerable for blood pressure and cardiac activity \| • Glucose value may change as per skin and motion • Suffers from tissue inhomogeneity Surface Plasma Resonance \| • Small glucose molecule can be detected due to high sensitivity \| • Long calibration process and size is bulky • Glucose value changes with variation in temperature,sweat and motion Time of flight and THz Time domain Spectroscopy \| • strong absorption and dispersion for glucose molecule \| • Lesser depth resolution and longer time for measurement Metabolic Heat Conformation \| • Uses the concept of well-known various physiological parameters for glucose prediction \| • Sensitive towards variation in temperature and sweat Electromagnetic sensing \| • low-cost and can be easily miniaturized • No risk of ionization \| • Lack of selectivity due to dielectric constant is mainly affected with other blood components • More sensitive for the slight change of temperature Ultrasound Technology \| • Well established technology with not much harm to tissue cell • Long penetration below the skin or tissue \| • Limited accuracy with ultrasound only hence mostly used with NIR as multi-model • costly technology for measurement and not useful for CGM Sonophoresis \| • Favourable technology as there is no side-effect to skin • Based on well known enzymatic method \| • Error prone due to environmental parameters ### IV-A Near-Infrared (NIR) Spectroscopy It is well known as Infrared spectroscopy (IR spectroscopy) or vibration spectroscopy where radiation of infrared type are incident on the matter [63, 64]. Various types of IR spectroscopy is shown in Fig. 13. In general, IR spectroscopy includes reflection, scattering and absorption spectroscopy [65]. The wave from IR absorption cause the molecular vibration and generate the spectrum band with wavelength number in $cm^{-1}$ [66]. Figure 13: Classification of vibrational spectroscopy [67]. In this case, the light in the wavelength range of 700nm to 2500nm for Near- infrared region is applied at the object (may be finger or ear lobe) [68]. The light may interact with blood components and it may scattered, absorbed and reflected [69, 70]. The intensity of received light varies as per glucose concentration as per Beer-Lambert law [71, 72] The receiver would help to measure the presence glucose molecule from the blood vessel [73]. Figure 14: Block diagram representation of IR spectroscopy. #### IV-A1 Long-Wave versus Short-Wave NIR Spectroscopy The optical detection is useful approach to have precise glucose measurement. FIR (Far infra-red) based optical technique help to get the resonance between OH and CH for first overtone. However, long wave NIR has good performance in vitro testing. In similar way, the fiber-optic sensor is used along with laser based mid-infrared spectroscopy for vitro based glucose measurement. The continuous glucose measurement has been achieved with multivariate calibration model for error analysis [74]. The FIR approach has limitation of shallow penetration in comparison with short wave NIR. The short NIR would help to detect the glucose molecule more accurately [75]. The concept of NIR spectroscopy for glucose detection is shown in Fig. 15. The specific wavelength of NIR spectroscopy has already been applied earlier for precise glucose measurement using non-invasive measurement [76]. Some specific wavelength such as 940 nm has been considered for the detection of glucose [77]. The vibration of CH molecule has been observed at 920 nm with NIR spectroscopy [76]. In some other works, the glucose absorption has been validated for the range 1300 to 1350 nm and stretching of glucose has been identified in NIR region [78, 79]. The presence of glucose component has been measured at 1300 nm in the work [80]. Figure 15: Penetration depth various Infrared Signals in Human Skin [25, 32]. #### IV-A2 NIR Spectroscopy Based Methods A method to estimate the non-invasive blood glucose with NIR spectroscopy using PPG has been proposed in literature [81]. This method is performed using NIR LED and photo detector with an optode pair. At NIR wavelengths(935nm, 950nm, 1070nm), PPG signal is obtained by implementation of analog front end system. The glucose levels has been estimated using Artificial Neural Network (ANN) running in FPGA. A microcontroller is used, for the painless and autonomous blood extraction [82]. The ideal system Blood Glucose Measurement (BGM) in which the microcontroller is used to display the blood glucose and for the transmission of blood glucose. A remote device is used for the tracking of the insulin pump which is needed for diabetes management. This type of measurement [83] method uses change in the pressure of the sensitive body part, because it generates the sound waves. The response of the photo acoustic signals will be stronger when glucose concentration is higher. In order to improve SNR and for the reduction of noise to transfer the signal to the computer for further processing,the signal is then amplified. Feature extraction and glucose estimation is estimated by photo acoustic amplitude. In order to gather the photo acoustic signals, two pulsed laser diodes and piezoelectric transducer is used. Utilization of the LASER makes the setup costly and bulky. #### IV-A3 Non-invasive Blood Glucose Measurement Device (iGLU) In this approach, “Intelligent Glucose Meter (iGLU)” [84] has been utilized for the acquisition of data. This device works on a combination of NIR spectroscopy and machine learning. This device has been implemented using three channels. It uses an Internet-of-Medical-Things platform for storage and remote monitoring of data. In the proposed device, an NIR Spectroscopy is used with multiple short wavelengths [85]. It uses three channels for data collection. Each channel has its own emitter and detector for optical detection. Then the data collection processed by a 16 bit ADC with the sampling rate of 128 samples/second. Regression techniques is used to calibrate and validate the data and analyse the optimized model. The data that is stored on cloud can be used and monitored by the patients and the doctors. Treatment can be given based on the stored data values. This is a low cost device with more than 90 % accuracy but it does not give real time results. #### IV-A4 Why NIR is Preferred Over other Noninvasive Approaches? Glucose measurement has been done using various non-invasive approaches such as impedance spectroscopy, NIR light spectroscopy, PPG signal analysis and so on. But, apart from optical detection, other techniques have not be able to provide the precise measurement. PPG is one of the promising alternative but the PPG signal varies according to blood concentration [86, 87]. It may not be useful to have precise prediction of the blood glucose. The saliva and sweat properties vary from one person to another person. Therefore, it could not be reliable glucose measurement method. The other spectroscopy have been also applied for glucose measurement. However, they are not able to provide portable, cost effective and accurate prediction of body glucose.The glucose measurement using optical detection using long NIR wave which is not capable to detect the glucose molecules beneath the skin as it has shallow penetration [75]. Therefore, small NIR wave has been considered as potential solution for real-time glucose detection [77, 88]. ### IV-B Mid Infrared (MIR) Spectroscopy The bending and stretching of glucose molecules would be observed very well with Mid Infrared (MIR) spectroscopy [89]. The depth of skin penetration is very less because it tends to have larger absorption of water. This technique helps to have ISF glucose value in vivo measurement. There are some attempts for precise glucose measurement through saliva and palm samples. ### IV-C Blood Glucose Level Measurement using PPG The change of blood volume with absorption of the light from tissue has been detected with PPG signal [87]. The change of the blood volume has been measured using pressure pulse with help of light detector [86]. The change in volume of blood would result as the change of light intensity hence it may not be occur due to glucose molecule. This may result as inaccurate glucose measurement. The difference of NIR against PPG has been shown in Fig 16. The intelligent glucose measurement device iGLU is mainly based on principle of NIR spectroscopy which helps to have precise glucose measurement. There have been several work for glucose detection based on PPG signal [90]. The data from patient body has logged to estimate the presence of glucose using PPG. Subsequently, various machine leaning models have been used for prediction of body glucose value [91]. The different parameters from total 70 subjects of healthy and diabetes have been considered for the prediction using Auto- Regressive Moving Average (ARMA) models [92]. There have been also several other smart solutions for glucose estimation using PPG signal with intelligent algorithms [93, 94, 95]. One of the optical based techniques is Photo-plethysmography (PPG) which is used in advanced health care. It is non-invasive glucose measurement technique. In NIR spectrum a sensor similar to a pulse oximeter is used to record the PPG signal [87]. Photo transmitter and receiver is used to build the sensor which will operate in near infrared region at 920nm. At wavelength 920nm, by measuring changes in the absorption of light, a PPG signal can be obtained. The veins in the finger grow and contract with every heartbeat. A method of measuring blood glucose using pulse oximeter and transmission of the PPG glucose monitoring system is available [90]. As the glucose concentration increases, there is decrease in the light absorbance in the blood. The obtained signal is in the form of photo current, and for the filtering of this signal is then changed into the measurable voltage values. For the processing of filtered signal, lab view is used to estimate the blood glucose level. A system using machine learning techniques and PPG system for the measurement of blood glucose level non-invasively has been prototyped [86]. In this model, a PPG sensor, an activity detector, and a signal processing module is used to extract the features of PPG waveform. It finds the shape of the PPG waveform and the blood pressure glucose levels, the functional relationship between these two can be obtained then. In PPG, the change in light intensity will be varied according to changes in blood volume. PPG signal analysis is not based on the principle of glucose molecule detection. Hence, the system has limited accuracy [25, 32]. Fig. 16 illustrates the differences. Figure 16: PPG Versus NIR for Non-invasive Glucose Measurement [25, 32]. ### IV-D Impedance Spectroscopy Impedance spectroscopy (IMPS) refers to the dielectric spectroscopy [96]. The steps of impedance spectroscopy (IMPS) is shown in Fig. 17. This technique finds the dielectric properties of skin [97]. The current is directed through the skin [98]. Due to directed small current at multiple wavelengths, the impedance range is obtained [99]. The range lies between 100 Hz to 100 MHz [100, 101]. Change in glucose concentration will reflect the change in sodium ions and potassium ions concentration [102]. So, the cell membrane potential difference will be changed [103]. Thus, the dielectric value will be changed which predicts the glucose value of human body [104]. Figure 17: The Steps of Impedance Spectroscopy (IMPS). An enzyme sensor in a flow cell has been explored for glucose measurement in saliva [105]. Polypyrrole (PPy) supported with copper (Cu) nanoparticles on alkali anodized steel (AS) electrode for glucose detection in human saliva is available in [106]. The high precision level cannot be possible through these methods as sweat and saliva properties vary according to person. Hence, this approach is not suitable for glucose measurement in smart healthcare. ### IV-E Raman Spectroscopy Due to the interaction of light with a glucose molecule, the polarization of the detected molecule will change [107]. In this technique, oscillation and rotation of molecules of the solution are possible through the incident of LASER light [108]. The vibration of the molecule affects the emission of scattered light [109]. Due to this principle, blood glucose concentration can be predicted as [110]. This technique provides more accuracy with compared to infra-red spectroscopy technique [111]. There has been several research based on Raman spectroscopy to have precise glucose measurement. The validation has been also carried out on using vivo testing. Fig. 18 presents basic framework of Raman spectroscopy, whereas Fig. 19 presents its usage for noninvasive glucose measurement. Figure 18: Building blocks of Raman spectroscopy. Figure 19: Noninvasive glucose measurement using Raman spectroscopy. ### IV-F Time of Flight and THz Domain The blood glucose estimation is adopted though Time of Flight (TOF) measurements for vitro testing [112]. The short pulse of laser light is inserted in the sample for photon migration measurement. This photon will experience scattering and absorption phenomenon while traveling from the sample. The optical analysis of the photons would be useful for precise glucose measurement. ### IV-G Photo Acoustic Spectroscopy Photoacoustic spectroscopy refers to the photoacoustic effect for the generation of the acoustic pressure wave from an object (refer Fig. 20) [113]. In this spectroscopy technique, the absorption of modulated optical input provides the estimation of blood glucose detection [114]. High intense optical light is absorbed by an object according to its optical conditions [115]. This process provided excitations of particular molecules according to its resonant frequency [116]. The absorbed light is considered as heat which provides rising in localized temperature and thermal expansion of the sample [117]. The expansion in volume generates pressure in acoustic form [118]. The generated photoacoustic wave can be used to predict the glucose concentration through specific excited wavelengths which are resonant for the vibration of glucose molecules [119]. At the specific resonance frequency, the glucose molecule changes own characteristic. This change is in the acoustic waveform [120]. In previous work, 905 nm wavelength optical light is used for excitation [121, 122]. Figure 20: Photo acoustic spectroscopy. ### IV-H Capacitance Spectroscopy In the capacitance spectroscopy technique, inductor stray capacitance varies according to body capacitance (Fig. 21) [123]. The body capacitance is used to estimate body glucose concentration [124]. Flexible inductor based sensor follows the coupling capacitance principle for body glucose detection. In this technique, there is not any interaction between the inductive sensor and body skin through the current [125]. This is the advantage of the impedance spectroscopy technique. The stray capacitance of the inductive sensor will vary according to body glucose. In this technique, the effect of fat and muscles will be negligible with respect to body glucose [126]. Figure 21: The typical steps of capacitance spectroscopy. ### IV-I Surface Plasmon Resonance (SPR) The Surface Plasmon Resonance (SPR) utilizes electron oscillation approach at dielectric and metal interface for glucose sensing [127]. It detects mainly the change in refractive index before as well as after the analytes interaction. The optical fiber based SPR has been used for point of care measurement for glucose due to its portability. ### IV-J Radio Frequency (RF) Technique and Microwave Sensing In the RF technique, the variation in the s-parameters response reflects the change in blood glucose [128, 129]. Fig. 22 shows typical steps of this technique. The response is determined through the antenna or resonator [130, 131]. They follow the changes in dielectric constant value through the transmission [132]. The change in dielectric constant can be found as the change in resonance frequency spectrum through the antenna or resonator [133, 134]. The dielectric of blood varies according to blood glucose concentration. The human finger is an appropriate measurement object but there are many factors that play a cardinal and dominant role in the accuracy of measurement and repeatability. These are; the skin thickness, fingerprints, the applied pressure by the fingertip during measurement and positioning of a finger on the sensor [135]. Figure 22: Glucose measurement using RF sensing technique. ### IV-K Ocular Spectroscopy In the Ocular Spectroscopy technique, glucose concentration is measured through the tears. A specific lens is used to predict the body glucose concentration [136]. A hydrogel wafer is deposited to the lens. This wafer is prepared by boronic acid with 7 $\mu m$ thickness. The wafer is deposited on lens and then optical rays are inserted on the lens. Then reflected light will change its wavelength. Change in wavelength will refer to a change in glucose concentration in tears. ### IV-L Iontophoresis In the Iontophoresis or Ionization technique, a small electric current passes through the skin diffusively. Three electrodes are used for the same [137]. A small potential is applied through the electrodes to the different behaviour electrodes. During this process, glucose is transferred towards the cathode. The working electrode can have the bio-sensing function by the generation of current during applied potential through electrodes. This biosensor determines passively body glucose. The measurement is possible through wrist frequently [138]. ### IV-M Optical Coherence Tomography The Optical Coherence Tomography technique is based on the principle followed by reflectance spectroscopy. In this technique, low coherent light is excited through the sample (sample is placed in an interferometer). In an interferometer, a moving mirror is placed in reference arc. A photodetector is placed on another side and it detects the interferometric signal. This signal contains backscattered and reflected light. Due to this process, we could get high-quality 2-D images. The glucose concentration increases with the increment of the refractive index in interstitial fluids. Change in the refractive index indicates the change in the scattering coefficient [107]. So, the scattering coefficient relates to glucose concentration indirectly. ### IV-N Polarimetry The Polarimetry technique is commonly used in a clinical laboratory with more accuracy. The optical linear polarization-based technique is used for glucose monitoring [139]. This technique is usually based on the rotation of vector due to thickness, temperature and concentration of blood glucose. Due to the process of prediction of glucose, the polarized light is transmitted through the medium containing glucose molecule. Due to high scattering through the skin, the depolarization of beam is possible. To overcome this drawback, a polarimetric test has been done through the eye. The light passes through the cornea. This technique is totally unaffected due to rotation of temperature and pH value of blood [140]. Figure 23: Non-invasive glucose measurement using Polarimetry. ### IV-O Thermal Emission Spectroscopy The Thermal Emission Spectroscopy based technique is based on the naturally generated IR wave from the body. The emitted IR waves will vary according to body glucose concentration. The usual mid-IR emission from tympanic membrane of human body is modulated with tissue emitting. The selectivity of this technique is same as the absorption spectroscopy. Due to this technique, glucose can be determined through the skin, fingers and earlobe. This technique is highly precise and accurate for glucose measurement [141]. It could provide the useful solution which is precise and acceptable at clinical with measurement of thermal emission from tympanic membrane. ### IV-P Ultrasound The Ultrasound method is based on low frequency components to extract the molecules from skin similar as reverse iontophoresis method [142]. It is also alike sonophoresis and has larger skin permeability than reverse iontophoresis. Few or several tens of minutes of ultrasound exposure are required to pull glucose outward through the skin. There are few attempts for such technology and there is not any commercial device with such type of technology. ### IV-Q Metabolic Heat Conformation (MHC) The Metabolic Heat Conformation (MHC) method helps to measure the glucose value with metabolic heat and oxygen level along various physiological parameters considerations [143]. The mathematical model for metabolic energy conservation has been modified by several physiological parameters consideration such as pulse rate, oxyhemoglobin saturation, heat metabolic rate and the blood flow volume. This method has shown good reproducibility and decent accuracy in humans. ### IV-R Fluorescence The Fluorescence technique is based on the excitation of blood vessels by UV rays at particular specified frequency ranges [144]. This is followed through the detection of fluorescence at a specified wavelength. The sensing of glucose using fluorescence through tear has been done by the diffraction of visible light. At 380 nm, an ultraviolet LASER was taken for excitation through the glucose solution medium. Fluorescence was estimated which is directly related to glucose concentration. In this technique, the signal is not affected by variation in light intensity through the environment. ### IV-S Kromoscopy The Kromoscopy technique uses the response from various spectroscopic of NIR light with four different detectors over different wavelength [145]. It employs the multi-channel approach with overlapped band-pass series filters to determine the glucose molecule. In this method, the radiation of IR are imparted on the sample and this will be divided among four detectors with band-pass filter. Each detector will detect the light of the similar structure of the tissue. Subsequently, the complex vector analysis has been utilised to measure the glucose concentration. ### IV-T Electromagnetic Sensing In the Electromagnetic Sensing method, the variations in blood sample conductivity is observed by change in blood glucose concentration [146]. The alternation of electric field would be measured by electromagnetic sensor whenever there will be change in blood glucose concentration. This method utilizes the dielectric parameter of the blood samples. The frequency range for electromagnetic sensing is in the range of 2.4 to 2.9 MHz. The glucose molecule has maximum sensitivity at particular optimal frequency for given temperature of the medium. ### IV-U Bioimpedance Spectroscopy and Dielectric Spectroscopy It is useful to measure the variation of the blood glucose with help of conductivity and permittivity from red blood cells membrane [147]. The spectrum of bioimpedance spectrum is measured from 0.1 to 100 MHz frequency range. It help to find the resistance with passing through electric current which is flowing from human biological tissue. The change of plasma glucose would allow the changes in potassium and sodium to have the change in conductivity of the membrane of the red blood cell. The multisensor approach is usually incorporated with this spectroscopy in order to measure sweat, moisture, movement and temperature for precise glucose measurement. ### IV-V Reverse Ionospheresis The small DC current is passed from anode to cathode on the skin surface to have small interstitial fluid (ISF). Iontophoresis is employed for ionized molecules penetration at skin surface by such low current [148]. The electric potential is passed from anode and cathode to electroosmotic flow across the skin. This would allow to extract the molecules through skin whereas the the molecules of glucose are moved towards the cathode. The enzyme method helps to sense the concentration of glucose molecules through oxidation process. the method has very widely accepted and has good potential to measure accurate glucose value. ### IV-W Sonophoresis The Sonophoresis technique is based on the cutaneous permittivity of the interstitial fluid (ISF) [149]. It also uses enzyme method for glucose measurement. The low frequency ultrasound wave has been applied in order to have glucose molcules at the skin surface. The cutaneous permittivity of the ISF is increased to enable glucose at the epidermis surface. The contraction and expansion occurs in stratum corneum that subsequently opens the ISF pathway. There has been some attempts with this method for glucose detection but it has been observed that it could be helpful in drug delivery in stead of glucose measurement. ### IV-X Occlusion Spectroscopy The Occlusion spectroscopy based methods depend on the concept of light scattering which is of inverse proportion of glucose concentration [150]. The flow is ceased for few seconds by applying pressure with pneumatic cuff. The volume of blood would change due to pulse generated from the pressure excursion. The light is transmitted through the sample and the variation of the intensity of in a received light defines the glucose concentration. The momentary blood flow cessation helps to get higher SNR value of the received signal. Hence, the sensitivity for glucose detection would be increases with good robustness for accurate glucose measurement. ### IV-Y Skin Suction Bluster Technique The Skin Suction Bluster technique uses the concept of blister generation through vacuum suction over limited skin area [151]. The glucose measurement is performed on fluid which is collected from the blister. It has lower glucose molecules than plasma but it is well enough to have the glucose measurement. This method has low risk of infection, painless and well- tolerated. It is actually useful to measure HbA1c value which represents three month average glucose value. ### IV-Z Multimodal approach based measurement A two modal spectroscopy combining IMPS and mNIR spectroscopy is explored for high-level reproducibility of non-invasive blood glucose measurement [152]. These two techniques are combined to overcome the limitation of individual employed technique [153]. Impedance spectroscopy based circuit measures the dielectric constant value of skin or tissue through RLC resonant frequency and impedance to predict glucose level [154]. To improve the accuracy of NIR spectroscopy, mNIR spectroscopy technique is used. In this technique, three wavelengths 850 nm, 950 nm and 1300 nm are used [155]. For precise and accurate measurement, IMPS and mNIR are joined by an ANN (Artificial Neural Network) through DSP processor [80]. Therefore multimodel approaches have been explored for precise glucose measurement in the literature [joo2020vivo, feng2019multi]. Figure 24: Multimodal IC based non invasive glucose measurement. Table II: Approaches Comparison with Noninvasive Works [25, 32]. Works \| Spectroscopy \| Spectra \| Specific \| Measurement \| Linearity ---\|---\|---\|---\|---\|--- \| technique \| \| wavelength \| range \| (%) Singh, et al. [160] \| Optical \| - \| - \| 32-516 mg/dl \| 80 Song, et al. [80] \| Impedance and Reflectance \| NIR \| 850-1300 nm \| 80-180 mg/dl \| - Pai, et al. [161] \| Photoacoustic \| NIR \| 905 nm \| upto 500 mg/dl \| - Dai, et al. [162] \| Bioimpedance \| - \| - \| - \| - Beach, et al. [163] \| Biosensing \| - \| - \| - \| - Ali, et al. [88] \| Transmittance and Refraction \| NIR \| 650 nm \| upto 450 mg/dl \| - Haxha, et al. [77] \| Transmission \| NIR \| 940 nm \| 70-120 mg/dl \| 96 Jain, et al. [164] \| Absorption and Reflectance \| NIR \| 940 nm \| 80-350 mg/dl \| 90 Jain, et al. (iGLU 1.0) [32, 28] \| Absorption and Reflectance \| NIR \| 940 and 1300 nm \| 80-420 mg/dl \| 95 Jain et al. (iGLU 2.0) [29, 25] \| Absorption and Reflectance \| NIR \| 940 and 1300 nm \| 80-420 mg/dl \| 97 Table III: Statistical and Parametrical Comparison with Noninvasive Works [25, 32]. Works \| R \| MARD \| AvgE \| MAD \| RMSE \| Samples \| Used \| Measurement \| Device ---\|---\|---\|---\|---\|---\|---\|---\|---\|--- \| value \| (%) \| (%) \| (mg/dl) \| (mg/dl) \| (100%) \| model \| sample \| cost Singh, et al. [160] \| 0.80 \| - \| - \| - \| - \| A&B \| Human \| Saliva \| Cheaper Song, et al. [80] \| - \| 8.3 \| 19 \| - \| - \| A&B \| Human \| Blood \| Cheaper Pai, et al. [161] \| - \| 7.01 \| - \| 5.23 \| 7.64 \| A&B \| in-vitro \| Blood \| Costly Dai, et al. [162] \| - \| 5.99 \| 5.58 \| - \| - \| - \| in-vivo \| Blood \| Cheaper Beach, et al. [163] \| - \| - \| 7.33 \| - \| - \| - \| in-vitro \| Solution \| - Ali, et al. [88] \| - \| 8.0 \| - \| - \| - \| A&B \| Human \| Blood \| Cheaper Haxha, et al. [77] \| 0.96 \| - \| - \| - \| 33.49 \| A&B \| Human \| Blood \| Cheaper Jain, et al. [164] \| 0.90 \| 5.20 \| 5.14 \| 5.82 \| 7.5 \| A&B \| Human \| Blood \| Cheaper Jain, et al. (iGLU 1.0) [32] \| 0.95 \| 6.65 \| 7.30 \| 12.67 \| 21.95 \| A&B \| Human \| Blood \| Cheaper Jain et al. (iGLU 2.0) [29] \| 0.97 \| 4.86 \| 4.88 \| 9.42 \| 13.57 \| Zone A \| Human \| Serum \| Cheaper ## V Post-processing and calibration techniques for non-invasive Glucose- Level measurement This Section presents various post-processing and calibration techniques which are deployed in various systems or frameworks for noninvasive glucose-level monitoring. ### V-A Post processing and calibration techniques Various calibration processes have been applied for a high level of accuracy and noise reduction from received signal. These post-processing techniques are used to design the model for errorless continuous monitoring [165, 166]. #### V-A1 Noise Minimization and Signal Conditioning The coherent averaging technique has been adapted to minimize the variance of random noise [167]. The impact of noise is minimized with averaging of N number of individual samples coming from the continuous frames [168]. Frames in the maximum count have been chosen for averaging to have SNR improvement [169]. This proposed coherent averaging has been used frequently through MATLAB and coherent averaged signal acquired. Golay code has been proposed as calibration of measured data. The filtering or cancellation of unusual measured data has been achieved through the implementation of Golay code [170, 171, 172]. #### V-A2 Computation Models for glucose Estimation The regression model of regularized least square is proposed by several researchers for measurement [173]. The estimated value is computed from photoacoustic signals. These photoacoustic signals are used to calibrate for estimation of glucose concentration [174]. This can be possible through multi- variable linear regression model [175]. With the objective of high-level accuracy, a post-processing SVM technique is proposed [176]. Support vector machine is a better option of correct measurement in glucose monitoring system [177]. Artificial Neural Network (ANN) has also been proposed for data combining [178]. The measured data from multiple techniques are combined through the proposed neural network model [179]. This artificial neural network has been implemented in DSP processor [180, pancholi2018portable]. This proposed data interpretation model has been used for combining and calibrating of data for final estimated glucose concentration [181]. ### V-B Metrics for Model Validation The calibration method is used to have precise blood glucose estimation for measurements [182]. The obtained glucose concentration values are used to compare with conventionally measured glucose concentrations [183]. The Clarke error grid analysis has been considered maximum measurement for analysis which is used to check the performance of any device for accuracy measurement [184]. The process flow is represented in Fig. 25. Figure 25: Representation of Metrics for model Validation. ### V-C Clinical Accuracy Evaluations using Clarke error grid analysis The Clarke Error Grid has been analysed as benchmark tool to examine the clinical precision for biomedical application. It has prediction of point as well as rate accuracy, and it amends for physiologic time lags inherent for measurement of body glucose. The exploitation of the Clarke error grid modelling will significantly make easy the development and refinement of a precise biomedical device. In 1970, this technique was developed by C.G. Clark to identify the accuracy of the clinical trials which helps to find the precision of estimated blood glucose with blood glucose value through the conventional method. A description of the Error Grid Analysis came into view in diabetes care in 1987. The grid is divided with five different zones mainly as zone A, zone B, zone C, zone D, and zone E. If the values residing in either zone A or zone B then it signifies satisfactory or accurate prediction of glucose results according to Beckman analyzer. The zone C values may prompt gratuitous corrections which may lead to a poor outcome. If the values are under zone D which actually defines a hazardous failure to sense. Zone E reflects the “erroneous treatment” [185]. Figure 26: Clarke error grid analysis. ## VI Consumer Products for Glucose-Level Measurement There have been several non-invasive glucometer at market(such as Freestyle Libre sensor, SugarBEAT from Nemaura medical) which used for proper medication. They would be like skin-patch with daily disposable feature and adhesive to have the continuous glucose monitoring. Most of the consumer products fail to provide precise glucose measurement and hence they are much popular for diabetes management. There are some products as DiaMon Tech, glucowise, glucotrack, glutrac and CNOGA medical device. Glutrac is smart healthcare device but it has accuracy issues for the blood glucose measurement. It has higher cost while precision is still not acceptable. The non-invasive stripless device known as Omelon B-2 has been used for the CGM. The fluorescent technique based Glucosense has been made for contonuous monitoring of the glucose value. The flexible textile-based biosensor has developed from Texas University to measure the glucose level. All the available device have accuracy issues and considerable higher cost. ### VI-A Wearable versus Non-wearable for Glucose Monitoring The glucose monitoring have been attempted using non-wearable and wearable solutions in the literature. Most of the non-wearable approaches are based on various spectroscopy such as photoacoustic spectroscopy, Raman spectroscopy etc. The implantable devices are of semi-invasive type and are mainly of biosensors in nature. Sweat patches, Glucowatch and Smart contact lenses are of wearable devices category. LifePlus has developed non-invasive and wearable device for CGM purpose and it is under consideration for commercialization. Most of non-invasive device are wearable and helpful for frequent glucose measurement. The continuous glucose monitoring would be more acceptable if they could measure the blood glucose values in day to day life. Therefore, the wearable devices are more state of art solutions then non wearable devices. ### VI-B Noninvasive Glucose Measurement Consumer Products There are variety of products such as GlucoTrack®, glucometer from Labiotech [186], and similar available solutions have accuracy issues and cost is also high. The glucowise is another non-invasive device for continuous glucose measurement from Medical Training Initiative (MTI) ™. The Raman scattering spectroscopy based non-invasive solution is also developed by 2M Engineering [187]. These devices are not much popular because of their cost and precision. Further for the high level of accuracy of glucose measurement, Glucotrack™ has been developed by integrity applications Ltd. [188]. This non-invasive glucose monitoring device employed three consecutive ultrasonic spectroscopy, thermal emission and electromagnetic techniques. This device is highly precise and accurate because of a combination of three techniques [189]. A comparative perspective of various consumer products for noninvasive glucose measurement has been summarized in Table VI-B. Table IV: A Comparative perspective of a selected consumer products for noninvasive glucose measurement. Company \| Device \| Technology \| Object \| Summary \| Snapshot ---\|---\|---\|---\|---\|--- Cygus Inc. (USA) \| GlucoWatch G2 Biographer \| Reverse iontophoresis \| Wrist skin \| It would be worn as watch is used with disposable component, autosensor which is to be attached at back of biographer that contact with the skin to provide frequent glucose monitoring \| CNOGA (Israel) \| Combo glucometer \| Tissue photography analysis \| Finger \| On basis of tissue photography analysis from fingertip capillaries, this device can analyze various bio parameters in very short time \| Pendragon Medical (Switzerland) \| Pendra \| Impedance Spectroscopy \| Wrist Skin \| It helps to measure the glucose with sodium transport of erythrocyte membrane, The change of fluxes of transmembraneous sodium occur due to impedance field which is detected by device to generate final glucose value \| OrSense Ltd. (Israel) \| OrSense NBM-200G \| Occlusion Spectroscopy \| Fingertip skin \| It is based on optical concept on finger which is attached to a ring-shaped sensor probe. The probe has red/near-infrared RNIR spectral region light source as well as detector. It has pneumatic cuffs which generates systolic pressure to produce optical signal for glucose monitoring \| C8 Medisensors (USA) \| C8 Medisensor Glucose detector \| Raman Spectroscopy \| Fingertip skin \| This technique is based on monochromatic light source passes through skin where scattered light is detected. The generated colors from Raman spectra helps to exact chemical structure of glucose molecule \| Integrity Applications (Israel) \| Glucotrack \| Combination of Electromagnetic, ultrasonic and Thermal \| Ear lobe tissue \| In this device, three different techniques are used concurrently to increase the accuracy and precision \| Tech4Life Enterprises (USA) \| Non invasive glucometer \| Infra red Spectroscopy \| Finger \| It is helpful for Hyperglycemia or Pre-Diabetic patients which allow for regular monitoring of precise blood glucose measurement at every 30 seconds \| MediWise Ltd. (United Kingdom) \| Glucowise \| Radio Wave Spectroscopy \| Forefinger skin/Earlobe \| This non invasive wireless device can measure glucose concentration in very short time. It is based on electromagnetic waves of specific frequencies for blood glucose detection. It uses a thin-film layer of metamaterial which increases the penetration for precise glucose measurement \| Nemaura Medical (united Kingdom) \| SugarBeat \| Reverse iontophoresis \| Arm,Leg and adbomen \| This has been proved accurate device, pain-free continuous blood glucose monitoring. SugarBEAT® provides real-time, needle-free glucose measurement. Generally, it needs one time finger-prick test for calibration. One time finger prick is used when new patch is required to insert \| Abott Ltd. (USA) \| Free Style Libre \| Glucose oxidase method \| Fore-arm skin \| It uses enzyme glucose sensing technology for the detection of glucose levels through interstitial fluid Glucose oxidase method is applied through sensor where electrical current proportional to the glucose concentration and glucose can be measured. \| C8 Medisensor \| Non invasive glucose monitor \| Raman Spectroscopy \| Fore arm skin \| Raman spectroscopy technique based this device can detect glucose in blood through returning spectrum from the skin \| ## VII Glucose Level Controls Approaches and Consumer Products Various models have been developed for diet control using various parameters for glucose-insulin balance. The parameters are mainly includes net hepatic glucose balance, renal excretion rate, glucose absorption rate and peripheral glucose utilization for the glucose consumption prediction for the diabetic patients. These are useful parameters to calculate the glucose level by proper insulin dosage along with scheduled diet plan. Therefore, the glucose-insulin control model was designed to balance glucose insulin level in the body for diabetes persons using proper medication. ### VII-A Glucose Controls Approaches The mathematical models for insulin delivery have been presented to determine the coefficients of blood regulation. The model has been proposed for insulin secretion with glycemic profile for type 2 diabetic person [190, 191]. The non-linear model is developed using differential equation with delay model with help of non-diabetic subjects [192]. Most poplar “Uva/Padova Simulator” was also explored which was approved from FDA to have the proper clinical trials. The parameter are extracted with type 1 diabetic virtual patients [193]. The intravenous test for glucose tolerance with Hovorka maximal model has explored for non-diabetic subjects [194]. The samples from type 1 diabetic persons were collected to explain the model with help of time monitoring. The model is proposed mathematically for blood glucose value prediction in the postprandial mode for type 1 diabetes patients [195, 196]. The mathematical model for glucose-insulin balance for longer period is explored using two days clinical information [197]. A algorithm was developed for T1DM patient meal detection for the purpose of frequent glucose measurement. The work has integrated bolus meal mathematical model for glucose-insulin delivery model [198]. Diabetic and healthy people were considered to acquire the values for the variable state dimension algorithm. The diet plan was examined in the absence of meal profile to have the glycemic profile balance, an intelligent PID controller (iPID) was developed to type 1 diabetic person [199, 200]. ### VII-B Glucose Controls Consumer Products Type-1 diabetic patients aren’t able to produce insulin. Insulin is a hormone that can balance body sugar (glucose) which is a prime source of energy that obtains from carbohydrates. If anybody has type 1 diabetes, it is necessary to be ready for insulin therapy. Insulin may be injected by self-injection, patients who take multiple injections daily of insulin may also think about use of an insulin pump. An insulin pump gives short-acting insulin all day long continuously. The insulin pump replaces the requirement of long-acting insulin. A pump also substitutes the requirement of multiple injections per day along with continuous insulin infusion and also serves to improve the glucose levels. Various types of insulin pumps are already available in the market as consumable product mainly as Animas, Medtronic, Roche, Tandem and Omnipod insulin pump are consumables. These insulin pumps are advanced to each other in terms of their upgraded features. A comparative perspective of a selected state of art approach for glucose measurement to have better glyncenic profile control is presented in Table VII-B. Table V: A comparative perspective of a selected state of art approaches for glucose measurement Work \| Technology \| Object \| Findings \| Observation ---\|---\|---\|---\|--- [86] \| photoplethy- -smography (PPG) \| Finger \| It helps to extract the features of PPG signal through machine learning models to estimate Systolic and diastolic blood pressure and blood glucose values \| machine learning models applied where random forest technique has best prediction results as $R^{2}_{SBP}$ = 0.91, $R^{2}_{DBP}$ = 0.89 and $R^{2}_{BGL}$ = 0.90. CEG has 87.7% observation in Zone A, 10.3 % in Zone B, and 1.9% in Zone D [201] \| mid-infrared attenuated total reflection (ATR) spectroscopy and trapezoidal multi-reflection ATR prism \| oral mucosa inner lips \| Using a multi-reflection prism brought about higher sensitivity, and the flat and wide contact surface of the prism resulted in higher measurement reproducibility & spectra around 1155 cm$-1$ for different blood glucose levels for fasting and before fasting \| the coefficient of determination $R^{2}$ is 0.75. The standard error is 12 mg/dl, and all the measured values are in Region A [202] \| Optical Coherence Tomography \| Fingertip \| It measures the optical rotation angle and depolarization index of aqueous glucose solutions with low and high scattering, respectively. The value of angle increases while depolarization index decreases with glucose value increases \| The correlation factor has a value of $R^{2}$ 0.9101, the average deviation is found around 0.027. [203] \| Contactlenses fluoresence \| Tears \| The fabrication of a soft, smart contact lens in which glucose sensors, wireless power transfer circuits, and display pixels to visualize sensing signals in real time are fully integrated using transparent and stretchable nanostructures \| The usage of smart and soft lens would provide the wireless operation at real-time for glucose monitoring in tears [204] \| transmission spectroscopy \| Sliva \| After completely absorbing the sufficient amount of saliva on the strip, the sample would reach detection zone via paper microfluidic movement and enzymatic reaction between GOx and salivary glucose would initiate a pH change, resulting in a change in strip color that was recorded by using RGB detector on the handheld instrument which helps for glucose detection \| The developed biosensor had a wide detection range of detection between 32- and 516-mg/dL glucose concentration while the sensitivity of detection was 1.0 mg/dL/count at a limit of detection (LOD) of 32 mg/dL within a response time of 15 s [205] \| impedance spectroscopy (IMPS) and multi-wavelength near-infrared spectroscopy (mNIRS) \| Left Handand wrist Hand \| IMPS and mNIRS use the indirect dielectric characteristics of the surrounding tissue around blood and the optical scattering characteristics of glucose itself in blood, respectively, the proposed IC can remove various systemic noises to enhance the glucose level estimation accuracy \| mean absolute relative differences (mARD) to 8.3% from 15.0% of the IMPS and 15.0–20.0% of the mNIRS in the blood glucose level range of 80–180 mg/dL. From the Clarke grid error (CGE) analysis, all of the measurement results are clinically acceptable and 90% of total samples can be used for clinical treatment [84] \| NIR Spectroscopy \| Fingertip \| short NIR waves with absorption and reflectance of light using specific wavelengths (940 and 1,300 nm) has been introduced \| The Pearson’s correlation coefficient (R) is 0.953 and MAD is 09.89 which is RMSE 11.56 [206] \| Microwave Detection \| earlobe \| The absorption spectrum of microwave signal helps to measure using two antenna. The sine wave of 500 MHz is for blood glucose measurement. \| It can measure blood glucose from 0 to 500 mg/dl with step size of 200 mg/dl used for the experiment for testing the resolution. It obtained 0.5226 mean standard deviation while the minimum value of standard deviation is 0.04119. [94] \| PPG \| Finger \| The prediction of blood glucose was with machine-learning using a smartphone camera. First the invalid data was separated and The system did not require any type of calibration \| The device was able to measure glucose only 70-130 mg/dl range. The results show accuracy of 98.2% for invalid single-period classification and and the overall accuracy is 86.2%. [46] \| MEMS \| Finger \| It is minimally invasive technique known as e-Mosquito which extracts blood sample with shape memory alloy (SMA)-based microactuator. It considered as first ever wearable device which performs the automatic situ blood extraction and performs the glucose analysis. \| The method provided linear correlation ($R^{2}$ = 0.9733) between standard measurements and the e-Mosquito prototype. [207] \| Visible NIR \| Wrist \| The paper developed biosensor which helps to exploit pulsation of arterial blood volume from the wrist tissue. The visible NIR spectroscopy was used for reflected optical signal to estimate blood glucose. \| The correlation coefficient (Rp) value after averaging all observation is 0.86, whereas the standard prediction error is around 6.16 mg/dl. [201] \| mid-infrared attenuated total reflection (ATR) spectroscopy \| inner lip mucosa \| Novel optical fiber probe was introduced using multireflection prism with ATR spectroscopy. The sensitivity increases with the number of reflections while measurement reproducibility was higher due to prism’s wide and flat & wide contact surface. \| The experimental results reveals the glucose signature at various spectra between fasting state and after the glucose injection. The plot for calibration defines peak for absorption at 1155 $cm^{-1}$ which has glucose measurement error less than 20% [chowdhury2016noninvasive] \| modulated ultrasound and infrared technique \| Finger \| the MATLAB toolbox is used with Fast Fourier Transform (FFT) for blood glucose extraction. The random blood glucose level test and oral glucose tolerance test was done for the human subjects for performance measurement \| The RMSE value of noninvasive and invasive measurement from both tests 28.20 mg/dl and 23.76 mg/dl. The pearson correlation coefficient was 0.85 and 0.76, respectively. At the same time MSE was 17.76 mg/dl and 15.92 mg/dl. ## VIII Glucose-Level Measurement and Controls - IoMT Perspectives The practical and sustainable mechanisms are the prime factors of smart and automated healthcare system. These are being optimized to support the population migration and quality of life in smart cities and smart villages [208, 209]. The features of smart healthcare system are continuous monitoring for critical care, ambient intelligence and quality of service for proper point of care mechanism [210, 211]. The non-invasive and precise glucose measurement is requirement for diabetic person and would also needed to store the information using IoMT for proper treatment [26]. The traditional method for glucose measurement has limited capability and is not able to assist the remotely located healthcare provider. The diabetic person would like to monitor their glycemic profile frequently in a day with support of storing at cloud server. The smart health care system would allow the point of care treatment for diabetes person with frequent monitoring. The internet of Medical Things (IoMT) has allowed to connect the patients with doctors remotely for rapid treatment and special assistance using smart healthcare [208]. The continuous monitoring of vital parameters have provided to awareness about the diet plan and routine activity management with contemporary healthcare consumers devices. With the active support of remote healthcare solution, the smart healthcare has potential to ameliorate the quality of service at reduced cost. The smart sensors would capture the patient data continuously and help to store the data on cloud data centre. It is also useful for the analysing the data and easy exchange of the information through mobile applications to doctors as well as patients. The healthcare Cyber-Physical System (H-CPS) has been used successfully to address the various challenges of healthcare sector with intelligent algorithms. The continuous glucose monitoring would certainly help the diabetic patients to plan their diet for the purpose of glucose control. The solution should be precise, low cost and easy to operate for rapid diagnosis [32, 28]. The serum glucose would always consider as accurate than capillary measurement. Therefore, the rapid serum glucose measurement solution with continuous monitoring is desired for the smart healthcare. The novel serum glucometer is portable device and is also integrated with IoMT to store the glucose values continuously at cloud. It would be useful for the healthcare provider to track the health records of remote located diabetes person. The smart healthcare management of continuous glucose measurement is defined in Fig. 27. Figure 27: Blood glucose diagnosis and Control in smart health care system. A detailed example of a closed-loop system that presents glucose-level monitoring and insulin release to control it is illustrated in Fig. 28 [2]. This IoMT framework can provide a better solution for evaluation of insulin doses through the closed-loop automated insulin secretion diabetes control. Such an integrated IoMT framework can be implemented to diagnose and for the treatment of diabetic patients in terms of controlling their blood glucose level in smart healthcare and be effective in smart village and smart cities for healthcare with limited medical personnel. Figure 28: A closed-loop automated insulin secretion diabetes control system in an IoMT framework [2]. The security and privacy issues of the medical devices are paramount aspect in any IoT network. The hardware security of wearable device is very crucial because control actions mainly occur in wireless media. The security vulnerabilities are defined for glucose measurement device and its control are shown in Fig. 29. The devices security are important due to connected health system in an insecure and unreliable IoMT framework [212]. The integrity of useful medical information is also crucial security aspect of smart healthcare. All patients medical records are stored over the server therefore the security of such data are also really important. The controlled access with proper authentication is required to have secure monitoring with proper patient treatment. Figure 29: Our Long-Term Vision of Security-Assured Non-invasive Glucose-Level Measurement and Control through our Proposed iGLU. ## IX Short-Comings of Existing Works and Open Problems This Section outlines the shortcomings and discusses some open problems of glucose level measurements and control. ### IX-A Limitations of the Existing Approaches and Products 1. 1. Photoacoustic spectroscopy has been implemented for glucose measurement. Real- time testing and validation have not been done from human blood. The artificial solution was prepared in the laboratory for glucose measurement. The prototype module with LASER and corresponding detector is costly and at the same time requires considerable bigger area and does not provide portable solution. Therefore, it is not much popular solution for continuous glucose monitoring. 2. 2. Raman spectroscopy is a nonlinear scattering which occurs when monochromatic light interacts with a certain sample. Raman spectroscopy based solution is applicable for a laboratory test and also occupies the significant larger area. Hence, the system based on this approach will not be applicable for frequent glucose measurement. 3. 3. The retina based glucose measurement is also one of the alternate non-invasive glucose detection approach, data has also been collected through retina for glucose measurement. Such technique is not useful for the glucose measurement all the time. 4. 4. In case of bio-capacitance spectroscopy, the slight difference in placing the sensor at the same location might affect the output of the sensor. Effect of pressure on the sensor, body temperature and sweat on the skin may also affect the output of the sensor. 5. 5. Glucose detection is performed with the impedance spectroscopy (IMPS) by electrodes connection to the skin which is affected with skin. The accuracy is always an issue as the saliva and sweat could change for each individual and that may reflect to the precision of glucose. Therefore, this technique is not best for reliable glucose measurement in smart healthcare. 6. 6. PPG signal has been used to extract features for blood glucose level prediction. But the PPG may be precise blood glucose measurement technique where the output value would vary according the blood volume only. Therefore, the glucose molecule has not been detected precisely in the blood sample using this technique. ### IX-B The Open Problems in Non-invasive Glucose Measurement There are lots of challenges for commercialization of non-invasive glucose measurement device. But, some open problems have been discussed which are prime challenges for precise non-invasive glucose measurement. These challenges have been represented in Fig. 30. The precise glucose measurement of hypoglycemic patient and long-time continuous glucose measurement without instantaneous error are the open problems which are focussed by the researchers recently. Figure 30: Open Challenges in Noninvasive Glucose-Level Measurement. * • The effect of blood pressure, body temperature and humidity have not been considered in the literature which affect the values of glucose measurement. * • The cost effective and portable solution of continuous glucose measurement device has also not been addressed properly. * • The accurate glucose measurement has been also been open challenge for full rage from 40 mg/dl to 450 mg/dl. * • The effective integration of glucometer with internet of medical things for continuously data logging to the cloud has still not potentially resolved. * • The mathematical model for automatic insulin secretion according to measured glucose value has to be address in better manner with internet framework. * • The privacy and security issues of insulin and blood glucose measurement system is still not resolved yet. * • The efficient power management mechanism has to be developed for continuous glucose measurement with insulin delivery system. ## X Conclusions and Future Research The paper presents survey of glucose measurement approaches along with overview of glucose control mechanism. Many techniques available in literature are only a proof of concept, showing good correlation between device estimated result and reference value of blood glucose. However, they are neither accurate nor cost effective solutions and not available for commercial purpose. The optical detection using short NIR has been potential solution to mitigate the drawbacks of all previous methods. In future, the multi-model approaches could be considered for precise glucose estimation. The device or prototype model should be more effective in different zones to support the continuous health monitoring. It should be implemented as a portable device for real time application with more frequently. This device should be developed as continuous health monitoring with minimum cost. The future research for upcoming noninvasive glucose monitoring device is mentioned in Fig. 31. The device is required to be integrated with advanced IoMT framework. This advanced IoMT framework will alow to connect the device with all nearest diabetic centers to get best treatment. Unification of glucose-level measurement and automatic diet quantification can have strong impact on smart healthcare domain [213]. The durability, portability and user- friendly device is also the future vision in this era. The device should have the feature of border-line cross indication. Because of this feature, any person will be aware to take own blood glucose level. A secured device with end to end users control and authentication is also necessary for future advancement. Physical Unclonable Function (PUF) based security of IoMT-devices can be effective for IoMT-devices which are intrinsically resource and battery constrained [212, 214]. Unified healthcare Cyber-Physical System (H-CPS) with blockchain based data and device management can be effective and needs research [215, 216]. 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Puthal, “PUFchain: A hardware-assisted blockchain for sustainable simultaneous device and data security in the internet of everything (IoE),” _IEEE Consumer Electronics Magazine_ , vol. 9, no. 2, pp. 8–16, 2020. ## About the Authors \| Prateek Jain (Member, IEEE) earned his B.E. degree in Electronics Engineering from Jiwaji University, India in 2010 and Master degree from ITM University Gwalior. Currently, he is an Assistant Professor in SENSE, VIT University, Amaravati (A.P.). His current research interest includes VLSI design, Biomedical Systems and Instrumentation. He is an author of 14 peer- reviewed publications. He is a regular reviewer of 12 journals and 10 conferences. ---\|--- \| Amit M. Joshi (Member, IEEE) received the Ph.D. degree from the NIT, Surat, India. He is currently an Assistant Professor at National Institute of Technology, Jaipur. His area of specialization is Biomedical signal processing, Smart healthcare, VLSI DSP Systems and embedded system design. He has also published papers in international peer reviewed journals with high impact factors. He has published six book chapters and also published more than 70 research articles in excellent peer reviewed international journals/conferences. He has worked as a reviewer of technical journals such as IEEE Transactions/ IEEE Access, Springer, Elsevier and also served as Technical Programme Committee member for IEEE conferences which are related to biomedical field. He also received honour of UGC Travel fellowship, the award of SERB DST Travel grant and CSIR fellowship to attend well known IEEE Conferences TENCON, ISCAS, MENACOMM etc across the world. He has served as session chair at various IEEE Conferences like TENCON -2016, iSES-2018, iSES-2019, ICCIC-14 etc. He has also supervised 19 PG Dissertations and 16 UG projects. He has completed supervision of 4 Ph.D thesis and six more research scholars are also working. ---\|--- \| Saraju P. Mohanty (Senior Member, IEEE) received the bachelor’s degree (Honors) in electrical engineering from the Orissa University of Agriculture and Technology, Bhubaneswar, in 1995, the master’s degree in Systems Science and Automation from the Indian Institute of Science, Bengaluru, in 1999, and the Ph.D. degree in Computer Science and Engineering from the University of South Florida, Tampa, in 2003. He is a Professor with the University of North Texas. His research is in “Smart Electronic Systems” which has been funded by National Science Foundations (NSF), Semiconductor Research Corporation (SRC), U.S. Air Force, IUSSTF, and Mission Innovation. He has authored 350 research articles, 4 books, and invented 4 granted and 1 pending patents. His Google Scholar h-index is 39 and i10-index is 149 with 6600 citations. He is regarded as a visionary researcher on Smart Cities technology in which his research deals with security and energy aware, and AI/ML-integrated smart components. He introduced the Secure Digital Camera (SDC) in 2004 with built-in security features designed using Hardware-Assisted Security (HAS) or Security by Design (SbD) principle. He is widely credited as the designer for the first digital watermarking chip in 2004 and first the low-power digital watermarking chip in 2006. He is a recipient of 12 best paper awards, Fulbright Specialist Award in 2020, IEEE Consumer Technology Society Outstanding Service Award in 2020, the IEEE-CS-TCVLSI Distinguished Leadership Award in 2018, and the PROSE Award for Best Textbook in Physical Sciences and Mathematics category in 2016. He has delivered 10 keynotes and served on 11 panels at various International Conferences. He has been serving on the editorial board of several peer- reviewed international journals, including IEEE Transactions on Consumer Electronics (TCE), and IEEE Transactions on Big Data (TBD). He is the Editor- in-Chief (EiC) of the IEEE Consumer Electronics Magazine (MCE). He has been serving on the Board of Governors (BoG) of the IEEE Consumer Technology Society, and has served as the Chair of Technical Committee on Very Large Scale Integration (TCVLSI), IEEE Computer Society (IEEE-CS) during 2014-2018. He is the founding steering committee chair for the IEEE International Symposium on Smart Electronic Systems (iSES), steering committee vice-chair of the IEEE-CS Symposium on VLSI (ISVLSI), and steering committee vice-chair of the OITS International Conference on Information Technology (ICIT). He has mentored 2 post-doctoral researchers, and supervised 12 Ph.D. dissertations, 26 M.S. theses, and 12 undergraduate projects. ---\|---
# Wet to dry self-transitions in dense emulsions: from order to disorder and back Andrea Montessori<EMAIL_ADDRESS>Istituto per le Applicazioni del Calcolo CNR, via dei Taurini 19, 00185, Rome, Italy Marco Lauricella Istituto per le Applicazioni del Calcolo CNR, via dei Taurini 19, 00185, Rome, Italy Adriano Tiribocchi Center for Life Nanoscience at la Sapienza, Istituto Italiano di Tecnologia, viale Regina Elena 295, 00161, Rome, Italy Istituto per le Applicazioni del Calcolo CNR, via dei Taurini 19, 00185, Rome, Italy Fabio Bonaccorso Center for Life Nanoscience at la Sapienza, Istituto Italiano di Tecnologia, viale Regina Elena 295, 00161, Rome, Italy Istituto per le Applicazioni del Calcolo CNR, via dei Taurini 19, 00185, Rome, Italy Università degi studi Roma ”Tor Vergata”, Via Cracovia, 50, 00133, Rome, Italy Sauro Succi Center for Life Nanoscience at la Sapienza, Istituto Italiano di Tecnologia, viale Regina Elena 295, 00161, Rome, Italy Istituto per le Applicazioni del Calcolo CNR, via dei Taurini 19, 00185, Rome, Italy Institute for Applied Computational Science, Harvard John A. Paulson School of Engineering and Applied Sciences, Cambridge, MA 02138, United States ###### Abstract One of the most distinctive hallmarks of many-body systems far from equilibrium is the spontaneous emergence of order under conditions where disorder would be plausibly expected. Here, we report on the self-transition between ordered and disordered emulsions in divergent microfluidic channels, i.e. from monodisperse assemblies to heterogeneous polydisperse foam-like structures, and back again to ordered ones. The transition is driven by the nonlinear competition between viscous dissipation and surface tension forces as controlled by the device geometry, particularly the aperture angle of the divergent microfluidic channel. An unexpected route back to order is observed in the regime of large opening angles, where a trend towards increasing disorder would be intuitively expected. ## I Introduction Self-organization can be broadly defined as the complex of processes which drives the emergence of spontaneous order in a given system, due to the action of local interactions between its elementary constituents [30]. This concept has provided a major paradigm to gain a deeper insight into a number of phenomena across a broad variety of complex systems in physics, engineering, biology and society [33, 2, 15, 21, 1]. Self-organization is usually triggered and sustained by competing processes far from equilibrium, as they occur in a gamut of different scientific endeavors, from natural sciences and biology to economics and anthropology [14, 35, 39, 40, 3, 10]. and often efficiently exploited to find innovative design solutions in a number of engineering applications [38, 5, 12]. From this standpoint, droplet-based microfluidics, namely the science of generating and manipulating large quantities of micron- sized droplets, offers a literal Pandora’s box of possibilities to investigate the physics of many-body systems out of equilibrium. In particular, the self- assembly between droplets and bubbles which results from the subtle multiscale competition between different forces and interactions, such as the external drive, interfacial (attractive) forces, near-contact (repulsive) interactions, viscous dissipation and inertia [26, 29]. Most importantly, in many instances, such competition is highly sensitive to geometrical factors, primarily the presence of confining boundaries. Among others, the ability to manipulate and control tiny volumes of droplets allows the generation of highly ordered porous matrices with finely tunable structural parameters [4]. This opens up the possibility of designing novel families of materials with potential use in a wide range of advanced applications, such as catalyst supports, ion-exchange modules, separation media and scaffolds in tissue engineering [20, 13, 25, 27]. Recently, Gai et al.[8], reported an unexpected ordering in the flow of a quasi-2D concentrated emulsion in a convergent microfluidic channel, and showed that confinement of the 2D soft crystal in the extrusion flow causes the reorganization of the crystal internal structure in a highly ordered pattern [8, 7]. The self-reorganization of the crystal is expected to bear major implications for the realization of confined low-dimensional materials, crucial for applications ranging from optoelectronics to energy conversion, which might be easier to control than previously thought, thus leading to novel flow control and mixing strategies in droplet microfluidics. In this paper, we report on the self-transition between wet and dry emulsions [32, 9], namely from ordered monodisperse assemblies to heterogeneous and polydisperse foam-like structures, in divergent microfluidic channel. Following the common terminology [24, 6], foams are wet when their droplets appears nearly round and the structures they form are organized according to ordered hexagonal patterns which flow basically deformation-free. In the dry regime instead, the droplets come closer and deform, assuming typical polyhedral shapes and giving rise to typical disordered foam-like structures. Such transition is driven by the capillary number, i.e. the competition between viscous dissipation and surface tension, which is in turn modulated by the device geometry. In particular, we observe a return to an ordered state in a parameter regime where a transition towards disorder would be intuitively expected. ## II Method In this section, we briefly describe the numerical model employed, namely an extended color-gradient lattice Boltzmann approach with repulsive near-contact interactions, previously introduced in [28]. In the multicomponent LB model, two sets of distribution functions evolve, according to the usual streaming- collision algorithm (see [36, 16]), to track the evolution of the two fluid components: $f_{i}^{k}\left(\vec{x}+\vec{c}_{i}\Delta t,\,t+\Delta t\right)=f_{i}^{k}\left(\vec{x},\,t\right)+\Omega_{i}^{k}[f_{i}^{k}\left(\vec{x},\,t\right)]+S_{i}(\vec{x},t),$ (1) where $f_{i}^{k}$ is the discrete distribution function, representing the probability of finding a particle of the $k-th$ component at position $\vec{x}$, time $t$ with discrete velocity $\vec{c}_{i}$, and $S_{i}$ is a source term coding for the effect of external forces (such as gravity, near- contact interactions, etc). In equation 1 the time step is taken equal to $1$, and the index $i$ spans over the discrete lattice directions $i=1,...,b$, being $b=9$ for a two dimensional nine speed lattice (D2Q9). The density $\rho^{k}$ of the $k-th$ component and the total linear momentum of the mixture $\rho\vec{u}=\sum_{k}\rho^{k}\vec{u^{k}}$ are obtained, respectively, via the zeroth and the first order moment of the lattice distributions $\rho^{k}\left(\vec{x},\,t\right)=\sum_{i}f_{i}^{k}\left(\vec{x},\,t\right)$ and $\rho\vec{u}=\sum_{i}\sum_{k}f_{i}^{k}\left(\vec{x},\,t\right)\vec{c}_{i}$. The collision operator splits into three components [11, 19, 18]: $\Omega_{i}^{k}=\left(\Omega_{i}^{k}\right)^{(3)}\left[\left(\Omega_{i}^{k}\right)^{(1)}+\left(\Omega_{i}^{k}\right)^{(2)}\right].$ (2) where $\left(\Omega_{i}^{k}\right)^{(1)}$, stands for the standard collisional relaxation [36], $\left(\Omega_{i}^{k}\right)^{(2)}$ code for the perturbation step [11], contributing to the buildup of the interfacial tension while $\left(\Omega_{i}^{k}\right)^{(3)}$ is the recoloring step [11, 17], which promotes the segregation between the two species, minimising their mutual diffusion. A Chapman-Enskog multiscale expansion can be employed to show that the hydrodynamic limit of Eq.1 is a set of equations for the conservation of mass and linear momentum (i.e. the Navier-Stokes equations), with a capillary stress tensor of the form: $\bm{\Sigma}=-\tau\sum_{i}\sum_{k}\left(\Omega_{i}^{k}\right)^{(2)}\vec{c}_{i}\vec{c_{i}}=\frac{\sigma}{2\|\nabla\rho\|}(\|\nabla\rho\|^{2}\mathbf{I}-\nabla\rho\otimes\nabla\rho)$ (3) being $\tau$ the collision relaxation time, controlling the kinematic viscosity via the relation $\nu=c_{s}^{2}(\tau-1/2)$ ( $c_{s}=1/\sqrt{3}$ the sound speed of the model) and $\sigma$ is the surface tension [36, 16]. In eq. 3, the symbol $\otimes$ denotes a dyadic tensor product. The stress-jump condition across a fluid interface is then augmented with an intra-component repulsive term aimed at condensing the effect of all the repulsive near- contact forces (i.e., Van der Waals, electrostatic, steric and hydration repulsion) acting on much smaller scales ($\sim O(1\;nm)$) than those resolved on the lattice (typically well above hundreds of nanometers). It takes the following form: $\mathbf{T}^{1}\cdot\vec{n}-\mathbf{T}^{2}\cdot\vec{n}=-\nabla(\sigma\mathbf{I}-\sigma(\vec{n}\otimes\vec{n}))-\pi\vec{n}$ (4) where $\pi[h(\vec{x})]$ is responsible for the repulsion between neighboring fluid interfaces, $h(\vec{x})$ being the distance between interacting interfaces along the normal $\vec{n}$. The additional, repulsive term can be readily included within the LB framework, by adding a forcing term acting only on the fluid interfaces in near-contact, namely: $\vec{F}_{rep}=\nabla\pi:=-A_{h}[h(\vec{x})]\vec{n}\delta_{I}$ (5) In the above, $A_{h}[h(\vec{x})]$ is the parameter controlling the strength (force per unit volume) of the near-contact interactions, $h(\vec{x})$ is the distance between the interfaces, $\vec{n}$ is a unit vector normal to the interface and $\delta_{I}\propto\nabla\phi$ is a function, proportional to the phase field $\phi=\frac{\rho^{1}-\rho^{2}}{\rho^{1}+\rho^{2}}$, employed to localize the force at the interface. The addition of the repulsive force (added to the right hand side of Eq. 1) naturally leads to the following (extended) conservation law for the momentum equation: $\frac{\partial\rho\vec{u}}{\partial t}+\nabla\cdot{\rho\vec{u}\vec{u}}=-\nabla p+\nabla\cdot[\rho\nu(\nabla\vec{u}+\nabla\vec{u}^{T})]+\nabla\cdot(\bm{\Sigma}+\pi\mathbf{I})$ (6) namely the Navier-Stokes equation for a multicomponent system, augmented with a surface-localized repulsive term, expressed through the gradient of the potential function $\pi$. ## III Results and Discussion The simulation set-up (see figure 1) consists of a microfluidic device composed by an inlet channel ($h_{c}$), a divergent channel with opening angle $\alpha$ and a main channel connected with the divergent ($h=10h_{c}$). The droplets are continuously generated in a buffer channel, placed upstream the inlet channel whose height is $h_{in}\sim 1.7h_{c}$, equal to the droplets diameter. This way, the droplets are forced to deform as they enter the narrower inlet channel, taking a typical oblate shape. The fluid motion is driven by a body force which mimics the effect of a pressure gradient across the device, which is set in such a way as to guarantee laminar flow conditions within both inlet and main channels. The main parameters employed (expressed in simulation (lattice) units) are the following. The microfluidic device is composed by a thin inlet channel (height $h_{c}=30$, length $l_{c}=220$) within which droplets are produced and a main, or self-assembly channel, height $h=10h_{c}$, length $l=900$, where the droplets are transported downstream by the main flow and self-assemble in clusters during their motion. The droplet diameter is set to $D=50$ lattice units, more than sufficient to capture the complex interfacial phenomena occurring in droplet microfluidics (50 lattice points per diameter means a Cahn number of the order $0.08$, typical in resolved diffuse interface simulations of complex interfaces (see [23])). The motion of the droplet is realized by imposing a constant body force $g=10^{-5}$. The viscosity of the two fluids has been set to $\nu=0.167$ while the near-contact force has been set to $A_{h}=0.1$. The choice of magnitude of the body force, along with the kinematic viscosity of the fluids is such to determine a droplet Reynolds number within the inlet channel $Re\sim 2.5$, small enough to guarantee laminar flow conditions. The surface tension has been varied in the range $\sigma=0.007\div 0.02$. Finally, the droplet generation is performed by implementing an internal periodic boundary condition whose short explanation is reported in appendix. All the simulations were performed in two-dimensions, being this a reasonable approximation for the simulation of droplets’ phenomena in shallow microfluidic channels. We wish to point out that the only parameter which has been varied throughout the simulation is the surface tension between the two components which, in turn, allowed us to tune the capillary number. The rate of injection of both the dispersed and the continuous phase were kept constant in all the simulations. In figure 1, we report two different assemblies of droplets within a microfluidic channel with a divergent opening angle $\alpha=45^{\circ}$. This figure shows that the tuning of the inlet Capillary number, ($Ca=U_{d}\mu_{d}/\sigma$, the $d$ subscript standing for droplet, $U_{d}$ is the average droplet velocity within the inlet channel, $\mu_{d}$ the dynamic viscosity and $\sigma$ the surface tension of the mixture), allows to switch between a closely packed, ordered, monodisperse emulsion (1(a)) characterized by regular hexagonal assemblies of droplets, traveling along the micro- channel, to a foam-like structures, formed by polyhedral-shaped droplets (see fig. 1 (c) (d)). The resulting structures appear to be irregular and polydispersed, as indicated by the distortion of the Delaunay triangulation and its dual Voronoi tessellation [22]. We wish to highlight that, both the dispersed and continuous phases’ discharges are kept constant in all the simulations. Thus, the observed transition is likely to by due to (i) the breakup processes promoting the formation of liquid films and (ii) the increased deformability of the droplets interface, due to the lower values of surface tension employed. Typically, droplet breakups increase the amount of interface, leading to an augment of the total length of the thin-film and to a redistribution of the dispersed phase in the system. Such dynamics is controlled by the Capillary number whose increase (within a quite wide range of aperture angles) leads to a spontaneous transition from an ordered state to a disordered one displaying the typical features of a dense foam, namely (i) polydispersity, (ii) formation of an interconnected web of plateaus, (iii) departure of the droplets shapes from the circular or spherical one and (iv) formation of droplets assemblies which are not regular as in the wet case. The transition between different droplets’ structures depends not only on the inlet capillary number but also on the geometrical details of the device, the latter being responsible for a counter-intuitive behavior, to be detailed shortly. Figure 1: Droplet assemblies within a microfluidic channel with a divergent opening angle $\alpha=45^{\circ}$ for two different Inlet channel Capillary number (a) $Ca=0.04$ (c) $Ca=0.16$. Panel (b) clearly shows the ordered, hexagonal packing typical of wet-state emulsions, while (d) the foam-like structure which results in a neat distortion of the Delauney triangulation (blue solid lines connecting the centers of neighboring droplets) and the associated Voronoi tesselation (dotted polygons enclosing the droplets) as well. The red lines are isocontour lines drawn for $\phi_{min}<\phi<\phi_{max}$ being $\phi$ the local phase field $\phi=(\rho_{1}-\rho_{2})/(\rho_{1}+\rho_{2})$ ($\rho_{i}$ the density of the $i$-th phase). The lines are superimposed to a density field. The thickness of the red isocontour line has been widening in order to better visualized the droplet contours. We begin with a phenomenological description of the droplets injection within the diverging channel (for $\alpha=45^{\circ}$), as influenced by the Capillary number. As shown in figure 2 (a-d), below a given value of the Capillary number at the inlet, $Ca\lesssim{0.05}$, every new droplet emerging in the divergent channel pushes away another immediately downstream, taking its place in the process. Indeed, as clearly sequenced in the figure, the yellow-triangle droplet comes out of the channel, pushes the orange-dotted one which, in turn, takes the place of its nearest-neighbor droplet (the red-star one). This process is metronomic i.e. it does not involve any breakup event and this rythmic push- and-slide mechanism reflects into the regular hexagonal crystal which forms downstream the main channel. Figure 2: (a-d) ($Ca\sim 0.04$) Push and slide mechanism of the outcoming droplet. The yellow-triangle droplet comes out of the channel, pushes the orange-dotted one which, in turn, takes the place of its nearest-neighbor droplet (the red-star one). (e-h) ($Ca\sim 0.16$) Droplet pinch-off process. The dotted-orange droplet undergoes a transversal stretching due to the squeezing between the outcoming droplet and the red-star drop. The stretched droplet finally reaches a critical elongation and thinning under the confinement of the neighbor drops before pinch-off. (i-n) Experimental sequence of the breakup mechanism at $Ca\sim 0.08$ (see [37]). The experimental and numerical critical capillary numbers above which droplets pinch-off can be observed are $Ca\geq\sim 0.04$ and $Ca\geq\sim 0.05$ respectively. As stated before, an increase of the Capillary number above a critical value, around $Ca\sim 0.1$, determines the transition to a heterogeneous, foam-like structure, as shown in fig. 1(b). This latter is due to the subsequent breakup events taking place immediately downstream the injection channel, a process highlighted in figure 2 (e-h). The dotted-orange droplet undergoes a transversal stretching due to the squeezing between the outcoming droplet (i.e. the hammer droplet ) and the red-star drop (i.e. the wall droplet). The stretched droplet finally reaches a critical elongation and thinning under the confinement of the neighbor drops before pinch-off. In the meantime, the yellow-triangle droplet, due to the rapid slow down determined by the channel expansion, gradually takes on a crescent shape, fills the area left free by the splitting of the orange drop and becomes a wall droplet in turn. The splitting mechanism just described is responsible for the formation of smaller droplets, which assemble in such a way as to form a heterogeneous foam-like structure within the main channel (fig.1(b)). Briefly, what we observe from the simulations is that, frequent and precise pinch-off requires sufficiently high capillary numbers to occur ($Ca>0.1$ for $\alpha=45^{\circ}$). This suggests that the ratio between the viscous forces (extensional force) and surface tension (retraction/restoring force), namely the Capillary number, is likely to govern the behavior of the droplet-droplet pinch-off process. Indeed, as the viscous force retard the expansion of the impinging droplet, the central one stretches and breaks at the midpoint due to the deformation arising from the normal stresses exerted by the impinging and wall droplet. The surface tension then acts so to contrast the effect of the normal stresses, since both the hammer and the wall droplets tend to retract to their undeformed circular state. It is worth noting that a similar pinching mechanism has been recently observed experimentally in [37] in the same range of capillary numbers as in our simulations.Incidentally, the transitional Capillary number of the experiments (i.e. $Ca$ above which the pinching mechanism is observed) was found to be in satisfactory agreement with the one predicted by the simulations (see caption fig. 2). Figure 3: Equivalent droplet diameter distributions for each pair of Capillary number and opening angle of the divergent channel ($Ca=0.04$(Dashed line), $Ca=0.1$ (Dotted line) and $Ca=0.16$ (full line)). The insets report snapshots of the droplet fields for different values of the Capillary number ($Ca$ increases from top to bottom). The equivalent droplet diameter is the diameter of the circular droplet with the same area of the deformed droplet and can be computed as $D_{e}=\sqrt{4(A_{d}/\pi)}$ being $A_{d}$ the area of the droplet. At this stage, a question naturally arises as to the role of geometrical details of the divergent channel on the wet to dry self-transition. To address this question, we performed a series of simulations by varying both the Capillary number and the opening angle of the divergent channel, so to systematically assess their combined effect on the final shape of the assemblies of droplets within the microfluidic channel. The results of this investigation are summarized in the histograms reported in figure 3. Each histogram shows the distribution of the equivalent droplet diameters within the microfluidic channel (i.e. the diameter of the circular droplet with the same area of the deformed droplet, computed as $\sqrt{4(A_{d}/\pi)}$ being $A_{d}$ the area of the droplet) for a given pair $Ca$ and $\alpha$. A number of comments is in order : i) Below $Ca\sim 0.05$, no breakup event is observed, regardless of the opening angle: the outcoming soft structures are monodisperse assemblies of droplets, as clearly suggested by the dashed-line histograms of fig. 3. ii) Upon raising the Capillary number, it is possible to trigger the breakup events which lead to the transition between ordered and disordered emulsions. By inspecting the histograms ($Ca\sim 0.1$ (dotted line) and $Ca\sim 0.16$ (solid line) ), it is evident that, for a given $\alpha$, the number of breakup events, and in turn, the structure of the resulting emulsion, depend on the inlet capillary number. Indeed, by increasing the Capillary number, the droplets structure increasingly takes the hallmarks of a dense-foam or highly packed dense emulsion (HIPE). For $\alpha$ in range $30^{\circ}-60^{\circ}$, $Ca\sim 0.1$ can be regarded as a critical value of the Capillary number, around which the emergent soft structure is a hybrid between a monodisperse (ordered) and polydisperse (disordered) emulsion, as also evidenced by the the central droplet fields reported in the insets of the histograms. iii) By further increasing the Capillary number, the assemblies of droplets take a typical foam-like structure, completely loosing memory of the structural hexagonal-ordering obtained at lower $Ca$. A more complex structure is found, due to the (a) higher degree of deformability of the droplets, an emergent effect due to the higher values of the Capillary numbers and (b) the presence of smaller droplets which fill the voids between groups of neighbor droplets. iv) Focusing on the highest value of the Capillary number, $Ca=0.16$, we note that the polydispersity, revealed by bimodal histograms, increases as $\alpha$ increases from $22.5^{\circ}$ to $45^{\circ}$. By further increasing the opening angle, the polydispersity starts to recede, nearly vanishing at $\alpha=90^{\circ}$. Figure 4: (a) Equivalent droplet diameter distributions for two couples of Capillary numbers and opening angles of the divergent channel, namely $Ca=0.04$, $Ca=0.16$ and $\alpha=45^{\circ}$, $\alpha=60^{\circ}$, $\alpha=90^{\circ}$. (b) Droplets’ field within the main channel for the case $Ca=0.16$ and $\alpha=90^{\circ}$ To better highlight the aforementioned return towards monodispersity for increasing values of $\alpha$, we directly compare the histograms for six cases, namely, (i) $Ca=0.04$ and $\alpha=45^{\circ}$, (ii) $Ca=0.16$ and $\alpha=45^{\circ}$, (iii)$Ca=0.04$ and $\alpha=60^{\circ}$,(iv)$Ca=0.16$ and $\alpha=60^{\circ}$, (v)$Ca=0.04$ and $\alpha=90^{\circ}$ and (iv)$Ca=0.16$ and $\alpha=90^{\circ}$ (panel (a)). A close inspection of the histograms leave no doubt as to the return to monodispersity for the case $Ca=0.16$ and $\alpha=90$. Indeed, at $Ca=0.16$ and $\alpha=45^{\circ}$ and $\alpha=60^{\circ}$ the emulsion is roughly bidisperse as evidenced by the two peaks at $D_{e}\sim 50$ and $D_{e}\sim 36$ (dashed circles) displayed in the two histograms, the latter one absent in the case $Ca=0.16$ and $\alpha=90^{\circ}$. The decreasing trend of the ratio between the peaks in the histograms at $Ca=0.16$ ($\sim 1.2$ for $\alpha=45^{\circ}$ and $\sim 2.5$ for $\alpha=60^{\circ}$) further points to a gradual return to an ordered structure as the aperture angle increases. This is also apparent from a visual inspection of the droplets’ field reported in panel (b) ($Ca=0.16$, $\alpha=90^{\circ}$) which shows an ensemble of flowing circular droplets of (approximately) the same size.The rare breakup events occurring at the outlet of the injection channel produces a limited number of smaller droplets, an effect evidenced by the small peaks in the upper right histogram. It is worth noting here that, by a plain argument of mass conservation, polydispersity can arise only as a result of droplet breakup via the droplet hammer mechanism since coalescence is frustrated due to the effect of the near contact forces. The counterintuitive behaviour described above, can be intuitively explained as follows: at high opening angles (approaching to $90^{\circ}$), each droplet exiting from the narrow channel experiences a sudden expansion, responsible for a fast recovery of their circular shape, just after their emergence within the main channel. The fast expansion, in turn, determines a strong deceleration (see plot in fig. 5), which forces the next outcoming droplet to loose its droplet hammer action, as the opening angle of the divergent increases above a critical value between $\alpha=45-60^{\circ}$. Further, the sharp deceleration favors the crossflow, transversal displacement of the outcoming droplets, which slide preferentially on the downstream, neighbour droplets rather than squeezing them. The process described above is reported in figure 5. Figure 5: (a-g) Droplets’ field at the outlet of the injection nozzle for $\alpha=90^{\circ}$ and $Ca=0.16$ with the normalized vector field superimposed. Even at high capillary numbers, the sharp deceleration, clearly evidenced by the velocity profiles reported in panel (h) taken in two distinct sections, within the nozzle(open circles) and at a downstream section, favors the transversal displacement of the outcoming droplets, which preferentially slide on the neighboring droplets rather than squeezing them. The velocity field is scaled with the maximum flow velocity at the inlet channel. The two sections, at which the velocity profiles are evaluated are at $x=190lu$ (inside the injection channel) and $x=340lu$ (within the main channel). The arrow within the plot indicates the average velocity drop between the inlet and the main channel. The axis of the plot are, $u/u_{M}$ (normalized magnitude of the velocity) versus $y$(crossflow coordinate). It is to note that, by varying the surface tension, and by keeping the other parameters fixed (so that the Reynolds number can be kept fixed), it is possible to vary viscous dissipation over surface forces independently of the inertia over viscous dissipation ratio. Indeed, the viscous vs surface tension forces ratio is responsible for the frequency of breakup events, hence, in turn, for the degree of order/disorder observed in the system. This competition strictly depends on the geometrical features of the microfluidic environment. To be noted that, the importance of Weber and Capillary number over the droplet breakup frequency in microchannels has been also highlighted in a very recent experimental work of Salari et al. [34] reporting power-law scaling of the dropplets’ breakup frequencies as a function of the product $WeCa^{2}$. To sum up, the simulations suggest that, the dependence of the crystal order on the geometrical feature of the device is not one-way since, once the monodispersity and the hexagonal order are lost, they can be reclaimed back by either decreasing or increasing the opening angle of the divergent channel below/above a critical angle. In other words, either ways, the system looses memory of the disordered configuration. As a note, we wish to stress out that, the detection of the specific regime of capillarity in which the transition occurs required an extensive set of simulations on a broad range of Capillary numbers, as such transition was found to occur indeed in a very narrow window of capillarity space. Thus, even though many more $Ca-\alpha$ combinations have been explored, it was found that the cases reported capture the essence of the phenomenon in point. To gain a quantitative insight into the order to disorder transition, we introduce a dispersity number, $\delta$, defined as the ratio between the number of droplets with a diameter below a critical value, $D_{crit}$, and the total number of droplets. This parameter has been evaluated for each pair of Capillary number $Ca$ and opening angle $\alpha$. The plot in figure 6 reports these data, made non-dimensional by the maximum opening angle $\tilde{\alpha}=\frac{\alpha}{\pi/2}$ and the maximum value of dispersity $\tilde{\delta}=\frac{\delta}{\delta_{M}}$, respectively. Figure 6: Non dimensional dispersity ($\tilde{\delta}$) as a function of the opening angle $\tilde{\alpha}$ for different values of $Ca$. Each dispersity set, $\delta(Ca,\tilde{\alpha})$, follows a Gaussian trend, with mean and variance depending on the Capillary number. Fitting function: $\tilde{\delta}(\tilde{\alpha})=e^{-\left((\tilde{\alpha}-\alpha_{M})/(2Ca)\right)^{2}}$ A few comments are in order: The first observation is that each dispersity set, $\delta(Ca,\tilde{\alpha})$, follows a Gaussian trend, with mean and variance depending on the Capillary number: $\tilde{\delta}(\tilde{\alpha})=e^{-\left((\tilde{\alpha}-\alpha_{M}(Ca))/(2Ca)\right)^{2}}$ (7) . In other words, the dispersity of the system features a ”temperature” which scales linearly with the inlet Capillary number, $T=2Ca$, and a mean value of the opening angle, slightly depending on the Capillary number and ranging between $45-60^{\circ}$. The analysis carried out in this paper should be of direct use for experimental research. Indeed, for each inlet Capillary number, which can be readily determined by evaluating the droplet velocity within the inlet channel, one can single out the channel geometry which allows to obtain the desired degree of polydispersity of the soft structure, by simply querying the gaussian curves. Reciprocally, given the channel geometry, the capillary number can be tuned in such a way as to modify the morphology of the droplet assembly, according again to the gaussian relation provided in this paper. The present findings are expected to help in defining experimental protocols for the development of novel, optimized, low-dimensional, soft porous matrices with tunable properties. We refer in particular to the so-called functionally graded materials [31], namely composite materials characterized by a controlled spatial variation of their microstructure, which are capturing mounting interest for a variety of material science, biology and medical applications. ## IV Conclusions In summary, we reported on order to disorder self-transition in dense emulsions in divergent microfluidic channels, as originated by a geometry- controlled competition between viscous dissipation and interfacial forces. We unveiled a counterintuitive mechanism, namely the spontaneous reordering of the emulsion at high Capillary numbers, obtained by increasing of the opening angle of the divergent channel. Such comeback of order is interpreted as the result of a subtle balance between viscous dissipation and interfacial forces, straight downstream the inlet channel. Moreover, We found that the dispersity of the droplet system follows a simple Gaussian law, whose temperature is directly proportional to the inlet Capillary number. The present findings are expected to offer valuable guidance for the future development of optimised functional materials with locally tunable properties. ## Acknowledgments A. M., M. L., A. T. and S. S. acknowledge funding from the European Research Council under the European Union’s Horizon 2020 Framework Programme (No. FP/2014-2020) ERC Grant Agreement No.739964 (COPMAT). A.M. acknowledges the ISCRA award SDROMOL (HP10CZXK6R) under the ISCRA initiative, for the availability of high performance computing resources and support. ## Appendix ### Droplets’ generation The droplet generation is performed by implementing an internal periodic boundary condition which is sketched in figure 7, for simplicity. Figure 7: Droplet generation via the internal periodic boundary conditions. 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Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). FIRE 2020: Forum for Information Retrieval Evaluation, December 16-20, 2020, Hyderabad, India [orcid=0000-0001-8336-195X, ]<EMAIL_ADDRESS> <EMAIL_ADDRESS> # CMSAOne@Dravidian-CodeMix-FIRE2020: A Meta Embedding and Transformer model for Code-Mixed Sentiment Analysis on Social Media Text Suman Dowlagar [ International Institute of Information Technology - Hyderabad (IIIT-Hyderabad), Gachibowli, Hyderabad, Telangana, India, 500032 Radhika Mamidi [ (2020) ###### Abstract Code-mixing(CM) is a frequently observed phenomenon that uses multiple languages in an utterance or sentence. CM is mostly practiced on various social media platforms and in informal conversations. Sentiment analysis (SA) is a fundamental step in NLP and is well studied in the monolingual text. Code-mixing adds a challenge to sentiment analysis due to its non-standard representations. This paper proposes a meta embedding with a transformer method for sentiment analysis on the Dravidian code-mixed dataset. In our method, we used meta embeddings to capture rich text representations. We used the proposed method for the Task: “Sentiment Analysis for Dravidian Languages in Code-Mixed Text”, and it achieved an F1 score of $0.58$ and $0.66$ for the given Dravidian code mixed data sets. The code is provided in the Github https://github.com/suman101112/fire-2020-Dravidian-CodeMix. ###### keywords: social media code-mixed sentiment analysis meta embedding Transformer GRU ## 1 Introduction Code-mixing(CM) of text is prevalent among social media users, where words of multiple languages are used in the sentence. Code-mixing occurs when conversant uses both languages together to the extent that they change from one language to another in the course of a single utterance [1]. The computational modeling of code-mixed text is challenging due to the linguistic complexity, nature of mixing, the presence of non-standard variations in spellings, grammar, and transliteration [2]. Because of such non-standard variations, CM poses several unseen difficulties in fundamental fields of natural language processing (NLP) tasks such as language identification, part- of-speech tagging, shallow parsing, Natural language understanding, sentiment analysis. Gysels [3] defined the Code-mixing as “the embedding of linguistic units of one language into an utterance of another language”. Code-mixing is broadly classified into two types, intra-sentential and inter-sentential. Intra- sentential code-mixing happens after every few words. Whereas, in inter- sentential code-mixing, one part of the sentence consists of Hindi words, and the other part is entirely English. The code-mixing helps people to express their emotions or opinions emphatically, thus leading to a phenomenal increase of use in code-mixed messages on social media platforms. With the increase in code-mixed data, the analysis of CMSM text has become an essential research challenge from the perspectives of both Natural Language Processing (NLP) and Information Retrieval (IR) communities. There have been some research works in this direction, such as GLUECoS, an evaluation benchmark in code-mixed text [4], automatic word-level language identification for CMSM text [5, 6], parsing pipeline for Hindi-English CMSM text [7, 8], and POS tagging for CMSM text [9]. To encourage research on code-mixing, the NLP community organizes several tasks and workshops such as Task9: SentiMix, SemEval 2020111https://competitions.codalab.org/competitions/20654, and 4th Workshop on Computational Approaches for Linguistic Code- Switching222https://www.aclweb.org/portal/content/fourth-workshop- computational-approaches-linguistic-code-switching. Similarly, the FIRE 2020’s Dravidian-CodeMix task333https://dravidian-codemix.github.io/2020/ was devoted to code-mixed sentiment analysis on Tamil and Malayalam languages. This task aims to classify the given CM youtube comments into one of the five predefined categories: positive, negative, mixed_feelings, not_<language>444the language might be Tamil and Malayalam, unknown_state. In this paper, we present a meta-embedding with a transformer model for Dravidian Code-Mixed Sentiment Analysis. Our work is similar to the meta embedding approach used for named entity recognition on code-mixed text [10]. The paper is organized as follows. Section 2 provides related work on code- mixed sentiment analysis. Section 3 describes the proposed work. Section 4 presents the experimental setup and the performance of the model. Section 5 concludes our work. ## 2 Related Work Sentiment analysis is one of the essential tasks in the field of NLP. Sentiment analysis is the process of understanding the polarity of the sentence. Sentiment analysis helps to attain the public’s attitude and mood, which can help us gather insightful information to make future decisions on large datasets [11]. Initially, sentiment analysis was used on government campaigns and news articles [12, 13]. Recently, due to social media prevalence, the research turned towards capturing the sentiment on social media texts in code-mixing scenarios [14]. The earlier approaches used syntactic rules and lexicons to extract features followed by traditional machine learning classifiers for sentiment analysis on code-mixed text. The process of rule extraction and defining lexicons is a time consuming, laborious process, and is domain-dependent. The recent work in the field of CMSA uses embeddings with deep learning and traditional classifiers [15]. The paper [16] used sub-word information for sentiment analysis on code-mixed text. The recent SentiMix 2020 task used BERT-like models and ensemble methods to capture the code-mixed texts’ sentiment [14]. We used meta embeddings with state of the art transformer model for this task. ## 3 Proposed Model This section presents our proposed code-mixed sentiment analysis framework. It has three main components: a sub-word level tokenizer, a text representation layer, and a transformer model. ### 3.1 Sub-word Level Tokenizer To deal with the non-standard variations in spellings, we used the SentencePiece [17]. SentencePiece is an unsupervised text tokenizer and de- tokenizer mainly used for neural network models. SentencePiece treats the sentences just as sequences of Unicode characters. It implements subword units by using byte-pair-encoding (BPE) [18] and unigram language model [19] . The byte pair encoding initializes the vocabulary to every character present in the corpus and progressively learn a given number of merge rules. The unigram language model trains the model with multiple subword segmentations probabilistically sampled during training. ### 3.2 Text Representation Layer Pre-trained embedding models do not perform well on the code-mixed corpus as they consider all the code-mixed words as OOV words [20]. Thus, we have to train word representations from the code-mixed corpus. Given the complexity of the code-mixed data, it is not easy to determine which embedding model to be used for better performance. Hence, we chose the combination of fastText [21], ELMO [22], and TF-IDF [23] embeddings. fastText captures efficient text representations and local dependencies at the word and sub-word level. ELMO captures contextual representations at the sentence level. TF-IDF captures the distribution of the words in the corpus. The use of TF-IDF for sentiment analysis helps in extracting a better correlation between words and their polarity. All these diverse text representations, when combined, proved beneficial in obtaining better embeddings for the downstream tasks. ### 3.3 Transformer model From [14], For the task of sentiment analysis, we saw that the attention mechanism works better in deciding which part of the sentence is essential for capturing the sentiment. Thus we chose the transformer model for our code- mixed sentiment analysis task. As the data is a classification type, we used only the encoder side from the Transformer. The encoder encodes the entire source sentence into a sequence of context vectors. First, the tokens are passed through a standard embedding layer, and the positional embeddings are concatenated with each source sequence. The embeddings are then passed through a series of encoder layers to get an encoded sequence. The encoder layers is an essential module where all the processing of the input sequence happens. We first pass the source sentence and its mask into the multi-head attention layer, then perform dropout, apply a residual connection, and pass it through a normalization layer. We later pass it through a position-wise feedforward layer and then, again, apply dropout, a residual connection, and layer normalization to get encoded output sequence. The output of this layer is fed into the next encoder layer. The Transformer model uses scaled dot-product attention given in equation 1, where the query $Q$ and key $K$ are combined by taking the dot product between them, then applying the softmax operation and scaled by a scaling factor $d_{k}$ then multiplied by the value $V$. Attention is a critical unit in the Transformer model as it helps in deciding which parts of the sequence are important. $Attention(Q,K,V)=Softmax(\frac{QK^{T}}{\sqrt[]{d_{k}}})V$ (1) The other main block inside the encoder layer is the position-wise feedforward layer. The input is transformed from hid_dim to pf_dim, where pf_dim is usually a lot larger than hid_dim. The ReLU activation function and dropout are applied before it is transformed back into a hid_dim representation. The intuition borrows from infinitely wide neural networks. The wide neural network grants more approximation power and helps to optimize the model faster. ### 3.4 Our Approach Figure 1: Meta Embedding with transformer and GRU model Initially, we tokenized the sentence using the SentencePiece model. After tokenization, we extracted local dependencies between embeddings at the subword level using the fastText model. The fastText model gave embeddings at the word level. We then applied the transformer model to obtain the encoded representations. We got the encoded representations at the word level. A GRU unit is used to get the encoded representation of all the words. We considered the representation of the last hidden layer of the GRU as the final encoded representation. We then obtained the ELMO contextual and TF-IDF representations at the sentence level. We concatenated the representations of the last hidden GRU layer, ELMO, and TF-IDF giving us the meta-embeddings. The meta embeddings are then passed to the output feed-forward network to predict the polarity of the sentence. ## 4 Experimental Setup ### 4.1 Data For Dravidian code-mixed sentiment analysis, we used the dataset provided by the organizers of Dravidian Code-mixed FIRE-2020. The training dataset consists of 15,744 Tamil CM and 6,739 Malayalam CM youtube video comments. The details of the dataset and the initial benchmarks on the corpus are given in [24, 25, 26, 27, 28] ### 4.2 Hyperparameters #### For Embedding models Embeddings play a vital role in improving the model’s performance. As mentioned above, we used fastText and ELMO embeddings. The dimensionality was set to 300 in-case of fastText embeddings. The embeddings are trained on training data using the parameters: learning rate = 0.05, context window = 5, epochs = 20. The ELMO model is obtained from tensorflow_hub555https://tfhub.dev/google/elmo/2, and the pre-set dimensionality of 1024 is used. #### For Transformer and GRU model After evaluating the model performance on the validation data, the optimal values of the hyper-parameters were set. We used the following list of hyper- parameters: learning rate = 0.0005, transformer encoder layer = 1, dropout rate = 0.1, optimizer = Adam, loss function = Cross-Entropy Loss, and batch size = 32, point wise feed forward dimension (pf_dim) = 2048. ### 4.3 Performance Table 1: Accuracy and weighted F1 score on Tamil Code-Mixed Text Method \| Accuracy \| weighted F1 ---\|---\|--- Fine Tuned BERT \| 0.65 \| 0.53 fastText + Tranformer \| 0.66 \| 0.57 fastText + ELMO + Transformer \| 0.66 \| 0.57 fastText + ELMO + TF-IDF + Transformer \| 0.67 \| 0.58 Table 2: Accuracy and weighted F1 score on Malayalam Code-Mixed Text Method \| Accuracy \| weighted F1 ---\|---\|--- Fine Tuned BERT \| 0.51 \| 0.46 fastText + Tranformer \| 0.47 \| 0.45 fastText + ELMO + Transformer \| 0.50 \| 0.47 fastText + ELMO + TF-IDF + Transformer \| 0.67 \| 0.66 We evaluated the performance of the method using weighted F1. The model performed well in classifying positive and not-language comments. The results are given in table 1 and 2 The positive comments had a lot of corpora to train. It made the classification of positive comments an easier task. The not_Malayalam and not_Tamil tweets had another language words in the data, as these language words had higher TF-IDF scores w.r.t the non-language label, their classification was straight-forward. We observed that the system could not identify the sentiment when sarcasm is used in the negative polarity comments. The words in the sarcasm are similar to those of positive comments. It made the sentiment analysis a difficult task. Mixed feelings had both positive and negative sentences. As the classifier was trained on a lot of positive corpora, it could not deduce the negative polarity sentences with sarcasm and irony imbibed in them. Thus the classifier labeled them as positive. It affected the performance of the classifier. More training data could help resolve such issues. ## 5 Conclusion This paper describes the approach we proposed for the Dravidian Code-Mixed FIRE-2020 task: Sentiment Analysis for Davidian Languages in Code-Mixed Text. We proposed meta embeddings with the transformer and GRU model for the sentiment analysis of Dravidian code mixed data set given in the shared task. Our model obtained 0.58 and 0.66 average-F1 for Tamil and Malayalam code-mixed datasets, respectively. We observed that the proposed model did a good job distinguishing positive and not_Malayalam and not_Tamil youtube comments. For future work, we will explore our model’s performance with larger corpora. As we observed sarcasm and irony in negative polarity sentences, we feel that it would be interesting to focus on techniques to detect irony and sarcasm in a code-mixed scenario. ## References * Wardhaugh [2011] R. Wardhaugh, An introduction to sociolinguistics, volume 28, John Wiley & Sons, 2011. * Bali et al. [2014] K. 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# High-efficient two-step entanglement purification using hyperentanglement Lan Zhou,1 Yu-Bo<EMAIL_ADDRESS>1 School of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210003, China 2Institute of Quantum Information and Technology, Nanjing University of Posts and Telecommunications, Nanjing, 210003, China ###### Abstract Entanglement purification is a powerful method to distill the high-quality entanglement from low-quality entanglement. In the paper, we propose an efficient two-step entanglement purification protocol (EPP) for the polarization entanglement by using only one copy of two-photon hyperentangled state in polarization, spatial-mode, and time-bin DOFs. We suppose that the entanglement in all DOFs suffer from channel noise. In two purification steps, the parties can reduce the bit-flip error and phase-flip error in polarization DOF by consuming the imperfect entanglement in the spatial-mode and time-bin DOFs, respectively. This EPP effectively reduces the consumption of entanglement pairs and the experimental difficulty. Moreover, if consider the practical photon transmission and detector efficiencies, our EPP has much higher purification efficiency than previous recurrence EPPs. Meanwhile, when one or two purification steps fail, the distilled mixed state may have residual entanglement. Taking use of the residual entanglement, the parties may still distill higher-quality polarization entanglement. Even if not, they can still reuse the residual entanglement in the next purification round. The existence of residual entanglement benefits for increasing the yield of the EPP. All the above advantages make our EPP have potential application in future quantum information processing. ###### pacs: 03.67.Pp, 03.67.Hk, 03.65.Ud ## I Introduction Entanglement is an indispensable resource which is widely applied in quantum communication field, such as quantum teleportationteleportation1 ; teleportation2 ; teleportation3 , quantum repeater repeater1 ; repeater2 ; repeater3 , quantum key distribution (QKD) qkd , quantum secret sharing (QSS) qss , and quantum secure direct communication (QSDC) qsdc1 ; qsdc2 ; qsdc3 ; qsdc4 . The above applications often require the maximal entanglement. However, entanglement is generally fragile due to the channel noise. During the practical applications, the degraded entanglement may decrease the quantum communication efficiency or even make quantum communication insecure. Entanglement purification which was first proposed by Bennett _et al._ in 1996 EPP0 is an efficient method to distill high quality entanglement from low quality entanglement with local operation and classical communications (LOCC). Recurrence entanglement purification is the most common entanglement purification form, which has been well developed in both theory and experiment EPP0 ; Deustch ; Murao ; Pan1 ; Pan2 ; Pan3 ; graph ; atom ; Cheong ; sheng1 ; sheng2 ; Zhou1 ; Wang ; Zhou2 ; Zhang ; Dur ; Rozpeedek ; Krastanov ; Wu ; zhouap ; zhouoe ; hu ; Du ; nest ; network ; shenghyper1 ; shenghyper2 . The recurrence entanglement purification protocols (EPPs) require two or more copies of low-quality entangled states from the same enables. After two communication parties in distant locations performing the controlled-not (CNOT) or other similar operations, one pair of low-quality entangled state is measured. If the purification is successful, the fidelity of left photon pair can be increased. For example, in 2001, Pan _et al._ presented an EPP of general mixed entangled states with linear optical elements Pan1 , and later they improved their EPP by adopting available parametric down conversion sources Pan2 . In 2003, Pan _et al._ demonstrated the experiment of the entanglement purification for general mixed states of polarization-entangled photons Pan3 . In 2008, Sheng _et al._ proposed an EPP based on nondestructive quantum nondemolition detectors sheng1 . In 2017, Pan _et al._ experimentally realized the nested purification for a linear optical quantum repeater nest . In addition, the experimental purification between two-atom entanglement and solid state quantum network nodes were also demonstrated atom ; network . In 2010, Sheng _et al._ proposed the deterministic EPPs (DEPPs) by adopting the hyperentanglement shenghyper1 ; shenghyper2 . Although the recurrence entanglement purification has been well studied, existing recurrence EPPs often have relatively low yield. The reason is that in each purification round, at least one pair of low-quality entangled states should be consumed. In practical applications, entanglement purification process often has to be iterated for many rounds to obtain high-fidelity entangled pairs, so that a large amount of low-quality entangled pairs have to be consumed. It is a big waste of the precious entanglement resources. In 2021, Hu _et al._ proposed and experimentally demonstrated the first long- distance polarization entanglement recurrence purification using only one polarization-spatial-mode hyperentangled photon pair hu . They supposed that the entanglement in both polarization and spatial-mode DOFs suffer from one kind of error, say, the bit-flip error or phase-flip error. After performing the EPP, they obtained a significant improvement in the fidelity of polarization entanglement. This protocol effectively reduces consumption of copies of entanglement pairs, especially in purification consisting of many rounds. Actually, in practical entanglement distribution process, the bit-flip error and phase-flip error may be occurred simultaneously. In the paper, we consider a more general recurrence EPP which can simultaneously reduce the bit-flip error and phase-flip error of the polarization entanglement by using only one pair of polarization-spatial-time-bin hyperentangled photon pair. We choose the spatial mode and time-bin entanglement for the entanglement in both two DOFs, especially in time-bin DOF being highly robust to the channel noise. The entanglement in time-bin DOF has been successfully used in the transmission of qubits over hundreds of kilometers Pan2 ; time1 ; time2 ; time3 ; time4 and in teleportation using real-world fiber networks time5 ; time6 . In 2005, Barreiro _et al._ experimentally demonstrated the generation of hyperentanglement in polarization, spatial-mode and time-energy DOFs of photon systems using pairs of photons produced in spontaneous parametric down- conversion generation5 . In our protocol, we suppose that the entanglement in all DOFs suffer from channel noise and degrade to mixed states. As the entanglement in above three DOFs have different noise robustness, after the photon transmission, the fidelities in three DOFs are naturally different. After performing our EPP, we can efficiently reduce both the bit-flip error and phase-flip error rate in polarization DOF by consuming the imperfect entanglement in the spatial-mode and time-bin DOFs. Moreover, we will prove that if a purification step fails, there may exist residual entanglement in the corresponding distilled mixed state. By using the residual entanglement, we may also increase the fidelity of the polarization entanglement after the whole purification process. The paper is organized as follows. In Sec. II, we describe our EPP in a simple case where the entanglement in spatial-mode and time-bin DOFs only suffer from a bit-flip error. In Sec. III, we extend our EPP to a general case where both the entanglement in spatial-mode and time-bin DOFs suffer from both bit-flip error and phase-flip error. In Sec. IV, we make a discussion. In Sec. V, we make a conclusion. ## II Entanglement purification principle In this section, we propose our long-distance EPP using a hyperentangled photon pair. Suppose the photon hyperentanglement source S generates a two- photon hyperentanglement in polarization, spatial-mode, and time-bin DOFs, which can be described as $\displaystyle\|\Phi^{+}_{p}\rangle\|\Phi^{+}_{s}\rangle\|\Phi^{+}_{t}\rangle=\frac{1}{\sqrt{2}}(\|HH\rangle+\|VV\rangle)$ (1) $\displaystyle\otimes$ $\displaystyle\frac{1}{\sqrt{2}}(\|a_{1}^{\prime}b_{1}^{\prime}\rangle+\|a_{2}^{\prime}b_{2}^{\prime}\rangle)\otimes\frac{1}{\sqrt{2}}(\|LL\rangle+\|SS\rangle).$ Here, $H$ ($V$) represents the horizontal (vertical) polarization, $a_{1}^{\prime}$, $a_{2}^{\prime}$, $b_{1}^{\prime}$ and $b_{2}^{\prime}$ are four different spatial modes, and $L$ ($S$) represents the long (short) time- bin. The photon in $a_{1}^{\prime}$ and $a_{2}^{\prime}$ are sent to Alice, while the photon in $b_{1}^{\prime}$ and $b_{2}^{\prime}$ are sent to Bob. After long distance transmission, the channel noise may degrade the entanglement in all DOFs. Here, we suppose that the entanglement in polarization DOF degrade to a Werner state with the form of $\displaystyle\rho_{p}$ $\displaystyle=$ $\displaystyle p_{p}\|\Phi^{+}_{p}\rangle\langle\Phi^{+}_{p}\|+\frac{1-p_{p}}{3}(\|\Psi^{+}_{p}\rangle\langle\Psi^{+}_{p}\|$ (2) $\displaystyle+$ $\displaystyle\|\Phi^{-}_{p}\rangle\langle\Phi^{-}_{p}\|+\|\Psi^{-}_{p}\rangle\langle\Psi^{-}_{p}\|),$ where $\displaystyle\|\Phi^{\pm}_{p}\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}(\|HH\rangle\pm\|VV\rangle),$ $\displaystyle\|\Psi^{\pm}_{p}\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}(\|HV\rangle\pm\|VH\rangle).$ (3) We notice that the state $\|\Phi^{+}_{p}\rangle$ becomes $\|\Psi^{+}_{p}\rangle$ when a bit-flip error occurs, $\|\Phi^{+}_{p}\rangle$ becomes $\|\Phi^{-}_{p}\rangle$ when a phase-flip error occurs, and $\|\Phi^{+}_{p}\rangle$ becomes $\|\Psi^{-}_{p}\rangle$ when bit-flip error and phase-flip error both occur. Considering the noise robustness of the entanglement in spatial-mode and time- bin DOFs are higher than that in the polarization DOF, we first focus on a simple case where the entanglement in spatial-mode and time-bin DOFs only suffer from bit-flip error. In this case, the entanglement in both DOFs degrade to $\displaystyle\rho_{s}$ $\displaystyle=$ $\displaystyle p_{s}\|\Phi^{+}_{s}\rangle\langle\Phi^{+}_{s}\|+(1-p_{s})\|\Psi^{+}_{s}\rangle\langle\Psi^{+}_{s}\|,$ (4) $\displaystyle\rho_{t}$ $\displaystyle=$ $\displaystyle p_{t}\|\Phi^{+}_{t}\rangle\langle\Phi^{+}_{t}\|+(1-p_{t})\|\Psi^{+}_{t}\rangle\langle\Psi^{+}_{t}\|,$ (5) where $\displaystyle\|\Phi^{\pm}_{s}\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}(\|a_{1}b_{1}\rangle\pm\|a_{2}b_{2}\rangle),$ $\displaystyle\|\Psi^{\pm}_{s}\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}(\|a_{1}b_{2}\rangle\pm\|a_{2}b_{1}\rangle),$ $\displaystyle\|\Phi^{\pm}_{t}\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}(\|LL\rangle\pm\|SS\rangle),$ $\displaystyle\|\Psi^{\pm}_{t}\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}(\|LS\rangle\pm\|SL\rangle).$ (6) Therefore, the initial state in Eq. (1) degrades to $\rho_{p}\otimes\rho_{s}\otimes\rho_{t}$. Here, we suppose that all the fidelity $p_{p}$, $p_{s}$, and $p_{t}$ are higher than $\frac{1}{2}$. The schematic principle of our EPP is shown in Fig. 1. The protocol includes two steps. In the first step, Alice and Bob correct the bit-flip error in polarization DOF by consuming the entanglement in spatial-mode DOF. In the second step, they correct the phase-flip error in polarization DOF with the help of the entanglement in time-bin DOF. Figure 1: The basic principle of our two-step EPP. Suppose a photon pair hyperentangled in polarization, spatial-mode, and time-bin DOFs suffer from channel noise, and the entanglement in all DOFs degrade to mixed states. Here, PBS means polarization beam splitter, which can totally transmit the photon in $\|H\rangle$ and reflect the photon in $\|V\rangle$. QWP means the $\lambda/4$-wave plate, which can make the Hadamard (H) operation in the polarization DOF. $PC_{l(s)}$ presents the Pockels cell, which can revers the polarization of a photon with the time-bin L (S). $D_{1}-D_{8}$ are the single photon detectors. In the first step, we only use the states in polarization and spatial-mode DOF and leave the state in time-bin DOF unchanged, so that we first neglect the state in time-bin DOF and only consider the mixed state as $\rho_{1}=\rho_{p}\otimes\rho_{s}$ for simplicity. There are totally eight possible cases. The photon system may be in $\|\Phi^{+}_{p}\rangle\|\Phi^{+}_{s}\rangle$ with the probability of $p_{p}p_{s}$, while it may be in $\|\Phi^{+}_{p}\rangle\|\Psi^{+}_{s}\rangle$ with the probability of $p_{p}(1-p_{s})$. Meanwhile, the photon system may be in $\|\Psi^{+}_{p}\rangle\|\Phi^{+}_{s}\rangle$, $\|\Phi^{-}_{p}\rangle\|\Phi^{+}_{s}\rangle$, or $\|\Psi^{-}_{p}\rangle\|\Phi^{+}_{s}\rangle$ with the equal probability of $\frac{(1-p_{p})p_{s}}{3}$, and it may be in the state $\|\Psi^{+}_{p}\rangle\|\Psi^{+}_{s}\rangle$, $\|\Phi^{-}_{p}\rangle\|\Psi^{+}_{s}\rangle$, or $\|\Psi^{-}_{p}\rangle\|\Psi^{+}_{s}\rangle$ with the probability of $\frac{(1-p_{p})(1-p_{s})}{3}$. Alice and Bob pass the photons in $a_{1}$ and $a_{2}$, $b_{1}$ and $b_{2}$ spatial modes through two polarization beam splitters (PBSs), which can totally transmit the photon in $\|H\rangle$ and reflect the photon in $\|V\rangle$. If the initial photon state is $\|\Phi^{\pm}_{p}\rangle\|\Phi^{+}_{s}\rangle$, after the PBS, the state will evolve to $\displaystyle\|\Phi^{\pm}_{p}\rangle\|\Phi^{+}_{s}\rangle=\frac{1}{\sqrt{2}}(\|HH\rangle\pm\|VV\rangle)\otimes\frac{1}{\sqrt{2}}(\|a_{1}b_{1}\rangle+\|a_{2}b_{2}\rangle)$ (7) $\displaystyle\rightarrow$ $\displaystyle\frac{1}{\sqrt{2}}(\|HH\rangle\pm\|VV\rangle)\otimes\frac{1}{\sqrt{2}}(\|a_{3}b_{3}\rangle+\|a_{4}b_{4}\rangle).$ When the initial state is $\|\Psi^{\pm}_{p}\rangle\|\Psi^{+}_{s}\rangle$, it will evolve to $\displaystyle\|\Psi^{\pm}_{p}\rangle\|\Psi^{+}_{s}\rangle=\frac{1}{\sqrt{2}}(\|HV\rangle\pm\|VH\rangle)\otimes\frac{1}{\sqrt{2}}(\|a_{1}b_{2}\rangle+\|a_{2}b_{1}\rangle)$ (8) $\displaystyle\rightarrow$ $\displaystyle\frac{1}{\sqrt{2}}(\|HV\rangle\pm\|VH\rangle)\otimes\frac{1}{\sqrt{2}}(\|a_{3}b_{3}\rangle+\|a_{4}b_{4}\rangle).$ All the four above states make the spatial modes $a_{3}b_{3}$ or $a_{4}b_{4}$ each have one photon. In these cases, the first step is successful. On the other hand, if the initial state is one of the other four cases, say, $\|\Phi^{\pm}_{p}\rangle\|\Psi^{+}_{s}\rangle$ and $\|\Psi^{\pm}_{p}\rangle\|\Phi^{+}_{s}\rangle$, after the PBSs, we can obtain the spatial-modes $a_{3}b_{4}$ or $a_{4}b_{3}$ each has one photon and the first step fails. As a result, when the first step is successful, we can distill a new mixed state as $\displaystyle\rho_{1p}$ $\displaystyle=$ $\displaystyle F_{1}\|\Phi^{+}_{p}\rangle\langle\Phi^{+}_{p}\|+F_{2}\|\Phi^{-}_{p}\rangle\langle\Phi^{-}_{p}\|+F_{3}(\|\Psi^{+}_{p}\rangle\langle\Psi^{+}_{p}\|$ (9) $\displaystyle+$ $\displaystyle\|\Psi^{-}_{p}\rangle\langle\Psi^{-}_{p}\|),$ in the spatial modes $a_{3}b_{3}$ or $a_{4}b_{4}$ with the success probability of $\displaystyle P_{1}$ $\displaystyle=$ $\displaystyle p_{p}p_{s}+\frac{1-p_{p}}{3}[p_{s}+2(1-p_{s})]$ (10) $\displaystyle=$ $\displaystyle\frac{1}{3}(4p_{p}p_{s}-p_{s}-2p_{p}+2).$ The four coefficients in Eq. (9) can be written as $\displaystyle F_{1}$ $\displaystyle=$ $\displaystyle\frac{p_{p}p_{s}}{P_{1}}=\frac{3p_{p}p_{s}}{4p_{p}p_{s}-p_{s}-2p_{p}+2},$ $\displaystyle F_{2}$ $\displaystyle=$ $\displaystyle\frac{(1-p_{p})p_{s}}{3P_{1}}=\frac{(1-p_{p})p_{s}}{4p_{p}p_{s}-p_{s}-2p_{p}+2},$ $\displaystyle F_{3}$ $\displaystyle=$ $\displaystyle\frac{(1-p_{p})(1-p_{s})}{3P_{1}}=\frac{(1-p_{p})(1-p_{s})}{4p_{p}p_{s}-p_{s}-2p_{p}+2}.$ (11) It is obvious that when $p_{p}>\frac{1}{2}$ and $p_{s}>\frac{1}{2}$, the rate of $\|\Psi^{\pm}_{p}\rangle$ ($F_{3}$) is smaller than their original rate $\frac{(1-p_{p})}{3}$. As a result, the first step can reduce the rate of $\|\Psi^{\pm}_{p}\rangle$. Moreover, the reduction of bit-flip error directly increases the fidelity of $\|\Phi^{+}_{p}\rangle$. We can obtain $F_{1}>p_{p}$ and $F_{1}>p_{s}$ when $p_{p}>\frac{1}{2}$ and $\frac{5p_{p}-2}{4p_{p}-1}>p_{s}>\frac{1}{2}$. However, the first step cannot deal with the phase-flip error ($\|\Phi^{-}_{p}\rangle$) and the rate of $\|\Phi^{-}_{p}\rangle$ is still in a relatively high level. Next, we try to correct the phase-flip error. In the second step, we require to consume the entanglement in the time-bin DOF. After the first step, the whole photon system collapse to $\rho_{1p}\otimes\rho_{t}$ in $a_{3}b_{3}$ or $a_{4}b_{4}$ modes. We first consider that the photons are in $a_{3}b_{3}$. Alice and Bob first pass the photons in $a_{3}b_{3}$ modes through two $\lambda/4$-wave plates (QWPs), respectively. The QWP performs a Hadamard (H) operation in the polarization DOF, which makes $\|H\rangle\rightarrow\frac{1}{\sqrt{2}}(\|H\rangle+\|V\rangle)$ and $\|V\rangle\rightarrow\frac{1}{\sqrt{2}}(\|H\rangle-\|V\rangle)$. After the H operation, $\|\Phi^{-}_{p}\rangle\leftrightarrow\|\Psi^{+}_{p}\rangle$, while $\|\Phi^{+}_{p}\rangle$ and $\|\Psi^{-}_{p}\rangle$ keep unchanged. In this way, they can transform $\rho_{1p}$ to $\displaystyle\rho_{2p}$ $\displaystyle=$ $\displaystyle F_{1}\|\Phi^{+}_{p}\rangle\langle\Phi^{+}_{p}\|+F_{2}\|\Psi^{+}_{p}\rangle\langle\Psi^{+}_{p}\|+F_{3}(\|\Phi^{-}_{p}\rangle\langle\Phi^{-}_{p}\|$ (12) $\displaystyle+$ $\displaystyle\|\Psi^{-}_{p}\rangle\langle\Psi^{-}_{p}\|)$ in the spatial modes $a_{5}b_{5}$. The whole photon system $\rho_{2p}\otimes\rho_{t}$ can be described as follows. With probability of $F_{1}p_{t}$ and $F_{1}(1-p_{t})$ the photon pair is in $\|\Phi^{+}_{p}\rangle\|\Phi^{+}_{t}\rangle$ and $\|\Phi^{+}_{p}\rangle\|\Psi^{+}_{t}\rangle$, respectively. With a probability of $F_{2}p_{t}$ and $F_{2}(1-p_{t})$, it is in $\|\Psi^{+}_{p}\rangle\|\Phi^{+}_{t}\rangle$ and $\|\Psi^{+}_{p}\rangle\|\Psi^{+}_{t}\rangle$, respectively. On the other hand, the whole system is in $\|\Phi^{-}_{p}\rangle\|\Phi^{+}_{t}\rangle$ or $\|\Psi^{-}_{p}\rangle\|\Phi^{+}_{t}\rangle$ with an equal probability of $F_{3}p_{t}$, and in $\|\Phi^{-}_{p}\rangle\|\Psi^{+}_{t}\rangle$ or $\|\Psi^{-}_{p}\rangle\|\Psi^{+}_{t}\rangle$ with an equal probability of $F_{3}(1-p_{t})$. Suppose that the photon pair is in $\|\Phi^{+}_{p}\rangle_{a_{5}b_{5}}\|\Phi^{+}_{t}\rangle$. As shown in Fig. 1, Alice (Bob) passes the photon in $a_{5}$ $(b_{5})$ through a Pockels cell ($PC_{S}$), which can flip the polarization feature of the incoming photon under the temporal mode $S$. After the $PC_{s}$, Alice (Bob) passes the photon through a PBS, which makes the photon in $\|H\rangle$ be in $a_{7}$ $(b_{7})$ and enter a $PC_{S}$ and the photon in $\|V\rangle$ be in $a_{8}$ $(b_{8})$ and enter a $PC_{L}$. As a result, the states $\|\Phi^{\pm}_{p}\rangle\|\Phi^{+}_{t}\rangle$ will finally evolve to $\displaystyle\|\Phi^{\pm}_{p}\rangle_{a_{3}b_{3}}\|\Phi^{+}_{t}\rangle$ (13) $\displaystyle\rightarrow$ $\displaystyle\frac{1}{2}(\|H^{L}H^{L}\rangle_{a_{9}b_{9}}+\|V^{S}V^{S}\rangle_{a_{10}b_{10}}$ $\displaystyle\pm$ $\displaystyle\|H^{L}H^{L}\rangle_{a_{10}b_{10}}\pm\|V^{S}V^{S}\rangle_{a_{9}b_{9}}).$ Then, with the help of two PBSs, Alice and Bob can make the photon in $\|H\rangle$ pass through the short (S) arm and photon in $\|V\rangle$ pass through the long (L) arm. By precisely controlling the length of long and short arms, they can adjust the time-bin feature of the photons in $\|H\rangle$ and $\|V\rangle$ to be the same. In this way, we can neglect the time-bin features of the photons, and the state in Eq. (13) evolves to $\displaystyle\|\Phi^{\pm}_{p}\rangle_{a_{3}b_{3}}\|\Phi^{+}_{t}\rangle$ (14) $\displaystyle\rightarrow$ $\displaystyle\frac{1}{\sqrt{2}}(\|HH\rangle\pm\|VV\rangle)\otimes\frac{1}{\sqrt{2}}(\|d_{2}d_{6}\rangle\pm\|d_{1}d_{5}\rangle),$ and will be detected by the single photon detector $D_{2}D_{6}$ or $D_{1}D_{5}$. In this case, the second purification step is successful. The polarization feature of the photon pair remains to be $\|\Phi^{\pm}_{p}\rangle$. If the initial state is $\|\Psi^{\pm}_{p}\rangle\|\Psi^{+}_{t}\rangle$, after the above operations, it will evolve to $\displaystyle\|\Psi^{\pm}_{p}\rangle\|\Psi^{+}_{t}\rangle$ (15) $\displaystyle\rightarrow$ $\displaystyle\frac{1}{\sqrt{2}}(\|HV\rangle\pm\|VH\rangle)\otimes\frac{1}{\sqrt{2}}(\|d_{2}d_{6}\rangle\pm\|d_{1}d_{5}\rangle),$ which will also lead to the successful detection results. The state in Eq. (15) will finally collapse to $\|\Psi^{\pm}_{p}\rangle$. If the initial state is $\|\Phi^{\pm}_{p}\rangle\|\Psi^{+}_{t}\rangle$ or $\|\Psi^{\pm}_{p}\rangle\|\Phi^{+}_{t}\rangle$, after the above operations, Alice and Bob would never obtain the successful measurement results. In detail, after above operations, we can obtain $\displaystyle\|\Phi^{\pm}_{p}\rangle\|\Psi^{+}_{t}\rangle$ (16) $\displaystyle\rightarrow$ $\displaystyle\frac{1}{\sqrt{2}}(\|HV\rangle\pm\|VH\rangle)\otimes\frac{1}{\sqrt{2}}(\|d_{2}d_{5}\rangle\pm\|d_{1}d_{6}\rangle),$ $\displaystyle\|\Psi^{\pm}_{p}\rangle\|\Phi^{+}_{t}\rangle$ $\displaystyle\rightarrow$ $\displaystyle\frac{1}{\sqrt{2}}(\|HH\rangle\pm\|VV\rangle)\otimes\frac{1}{\sqrt{2}}(\|d_{2}d_{5}\rangle\pm\|d_{1}d_{6}\rangle),$ which makes the single photon detectors $D_{1}D_{6}$ or $D_{2}D_{5}$ each register a single photon. In this case, the second purification step fails. On the other hand, if the photon state is in $a_{4}b_{4}$ modes, we can obtain when the photon detectors $D_{3}D_{7}$ or $D_{4}D_{8}$ each register one photon, the second purification step is successful. Therefore, when the second purification step is successful, we can distill a new mixed state in polarization DOF as $\displaystyle\rho_{3p}$ $\displaystyle=$ $\displaystyle F^{\prime}_{1}\|\Phi^{+}_{p}\rangle\langle\Phi^{+}_{p}\|+F^{\prime}_{2}\|\Phi^{-}_{p}\rangle\langle\Phi^{-}_{p}\|$ (17) $\displaystyle+$ $\displaystyle F^{\prime}_{3}\|\Psi^{+}_{p}\rangle\langle\Psi^{+}_{p}\|+F^{\prime}_{4}\|\Psi^{-}_{p}\rangle\langle\Psi^{-}_{p}\|,$ with a success probability of $\displaystyle P_{2}$ $\displaystyle=$ $\displaystyle(F_{1}+F_{3})p_{t}+(F_{2}+F_{3})(1-p_{t})$ (18) $\displaystyle=$ $\displaystyle\frac{1-p_{p}-p_{s}p_{t}+4p_{p}p_{s}p_{t}}{2-2p_{p}-p_{s}+4p_{p}p_{s}}.$ The four coefficients in Eq. (17) can be written as $\displaystyle F^{\prime}_{1}$ $\displaystyle=$ $\displaystyle\frac{F_{1}p_{t}}{P_{2}}=\frac{3p_{p}p_{s}p_{t}}{1-p_{t}p_{s}+p_{p}(4p_{t}p_{s}-1)},$ $\displaystyle F^{\prime}_{2}$ $\displaystyle=$ $\displaystyle\frac{F_{3}p_{t}}{P_{2}}=\frac{(1-p_{p})(1-p_{s})p_{t}}{1-p_{t}p_{s}+p_{p}(4p_{t}p_{s}-1)},$ $\displaystyle F^{\prime}_{3}$ $\displaystyle=$ $\displaystyle\frac{F_{2}(1-p_{t})}{P_{2}}=\frac{(1-p_{p})(1-p_{t})p_{s}}{1-p_{t}p_{s}+p_{p}(4p_{t}p_{s}-1)},$ $\displaystyle F^{\prime}_{4}$ $\displaystyle=$ $\displaystyle\frac{F_{3}(1-p_{t})}{P_{2}}=\frac{(1-p_{p})(1-p_{s})(1-p_{t})}{1-p_{t}p_{s}+p_{p}(4p_{t}p_{s}-1)}.$ (19) Similarly as the first step, the second step can reduce the rate of $\|\Psi^{\pm}_{p}\rangle$ ($F^{\prime}_{3}<F_{2}$ and $F^{\prime}_{4}<F_{3}$). It can be calculated that $F^{\prime}_{1}>F_{1}$ and $F^{\prime}_{1}>p_{t}$ when $\frac{3p_{p}p_{s}}{+}p_{p}-1{p_{s}(4p_{p}-1)}>p_{t}>\frac{1}{2}$. Comparing with original mixed state $\rho_{p}$ in Eq. (2), after two steps of purification, the rates of $\|\Psi^{\pm}\rangle$ and $\|\Phi^{-}\rangle$ can be all reduced, so that the fidelity of $\|\Phi^{+}_{p}\rangle$ can be efficiently increased. ## III General entanglement purification In this section, we consider a general case that after the long-distance transmission in noisy channel, the entanglement in the spatial-mode and time- bin DOFs also degrade to Werner states. In this way, the mixed states in above two DOFs can be written as $\displaystyle\rho_{sn}$ $\displaystyle=$ $\displaystyle p_{s}\|\Phi^{+}_{s}\rangle\langle\Phi^{+}_{s}\|+\frac{1-p_{s}}{3}(\|\Psi^{+}_{s}\rangle\langle\Psi^{+}_{s}\|$ $\displaystyle+$ $\displaystyle\|\Phi^{-}_{s}\rangle\langle\Phi^{-}_{s}\|+\|\Psi^{-}_{s}\rangle\langle\Psi^{-}_{s}\|),$ $\displaystyle\rho_{tn}$ $\displaystyle=$ $\displaystyle p_{t}\|\Phi^{+}_{t}\rangle\langle\Phi^{+}_{t}\|+\frac{1-p_{t}}{3}(\|\Psi^{+}_{t}\rangle\langle\Psi^{+}_{t}\|$ (20) $\displaystyle+$ $\displaystyle\|\Phi^{-}_{t}\rangle\langle\Phi^{-}_{t}\|+\|\Psi^{-}_{t}\rangle\langle\Psi^{-}_{t}\|).$ In Eq. (2) and Eq. (20), we also suppose that $p_{p(t,s)}>\frac{1}{2}$. In the first step, we only consider $\rho_{p}\otimes\rho_{sn}$, which has 16 possible cases, say $\|\Phi^{\pm}_{p}\rangle\|\Phi^{\pm}_{s}\rangle$, $\|\Phi^{\pm}_{p}\rangle\|\Psi^{\pm}_{s}\rangle$, $\|\Psi^{\pm}_{p}\rangle\|\Phi^{\pm}_{s}\rangle$, and $\|\Psi^{\pm}_{p}\rangle\|\Psi^{\pm}_{s}\rangle$. By passing the photons in $a_{1}a_{2}$ and $b_{1}b_{2}$ modes through the PBSs, we also select the items which make the spatial modes $a_{3}b_{3}$ or $a_{4}b_{4}$ each have one photon. All the 8 initial states $\|\Phi^{\pm}_{p}\rangle\|\Phi^{\pm}_{s}\rangle$ and $\|\Psi^{\pm}_{p}\rangle\|\Psi^{\pm}_{s}\rangle$ can lead to the successful cases. As a result, when the first step is successful, the parties can distill a new mixed state in polarization DOF as $\displaystyle\rho_{1pn}$ $\displaystyle=$ $\displaystyle F_{1n}\|\Phi^{+}_{p}\rangle\langle\Phi^{+}_{p}\|+F_{2n}\|\Phi^{-}_{p}\rangle\langle\Phi^{-}_{p}\|$ (21) $\displaystyle+$ $\displaystyle F_{3n}(\|\Psi^{+}_{p}\rangle\langle\Psi^{+}_{p}\|+\|\Psi^{-}_{p}\rangle\langle\Psi^{-}_{p}\|).$ with a probability of $\displaystyle P_{1n}$ $\displaystyle=$ $\displaystyle p_{p}p_{s}+\frac{(1-p_{p})(1-p_{s})}{9}$ (22) $\displaystyle+$ $\displaystyle\frac{p_{p}(1-p_{s})+p_{s}(1-p_{p})}{3}+\frac{4(1-p_{p})(1-p_{s})}{9}$ $\displaystyle=$ $\displaystyle\frac{8p_{p}p_{s}-2p_{p}-2p_{s}+5}{9}.$ In $\rho_{1pn}$, the rates of $\|\Phi^{\pm}_{p}\rangle$ and $\|\Psi^{\pm}_{p}\rangle$ can be written as $\displaystyle F_{1n}$ $\displaystyle=$ $\displaystyle\frac{p_{p}p_{s}+\frac{(1-p_{p})(1-p_{s})}{9}}{P_{1n}}$ $\displaystyle=$ $\displaystyle\frac{10p_{p}p_{s}-p_{p}-p_{s}+1}{8p_{p}p_{s}-2p_{p}-2p_{s}+5},$ $\displaystyle F_{2n}$ $\displaystyle=$ $\displaystyle\frac{p_{p}(1-p_{s})+(1-p_{p})p_{s}}{3P_{1n}}$ $\displaystyle=$ $\displaystyle\frac{3p_{p}+3p_{s}-6p_{p}p_{s}}{8p_{p}p_{s}-2p_{p}-2p_{s}+5},$ $\displaystyle F_{3n}$ $\displaystyle=$ $\displaystyle\frac{2(1-p_{p})(1-p_{s})}{9P_{1n}}$ (23) $\displaystyle=$ $\displaystyle\frac{2(1-p_{p})(1-p_{s})}{8p_{p}p_{s}-2p_{p}-2p_{s}+5}.$ Figure 2: The value of $F_{1n}$, $F_{2n}$, and $F_{3n}$ as a function of $p_{s}$. Here, we control $p_{p}=0.6$ and adjust $p_{s}$ from 0.505 to 1. In Fig. 2, we show the values of $F_{1n}$, $F_{2n}$, and $F_{3n}$ as a function of $p_{s}$. Here, we control $p_{p}=0.6$ and adjust $p_{s}$ from 0.505 to 1. It is obvious that both $F_{2n}$ and $F_{3n}$ reduce with the growth of $p_{s}$, which makes $F_{1n}$ increase with the growth of $p_{s}$. We also obtain that $F_{3n}<\frac{1-p_{p}}{3}$ when $p_{s}>\frac{1}{2}$, so that we can reduce the rate of bit-flip error. However, in this general case, as the rate of $\|\Phi^{-}_{p}\rangle$ ($F_{2n}$) may be relatively high, and we cannot simply obtain $F_{1n}>p_{p}$ or $F_{1n}>p_{s}$ when $p_{s}>\frac{1}{2}$ and $p_{p}>\frac{1}{2}$. Here, we provide the criterion of $F_{1n}>p_{p}$ as $\displaystyle p_{s}>\frac{6p_{p}-2p_{p}^{2}-1}{12p_{p}-8p_{p}^{2}-1},$ (24) and the criterion of $F_{1n}>p_{s}$ as $\displaystyle(8p_{p}-2)p_{s}^{2}+6(1-2p_{p})p_{s}+p_{p}-1<0.$ (25) Figure 3: The minimum threshold of $p_{s}$ corresponding to $F_{1n}>p_{p}$, and the maximum threshold of $p_{s}$ corresponding to $F_{1n}>p_{s}$ as a function of $p_{p}$. Here, we control $p_{p}\in[0.505,0.95]$. Fig. 3 provides the minimum threshold of $p_{s}$ corresponding to $F_{1n}>p_{p}$, and the maximum threshold of $p_{s}$ corresponding to $F_{1n}>p_{s}$ under different values of $p_{p}$. It can be found that both the minimum and maximum thresholds of $p_{s}$ increase with the growth of $p_{p}$. Combined with Fig. 2 and Fig. 3, we can obtain that a high $p_{s}$ can lead to a small $F_{3n}$ and relatively high $F_{1n}$. However, when the value of $p_{s}$ is too high, we cannot ensure that $F_{1n}>p_{s}$. In this way, for satisfying both the criterions in Eq. (24) and Eq. (30), the practical value of $p_{s}$ should be between two thresholds. On the other hand, even when $p_{s}$ is relatively high, $F_{2n}$ can be still in a relatively high level, so that we need to perform the second purification step to further reduce $F_{2n}$ and increase the fidelity of $\|\Phi^{+}_{p}\rangle$. In the second purification step, with the help of H operation, we can transform $\rho_{1pn}$ to $\rho_{2pn}$ with the form of $\displaystyle\rho_{2pn}$ $\displaystyle=$ $\displaystyle F_{1n}\|\Phi^{+}_{p}\rangle\langle\Phi^{+}_{p}\|+F_{3n}\|\Phi^{-}_{p}\rangle\langle\Phi^{-}_{p}\|$ (26) $\displaystyle+$ $\displaystyle F_{2n}\|\Psi^{+}_{p}\rangle\langle\Psi^{+}_{p}\|+F_{3n}\|\Psi^{-}_{p}\rangle\langle\Psi^{-}_{p}\|,$ and $F_{2n}$ transforms to the rate of $\|\Psi^{+}_{p}\rangle$. In this way, the whole photon system is $\rho_{2pn}\otimes\rho_{tn}$, which also has 16 possible cases. Alice and Bob pass the photons in $a_{3}b_{3}$ and $a_{4}b_{4}$ modes through the purification units. When the detection result is $D_{2}D_{6}$, $D_{1}D_{5}$, $D_{3}D_{7}$ or $D_{4}D_{8}$ each registering one photon, the second purification step will be successful. According to the description in Sec. II, all the states $\|\Phi^{\pm}_{p}\rangle\|\Phi^{\pm}_{t}\rangle$ and $\|\Psi^{\pm}_{p}\rangle\|\Psi^{\pm}_{t}\rangle$ can lead to above successful detection. In this way, the success probability of the second step is $\displaystyle P_{2n}$ $\displaystyle=$ $\displaystyle F_{1n}p_{t}+F_{1n}\frac{1-p_{t}}{3}+F_{3n}p_{t}$ $\displaystyle+$ $\displaystyle 3F_{3n}\frac{1-p_{t}}{3}+2F_{2n}\frac{1-p_{t}}{3}$ $\displaystyle=$ $\displaystyle\frac{7-p_{s}+p_{p}(4p_{s}-1)+2p_{t}(4p_{p}-1)(4p_{s}-1)}{3(8p_{p}p_{s}-2p_{p}-2p_{s}+5)}.$ When the successful detection is obtained, we can distill a new mixed state as $\displaystyle\rho_{3pn}$ $\displaystyle=$ $\displaystyle F^{\prime}_{1n}\|\Phi^{+}_{p}\rangle\langle\Phi^{+}_{p}\|+F^{\prime}_{2n}\|\Phi^{-}_{p}\rangle\langle\Phi^{-}_{p}\|$ (28) $\displaystyle+$ $\displaystyle F^{\prime}_{3n}(\|\Psi^{+}_{p}\rangle\langle\Psi^{+}_{p}\|+\|\Psi^{-}_{p}\rangle\langle\Psi^{-}_{p}\|),$ where $\displaystyle F^{\prime}_{1n}$ $\displaystyle=$ $\displaystyle\frac{F_{1n}p_{t}+F_{3n}\frac{1-p_{t}}{3}}{P_{2n}}$ $\displaystyle=$ $\displaystyle\frac{2(1-p_{p})(1-p_{s})+p_{t}(1-p_{s}-p_{p}+28p_{p}p_{s})}{7-p_{s}-p_{p}+4p_{p}p_{s}+2p_{t}(4p_{p}-1)(4p_{s}-1)},$ $\displaystyle F^{\prime}_{2n}$ $\displaystyle=$ $\displaystyle\frac{F_{1n}\frac{1-p_{t}}{3}+F_{3n}p_{t}}{P_{2n}}$ $\displaystyle=$ $\displaystyle\frac{p_{s}+p_{p}-10p_{p}p_{s}-1+p_{t}(5p_{s}+5p_{p}+4p_{p}p_{s}-5)}{7-p_{s}-p_{p}+4p_{p}p_{s}+2p_{t}(4p_{p}-1)(4p_{s}-1)},$ $\displaystyle F^{\prime}_{3n}$ $\displaystyle=$ $\displaystyle\frac{(F_{2n}+F_{3n})\frac{1-p_{t}}{3}}{P_{2n}}$ (29) $\displaystyle=$ $\displaystyle\frac{(1-p_{t})(p_{s}+p_{p}+2-4p_{p}p_{s})}{7-p_{s}-p_{p}+4p_{p}p_{s}+2p_{t}(4p_{p}-1)(4p_{s}-1)}.$ Figure 4: The value of $F^{\prime}_{1n}$, $F^{\prime}_{2n}$, and $F^{\prime}_{3n}$ as a function of $p_{t}$. Here, we control $p_{p}=0.6$ and $p_{s}=0.8$, and adjust $p_{t}$ from 0.505 to 1. In Fig. 4, we control $p_{p}=0.6$ and $p_{s}=0.8$ (In this case, we can obtain $F_{1n}\approx 0.7285$), and show the values of $F^{\prime}_{1n}$, $F^{\prime}_{2n}$, and $F^{\prime}_{3n}$ as a function of $p_{t}$. It can be found that both $F^{\prime}_{2n}$ and $F^{\prime}_{3n}$ reduce with the growth of $p_{t}$, which makes $F^{\prime}_{1n}$ increase. The higher value of $p_{t}$ leads to higher $F^{\prime}_{1n}$ and lower $F^{\prime}_{2n}$ and $F^{\prime}_{3n}$. It is important to compare $F^{\prime}_{1n}$ with $F_{1n}$ and $p_{t}$. We can also calculate the criterion for $F^{\prime}_{1n}>F_{1n}$ as $\displaystyle\frac{3(p_{t}-\frac{1}{2})}{1-p_{t}}>\frac{(F_{1n}-\frac{1}{2})(F_{1n}+F_{3n})}{F_{1n}[1-(F_{1n}+F_{3n})]}.$ (30) and the criterion for $F^{\prime}_{1n}>p_{t}$ as $\displaystyle p_{t}^{2}(4p_{p}-1)(4p_{s}-1)+3p_{t}(1-4p_{p}p_{s})<(1-p_{p})(1-p_{s}).$ (31) Figure 5: The minimum threshold of $p_{t}$ corresponding to $F^{\prime}_{1n}>F_{1n}$, and the maximum threshold of $p_{t}$ corresponding to $F^{\prime}_{1n}>p_{t}$ as a function of $p_{s}$. Here, we control $p_{p}=0.65$, and change $p_{s}$ from 0.61 to 0.71. Similar as the fist step, Eq. (30) and Eq. (31) provide the minimum threshold and maximum threshold of $p_{t}$. The practical value of $p_{t}$ should be between the two thresholds. For example, we suppose that $p_{p}=0.65$, where the suitable value of $p_{s}$ should be in the scale $(0.601,0.715)$. Under this case, we control the $p_{s}$ in the scale of $[0.61,0.71]$ and provide the minimum value and maximal value of $p_{t}$ altered with the value of $p_{s}$ in Fig. 5. ## IV Discussion In the paper, we demonstrate an efficient and simple two-step EPP assisted with hyperentanglement. This EPP requires only one copy of photon pair hyperentangled in polarization, spatial-mode, and time-bin DOFs. We consider a general degradation model that the entanglement in all DOFs suffer from channel noise. By consuming the imperfect entanglement in spatial-mode and time-bin DOFs, we can reduce both the bit-flip error and phase-flip error in polarization DOF and increase the fidelity of the target polarization state. Our two-step EPP has some attractive advantages. First, comparing with previous recurrence EPPs which require two or more copies of low-quality entangled pairs, our EPP uses the spatial and time-bin information to complete the measurement pointer, which avoids consuming an entangled photon copy. In this way, our EPP protocol effectively reduces consumption of entanglement pairs, especially in purification consisting of many rounds. Second, using only one pair of hyperentangled state also reduces the experimental difficulty, for it is hard to generate two pairs of hyperentangled states simultaneously. Third, all the devices in our EPP are available under current experimental condition, so that our EPP is feasible for experiment. Figure 6: The value of $LgR$ as a function of the photon distribution length. We set $\eta_{d}=0.9$ and $\eta_{c}=0.95$ L , and consider the low initial fidelity case ($p_{p}=0.52$, $p_{s}=0.56$, $p_{t}=0.60$) and high initial fidelity case ($p_{p}=0.8$, $p_{s}=0.82$, $p_{t}=0.85$), respectively. It is important to compare the purification efficiency of our two-step EPP with previous recurrence EPPs Pan1 in linear optics in a practical environment. In previous EPPs, for reducing both the bit-flip and phase-flip error in polarization DOF, four pairs of identical low-quality mixed states should be distributed to Alice and Bob (Suppose that the low quality mixed states are the Werner states with the form of Eq. (2)). The transmission efficiency of each photon is $\eta_{t}=e^{-\frac{d}{d_{0}}}$, where $d_{0}$ is the attenuation length of the channel (25 km for commercial fibre L ) and $d$ is the practical photon transmission distance. They first perform the purification operation on each two identical pairs to reduce the bit-flip error. Only when both the purification operations are successful, they perform the H operation on the distilled two photon pairs and further correct the phase-flip error. We suppose that $\eta_{d}$ and $\eta_{c}$ are the detection efficiency of the practical photon detector and the coupling efficiency of a photon to the photon detector, respectively. The total purification efficiency of previous EPP Pan1 can be calculated as $\displaystyle E_{o}=\frac{1}{4}P_{1t}^{2}P_{2t}\eta_{t}^{8}\eta_{d}^{8}\eta_{c}^{8}.$ (32) Here, $P_{1t}$ and $P_{2t}$ represent the success probability of the first and second purification rounds, respectively, which can be calculated as $\displaystyle P_{1t}$ $\displaystyle=$ $\displaystyle\frac{1}{2}[p_{p}^{2}+\frac{2p_{p}(1-p_{p})}{3}+\frac{5(1-p_{p})^{2}}{9}],$ $\displaystyle P_{2t}$ $\displaystyle=$ $\displaystyle\frac{1}{8P_{1t}^{2}}\\{[p_{p}^{2}+\frac{(1-p_{p})^{2}}{9}]^{2}+\frac{8}{81}(1-p_{p})^{4}$ (33) $\displaystyle+$ $\displaystyle\frac{4}{9}[p_{p}^{2}+\frac{(1-p_{p})^{2}}{9}](1-p_{p})^{2}+\frac{4}{9}p_{p}^{2}(1-p_{p})^{2}$ $\displaystyle+$ $\displaystyle\frac{8}{27}p_{p}(1-p_{p})^{3}\\}.$ On the other hand, according to above description, the total purification efficiency of our two-step EPP is $\displaystyle E_{n}=P_{1n}P_{2n}\eta_{t}^{2}\eta_{d}^{2}\eta_{c}^{2}.$ (34) In this way, the ratio of $E_{n}$ and $E_{o}$ can be defined as $\displaystyle R=\frac{E_{n}}{E_{o}}=\frac{4P_{1n}P_{2n}}{P_{1t}^{2}P_{2t}\eta_{t}^{6}\eta_{d}^{6}\eta_{c}^{6}}$ (35) Fig. 6 shows the value of $LgR$ as a function of the photon transmission length $d$. Here, we set $d_{0}=25$ $km$, $\eta_{d}=0.9$ and $\eta_{c}=0.95$ L , and change the distance $d$ from 0 to 100 $km$. Here, we select the low initial fidelity case ($p_{p}=0.52$, $p_{s}=0.56$, $p_{t}=0.60$) and high initial fidelity case ($p_{p}=0.8$, $p_{s}=0.82$, $p_{t}=0.85$), respectively. It can be found that the influences from the initial fidelities in three DOFs on $LgR$ are slight, and $LgR$ increases linearly with the growth of $d$. In this way, our EPP is extremely useful in the long-distance entanglement distribution. Next, we discuss the residual entanglement when the purification steps fail. Here, we consider the case in Sec. II for simplicity. We first consider the case that the first purification step fails, but the second step is successful. When the first step fails, say, the spatial modes $a_{3}b_{4}$ or $a_{4}b_{3}$ each have a photon, the parties can distill a new mixed state with the form of $\displaystyle\rho_{fail1}$ $\displaystyle=$ $\displaystyle F_{fail1}\|\Phi^{+}_{p}\rangle\langle\Phi^{+}_{p}\|+A_{fail1}\|\Phi^{-}_{p}\rangle\langle\Phi^{-}_{p}\|$ (36) $\displaystyle+$ $\displaystyle B_{fail1}(\|\Psi^{+}_{p}\rangle\langle\Psi^{+}_{p}\|+\|\Psi^{-}_{p}\rangle\langle\Psi^{-}_{p}\|),$ with the probability of $\displaystyle P_{fail1}$ $\displaystyle=$ $\displaystyle p_{p}(1-p_{s})+\frac{(1-p_{p})(1-p_{s})}{3}+\frac{2(1-p_{p})p_{s}}{3}$ (37) $\displaystyle=$ $\displaystyle\frac{1+p_{s}+2p_{p}-4p_{p}p_{s}}{3}.$ The three coefficients can be calculated as $\displaystyle F_{fail1}$ $\displaystyle=$ $\displaystyle\frac{p_{p}(1-p_{s})}{P_{fail1}}=\frac{3p_{p}(1-p_{s})}{1+p_{s}+2p_{p}-4p_{p}p_{s}},$ $\displaystyle A_{fail1}$ $\displaystyle=$ $\displaystyle\frac{\frac{(1-p_{p})(1-p_{s})}{3}}{P_{fail1}}=\frac{(1-p_{p})(1-p_{s})}{1+p_{s}+2p_{p}-4p_{p}p_{s}},$ $\displaystyle B_{fail1}$ $\displaystyle=$ $\displaystyle\frac{\frac{(1-p_{p})p_{s}}{3}}{P_{fail1}}=\frac{(1-p_{p})p_{s}}{1+p_{s}+2p_{p}-4p_{p}p_{s}}.$ (38) In order to make the distilled mixed state have residual entanglement, we require $F_{fail1}>\frac{1}{2}$. This requirement can be satisfied when $p_{s}<\frac{4p_{p}-1}{1+2p_{p}}$. Under this case, when the second purification step is successful, say, the photon detectors $D_{2}D_{7}$, $D_{1}D_{8}$, $D_{3}D_{6}$, or $D_{4}D_{5}$ each registering a single photon, the parties can obtain a new mixed state as $\displaystyle\rho^{\prime}_{fail1}$ $\displaystyle=$ $\displaystyle F^{\prime}_{fail1}\|\Phi^{+}_{p}\rangle\langle\Phi^{+}_{p}\|+A^{\prime}_{fail1}\|\Phi^{-}_{p}\rangle\langle\Phi^{-}_{p}\|$ (39) $\displaystyle+$ $\displaystyle B^{\prime}_{fail1}\|\Psi^{+}_{p}\rangle\langle\Psi^{+}_{p}\|+C^{\prime}_{fail1}\|\Psi^{-}_{p}\rangle\langle\Psi^{-}_{p}\|,$ with the probability of $\displaystyle P^{\prime}_{fail1}$ $\displaystyle=$ $\displaystyle(F_{fail1}+B_{fail1})p_{t}+(A_{fail1}+B_{fail1})(1-p_{t})$ (40) $\displaystyle=$ $\displaystyle\frac{1-p_{p}+p_{t}(4p_{p}-1)(1-p_{s})}{1+p_{s}-2p_{p}(2p_{s}-1)}.$ The coefficients in Eq. (39) can be written as $\displaystyle F^{\prime}_{fail1}$ $\displaystyle=$ $\displaystyle\frac{F_{fail1}p_{t}}{P^{\prime}_{fail1}}$ $\displaystyle=$ $\displaystyle\frac{3p_{t}p_{p}(1-p_{s})}{1-p_{p}+p_{t}(4p_{p}-1)(1-p_{s})},$ $\displaystyle A^{\prime}_{fail1}$ $\displaystyle=$ $\displaystyle\frac{B_{fail1}p_{t}}{P^{\prime}_{fail1}}$ $\displaystyle=$ $\displaystyle\frac{p_{t}p_{s}(1-p_{p})}{1-p_{p}+p_{t}(4p_{p}-1)(1-p_{s})},$ $\displaystyle B^{\prime}_{fail1}$ $\displaystyle=$ $\displaystyle\frac{A_{fail1}(1-p_{t})}{P^{\prime}_{fail1}}$ $\displaystyle=$ $\displaystyle\frac{(1-p_{t})(1-p_{s})(1-p_{p})}{1-p_{p}+p_{t}(4p_{p}-1)(1-p_{s})},$ $\displaystyle C^{\prime}_{fail1}$ $\displaystyle=$ $\displaystyle\frac{B_{fail1}(1-p_{t})}{P^{\prime}_{fail1}}$ (41) $\displaystyle=$ $\displaystyle\frac{p_{s}(1-p_{t})(1-p_{p})}{1-p_{p}+p_{t}(4p_{p}-1)(1-p_{s})}.$ Figure 7: The value of $F^{\prime}_{fail}$ as a function of $p_{t}$. Here, we control $p_{p}=0.65$, so that maximal value of $p_{s}$ for ensuring $F_{fail1}>\frac{1}{2}$ is 0.70. In this way, we control $p_{s}=$ 0.52, 0.62, 0.68, respectively. Fig. 7 shows the value of $F^{\prime}_{fail1}$ as a function of $p_{t}$ when $p_{p}=0.65$. For ensuring $F_{fail1}>\frac{1}{2}$, we require $p_{s}<0.7$. In this way, we control $p_{s}=$ 0.52, 0.62, 0.68, respectively. It can be found that $F^{\prime}_{fail1}$ increases with the growth of $p_{t}$, but reduces with the growth of $p_{s}$. With suitable value of $p_{t}$, we can obtain $F^{\prime}_{fail1}>p_{p}$ and $F^{\prime}_{fail1}>p_{t}$. Second, we consider the case that the first purification is successful, but the second purification fails. After the first step, the parties share a mixed state as $\rho_{1p}$ in Eq. (9). When the second purification step fails, say, the photon detectors $D_{1}D_{6}$, $D_{2}D_{5}$, $D_{3}D_{8}$, or $D_{4}D_{7}$ each registering a single photon, they can distill a new mixed state with the form of $\displaystyle\rho_{fail2}$ $\displaystyle=$ $\displaystyle F_{fail2}\|\Phi^{+}_{p}\rangle\langle\Phi^{+}_{p}\|+A_{fail2}\|\Phi^{-}_{p}\rangle\langle\Phi^{-}_{p}\|$ (42) $\displaystyle+$ $\displaystyle B_{fail2}\|\Psi^{+}_{p}\rangle\langle\Psi^{+}_{p}\|+C_{fail2}\|\Psi^{-}_{p}\rangle\langle\Psi^{-}_{p}\|,$ with the probability of $\displaystyle P_{fail2}$ $\displaystyle=$ $\displaystyle(F_{1}+F_{3})(1-p_{t})+(F_{2}+F_{3})p_{t}$ (43) $\displaystyle=$ $\displaystyle\frac{1-(1-p_{t})p_{s}-p_{p}+4p_{p}p_{s}(1-p_{t})}{2-p_{s}-2p_{p}(1-2p_{s})}.$ The coefficients in Eq. (42) can be written as $\displaystyle F_{fail2}$ $\displaystyle=$ $\displaystyle\frac{F_{1}(1-p_{t})}{P_{fail2}}$ $\displaystyle=$ $\displaystyle\frac{3(1-p_{t})p_{p}p_{s}}{1-p_{s}+p_{t}p_{s}-p_{p}+4p_{p}p_{s}(1-p_{t})},$ $\displaystyle A_{fail2}$ $\displaystyle=$ $\displaystyle\frac{F_{3}(1-p_{t})}{P_{fail2}}$ $\displaystyle=$ $\displaystyle\frac{(1-p_{p})(1-p_{s})(1-p_{t})}{1-p_{s}+p_{t}p_{s}-p_{p}+4p_{p}p_{s}(1-p_{t})},$ $\displaystyle B_{fail2}$ $\displaystyle=$ $\displaystyle\frac{F_{2}p_{t}}{P_{fail2}},$ $\displaystyle=$ $\displaystyle\frac{(1-p_{p})p_{s}p_{t}}{1-p_{s}+p_{t}p_{s}-p_{p}+4p_{p}p_{s}(1-p_{t})},$ $\displaystyle C_{fail2}$ $\displaystyle=$ $\displaystyle\frac{F_{3}p_{t}}{P_{fail2}},$ (44) $\displaystyle=$ $\displaystyle\frac{(1-p_{p})(1-p_{s})p_{t}}{1-p_{s}+p_{t}p_{s}-p_{p}+4p_{p}p_{s}(1-p_{t})}.$ Figure 8: The value of $F_{fail2}$ as a function of $p_{t}$. Here, we control $p_{p}=0.65$ and $p_{s}=$ 0.52, 0.62, 0.7, 0.8, respectively, and adjust $p_{t}$ from 0.51 to 1. In Fig. 8, we show the value of $F_{fail2}$ as a function of $p_{t}$ by controlling $p_{p}=0.65$ and $p_{s}=$ 0.52, 0.62, 0.7, 0.8. It can be found that $F_{fail2}$ increases with the growth of $p_{s}$ but reduces with the growth of $p_{t}$. In this way, for obtaining $F_{fail2}>p_{p}$, we require $p_{t}$ to be relatively low. With the growth of $p_{s}$, the maximal values of $p_{t}$ which make $F_{fail2}>p_{p}$ increase. In detail, when $p_{p}=0.65$, and $p_{s}=$ 0.52, 0.62, 0.7, 0.8, the maximal values of $p_{t}$ are 0.519, 0.596, 0.642, and 0.687, respectively. Figure 9: The values of $F_{fail3}$ as a function of $p_{t}$. Here, we control $p_{p}=0.65$ and $p_{s}=$ 0.52, 0.62, 0.68, respectively, and adjust $p_{t}$ from 0.51 to 1. Finally, we will discuss the case that both two purification steps fail. This case corresponds to the photon detector $D_{2}D_{8}$, $D_{1}D_{7}$, $D_{3}D_{5}$, or $D_{4}D_{6}$ each registering one photon. After the first purification step, the parties share a new mixed state with the form of $\rho_{fail1}$ in Eq. (36). Then, when the second purification step fails, they can finally obtain a new mixed state as $\displaystyle\rho_{fail3}$ $\displaystyle=$ $\displaystyle F_{fail3}\|\Phi^{+}_{p}\rangle\langle\Phi^{+}_{p}\|+A_{fail3}\|\Phi^{-}_{p}\rangle\langle\Phi^{-}_{p}\|$ (45) $\displaystyle+$ $\displaystyle B_{fail3}\|\Psi^{+}_{p}\rangle\langle\Psi^{+}_{p}\|+C_{fail3}\|\Psi^{-}_{p}\rangle\langle\Psi^{-}_{p}\|,$ with the probability of $\displaystyle P_{fail3}$ $\displaystyle=$ $\displaystyle(F_{fail1}+B_{fail1})(1-p_{t})+(A_{fail1}+B_{fail1})p_{t},$ (46) $\displaystyle=$ $\displaystyle\frac{p_{p}(3-4p_{s})-p_{t}(4p_{p}-1)(1-p_{s})+p_{s}}{1-2p_{p}(2p_{s}-1)+p_{s}}.$ The four coefficients can be calculated as $\displaystyle F_{fail3}$ $\displaystyle=$ $\displaystyle\frac{F_{fail1}(1-p_{t})}{P_{fail3}}$ $\displaystyle=$ $\displaystyle\frac{3p_{p}(1-p_{t})(1-p_{s})}{p_{p}(3-4p_{s})-p_{t}(4p_{p}-1)(1-p_{s})+p_{s}},$ $\displaystyle A_{fail3}$ $\displaystyle=$ $\displaystyle\frac{B_{fail1}(1-p_{t})}{P_{fail3}}$ $\displaystyle=$ $\displaystyle\frac{(1-p_{p})(1-p_{t})p_{s}}{p_{p}(3-4p_{s})-p_{t}(4p_{p}-1)(1-p_{s})+p_{s}},$ $\displaystyle B_{fail3}$ $\displaystyle=$ $\displaystyle\frac{A_{fail1}p_{t}}{P_{fail3}}$ $\displaystyle=$ $\displaystyle\frac{(1-p_{p})(1-p_{s})p_{t}}{p_{p}(3-4p_{s})-p_{t}(4p_{p}-1)(1-p_{s})+p_{s}},$ $\displaystyle C_{fail3}$ $\displaystyle=$ $\displaystyle\frac{B_{fail1}p_{t}}{P_{fail3}}$ (47) $\displaystyle=$ $\displaystyle\frac{(1-p_{p})p_{s}p_{t}}{p_{p}(3-4p_{s})-p_{t}(4p_{p}-1)(1-p_{s})+p_{s}}.$ In Fig. 9, we show the value of $F_{fail3}$ as a function of $p_{t}$ with $p_{p}=0.65$. For ensuring the existence of residual entanglement after the first purification step, we require $p_{s}<0.7$. In this way, we control $p_{s}=$ 0.52, 0.62, and 0.68, respectively. It can be found that $F_{fail3}$ reduces with the growth of $p_{s}$ and $p_{t}$. $F_{fail3}$ is lower than $p_{p}$ under arbitrary value of $p_{s}$ and $p_{t}$. However, with relatively low $p_{s}$ and $p_{t}$, there may still exist residual entanglement ($F_{fail3}>\frac{1}{2}$) in $\rho_{fail3}$. For example, for ensuring the existence of residual entanglement in $\rho_{fail3}$, we can calculate the maximal values of $p_{t}$ to be 0.682, 0.599, and 0.523, corresponding to $p_{s}=$ 0.52, 0.62, and 0.68, respectively. From above discussion, there may residual entanglement exist when the purification steps fail. When only one purification step fails, the parties may still distill high-quality entanglement with suitable $p_{s}$ and $p_{t}$ by using of the residual entanglement. Even if the parties can not directly obtain high-quality polarized mixed state, they can still reuse the residual entanglement to distill high-quality entanglement in the next purification round. The existence of residual entanglement provides us a possibility to increase the fidelity of polarization entanglement, thus increase the yield of our EPP. Finally, it is interesting to compare out two-step EPP with the deterministic EPPs (DEPPs), which also adopt the hyperentanglement to realize the purification shenghyper1 ; shenghyper2 . The DEPPs require one pair of hyperentangled state, i.e., polarization-spatial-mode and polarization- spatial-mode-frequency hyperentanglement, respectively. Actually, the DEPPs can completely transform the entanglement in the other DOF to the target DOF, and they do not require the initial target DOF to be entangled. The upper bound of the fidelity in the target DOF is the initial fidelity of the consumed entanglement. On the other hand, our current two-step EPP belongs to recurrence EPP. In our two-step EPP, we suppose that the initial fidelities in three DOFs are larger than $\frac{1}{2}$. After two purification steps, the fidelity of the target polarization state can be higher than $p_{p}$, $p_{s}$, and even $p_{t}$. Actually, in the first purification step, if $p_{p}<\frac{1}{2}$, we can obtain the fidelity $p_{p}<F_{1n}<p_{s}$. Under this case, when the $p_{p}$ and $p_{s}$ satisfy $p_{p}p_{s}>\frac{1}{4}$, we can obtain $F_{1n}>\frac{1}{2}$, say, the distilled new mixed state in polarization DOF has entanglement. In the second purification step, the fidelity of the target polarization DOF can be increased to be higher than $F_{1n}$ and even $p_{t}$. If $p_{p}$ is so low that $p_{p}p_{s}>\frac{1}{4}$ can not be satisfied, our two-step EPP can not work. Based on above comparison, if the initial fidelity in the polarization DOF is relatively high, the current two-step EPP may be more advantageous, and while if that of the polarization DOF is low, the DEPP may be more advantageous. ## V Conclusion In conclusion, we present an efficient two-step recurrence EPP for purifying the entanglement in polarization DOF. In the protocol, we only require one copy of two-photon pair, which is hyperentangled in polarization, spatial- mode, and time-bin DOFs. We suppose that after the photon transmission, the entanglement in all DOFs suffer from the channel noise and degrade to mixed states. As the entanglement in different DOFs have different noisy robustness, the initial mixed states in three DOFs have different fidelities. In the first purification step, the bit-flip error in polarization DOF can be reduced by consuming the imperfect spatial-mode entanglement, while in the second step, the phase-flip error in polarization DOF can be reduced by consuming the imperfect time-bin entanglement. As a result, the fidelity of the target polarization state can be efficiently increased. Our EPP has some attractive advantages. First, comparing with previous two-step recurrence EPPs, which require two or more same copies of nonlocal entangled pairs, our EPP largely reduces the consumption of entanglement pairs. Second, using only one pair of hyperentangled state also reduces the experimental difficulty, for it is hard to generate two pairs of hyperentangled states simultaneously. Third, if we consider the practical photon transmission and detector efficiency, our EPP has much higher purification efficiency. Forth, all the devices in our EPP are available under current experimental condition, so that our EPP is feasible for experiment. Moreover, in traditional two-step recurrent EPP, the parties can distill a high-quality entanglement only when both two steps are successful. If any one step fails, there are no residual entanglement in the distilled mixed state and the distilled photon states have to be discarded. However, when a purification step of our EPP fails, there may exist residual entanglement in the distilled mixed state. The existence of residual entanglement may make the parties distill higher-fidelity polarization entanglement. Even not, the residual entanglement may be reused in the next purification round. In this way, the existence of residual entanglement benefits for further increasing the yield of our EPP. 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Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). Hate Speech and Offensive Content Identification in Indo-European Languages (HASOC) at Forum for Information Retrieval and Evaluation (FIRE), 16th-20th December, 2020, Hyderabad, IN [orcid=0000-0001-8336-195X, ]<EMAIL_ADDRESS> <EMAIL_ADDRESS> # HASOCOne@FIRE-HASOC2020: Using BERT and Multilingual BERT models for Hate Speech Detection Suman Dowlagar [ International Institute of Information Technology - Hyderabad (IIIT-Hyderabad), Gachibowli, Hyderabad, Telangana, India, 500032 Radhika Mamidi [ (2020) ###### Abstract Hateful and Toxic content has become a significant concern in today’s world due to an exponential rise in social media. The increase in hate speech and harmful content motivated researchers to dedicate substantial efforts to the challenging direction of hateful content identification. In this task, we propose an approach to automatically classify hate speech and offensive content. We have used the datasets obtained from FIRE 2019 and 2020 shared tasks. We perform experiments by taking advantage of transfer learning models. We observed that the pre-trained BERT model and the multilingual-BERT model gave the best results. The code is made publically available at https://github.com/suman101112/hasoc-fire-2020 ###### keywords: Hate speech offensive content label classification transfer learning BERT ## 1 Introduction Nowadays, people are frequently using social media platforms to communicate their opinions and share information. Although the communication among users can lead to constructive conversations, the people have been increasingly hit by hateful and offensive content due to these platforms’ anonymity features. It has become a significant issue. The threat of abuse and harassment made many people stop expressing themselves. According to the Cambridge dictionary, Hate speech and offensive content is defined as, * • To harass and cause lasting pain by attacking something uniquely dear to the target. * • To use words that are considered insulting by most people. The main obstacle with hate speech is, it is difficult to classify based on a single sentence because most of the hate speech has context attached to it, and it can morph into many different shapes depending on the context. Another obstacle is that humans cannot always agree on what can be classified as hate speech. Hence it is not very easy to create a universal machine learning algorithm that would detect it. Also, the datasets used to train models tend to "reflect the majority view of the people who collected or labeled the data". To deal with the above scenarios and to encourage research on hate speech and offensive content, the NLP community organized several tasks and workshops such as Task 12: OffensEval 2: Multilingual Offensive content identification in Social Media text 111https://sites.google.com/site/offensevalsharedtask/, OSATC4 shared task on offensive content detection 222http://edinburghnlp.inf.ed.ac.uk/workshops/OSACT4/. Similarly, the FIRE 2020’s HASOC shared task was devoted to the Hate Speech and Offensive Content Identification in Indo-European Languages. This task aims to classify the given annotated tweets. This paper presents the state-of-the-art BERT transfer learning models for automated detection of hate speech and offensive content. The paper is organized as follows. Section 2 provides related work on hate speech and offensive content detection. Section 3 describes the methodology used for this task. Section 4 presents the experimental setup and the performance of the model. Section 5 concludes our work. ## 2 Related Work Machine learning and natural language processing approaches have made a breakthrough in detecting hate speech on web platforms. Many scientific studies have been dedicated to using Machine Learning (ML) [1, 2] and Deep Learning (DL) [3, 4] methods for automated hate speech and offensive content detection. The features used in traditional machine learning approaches are word-level and character-level n-grams, etc. Although supervised machine learning-based approaches have used different text mining-based features such as surface features, sentiment analysis, lexical resources, linguistic features, knowledge-based features, or user-based and platform-based metadata, they necessitate a well-defined feature extraction approach. Nowadays, the neural network models apply text representation and deep learning approaches such as Convolutional Neural Networks (CNNs) [5], Bi-directional Long Short- Term Memory Networks (LSTMs) [6], and BERT [7] to improve the performance of hate speech and offensive content detection models. ## 3 Methodology Figure 1: BERT model for sequence classification on Hate Speech Data. Here, we use the pre-trained BERT transformer model for hate speech and offensive content detection. Figure 1 depicts the abstract view of BERT model that is used for hate speech detection and offensive language identification. Bidirectional Encoder Representations from Transformers (BERT) is a transformer Encoder stack trained on the large English corpus. It has 2 models, $BERT_{base}$ and $BERT_{large}$. These model sizes have a large number of transformer layers. The $BERT_{base}$ version has 12 transformer layers and the $BERT_{large}$ has 24. These also have larger feed-forward networks with 768 and 1024 hidden representations, and attention heads are 12 and 16 for the respective models. Like the vanilla transformer model [8], BERT takes a sequence of words as input. Each layer applies self-attention, passes its results through a feed-forward network, and then hands it off to the next encoder. Embeddings from $BERT_{base}$ have 768 hidden units. The BERT configuration model takes a sequence of words/tokens at a maximum length of 512 and produces an encoded representation of dimensionality 768. The pre-trained BERT models have a better word representation as they are trained on a large Wikipedia and book corpus. As the pre-trained BERT model is trained on generic corpora, we need to fine-tune the model for the downstream tasks. During fine-tuning, the pre-trained BERT model parameters are updated when trained on the labeled hate speech and offensive content dataset. When fine-tuned on the downstream sentence classification task, a very few changes are applied to the $BERT_{base}$ configuration. In this architecture, only the [CLS] (classification) token output provided by BERT is used. The [CLS] output is the output of the 12th transformer encoder with a dimensionality of 768. It is given as input to a fully connected neural network, and the softmax activation function is applied to the neural network to classify the given sentence. Thus, BERT learns to predict whether a tweet can be classified as a hate speech or offensive content. Apart from $BERT_{base}$ model, we used the pre-trained multilingual $BERT_{base}$ model, as our data consisted of German and Hindi multilingual languages. The multilingual BERT and vanilla BERT models’ architecture is the same, but the pre-trained multilingual BERT model is trained on multilingual Wikipedia language sources. ## 4 Experiment Initially, we introduce datasets used, the task description, and then review the BERT model’s performance on hate speech and offensive content detection. We also include our implementation details and error analysis in the subsequent sections. ### 4.1 Dataset Table 1: Data Statistics Language \| Train Sentences \| Test Sentences ---\|---\|--- English (HASOC 2019) \| 5852 \| 1153 German (HASOC 2019) \| 3819 \| 850 Hindi (HASOC 2019) \| 4665 \| 1318 English (HASOC 2020) \| 3708 \| 814 German (HASOC 2020) \| 2373 \| 526 Hindi (HASOC 2020) \| 2963 \| 663 We used the dataset provided by the organizers of HASOC FIRE-2020 [hasoc2020overview] and FIRE-2019 [9]. The HASOC dataset was subsequently sampled from Twitter and partially from Facebook for English, German, and Hindi languages. The tweets were acquired using hashtags and keywords that contained offensive content. The statistics of FIRE 2020 and 2019 datasets are given in the Table 1. ### 4.2 Task description The following tasks are in HASOC 2020. Sub-task A focuses on coarse-grained Hate speech detection in all three languages. The task is to classify tweets into two classes: * • (NOT) Non Hate-Offensive - Post does not contain any Hate speech, profane, offensive content. * • (HOF) Hate and Offensive - Post contains Hate, offensive, and profane content. Sub-task B represents a fine-grained classification. Hate-speech and offensive posts from the sub-task A are further classified into three categories. The task is to classify the tweets into three classes: * • (HATE) Hate speech - Post contains Hate speech content. * • (OFFN) Offenive - Post contains offensive content such as insulting, degrading, dehumanizing and threatening. * • (PRFN) Profane - Post contains profane words. This typically concerns the usage of swearwords and cursing. ### 4.3 Implementation For the implementation, we used the transformers library provided by HuggingFace [10]. The HuggingFace transformers package is a python library providing pre-trained and configurable transformer models useful for a variety of NLP tasks. It contains the pre-trained BERT and multilingual BERT, and other models suitable for downstream tasks. As the implementation environment, we use the PyTorch library that supports GPU processing. The BERT models were run on NVIDIA RTX 2070 graphics card with an 8 GB graphics card. We trained our classifier with a batch size of 64 for 5 to 10 epochs based on our experiments. The dropout is set to 0.1, and the Adam optimizer is used with a learning rate of 2e-5. We used the hugging face transformers pre-trained BERT tokenizer for tokenization. We used the BertForSequenceClassification module provided by the HuggingFace library during finetuning and sequence classification. ### 4.4 Baseline models Here, we compared the BERT model with other machine learning algorithms. #### 4.4.1 SVM with TF_IDF text representation We chose Support Vector Machines (SVM) for hate speech and offensive content detection. The tokenizer used is SentencePiece [11]. SentencePiece is a commonly used technique to segment words into a subword-level. In both cases, the vocabulary is initialized with all the individual characters in the language, and then the most frequent or likely combinations of the symbols are iteratively added to the vocabulary. #### 4.4.2 ELMO embeddings with SVM model ELMO(Embeddings from Language Models) [12] deals with contextual embeddings. Contextual word-embeddings are born to capture the word meaning in its context. Instead of using a fixed embedding for each word, ELMO looks at the word’s context, i.e., the word’s entire sentence, before assigning embedding to the word. It uses a bi-LSTM trained on a specific task to be able to create those embeddings. We used the ELMO model present on tensorflow hub (https://tfhub.dev/google/elmo/2) to obtain the ELMO embeddings on the hate speech data for all the languages. After obtaining the embeddings, we take the mean of embeddings and apply an SVM classifier to classify the given sentence into hate speech or offensive content. We used the SentencePiece tokenizer. ## 5 Results Table 2: macro F1 and Accuracy on English Subtasks A and B \| Hate speech Detection \| Offensive Content Identification ---\|---\|--- Model \| macro F1 \| Accuracy \| macro F1 \| Accuracy SVM \| 81.56% \| 81.57% \| 47.49% \| 76.78% ELMO + SVM \| 82.43% \| 83.78% \| 49.62% \| 79.54% BERT \| 88.33% \| 88.33% \| 54.44% \| 81.57% Table 3: macro F1 and Accuracy on German Subtasks A and B \| Hate speech Detection \| Offensive Content Identification ---\|---\|--- Model \| macro F1 \| Accuracy \| macro F1 \| Accuracy SVM \| 73.29% \| 79.27% \| 45.54% \| 77.94% ELMO + SVM \| 71.73% \| 80.42% \| 45.94% \| 78.21% multilingual-BERT \| 77.91% \| 82.51% \| 47.78% \| 80.42% Table 4: macro F1 and Accuracy on Hindi Subtasks A and B \| Hate speech Detection \| Offensive Content Identification ---\|---\|--- Model \| macro F1 \| Accuracy \| macro F1 \| Accuracy SVM \| 59.73% \| 70.13% \| 36.78% \| 72.39% ELMO + SVM \| 60.91% \| 71.47% \| 39.89% \| 72.76% multilingual-BERT \| 63.54% \| 74.96% \| 49.71% \| 73.15% The results are tabulated in Tables 2, 3 and 4. We evaluated the performance of the method using macro F1 and accuracy. The BERT model performed well when compared to the other SVM with TF-IDF and ELMO text representations. Given all the languages and both the subtasks A and B, we have observed an increase of 1-2% in classification metrics for ELMO embeddings + SVM classifier compared to the baseline SVM classifier. However, BERT showed an increase of 5-7% in classification metrics compared to ELMO and SVM models. It shows the pre- trained BERT model’s capability, which learnt better text representations from the generic data. The state of the art transformer architecture used in the BERT model helped the model learn better parameter weights in hate speech and offensive content detection. Figure 2: Confusion matrix on the given test data for the English, German and Hindi languages given subtask A: Hate Speech Detection and subtask B: Offensive Content Identification ## 6 Error Analysis The confusion matrix of BERT model for subtasks A and B for the english, german and hindi datasets is given in the Figure 2. For the binary classification, the best-performed model was for English subtask A. The binary classification for the Hindi model is not helpful. The model misclassified most of the hate-speech labels. It can be seen in subfigure 2. For offensive content evaluation, the model performed better on English subtask B. It correctly classified "NONE (not offensive)" and "PROF (profane)" but was unable to classify "HATE (hate speech)" and "OFFN (offensive)" and misunderstood most of them as "PROF". The multilingual-BERT model misclassified most of the hate speech and offensive content labels for the German and Hindi languages as "NONE" and didn’t perform well on those datasets. ## 7 Conclusion and Future work We used pre-trained bi-directional encoder representations using transformers (BERT) and multilingual-BERT for hate speech and offensive content detection for English, German, and Hindi languages. We compared the BERT with other machine learning and neural network classification methods. Our analysis showed that using the pre-trained BERT and multilingual BERT models and finetuning it for downstream hate-speech text classification tasks showed an increase in macro F1 score and accuracy metrics compared to traditional word- based machine learning approaches. The given data has both hate speech and offensive content labeled for a given same sentence. It implies that both tasks are related. 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# Strong gravitational lensing by Kerr and Kerr-Newman black holes Tien Hsieh Da-Shin Lee<EMAIL_ADDRESS>Chi-Yong Lin <EMAIL_ADDRESS>Department of Physics, National Dong Hwa University, Hualien 97401, Taiwan, Republic of China ###### Abstract We study the strong gravitational lensing due to the Kerr black holes with angular momentum $a$ and the Kerr-Newman black holes with additional charge $Q$. We first derive the analytical expressions of the deflection angles of light rays that particularly diverge as they travel near the photon sphere. In this strong deflection limit, the light rays can circle around the black hole multiple times before reaching the observer, giving relativistic images. The obtained analytical expressions are then applied to compute the angular positions of relativistic images due to the supermassive galactic black holes. In this work, we focus on the outermost image with reference to the optical axis. We find that its angular separation from the one closest to the optical axis increases with the increase of angular momentum $a$ of the black holes for light rays in direct orbits. Additionally, the effects of the charge $Q$ of black holes also increase the angular separation of the outermost image from the others for both direct and retrograde orbits. The potentially increasing observability of the relativistic images from the effects of angular momentum and charge of the black holes will be discussed. ###### pacs: 04.70.-s, 04.70.Bw, 04.80.Cc ## I Introduction Gravitational lensing is one of the powerful tools to test general relativity (GR) MIS ; HAR . Weak lensing has been fully studied in the formalism of weak field approximations, which can be used to successfully explain various lensing phenomena in a broad array of astrophysical contexts SEF1992 . Nevertheless, in recent years, there have been significant theoretical studies looking into lensing phenomena from strong field perspectives Vir ; Fri ; Bozza1 ; Bozza2 ; Bozza_2003 ; Bozza3 ; Eiroa ; Iyer1 ; Tsuka1 ; Tsuka2 ; Vir3 ; Sha . Through the gravitational lensing in the vicinity of the compact massive objects such as a black hole would provide another avenue to test GR. So far, observational evidence has shown that almost every large galaxy has a supermassive black hole at the galaxy’s center Ric . The Milky Way has a supermassive black hole in its Galactic Center with the location of Sagittarius A* Ghez ; Sch . Together with the first image of the black hole captured by the Event Horizon Telescope EHT1 ; EHT2 ; EHT3 , gravitational lensing will also become an important probe to study the isolated dim black hole. Recently, Virbhadra and Ellis have developed a new gravitational lens equation, which allows us to study large deflection of light rays, resulting in the strong gravitational lensing Vir . This lens equation is then applied to analyze the lensing by a Schwarzschild black hole in the center of the galaxy using numerical methods. Later, Frittelli et al. propose the definition of an exact lens equation without reference to the background spacetime, and construct the exact lens equation explicitly in the Schwarzschild spacetime Fri . Strong field lensing in the general spherically symmetric and static spacetime is first studied analytically by Bozza in Bozza1 ; Bozza2 ; Bozza3 and later by Tsukamoto in Tsuka1 ; Tsuka2 . These works show that the deflection angle $\hat{\alpha}(b)$ of light rays for a given impact parameter $b$, which in the strong deflection limit (SDL) as $b\to b_{c}$, can be approximated in the form $\hat{\alpha}(b)\approx-\bar{a}\log{\left(\frac{b}{b_{c}}-1\right)}+\bar{b}+O((b-b_{c})\log(b-b_{c}))$ (1) with two parameters $\bar{a}$ and $\bar{b}$ as a function of the black hole’s parameters. Then, in Bozza_2003 , the Kerr black hole of the nonspherically symmetric black holes is considered, exploring $\bar{a}$ and $\bar{b}$ numerically. In this paper, we extend the works of Bozza1 and Tsuka1 ; Tsuka2 and find the analytic form of $\bar{a}$ and $\bar{b}$ for nonspherically symmetric Kerr and Kerr-Newman black holes, respectively, using the analytical closed-form expressions of the deflection angles in Iyer2 and Hsiao . Although one might not expect that astrophysical black holes have large residue electric charge, some accretion scenarios are proposed to investigate the possibility of the spinning charged back holes Wilson_1975 ; Dam . It is then still of great interest to extend the studies to the Kerr-Newman black holes Liu ; Jiang_2018 ; Kraniotis_2014 . The analytical expressions can be applied to examine the lensing effects due to the supermassive galactic black holes as illustrated in Fig.(1). The light rays are emitted from the source, and circle around the black hole multiple times in the SDL along a direct orbit (red line) or a retrograde orbit (blue line), giving two sets of the relativistic images. Following the approach of Bozza2 enables us to study the observational consequences. Figure 1: Gravitational lens about relativistic images. Considering the Kerr or the Kerr-Newman black hole with angular momentum of the clockwise rotation, the light rays are emitted from the source, and circle around the black hole multiple times in the SDL along a direct orbit (red line) or a retrograde orbit (blue line). The graph illustrates two sets of the relativistic images. The layout of the paper is as follows. In Sec.II, we first review the closed- form expression of the deflection angle due to the Kerr and/or the Kerr-Newman black holes. In particular, we discuss the results of the radius of the innermost circular motion of light rays as well as the associated critical impact parameters as a function of the black hole’s parameters. These will serve as the important inputs to find the values of the coefficients $\bar{a}$ and $\bar{b}$ in the SDL deflection angle. Then we derive the analytic form of $\bar{a}$ and $\bar{b}$ in the cases of Kerr and Kerr-Newman black holes, respectively, and check the consistency with the known results from taking the proper limits of the black holes’s parameters. In Sec. III, the analytical expressions on the equatorial gravitational lensing are then applied to compute the angular positions of relativistic images due to the supermassive galactic black holes. When the light rays travel on the quasiequatorial plane, the obtained results for $\theta=\frac{\pi}{2}$ can also be used to estimate the magnification of relativistic images, as the light sources are near one of the caustic points with the additional inputs from the dynamics of the light rays in the angle $\theta$. The potentially increasing observability of the relativistic images from the effects of angular momentum and charge of the black holes will be summarized in the closing section. ## II Deflection angle due to black holes in the strong deflection limit We consider nonspherically symmetric spacetimes of the Kerr and Kerr-Newman metrics respectively to obtain the deflection angle $\hat{\alpha}(b)$ of light rays for a given impact parameter $b$. In the SDL, as $b\to b_{c}$, $\hat{\alpha}(b)$ can be approximated in the form of (1). In what follows, we will consider the above two types of the black holes separately. ### II.1 Kerr black holes The line element of the Kerr black hole in which spacetime outside a black hole with the gravitational mass $M$ and angular momentum per unit mass $a=J/M$ is described by $\displaystyle{ds}^{2}$ $\displaystyle=$ $\displaystyle g_{\mu\nu}dx^{\mu}dx^{\nu}$ (2) $\displaystyle=$ $\displaystyle-\frac{\left(\Delta-a^{2}\sin^{2}\theta\right)}{\Sigma}{dt}^{2}-\frac{a\sin^{2}\theta\left(2Mr\right)}{\Sigma}({dt}{d\phi+d\phi dt)}$ $\displaystyle+\frac{\Sigma}{\Delta}dr^{2}+\Sigma{\,d\theta}^{2}+\frac{\sin^{2}\theta}{\Sigma}\left((r^{2}+a^{2})^{2}-a^{2}\Delta\sin^{2}\theta\right){d\phi}^{2}\,$ with $\Sigma=r^{2}+a^{2}\cos^{2}\theta\,,\quad\Delta=r^{2}+a^{2}-2Mr\,.$ (3) The outer (inner) event horizon $r_{+}$ ($r_{-}$) can be found by solving $\Delta(r)=0$, and is given by $r_{\pm}=M\pm\sqrt{M^{2}-a^{2}}\,$ (4) with the condition $M^{2}>a^{2}$. Notice that we just adopt the notation of $r_{-}$ where the light rays traveling outside the horizon are considered. The Lagrangian of a particle is then $\displaystyle\mathcal{L}=\frac{1}{2}g_{\mu\nu}u^{\mu}u^{\nu}\,$ (5) with the 4-velocity $u^{\mu}=dx^{\mu}/d\lambda$ defined in terms of an affine parameter $\lambda$. The metric of the Kerr black hole, which is independent on $t$ and $\phi$, gives the associated Killing vectors $\xi_{(t)}^{\mu}$ and $\xi_{(\phi)}^{\mu}$ $\displaystyle\xi_{(t)}^{\mu}=\delta_{t}^{\mu}\;,\quad\xi_{\phi}^{\mu}=\delta_{\phi}^{\mu}\,.$ (6) Then, together with 4-velocity of light rays, the conserved quantities along a geodesic, can be constructed by the above Killing vectors $\varepsilon\equiv-\xi_{(t)}^{\mu}u_{\mu}$ and $\ell\equiv\xi_{\phi}^{\mu}u_{\mu}$, where $\varepsilon$ and $\ell$ are the light ray’s energy and azimuthal angular momentum at spatial infinity. Light rays traveling along null world lines obey the condition $u^{\mu}u_{\mu}=0$. To indicate whether the light rays are traversing along the direction of frame dragging or opposite to it, we define the following impact parameter : $\displaystyle b_{s}=s\left\|\frac{\ell}{\varepsilon}\right\|\equiv s\,b\;,$ (7) where $s=\text{Sign$(\ell/\varepsilon)$}$ and $b$ is the positive magnitude. The parameter $s=+1$ for $b_{s}>0$ will be referred to as direct orbits, and those with $s=-1$ for $b_{s}<0$ as retrograde orbits (see Fig.(1) for the sign convention). Here we restrict the light rays traveling on the equatorial plane of the black hole by choosing $\theta={\pi}/{2}$, and $\dot{\theta}=0$. The equation of motion along the radial direction can be cast in the form Hsiao $\displaystyle\frac{1}{b^{2}}$ $\displaystyle=\frac{\dot{r}^{2}}{\ell^{2}}+W_{\text{eff}}(r)\;,$ (8) from which we define the function $W_{\text{eff}}$ as $\displaystyle W_{\text{eff}}(r)=\frac{1}{r^{2}}\left[1-\frac{a^{2}}{b^{2}}-\frac{2M}{r}\left(1-\frac{a}{b_{s}}\right)^{2}\right]\,.$ (9) The above equation is analogous to that of particle motion in the effective potential $W_{\text{eff}}(r)$ with the kinetic energy ${\dot{r}}^{2}/\ell^{2}$ and constant total energy $1/b^{2}$. Let us consider that a light ray starts in the asymptotic region to approach the black hole, and then turns back to the asymptotic region to reach the observer. Such light rays have a turning point, the closest approach distance to a black hole $r_{0}$, which crucially depends on the impact parameter $b$, determined by $\displaystyle\left.\frac{\dot{r}^{2}}{\ell^{2}}\right\|_{r=r_{0}}=\frac{1}{b^{2}}-W_{\text{eff}}(r_{0})=0\,.$ (10) From (10), also shown in Hsiao ; Iyer2 , one can find the impact parameter $b$ for a given $r_{0}$, which becomes the important input for the analytical expressions of the deflection angle in the SDL, as $b(r_{0})=\frac{2sMa-r_{0}\sqrt{a^{2}-2r_{0}M+r_{0}^{2}}}{2M-r_{0}}\,.$ (11) The behavior of the light ray trajectories depends on whether $1/b^{2}$ is greater or less than the maximum height of $W_{\text{eff}}(r)$. The innermost trajectories of light rays have a direct consequence on the apparent shape of the black hole. The smallest radius ${r_{sc}}$, when the turning point $r_{0}$ is located at the maximum of $W_{\text{eff}}(r)$, with the critical impact parameter ${b_{sc}}$, obeys $\displaystyle\left.\frac{d\,W_{\text{eff}}(r)}{dr}\right\|_{r=r_{sc}}$ $\displaystyle=0\,.$ (12) Then the radius of the circular motion forming the photon sphere is given by (See Hsiao ; Iyer2 ). $\displaystyle r_{sc}=2M\bigg{\\{}1+\cos\bigg{[}\frac{2}{3}\cos^{-1}\bigg{(}\frac{-sa}{M}\bigg{)}\bigg{]}\bigg{\\}}\;$ (13) with the corresponding impact parameter $\displaystyle b_{sc}$ $\displaystyle=-a+s6M\cos\bigg{[}\frac{1}{3}\cos^{-1}\bigg{(}\frac{-sa}{M}\bigg{)}\bigg{]}\,.$ (14) In the case of a Kerr black hole, the nonzero spin of the black hole is found to give more repulsive effects to the light rays in the direct orbits than those in the retrograde orbits due to the $1/r^{3}$ term in the effective potential. The repulsive effects in turn affect light rays in the direct orbits in a way to prevent them from collapsing into the event horizon. As a result, this shifts the innermost circular trajectories of the light rays toward the black hole with the smaller critical impact parameter $b_{+c}$ than $b_{-c}$ in the retrograde orbits as shown in Fig.(2). As such, when $a$ increases, the impact parameter $b_{+c}$ decreases whereas $\|b_{-c}\|$ increases instead Iyer2 ; Hsiao . It will be shown in the next section that the value of $b_{sc}$ is a key quantity to determine the features of the angular position of the induced images of the distant light sources due to the strong gravitational lensing effects. Also, the presence of black hole’s spin is to give the smaller deflection angle in the direct orbits as compared with the retrograde orbits with the same impact parameter $b$ Hsiao ; Iyer2 . We proceed by introducing the variable $z\equiv 1-\frac{r_{0}}{r}\,.$ (15) The geodesic equations for $r$ and $\phi$ found in Hsiao can be rewritten in terms of $z$ as Tsuka1 $\frac{dz}{d\phi}=\frac{1}{r_{0}}\frac{1-\frac{2{M}}{r_{0}}(1-z)+\frac{a^{2}}{r_{0}^{2}}(1-z)^{2}}{1-\frac{2{M}}{r_{0}}(1-z)(1-\frac{a}{b_{s}})}\sqrt{B(z,r_{0})}\;,$ (16) where the function $B(z,r_{0})$ has the trinomial form in $z$ $B(z,r_{0})=c_{1}(r_{0})z+c_{2}(r_{0})z^{2}+c_{3}(r_{0})z^{3}$ (17) with the coefficients $\begin{split}c_{1}(r_{0})=&-6Mr_{0}\left(1-\frac{a}{b_{s}}\right)^{2}+2r_{0}^{2}\left(1-\frac{a^{2}}{b^{2}}\right)\,,\\\ c_{2}(r_{0})=&6Mr_{0}\left(1-\frac{a}{b_{s}}\right)^{2}-r_{0}^{2}\left(1-\frac{a^{2}}{b^{2}}\right)\,,\\\ c_{3}(r_{0})=&-2Mr_{0}\left(1-\frac{a}{b_{s}}\right)^{2}\,.\end{split}$ (18) Next we rewrite $\frac{1-\frac{2{M}}{r_{0}}(1-\frac{a}{b_{s}})+\frac{2{M}}{r_{0}}(1-\frac{a}{b_{s}})z}{1-\frac{2M}{r_{0}}+\frac{a^{2}}{r_{0}^{2}}+(\frac{2{M}}{r_{0}}-\frac{2a^{2}}{r_{0}^{2}})z+\frac{a^{2}}{r_{0}^{2}}z^{2}}=\frac{r_{0}^{2}}{a^{2}}\left(\frac{C_{-}}{z-z_{-}}+\frac{C_{+}}{z-z_{+}}\right)\;,$ (19) where the roots $z_{-}$, $z_{+}$, and the coefficients $C_{-}$, $C_{+}$ are $\begin{split}z_{-}=&1-\frac{r_{0}r_{-}}{a^{2}}\,,\\\ z_{+}=&1-\frac{r_{0}r_{+}}{a^{2}}\,,\end{split}$ (20) $\begin{split}C_{-}=&\frac{a^{2}-2Mr_{-}(1-\frac{a}{b_{s}})}{2r_{0}\sqrt{M^{2}-a^{2}}}\,,\\\ C_{+}=&\frac{-a^{2}+2Mr_{+}(1-\frac{a}{b_{s}})}{2r_{0}\sqrt{M^{2}-a^{2}}}\end{split}$ (21) with $r_{+}$ ($r_{-}$) being the outer (inner) horizon of a Kerr black hole defined in (4). Also note that $z_{-}$, $z_{+}\leq 0$, for all spin $a$. Then the deflection angle can be calculated as a function of the closest approach distance $r_{0}$ from (16) giving $\hat{\alpha}(r_{0})=I(r_{0})-\pi\,,\quad I(r_{0})=\int_{0}^{1}f(z,r_{0})dz\,,$ (22) where the integrand becomes $f(z,r_{0})=\frac{r_{0}^{2}}{a^{2}}\left(\frac{C_{-}}{z-z_{-}}+\frac{C_{+}}{z-z_{+}}\right)\frac{2{r_{0}}}{\sqrt{c_{1}(r_{0})z+c_{2}(r_{0})z^{2}+c_{3}(r_{0})z^{3}}}\,.$ (23) In the SDL of our interest, when the closest approach distance reaches its critical limit, namely $r_{0}\to r_{sc}$, and $c_{1}(r_{0})\to 0$ in (18) obtained from (12), the integrand $f(z,r_{0})\rightarrow\frac{1}{z}$ for small $z$ leads to the logarithmic divergence as $r_{0}\to r_{sc}$. Let us now define a new function $f_{D}(z,r_{0})$ $f_{D}(z,r_{0})=\frac{r_{0}^{2}}{a^{2}}\left(\frac{C_{-}}{z-z_{-}}+\frac{C_{+}}{z-z_{+}}\right)\frac{2{r_{0}}}{\sqrt{c_{1}(r_{0})z+c_{2}(r_{0})z^{2}}}\;,$ (24) that separates the divergent part from the regular part given by $f_{R}(z,r_{0})=f(z,r_{0})-f_{D}(z,r_{0})$. The integral of $f_{R}$ is thus finite. The divergent part comes from an integral of the function $f_{D}(z,r_{0})$, which contributes not only to $\bar{a}$ for the logarithmic term but also $\bar{b}$ for the regular part in (1), giving $\begin{split}I_{D}(r_{0})=&\int_{0}^{1}f_{D}(z,r_{0})dz\\\ =&\frac{2r_{0}^{3}}{a^{2}}\frac{C_{-}}{\sqrt{c_{1}(r_{0})z_{-}+c_{2}(r_{0})z_{-}^{2}}}\log{\left(\frac{\sqrt{c_{1}(r_{0})z_{-}+c_{2}(r_{0})z_{-}}+\sqrt{c_{1}(r_{0})+c_{2}(r_{0})z_{-}}}{\sqrt{c_{1}(r_{0})z_{-}+c_{2}(r_{0})z_{-}}-\sqrt{c_{1}(r_{0})+c_{2}(r_{0})z_{-}}}\right)}\\\ +&\frac{2r_{0}^{3}}{a^{2}}\frac{C_{+}}{\sqrt{c_{1}(r_{0})z_{+}+c_{2}(r_{0})z_{+}^{2}}}\log{\left(\frac{\sqrt{c_{1}(r_{0})z_{+}+c_{2}(r_{0})z_{+}}+\sqrt{c_{1}(r_{0})+c_{2}(r_{0})z_{+}}}{\sqrt{c_{1}(r_{0})z_{+}+c_{2}(r_{0})z_{+}}-\sqrt{c_{1}(r_{0})+c_{2}(r_{0})z_{+}}}\right)}\,.\end{split}$ (25) In the SDL, the expansions of the coefficient $c_{1}(r_{0})$ (18) and the impact parameter $b(r_{0})$ in powers of small $r_{0}-r_{sc}$ read $c_{1}(r_{0})=c_{1sc}^{\prime}(r_{0}-r_{sc})+{O}(r_{0}-r_{sc})^{2}\,,$ (26) $b(r_{0})=b_{sc}+\frac{b_{sc}^{\prime\prime}}{2!}(r_{0}-r_{sc})^{2}+{O}(r_{0}-r_{sc})^{3}\,,$ (27) where $c_{1}(r_{sc})\equiv c_{1sc}=0$ and $b(r_{sc})\equiv b_{sc}$ is the critical impact parameter given by (14). The subscript $sc$ denotes evaluating the function at $r=r_{sc}$. The prime means the derivative with respect to $r_{0}$. Notice that using $c_{1sc}=0$ in (18), one finds $c_{3sc}=-\frac{2}{3}c_{2sc}\;.$ (28) Combining (26) with (27), we can write $c_{1}(r_{0})$ in terms of small $b-b_{sc}$ as $\lim_{r_{0}\to r_{sc}}c_{1}(r_{0})=\lim_{b\to b_{sc}}c_{1sc}^{\prime}\sqrt{\frac{2b_{sc}}{b_{sc}^{\prime\prime}}}\left(\frac{b}{b_{sc}}-1\right)^{1/2}\;.$ (29) In the SDL, substituting (29) into (25), $I_{D}$ becomes $\begin{split}I_{D}(b)\simeq&-\left(\frac{r_{sc}^{3}}{a^{2}}\frac{C_{-sc}}{\sqrt{c_{2sc}\,z_{-sc}^{2}}}+\frac{r_{sc}^{3}}{a^{2}}\frac{C_{+sc}}{\sqrt{c_{2sc}\,z_{+sc}^{2}}}\right)\log{\left(\frac{b}{b_{sc}}-1\right)}\\\ &+\frac{r_{sc}^{3}}{a^{2}}\frac{C_{-sc}}{\sqrt{c_{2sc}\,z_{-sc}^{2}}}\log{\left(\frac{16\,c^{2}_{2sc}\,z_{-sc}^{2}b_{sc}^{\prime\prime}}{c_{1sc}^{\prime 2}2b_{sc}(z_{-sc}-1)^{2}}\right)}+\frac{r_{sc}^{3}}{a^{2}}\frac{C_{+sc}}{\sqrt{c_{2sc}z_{+sc}^{2}}}\log{\left(\frac{16\,c^{2}_{2sc}\,z_{+sc}^{2}\,b_{sc}^{\prime\prime}}{c_{1sc}^{\prime 2}2b_{sc}(z_{+sc}-1)^{2}}\right)}\,.\end{split}$ (30) Finally, the coefficients $\bar{a}$ and the contribution from $I_{D}(b)$ to $\bar{b}$ denoted by $b_{D}$ in (1) are $\begin{split}\bar{a}=&\frac{r_{sc}^{3}}{\sqrt{c_{2sc}}}\left[\frac{C_{-sc}}{r_{sc}r_{-}-a^{2}}+\frac{C_{+sc}}{r_{sc}r_{+}-a^{2}}\right]\end{split}$ (31) and $\begin{split}b_{D}=\bar{a}\log{\left[\frac{8c_{2sc}^{2}b_{sc}^{\prime\prime}}{c_{1sc}^{\prime 2}b_{sc}}\right]}+\frac{2r_{sc}^{3}}{\sqrt{c_{2sc}}}\left[\frac{C_{-sc}}{r_{sc}r_{-}-a^{2}}\log{\left(1-\frac{a^{2}}{r_{sc}r_{-}}\right)}+\frac{C_{+sc}}{r_{sc}r_{+}-a^{2}}\log{\left(1-\frac{a^{2}}{r_{sc}r_{+}}\right)}\right]\,,\end{split}$ (32) where $z_{\pm}$ are replaced by $r_{\pm}$ through (20). The leading order result in the SDL from the integration of $f_{R}(z,r_{sc})$, which contributes the coefficient $\bar{b}$, is denoted by $b_{R}$, and is obtained as $\begin{split}b_{R}&=I_{R}(r_{sc})=\int_{0}^{1}f_{R}(z,r_{sc})dz\\\ =&\frac{2r_{0}^{3}}{a^{2}}\frac{C_{-}}{\sqrt{c_{2}}z_{-}}\log{\left(\frac{z_{-}}{z_{-}-1}\frac{\sqrt{c_{2}+c_{3}}+\sqrt{c_{2}}}{\sqrt{c_{2}+c_{3}}-\sqrt{c_{2}}}\frac{c_{3}}{4c_{2}}\right)}\\\ &+\frac{2r_{0}^{3}}{a^{2}}\frac{C_{-}}{\sqrt{c_{2}+c_{3}z_{-}}z_{-}}\log{\left(\frac{\sqrt{c_{2}+c_{3}z_{-}}-\sqrt{c_{2}+c_{3}}}{\sqrt{c_{2}+c_{3}z_{-}}+\sqrt{c_{2}+c_{3}}}\frac{\sqrt{c_{2}+c_{3}z_{-}}+\sqrt{c_{2}}}{\sqrt{c_{2}+c_{3}z_{-}}-\sqrt{c_{2}}}\right)}\\\ &+\frac{2r_{0}^{3}}{a^{2}}\frac{C_{+}}{\sqrt{c_{2}}z_{+}}\log{\left(\frac{z_{+}}{z_{+}-1}\frac{\sqrt{c_{2}+c_{3}}+\sqrt{c_{2}}}{\sqrt{c_{2}+c_{3}}-\sqrt{c_{2}}}\frac{c_{3}}{4c_{2}}\right)}\\\ &+\frac{2r_{0}^{3}}{a^{2}}\frac{C_{+}}{\sqrt{c_{2}+c_{3}z_{+}}z_{+}}\log{\left(\frac{\sqrt{c_{2}+c_{3}z_{+}}-\sqrt{c_{2}+c_{3}}}{\sqrt{c_{2}+c_{3}z_{+}}+\sqrt{c_{2}+c_{3}}}\frac{\sqrt{c_{2}+c_{3}z_{+}}+\sqrt{c_{2}}}{\sqrt{c_{2}+c_{3}z_{+}}-\sqrt{c_{2}}}\right)}\Big{\|}_{r_{0}=r_{sc}}\;.\end{split}$ (33) Thus, the coefficient $\bar{b}$ can be computed from the sum of $b_{D}$ and $b_{R}$ $\bar{b}=-\pi+b_{D}+b_{R}\,$ (34) with the help of (32) and (33). In (33) we again use (28) and (20) to replace $c_{3sc}$ by $c_{2sc}=-\frac{2}{3}c_{3sc}$ and $z_{\pm}$ by $r_{\pm}$. After some straightforward algebra we find $\begin{split}\bar{b}=&-\pi+\bar{a}\log{\left(\frac{36}{7+4\sqrt{3}}\frac{8c_{2sc}^{2}b_{sc}^{\prime\prime}}{c_{1sc}^{\prime 2}b_{sc}}\right)}\\\ &+\frac{r_{sc}^{3}}{\sqrt{c_{2sc}}}\frac{2aC_{-sc}}{a^{2}-r_{sc}r_{-}}\frac{\sqrt{3}}{\sqrt{a^{2}+2r_{sc}r_{-}}}\log{\left(\frac{\sqrt{a^{2}+2r_{sc}r_{-}}-a}{\sqrt{a^{2}+2r_{sc}r_{-}}+a}\frac{\sqrt{a^{2}+2r_{sc}r_{-}}+\sqrt{3}a}{\sqrt{a^{2}+2r_{sc}r_{-}}-\sqrt{3}a}\right)}\\\ &+\frac{r_{sc}^{3}}{\sqrt{c_{2sc}}}\frac{2aC_{+sc}}{a^{2}-r_{sc}r_{+}}\frac{\sqrt{3}}{\sqrt{a^{2}+2r_{sc}r_{+}}}\log{\left(\frac{\sqrt{a^{2}+2r_{sc}r_{+}}-a}{\sqrt{a^{2}+2r_{sc}r_{+}}+a}\frac{\sqrt{a^{2}+2r_{sc}r_{+}}+\sqrt{3}a}{\sqrt{a^{2}+2r_{sc}r_{+}}-\sqrt{3}a}\right)}\,.\end{split}$ (35) Using the results of $r_{sc}$ (13), $b_{sc}$ (14) and the expression of $b(r_{0})$ (11), together with the definitions of $C_{\pm}$ and $c_{2}$ in (21) and (18) respectively, one can compute the coefficients $\bar{a}$ and $\bar{b}$ given by (31) and (35) in the form of (1). Notice that with the parameters under investigation $\bar{a}>0$, but $\bar{b}<0$. Our results are shown in Fig.(3), where both $\bar{a}$ and $\|\bar{b}\|$ increase (decrease) in $a$ in direct (retrograde) orbits, giving the fact that the deflection angle $\hat{\alpha}$ decreases (increases) with the increase of the black hole’s spin for a given impact parameter. Later in Sec. III we will compare with the full numerical computations from (22) in the SDL. The results of $\bar{a}$ and $\bar{b}$ due to the Schwarzschild black hole in Bozza2 ; Tsuka1 can be reproduced by sending $a\to 0$ where $r_{+}\rightarrow 2M$, $r_{-}\to a^{2}/2M$, $C_{+sc}\to 2M/r_{sc}$, $C_{-sc}\to a^{3}/2b_{sc}Mr_{sc}$, and $c_{2sc}\to r_{sc}^{2}$ using $c_{1sc}=0$ in (4) and (21). We can check that $\bar{a}=1$ in (31) and $\bar{b}$ in (35) reduces to the expression proportional to $\bar{a}$ given by $\begin{split}\bar{b}=&-\pi+\bar{a}\log{\left(36(7-4\sqrt{3})\frac{8c_{2sc}^{2}b_{sc}^{\prime\prime}}{c_{1sc}^{\prime 2}b_{sc}}\right)}\\\ =&-\pi+\log{\left(216(7-4\sqrt{3})\right)}\,.\end{split}$ (36) In the second equality above we have further used substitutions $b_{sc}\to 3\sqrt{3}M$, $b_{sc}^{\prime\prime}\to\sqrt{3}/M$, $c_{1sc}^{\prime}\to 6M$, and $c_{2sc}\to 9M^{2}$ obtained from $r_{sc}=3M$ in the Schwarzschild black hole. In Fig.(5), we compare the approximate results of the deflection angle in the SDL with the exact ones in Iyer2 and Hsiao , and find that they are in good agreement when $b\to b_{sc}$. The analytical expressions of the coefficient $\bar{a}$ and $\bar{b}$ in the form of the SDL deflection angle due to the Kerr black hole are successfully achieved. They are an extension of the works in Bozza3 and Tsuka1 where the spherically symmetric black holes are considered. This is one of the main results in this work. ### II.2 Kerr-Newman black holes We now consider another example with the nonspherically symmetric metric of a charged spinning black hole. With an addition of charge $Q$ comparing with the Kerr case, the line element associated with the Kerr-Newman metric is $\displaystyle{ds}^{2}$ $\displaystyle=$ $\displaystyle g_{\mu\nu}dx^{\mu}dx^{\nu}$ (37) $\displaystyle=$ $\displaystyle-\frac{\left(\Delta-a^{2}\sin^{2}\theta\right)}{\Sigma}{dt}^{2}+\frac{a\sin^{2}\theta\left(Q^{2}-2Mr\right)}{\Sigma}({dt}{d\phi+d\phi dt)}$ $\displaystyle+\frac{\Sigma}{\Delta}dr^{2}+\Sigma{\,d\theta}^{2}+\frac{\sin^{2}\theta}{\Sigma}\left((r^{2}+a^{2})^{2}-a^{2}\Delta\sin^{2}\theta\right){d\phi}^{2}\,,$ where $\Sigma=r^{2}+a^{2}\cos^{2}\theta\,,\quad\Delta=r^{2}+a^{2}+Q^{2}-2Mr\,.$ (38) The outer (inner) event horizon $r_{+}$ ($r_{-}$) is $r_{\pm}=M\pm\sqrt{M^{2}-(Q^{2}+a^{2})}\,$ (39) with $M^{2}>Q^{2}+a^{2}$. The light rays traveling on the equatorial plane of the black hole have been studied analytically in our previous work in Hsiao , in which the function $W_{\text{eff}}$ from the equation of motion along the radial direction in (8) can be regarded as an effective potential given by $\displaystyle W_{\text{eff}}(r)=\frac{1}{r^{2}}\left[1-\frac{a^{2}}{b^{2}}+\left(-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)\left(1-\frac{a}{b_{s}}\right)^{2}\right]\,.$ (40) For the Kerr-Newman black hole, the nonzero charge of the black hole is found to give repulsive effects to light rays as seen from its contributions to the function $W_{\text{eff}}$ of the $1/r^{4}$ term, which shifts the innermost circular trajectories of the light rays toward the black holes with the smaller critical impact parameter $b_{sc}$ for both direct and retrograde orbits, as illustrated in Fig.(2). Also, the presence of black hole’s charge is to decrease the deflection angle due to the additional repulsive effects on the light rays, as compared with the Kerr case with the same impact parameter $b$ Hsiao . As we will discuss in the next section, the angular positions of the relativistic images of the distant light sources due to the gravitational lensing of the black holes critically depends on the critical impact parameter $b_{sc}$. The impact parameter $b$ as a function of the radius of the circular motion $r_{0}$ is obtained as $\displaystyle b(r_{0})=\frac{s(2aM-a\frac{Q^{2}}{r_{0}})-r_{0}\sqrt{(a\frac{Q^{2}}{r_{0}^{2}}-a\frac{2M}{r_{0}})^{2}+(1-\frac{2M}{r_{0}}+\frac{Q^{2}}{r_{0}^{2}})[a^{2}(1+\frac{2M}{r_{0}}-\frac{Q^{2}}{r_{0}^{2}})+r_{0}^{2}]}}{2M-r_{0}-\frac{Q^{2}}{r_{0}}}\,.$ (41) The solution of $r_{sc}$ of the radius of the innermost circular motion has been found in Hsiao as $\displaystyle r_{sc}$ $\displaystyle=\frac{3M}{2}+\frac{1}{2\sqrt{3}}\sqrt{9M^{2}-8Q^{2}+U_{c}+\frac{P_{c}}{U_{c}}}$ $\displaystyle\quad-\frac{s}{2}\sqrt{6M^{2}-\frac{16Q^{2}}{3}-\frac{1}{3}\left(U_{c}+\frac{P_{c}}{U_{c}}\right)+\frac{8\sqrt{3}Ma^{2}}{\sqrt{9M^{2}-8Q^{2}+U_{c}+\frac{P_{c}}{U_{c}}}}}\;\;,$ (42) where $\displaystyle P_{c}$ $\displaystyle=(9M^{2}-8Q^{2})^{2}-24a^{2}(3M^{2}-2Q^{2})\,,$ $\displaystyle U_{c}$ $\displaystyle=\bigg{\\{}(9M^{2}-8Q^{2})^{3}-36a^{2}(9M^{2}-8Q^{2})(3M^{2}-2Q^{2})+216M^{2}a^{4}$ $\displaystyle\quad\quad+24\sqrt{3}a^{2}\sqrt{(M^{2}-a^{2}-Q^{2})\left[Q^{2}(9M^{2}-8Q^{2})^{2}-27M^{4}a^{2}\right]}\bigg{\\}}^{\frac{1}{3}}\,.$ (43) The analytical expression of the critical value of the impact parameter ${b_{sc}}$ can be written as a function of black hole’s parameters Hsiao , $\displaystyle b_{sc}$ $\displaystyle=-a+\frac{M^{2}a}{2(M^{2}-Q^{2})}+\frac{s}{2\sqrt{3}(M^{2}-Q^{2})}\Bigg{[}\sqrt{V+(M^{2}-Q^{2})\left(U+\frac{P}{U}\right)}$ $\displaystyle+\sqrt{2V-(M^{2}-Q^{2})\left(U+\frac{P}{U}\right)-\frac{s6\sqrt{3}M^{2}a\left[(M^{2}-Q^{2})(9M^{2}-8Q^{2})^{2}-M^{4}a^{2}\right]}{\sqrt{V+(M^{2}-Q^{2})\left(U+\frac{P}{U}\right)}}}\Bigg{]}\;,$ (44) where $\displaystyle P$ $\displaystyle=(3M^{2}-4Q^{2})\left[9(3M^{2}-4Q^{2})^{3}+8Q^{2}(9M^{2}-8Q^{2})^{2}-216M^{4}a^{2}\right]\,,$ $\displaystyle U$ $\displaystyle=\bigg{\\{}-\left[3(3M^{2}-2Q^{2})^{2}-4Q^{4}\right]\left[9M^{2}(9M^{2}-8Q^{2})^{3}-8\left[3(3M^{2}-2Q^{2})^{2}-4Q^{4}\right]^{2}\right]\,$ $\displaystyle\qquad+108M^{4}a^{2}\left[9(3M^{2}-4Q^{2})^{3}+4Q^{2}(9M^{2}-8Q^{2})^{2}-54M^{4}a^{2}\right]\,$ $\displaystyle\qquad+24\sqrt{3}M^{2}\sqrt{(M^{2}-a^{2}-Q^{2})\left[Q^{2}(9M^{2}-8Q^{2})^{2}-27M^{4}a^{2}\right]^{3}}\bigg{\\}}^{\frac{1}{3}}\,,$ $\displaystyle V$ $\displaystyle=3M^{4}a^{2}+(M^{2}-Q^{2})\left[6(3M^{2}-2Q^{2})^{2}-8Q^{4}\right]\,.$ (45) These will serve as the important inputs for the analytical expressions of the coefficients $\bar{a}$ and $\bar{b}$ in (1). Figure 2: The critical impact parameter $b_{sc}/M$ as a function of the spin parameter $a/M$ for (a) $Q/M=0.3$, (b) $Q/M=0.6$. Also, the critical impact parameter $b_{sc}/M$ as a function of charge $Q/M$ for (c) ${a/M}=0.3$, (d) ${a/M}=0.6$. The plots show the Schwarzschild, Reissner-Nordström, Kerr and Kerr-Newman black holes for comparison. The plot convention used henceforth: Kerr-Newman direct (solid red line), Kerr-Newman retrograde (solid blue line), Kerr direct (black dashed line with $Q=0$), Kerr retrograde (black dotted line, with $Q=0$), Reissner-Nordström (solid purple line, with $a=0$), and Schwarzschild (solid black line, with $Q=0,a=0$). The counterpart of (16) for the Kerr-Newman case as a function of $z$ in (15) can be easily derived giving $\frac{dz}{d\phi}=\frac{1}{r_{0}}\frac{1-\frac{2{M}}{r_{0}}(1-z)+\frac{a^{2}+Q^{2}}{r_{0}^{2}}(1-z)^{2}}{1-\frac{2{M}}{r_{0}}(1-\frac{a}{b_{s}})(1-z)+\frac{Q^{2}}{r_{0}^{2}}(1-\frac{a}{b_{s}})(1-z)^{2}}\sqrt{B(z,r_{0})}\;,$ (46) where $B(z,r_{0})=c_{1}(r_{0})z+c_{2}(r_{0})z^{2}+c_{3}(r_{0})z^{3}+c_{4}(r_{0})z^{4}\,.$ (47) The function $B(z,r_{0})$ is then the quartic polynomial in $z$ with the coefficients $\begin{split}c_{1}(r_{0})=&4Q^{2}\left(1-\frac{a}{b_{s}}\right)^{2}-6Mr_{0}\left(1-\frac{a}{b_{s}}\right)^{2}+2r_{0}^{2}\left(1-\frac{a^{2}}{b^{2}}\right)\,,\\\ c_{2}(r_{0})=&-6Q^{2}\left(1-\frac{a}{b_{s}}\right)^{2}+6Mr_{0}\left(1-\frac{a}{b_{s}}\right)^{2}-r_{0}^{2}\left(1-\frac{a^{2}}{b^{2}}\right)\,,\\\ c_{3}(r_{0})=&4Q^{2}\left(1-\frac{a}{b_{s}}\right)^{2}-2Mr_{0}\left(1-\frac{a}{b_{s}}\right)^{2}\,,\\\ c_{4}(r_{0})=&-Q^{2}\left(1-\frac{a}{b_{s}}\right)^{2}\,.\end{split}$ (48) All coefficients have the additional contributions from the charge $Q$. In particular, the presence of the $z^{4}$ term with the coefficient $c_{4}(r_{0})$ in $B$, which vanishes in the Kerr case, makes the calculations of $\bar{a}$ and $\bar{b}$ more involved. The integrant function $f(z,r_{0})$ in (22) now takes the form $f(z,r_{0})=\frac{r_{0}^{2}}{a^{2}+Q^{2}}\left(\frac{C_{-}}{z-z_{-}}+\frac{C_{Q}z+C_{+}}{z-z_{+}}\right)\frac{2{r_{0}}}{\sqrt{c_{1}(r_{0})z+c_{2}(r_{0})z^{2}+c_{3}(r_{0})z^{3}+c_{4}(r_{0})z^{4}}}\,.$ (49) The corresponding coefficients $C_{-}$, $C_{Q}$, and $C_{+}$ in the Kerr- Newman case are $\begin{split}C_{-}=&\frac{a^{2}+Q^{2}-2Mr_{-}(1-\frac{a}{b_{s}})+\frac{Q^{2}r_{-}^{2}}{a^{2}+Q^{2}}(1-\frac{a}{b_{s}})}{2r_{0}\sqrt{M^{2}-a^{2}-Q^{2}}}\,,\\\ C_{Q}=&\frac{Q^{2}}{r_{0}^{2}}\left(1-\frac{a}{b_{s}}\right)\,,\\\ C_{+}=&\frac{a^{2}+Q^{2}-2Mr_{-}(1-\frac{a}{b_{s}})+\frac{Q^{2}}{r_{0}}(r_{+}-r_{-})(1-\frac{a}{b_{s}})+Q^{2}(1-\frac{a}{b_{s}})}{-2r_{0}\sqrt{M^{2}-a^{2}-Q^{2}}}\,,\end{split}$ (50) where $z_{+}$, $z_{-}$ then become $\begin{split}z_{-}=&1-\frac{r_{0}r_{-}}{a^{2}+Q^{2}}\;,\\\ z_{+}=&1-\frac{r_{0}r_{+}}{a^{2}+Q^{2}}\;,\end{split}$ (51) defined in terms of the outer(inner) black hole horizon $r_{+}$ ($r_{-}$). Again, $z_{\pm}\leq 0$ for all $a$ and $Q$ with the nonzero $r_{+}$. Note that, for charge $Q\to 0$, $C_{Q}$ vanishes. Analogous to the previous subsection of the Kerr case, we define the function $f_{D}(z,r_{0})$ as $f_{D}(z,r_{0})=\frac{r_{0}^{2}}{a^{2}+Q^{2}}\left(\frac{C_{-}}{z-z_{-}}+\frac{C_{Q}z+C_{+}}{z-z_{+}}\right)\frac{2{r_{0}}}{\sqrt{c_{1}(r_{0})z+c_{2}(r_{0})z^{2}}}\,.$ (52) As $z\to 0$, $f_{D}(z,r_{0})\to 1/z$. Its integration over $z$ gives the divergent part of $I_{D}(r_{0})$ when $b\to b_{c}$. Here we find $\begin{split}I_{D}(r_{0})=&\int_{0}^{1}f_{D}(z,r_{0})dz\\\ =&\frac{2r_{0}^{3}}{a^{2}+Q^{2}}\frac{C_{-}}{\sqrt{c_{1}(r_{0})z_{-}+c_{2}(r_{0})z_{-}^{2}}}\log{\left(\frac{\sqrt{c_{1}(r_{0})z_{-}+c_{2}(r_{0})z_{-}}+\sqrt{c_{1}(r_{0})+c_{2}(r_{0})z_{-}}}{\sqrt{c_{1}(r_{0})z_{-}+c_{2}(r_{0})z_{-}}-\sqrt{c_{1}(r_{0})+c_{2}(r_{0})z_{-}}}\right)}\\\ &+\frac{2r_{0}^{3}}{a^{2}+Q^{2}}\frac{C_{+}+C_{Q}z_{+}}{\sqrt{c_{1}(r_{0})z_{+}+c_{2}(r_{0})z_{+}^{2}}}\log{\left(\frac{\sqrt{c_{1}(r_{0})z_{+}+c_{2}(r_{0})z_{+}}+\sqrt{c_{1}(r_{0})+c_{2}(r_{0})z_{+}}}{\sqrt{c_{1}(r_{0})z_{+}+c_{2}(r_{0})z_{+}}-\sqrt{c_{1}(r_{0})+c_{2}(r_{0})z_{+}}}\right)}\\\ &-\frac{2r_{0}^{3}}{a^{2}+Q^{2}}\frac{2C_{Q}}{\sqrt{c_{2}(r_{0})}}\log{\left(\sqrt{c_{1}(r_{0})}\sqrt{c_{2}(r_{0})}\right)}\\\ &+\frac{2r_{0}^{3}}{a^{2}+Q^{2}}\frac{2C_{Q}}{\sqrt{c_{2}(r_{0})}}\log{\left(c_{2}(r_{0})+\sqrt{c_{2}(r_{0})}\sqrt{c_{1}(r_{0})+c_{2}(r_{0})}\right)}\,.\end{split}$ (53) In the SDL, by substituting (29), $I_{D}(b)$ becomes $\begin{split}I_{D}(b)\approx&-\frac{r_{sc}^{3}}{a^{2}+Q^{2}}\left(\frac{C_{-sc}}{\sqrt{c_{2sc}z_{-sc}^{2}}}+\frac{C_{+sc}+C_{Qsc}z_{+sc}}{\sqrt{c_{2sc}z_{+sc}^{2}}}+\frac{C_{Qsc}}{\sqrt{c_{2sc}}}\right)\log{\left(\frac{b}{b_{sc}}-1\right)}\\\ &+\frac{r_{sc}^{3}}{a^{2}+Q^{2}}\frac{C_{-sc}}{\sqrt{c_{2sc}z_{-sc}^{2}}}\log{\left[\frac{16c_{2sc}^{2}z_{-sc}^{2}b_{sc}^{\prime\prime}}{c_{1sc}^{\prime 2}2b_{sc}(z_{-sc}-1)^{2}}\right]}\\\ &+\frac{r_{sc}^{3}}{a^{2}+Q^{2}}\frac{C_{+sc}+C_{Qsc}z_{+sc}}{\sqrt{c_{2}z_{+}^{2}}}\log{\left[\frac{16c_{2sc}^{2}z_{+sc}^{2}b_{sc}^{\prime\prime}}{c_{1}^{\prime 2}2b_{sc}(z_{+sc}-1)^{2}}\right]}+\frac{r_{sc}^{3}}{a^{2}+Q^{2}}\frac{C_{Qsc}}{\sqrt{c_{2sc}}}\log{\left[\frac{16c_{2sc}^{2}b_{sc}^{\prime\prime})}{c_{1sc}^{\prime 2}2b_{sc}}\right]}\,,\end{split}$ (54) from which we can read off the coefficients $\bar{a}$ and $b_{D}$ as follows $\begin{split}\bar{a}=&\frac{r_{sc}^{3}}{\sqrt{c_{2sc}}}\left[\frac{C_{-sc}}{r_{sc}r_{-}-(a^{2}+Q^{2})}+\frac{C_{+sc}}{r_{sc}r_{+}-(a^{2}+Q^{2})}\right]\end{split}$ (55) $\begin{split}b_{D}=&\bar{a}\log{\left[\frac{8c_{2sc}^{2}b_{sc}^{\prime\prime}}{c_{1sc}^{\prime 2}b_{sc}}\right]}\\\ &+\frac{2r_{sc}^{3}}{\sqrt{c_{2sc}}}\left[\frac{C_{-sc}}{r_{sc}r_{-}-(a^{2}+Q^{2})}\log{\left(1-\frac{a^{2}+Q^{2}}{r_{sc}r_{-}}\right)}+\frac{C_{+sc}+C_{Qsc}z_{+sc}}{r_{sc}r_{+}-(a^{2}+Q^{2})}\log{\left(1-\frac{a^{2}+Q^{2}}{r_{sc}r_{+}}\right)}\right]\,.\end{split}$ (56) They reduce to their counterparts in (31) and (32) respectively as $Q\to 0$. As for the remaining contributions to the regular part, and in the SDL, we have $\begin{split}&b_{R}=I_{R}(r_{sc})=\int_{0}^{1}f_{R}(z,r_{sc})dz\\\ &=\frac{2r_{0}^{3}}{a^{2}+Q^{2}}\frac{C_{-}}{\sqrt{c_{2}}z_{-}}\log{\left(\frac{z_{-}}{z_{-}-1}\frac{2c_{2}+c_{3}+2\sqrt{c_{2}+c_{3}+c_{4}}\sqrt{c_{2}}}{4c_{2}}\right)}\\\ &+\frac{2r_{0}^{3}}{a^{2}+Q^{2}}\frac{C_{-}}{z_{-}\sqrt{c_{2}+c_{3}z_{-}+c_{4}z^{2}_{-}}}\log{\left(\frac{z_{-}-1}{z_{-}}\frac{\left(\sqrt{c_{2}+c_{3}z_{-}+c_{4}z_{-}^{2}}+\sqrt{c_{2}}\right)^{2}-c_{4}z_{-}^{2}}{\left(\sqrt{c_{2}+c_{3}z_{-}+c_{4}z_{-}^{2}}+\sqrt{c_{2}+c_{3}+c_{4}}\right)^{2}-c_{4}(z_{-}-1)^{2}}\right)}\\\ &+\frac{2r_{0}^{3}}{a^{2}+Q^{2}}\frac{C_{+}}{\sqrt{c_{2}}z_{+}}\log{\left(\frac{z_{+}}{z_{+}-1}\frac{2c_{2}+c_{3}+2\sqrt{c_{2}+c_{3}+c_{4}}\sqrt{c_{2}}}{4c_{2}}\right)}\\\ &+\frac{2r_{0}^{3}}{a^{2}+Q^{2}}\frac{C_{+}+C_{Q}z_{+}}{z_{+}\sqrt{{c_{2}+c_{3}z_{-}+c_{4}z^{2}_{-}}}}\log{\left(\frac{z_{+}-1}{z_{-}}\frac{\left(\sqrt{c_{2}+c_{3}z_{+}+c_{4}z_{+}^{2}}+\sqrt{c_{2}}\right)^{2}-c_{4}z_{+}^{2}}{\left(\sqrt{c_{2}+c_{3}z_{+}+c_{4}z_{+}^{2}}+\sqrt{c_{2}+c_{3}+c_{4}}\right)^{2}-c_{4}(z_{+}-1)^{2}}\right)}\\\ &+\frac{2r_{0}^{3}}{a^{2}+Q^{2}}\frac{C_{Q}}{\sqrt{c_{2}}}\log{\left(\frac{z_{+}}{z_{+}-1}\right)}\Big{\|}_{r_{0}=r_{sc}}\end{split}$ (57) In the limit of $Q\to 0$, where $C_{Q}$ and $c_{4}$ go to zero, the above expression of $b_{R}$ reduces to (33) in the Kerr case after implementing straightforward algebra. The coefficient $\bar{b}$ is obtained using (56) and (57) as $\begin{split}\bar{b}=&-\pi+\bar{a}\log{\left[\frac{36}{4(1-c_{4sc}/c_{2sc})^{2}+4\sqrt{3}(1-c_{4sc}/c_{2sc})^{3/2}+3(1-c_{4sc}/c_{2sc})}\frac{8c_{2sc}^{2}b_{sc}^{\prime\prime}}{c_{1sc}^{\prime 2}b_{sc}}\right]}\\\ &+\frac{r_{sc}^{3}}{\sqrt{c_{2sc}}}\frac{2(a^{2}+Q^{2})C_{-sc}}{(a^{2}+Q^{2}-r_{sc}r_{-})}\frac{\sqrt{3}}{{P_{-}}}\\\ &\quad\quad\times\log\left[\frac{-r_{sc}r_{-}}{a^{2}+Q^{2}-r_{sc}r_{-}}\frac{\left({P_{-}}+\sqrt{3}({a^{2}+Q^{2}})\right)^{2}-3\left({a^{2}+Q^{2}}-{r_{sc}r_{-}}\right)^{2}({c_{4sc}}/{c_{2sc}})}{\left(P_{-}+(a^{2}+Q^{2})(1-c_{4sc}/c_{2sc})^{1/2}\right)^{2}-3{r^{2}_{sc}r^{2}_{-}}({c_{4sc}}/{c_{2sc}})}\right]\\\ &+\frac{r_{sc}^{3}}{\sqrt{c_{2sc}}}\frac{2[(a^{2}+Q^{2})C_{+sc}+(a^{2}+Q^{2}-r_{sc}r_{+})C_{Qsc}]}{(a^{2}+Q^{2}-r_{sc}r_{+})}\frac{\sqrt{3}}{{P_{+}}}\\\ &\quad\quad\times\log{\left[\frac{-r_{sc}r_{+}}{a^{2}+Q^{2}-r_{sc}r_{+}}\frac{\left({P_{+}}+\sqrt{3}({a^{2}+Q^{2}})\right)^{2}-3\left({a^{2}+Q^{2}}-{r_{sc}r_{+}}\right)^{2}({c_{4sc}}/{c_{2sc}})}{\left(P_{+}+(a^{2}+Q^{2})(1-c_{4sc}/c_{2sc})^{1/2}\right)^{2}-3{r^{2}_{sc}r^{2}_{+}}({c_{4sc}}/{c_{2sc}})}\right]}\,.\end{split}$ (58) In the equation above, we have replaced $c_{3sc}$ by the linear combination of $c_{2sc}$ and $c_{4sc}$ in (48), given by $\begin{split}c_{3sc}=&-\frac{2}{3}c_{2sc}-\frac{4}{3}c_{4sc}\,.\end{split}$ (59) We also have $\begin{split}P^{2}_{\pm}&=(a^{2}+Q^{2}+2r_{sc}r_{\pm})(a^{2}+Q^{2})-(a^{2}+Q^{2}+r_{sc}r_{\pm})(a^{2}+Q^{2}-r_{sc}r_{\pm})(c_{4sc}/c_{2sc})\,.\end{split}$ (60) Combining (41),(II.2) and (II.2), the coefficients $\bar{a}$ and $\bar{b}$ in (55) and (58) can be analytically expressed as a function of the black hole’s parameters in the SDL. Our results are ploted in Fig.(4). Again, notice that $\bar{a}>0$ but $\bar{b}<0$ with the parameters in the figure. Due to the fact that the bending angle for light rays resulting from the charged black hole is suppressed as compared with the neutral black hole with the same impact parameter $b$, both $\bar{a}$ and $\|\bar{b}\|$ are found to increase with the charge $Q$. Figure 3: The coefficients $\bar{a}$ and $\bar{b}$ as a function of the spin parameter $a/M$ for the Kerr black hole with $Q/M=0$ and the Kerr-Newman black hole with $Q/M=0.6$: (a) the coefficient $\bar{a}$, (b) the coefficient $\bar{b}$. The display of the plot follows the convention in Fig.(2). Figure 4: The coefficients of $\bar{a}$ and $\bar{b}$ as a function of charge $Q/M$ for the Schwarzschild black hole with ${a/M}=0$, ${Q/M}=0$, the the Reissner- Nordström blck hole with ${a/M}=0$ and the Kerr-Newman black hole with ${a/M}=0.6$: (a) the coefficient $\bar{a}$, (b) the coefficient $\bar{b}$. The display of the plot follows the convention in Fig.(2). Figure 5: The SDL deflection angle (dotted lines) and the exact one (solid lines): (a)${a/M}=0.5$ and ${Q/M}=0.6$, (b) Error between them defined by $(\hat{\alpha}_{exact}-\hat{\alpha})/\hat{\alpha}_{exact}\times 100\%$; (c) ${a/M}=0.5$ and ${Q/M}=0.3$,(d) Error; (e) ${a/M}=0.9$ and ${Q/M}=0.3$, (f) Error; (g) ${a/M}=0.5$ and ${Q/M}=0.8$, (h) Error. It is then quite straightforward to check that the coefficients $\bar{a}$ and $\bar{b}$ in the Kerr-Newmann case can reduce to those in (31) and (35) in the Kerr case by setting $c_{4}\to 0$ in the limit of $Q\to 0$, also leading to $P_{\pm}\to a\sqrt{a^{2}+2r_{sc}r_{\pm}}$. To compare with the Reissner- Nordström black hole in Tsuka1 ; Tsuka2 , it is known that the impact parameter $b$ as a function of $r_{0}$ is $b(r_{0})=\frac{r_{0}^{2}}{\sqrt{Q^{2}-2Mr_{0}+r_{0}^{2}}}$ (61) and the circular motion of light rays forms the photon sphere with the radius $r_{c}=\frac{3M+\sqrt{9M^{2}-8Q^{2}}}{2}\;.$ (62) The critical impact parameter as a function of $r_{c}$ is given by $b_{c}=\frac{r_{c}^{2}}{\sqrt{Mr_{c}-Q^{2}}}\,.$ (63) Notice the subscript is changed from $sc$ to $c$ since the same critical impact parameters are obtained for light rays in direct orbits and retrograde orbits in the case of the nonspinning black holes. In the limit of $a\to 0$, we have $C_{-sc}\to 0$ in (50) using the definition of $r_{-}$ in (39). Thus, the coefficient $\bar{a}$ in (55) can be further simplified using (50), (39) and $c_{1sc}=0$ giving $\bar{a}=\frac{r_{c}}{\sqrt{3Mr_{c}-4Q^{2}}}\,,$ (64) which reproduces the expression in Tsuka1 ; Tsuka2 . As for the coefficient $\bar{b}$, in the limit of $a\to 0$, apart from $C_{-sc}\to 0$, $(a^{2}+Q^{2})C_{+sc}+(a^{2}+Q^{2}-r_{sc}r_{+})C_{Qsc}\to 0$ as well. So, the coefficient $\bar{b}$ in (58) has the contribution only from the term proportional to $\bar{a}$. After substituting (61) and (48) in the limit of $a\to 0$ to (58) and going through nontrivial algebra, we indeed recover the compact analytical expression in Tsuka1 ; Tsuka2 : $\bar{b}=-\pi+\bar{a}\log{\left[\frac{8(3Mr_{c}-4Q^{2})^{3}}{M^{2}r_{c}^{2}(Mr_{c}-Q^{2})^{2}}\left(2\sqrt{Mr_{c}-Q^{2}}-\sqrt{3Mr_{c}-4Q^{2}}\right)^{2}\right]}\,.$ (65) Figure 5 shows good agreement between the obtained SDL expression and the exact one in Hsiao computed numerically when $b$ approaches $b_{c}$ for some values of $a$ and $Q$. In conclusion, we have successfully achieved the analytical expression of the coefficient $\bar{a}$ and $\bar{b}$ in the form (1) of the SDL deflection angle due to the spherically nonsymmetric black holes, although they look not as simple as in the cases of the spherically symmetric black holes. Additionally, the obtained expressions can reproduce the respective ones due to the Kerr, Reissner-Nordström black holes and also due to the Schwarzschild black hole by taking the appropriate limits of the black hole’s parameters. ## III Relativistic Images of Gravitational lens and applications to supermassive galactic black holes We consider the cases of the planar light rays with the lens diagram shown in Fig.(1), where $d_{L}$ and $d_{S}$ are the distances of the lens (black hole) and the light source from the observer, and also $d_{LS}$ represents the distance between the lens and the source. The line joining the observer and the lens is considered as a reference optical axis. The angular positions of the source and the image are measured from the optical axis, and are denoted by $\beta$ and $\theta$, respectively. The lens equation is given by $\tan s\beta=\tan\theta-\frac{d_{LS}}{d_{S}}[\tan\theta+\tan(\hat{\alpha}-\theta)]\;,$ (66) where $\hat{\alpha}$ is the deflection angle of light rays obtained from (22) that can be expressed in terms of the impact parameter $b$ as the light rays approach to the black holes. In Eiroa , it is mentioned that the lens equations are applied for the observer and the source immersed in the asymptotically flat spacetime, where the Kerr and Kerr-Newman black holes have the asymptotically flat metric. Also, in the small $\beta$ and $\theta$ limits, we will see that the approximate lens equations to be found later are the same ones in Bozza_2003 , in which the Kerr black holes are considered. In the SDL of our interest, when the light rays wind around the black hole $n$ times, the deflection angle $\hat{\alpha}$ can be approximately by (1). The angle appearing in the lens equation should be within $2\pi$ and can be obtained from the deflection angle $\hat{\alpha}$ subtracting $2n\pi$. Together with the relation between the impact parameter $b$ and the angular position of the image given by $b=d_{L}\sin\theta\;,$ (67) in Fig.(1), we can solve the lens equation (66) with a given angular position of the source $\beta$ for the angular position of the observed image $\theta$. In the SDL, when the angular position of the source is small, $\theta$ is expectedly small with the small impact parameter $b$. Then the lens equation (66) can be further simplified by $s\beta\simeq\theta-\frac{d_{LS}}{d_{S}}[\hat{\alpha}(\theta)-2n\pi]\,$ (68) and (67) can be approximated by $b\simeq d_{L}\theta$. This can reduce to the lens equations in Bozza_2003 , in which the small angle limits are considered. According to Bozza3 , the zeroth order solution $\theta_{sn}^{0}$ is obtained from $\hat{\alpha}(\theta_{sn}^{0})=2n\pi$. Using the SDL deflection angle in (1) we have then $\theta_{sn}^{0}=\frac{\|b_{sc}\|}{d_{L}}\left(1+e^{\frac{\bar{b}-2n\pi}{\bar{a}}}\right)$ (69) for $n=1,2,\cdots$. The angular position $\theta_{sn}$ decrease in $n$ and reaches the asymptotic angular position given by $\theta_{s\infty}=\|b_{sc}\|/{d_{L}}$ as $n\to\infty$. With the zeroth order solution (69), the expansion of $\hat{\alpha}(\theta)$ around $\theta=\theta_{sn}^{0}$ is written explicitly as $\hat{\alpha}(\theta)=\hat{\alpha}(\theta_{sn}^{0})-\frac{\bar{a}}{e^{(\bar{b}-2n\pi)/\bar{a}}}\frac{d_{L}d_{LS}}{\|b_{sc}\|d_{S}}(\theta-\theta_{sn}^{0})+{O}(\theta-\theta_{sn}^{0})^{2}\,.$ (70) Then the approximate lens equation (68) to the order $(\theta-\theta_{sn}^{0})$ becomes $s\beta\simeq\theta_{sn}^{0}+\left(1+\frac{\bar{a}}{e^{(\bar{b}-2n\pi)/\bar{a}}}\frac{d_{L}d_{LS}}{\|b_{sc}\|d_{S}}\right)(\theta-\theta_{sn}^{0})\,.$ (71) Solving for $\theta$, by keeping the lowest order term in $\|b_{sc}\|/d_{L}\ll 1$, we find the angular position of the image as Bozza2 $\begin{split}\theta_{sn}\simeq\theta_{sn}^{0}+\frac{e^{(\bar{b}-2n\pi)/\bar{a}}}{\bar{a}}\frac{\|b_{sc}\|d_{S}}{d_{LS}d_{L}}(s\beta-\theta_{sn}^{0})\;.\end{split}$ (72) We assume that either Kerr or Kerr-Newman black holes have the clockwise rotation shown in Fig.(1). The light rays emitted from the source circle around the black hole multiple times in the SDL along a direct orbit (red line) with $s=+1$, where both the image and the source end up in the same sides of the optical axis with the angular position $\theta_{+n}$ and/or along a retrograde orbit (blue line) with $s=-1$, where the image and the source are in the opposite sides with the angular position $\theta_{-n}$. We also define the angular position difference between the outermost image $\theta_{1\pm}$ and the asymptotic one near the optical axis as $\Delta\theta_{s}=\theta_{s1}-\theta_{s\infty}\,\,,$ (73) which is the value to compare with the resolution of the observation that allows to distinguish among a set of the relativistic images. We now compute the angular positions of the relativistic images of the sources for $n=1$ (the outermost image) due to either Kerr or Kerr-Newman black holes with the mass $M=4.1\times 10^{6}M_{\odot}$ and the distance $d_{L}=26000\;{\rm ly}$ of the supermassive black hole Sagittarius A* at the center of our Galaxy as an example. We also take the ratio to be $d_{LS}/d_{S}=1/2$. In Table 1 (2), we consider both the image and the source are in the same (opposite) sides of the optical axis, where the light rays travel along the direct (retrograde) orbits, and choose $\beta\sim\theta_{\pm 1}$. The angular positions of the relativistic images are computed by (72). In the case of $\|b_{sc}\|\ll d_{L}$, $\theta_{sn}$ is not sensitive to $\beta$ but mainly determined by $\theta_{sn}^{0}$ in (69). Given $\bar{a}$ and $\|\bar{b}\|$ of the magnitudes shown in Fig.(3) and (4), $e^{-\frac{\|\bar{b}\|+2n\pi}{\bar{a}}}\ll 1$. The behavior of $\theta_{sn}$ thus depends mainly on $\|b_{sc}\|$ as a function of angular momentum $a$ and charge $Q$ of the black holes. As discussed in the previous section, since the effects from the angular momentum of the black hole for direct orbits effectively induces more repulsive effects compared with the retrograde orbits, clearly shown in their effective potential $W_{\text{eff}}$ (9), the resulting $b_{+c}<\|b_{-c}\|$ yields asymmetric values of $\theta_{+1}<\theta_{-1}$ for the same $a$ and $Q$. These features are shown in the Tables 1 and 2. Additionally, we notice that $\theta_{+1}$ ($\theta_{-1}$) decreases (increase) in $a$ for fixed $Q$ resulting from the decrease (increase) of $b_{+c}$ ($\|b_{-c}\|$) as $a$ increases. As for $\Delta\theta$, for the same $Q$, $\Delta\theta_{+}$ increases with $a$ whereas $\Delta\theta_{-}$ decreases with $a$. In particular, $\Delta\theta_{+}$ can be increased from about $10^{-2}\mu\rm{as}$ with $a/M\sim 10^{-3}$ and $Q/M=10^{-3}$ to the value as high as $0.6\mu{\rm as}$ with $a/M=0.9$ and $Q/M=10^{-3}$, which certainly increases their observability by the current very long baseline interferometry (VLBI) Ulv ; Johnson_2020 . As for the finite $Q$ effects, also showing the repulsion to the light rays seen in the effective potential (40), both $\theta_{+1}$ and $\theta_{-1}$ decrease in $Q$ for fixed $a$, resulting in the slightly increase of $\Delta\theta_{\pm}$ as $Q$ increases. $a/{M}$ \| $Q/{M}$ \| $\theta_{+1}$ ($\mu$as) \| $\hat{\alpha}$ \| $b/M$ \| $\theta_{+\infty}$ ($\mu$as) \| $\Delta\theta_{+}$ ($\mu$as) ---\|---\|---\|---\|---\|---\|--- $10^{\tiny-3}$ \| $10^{\tiny-3}$ \| 26.4231 \| $2\pi+32.8135$ ($\mu$as) \| $5.2007$ \| 26.3900 \| 0.0331 \| $0.3$ \| 26.0217 \| $2\pi+32.0563$ ($\mu$as) \| $5.1217$ \| 25.9866 \| 0.0351 \| $0.6$ \| 24.7179 \| $2\pi+29.4336$ ($\mu$as) \| $4.8651$ \| 24.6747 \| 0.0432 \| $0.8$ \| 23.1445 \| $2\pi+26.2837$ ($\mu$as) \| $4.5554$ \| 23.0849 \| 0.0596 $0.5$ \| $10^{\tiny-3}$ \| 20.9290 \| $2\pi+21.8561$ ($\mu$as) \| $4.1193$ \| 20.8119 \| 0.1171 \| $0.3$ \| 20.4085 \| $2\pi+20.8203$ ($\mu$as) \| $4.0169$ \| 20.2758 \| 0.1327 \| $0.6$ \| 18.6189 \| $2\pi+17.2398$ ($\mu$as) \| $3.6646$ \| 18.4049 \| 0.2140 \| $0.8$ \| 16.0922 \| $2\pi+12.1835$ ($\mu$as) \| $3.1673$ \| 15.5372 \| 0.5550 $0.9$ \| $10^{\tiny-3}$ \| 15.1170 \| $2\pi+10.2354$ ($\mu$as) \| $2.9754$ \| 14.4517 \| 0.6653 \| $0.3$ \| 14.1818 \| $2\pi+8.36638$ ($\mu$as) \| $2.7913$ \| 13.2701 \| 0.9117 \| $0.6$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $0.8$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ Table 1: Relativistic images on the same side of the source with the angular position $\beta=10$ ($\mu$as) where the light rays are along direct orbits seen in Fig.(1). $a/{M}$ \| $Q/{M}$ \| $\theta_{-1}$ ($\mu$as) \| $\hat{\alpha}$ \| $b/M$ \| $\theta_{-\infty}$ ($\mu$as) \| $\Delta\theta_{-}$ ($\mu$as) ---\|---\|---\|---\|---\|---\|--- $10^{\tiny-3}$ \| $10^{\tiny-3}$ \| 26.4433 \| $2\pi+72.8286$ ($\mu$as) \| $5.20464$ \| 26.4103 \| 0.0330 \| $0.3$ \| 26.0422 \| $2\pi+72.0654$ ($\mu$as) \| $5.12569$ \| 26.0073 \| 0.0349 \| $0.6$ \| 24.7395 \| $2\pi+69.4844$ ($\mu$as) \| $4.86931$ \| 24.6966 \| 0.0429 \| $0.8$ \| 23.1680 \| $2\pi+66.3165$ ($\mu$as) \| $4.56000$ \| 23.1088 \| 0.0592 $0.5$ \| $10^{\tiny-3}$ \| 31.1994 \| $2\pi+82.4458$ ($\mu$as) \| $6.14075$ \| 31.1862 \| 0.0132 \| $0.3$ \| 30.8561 \| $2\pi+81.6482$ ($\mu$as) \| $6.07318$ \| 30.8422 \| 0.0139 \| $0.6$ \| 29.7638 \| $2\pi+79.5352$ ($\mu$as) \| $5.85820$ \| 29.7479 \| 0.0159 \| $0.8$ \| 28.5058 \| $2\pi+76.9916$ ($\mu$as) \| $5.61059$ \| 28.4866 \| 0.0192 $0.9$ \| $10^{\tiny-3}$ \| 34.7203 \| $2\pi+89.4397$($\mu$as) \| $6.83374$ \| 34.7130 \| 0.0073 \| $0.3$ \| 34.4063 \| $2\pi+88.5984$ ($\mu$as) \| $6.77195$ \| 34.3988 \| 0.0075 \| $0.6$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $0.8$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ Table 2: Relativistic images on the opposite side of the source with the angular position $\beta=10$ ($\mu$as) where the light rays are in retrograde orbits seen in Fig.(1). Another application of the analytical expression of the deflection angle on the equatorial plane is to consider the quasiequatorial gravitational lensing based upon the works of Bozza_2003 ; Gyu_2007 . In this situation, the polar angle $\theta$ is set to be slightly away from $\theta=\frac{\pi}{2}$ and now becomes time dependent. In the SDL, the deflection angle of light rays with the additional initial declination can also be cast into the form of (1) where the coefficients are replaced by $\hat{a}$ and $\hat{b}$. In particular, the coefficient $\hat{a}$ obtained from the slightly off the equatorial plane can be related by the coefficient $\bar{a}$ on the equatorial plane through the $\omega$ function as $\hat{a}=\omega(r_{sc})\,\bar{a}\;,$ (74) where $\omega$ depends on $r$, and in turn depends on the deflection angle $\phi(r)$. Notice that the above relation (74) involves $\omega$, which is evaluated at $r=r_{sc}$. In the case of the Kerr black hole, it is found that Bozza_2003 $\omega(r_{sc})=\frac{(r_{sc}^{2}+a^{2}-2Mr_{sc})\sqrt{b_{sc}^{2}-a^{2}}}{2Mar_{sc}+b_{sc}(r_{sc}^{2}-2Mr_{sc})}\,,$ (75) and thus for the Schwarzschild case we have $a\rightarrow 0$, $\omega\rightarrow 1$. Then, substituting (13) and (14) into (75), together with the expression of $\bar{a}$ in (31), through (74) gives $\hat{a}=1$ for the Kerr case. However, in the Kerr-Newman black hole, the straightforward calculations show that the above relation (74) still holds true. The detailed derivations will appear in our future publication. Thus, the coefficient $\hat{a}$ can be analytically given by the coefficient $\bar{a}$ in (55), together with the $\omega$ function in the Kerr-Newman case in below $\omega(r_{sc})=\frac{(r_{sc}^{2}+a^{2}+Q^{2}-2Mr_{sc})\sqrt{b_{sc}^{2}-a^{2}}}{-a(Q^{2}-2Mr_{sc})+b_{sc}(r_{sc}^{2}+Q^{2}-2Mr_{sc})}\,.$ (76) As $Q\rightarrow 0$, (76) reduces to (75) in the Kerr case. The behavior of $\hat{a}$ as a function of the charge $Q$ with the choices of the angular momentum $a$ for direct and retrograde orbits is displayed in Fig.(6). The value of $\hat{a}$ ($\hat{a}>1$) increases with $Q$ for both direct and retrograde orbits. According to Bozza_2003 ; Gyu_2007 , the magnification of relativistic images might formally diverge when the angular positions of the sources are at caustic points. The corresponding magnifying power close to caustic points due to the light rays winding around the black hole $n$ times is given by $\bar{\mu}_{n}$ with the ratio between two neighboring caustic points $\frac{\bar{\mu}_{n+1}}{\bar{\mu}_{n}}\propto e^{-\pi/{\hat{a}}}\,$ (77) depending only on $\hat{a}$. In the Kerr case with $\hat{a}=1$, this ratio is independent of the black hole angular momentum $a$, whereas in the Kerr-Newman case with $\hat{a}>1$ shown in the Fig.(6), the ratio decreases with $Q$ for both direct and retrograde orbits Gyu_2007 . Here we just sketch some of the effects from the charge $Q$ of the black hole on the magnification of relativistic images. To have the full pictures of the caustic points and find the magnification of relativistic images, it in fact deserves the extensive study to compute not only $\hat{a}$ but also $\hat{b}$ by following Bozza_2003 ; Gyu_2007 . The further extension from quasiequatorial plane to the full sky is also of great interest Gralla_2020a ; Gralla_2020b ; Johnson_2020 . Figure 6: The coefficient $\hat{a}$ as a function of the black hole charge $Q/M$ for the direct (retrograde) orbits with (a) $a/M=0.3$, (b) $a/M=0.6$. The display of the plot follows the convention in Fig.(2). ## IV Summary and outlook In summary, the dynamics of light rays traveling around the Kerr black hole and the Kerr-Newman black hole, respectively, is studied with the detailed derivations on achieving analytical expressions of $\bar{a}$ and $\bar{b}$ in the approximate form of the deflection angle in the SDL. Various known results are checked by taking the proper limits of the black hole’s parameters. The analytical expressions are then applied to compute the angular positions of relativistic images due to the supermassive galactic black holes. We find that the effects from the angular momentum $a$ for direct orbits of light rays and the charge $Q$ for both direct and retrograde orbits increase the angular separation of the outermost images from the others. Although the observation of relativistic images is a very difficult task Ulv , our studies show potentially increasing observability of the relativistic images from the effects of angular momentum and charge of the black holes. Hopefully, relativistic images will be observed in the near future. Through the analytical results we present in this work, one can reconstruct the black hole’s parameters that give strong lensing effects. As light rays travel on the quasiequatorial plane, our analytical results on the equatorial plane can also be applied to roughly estimate the relative magnifications of relativistic images with the sources near one of the caustic points by taking account of the dynamics of the light rays in the polar angle. The work of investigating the structure of the caustic points from the effects of the charge $Q$ of the Kerr-Newman black holes and the magnification of relativistic images is in progress. 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# Does a Hybrid Neural Network based Feature Selection Model Improve Text Classification? Suman Dowlagar LTRC IIIT-Hyderabad suman.dowlagar@ research.iiit.ac.in &Radhika Mamidi LTRC IIIT-Hyderabad radhika.mamidi@ iiit.ac.in ###### Abstract Text classification is a fundamental problem in the field of natural language processing. Text classification mainly focuses on giving more importance to all the relevant features that help classify the textual data. Apart from these, the text can have redundant or highly correlated features. These features increase the complexity of the classification algorithm. Thus, many dimensionality reduction methods were proposed with the traditional machine learning classifiers. The use of dimensionality reduction methods with machine learning classifiers has achieved good results. In this paper, we propose a hybrid feature selection method for obtaining relevant features by combining various filter-based feature selection methods and fastText classifier. We then present three ways of implementing a feature selection and neural network pipeline. We observed a reduction in training time when feature selection methods are used along with neural networks. We also observed a slight increase in accuracy on some datasets. ## 1 Introduction Text classification assigns one or more class labels from a predefined set to a document based on its content. Text classification has broad applications in real-world scenarios such as document categorization, news filtering, spam detection, Optical character recognition (OCR), and intent recognition. Giving high weights to relevant features is the objective of text classification. The field of text classification has gained more interest during the machine learning (ML) era. Many discriminative and generative machine learning classifiers have achieved excellent results in the field of text classification Deng et al. (2019). Feature selection and feature extraction methods are often used to reduce high dimensionality Bharti and Singh (2015). Feature extraction generates features from text Agarwal and Mittal (2014). Feature selection (FS) selects the most prominent features Saleh and El- Sonbaty (2007). These feature selection and extraction methods are used along with traditional classification algorithms. These methods reduced the curse of dimensionality and increased the classification accuracy Deng et al. (2019). Recently, deep learning models are used to learn better text representations and to classify the text Minaee et al. (2020). Such models include convolutional neural networks (CNN) Kim (2014), recurrent neural networks (RNN) Hochreiter and Schmidhuber (1997), Transformer models Adhikari et al. (2019), and graph convolutional networks (GCN) Yao et al. (2019). These NN models capture semantic and syntactic information in local and global word sequences. Even though the neural networks capture a complex and dense representation of data, the set of words introducing noise in the classifier is still present. Such words add the burden of increased vocabulary, which results in increased textual representation and an increase in the training time of the classifiers Song et al. (2011). Similar to the traditional approaches, we want to understand the effects of using statistical feature selection algorithms beforehand to calculate the features’ relevance and then train a fastText text-classification algorithm on those relevant features. Using this feature selection and neural network pipeline, we assume that the complexity of dealing with larger vocabulary decreases. Including feature selection with fastText text-classification helps reduce the classifier’s training time and helps the classifier reach better local optima, showing a significant increase in classification accuracy. In this work, we analyzed a feature selection and neural network pipeline for text classification. We used a hybrid feature selection method to get a score on relevant features. Using this score, we formulated three methods. The first and second methods deal with modifying the original text by extracting the relevant features. The third method deals with using the feature selection scores and pass it along with the word embeddings. We then observed the effect of feature selection on various neural networks. The rest of the paper is organized as follows. Section 2 gives a brief review of previous works in the field of feature selection and text classification. Section 3 presents a detailed procedure of the proposed pipeline and presents the experiments and datasets used for our study. Section 4 reports the performance of text classifiers with and without feature selection methods. Section 5 concludes the paper. ## 2 Literature Survey This section presents a brief description of the neural network (NN) classification algorithms and various feature selection methods. ### 2.1 Deep Learning for Text Classification Nowadays, various NNs such as CNN, RNN, BERT, and Text GCN achieve state-of- the-art results on text classification. CNN uses 1d convolutions Zhang et al. (2015) and character level convolutions Conneau et al. (2016) to learn the semantic similarity of words or characters, which helps in classifying the text. RNN models such as GRU, LSTM, and BiLSTM Liu et al. (2016) take word to word sequences to learn a better textual representation of a document that helps in text classification. Attention mechanisms have been introduced in these LSTM models, which increased the representativeness of the text for better classification Yang et al. (2016). Transformer models such as BERT Devlin et al. (2018) uses the attention mechanism that learns contextual relations between words or sub-words in a text Adhikari et al. (2019). Text GCN Yao et al. (2019) uses a graph-convolutional network to learn a heterogeneous word document graph on the whole corpus. Text GCN can capture global word co-occurrence information and use graph convolutions to learn a global representation, which helps classify the documents. ### 2.2 Feature Selection on Text Data The text classification often involves extensive data with thousands of features. Although tens of thousands of words are in a typical text collection, most of them contain little or no information to predict the text label. These features introduce complexity and increase the training time of an ML classifier. Feature selection is one method for giving high scores to relevant features (Deng et al., 2019). The goal of feature selection is to select highly-relevant features with minimum redundancy. The relevance of a feature indicates that the feature is always necessary to predict the class label. There are various text feature selection methods in the literature, each being filter, wrapper, hybrid, and embedded methods. The filter method evaluates the quality of a feature using a scoring function. Some filter methods evaluate the goodness of a term based on how frequently it appears in a text corpus. Document Frequency (DF) Lam and Lee (1999) and Term Frequency - Inverse Document Frequency (TFIDF) Rajaraman and Ullman (2011) comes under this category. Other filter methods that originate from information theory are, Mutual Information (MI) Taira and Haruno (1999); Tang et al. (2019), Information Gain (IG) Yang and Pedersen (1997), CHI Rogati and Yang (2002), ANOVA F measure Elssied et al. (2014), Bi-Normal Separation (BNS) Forman (2003) and the GINI method Shang et al. (2013). They use hypothesis testing, contingency tables, mean and variance scores, conditional and posterior probabilities for selecting the features. The wrapper method Maldonado and Weber (2009) use a search strategy to construct each possible subset, feeds each subset to the chosen classifier, and then evaluates the classifier’s performance. These two steps are repeated until the desired quality of the feature subset is reached. The wrapper approach achieves better classification accuracy than filter methods. However, the time taken by the wrapper method is very high when compared to filter methods. Embedded methods complete the FS process within the construction of the machine learning algorithm itself. In other words, they perform feature selection during the model training. An embedded method is Decision Tree (DT) Quinlan (1986). In DT, while constructing the classifier, DT selects the best features/attributes that may give the best discriminative power. Hybrid methods are robust and take less time when compared to the wrapper and embedded methods. They combine a filter method with a wrapper method during the feature selection process. The HYBRID model Günal (2012) employs a combination of filter methods to select to rank the features and then a wrapper method to the obtained final features set. Our FS method is similar to the HYBRID model. A detailed report on the benefits of using the feature selection methods in the pipeline with traditional classifiers is presented in Deng et al. (2019); Forman (2003). Apart from using traditional classification methods, deep feature selection using neural networks were also proposed. These models use deep neural network autoencoders for the feature set reduction and text generation Mirzaei et al. (2019); Han et al. (2018). Lam and Lee (1999) studies the effect of feature set reduction before applying the neural network classifiers. The paper uses a multi-layer perceptron (MLP) classifier in combination with filter-based FS method. Alkhatib et al. (2017) proposes the use of neural network-based feature selection and text classification. Our work comes under this category. ## 3 Proposed Pipeline In this section, we present our feature selection and neural network pipeline. The feature selection and neural network pipeline start with selecting a good tokenizer to tokenize the data and create a feature set. The tokenizer used for our feature selection is the Sentencepiece tokenizer Kudo and Richardson (2018). Sentencepiece tokenizer implements subword units by using byte-pair- encoding (BPE) Sennrich et al. (2015) and unigram language model Kudo (2018). In the feature subset generation, we considered a hybrid feature selection method known as HYBRID Günal (2012). It has proved that a combination of the features selected by various methods is more effective and computationally faster than the features selected by individual filter and wrapper methods. Similar to the HYBRID model, we used three filters to obtain the relevancy score. The filters we considered were CHI2, ANOVA-F, and MI. These filters calculate the relevancy between the word and the class labels. Before feature selection, we used the Bag-of-Words(BoW) model to vectorize the data. In the BoW model, each feature vector is represented by $TF\\_IDF$ scores. Then we used statistical measures such as $\chi^{2}$, ANOVA-F, and MI for obtaining feature scores. $\chi^{2}$ 111A detailed explanation and a simple example of $\chi^{2}$ is given at https://www.mathsisfun.com/data/chi-square-test.html is a statistic to measure a relationship between feature vectors and a label vector. Analysis of Variance (ANOVA222A detailed explanation for ANOVA is given in https://towardsdatascience.com/anova-for-feature-selection-in-machine- learning-d9305e228476.) is a statistical method used to check the means of two or more groups that are significantly different from each other. Mutual Information (MI333A simple explanation and working python example of MI is available at https://machinelearningmastery.com/information-gain-and- mutual-information/) is frequently used to measure the mutual dependency between two variables. Using different statistical methods, we measured the relevance of each feature. We then aggregate the relevance scores of all satistical methods for each feature. The relevance of a feature $x_{i}$ is given by, $Relevancy(x_{i})=\\\ \chi^{2}(x_{i})+ANOVA(x_{i})+MI(x_{i})$ (1) Figure 1: Modifying text by masking the low ranked words Figure 2: Meta-Embeddings, including feature scores along with word embeddings Instead of an LR classifier given in the HYBRID model, we used the fastText classifier Joulin et al. (2016) for the feature selection. We used the fastText classifier as it is often on par with deep learning classifiers in terms of accuracy and performs faster computations. The fastText classifier treats the average of word embeddings as document embeddings, then feeds document embeddings into a feed-forward NN or a multinomial LR classifier. We used pre-trained fastText word embeddings Grave et al. (2018) while training a classifier. To get the final features list, we sorted the normalized, aggregated value in descending order and divided the entire feature space into k sets. In our model, we divided the sorted feature space into 20 sets. The value of k is fixed to 20 using a trial and error basis. We take the first set as the vocabulary of the classifier. We then trained the classifier and noted its accuracy. In the second iteration, we considered the vocabulary as the combination of first and second sets. Similarly, the third set has the vocabulary of the first three sets combined. We repeated the process until all the lists are exhausted. The set of features that resulted in a better classification metric is considered as the final feature set. According to the proposed FS method, the final feature set is considered relevant, and they are necessary to perform the text classification. In contrast, the other features have little to no effect on the text classification or might degrade the classifier’s performance. After feature subset generation, we propose three methods for including the feature selection information before training the neural network classifiers. 1. 1. Method 1 (Selecting only the relevant features)444This method is already used while selecting the final features set by the fastText classifier.: Like traditional classification algorithms, we select only the relevant features that are estimated to be important by the feature selection method before training a neural network classifier. 2. 2. Method 2 (Masking the features that were given low importance by our FS method): We felt that removing the features given low rank by our FS method might disturb the original data’s grammatical structure, thus disturbing the word to word dependencies. We masked the low ranked words with the help of $<MASK>+POS(word)$ tag. $<MASK>$ word masks the low ranked word, and POS preserves the word’s part of speech. The visual representation of method 2 is shown in figure 1 3. 3. Method 3 (Meta Embeddings): As shown in figure 2, we pass the relevancy and feature selection information along with embeddings in this method. Each slot holds the filter scores, i.e., CHI, ANOVA, MI scores of each feature. The last slot holds a 1 or 0 value. 1 is used for the selected features, and 0 is used for low ranked features that were not selected by our hybrid feature selection approach. We analyzed and evaluated the above methods with various state-of-the-art NN classifiers on the benchmark datasets. ### 3.1 Experiment In this section, we evaluated our feature selection and neural network pipeline on two tasks. We wanted to determine: * • If the pipeline decreases the training time of the classifier * • If it helps in obtaining better local optima, thus improving the classification accuracy. We tested our pipeline across multiple state-of-the-art text classification algorithms. 1. 1. CNN: Kim (2014) This convolutional neural network-based text classifier is trained by considering pre-trained word vectors. 2. 2. Bi-LSTM: Liu et al. (2016) A two-layer, bi-directional LSTM text classifier with pre-trained word embeddings as input was considered for the task of text classification. 3. 3. fastText: Joulin et al. (2016) This is a simple, efficient, and the fastest text classification method. It treats the average of word/n-grams embeddings as document embeddings, then feeds document embeddings into a linear classifier. 4. 4. Text GCN: Yao et al. (2019) Builds a heterogeneous word document graph for a whole corpus and turns document classification into a node classification problem. It uses GCN Kipf and Welling (2017) to learn word and document embeddings. 5. 5. DocBERT: Adhikari et al. (2019) A fine-tuned BERT model for document classification. The BERT model Devlin et al. (2018) uses a series of multiheaded attention and feedforward networks for various NLP tasks. ### 3.2 Datasets We ran our experiments on three widely used benchmark corpora and multilingual corpora. They are 20Newsgroups(20NG), R8, and R52 of Reuters 21578 and MLMRD. * • The 20NG dataset contains 18,846 documents divided into 20 different categories. 11,314 documents were used for training, and 7,532 documents were used for testing. * • R52 and R8 are two subsets of the Reuters 21578 dataset. R8 has 8 categories of the top eight document classes. It was split into 5,485 training and 2,189 test documents. R52 has 52 categories and was split into 6,532 training and 2,568 test documents. * • MLMRD is a Multilingual Movie Review Dataset. It consists of the genre and synopsis of movies across multiple languages, namely Hindi, Telugu, Tamil, Malayalam, Korean, French, and Japanese. The data set is minimal and unbalanced. It has 9 classes and a total of 14,997 documents. The data was split into 10,493 training and 4,504 test documents. We first preprocessed all the datasets by cleaning and tokenizing. The tokenizer used is the fastText tokenizer. For baseline 1 models, we used multilingual fastText embeddings Grave et al. (2018) of dimensionality 300, and baseline 2 models had the dimensionality of 304. We used default parameter settings as in their original papers for implementations. For calculating TFIDF, CHI2, ANOVA-F, MI scores, we used the scikit-learn library Pedregosa et al. (2011). For POS tagging, we used the NLTK Bird et al. (2009) pos tagger. All the neural network models were run on the GPU processor on the Windows platform with NVIDIA RTX 2070 graphics card. ## 4 Performance Datasets \| Our FS \| HYBRID FS ---\|---\|--- 20Newsgroups \| 81.27% \| 77.34% R8 \| 96.94% \| 93.79% R52 \| 92.72% \| 86.43% MLMRD \| 47.09% \| 42.98% Table 1: The classification accuracy of our FS model when compared to the HYBRID model. In our work, we modified the HYBRID Günal (2012) feature selection model by changing the LR classifier to the fastText classifier. We selected the fastText classifier in the feature selection process because of its fast learning ability of a NN model compared to the traditional ML classifiers and other neural network classifiers Joulin et al. (2016) without any decrease in classification accuracy. The neural network classifiers such as MLP, CNN, RNN, transformer, and GCN models achieve better classification accuracy when compared to traditional ML classifiers, but their training time is very high. Using a fastText classifier during feature selection, we observed that our model performed better on all the benchmark datasets than the HYBRID model. The results are shown in table 1. The fastText classifier’s use helped the model obtain better relevant features, increasing the current feature selection model’s accuracy compared to the HYBRID model. Datasets \| 20Newsgroups \| R8 \| R52 \| MLMRD ---\|---\|---\|---\|--- Baseline 1 & 2 \| 1,01,631 (V) \| 19,956 (V) \| 26287 (V) \| 94073 (V) Method 1 \| 25732 (0.25V) \| 17364 (0.87V) \| 22372 (0.85V) \| 52015 (0.55V) Method 2 \| 25732+30 (0.25V) \| 17364+30 (0.87V) \| 22372+30 (0.85V) \| 52015+143 (0.55V) Method 3 \| 1,01,631 (V) \| 19,956 (V) \| 26287 (V) \| 94073 (V) Table 2: The vocabulary size in all the FS inclusion methods when compared to the baselines. “V” is denoted as the vocabulary size of the actual data. Baselines 1,2, and method 3 have no change in vocabulary. However, using our FS method, the vocabulary is reduced to a maximum of 75% (for 20Newsgroups data). Other datasets have seen a 13% to 45% decrease in vocabulary size. We can see an increase in vocabulary from method 1 to method 2. It is due to the additional vocabulary resulted from the mask words when they are accompanied by pos tags. Here Penn Treebank POS tagset is used. Datasets \| Method \| Classifier(s) ---\|---\|--- \| \| CNN \| Bi-LSTM \| fastText \| DocBERT \| Text GCN 20Newsgroups \| Baseline 1 \| 79.31% \| 73.60% \| 81.04% \| 90.19% \| 86.13% \| Baseline 2 \| 79.46% \| 74.25% \| 82.44% \| NA \| 86.23% \| Method 1 \| 78.27% \| 73.44% \| 81.27% \| 89.37% \| 86.25% \| Method 2 \| 77.29% \| 70.48% \| 80.14% \| 88.43% \| 85.65% \| Method 3 \| 80.59% \| 76.57% \| 84.48% \| NA \| 86.15% R8 \| Baseline 1 \| 97.24% \| 92.70% \| 96.13% \| 97.62% \| 96.80% \| Baseline 2 \| 97.37% \| 93.82% \| 96.50% \| NA \| 96.94% \| Method 1 \| 97.39% \| 93.74% \| 96.94% \| 97.44% \| 96.28% \| Method 2 \| 96.57% \| 94.34% \| 96.07% \| 97.44% \| 96.85% \| Method 3 \| 97.39% \| 96.74% \| 97.18% \| NA \| 96.94% R52 \| Baseline 1 \| 94.78% \| 87.53% \| 92.02% \| 92.95% \| 93.56% \| Baseline 2 \| 94.84% \| 90.79% \| 92.76% \| NA \| 93.64% \| Method 1 \| 94.29% \| 87.47% \| 92.72% \| 93.10% \| 92.97% \| Method 2 \| 91.71% \| 91.90% \| 90.30% \| 92.10% \| 93.19% \| Method 3 \| 94.84% \| 91.48% \| 92.83% \| NA \| 93.74% MLMRD \| Baseline 1 \| 47.63% \| 46.43% \| 46.92% \| 53.11% \| 47.62% \| Baseline 2 \| 47.79% \| 47.43% \| 48.92% \| NA \| 49.62% \| Method 1 \| 44.98% \| 44.82% \| 47.09% \| 51.90% \| 46.58% \| Method 2 \| 44.63% \| 44.05% \| 46.61% \| 50.90% \| 46.98% \| Method 3 \| 48.44% \| 49.13% \| 49.55% \| NA \| 51.50% Table 3: Test accuracy on various neural network classifiers for the task of document classification. As the BERT model used is a fine-tuned one, we did not modify the model. As mentioned above, we used the training time-taken and test accuracy as the metrics to evaluate our approach. The accuracy and training time are recorded by running the model 10 times, and the average of the metrics was presented. ### 4.1 Effects of our methods on classification accuracy Table 3 demonstrates the accuracy of feature selection methods on NN classifiers. When methods 1 and 2 were used, there is a slight decrease in classification accuracy because the first two methods lost semantic connection among words. Thus, the classification performance is degraded. Also, some words which were relevant to the classifier were masked out during the FS method. Whereas in method 3, including the feature selection scores with word-embeddings, has shown a significant improvement in accuracy on all the datasets. Compared to the other datasets, the 20NG dataset has seen a significant decrease in vocabulary size. The vocabulary was decreased by 75%. However, eliminating those features did not affect the accuracy of the classifier for methods 1 and 2. Introducing the masked features in method 2 shown an increase in accuracy only in the Bi-LSTM method as this method considers word dependencies while training a classifier. Including the feature selection scores along with the word-embeddings improved the classification accuracy on all the datasets. The feature selection metadata helped the neural network classifier learn a better relationship between the words and classes and improve the classifier’s accuracy by reaching better local optima. In R8 and R52 datasets, we have seen an increase in accuracy using method 1 because our hybrid FS method worked better on these datasets by removing the noisy words without disturbing the relevant words. The maximum improvement in accuracy is shown in the R8 dataset, with a +4% increase in classification accuracy. Our approach did not show any better results on MLMRD datasets as this dataset has a limited number of documents to train and test the data for some languages (Telugu, Tamil, Malayalam, Korean). Reducing vocabulary size by the FS method decreased the classification accuracy. ### 4.2 Effects of our methods on training time The pictorial representation of time taken by the classifiers for all the datasets is given in appendix B of the supplementary material. The time taken by method 1 is lower than in all baseline models. In method 1, as the text is modified by considering only relevant features, the vocabulary size is reduced, and the sentence length is reduced. It resulted in the more accelerated training of the neural network. The time taken by baseline 2 and method 3 is similar because of the same embedding dimensionality of 304, but method 3 has achieved local optima a few epochs before compared to baseline 2, resulting in a time decrease of a few seconds. This phenomenon is attributed to the use of feature selection scores along with word embeddings. Method 2 has shown an increase in training time even though the vocabulary is decreased because of 2 factors. 1. 1. The masking of features created unknown words in the data, and the classifier has to be trained to learn the representation of masked words, whereas the other words had pre-trained embeddings. 2. 2. Apart from vocabulary, the neural network training time also depends on the input batch size given to the network and the length of the sentence in each batch. Because of the masked words, there is no decrease in either batch size or the sentence length. So the masking of data did not decrease the training time of the classifier. On the contrary, the Text GCN model has shown a decrease in training time because the classifier computes heterogeneous graph embeddings of each word based on the textual data before classification. It did not use any pre- trained embeddings. In method 3, there is a slight increase in training time because of increased vocabulary size due to the inclusion of feature selection metadata. Of all the NN classifiers, the Text GCN model had shown a maximum decrease in training time by 488 sec when method 1 was used on 20NG data. As the Text GCN operates on building a graph on the complete vocabulary of data, the time taken by the method to build the graph is reduced significantly by reducing the vocabulary size. It is followed by the DocBERT and Bi-LSTM on 20NG data with a decrease in training time by 480 and 394 sec. Text GCN and Bi-LSTM have shown a significant decrease in training time on all the datasets. On the contrary, fastText and CNN are very fast while training the NN model. The training time of such models was unchanged when our method 1 was used. When compared to all the classifiers, DocBERT achieved better results because of its evolutionary multi-headed attention and transformer models. As the Text GCN captures both local and word embeddings by constructing a heterogeneous graph, their results were better than those of the CNN and Bi-LSTM models, which work only on local word dependencies. As we increased the size of the embedding in FS method 2, this increased the dimensionality of vocabulary, resulting in the classifier’s increased training time. ## 5 Conclusion In our work, ”Does a Hybrid NN FS Model Improve Text Classification?”, we used the NN based hybrid FS method to extract relevant features and used NN classifiers for text classification. We extracted the relevant or high ranked features using filter-based methods and a fastText classifier. We then proposed three methods on how the feature selection can be included in the NN classification process. First, modifying the corpus by considering only relevant features. Second, modifying the data by masking the low ranked features, and the third method introduces feature selection information along with word embeddings. We observed that method 1 had shown a significant reduction in training time when large datasets or slower models are used, accompanied by a minimal change in classification accuracy. By introducing $MASK+P0S(word)$, we inferred that the masked word was a burden to the classifier, and it always tried to adjust the word embeddings, which resulted in increased epoch time during training and a slightly negative effect on classification accuracy. Whereas method 3 has shown no effect on decreasing the training time, it has shown a maximum of $4\%$ increase in the classification accuracy compared to baseline. It proved that introducing feature scores along with pre-trained word embeddings while training the NN classifier is beneficial. Instead of opting for random naive vocabulary reduction techniques such as using min_df and max_df (minimum and maximum document frequency) for selecting features, by using FS methods, we can calculate the relevance of the word beforehand and use that as metadata to the NN classifier. When the datasets are huge, these methods are of more significance. We can use the modified data while tuning the hyperparameters. Then we can use the real data to train and evaluate the model. Even in the critical domain datasets such as “medical”, we cannot rely on removing a word based on min_df and max_df scores. Each word in those datasets should be treated with utmost significance. FS methods help in such scenarios by calculating the word’s relevance and helps maintain better vocabulary before training neural network classifiers. ## References * Adhikari et al. (2019) Ashutosh Adhikari, Achyudh Ram, Raphael Tang, and Jimmy Lin. 2019. 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# Multilingual Pre-Trained Transformers and Convolutional NN Classification Models for Technical Domain Identification Suman Dowlagar LTRC IIIT-Hyderabad suman.dowlagar@ research.iiit.ac.in &Radhika Mamidi LTRC IIIT-Hyderabad radhika.mamidi@ iiit.ac.in ###### Abstract In this paper, we present a transfer learning system to perform technical domain identification on multilingual text data. We have submitted two runs, one uses the transformer model BERT, and the other uses XLM-ROBERTa with the CNN model for text classification. These models allowed us to identify the domain of the given sentences for the ICON 2020 shared Task, TechDOfication: Technical Domain Identification. Our system ranked the best for the subtasks 1d, 1g for the given TechDOfication dataset. ## 1 Introduction Automated technical domain identification is a categorization/classification task where the given text is categorized into a set of predefined domains. It is employed in tasks like Machine Translation, Information Retrieval, Question Answering, Summarization, and so on. In Machine Translation, Summarization, Question Answering, and Information Retrieval, the domain classification model will help leverage the contents of technical documents, select the appropriate domain-dependent resources, and provide personalized processing of the given text. Technical domain identification comes under text classification or categorization. Text classification is one of the fundamental tasks in the field of NLP. Text classification is the process of identifying the category where the given text belongs. Automated text classification helps to organize unstructured data, which can help us gather insightful information to make future decisions on downstream tasks. Traditional text classification approaches mainly focus on feature engineering techniques such as bag-of-words and classification algorithms Yang (1999). Nowadays, the sate-of-the-art results on text classification are achieved by various NNs such as CNN Kim (2014), LSTM Hochreiter and Schmidhuber (1997), BERT Adhikari et al. (2019), and Text GCN Adhikari et al. (2019). Attention mechanisms Vaswani et al. (2017) have been introduced in these models, which increased the representativeness of the text for better classification. Transformer models such as BERT Devlin et al. (2018) uses the attention mechanism that learns contextual relations between words or sub-words in a text. Text GCN Yao et al. (2019) uses a graph-convolutional network to learn a heterogeneous word document graph on the whole corpus, which helped classify the text. However, of all the deep learning approaches, transformer models provided SOTA results in text classification. In this paper, We present two approaches for technical domain identification. One approach uses the pre-trained Multilingual BERT model, and the other uses XLM-ROBERTa with CNN model. The rest of the paper is structured as follows. Section 2 describes our approach in detail. In Section 3, we provide the analysis and evaluation of results for our system, and Section 4 concludes our work. ## 2 Our Approach Here we present two approaches for the TechDOfication task. ### 2.1 BERT for TechDOfication Figure 1: The architecture of the BERT model for sentence classification. In the first approach, we use the pre-trained multilingual BERT model for domain identification of the given text. Bidirectional Encoder Representations from Transformers (BERT) is a transformer encoder stack trained on the large corpora. Like the vanilla transformer model Vaswani et al. (2017), BERT takes a sequence of words as input. Each layer applies self-attention, passes its results through a feed-forward network, and then hands it off to the next encoder. The BERT configuration model takes a sequence of words/tokens at a maximum length of 512 and produces an encoded representation of dimensionality 768. The pre-trained multilingual BERT models have a better word representation as they are trained on a large multilingual Wikipedia and book corpus. As the pre-trained BERT model is trained on generic corpora, we need to finetune the model for the given domain identification tasks. During finetuning, the pre- trained BERT model parameters are updated. In this architecture, only the [CLS] (classification) token output provided by BERT is used. The [CLS] output is the output of the 12th transformer encoder with a dimensionality of 768. It is given as input to a fully connected neural network, and the softmax activation function is applied to the neural network to classify the given sentence. ### 2.2 XLM-ROBERTa with CNN for TechDOfication Figure 2: The architecture of the XLM-ROBERTa with CNN for sentence classification. XLM-ROBERTa Conneau et al. (2019) is a transformer-based multilingual masked language model pre-trained on the text in 100 languages, which obtains state- of-the-art performance on cross-lingual classification, sequence labeling, and question answering. XLM-ROBERTa improves upon BERT by adding a few changes to the BERT model such as training on a larger dataset, dynamically masking out tokens compared to the original static masking, and uses a known pre- processing technique (Byte-Pair-Encoding) and a dual-language training mechanism with BERT in order to learn better relations between words in different languages. The given model is trained for the language modeling task, and the output is of dimensionality 768. It is given as input to a CNN Kim (2014) because convolution layers can extract better data representations than Feed Forward layers, which indirectly helps in better domain identification. ## 3 Experiment This section presents the datasets used, the task description, and two models’ performance on technical domain identification. We also include our implementation details and error analysis in the subsequent sections. ### 3.1 Dataset We used the dataset provided by the organizers of TechDOfication ICON-2020. There are two subtasks, one is coarse-grained, and the other is fine-grained. The coarse-grained TechDOfication dataset contains sentences about Chemistry, Communication Technology, Computer Science, Law, Math, and Physics domains in different languages such as English, Bengali, Gujarati, Hindi, Malayalam, Marathi, Tamil, and Telugu. Whereas the fine-grained English dataset focuses on the Computer-Science domain with sub-domain labels as Artificial Intelligence, Algorithm, Computer Architecture, Computer Networks, Database Management system, Programming, and Software Engineering. ### 3.2 Implementation For the implementation, we used the transformers library provided by HuggingFace111https://huggingface.co/. The HuggingFace contains the pre- trained multilingual BERT, XLM-ROBERTa, and other models suitable for downstream tasks. The pre-trained multilingual BERT model used is _“bert-base- multilingual-cased”_ and pre-trained XLM-R model used is _“xlm-roberta-base”_. We programmed the CNN architecture as given in the paper Kim (2014). We used the PyTorch library that supports GPU processing for implementing deep neural nets. The BERT models were run on the Google Colab and Kaggle GPU notebooks. We trained our classifier with a batch size of 128 for 10 to 30 epochs based on our experiments. The dropout is set to 0.1, and the Adam optimizer is used with a learning rate of 2e-5. We used the hugging face transformers pre- trained BERT tokenizer for tokenization. We used the BertForSequenceClassification module provided by the HuggingFace library during finetuning and sequence classification for the multilingual-BERT based approach. ### 3.3 Baseline models Here, we compared the BERT model with other machine learning algorithms. #### SVM with TF_IDF text representation We chose Support Vector Machines (SVM) with TF_IDF text representation for technical domain identification. SVM classifier and TF_IDF vector representation is obtained from the scikit-learn library Pedregosa et al. (2011). #### CNN: Convolutional Neural Network Kim (2014). We explored CNN-non-static, which uses pre-trained word embeddings. ### 3.4 Results The results are tabulated in Table 1. We evaluated the performance of the method using macro F1. The multilingual-BERT model performed well when compared to the other SVM with TF-IDF and CNN models. Given all the languages, we have observed an increase of 7 to 25% in classification metrics for BERT compared to the baseline SVM classifier, it showed a 2 to 5% increase in classification metrics compared to the CNN classifier on the validation data. On the test data, multilingual BERT showed better performance in subtasks 1a, 1b, 1c, 1h and 2a whereas XLM-ROBERTa with CNN showed better performance in the subtasks 1d, 1e, 1f, 1g. This increase in classification metrics is due to the transformer model’s and convolutional NN’s capability, which learned better text representations from the generic data than other models. \| Classifier Models ---\|--- Dataset \| Validation \| Test \| SVM \| CNN \| M-Bert \| XLM-R+CNN \| M-Bert \| XLM-R+CNN English subtask-1a \| 81.48 \| 83.05 \| 88.87 \| 87.09 \| 79.84 \| 73.57 Bengali subtask-1b \| 66.35 \| 85.78 \| 86.81 \| 85.71 \| 80.35 \| 78.17 Gujarati subtask-1c \| 69.63 \| 86.27 \| 87.21 \| 86.89 \| 68.67 \| 66.73 Hindi subtask-1d \| 58.21 \| 81.03 \| 83.40 \| 82.13 \| 59.89 \| 60.44 Malayalam subtask-1e \| 80.60 \| 92.51 \| 94.72 \| 93.40 \| 34.47 \| 34.86 Marathi subtask-1f \| 73.32 \| 86.89 \| 87.42 \| 86.37 \| 59.52 \| 59.89 Tamil subtask-1g \| 65.95 \| 85.75 \| 87.50 \| 86.54 \| 49.24 \| 51.34 Telugu subtask-1h \| 71.98 \| 88.07 \| 90.28 \| 89.43 \| 67.17 \| 62.26 English subtask-2a \| 70.24 \| 72.53 \| 77.36 \| 76.77 \| 78.98 \| 78.07 Table 1: macro F1 on validation and test data for all the subtasks ## 4 Error Analysis The multilingual-BERT model’s confusion matrix is compared with the poorly performed model for languages, Hindi, and Tamil languages are shown in Figure 3. We chose Hindi and Tamil languages because, here, the difference in performance is more significant. For the Hindi subtask, the SVM classifier confused between “cse”, “com_tech”, and “mgmt” labels, whereas the BERT model performed better. For the Tamil subtask, the SVM classifier confused between “com_tech” and “mgmt” labels, whereas the BERT model performed better than the other models. This is because both the approaches (pre-trained multilingual- BERT and pre-trained XLM-ROBERTa with CNN) learned better representation of the above data than the other models that helped in technical document identification. (a) (b) (c) (d) Figure 3: Confusion matrix on the given validation data for the Hindi and Tamil languages ## 5 Conclusion and Future work We used pre-trained bi-directional encoder representations using multilingual- BERT and XLM-ROBERTa with CNN technical domain identification for English, Bengali, Gujarati, Hindi, Malayalam, Marathi, Tamil, and Telugu languages. We compared the approaches with the baseline methods. Our analysis showed that pre-trained multilingual BERT and XLM-ROBERTa with CNN models and finetuning it for text classification tasks showed an increase in macro F1 score and accuracy metrics compared to baseline approaches. Some datasets are large, like for the Hindi, Tamil, and Telugu, we can train the BERT and XLM-ROBERTa models from scratch and consider its hidden layer representation, and concatenate this with the representation of the pre- trained model. It might help to classify the datasets even better. ## References * Adhikari et al. (2019) Ashutosh Adhikari, Achyudh Ram, Raphael Tang, and Jimmy Lin. 2019. Docbert: Bert for document classification. _arXiv preprint arXiv:1904.08398_. * Conneau et al. (2019) Alexis Conneau, Kartikay Khandelwal, Naman Goyal, Vishrav Chaudhary, Guillaume Wenzek, Francisco Guzmán, Edouard Grave, Myle Ott, Luke Zettlemoyer, and Veselin Stoyanov. 2019. 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# Personal Fixations-Based Object Segmentation with Object Localization and Boundary Preservation Gongyang Li, Zhi Liu, , Ran Shi, Zheng Hu, Weijie Wei, Yong Wu, Mengke Huang, and Haibin Ling Gongyang Li, Zhi Liu, Zheng Hu, Weijie Wei, Yong Wu, and Mengke Huang are with Shanghai Institute for Advanced Communication and Data Science, Shanghai University, Shanghai 200444, China, and School of Communication and Information Engineering, Shanghai University, Shanghai 200444, China (email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>huangmengke@shu.edu.cn).Ran Shi is with School of Computer Science and Engineering, Nanjing University of Science and Technology, Nanjing 210094, China (email: rshi@njust.edu.cn).Haibin Ling is with the Department of Computer Science, Stony Brook University, Stony Brook, NY 11794, USA (email: hling@cs.stonybrook.edu).Corresponding author: Zhi Liu ###### Abstract As a natural way for human-computer interaction, fixation provides a promising solution for interactive image segmentation. In this paper, we focus on Personal Fixations-based Object Segmentation (PFOS) to address issues in previous studies, such as the lack of appropriate dataset and the ambiguity in fixations-based interaction. In particular, we first construct a new PFOS dataset by carefully collecting pixel-level binary annotation data over an existing fixation prediction dataset, such dataset is expected to greatly facilitate the study along the line. Then, considering characteristics of personal fixations, we propose a novel network based on Object Localization and Boundary Preservation (OLBP) to segment the gazed objects. Specifically, the OLBP network utilizes an Object Localization Module (OLM) to analyze personal fixations and locates the gazed objects based on the interpretation. Then, a Boundary Preservation Module (BPM) is designed to introduce additional boundary information to guard the completeness of the gazed objects. Moreover, OLBP is organized in the mixed bottom-up and top-down manner with multiple types of deep supervision. Extensive experiments on the constructed PFOS dataset show the superiority of the proposed OLBP network over 17 state-of- the-art methods, and demonstrate the effectiveness of the proposed OLM and BPM components. The constructed PFOS dataset and the proposed OLBP network are available at https://github.com/MathLee/OLBPNet4PFOS. ###### Index Terms: Personal fixations, interactive image segmentation, object localization, boundary preservation. ## I Introduction Fixation is a flexible interaction mechanism of the human visual system. Compared with scribble, click and bounding box, fixation provides the most convenient interaction for patients with hand disability, amyotrophic lateral sclerosis (ALS) and polio. This kind of eye control interaction, i.e. fixation, can greatly improve the interaction efficiency of these patients. In addition, fixation is closely related to personal information such as age [1, 2] and gender [3, 4]. This means that different individuals may have different perceptions and preferences of a scene [5, 6]. Thus motivated, in this paper, we pay close attention to personal fixations-based object segmentation, which is a more natural manner for interactive image segmentation. \begin{overpic}[width=433.62pt]{Figs/Fig1_background.png} \end{overpic} Figure 1: Examples of image with ambiguous fixations. Green dots in each image indicate fixations. Some fixations fall in the background. The typical manners of interaction, such as scribbles [7, 8, 9, 10, 11], clicks [12, 13, 14, 15, 16, 17] and bounding boxes [18, 19, 20, 21] for interactive image segmentation, are explicit behaviors without interference. By contrast, fixations are implicit [22, 23, 24, 25], and their convenience comes with interaction ambiguity. Concretely, the positive and negative labels of scribbles and clicks are deterministic. However, fixations are unlabeled when collected. They do not distinguish between positive labels and negative labels (i.e. some fixations may fall in the background as shown in Fig. 1), resulting in a few noise in the fixations. Such ambiguous interaction makes the fixations-based object segmentation task difficult. Recently, with the rise of convolutional neural networks (CNNs), the clicks-based interactive image segmentation has been greatly developed. Even though fixation points and clicking points are similar to some extent, clicks-based methods [12, 13, 14, 16, 17] cannot be directly applied to fixations-based object segmentation. The above observations suggest that there are two main reasons that limit the development of fixations-based object segmentation. First, there is not a suitable dataset for the fixations-based object segmentation task, let alone dataset based on the personal fixations. Second, as aforementioned, the ambiguous representation of fixations makes this type of interaction difficult to handle by other methods which are based on clicks and scribbles. To address the first crucial issue, we construct a Personal Fixations-based Object Segmentation (PFOS) dataset, which is extended from the fixation prediction dataset OSIE [26]. The PFOS dataset contains 700 images, and each image has 15 personal fixation maps collected from 15 subjects with corresponding pixel-level annotations of objects. To overcome the ambiguity of fixations, we propose an effective network based on Object Localization and Boundary Preservation (OLBP). The key idea of OLBP is to locate the gazed objects based on the analysis of fixations, and then the boundary information is introduced to guard the completeness of the gazed objects and to filter the background. In particular, the overall structure of OLBP network is a mixture of bottom-up and top-down architectures. To narrow the gap between fixations and objects, we propose the Object Localization Module (OLM) to analyze personal fixations in detail and grasp location information of the gazed objects of different individuals. Based on the interpretation of location information, OLM modulates CNN features of image in a bottom-up way. Moreover, considering that the object location information may involve confusing noise, we propose a Boundary Preservation Module (BPM) to exploit boundary information to enforce object completeness and filter the background of erroneous localization. BPM is integrated into the top-down prediction. Both OLMs and BPMs employ deep supervision to further improve the capabilities of feature representation. In this way, the scheme of object localization and boundary preservation is successfully applied to the bottom-up and top-down structure, and the proposed OLBP network greatly promotes the performance of the personal fixations-based object segmentation task. Experimental results on the challenging PFOS dataset demonstrate that OLBP outperforms 17 state-of-the-art methods under various evaluation metrics. The contributions of this work are summarized as follows: * • We construct a new dataset for Personal Fixations-based Object Segmentation (PFOS), which focuses on the natural interaction (i.e. fixation). This dataset contains free-view personal fixations without any constraints, expanding its applicability. We believe that the PFOS dataset will boost the research of fixations-based human-computer interaction. * • We propose a novel Object Localization and Boundary Preservation (OLBP) network to segment the gazed objects based on personal fixations. The OLBP network, equipped with the Object Localization Module and the Boundary Preservation Module, effectively overcomes the difficulties from ambiguous fixations. * • We conduct extensive experiments to evaluate our OLBP network and other state- of-the-art methods on the PFOS dataset. Comprehensive results demonstrate the superiority of our OLBP network, and also reveal the difficulties and challenges of the constructed PFOS dataset. The rest of the paper is organized as follows: Sec. II reviews related previous works. Then, we formulate the PFOS task in Sec. III. After that, in Sec. IV, we construct the PFOS dataset. Sec. V presents the proposed OLBP network in detail. In Sec. VI, we evaluate the performance of the proposed OLBP network and other methods on the constructed PFOS dataset. Finally, the conclusion is drawn in Sec. VII. ## II Related Work In this section, we first give an overview of previous works of interactive image segmentation in Sec. II-A. Then, we introduce related works of fixations-based object segmentation in Sec. II-B. Finally, we review some related works on boundary-aware segmentation in Sec. II-C. ### II-A Interactive Image Segmentation 1) Scribbles-based interactive image segmentation. Scribble is a traditional manner of interaction. Most of scribbles-based methods are built on graph structures. GraphCut [7] is one of the most representative methods. It uses the max-flow/min-cut theorem to minimize energy function with hard constraints (i.e. labeled scribbles) and soft constraints. Grady et al. [8] adopted the random walk algorithm to assign a label to each unlabeled pixel based on the predefined seed pixels in discrete space. In [9], Bai et al. proposed a weighted geodesic distance based framework, which is fast for image and video segmentation and matting. Nguyen et al. [10] proposed a convex active contour model to segment objects, and their results were with smooth and accurate boundary contour. Spina et al. [11] presented a live markers methodology to reduce the user intervention for effective segmentation of target objects. Following the seed propagation strategy, Jian et al. [27] employed the adaptive constraint propagation to adaptively propagate the scribbles information into the whole image. Recently, Wang et al. [28] changed their view on interactive image segmentation and formulated it as a probabilistic estimation problem, proposing a pairwise likelihood learning based framework. These methods are friendly to clearly defined scribbles, but they cannot solve the ambiguity of fixations and their inference speed is usually slow. 2) Clicks-based interactive image segmentation. Click is a classical manner of interaction. It has been deeply studied in the deep learning era. The positive and negative clicks are transformed into two separate Euclidean distance maps for network input. Xu et al. [12] directly sent RGB image and two distance maps into a fully convolutional network. Liew et al. [13] proposed a two- branch fusion network with global prediction and local regional refinement. In addition to the RGB image and distance maps, Li et al. [14] included clicks in their network input and proposed an end-to-end segmentation-selection network. In [16], Jang et al. introduced the backpropagating refinement scheme to correct mislabeled locations in the initial segmentation map. Different from the direct concatenation of RGB image and interaction maps of the above methods, Hu et al. [17] separately input RGB image and interaction maps into two networks, and designed a fusion network for feature interactions. CNNs have greatly improved the performance of clicks-based interactive image segmentation, but when these methods are applied to fixations-based object segmentation, some background regions will be mistakenly segmented. To address the problem of erroneous localization, we explore the boundary information in our BPM to filter redundant background regions and guard the gazed object. 3) Bounding boxes-based interactive image segmentation. In a bounding box, the target object and background coexist, which is different from scribble and click. Rother et al. [18] extended the graph-cut approach, and segmented object with a rectangle, namely GrabCut. To overcome the looseness of the bounding box, Lempitsky et al. [19] incorporated the tightness prior into the global energy minimization function as hard constraints to further completed target object. Shi et al. [21] proposed a coarse-to-fine method with region- level and pixel-level segmentation. Similar to [12], Xu et al. [20] transformed the bounding box to a distance map and concatenated it with the RGB image to input into an encoder-decoder network. Although bounding box and fixation are similar (i.e. target object and background coexist in both interactions), the bounding box-based methods are difficult to transfer to fixations-based object segmentation. ### II-B Fixations-Based Object Segmentation Fixation plays an integral role in the human visual system and it is convenient for interaction. In an early study, Sadeghi et al. [29] constructed an eyegaze-based interactive segmentation system which adopts random walker to segment objects. Meanwhile, Mishra et al. [22] gave the definition of fixations-based object segmentation, that is, segmenting regions containing fixation points. They transformed the image to polar coordinate system, and found the optimal contour to fit the target object. Based on the interpretation of visual receptive field, Kootstra et al. [30] used symmetry to select fixations closer to the center of the object to obtain more complete segmentation. Differently, Li et al. [23] focused on selecting the most salient objects, and they ranked object proposals based on fixations. Similar to [23], Shi et al. [24] analyzed the fixation distribution and proposed three metrics to evaluate the score of each candidate region. In [31], Tian et al. first determined the uninterested regions, and then used superpixel-based random walk model to segment the gazed objects. Khosravan et al. [32] integrated fixations into the medical image segmentation and proposed a Gaze2Segment system. Li et al. [25] constructed a dataset where all fixations fall in objects (i.e. constrained fixations), and proposed a CNN-based model to simulate the human visual system to segment objects based on fixations. These studies have promoted the development of fixations-based object segmentation. However, all the fixations in [22, 30, 31, 25] fall in objects, which are hardly guaranteed in practice. These methods [22, 30, 31, 25] will get stuck in the ambiguity of unconstrained fixations, especially of personal fixations. For [23, 24], they are based on region proposal and cannot obtain accurate results. In summary, the above methods cannot solve the problem of ambiguous fixations, as shown in Fig. 1. In this paper, we take advantage of CNNs, and propose a bottom-up and top-down network to locate objects and preserve objects’ boundaries. Moreover, we construct a dataset to promote this special direction of interactive image segmentation, i.e. personal fixations- based object segmentation. ### II-C Boundary-Aware Segmentation The boundary/edge-aware segmentation idea is widely-used in salient object detection [33, 34, 35, 36] and semantic segmentation [37]. In [33], Wang et al. modeled the boundary information as an edge-preserving constraint, and included it as an additional supervision in loss function. In [34], Wang et al. proposed a two-branch network, including boundary and mask sub-networks, for jointly predicting masks of salient objects and detecting object boundaries. In [35], Wu et al. explored the logical interrelations between binary segmentation and edge maps in a multi-task network, and proposed a cross refinement unit in which the segmentation features and edge features are fused in a cross-task manner. In [36], Zhao et al. focused on the complementarity between salient edge information and salient object information. They integrated the local edge information of shallow layers and global location information of deep layers to obtain the salient edge features, and then the edge features were fed to the one-to-one guidance module to fuse the complementary region and edge information. In [37], Ding et al. first introduced the boundary information as an additional semantic class to enable the network to be aware of the boundary layout, and then proposed a boundary-aware feature propagation network to control the feature propagation based on the learned boundary information. In our method, we use the boundary information in two aspects: the multi-task structure (i.e. segmentation and boundary predictions) and the Boundary Preservation Module. Different from [34, 35], we integrate the learned boundary map into the prediction network in BPMs to preserve the completeness of the gazed objects, rather than fuse the segmentation features and boundary features. Compared with [36], our segmentation prediction is accompanied by the boundary prediction in a uniform prediction network, and the boundary supervision is employed at multiple scales. Different from [37], which uses the boundary map to control the region of feature propagation, our method uses the boundary map to filter the background of erroneous localization in features. In short, our use of boundary information is diverse and in-depth, which is suitable for the personal fixations-based object segmentation task. ## III Personal Fixations-Based Object Segmentation Problem Statement. Given an image ${\bf I}$ and a fixation map ${\bf FM}$ of a person, personal fixations-based object segmentation aims to segment the gazed objects of this person according to his/her personal ${\bf FM}$, producing a binary segmentation map. In general, different individuals generate different fixation maps when observing the same image, which means that individuals may be interested in different objects. In other words, segmentation results of different individuals on the same image vary with the observer. So, the special characteristic of this task is that an image has multiple binary segmentation maps due to multiple fixation maps. Although the ambiguity of fixations makes this task difficulty, the personal fixation map is the only information that can determine the gazed objects. Applications. This task has several meaningful applications. First, such a convenient manner of interaction is conducive to the development of special eye-control devices for patients with hand disability, ALS and polio, facilitating their lives and improving their quality of life. Second, fixation is advantageous to diagnose certain mental illnesses, such as autism spectrum disorder (ASD) [38, 39] and schizophrenia spectrum disorders (SSD) [40, 41]. This task understands personal fixations at the object level, which is helpful to improve the accuracy of disease diagnosis. For example, patients with ASD prefer to pay attention to background rather than foreground, so the proportion of foreground in their segmentation results will be less than that of healthy people. TABLE I: Categories of fixation map (FM) in the PFOS dataset. Constrained FM means that all fixations fall in the objects/foreground. Unconstrained FM represents that some fixations fall in the background. PFOS dataset \| Constrained FM \| Unconstrained FM ---\|---\|--- 10,500 \| 3,683 (35.1%) \| 6,817 (64.9%) ## IV Dataset Construction and Transformation Currently, there are many prevalently used datasets for fixation prediction, such as MIT1003 [42], OSIE [26] and SALICON [43], and for interactive image segmentation, such as GrabCut [18], Berkeley [44] and PASCAL VOC [45]. However, there is no dataset for the personal fixations-based object segmentation task. Considering that it is time-consuming for dataset annotations, we propose a convenient way to collect suitable data from existing datasets for this task. Obviously, the PFOS dataset must contain fixation data and pixel-level annotations for objects. Among the existing datasets, some datasets, such as DUTS-OMRON [46], PASCAL-S [23] and OSIE [26], are potential candidates. The pixel-level annotations of DUTS-OMRON and PASCAL-S are for salient object detection [47, 48, 49], that is, these annotations only focus on the most visually attractive objects but ignore other objects, which could be fixated by different individuals, in a scene. Therefore, they are not perfect for constructing a PFOS dataset. Fortunately, the pixel-level annotations of OSIE have semantic attributes. This means that we can select objects, which the user is interested in, based on personal fixations. In other words, we can create the pixel-level binary ground truths (GTs) for personal fixations-based object segmentation. So, we transform the fixation prediction dataset OSIE to our PFOS dataset. For each image in the OSIE dataset, it has corresponding fixation maps and semantic GTs of different subjects. The detailed steps for dataset transformation are as follows: 1) Semantic labels collection. We get the position of each fixation point from the fixation map, and we collect the semantic label of each position in the corresponding semantic GT. 2) Semantic labels distillation. As mentioned in Sec. I and shown in Fig. 1, some fixation points fall in the background or the same object. For semantic labels collected from Step 1, we discard the semantic label “0” which indicates background. Then, if there are several same semantic labels, we keep only one. 3) Binary GT creation. Based on the distilled semantic labels from Step 2, we can determine the gazed objects and create the binary GT. We reserve the regions with the distilled semantic labels in the semantic GT, and set them as foreground. We set the regions with the other unrelated semantic labels as background. In this convenient way, we efficiently create the binary GTs and successfully construct the PFOS dataset. The PFOS dataset retains all 700 images and 10,500 free-view personal fixation maps from the OSIE dataset. In the PFOS dataset, the image resolution is $800\times 600$. Each image has 15 personal fixation maps from 15 subjects and the transformed binary GTs. In the constructed PFOS dataset, there are two categories of fixation maps. The first category is that all fixations fall in the objects/foreground, i.e. the constrained fixation map in [25]. The second category is that some fixations fall in the background, namely the unconstrained fixation map. We present the details of them in Tab. I. In our PFOS dataset, the unconstrained fixation maps account for 64.9% and the constrained fixation maps hold 35.1%. The large proportion of unconstrained fixation maps increase the ambiguity of our PFOS dataset and make this dataset challenging. \begin{overpic}[width=432.75322pt]{Figs/Fig2_example.pdf} \put(4.5,0.0){ Image with fixations } \put(45.5,0.0){ FDM } \put(79.5,0.0){ GT } \put(-1.0,57.0){ \begin{sideways}{Subject A}\end{sideways} } \put(-1.0,33.2){ \begin{sideways}{Subject B}\end{sideways} } \put(-1.0,8.5){ \begin{sideways}{Subject C}\end{sideways} } \end{overpic} Figure 2: Examples of the PFOS dataset. Green dots in each image indicate fixations, FDM is fixation density map, and GT represents ground truth. ## V Methodology In this section, we first conduct data preprocessing which transforms the fixation points into fixation density maps in Sec. V-A. Then, we present the overview and motivation of the proposed Object Localization and Boundary Preservation (OLBP) network in Sec. V-B. Next, we give the detailed formulas of the Object Localization Module (OLM) and the Boundary Preservation Module (BPM) in Sec. V-C and Sec. V-D, respectively. Finally, we clarify the implementation details of OLBP network in Sec. V-E. Figure 3: The overall architecture of the proposed OLBP network. OLBP network is organized in the mixed bottom-up and top-down manner. We employ the modified VGG-16 to extract five blocks of features from an input image. Then in each OLM, FDM is analyzed by several dilated and normal convolutional layers to determine the location of objects in the corresponding block features. Based on the object localization in each feature block, the top-down prediction is established. During the prediction process, the boundary information is introduced into BPMs to guard the completeness of objects and to filter background of erroneous localization. We also construct a multi-task prediction structure, which contains object segmentation branch and boundary prediction branch, to exploit the complementarity between regions and boundaries. ### V-A Data Preprocessing The fixation points in each fixation map are sparse. With only a few pixels per fixation map, there is too little valuable information to supply. The similar problem arises in the clicks-based interactive image segmentation. Xu et al. [12] transformed the clicks into Euclidean distance maps. Inspired by this, we employ the Gaussian blur to transform the sparse fixation map (i.e. FM) into the fixation density map (i.e. FDM): $\mathbf{FDM}={\mathrm{nor}_{\mathrm{min-max}}}(\mathbf{FM}\circledast G_{\sigma}(x,y;\sigma)),$ (1) where ${\mathrm{nor}_{\mathrm{min-max}}}(\cdot)$ is the min-max normalization, $\circledast$ denotes convolution operator, and $G_{\sigma}(\cdot)$ is a Gaussian filter with parameter $\sigma$ which is the standard deviation. $\sigma$ is set corresponding to 1∘ visual angle in the OSIE dataset [26]. It is 24 pixels of an $800\times 600$ image by default. The effect of Gaussian blur is similar to the receptive field of eye, that is, the center of fixation is with a high resolution and the surrounding of fixation is with a low resolution. Thus, after performing Gaussian blur and linear transformation on FM, the dense FDM contains more prior information of objects. In this paper, we adopt the dense FDM rather than the raw FM. We present an image with the personal fixations of three subjects of the PFOS dataset in Fig. 2. The fixation maps of Subject A and Subject B are constrained fixation maps, while the fixation map of Subject C is an unconstrained fixation map. ### V-B Network Overview and Motivation The proposed OLBP network has three critical components: the feature extractor, the object locator and the prediction network with boundary preservation. The overall architecture of OLBP network is illustrated in Fig. 3. Feature Extractor. In the OLBP network, we adopt the modified VGG-16 [50], from which the last three fully connected layers have been removed, as the feature extractor. We denote its input image as ${\bf I}\\!\in\\!\mathbb{R}^{H\\!\times\\!W\\!\times\\!C}$, and initialize its parameters by the image classification model [50]. The feature extractor has five convolutional blocks, as shown in Fig. 3. We operate on the feature map of the last convolutional layer in each block, i.e. conv1-2, conv2-2, conv3-3, conv4-3 and conv5-3, which are denoted as $\\{{\bf F}^{(i)}_{r}$: ${\bf F}^{(i)}_{r}\\!\in\\!\mathbb{R}^{\textit{h}_{i}\\!\times\\!\textit{w}_{i}\\!\times\\!\textit{c}_{i}},i=1,2,...,5\\}$. Notably, the feature resolution at the i-th block, i.e. $[\textit{h}_{i},\textit{w}_{i}]$, is $[\frac{\textit{H}}{2^{i-1}},\frac{\textit{W}}{2^{i-1}}]$ and $\textit{c}_{i\in\\{1,2,3,4,5\\}}=\\{64,128,256,512,512\\}$. In reality, the input resolution $[\textit{H},\textit{W},\textit{C}]$ of ${\bf I}$ is set to $288\times 288\times 3$. Object Localization Module. Although FDM is a probability map, it is a critical interaction that reflects the intention of the user. It is important to effectively explore the object location information of FDM. However, when we construct a CNN-based model for the personal fixations-based object segmentation task, it is natural to directly concatenate FDM and the input image for the network input. Since there are three channels for image and only one channel for FDM, the direct concatenation operation may drown out the critical interaction information of FDM. Based on the above analysis, we propose the Object Localization Module to process FDM. The parallel convolution structure is effective to explore meaningful information in CNN features [51], especially with the dilated convolution [52]. Thus, in OLM, we employ several parallel dilated convolutions with different dilation rates to profoundly analyze the personal FDM to obtain object location information, which are a group of response maps. These response maps belong to ${[0,1]}^{\textit{h}_{i}\\!\times\\!\textit{w}_{i}\\!\times\\!\textit{c}_{i}}$, which shows they have the same number of channels as the features of image at the i-th block. They are applied to re-weight features of image to highlight the gazed objects at channel-wise and spatial-wise. To enhance the location presentation of the response maps, we apply deep supervision [53] in OLM. As presented in Fig. 3, the OLM is performed in a bottom-up manner, and it is assembled after each block of feature extractor for strong object localization. The detailed description of OLM is presented in Sec. V-C. We show the ablation study of OLM in Sec. VI-C, including a variant of direct concatenation of image and FDM. Boundary Preservation Module and Prediction Network. Since some fixations fall in the background, there may be some noise on the re-weighted feature of OLM. The ambiguity over the fixations causes great disturbance to the segmentation result. Fortunately, there is a priori knowledge that the background usually does not have a regular boundary. Thus, we introduce the boundary information into the prediction network, and propose the Boundary Preservation Module to filter the background of erroneous localization and preserve the completeness of the gazed objects. BPM is a momentous component to purify the segmentation result. We also attach the pixel-level segmentation supervision and boundary supervision to BPM. As shown in Fig. 3, BPMs are equipped between convolutional blocks in the prediction network from top to down. To make full use of the boundary information, we also construct a multi-task structure in the prediction network. We elaborate the formulation and ablation study of BPM in Sec. V-D and Sec. VI-C, respectively. TABLE II: Detailed parameters of each OLM. We present the kernel size and channel number of each dilated/normal convolutions. Besides, we also present the dilation rates and the size of output feature. For instance, $(3\times 3,32)$ denotes that the kernel size is $3\times 3$ and the channel number is 32. Aspects \| \| Dilation --- conv \| Dilation --- rate 2$\times$Conv \| Output size OLM-1 \| $(3\times 3,32)$ \| $1/3/5/7$ \| $(7\times 7,128)$ \| $[288\times 288\times 128]$ OLM-2 \| $(3\times 3,64)$ \| $1/3/5/7$ \| $(5\times 5,256)$ \| $[144\times 144\times 256]$ OLM-3 \| $(3\times 3,128)$ \| $1/3/5/7$ \| $(5\times 5,512)$ \| $[72\times 72\times 512]$ OLM-4 \| $(3\times 3,256)$ \| $1/2/3/4$ \| $(3\times 3,1024)$ \| $[36\times 36\times 1024]$ OLM-5 \| $(3\times 3,256)$ \| $1/2/3/4$ \| $(3\times 3,1024)$ \| $[18\times 18\times 1024]$ ### V-C Object Localization Module As the OLM-5 shown in Fig. 3, there are three main parts in the Object Localization Module: location analysis unit, feature re-weighting (i.e. Re- wei) and segmentation supervision (i.e. Seg sup). Its objective is to extract object location information of personal FDM and to highlight objects in feature of image ${\bf F}^{(i)}_{r}$. OLM is the most indispensable part of the whole OLBP network. Concretely, in OLM-i, the ${\bf FDM}\\!\in\\!\mathbb{R}^{H\\!\times\\!W\\!\times\\!1}$ is first downsampled to fit the resolution of ${\bf F}^{(i)}_{r}$ and to generate $\mathbf{F}^{(i)}_{fdm}\\!\in\\!\mathbb{R}^{\textit{h}_{i}\\!\times\\!\textit{w}_{i}\\!\times\\!1}$ which is formulated as: $\displaystyle\mathbf{F}^{(i)}_{fdm}=\mathrm{MaxPool}(\mathbf{FDM};{W}^{(i)}_{ks}),$ (2) where $\mathrm{MaxPool}(\cdot)$ is the max pooling with parameters ${W}^{(i)}_{ks}$, which are $2^{i-1}\times 2^{i-1}$ kernel with $2^{i-1}$ stride. Then, we design the location analysis unit, which contains four parallel dilated convolutions [52] with different dilation rates, to analyze $\mathbf{F}^{(i)}_{fdm}$, and obtain the multi-interpretation feature $\mathbf{F}^{(i)}_{mi}$. The process in this unit can be formulated as: $\displaystyle\mathbf{F}^{(i)}_{mi}=\mathrm{concat}\big{(}$ $\displaystyle C_{d}(\mathbf{F}^{(i)}_{fdm};{W}^{(i_{1})}_{d}),C_{d}(\mathbf{F}^{(i)}_{fdm};{W}^{(i_{2})}_{d}),$ (3) $\displaystyle C_{d}(\mathbf{F}^{(i)}_{fdm};{W}^{(i_{3})}_{d}),C_{d}(\mathbf{F}^{(i)}_{fdm};{W}^{(i_{4})}_{d})\big{)},$ where $\mathrm{concat}(\cdot)$ is the cross-channel concatenation, and $C_{d}(\cdot;{W}^{(i_{n})}_{d})$ is the dilated convolution with parameters ${W}^{(i_{n})}_{d}$ for $n\in\\{1,2,3,4\\}$. Notably, ${W}^{(i_{n})}_{d}$ are comprised of kernel size, channel number and dilation rate. Considering the resolution difference of each $\mathbf{F}^{(i)}_{r}$, the dilation rates of each unit are different and the details are presented in Tab. II. In this unit, the dilated convolutions large the receptive field without increasing the computation. They are performed in a parallel manner, which makes $\mathbf{F}^{(i)}_{mi}$ effectively capture the local and global location information of the gazed objects. The multi-scale features in $\mathbf{F}^{(i)}_{mi}$ are complementary to each other. They are blended to produce the location response maps $\mathbf{r}^{(i)}_{loc}\\!\in\\!{[0,1]}^{\textit{h}_{i}\\!\times\\!\textit{w}_{i}\\!\times\\!\textit{c}_{i}}$ via: $\displaystyle\mathbf{F}^{(i)}_{int}={2C}(\mathbf{F}^{(i)}_{mi};{W}^{(i)}_{2c}),$ (4) $\displaystyle\mathbf{r}^{(i)}_{loc}=\psi(C(\mathbf{F}^{(i)}_{int};{W}^{(i)}_{c})),$ (5) where $\mathbf{F}^{(i)}_{int}$ is the interim feature, $2C(\ast;{W}^{(i)}_{2c})$ are two convolutional layers with the same parameters ${W}^{(i)}_{2c}$, $\psi(\cdot)$ is the sigmoid function, and $C(\ast;{W}^{(i)}_{c})$ is the convolutional layer with parameters ${W}^{(i)}_{c}$ which are $3\times 3$ kernel with ${c}_{i}$ channels. ${W}^{(i)}_{2c}$ contain kernel size and channel number, which are different in different OLMs. Their details are shown in the column with “2$\times$Conv” of Tab. II. \begin{overpic}[width=433.62pt]{Figs/Fig4_enhanced_Feature.png} \put(5.9,-2.4){ Image } \put(25.9,-2.4){ FDM } \put(46.6,-2.4){ GT } \put(65.2,-3.1){ $\mathbf{r}^{(2)}_{loc}$ } \put(84.4,-3.1){ $\mathbf{F}^{(2)}_{loc}$} \end{overpic} Figure 4: Feature visualization in OLM-2. $\mathbf{r}^{(2)}_{loc}$ is the location response map, and $\mathbf{F}^{(2)}_{loc}$ is the location-enhanced feature. After completing the FDM interpretation in location analysis unit, we successfully obtain $\mathbf{r}^{(i)}_{loc}$, which are the protagonists of the feature re-weighting (i.e. Re-wei) part. We employ $\mathbf{r}^{(i)}_{loc}$ to re-weight $\mathbf{F}^{(i)}_{r}$ at channel-wise and spatial-wise, and receive the location-enhanced feature $\mathbf{F}^{(i)}_{loc}\\!\in\\!\mathbb{R}^{\textit{h}_{i}\\!\times\\!\textit{w}_{i}\\!\times\\!\textit{c}_{i}}$, which is computed as: $\displaystyle\mathbf{F}^{(i)}_{loc}=\mathbf{F}^{(i)}_{r}\otimes\mathbf{r}^{(i)}_{loc},$ (6) where $\otimes$ is element-wise multiplication. Besides, in Re-wei, to balance the information of image and location, we concatenate $\mathbf{F}^{(i)}_{r}$ to $\mathbf{F}^{(i)}_{loc}$ and obtain the output feature $\mathbf{F}^{(i)}_{olm}$ of OLM. The size of $\mathbf{F}^{(i)}_{olm}$ is shown in Tab. II. Notably, at the training phase, we apply the pixel-level segmentation supervision (i.e. Seg sup) to each OLM. In Fig. 4, we visualize feature in OLM-2 to verify the effectiveness of the location enhancement. Concretely, in OLM-2, the conv2-2 is re-weighted by the location response map. As shown in Fig. 4, the location response map $\mathbf{r}^{(2)}_{loc}$ contains rich location information of the gazed objects. After using Eq. 6 to perform the location enhancement operation on conv2-2, we observe that the gazed objects are highlighted in $\mathbf{F}^{(2)}_{loc}$ (with darker color). In summary, the location- enhanced feature $\mathbf{F}^{(i)}_{loc}$ of OLM has strong location expression ability and contributes to the subsequent segmentation prediction network. ### V-D Boundary Preservation Module The Boundary Preservation Module is built to restrain the falsely highlighted part of the re-weighted feature of OLM and to preserve the completeness of the gazed objects for the segmentation prediction. As the BPM-5 shown in Fig. 3, the structure of BPM is succinct, but it is a key bridge to connect convolutional blocks of the prediction network. Let $\\{{\bf F}^{(i)}_{p}$: ${\bf F}^{(i)}_{p}\\!\in\\!\mathbb{R}^{\textit{h}_{i-1}\\!\times\\!\textit{w}_{i-1}\\!\times\\!\textit{c}_{i-1}},i=2,3,4,5\\}$ denote the output feature of each deconvolutional layer in the prediction network. In BMP, $\mathbf{F}^{(i)}_{p}$ is processed by a convolutional layer to generate the boundary mask $\mathbf{B}^{(i)}$, which is defined as: $\displaystyle\mathbf{B}^{(i)}=C(\mathbf{F}^{(i)}_{p},{W}^{(i)}_{c}).$ (7) To increase the accuracy of $\mathbf{B}^{(i)}\\!_{i\in\\{2,3,4,5\\}}$, we introduce the pixel-level boundary supervision (i.e. “Bound sup” on BPM-5 in Fig. 3) in BPM. Since that there are no pixel-level boundary annotations in the PFOS dataset, we employ the morphological operation on binary segmentation GT $\mathbf{G}_{s}$ to produce the boundary GT $\mathbf{G}_{b}$, as follow: $\displaystyle\mathbf{G}_{b}=\mathrm{Dilate}(\mathbf{G}_{s};\theta)-\mathbf{G}_{s},$ (8) where $\mathrm{Dilate}(\ast;\theta)$ is the morphological dilation operation with dilation coefficient $\theta$ which is 2 pixels. Then, $\mathbf{B}^{(i)}$ is concatenated to $\mathbf{F}^{(i)}_{p}$ to generate the output feature $\mathbf{F}^{(i)}_{bpm}$ of BPM. We also put the pixel- level segmentation supervision behind $\mathbf{F}^{(i)}_{bpm}$, such as “Seg sup” on BPM-5 in Fig. 3. The segmentation supervision and the boundary supervision cooperate well with each other, improving the feature representation of the gazed objects. In this way, we novelly introduce boundary information into the BPM, and $\mathbf{F}^{(i)}_{bpm}$ carries the feature de-noising and boundary preservation capabilities into the prediction network. ### V-E Implementation Details TABLE III: Quantitative results including Jaccard index, S-measure, weighted F-measure, E-measure and F-measure on the PFOS dataset (in percentage %). Semantic Segmentation means semantic segmentation-based method. Clicks means clicks-based interactive image segmentation method. Fixations means fixations-based object segmentation method. FDM-Guided Semantic Segmentation means embedding FDM into semantic segmentation method. FDM-Guided Salient Object Detection means embedding FDM into salient object detection method. The best three results are shown in red, blue, and green. $\uparrow$ denotes larger is better. The subscript of each method represents the publication year. † means CNNs-based method. Aspects \| Methods \| PFOS Dataset ---\|---\|--- $\mathcal{J}\uparrow$ \| $\mathcal{S}_{\lambda}\uparrow$ \| $w\mathcal{F}_{\beta}\uparrow$ \| $\mathcal{E}_{\xi}\uparrow$ \| $\mathcal{F}_{\beta}\uparrow$ Semantic Segmentation \| PSPNet17† [54] \| 51.0 \| 58.9 \| 55.5 \| 64.2 \| 60.2 SegNet17† [55] \| 58.7 \| 70.4 \| 66.6 \| 78.4 \| 72.5 DeepLab18† [51] \| 52.8 \| 65.7 \| 60.5 \| 72.9 \| 66.9 EncNet18† [56] \| 55.5 \| 62.2 \| 60.5 \| 69.0 \| 65.3 DeepLabV3+18† [57] \| 45.6 \| 61.4 \| 53.1 \| 67.8 \| 59.3 HRNetV219† [58] \| 46.1 \| 50.7 \| 49.0 \| 53.8 \| 53.2 Clicks \| ISLD18† [14] \| 61.2 \| 73.4 \| 71.2 \| 82.5 \| 77.9 FCTSFN19† [17] \| 62.4 \| 72.9 \| 69.9 \| 82.8 \| 75.1 BRS19† [16] \| 62.1 \| 73.0 \| 69.1 \| 82.3 \| 74.6 Fixations \| AVS12 [22] \| 40.9 \| 56.0 \| 48.7 \| 65.1 \| 56.6 SOS14 [23] \| 42.6 \| 57.5 \| 51.4 \| 67.8 \| 60.0 GBOS17 [24] \| 38.0 \| 56.7 \| 48.0 \| 63.9 \| 58.1 CFPS19† [25] \| 70.5 \| 78.9 \| 76.7 \| 87.4 \| 81.3 FDM-Guided Semantic Segmentation \| DeepLabV3+18† [57] \| 71.0 \| 79.5 \| 78.3 \| 87.6 \| 83.2 HRNetV219† [58] \| 58.8 \| 71.3 \| 68.6 \| 80.4 \| 75.7 FDM-Guided Salient Object Detection \| CPD19† [59] \| 69.2 \| 78.4 \| 76.4 \| 86.2 \| 81.7 GCPA20† [60] \| 72.3 \| 80.3 \| 78.9 \| 88.1 \| 83.6 Personal Fixations \| OLBP (Ours) \| 73.7 \| 81.1 \| 80.0 \| 88.7 \| 84.3 Prediction Network. The prediction network is constructed in the top-down manner to gradually restore resolution. It consists of five convolutional blocks, four BPMs and four deconvolutional layers. A dropout layer [61] is placed before each deconvolutional layer to prevent the prediction network from overfitting. In addition, we attach the boundary prediction branch to the prediction network to assist the object segmentation branch. We initialize parameters of the prediction network by xavier method [62]. Overall Loss. As shown in Fig. 3, there are totally 15 losses in the OLBP network, including 10 segmentation losses and 5 boundary losses. The overall loss $\mathbb{L}$ can be divided into three parts: losses of multi-task prediction, losses on OLMs and losses on BPMs. $\mathbb{L}$ is calculated as: $\displaystyle\mathbb{L}=$ $\displaystyle[\mathcal{L}_{s}(\mathbf{S}^{(1)},\mathbf{G}_{s})+\mathcal{L}_{s}(\mathbf{B}^{(1)},\mathbf{G}_{b})]+\sum\limits_{i=1}^{5}\mathcal{L}_{s}(\mathbf{S}^{(i)}_{olm},\mathbf{G}_{s})$ (9) $\displaystyle+\sum\limits_{i=2}^{5}[\mathcal{L}_{s}(\mathbf{S}^{(i)}_{bpm},\mathbf{G}_{s})+\mathcal{L}_{s}(\mathbf{B}^{(i)},\mathbf{G}_{b})],$ where $\mathcal{L}_{s}(\cdot,\cdot)$ is the softmax loss, $\mathbf{S}^{(1)}$ is the predicted segmentation map, and $\mathbf{B}^{(1)}$ is the predicted boundary map. $\mathbf{S}^{(i)}_{olm}$ and $\mathbf{S}^{(i)}_{bpm}$ present the side output segmentation results in OLM and BPM, respectively. $\mathbf{B}^{(i)}\\!_{i\in\\{2,3,4,5\\}}$ is the boundary mask in BPM. Notably, for each softmax loss, we resize the resolutions of $\mathbf{G}_{s}$ and $\mathbf{G}_{b}$ to fit the resolutions of corresponding $\mathbf{S}^{(i)}_{olm}$, $\mathbf{S}^{(i)}_{bpm}$ and $\mathbf{B}^{(i)}$. Network Training. The PFOS dataset is separated into training set and testing set. The training set contains 600 images with 9,000 personal fixation maps, including 3,075 constrained fixation maps and 5,925 unconstrained fixation maps. The testing set consists of 100 images with 1,500 personal fixations, including 608 constrained fixation maps and 892 unconstrained fixation maps. The OLBP network is implemented on Caffe [63] and experimented using a NVIDIA Titan X GPU. The data of training set and testing set are resized to $288\times 288$ for training and inference. We adopt the standard stochastic gradient descent (SGD) method [64] to optimize our OLBP network for 30,000 iterations. The learning rate is set to $8\times 10^{-8}$, and it will be divided by 10 after 14,000 iterations. The dropout ratio, batch size, iteration size, momentum and weight decay are set to 0.5, 1, 8, 0.9 and 0.0001, respectively. ## VI Experiments In this section, we present comprehensive experiments on the proposed PFOS dataset. We introduce evaluation metrics in Sec. VI-A. In Sec. VI-B, we compare the proposed OLBP network with state-of-the-art methods. Then, we conduct ablation studies in Sec. VI-C and show some personal segmentation results in Sec. VI-D. Finally, we present some discussions on the connections between fixation-based object segmentation and salient object detection in Sec. VI-E. ### VI-A Evaluation Metrics We use five evaluation metrics, i.e. Jaccard index ($\mathcal{J}$), S-measure ($\mathcal{S}_{\lambda}$) [65], F-measure ($\mathcal{F}_{\beta}$), weighted F-measure ($w\mathcal{F}_{\beta}$) [66], and E-measure ($\mathcal{E}_{\xi}$) [67], to evaluate the performance of different methods. Jaccard Index $\mathcal{J}$. Jaccard index is also called intersection-over- union (IoU), which can compare similarities and differences between two binary maps. It is defined as: $\mathcal{J}=\frac{\|\mathbf{S}\cap\mathbf{G}_{s}\|}{\|\mathbf{S}\cup\mathbf{G}_{s}\|},$ (10) where $\mathbf{S}$ is the predicted segmentation map, and $\mathbf{G}_{s}$ is the binary segmentation GT. S-measure $\mathcal{S}_{\lambda}$. S-measure focuses on the structural similarity between the predicted segmentation map and the binary segmentation GT. It evaluates the structural similarity of region-aware ($S_{r}$) and object-aware ($S_{o}$) simultaneously. S-measure is defined as: $\displaystyle\mathcal{S}_{\lambda}=\lambda\ast S_{o}+(1-\lambda)\ast S_{r},$ (11) where $\lambda$ is set to 0.5 by default. F-measure $\mathcal{F}_{\beta}$. F-measure is a weighted harmonic mean of precision and recall, which considers precision and recall comprehensively. It is defined as: $\displaystyle\mathcal{F}_{\beta}=\frac{(1+\beta^{2})\times Precision\times Recall}{\beta^{2}\times Precision+Recall},$ (12) where $\beta^{2}$ is set to 0.3 following previous studies [47, 48]. Weighted F-measure $w\mathcal{F}_{\beta}$. Weighted F-measure has the ability to evaluate the non-binary and binary map. It focuses on evaluating the weights errors of predicted pixels according to their location and their neighborhood, which is formulated as: $\displaystyle w\mathcal{F}_{\beta}=\frac{(1+\beta^{2})\times Precision^{w}\times Recall^{w}}{\beta^{2}\times Precision^{w}+Recall^{w}},$ (13) where $\beta^{2}$ is set to 1 following previous studies [68, 69]. \begin{overpic}[width=888.9223pt]{Figs/Fig5_Visual_example.png} \put(1.9,-1.3){ Image} \put(10.4,-1.3){ GT } \put(17.3,-1.3){ {Ours} } \put(24.75,-1.3){ CFPS } \put(32.1,-1.3){ GBOS } \put(40.3,-1.3){ SOS } \put(48.0,-1.3){ AVS } \put(55.75,-1.3){ BRS } \put(62.0,-1.3){ FCTSFN } \put(70.65,-1.3){ ISLD } \put(77.8,-1.3){ EncNet } \put(85.2,-1.3){ Deeplab } \put(93.1,-1.3){ SegNet } \end{overpic} Figure 5: Visualization comparison to some representative methods on the PFOS dataset. Zoom-in for the best view. E-measure $\mathcal{E}_{\xi}$. E-measure is based on cognitive vision studies. It evaluates the local errors (i.e. pixel-level) and the global errors (i.e. image-level) together. We introduce it to provide a more comprehensive evaluation. It could be computed as: $\displaystyle\mathcal{E}_{\xi}=\frac{1}{W\times H}\sum\limits_{x=1}^{W}\sum\limits_{y=1}^{H}f\Big{(}\frac{2\varphi_{\mathbf{G}_{s}}\circ\varphi_{\mathbf{s}}}{\varphi_{\mathbf{G}_{s}}\circ\varphi_{\mathbf{G}_{s}}+\varphi_{\mathbf{s}}\circ\varphi_{\mathbf{s}}}\Big{)},$ (14) where $\varphi_{\mathbf{G}_{s}}$ and $\varphi_{\mathbf{s}}$ are distance bias matrices for binary segmentation GT and predicted segmentation map, respectively, $\circ$ is the Hadamard product, and $f(\cdot)$ is the quadratic form. ### VI-B Comparison with the State-of-the-arts Comparison Methods. We compare our OLBP network against three types of state- of-the-art methods, including semantic segmentation-based methods, clicks- based interactive image segmentation methods and fixations-based object segmentation methods. For a reasonable comparison of the first type of method, we follow [12, 25], which convert the segmentation problem into the selection problem. Concretely, we first apply semantic segmentation methods, i.e. PSPNet [54], SegNet [55], DeepLab [51], EncNet [56], DeepLabV3+ [57], and HRNetV2 [58], to image, and then use the fixations to select the gazed objects. The second type of method includes ISLD [14], FCTSFN [17], and BRS [16]. The last type of method includes AVS [22], SOS [23], GBOS [24] and CFPS [25]. For all the above compared methods, we use the implementations with recommend parameter settings for a fair comparison. In addition, we modify several semantic segmentation methods (i.e. DeepLabV3+ [57] and HRNetV2 [58]) and recent salient object detection methods (i.e. CPD [59] and GCPA [60]) by embedding FDM in them to guide object segmentation. Two types of comparison methods are thus generated, namely FDM-guided semantic segmentation and FDM-guided salient object detection, respectively. Specifically, for DeepLabV3+, we embed FDM into features (i.e. low-level features and features generated from the ASPP) to bridge the encoder and decoder; for HRNetV2, we embed FDM between the second stage and the third stage; for CPD, we embed FDM into two partial decoders; and, for GCPA, we embed FDM into four self refinement modules. We retrain these modified methods with the same training dataset as our method, and their parameters are adjusted for better convergence. Notably, we use the well-known OTSU method [70] to binarize the generated probability map of our method and other CNNs- based methods. TABLE IV: Robustness evaluation of our method and several representative methods, such as the modified GCPA [60], CFPS [25] and the modified CPD [59], on the test part of the PFOS dataset in terms of Jaccard Index. The best result of each row is shown in bold. Notably, “+15% noise” means an additional 15% increase in the number of unconstrained fixations of the total number of fixations in a fixation map. We add the noise (i.e. unconstrained fixations) at three levels, i.e. 15%, 30%, and 45%. \| OLBP \| GCPA20 \| CFPS19 \| CPD19 ---\|---\|---\|---\|--- Dataset \| (Ours) \| [60] \| [25] \| [59] PFOS \| 73.7 \| 72.3 \| 70.5 \| 69.2 +15% noise \| 72.2 \| 70.9 \| 69.6 \| 68.7 +30% noise \| 71.3 \| 70.1 \| 69.2 \| 68.4 +45% noise \| 70.3 \| 69.7 \| 68.8 \| 68.1 Quantitative Performance Evaluation. We evaluate our OLBP network and other 17 state-of-the-art methods on the PFOS dataset using above five evaluation metrics. The quantitative results are presented in Table III. Our OLBP network favorably outperforms all the compared methods in terms of different metrics. Concretely, compared with the best method CFPS [25] in fixations-based object segmentation methods, the performance of our method is improved by $3.2\%$, $2.2\%$ and $3.0\%$ in $\mathcal{J}$, $\mathcal{S}_{\lambda}$ and $w\mathcal{F}_{\beta}$, respectively. The performance of our method is $5.9\%$ better than FCTSFN [17] in $\mathcal{E}_{\xi}$, and is $6.4\%$ better than ISLD [14] in $\mathcal{F}_{\beta}$. Note that the performance of our method is far better than that of three traditional methods AVS [22], SOS [24] and GBOS [24]. We attribute the performance superiority of the proposed OLBP network to the scheme of object localization and boundary preservation. In addition, semantic segmentation-based methods get an average of $51.6\%$ in $\mathcal{J}$. This may be due to the fact that semantic segmentation methods cannot accurately segment all objects, resulting in the failure of the object selection process. Clicks-based interactive image segmentation methods achieve an average of $61.9\%$ in $\mathcal{J}$, while our OLBP network obtains $73.7\%$ in $\mathcal{J}$. This demonstrates that our method is more robust than clicks-based interactive image segmentation methods in adapting the ambiguity of fixations. Fixations-based object segmentation methods contain three traditional methods and one CNN-based method, obtaining an average of $48.0\%$ in $\mathcal{J}$. Specifically, we present the results of the FDM-guided semantic segmentation methods, including the modified DeepLabV3+ and HRNetV2, in Table III. The modified DeepLabV3+ achieves a promising performance, but does not exceed our OLBP network (e.g. 71.0% vs 73.7% in $\mathcal{J}$). Although the FDM guidance brings some advantages to HRNetV2, but the modified HRNetV2 still does not perform well. For the FDM-guided salient object detection, both modified CPD and GCPA perform well, though our OLBP still outperforms them (e.g. 4.5% and 1.4% better than the modified CPD and GCPA in $\mathcal{J}$, respectively). In summary, there is a large room for performance improvement on the proposed PFOS dataset, suggesting that the PFOS dataset is challenging to all compared methods including OLBP. TABLE V: Ablation analyses for the proposed OLBP network on the PFOS dataset (in percentage %). As can be observed, each component in OLBP network plays an important role and contributes to the performance. The best result in each column is bold. Baseline: encoder-decoder network, OLM: object localization module, and BPM: boundary preservation module. \| Baseline \| OLM \| BPM \| $\mathcal{J}\uparrow$ \| $\mathcal{S}_{\lambda}\uparrow$ \| $w\mathcal{F}_{\beta}\uparrow$ ---\|---\|---\|---\|---\|---\|--- 1 \| ✓∗ \| \| \| 67.2 \| 75.9 \| 72.2 2 \| ✓∗ \| \| ✓ \| 68.0 \| 76.4 \| 72.5 3 \| ✓ \| \| \| 70.7 \| 78.3 \| 75.0 4 \| ✓ \| ✓ \| \| 73.0 \| 80.7 \| 79.5 5 \| ✓ \| \| ✓ \| 71.4 \| 78.7 \| 75.6 6 \| ✓ \| ✓ \| ✓ \| 73.7 \| 81.1 \| 80.0 ✓∗ means the image and FDM are concatenated. ✓means the image and FDM are fed to network separately. \begin{overpic}[width=429.28616pt]{Figs/Fig6_Ablation.pdf} \put(4.0,-1.0){ Image } \put(21.9,-1.0){ GT } \put(38.5,-1.0){ Ba${}^{}$ } \put(51.1,-1.0){ Ba+OLM } \put(66.65,-1.0){ Ba${}^{}$+BPM } \put(87.0,-1.0){ Ours } \end{overpic} Figure 6: Visual comparisons of different variants. “Ba∗” is the baseline network, whose input is the concatenated image and FDM. Qualitative Performance Evaluation. In Fig. 5, we show some representative visualization results of our OLBP network and other methods. Obviously, the visual segmentation maps of three traditional methods GBOS [24], SOS [24] and AVS [22] are rough. However, the CNN-based method CFPS [25], which belongs to the same type as GBOS, SOS and AVS, basically captures the gazed objects and brings in less background regions. The gazed objects in the segmentation results of clicks-based interactive image segmentation methods BRS [16], FCTSFN [17], and ISLD [14] are partially segmented and the details are relatively coarse. As for the EncNet [56], DeepLab [51] and SegNet [55], the object segmentation maps of them depend on the semantic segmentation results, which are great uncertainty. This results in their object segmentation maps that are sometimes accurate and sometimes bad. In contrast, our OLBP network is equipped with the scheme of object localization and boundary preservation, which precisely analyzes the location information of fixations and completes the gazed objects. The segmentation maps of “Ours” in Fig. 5 are very localized in the gazed objects with pretty fine details, even under the interference of some ambiguous fixations. Robustness Evaluation. We provide a robustness evaluation of our method and several representative methods, including the modified GCPA [60], CFPS [25] and the modified CPD [59], on the test dataset of the PFOS dataset. Concretely, we add the noise, i.e. unconstrained fixations, to the fixation map by random sampling on the background regions at three levels, i.e. different percentages (15%, 30%, 45%) increase in the number of unconstrained fixations of the total number of fixations. The performance of above methods after adding noise are presented in Table IV. Our method consistently outperforms the compared methods under three challenging situations, showing excellent robustness. ### VI-C Ablation Studies We comprehensively evaluate the contribution of each vital component to performance in our OLBP network. Specifically, we assess 1) the overall contributions of OLM and BPM; 2) the effectiveness of the three parts in OLM; and 3) the usefulness of BPM and the top-down manner in prediction network. The variants are retrained with the same hyper-parameters and training set as aforementioned settings in Sec. V-E, and the experiments are conducted on the PFOS dataset. TABLE VI: Ablation results of the OLM on the PFOS dataset (in percentage %). The best result in each column is bold. The corresponding structures of the listed variants are presented in Fig. 7. OLM variants \| $\mathcal{J}\uparrow$ \| $\mathcal{S}_{\lambda}\uparrow$ \| $w\mathcal{F}_{\beta}\uparrow$ ---\|---\|---\|--- w/o dilated convs \| 72.7 -1.0 \| 80.7 -0.4 \| 79.0 -1.0 w/o multiply \| 72.6 -1.1 \| 80.5 -0.6 \| 79.2 -0.8 w/o concat \| 72.8 -0.9 \| 80.8 -0.3 \| 79.6 -0.4 w/o Seg sup \| 72.9 -0.8 \| 80.9 -0.2 \| 79.4 -0.6 Ours \| 73.7 \| 81.1 \| 80.0 \begin{overpic}[width=429.28616pt]{Figs/Fig7_OLM.pdf} \end{overpic} Figure 7: Structures of four OLM variants. w/o dilated convs: the four dilated convolutions are replaced by one convolutional layer; w/o multiply: without using response maps to re-weight image feature in Re-wei; w/o concat: without concatenating re-weighted feature and image feature in Re-wei; w/o Seg sup: without segmentation supervision. TABLE VII: The performance of side output segmentation maps of with/without BPM on PFOS dataset (in percentage %). The number in the lower right corner of the performance of w/o BPM is the difference between it and the performance of w/ BPM. The best result in each column is bold. Side outputs \| w/ BPM (Ours) \| w/o BPM ---\|---\|--- $\mathcal{J}\uparrow$ \| $\mathcal{S}_{\lambda}\uparrow$ \| $w\mathcal{F}_{\beta}\uparrow$ \| $\mathcal{J}\uparrow$ \| $\mathcal{S}_{\lambda}\uparrow$ \| $w\mathcal{F}_{\beta}\uparrow$ $\mathbf{S}^{(5)}_{bpm}$ \| 62.0 \| 71.6 \| 67.8 \| 60.2 -1.8 \| 70.4 -1.2 \| 66.0 -1.8 $\mathbf{S}^{(4)}_{bpm}$ \| 69.2 \| 77.5 \| 75.5 \| 68.2 -1.0 \| 76.9 -0.6 \| 74.8 -0.7 $\mathbf{S}^{(3)}_{bpm}$ \| 72.6 \| 80.2 \| 78.9 \| 71.9 -0.7 \| 79.8 -0.4 \| 78.3 -0.6 $\mathbf{S}^{(2)}_{bpm}$ \| 73.7 \| 81.1 \| 79.9 \| 72.9 -0.8 \| 80.6 -0.5 \| 79.4 -0.5 $\mathbf{S}^{(1)}$ \| 73.7 \| 81.1 \| 80.0 \| 73.0 -0.7 \| 80.7 -0.4 \| 79.5 -0.5 1\. Does the proposed OLM and BPM contribute to OLBP network? To evaluate the contribution of the proposed OLM and BPM to OLBP network, we derive three variants: baseline network (denoted by “Ba”/“Ba∗”), baseline network with only OLMs (“Ba+OLM”), and baseline network with only BPMs (“Ba/Ba∗+BPM”). In particular, we provide two types of baseline network: the first one is an encoder-decoder network, whose input is the concatenated image and FDM (denoted by “Ba∗”); the second one is an encoder-decoder network with the down-sampled FDMs being concatenated to each skip-layer (denoted by “Ba”), i.e. the image and FDM are fed to network separately. We report the quantitative results in Tab. V. We observe that the first baseline network “Ba∗” (the 1st line in Tab. V) only obtains $67.2\%$ in $\mathcal{J}$, and the second baseline network “Ba” (the 3rd line in Tab. V) obtains $70.7\%$ in $\mathcal{J}$. This confirms that direct concatenation of the image and FDM results in the location information of FDM being submerged by image information; by contrast, concatenating FDM with image features at each scale benefits object location. OLM significantly improves the performance of the baseline network (e.g. $\mathcal{J}\\!:67.2\%/70.7\%\\!\rightarrow\\!73.0\%$ and $w\mathcal{F}_{\beta}\\!:72.2\%/75.0\%\\!\rightarrow\\!79.5\%$). This shows that the contribution of OLM is remarkable, and OLM does capture the location information. Comparing with OLM, the contribution of BPM to baseline networks is slightly inferior (e.g. $\mathcal{J}\\!:67.2\%\\!\rightarrow\\!68.0\%$; $70.7\%\\!\rightarrow\\!71.4\%$), but BPM also shows its effectiveness to improve performance of “Ba+OLM” (e.g. $w\mathcal{F}_{\beta}\\!:79.5\%\\!\rightarrow\\!80.0\%$). This demonstrates that BPM can further complete the objects and filter background of erroneous localization. With the cooperation between OLM and BPM, the performance of the whole OLBP network is improved by $6.5\%/3.0\%$ in $\mathcal{J}$, $5.2\%/2.8\%$ in $\mathcal{S}_{\lambda}$ and $7.8\%/5.0\%$ in $w\mathcal{F}_{\beta}$ compared with the baseline network “Ba∗”/“Ba”. This demonstrates that the scheme of bottom-up object localization and top-down boundary preservation is successfully embedded into the baseline network. Additionally, the segmentation maps of variants based on the first baseline network “Ba∗” are shown in Fig. 6. We observe that “Ba∗” almost segments all the objects in images. With the assistance of OLM, “Ba∗+OLM” determines the location of the gazed objects, and the gazed objects on the segmentation maps of “Ba+OLM” are much clearer. Finally, with the help of BPM, the segmentation maps of ours (i.e. OLBP network) are satisfactory. 2\. How effective are the three parts in OLM? As described in Sec. V-C, OLM consists of location analysis unit, feature re-weighting (i.e. Re-wei) and segmentation supervision (i.e. Seg sup). To validate the effectiveness of the three parts in OLM, we modify the structure of OLM and provide four variants: a) the four dilated convolutions are replaced by one convolutional layer in the location analysis unit (w/o dilated convs); b) without using response maps to re-weight image feature in Re-wei (w/o multiply); c) without concatenating re-weighted feature and image feature in Re-wei (w/o concat); and d) without segmentation supervision (w/o Seg sup). The ablation results are reported in Tab. VI, and the detailed structures of the above four OLM variants are presented in Fig. 7. \begin{overpic}[width=429.28616pt]{Figs/Fig8_PersonlVS.pdf} \put(29.5,66.3){ Visual individuation } \put(17.8,68.5){ 0.341 } \put(56.3,68.5){ 0.400 } \put(17.8,33.3){ 0.222 } \put(56.3,33.3){ 0.123 } \put(30.5,-2.3){ Visual consistency } \put(17.8,-0.1){ 0.126 } \put(56.3,-0.1){ 0.219 } \par\put(-1.0,91.8){ \begin{sideways}{Image}\end{sideways} } \put(-1.0,83.7){ \begin{sideways}{GT}\end{sideways} } \put(-1.0,73.05){ \begin{sideways}{Ours}\end{sideways} } \par\put(-1.0,56.45){ \begin{sideways}{Image}\end{sideways} } \put(-1.0,48.35){ \begin{sideways}{GT}\end{sideways} } \put(-1.0,37.7){ \begin{sideways}{Ours}\end{sideways} } \par\put(-1.0,23.0){ \begin{sideways}{Image}\end{sideways} } \put(-1.0,14.9){ \begin{sideways}{GT}\end{sideways} } \put(-1.0,4.25){ \begin{sideways}{Ours}\end{sideways} } \end{overpic} Figure 8: Visual examples of personal segmentation results. There are two basic properties of personal visual systems: visual individuation and visual consistency. The value of each image is the mean JS score. We discover that the performances of the four variants are worse than ours. Concretely, the performance degradation of w/o dilated convs (e.g. $\mathcal{J}\\!:73.7\%\\!\rightarrow\\!72.7\%$) validates that the parallel dilated convolutions analyze FDM thoroughly and one convolutional layer cannot mine sufficient location information from FDM. The performance drop of w/o multiply (e.g. $\mathcal{S}_{\lambda}\\!:81.1\%\\!\rightarrow\\!80.5\%$) confirms that the location response maps are more suitable to highlight objects on CNN feature of image than using them directly. The reason behind this is that location response maps are a group of probability maps, without rich object, texture and color information. Besides, w/o concat brings $0.9\%$ performance penalty in $\mathcal{J}$, which shows that the information balance between image and location is important. w/o Seg sup carries $0.6\%$ performance drop in $w\mathcal{F}_{\beta}$. This demonstrates that the segmentation supervision can enhance representation of the gazed objects. 3\. Is it useful to adopt BPM and the top-down manner in prediction network? To investigate the usefulness of the top-down manner in prediction network, we report the performance of side output segmentation maps of BPM in Tab. VII. Besides, we also report the side output performance of w/o BPM in Tab. VII to evaluate the importance of BPM. We observe that the quantitative results of side outputs ($\mathbf{S}^{(5)}_{bpm}$, $\mathbf{S}^{(4)}_{bpm}$, $\mathbf{S}^{(3)}_{bpm}$, $\mathbf{S}^{(2)}_{bpm}$ and $\mathbf{S}^{(1)}$) are incremental in terms of both w/ BPM (e.g. $w\mathcal{F}_{\beta}\\!:67.8\%\\!\rightarrow\\!75.5\%\\!\rightarrow\\!78.9\%\\!\rightarrow\\!79.9\%\\!\rightarrow\\!80.0\%$) and w/o BPM (e.g. $\mathcal{S}_{\lambda}\\!:70.4\%\\!\rightarrow\\!76.9\%\\!\rightarrow\\!79.8\%\\!\rightarrow\\!80.6\%\\!\rightarrow\\!80.7\%$). This confirms that the top-down manner is useful for the prediction network. The differences between the performance of w/o BPM and w/ BPM are also reported in Tab. VII. We discover that all the differences are negative, which shows that BPM works well for each side output of the top-down prediction network. ### VI-D Personal Segmentation Results Due that the personal fixations are closely related to age and gender, different users are interested in different objects when observing the same scene. We define the visual difference of different personal visual systems as visual individuation. Some examples of visual individuation are presented in the first part of Fig. 8. We can observe that there are multiple different types of objects and complex backgrounds in these images. The personal fixations of different users are located on different objects, which correspond to the distinctive GTs. \begin{overpic}[width=429.28616pt]{Figs/Fig9.pdf} \put(4.9,-2.5){ Image} \put(21.3,-2.5){ {Ours} } \put(34.8,-2.5){GT of SOD } \put(52.1,-2.5){ CPD${}_{\mathrm{sod}}$ } \put(67.3,-2.5){ GCPA${}_{\mathrm{sod}}$ } \end{overpic} Figure 9: Visual comparisons between our method, which is proposed for fixation-based object segmentation, and recent state-of-the-art salient object detection methods, including CPD [59] and GCPA [60], on the DUTS-OMRON [46] and PASCAL-S [23] datasets. “GT of SOD” means that the GT is for SOD task. “CPDsod” means the original CPD method for SOD. “GCPAsod” means the original GCPA method for SOD. In addition, we discover that personal visual systems are also consistent in some scenes, which is denoted as visual consistency. We show some examples of visual consistency in the second and third parts of Fig. 8. The images in the second part contain simple backgrounds and sparse objects, and the images in the third part contain more competitive situation, i.e. complex background and partially selected objects. In both parts, we observe that the locations of different personal fixations are similar, resulting in the identical GTs of different users. Notably, in either case, our method show the ability to segment the gazed objects consistent with the corresponding GT. We also provide the quantitative analysis of visual individuation and visual consistency with Jensen-Shannon (JS) divergence. JS divergence evaluates the similarity of two probability distributions $\mathbf{S}^{1}$ and $\mathbf{S}^{2}$, and it is based on Kullback-Leibler (KL) divergence. Its value belongs to [0, 1]. The closer its value is to zero, the smaller the difference between $\mathbf{S}^{1}$ and $\mathbf{S}^{2}$ is and the more similar they are. It can be expressed as follows: $\displaystyle\mathrm{JS}(\mathbf{S}^{1},\mathbf{S}^{2})=\frac{1}{2}\mathrm{KL}(\mathbf{S}^{1},\frac{\mathbf{S}^{1}+\mathbf{S}^{2}}{2})+\frac{1}{2}\mathrm{KL}(\mathbf{S}^{2},\frac{\mathbf{S}^{1}+\mathbf{S}^{2}}{2}),$ (15) $\displaystyle\mathrm{KL}(\mathbf{P},\mathbf{Q})=\sum^{N}_{i=1}\mathbf{P}_{\textit{i}}\mathrm{log}\left(\epsilon+\frac{\mathbf{P}_{\textit{i}}}{\epsilon+\mathbf{Q}_{\textit{i}}}\right),$ (16) where KL($\cdot$) is Kullback-Leibler divergence, which is often used as an evaluation metric in fixation prediction [71, 72, 73, 74], i indicates the ith pixel in the probability distribution, $N$ is the total number of pixels, and $\epsilon$ is a regularization constant. We introduce JS to measure the similarity of fixation points maps of each image in Fig. 8. First, we transform the fixation points map (green dots in each image) to FDM using Eq. 1; then we compute the JS score of each two FDMs; finally we report the mean JS score for each image in Fig. 8. It is obvious that the mean JS scores (i.e. 0.222, 0.123, 0.126, and 0.219) of images which belong to visual consistency are relatively smaller than those (i.e. 0.341 and 0.400) of images which belong to visual individuation. And the mean JS scores of images which belong to visual consistency are close to zero, which indicates that the distributions of FDMs are very similar, i.e. people may look at the same object(s). ### VI-E Discussions Salient Object Detection (SOD) is widely explored in color images [59, 60, 75, 76, 77], RGB-D images [78, 79] and videos [80, 81, 82], and it is closely related to our fixation-based object segmentation task. In this section, we discuss the connections between fixation-based object segmentation and SOD. SOD aims to highlight the most visually attractive object(s) in a scene, while fixation-based object segmentation aims to segment the gazed objects according to the fixation map, as defined in Sec. III. To illustrate the differences and connections between these two tasks, we conduct experiments on two SOD datasets, i.e. DUTS-OMRON [46] and PASCAL-S [23], and show visual comparisons with two state-of-the-art SOD methods, i.e. CPD [59] and GCPA [60], in Fig. 9, which summarizes three situations. First, in the $1^{\mathrm{st}}$ and $2^{\mathrm{nd}}$ rows, we present the differences of these two tasks: our method not only segments the salient objects, such as the bird and the big tent, but also segments the gazed wood stake and cloth that are not found in the GT of SOD and the results of CPD and GCPA. Second, in the $3^{\mathrm{rd}}$ and $4^{\mathrm{th}}$ rows, we find that the results of CPD and GCPA are similar to ours, but different from the GT of SOD. This shows that to some extent, the results of SOD methods CPD and GCPA are consistent with the fixation maps, even if the fixation maps are not exploited in these methods. Third, in the $5^{\mathrm{th}}$ and $6^{\mathrm{th}}$ rows, we can clearly observe that our results are consistent with the fixation points in images, while the other three maps are different. This shows that different SOD methods may cause confusion in some complicated scenes, resulting in inaccurate saliency maps. Furthermore, we find that the salient objects always appear in the results of our method, while there is ambiguity among different SOD methods, which may highlight different salient objects. So, to improve the accuracy of different SOD methods, we believe that the fixation-based object segmentation can be a pre-processing operation for SOD to determine the salient object proposals. ## VII Conclusion In this paper, we propose a three-step approach to transform the available fixation prediction dataset OSIE to the PFOS dataset for personal fixations- based object segmentation. The PFOS dataset is meaningful to promote the development of fixations-based object segmentation. Moreover, we present a novel OLBP network with the scheme of bottom-up object localization and top- down boundary preservation to segment the gazed objects. Our OLBP network is equipped with two essential components: the object localization module and the boundary preservation module. OLM is object locator, which is in charge of location analysis of fixations and object enhancement. BPM emphasizes erroneous localization distillation and object completeness preservation. 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# Unsupervised Technical Domain Terms Extraction using Term Extractor Suman Dowlagar LTRC IIIT-Hyderabad suman.dowlagar@ research.iiit.ac.in Radhika Mamidi LTRC IIIT-Hyderabad radhika.mamidi@ iiit.ac.in ###### Abstract Terminology extraction, also known as term extraction, is a subtask of information extraction. The goal of terminology extraction is to extract relevant words or phrases from a given corpus automatically. This paper focuses on the unsupervised automated domain term extraction method that considers chunking, preprocessing, and ranking domain-specific terms using relevance and cohesion functions for ICON 2020 shared task 2: TermTraction. ## 1 Introduction The aim of Automatic Term Extraction (ATE) is to extract terms such as words, phrases, or multi-word expressions from the given corpus. ATE is widely used in many NLP tasks, such as machine translation, summarization, clustering the documents, and information retrieval. Unsupervised algorithms for domain term extraction are not labeled and trained on the corpus and do not have any pre-defined rules or dictionaries. They often use statistical information from the text. Most of these algorithms use stop word lists and can be applied to any text datasets. The standard unsupervised automated term extraction pipeline consists of * • Simple Rules: using chunking or POS tagging to extract Noun phrases for multi- word extraction. * • Naive counting: that counts how many terms each word occurs in the corpus. * • Preprocessing: Removing punctuation and common words such as stop words from the text. * • Candidate generation and scoring: using statistical measures and ranking algorithms to generate the possible set of domain terms * • Final set: Arrange the ranked terms in descending order based on the scores and take the top N keywords as the output. Currently, there are many methods for automatic term recognition. Evans and Lefferts (1995) used TF-IDF measure for term extraction.Navigli and Velardi (2002) used domain consensus which is designed to recognize the terms uniformly distributed over the whole corpus. The most popular method C-value Frantzi et al. (2000) is also a statistical measure that extracts a term based on the term’s frequency, length of the term, and the set of the candidates that enclose the term such that the term is in their substring. Bordea et al. (2013) proposed the method called Basic, which is a modification of the C-value for recognizing terms of average specificity. The successor of C-value statistic called the NC value Frantzi et al. (2000) considered scored the term based on the condition if it exists in a group of common words or if it contains nouns, verbs, or adjectives that immediately precede or follow the term. The methods proposed by Ahmad et al. (1999); Kozakov et al. (2004); Sclano and Velardi (2007) are based on extracting the terms of a text by considering the frequency of occurrence of terms in the general domain. A detailed survey of the existing automated term extraction algorithms and their evaluation are presented in papers by Astrakhantsev et al. (2015); Šajatović et al. (2019) In this paper, we used the term extractor algorithm Sclano and Velardi (2007) present in the pyate111https://pypi.org/project/pyate/ library for domain term extraction. The term extractor algorithm is developed initially for ontology extraction from large corpora. It uses domain pertinence/relevance, domain consensus, and lexical cohesion for extracting terms. A detailed description of the modules is given in the next section. The paper is organized as follows. Section 2 gives a detailed description of the term extraction algorithm used. Section 3 gives information about the datasets used and results. Section 4 concludes the paper. ## 2 Our Approach In this section, we describe in detail the methods used in the term extractor algorithm. Initially, TermExtractor performs chunking and proper name recognition and then extracts structures based on linguistic rules and patterns, including stop words, detection of misspellings, and acronyms. The extraction algorithm uses Domain Pertinence, Domain Cohesion, and Lexical Cohesion to decide if a term is considered a domain term. Domain Pertinence, or Domain Relevance (DR), requires a contrastive corpus and compares a candidate’s occurrence in the documents belonging to the target domain to its occurrence in other domains, but the measure only depends on the contrastive domain where the candidate has the highest frequency. The Domain Pertinence is based on a simple formula, $DR_{D_{i}}(t)=\frac{tf_{i}}{max_{j}(tf_{j})}$ (1) Where $tf_{i}$ is the frequency of the candidate term in the input domain- specific document collection and $max_{j}(tf_{j})$ is the general corpus domain, where the candidate has the highest frequency, and $D_{i}$ is the domain in consideration. Domain Consensus (DC) assumes that several documents represent a domain. It measures the extent to which the candidate is evenly distributed on these documents by considering normalized term frequencies $(\phi)$, $DC_{D_{i}}(t)=\sum_{d_{k}\epsilon D_{i}}\phi_{k}log\phi_{k}$ (2) Here, we assume $k$ distinct documents for the domain $D_{i}$. Lexical cohesion involves the choice of vocabulary. It is concerned with the relationship that exists between lexical items in a text, such as words and phrases. It compares the in-term distribution of words that make up a term with their out-of-term distribution. $LC_{D_{i}}(t)=\frac{ntf_{i}logtf_{i}}{\sum_{j}tf_{w_{j}}i}$ (3) Where $n$ is the number of documents in which the term $t$ occurs. The final weight of a term is computed as a weighted average of the three filters above, $score(t,D_{i})=\alphaDR+\betaDC+\gammaLC$ (4) where $\alpha$, $\beta$, $\gamma$ are the weights, and they are equal to $1/3$ ## 3 Experiments This section describes the dataset used for domain terms extraction, implementation of the above approach, followed by results, and error analysis. ### 3.1 Dataset We used the dataset provided by the organizers of TermTraction ICON-2020. The task is to extract domain terms from the given English documents from the four technical domains like Computer Science, Physics, Life Science, Law. The data statistics of the documents in the respective domains are shown in the table 1. Domain \| #Train docs \| #Test docs ---\|---\|--- Bio-Chemistry \| 229 \| 10 Communication \| 127 \| 10 Computer-Science \| 201 \| 8 Law \| 70 \| 16 Table 1: Data statistics Biochemistry \| Communication \| Computer Science \| Law ---\|---\|---\|--- Data \| run 1 \| run 2 \| Data \| run 1 \| run 2 \| Data \| run 1 \| run 2 \| Data \| run 1 \| run 2 M12S1 \| 0.247 \| 0.222 \| M2-1 \| 0.109 \| 0.086 \| KL2 \| 0.220 \| 0.225 \| A01 \| 0.079 \| 0.077 M15S2 \| 0.208 \| 0.195 \| M2-2 \| 0.102 \| 0.104 \| KL4 \| 0.241 \| 0.246 \| A02 \| 0.099 \| 0.066 M16S2 \| 0.224 \| 0.207 \| M2-3 \| 0.094 \| 0.074 \| KL8 \| 0.138 \| 0.146 \| A03 \| 0.144 \| 0.126 M23S3 \| 0.266 \| 0.233 \| M3-1 \| 0.240 \| 0.236 \| W12 \| 0.143 \| 0.122 \| FA1 \| 0.104 \| 0.116 M26S2 \| 0.096 \| 0.081 \| M3-2 \| 0.159 \| 0.148 \| W1332 \| 0.216 \| 0.195 \| FA2 \| 0.077 \| 0.067 T18 \| 0.463 \| 0.427 \| M3-3 \| 0.140 \| 0.132 \| W13 \| 0.108 \| 0.089 \| FC1 \| 0.082 \| 0.073 T25 \| 0.310 \| 0.282 \| RM16 \| 0.101 \| 0.088 \| W1436 \| 0.181 \| 0.165 \| FC2 \| 0.032 \| 0.021 T39 \| 0.265 \| 0.247 \| RM17 \| 0.067 \| 0.065 \| W921 \| 0.221 \| 0.188 \| FC3 \| 0.016 \| 0.014 T4 \| 0.271 \| 0.234 \| RM18 \| 0.098 \| 0.115 \| \| \| \| FR1 \| 0.149 \| 0.113 T9 \| 0.323 \| 0.315 \| SW1AW \| 0.120 \| 0.113 \| \| \| \| FR2 \| 0.144 \| 0.112 \| \| \| \| \| \| \| \| \| FR3 \| 0.073 \| 0.062 \| \| \| \| \| \| \| \| \| G3 \| 0.103 \| 0.098 \| \| \| \| \| \| \| \| \| G4 \| 0.056 \| 0.052 \| \| \| \| \| \| \| \| \| R1 \| 0.022 \| 0.055 \| \| \| \| \| \| \| \| \| R2 \| 0.033 \| 0.026 \| \| \| \| \| \| \| \| \| R3 \| 0.044 \| 0.048 Table 2: Term Extraction macro-F1 score. Template Sentence \| Domain terms identified ---\|--- We are not going to that , remove it completely, but nevertheless this is an indication that , NO plus is going to be a poorer donor , compared to carbon monoxide . So , this drastic reduction in the stretching frequency can only happen if you have , a large population of the anti - bonding orbitals of NO plus . And it has got a structure , which is very similar , a structure which is very similar to the structure of nickel tetra carbonyl . You will see that , while carbon monoxide is ionized with 15 electron volts , if you supply 15 electron volts , carbon monoxide can be oxidized or ionized . \| \| large population --- similar ionized carbonyl frequency poorer donor anti - bonding orbitals indication carbon monoxide electron volts nickel plus drastic reduction structure Table 3: Error analysis on the template sentence ### 3.2 Implementation We used Pyate (python automated term extraction library) that contains the term extractor method and is trained on the general corpus. With the help of the term extraction method, we extracted the relevant terms from the given corpus. We have submitted two runs, one run (run 1) is the term extractor function itself, and the other run (run 2) is term extractor combined with NP chunks of phrase length ¿ 2 obtained from NLTK ConsecutiveNPChunkTagger222ConsecutiveNPChunkTagger . ### 3.3 Results and Error Analysis We evaluated the performance of the method using average precision. The results are tabulated in Table 2. For the template sentence given in Table 3, our algorithm failed to recognize the domain terms NO plus and nickel tetra carbonyl. It considered NO as the stop word (no or negation) and discarded it while preprocessing. The algorithm also misunderstood words like “similar” as domain terms and failed to identify nickel tetra carbonyl as a domain term. It indicates that further study is necessary, which considers the candidate terms’ capitalization and uses better methods that support the more reliable form of compound words or multi-word expressions. ## 4 Conclusion For domain term extraction from technical domains like Bio-Chemistry, Law, Computer-Science, and communication, We used the term extractor method from pyate library for obtaining technical terms. The term extractor method uses keywords from the general corpora, and it considers Domain Pertinence, Domain Cohesion, and Lexical Cohesion methods for extracting domain terms in the given corpus. As mentioned above, it did not give preference to capitalized terms and did not consider some compound words. So we have to work towards better methods that consider capitalization, better formation of compound words for the more reliable performance of the automated domain term extractor. ## References Ahmad et al. (1999) Khurshid Ahmad, Lee Gillam, Lena Tostevin, et al. 1999. University of surrey participation in trec8: Weirdness indexing for logical document extrapolation and retrieval (wilder). In _TREC_ , pages 1–8. * Astrakhantsev et al. (2015) Nikita A Astrakhantsev, Denis G Fedorenko, and D Yu Turdakov. 2015. Methods for automatic term recognition in domain-specific text collections: A survey. _Programming and Computer Software_ , 41(6):336–349. * Bordea et al. (2013) Georgeta Bordea, Paul Buitelaar, and Tamara Polajnar. 2013. Domain-independent term extraction through domain modelling. In _The 10th international conference on terminology and artificial intelligence (TIA 2013), Paris, France_. 10th International Conference on Terminology and Artificial Intelligence. * Evans and Lefferts (1995) David A Evans and Robert G Lefferts. 1995. Clarit-trec experiments. _Information processing & management_, 31(3):385–395. * Frantzi et al. (2000) Katerina Frantzi, Sophia Ananiadou, and Hideki Mima. 2000. Automatic recognition of multi-word terms:. the c-value/nc-value method. _International journal on digital libraries_ , 3(2):115–130. * Kozakov et al. (2004) Lev Kozakov, Youngja Park, T Fin, Youssef Drissi, Yurdaer Doganata, and Thomas Cofino. 2004. Glossary extraction and utilization in the information search and delivery system for ibm technical support. _IBM Systems Journal_ , 43(3):546–563. * Navigli and Velardi (2002) Roberto Navigli and Paola Velardi. 2002. Semantic interpretation of terminological strings. In _Proc. 6th Int’l Conf. Terminology and Knowledge Eng_ , pages 95–100. * Šajatović et al. (2019) Antonio Šajatović, Maja Buljan, Jan Šnajder, and Bojana Dalbelo Bašić. 2019. Evaluating automatic term extraction methods on individual documents. In _Proceedings of the Joint Workshop on Multiword Expressions and WordNet (MWE-WN 2019)_ , pages 149–154. * Sclano and Velardi (2007) Francesco Sclano and Paola Velardi. 2007. Termextractor: a web application to learn the shared terminology of emergent web communities. In _Enterprise Interoperability II_ , pages 287–290. Springer.
ﬁfi # Experimentally Realizing Efficient Quantum Control with Reinforcement Learning Ming-Zhong Ai These two authors contributed equally to this work. CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China Yongcheng Ding These two authors contributed equally to this work. International Center of Quantum Artiﬁcial Intelligence for Science and Technology (QuArtist) and Department of Physics, Shanghai University, 200444 Shanghai, China Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain Yue Ban Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain School of Materials Science and Engineering, Shanghai University, 200444 Shanghai, China José D. Martín-Guerrero IDAL, Electronic Engineering Department, University of Valencia, Avgda. Universitat s/n, 46100 Burjassot, Valencia, Spain Jorge Casanova Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain IKERBASQUE, Basque Foundation for Science, Plaza Euskadi 5, 48009 Bilbao, Spain Jin-Ming Cui<EMAIL_ADDRESS>CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China Yun-Feng Huang<EMAIL_ADDRESS>CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China Xi Chen<EMAIL_ADDRESS>International Center of Quantum Artiﬁcial Intelligence for Science and Technology (QuArtist) and Department of Physics, Shanghai University, 200444 Shanghai, China Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain Chuan-Feng Li<EMAIL_ADDRESS>CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China Guang-Can Guo CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China ###### Abstract Robust and high-precision quantum control is crucial but challenging for scalable quantum computation and quantum information processing. Traditional adiabatic control suffers severe limitations on gate performance imposed by environmentally induced noise because of a quantum system’s limited coherence time. In this work, we experimentally demonstrate an alternative approach to quantum control based on deep reinforcement learning (DRL) on a trapped ${}^{171}\mathrm{Yb}^{+}$ ion. In particular, we find that DRL leads to fast and robust digital quantum operations with running time bounded by shortcuts to adiabaticity (STA). Besides, we demonstrate that DRL’s robustness against both Rabi and detuning errors can be achieved simultaneously without any input from STA. Our experiments reveal a general framework of digital quantum control, leading to a promising enhancement in quantum information processing. ††preprint: APS/123-QED ## I INTRODUCTION Two-level systems physically realize qubits, which are the basic units of digital quantum computing. In this paradigm, externally controllable parameters should be designed to manipulate the qubits, implementing fast and robust gate operations. Thus, one can construct a universal fault-tolerant quantum computer with physical platforms based on trapped ions and superconducting circuits Nielsen and Chuang (2010). In this way, quantum error correction can also be realized physically to reduce the effects of quantum noises and systematic errors. From this perspective, quantum control is bridged to quantum information processing and quantum computing. This connection leads to enormous researches devoted to producing precise quantum control of qubits with driving fields, including adiabatic passages Král _et al._ (2007), optimized resonant $\pi$ pulses Remizov _et al._ (2015), composite pulses Brown _et al._ (2004); Torosov _et al._ (2011); Rong _et al._ (2015), pulse-shape engineering Steffen and Koch (2007); Barnes and Sarma (2012); Daems _et al._ (2013), and other optimizations Glaser _et al._ (2015); Caneva _et al._ (2009); Guérin _et al._ (2011); Hegerfeldt (2013); Garon _et al._ (2013); Van Damme _et al._ (2017); Arenz _et al._ (2017). A most straightforward approach to transit less dynamics obeys the adiabatic theorem by tuning the time-dependent parameter sufficiently slow. However, prolonged operation time destructs the quantum information by induced decoherence, affecting information processing efficiency. The concept of shortcuts to adiabaticity (STA) Guéry-Odelin _et al._ (2019); Torrontegui _et al._ (2013) is proposed, which combines the advantages of both adiabatic passages and resonant pulses. It breaks the adiabatic regime by various techniques, including inverse engineering Chen _et al._ (2010), counter-diabatic driving Deffner _et al._ (2014); An _et al._ (2016), fast- forward scaling Masuda and Nakamura (2010); Masuda (2012), which has been well developed over the past decade. Specifically, inverse engineering emanates from the Lewis-Riesenfeld theory, allowing superadiabatic state evolution on dynamical modes with boundary conditions. In addition, inverse engineering leaves enough freedom to further allow other tasks such as, e.g., suppressing systematic errors by collaborating with optimal control theory Daems _et al._ (2013); Ruschhaupt _et al._ (2012); Lu _et al._ (2013), dynamical decoupling techniques Munuera-Javaloy _et al._ (2020), and machine learning methods Zahedinejad _et al._ (2016); Liu _et al._ (2019); Ding _et al._ (2020). However, invariant-based STA requires continuously tunable parameters, limiting the genre of quantum control as analog-only. We consider a more complicated task: designing digital pulses instead of an analog controller with the same output and similar features. In this manner, we would deliver a framework that can be naturally integrated in current quantum computing paradigms based on the application of several digital quantum gates. We look for the optimal digital pulses design, which is similar to invariant- based STA for realizing robust quantum control. The optimal design is indeed a combinational optimization problem, being equivalent to dynamic programming, which is no longer analytically solvable. As artificial intelligence approach, Reinforcement Learning is a well-known tool for system control Sutton and Barto (2018), and deep learning has been developed for conquering complicated tasks in many areas Mnih _et al._ (2015, 2013); Silver _et al._ (2016, 2017), later applied in studying physics Carleo and Troyer (2017); Nagy and Savona (2019); Hartmann and Carleo (2019); Vicentini _et al._ (2019); Yoshioka and Hamazaki (2019); Iten _et al._ (2020). The framework of deep learning can be combined with reinforcement learning, searching control pulses for quantum state preparation Henson _et al._ (2018); Zhang _et al._ (2019), gate operation An and Zhou (2019), and quantum Szilard engine Sørdal and Bergli (2019). Since recent researches have employed Deep Reinforcement Learning (DRL) for quantum control Bukov _et al._ (2018); Porotti _et al._ (2019); Niu _et al._ (2019); Zhang _et al._ (2018); Wu _et al._ (2019); Wang _et al._ (2020), we are inspired to investigate the connection between DRL and STA. An optimistic expectation is that one can extend STA’s concept, introducing DRL as a new technique if it learns the features of STA protocols. In this paper, we present an experimental demonstration of a robust and high- precision quantum control task based on the deep reinforcement learning method on a trapped ${}^{171}\mathrm{Yb}^{+}$ ion. To be more specific, we train an Agent in a computer through DRL to achieve a single qubit X gate with time prior information bounded by STA. The multi-pulses control sequences produced by the DRL model is more robust than the standard $\pi$ pulse method (interacting with a constant amplitude for a period of time) with constant Rabi frequency in the presence of system noise. Besides, the robustness against both Rabi and detuning errors at the same time by DRL sequences is also verified. To demonstrate the application in the real laboratory noise environment, we examine the DRL models in the Zeeman energy level of the ion, which is sensitive to magnetic field noise. The results show that these DRL models can combat real system noises. Figure 1: (color online) Experimental sequences and model wave-forms. (a) Optimized detuning with time under STA method. The time is normalized to $[0,1]$. (b) Optimized detuning with time under DRL method. The time is normalized to $[0,1]$. (c) Energy level of ${}^{171}\textrm{Yb}^{+}$ ion. (d) Evolution of state in Bloch sphere under the driving of DRL model. Red solid line represents the trajectory optimized for $\Omega$ errors while blue solid line represents the trajectory optimized for $\Delta$ errors. Hollow circle and hollow triangle represent the state at the end of each driving step. (e) Experimental sequences in DRL model. After laser cooling and pumping, the ion is initialized to $\|0\rangle$ state. Then a 20-steps microwave which contains DRL driving information is transmitted to the ion. finally a detecting laser is used to detect the probability in $\|1\rangle$ state of the ion. ## II THEORETICAL MODELS Consider the coherent manipulation of a single qubit, whose Hamiltonian reads $H=\frac{\hbar}{2}\left[\Omega\sigma_{x}+\Delta(t)\sigma_{z}\right],$ (1) where the Rabi frequency $\Omega$ is fixed, while the detuning $\Delta(t)$ is time-varying. To achieve a robust qubit flipping from $\|0\rangle$ to $\|1\rangle$, a standard $\pi$ pulse, which corresponds to the Hamiltonian $\frac{\hbar}{2}\Omega\sigma_{x}$, is convenient and adequate. However, this operation is sensitive to systematic noise and decoherence. The invariant-based STA suggests that, one can achieve nonadiabatic quantum control of high fidelity and robustness by designed protocols, which satisfy the auxiliary equations derived from Lewis-Riesenfeld (LR) invariant. The LR invariant of a two-level system is constructed by $I(t)=\frac{\hbar}{2}\Omega_{0}\sum_{\pm}\|\psi_{\pm}(t)\rangle\langle\psi_{\pm}(t)\|$, where the eigenstates are $\|\psi_{+}(t)\rangle=\left(\cos\frac{\theta}{2}e^{-i\frac{\beta}{2}},\sin\frac{\theta}{2}e^{i\frac{\beta}{2}}\right)^{\text{T}}$ and $\|\psi_{-}(t)\rangle=\left(\sin\frac{\theta}{2}e^{-i\frac{\beta}{2}},-\cos\frac{\theta}{2}e^{i\frac{\beta}{2}}\right)^{\text{T}}$. The dynamics of the Hamiltonian is governed by time-dependent Schrödinger’s equation, whose solution is in superposition of these eigenstates as $\|\Psi(t)\rangle=\sum_{\pm}c_{\pm}\exp(i\gamma_{\pm})\|\psi_{\pm}(t)\rangle$, with LR phase calculated as $\gamma_{\pm}=\pm\frac{1}{2}\int_{0}^{t}\left(\frac{\dot{\theta}\cot\beta}{\sin\theta}\right)dt^{\prime}.$ (2) According to the condition for invariant $dI(t)/dt=\partial I(t)/\partial t+(1/i\hbar)[I(t),H(t)]=0$, we have the auxiliary equations $\displaystyle\dot{\theta}$ $\displaystyle=$ $\displaystyle-\Omega\sin\beta,$ (3) $\displaystyle\dot{\beta}$ $\displaystyle=$ $\displaystyle-\Omega\cot\theta\cos\beta+\Delta(t),$ (4) describing the state evolution along the dynamical modes with angular parameters $\theta$ and $\beta$, which characterize the trajectory on the Bloch sphere. As proposed in Ref. Ding _et al._ (2020), the framework can be applied to design robust quantum control, e.g., qubit flipping, against systematic errors with an adequate ansatz of free parameter $a$, such that $\theta(t)=\frac{\Omega T}{a}\left[as-\frac{\pi^{2}}{2}(1-s)^{2}+\frac{\pi^{3}}{3}(1-s)^{3}+\cos(\pi s)+A\right],$ (5) where $T=-\pi a/[(2-a-\pi^{2}/6)\Omega]$, $s=t/T$, and $A=\pi^{2}/6-1$ determined by boundary conditions $\theta(0)=0,~{}\dot{\theta}(0)=\Omega,~{}\ddot{\theta}(0)=0$ and $\theta(T)=\pi,~{}\dot{\theta}(T)=\Omega,~{}\ddot{\theta}(T)=0$. Specifically, one can nullify the probability of the first-order transition $P=\frac{\hbar^{2}}{4}\left\|\int_{0}^{T}\langle\Psi_{-}(t)\|\left(\delta_{\Omega}\Omega\sigma_{x}+\delta_{\Delta}\sigma_{z}\right)\|\Psi_{+}(t)\rangle\right\|^{2},$ (6) which yields the condition for error cancellation $\left\|\int_{0}^{T}dte^{i2\gamma_{+}(t)}\left(\delta_{\Delta}\sin\theta-i2\delta_{\Omega}\dot{\theta}\sin^{2}\theta\right)\right\|=0,$ (7) where systematic errors are characterized by $\Delta(t)\rightarrow\Delta(t)+\delta_{\Delta}$ and $\Omega\rightarrow\Omega(1+\delta_{\Omega})$, resulting in the configuration $a=0.604$ and $0.728$ for eliminating $\Delta$ and $\Omega$-error, respectively. Indeed, smooth detuning pulse $\Delta(t)$ as analog control of single-component is inversely engineered by substituting the ansatz into the following expression $\Delta(t)=-\frac{\ddot{\theta}}{\Omega\sqrt{1-\left(\frac{\dot{\theta}}{\Omega}\right)^{2}}}+\Omega\cot\sqrt{1-\left(\frac{\dot{\theta}}{\Omega}\right)^{2}}.$ (8) which is derived from combining auxiliary equations. The wave-forms of $\Delta(t)$ optimized for different systematic errors in STA are shown in fig. 1(a) and the maximum detuning $\Delta_{\max}$ for $\Delta$ and $\Omega$ errors are $1.5\Omega$ and $1.7\Omega$, respectively. Concerning our physical realization in trapped ions, the Rabi frequency $\Omega=(2\pi)3.3$ kHz is fixed, where we calculate the corresponding operation time for robust qubit flipping against $\Delta$ and $\Omega$-errors as $T_{\Delta}=364$ $\mu$s and $T_{\Omega}=293$ $\mu$s. Since an analog quantum control can be derived from the STA framework, it is more challenging to consider the digital quantum control of Landau-Zener problem. The problem is reformulated to the following expression: how should we manipulate a quantum system for a certain target with a step controller of $N$ intervals within a fixed time? The combinational optimization problem is equivalent to dynamic programming, i.e., a multi-step decision problem whose complexity grows exponentially with step number, allowing an approximation solution by artificial neural networks (ANN) or other universal function approximators; the use of deep ANN architectures with many layers leads to the concept of deep learning, and this, in turn, to DRL. In the framework of DRL, one assumes that there exists an unknown global optimal policy $\pi$ for a task, which gives an action $\textbf{a}(t_{i})$ once observing an arbitrary state $\textbf{s}(t_{i})$ at time $t_{i}$. The state-action relation $\pi(\textbf{s}\|\textbf{a})$ is approximated by an Agent ANN, containing propagation of information between layers and nonlinear activation of neurons, whose parameters are tuned by optimizing algorithms for maximizing the accumulated reward. Details about the implementation of deep reinforcement learning can be found in supplementary materials. In our numerical experiments, the tunable range of detuning $[-\Delta_{\max},\Delta_{\max}]$ is renormalized into $\tilde{\Delta}\in[0,1]$ with $\Delta_{\max}$ being the maximal reachable value of $\Delta(t)$ in STA, which is the output of ANN as the encoded action at time step $t_{i}$: $\tilde{\Delta}(t_{i})=[\Delta(t_{i})+\Delta_{\max}]/2\Delta_{\max}$. Information of the two-level system, specifically, the expectation of spin on Z direction $\langle\sigma_{z}\rangle$, the renormalized detuning $\tilde{\Delta}(t_{i-1})$ that drives the system to the current state, and the system time $i/N$, are fed to the input layer of the ANN. The quantum dynamics are simulated by Liouville-von Neumann equation, which can be generalized to the Lindblad master equation for taking quantum noises into consideration. While network configuration, hyperparameters, and training details are explained in the literature Ding _et al._ (2020), we introduce the reward functions that we artificially design, which are similar to invariant-based STA that chooses an ansatz for obtaining quantum control. For converging the Agent to robust control of LZ-type, we firstly pre-train the Agent with $r(t_{i})=-\|\tilde{\Delta}(t_{i})-\frac{i-1}{N-1}\|$, punishing the deviations from linear growth of detuning, later rewarding a constant if $\langle\sigma_{z}\rangle>0.997$ at the final time step for fine-tuning under random systematic errors. For evaluating the DRL-inspired robust quantum control, we perform two numerical experiments as follows: (i) We set the operation time as $T_{\Delta}=364$ $\mu$s and $T_{\Omega}=293$ $\mu$s, being split uniformly by 20 pulses as the only hint from STA. The digital wave-forms output from our DRL model optimized for different systematic errors are shown in fig. 1(b). We emphasize that the STA framework clarifies the upper bound of robustness in Landau-Zener problems, which could be employed for benchmarking the capability of the Agent, as an artificial intelligence approach to digital quantum control with the alike feature. (ii) The operation time is arbitrarily set to be $T=300$ $\mu$s for checking if the Agent can explore desired protocols against hybrid systematic errors without any field knowledge of STA. We clarify that DRL is more general for this task since invariant-based STA no longer eliminates the hybrid errors perfectly but on certain proportion of $\delta_{\Delta}$ and $\delta_{\Omega}$ instead. All wave-forms used in real experiments are from these two numerical experiment models. ## III EXPERIMENTAL REALIZATION Our experiments are performed on a ${}^{171}{\rm Yb}^{+}$ ion trapped in a harmonic Paul trap, with the simplified structure being described in detail in supplementary materials. As shown in fig. 1(c), the two level system (TLS) is encoded in the ${}^{2}{\rm S}_{1/2}$ ground state of the ion, with $\left\|0\right\rangle=\left\|{}^{2}{\rm S}_{1/2},F=0,m_{F}=0\right\rangle$ and $\left\|1\right\rangle=\left\|{}^{2}{\rm S}_{1/2},F=1,m_{F}=0\right\rangle$. The difference of energy level $\left\|0\right\rangle$ and $\left\|1\right\rangle$ is about $\omega_{01}=12.6428$ GHz. The microwaves used to drive the TLS are generated through mixing method. More specifically, a microwave around 12.4428 GHz generated from signal generator (Agilent E8257D) is mixed with a 200 MHz microwave signal which is generated from a arbitrary waveform generator (AWG) and is used to modulate the microwave. After a high pass filter (HPF), this signal will be amplified to about 10 W and then transmitted to the ion with a microwave horn (Cui _et al._ , 2016). Our trap device is shielded with a 1.5 mm thick single layer Mu-metal (Farolfi _et al._ , 2019), making the final coherence time about 200 ms for $\left\|0\right\rangle\leftrightarrow\left\|1\right\rangle$ transition, which is characterized by Ramsey experiments. In each cycle, the experiment takes the following process: after 1 ms Doppler cooling, the state of the ion is initialized to $\left\|0\right\rangle$ state through 20 $\mu$s optical pumping with $99.5\%$ fidelity. The wave-form output from DRL model is transformed into driving microwave through modulating the detuning, which is shown in fig. 1(e). Then the driving microwave is transmitted to the ion to drive the TLS. finally, a NA (numerical aperture) = 0.4 objective is used for state dependent fluorescence detection to determine the probability in state $\left\|1\right\rangle$. In all of our experiments we set the Rabi frequency to $\Omega=(2\pi)$ 3.3 kHz, that is to say, the corresponding $2\pi$ time is about 300 $\mu$s. Figure 2: (color online) Noise robustness comparison of $\pi$ pulse, STA and DRL methods in single-qubit X gate task. (a) and (b) The performance of three control methods under different $\Omega$ and $\Delta$ errors respectively. The DRL method is as robust as STA method in most cases, except in big $\Omega$ and $\Delta$ errors. But they are all more robust than $\pi$ pulse in both kinds of errors. (c) and (d) The performance of feedback DRL. The feedback DRL agrees well with theoretical DRL in both $\Omega$ and $\Delta$ errors, which indicates our DRL model is robust to the disturbance of control pulses. The error bars indicate the standard deviation, and each data point is averaged over 2000 realizations. To verify the robustness of the DRL control method against systematic errors, we compare the performance of STA, DRL, and standard $\pi$ pulse method in the single-qubit X gate task under different $\Delta$ and $\Omega$ errors. The DRL models are pre-trained according to the time preliminary information provided by STA methods optimized in $\Delta$ and $\Omega$ errors, respectively. The state evolution under STA driving in Bloch sphere is shown in fig. 1(d). As shown in fig. 2 (a) and 2 (b), the DRL method performs as well as STA in most cases, in addition to the case that $\Omega$ error or $\Delta$ error is too large. Meanwhile, they are all more robust than the $\pi$ pulse method under system errors. To further explore our DRL model’s robustness, we also perform a feedback DRL experiment. In this experiment, 19 cycles are carried out. In cycle $n$, where $n$ belongs to [1,19], we measure the experimental result after $n$ control pulses, and feedback this result to the DRL model to obtain the next control pulses. After 19 cycles, we get the final 20 control pulses, and these pulses are only a little different from theoretical DRL pulses. As shown in fig. 2 (c) and 2 (d), the experimental feedback DRL results agree well with theoretical DRL, which means that our DRL model is robust to the disturbance of control pulses. Figure 3: (color online) The performance of $\pi$ pulse and DRL model under hybrid errors. The $\pi$ pulse method performs a little better than DRL in the case of almost no errors, because the pulses of DRL are more complex that it is easy to accumulate operation errors. However, with the increase of hybrid errors, the performance of DRL model is much better than $\pi$ pulse control. Then we examine the DRL model under $\Delta$ and $\Omega$ hybrid errors. It is worthwhile to mention that we set operation time and tunable range of detuning without any knowledge from STA when pre-training the DRL model locally. The performance of $\pi$ pulse and DRL method under hybrid errors is shown in fig. 3, in which the probabilities are taken logarithm to better distinguish the difference between these two methods The DRL method is more likely to accumulate errors than $\pi$ pulse due to the multi pulses driving operation on the one hand, on the other hand we just stop our training once $\langle\sigma_{z}\rangle>0.997$, which can be further improved theoretically. As we can expect, the $\pi$ pulse method performs a little better than DRL in the case of almost no errors. Nevertheless, with the increase of hybrid errors, the DRL method’s performance is much better than $\pi$ pulse in most cases, which is essential in precise quantum manipulation. Besides, we also examine the DRL model in the Zeeman energy level of the ion with $\left\|1\right\rangle_{\textrm{z}}=\left\|{}^{2}{\rm S}_{1/2},F=1,m_{F}=1\right\rangle$. The Zeeman energy level is first-order sensitive to the disturbance of the magnetic field, which could induce the realistic laboratory noise into TLS, and the corresponding coherence time is about 0.35 ms for $\left\|0\right\rangle\leftrightarrow\left\|1\right\rangle_{\textrm{z}}$ transition. The experimental results demonstrate that DRL’s performance is a little worse than theoretical expectation both in $\Omega$ and $\Delta$ errors due to extra decoherence, which is shown in fig. 4 (a) and 4 (b). We also compare the performance of $\pi$ pulse and DRL method in the single-qubit X gate under only the magnetic field noise with different Rabi time and different number of $\pi$ flips. As shown in fig. 4 (c) and 4 (d), the final probability decreases rapidly with the Rabi time and number of $\pi$ flips in $\pi$ pulse method owing to inevitable decoherence. However, the DRL method is more robust with the increase of Rabi time and number of $\pi$ flips, which is important in noisy quantum information processing. Figure 4: (color online) Noise-resilient feature of DRL method in Zeeman energy level. (a) and (b) The performance of DRL in Zeeman energy level. The DRL performs a little worse than theoretical values both in $\Omega$ and $\Delta$ errors due to disturbance of laboratory noise. (c) The comparison of $\pi$ pulse and DRL under different Rabi time. With the increase of Rabi time, the performance of $\pi$ pulse decreases rapidly while the DRL is more robust against decoherence. (d) The comparison of $\pi$ pulse and DRL under different number of $\pi$ flips. With the increase of $\pi$ flips, the performance of $\pi$ pulse decreases rapidly while the DRL is more robust against decoherence. The error bars indicate the standard deviation, and each data point is averaged over 2000 realizations ## IV CONCLUSION In summary, we experimentally demonstrate a robust quantum control task based on deep reinforcement learning. The DRL model’s multi-pulse control sequences are more robust than $\pi$ pulse in the presence of systematic errors. We also verify that robustness against both Rabi and detuning errors simultaneously can be achieved by DRL without any input from STA. In addition, we confirm that these DRL models can be significant in the real laboratory environment, which will lead to a promising enhancement in quantum information processing. ###### Acknowledgements. This work was supported by the National Key Research and Development Program of China (Nos. 2017YFA0304100, 2016YFA0302700), the National Natural Science Foundation of China (Nos. 11874343, 61327901, 11774335, 11474270, 11734015, 11874343), Key Research Program of Frontier Sciences, CAS (No. QYZDY-SSW- SLH003), the Fundamental Research Funds for the Central Universities (Nos. WK2470000026, WK2470000018), An-hui Initiative in Quantum Information Technologies (AHY020100, AHY070000), the National Program for Support of Topnotch Young Professionals (Grant No. BB2470000005). The theoretical part of the work is also partially supported from NSFC (12075145), STCSM (2019SHZDZX01-ZX04, 18010500400 and 18ZR1415500), Program for Eastern Scholar, HiQ funding for developing STA (YBN2019115204), QMiCS (820505) and OpenSuperQ (820363) of the EU Flagship on Quantum Technologies, Spanish Government PGC2018-095113-B-I00 (MCIU/AEI/FEDER, UE), Basque Government IT986-16, EU FET Open Grant Quromorphic (828826) as well as EPIQUS (899368). X. C. acknowledges Ramón y Cajal program (RYC-2017-22482). J. C. acknowledges the Ramón y Cajal program (RYC2018-025197-I) and the EUR2020-112117 project of the Spanish MICINN, as well as support from the UPV/EHU through the grant EHUrOPE. ## References * Nielsen and Chuang (2010) M. A. Nielsen and I. Chuang, _Quantum computation and quantum information_ (Cambridge University Press, 2010). * Král _et al._ (2007) P. Král, I. Thanopulos, and M. Shapiro, Reviews of modern physics 79, 53 (2007). * Remizov _et al._ (2015) S. V. Remizov, D. S. 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The size of the needle trap depends mainly on the distance between the two needles tips near the trap center, which is set to 180 $\mu$m in our experiment. The trap is installed in an ultrahigh vacuum below $10^{-11}$ torr, and a helical resonator provides the RF signal with frequency 24 MHz and amplitude of 180 V to the trap. Ion fluorescence is collected by an objective lens with 0.4 numerical aperture, and detected by a photo-multiplier tube (PMT). The total fluorescence detection efficiency is about 2$\%$. We generate required waveform of the microwave field through setting the waveform of AWG for modulation. The carrier microwave $B_{c}(t)=A_{1}\mathrm{sin}(\omega_{c}t)$, where $A_{1}$ is amplitude and $f_{c}=\omega_{c}/2\pi=12.4$ GHz is the frequency. The waveform generated by AWG for modulation is $I(t)=A_{2}\mathrm{sin}(\phi(t))$. After mixing, the microwave field will be $B(t)=\frac{A_{1}A_{2}}{2}(\mathrm{sin}(\omega_{c}t+\phi(t))+\mathrm{sin}(\omega_{c}t-\phi(t)))$, where the phase function $\phi(t)$ can be expressed in a piece-wise function for the microwave composed of 20 steps in our DRL experiments. With the qubit resonance frequency $f_{0}=\omega_{0}/2\pi=12.6$ GHz, we filter out the low frequency components of the microwave through a high pass filter. In our experiments, we only adjust $\Delta(t)$ with discrete steps by changing phase $\phi(t)$ as follows: $\phi(t)=\begin{cases}(\omega_{0}-\omega_{c})t+\Delta_{1}t,&(0,t_{1})\\\ (\omega_{0}-\omega_{c})t+\Delta_{2}t+\phi_{1},&(0,t_{2}-t_{1})\\\ (\omega_{0}-\omega_{c})t+\Delta_{3}t+\phi_{2},&(0,t_{3}-t_{2})\\\ \cdots\\\ (\omega_{0}-\omega_{c})t+\Delta_{20}t+\phi_{19},&(0,t_{20}-t_{19})\end{cases}$ (1) where $\Delta_{n}(n\in[1,20])$ is the step-wise detuning and $\phi_{1}=(\omega_{0}-\omega_{c})t_{1}+\Delta_{1}t_{1}$, $\phi_{2}=(\omega_{0}-\omega_{c})(t_{2}-t_{1})+\Delta_{2}(t_{2}-t_{1})+\phi_{1}$ and so on. Figure S1: (color online) Experimental setup. A single ${}^{171}\rm{Yb}^{+}$ ion is trapped in center of the needle trap. Two 369 nm and 935 nm lasers are used to cooling the ion and 369 nm laser is also used to detect the state of ion. The microwave used to drive the ion is generated through mixing method. The whole experimental sequences are controlled by a TTL sequences board based on Field Programmable Gate Array (FPGA). ## II QUBIT STATE PREPARATION AND MEASUREMENT In ion trap experiments, the state preparation and measurement cannot be perfect and there will always be some limitations. We characterize these errors as follows. The ion is prepared in $\|0\rangle$ state through optical pumping and ideally, no photon should be detected as the ion is in dark state. However, due to the dark counts of the photon detector as well as photons scattered from the environment, we will collect some photons sometimes. Then we apply a $\pi$ pulse to flip the $\|0\rangle$ state to $\|1\rangle$ state and detect the fluorescence. Because the collection efficiency problem, no photon will be collected sometimes. The histograms of dark and bright state is shown in Fig. S2. The threshold is selected as 2 in our experiments. When the photon number is $>2$, the qubit is identified as bright state and the probability of being mistaken as dark state is $\epsilon_{D}$. By contrary, the probability of being mistaken as bright state when photon number is $\leq 2$ is $\epsilon_{B}$. The total error can be taken as $\epsilon=(\epsilon_{B}+\epsilon_{D})/2$. Figure S2: (color online) Histogram for photon counts in state preparation and detection experiments. The distribution of photon counts is shown when the qubit state is prepared in $\|0\rangle$ (dark state) and $\|1\rangle$ (bright state). ## III IMPLEMENTATION OF DEEP REINFORCEMENT LEARNING By combining reinforcement learning and deep learning, deep reinforcement learning (DRL) aims to solve decision-making problems, allowing a computational agent to make decisions from input data by trial. A mathematical model called Markov decision process describes the problem, where an agent at every time step $t$ observes a state $s_{t}$, takes an action $a_{t}$, receives a reward $r_{t}$ and transits to the state at the next time step $s_{t+1}$ according to the dynamics of the environment $P(s_{t+1}\|s_{t},a_{t})$. The agent’s goal is to learn a policy $\pi(a\|s)$ that maximizes the total reward $\sum_{i}\gamma^{i}r_{i}$, with $\gamma$ being the discount rate and $r_{i}$ being the scalar rewards. DRL employs an artificial neural network (ANN) as a general function approximator for the policy $\pi(a\|s)$, leading to specialized algorithms for obtaining an optimal approximation. The simplified model framework can be found in Fig. S3. Figure S3: (color online) DRL framework for quantum control with qubit of one time step in training. The agent (DNN), consists of three hidden layers, observes a state from environment. After propagation between layers and nonlinear activations of the DNN nodes, output layer gives an action $a_{t}$. The environment rewards $r_{t}$ enable the agent to learn how to achieve the goal. Here we use Proximal Policy Optimization (PPO), which performs comparably state-of-the-art, remaining simplicity for implementation and tuning. It is worthwhile to mention that it is also the default RL algorithm at OpenAI. As an on-policy algorithm, PPO attempts to evaluate and improve the behavior policy that is used to make decisions. Its objective function is $L_{\text{clip}}(\theta)=\hat{E}_{t}\left\\{\min\left[r_{t}(\theta)\hat{A}_{t},\text{clip}\left(r_{t}(\theta),1+\epsilon,1-\epsilon\right)\hat{A}_{t}\right]\right\\},$ (2) , where $\theta$ is the policy parameter (the set that contains all weights and biases), $\hat{E}_{t}$ is the expectation over time steps, $r_{t}$ is the ratio of the probability under the new and old policies, $\hat{A}_{t}$ is the estimated advantage, and $\epsilon$ is the hyperparameter for bounding the clipping range. There is also a variant of PPO based on an adaptive Kullback- Leiber penalty, which controls the change of $\pi(a\|s)$ at each iteration. A detailed explanation of PPO, as well as its pseudocodes, are already clearly presented in the original paper Schulman _et al._ (2017). Although there are arguments about the origin of performance enhancement from Trust Region Policy Optimization Schulman _et al._ (2015) (whether it is from the clipping or code-level tricks Engstrom _et al._ (2019)), we reckon these topics, including if PPO-like algorithms can be further optimized, go beyond the scope of this work. Thus, we implement a minimal PPO for our quantum control task. We use an open-source Python library, TensorForce (version 0.5.2) Schaarschmidt _et al._ (2017), for a quick implementation. The library is based on TensorFlow, a well-known framework for deep learning with GPU acceleration. The two-level system’s quantum dynamics in our training environment are numerically simulated by QuTiP (version 4.4.1) Johansson _et al._ (2012). We set a batch size of 16, and the learning rate is 1e-4 for both pre-training and fine-tuning. The ANN contains three hidden layers, where each of the layers consists of 32 fully-connected neurons activated by ReLU. Other hyperparameters and settings are the default configuration of the PPO Agent provided by TensorForce. Another evaluation environment can interact with the trapped ion system for verifying quantum control with feedback. Codes are compatible with both CPU and GPU versions of TensorFlow 1.13.1., which are available from the corresponding authors upon reasonable request. ## References * Schulman _et al._ (2017) J. Schulman, F. Wolski, P. Dhariwal, A. Radford, and O. Klimov, arXiv preprint arXiv:1707.06347 (2017). * Schulman _et al._ (2015) J. Schulman, S. Levine, P. Abbeel, M. Jordan, and P. Moritz, in _International conference on machine learning_ (2015) pp. 1889–1897. * Engstrom _et al._ (2019) L. Engstrom, A. Ilyas, S. Santurkar, D. Tsipras, F. Janoos, L. Rudolph, and A. Madry, in _International Conference on Learning Representations_ (2019). * Schaarschmidt _et al._ (2017) M. Schaarschmidt, A. Kuhnle, and K. Fricke, https:// github.com/tensorforce/tensorforce (2017). * Johansson _et al._ (2012) J. R. Johansson, P. D. Nation, and F. Nori, Computer Physics Communications 183, 1760 (2012).
# Online Packing to Minimize Area or Perimeter Mikkel Abrahamsen Lorenzo Beretta∗ Basic Algorithms Research Copenhagen (BARC), University of Copenhagen. BARC is supported by the VILLUM Foundation grant 16582. Lorenzo Beretta received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No. 801199. (January 21, 2021) ###### Abstract We consider online packing problems where we get a stream of axis-parallel rectangles. The rectangles have to be placed in the plane without overlapping, and each rectangle must be placed without knowing the subsequent rectangles. The goal is to minimize the perimeter or the area of the axis-parallel bounding box of the rectangles. We either allow rotations by $90^{\circ}$ or translations only. For the perimeter version we give algorithms with an absolute competitive ratio slightly less than $4$ when only translations are allowed and when rotations are also allowed. We then turn our attention to minimizing the area and show that the asymptotic competitive ratio of any algorithm is at least $\Omega(\sqrt{n})$, where $n$ is the number of rectangles in the stream, and this holds with and without rotations. We then present algorithms that match this bound in both cases and the competitive ratio is thus optimal to within a constant factor. We also show that the competitive ratio cannot be bounded as a function of Opt. We then consider two special cases. The first is when all the given rectangles have aspect ratios bounded by some constant. The particular variant where all the rectangles are squares and we want to minimize the area of the bounding square has been studied before and an algorithm with a competitive ratio of $8$ has been given [Fekete and Hoffmann, Algorithmica, 2017]. We improve the analysis of the algorithm and show that the ratio is at most $6$, which is tight. The second special case is when all edges have length at least $1$. Here, the $\Omega(\sqrt{n})$ lower bound still holds, and we turn our attention to lower bounds depending on Opt. We show that any algorithm for the translational case has an asymptotic competitive ratio of at least $\Omega(\sqrt{\textsc{Opt}})$. If rotations are allowed, we show a lower bound of $\Omega(\sqrt[4]{\textsc{Opt}})$. For both versions, we give algorithms that match the respective lower bounds: With translations only, this is just the algorithm from the general case with competitive ratio $O(\sqrt{n})=O(\sqrt{\textsc{Opt}})$. If rotations are allowed, we give an algorithm with competitive ratio $O(\min\\{\sqrt{n},\sqrt[4]{\textsc{Opt}}\\})$, thus matching both lower bounds simultaneously. ## 1 Introduction Problems related to packing appear in a plethora of big industries. For instance, two-dimensional versions of packing arise when a given set of pieces have to be cut out from a large piece of material so as to minimize waste. This is relevant to clothing production where cutting patterns are cut out from a roll of fabric, and similarly in leather, glass, wood, and sheet metal cutting. In some applications, it is important that the pieces are placed in an _online_ fashion. This means that the pieces arrive one by one and we need to decide the placement of one piece before we know the ones that will come in the future. This is in contrast to _offline_ problems, where all the pieces are known in advance. Problems related to packing were some of the first for which online algorithms were described and analyzed. Indeed, the first use of the terms “online” and “offline” in the context of approximation algorithms was in the early 1970s and used for algorithms for bin-packing problems [14]. In this paper, we study online packing problems where the pieces can be placed anywhere in the plane as long as they do not overlap. The goal is to minimize the region occupied by the pieces. The pieces are axis-parallel rectangles, and they may or may not be rotated by $90^{\circ}$. We want to minimize the size of the axis-parallel bounding box of the pieces, and the size of the box is either the perimeter or the area. This results in four problems: PerimeterRotation, PerimeterTranslation, AreaRotation, and AreaTranslation. #### Competitive analysis The _competitive ratio_ of an online algorithm is the equivalent of the _approximation ratio_ of an (offline) approximation algorithm. The usual definitions [7, 9, 11] of competitive ratio (or _worst case ratio_ , as it may also be called [11]) can only be used to describe that the cost of the solution produced by an online algorithm is at most some constant factor higher than the cost Opt of the optimal (offline) solution. In the study of approximation algorithms, it is often the case that the approximation ratio is described not just as a constant, but as a more general function of the input. In the same way, we generalize the definition of competitive ratios to support such statements about online algorithms. Consider an algorithm $A$ for one of the packing problems studied in this paper. Let $\mathcal{L}$ be the set of non-empty streams of rectangular pieces. For a stream $L\in\mathcal{L}$, we define $A(L)$ to be the cost of the packing produced by $A$ and let $\textsc{Opt}(L)$ be the cost of the optimal (offline) packing. We say that $A$ has an _absolute competitive ratio_ of $f(L)$, for some function $f:\mathcal{L}\longrightarrow\mathbb{R}^{+}$ which may just be a constant, if $\sup_{L\in\mathcal{L}}\frac{A(L)}{\textsc{Opt}(L)f(L)}\leq 1.$ We say that $A$ has an _asymptotic competitive ratio_ of $f(L)$ if $\limsup_{c\longrightarrow\infty}\left(\sup\left\\{\frac{A(L)}{\textsc{Opt}(L)f(L)}\mid L\in\mathcal{L}\text{ and }\textsc{Opt}(L)=c\right\\}\right)\leq 1.$ In this paper, the functions $f(L)$ that we consider will be (i) constants, (ii) functions of the number of pieces $n=\|L\|$, (iii) functions of $\textsc{Opt}(L)$. By definition, if $A$ has an absolute competitive ratio of $f(L)$, then $A$ also has an asymptotic competitive ratio of $f(L)$, but $A$ may also have a smaller asymptotic competitive ratio $g(L)<f(L)$. However, the following easy lemma shows that for the problems studied in this paper, any constant asymptotic competitive ratio can be matched to within an arbitrarily small difference by an absolute competitive ratio. ###### Lemma 1. For the problems studied in this paper, if an algorithm $A$ has an asymptotic competitive ratio of some constant $c>1$, then for every $\varepsilon>0$, there is an algorithm $A^{\prime}$ with absolute competitive ratio $c+\varepsilon$. It follows that any constant lower bound on the absolute competitive ratio is also a lower bound on the asymptotic competitive ratio. ###### Proof. Let $n>0$ be so large that when $\textsc{Opt}(L)\geq n$, we have $\frac{A(L)}{c\textsc{Opt}(L)}\leq 1+\varepsilon/c$. When the first piece $p$ of a stream $L$ is given, $A^{\prime}$ chooses a scale factor $\lambda>0$ big enough that when $p$ is scaled up by $\lambda$, the resulting piece $p^{\prime}:=\lambda p$ alone has cost $n$ (i.e., the area or the perimeter of $p^{\prime}$ is $n$). The algorithm $A^{\prime}$ now imitates the strategy of $A$ on the stream $\lambda L$ we get by scaling up all pieces of $L$ by $\lambda$. We then get that $\frac{A^{\prime}(L)}{(c+\varepsilon)\textsc{Opt}(L)}=\frac{A(\lambda L)}{(c+\varepsilon)\textsc{Opt}(\lambda L)}\leq\frac{(1+\varepsilon/c)c}{c+\varepsilon}=1.\qed$ For this reason, we do not distinguish between absolute and asymptotic competitive ratios when the ratio is a constant. Note that the argument does not work when the competitive ratio is a non-constant function of Opt. #### Results and structure of the paper We develop online algorithms for the perimeter versions PerimeterRotation and PerimeterTranslation, both with a competitive ratio slightly less than $4$. These algorithms are described in Section 2. The idea is to partition the positive quadrant into _bricks_ , which are axis-parallel rectangles with aspect ratio $\sqrt{2}$. In each brick, we build a stack of pieces which would be too large to place in a brick of smaller size. Online packing algorithms using higher-dimensional bricks were described by Januszewski and Lassak [15] and our algorithms are inspired by an algorithm of Fekete and Hoffmann [13] that we will get back to. Interestingly, we show in Section 2.2 that a more direct adaptation of the algorithm of Fekete and Hoffmann has a competitive ratio of at least $4$, and is thus inferior to the algorithm we describe. We also give a lower bound of $4/3$ for the version with translations and $5/4$ for the version with rotations. In Section 3, we study the area versions AreaRotation and AreaTranslation. We show in Section 3.1 that for any algorithm $A$ processing a stream of $n$ pieces cannot achieve a better competitive ratio than $\Omega(\sqrt{n})$, and this holds for all online algorithms and with and without rotations allowed. It also holds in the special case where all the edges of pieces have length at least $1$. We furthermore show that when the pieces can be arbitrary, there can be given no bound on the competitive ratio as a function of Opt for AreaRotation nor AreaTranslation. In Section 3.2 we describe the algorithms DynBoxTrans and DynBoxRot, which achieve a $O(\sqrt{n})$ competitive ratio for AreaTranslation and AreaRotation, respectively, for an arbitrary stream of $n$ pieces. This is thus optimal up to a constant factor when measuring the competitive ratio as a function of $n$. Both algorithms use a row of boxes of exponentially increasing width and dynamically adjusted height. In these boxes, we pack pieces using a next-fit shelf algorithm, which is a classic online strip packing algorithm first described by Baker and Schwartz [6]. We then turn our attention to two special cases. The first special case is when the aspect ratio is bounded by a constant $\alpha\geq 1$. A case of particular interest is when all pieces are squares, i.e., $\alpha=1$. It is natural to have the same requirement to the container as to the pieces, so let us assume that the goal is to minimize the area of the axis-parallel bounding square of the pieces, and call the problem SquareInSquareArea. This problem was studied by Fekete and Hoffmann [13], and they gave an algorithm for the problem and proved that it was $8$-competitive. We prove that the same algorithm is in fact $6$-competitive and that this is tight. It easily follows that if the aspect ratio is bounded by an arbitrary constant $\alpha\geq 1$ or if the goal is to minimize the area of the axis- parallel bounding rectangle, we also get a $O(1)$-competitive algorithm. The second special case is when all edges are _long_ , that is, when they have length at least $1$ (any other constant will work too). In Section 3.4, we show that under this assumption, there is a lower bound of $\Omega(\sqrt{\textsc{Opt}})$ for the asymptotic competitive ratio of AreaTranslation, whereas for AreaRotation, we get the lower bound $\Omega(\sqrt[4]{\textsc{Opt}})$. In Section 3.5, we provide algorithms for the area versions when the edges are long. For both problems AreaRotation and AreaTranslation, we give algorithms that match the lower bounds of Section 3.4 to within a constant factor. With translations only, this is just the algorithm from the general case with competitive ratio $O(\sqrt{n})=O(\sqrt{\textsc{Opt}})$. The algorithm with ratio $O(\sqrt[4]{\textsc{Opt}})$ for the rotational case follows the same scheme as the algorithms for arbitrary rectangles of Section 3.2, but differ in the way we dynamically increase boxes’ heights. We finally describe an algorithm for the rotational case with competitive ratio $O(\min\\{\sqrt{n},\sqrt[4]{\textsc{Opt}}\\})$, thus matching the lower bounds $\Omega(\sqrt{n})$ and $\Omega(\sqrt[4]{\textsc{Opt}})$ simultaneously. Actually, the two lower bounds for AreaRotation can be summarized by $\Omega(\max\\{\sqrt{n},\sqrt[4]{\textsc{Opt}}\\})$, while we manage to achieve a competitive ratio of $O(\min\\{\sqrt{n},\sqrt[4]{\textsc{Opt}}\\})$. However, this gives no contradiction, it simply proves that the _edge cases_ that have a competitive ratio of at least $\Omega(\sqrt[4]{\textsc{Opt}})$ must satisfy $\textsc{Opt}=O(n^{2})$, and those for which the competitive ratio is at least $\Omega(\sqrt{n})$ satisfy $n=O(\sqrt{\textsc{Opt}})$. We summarize the results in Table 1. Measure \| Version \| Trans./Rot. \| Lower bound \| Upper bound ---\|---\|---\|---\|--- Perimeter \| General \| Translation \| $4/3$, Sec. 2.3 \| $4-\varepsilon$, Sec. 2.1 Rotation \| $5/4$, Sec. 2.3 \| $4-\varepsilon$, Sec. 2.1 Area \| General \| Translation \| $\Omega(\sqrt{n})$ & $\forall f:\Omega(f(\textsc{Opt}))$, Sec. 3.1 \| $O(\sqrt{n})$, Sec. 3.2 Rotation \| $\Omega(\sqrt{n})$ & $\forall f:\Omega(f(\textsc{Opt}))$, Sec. 3.1 \| $O(\sqrt{n})$, Sec. 3.2 Sq.-in-sq. \| N/A \| 16/9, Sec. 3.3 \| $6$, Sec. 3.3 Long edges \| Translation \| $\Omega(\sqrt{\textsc{Opt}})$, Sec. 3.4 \| $O(\sqrt{n})=O(\sqrt{\textsc{Opt}})$, Sec. 3.5 Rotation \| $\Omega(\max\\{\sqrt{n},\sqrt[4]{\textsc{Opt}}\\})$, Sec. 3.1 and 3.4 \| $O(\min\\{\sqrt{n},\sqrt[4]{\textsc{Opt}}\\})$, Sec. 3.5 Table 1: Results of this paper. #### Related work The literature on online packing problems is rich. See the surveys of Christensen, Khan, Pokutta, and Tetali [9], van Stee [25, 26], and Csirik and Woeginger [11] for an overview. It seems that the vast majority of previous work on online versions of two-dimensional packing problems is concerned with either bin packing (packing the pieces into a minimum number of unit squares) or strip packing (packing the pieces into a strip of unit width so as to minimize the total height of the pieces). From a mathematical point of view, we find the problems studied in this paper perhaps even more fundamental than these important problems in the sense that we give no restrictions on where to place the pieces, whereas the pieces are restricted by the boundaries of the bins and the strip in bin and strip packing. Another related problem is to find the critical density of online packing squares into a square. In other words, what is the maximum $\Sigma\leq 1$ such that there is an online algorithm that packs any stream of squares of total area at most $\Sigma$ into the unit square? This was studied, among others, by Fekete and Hoffmann [13] and Brubach [8]. Lassak [16] and Januszewski and Lassak [15] studied higher-dimensional versions of this problem. Milenkovich [20] studied generalized offline versions of the minimum area problem: Translate $k$ given $m$-gons into a convex container of minimum area with edges in $n$ fixed directions. When the $m$-gons can be non-convex, the running time is $O((m^{2}+n)^{2k-2}(n+\log m))$, and when they are convex, the running times are $O((m+n)^{2k}(n+\log m))$ or $O(m^{k-1}(n^{2k+1}+\log m))$. Milenkovich and Daniels [22] described different algorithms for the same problems. Milenkovich [21] also studied the same problem when arbitrary rotations are allowed and the container is either a strip with a fixed width, a homothet of a given convex polygon, or an arbitrary rectangle (as in our work). He gave $(1+\varepsilon)$-approximation algorithms (no explicit running times are given, but they are apparently also exponential). Some algorithms have been described for computing the packing of two or three convex polygons that minimizes the perimeter or area of the convex hull or the bounding box [1, 5, 17, 23]. Alt [2] demonstrated how a $\rho$-approximation algorithm for strip packing (axis-parallel rectangles with translations) can be turned into a $(1+\varepsilon)\rho$-approximation algorithm for the offline version of AreaTranslation, for any constant $\varepsilon>0$. The same technique works for AreaRotation. The idea is to apply the strip packing algorithm to strips of increasing widths and in the end choose the packing that resulted in the smallest area. Therefore, the same technique cannot be applied in the online setting, where we need to choose a placement for each piece and stick with it. Alt also mentioned that finding a minimum area bounding box of a set of convex polygons with arbitrary rotations allowed can be reduced to the problem where the pieces are axis-parallel rectangles with only translations allowed. This reduction increases the approximation ratio by a factor by $2$. The reduction does not work when the pieces can be only translated, but Alt, de Berg, and Knauer [4] gave a $17.45$-approximation algorithm for this problem using different techniques. Lubachevsky and Graham [18] used computational experiments to find the rectangles of minimum area into which a given number $n\leq 5000$ of congruent circles can be packed; see also the follow-up work by Specht [24]. In another paper, Lubachevsky and Graham [19] studied the problem of minimizing the perimeter instead of the area. Another fundamental packing problem is to find the smallest square containing a given number of _unit_ squares, with arbitrary rotations allowed. A long line of mathematical research has been devoted to this problem, initiated by Erdős and Graham [12] in 1975, and it is still an active research area [10]. ## 2 The perimeter versions In Section 2.1, we present two online algorithms to minimize the perimeter of the bounding box: the algorithm BrickTranslation solves the problem PerimeterTranslation, where we can only translate pieces; the algorithm BrickRotation solves the problem PerimeterRotation, where also rotations are allowed. Both algorithms achieve a competitive ratio of $4$. In Section 2.3, we show a lower bound of $4/3$ for the version with translations and $5/4$ for the version with rotations. ### 2.1 Algorithms to minimize perimeter #### Algorithm for translations We pack the pieces into non-overlapping _bricks_ ; a technique first described by Januszewski and Lassak [15] which was also used by Fekete and Hoffmann [13] for the problem SquareInSquareArea. Let a _$k$ -brick_ be a rectangle of size $\sqrt{2}^{-k}\times\sqrt{2}^{-k-1}$ if $k$ is even and $\sqrt{2}^{-k-1}\times\sqrt{2}^{-k}$ if $k$ is odd. A _brick_ is a $k$-brick for some integer $k$. We tile the positive quadrant using one $k$-brick $B_{k}$ for each integer $k$ as in Figure 1 (left): if $k$ is even, $B_{k}$ is the $k$-brick with lower left corner $(0,\sqrt{2}^{-k-1})$ and otherwise, $B_{k}$ is the $k$-brick with lower left corner $(\sqrt{2}^{-k-1},0)$. The bricks $B_{k}$ are called the _fundamental_ bricks. We define $B_{>k}:=\bigcup_{i>k}B_{i}$ and $B_{\geq k}:=B_{>k-1}$, so that $B_{>k}$ is the $k$-brick immediately below (if $k$ is even) or to the left (if $k$ is odd) of $B_{k}$. An important property of a $k$-brick $B$ is that it can be split into two $(k+1)$-bricks: $B\dagger 1$ and $B\dagger 2$; see Figure 1 (middle). We introduce a uniform naming and define $B\dagger 1$ to be the left half of $B$ if $k$ is even and the lower half of $B$ if $k$ is odd. We define a _derived_ brick recursively as follows: a derived brick is either (i) a fundamental brick $B_{k}$ or (ii) $B\dagger 1$ or $B\dagger 2$, where $B$ is a derived brick. We introduce an ordering $\prec$ of the derived $k$-bricks as follows. Consider two derived $k$-bricks $D_{1}$ and $D_{2}$ such that $D_{1}\subset B_{i}$ and $D_{2}\subset B_{j}$. If $i>j$, then $D_{1}\prec D_{2}$. Else, if $i=j$ then the bricks $D_{1}$ and $D_{2}$ are both obtained by splitting the fundamental brick $B_{i}$, and the number of splits is $\ell:=i-k$. Hence the bricks have the forms $D_{1}=B_{i}\dagger b_{11}\dagger b_{12}\dagger\ldots\dagger b_{1\ell}$ and $D_{2}=B_{i}\dagger b_{21}\,b_{22}\,\ldots\,b_{2\ell}$, where $b_{ij}\in\\{1,2\\}$ for $i\in\\{1,2\\}$ and $j\in\\{1,\ldots,\ell\\}$. We then define $D_{1}\prec D_{2}$ if $(b_{11},b_{12},\ldots,b_{1\ell})$ precedes $(b_{21},b_{22},\ldots,b_{2\ell})$ in the lexicographic ordering. We say that a $k$-brick is _suitable_ for a piece $p$ of size $w\times h$ if the width and height of the brick are at least $w$ and $h$, respectively, and if that is not the case for a $(k+1)$-brick. We will always pack a given piece $p$ in a derived $k$-brick that is suitable for $p$. Figure 1: Left: Fundamental bricks. Middle: Splitting a brick. Right: Rectangular pieces packed in a brick. We now explain how we pack pieces into one specific brick; see Figure 1 (right). The first piece $p$ that is packed in a brick $B$ is placed with the lower left corner of $p$ at the lower left corner of $B$. Suppose now that some other pieces $p_{1},\ldots,p_{i}$ have been packed in $B$. If $k$ is even, then $p_{1},\ldots,p_{i}$ form a stack with the left edges contained in the left edge of $B$, and we place $p$ on top of $p_{i}$ (again, with the left edge of $p$ contained in the left edge of $B$). Otherwise, $p_{1},\ldots,p_{i}$ form a stack with the bottom edges contained in the bottom edge of $B$, and we place $p$ to the right of $p_{i}$ (again, with the bottom edge of $p$ contained in the bottom edge of $B$). We say that a brick _has room_ for a piece $p$ if the packing scheme above places $p$ within $B$, and it is apparent that an empty suitable brick for $p$ has room for $p$. Figure 2: Left: Some pieces have been packed by the algorithm. The bricks in $\mathcal{D}$ are drawn with fat edges. Right: A new piece arrives. There is already a brick of the suitable size in $\mathcal{D}$, but there is not enough room, so a new brick of the same size is added to $\mathcal{D}$ where the piece is placed. The algorithm BrickTranslation maintains the collection $\mathcal{D}$ of non- overlapping derived bricks, such that one or more pieces have been placed in each brick in $\mathcal{D}$; see Figure 2. Before the first piece arrives, we set $\mathcal{D}:=\emptyset$. Suppose that some stream of pieces have been packed, and that a new piece $p$ appears. Choose $k$ such that a $k$-brick is suitable for $p$. If there exists a derived $k$-brick $D\in\mathcal{D}$ such that $D$ has room for $p$, then we pack $p$ in $D$. Else, let $D$ be the minimum derived $k$-brick (with respect to the ordering $\prec$ described before) such that $D$ is interior-disjoint from each brick in $\mathcal{D}$; we then add $D$ to $\mathcal{D}$ and pack $p$ in $D$. ###### Theorem 2. The algorithm BrickTranslation has a competitive ratio strictly less than 4 for PerimeterTranslation. ###### Proof. We can assume, without loss of generality, that after we have packed the last rectangle, we have $\bigcup\mathcal{D}\subseteq B_{\geq 0}$ and $\bigcup\mathcal{D}\not\subseteq B_{\geq 1}$. As shown in Figure 3, we define a derived $k$-brick $B\subseteq B_{\geq 0}$ to be * • _sparse_ if $B\in\mathcal{D}$ and the total height (if $k$ is even, else width) of pieces stacked in $B$ is less than half of the height (if $k$ is even, else width) of $B$, * • _dense_ if $B\in\mathcal{D}$ and $B$ it is not sparse, * • _free_ if $B$ is interior-disjoint from each brick in $\mathcal{D}$, and * • _empty_ if $B$ is a maximal (w.r.t. inclusion) free brick. Figure 3: Brick $B_{3}$ is _sparse_ , brick $B_{2}\dagger 2$ is _empty_ , brick $B_{1}\dagger 1$ is _dense_. Brick $B_{1}\dagger 2\dagger 2$ is free, but not empty, since it is contained in $B_{1}\dagger 2$. ###### Remark 3. Sparse, dense and empty bricks together cover $B_{\geq 0}$, in fact every brick in $\mathcal{D}$ is either sparse or dense, and any brick in $B_{\geq 0}$ that is interior-disjoint from bricks in $\mathcal{D}$ is contained in some empty brick. ###### Remark 4. Every $k$-brick $D\in\mathcal{D}$ contains pieces for which it is suitable. Therefore, if $k$ is odd $D$ contains a piece of height at least $\sqrt{2}^{-k}/2$, and if $k$ is even $D$ contains a piece of width at least $\sqrt{2}^{-k}/2$. ###### Remark 5. Every $k$-brick $D\in\mathcal{D}$ that is dense contains pieces with total area at least $1/4$ of the area of $D$. To see this, suppose that $k$ is even, so that $D$ is $\sqrt{2}^{-k}\times\sqrt{2}^{-k-1}$, then thanks to density the total height of pieces in $D$ is at least half of its height, moreover thanks to Remark 4 all the pieces contained in $D$ have width at least $\sqrt{2}^{-k}/2$. If $k$ is odd we prove it analogously. ###### Remark 6. Consider two $k$-bricks $M$ and $N$. If $M\prec N$ and $M$ is free, then $N$ is free. To prove this it is sufficient to consider the step in which the first piece $p$ is placed within $N$ and $N\dagger b_{1}\dagger\ldots\dagger b_{\ell}$ is added to $\mathcal{D}$. Then, $N\dagger b_{1}\dagger\ldots\dagger b_{\ell}$ should be the $\prec$-minimum free suitable $k$-brick, but $M\dagger b_{1}\dagger\ldots\dagger b_{\ell}\prec N\dagger b_{1}\dagger\ldots\dagger b_{\ell}$ gives a contradiction. It follows that whenever we have a set $S$ of $k$-bricks that contains a free $k$-brick, then also $\max_{\prec}S$ is free. This turns out to be useful multiple times along the proof, choosing $S$ to be the set of $k$-bricks not contained in a strictly larger empty brick. ###### Remark 7. There exists no empty $0$-brick, otherwise $D\subseteq B_{\geq 1}$. Moreover, for every $k\geq 1$ we can have at most one sparse $k$-brick and one empty $k$-brick. In fact, a new empty (resp. sparse) $k$-brick is created only when no empty (resp. sparse) $k$-brick exists. In the following we prove an upper bound on the competitive ratio $\textsc{Alg}/\textsc{Opt}$, where Alg is the perimeter of the bounding box achieved by our online algorithm and Opt is the optimal perimeter computed offline. Hence, we need some techniques to provide an upper bound on Alg and a lower bound on Opt. For Alg, we will simply show a bounding box, in fact the perimeter of any bounding box containing all the pieces provides an upper bound to the minimum perimeter bounding box. For Opt, let $A$ be the total area of pieces and $L$ be the maximum length of an edge of a piece. If $L^{2}>A$ then the minimum perimeter bounding box cannot have a smaller perimeter than a box of size $L\times A/L$. Otherwise, if $L^{2}\leq A$ we have a weaker lower bound given by the box $\sqrt{A}\times\sqrt{A}$. Throughout the analysis we consider semiperimeters instead of perimeters to improve readability. We denote with $A(empty),A(sparse),A(dense)$ the total area of empty, sparse and dense bricks respectively. Thanks to Remark 3, we have that $A(empty)+A(sparse)+A(dense)=A(B_{\leq 0})=\sqrt{2}$. We denote with $A_{pcs}$ the total area of pieces in the stream. Thanks to Remark 5, we have $A_{pcs}\geq A(dense)/4$. From now on the proof branches in many cases and subcases. We will perform a depth-first visit of the case tree, and for each leaf of this tree we will prove that the competitive ratio is strictly less than $4$. Let $k$ be the smallest integer such that there exists a $k$-brick in $\mathcal{D}$, and let $M\in\mathcal{D}$ be the $\prec$-maximal $k$-brick. From our assumptions, it follows that $k\geq 0$. #### Case Tree Figure 4: Some of the cases listed in the proof of Theorem 2 are shown. The grey area must fit within the bounding box considered in the case analysis. Case (1) [_$M$ is a $0$-brick_] Thanks to Remark 4 every piece in $M$ has width at least $1/2$. Let $h$ be the total height of pieces stacked in $M$, then a bounding box of size size $1\times(\sqrt{2}/2+h)$ is obtained cutting the topmost part of $M$; see Figure 4. We can easily bound Opt with $1/2\times h$, and we get $\frac{\textsc{Alg}}{\textsc{Opt}}\leq\frac{1+\frac{\sqrt{2}}{2}+h}{\frac{1}{2}+h}\leq 2+\sqrt{2}<4.$ Case (2) [_$M$ is a $k$-brick for $k\geq 2$_] Here we have two cases. Case (2.1) [_There exist a $1$-brick $N_{1}$ and a $2$-brick $N_{2}$ that are empty_] Thanks to Remark 6 we can choose $N_{1}$ = $B_{0}\dagger 2$ and $N_{2}=B_{0}\dagger 1\dagger 2$. In fact, for $B_{0}\dagger 2$ it is sufficient to choose $S$ as the set of all $1$-bricks, while for $B_{0}\dagger 1\dagger 2$ we can choose $S$ to be the set of all $2$-bricks that are not contained in a larger free brick. Thus, we can cut the topmost half of $B_{0}$ and get $\textsc{Alg}\leq 1+3/4\cdot\sqrt{2}$; see Figure 4. We have $\displaystyle A(empty)\leq\sum_{i\geq 1}A(B_{i})\leq\frac{\sqrt{2}}{2}$ $\displaystyle A(sparse)\leq\sum_{i\geq 2}A(B_{i})=\frac{\sqrt{2}}{4}\quad\text{(thanks to case (2) clause there is no sparse $1$-brick)}$ $\displaystyle A_{pcs}\geq\frac{A(dense)}{4}\geq\frac{A(B_{\geq 0})-A(sparse)-A(empty)}{4}\geq\frac{\sqrt{2}}{16}$ Now we are ready to bound Opt: $\displaystyle\textsc{Opt}\geq 2\cdot\sqrt{A_{pcs}}=\sqrt{\frac{\sqrt{2}}{4}}$ $\displaystyle\frac{\textsc{Alg}}{\textsc{Opt}}\leq\frac{1+\frac{3}{4}\sqrt{2}}{\sqrt{\frac{\sqrt{2}}{4}}}\approx 3.47<4.$ Case (2.2) [_For $j=1$ or $j=2$ there does not exist an empty $j$-brick_] In this case we just use $\textsc{Alg}\leq 1+\sqrt{2}$. Then we have $\displaystyle A(empty)\leq\sum_{i\geq 1\land i\neq j}A(B_{i})\leq\frac{3}{8}\sqrt{2}\quad\text{(worst case is when $j=2$)}$ $\displaystyle A(sparse)\leq\sum_{i\geq 2}A(B_{i})=\frac{\sqrt{2}}{4}$ therefore performing the same computations of case (2.1), $A_{pcs}\geq 3/32\cdot\sqrt{2}$, and finally $\displaystyle\textsc{Opt}\geq 2\cdot\sqrt{\frac{3}{32}\sqrt{2}}=\sqrt{\frac{3}{8}\sqrt{2}}$ $\displaystyle\frac{\textsc{Alg}}{\textsc{Opt}}\leq\frac{1+\sqrt{2}}{\sqrt{\frac{3}{8}\sqrt{2}}}\approx 3.32<4.$ Case (3) [_$M$ is a $1$-brick_] For the rest of the proof $L$ will be the length of the longest edge among all pieces. Since $M$ is a $1$-brick, we have $\sqrt{2}/4<L\leq\sqrt{2}/2$. Here we have two cases. Case (3.1) [_There does not exists an empty $1$-brick_] Here we have two cases. Case (3.1.1) [_For $j=2$ and $j=3$ there exists an empty $j$-brick_] Here we have three cases. Case (3.1.1.1) [_$M$ is the fundamental brick $B_{1}$_] Thanks to Remark 6 we can assume $B_{0}\dagger 2\dagger 2$ and $B_{0}\dagger 2\dagger 1\dagger 2$ to be empty. Here we have two cases. Case (3.1.1.1.1) [_$M$ is dense_] Since $M=B_{1}$ is the $\prec$-maximal $k$-brick in $\mathcal{D}$, then there does not exist a sparse $1$-brick. $\displaystyle A(empty)\leq\sum_{i\geq 2}A(B_{i})\leq\frac{\sqrt{2}}{4}$ $\displaystyle A(sparse)\leq\sum_{i\geq 2}A(B_{i})=\frac{\sqrt{2}}{4}$ therefore $A_{pcs}\geq\frac{\sqrt{2}}{8}$, and finally $\displaystyle\textsc{Opt}\geq 2\cdot\sqrt{\frac{\sqrt{2}}{8}}=\sqrt{\frac{\sqrt{2}}{2}}$ $\displaystyle\frac{\textsc{Alg}}{\textsc{Opt}}\leq\frac{1+\sqrt{2}}{\sqrt{\frac{\sqrt{2}}{2}}}\approx 2.87<4.$ Case (3.1.1.1.2) [_$M$ is sparse_] Then, we can cut the rightmost part of $B_{\geq 0}$ and get a $3/4\times\sqrt{2}$ bounding box; see Figure 4. We have $\displaystyle A(empty)\leq\sum_{i\geq 2}A(B_{i})\leq\frac{\sqrt{2}}{4}$ $\displaystyle A(sparse)\leq\sum_{i\geq 1}A(B_{i})\leq\frac{\sqrt{2}}{2}$ hence $A_{pcs}\geq\sqrt{2}/16$. Since $L^{2}>1/8>\sqrt{2}/16$ we finally have $\displaystyle\textsc{Opt}\geq L+\frac{A_{pcs}}{L}\geq\frac{\sqrt{2}}{4}+\frac{1}{4}\quad\text{(minimizing over $L\in[\sqrt{2}/4,\sqrt{2}/2]$)}$ $\displaystyle\frac{\textsc{Alg}}{\textsc{Opt}}\leq\frac{3/4+\sqrt{2}}{\frac{\sqrt{2}}{4}+\frac{1}{4}}\approx 3.59<4.$ Case (3.1.1.2) [_$M=B_{0}\dagger 1$_] Thanks to Remark 6 we can assume $B_{0}\dagger 2\dagger 2$ to be empty. Then, we can cut the topmost part of $B_{\geq 0}$ and get a $1\times\sqrt{2}/2+L$ bounding box; see Figure 4. We have $\displaystyle A(empty)\leq\sum_{i\geq 2}A(B_{i})\leq\frac{\sqrt{2}}{4}$ $\displaystyle A(sparse)\leq\sum_{i\geq 1}A(B_{i})\leq\frac{\sqrt{2}}{2}$ hence $A_{pcs}\geq\sqrt{2}/16$. Since $L^{2}>1/8>\sqrt{2}/16$ we finally have $\displaystyle\textsc{Opt}\geq L+\frac{A_{pcs}}{L}\geq L+\frac{\sqrt{2}}{16L}$ $\displaystyle\frac{\textsc{Alg}}{\textsc{Opt}}\leq\frac{1+\sqrt{2}/2+L}{L+\frac{\sqrt{2}}{16L}}\leq 2+\sqrt{2}<4.\quad\text{(maximizing over $L\in[\sqrt{2}/4,\sqrt{2}/2]$)}$ Case (3.1.1.3) [_$M=B_{0}\dagger 2$_] This case is analogous to the previous one, in fact thanks to Remark 6 we can assume $B_{0}\dagger 1\dagger 2$ to be empty and cut the topmost part of $B_{\geq 0}$. Case (3.1.2) [_For $j=2$ or $j=3$ there does not exist an empty $j$-brick_] $\displaystyle A(empty)\leq\sum_{i\geq 2\land i\neq j}A(B_{i})\leq\frac{3}{16}\sqrt{2}\quad\text{(worst case is when $j=3$)}$ $\displaystyle A(sparse)\leq\sum_{i\geq 1}A(B_{i})\leq\frac{\sqrt{2}}{2}$ hence $A_{pcs}\geq\frac{5}{64}\cdot\sqrt{2}$. Since $L^{2}>1/8>\frac{5}{64}\cdot\sqrt{2}$ we finally have $\displaystyle\textsc{Opt}\geq L+\frac{A_{pcs}}{L}\geq\frac{\sqrt{2}}{4}+\frac{5}{16}\quad\text{(minimizing over $L\in[\sqrt{2}/4,\sqrt{2}/2]$)}$ $\displaystyle\frac{\textsc{Alg}}{\textsc{Opt}}\leq\frac{1+\sqrt{2}}{\frac{\sqrt{2}}{4}+\frac{5}{16}}\approx 3.62<4.$ Case (3.2) [_There exists an empty $1$-brick_] Thanks to Remark 6 we can assume $B_{0}\dagger 2$ to be empty. Here we have two cases. Case (3.2.1) [_$M$ is the fundamental brick $B_{1}$_] Let $w$ be the total width of pieces stacked in $M$. Since $B_{0}\dagger 2$ is empty, we can cut the rightmost part of $B_{\geq 0}$ and get a $(1/2+w)\times\sqrt{2}$ bounding box; see Figure 4. Since increasing $w$ only improves our estimates, we consider the corner case $w=0$. Now we have two cases. Case (3.2.1.1) [_There does not exist an empty $2$-brick_] Here we have two cases. Case (3.2.1.1.1) [_For $j=3$ and $j=4$ there exists an empty $j$-brick_] Thanks to Remark 6 we can assume $B_{0}\dagger 1\dagger 2\dagger 2$ and $B_{0}\dagger 1\dagger 2\dagger 1\dagger 2$ to be empty. Thus, we can cut the topmost part of $B_{\geq 0}$ and get a $1/2\times(7/8\cdot\sqrt{2})$ bounding box; see Figure 4. We have $\displaystyle A(empty)\leq\sum_{i\geq 1\land i\neq 2}A(B_{i})\leq\frac{3}{8}\sqrt{2}$ $\displaystyle A(sparse)\leq\sum_{i\geq 1}A(B_{i})\leq\frac{\sqrt{2}}{2}$ hence $A_{pcs}\geq\sqrt{2}/32$. Since $L^{2}>1/8>\sqrt{2}/32$ we finally have $\displaystyle\textsc{Opt}\geq L+\frac{A_{pcs}}{L}\geq\frac{\sqrt{2}}{4}+\frac{1}{8}$ $\displaystyle\frac{\textsc{Alg}}{\textsc{Opt}}\leq\frac{1/2+(7/8\cdot\sqrt{2})}{\frac{\sqrt{2}}{4}+\frac{1}{8}}\approx 3.63<4.$ Case (3.2.1.1.2) [_For $j=3$ or $j=4$ there does not exist an empty $j$-brick_] $\displaystyle A(empty)\leq\sum_{i\geq 1\land i\neq 2,j}A(B_{i})\leq\frac{11}{32}\sqrt{2}\quad\text{(worst case is when $j=4$)}$ $\displaystyle A(sparse)\leq\sum_{i\geq 1}A(B_{i})\leq\frac{\sqrt{2}}{2}$ hence $A_{pcs}\geq 5/128\cdot\sqrt{2}$. Since $L^{2}>1/8>5/128\cdot\sqrt{2}$ we finally have $\displaystyle\textsc{Opt}\geq L+\frac{A_{pcs}}{L}\geq\frac{\sqrt{2}}{4}+\frac{5}{32}$ $\displaystyle\frac{\textsc{Alg}}{\textsc{Opt}}\leq\frac{1/2+\sqrt{2}}{\frac{\sqrt{2}}{4}+\frac{5}{32}}\approx 3.75<4.$ Case (3.2.1.2) [_There exists an empty $2$-brick_] Thanks to Remark 6 we can assume $B_{0}\dagger 1\dagger 2$ to be empty. Thus, we can cut the topmost part of $B_{\geq 0}$ and get a $1/2\times(3/4\cdot\sqrt{2})$ bounding box; see Figure 4. Here we have two cases. Case (3.2.1.2.1) [_There does not exist an empty $3$-brick_] $\displaystyle A(empty)\leq\sum_{i\geq 1\land i\neq 3}A(B_{i})\leq\frac{7}{16}\sqrt{2}$ $\displaystyle A(sparse)\leq\sum_{i\geq 1}A(B_{i})\leq\frac{\sqrt{2}}{2}$ hence $A_{pcs}\geq\sqrt{2}/64$. Since $L^{2}>1/8>\sqrt{2}/64$ we finally have $\displaystyle\textsc{Opt}\geq L+\frac{A_{pcs}}{L}\geq\frac{\sqrt{2}}{4}+\frac{1}{16}$ $\displaystyle\frac{\textsc{Alg}}{\textsc{Opt}}\leq\frac{\frac{1}{2}+\frac{3}{4}\sqrt{2}}{\frac{\sqrt{2}}{4}+\frac{1}{16}}\approx 3.75<4.$ Case (3.2.1.2.2) [_There exists an empty $3$-brick_] Thanks to Remark 6 we can assume $B_{0}\dagger 1\dagger 1\dagger 2$ to be empty. Here we have two cases. Case (3.2.1.2.2.1) [_There does not exist an empty $4$-brick_] Here we have two cases. Case (3.2.1.2.2.1.1) [_For $j=5$ or $j=6$ there does not exist an empty $j$-brick_] $\displaystyle A(empty)\leq\sum_{i\geq 1\land i\neq 4,j}A(B_{i})\leq\frac{59}{128}\sqrt{2}\quad\text{(worst case is when $j=6$)}$ $\displaystyle A(sparse)\leq\sum_{i\geq 1}A(B_{i})\leq\frac{\sqrt{2}}{2}$ hence $A_{pcs}\geq 5/512\cdot\sqrt{2}$. Since $L^{2}>1/8>5/512\cdot\sqrt{2}$ we finally have $\displaystyle\textsc{Opt}\geq L+\frac{A_{pcs}}{L}\geq\frac{\sqrt{2}}{4}+\frac{5}{128}$ $\displaystyle\frac{\textsc{Alg}}{\textsc{Opt}}\leq\frac{\frac{1}{2}+\frac{3}{4}\sqrt{2}}{\frac{\sqrt{2}}{4}+\frac{5}{128}}\approx 3.98<4.$ Case (3.2.1.2.2.1.2) [_For $j=5$ and $j=6$ there exists an empty $j$-brick_] Thanks to Remark 6 we can assume $B_{0}\dagger 1\dagger 1\dagger 1\dagger 2\dagger 2$ and $B_{0}\dagger 1\dagger 1\dagger 1\dagger 2\dagger 1\dagger 2$ to be empty. Then, we can cut the topmost part of $B_{\geq 0}$ and get a $1/2\times(11/16\cdot\sqrt{2})$ bounding box; see Figure 4. We have $\displaystyle A(empty)\leq\sum_{i\geq 1\land i\neq 4}A(B_{i})\leq\frac{15}{32}\sqrt{2}$ $\displaystyle A(sparse)\leq\sum_{i\geq 1}A(B_{i})\leq\frac{\sqrt{2}}{2}$ hence $A_{pcs}\geq\sqrt{2}/128$. Since $L^{2}>1/8>\sqrt{2}/128$ we finally have $\displaystyle\textsc{Opt}\geq L+\frac{A_{pcs}}{L}\geq\frac{\sqrt{2}}{4}+\frac{1}{32}$ $\displaystyle\frac{\textsc{Alg}}{\textsc{Opt}}\leq\frac{\frac{1}{2}+\frac{11}{16}\sqrt{2}}{\frac{\sqrt{2}}{4}+\frac{1}{32}}\approx 3.83<4.$ Case (3.2.1.2.2.2) [_There exists an empty $4$-brick_] Thanks to Remark 6 we can assume $B_{0}\dagger 1\dagger 1\dagger 1\dagger 2$ to be empty. Then, we can cut the topmost part of $B_{\geq 0}$ and get a $1/2\times(5/8\cdot\sqrt{2})$ bounding box; see Figure 4. Now it remains to bound Opt, and we just assume $\textsc{Opt}\geq L\geq\sqrt{2}/4$, finally $\frac{\textsc{Alg}}{\textsc{Opt}}\leq\frac{\frac{1}{2}+\frac{5}{8}\sqrt{2}}{\frac{\sqrt{2}}{4}}\approx 3.92<4.$ Case (3.2.2) [_$M=B_{0}\dagger 1$_] For the rest of the proof let $L$ be the length of the longest of pieces’ edges then, according to Remark 4, $\sqrt{2}/4\leq L\leq\sqrt{2}/2$. We can cut the topmost part of $B_{\geq 0}$ and get a $1\times(\sqrt{2}/2+L)$ bounding box; see Figure 4. Here we have two cases. Case (3.2.2.1) [_There exists a $2$-brick in $\mathcal{D}$_] Thanks to Remark 4, we have a piece of width at least $1/4$, and combining this with the fact that we have a piece of height $L$, it is apparent that $\textsc{Opt}\geq 1/4+L$; see Figure 4. Thus, $\frac{\textsc{Alg}}{\textsc{Opt}}\leq\frac{1+\frac{\sqrt{2}}{2}+L}{\frac{1}{4}+L}\leq 2+\sqrt{2}<4\quad\text{(maximizing over $L\in[\sqrt{2}/4,\sqrt{2}/2]$).}$ Case (3.2.2.2) [_There does not exist a $2$-brick in $\mathcal{D}$_] Here we have two cases. Case (3.2.2.2.1) [_There does not exist an empty $2$-brick_] $\displaystyle A(empty)\leq\sum_{i\geq 1\land i\neq 2}A(B_{i})\leq\frac{3}{8}\sqrt{2}$ $\displaystyle A(sparse)\leq\sum_{i\geq 1}A(B_{i})\leq\frac{3}{8}\sqrt{2}$ hence $A_{pcs}\geq\sqrt{2}/16$. Since $L^{2}>1/8>\sqrt{2}/16$ we finally have $\displaystyle\textsc{Opt}\geq L+\frac{A_{pcs}}{L}\geq L+\frac{\sqrt{2}}{16L}$ $\displaystyle\frac{\textsc{Alg}}{\textsc{Opt}}\leq\frac{1+\frac{\sqrt{2}}{2}+L}{L+\frac{\sqrt{2}}{16L}}\leq 2+\sqrt{2}<4\quad\text{(maximizing over $L\in[\sqrt{2}/4,\sqrt{2}/2]$).}$ Case (3.2.2.2.2) [_There exists an empty $2$-brick_] Thanks to Remark 6 we can assume $B_{1}\dagger 2$ to be empty. Here we have two cases. Case (3.2.2.2.2.1) [_There exists an empty $3$-brick_] Thanks to Remark 6 we can assume $B_{1}\dagger 1\dagger 2$ to be empty. Then, we can cut the rightmost part of $B_{\geq 0}$ and get a $3/4\times(\sqrt{2}/2+L)$ bounding box; see Figure 4. Now it remains to bound Opt. We have $\displaystyle A(empty)\leq\sum_{i\geq 1}A(B_{i})\leq\frac{\sqrt{2}}{2}$ $\displaystyle A(sparse)\leq\sum_{i\geq 1\land i\neq 2}A(B_{i})\leq\frac{3}{8}\sqrt{2}$ hence $A_{pcs}\geq\sqrt{2}/32$. Since $L^{2}>1/8>\sqrt{2}/32$ we finally have $\displaystyle\textsc{Opt}\geq L+\frac{A_{pcs}}{L}\geq L+\frac{\sqrt{2}}{32L}$ $\displaystyle\frac{\textsc{Alg}}{\textsc{Opt}}\leq\frac{\frac{3}{4}+\frac{\sqrt{2}}{2}+L}{L+\frac{\sqrt{2}}{32L}}\leq 3.79<4\quad\text{(maximizing over $L\in[\sqrt{2}/4,\sqrt{2}/2]$).}$ Case (3.2.2.2.2.2) [_There does not exist an empty $3$-brick_] $\displaystyle A(empty)\leq\sum_{i\geq 1\land i\neq 3}A(B_{i})\leq\frac{7}{16}\sqrt{2}$ $\displaystyle A(sparse)\leq\sum_{i\geq 2\land i\neq 2}A(B_{i})\leq\frac{3}{8}\sqrt{2}$ hence $A_{pcs}\geq 3/64\cdot\sqrt{2}$. Since $L^{2}>1/8>3/64\cdot\sqrt{2}$ we finally have $\displaystyle\textsc{Opt}\geq L+\frac{A_{pcs}}{L}\geq L+\frac{3\sqrt{2}}{64L}$ $\displaystyle\frac{\textsc{Alg}}{\textsc{Opt}}\leq\frac{1+\frac{\sqrt{2}}{2}+L}{L+\frac{3\sqrt{2}}{64L}}\leq 3.82<4\quad\text{(maximizing over $L\in[\sqrt{2}/4,\sqrt{2}/2]$).}$ ∎ #### Algorithm using rotations The algorithm BrickRotation is almost identical to BrickTranslation, but with the difference that we rotate each piece so that its height is at least its width. ###### Theorem 8. The algorithm BrickRotation has a competitive ratio of strictly less than 4 for PerimeterRotation. ###### Proof. The analysis of BrickTranslation carried out in the proof of Theorem 2 still holds, in fact all the estimates on Opt derived from consideration about area are still valid, and the only delicate spot is case (3.2.2.1). In that case we assume to have a piece $p$ having an edge of length $L\in[\sqrt{2}/4,\sqrt{2}/2]$, and that there exists a $2$-brick in $\mathcal{D}$. Thanks to Remark 4 there exists a piece $q$ of size $w_{q}\times h_{q}$ with $w_{q}\geq 1/4$, moreover we rotate every piece so that $1/4\leq w_{q}\leq h_{q}$. Finally, a box that contains both $p$ and $q$ must have size at least $L\times w_{q}$ or $L\times h_{q}$, hence $\textsc{Opt}\geq\min\left\\{L+w_{q},L+h_{q}\right\\}\geq L+\frac{1}{4}.$ This gives exactly the same bound showed in case (3.2.2.1) and completes the proof. ∎ ### 2.2 A similar but inferior algorithm Here we consider the algorithm we get by making a slight change to BrickTranslation. Suppose that the very first piece $p$ arrives and that a $k$-brick is suitable for $p$. Instead of placing $p$ in $B_{k}$ (as BrickTranslation would do), we consider the brick $B_{>k}$ to be a fundamental brick (although in the original algorithm, it was an infinite union of fundamental bricks) and we place $p$ in $B_{>k}$. Thus, we are never going to use the fundamental bricks $B_{i}$ individually, for $i>k$. From here on, the algorithm does as BrickTranslation: Whenever a new piece arrives, we place it in the first derived brick of the suitable size that has room. This behavior is similar to the algorithm for the problem SquareInSquareArea that was described by Fekete and Hoffmann [13]. That problem is studied in more detail in Section 3.3, and for that problem, the algorithm seems to be no worse than ours. Interestingly, the following theorem together with Theorem 2 implies that the modified algorithm is worse for the problem PerimeterTranslation. Figure 5: Left: A configuration produced by the modified version of BrickTranslation. Right: The configuration produced by the original algorithm BrickTranslation. ###### Theorem 9. The modified version of BrickTranslation has a competitive ratio of at least $4$ for the problem PerimeterTranslation. ###### Proof. For any $\varepsilon^{\prime}>0$, we can make an instance realizing a competitive ratio of more than $4-\varepsilon^{\prime}$ as follows. Figure 5 shows the packing produced by the modified and the original algorithm. We first give the algorithm the rectangle $(1/2\sqrt{2}+\varepsilon)\times\varepsilon$ for an infinitesimal $\varepsilon>0$. The rectangle is placed in $B_{>1}$ by the modified algorithm. For a large odd integer $k$, we then feed the algorithm with small rectangles of size $(\sqrt{2}^{-k-1}+\varepsilon)\times(\sqrt{2}^{-k}+\varepsilon)$ until $B_{1}\dagger 1$ has been completely split into $(k-2)$-bricks, each of which contains one small rectangle. We now give the algorithm a piece of size $\varepsilon\times(1/4+\varepsilon)$, which is placed in $B_{1}\dagger 2$. We again give the algorithm many small rectangles until $B_{0}\dagger 1\dagger 1\dagger 1$ has been split into $(k-2)$-bricks. Now follows a rectangle of size $(1/4\sqrt{2}+\varepsilon)\times\varepsilon$, which is placed in $B_{0}\dagger 1\dagger 1\dagger 2$. Finally, we fill $B_{0}\dagger 1\dagger 2\dagger 1$ with small rectangles. Note that as $k\longrightarrow\infty$, the bounding box of the produced packing converges to $B_{\geq 0}$, so it has a perimeter as $B_{-1}$. On the other hand, observe that as $\varepsilon\longrightarrow 0$, we have $\Sigma\longrightarrow A(B_{1})/4=A(B_{3})$, since the small rectangles fill out bricks with a total area of $A(B_{1})$ and with density $1/4$. In the limit, all the pieces can actually be packed into $B_{3}$, so Opt is at most the perimeter of $B_{3}$. But the perimeter of $B_{-1}$ is $4$ times that of $B_{3}$, which finishes the proof. ∎ ### 2.3 Lower bounds ###### Lemma 10. Consider any algorithm $A$ for the problem PerimeterTranslation. Then the competitive ratio of $A$ is at least $4/3$. ###### Proof. We first feed $A$ with two unit squares. Let the bounding box of the two squares have size $a\times b$ and suppose without loss of generality that $a\leq b$. Then $a\geq 1$ and $b\geq 2$. We now give $A$ a rectangle of size $2\times\varepsilon$ for a small value $\varepsilon>0$. The produced packing has a bounding box of perimeter more than $8$, whereas the optimal has perimeter $6+2\varepsilon$. Therefore, the competitive ratio is $\frac{8}{6+2\varepsilon}=\frac{4}{3+\varepsilon}$. By letting $\varepsilon\longrightarrow 0$, we get that the ratio is at least $4/3$. ∎ ###### Lemma 11. Consider any algorithm $A$ for the problem PerimeterRotation. Then the competitive ratio of $A$ is at least $5/4$. ###### Proof. We first feed $A$ with three unit squares. Let the bounding box of the three squares have size $a\times b$ and suppose without loss of generality that $a\leq b$. Suppose first that $b<3$. Then we must have $a\geq 2$ for the box to contain the squares. We then give the algorithm the rectangle $\varepsilon\times 3$ for a small value $\varepsilon>0$. The produced packing has a bounding box of size at least $(2+\varepsilon)\times 3$ and perimeter more than $10$, while the optimal solution has size $(1+\varepsilon)\times 3$ and perimeter $8+2\varepsilon$. On the other hand, if $b\geq 3$, we give the algorithm one more unit square. The produced packing has a bounding box of size at least $2\times 3$ or at least $1\times 4$, and thus perimeter at least $10$, while the optimal packing has size $2\times 2$ and perimeter $8$. We get that the competitive ratio is at least $\frac{10}{8+2\varepsilon}=\frac{5}{4+\varepsilon}$, and by letting $\varepsilon\longrightarrow 0$, we get that the ratio is at least $5/4$. ∎ ## 3 Area versions ### 3.1 General lower bounds In this section we show that, if we allow pieces to be arbitrary rectangles, we cannot bound the competitive ratio for neither AreaTranslation nor AreaRotation as a function of the area Opt of the optimal packing. However we will be able to bound the competitive ratio as a function of the total number $n$ of pieces in the stream. ###### Lemma 12. Consider any algorithm $A$ solving AreaTranslation or AreaRotation and let any $m\in\mathbb{N}$ and $p\in\mathbb{R}$ be given. There exists a stream of $n=m^{2}+1$ rectangles such that (i) the rectangles can be packed into a bounding box of area $2p^{2}$, and (ii) algorithm $A$ produces a packing with a bounding box of area at least $mp^{2}$. ###### Proof. We first feed $A$ with $m^{2}$ rectangles of size $p\times\frac{p}{m^{2}}$. These rectangles have total area $p^{2}$. Let $a\times b$ be the size of the bounding box of the produced packing. Suppose first that $a\geq\frac{p}{m}$ and $b\geq\frac{p}{m}$ hold. We then feed $A$ with a long rectangle of size $pm^{2}\times\frac{p}{m^{2}}$. The produced packing has a bounding box of area at least $\frac{p}{m}\cdot pm^{2}=mp^{2}$. The optimal packing is to pack the $m^{2}$ small rectangles along the long rectangle, which would produce a packing with bounding box of size $pm^{2}\times\frac{2p}{m^{2}}=2p^{2}$. Otherwise, we must have $b>pm$ or $a>pm$, since $ab\geq p^{2}$. We then feed $A$ with a square of size $p\times p$. The produced packing has a bounding box of area at least $p\cdot pm=mp^{2}$. The optimal packing is obtained stacking the $m^{2}$ thin rectangles on top of the big square, which produces a packing with bounding box of size $p\times 2p=2p^{2}$. ∎ ###### Corollary 13. Let $A$ be an algorithm for AreaTranslation or AreaRotation. Then $A$ does not have an asymptotic, and hence also absolute, competitive ratio which is a function of Opt. ###### Proof. Let $f$ be any function of Opt. For any value $\textsc{Opt}=c$, we choose $p:=\sqrt{c/2}$. We now choose $m>2f(c)$ and obtain that the competitive ratio is at least $\frac{mp^{2}}{2p^{2}}=m/2>f(c)=f(\textsc{Opt})$. ∎ ###### Corollary 14. Let $A$ be an algorithm for AreaTranslation or AreaRotation. If $A$ has an asymptotic competitive ratio of $f(n)$, where $n=\|L\|$ is the number of pieces in the stream, then $f(n)=\Omega(\sqrt{n})$. This holds even when all edges of the pieces are required to have length at least $1$. ###### Proof. We choose $p:=m^{2}$. Then all edges have length at least $1$, and the competitive ratio is at least $\frac{mp^{2}}{2p^{2}}=m/2=\Omega(\sqrt{n})$. Here, Opt can be arbitrarily big by choosing $m$ big enough, so it is a lower bound on the asymptotic competitive ratio. ∎ ### 3.2 Algorithms for arbitrary pieces In this section we provide algorithms that solve AreaTranslation and AreaRotation with a competitive ratio of $O(\sqrt{n})$, where $n$ is the total number of pieces. Thus we match the bounds provided in the previous section. We first describe the algorithm DynBoxTrans that solves AreaTranslation. We assume to receive a stream of pieces $p_{1},\dots,p_{n}$ of unknown length $n$, such that piece $p_{i}$ has size $w_{i}\times h_{i}$. For each $k\in\mathbb{Z}$, we define a rectangular box $B_{k}$ with a size varying dynamically. After pieces $p_{1},\dots,p_{j}$ have been processed $B_{k}$ has size $2^{k}\times T_{j}$, where $T_{j}:=H_{j}\sqrt{j}+7H_{j}$ and $H_{j}:=\max_{i=1,\dots,j}h_{i}$. We place the boxes with their bottom edges on the $x$-axis and in order such that the right edge of $B_{k-1}$ is contained in the left edge of $B_{k}$; see Figure 6. Furthermore, we place the lower left corner of box $B_{0}$ at the point $(1,0)$. It then holds that all the boxes are to the right of the point $(0,0)$. Figure 6: The algorithm DynBoxTrans packs pieces into the boxes $B_{k}$ that form a row. Every box has height $T_{j}$ that is dynamically updated. We say that the box $B_{k}$ is _wide enough_ for a piece $p_{i}=w_{i}\times h_{i}$ if $w_{i}\leq 2^{k}$. If a box $B_{k}$ is wide enough for $p_{i}$, we can pack $p_{i}$ in $B_{k}$ using the online strip packing algorithm $\textsc{NFS}_{k}$ that packs rectangles into a strip of width $2^{k}$. The algorithm $\textsc{NFS}_{k}$ is the _next-fit shelf algorithm_ first described by Baker and Schwartz [6]. The algorithm packs pieces in _shelves_ (rows), and each shelf is given a fixed height of $2^{j}$ for some $j\in\mathbb{Z}$ when it is created; see Figure 7. The width of each shelf is $2^{k}$, since this is the width of the box $B_{k}$. Figure 7: A packing produced by the next-fit shelf algorithm using four shelves. A piece of height $h$, where $2^{j-1}<h\leq 2^{j}$, is packed in a shelf of height $2^{j}$. We divide the shelves into two types. If the total width of pieces in a shelf is more than $2^{k-1}$ we call that shelf _dense_ , otherwise we say it is _sparse_. The algorithm $\textsc{NFS}_{k}$ places each piece as far left as possible into the currently sparse shelf of the proper height. If there is no sparse shelf of this height or the sparse shelf has not room for the piece, a new shelf of the appropriate height is created on top of the top shelf, and the piece is placed there at the left end of this new shelf. This ensures that at any point in time there exists at most one sparse shelf for each height $2^{j}$. If we allow the height of the box $B_{k}$ to grow large enough with respect to shelves’ heights, the space wasted by sparse shelves becomes negligible and we obtain a constant density strip packing, as stated in the following lemma. ###### Lemma 15. Let $\widetilde{H}$ be the total height of shelves in $B_{k}$, and $H_{max}$ be the maximum height among pieces in $B_{k}$. If $\widetilde{H}\geq 6H_{max}$, then the pieces in $B_{k}$ are packed with density at least $1/12$. ###### Proof. Let $2^{m-1}<H_{max}\leq 2^{m}$, so that $\widetilde{H}\geq 3\cdot 2^{m}$. For each $i\leq m$ we have at most one sparse shelf of height $2^{i}$ and each shelf of $B_{k}$ has height at most $2^{m}$, hence the total height of sparse shelves is at most $\sum_{i\leq m}2^{i}=2^{m+1}$, so the total height of dense shelves is at least $\widetilde{H}-2^{m+1}\geq\widetilde{H}/3$. Thus, the total area of the dense shelves is at least $2^{k}\cdot\widetilde{H}/3$. Consider a dense shelf of height $2^{i}$. Into that shelf, we have packed pieces of height at least $2^{i-1}$, and the total width of these pieces is at least $2^{k-1}$. Hence, the density of pieces in the shelf is at least $1/4$. Therefore, the total area of pieces in $B_{k}$ is at least $2^{k}\cdot\widetilde{H}/12$. On the other hand, the area of the bounding box is $2^{k}\cdot\widetilde{H}$, that yields the desired density. ∎ Now we are ready to describe how the algorithm works. When the first piece $p_{1}$ arrives, let $2^{k-1}<w_{1}\leq 2^{k}$, then we pack it in the box $B_{k}$ according to $\textsc{NFS}_{k}$ and define $B_{k}$ to be the _active box_. Suppose now that $B_{i}$ is the active box when the piece $p_{j}$ arrives, first we update the value of the threshold $T_{j-1}$ to $T_{j}$, then we have two cases. If $w_{j}>2^{i}$ we choose $\ell$ such that $2^{\ell-1}<w_{j}\leq 2^{\ell}$, pack $p_{j}$ in $B_{\ell}$ and define $B_{\ell}$ to be the active box. Else, $B_{i}$ is wide enough for $p_{j}$ and we try to pack $p_{j}$ into $B_{i}$. Since $B_{i}$ has size $2^{i}\times T_{j}$ it may happen that $\textsc{NFS}_{i}$ exceeds the threshold $T_{j}$ while packing $p_{j}$, generating an overflow. In this case, instead of packing $p_{j}$ in $B_{i}$, we pack $p_{j}$ into $B_{i+1}$ and define that to be the active box. ###### Theorem 16. The algorithm DynBoxTrans has an absolute competitive ratio of $O(\sqrt{n})$ for the problem AreaTranslation on a stream of $n$ pieces. ###### Proof. First, define $\Sigma_{j}$ as the total area of the first $j$ pieces, $W:=\max_{i=1,\dots,n}w_{i}$ and recall that $H_{j}=\max_{i=1,\dots,j}h_{i}$ and $T_{j}=H_{j}\sqrt{n}+7H_{j}$. Let $B_{k}$ be the last active box, so that we can enclose all the pieces in a bounding box of size $2^{k+1}\times T_{n}$, and bound the area returned by the algorithm as $\textsc{Alg}=O(2^{k}H_{n}\sqrt{n})$. On the other hand we are able to bound the optimal offline packing as $\textsc{Opt}=\Omega(\Sigma_{n}+WH_{n})$. If the active box never changed, then we have $2^{k}<2W$ that implies $\textsc{Alg}=O(WH_{n}\sqrt{n})=\textsc{Opt}\cdot O(\sqrt{n})$. Otherwise, let $B_{\ell}$ be the last active box before $B_{k}$, and $p_{j}$ be the first piece put in $B_{k}$. Here we have two cases. Case (1) [_$w_{j} >2^{\ell}$_] In this case we have $2^{k}<2W$ that implies $\textsc{Alg}=O(WH_{n}\sqrt{n})=\textsc{Opt}\cdot O(\sqrt{n})$. Case (2) [_$w_{j}\leq 2^{\ell}$_] In this case we have $k=\ell+1$. Denote with $\widetilde{H_{i}}$ the total height of shelves in $B_{i}$. Then we have $\widetilde{H_{\ell}}\geq T_{j}-H_{j}=H_{j}\sqrt{n}+6H_{j}$, otherwise we could pack $p_{j}$ in $B_{\ell}$. Thus, we can apply Lemma 15 and conclude that the box $B_{\ell}$ of size $2^{\ell}\times T_{j}$ is filled with constant density. Here we have two cases. Case (2.1) [_$\widetilde{H_{k}}\leq T_{j}$_] In this case we have $\textsc{Alg}=O(2^{k}T_{j})$ and, thanks to the constant density packing of $B_{\ell}$ we have $\Sigma_{j}=\Theta(2^{\ell}\widetilde{H_{\ell}})=\Theta(2^{k}T_{j})$. Since $\textsc{Opt}\geq\Sigma_{j}$, we get $\textsc{Alg}=O(\textsc{Opt})$. Case (2.2) [_$\widetilde{H_{k}} >T_{j}$_] In this case we have $\textsc{Alg}=O(2^{k}\widetilde{H_{k}})$. Moreover, $\widetilde{H_{k}}=O(H_{n}+\Sigma_{n}/2^{k})$, in fact if $2^{s-1}<H_{n}\leq 2^{s}$, then the total height of sparse shelves is $\sum_{i\leq s}2^{i}=2^{s+1}=O(H_{n})$. Furthermore, dense shelves are filled with constant density, therefore their total height is at most $O(\Sigma_{n}/2^{k})$. Finally, we need to show that $2^{k}=O(W\sqrt{n})$. Thanks to the constant density packing of $B_{\ell}$, we have $2^{k}H_{j}\sqrt{j}=O(2^{\ell}T_{j})=O(\Sigma_{j})$. We can upper bound the size of every piece $p_{i}$ for $i\leq j$ with $W\times H_{j}$ and obtain $\Sigma_{j}\leq n\cdot WH_{j}$. Plugging it in the previous estimate and dividing both sides by $H_{j}\sqrt{n}$ we get $2^{k}=O(W\sqrt{n})$. Now we have $\textsc{Alg}=O(2^{k}\widetilde{H_{k}})=O(2^{k}H_{n}+\Sigma_{n})=O(WH_{n}\sqrt{n}+\Sigma_{n})=\textsc{Opt}\cdot O(\sqrt{n})$. ∎ The algorithm DynBoxRot is obtained from DynBoxTrans with a slight modification: before processing any piece $p_{i}$ we rotate it so that $w_{i}\leq h_{i}$. In this way, it still holds that $\textsc{Opt}=\Omega(\Sigma_{n}+WH_{n})$ and the proof of Theorem 16 works also for the following. ###### Theorem 17. The algorithm DynBoxRot has an absolute competitive ratio of $O(\sqrt{n})$ for the problem AreaRotation on a stream of $n$ pieces. ### 3.3 Bounded aspect ratio In this section, we will consider the special case where the aspect ratio of all pieces is $\alpha=1$, i.e., all the pieces are squares. Furthermore, we will measure the size of the packing as the area of the minimum axis-parallel bounding _square_ , and we call the resulting problem SquareInSquareArea. Since we get a constant competitive ratio in this case, it follows that for other values of $\alpha$ and when allowing the bounding box to be a general rectangle, one can likewise achieve a constant competitive ratio. We first give a lower bound. ###### Lemma 18. Consider any algorithm $A$ for the problem SquareInSquareArea. Then the competitive ratio of $A$ is at least $16/9$. ###### Proof. We first give $A$ four $1\times 1$ squares. Let the bounding square have size $\ell\times\ell$. If $\ell\geq 3$, the bounding square of the four $1\times 1$ squares has size at least $3\times 3$, while the optimal packing has size $2\times 2$, which gives ratio at least $9/4$. Otherwise, if $\ell<3$, we give a $2\times 2$ square and we will prove that the bounding square has size at least $4\times 4$ while the optimal packing fits in a $3\times 3$ square, so the ratio is at least $16/9$. Let us assume by contradiction that there exists a $(4-\varepsilon)\times(4-\varepsilon)$ bounding square containing both a $2\times 2$ square and four $1\times 1$ squares, with the additional hypothesis that the $1\times 1$ squares fit in a $(3-\delta)\times(3-\delta)$ bounding box. We refer to notation in Figure 8 (left) and notice that we have $a<1$ or $b<1$, and analogously $c<1$ or $d<1$. Without loss of generality, we can assume $a,d<1$. Hence, starting from the configuration in Figure 8 (left) we can drag the $2\times 2$ square to the bottom left corner and obtain the configuration in Figure 8 (right), that still fulfill the hypotheses we assumed by contradiction. Figure 8: Left: A $2\times 2$ square inside a bounding square having edges shorter than $4$. Right: The $2\times 2$ square has been dragged in the bottom left corner of the bounding square. Four $1\times 1$ squares $Q_{1},\dots,Q_{4}$ are placed within the bounding square. From now on we employ the notation of Figure 8 (right). Let $(x_{i},y_{i})$ be the coordinates of the bottom left corner of square $Q_{i}$. Stating that $Q_{i}$ and $Q_{j}$ are disjoint is equivalent to $\max\\{\|x_{i}-x_{j}\|,\|y_{i}-y_{j}\|\\}\geq 1$. Consider now the two rectangular regions $ABDE$ and $GCDF$: note that each of them can contain at most two squares. Indeed, given $Q_{i}$ and $Q_{j}$ completely contained in $ABDE$, it holds $\|y_{i}-y_{j}\|\leq 1-\varepsilon$ thus $\|x_{i}-x_{j}\|\geq 1$. If three squares $Q_{1},Q_{2},Q_{3}$ are completely contain in $ABDE$ then we have, without loss of generality, $x_{1}\leq x_{2}-1\leq x_{3}-2$ and the minimal bounding square of $Q_{1},Q_{2},Q_{3}$ has size at least $3\times 3$, that gives a contradiction. The same holds for $GCDF$. Finally, every $Q_{i}$ is either fully contained in $ABDE$ or $GCDF$ hence, without loss of generality, we can assume that $Q_{1},Q_{2}$ are contained in $ABDE$ and $Q_{3},Q_{4}$ are contained in $GCDF$. This implies that $x_{1}\leq x_{2}-1$ and $y_{4}\leq y_{3}-1$, again without loss of generality. Observe that $x_{2}\leq x_{3}+1-\varepsilon$ and $y_{3}\leq y_{2}+1-\varepsilon$. $Q_{2}$ and $Q_{3}$ are disjoint, using the previous characterization we have two cases. First, $\|x_{2}-x_{3}\|\geq 1$ and thanks to the observation above it cannot be $x_{2}>x_{3}$, therefore we have $x_{1}\leq x_{2}-1\leq x_{3}-2$. Else, $\|y_{2}-y_{3}\|\geq 1$ and thanks to the observation above we have $y_{4}\leq y_{3}-1\leq y_{2}-2$. In both cases that gives a contradiction since we cannot pack all $Q_{i}$s in a $(3-\delta)\times(3-\delta)$ bounding square. ∎ We are now going to analyze the competitive ratio of the algorithm BrickTranslation (in fact, the algorithm BrickRotation has the exact same behavior when the pieces are squares). Note that a brick can never contain more than one piece. The algorithm is almost the same as the one described by Fekete and Hoffmann [13]. The slight difference is addressed in Section 2.2 and it is shown there that the behavior as described by Fekete and Hoffmann makes a worse algorithm for the problem PerimeterTranslation. However, even though the two algorithms will not always produce identical packings for the problem SquareInSquareArea, the analysis of the following theorem seems to hold for both versions, so for the problem SquareInSquareArea, the algorithms are equally good. ###### Theorem 19. The algorithm BrickTranslation has a competitive ratio of $6$ for SquareInSquareArea. The analysis is tight. ###### Proof. Suppose a stream of squares have been packed by BrickTranslation, and let Alg be the area of the bounding square of the resulting packing. Let $B_{k}$ be the largest elementary brick in which a square has been placed. Suppose without loss of generality that $k=0$, so that $B_{k}$ has size $1\times 1/\sqrt{2}$ and $B_{\geq k}$, which contains all the packed squares, has size $1\times\sqrt{2}$. Figure 9: Left: A $2k$-packing. The grey bricks are non-empty and may have been split into smaller bricks. Right: The $2k$-packing produced by BrickTranslation when providing the algorithm with enough copies of the square $S_{k}$ (the small grey squares), showing that the competitive ratio can be arbitrarily close to $6$. We now recursively define a type of packing that we call a $2k$-packing, for a non-negative integer $k$; see Figure 9 (left). As $k$ increases, so do the requirements to a $2k$-packing, in the sense that a $(2k+2)$-packing is also a $2k$-packing, but the other way is in general not the case. Define $F_{0}:=B_{\geq 1}$ and $U_{0}:=B_{0}$. A packing is a $0$-packing if pieces have been placed in $U_{0}$ (the brick $U_{0}$ may or may not have been split in smaller bricks). Hence, the considered packing is a $0$-packing by the assumption that a piece has been placed in $B_{0}$. Suppose that we have defined a $2k$-packing for some integer $k$. A $(2k+2)$-packing is a $2k$-packing with the additional requirements that * • the brick $U_{2k}$ has been split into $L:=U_{2k}\dagger 1$ and $E_{2k+1}:=U_{2k}\dagger 2$, * • the right brick $E_{2k+1}$ is empty, * • the left brick $L$ has been split into $F_{2k+2}:=L\dagger 1$ and $U_{2k+2}:=L\dagger 2$, and * • $U_{2k+2}$ is non-empty, and thus also $F_{2k+2}$ is non-empty. The symbols $U_{j},E_{j},F_{j}$ have been chosen such that the brick is a $j$-brick, i.e., the index tells the size of the brick. Consider a $2k$-packing. It follows from the definition that along the top edge of $B_{\geq 0}$ from the right corner $(1,\sqrt{2})$ to the left corner $(0,\sqrt{2})$, we meet a sequence $E_{1},E_{3},\ldots,E_{2k-1}$ of empty bricks of decreasing size, and finally meet a non-empty brick $U_{2k}$ which may have been split into smaller bricks. ###### Claim 20. If the packing is a $2k$-packing and not a $(2k+2)$-packing, then $\textsc{Alg}/\textsc{Opt}<6$. Since we pack a finite number of squares, the produced packing is a $2k$-packing but not a $(2k+2)$-packing for some sufficiently large $k$, so Claim 20 implies Theorem 19. Let us now prove Claim 20. We first compute the area of the brick $U_{2k}$ and the total areas of the bricks $F_{0},F_{2},\ldots,F_{2k}$, as these areas will be used often: $\displaystyle u_{k}$ $\displaystyle:=\|U_{2k}\|=2^{-2k}/\sqrt{2}$ (1) $\displaystyle f_{k}$ $\displaystyle:=\sum_{i=0}^{k}\|F_{2i}\|=\frac{2\|B_{\geq 0}\|-u_{k}}{3}=\frac{4-4^{-k}}{3\sqrt{2}}.$ (2) * 1) Suppose first that $U_{2k}$ has not been split into smaller bricks. Then, since $U_{2k}$ is non-empty by assumption, we know that $U_{2k}$ contains a square $S$ of size $s\times s$ where $s\in(s_{l},s_{h}]=\left(\sqrt{2}^{-2k-2},\sqrt{2}^{-2k-1}\right]$. Since the bricks $E_{1},E_{3},\ldots,E_{2k-1}$ are all empty, we get that the upper edge of the bounding square coincides with the upper edge of $S$, and we thus have $\textsc{Alg}\leq\textsc{Alg}(s):=(\sqrt{2}-(\sqrt{2}^{-2k-1}-s))^{2}.$ The largest empty brick in the bricks $F_{2i}$ can have size $\|U_{2k}\|/2$, so the total size of empty bricks in $F_{0},F_{2},\ldots,F_{2k}$ is $\|U_{2k}\|$. Moreover, the density of squares into bricks is at least $1/2\sqrt{2}$ and by (2), we get that $\textsc{Opt}\geq\textsc{Opt}(s):=\frac{f_{k}-u_{k}}{2\sqrt{2}}+s^{2}=\frac{1-4^{-k}}{3}+s^{2}.$ In the case that $k=0$, we get $\frac{\textsc{Alg}}{\textsc{Opt}}\leq\frac{\textsc{Alg}(s)}{\textsc{Opt}(s)}=\frac{2s\sqrt{2}+2s^{2}+1}{2s^{2}}.$ A simple analysis shows that the fraction is largest when $s=s_{l}$, so we get the bound $\frac{\textsc{Alg}}{\textsc{Opt}}\leq\frac{2s_{l}\sqrt{2}+2s_{l}^{2}+1}{2s_{l}^{2}}=3+2\sqrt{2}<5.83$ Suppose now that $k>0$. We divide into two cases of whether $s$ is in the lower or the upper half of the range $(s_{l},s_{h}]$. For the lower half, that is, $s\in(s_{l},\frac{s_{l}+s_{h}}{2}]$, we get $\frac{\textsc{Alg}}{\textsc{Opt}}\leq\frac{\textsc{Alg}(\frac{s_{l}+s_{h}}{2})}{\textsc{Opt}(s_{l})}=\frac{96\cdot 4^{k}+(24\sqrt{2}-48)\cdot 2^{k}-6\sqrt{2}+9}{16\cdot 4^{k}-4}.$ It is straightforward to check that $(24\sqrt{2}-48)\cdot 2^{k}-6\sqrt{2}+9<6\cdot(-4)$ for all $k\geq 1$, so it follows that the ratio is less than $6$. For the upper half, that is, $s\in[\frac{s_{l}+s_{h}}{2},s_{h}]$, we get $\frac{\textsc{Alg}}{\textsc{Opt}}\leq\frac{\textsc{Alg}(s_{h})}{\textsc{Opt}(\frac{s_{l}+s_{h}}{2})}=\frac{96\cdot 4^{k}}{16\cdot 4^{k}+6\sqrt{2}-7}.$ As $6\sqrt{2}-7>0$, the ratio is less than $6$. * 2) We now assume that $U_{2k}$ has been split into a $L$ and $E_{2k+1}$, which are the left and right halfs of $U_{2k}$, respectively. * 2.1) We first suppose that $E_{2k+1}$ is not empty. This implies that there is no empty $(2k+1)$-brick in $F_{0},F_{2},\ldots,F_{2k},U_{2k}$. Hence, each empty brick in the bricks $F_{0},F_{2},\ldots,F_{2k},U_{2k}$ is a $(2k+2)$-brick or smaller, so these empty bricks have total size at most $u_{k}/2$. We then get $\textsc{Opt}\geq\frac{f_{k}+u_{k}-u_{k}/2}{2\sqrt{2}}=\frac{1}{3}+\frac{4^{-k}}{24}>\frac{1}{3}.$ Since $\textsc{Alg}\leq 2$, it follows that $\frac{\textsc{Alg}}{\textsc{Opt}}<6$. * 2.2) We now suppose that $E_{2k+1}$ is empty. * 2.2.1) Suppose now that $L$ has not been split into smaller bricks. Then $L$ contains a square $S$ of size $s\times s$ for $s\in(s_{l},s_{h}]=\left(\sqrt{2}^{-2k-3},\sqrt{2}^{-2k-2}\right]$. As in case 1, we get $\textsc{Alg}\leq\textsc{Alg}(s):=(\sqrt{2}-(\sqrt{2}^{-2k-1}-s))^{2}.$ Note that there is no empty $(2k+1)$-brick in the bricks $F_{0},F_{2},\ldots,F_{2k}$, so these bricks contain a total area of at most $u_{k}/2$ empty bricks. We then get $\textsc{Opt}\geq\textsc{Opt}(s):=\frac{f_{k}-u_{k}/2}{2\sqrt{2}}+s^{2}.$ We then get the bound $\frac{\textsc{Alg}}{\textsc{Opt}}\leq\frac{\textsc{Alg}(s_{h})}{\textsc{Opt}(s_{l})}={\frac{24\cdot{4}^{k}+(12\sqrt{2}-24)\cdot{2}^{k}-6\,\sqrt{2}+9}{4\cdot{4}^{k}-1}}.$ Here, it is straightforward to verify that $(12\sqrt{2}-24)\cdot{2}^{k}-6\,\sqrt{2}+9<6\cdot(-1)$ for all $k\geq 0$, and hence the ratio is less than $6$. * 2.2.2) We now assume that $L$ has been split into $F_{2k+2}$ and $U_{2k+2}$, which are the bottom and top parts, respectively. * 2.2.2.1) Suppose that $U_{2k+2}$ is empty. Since also $E_{1},E_{3},\ldots,E_{2k+1}$ are empty, we get that $\textsc{Alg}\leq(\sqrt{2}-\sqrt{2}^{-2k-1}/2)^{2}$. Note that each empty bricks in the bricks $F_{0},F_{2},\ldots,F_{2k+2}$ can have size at most $u_{k}/8$, so the total size of the empty bricks is at most $u_{k}/4=u_{k+1}$, and we get $\textsc{Opt}\geq\frac{f_{k+1}-u_{k+1}}{2\sqrt{2}}.$ We therefore get $\frac{\textsc{Alg}}{\textsc{Opt}}\leq{\frac{48\cdot{4}^{k}-24\cdot{2}^{k}+3}{8\cdot{2}^{2\,k}-2}}.$ Here, it is straightforward to check that $-24\cdot{2}^{k}+3<6\cdot(-2)$ for all $k\geq 0$, so the ratio is less than $6$. * 2.2.2.2) We are finally left with the case that $U_{2k+2}$ is not empty. But then all the requirements are satisfied for the packing to be a $(2k+2)$-packing. We now observe that the analysis is tight. To this end, we show that for any given $k$ and a small $\varepsilon>0$, we can force the algorithm to produce a $2k$-packing, such that as $k\longrightarrow\infty$ and $\varepsilon\longrightarrow 0$, the ratio $\frac{\textsc{Alg}}{\Sigma}$ tends to $6$, where $\Sigma$ is the total area of the packed squares. Let $\varepsilon_{k}:=\varepsilon\sqrt{2}^{-k}$, $\ell_{k}:=\sqrt{2}^{-k}/2+\varepsilon_{k}$, and let $S_{k}$ be a square of size $\ell_{k}\times\ell_{k}$. We now feed the algorithm with copies of $S_{k}$. This will eventually result in a $2k$-packing, where each non-empty brick is a $2k$-brick; see Figure 9 (right). Let $n_{k}$ be the number needed to produce the $2k$-packing. The density in each non-empty brick is $\rho_{\varepsilon}:=\frac{\|S_{k}\|}{\|B_{2k}\|}$. As $\varepsilon\longrightarrow 0$, we get that $\rho_{\varepsilon}\longrightarrow\frac{1}{2\sqrt{2}}$. As $k\longrightarrow\infty$, the area of non-empty bricks converges to $\frac{2\|B_{\leq 0}\|}{3}=\frac{2\sqrt{2}}{3}$. Hence, we have $\Sigma\longrightarrow\frac{1}{2\sqrt{2}}\cdot\frac{2\sqrt{2}}{3}=\frac{1}{3}$. We then get $\frac{\textsc{Alg}}{\Sigma}\longrightarrow\frac{2}{1/3}=6$. Furthermore, the optimal packing of the squares is to place them so that their bounding box is a square of size $\lceil\sqrt{n}_{k}\rceil\ell_{k}\times\lceil\sqrt{n}_{k}\rceil\ell_{k}$. As $k\longrightarrow\infty$, we then have $\frac{\Sigma}{\textsc{Opt}}\longrightarrow 1$. Hence, we have $\frac{\textsc{Alg}}{\textsc{Opt}}\longrightarrow 6$. ∎ ### 3.4 More lower bounds when edges are long We already saw in Corollary 14 that as a function of $n$, the competitive ratio of an algorithm for AreaTranslation or AreaRotation must be at least $\Omega(\sqrt{n})$, even when all edges have length $1$. In this section, we give lower bounds in terms of Opt for the same case. Note that the assumption that the edges are long is needed for these bounds to be matched by actual algorithms, since Corollary 13 states that without the assumption, the competitive ratio cannot be bounded as a function of Opt. ###### Theorem 21. Consider any algorithm $A$ for the problem AreaTranslation with the restriction that all edges of the given rectangles have length at least $1$. If $A$ has an asymptotic competitive ratio $f(\textsc{Opt})$ as a function of Opt, then $f(\textsc{Opt})=\Omega(\sqrt{\textsc{Opt}})$. ###### Remark 22. Note that when the edges are long, $\Omega(\sqrt{\textsc{Opt}})=\Omega(\sqrt{n})$, so this bound is stronger than the $\Omega(\sqrt{n})$ bound of Corollary 14. ###### Proof of Theorem 21.. For any $n\in\mathbb{N}$, we do as follows. We first provide $A$ with $n^{2}$ unit squares. Let the bounding box of the produced packing of these squares have size $a\times b$. Assume without loss of generality that $a\leq b$, so that $b\geq n$. We now give $A$ the rectangle $n^{2}\times 1$. The optimal offline solution to this set of rectangles has a bounding box of size $n^{2}\times 2$. The packing produced by $A$ has a bounding box of size at least $n^{2}\times n=\Omega(\sqrt{\textsc{Opt}})\cdot\textsc{Opt}$. ∎ ###### Theorem 23. Consider any algorithm $A$ for the problem AreaRotation with the restriction that all edges of the given rectangles have length at least $1$. If $A$ has a competitive ratio $f(\textsc{Opt})$ as a function of Opt, then $f(\textsc{Opt})=\Omega(\sqrt[4]{\textsc{Opt}})$. ###### Proof. For any $n\in\mathbb{N}$, we do as follows. We first provide $A$ with $n^{2}$ unit squares. Let the bounding box of the produced packing of these squares have size $a\times b$. Assume without loss of generality that $a\leq b$. If $a\geq n^{1/2}$, we give $A$ the rectangle $1\times n^{2}$. Otherwise, we have $b>n^{3/2}$, and then we give $A$ the square $n\times n$. In either case, there is an optimal offline solution of area $2n^{2}$, but the bounding box of the packing produced by $A$ has area at least $n^{5/2}=\Omega(\sqrt[4]{\textsc{Opt}})\cdot\textsc{Opt}$. ∎ ### 3.5 Algorithms when edges are long In this section, we describe algorithms that match lower bounds of Section 3.4. We analyze these algorithms under the assumption that we feed them with rectangles with edges of length at least $1$ (of course, any other positive constant will also work), but we require no bound on the aspect ratio. Under this assumption, we observe that DynBoxTrans has absolute competitive ratio $O(\sqrt{\textsc{Opt}})$ for AreaTranslation. We then describe the algorithm DynBoxRot${}_{\sqrt[4]{\textsc{Opt}}}$, which we prove to have absolute competitive ratio $O(\sqrt[4]{\textsc{Opt}})$ for AreaRotation. By Theorems 21 and 23, both algorithms are optimal to within a constant factor. In previous sections we proved lower bounds of $\Omega(\sqrt{n})$ and $\Omega(\sqrt[4]{\textsc{Opt}})$ for AreaRotation. They can be summarized stating that AreaRotation has a competitive ratio of $\Omega(\max\\{\sqrt{n},\sqrt[4]{\textsc{Opt}}\\})$. The last theorem of this section, describes the algorithm $\textsc{DynBoxRot}_{\sqrt{n}\,\wedge\sqrt[4]{\textsc{Opt}}}$ that simultaneously matches both lower bounds achieving a competitive ratio of $O(\min\\{\sqrt{n},\sqrt[4]{\textsc{Opt}}\\}$. At a first sight it may seem that this algorithm contradicts the lower bound of $\Omega(\max\\{\sqrt{n},\sqrt[4]{\textsc{Opt}}\\})$; however this simply proves that the _edge cases_ that have a competitive ratio of at least $\Omega(\sqrt[4]{\textsc{Opt}})$ must satisfy $\textsc{Opt}=O(n^{2})$. Likewise, those for which the competitive ratio is at least $\Omega(\sqrt{n})$ satisfy $n=O(\sqrt{\textsc{Opt}})$. #### Translations only Under the long edge assumption, we have $n\leq\textsc{Opt}$. Therefore, DynBoxTrans achieves a competitive ratio of $O(\sqrt{n})=O(\sqrt{\textsc{Opt}})$ for AreaTranslation and matches the bound stated in Theorem 21. #### Rotations allowed Now we tackle the AreaRotation problem and describe the algorithm DynBoxRot${}_{\sqrt[4]{\textsc{Opt}}}$. We define the threshold function $T_{j}=\Sigma_{j}^{3/4}+7H_{j}$, where $H_{j}=\max_{i=1,\dots,j}h_{i}$ and $\Sigma_{j}$ is the total area of pieces $p_{1},\dots,p_{j}$. $\textsc{DynBoxRot}_{\sqrt[4]{\textsc{Opt}}}$ is obtained by running DynBoxRot, as described in Section 3.2, employing this new threshold $T_{j}$. ###### Theorem 24. The algorithm DynBoxRot${}_{\sqrt[4]{\textsc{Opt}}}$ has an absolute competitive ratio of $O(\sqrt[4]{\textsc{Opt}})$ for the problem AreaRotation, where Opt is the area of the optimal offline packing. ###### Proof. This proof is similar to the one of Theorem 16. Define $W:=\max_{i=1,\dots,n}w_{i}$. Recall that in DynBoxRot we preprocess every piece $p$ rotating it so the $w_{p}\leq h_{p}$, hence $W\leq\sqrt{\Sigma_{n}}$. Let $B_{k}$ be the last active box, so that we can enclose all the pieces in a bounding box of size $2^{k+1}\times T_{n}$, and bound the area returned by the algorithm as $\textsc{Alg}=O(2^{k}H_{n}+2^{k}\Sigma_{n}^{3/4})$. On the other hand we are able to bound the optimal offline packing as $\textsc{Opt}=\Omega(\Sigma_{n}+WH_{n})$. If the active box never changed, then we have $2^{k}<2W$ that implies $\textsc{Alg}=O(WH_{n}+\Sigma_{n}^{5/4})=\textsc{Opt}\cdot O(\sqrt[4]{\textsc{Opt}})$. Otherwise, let $B_{\ell}$ be the last active box before $B_{k}$, and $p_{j}$ be the first piece put in $B_{k}$. Here we have two cases. Case (1) [_$w_{j} >2^{\ell}$_] In this case we have $2^{k}<2W$ that implies $\textsc{Alg}=O(WH_{n}+\Sigma_{n}^{5/4})=\textsc{Opt}\cdot O(\sqrt[4]{\textsc{Opt}})$. Case (2) [_$w_{j}\leq 2^{\ell}$_] In this case we have $k=\ell+1$. Denote with $\widetilde{H_{i}}$ the total height of shelves in $B_{i}$. Then we have $\widetilde{H_{\ell}}\geq T_{j}-H_{j}=\Sigma_{j}^{3/4}+6H_{j}$, otherwise we could pack $p_{j}$ in $B_{\ell}$. Thus, we can apply Lemma 15 and conclude that the box $B_{\ell}$ of size $2^{\ell}\times T_{j}$ is filled with constant density. Here we have two cases. Case (2.1) [_$\widetilde{H_{k}}\leq T_{j}$_] In this case we have $\textsc{Alg}=O(2^{k}T_{j})$ and, thanks to the constant density packing of $B_{\ell}$ we have $\Sigma_{j}=\Theta(2^{\ell}\widetilde{H_{\ell}})=\Theta(2^{k}T_{j})$. Since $\textsc{Opt}\geq\Sigma_{j}$, we get $\textsc{Alg}=O(\textsc{Opt})$. Case (2.2) [_$\widetilde{H_{k}} >T_{j}$_] In this case we have $\textsc{Alg}=O(2^{k}\widetilde{H_{k}})$. Moreover, $\widetilde{H_{k}}=O(H_{n}+\Sigma_{n}/2^{k})$, in fact if $2^{s-1}<H_{n}\leq 2^{s}$, then the total height of sparse shelves is $\sum_{i\leq s}2^{i}=2^{s+1}=O(H_{n})$. Furthermore, dense shelves are filled with constant density, therefore their total height is at most $O(\Sigma_{n}/2^{k})$. Finally, we need to show that $2^{k}=O(\sqrt[4]{\Sigma_{n}})$. Thanks to the constant density packing of $B_{\ell}$, we have $2^{k}\Sigma_{j}^{3/4}=O(2^{\ell}T_{j})=O(\Sigma_{j})$. Dividing both sides by $\Sigma_{j}^{3/4}$ we get $2^{k}=O(\Sigma_{j}^{1/4})$. In the end notice that, thanks to the long edge hypotheses $H_{n}\leq\Sigma_{n}$ and we have $\textsc{Alg}=O(2^{k}\widetilde{H_{k}})=O(2^{k}H_{n}+\Sigma_{n})=O(\Sigma_{n}^{5/4})=\textsc{Opt}\cdot O(\sqrt[4]{\textsc{Opt}})$. ∎ So far we managed to match the competitive ratio lower bounds of $\Omega(\sqrt{n})$ and $\Omega(\sqrt[4]{\textsc{Opt}})$ employing two different algorithms: DynBoxRot and $\textsc{DynBoxRot}_{\sqrt[4]{\textsc{Opt}}}$. A natural question is whether is it possible to match the performance of these algorithms simultaneously, having an algorithm that achieves a competitive ratio of $O(\min\\{\sqrt{n},\sqrt[4]{\textsc{Opt}}\\})$. We give an affirmative answer by describing the algorithm $\textsc{DynBoxRot}_{\sqrt{n}\,\wedge\sqrt[4]{\textsc{Opt}}}$. Again, we employ the same scheme of DynBoxRot with a different threshold function. This time the definition of $T_{j}$ is slightly more involved. First define $\widetilde{T}_{j}=\begin{cases}\Sigma_{j}^{3/4}+7H_{j},&\text{ if }\Sigma_{j}<j^{2}\\\ H_{j}\sqrt{n}+7H_{j},&\text{ otherwise.}\end{cases}$ Later we will write $\widetilde{T}_{j}$ as $\widetilde{T}_{j}=\mathbbm{1}_{\\{\Sigma_{j}<j^{2}\\}}\cdot\Sigma_{j}^{3/4}+\mathbbm{1}_{\\{\Sigma_{j}\geq{j}^{2}\\}}\cdot H_{j}\sqrt{n}+7H_{j}$. We now define $T_{j}=\begin{cases}0,&\text{ if }j=0\\\ \max\left\\{T_{j-1},\widetilde{T_{j}}\right\\},&\text{ if }j\geq 1.\end{cases}$ This two-step definition is necessary for a correct implementation of the algorithm because we must guarantee that $T_{j}$ does not decrease. ###### Theorem 25. When used on the problem AreaRotation, the algorithm $\textsc{DynBoxRot}_{\sqrt{n}\,\wedge\sqrt[4]{\textsc{Opt}}}$ has an absolute competitive ratio of $O(\min\\{\sqrt{n},\sqrt[4]{\textsc{Opt}}\\})$, where Opt is the area of the optimal offline packing and $n$ is the total number of pieces in the stream. ###### Proof. Again, we define $W:=\max_{i=1,\dots,n}w_{i}$. Recall that in DynBoxRot we preprocess every piece $p$ rotating it so the $w_{p}\leq h_{p}$, hence $W\leq\sqrt{\Sigma_{n}}$. Let $B_{k}$ be the last active box, so that we can enclose all the pieces in a bounding box of size $2^{k+1}\times T_{n}$. There exists a $n^{\prime}\leq n$ such that $T_{n}=\widetilde{T}_{n^{\prime}}$. We can bound the area returned by the algorithm as $\textsc{Alg}=O\left(2^{k}\widetilde{T}_{n^{\prime}}\right)=O\left(2^{k}H_{n^{\prime}}+\mathbbm{1}_{\\{\Sigma_{n^{\prime}}<{n^{\prime}}^{2}\\}}\cdot 2^{k}\Sigma_{n^{\prime}}^{3/4}+\mathbbm{1}_{\\{\Sigma_{n^{\prime}}\geq{n^{\prime}}^{2}\\}}\cdot 2^{k}H_{n^{\prime}}\sqrt{{n^{\prime}}}\right).$ We bound the optimal offline packing as $\textsc{Opt}=\Omega(\Sigma_{n}+WH_{n})$. If the active box never changed, then we have $2^{k}<2W$ that implies $\displaystyle ALG$ $\displaystyle=O\left(WH_{n}+\mathbbm{1}_{\\{\Sigma_{n^{\prime}}<{n^{\prime}}^{2}\\}}\cdot W\Sigma_{n^{\prime}}^{3/4}+\mathbbm{1}_{\\{\Sigma_{n^{\prime}}\geq{n^{\prime}}^{2}\\}}\cdot WH_{n^{\prime}}\sqrt{{n^{\prime}}}\right)$ $\displaystyle=O\left(WH_{n}+\mathbbm{1}_{\\{\Sigma_{n^{\prime}}<{n^{\prime}}^{2}\\}}\cdot\Sigma_{n}\sqrt[4]{\Sigma_{n^{\prime}}}+\mathbbm{1}_{\\{\Sigma_{n^{\prime}}\geq{n^{\prime}}^{2}\\}}\cdot WH_{n}\sqrt{{n^{\prime}}}\right)$ $\displaystyle\leq\textsc{Opt}\cdot O\left(\min\left\\{\sqrt[4]{\Sigma_{n^{\prime}}},\sqrt{n^{\prime}}\right\\}\right)=\textsc{Opt}\cdot O\left(\min\left\\{\sqrt[4]{\textsc{Opt}},\sqrt{n}\right\\}\right).$ Otherwise, let $B_{\ell}$ be the last active box before $B_{k}$, and $p_{j}$ be the first piece put in $B_{k}$. Here we have two cases. Case (1) [_$w_{j} >2^{\ell}$_] In this case we have, again, $2^{k}<2W$ and we use the same argument employed above. Case (2) [_$w_{j}\leq 2^{\ell}$_] In this case we have $k=\ell+1$. Denote with $\widetilde{H_{i}}$ the total height of shelves in $B_{i}$. Then we have $\widetilde{H_{\ell}}\geq T_{j}-H_{j}\geq\widetilde{T}_{j}-H_{j}\geq 6H_{j}$, otherwise we could pack $p_{j}$ in $B_{\ell}$. Thus, we can apply Lemma 15 and conclude that the box $B_{\ell}$ of size $2^{\ell}\times T_{j}$ is filled with constant density. Here we have two cases. Case (2.1) [_$\widetilde{H_{k}}\leq T_{j}$_] In this case we have $\textsc{Alg}=O(2^{k}T_{j})$ and, thanks to the constant density packing of $B_{\ell}$ we have $\Sigma_{j}=\Theta(2^{\ell}\widetilde{H_{\ell}})=\Theta(2^{k}T_{j})$. Since $\textsc{Opt}\geq\Sigma_{j}$, we get $\textsc{Alg}=O(\textsc{Opt})$. Case (2.2) [_$\widetilde{H_{k}} >T_{j}$_] In this case we still have $\textsc{Alg}=O(2^{k}\widetilde{H_{k}})$. Moreover, $\widetilde{H_{k}}=O(H_{n}+\Sigma_{n}/2^{k})$, in fact if $2^{s-1}<H_{n}\leq 2^{s}$, then the total height of sparse shelves is $\sum_{i\leq s}2^{i}=2^{s+1}=O(H_{n})$. Furthermore, dense shelves are filled with constant density, therefore their total height is at most $O(\Sigma_{n}/2^{k})$. Finally, we need to show that $2^{k}=O(\min\\{\sqrt[4]{\Sigma_{n}},\sqrt{n}\\})$. Let $T_{j}=\widetilde{T}_{j^{\prime}}$, we have two cases. Case (2.2.1) [_$\Sigma_{j^{\prime}} <{j^{\prime}}^{2}$_] We have $\widetilde{T}_{j^{\prime}}\geq\Sigma_{j^{\prime}}^{3/4}$. And thanks to the constant density packing of $B_{\ell}$, we have also $2^{k}\Sigma_{j^{\prime}}^{3/4}=O(2^{\ell}T_{j})=O(\Sigma_{j^{\prime}})$. Dividing both sides by $\Sigma_{j^{\prime}}^{3/4}$ we get $2^{k}=O(\sqrt[4]{\Sigma_{j^{\prime}}})$. Case (2.2.2) [_$\Sigma_{j^{\prime}}\geq{j^{\prime}}^{2}$_] In this case we have $\widetilde{T}_{j^{\prime}}\geq H_{j^{\prime}}\sqrt{j^{\prime}}$. Using the constant density argument we get $2^{k}H_{j^{\prime}}\sqrt{j^{\prime}}=O(2^{k}\widetilde{T}_{j^{\prime}})=O(\Sigma_{j^{\prime}})\leq O(j^{\prime}\cdot WH_{j^{\prime}})$. Dividing both sides by $H_{j^{\prime}}\sqrt{j^{\prime}}$ we obtain $2^{k}=O(W\sqrt{j^{\prime}})$. Therefore, we have $2^{k}=\begin{cases}O(\sqrt[4]{\Sigma_{j^{\prime}}})&\text{ if }\Sigma_{j^{\prime}}<{j^{\prime}}^{2}\\\ W\sqrt{j^{\prime}}&\text{otherwise.}\end{cases}$ In the end notice that, thanks to the long edge hypotheses $H_{n}\leq\Sigma_{n}$, thus Alg $\displaystyle=O\left(2^{k}\widetilde{H_{k}}\right)=O\left(2^{k}H_{n}+\Sigma_{n}\right)$ $\displaystyle=O\left(\mathbbm{1}_{\\{\Sigma_{j^{\prime}}<{j^{\prime}}^{2}\\}}\cdot H_{n}\sqrt[4]{\Sigma_{j^{\prime}}}+\mathbbm{1}_{\\{\Sigma_{j^{\prime}}\geq{j^{\prime}}^{2}\\}}\cdot WH_{n}\sqrt{j^{\prime}}+\Sigma_{n}\right)$ $\displaystyle\leq\textsc{Opt}\cdot O\left(\min\left\\{\sqrt[4]{\Sigma_{j^{\prime}}},\sqrt{j^{\prime}}\right\\}\right)=\textsc{Opt}\cdot O\left(\min\left\\{\sqrt[4]{\textsc{Opt}},\sqrt{n}\right\\}\right).$ ∎ ## 4 Further questions It is natural to consider problems where the given pieces are more general, such as convex polygons. Here, we may allow the pieces to be rotated by arbitrary angles. In that case, it follows from the technique described by Alt [2] that one can obtain a constant competitive ratio for computing a packing with a minimum perimeter bounding box: For each new piece, we rotate the piece so that a diameter of the piece is horizontal. We then use the algorithm BrickRotation to pack the bounding boxes of the pieces. Since the area of each piece is at least half of the area of its bounding box, the density of the produced packing is at least half of the density of the packing of the bounding boxes. This results in an increase of the competitive ratio by a factor of at most $\sqrt{2}$. For the problem of minimizing the perimeter of the bounding box (or convex hull) with convex polygons as pieces and only translations allowed, we do not know if it is possible to get a competitive ratio of $O(1)$, and this seems to be a very interesting question for future research. 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Let $(X,g)$ be a compact manifold with boundary $M^n$ and $\sigma$ a defining function of $M$. To these data, we associate natural conformally covariant polynomial one-parameter families of differential operators $C^\infty(X) \to C^\infty(M)$. They arise through a residue construction which generalizes an earlier construction in the framework of Poincaré-Einstein metrics and are referred to as residue families. Residue families may be viewed as curved analogs of conformal symmetry breaking differential operators. The main ingredient of the definition of residue families are eigenfunctions of the Laplacian of the singular metric $\sigma^{-2}g$. We prove that if $\sigma$ is an approximate solution of a singular Yamabe problem, i.e., if $\sigma^{-2}g$ has constant scalar curvature $-n(n+1)$, up to a sufficiently small remainder, these families can be written as compositions of certain degenerate Laplacians (Laplace-Robin operators). This result implies that the notions of extrinsic conformal Laplacians and extrinsic $Q$-curvature introduced in recent works by Gover and Waldron can naturally be rephrased in terms of residue families. This new spectral theoretical perspective allows easy new proofs of several results of Gover and Waldron. Moreover, it enables us to relate the extrinsic conformal Laplacians and the critical extrinsic $Q$-curvature to the scattering operator of the asymptotically hyperbolic metric $\sigma^{-2}g$ extending the work of Graham and Zworski. The relation to the scattering operators implies that the extrinsic conformal Laplacians are self-adjoint. We describe the asymptotic expansion of the volume of a singular Yamabe metric in terms of Laplace-Robin operators (reproving results of Gover and Waldron). We also derive new local holographic formulas for all extrinsic $Q$-curvatures (critical and sub-critical ones) in terms of renormalized volume coefficients, the scalar curvature of the background metric, and the asymptotic expansions of eigenfunctions of the Laplacian of the singular metric $\sigma^{-2}g$. These results naturally extend earlier results in the Poincaré-Einstein case. Furthermore, we prove a new formula for the singular Yamabe obstruction $\B_n$. The simple structure of these formulas shows the benefit of the systematic use of so-called adapted coordinates. We use the latter formula for $\B_n$ to derive explicit expressions for the obstructions in low-order cases (confirming earlier results). Finally, we relate the obstruction $\B_n$ to the supercritical $Q$-curvature $\QC_{n+1}$. [2020]Primary 35J30 53B20 53B25 53C18; Secondary 35J70 35Q76 53C25 58J50 August 27, 2024 § INTRODUCTION Conformal differential geometry studies natural geometric quantities associated to Riemannian (and pseudo-Riemannian) manifolds (such as curvature invariants and natural differential operators) which transform nicely under conformal changes of the metric. In recent years, it has developed in profound and surprising ways, which are connected to the spectral theory of Laplace-type operators, scattering theory, holographic principles (AdS-CFT correspondence), Cartan geometry, non-linear partial differential equations, and representation theory. Particular powerful tools are tractor calculus and the Poincaré-Einstein and ambient metrics in the sense of Fefferman-Graham <cit.>. A central role in those parts of geometric analysis which are related to conformal differential geometry play the conformally invariant powers of the Laplacian, which are known as GJMS-operators <cit.> and the related Branson's $Q$-curvatures <cit.>. The GJMS-operators $P_{2N}(h)$ act on the space $C^\infty(M)$ on any Riemannian manifold $(M,h)$. They are of the form $\Delta_h^N + \mbox{lower-order terms}$, where the lower-order terms are given in terms of the metric and covariant derivatives of its curvature. We recall that for general $h$, the operators $P_{2N}(h)$ exist for all orders $2N$ if $n$ is odd but only for $2N \le n$ if $n$ is even. The GJMS-operators $P_{2N}$ generalize the conformal Laplacian (Yamabe P_2 = \Delta - \left(\frac{n}{2}-1\right) \J, \quad 2 (n-1) \J = \scal and the Paneitz operator P_4 = \Delta^2 - \delta ((n-2) \J - 4 \Rho) d + \left(\frac{n}{2}-2\right) \left(\frac{n}{2}\J^2 - 2 \|\Rho\|^2 - \Delta (\J)\right), where $\Rho$ is the Schouten tensor of $h$ (for the notation, we refer to Section <ref>). The scalar curvature quantities Q_2 = \J \quad \mbox{and} \quad Q_4 = \frac{n}{2} \J^2 - 2 \|\Rho\|^2 - \Delta(\J) are the lowest-order cases of Branson's $Q$-curvatures. In particular, the quantity $Q_4$ plays a central role in geometric analysis <cit.>. In <cit.>, one of the authors analyzed the structure of GJMS-operators and Branson's $Q$-curvatures of a Riemannian manifold $(M^n,h)$ through a theory of conformally covariant polynomial one-parameter families of differential operators $C^\infty(X) \to C^\infty(M)$, where $X^{n+1}$ is a manifold of one more dimension. Such an extrinsic perspective on objects living on $M$ is sometimes referred to as a holographic point of view following <cit.>. These families of differential operators can be regarded as curved versions of certain intertwining operators in representation theory. Following the framework introduced in a series of works by T. Kobayashi and his coauthors, these intertwining operators are now known as conformal symmetry breaking operators $C^\infty(S^{n+1}) \to C^\infty(S^n)$. The notion is motivated by the fact that they are equivariant only with respect to the subgroup of the conformal group of $S^{n+1}$ consisting of diffeomorphisms that leave the equatorial subsphere $S^n \hookrightarrow S^{n+1}$ invariant. Such intertwining operators acting on functions have analogs acting on sections of homogeneous vector bundles on spheres as well as in contexts where other groups replace the conformal group of the sphere. We refer the interested reader to <cit.>. In the curved case, the conformal covariance property takes the role of the intertwining property, and the definition of the operators uses a Poincaré-Einstein metric in the sense of Fefferman and Graham <cit.>. The curved versions of the symmetry breaking operators then are defined in terms of the asymptotic expansions of eigenfunctions of the Laplacian of a Poincaré-Einstein metric on $X$ and the resulting so-called renormalized volume coefficients. The latter quantities are curvature invariants of a metric on $M$, the study of which originally was motivated by the AdS/CFT-correspondence <cit.>. Since the construction of the curved versions of the symmetry breaking differential operators can be expressed as a residue construction, these were termed residue families in <cit.>. A key feature of the theory of residue families is that each GJMS-operator $P_{2N}(h)$ comes along with a residue family (see (<ref>)), and one can effectively utilize the family parameter to study the structure of these special values <cit.>. Another basic perspective on GJMS-operators and $Q$-curvatures of a metric $h$ on $M$ was developed in <cit.> by showing how the scattering operator $\Sc(\lambda)$ of the Laplacian of a Poincaré-Einstein metric on $X$ in normal form relative to $h$ naturally encodes both quantities. The scattering operator $\Sc(\lambda)$ is a one-parameter meromorphic family of conformally covariant pseudo-differential operators, which may be regarded as a curved version of the Knapp-Stein intertwining operator for spherical principal series representations. Thus, both $\Sc(\lambda)$ and residue families describe GJMS-operators and $Q$-curvatures. But whereas the former are meromorphic families of pseudo-differential operators, the latter are polynomial families of differential In <cit.>, Gover and Waldron developed an alternative perspective on the above holographic descriptions of GJMS-operators and $Q$-curvatures. In this approach, the conformal geometry of the metric $h$ on $M$ and the associated Poincaré-Einstein metric $g_+$ on $X$ play a central role. In particular, this leads to a deeper understanding of the role of the Einstein property of $g_+$ in constructions of GJMS-operators. The conformal tractor calculus on $X$ naturally associates a conformally covariant polynomial one-parameter family of second-order differential operators on functions on $X$ to a metric $g$ on $X$ and a defining function $\sigma$ of the hypersurface $\iota^: M \hookrightarrow X$. Later, Gover and Waldron termed this operator the Laplace-Robin operator <cit.>.[In <cit.>, the Laplace-Robin operators are regarded as operators acting on densities.] We shall follow this terminology here. The Laplace-Robin operators degenerate on $M$ in the sense that the coefficients of its leading parts vanish on $M$. The data $(g,\sigma)$ induce a metric $\iota^(g)$ on $M$ and a singular metric $\sigma^{-2} g$ on the complement of $M$ in $X$. It is the latter metric that generalizes the Poincaré-Einstein metric. Now, if $\sigma^{-2} g$ is Einstein, then certain compositions of Laplace-Robin operators on $X$ reduce to GJMS-operators on $M$. Even if the Einstein condition is violated, analogous compositions of (renormalized) Laplace-Robin operators still reduce to conformally covariant operators on $M$. We emphasize that the notion of conformal covariance in the latter statement concerns constructions that depend on a metric $g$ on $X$ and a defining function $\sigma$ of the boundary $M$ of $X$, i.e., constructions which are invariant under conformal changes of both $g$ and $\sigma$: $\hat{g} = e^{2\varphi} g$ and $\hat{\sigma} = e^\varphi \sigma$. Note that $\sigma^{-2} g$ is invariant under such substitutions. Now a version of the singular Yamabe problem asks to find (for a given metric $g$ on $X$) a defining function $\sigma$ so that the scalar curvature of $\sigma^{-2}g$ is a negative constant. This conformally invariant condition for pairs $(g,\sigma)$ actually determines the defining function $\sigma$ by the background metric $g$ (at least to some extent) and the embedding $\iota$. If $\sigma$ solves the singular Yamabe problem, the above constructions of conformally covariant operators on $M$ only depend on the metric $g$ and the embedding $\iota$. In other words, these operators live on $M$, their definition depends on the embedding of $M \hookrightarrow X$, and they are conformally covariant with respect to conformal changes of the background metric $g$ on $X$. Gover, Waldron, and coauthors used this idea in a series of works to develop a theory of conformal invariants of hypersurfaces. To a large extent, this theory rests on the conformal tractor calculus <cit.>. In recent years, Laplace-Robin operators independently also appeared in other contexts, albeit not under this name. As described above, their original and most general definition comes from conformal tractor calculus. But already in <cit.>, it was observed that a special case was crucial in <cit.> in the setting of Poincaré-Einstein metrics. Some other special cases were found to be interesting in representation theory. Clerc <cit.> proved that the symmetry breaking differential operators $C^\infty(\R^{n+1}) \to C^\infty(\R^n)$ introduced in <cit.> also arise as compositions of some second-order intertwining operator for spherical principal series representations on functions on $\R^{n+1}$. In <cit.>, an attempt to generalize the result of Clerc led to a description of the general residue families of <cit.> as compositions of some second-order operators, which turned out to be Laplace-Robin operators. In other words, the latter result could be viewed as a curved version of the quoted result of Clerc. In <cit.>, the notions of spectral shift operators and Bernstein-Sato operators are used in the context of symmetry breaking operators on functions, differential forms, and spinors. These results suggest extensions of the notion of Laplace-Robin operators on forms. Such extensions on forms using tractor calculus have been developed in the monograph <cit.>. The present work sheds new light on the above results of Gover and Waldron. We connect the various strands of development by introducing an extension of the notion of residue families. Following the principles of <cit.>, we prove a series of new results, and - as by-products - we confirm and give alternative proofs of some already known results. The treatment replaces tractor calculus and distributional calculus by more classical arguments. Among other things, we stress the role of the scattering operator extending <cit.>. We hope that our methods enhance the understanding of the We continue with a more detailed description of the main new results. We consider a compact Riemannian manifold $(X,g)$ with boundary $M$. Let $\iota: M \hookrightarrow X$ be the canonical embedding. Let $h = \iota^(g)$ be the induced metric on $M$ and $\sigma \in C^\infty(X)$ a boundary defining function, i.e., $\sigma \ge 0$ on $X$, $\sigma^{-1}(0)=M$ and $d\sigma\|_M \ne 0$. The data $(g,\sigma)$ define a Laplace-Robin operator <cit.> \begin{equation} L(g,\sigma;\lambda) \st (n+2\lambda-1) (\nabla_{\grad_g (\sigma)} + \lambda \rho) - \sigma (\Delta_g + \lambda \J), \; \lambda \in \C \end{equation} on $C^\infty(X)$. Here $2n \J = 2n \J^g = \scal^g$ and $(n+1) \rho(g,\sigma) = -\Delta_g (\sigma) - \sigma \J$. The significance of this operator rests on its conformal covariance property \begin{equation}\label{CTL-basic} L(\hat{g}, \hat{\sigma}; \lambda) \circ e^{\lambda\varphi} = e^{(\lambda-1)\varphi} \circ L(g,\sigma;\lambda), \; \lambda \in \C \end{equation} for all conformal changes $(\hat{g},\hat{\sigma})=(e^{2\varphi}g,e^{\varphi}\sigma)$ with $\varphi \in C^\infty(X)$. This property implies that all \begin{equation}\label{LR-basic} \st L(g,\sigma;\lambda\!-\!N\!+\!1) \circ \cdots \circ L(g,\sigma;\lambda), \; N \in \N \end{equation} are conformally covariant: $L_N(\lambda)$ shifts the conformal weight $\lambda$ into $\lambda-N$. The notion of residue families has its origin in a far-reaching generalization of Gelfand's distribution $r_+^\lambda$ <cit.>. As the first generalization of $r_+^\lambda$ we consider the distribution \left\langle M(\lambda),\psi \right\rangle = \int_X \sigma^\lambda \psi dvol_g, \; \Re(\lambda) > -1 with $\psi \in C_c^\infty(X)$. Later $\sigma$ will be a solution of the singular Yamabe problem, i.e., $\sigma^{-2}g$ has constant scalar curvature $-n(n+1)$.[Then $\sigma$ is not necessarily $C^\infty$ up to the boundary.] Note that $\left\langle M(0),1 \right\rangle = \int_X dvol_g$. The one-parameter family of distributions $M(\lambda)$ admits a meromorphic continuation to $\C$ with simple poles in $-\N$. In order to describe the residues of $M(\lambda)$, we introduce coordinates on $X$ near its boundary as follows. Let $\mathfrak{X} \st \NV/\|\NV\|^2$ with $\NV = \grad (\sigma)$ and define a local diffeomorphism \eta: I \times M \to X, \; (s,x) \mapsto \Phi^s_{\mathfrak{X}}(x), \quad I = (0,\varepsilon) using the flow $\Phi^s_\mathfrak{X}$ of $\mathfrak{X}$. Such coordinates will be called adapted coordinates. Then $\eta^(\sigma) = s$ and $\sigma^\lambda$ pulls back to $s^\lambda$. Moreover, it holds dvol_{\eta^(g)} = v(s) ds dvol_h with some $v \in C^\infty(I \times M)$. Hence studying the residues of $M(\lambda)$ reduces to studying the residues of \left \langle M(\lambda),\psi \right\rangle = \int_I \int_M s^\lambda v(s) \psi(s,x) ds dvol_h. Now Gelfand's formula \Res_{\lambda=-N-1} \left(\int_0^\infty r^\lambda \psi dr \right) = \frac{\psi^{(N)}(0)}{N!} shows that \begin{equation}\label{M-res-g} \Res_{\lambda=-N-1} \left \langle M(\lambda),\psi \right \rangle = \frac{1}{N!} \int_M (v\psi)^{(N)}(0) dvol_h. \end{equation} In particular, if $\psi = 1$ near the boundary, then \begin{equation}\label{M-res} \Res_{\lambda=-N-1} \left \langle M(\lambda),\psi \right \rangle = \int_M v_N dvol_h, \end{equation} where the expansion $v(s) = \sum_{j \ge 0} s^j v_j$ defines the coefficients $v_j(g,\sigma) \in C^\infty(M)$. The above definitions immediately imply that the residue of $\left \langle M(\lambda),1\right\rangle$ at $\lambda=-n-1$ is a conformal invariant in the sense that I(\hat{g},\hat{\sigma}) = I(g,\sigma), \quad I(g,\sigma) \st \int_M v_n dvol_g. where $\hat{g} = e^{2\varphi}g$ and $\hat{\sigma} = e^{\varphi} \sigma$. Moreover, the formula \left\langle M(\lambda),1 \right\rangle = \int_X \sigma^{\lambda+n+1} dvol_{\sigma^{-2}g}, \; \Re(\lambda) > -1 suggests to regard the finite part of $\left\langle M(\lambda),1\right\rangle$ at $\lambda = -n-1$ as a renormalized volume of the singular metric $\sigma^{-2}g$. In fact, this should be called the Riesz renormalization of the volume of $\sigma^{-2} g$ <cit.>. In Theorem <ref> and Theorem <ref>, we shall return to the issue of renormalized volumes. We shall see that the residues in (<ref>) for $N \le n$ determine the singular terms in the Hadamard renormalization of the volume of $\sigma^{-2}g$. Similar renormalization techniques have also been used in the context of Möbius invariant energies of knots and their generalizations <cit.>. Now we continue with a definition of residue families. Here we assume that $\|d\sigma\|^2_g = 1$ on $M$. This assumption implies that the metric $\sigma^{-2}g$ is asymptotically hyperbolic. The data $(g,\sigma)$ give rise to a holomorphic one-parameter family \begin{equation}\label{MU} \lambda \mapsto \langle M_u(\lambda),\psi \rangle \st \int_X \sigma^\lambda u \psi dvol_g, \; \Re(\lambda) \gg 0 \end{equation} of distributions $M_u(\lambda)$ on $X$. Here $\psi$ is a test function on $X$ with support up to the boundary, and the additional datum $u$ is an eigenfunction of the Laplacian of the singular metric $\sigma^{-2} g$: -\Delta_{\sigma^{-2} g} u = \mu (n-\mu) u, \; \Re(\mu) = n/2, \, \mu \ne n/2 with boundary value $f \in C^\infty(M)$. In particular, it holds $M(\lambda) = M_1(\lambda)$ since $u=1$ is an eigenfunction of $\Delta_{\sigma^{-2}g}$ with eigenvalue $0$ and boundary value $1$. Since $\sigma^{-2}g$ is asymptotically hyperbolic, there is an eigenfunction $u$ for any $f \in C^\infty(M)$ so that in its asymptotic expansion $f$ defines one of the leading terms. Now $M_u(\lambda)$ admits a meromorphic continuation to $\C$ with simple poles in $\{ -\mu-N-1, \, \|\, N \in \N \}$. For $N \in \N_0$, we set \begin{equation}\label{res-fam-basic} \D_N^{res}(g,\sigma;\lambda) \st N!(2\lambda\!+\!n\!-\!2N\!+\!1)_N \delta_N(g,\sigma;\lambda\!+\!n\!-\!N), \end{equation} where $(a)_N = a (a+1) \cdots (a+N-1)$ is the Pochhammer symbol and \Res_{\lambda=-\mu-1-N} (\langle M_u(\lambda), \psi \rangle) = \int_M f \delta_N(g,\sigma;\mu)(\psi) dvol_h for a meromorphic family $\delta_N(g,\sigma;\mu)$ of differential operators $C^\infty(X) \to C^\infty(M)$ of order $\le N$. $M_u(\lambda)$ should be regarded as a further generalization of Gelfand's distribution $r_+^\lambda$ <cit.>. The factor in (<ref>) guarantees that the residue family $\D_N^{res}(\lambda)$ is polynomial in $\lambda$. Its degree equals $2N$. It easily follows from these definitions that $\D_N^{res}(\lambda)$ satisfies a conformal transformation law which is analogous to that of $L_N(\lambda)$. In view of $M(\lambda) = M_1(\lambda)$, the residue formula (<ref>) can be restated as \int_M \D_N^{res}(g,\sigma;0)(\psi) dvol_h \sim \int_M (v \psi)^{(N)}(0) dvol_h (for $N < n$). The above definition generalizes the notion of residue families introduced in <cit.>. For the details, we refer to Section <ref>. We also emphasize that while $L_N(\lambda)$ maps functions on $X$ to functions on $X$, residue families map functions on $X$ to functions on the boundary $M$. The families $L_N(g,\sigma;\lambda)$ and $D_N^{res}(g,\sigma;\lambda)$ are the main objects of the present paper. These objects and most related discussions will only depend on approximations of the data $(g,\sigma)$ in a sufficiently small neighborhood of the boundary $M$. As a preparation for the following discussion, we briefly recall the role of residue families for Poincaré-Einstein metrics in <cit.>. These polynomial families of differential operators are defined through a residue construction as in (<ref>), where $g_+ = r^{-2}(dr^2 + h_r)$ is a Poincaré-Einstein metric on $X = (0,\varepsilon) \times M$ in normal form relative to a given metric $h$ on $M$. Let $\iota: M \hookrightarrow X$ be the embedding $m \mapsto (0,m)$. The theory in <cit.> deals with approximations of Poincaré-Einstein metrics which are completely determined by the metric $h$. In particular, for even $n$, these define even-order residue families $D_{2N}^{res}(h;\lambda)$ of order $2N \le n$. Since \begin{equation}\label{GJMS-res} D_{2N}^{res}\left(h;-\frac{n}{2}+N\right) = P_{2N}(h) \iota^, \end{equation} the residue family $D_{2N}^{res}(h;\lambda)$ can be regarded as a perturbation of the GJMS-operator $P_{2N}(h)$. Moreover, residue families satisfy systems of recursive relations that involve lower-order residue families and GJMS-operators. These imply recursive relations among GJMS-operators and recursive relations for $Q$-curvatures. In other words, residue families may be viewed as a device to study the GJMS-operators and $Q$-curvatures on the boundary $M$. The above more general construction deals with a general metric $g$ on $X$. It induces a metric $h = \iota^(g)$ on $M$, but of course is not determined by $h$. In that case, the resulting residue families depend on the metric $g$ and a boundary defining function $\sigma$. To get residue families, which are determined only by the metric $g$ and the embedding $\iota$, we put an extra condition on $\sigma$. At the same time, we restrict considerations to residue families for $N \le n$. Similarly, as in the Poincaré-Einstein case, this restriction means that it suffices to deal with finite approximations of true eigenfunctions. This way, we obtain specific conformally covariant polynomial one-parameter families of differential operators $C^\infty(X) \to C^\infty(M)$. However, we emphasize that the problem of describing all such conformally covariant families is more complicated <cit.>. Now, following <cit.>, the extra condition which we pose is that $\sigma$ solves a singular Yamabe problem. The version of that problem of interest here asks to find a defining function $\sigma$ of $M$ so that $\scal(\sigma^{-2}g) = -n(n+1)$. By <cit.> such $\sigma$ exist and are unique. However, in general, $\sigma$ is not smooth up to the boundary. More precisely, the existence of smooth solutions is obstructed by a conformally covariant scalar curvature invariant $\B_n$ called the singular Yamabe obstruction. Although this lack of smoothness will only play a minor role for our main purposes, we also have some new results for the invariant $\B_n$. By the conformal transformation law of scalar curvature, the condition that $\sigma$ solves the singular Yamabe problem can be restated as \SC(g,\sigma) \st \|d\sigma\|_g^2 + 2 \sigma \rho \stackrel{!}{=} 1. This role of the functional $\SC(g,\sigma)$ also implies its conformal invariance: \SC(\hat{g},\hat{\sigma}) = \SC(g,\sigma). The main results of the present work only require us to assume that $\sigma$ satisfies the weaker condition \begin{equation}\label{condition-Y} \SCY: \hspace{0.5cm} \SC(g,\sigma) = 1 + \sigma^{n+1} R_{n+1} \qquad \qquad \end{equation} with a smooth remainder term $R_{n+1}$. The restriction of the remainder term to $\sigma = 0$ then defines the singular Yamabe obstruction $\B_n$. For more details and background on the singular Yamabe problems, we refer to Section <ref>. Now we are ready to state the first main result. Let $\iota: M \hookrightarrow X$ be the embedding of $M$ into $X$. Let $\N \ni N \le n$ and assume that the condition $\SCY$ is satisfied. Then \begin{equation}\label{equivalence} \iota^ L_N(g,\sigma;\lambda) = \D_N^{res}(g,\sigma;\lambda), \; \lambda \in \C. \end{equation} Some comments on this result are in order. The identity (<ref>) is to be interpreted as an identity of operators acting on smooth functions with support in a sufficiently small neighborhood of $M$. Although the assumptions guarantee that the operators on both sides of (<ref>) only depend on the metric $g$ and the embedding $\iota$, we keep the notation of the general case indicating the dependence of the operators on $g$ and $\sigma$. If $g = r^2 g_+$ is a Poincaré-Einstein metric in normal form relative to the metric $h$ on $M$, then the equality (<ref>) was proven in <cit.>.[<cit.> uses a different normalization of residue families.] In particular, if $g$ is the Euclidean metric on $\R^{n+1}$, then Theorem <ref> reduces to the main result of <cit.> (for more details, we refer to <cit.>). Theorem <ref> is a consequence of the following identity. For any boundary defining function $\sigma \in C^\infty(X)$, it \begin{equation}\label{M-CF} L(g,\sigma;\lambda) + \sigma^{\lambda-1} \circ \left(\Delta_{\sigma^{-2}g} - \lambda(n\!+\!\lambda) \id \right) \circ \sigma^{-\lambda} = \lambda (n\!+\!\lambda) \sigma^{-1}(\SC(g,\sigma)-1) \id,\; \lambda \in \C. \end{equation} Again, Theorem <ref> is to be interpreted as an identity near the boundary of $X$. We apply Theorem <ref> to prove the existence of a meromorphic continuation of the family $M_u(\lambda)$ using a Bernstein-Sato argument. We recall that for any polynomial $p$, the classical Bernstein-Sato argument <cit.> proves the existence of a meromorphic distribution-valued function $p^\lambda$ by using the functional equation D(\lambda) (p^{\lambda+1}) = b(\lambda) p^\lambda for some polynomial $b(\lambda)$ and a family $D(\lambda)$ of differential operators. Here $\sigma^\lambda u$ takes the role of $p^\lambda$ and $L(\lambda)$ takes the role of the family $D(\lambda)$. The Bernstein-Sato argument yields formulas for the residues and proves Theorem <ref>. The conjugation formula in Theorem <ref> easily implies the commutator relations \begin{equation}\label{sl2} L_N(g,\sigma;\lambda) \circ \sigma - \sigma \circ L_N(g,\sigma;\lambda\!-\!1) = N(2\lambda\!+\!n\!-\!N) L_{N-1}(g,\sigma;\lambda\!-\!1) \end{equation} if $\SC(g,\sigma)=1$. The relation (<ref>) was discovered in <cit.>, where the special case $N=1$ is regarded as one commutator relation in a basic $sl(2)$-structure. In turn, it follows that for $2\lambda = -n+N$ the operator $L_N(g,\sigma;\lambda)$ reduces to a tangential operator $\PO_N(g,\sigma)$ on $M$. By Theorem <ref>, we obtain \begin{equation}\label{PO-Dres} \PO_N(g,\sigma) \iota^* = \D_N^{res}\left(g,\sigma;\frac{-n+N}{2}\right). \end{equation} In other words, each operator $\PO_N(g,\sigma)$ on $M$ comes with a polynomial one-parameter family. The definition of $\PO_N$ in terms of $L_N$ is due to <cit.>. The spectral theoretic description (<ref>) of $\PO_N$ will be shown to have a series of significant consequences. The conformal covariance of the families $L_N(\lambda)$ implies that the operators $\PO_N$ on $C^\infty(M)$ are conformally covariant in the sense that e^{\frac{n+N}{2} \iota^(\varphi)} \circ \PO_N(\hat{g},\hat{\sigma}) = \PO_N(g,\sigma) \circ e^{\frac{n-N}{2} \iota^ (\varphi)}, \; \varphi \in C^\infty(X). For $N \le n$, these conformally covariant operators are completely determined by the metric $g$ and the embedding $M \hookrightarrow X$. Following <cit.>, we call them extrinsic conformal Laplacians. The notion is motivated by the fact that for even $N$ the leading term of $\PO_N(g,\sigma)$ is given by a constant multiple of $\Delta_h^{N/2}$, where $h =\iota^(g)$ (see Theorem <ref>). If the background metric $g$ is the conformal compactification $r^2 g_+$ of a Poincaré-Einstein metric in normal form relative to $h$, then these even-order operators actually are constant multiples of GJMS-operators of $h$. But, in general, the operators $\PO_N(g,\sigma)$ depend on the metric $g$ in a neighborhood of $M$ and the embedding $M \hookrightarrow X$. The following result makes the leading parts of all extrinsic conformal Laplacians explicit. Its proof rests on the spectral theoretic interpretation (<ref>) of $\PO_N$. It holds \begin{equation}\label{LT-P-even-M} \PO_{2N} = (2N\!-\!1)!!^2 \Delta^N + LOT \end{equation} for $2N \le n$ and \begin{equation}\label{LT-P-odd-M} \PO_N = (2N\!-\!2) (N\!-\!1)! \sum_{r=0}^{\frac{N-3}{2}} m_N(r) \Delta^r \delta(\lo d) \Delta^{\frac{N-3}{2}-r} + LOT \end{equation} for odd $N$ with $n \ge N \ge 3$ with rational coefficients given by (<ref>). Here $\Delta$ is the Laplacian of $h$, and $\lo$ is the trace-free part of the second fundamental form $L$. LOT refers to terms of order $2N-2$ and $N-3$, respectively. In the Poincaré-Einstein case, <cit.> identifies the operator $\PO_{2N}$ with a constant multiple of the GJMS operator $P_{2N}$. An alternative proof of that result is given in <cit.>. In the general case, formula (<ref>) is also stated in <cit.> although the proof only refers to results in the Poincaré-Einstein case in <cit.>. Note also that in the Poincaré-Einstein case, the operators $\PO_N$ for odd $N$ vanish. Note that the order of $\PO_{2N}$ equals $2N$ and the order of $\PO_N$ for odd $N$ equals $N-1$ in the generic case. Now we use the operators $\PO_N$ to define analogs of Branson's $Q$-curvatures. Theorem <ref> implies that for $N < n$ the identity \PO_N(g,\sigma) (1) = \left(\frac{n-N}{2}\right) \QC_N(g,\sigma) defines a function $\QC_N(g,\sigma)$. These functions will be called the subcritical extrinsic $Q$-curvatures. The critical extrinsic $Q$-curvature $\QC_n(g,\sigma)$ can be defined either through a limiting argument from the subcritical ones or more elegantly by \begin{equation}\label{Q-crit-M} \QC_n^{res}(g,\sigma) = - \dot{\D}_n^{res}(g,\sigma;0)(1). \end{equation} We recall that the condition $\SCY$ guarantees that the quantities in(<ref>) are determined by the metric $g$ and the embedding $M \hookrightarrow X$. In the Poincaré-Einstein case, these definitions reduce to constant multiples of Branson's $Q$-curvatures. If $\SC(g,\sigma)=1$, it follows from Theorem <ref> that the definition (<ref>) is a special case of the notion of $Q$-curvature defined in <cit.>.[The numbering in the published version differs from that in the arXiv version.] By differentiating the conformal transformation law of $\D_n^{res}(g,\sigma;\lambda)$ (see Theorem <ref>) at $\lambda=0$, it follows that \begin{equation}\label{fundamental-M} e^{n\iota^(\varphi)} \QC_n(\hat{g},\hat{\sigma}) = \QC_n(g,\sigma) + \PO_n(g,\sigma)(\iota^(\varphi)) \end{equation} for $\varphi \in C^\infty(X)$. As noticed above, the metric $\sigma^{-2}g$ is asymptotically hyperbolic if $\|d\sigma\|^2_g=1$ on $M$. Associated to such metrics, there is a scattering operator $\Sc(\lambda)$ acting on $C^\infty(M)$. It describes the asymptotic expansion of eigenfunctions of the Laplacian of $\sigma^{-2}g$. In <cit.>, Graham and Zworski showed how the GJMS-operators $P_{2N}$ and the critical Branson $Q$-curvature $Q_n$ of a metric $h$ on $M$ are encoded in the scattering operator of a Poincaré-Einstein metric with the conformal class of $h$ as conformal infinity. The following result extends these interpretations to the present framework. Suppose that condition $\SCY$ is satisfied. Let $N \in \N$ with $2 \le N \le n$ and suppose that $(n/2)^2-(N/2)^2$ is not in the discrete spectrum of $-\Delta_{\sigma^{-2}g}$. Then \PO_N = 2 (-1)^N (N-1)! N! \Res_{\frac{n-N}{2}}(\Sc(\lambda)). The function $\Sc(\lambda)(1)$ is regular at $\lambda=0$ and its value at $\lambda=0$ is denoted by $\Sc(0)(1)$. Then \QC_n = 2 (-1)^n (n-1)! n! \Sc(0)(1). Note that the scattering operator $\Sc(\lambda)$ is a pseudo-differential operator with principal symbol 2^{2\lambda-n} \frac{\Gamma(\lambda-\frac{n}{2})}{\Gamma(-\lambda+\frac{n}{2})}\|\xi\|_h^{n-2\lambda}. This shows that the residues of $\Sc(\lambda)$ at $\lambda=\frac{n}{2}-N$ are caused by the $\Gamma$-factors and that the residue of its pole at $\lambda = \frac{n}{2}-N$ is an operator with principal part given by a constant multiple of $\Delta^N$. Moreover, Theorem <ref> yields an identification of this residue. In particular, these residues are elliptic operators. On the other hand, the above formula yields no information on the structure of the residues of $\Sc(\lambda)$ at $\frac{n-N}{2}$ for odd $N$. This makes the second part of Theorem <ref> interesting. Similarly as in <cit.>, the self-adjointness of $\Sc(\lambda)$ for $\lambda \in \R$ combined with Theorem <ref> implies Let $\SCY$ be satisfied. Then the operators $\PO_N(g)$ are formally self-adjoint with respect to the scalar product on $C^\infty(M)$ defined by $h$. For closed $M$, combining (<ref>) with the self-adjointness of $\PO_n$ and $\PO_n(1)=0$ shows that the integral \int_M \QC_n(g) dvol_h is a global conformal invariant. Next, we describe a local formula for the critical extrinsic $Q$-curvature. It extends the holographic formula for Branson's critical $Q$-curvature proved in <cit.>. The formulation of that result requires two more ingredients: renormalized volume coefficients $v_k$ and solution operators $\T_k(0)$. These data are defined in local coordinates. We choose a diffeomorphism $\eta$ between the product $[0,\varepsilon) \times M$ with coordinates $(s,x)$ and a neighborhood of $M$ in $X$. $\eta$ is defined by a renormalized gradient flow of $\sigma$. It satisfies $\eta^(\sigma) = s$. These coordinates will be called adapted coordinates. Then \eta^* (dvol_{\sigma^{-2}g}) = s^{-n-1} v(s,x) ds dvol_h, and the expansion $v(s,x) = \sum_{k\ge 0} v_k(x) s^k$ defines the coefficients $v_k$. Let $(v\J)_k$ be the coefficients in the analogous expansion of $v \eta^(\J)$, where $\J = \J^g = \scal^g/2n$. Secondly, the differential operators $\T_k(0)$ describe the asymptotic expansions of harmonic functions of the Laplacian of $\sigma^{-2} g$. The details of these definitions are given in Sections [Holographic formula for $\QC_n$] Let $n$ be even. If $\SCY$ is satisfied, then it holds \begin{equation}\label{Q-holo-form-M} \QC_n(g,\sigma) = (n\!-\!1)!^2 \sum_{k=0}^{n-1} \frac{1}{n\!-\!1\!-\!2k} \T_k^(g,\sigma;0) \left( (n\!-\!1)(n\!-\!k) v_{n-k} + 2k (v \J)_{n-k-2} \right), \end{equation} where $\J = \J^g$. For $k=n-1$, the second term in the sum is defined as $0$. The proof of Theorem <ref> generalizes a proof of the holographic formula for $Q_n$ for Poincaré-Einstein metrics given in <cit.>. The arguments rest on Theorem <ref> and an alternative description of residue families in terms of the coefficients $v_k$ and the operators $\T_k(\lambda)$. We conjecture that the result extends to odd $n$ (Conjecture <ref>). There are analogous formulas for all subcritical extrinsic $Q$-curvatures which extend <cit.>. Theorem <ref> implies the representation \QC_n = n! (n\!-\!1)! v_n + \sum_{k=1}^{n-1} \T_k^(0)(\cdots). Integration of this identity for closed $M$ implies the equality \begin{equation}\label{GI} \int_M \QC_n dvol_h = n! (n\!-\!1)! \int_M v_n dvol_h \end{equation} of global conformal invariants. This extends a result of <cit.> for Poincaré-Einstein metrics and reproves a special case of <cit.>. We shall also give a second proof of the identity (<ref>) which utilizes residue families and extends to odd $n$. We emphasize again that the integrands on both sides depend on the embedding of $M$ in $X$, i.e., these are global conformal invariants of the embedding. The integral on the right-hand side of (<ref>) appears as a coefficient of $\log \varepsilon$ in the asymptotic expansion of the volume \begin{equation}\label{reno-volume-M} vol_{\sigma^{-2}g}(\{\sigma > \varepsilon \}) = \int_{\sigma > \varepsilon} dvol_{\sigma^{-2}g} \end{equation} of the singular metric $\sigma^{-2}g$ for $\varepsilon \to 0$. This implies its conformal invariance. Through that interpretation of the right-hand side of (<ref>), this identity is a special case of <cit.>. The singular terms in this expansion of the volume are given by the integrals of the renormalized volume coefficients $v_k$ for $k <n$. All these integrals admit the uniform expressions \begin{equation}\label{HF-v} \int_{M} v_{k} dvol_h = (-1)^k \frac{(n\!-\!1\!-\!k)!}{ (n-1)! k!} \int_{M} \iota^ L_k(-n\!+\!k)(1) dvol_h \end{equation} in terms of Laplace-Robin operators. This reproves a special case of <cit.>. The above results for the expansion of the volumes (<ref>) have analogs for general defining functions $\sigma$. In that case, the operators $L_k$ are replaced by composition $\tilde{L}_k = (L \circ \SC^{-1})_k$ and the coefficient of $\log \varepsilon$ is a constant multiple of the integral of a curvature quantity $\tilde{\QC}_n(g,\sigma)$ (Theorem <ref>, (<ref>)). The total integral of $\tilde{\QC}_n$ is a conformal invariant of the pair $(g,\sigma)$ (Lemma <ref>). The integrals in (<ref>) are generalizations of the Willmore energy of a closed surface in $\R^3$. In fact, for a surface $M^2 \hookrightarrow X^3$ with Gauss curvature $K$, it holds 2 v_2 = \QC_2 = -K + \frac{1}{2} \|\lo\|^2 (by (<ref>), (<ref>) and $\J^h = K$). But if $X$ is the flat $\R^3$, then $H^2 - K = \|\lo\|^2$. Hence for closed $M \hookrightarrow \R^3$ the integral $\int_M v_2 dvol_h$ is a linear combination of the Euler characteristic $\chi(M)$ and the Willmore energy $\W$ Willmore energy \W = \int_M H^2 dvol_h. The Willmore energy also contributes to the rigid string action introduced in <cit.>, Finally, the usage of adapted coordinates leads to a new formula for the singular Yamabe obstruction $\B_n$ (see Section <ref> for its definition). In order to state that formula, we note that the pull-back $\eta^(g)$ of $g$ takes the form $a^{-1} ds^2 + h_s$ with some coefficient $a \in C^\infty ((0,\varepsilon) \times M)$ and a one-parameter family $h_s$ of metrics on $M$. For \mathring{v} \st dvol_{h_s}/dvol_h, it holds \frac{\mathring{v}'}{\mathring{v}} = \frac{1}{2} \tr (h_s^{-1} h_s'). Let $\J$ and $\rho$ denote the pull-backs of $\J^g$ and $\rho(g,\sigma)$ by $\eta$, respectively. The following result restates Theorem <ref>. [The obstruction $\B_n$] If $\sigma$ satisfies $\SCY$, then \begin{equation}\label{ob-magic-2} (n\!+\!1)! \B_n = -2\partial_s^n \left( \frac{\mathring{v}'}{\mathring{v}}\right)\|_0 + 4 \sum_{j=1}^n j \binom{n}{j} \partial_s^{j-1}(\rho)\|_0 \partial_s^{n-j} \left( \frac{\mathring{v}'}{\mathring{v}}\right)\|_0 - 2n \partial_s^{n-1}(\J)\|_0 \end{equation} for $n \ge 1$. For $n=1$, the identity (<ref>) yields $\B_1 = 0$. Moreover, the Taylor coefficients of $\rho$ obey a recursive relation in terms of the Taylor coefficients of $\mathring{v}'/\mathring{v}$ and $\J$ (Proposition <ref>). This result has several consequences of independent interest. For a flat background metric, the scalar curvature term on the right-hand side of (<ref>) vanishes, and the recursive structure of all terms implies that for even $n$ \begin{equation} (n\!+\!1)! \B_n = - 2 (n\!-\!1)!!/(n\!-\!2)!! \Delta^\frac{n}{2} (H) + LOT \end{equation} (Theorem <ref>), where LOT refers to terms of lower differential order. Up to the numerical coefficients, that description of the leading term of $\B_n$ was first formulated in <cit.> <cit.> (for general backgrounds). We also use Theorem <ref> to reproduce the known explicit formulas for $\B_2$ (for general backgrounds) and to calculate $\B_3$ for (conformally) flat backgrounds confirming earlier results. it is unclear what LOT means in GW we also can prove that the LOT contains at most n-1 derivatives of the curvature tensor of the background Moreover, Theorem <ref> and Theorem <ref> play a key role in the proof of the residue formula \begin{equation} \Res_{n=N-1} (\QC_N) = (-1)^{n-1} n! (n\!+\!1)! \frac{n}{2} \B_n \end{equation} for even $n$ (Theorem <ref>), where we regard $\QC_N$ as a function of $n$ which is singular at $n=N-1$. This formula relates $\B_n$ for even $n$ to the super-critical $Q$-curvature $\QC_{n+1}$. It proves a conjecture in <cit.>. We conjecture that this result extends to all dimensions. <cit.> established a coordinate-free formula for the Yamabe obstructions $\B_n$ in terms of tractor calculus constructions. This formula involves an analog of the critical extrinsic conformal Laplacian acting on the so-called normal tractor of the hypersurface $M$ as a key ingredient.[This result coincides with Theorem 7.7 in <arXiv:150602723v4>.] We finish this section with an outline of the content of the paper. We combine this review with additional comments on the relations of the new results to the literature. Section <ref> briefly recalls basic results on the version of the singular Yamabe problem of interest here. In Section <ref>, we establish Theorem <ref> and describe first consequences. This result is the basic link that connects the Laplace-Robin operators of Gover and Waldron with the spectral theory of the Laplacian of the singular metric $\sigma^{-2}g$. In Section <ref>, we describe some of the representation theoretical aspects of the Laplace-Robin operators. Section <ref> introduces the notion of adapted coordinates. This notion is basic for the whole work. Adapted coordinates give rise to the definition of renormalized volume coefficients $v_k$, which enables us to prove local formulas for $Q$-curvatures (Theorem <ref>) and to prove the formula for the obstruction $\B_n$ in Theorem <ref>. In this connection, we show that in adapted coordinates, the function $\rho$ obeys an extremely beneficial ordinary differential equation in the variable $s$, which implies recursive relations for the Taylor coefficients of $\rho$. We emphasize that Graham <cit.> utilizes a different notion of renormalized volume coefficients which are defined in terms of the normal exponential map. This notion seems to be less appropriate from the present point of view. Section <ref> introduces the notion of residue families. Here we extend earlier definitions in the context of Poincaré-Einstein metrics <cit.>. The constructions are built on basic facts in the spectral theory of asymptotically hyperbolic metrics, which are detailed in <cit.>. In particular, we use a version of a Poisson transform and introduce the scattering operator $\Sc(\lambda)$. We derive a formula for residue families in terms of the coefficients $v_k$ and the operators $\T_k(\lambda)$. Section <ref> contains the proof of Theorem <ref>. The identity (<ref>) is a direct consequence. Here we also prove that these identities can be regarded as consequences of a beautiful formula for the action of $L_N(-1)$ on the distribution $\sigma^(\delta) = \delta_M$ of $M$ (Theorem <ref>). This distributional formula reproves a basic technical result in <cit.>. In Section <ref>, we introduce the notion of extrinsic conformal Laplace operators $\PO_N$. Here we derive the spectral theoretical interpretation (<ref>) of $\PO_N$, use it to determine the leading terms of $\PO_N$ (Theorem <ref>) and recognize these operators as residues of the scattering operator (first part of Theorem <ref>). These results extend results of <cit.>. Section <ref> defines extrinsic $Q$-curvatures and establishes the second part of Theorem <ref> which extends another result of <cit.>. Moreover, we supply two proofs of the equality (<ref>). The integrated coefficients $v_k$ are shown to describe the singular coefficients in the asymptotic expansion of (<ref>). These expansions and their generalizations (Theorem <ref>) reprove results of <cit.>. In Section <ref>, we establish extensions of the holographic formulas for $Q$-curvatures in <cit.>. In particular, we prove Theorem <ref>. Section <ref> contains comments on further perspectives. In the main body of the text, we usually suppress detailed calculations and the discussion of examples and special cases. However, the reader can find this material in Section <ref>. It starts with an overview of its own. The results presented here may be used to gain a deeper understanding of the material. In particular, this section contains full details on low-order Yamabe obstructions, low-order cases of conformal Laplacians, extrinsic $Q$-curvatures, and renormalized volume coefficients. All proofs are independent of the literature. Hopefully, the attached list of symbols facilitates reading. Finally, we like to emphasize that, although the present work is deeply inspired by the pioneering works of Gover and Waldron, the current treatment is fully independent and self-contained. We also stress again that our perspective is (via the residue families and their applications) one of a spectral-theoretic nature exploiting the structure of eigenfunctions of the Laplacian. After the present work had been posted, the paper <cit.> again discussed extrinsic conformal Laplacians from the perspective of scattering theory. It defines extrinsic conformal Laplacians in terms of the scattering operator of the singular Yamabe metric $\sigma^{-2}g$ by mimicking the known relations between GJMS-operators and the scattering operator of Poincaré-Einstein metrics <cit.>. However, the relations between these definitions and the notions introduced by Gover and Waldron are established only here. Acknowledgments. The work on this project started during a visit of the first author at the University of Århus in autumn 2019. A large part of the paper's final version was written during a stay of the first author at IHES in early 2020. He is grateful to both organizations for financial support and very stimulating atmospheres. Finally, we thank the anonymous referee who provided valuable detailed comments on an earlier version of the manuscript. § GENERAL NOTATION $\N$ natural numbers $\N_0$ non-negative integers $(a)_N$ Pochhammer symbol $\mathfrak{X}(X)$ space of vector fields on $X$ $\N$ is the set of natural numbers, and $\N_0$ is the set of non-negative integers. For a complex number $a\in\C$ and an integer $N\in\N$, the Pochhammer symbol $(a)_N$ is defined by $(a)_N \st a(a+1) \cdots (a+N-1)$. We set $(a)_0 \st 1$. $R$ curvature tensor $C^{-\infty}(X)$ space of distributions on $X$ All manifolds are smooth. For a manifold $X$, $C^\infty(X)$ and $C_c^\infty(X)$ denote the respective spaces of smooth functions and smooth functions with compact support on $X$. If $X$ is a manifold with boundary, then $C^\infty(X)$ is the space of functions that are smooth up to the boundary. $C^{-\infty}(X)$ is the space of distributions on $X$. Let $\mathfrak{X}(X)$ be the space smooth vector fields on $X$. Metrics on $X$ usually are denoted by $g$. $dvol_g$ is the Riemannian volume element defined by $g$. The Levi-Civita connection of $g$ is denoted by $\nabla_X^g$ or simply $\nabla_X$ for $X \in \mathfrak{X}(X)$ if $g$ is understood. In these terms, the curvature tensor $R$ of the Riemannian manifold $(X,g)$ is defined by $R(X,Y)Z =\nabla_X \nabla_Y (Z) - \nabla_Y \nabla_X (Z) - \nabla_{[X.Y]}(Z)$ for vector fields $X,Y,Z \in \mathfrak{X}(X)$. We also set $\nabla_X (u) = \langle du,X \rangle$ for $X \in \mathfrak{X}(X)$ and $u \in C^\infty(X)$. $dvol_g$ volume element $\Omega^p$ space of $p$-forms $\nabla^g$ Levi-Civita connection of $g$ $\grad_g(u)$ gradient field $\delta^g$ divergence operator $\Delta_g$ Laplacian of $g$ For a metric $g$ on $X$ and $\sigma \in C^\infty(X)$, let $\grad_g(\sigma)$ be the gradient of $\sigma$ with respect to $g$, i.e., it holds $g(\grad_g(\sigma),V) = \langle d\sigma,V \rangle$ for all vector fields $V \in \mathfrak{X}(X)$. $g$ defines pointwise scalar products and norms $\|\cdot\|_g$ on $\mathfrak{X}(X)$ and on forms $\Omega^(X)$. Then $\|\grad_g(\sigma)\|_g^2 = \|d\sigma\|_g^2$. $\delta^g$ is the divergence operator on differential forms or symmetric bilinear forms. On forms, it coincides with the negative adjoint $-d^$ of the enaböe differential $d$ with respect to the Hodge scalar product defined by $g$. Let $\Delta_g = \delta^g d$ be the non-positive Laplacian on $C^\infty(X)$. On the Euclidean space $\R^n$, it equals $\sum_i \partial_i^2$. $\Ric^g$ Ricci tensor of $g$ $\scal^g$ scalar curvature of $g$ $\scal(g)$ scalar curvature of $g$ $\Rho^g$ Schouten tensor of $g$ A metric $g$ on a manifold $X$ with boundary $M$ induces a metric $h$ on $M$. Curvature quantities of $g$ and $h$ have the respective metric as an index if required by clarity. In particular, the scalar curvature of the metric $g$ on $X$ is denoted by $\scal^g$ or $\scal(g)$. $\Ric^g$ denote the Ricci tensor of $g$. On a manifold $(M,g)$ of dimension $n$, we set $\J^g =\frac{1}{2(n-1)} \scal^g$ if $n \ge 2$ and define the Schouten tensor $\Rho^g$ of $g$ by $(n-2)\Rho^g = \Ric^g - \J^g g$ if $n \ge 3$. $L$ second fundamental form Let $M$ be a hypersurface in $(X,g)$ with the induced metric $h$. The second fundamental form $L$ of $M$ is defined by $L(X,Y) = -g(\nabla_X(Y),N)$ for vector fields $X,Y \in \mathfrak{X}(M)$ and a unit normal vector field $N$. In particular, if $X$ is a manifold with boundary $M$ with defining function $\sigma$ so that $\|d\sigma\|_g=1$ on $M$, then we set $L(X,Y) = -g(\nabla_X(Y),\NV)$ with $\NV \st \grad_g(\sigma)$. With these conventions, $L=g$ for the round sphere $S^n \subset \R^{n+1}$ if $\sigma$ is the distance function of $S^n$. We set $n H = \tr_h(L)$ if $M$ has dimension $n$. $H$ is the mean curvature of $M$. Let $\lo = L - H h$ be the trace-free part of $L$. We sometimes identify $L$ with the shape operator $S$ defined by $h(X,S(Y)) = L(X,Y)$. For a bilinear form $b$ on $T(M)$, we denote its trace-free part with respect to a metric known by context by $\mathring{b}$ or $b_\circ$. $\NV$ gradient of $\sigma$ $H$ mean curvature $\lo$ trace-free part of $L$ $\iota$ embedding $b_\circ$ trace-free part of $b$ $\iota$ denotes various canonical embeddings such as $\iota: M \hookrightarrow X$. The symbol $\iota^$ will be used for the induced pull-back of functions, forms, and metrics. For any diffeomorphism $f$, the symbol $f_ = (f^{-1})^$ denotes the push-forward by $f$. Differentiation with respect to the variable $\lambda$ will often be denoted by $^\cdot$. In contrast, differentiation with respect to the variables $r$ and $s$ will usually be denoted by $'$. The symbol $\circ$ denotes compositions of operators. The symbol $\sim$ indicates a proportionality. § THE SINGULAR YAMABE PROBLEM Let $(X,g)$ be a compact manifold with boundary $M$ of dimension $n$. The problem to ask for a defining function $\sigma$ of $M$ so that \begin{equation}\label{syp} \scal (\sigma^{-2}g) = -n(n+1) \end{equation} is known as (a version of) the singular Yamabe problem <cit.>. The metric $\sigma^{-2} g$ is called a singular Yamabe metric if (<ref>) is true.[Later we shall often deal with a weaker condition.] The conformal transformation law of scalar curvature shows that \scal(\sigma^{-2}g) = -n(n+1) \|d\sigma\|_g^2 + 2n \sigma \Delta_g(\sigma) + \sigma^2 \scal(g). Following <cit.>, we write this equation in the form \scal(\sigma^{-2}g) = -n(n+1) \SC(g,\sigma), \SC(g,\sigma) = \|d\sigma\|_g^2 + 2 \rho \sigma (see Definition (<ref>)).[In <cit.>, the quantity $\SC(g,\sigma)$ is termed the $\SC$-curvature of $(g,\sigma)$.] In these terms, $\sigma$ is a solution of (<ref>) iff $\SC(g,\sigma)=1$. Such $\sigma$ exist <cit.> and are unique <cit.>. However, in general, $\sigma$ is not smooth up to the boundary. The smoothness is obstructed by a locally determined conformally invariant scalar function on $M$, called the singular Yamabe obstruction. Moreover, the solution is smooth up to the boundary iff the obstruction vanishes In order to describe the structure of $\sigma$ more precisely, we follow <cit.> and <cit.>. We use geodesic normal coordinates (see Section <ref>). Let $r = d_M$ be the distance function of $M$ for the background metric $g$. Then there are uniquely determined coefficients $\sigma_{(k)} \in C^\infty(M)$ for $2 \le k \le n+1$ so that the smooth defining function $\sigma_F$ \begin{equation}\label{sigma-finite} \sigma_F \st r + \sigma_{(2)} r^2 + \dots + \sigma_{(n+1)} r^{n+1} \end{equation} \begin{equation}\label{Yamabe-finite} \SC(g,\sigma_F) = 1 + R r^{n+1} \end{equation} with a smooth remainder term $R$. We briefly describe how these coefficients are recursively determined. In geodesic normal coordinates, the metric $g$ takes the form $dr^2 + h_r$ with a one-parameter family $h_r$ of metrics on $M$. The condition (<ref>) is equivalent to \|d\sigma_F\|_g^2 - \frac{2}{n+1} \sigma_F \Delta_g(\sigma_F) - \frac{1}{n(n+1)} \sigma_F^2 \scal^g = 1 + R r^{n+1}. We write the left-hand side of this equation in the form \begin{align}\label{Y-F} & \partial_r(\sigma_F)^2 + h_r^{ij} \partial_i (\sigma_F) \partial_j (\sigma_F) \notag \\ & - \frac{2}{n+1} \sigma_F \left (\partial_r^2 (\sigma_F) + \frac{1}{2} \tr (h_r^{-1} h_r') \partial_r (\sigma_F) + \Delta_{h_r} (\sigma_F) \right) - \frac{1}{n(n+1)} \sigma_F^2 \scal^g \end{align} and expand this sum into a Taylor series in the variable $r$. Then the vanishing of the coefficient of $r^k$ for $k \le n$ is equivalent to an identity of the form (k-1-n) \sigma_{(k+1)} = LOT, where $LOT$ involves only lower-order Taylor coefficients of $\sigma$. The latter relation also shows that there is a possible obstruction to the existence of an improved solution $\sigma_F'$ which contains a term $\sigma_{(n+2)} r^{n+2}$ and satisfies $\SC(g,\sigma_F') = 1 + R r^{n+2}$. However, by setting $\LO_n$ \begin{equation}\label{sol-log-d} \sigma = \sigma_F + \LO_n r^{n+2} \log r \end{equation} with an appropriate coefficient $\LO_n \in C^\infty(M)$ one may get a solution of \begin{equation}\label{sol-log} \SC(g,\sigma) = 1 + O(r^{n+2} \log r). \end{equation} The coefficient $\LO_n$ is determined by the condition that the coefficient of $r^{n+1}$ in the expansion of (<ref>) vanishes. But that coefficient equals \left( r^{-n-1} (\SC(g,\sigma_F) - 1) \right)\|_{r=0} - 2 \frac{n+2}{n+1} \LO_n. The first term exists by the construction of $\sigma_F$ and the second term is generated by the action of the terms $\partial_r(\sigma)^2$ and $\sigma \partial_r^2(\sigma)$ in (<ref>) on the log-term in $\sigma$.[In fact, $\partial_r (r^{n+2} \log r) = r^{n+1} + \cdots$ and $\partial_r^2 (r^{n+2} \log r) = (2n+3) r^n + \cdots$.] Hence for \LO_n \st \frac{1}{2} \frac{n+1}{n+2} \left( r^{-n-1} (\SC(g,\sigma_F) - 1) \right)\|_{r=0} the condition (<ref>) is satisfied. Following <cit.>, we define the singular Yamabe obstruction by $\B_n$ singular Yamabe obstruction \begin{equation}\label{B-def} \B_n \st \left( r^{-n-1} (\SC(g,\sigma_F) - 1) \right)\|_{r=0} . \end{equation} In these terms, we see that with \begin{equation}\label{obstruction-two} \LO_n = \frac{1}{2} \frac{n+1}{n+2} \B_n \end{equation} the improved $\sigma$ defined in (<ref>) satisfies (<ref>). By <cit.>, the unique solution $\sigma$ of the singular Yamabe problem has an expansion of the form \sigma = r + \sigma_{(2)} r^2 + \dots + \sigma_{(n+1)} r^{n+1} + \LO_n r^{n+2} \log r + \dots. Graham <cit.> calls $\LO_n$ the singular Yamabe obstruction. Since $\sigma_F$ is determined by $g$ (and the embedding $M \hookrightarrow X$), we regard $\B_n$ as a functional of $g$ (and the embedding). It is a key result that $\B_n$ is a conformal invariant of $g$. More precisely, we write $\hat{\B}_n$ for the obstruction defined by $\hat{g}=e^{2\varphi} g$ with $\varphi \in C^\infty(X)$. Then $e^{(n+1) \iota^(\varphi)} \hat{\B}_n = \B_n$. Let $r$ be the distance function of $M$ for $g$. Let S_N \st 1 + \sum_{j=2} r^j \sigma_{(j)}. Then for $N \le n+1$ the condition \begin{equation}\label{scalar-exp} \SC (g,S_N) = 1 + 0 r + \cdots + 0 r^{N-1} + O(r^N) = 1 + O(r^N) \end{equation} determines the coefficient $\sigma_{(N)}$ in terms of lower-order coefficients $\sigma_{(2)}, \dots, \sigma_{(N-1)}$. In that case, we also write $S_N = S_N(g)$. Moreover, we recall that the coefficient of $r^k$ ($k \le N-1$) in the expansion of $\SC(g,S_N)$ depends only on $\sigma_{(2)}, \dots, \sigma_{(k+1)}$. Now we have the obvious relation \SC(\hat{g},\hat{\sigma}) = \SC(g,\sigma) for any $\sigma \in C^\infty(X)$ and $\hat{\sigma} \st e^{\varphi} \sigma$. We write the expansion (<ref>) (for $N \le n+1$) as an expansion in terms of the distance function $\hat{r}$ of $M$ for $\hat{g}$. Hence \begin{equation}\label{scalar-exp-hat} \SC(\hat{g},e^\varphi S_N(g)) = 1 + O(\hat{r}^N). \end{equation} This expansion determines the coefficients $\hat{\sigma}_{(2)}, \dots, \hat{\sigma}_{(N)}$ in the expansion e^\varphi S_N(g) = \hat{r} + \hat{r}^2 \hat{\sigma}_{(2)} + \dots + \hat{r}^{N} \hat{\sigma}_{(N)} + \cdots = S_N(\hat{g}) + \mbox{higher-order terms}. In general, this expansion involves higher-order terms, i.e., $e^\varphi S_N(g) \ne S_N(\hat{g})$. In particular, the remainder term in (<ref>) depends on $\hat{\sigma}_{N+1}$. However, for $N=n+1$, the coefficient of $\hat{r}^{n+1}$ in (<ref>) does not depend on $\hat{\sigma}_{(n+2)}$ since it appears with a prefactor $0$. Therefore, \begin{align} (\hat{r}^{-n-1} \SC(\hat{g},S_{n+1}(\hat{g})))\|_{\hat{r}=0} & = (\hat{r}^{-n-1} \SC(\hat{g},e^\varphi S_{n+1}(g)))\|_{\hat{r}=0} \\ &= e^{-(n+1) \iota^(\varphi)} (r^{-n-1} \SC(g,S_{n+1}(g)))\|_{r=0}. \end{align} This relation implies the assertion. In <cit.>, Gover and Waldron describe an elegant algorithm that recursively determines the solution $\sigma$ of the singular Yamabe problem as a power series of some boundary defining function $\sigma_0$ (like the distance function $\sigma_0 = d_M$). Note that these power series are not Taylor series: their coefficients still live on the ambient space $X$. This algorithm rests on the interpretation of the quantity $\SC(g,\sigma)$ as the squared length of the scale tractor associated to $\sigma$. For our purposes, it will be enough to consider smooth approximate solutions of the singular Yamabe problem. This motivates the following definition. The defining function $\sigma \in C^\infty(X)$ of $M$ is said to satisfy the condition $\SCY$ iff \sigma = r + \sigma_{(2)} r^2 + \dots + \sigma_{(n+1)} r^{n+1} + O(r^{n+2}) \SC(g,\sigma) = 1 + R_{n+1} r^{n+1} for a smooth remainder term $R_{n+1}$. Equivalently, it holds \SC(g,\sigma) = 1 + R_{n+1} \sigma^{n+1} for another smooth remainder term $R_{n+1}$. The restriction of either remainder terms to $M$ is the singular Yamabe obstruction: $\B_n = \iota^ R_{n+1}$. We recall that the obstruction $\B_n$ satisfies \begin{equation}\label{B-CI} e^{(n+1) \iota^(\varphi)} \B_n(\hat{g}) = \B_n(g). \end{equation} Graham <cit.> determined the first two non-trivial coefficients $\sigma_{(2)}$ and $\sigma_{(3)}$ in the expansion of $\sigma$ (Lemma <ref>). An explicit formula for the next coefficient $\sigma_{(4)}$ is given in <cit.>. We reproduce these results in a slightly different form in Section <ref> and also display a formula for $\sigma_{(5)}$. Graham <cit.> shows that the obstruction $\B_1$ vanishes. The obstruction for surfaces in a three-manifold is given by the formula $\B_2$ $\B_3$ \begin{align}\label{B2} \B_2 & = -\frac{1}{3} (\delta^h \delta^h (\lo) + H \|\lo\|^2 + (\lo,\iota^(\Rho^g))) \notag \\ & = -\frac{1}{3} (\Delta_h (H) + \delta^h (\Ric^g (\NV,\cdot)) + H \|\lo\|^2 + (\lo,\iota^(\Rho^g))) \end{align} (<cit.>). The equivalence of both expressions follows from the Codazzi-Mainardi equation. For the details of that argument and a derivation of these formulas from Theorem <ref>, we refer to Section <ref>. The first formula reproduces a result in <cit.>. Explicit formulas for $\B_3$ were first derived in <cit.> and [3] from a general universal tractor formula found in <cit.>. In particular, it was proved that for a conformally flat background metric $g$ the obstruction $\B_3$ is given by the closed formula \begin{equation}\label{B3-closed} 6 \B_3 = 3 (\delta^h \delta^h + (\Rho^h,\cdot))((\lo^2)_\circ) + \|\lo\|^4, \end{equation} where $b_\circ$ denotes the trace-free part of the symmetric bilinear form $b$. For general background metrics $g$, the formula for $\B_3$ contains additional terms defined by the Weyl tensor of $g$. For full details, we refer to [3]. In the more recent work <cit.>, these formulas for $\B_3$ were derived without utilizing tractor calculus. In Section <ref>, we shall deduce (<ref>) from the general formula in Theorem <ref>. Formula (<ref>) manifestly implies the conformal invariance $e^{4 \varphi} \hat{\B}_3 = \B_3$ since the operator $b \mapsto \delta \delta (b) + (\Rho,b)$ is conformally covariant on trace-free symmetric bilinear forms on $M$. Up to constant multiples, $\B_2$ and $\B_3$ are the respective variations of the functionals \int_M \|\lo\|_h^2 dvol_h \quad \mbox{and} \quad \int_M (\lo,\JF)_h dvol_h (for the definition of the Fialkov tensor $\JF$ we refer to Section <ref>). These are special cases of the variation formulas of the functional $\A \st \int_M v_n dvol_h$ (with respect to a one-parameter family of hypersurfaces $M \hookrightarrow X$) which were proved in <cit.> and <cit.>. They state that the variation of $\A$ is proportional to the obstruction $\B_n$. The equivalence of both results follows from (<ref>). This result may be regarded as an analog of the result that the metric variation of the total critical $Q$-curvature of an even-dimensional closed manifold is given by the corresponding Fefferman-Graham obstruction tensor <cit.>. Poincaré-Einstein metrics are an important special class of singular Yamabe metrics. In fact, if $g_+ = r^{-2} (dr^2 + h_r)$ is a Poincaré-Einstein metric in normal form relative to $h=h_0$ <cit.>, then $\scal(g_+) = -n(n+1)$, the background metric is $dr^2+h_r$ and the corresponding defining function is $\sigma=r$. In this case, the singular Yamabe obstruction vanishes. We shall refer to this case as the Poincaré-Einstein case. § THE CONJUGATION FORMULA In this section, we introduce Laplace-Robin operators (or degenerate Laplacians) following <cit.>. We relate them to the spectral theory of the Laplacian of singular metrics $\sigma^{-2}g$ and use this relation to prove basic properties of the Laplace-Robin operators. Let $X$ be a manifold of dimension $n+1$. $L(g,\sigma)$ Laplace-Robin operator For any pair $(g,\sigma)$ consisting of a metric $g$ on $X$ and $\sigma \in C^\infty(X)$, the one-parameter family \begin{equation}\label{LR-OP} L(g,\sigma;\lambda) \st (n\!+\!2\lambda\!-\!1) (\nabla_{\grad_g(\sigma)} + \lambda \rho) - \sigma (\Delta_g + \lambda \J): C^\infty(X) \to C^\infty(X) \end{equation} of differential operators is called the Laplace-Robin operator of the pair $(g,\sigma)$. Here $\lambda \in \C$, 2 n \J \st \scal(g) \quad \mbox{and} \quad (n+1) \rho(g,\sigma) \st - \Delta_g (\sigma) - \sigma \J. Moreover, we set $\SC(g,\sigma)$ \begin{equation}\label{SC} \SC(g,\sigma) \st \|d\sigma\|_g^2 + 2 \sigma \rho. \end{equation} Similarly, we define $L(g,\sigma;\lambda)$ for a manifold $X$ with boundary $M$. In this case, $g$ and $\sigma$ are assumed to be smooth up to the boundary. Then $L(\lambda)$ acts on the space $C^\infty(X)$ of smooth functions up to the boundary and on the space $C^\infty(X^\circ)$ of smooth functions on the open interior $X^\circ = X \setminus M$ of $X$. From now on, we assume that $(X,g)$ is a compact manifold with boundary $M$ and $\sigma$ is a defining function of $M$. We recall that $\sigma$ is a defining function of $M$ if $\sigma^{-1}(0)=M$, $\sigma > 0$ on $X^\circ$ and $d\sigma\|_M \ne 0$. Let $\iota: M \hookrightarrow X$ be the embedding and set $h \st \iota^(g)$. Then the operator $\iota^* L(g,\sigma;\lambda)$ degenerates to the first-order operator \begin{equation}\label{bv-op} C^\infty(X) \ni u \mapsto (n\!+\!2\lambda\!-\!1) \iota^* \left[\nabla_{\grad_g(\sigma)}(u) - \frac{\lambda}{n\!+\!1}\Delta_g(\sigma) u \right] \in C^\infty(M). \end{equation} Since certain linear combinations of Dirichlet and Neumann boundary values are also known as Robin boundary values, this naturally motivates the above notion of a Laplace-Robin operator. If $\sigma^{-2} g$ has constant scalar curvature $-n(n+1)$, the boundary operator (<ref>) reduces to the conformally covariant boundary operator u \mapsto (n\!+\!2\lambda\!-\!1) \iota^* (\nabla_{\grad_g(\sigma)} - \lambda H) u, where $H$ is the mean curvature of $M$ (<cit.>, <cit.>). If $g_+ = r^{-2} g$ is Poincaré-Einstein in the sense that $\Ric(g_+) = - n g_+$, then $L(g,r;\lambda-n+1)$ equals the shift operator $S(g_+;\lambda)$ of <cit.>. Assume that $\sigma \in C^\infty(X)$ is a defining function of the boundary $M$ of $X$. Then it holds \begin{equation}\label{CF} L(g,\sigma;\lambda) + \sigma^{\lambda-1} \circ \left(\Delta_{\sigma^{-2}g} - \lambda(n\!+\!\lambda) \id \right) \circ \sigma^{-\lambda} = \lambda (n\!+\!\lambda) \sigma^{-1}(\SC(g,\sigma)-1) \id \end{equation} as an identity of operators acting on $C^\infty(X^\circ)$. Let $\Con(g,\sigma;\lambda)$ denote the operator \sigma^{\lambda-1} \circ \left(\Delta_{\sigma^{-2}g} - \lambda(n\!+\!\lambda) \id \right) \circ \sigma^{-\lambda}. The relation \Delta_{\sigma^{-2}g} = \sigma^2 \Delta_g - (n\!-\!1) \sigma \nabla_{\grad_g(\sigma)} shows that \begin{align} \Con(g,\sigma;\lambda) (u) & = \sigma^{\lambda+1} \Delta_g (\sigma^{-\lambda} u) - (n\!-\!1) \sigma^\lambda \nabla_{\grad_g(\sigma)}(\sigma^{-\lambda} u) - \lambda(n\!+\!\lambda) \sigma^{-1} u \\ & = \sigma \Delta_g (u) - (n\!+\!2\lambda\!-\!1) \nabla_{\grad_g(\sigma)}(u) + \Con(g,\sigma;\lambda)(1) u \end{align} for $u \in C^\infty(X^\circ)$. Thus, it only remains to calculate the constant term $\CT(\Con)(\lambda) \st \Con(\lambda)(1) \in C^\infty(X^\circ)$ of $\Con(\lambda)$. Note that \begin{align} \CT(\Con)(\lambda) & = \sigma^{\lambda-1}(\Delta_{\sigma^{-2}g} - \lambda(n\!+\!\lambda) \id )(\sigma^{-\lambda}) \\ & = \sigma^{\lambda+1} \Delta_{g} (\sigma^{-\lambda}) + \lambda (n\!-\!1) \sigma^{-1} \|\grad_g(\sigma)\|^2 - \lambda(n\!+\!\lambda) \sigma^{-1} \end{align} shows that $\CT(\Con)(\lambda)$ is a quadratic polynomial in $\lambda$. It is obvious that $\CT(\Con)(0)=0$. Next, we determine the leading coefficient of that polynomial. We choose orthonormal bases $\left\{\partial_i \right\}$ on the tangent spaces of the level hypersurfaces $\sigma^{-1}(c)$ of $\sigma$. The sets $\sigma^{-1}(c)$ are smooth manifolds if $c$ is sufficiently small. $\NV$ is perpendicular to these hypersurfaces. Let $\left\{ \alpha, dx^i \right\}$ be the dual basis of $\left\{ \NV, \partial_i \right\}$. Then \begin{equation}\label{Laplace-basis} \Delta_g (u) = \frac{1}{\|\NV\|^2} \nabla_{\NV}^2 (u) + \Delta_{g_\sigma}(u) + \frac{1}{\|\NV\|} H_\sigma \nabla_\NV (u) - \frac{1}{\|\NV\|^2} \langle du, \nabla_{\NV}(\NV) \rangle, \end{equation} where $-H_\sigma = \langle \alpha,\nabla_{\partial_i} (\partial_i) \rangle$ and $\Delta_{g_\sigma}$ denotes the tangential Laplacians for the induced metrics on the leaves $\sigma^{-1}(c)$; for more details, see Section <ref>. Since $\Delta_{g_\sigma} (\sigma^{-\lambda}) = 0$, it follows that the coefficient of $\lambda^2$ in $\sigma^{\lambda+1} \Delta_{g}(\sigma^{-\lambda})$ is given by $\sigma^{-1} \|\NV\|^{-2} \nabla_\NV(\sigma)^2 = \sigma^{-1} \|\NV\|^2$. Hence the leading coefficient of the quadratic polynomial $\CT(\Con)(\lambda)$ equals \sigma^{-1 }\|\NV\|^2 - \sigma^{-1}. Finally, we calculate \CT(\Con)(-1) = \Delta_g(\sigma) - (n-1) \sigma^{-1} \|\NV\|^2 + (n-1) \sigma^{-1}. These arguments prove that \begin{equation}\label{C-CT} \CT(\Con)(\lambda) = \lambda \left((n\!+\!\lambda) \sigma^{-1} (\|\NV\|^2-1) - \Delta_g(\sigma)\right). \end{equation} \begin{align} \CT(L)(\lambda) + \CT(\Con)(\lambda) & = \lambda (n\!+\!2\lambda\!-\!1) \rho - \lambda \sigma \J + \lambda \left((n\!+\!\lambda) \sigma^{-1} (\|\NV\|^2-1) - \Delta_g(\sigma)\right) \\ & = \lambda (n\!+\!\lambda) \sigma^{-1}(2 \rho \sigma + \|\NV\|^2-1) \\ & = \lambda (n\!+\!\lambda) \sigma^{-1} (\SC(g,\sigma)-1) \end{align} by the definition of $\rho$. This completes the proof. The above proof shows the identity (<ref>) only for functions with support near the boundary $M$. This will be enough for all later applications. For simplicity, we shall interpret (<ref>) and similar identities in this way without further mentioning. Let $g_+ = r^{-2} (dr^2 + h_r)$ be a Poincaré-Einstein metric in normal form relative to the metric $h$ on $M$. It lives on the space $(0,\varepsilon) \times M$ and satisfies $\Ric(g_+) = -n g_+$. Hence $\scal(g_+)=-n(n+1)$. Then $\SC(dr^2+h_r,r;\lambda) = \SC(g_+,1;\lambda)=1$, and the conjugation formula reads - L(dr^2+h_r,r;\lambda) = r^{\lambda-1} \circ (\Delta_{g_+} - \lambda(n+\lambda) \id ) \circ r^{-\lambda}. We refer to <cit.> for the discussion of the relation between this conjugation formula and a formula in <cit.>. The conjugation formula is equivalent to the identity \begin{equation}\label{CF-s} L(g,\sigma;\lambda) + \sigma^{\lambda-1} \circ \Delta_{\sigma^{-2}g} \circ \sigma^{-\lambda} = \lambda (n\!+\!\lambda) \sigma^{-1} \SC(g,\sigma) \id \end{equation} of operators acting on $C^\infty(X^\circ)$. The conformal covariance of the Laplace-Robin operator is an immediate consequence of these identities. More precisely, we have The Laplace-Robin operator satisfies \begin{equation}\label{CT-ID} L(\hat{g}, \hat{\sigma}; \lambda) \circ e^{\lambda\varphi} = e^{(\lambda-1)\varphi} \circ L(g,\sigma;\lambda), \; \lambda \in \C \end{equation} for all conformal changes $(\hat{g},\hat{\sigma}) = (e^{2\varphi}g,e^{\varphi}\sigma)$, $\varphi \in It suffices to note that $\hat{\sigma}^{-2} \hat{g} = \sigma^{-2} g$ and $\SC(\hat{g},\hat{\sigma}) = \SC(g,\sigma)$. Strictly speaking, the above arguments prove the conformal covariance of the operator $L(g,\sigma;\lambda)$ for boundary defining $\sigma$ when acting on $C^\infty(X^\circ)$. In <cit.>, the conformal covariance of the operator $L(g,\sigma;\lambda)$ for any pair $(g,\sigma)$ acting on $C^\infty(X)$ follows from its interpretation in terms of tractor calculus. For a direct proof, see <cit.>. The following consequence of the conjugation formula will be of central significance in the rest of the paper. We continue to assume that $\sigma$ is a boundary defining function and statements are valid near $M$. It holds \begin{equation}\label{L-Delta} - L(g,\sigma;\lambda) = \sigma^{\lambda-1} \circ (\Delta_{\sigma^{-2}g} - \lambda(n+\lambda) \id ) \circ \sigma^{-\lambda} \end{equation} for $\lambda \in \C$ iff $\SC(g,\sigma) = 1$. More generally, it holds -L(g,\sigma;\lambda) = \sigma^{\lambda-1} \circ (\Delta_{\sigma^{-2}g} - \lambda(n+\lambda) \id ) \circ \sigma^{-\lambda} + O(\sigma^n) if $\sigma$ satisfies $\SCY$. These identities are identities of operators acting on $C^\infty(X^\circ)$. For $N \in \N$, we define \begin{equation}\label{LN} L_N(g,\sigma;\lambda) \st L(g,\sigma;\lambda\!-\!N\!+\!1) \circ \cdots \circ L(g,\sigma;\lambda). \end{equation} In these terms, iterated application of (<ref>) implies Assume that $\SC(g,\sigma)=1$. Then it holds \begin{equation} \sigma^{-\frac{n}{2}-N} \circ \prod_{j=0}^{2N-1} \left(\Delta_{\sigma^{-2}g} + \left(\frac{n}{2}\!+\!N\!-\!j\right) \left(\frac{n}{2}\!-\!N\!+\!j\right) \id \right) \circ \sigma^{\frac{n}{2}-N} = L_{2N} \left(g,\sigma;-\frac{n}{2}\!+\!N\right) \end{equation} for $2N \le n$. In particular, we have \begin{equation} \sigma^{-n} \circ \prod_{j=0}^{n-1} \left(\Delta_{\sigma^{-2}g} + (n\!-\!j)j \id \right) = L_n (g,\sigma;0) \end{equation} The conjugation formula also sheds new light on the fact that the formal adjoint of a Laplace-Robin operator is another Laplace-Robin operator. The Laplace-Robin operator satisfies \begin{equation}\label{L-ad} L(g,\sigma;\lambda)^* = L(g,\sigma;-\lambda\!-\!n), \; \lambda \in \C, \end{equation} where $^$ denotes the adjoint operator with respect to the Riemannian volume of $g$. More precisely, it holds \begin{equation}\label{adjoint} \int_X L(g,\sigma;\lambda)(\varphi) \psi dvol_g = \int_X \varphi L(g,\sigma;-\lambda\!-\!n)(\psi) dvol_g \end{equation} for $\varphi, \psi \in C_c^\infty(X^\circ)$. Let $\varphi,\psi \in C_c^\infty(X^\circ)$. The identity (<ref>) yields \begin{align} & \int_X L(g,\sigma;\lambda)(\varphi) \psi dvol_g \\ & = - \int_X \sigma^{\lambda-1} \Delta_{\sigma^{-2}g} (\sigma^{-\lambda} \varphi) \psi dvol_g + \lambda (n\!+\!\lambda) \int_X \sigma^{-1} \SC(g,\sigma) \varphi \psi dvol_g. \end{align} Note that $\lambda(n+\lambda)$ is invariant under the substitution $\lambda \mapsto -\lambda-n$. We rewrite the first integral in terms of volumes with respect to the metric $\sigma^{-2}g$ and apply the self-adjointness of $\Delta_{\sigma^{-2}g}$ with respect to the volume of the metric $\sigma^{-2}g$. Using $dvol_{\sigma^{-2}g} = \sigma^{-n-1} dvol_g$, we find - \int_X \sigma^{-\lambda} \varphi \Delta_{\sigma^{-2}g} (\sigma^{\lambda+n} \psi) dvol_{\sigma^{-2} g} = - \int_X \varphi \sigma^{-\lambda-n-1} \Delta_{\sigma^{-2}g} (\sigma^{\lambda+n} \psi) dvol_g. Now another application of (<ref>) implies the assertion (<ref>). For later applications, we need an extension of Corollary <ref> to another class of test functions. The proof of the following result will be given in Section <ref>. The identity (<ref>) continues to be true for $\psi \in C^\infty(X)$ and $\varphi \in C^2(X)$ so that $\iota^(\varphi)=0$. The proof of this result actually shows that \begin{align}\label{adjoint-g} &\int_X L(g,\sigma;\lambda)(\varphi) \psi dvol_g - \int_X \varphi L(g,\sigma;-\lambda\!-\!n)(\psi) dvol_g \notag \\ & = (n\!+\!2\lambda) \int_M \iota^(\varphi \psi \|\NV\|) dvol_h \end{align} for $\varphi , \psi \in C^2(X)$. The proof of this identity rests on a calculation of the left-hand side (see also <cit.>). Finally, we derive some basic commutator relations. For any $N \in \N$, it holds L(g,\sigma;\lambda\!+\!N) \circ \sigma^N - \sigma^N \circ L(g,\sigma;\lambda) = N(n\!+\!2\lambda\!+\!N) \sigma^{N-1} \SC(g,\sigma) \id. In particular, it holds \begin{equation}\label{sl2-comm} L(g,\sigma;\lambda\!+\!1) \circ \sigma - \sigma \circ L(g,\sigma;\lambda) = (n\!+\!2\lambda\!+\!1) \SC(g,\sigma) \id. \end{equation} \begin{align}\label{LN-gen-b} & L_N(g,\sigma;\lambda) \circ \sigma - \sigma \circ L_N(g,\sigma;\lambda\!-\!1) = N(n\!+\!2\lambda\!-\!N) L_{N-1}(g,\sigma;\lambda-1) \notag \\ & + \sum_{j=1}^N (n\!+\!2\lambda\!-\!2j\!+\!1) \underbrace{L(g,\sigma;\lambda\!-\!N\!+\!1) \circ \cdots \circ (\SC(g,\sigma)-1) \circ \cdots \circ L(g,\sigma;\lambda\!-\!1)}_{N factors}. \end{align} Moreover, if $\SC(g,\sigma)$ is nowhere zero, then for any $N \in \N$ it holds \begin{equation}\label{LN-gen-tilde} \tilde{L}_N(g,\sigma;\lambda\!+\!1) \circ \sigma - \sigma \circ \tilde{L}_N(g,\sigma;\lambda) = N(n\!+\!2\lambda\!-\!N\!+\!2) \tilde{L}_{N-1}(g,\sigma;\lambda), \end{equation} \begin{equation}\label{LR-general} \index{$\tilde{L}(g,\sigma)$} \tilde{L}(g,\sigma;\lambda) \st L(g,\sigma;\lambda) \circ \SC(g,\sigma)^{-1} \end{equation} \begin{equation}\label{LN-tilde} \index{$\tilde{L}_N(g,\sigma)$} \tilde{L}_N(g,\sigma;\lambda) \st \tilde{L}(g,\sigma;\lambda\!-\!N\!+\!1) \circ \cdots \circ \tilde{L}(g,\sigma;\lambda). \end{equation} The identity (<ref>) implies \begin{align} & L(g,\sigma;\lambda) \circ \sigma^N - \sigma^N \circ L(g,\sigma;\lambda-N) \\ & = \lambda (n\!+\!\lambda) \sigma^{N-1} \SC(g,\sigma) \id - (\lambda\!-\!N) (n\!+\!\lambda\!-\!N) \sigma^{N-1} \SC(g,\sigma) \id \\ & = N(n\!+\!2\lambda\!-\!N) \sigma^{N-1} \SC(g,\sigma) \id. \end{align} This proves the first commutator relation. The remaining claims are consequences. This completes the proof. The above commutator relations substantially simplify if $\SC=1$. Although we proved the identities in Corollary <ref> as identities of operators acting on $C^\infty(X^\circ)$, they are also valid for operators acting on $C^\infty(X)$ (<cit.>). § SYMMETRY BREAKING OPERATORS In the present section, we discuss some representation theoretical aspects of the results in Section <ref>. The simplest special case of the Laplace-Robin operator $L$ appears for the hyperplane $M=\R^n$ in $X=\R^{n+1}$ with the flat Euclidean metric $g_0$. Let $M$ be given by the zero locus of the defining function $\sigma_0 = x_{n+1}$. We shall also write $\sigma_0=r$ and $g_+ = r^{-2} g_0$. Then $\J = \rho = 0$ and we obtain \begin{equation}\label{L-flat} L(g_0,\sigma_0;\lambda) = (n+2\lambda-1) \partial_{n+1} - x_{n+1} \Delta_{\R^{n+1}}: C^\infty(\R^{n+1}) \to C^\infty(\R^{n+1}). \end{equation} An easy calculation shows the conjugation formula L(g_0,r;\lambda) = r^{\lambda-1} \circ (-\Delta_{g_+} + \lambda(n+\lambda)) \circ r^{-\lambda}. It implies that the operator $L(g_0,r;\lambda)$ is an intertwining operator for spherical principal series representations. Indeed, let $\gamma \in SO(1,n+1)$ be an isometry of the hyperbolic metric $g_+=r^{-2} g_0$ acting on the upper-half space $r > 0$. Then we calculate \begin{align} & L(g_0,r;\lambda) \left( \left(\frac{\gamma_(r)}{r}\right)^{-\lambda} \gamma_(u)\right) \\ & = r^{\lambda-1} (-\Delta_{g_+} + \lambda(n+\lambda)) \left(r^{-\lambda} \left(\frac{\gamma_(r)}{r}\right)^{-\lambda} \gamma_(u)\right) \\ & = r^{\lambda-1} (-\Delta_{g_+} + \lambda(n+\lambda)) (\gamma_(r^{-\lambda} u)) \\ & = r^{\lambda-1}\gamma_ (-\Delta_{g_+} + \lambda(n+\lambda))(r^{-\lambda} u)) \\ & = \left(\frac{\gamma_(r)}{r}\right)^{-\lambda+1} \gamma_ (r^{\lambda-1} (-\Delta_{g_+}+\lambda(n+\lambda))(r^{-\lambda} u) \\ & = \left(\frac{\gamma_(r)}{r}\right)^{-\lambda+1} \gamma_ L(g_0,r;\lambda)(u). \end{align} In other words, it holds \begin{equation}\label{rep-flat} L(g_0,r;\lambda) \circ \pi^0_{-\lambda}(\gamma) = \pi^0_{-\lambda+1}(\gamma) \circ L(g_0,r;\lambda) \end{equation} with $\pi_\lambda^0$ non-compact model of spherical principal series representation \pi^0_\lambda(\gamma) \st \left( \frac{\gamma_(r)}{r} \right)^\lambda \gamma_. Note that \frac{\gamma_(r)}{r} = e^{\Phi_\gamma}, where $\Phi_\gamma$ is the conformal factor of the conformal transformation induced by $\gamma$ with respect to the Euclidean metric, i.e., $\gamma_(g_0) = e^{2\Phi_\gamma} g_0$. The representation $\pi_\lambda^0(\gamma)$ is actually well-defined for all $\gamma \in SO(1,n+2)$ acting on $\R^{n+1}$ (viewed as the boundary of hyperbolic space of dimension $n+2$). However, the intertwining property (<ref>) holds true only for the subgroup of $SO(1,n+1)$ leaving the boundary $r=0$ of the upper half-space invariant. The fact that $L(g_0,r;\lambda)$ is an intertwining operator for a subgroup of the conformal group of the Euclidean metric on $\R^{n+1}$ connects it with the theory of symmetry breaking operators. In fact, it follows from the above that the compositions D_N(\lambda) \st \iota^ L(\lambda-N+1) \circ \cdots \circ L(\lambda): C^\infty(\R^{n+1}) \to C^\infty(\R^n), \; N \in \N D_N(\lambda) \circ \pi_{-\lambda}^0(\gamma) = \pi_{-\lambda+N}^{0 \prime}(\gamma) \circ D_N(\lambda), \; \gamma \in SO(1,n+1) and Clerc <cit.> proved that $D_N(\lambda)$ coincides with the symmetry breaking operator introduced in <cit.>.[$\pi_\lambda^{0 \prime}$ denotes the analogous representation on functions on the subspace $\R^n$.] Similarly, let $M = S^n$ be an equatorial subsphere of $X=S^{n+1}$ with the round metric $g$. Let $M$ be defined as the zero locus of the height function $\sigma=\He$ being defined as the restriction of $x_{n+2}$ to $S^{n+1}$. Then $\J = \frac{n+1}{2}$ and $\He$ height (n+1) \rho = - \Delta_{S^{n+1}} \He + \J \He = -(n+1) \He + \frac{n+1}{2} \He = - \frac{n+1}{2} \He using the fact that $\He$ is an eigenfunction of the Laplacian on the sphere $S^{n+1}$. Thus, $\rho = \frac{1}{2} \He$ and we obtain \begin{equation}\label{L-sphere} L(g,\He;\lambda) = (n+2\lambda-1) \nabla_{\grad (\He)} - \He \Delta_{S^{n+1}} + \lambda(\lambda+1) \He: C^\infty(S^{n+1}) \to C^\infty(S^{n+1}). \end{equation} A calculation shows that L(g,\He;\lambda) = \He^{\lambda-1} \circ (-\Delta_{\He^{-2}g} + \lambda(n+\lambda)) \circ \He^{-\lambda}. Again, the operator $L(g,\He;\lambda)$ is an intertwining operator for spherical principal series representations. Indeed, it holds L(g,\He;\lambda) \circ \pi_{-\lambda}(\gamma) = \pi_{-\lambda+1}(\gamma) \circ L(g,\He;\lambda) for all $\gamma \in SO(1,n+1)$ acting on the upper hemisphere $\He > 0$ of $S^{n+1}$. Here $\pi_\lambda$ spherical principal series representation \pi_\lambda(\gamma) \st \left( \frac{\gamma_(\He)}{\He} \right)^\lambda \gamma_. The latter representations are well-defined for $\gamma \in SO(1,n+2)$ acting on $S^{n+1}$. Note that \frac{\gamma_(\He)}{\He} = e^{\Phi_\gamma}, where $\Phi_\gamma$ is the conformal factor of the conformal transformation induced by $\gamma$ with respect to the round metric $g$, i.e., $\gamma_(g) = e^{2\Phi_\gamma} g$. We also note that the operator (<ref>) is equivalent to the intertwining operator displayed in <cit.>. We omit the details of that calculation. Finally, we observe that the above two models of the Laplace-Robin operator are conformally equivalent. In fact, let $\kappa: S^{n+1} \to \R^{n+1}$ be the stereographic projection. Then \kappa^* (\He) = \Phi x_{n+1} \quad \mbox{and} \quad \kappa^(g) = \Phi^2 \sum_{i=1}^{n+1} dx_i^2 = \Phi^2 g_0 with $\Phi = 2/(1+\|x\|^2)$ <cit.>. Hence \begin{align} \kappa^* L(g,\He;\lambda) \kappa_* & = L(\kappa^(g),\kappa^(\He);\lambda) \\ & = L(\Phi^2 g_0,\Phi x_{n+1};\lambda) \\ & = \Phi^{\lambda-1} L(g_0,x_{n+1};\lambda) \Phi^{-\lambda} \end{align} using a very special case of the conformal invariance of the Laplace-Robin operator (Corollary <ref>). By combining this conjugation formula with the results in the later sections, it follows that the equivariant families $D_{2N}^c(\lambda): C^\infty(S^{n+1}) \to C^\infty (S^n)$ constructed in <cit.> can be regarded as residue families $\D_{2N}^{res}(g,\He;\lambda)$ (as defined in Section <ref>). § ADAPTED COORDINATES, RENORMALIZED VOLUME COEFFICIENTS AND A FORMULA FOR $\B_N$ Let $X$ be compact with closed boundary $M$ and let $\sigma$ be a defining function of $M$, i.e., $\sigma^{-1}(0)=M$, $\sigma > 0$ on $X \setminus M$ and $d\sigma\|_M \ne 0$. Let $\iota: M \hookrightarrow X$ be the embedding and $h = \iota^(g)$. We start with the definition of two different types of local coordinates of $X$ near the boundary: geodesic normal coordinates and adapted coordinates. $w_j$ renormalized volume coefficients (normal coordinates) Geodesic normal coordinates are defined by the normal geodesic flow of the hypersurface $M$, i.e., we consider a diffeomorphism of $I \times M$ (with a small interval $I = [0,\varepsilon)$) onto a neighborhood of $M$ in $X$, which is defined by $\Phi^r$ geodesic flow \kappa: I \times M \ni (r,x) \mapsto \Phi^r(x) \in X, where $\Phi^r$ is the geodesic flow with initial speed given by a unit normal field on $M$. Then $\kappa^(g)$ has the form $dr^2 + h_r$ for a one-parameter family $h_r$ on $M$. Let $u(r)$ $u_j$ volume coefficients \begin{equation}\label{v-geo} u(r) \st dvol_{h_r}/dvol_h, \quad u(r) = \sum_{j \ge 0} r^j u_j, \quad u_j \in C^\infty(M). \end{equation} Now, if $\SC(g,\sigma) = 1$, then the volume form of the singular metric $\sigma^{-2} g$ has the form \begin{align}\label{def-w} dvol_{\kappa^(\sigma^{-2} g)} & = \sigma(r)^{-n-1} u(r) dr dvol_h \notag \\ & = r^{-n-1} w(r) dr dvol_h. \end{align} for $\sigma(r) = \kappa^(\sigma)$ and some $w \in C^\infty(I \times M)$. Moreover, we have expansions \begin{equation}\label{RVC-A} w(r) = 1 + \sum_{j \ge 1} r^j w_j \end{equation} with $w_j \in C^\infty(M)$ and dvol_{\kappa^(\sigma^{-2} g)} = \sum_{j \ge 0} r^{-n-1+j} w_j dr dvol_h. Following <cit.>, the coefficients $w_j \in C^\infty(M)$ for $j \le n$ are called singular Yamabe renormalized volume coefficients. Note that the definition of the coefficients $w_j \in C^\infty(M)$ involves the Taylor expansion of $\sigma(r)$ in $r$. Special interest deserves the critical coefficient $w_n$ since for closed $M$, the total integral \int_M w_n dvol_h is conformally invariant <cit.>. $\eta$ adapted coordinates Similarly, adapted coordinates are associated to the data $(g,\sigma)$ through a diffeomorphism \eta: I \times M \ni (s,x) \mapsto \Phi_\mathfrak{X}^s(x) \in X onto a open neighborhood of $M$ in $X$ (with some small interval $I = [0,\varepsilon)$), where $\Phi_\mathfrak{X}^s$ denotes the flow of the vector field $\mathfrak{X}$ \begin{equation}\label{X-field} \mathfrak{X} \st \NV /\|\NV\|^2, \quad \NV = \grad_g(\sigma) \end{equation} with $\Phi^0_\mathfrak{X} = \id$. We shall also use the notation $\mathfrak{X}_\sigma$ in cases where the dependence on $\sigma$ is important. Note that $\|\mathfrak{X}\| = 1/\|\NV\|$. Then (d/ds)(\sigma \circ \eta) = \langle d\sigma, \mathfrak{X} \rangle = \langle d \sigma, \NV \rangle / \|\NV\|^2 \stackrel{!}{=} 1 and the differential of $\eta$ maps the vector field $\partial_s$ to the vector field $\mathfrak{X}$ (see Section <ref>). This implies the important relation \begin{equation}\label{key-pb} \eta^(\sigma) = s \end{equation} and the intertwining property \begin{equation}\label{intertwine} \eta^ \circ \mathfrak{X} = \partial_s \circ \eta^, \end{equation} where $\partial_s$ and $\mathfrak{X}$ are viewed as first-order differential operators. Therefore, \eta^ \circ \mathfrak{X}^k = \partial_s^k \circ \eta^, and by composition with $\iota^$, we obtain \begin{equation}\label{translate} \iota^* \partial_s^k \circ \eta^* = \iota^* \mathfrak{X}^k = \iota^* \left(\|\NV\|^{-2} \nabla_\NV\right)^k. \end{equation} Now if $\SC(g,\sigma)=1$, i.e., if $\|\NV\|^2=1-2\sigma \rho$, then it follows from (<ref>) that the Taylor coefficients in the variable $s$ of any function $\eta^(u)$ with $u \in C^\infty(X)$ can be written as linear combinations of iterated gradients $\iota^ \nabla_\NV^k (u)$ with coefficients that are polynomials in the quantities $\iota^* \nabla_\NV^k(\rho) \in C^\infty(M)$. In particular, it \begin{equation}\label{trans-1-2} \iota^* \partial_s \circ \eta^* = \iota^* \nabla_\NV \quad \mbox{and} \quad \iota^* \partial^2_s \circ \eta^* = \iota^* \nabla_\NV^2 - 2 H \nabla_\NV. \end{equation} For more details, we refer to Section <ref>. If $\sigma$ satisfies only the weaker condition $\SCY$, then $\|\NV\|^2 = 1 - 2\sigma \rho + O(\sigma^{n+1})$ and the same conclusions are true for sufficiently small $k$. Note that the metric $\eta^(g)$ has the form \begin{equation}\label{normal-adapted} \eta^(\|\NV\|^{-2}) ds^2 + h_s \end{equation} with a one-parameter family $h_s$ of metrics on $M$ so that $h_0 = h = \iota^(g)$. We shall refer to (<ref>) as the normal form of $g$ in adapted coordinates. We expand $h_s = \sum_{j \ge 0} h_{(j)} s^j$. It follows from (<ref>) that \iota^ \partial_s^k (\eta^(\|\NV\|^{-2})) = \iota^ (\nabla_\NV(\|\NV\|^{-2}))^k. Thus, if $\sigma$ satisfies $\SCY$, the Taylor coefficients of the coefficient $\eta^(\|\NV\|^{-2})$ in the variable $s$ are polynomials in the quantities In later calculations in adapted coordinates, we shall often use the same notation for quantities like $\rho$ and $\J$ and their pull-backs by $\eta$ without further mentioning. $v_j$ renormalized volume coefficients (adapted coordinates) Now (<ref>) implies dvol_{\eta^(g)} = \eta^(\|\NV\|)^{-1} ds dvol_{h_s} = v(s) ds dvol_h for some $v(s) \in C^\infty(I \times M)$. Since the condition $\SCY$ implies $\|\NV\|=1$ on $M$, we get $v(0,x)=1$, and we have an expansion \begin{equation}\label{RVC-B} v(s) = 1 + \sum_{j \ge 1} s^j v_j \quad \mbox{with $v_j \in C^\infty(M)$}. \end{equation} The coefficients $v_j$ for $j \le n$ also will be called singular Yamabe renormalized volume coefficients. They describe the volume of the singular metric $\sigma^{-2} g$ through the expansion dvol_{\eta^(\sigma^{-2} g)} = s^{-n-1} v(s) ds dvol_h = \sum_{j \ge 0} s^{-n-1+j} v_j ds dvol_h. Again, special interest deserves the critical coefficient $v_n$ since \int_M v_n dvol_h is conformally invariant for closed $M$. This follows from the equality \begin{equation}\label{w=v} \int_M v_n dvol_h = \int_M w_n dvol_h \end{equation} which can be proved by the following argument of <cit.>. The identity $\|d\sigma\|^2_{\tilde{g}}=1$ for $\tilde{g} = \|d\sigma\|_g^2 g$ shows that $\sigma$ can be viewed as the distance function $d_M^{\tilde{g}}$ of $M$ in the metric $\tilde{g}$. Hence it holds vol_{\sigma^{-2}g} (\{ \sigma > \varepsilon \}) = vol_{\sigma^{-2} g} (\{ d_M^{\tilde{g}} > \varepsilon \}) = vol_{\tilde{\sigma}^{-2} \tilde{g}} (\{ d_M^{\tilde{g}} > \varepsilon \}) with $\tilde{\sigma}= \|d\sigma\|_g \sigma$. By comparing the coefficients of $\log \varepsilon$ in the expansions of both sides, we find \int_M v_n(g) dvol_{\iota^(g)} = \int_M w_n(\tilde{g}) dvol_{\iota^(\tilde{g})}. Now the conformal invariance of the latter integral implies the equality (<ref>). Following <cit.>, the integral $\A$ anomaly \begin{equation}\label{def-A} \A \st \int_{M^n} v_n dvol_h \end{equation} for a closed $M$ is called the singular Yamabe energy of $M$. The quantity $\A$ appears in the asymptotic expansion of the volume of the singular metric $\sigma^{-2}g$ (Theorem <ref>). Now we continue with the We use adapted coordinates. It suffices to prove that the operator $\mathbf{L}(\lambda) \st L(\eta^(g),s;\lambda)$ satisfies \int_X \mathbf{L}(\lambda)(\varphi) \psi dvol_{\eta^(g)} = \int_X \varphi \mathbf{L}(-\lambda-n)(\psi) dvol_{\eta^(g)} if $X=[0,\varepsilon) \times M$, $\psi \in C_c^2([0,\varepsilon) \times M)$ and $\varphi \in C^2([0,\varepsilon) \times M)$ so that $\varphi(0,x)=0$. Now, by definition \mathbf{L}(\lambda) = (n\!+\!2\lambda\!-\!1) (\eta^(\|\NV\|^2) \partial_s + \lambda \eta^(\rho)) - s (\Delta_{\eta^(g)} + \lambda \eta^(\J)). In the following, we simplify the notation by writing $g$, $\rho$, and $\J$ instead of the pull-backs of these quantities by $\eta$. Then $a$ $h_s$ \begin{equation}\label{LR-adapted} \mathbf{L}(\lambda) = (n\!+\!2\lambda\!-\!1) (a \partial_s + \lambda \rho) - s (\Delta_g + \lambda \J), \end{equation} where $a \st \|\NV\|^2$ . In these terms, the background metric reads $g = a^{-1} ds^2 + h_s$ and we obtain $dvol_g = a^{-1/2} ds dvol_{h_s}$. Hence \begin{equation} v = a^{-1/2} (\det (h_s)/\det(h))^{1/2} \end{equation} \begin{equation}\label{vol-g} \frac{v'}{v} = -\frac{1}{2} \frac{a'}{a} + \frac{1}{2} \tr (h_s^{-1} h'_s), \end{equation} where $'$ denotes the derivative in the variable $s$. An easy calculation shows that \begin{equation}\label{Laplace-adapted} \Delta_g = a \partial_s^2 + \frac{a}{2} \tr (h_s^{-1} h'_s) \partial_s + \frac{1}{2} a' \partial_s - \frac{1}{2} (d\log a,d \cdot)_{h_s} + \Delta_{h_s}. \end{equation} Now we observe that \begin{align} & \int_X a \varphi' \psi dvol_g = \int_X a \varphi' \psi v ds dvol_h \\ & = - \int_X \varphi \left[a \psi' + a \frac{v'}{v} \psi + a' \psi \right] v ds dvol_h = - \int_X \varphi \left[a \psi' + a \frac{v'}{v} \psi + a' \psi \right] dvol_g \end{align} \int_X s \Delta_g(\varphi) \psi dvol_g = \int_X \varphi \Delta_g(s\psi) dvol_g using Green's formula and the assumptions. The expression (<ref>) shows that \Delta_g(s\psi) = s \Delta_g(\psi) + 2 a \psi' + \frac{a}{2} \tr (h_s^{-1} h'_s) \psi + \frac{1}{2} a' \psi. \begin{align} \int_X \mathbf{L}(\lambda)(\varphi) \psi dvol_g & = -(n\!+\!2\lambda\!-\!1) \int_X \varphi \left( a \psi' + a \frac{v'}{v} \psi + a' \psi \right) dvol_g \\ & - \int_X \varphi \left(s \Delta_g(\psi) + 2 a \psi' + \frac{a}{2} \tr (h_s^{-1} h'_s) \psi + \frac{1}{2} a' \psi \right) dvol_g \\ & + \int_X \varphi \psi (\lambda(n\!+\!2\lambda\!-\!1)\rho - \lambda s \J) dvol_g. \end{align} On the other hand, we have \begin{align} & \int_X \varphi \mathbf{L}(-\lambda\!-\!n)(\psi) dvol_g \\ & = \int_X \varphi \left(-(n\!+\!2\lambda\!+\!1) a \psi' - s \Delta_g (\psi) + (n\!+\!2\lambda\!+\!1)(\lambda\!+\!n) \rho \psi + (\lambda\!+\!n) s \J \psi \right) dvol_g. \end{align} It follows that the assertion is equivalent to the identity \begin{align} & -(n\!+\!2\lambda\!-\!1) a \frac{v'}{v} - (n\!+\!2\lambda\!-\!1) a' - \frac{a}{2} \tr (h_s^{-1} h'_s) - \frac{1}{2} a' + \lambda(n\!+\!2\lambda\!-\!1) \rho - \lambda s \J \\ & = (n\!+\!2\lambda\!+\!1)(\lambda\!+\!n) \rho + (\lambda \!+\!n) s\J. \end{align} By (<ref>), this identity is equivalent to -a \frac{v'}{v} - a' = (n+1) \rho + s \J. The identities (<ref>) and (<ref>) also show that \begin{equation}\label{Laplace-s} \Delta_g(s) = \frac{a}{2} \tr (h_s^{-1} h'_s) + \frac{1}{2} a' \stackrel{!}{=} a \frac{v'}{v} + a'. \end{equation} Thus, we have reduced the assertion to the identity - \Delta_g(s) = (n+1) \rho + s \J. But this is just the definition of $\rho$. The proof is complete. The above arguments also prove the relation (<ref>). In fact, partial integration and Green's formula yield the additional terms (n\!+\!2\lambda\!-\!1) \int_M \iota^(a \varphi \psi v) dvol_h + \int_M \iota^* (\varphi \psi \|\NV\|) dvol_h = (n+2\lambda) \int_M \iota^(\varphi \psi \|\NV\|) dvol_h since the unit normal field on $M$ is $\|\NV\| \partial_s$ and $v_0 = \|\NV\|^{-1}$ Note that equation (<ref>) can be written in the form \begin{equation}\label{bL2} v(s) \Delta_{g} (s) = \partial_s (v(s) a); \end{equation} we recall that $a = \eta^(\|\NV\|^2)$. As a corollary of this formula, we obtain a useful formula for $v(s)$ in terms of $\rho$ and $\J$. If $\sigma$ satisfies $\SCY$, then it holds \begin{equation}\label{bL} \frac{v'}{v} = \frac{-(n-1)\rho + 2 s \rho' - s \J}{1-2s\rho} + O(s^n). \end{equation} Here $\rho$ and $\J$ are identified with their pull-backs by $\eta$. If $\rho=0$, then it holds $v'/v = - s \J + O(s^n)$. The latter case contains the Poincaré-Einstein case. We write (<ref>) in the form v(s) (-(n+1) \rho - s \J) = \partial_s (v(s) a). Now the assumption implies $a = \SC - 2s \rho = 1- 2s \rho + O(s^{n+1})$. Hence \begin{align} v(s) (-(n+1) \rho - s \J) & = \partial_s (v(s) (1-2s \rho+ O(s^{n+1}))) \\ & = v'(s) (1-2s\rho) - 2 v(s) \rho - 2 v(s) s \rho' + v O(s^n). \end{align} Now simplification proves the claim. In the Poincaré-Einstein case, one easily shows that $\rho=0$ and $v'/v = - s \J$ (see Example <ref>). Lemma <ref> can be used to derive formulas for the coefficients $v_k$ with $k \le n$ in terms of the Taylor coefficients of $\rho$ and $\J$. In particular, we obtain For $\N \ni k \le n$, we have v_k = -(n\!-\!2k\!+\!1) \iota^* \frac{1}{k!} \partial_s^{k-1}(\rho) + LOT, where LOT refers to terms with lower-order derivatives of $\rho$ and $\J$. We consider the coefficient of $s^{k-1}$ in the expansion of $v'/v$. On the one hand, it equals $k v_k$. On the other hand, (<ref>) yields the expression \iota^* \partial_s^{k-1} (\rho) \left( - \frac{n-1}{(k-1)!} + \frac{2}{(k-2)!} \right) for this coefficient. This implies the assertion. In particular, the critical coefficient $v_n$ involves the quantity $\iota^* \partial_s^{n-1}(\rho)$. Conversely, a version of Lemma <ref> implies a recursive formula for the Taylor coefficients of $\rho$. For the discussion of that formula, we introduce the notation $\mathring{v}(s)$ \begin{equation}\label{ring-v} \mathring{v}(s) \st dvol_{h_s} / dvol_h. \end{equation} \frac{\mathring{v}'}{\mathring{v}} = \frac{1}{2} \tr (h_s^{-1} h_s'). We also recall that $a = 1- 2s \rho + O(s^{n+1})$ if $\sigma$ satisfies $\SCY$. The recursive formula for the Taylor coefficients of $\rho$ will be a consequence of a first-order differential equation. If $\sigma$ satisfies $\SCY$, then $\rho$ solves the differential equation \begin{equation}\label{magic-rec} - s \rho' + n \rho + a \frac{\mathring{v}'}{\mathring{v}} + s \J = O(s^n) \end{equation} with the initial condition $\rho(0) = - H$. The identity (<ref>) implies \begin{equation}\label{v-deco} \frac{v'}{v} = - \frac{1}{2} \frac{a'}{a} + \frac{\mathring{v}'}{\mathring{v}}. \end{equation} We use this decomposition on the left-hand side of (<ref>) and multiply the resulting identity with $a$. Then (n-1) \rho - 2 s \rho' + s \J - \frac{1}{2} a' + a \frac{\mathring{v}'}{\mathring{v}} = O(s^n). By $a' = - 2\rho - 2 s\rho' + O(s^n)$, this identity simplifies to (<ref>). By restriction of (<ref>) to $s=0$, we obtain $n \rho(0) + \tr (L) = 0$ using $h_{(1)} = 2L$ (see (<ref>)). Hence $\rho(0) = - H$. Another proof of $\rho(0) = -H$ will be given in Lemma <ref>. By repeated differentiation of the identity (<ref>) in the variable $s$, it follows that the Taylor coefficients of $\rho$ can be determined recursively using the Taylor coefficients of $h_s$ and $\J$. More precisely, we obtain Assume that $\sigma$ satisfies $\SCY$. Then \begin{equation}\label{rec-rho-full} (n-k) \partial_s^k(\rho)\|_0 = -\partial_s^k \left( \frac{\mathring{v}'}{\mathring{v}}\right)\|_0 + 2 \sum_{j=1}^k j \binom{k}{j} \partial_s^{j-1}(\rho)\|_0 \partial_s^{k-j} \left( \frac{\mathring{v}'}{\mathring{v}}\right)\|_0 - k \partial_s^{k-1}(\J)\|_0 \end{equation} for $1 \le k \le n-1$. For a discussion of more details of such types of formulas in low-order cases, we refer to Section <ref>. In particular, we use (<ref>) to derive explicit formulas for the first two derivatives of $\rho$ in $s$ at $s=0$. Let $g_+ = s^{-2} (ds^2 + h_s)$ be a Poincaré-Einstein metric. Assume that $g=ds^2+h_s$ is smooth up to the boundary. In particular, the obstruction tensor vanishes. In that case, adapted coordinates coincide with geodesic normal coordinates. Now it holds $\rho=0$. We show that the vanishing of the Taylor coefficients of $\rho$ (in the variable $s$) up to order $n-1$ recursively follows from (<ref>). In fact, assume that we know that $\partial_s^j (\rho)\|_0=0$ for $j=0,\dots,k-1$. Then the right-hand side of (<ref>) simplifies to -\partial_s^k \left( \frac{\mathring{v}'}{\mathring{v}}\right)\|_0 - k \partial_s^{k-1}(\J)\|_0. k \partial_s^{k-1}(\J)\|_0 = \partial_s^k(s \J)\|_0 = - \frac{1}{2} \partial_s^k (\tr (h_s^{-1}h_s')\|_0 = - \partial_s^k\left( \frac{\mathring{v}'}{\mathring{v}}\right)\|_0 \begin{equation}\label{J-trace} \J = -\frac{1}{2s} \tr (h_s^{-1}h_s') \end{equation} (which follows by combining the Einstein condition with the conformal transformation law for scalar curvature - for the details see <cit.>). Hence (<ref>) implies $\partial_s^k(\rho)\|_0=0$. Alternatively, we could note that the relation (<ref>) transforms the differential equation (<ref>) into -s \rho' + n\rho - 2s \rho \frac{\mathring{v}'}{\mathring{v}} = O(s^n) with the initial condition $\rho(0) = 0$. Then $\rho=0$ is the unique solution of this initial value problem. For $k=n$, the coefficient on the left-hand side of (<ref>) vanishes. This suggests the following formula for the singular Yamabe obstruction. Assume that $\sigma$ satisfies $\SCY$. Then \begin{equation}\label{obstruction-magic} (n\!+\!1)! \B_n = -2\partial_s^n \left( \frac{\mathring{v}'}{\mathring{v}}\right)\|_0 + 4 \sum_{j=1}^n j \binom{n}{j} \partial_s^{j-1}(\rho)\|_0 \partial_s^{n-j} \left( \frac{\mathring{v}'}{\mathring{v}}\right)\|_0 - 2n \partial_s^{n-1}(\J)\|_0. \end{equation} We start with general data $(g,\sigma)$. As before, we identify $\|\NV\|^2$ and $\SC$ with their respective pull-backs by $\eta$. Then $\SC = \|\NV\|^2 + 2 s \rho = a + 2s\rho$. In these terms, the identity (<ref>) reads v (-(n+1) \rho - s \J) = v' a + v a'. \begin{align} a \frac{v'}{v} & = - (n+1) \rho - s \J - \partial_s(a) \\ & = -(n+1) \rho - s \J + 2 \partial_s (s\rho) - \partial_s (\SC-1) \\ & = -(n-1) \rho - s \J + 2 s \rho' - \partial_s(\SC-1). \end{align} We decompose the left-hand side using (<ref>) and reorder. This gives a \frac{\mathring{v}'}{\mathring{v}} - \frac{1}{2} a' + (n-1) \rho + s \J - 2 s \rho' + \partial_s(\SC-1) = 0. Now, using $a = (1-2s\rho) + (\SC-1)$, we obtain the relation \begin{equation}\label{Basic-R} - s \rho' + n \rho + a \frac{\mathring{v}'}{\mathring{v}} + s \J = -\frac{1}{2} \partial_s (\SC-1) \end{equation} which improves (<ref>). Now, assuming that $\sigma$ satisfies $\SCY$, differentiate (<ref>) $n$ times in $s$. By $\partial_s^j(a)\|_0 = -2j \partial_s^{j-1}(\rho)\|_0$ for $1 \le j \le n$ and \partial_s^{n+1}(\SC -1)\|_0 = (n+1)! \B_n, this proves the assertion. The basic relation (<ref>) will be confirmed in a number of special cases with $\SC = 1$ in Examples <ref>–<ref>. Proposition <ref> and Theorem <ref> should be compared with <cit.>. The latter result establishes formulas for the restrictions of normal derivatives $\nabla_\NV^k(\rho)$ of $\rho$ to $M$ and for the obstruction $\B_n$ in terms of lower-order normal derivatives of $\rho$ and additional terms. The above results clarify the structure of all such additional terms. Here it is crucial to work in adapted coordinates. Note that the formula (<ref>) shows that the obstruction $\B_n$ involves the Taylor coefficients $h_{(k)}$ of $h_s$ (in the normal form (<ref>) of $g$ in adapted coordinates) for $k \le n+1$. In Section <ref>, we shall derive the classical formula for $\B_2$ (see (<ref>) and <cit.>) from (<ref>). Similarly, in Section <ref> we evaluate the formula (<ref>) for the obstruction $\B_3$ in case of a (conformally) flat background. Finally, we apply the above results to determine the leading term of the obstruction $\B_n$ for an embedding $M^n \hookrightarrow \R^{n+1}$ if $n$ is even. First, we note that Theorem <ref> and Proposition <ref> show that $\B_n$ is a functional of the second fundamental form $L$. Here we also use the formulas which recursively determine the coefficients of $h_s$ and the formula for the first two coefficients. The claim for the Taylor coefficients of $h_s$ is a bit more complicated since it involves the proof that the variations of the Christoffel symbols also lead to functionals of $L$. For a flat background metric and even $n$, it holds \begin{equation}\label{Bn-deco} (n+1)! \B_n = c_n \Delta^\frac{n}{2} (H) + nl \end{equation} c_n = - 2 \frac{(n-1)!!}{(n-2)!!} and a non-linear functional $nl$ of $L$. For odd $n$, the obstruction $\B_n$ is non-linear in $L$. The non-linear part in (<ref>) can also be described as a term of lower differential order. In fact, we can write $\B_n$ as a sum of terms that are homogeneous in $L$. In each such term, the sum of the number of derivatives and the homogeneous degree in $L$ is $n+1$. But the non-linear terms in (<ref>) consist of homogeneous terms of degree at least $2$. One should compare that version of the structural result for $\B_n$ with <cit.>. Theorem <ref> extends the following observations. By the second formula in (<ref>), $\B_2$ is the sum of a constant multiple of $\Delta(H)$ being linear in $L$ and a term that is cubic in $L$ and does not contain derivatives. Similarly, the first three terms in (<ref>) are homogeneous of degree $2$ in $L$ and each such term involves $2$ derivatives. Let $n$ be even. We extract from the formula \begin{equation}\label{ob-magic} (n\!+\!1)! \B_n = -2 \partial_s^n \left( \frac{\mathring{v}'}{\mathring{v}}\right)\|_0 + 4 \sum_{j=1}^n j \binom{n}{j} \partial_s^{j-1}(\rho)\|_0 \partial_s^{n-j} \left( \frac{\mathring{v}'}{\mathring{v}}\right)\|_0 \end{equation} the contributions which are linear in $L$. In the following, the symbol $nl$ indicates non-linear terms. First, we ignore in this sum all products with at least two factors. Hence (n+1)! \B_n = - 2 \partial_s^n \left( \frac{\mathring{v}'}{\mathring{v}}\right)\|_0 + nl. Moreover, the expansion \frac{\mathring{v}'}{\mathring{v}} = \frac{1}{2} \tr (h_s^{-1} h_s') = \frac{1}{2} \sum_{k \ge 1} ( k \tr (h_{(k)}) + nl ) s^{k-1} \partial_s^n \left( \frac{\mathring{v}'}{\mathring{v}}\right)\|_0 = \frac{1}{2} (n+1)! \tr (h_{(n+1)}) + nl. In order to evaluate $h_{(n+1)}$, we $(n-1)$-times differentiate in $s$ the determining relation (<ref>) for $h_s$. Then \frac{1}{2}\partial_s^{n+1}(h_s)\|_0 = - \Hess (\partial_s^{n-1}(s\rho))\|_0) + nl using $g^{00} = a = 1 - 2s \rho$. Hence \frac{1}{2} (n+1)! h_{(n+1)} = - (n-1) \Hess (\partial_s^{n-2}(\rho)\|_0 )+ nl. These results imply \begin{align}\label{Bn-leading} (n+1)! \B_n & = - (n+1)! \tr (h_{(n+1)}) + nl \notag \\ & = 2 (n-1) \Delta (\partial_s^{n-2}(\rho)\|_0) + nl. \end{align} Now Proposition <ref> shows that \begin{align} (n-k) \partial_s^k(\rho)\|_0 & = - \partial_s^k \left(\frac{\mathring{v}'}{\mathring{v}}\right)\|_0 + nl \\ & = - \frac{1}{2} (k+1)! \tr (h_{(k+1)}) + nl. \end{align} But $(k-1)$-times differentiating in $s$ the determining relation for $h_s$, shows that \begin{align} \frac{1}{2} \partial_s^{k+1} (h_s)\|_0 & = - \Hess(\partial_s^{k-1}(s \rho)\|_0) + nl \\ & = - (k-1) \Hess (\partial_s^{k-2}(\rho)\|_0) +nl. \end{align} \begin{equation}\label{rho-leading} (n-k) \partial_s^k(\rho)\|_0 = (k-1) \Delta (\partial_s^{k-2}(\rho)\|_0) + nl. \end{equation} Combining (<ref>) and (<ref>) gives \begin{align} (n+1)! \B_n & = 2 (n-1) \Delta (\partial_s^{n-2}(\rho)\|_0) + nl \\ & = (n-1)(n-3) \Delta^2 (\partial_s^{n-4}(\rho)\|_0) + nl \\ & = \cdots = 2 (n-1)!!/(n-2)!! \Delta^\frac{n}{2}(\rho\|_0) + nl. \end{align} This implies the assertion using $\rho\|_0 = -H$ (Lemma <ref>). For odd $n$, the same arguments show that $\B_n$ is a constant multiple of $\Delta^{\frac{n-1}{2}}(\partial_s(\rho)\|_0) + nl$. Since $\partial_s(\rho)\|_0$ is a constant multiple of $\|\lo\|^2$ (Lemma <ref>), this completes the proof. For $n=2$ and $n=4$, we find 3! \B_2 = - 2 \Delta(H) + nl \quad \mbox{and} \quad 5! \B_4 = - 3 \Delta^2 (H) + nl, respectively. The first decomposition fits with (<ref>). For a flat background and odd $n$, the proof of Theorem <ref> shows that one contribution to $\B_n$ is a constant multiple of $\Delta^{\frac{n-1}{2}}(\|\lo\|^2)$. But $\B_n$ has further contributions of the same differential order, which are quadratic in $L$. For instance, (<ref>) shows that in addition to $\Delta(\|\lo\|^2)$, $\B_3$ contains the contributions $\|dH\|^2$ and $(\lo,\Hess(H))$ of differential order $2$. We finish this section with a representation theoretical argument proving the vanishing of the obstruction $\B_n$ for the equatorial subsphere $S^n \hookrightarrow S^{n+1}$. In the following, we use the notation of Section <ref>. First, assume that $M^n \hookrightarrow \R^{n+1}$ (with the flat metric $g_0$). Let $\B_n^M(g_0)$ be the singular Yamabe obstruction of $M$. Let $\gamma \in SO(1,n+2)$ be a conformal diffeomorphism of $g_0$, i.e., $\gamma_(g_0) = e^{2 \Phi_\gamma} g_0$. Then e^{(n+1) \iota^ \Phi_\gamma} \B_n^M (\gamma_(g_0)) = \B_n^M(g_0) by Lemma <ref>. This relation is equivalent to \begin{equation}\label{B-diffeo} e^{(n+1) \iota^ \Phi_\gamma} \gamma_* (\B_n^{\gamma(M)}(g_0)) = \B_n^M(g_0), \; \gamma \in SO(1,n+2). \end{equation} In particular, all $\gamma \in SO(1,n+1) \hookrightarrow SO(1,n+2)$ leave invariant the hypersurface $M^n = \R^n \hookrightarrow \R^{n+1}$, and it holds \iota^* \left(\frac{\gamma_(r)}{r}\right)^{n+1} \gamma_ (\B_n^M (g_0) ) = \B_n^M(g_0), \pi_{n+1}^0(\gamma) (\B_n^M(g_0)) = \B_n^M(g_0). But since the identical representation is not a subrepresentation of $\pi_{n+1}^0$, it follows that $\B_n^M(g_0)=0$. A similar argument proves the vanishing of $\B_n$ for the equatorial subsphere $S^n \hookrightarrow S^{n+1}$. By the analog of (<ref>) for hypersurfaces of $S^{n+1}$, the obstruction of any $\gamma(S^n)$ vanishes, too. § RESIDUE FAMILIES In the present section, we associate residue families to any pair $(g,\sigma)$ which satisfies the condition $\SCY$ (see Section <ref>). Residue families are defined in terms of the residues of one-parameter families \lambda \mapsto \langle M_u(\lambda),\psi \rangle = \int_X \sigma^\lambda u \psi dvol_g, \; \Re(\lambda) \gg 0 of distributions on $X$, where $u$ are eigenfunctions of the Laplacian $\Delta_{\sigma^{-2}g}$ of the singular metric $\sigma^{-2}g$. These residues express the obstruction extending $u$ as a distribution up to the boundary of $X$. The restriction of $\SC(g,\sigma)$ to the boundary $M$ equals $\|\grad_g(\sigma)\|_g^2 = \|d\sigma\|^2_g$. Therefore, the condition $\SCY$ implies that $\|d\sigma\|_g^2=1$ on $M$. That property is equivalent to the property that the sectional curvatures of $\sigma^{-2}g$ tend to $-1$ at the boundary $M$, i.e., the metric $\sigma^{-2}g$ is asymptotically hyperbolic. Next, we recall some basic results in the spectral theory of the Laplacian of asymptotically hyperbolic metrics. For more details, we refer to <cit.>. The spectrum of $-\Delta_{\sigma^{-2}g}$ is the union of a finite pure point spectrum $\sigma_{pp} \subset (0,(n/2)^2)$ and an absolutely continuous spectrum $\sigma_{ac} = [(n/2)^2,\infty)$ of infinite multiplicity. The generalized eigenfunctions with smooth functions on $M$ as boundary values are described by a Poisson operator. This operator is a far-reaching generalization of the well-known Poisson transform of Helgason <cit.> which relates generalized eigenfunctions of the commutative algebra of invariant differential operators on a symmetric space of the non-compact type to hyperfunctions on a naturally associated boundary. It is defined by an integral transform. In the present situation, the family $\Po(\lambda)$ of Poisson operators is meromorphic for $\Re(\lambda)<n/2$, $\lambda \ne n/2$ with poles in $\lambda$ iff $\lambda(n-\lambda) \in \sigma_{pp}$ such that $\Po(\lambda)$ Poisson operator (\Delta_{\sigma^{-2}g} + \lambda(n-\lambda)) \Po(\lambda) (f) = 0 for any $f \in C^\infty(M)$. In contrast to Helgason's definition of a Poisson operator by an integral transform, it is defined in terms of the resolvent of the Laplacian. In both theories, the argument $f$ is seen in the leading terms of the asymptotic expansion of the eigenfunction $\Po(\lambda)(f)$. To describe the asymptotic expansion of eigenfunctions in the range of the Poisson operator, we choose coordinates on $X$ near the boundary. Indeed, there is a unique defining function $\theta$ and a diffeomorphism $\tau$ mapping $[0,\varepsilon] \times M$ with coordinates $(t,x)$ to a neighborhood of $M$ in $X$ so that $\tau$ $\theta$ \tau^* (\sigma^{-2} g) = t^{-2} (dt^2 + h_t) \quad \mbox{and} \quad \; \iota^(\theta^2 \sigma^{-2}g) = h_0 \stackrel{!}{=} h, \; \tau^(\theta) = t. This is the normal form of an asymptotically hyperbolic metric with prescribed conformal infinity as used in <cit.>. The function $\theta$ is a solution of the eikonal equation $\|d\theta\|_{\theta^2 \sigma^{-2} g}=1$ near $M$ and the gradient flow of $\theta$ with respect to the metric $\theta^2 \sigma^{-2} g$ defines the diffeomorphism $\tau$. Note that $\iota^(\theta^2 \sigma^{-2}) = 1$. In these terms, the eigenfunctions $\Po(\lambda)(f)$ have the following properties. (i) $ \tau^ \Po(\lambda) (f) = t^{\lambda} f + t^{n-\lambda} g + O(t^{n/2+1})$ for some $g \in C^\infty(M)$ if $\Re(\lambda)=n/2$, $\lambda \ne n/2$.[Of course, the function $g$ should not be confused with the metric $g$.] (ii) $\tau^* \Po(\lambda) (f) = t^\lambda F + t^{n-\lambda} G$ with smooth $F$ and $G$ on $[0,\varepsilon) \times M$ so that $\iota^(F)=f$ if $\Re(\lambda) \le n/2$ with $\lambda \not\in \{n/2-N/2 \,\|\, N \in \N_0\}$ and $\lambda(n-\lambda) \notin \sigma_{pp}$. The function $f$ is called the boundary value of $u = \Po(\lambda)(f)$. The function $F$ in (ii) depends on $\lambda$ and has poles in $\lambda \in \{n/2-N/2 \,\|\, N \in \N\}$. But these poles cancel against poles of the second term $G= G(\lambda)$ in the decomposition of $\Po(\lambda)$. If $\lambda$ is as in (ii), we define $\SC(\lambda)$ scattering operator \begin{equation}\label{scatt-def} \Sc(\lambda)(f) = \iota^(G). \end{equation} The operator $\Sc(\lambda)$ is called the scattering operator of the asymptotically hyperbolic metric $\sigma^{-2}g$.[The substitution $\lambda \mapsto n-\lambda$ maps the operator $\Sc(\lambda)$ to the scattering operator in <cit.>.] It is a family of pseudo-differential operators with principal symbol being a constant multiple of $\|\xi\|^{n-2\lambda}$. It is meromorphic in $\Re(\lambda) < n/2$ with poles in the set $\{\frac{n-N}{2} \,\|\, N \in \N\}$ and if $\lambda(n-\lambda) \in \sigma_{pp}$. The poles in $\lambda = \frac{n-N}{2}$ are sometimes referred to as its trivial poles. Its nontrivial poles in $\Re(\lambda) > n/2$ will not be of interest here. Now let $u = \Po(\mu)(f) $ be a solution of -\Delta_{\sigma^{-2}g} u = \mu (n-\mu)u, \quad \Re(\mu) = n/2, \; \mu \ne n/2 with boundary value $f \in C^\infty(M)$. Instead of the asymptotic expansions of $u$ as above, we will consider asymptotic expansion in terms of powers of $\sigma$. The following formal arguments describe the expansion of $u$ using the Laplace-Robin operators $L(g,\sigma;\mu)$.[The arguments are only formal since the products $\sigma^j f_j$ are not functions on $X$ without a specification of coordinates.] In the Poincaré-Einstein case, the following algorithm is contained in the proof of <cit.>. We start with $f_0 = f$ and define $f_N \in C^\infty(M)$ recursively by N(n-2\mu-N) f_N = \iota^\left(\sigma^{-N+1} L(-\mu) \left(\sum_{j=0}^{N-1} \sigma^j f_j \right)\right). But the definition of $L(\lambda)$ implies L(\lambda) (\sigma^j f_j) = j(n+2\lambda-j) \sigma^{j-1} f_j + O(\sigma^j), \; \lambda \in \C. L(-\mu) \left(\sum_{j=0}^N \sigma^j f_j\right) = O(\sigma^N) and the conjugation formula yields -(\Delta_{\sigma^{-2}g} + \mu(n-\mu)) \left(\sigma^\mu \sum_{j=0}^{N} \sigma^j f_j \right) = \sigma^{\mu+1} L(-\mu) \left(\sum_{j=0}^{N} \sigma^j f_j \right) = \sigma^{\mu} The coefficients $f_j$ are given by differential operators $f \mapsto \T_j(\mu)(f)$. The construction shows that the operator $\T_N(\mu)$ has simple poles in the set \left \{ \frac{n-j}{2} \; \|\, j \le N \right \}. These are the poles that appeared above in (ii). Note that, using $\iota^(\rho)=-H$, it easily follows that $\T_1(\mu)=-\mu H$ (see also Lemma <ref>). We also observe that $L(\mu)(1) \in C^\infty(X)$ is a multiple of $\mu$. By an easy induction, this implies that \begin{equation}\label{T-1} \T_N(\mu)(1) = \frac{\mu}{N! (n\!-\!2\mu\!-\!1) \cdots (n\!-\!2\mu\!-\!N)} \QC_{N}(\mu) \end{equation} for some polynomial $\QC_{N}(\mu) \in C^\infty(M)$. In particular, the function $\T_n(\mu)(1)$ is regular at $\mu=0$. Now, in order to justify the above arguments, we use adapted coordinates. Let $u$ be as above. Then $\tau^(u)$ has the form $t^\mu F + t^{n-\mu} G$ with $\iota^(F)=f$ and $\iota^(G) = \SC(\mu)(f)$. Let $\zeta \st \tau^{-1} \circ \eta$. Then $\zeta$ \zeta^(t) = \eta^* \tau_* (t) = \eta^* \left(\sigma \frac{\tau_(t)}{\sigma}\right) = s \eta^ \left(\frac{\tau_(t)}{\sigma}\right) by $\eta^(\sigma)=s$. It follows that the pull-back by $\zeta$ of $\tau^(u)$ equals \eta^(u) = s^\mu \Omega^\mu \zeta^(F) + s^{n-\mu} \Omega^{n-\mu} \zeta^(G) with $\Omega \st \tau_(t)/\sigma$. But $\iota^ \zeta^* (F) = \iota^(F)$, $\iota^ \zeta^(G) = \iota^(G)$ \begin{equation}\label{Omega-rest} \iota^* (\Omega) = 1 \end{equation} imply that the leading terms in the expansion of $\eta^(u)$ are $\iota^(F) = f$ and $\iota^(G) = \Sc(\mu)(f)$. In order to prove the restriction property (<ref>), we recall that \tau_ (t^{-2} (dt^2+h_t)) = \sigma^{-2} g. \Omega^2 g = dt^2 + h_t. In particular, $\iota^* (\Omega^2) h = h_0$. But in the construction of $(\tau,\theta)$ we required that $h_0=h$. This proves (<ref>). Thus, the asymptotic expansion of the eigenfunction $\eta^(u)$ of $\Delta_{\eta^(\sigma^{-2}g)}$ takes the form $\T_j(\lambda)$ solution operators \begin{equation}\label{asymp} \sum_{j \ge 0} s^{\mu+j} \T_j(\mu)(f)(x) + \sum_{j \ge 0} s^{n-\mu+j} \T_j(n\!-\!\mu) \Sc(\mu)(f)(x), \quad s \to 0, \end{equation} where $\T_j(\mu)$ are families of differential operators on $M$; we shall refer to these operators as solution operators. One easily find that the order of $\T_j(\mu)$ is $\le 2 [\frac{j}{2}]$. The above formal arguments show that the families $\T_N(\mu)$ are rational in $\mu$ with simple poles in the set \left\{ \frac{n-j}{2} \; \|\, j \le N \right\}. Of course, the coefficients $\T_j(\mu)$ can be determined recursively in terms of the Laplace-Robin operator in adapted coordinates. In the following, it will often suffice to work with a finite version of the expansion The solution operators $\T_j(\lambda): C^\infty(M) \to C^\infty(M)$ describe formal asymptotic expansions of eigenfunctions (with smooth boundary value) of the Laplacian of asymptotically hyperbolic metrics $\sigma^{-2}g$. Another type of asymptotic expansions of eigenfunctions appears in <cit.>. In an even more general setting, these are expansions, say in powers of a defining function, the coefficients of which are functions on the space $X$ but not on its boundary $M$. Comparing both types of expansions would require additional expansions of the coefficients. Later, we shall use the fact that the scattering operator $\Sc(\lambda)$ for $\lambda \in \R$ is formally selfadjoint with respect to the scalar product on $C^\infty(M)$ defined by $h$. For the convenience of the reader, we include a proof that directly derives this property from the expansion (<ref>) (without invoking the definition of $\Sc(\lambda)$ in terms of expansions in power series of $t$) (compare with <cit.>). $\Sc(\lambda)^* = \Sc(\lambda)$ for $\lambda \in \R$, $\lambda < n/2$ such that $\lambda \notin \left\{ \frac{n-N}{2} \,\|\, N \in \N \right\}$ and $\lambda(n-\lambda) \notin \sigma_{pp}$. Note that the assumptions guarantee that the ladders $\lambda+\N$ and $n-\lambda +\N$ are disjoint. We recall Green's formula \begin{equation}\label{Green} \int_X (du,dv)_g dvol_g + \int_X u \Delta_g(v) dvol_g = \int_{\partial X} u \star_g dv = \int_{\partial X} u i_{N(v) N} (dvol_g) \end{equation} on a compact Riemannian manifold $(X,g)$ with boundary $\partial X$. Here $N$ denotes a unit normal field. Now let $\R \ni \lambda < n/2$ be as above. Let $u_1$ and $u_2$ be real solutions of -\Delta_{\eta^(\sigma^{-2}g)} u = \lambda(n-\lambda) u on $(0,\varepsilon) \times M$ of the form $u = s^{\lambda} F + s^{n-\lambda} G$ with smooth $F,G$. These are defined by the Poisson transforms of smooth boundary functions $f_1$ and $f_2$. Let k \st \eta^(\sigma^{-2}g) = s^{-2}( a^{-1} ds^2 + h_s). Then $dvol_k = s^{-n-1} a^{-1/2} ds dvol_{h_s}$ and the restriction of $\nu = s \sqrt{a} \partial_s$ to the boundary $s=\varepsilon$ defines a unit normal field. By (<ref>), we find \begin{align}\label{Green-a} & \int_{s > \varepsilon} ((du_1,du_2)_k - \lambda(n-\lambda) u_1 u_2) dvol_k \notag \\ & = \varepsilon^{-n} \int_{s=\varepsilon} u_1 \partial_{\nu}(u_2) dvol_{h_\varepsilon} = \varepsilon^{-n+1} \int_{s=\varepsilon} u_1 \partial_s(u_2) \sqrt{a} dvol_{h_\varepsilon}. \end{align} The finite part of the expansion of the last integral in $\varepsilon$ is the coefficient of $\varepsilon^{n-1}$ in the expansion of \int_{s=\varepsilon} u_1 \partial_s(u_2) \sqrt{a} dvol_{h_\varepsilon}. By plugging in the expansions of $u_1$, $u_2$, $\sqrt{a}$ and $dvol_{h_\varepsilon}$, we obtain the expression \int_M \iota^* (\lambda F_2 G_1 + (n-\lambda) F_1 G_2) dvol_h for this coefficient. Since the left-hand side of (<ref>) is symmetric in $u_1$ and $u_2$, the latter result equals \int_M \iota^* (\lambda F_1 G_2+ (n-\lambda) F_2 G_1) dvol_h. It follows that \int_M \iota^* (F_1 G_2) dvol_h = \int_M \iota^* (F_2 G_1) dvol_h. In terms of the expansion (<ref>), this means that \int_M f_1 \SC(\lambda) (f_2) dvol_h = \int_M f_2 \SC(\lambda) (f_1) dvol_h, i.e., $\Sc(\lambda)^* = \Sc(\lambda)$. Now we are ready to define residue families. $\D_N^{res}(g,\sigma;\lambda)$ residue family Let $\N \ni N \le n$ and assume that the condition $\SCY$ is satisfied. We consider an eigenfunction $u$ of $\Delta_{\sigma^{-2}} g$ with boundary value $f \in C^\infty(M)$ satisfying \begin{equation}\label{eigen} \Delta_{\sigma^{-2}g} u + \mu (n-\mu) u = 0 \quad \mbox{for $\Re(\mu) = n/2$, $\mu \ne n/2$}. \end{equation} Such eigenfunctions have asymptotic expansions (as above) near the boundary. We consider the integral \begin{equation}\label{Mu-L} \left\langle M_u(\lambda),\psi \right\rangle = \int_X \sigma^\lambda u \psi dvol_g \end{equation} with $\psi \in C_c^\infty(X)$ and $\lambda \in \C$. The function $\lambda \mapsto \left\langle M_u(\lambda),\psi \right\rangle$ is holomorphic if $\Re(\lambda) \gg 0$ and we regard $M_u(\lambda)$ as a holomorphic family of distributions on $X$. If $\supp(\psi) \cap M = \emptyset$, then $M_u(\lambda)$ admits a holomorphic continuation to $\C$. $M_u(\lambda)$ generalizes the meromorphic family of distributions $M(\lambda) = M_1(\lambda)$ discussed in Section <ref>. Likewise as $M(\lambda)$, the family $M_u(\lambda)$ admits a meromorphic continuation with simple poles in $\{-\mu - N - 1 \}$. The proof of this fact is similar as for $u=1$ and follows by expanding the integrand near the boundary. In addition to these poles, $M_u(\lambda)$ has simple poles in the set $\{\mu + n - N - 1\}$. However, this second ladder of poles will be ignored in the following. The details are given in the proof of Theorem <ref>. This proof also shows that the residues have the form \begin{equation}\label{delta-def} \Res_{\lambda=-\mu-1-N} \left( \int_X \sigma^\lambda u \psi dvol_g \right) = \int_M f \delta_N(g,\sigma;\mu)(\eta^(\psi)) dvol_h \end{equation} with some meromorphic families $\delta_N(g,\sigma;\mu)$ of differential operators $C^\infty([0,\varepsilon)\times M) \to C^\infty(M)$ of order $\le N$. The residues of $M_u(\lambda)$ are distributions on $X$ with support on the boundary $M$ of $X$. They may be regarded as obstructions to extending $M_u(\lambda)$ as a distribution to $X$. Let $\N \ni N \le n$ and assume that the condition $\SCY$ is satisfied. Then the one-parameter family \begin{equation}\label{D-res-def} \D_N^{res}(g,\sigma;\lambda) = N!(2\lambda\!+\!n\!-\!2N\!+\!1)_N \delta_N(g,\sigma;\lambda\!+\!n\!-\!N), \; \lambda \in \C \end{equation} is called the residue family of order $N$. The family $\D_n^{res}(\lambda)$ will be called the critical residue family. residue families are not conformally covariant: we need η^ for that! we distinguish two versions: one without and one with η^* Some comments are in order. The families $\delta_N(\mu)$ are defined for $\Re(\mu) = \frac{n}{2}$ (with $\mu \ne \frac{n}{2}$). Hence $\D_N^{res}(\lambda)$ is defined for $\Re (\lambda) = -\frac{n}{2}+N$ (with $\lambda \ne -\frac{n}{2}+N$). But the normalizing coefficient in (<ref>) is chosen so that $\D_N^{res}(g,\sigma;\lambda)$ actually extends to a polynomial family of order $N$ and degree $2N$ in $\lambda \in \C$. This fact will follow from Theorem <ref>. The proof of the latter result actually contains a second proof of the existence of the meromorphic continuation of $\langle M_u(\lambda),\psi \rangle$. The method is a version of a Bernstein-Sato-type argument. It provides explicit knowledge of the position of poles of $M_u(\lambda)$ and formulas for the residues. However, the following result gives a more direct description of the operator $\delta_N(g,\sigma;\lambda)$ in terms of solution operators. The equivalence of both descriptions of residues will have interesting consequences. Let $\N \ni N \le n$. Then \begin{align}\label{D-res-sol} \D_N^{res}(g,\sigma;\lambda) & = N! (2\lambda\!+\!n\!-\!2N\!+\!1)_N \\ & \times \sum_{j=0}^{N} \frac{1}{j!} \left[ \T_{N-j}^(g,\sigma;\lambda\!+\!n\!-\!N) v_0 + \cdots + \T_0^(g,\sigma;\lambda\!+\!n\!-\!N) v_{N-j} \right] \iota^* \partial_s^j. \notag \end{align} The relation (<ref>) implies \begin{equation}\label{M-adapted} \langle M_u(\lambda),\psi \rangle = \int_X \sigma^\lambda u \psi dvol_g = \int_{(0,\varepsilon) \times M} s^\lambda \eta^(u) \eta^(\psi) dvol_{\eta^(g)}, \; \Re(\lambda) \gg 0. \end{equation} Here $\eta^(u) $ is an eigenfunction of the Laplacian of the metric $s^{-2} \eta^(g)$. In order to simplify the notation, we write the latter integral as \int_{[0,\varepsilon) \times M} s^\lambda u \psi dvol_{\eta^(g)} = \int_0^\infty \int_M s^\lambda u \psi v ds dvol_h with an appropriate eigenfunction $u$ and test functions $\psi$ on the space $[0,\varepsilon) \times M$. Now we expand $v$ according to (<ref>) and $u$ according to (<ref>). The classical formula <cit.> \begin{equation}\label{Gelfand} \Res_{\lambda=-N-1} \left( \int_0^\infty s^\lambda \psi(s) ds \right) = \frac{\psi^{(N)}(0)}{N!}, \; N \in \N_0 \end{equation} for test functions $\psi \in C_c^\infty(\overline{\R_+})$ shows that \begin{align} & \Res_{\lambda=-\mu-1-N} \left( \int_X \sigma^\lambda u \psi dvol_g \right) \\ & = \Res_{\lambda=-\mu-1-N} \left( \int_0^\infty \sum_{a=0}^N \sum_{j+k=a} \int_M s^{\lambda+\mu+a} \T_k(\mu)(f) v_j \psi dvol_h \right) \\ & = \sum_{a=0}^N \frac{1}{(N-a)!} \sum_{j+k=a} \int_M \T_k(\mu)(f) v_j \iota^ \partial_s^{N-a}(\psi) dvol_h. \end{align} Since $f$ is arbitrary, taking adjoints proves the assertion. Definition <ref> generalizes the notion of residue families $D_N^{res}(h;\lambda)$ introduced in <cit.>. In that case, $g = r^2 g_+$ is the conformal compactification of a Poincaré-Einstein metric $g_+$. However, the definitions in <cit.> use a different normalizing coefficient. That choice is motivated by the fact that in these references, the expansion of $g$ involves only even powers of $r$. More precisely, if $g_+$ is in normal form relative to $h$, i.e., $\iota^(r^2g_+)=h$, then it holds \begin{align} \D^{res}_{2N}(g,r;\lambda) & = (-2N)_N \left(\lambda\!+\!\frac{n}{2}\!-\!2N\!+\!\frac{1}{2}\right)_N D^{res}_{2N}(h;\lambda), \\ \D^{res}_{2N+1}(g,r;\lambda) & = -2(-2N\!-\!1)_{N+1}\left(\lambda\!+\!\frac{n}{2}\!-\!2N\!-\!\frac{1}{2}\right)_{N+1} D^{res}_{2N+1}(h;\lambda). \end{align} If $\lambda$ is a zero of the prefactors in these factorization identities, then the family $\D_N^{res}(g,r;\lambda)$ vanishes and therefore hides the non-trivial operator $D_N^{res}(h;\lambda)$. In particular, if $2N+1=n$, then \D_n^{res} (g,r;0) = 0. However, for general $g$ and $\sigma$ satisfying $\SCY$, the critical value $\D_n^{res}(g,\sigma;0)$ for odd $n$ need not vanish. For the case $n=3$, we refer to Section <ref>. The degrees of $D_{2N}^{res}(h;\lambda)$ and $D_{2N+1}^{res}(h;\lambda)$ both equal $N$. Hence the above relations show that the respective degrees of $\D_{2N}^{res}(g,r;\lambda)$ and $\D_{2N+1}^{res}(g,r,\lambda)$ are $2N$ and $2N+1$. Since in the generic case $\D_N^{res}(\lambda)$ has degree $2N$, it follows that in the Poincaré-Einstein case the degrees fall on half. This drop in degree reflects the vanishing of curvature invariants in the Poincaré-Einstein case. For instance, (<ref>) and Lemma <ref> show that $\D_1^{res}(g,r;\lambda)$ and $\D_2^{res}(g,r;\lambda)$ have respective degrees $1$ and $2$. In these cases, the vanishing of $H$, $\lo$, and $\Rho_{00}$ are responsible for the drop in degree. The following result implies that residue families of order $N \le n$ are completely determined by the metric $g$ and the embedding $M \hookrightarrow X$. Let $\N \ni N \le n$. Then $\D_N^{res}(g,\sigma;\lambda)$ is determined only by the coefficients $\sigma_{(j)}$ for $j \le N+1$ in the expansion of $\sigma$ in geodesic normal coordinates. The claim follows by evaluating the residue definition of residue families in terms of geodesic normal coordinates. We use the diffeomorphism $\kappa$ to write \begin{equation}\label{M-geodesic} \langle M_u(\lambda),\psi \rangle = \int_X \sigma^\lambda u \psi dvol_g = \int_{[0,\varepsilon)} \int_M \left( \frac{\kappa^(\sigma)}{r} \right)^\lambda r^\lambda \kappa^(u) \kappa^(\psi) dr dvol_{h_r} \end{equation} for $\Re(\lambda) \gg 0$ and test functions $\psi$ with sufficiently small support. The eigenfunction $\kappa^(u)$ of $\Delta_{\kappa^(\sigma)^{-2} g}$ has an asymptotic expansion in $r$ of the form \sum_{j \ge 0} r^{\mu+j} \T_j(\mu)(f) + \cdots, where the dots indicate an asymptotic expansion with exponents $n-\mu+j$. In that expansion, the operators $\T_j(\mu)$ are determined by recursive relations. An induction argument using the formula \Delta_{\kappa^(\sigma)^{-2} g} = (\kappa^(\sigma))^2 \Delta_{dr^2 + h_r} - (n-1) \kappa^(\sigma) \nabla_{\grad(\kappa^(\sigma))} shows that $\T_N(\mu)$ is determined only by the coefficients of $r^j$ for $j \le N+1$ in the expansion of $\kappa^(\sigma)$, i.e., by $\sigma_{(j)}$ for $j \le N+1$. Now the residue of the left-hand side of (<ref>) at $\lambda=-\mu-1-N$ is determined by the coefficient of $r^{\lambda+\mu+N}$ in the expansion of the integrand. That coefficient involves the operators $\T_j(\mu)$ with $j \le N$ and the coefficients of $r^{j}$ for $j \le N$ in the expansion of $\kappa^(\sigma)/r$. The latter are determined by $\sigma_{(j)}$ for $j \le N+1$. All other ingredients of the integrand do not depend on $\sigma$. The proof is complete. Since the coefficients $\sigma_{(j)}$ for $j \le n+1$ are determined by the metric $g$ and the embedding $\iota$, the residue families $\D_N^{res}(g,\sigma;\lambda)$ for $N \le n$ are completely determined by the metric $g$ and the embedding $\iota$, and it is justified to use the simplified notation to $\D_N^{res}(g;\lambda)$. The definition of residue families can be extended to a wider setting. Let $\sigma \in C^\infty(X)$ be a boundary defining function so that $\|d\sigma\|_g^2=1$ on the boundary $M$. Then the singular metric $\sigma^{-2}g$ is asymptotically hyperbolic. This implies the existence of a Poisson operator and the existence of an eigenfunction $u$ of $\Delta_{\sigma^{-2}g}$ with eigenvalue $-\mu(n-\mu)$ and arbitrary given boundary value $f\in C^\infty(M)$. The asymptotic expansion of $u$ in terms of adapted coordinates can be stated as an asymptotic expansion in powers of $\sigma$. For $N \in \N$, we define an operator \delta_N(g,\sigma;\mu): C^\infty(X) \to C^\infty(M) \Res_{\lambda=-\mu-1-N} \left(\int_X \sigma^\lambda u \psi dvol_g \right) = \int_M f \delta_N(g,\sigma;\mu) (\psi) dvol_{\iota^(g)} and let \D_N^{res}(g,\sigma;\lambda) \st N! (2\lambda\!+\!n\!-\!2N\!+\!1)_N \delta_N(g,\sigma;\lambda\!+\!n\!-\!N). These general residue families are conformally covariant in the following sense. The residue family $\D_N^{res}(g,\sigma;\lambda)$ is conformally covariant in the sense that \D_N^{res}(\hat{g},\hat{\sigma};\lambda) \circ e^{\lambda \varphi} = e^{(\lambda-N) \iota^* (\varphi)} \circ \D_N^{res}(g,\sigma;\lambda) for all conformal changes $(\hat{g},\hat{\sigma}) = (e^{2\varphi}g,e^{\varphi}\sigma)$, $\varphi \in C^\infty(X)$. Let $u \in \ker(\Delta_{\sigma^{-2}g}+\mu(n-\mu))$ be an eigenfunction with leading term $f \in C^\infty(M)$ in its expansion into powers of $\sigma$. We calculate the \Res_{\lambda=-\mu-1-N} \left(\int_X \sigma^\lambda u \psi dvol_g \right) in two ways. On the one hand, it equals \int_M f \delta_N(g,\sigma;\mu) (\psi) dvol_{\iota^(g)}. On the other hand, for $\Re(\lambda) \gg 0$, we have \int_X \sigma^\lambda u \psi dvol_g = \int_X \hat{\sigma}^\lambda u (e^{(-\lambda-n-1)\varphi} \psi) dvol_{\hat{g}} and the leading term in the $\hat{\sigma}$-expansion of $u$ equals $e^{-\mu \iota^(\varphi)}f$. Hence the residue equals \int_M e^{-\mu \iota^(\varphi)} f \delta_N(\hat{g},\hat{\sigma};\mu)(e^{(\mu-n+N)\varphi} \psi) But $dvol_{\iota^(\hat{g})} = e^{n \iota^* \varphi} dvol_{\iota^(g)}$. Since $f \in C^\infty(M)$ is arbitrary, we find \delta_N(g,\sigma;\mu) = e^{(-\mu+n)\iota^(\varphi)} \circ \delta_N(\hat{g},\hat{\sigma};\mu) \circ e^{(\mu+N-n)\varphi}. This result implies the assertion. As a special case, it follows that the compositions of residue families of order $N \le n$ in the sense of Definition <ref> with $\eta^$ (and $\sigma$ being a solution of the Yamabe problem) are conformally covariant. Whereas the general residue families in Theorem <ref> depend on $g$ and $\sigma$, Proposition <ref> shows that the special cases in Definition <ref> only depend on $g$ (and the embedding $\iota$). Although Definition <ref> breaks the conformal covariance (by omitting $\eta^$), for those values of $\lambda$ for which residue families are tangential, the resulting operators on $M$ are still conformally covariant. This observation will play a central role in Section <ref>. In the following sections, it will always be clear from the context which notion of residue families is being used. § RESIDUE FAMILIES AS COMPOSITIONS OF $L$-OPERATORS In the present section, we show that the composition of residue families as defined in Definition <ref> with $\eta^$ (defining adapted coordinates) can be identified with compositions of Laplace-Robin operators and the restriction operator $\iota^$. We recall the notation $L_N(g,\sigma;\lambda) \st L(g,\sigma;\lambda\!-\!N\!+\!1) \circ \cdots \circ L(g,\sigma;\lambda)$ and set $L_0 = \id$ (see (<ref>)). Let $\N \ni N \le n$ and assume that $\sigma$ satisfies the condition $\SCY$. Then \D_N^{res}(g,\sigma;\lambda) \circ \eta^* = \iota^* L_N(g,\sigma;\lambda). It suffices to prove that \begin{equation}\label{red-main} \delta_N(g,\sigma;\lambda) \circ \eta^* = \frac{1}{N! (2\lambda\!-\!n\!+\!1)_N} \iota^* L(g,\sigma;\lambda\!-\!n\!+\!1) \circ \cdots \circ L(g,\sigma;\lambda\!-\!n\!+\!N). \end{equation} Let $u$ be an eigenfunction with boundary value $f \in C^\infty(M)$ satisfying (<ref>) with $\Re(\mu) = n/2$, $\mu \ne n/2$. In the following, it will be convenient to use the notation \begin{equation} A((\lambda)_N) \st A(\lambda) \circ A(\lambda+1) \circ \cdots \circ A(\lambda+N-1) \end{equation} for any $\lambda$-dependent family $A(\lambda)$ of operators. Then $L_N(\lambda) = L((\lambda-N+1)_N)$. On the one hand, (<ref>) states that \begin{equation}\label{eq:h1} \Res_{\lambda=-\mu-1-N} \left(\int_{X} \sigma^{\lambda} u \psi dvol_g \right) = \int_M f \delta_N(g,\sigma;\mu)(\eta^(\psi)) dvol_h. \end{equation} Now we first assume that $\sigma$ satisfies the stronger assumption $\SC(g,\sigma)=1$. We apply Corollary <ref> to calculate \begin{align} - L(g,\sigma;\lambda+1) (\sigma^{\lambda+1} u) & = \sigma^\lambda \left(\Delta_{\sigma^{-2}g} - (\lambda\!+\!1)(n\!+\!\lambda\!+\!1) \id \right) u \\ & = \sigma^\lambda (-\mu(n\!-\!\mu) - (\lambda\!+\!1)(n\!+\!\lambda\!+\!1)) u \\ & = -(\lambda\!+\!\mu\!+\!1)(\lambda\!-\!\mu\!+\!n\!+\!1) \sigma^\lambda u. \end{align} We regard this relation as a Bernstein-Sato-type functional equation. Hence for $\Re(\lambda) \notin -\frac{n}{2}-\N$ we obtain \begin{align} \sigma^\lambda u & = \frac{L(g,\sigma;\lambda\!+\!1)(\sigma^{\lambda+1} u)} {(\lambda\!+\!\mu\!+\!1)(\lambda\!-\!\mu\!+\!n\!+\!1)} \\ & = \frac{L(g,\sigma;\lambda\!+\!1)L(g,\sigma;\lambda\!+\!2)(\sigma^{\lambda+2} u)} {(\lambda\!+\!\mu\!+\!1)(\lambda\!+\!\mu\!+\!2)(\lambda\!-\!\mu\!+\!n\!+\!1)(\lambda\!-\!\mu\!+\!n\!+\!2)} \\ & = \dots = \frac{L(g,\sigma;(\lambda\!+\!1)_N) (\sigma^{\lambda+N} u)} {(\lambda\!+\!\mu\!+\!1)_N \end{align} It follows that the integral \lambda \mapsto \int_X \sigma^\lambda u \psi dvol_g, \; \Re(\lambda) \gg 0 admits a meromorphic continuation to $\C$ with simple poles in the set -\mu-1-\N_0 \cup \mu-n-1-\N_0. More precisely, we get \begin{equation}\label{int-N} \int_X \sigma^\lambda u \psi dvol_g = \frac{1}{(\lambda\!+\!\mu\!+\!1)_N (\lambda\!-\!\mu\!+\!n\!+\!1)_N} \int_X L(g,\sigma;(\lambda\!+\!1)_N)(\sigma^{\lambda+N} u) \psi dvol_g \end{equation} for $\Re(\lambda) > - \frac{n}{2}-1$ and $N \ge 1$. In the following, it will be convenient to choose $\lambda$ so that $\Re(\lambda) > - \frac{n}{2}+1$. Now we note that a function in $C^\infty(X^\circ)$ with an asymptotic expansion of the form $\sum_{j\ge 0} \sigma^{\nu + j} a_j$ with $\Re(\nu) > 2$ and $a_j \in C^\infty(M)$ satisfies the assumptions in Proposition <ref>. Thus, by a repeated application of Proposition <ref>, the right-hand side of (<ref>) equals \frac{ 1}{(\lambda\!+\!\mu\!+\!1)_N (\lambda\!-\!\mu\!+\!n\!+\!1)_N} \int_X \sigma^{\lambda+N} u L(g,\sigma;(-\lambda\!-\!n\!-\!N)_N)(\psi) dvol_g. By the assumptions, the zeros of the product \mu \mapsto (\lambda\!+\!\mu\!+\!1)_N (\lambda\!-\!\mu\!+\!n\!+\!1)_N are simple for $\Re(\lambda) > - \frac{n}{2}+1$. Thus, using the residue formula \begin{equation}\label{Delta0} \Res_{\lambda=-\mu-1} \left(\int_{X} \sigma^\lambda u \psi dvol_g \right) = \int_M f \iota^(\psi) dvol_h, \end{equation} we find \begin{align}\label{residue-2} & \Res_{\lambda=-\mu-1-N} \left(\int_{X} \sigma^\lambda u \psi dvol_g \right) \notag \\ & = \frac{(-1)^N}{(-N)_N (2\mu\!-\!n\!+\!1)_N} \int_M f \iota^* L(g,\sigma;(\mu\!-\!n\!+\!1)_N)(\psi) dvol_h \notag \\ & = \frac{1}{N!(2\mu\!-\!n\!+\!1)_N} \int_M f \iota^* L_N(g,\sigma;\mu\!-\!n\!+\!N)(\psi) dvol_h \end{align} for $N \ge 1$. Comparing this result with (<ref>), completes the proof of (<ref>) for $\Re(\mu) = n/2$, $\mu \ne n/2$. The assertion then follows by meromorphic continuation. If $\sigma$ satisfies only the assumption $\SCY$, analogous arguments show that the right-hand side of (<ref>) contains an additional integral \int_X u \psi R_{n+1} \sigma^{\lambda+n+1} dvol_g. Since $R_{n+1}$ is smooth up to the boundary and $N \le n$, this integral is regular at $\lambda=-\mu-1-N$, i.e., does not contribute to the residue. The proof is Let $N \in \N$ with $N \le n$ and assume that $\sigma$ satisfies $\SCY$. Then \D_N^{res}(g,\sigma;\lambda) = \D_{N-1}^{res}(g,\sigma;\lambda-1) \circ L(g,\sigma;\lambda). Theorem <ref> identifies the composition of residue families with $\eta^$ with compositions of $L$-operators if $\sigma$ satisfies $\SCY$. By Theorem <ref>, residue families are linear combinations of compositions of tangential operators and iterated normal derivatives $\iota^ \partial^k_s$. We may use formula (<ref>) to write their composition with $\eta^$ in terms of iterated gradients $\nabla_\NV^k$. This yields a formula for the composition of residue families with $\eta^$ in terms of iterated gradients and tangential For closed $M$, Theorem <ref> implies formulas for integrated renormalized volume coefficients in terms of compositions of Laplace-Robin operators. First, we observe that, for $N \le n-1$, Theorem <ref> implies \begin{align} \int_M \D_N^{res}(g,\sigma;-n\!+\!N)(1) dvol_h & = N! (-n\!+\!1)_N \sum_{j=0}^{N} \int_M \T_{N-j}^(g,\sigma;0) (v_{j}) dvol_h \\ & = (-1)^N \frac{(n\!-\!1)! N!}{(n\!-\!1\!-\!N)!} \sum_{j=0}^{N} \int_M v_{j} \T_{N-j}(g,\sigma;0)(1) dvol_h \\ & = (-1)^N \frac{(n\!-\!1)! N!}{(n\!-\!1\!-\!N)!} \int_M v_{N} dvol_h. \end{align} In the last equality, we used the fact that all coefficients except the leading one in the expansion of the harmonic function $u=1$ vanish. Combining this identity with Theorem <ref> we obtain Let $\N \ni N \le n-1$ and assume that $\sigma$ satisfies the condition $\SCY$. Then \begin{equation}\label{HF-volume} \int_{M} v_N dvol_h = (-1)^N \frac{(n\!-\!1\!-\!N)!}{ (n-1)!N!} \int_{M} \iota^ L_{N}(g,\sigma;-n\!+\!N)(1) dvol_h. \end{equation} We shall see later in Theorem <ref> that this reproves the special case $\tau=1$ of <cit.>. The critical case $N = n$ will be discussed in Section <ref>. These identities for integrated renormalized volume coefficients admit a natural interpretation as special cases of an interesting identity for distributions. In order to describe that point of view, we smoothly extend $g$ and $\sigma$ to a sufficiently small neighborhood $\tilde{X}$ of $M$ so that $\|\NV\| \ne 0$ on $\tilde{X}$ (this is always possible <cit.>). It will be convenient to assume that $\tilde{X} = \eta (I \times M)$ with a sufficiently small interval $I=(-\varepsilon,\varepsilon)$ around $0$. For any $u \in C^\infty(\R)$, the pull-back $\sigma^(u) \in C^\infty(X)$ defines a current by \left\langle \sigma^(u),\psi dvol_g \right\rangle = \int_{\tilde{X}} \sigma^(u) \psi dvol_g, \; \psi \in C_c^\infty(\tilde{X}). By approximating the delta distribution $\delta$ at $0$ by test functions, we obtain a current $\sigma^(\delta)$. We recall that the pull-back $\sigma^(u)$ of a distribution $u$ on the real line by $\sigma$ exists since the differential of $\sigma$ is surjective. The pull-back operation itself then is continuous on distributions (and currents) <cit.>. Since $\|\NV\|=1$ on $M$, it holds \begin{equation}\label{Ho-simple} \left\langle \sigma^(\delta),\psi dvol_g \right\rangle = \int_M \iota^(\psi) dvol_h \end{equation} by an extension of <cit.>. We also use the notation $\delta_M$ for the latter distribution and call it the delta distribution of $M$. $\delta_M$ delta distribution of $M$ We define the action of a differential operator $D$ on currents $u$ on $\tilde{X}$ by \langle D (u), \psi dvol_g \rangle = \langle u, D^ (\psi) dvol_g \rangle, \; \psi \in C_c^\infty(\tilde{X}). Here the formal adjoint $D^$ of $D$ is determined by the relation \begin{equation}\label{pair} \int_{\tilde{X}} D(\varphi) \psi dvol_g = \int_{\tilde{X}} \varphi D^(\psi) dvol_g, \; \varphi \in C^\infty(\tilde{X}), \psi \in C_c^\infty(\tilde{X}). \end{equation} Assume that $N \le n-1$ and assume that $\sigma$ satisfies $\SCY$. Then \begin{equation}\label{Shift-delta} L(g,\sigma;-N) \circ \cdots \circ L(g,\sigma;-1) (\sigma^(\delta)) = a_N\mathfrak{X}^N (\sigma^(\delta)), \end{equation} where $a_N = (n\!-\!1)!/(n\!-\!1\!-\!N)!$ and $\mathfrak{X} = \NV / \|\NV\|^2$ is defined in Section <ref>. Corollary <ref> implies that $L_N(\lambda)$ acts on $\sigma^(\delta)$ by \begin{equation}\label{dual-1} \langle L_N(\lambda)(\sigma^(\delta)),\psi dvol_g \rangle = \langle \sigma^(\delta), L_N(-n\!-\!\lambda\!+\!N\!-\!1) (\psi) dvol_g \rangle. \end{equation} \begin{equation} \langle L_N(\lambda)(\sigma^(\delta)), \psi dvol_g \rangle = \int_M \iota^ L_N(-n\!-\!\lambda\!+\!N\!-\!1) (\psi) dvol_h. \end{equation} Thus, Theorem <ref> yields \langle L_{N}(\lambda)(\sigma^(\delta)), \psi dvol_g \rangle = \int_M \D_{N}^{res}(-\lambda\!-\!n+\!N\!-\!1)(\eta^(\psi)) dvol_h. Note that this formula implies that $\langle L_{N}(\lambda)(\sigma^(\delta)), \psi dvol_g \rangle$ only depends on the first $N$ terms in the expansion of $\sigma$. Now, by Theorem <ref>, the latter integral equals \begin{align} & N!(-2\lambda\!-\!n\!-\!1)_N \\ & \times \sum_{j=0}^{N} \frac{1}{(N\!-\!j)!} \int_M [\T_{j}^(-\lambda\!-\!1) \circ v_0 + \cdots + \T_0^(-\lambda\!-\!1) \circ v_j ] (\iota^ \partial_s^{N-j}(\eta^(\psi))) dvol_h. \end{align} Hence using partial integration, we obtain \begin{align} & \langle L_{N}(\lambda)(\sigma^(\delta)), \psi dvol_g \rangle = N!(-2\lambda\!-\!n\!-\!1)_N \\ & \times \sum_{j=0}^{N} \frac{1}{(N\!-\!j)!} \int_M [\T_{j}(-\lambda\!-\!1)(1) + \cdots + \T_0(-\lambda\!-\!1)(1) v_{j}] \iota^* \partial_s^{N-j}(\eta^(\psi)) dvol_h. \end{align} Now let $\lambda = -1$. Since $\T_{j}(0)(1) = 0$ for $j \ge 1$ and $\T_0 = \id$, it follows that \begin{align}\label{int-L} \langle L_N(-1)(\sigma^(\delta)), \psi dvol_g \rangle & = N!(-n\!+\!1)_N \sum_{j=0}^{N} \frac{1}{(N\!-\!j)!} \int_M v_{j} \iota^ \partial_s^{N-j}(\eta^(\psi)) dvol_h \notag \\ & = (-n\!+\!1)_N \int_M \iota^ \partial_s^{N} (v \eta^(\psi)) dvol_h. \end{align} On the other hand, partial integration shows \begin{align} \langle \mathfrak{X}^N(u),\psi dvol_g \rangle & = \int_{\tilde{X}} \mathfrak{X}^N(u) \psi dvol_g \\ & = \int_{I \times M} \eta^* \mathfrak{X}^N(u) \eta^(\psi) v ds dvol_h \\ & = \int_{I \times M} \partial_s^N \eta^(u) \eta^(\psi) v ds dvol_h & \mbox{(by \eqref{intertwine})} \\ & = (-1)^N \int_{I \times M} \eta^(u) v^{-1} \partial_s ^N(\eta^(\psi) v) v ds dvol_h \\ & = (-1)^N \int_{\tilde{X}} u \eta_(v^{-1} \partial_s^N (v \eta^(\psi)) dvol_g \end{align} for $u \in C^\infty(\tilde{X})$ and $\psi \in C^\infty_c (\tilde{X})$. Hence $(\mathfrak{X}^N)^* (\psi) = (-1)^N \eta_(v^{-1} \partial_s^N (v \eta^(\psi))$ and \begin{equation}\label{fund-dist} \langle \mathfrak{X}^N(\sigma^(\delta)),\psi dvol_g \rangle = (-1)^N \int_M \iota^ \partial_s^N (v \eta^(\psi)) dvol_h. \end{equation} The proof is complete. Note that, for $\psi = 1$ near $M$, the arguments in the above proof show that \begin{align} \int_M \iota^* L_N(-n\!+\!N) (1) dvol_h & = \frac{(n\!-\!1)!}{(n\!-\!1\!-\!N)!} \langle \mathfrak{X}^N (\sigma^(\delta)), dvol_g \rangle \\ & = (-1)^N \frac{(n\!-\!1)!}{(n\!-\!1\!-\!N)!}\int_M \iota^ \partial_s^N (v) dvol_h \end{align} for $N \le n-1$. This proves that the identity (<ref>) is a special case of Theorem <ref>. Theorem <ref> results from our attempt to understand the distributional formula in <cit.>. In <cit.>, this distributional identity is the key to prove formulas like (<ref>). Here we follow a reverse logic. The following shift-property of residue families either follows by combining Theorem <ref> with Corollary <ref> or directly from the residue definition of residue families. Let $1 \le N \le n$ and assume that $\sigma$ satisfies the condition $\SCY$. Then \D_N^{res}(g,\sigma;\lambda) \circ s = N (2\lambda\!+\!n\!-\!N) \D_{N-1}^{res}(g,\sigma;\lambda\!-\!1). We use the residue definition of residue families (Definition <ref>). In particular, $u$ is an eigenfunction of the Laplacian of $\sigma^{-2}g$ for the eigenvalue $-\mu(n-\mu)$. Now the calculation \begin{align} \int_M f \delta_N(\mu) (\eta^(\sigma \psi)) dvol_h & = \Res_{\lambda=-\mu-1-N} \left( \int_X \sigma^\lambda u (\sigma \psi) dvol_g \right) \\ & = \Res_{\lambda=-\mu-1-N} \left( \int_X \sigma^{\lambda+1} u \psi dvol_g \right) \\ & = \Res_{\lambda=-\mu-N} \left( \int_X \sigma^\lambda u \psi dvol_g \right) \\ & = \int_M f \delta_{N-1}(\mu) (\eta^ (\psi)) dvol_h \end{align} shows that $\delta_N(\mu) \circ \eta^ \circ \sigma = \delta_{N-1}(\mu) \circ \eta^$, i.e., $\delta_N(\mu) \circ s = \delta_{N-1}(\mu)$. The claim is a direct consequence. Finally, we note that the relation $\delta_N(\mu) \circ s = \delta_{N-1}(\mu)$ also is an immediate consequence of Theorem <ref>. § EXTRINSIC CONFORMAL LAPLACIANS If $\SC(g,\sigma) \ne 0$ and $N \in \N$, the commutator relation (<ref>) in Corollary <ref> shows that the composition \tilde{L}_N(g,\sigma) \st \tilde{L}_N \left(g,\sigma;\frac{-n\!+\!N}{2}\right) = \tilde{L}\left(g,\sigma;\frac{-n\!-\!N}{2}\!+\!1\right) \circ \cdots \circ \tilde{L}\left(g,\sigma;\frac{-n\!+\!N}{2}\right) is a tangential operator, i.e., it holds $\tilde{L}_N \circ \sigma = \sigma \circ \tilde{L}'_N$ for some operator $\tilde{L}'_N$. Thus, the operator $\tilde{L}_N(g,\sigma)$ on $C^\infty(X)$ induces an operator $\tilde{\PO}_N(g,\sigma)$ on $C^\infty(M)$ according to \begin{equation}\label{conformal-power} \iota^ \tilde{L}_N(g,\sigma) = \tilde{\PO}_N(g,\sigma) \iota^. \end{equation} This observation is a special case of <cit.>. If additionally $\sigma$ satisfies the condition $\SCY$ and $N \le n$, then the resulting operator on $C^\infty(M)$ will be denoted by $\PO_N(g,\sigma)$. In the latter case, one may also directly apply (<ref>). Now, following Gover and Waldron, we define Let $\N \ni N \le n$. Assume that $\sigma$ satisfies the condition $\SCY$. Then the operators $\PO_N(g,\sigma)$ are called extrinsic conformal Laplacians. The operator $\PO_n(g,\sigma)$ is called the critical extrinsic conformal Laplacian. The notion is justified by the fact that for even $N$ these operators generalize the GJMS-operators $P_{2N}$ which are of the form $\Delta^N + LOT$. More precisely, let $g_+$ be a Poincaré-Einstein metric in normal form relative to $h$. Then \begin{equation}\label{P-GJMS} \PO_{2N}(r^2g_+,r) = (2N\!-\!1)!!^2 P_{2N}(h). \end{equation} In the general case, the operator $\PO_{2N}(g,\sigma)$ is of the form (2N\!-\!1)!!^2 \Delta^N_M + LOT. Here the lower order terms depend on the embedding $M \hookrightarrow X$ (Proposition <ref>). For odd $N$, the leading part of $\PO_N(g)$ is not given by a power of the Laplacian but involves $\lo$ (Proposition <ref>). Note that $\D_1^{res}(\frac{-n+1}{2})=0$ shows that $\PO_1 = 0$. Note also that the L(g,\sigma;0)(1) = 0 \begin{equation}\label{crit-van} \PO_n(g,\sigma)(1) = 0. \end{equation} Let $\N \ni N \le n$. Assume that $\sigma$ satisfies $\SCY$. \begin{equation}\label{D-res-power} \D_N^{res}\left(g,\sigma;\frac{-n\!+\!N}{2}\right) = \PO_N(g,\sigma) \iota^. \end{equation} More generally, the factorization identities \D_N^{res}\left(g,\sigma;\frac{-n\!-\!k}{2}\!+\!N\right) = \PO_k(g,\sigma) \circ \D_{N-k}^{res}\left(g,\sigma;\frac{-n\!-\!k}{2}\!+\!N\right) for $1\le k \le N$ hold true. Theorem <ref> implies \D_N^{res}\left(\frac{-n\!-\!k}{2}\!+\!N\right) = \iota^* L\left(\frac{-n\!-\!k}{2}\!+\!1\right) \circ \cdots \circ L\left(\frac{-n\!-\!k}{2}\!+\!N\right). We decompose this product as \left( \iota^* L\left(\frac{-n\!-\!k}{2}\!+\!1\right) \circ \cdots \circ L\left(\frac{-n\!+\!k}{2}\right) \right) \circ \left(L\left(\frac{-n\!+\!k}{2}+1\right) \circ \cdots \circ L\left(\frac{-n\!-\!k}{2}\!+\!N\right) \right). By the definition of $\PO_k$ and Theorem <ref>, this composition equals \PO_k \circ \D_{N-k}^{res}\left(\frac{-n\!-\!k}{2}\!+\!N\right). The proof is complete. Note that $\PO_1=0$ implies the vanishing property $\D_N^{res}\left(\frac{-n-1}{2}\!+\!N\right)=0$. The trivial zero of $D_N^{res}(\lambda)$ at $\lambda=-\frac{n+1}{2}\!+\!N$ is actually one of the zeros in the prefactor in the definition (<ref>). It appears here as a trivial zero since the solution operator $\T_1(\lambda)$ is regular (Lemma <ref>). Theorem <ref> extends factorization formulas in the setting of Poincaré-Einstein metrics. In that case, the left factors are GJMS-operators on the boundary. For details, we refer to <cit.>. Finally, we notice an interesting direct formula for the critical extrinsic conformal Laplacian $\PO_n(g,\sigma)$ in terms of the Laplacian of the singular Yamabe metric $\sigma^{-2} g$. By composition with $\iota^$, Corollary <ref> shows that \PO_n(g,\sigma) \iota^(f) = \iota^* \left( \sigma^{-n} \prod_{j=0}^{n-1} \left(\Delta_{\sigma^{-2}g} + (n\!-\!j)j \right) \right) (f) for any $f \in C^\infty(X)$. We omit the analogous formulas in the subcritical cases. Next, we provide a spectral theoretical description of the extrinsic conformal Laplacians. Let $N \in \N$ with $2 \le N \le n$. Assume that $\sigma$ satisfies the condition $\SCY$. Then \PO_N = (-1)^{N-1} 2 (N\!-\!1)! N! \Res_{\frac{n-N}{2}}(\T_N^(\lambda)). On the right-hand side of formula (<ref>) in Theorem <ref>, all solution operators except $\T_N(\lambda)$ are regular at $\lambda=\frac{n-N}{2}$. The result follows by combining that observation with the identity (<ref>) in Theorem <ref>. We use the spectral theoretical interpretation of the operators $\PO_N$ to separate their leading parts. Let $\N \ni N \le n$. Assume that $\sigma$ satisfied $\SCY$. Then \LT( \Res_{\lambda=\frac{n}{2}-N}(\T_{2N}(\lambda)) = - \frac{1}{2^{2N} (N-1)! N!} \Delta^N_h. \LT(\PO_{2N}) = (2N-1)!!^2 \Delta^N_h. The remaining terms are of order $\le N-2$. As a preparation for the proof, we observe that formula (<ref>) and the identity \Delta_{\sigma^{-2}g} = \sigma^2 \Delta_g - (n\!-\!1) \sigma \nabla_{\grad(\sigma)} imply that the Laplacian of the metric $\eta^(\sigma^{-2}g) = s^{-2} \eta^(g)$ takes the form \begin{align}\label{L-op} & s^2 a \partial_s^2 + \frac{1}{2} s^2 a \tr (h_s^{-1} h'_s) \partial_s + \frac{1}{2} s^2 a' \partial_s - (n\!-\!1) s a \partial_s - \frac{1}{2} s^2 (d \log a, d \cdot)_{h_s} + s^2 \Delta_{h_s}, \end{align} where $a = \eta^(\|\NV\|^2)$. In the special case $\|\NV\|=1$, the formula (<ref>) simplifies to s^2 \partial_s^2 + \frac{1}{2} s^2 \tr (h_s^{-1} h'_s) \partial_s - (n\!-\!1) s \partial_s + s^2 \Delta_{h_s}. In particular, this reproduces the formula for the Laplacian of a Poincaré-Einstein metric <cit.>. By combining the formula (<ref>) with (<ref>), we can give another proof of the conjugation formula (<ref>). Here the identities (<ref>) and (<ref>) are useful. We omit the details. The solution operators $\T_N(\lambda)$ are determined by the ansatz \sum_{N\ge 0} s^{\lambda+N} \T_N(\lambda)(f), \; \T_0 = \id for an approximate solution $u$ of the equation $-\Delta_{s^{-2} \eta^(g)} u = \lambda(n\!-\!\lambda) u$ with boundary value $f$. By formula (<ref>), this means that the sum \begin{align}\label{L-A-Sol} & a \sum_{N \ge 0} (\lambda\!+\!N)(\lambda\!+\!N\!-\!1) \T_N(\lambda) s^{\lambda+N} \notag \\ + & \frac{a}{2} \sum_{N \ge 0} (\lambda\!+\!N) \tr (h_s^{-1} h'_s) \T_N(\lambda) s^{\lambda+N+1} \notag \\ + & \frac{a'}{2} \sum_{N \ge 0} (\lambda\!+\!N) \T_N(\lambda) s^{\lambda+N+1} \notag \\ - & (n\!-\!1) a \sum_{N \ge 0} (\lambda\!+\!N) \T_N(\lambda) s^{\lambda+N} \notag \\ - & \frac{1}{2} \sum_{N \ge 0} (d \log a,d \T_N(\lambda))_{h_s} s^{\lambda+N+2} \notag \\ + & \sum_{N \ge 0} \Delta_{h_s} \T_N(\lambda) s^{\lambda+N+2} \end{align} coincides with the sum \begin{equation}\label{L-A-Sol-2} -\lambda(n\!-\!\lambda) \sum_{N\ge 0} \T_N(\lambda) s^{\lambda+N}. \end{equation} In order to compare coefficients of powers of $s$ in (<ref>) and (<ref>), we also insert the expansions of $a$ and of $h_s$. Note that the equality of the coefficients of $s^\lambda$ in (<ref>) and (<ref>) is trivially satisfied using $\iota^a = 1$. The equality of coefficients of $s^{\lambda+N}$ in (<ref>) and (<ref>) yields a recursive formula for $\T_N(\lambda)$. For the coefficient $\T_1(\lambda)$, we find $\T_1(\lambda) = -H \id$ (Lemma <ref>). An easy induction shows that the order of $\T_{2N}(\lambda)$ and $\T_{2N+1}(\lambda)$ is $2N$. Now we give the proof of Proposition <ref>. Let $N$ be even. Then (<ref>) implies that N(2\lambda-n+N) \LT(\T_N(\lambda)) + \Delta_h \LT(\T_{N-2}(\lambda)) = 0. It follows that the leading term of $\T_{2N}(\lambda)$ is given by \begin{equation}\label{LT-T} \prod_{j=1}^N \frac{1}{2j (n-2\lambda-2j)} \Delta_h^N = \frac{1}{N! 2^{2N}} \frac{\Gamma(\frac{n}{2}-\lambda-N)}{\Gamma(\frac{n}{2}-\lambda)} \Delta_h^N. \end{equation} Note that this observation fits with <cit.>. Hence \LT( \Res_{\lambda=\frac{n}{2}-N}(\T_{2N}(\lambda)) = - \frac{1}{2^{2N} (N-1)! N!} \Delta_h^N. The second claim follows from that result by combining it with Theorem <ref>. For odd $N$, the operator $\PO_N$ is of order $N-1$ for general metrics. In the following result, we separate from the residue $\Res_{\lambda=\frac{n-N}{2}}(\T_N(\lambda))$ and from $\PO_N$ respective self-adjoint leading terms $\LT(\cdot)$ so that the remaining terms are of order $N-3$ for general metrics. Let $3 \le N \in \N$ be odd. Assume that $\sigma$ satisfies $\SCY$. Then \LT( \Res_{\lambda=\frac{n-N}{2}}(\T_N(\lambda)) = \frac{1}{N (N\!-\!2)!} \sum_{r=0}^{\frac{N-3}{2}} m_N(r) \Delta^r \delta(\lo d) \Delta^{\frac{N-3}{2}-r} \begin{equation}\label{m-coeff} m_N(r) \st \binom{N-1}{r} \prod_{j=1}^{\frac{n-3}{2}} (N-j) \prod_{j=1}^r \frac{1}{(N-2j)} \prod_{j=1}^{\frac{n-3}{2}-r} \frac{1}{(N-2j)}. \end{equation} \begin{equation}\label{LT-P-odd} \LT(\PO_N) = (2N\!-\!2) (N\!-\!1)! \sum_{r=0}^{\frac{N-3}{2}} m_N(r) \Delta^r \delta(\lo d) \Delta^{\frac{N-3}{2}-r}. \end{equation} Let $N$ be odd. Note that $a = 1 + 2 H s + \cdots$ and $\tr (h_s^{-1} h'_s ) = 2 \tr(L) + \cdots$ by Lemma <ref> and (<ref>). Comparing the coefficients of $s^{\lambda+N}$ in (<ref>) and (<ref>) yields the relation \begin{align} & N (2\lambda\!-\!n\!+\!N) \T_N(\lambda) \\ & + 2 (\lambda\!+\!N\!-\!1) (\lambda\!+\!N\!-\!2) H \T_{N-1}(\lambda) + (\lambda\!+\!N\!-\!1) \tr (L) \T_{N-1}(\lambda) \\ & + (\lambda\!+\!N\!-\!1) H \T_{N-1}(\lambda) - 2(n\!-\!1)(\lambda\!+\!N\!-\!1) \T_{N-1}(\lambda) \\ & - (dH,d\T_{N-3}(\lambda))_h \\ & + \Delta_h \T_{N-2}(\lambda) + \Delta_h' \T_{N-3}(\lambda) = 0, \end{align} up to operators of order $\le N-3$. Here $\Delta_{h_s} = \Delta_h + s \Delta_h' + \cdots$. Simplification shows that \begin{align}\label{RR-odd} & N(2\lambda\!-\!n\!+\!N) \T_N(\lambda) + (\lambda\!+\!N\!-\!1)(2\lambda\!+\!2N\!-\!n\!-\!1) H \T_{N-1}(\lambda) - (dH,d\T_{N-3}(\lambda))_h \notag \\ & + \Delta \T_{N-2}(\lambda) + \Delta' \T_{N-3}(\lambda) = 0, \end{align} up to operators of order $\le N-3$.[From now on, $\Delta$ is the Laplacian of $h$.] The leading terms of the solution operators $\T_{N-1}(\lambda)$ and $\T_{N-3}(\lambda)$ are multiplies of powers of $\Delta$ (see (<ref>)). Moreover, we recall the variation formula \begin{equation}\label{Laplace-var-g} (d/dt)\|_0(\Delta_{g+sh}) = - (\Hess_g (\cdot), h)_g - (\delta^g (h),d \cdot)_g + \frac{1}{2} (d \tr_g(h),d\cdot)_g \end{equation} (<cit.>). Hence $\Delta'$ \begin{align}\label{Delta-var} \Delta'_h \st (d/dt)\|_0(\Delta_{h+2sL}) & = - (\Hess_h(\cdot),2L)_h - 2(\delta^h (L),d \cdot)_h + n (dH,d\cdot)_h \notag \\ & = - 2\delta^h ( L d) + n (dH,d \cdot)_h \notag \\ & = - 2 H \Delta_h + (n-2) (dH,d\cdot)_h - 2 \delta^h (\lo d). \end{align} In other words, the first variation $\Delta_h'$ of the Laplacian with respect to the variation $h_s$ of $h$ is given by \begin{equation}\label{var-1} -2H \Delta + (n-2) (dH,d\cdot)_h - 2 \delta (\lo d). \end{equation} It follows that $N(2\lambda\!-\!n\!+\!N)\T_N(\lambda)$ is a linear combination of terms of the form \begin{equation}\label{H-terms} H \Delta^{\frac{N-1}{2}} \quad \mbox{and} \quad (dH,d\Delta^{\frac{N-3}{2}}), \end{equation} \begin{equation}\label{L-terms} \Delta^r \delta ( \lo d) \Delta^{\frac{N-3}{2}-r}, \; r=0,\dots,\tfrac{N-3}{2} \end{equation} and terms of order $\le N-3$. In order to determine the coefficients of the terms (<ref>) in $\T_N(\lambda)$, we let $\mathring{\T}_N(\lambda)$ be the sum of these contributions to $\T_N(\lambda)$. Then (<ref>) implies \begin{equation}\label{RR-odd-L} a_N(\lambda) \mathring{\T}_N(\lambda) + \Delta \mathring{\T}_{N-2}(\lambda) - 2 (-1)^{\frac{N-3}{2}} \prod_{j=1}^\frac{N-3}{2} \frac{1}{a_{2j}(\lambda)} \delta (\lo d) \Delta^{\frac{N-3}{2}} = 0, \end{equation} where $a_N(\lambda) \st N(2\lambda-n+N)$. That recursive relation is solved by (-1)^\frac{N-3}{2} a_N(\lambda) \mathring{\T}_N(\lambda) = 2 \sum_{r=0}^{\frac{N-3}{2}} \prod_{j=1}^{\frac{N-3}{2}-r} \frac{1}{a_{2j}(\lambda)} \prod_{j=1}^{r} \frac{1}{a_{N-2j}(\lambda)} \Delta^r \delta (\lo d) \Delta^{\frac{N-3}{2}-r}. \begin{align} & N \Res_{\lambda=\frac{n-N}{2}}(\mathring{\T}_N(\lambda)) \\ & = 2^{-\frac{N-3}{2}} \sum_{r=0}^{\frac{N-3}{2}} \frac{1}{(\frac{N-3}{2}\!-\!r)! r!} \prod_{j=1}^{\frac{N-3}{2}-r} \frac{1}{(N-2j)} \prod_{j=1}^r \frac{1}{(N-2j)} \Delta^r \delta (\lo d) \Delta^{\frac{N-3}{2}-r}. \end{align} Now the inverse of the coefficient of the term for $r=0$ equals 2^{\frac{N-3}{2}} \left(\frac{N-3}{2}\right)! \prod_{j=1}^{\frac{N-3}{2}} (N-2j) = (N-2)!. Therefore, we obtain \Res_{\lambda=\frac{n-N}{2}} \left(\T_N(\lambda)\right) = \frac{1}{N (N-2)!} \sum_{r=0}^{\frac{N-3}{2}} m_N(r) \Delta^r \delta (\lo d) \Delta^{\frac{N-3}{2}-r}, up to contributions by the terms in (<ref>) (containing $H$) and lower-order terms. However, the terms in (<ref>) do not contribute. This is a consequence of the conformal covariance of $\PO_N$. The proof is complete. The formula (<ref>) also makes clear that, if $\lo$ vanishes, the operator $\PO_N$ is of order $< N-2$. For a discussion of the special cases of $\PO_3$ and $\PO_5$, we refer to Lemma <ref> and Lemma <ref>. In particular, the proof of Lemma <ref> confirms the vanishing of the terms $H \Delta$ and Next, we relate the operators $\PO_N$ to the scattering operator $\Sc(\lambda)$ generalizing a result of <cit.>. Let $N \in \N$ with $2 \le N \le n$. Assume that $\sigma$ satisfies $\SCY$ and that $(n/2)^2 - (N/2)^2 \notin \sigma_{pp}$. Then \begin{equation}\label{P-S} \PO_N = (-1)^N 2 (N-1)! N! \Res_{\frac{n-N}{2}}(\Sc(\lambda)). \end{equation} The assumptions guarantee that the scattering operator is well-defined and that $\lambda(n-\lambda) \notin \sigma_{pp}$ for $\lambda=\frac{n-N}{2}$. If $\Re(\lambda) < \frac{n}{2}$ so that $\lambda(n-\lambda) \notin \sigma_{pp}$ and $\lambda \notin \frac{n}{2} - \N$, the Poisson transform $\Po(\lambda)(f)$ yields an eigenfunction $u$ of the Laplacian of the metric $\eta^(\sigma^{-2}g) = s^{-2} \eta^(g)$ with boundary value $f \in C^\infty(M)$ and with an asymptotic expansion of the form \begin{equation} \sum_{j\ge 0} s^{\lambda+j} \T_j(\lambda)(f) + \sum_{j \ge 0} s^{n-\lambda+j} \T_j(n-\lambda) \Sc(\lambda)(f). \end{equation} Although the families $\T_N(\lambda)$ and $\Sc(\lambda)$ have simple poles at $\lambda=\frac{n-N}{2}$, the Poisson transform $\Po(\lambda)(f)$ is holomorphic at $\lambda=\frac{n-N}{2}$ <cit.>. That means that \frac{\Res_{\frac{n-N}{2}}(\T_N(\lambda)) s^{\lambda+N}}{\lambda - \frac{n-N}{2}} + \frac{\Res_{{\frac{n-N}{2}}}(\Sc(\lambda)) s^{n-\lambda}}{\lambda-\frac{n-N}{2}} is regular at $\lambda = \frac{n-N}{2}$. Hence \Res_{\frac{n-N}{2}}(\T_N(\lambda)) + \Res_{{\frac{n-N}{2}}}(\Sc(\lambda)) = 0 and the asymptotic expansion of $u$ involves a $\log$-term 2 \Res_{\frac{n-N}{2}}(\T_N(\lambda))(f) s^{\frac{n+N}{2}} \log (s). Now the claim follows from Theorem <ref>. In <cit.>, the scattering operator is defined as $\Sc(n-\lambda)$ and the GJMS-operators $P_{2N}$ have leading part $(-\Delta)^N$. Now \begin{align} P_{2N}(h) & = \frac{1}{(2N\!-\!1)!!^2} \PO_{2N}(r^2 g_+,r) & \mbox{(by \eqref{P-GJMS})} \\ & = \frac{2(2N\!-\!1)! (2N)!} {(2N\!-\!1)!!^2} \Res_{\frac{n}{2}-N}(\Sc(\lambda)) & \mbox{(by Theorem \ref{LS})} \\ & = 2^{2N} N!(N\!-\!1)! \Res_{\frac{n}{2}-N}(\Sc(\lambda)). \end{align} This shows that Theorem <ref> extends <cit.>. Let $\N \ni N \le n$. Assume that $\sigma$ satisfied $\SCY$. Then the operators $\PO_N(g,\sigma)$ are formally self-adjoint as operators on $C^\infty(M)$ with respect to the scalar product defined by $h$. Lemma <ref> shows that $\Sc(\lambda)$ is self-adjoint on $\R \setminus \left\{\frac{n-N}{2}\,\|\, N \in \Z \right\}$. Since $\Sc(\lambda)$ is meromorphic, its residue at $\lambda = \frac{n-N}{2}$ is self-adjoint, too. Then Theorem <ref> proves the assertion. § EXTRINSIC $Q$-CURVATURES AND RENORMALIZED VOLUME COEFFICIENTS theoremsection equationsection The zeroth-order terms of the GJMS-operators $P_{2N}$ led to the notion of Branson's $Q$-curvature <cit.>. In the present section, we use residue families to extend the notion of Branson's $Q$-curvatures to the framework of the singular Yamabe problem. This extends the discussion of $Q$-curvatures in <cit.>. The resulting curvature quantities will be called extrinsic $Q$-curvatures.[In <cit.>, the critical extrinsic $Q$-curvature is called the extrinsically coupled $Q$-curvature.] We relate the integrated critical renormalized volume coefficient $v_n$ to the integrated critical extrinsic $Q$-curvature. Moreover, we discuss the Hadamard renormalization of the volume of singular Yamabe metrics. Here the total critical extrinsic $Q$-curvature plays an important role. Although the treatment is inspired by <cit.>, our arguments differ and may continue to illuminate these topics. By Theorem <ref>, it holds $\PO_n(g,\sigma) \iota^* = \D_n^{res}(g,\sigma;0)$ if $\sigma$ satisfies $\SCY$. Now, following the philosophy of <cit.>, it is natural to consider the pair $(\D_n^{res}(g,\sigma;0),\dot{\D}_n^{res}(g,\sigma;0)(1))$. $\QC_n(g,\sigma)$ critical extrinsic $Q$-curvature Assume that $\sigma$ satisfies $\SCY$. Then the function \begin{equation}\label{Q-critical} \QC_n(g,\sigma) \st - \dot{\D}_n^{res}(g,\sigma;0)(1) \in C^\infty(M) \end{equation} is called the critical extrinsic $Q$-curvature of $g$. Since $L(g,\sigma;0)(1) = 0$, the identification of residue families with products of $L$-operators (Theorem <ref>) implies that \begin{equation}\label{QL} \QC_n(g,\sigma) = - \iota^* L(g,\sigma;-n\!+\!1) \circ \cdots \circ L(g,\sigma;-1) \circ \dot{L}(g,\sigma;0)(1). \end{equation} Moreover, the definition of $L$ yields \begin{equation}\label{b} \dot{L}(g,\sigma;0)(1) = (n-1) \rho - \sigma \J. \end{equation} Therefore, we obtain \begin{equation}\label{QL-2} \QC_n(g,\sigma) = - \iota^* L(g,\sigma;-n\!+\!1) \circ \cdots \circ L(g,\sigma;-1) ((n-1) \rho - \sigma \J). \end{equation} Next, we define subcritical versions of $\QC_n$. Let $N < n$. The definition of $L$ implies that \begin{equation}\label{aa} L\left(g,\sigma;\frac{-n+N}{2}\right)(1) = \left(\frac{n-N}{2}\right) (-(N\!-\!1) \rho + \sigma \J). \end{equation} Hence the function $\PO_N(g,\sigma)(1)$ is of the form \left(\frac{n-N}{2}\right) \QC_N(g,\sigma) with a scalar curvature quantity $\QC_N(g,\sigma) \in C^\infty(M)$. It follows that \begin{equation}\label{a} \QC_N(g,\sigma) = -\iota^* L\left(g,\sigma;\frac{-n\!-\!N}{2}\!+\!1\right) \circ \cdots \circ L\left(g,\sigma;\frac{-n\!+\!N}{2}\!-\!1\right) ((N\!-\!1)\rho - \sigma \J) \end{equation} We shall call these quantities subcritical extrinsic $Q$-curvatures. In terms of residue families, these definitions are equivalent to the following definition. $\QC_N(g,\sigma)$ extrinsic $Q$-curvature Let $\N \ni N < n$. Assume that $\sigma$ satisfies $\SCY$. The functions $\QC_N(g,\sigma) \in C^\infty(M)$ which are determined by the equation \begin{equation}\label{Q-sub-residue} \D_N^{res}\left(g,\sigma;\frac{-n+N}{2}\right) (1) = \left(\frac{n-N}{2}\right) \QC_N(g,\sigma) \end{equation} are called subcritical extrinsic $Q$-curvatures. Since the residue families $\D_N^{res}(g,\sigma;\lambda)$ for $N \le n$ are completely determined by $g$ (and $\iota$) (Proposition <ref>), the quantities $\QC_N(g,\sigma)$ for $N \le n$ are also completely determined by $g$ (and $\iota$). Therefore, we shall also use the notation $\QC_N(g)$. The identities (<ref>) and (<ref>) show that the critical extrinsic $Q$-curvature is a limiting case $\lim_{n \to N}$ of the subcritical extrinsic $Q$-curvatures (continuation in dimension). The subcritical $Q$-curvature $\QC_N$ is directly linked to the solution operator $\T_N(\lambda)$ through the relation \begin{equation}\label{QT} \QC_N = (-1)^N \QC_N\left(\frac{n-N}{2}\right), \end{equation} where the polynomial $\QC_N(\lambda)$ is defined in (<ref>). In fact, (<ref>) implies \Res_{\frac{n-N}{2}}(\T_N)(1) = -\frac{\frac{n-N}{2}}{2 (N-1)! N!} \QC_N\left(\frac{n-N}{2}\right). On the other hand, Theorem <ref> shows that \Res_{\frac{n-N}{2}}(\T_N)(1) = - (-1)^{N} \frac{1}{2 (N-1)!N!} \PO_N (1) = - (-1)^N \frac{\frac{n-N}{2}}{2 (N-1)!N!} \QC_N. The identity (<ref>) follows by combining both relations. Continuation in $n$ also gives the relation \begin{equation}\label{QT-critical} \QC_n = (-1)^n \QC_n(0) \end{equation} in the critical case. The extrinsic $Q$-curvatures $\QC_N$ are related to Branson's $Q$-curvatures $Q_{2N}$ as follows. We use the convention that $Q_{2N}$ for even $n$ and $2N <n$ is defined by P_{2N} (g) (1) = \left({\frac{n}2}-N\right) (-1)^N Q_{2N}(g), where $P_{2N} = \Delta^N + LOT$ is the GJMS-operator of order $2N$ with $\Delta$ denoting the non-positive Laplacian. Now let $g_+$ be a Poincaré-Einstein metric in normal form relative to $h$. Then (<ref>) implies \QC_{2N}(r^2 g_+,r) = (-1)^N(2N-1)!!^2 Q_{2N}(h). Formula (<ref>) for the critical extrinsic $Q$-curvature $\QC_n$ is closely related to the definition of the critical extrinsic $Q$-curvature used in <cit.>. In fact, under the assumption $\SC(g,\sigma)=1$, <cit.> defines $\QC_n(g,\sigma)$ by (-1)^{n} \iota^* L(g,\sigma;-n+1) \circ \cdots \circ L(g,\sigma;-1) \circ L_{\log}(g,\sigma;1) (\log (1)). Here $L_{\log}(g,\sigma;\lambda)$ is a version of $L$ which acts on log-densities by[For the definition of the notion of log-densities, we refer to <cit.>.] L_{\log}(g,\sigma;\lambda)(u) \st (n-1) (\nabla_{\grad(\sigma)}(u) + \lambda \rho) - \sigma (\Delta_g (u) + \lambda \J). In particular, we have $L_{\log}(g,\sigma;\lambda)(\log(1)) = \lambda ((n-1) \rho -\sigma \J)$ and hence L_{\log}(g,\sigma;1)(\log(1)) = (n-1) \rho - \sigma \J = \dot{L}(g,\sigma;0)(1) using (<ref>). Differentiation of the conformal transformation law for the critical residue family $\D_n^{res}(g,\sigma;\lambda)$ at $\lambda=0$ yields the following result. Assume that $\sigma$ satisfies $\SCY$. Then \begin{equation}\label{CTL-Q2} e^{n \iota^(\varphi)} \QC_n(\hat{g},\hat{\sigma}) = \QC_n(g,\sigma) + \PO_n(g,\sigma)(\iota^(\varphi)) \end{equation} for all conformal changes $(\hat{g},\hat{\sigma}) = (e^{2\varphi}g,e^\varphi \sigma)$, $\varphi \in C^\infty(X)$. Theorem <ref> implies e^{-(\lambda-n)\iota^(\varphi)} \circ \D_n^{res}(\hat{g},\hat{\sigma};\lambda) = \D_n^{res}(g,\sigma;\lambda) \circ e^{-\lambda \varphi}. By differentiating this identity with respect to $\lambda$ at $\lambda=0$, we obtain -e^{n \iota^(\varphi)} \iota^(\varphi) \circ \D_n^{res}(\hat{g},\hat{\sigma};0) + e^{n \iota^(\varphi)} \circ \dot{\D}_n^{res}(\hat{g},\hat{\sigma};0) = \dot{\D}_n^{res}(g,\sigma;0) - \D_n^{res}(g,\sigma;0) \circ \varphi. -\iota^(\varphi) \D_n^{res}(g,\sigma;0)(1) + e^{n \iota^(\varphi)} \dot{\D}_n^{res}(\hat{g},\hat{\sigma};0)(1) = \dot{\D}_n^{res}(g,\sigma;0)(1) - \D_n^{res}(g,\sigma;0) (\varphi). Now $\D_n^{res}(g,\sigma;0) =\PO_n(g,\sigma) \iota^$ and (<ref>) imply the For closed $M^n$, integration of(<ref>) implies \int_{M} \QC_n(\hat{g},\hat{\sigma}) dvol_{\hat{h}} = \int_{M} \QC_n(g,\sigma) dvol_h + \int_{M} \PO_n(g,\sigma)(\iota^(\varphi)) dvol_h. Later we shall prove that $\PO_n$ is self-adjoint. Hence the second integral on the right-hand side equals $\int_M \PO_n (1) \iota^(\varphi) dvol_h$. By (<ref>), this integral vanishes. This shows Let $M^n$ be closed. Assume that $\sigma$ satisfies $\SCY$. Then the integral \int_M \QC_n(g) dvol_h is conformally invariant as a functional of $g$ and the embedding $M \hookrightarrow X$. An alternative argument proving this invariance will be given in Theorem <ref>. A generalization of Corollary <ref> to general $\sigma$ will be discussed in Lemma <ref>. Next, we recall that Theorem <ref> relates the operators $\PO_N$ to the scattering operator $\Sc(\lambda)$. In particular, it holds \PO_n = (-1)^n 2 (n-1)! n! \Res_0(\Sc(\lambda)). Since $\PO_n(1)=0$, it follows that the function $\Sc(\lambda)(1)$ is regular at $\lambda=0$. Its value at $\lambda=0$ will be denoted by $\Sc(0)(1)$. Assume that $\sigma$ satisfies $\SCY$. Then \QC_n = 2 (-1)^n (n-1)! n! \Sc(0)(1). We consider the coefficient of $s^n$ in the asymptotic expansion (<ref>) of the eigenfunction $\Po(\lambda)(1)$ for $\lambda=0$. Since $\Po(0)(1)=1$, that coefficient vanishes. On the other hand, it is given by \T_n(0)(1) + \Sc(0)(1); we recall that the function $\T_n(\lambda)(1)$ is regular at $\lambda=0$. Hence $\T_n(0)(1) = - \Sc(0)(1)$. But \begin{equation}\label{T-Q} \T_n(0)(1) = -\frac{1}{2 (n-1)! n!} \QC_n(0) = - (-1)^n \frac{1}{2 (n-1)!n!} \QC_n \end{equation} using (<ref>) and (<ref>). This implies the assertion. Theorem <ref> extends <cit.> for the scattering operator of Poincaré-Einstein metrics. In fact, \begin{align} Q_n & = (-1)^\frac{n}{2} \frac{1}{(n-1)!!^2} \QC_n & \mbox{(by Remark \ref{Q-GJMS})} \\ & = (-1)^\frac{n}{2} \frac{2 (n-1)! n!}{(n-1)!!^2} \Sc(0)(1) & \mbox{(by Theorem \ref{Q-S})} \\ & = (-1)^\frac{n}{2} 2^n \left(\frac{n}{2}\right)! \left(\frac{n}{2}-1\right)! \Sc(0)(1). \end{align} In that case, the crucial relation between the values of $\Sc(\lambda)(1)$ and $\T_n(\lambda)(1)$ at $\lambda=0$ is provided by <cit.>. The following result (<cit.> for $\tau=1$ and $\SC=1$) is an analog of the identity (<ref>) in the critical case. Let $M$ be closed and assume that $\sigma$ satisfies $\SCY$. Let $n\ge 2$ be even. Then it holds the equality \begin{equation}\label{vQ} \LA = \int_M v_n(g) dvol_h = \frac{(-1)^{n} }{(n-1)! n!} \int_M \QC_n(g) dvol_h \end{equation} of conformal invariants. The quantity $\LA$ is sometimes referred to as an anomaly. This is motivated by the fact that, in the Poincaré-Einstein case, the function $v_n$ is the infinitesimal conformal anomaly of the renormalized volume <cit.>. We shall give two proofs of that result. The arguments in the first proof will also play a role in Section <ref>. The second proof resembles the proof of the analogous result in the subcritical cases. We work in adapted coordinates. In particular, the notation will not distinguish between objects on $X$ and their pull-backs by $\eta$. First, we note that \begin{equation}\label{help-2} \iota^ \partial_s^{n-1} (v \dot{L}(0)(1)) = \iota^* \partial_s^n (v). \end{equation} Since $\dot{L}(0)(1) = (n-1) \rho - s \J$ (see (<ref>)), this local identity is a special case of the local identity (<ref>). The current assumption implies $\|\NV\|^2 = 1 - 2 s \rho + O(s^{n+1})$ and the same arguments as in the proof of Theorem <ref> yield the assertion. Now we integrate (<ref>). It holds \int_M \iota^* \partial_s^n (v) dvol_h = n! \int_M v_n dvol_h and the integral of the left-hand side of (<ref>) equals \begin{align} & (-1)^{n-1} \langle \mathfrak{X}^{n-1} (\sigma^(\delta)), \dot{L}(0)(1) dvol_g \rangle & \mbox{(by \eqref{fund-dist})} \\ & = c_n \langle L(-n\!+\!1) \circ \cdots \circ L(-1) (\sigma^(\delta)), \dot{L}(0)(1) dvol_g \rangle & \mbox{(by Theorem \ref{dist-volume})} \\ & = c_n \langle \sigma^(\delta), L(-n\!+\!1) \circ \cdots \circ L(-1) \circ \dot{L}(0)(1) dvol_g \rangle & \mbox{(by \eqref{dual-1})} \\ & = c_n \int_M \iota^* L(-n\!+\!1) \circ \cdots \circ L(-1) \circ \dot{L}(0)(1) dvol_h & \mbox{(by \eqref{Ho-simple})} \\ & = -c_n \int_M \QC_n dvol_h & \mbox{(by \eqref{QL})} \end{align} with $c_n = (-1)^{n-1}/(n\!-\!1)!$. This completes the proof. In the Poincaré-Einstein case, it holds $v' = v \dot{L}(0)(1)$. This immediately proves (<ref>). The proof of Theorem <ref> rests on the local identity (<ref>). This identity will also play a role in Section <ref>. We continue with a Theorem <ref> yields the identity \begin{equation}\label{critical-c} \int_M \D_n^{res}(\lambda)(1) dvol_h = n! (2\lambda\!-\!n\!+\!1)_n \sum_{j=0}^n \int_M v_j \T_{n-j}(\lambda)(1) dvol_h. \end{equation} We split the sum on the right-hand side as \sum_{j=1}^{n-1} \int_M v_j \T_{n-j}(\lambda)(1) dvol_h + \int_M v_n dvol_h + \int_M \T_n(\lambda)(1) dvol_h. Now we differentiate (<ref>) at $\lambda=0$. Since, $\T_j(0)(1)=0$ for $j=1,\dots,n-1$, we find \int_M \dot{\D}_n^{res}(0)(1) dvol_h = 2 (-1)^{n-1} (n-1)! n! \left( \int_M v_n dvol_h + \int_M \T_n(0)(1) dvol_h \right) using that $\T_n(\lambda)(1)$ is regular at $\lambda=0$. Now (<ref>) and (<ref>) imply -2 \int_M \QC_n dvol_h = 2 (-1)^{n-1} (n-1)!n! \int_M v_n dvol_h. This implies (<ref>). We finish this section with a discussion of renormalized volumes of singular metrics $\sigma^{-2} g$. First, we combine the above results to prove Let $M$ be closed and assume that $\sigma$ satisfied $\SCY$. Then the volume vol_{\sigma^{-2}g}(\{\sigma > \varepsilon \}) = \int_{\sigma > \varepsilon} dvol_{\sigma^{-2}g} admits the expansion \sum_{k=0}^{n-1} \frac{c_k}{n-k} \varepsilon^{-n+k} - \LA \log \varepsilon + V + o(1), \; \varepsilon \to 0, c_k = \int_M v_k(g) dvol_h \quad \mbox{and} \quad \LA = \int_M v_n(g) dvol_h. $V$ is called the renormalized volume. The coefficients in the expansion are natural functionals of the metric background $g$, which can be written in the form c_k = (-1)^k \frac{(n\!-\!1\!-\!k)!}{(n\!-\!1)! k!} \int_M \iota^ L_k(-n\!+\!k) (1) dvol_h for $k=0,\dots,n-1$ and \begin{equation}\label{anomaly-SY} \LA = (-1)^{n-1} \frac{1}{(n\!-\!1)! n!} \int_M \iota^* \dot{L}_n(0) (1) dvol_h. \end{equation} Using $\eta^(dvol_{\sigma^{-2} g}) = dvol_{s^{-2} \eta^(g)} = s^{-n-1} v(s) ds dvol_h$, we obtain the asymptotic expansion \int_{\sigma > \varepsilon} dvol_{\sigma^{-2}g} = \sum_{k=0}^{n-1} \frac{1}{n-k} \varepsilon^{-n+k} \int_M v_k dvol_h - \log \varepsilon \int_M v_n dvol_h + V + o(1). Now the expressions for the coefficients in terms of Laplace-Robin operators follow from Corollary <ref> and Theorem <ref>. The above definition of the renormalized volume is also known as the Hadamard renormalization <cit.>. In the proof of Theorem <ref>, we deduced the formulas for the coefficients $c_k$ from the relation between residue families and iterated Laplace-Robin operators (Theorem <ref>). We recall that this relation requires assuming that $\sigma$ solves the singular Yamabe problem. However, the only consequences of Theorem <ref> which are relevant in this context already follow from a study of the residues of the meromorphic continuation of the integral \begin{equation}\label{basic-int} \lambda \mapsto \int_X \sigma^\lambda dvol_g, \; \Re(\lambda) > -1. \end{equation} Therefore it is of interest to include a discussion of an extension of Theorem <ref> to general $\sigma$ which only rests on the study of that integral. First, we note that the coefficients $c_k$ in Theorem <ref> are related to the residues of (<ref>): $c_k = \int_M v_k dvol_h$. Moreover, the Hadamard renormalization $V$ of the volume of $\sigma^{-2}g$ is related to the Riesz renormalization of the volume of $\sigma^{-2} g$ which is defined as the constant term in the Laurent series of (<ref>) at $\lambda=-n-1$ <cit.>. We first observe that Remark <ref> implies \begin{equation} \SC(g,\sigma)^{-1} \circ L(g,\sigma;\lambda) + \sigma^{\lambda-1} \circ \SC(g,\sigma)^{-1} \circ \Delta_{\sigma^{-2}g} \circ \sigma^{-\lambda} = \lambda (n\!+\!\lambda) \sigma^{-1} \id \end{equation} in a sufficiently small neighborhood of the boundary, where $\SC(g,\sigma) \ne 0$. Hence \begin{equation}\label{BS-shift} \SC(g,\sigma)^{-1} L(g,\sigma;\lambda) (\sigma^{\lambda}) = \lambda (n\!+\!\lambda) \sigma^{\lambda-1}. \end{equation} This is a Bernstein-Sato-type functional equation. Let $\chi \in C_c^\infty(X)$ be a cut-off function of the boundary $M$ so that $\SC \ne 0$ on the support of $\chi$. In the following, we shall simplify the notation by suppressing the arguments $(g,\sigma)$ of $L$ and $\SC$. The second integral on the right-hand side of the \int_X \sigma^\lambda dvol_g = \int_X \chi \sigma^\lambda dvol_g + \int_X (1-\chi) \sigma^\lambda dvol_g is holomorphic on $\C$. Now (<ref>) implies \int_X \chi \sigma^\lambda dvol_g = \frac{1}{(\lambda\!+\!1) (n\!+\!\lambda\!+\!1)} \int_X \chi \SC^{-1} L(\lambda\!+\!1)(\sigma^{\lambda+1}) dvol_g for $\Re(\lambda) > -1$. Partial integration using Proposition <ref> shows that for $\Re(\lambda) \gg 0$ the latter integral equals \int_X \sigma^{\lambda+1} L(-\lambda\!-\!n\!-\!1)(\chi \SC^{-1}) dvol_g. Next, we note that $v_0 = \iota^* \|\NV\|^{-1}$ implies the residue formula \begin{equation}\label{res-formula} \Res_{\lambda=-1} \left( \int_X \sigma^\lambda \psi dvol_g \right) = \int_M \iota^* \frac{1}{\|\NV\|} \psi dvol_h. \end{equation} Now combining the above result with the residue formula (<ref>) yields \Res_{\lambda=-2} \left( \int_X \sigma^\lambda dvol_g \right) = - \frac{1}{n\!-\!1} \int_M \iota^* \frac{1}{\|\NV\|} L(-n\!+\!1)(\SC^{-1}) dvol_h. More generally, arguments as in the proof of Theorem <ref> resting on a repeated application of the functional equation (<ref>) and Proposition <ref> show that \begin{equation}\label{vol-pi} \int_X \chi \sigma^\lambda dvol_g = \frac{1}{(\lambda\!+\!1)_k (\lambda\!+\!n\!+\!1)_k} \int_X \sigma^{\lambda+k} \tilde{L}_k (-\lambda\!-\!n\!-\!1)(\chi) dvol_g, \end{equation} for $\Re(\lambda) \gg 0$, and it follows that \begin{equation}\label{res-vol-g} \Res_{\lambda=-k-1} \left( \int_X \sigma^\lambda dvol_g \right) = \frac{1}{(-n\!+\!1)_k k!} \int_M \iota^* \frac{1}{\|\NV\|} \tilde{L}_k (-n\!+\!k) (1) dvol_h \end{equation} for $k < n$ using the relation $\iota^* \tilde{L}_k(\lambda)(\chi) = \iota^* \tilde{L}_k(\lambda)(1)$. The assumption $k < n$ guarantees that the prefactor on the right-hand side of (<ref>) is regular at $\lambda = -k-1$. Here we use the notation \tilde{L}(\lambda) \st L(\lambda) \circ \SC^{-1} \quad \mbox{and} \quad \tilde{L}_k(\lambda) \st \tilde{L}(\lambda\!-\!k\!+\!1) \circ \cdots \circ \tilde{L}(\lambda) (see (<ref>) and (<ref>)). If $\SC =1$, then $\iota^* \NV=1$ and \begin{align} \Res_{\lambda=-2} \left( \int_X \sigma^\lambda dvol_g \right) & = - \frac{1}{n\!-\!1} \int_M \iota^ L(-n\!+\!1)(1) dvol_h \\ & = -(n-1) \int_M \iota^* \rho dvol_h = (n-1) \int_M H dvol_h \end{align} using $\iota^ \rho = - H$ (Lemma <ref>). This fits with the fact that $v_1 = (n-1)H$ (Example <ref>). Similarly, if $\sigma = d_M$ is the distance function of $M$, then $\SC = 1+2\sigma \rho$ implies $\iota^* L(-n+1)(\SC^{-1}) = (n-1)^2 \iota^* \rho + 2(n-1) \iota^* \rho = (n-1)(n+1) \iota^* \rho$. Hence \begin{align} \Res_{\lambda=-2} \left( \int_X \sigma^\lambda dvol_g \right) = \int_M \iota^ \Delta_g(\sigma) dvol_h = n \int_M H dvol_h. \end{align} These results prove the first part of the following theorem. Let $M$ be closed and $\sigma$ be a defining function of $M$. Then the volume vol_{\sigma^{-2}g}(\{\sigma > \varepsilon \}) = \int_{\sigma > \varepsilon} dvol_{\sigma^{-2}g} admits the expansion \sum_{k=0}^{n-1} \frac{c_k}{n-k} \varepsilon^{-n+k} - \LA \log \varepsilon + V + o(1), \; \varepsilon \to 0, c_k = \int_M v_k dvol_h = (-1)^k \frac{(n\!-\!1\!-\!k)!}{(n\!-\!1)! k!} \int_M \iota^ \frac{1}{\|\NV\|} \tilde{L}_k(-n\!+\!k)(1) dvol_h for $k=0,\dots,n-1$. Moreover, it holds \begin{equation}\label{anomaly-g} \LA = (-1)^{n-1} \frac{1}{(n\!-\!1)! n!} \int_M \iota^* \frac{1}{\|\NV\|} \tilde{L}_{n-1}(-1) \left( \dot{L}(0)(1) \SC^{-1} + (d \SC^{-1}, d\sigma)_g \right) dvol_h. \end{equation} Theorem <ref> is due to <cit.>. For a discussion of the relations between the current arguments and the proofs in these references, we refer to Remark <ref>. We work in adapted coordinates. The form of the expansion follows as in the proof of Theorem <ref>. Since $c_k = \int_M v_k dvol_h$ is the residue of $\int_X\sigma^\lambda dvol_g$ at $\lambda=-k-1$, the asserted formula for $c_k$ follows by a calculation of these residues using the functional equation (<ref>). It only remains to prove the formula for the anomaly $\LA$. Note that the proof for $c_k$ with $k \le n-1$ does not extend to the present case since for $k=n$ the right-hand side of (<ref>) has a double pole at $\lambda=-n-1$. To bypass this difficulty, we first prove the local relation[This relation can be interpreted as a local version of the global relation in <cit.>.] \begin{equation}\label{Myst-a} \iota^* \partial_s^{n-1} \left(v \left[((n-1) \rho - s\J) \SC^{-1} + \|\NV\|^2 \partial_s(\SC^{-1})\right] \right) = \iota^* \partial_s^n(v) \end{equation} near $M$. Let I \st \iota^* \partial_s^{n-1} (v \left[(n-1) \rho - s\J \right] \SC^{-1}). The definition $\rho = -\frac{1}{n+1}(\Delta (s) + s \J)$ implies that \begin{equation}\label{help-1} I = - \frac{n-1}{n+1} \left(\iota^* \partial_s^{n-1} (v \Delta (s) \SC^{-1}) + 2n \iota^* \partial_s^{n-2} (v \J \SC^{-1}) \right). \end{equation} Now we consider the first term on the right-hand side of (<ref>). By (6.17), it holds v \Delta(s) = \partial_s (v \|\NV\|^2). Hence we obtain \iota^* \partial_s^{n-1} (v \Delta (s) \SC^{-1}) = \iota^* \partial_s^n (v \|\NV\|^2 \SC^{-1}) - \iota^* \partial_s^{n-1} (v \|\NV\|^2 \partial_s(\SC^{-1})). But $\SC = \|\NV\|^2 + 2 s \rho$ implies $\|\NV\|^2 = \SC -2s \rho$. Hence \iota^* \partial_s^{n-1} (v \Delta (s) \SC^{-1}) = \iota^* \partial_s^n (v) - 2 n \iota^* \partial_s^{n-1} (v \rho \SC^{-1}) - \iota^* \partial_s^{n-1} (v \|\NV\|^2 \partial_s(\SC^{-1})). Now another application of the definition of $\rho$ shows that the middle term on the right-hand side of the last equation \frac{2n}{n+1} \iota^* \partial_s^{n-1} ((v \Delta(s) + v s \J)\SC^{-1}). Simplification of the resulting equation yields \begin{align}\label{red-id} & -\frac{n-1}{n+1} \iota^* \partial_s^{n-1} (v \Delta (s) \SC^{-1}) \notag \\ & = \iota^* \partial_s^n (v) + \frac{2n(n-1)}{n+1} \iota^* \partial_s^{n-2} (v \J) - \iota^* \partial_s^{n-1} (v \|\NV\|^2 \partial_s(\SC^{-1})). \end{align} By combining this result with (<ref>), we have proved I = \iota^* \partial_s^n (v) - \iota^* \partial_s^{n-1} (v \|\NV\|^2 \partial_s(\SC^{-1})). This implies (<ref>). Now the relation (<ref>) shows that \begin{align} \int_M v_n dvol_h = \frac{1}{n!} \int_M \iota^ \partial_s^n(v) dvol_h = \frac{1}{n!} \int_M \iota^* \partial_s^{n-1} \left( v \E \right) dvol_h, \end{align} \begin{equation}\label{E} \E \st ((n-1) \rho - s\J) \SC^{-1} + \|\NV\|^2 \partial_s(\SC^{-1}). \end{equation} \int_M v_n dvol_h = \frac{(n-1)!}{n!} \Res_{\lambda=-n} \left(\int_X \sigma^\lambda \E dvol_g \right) = \frac{1}{n} \Res_{\lambda=-n} \left(\int_X \sigma^\lambda \E dvol_g \right). Now we apply the functional equation (<ref>). We assume that $\Re(\lambda) \gg 0$ and let $\chi \in C_c^\infty(X)$ be a cut-off function as in the discussion following (<ref>). First, the relation \sigma^\lambda = \SC^{-1} \frac{1}{(\lambda\!+\!1)(n\!+\!\lambda\!+\!1)} L(\lambda\!+\!1)(\sigma^{\lambda+1}) \begin{align} \int_X \chi \sigma^\lambda \E dvol_g & = \frac{1}{(\lambda\!+\!1)(n\!+\!\lambda\!+\!1)} \int_X \chi \SC^{-1} L(\lambda\!+1\!)(\sigma^{\lambda+1}) \E dvol_g \\ & = \frac{1}{(\lambda\!+\!1)(n\!+\!\lambda\!+\!1)} \int_X \sigma^{\lambda+1} \tilde{L}(-n\!-\!\lambda\!-\!1) (\chi \E) dvol_g \end{align} by partial integration using Proposition <ref>. We continue to apply this argument and find \int_X \chi \sigma^\lambda \E dvol_g = \frac{1}{(\lambda\!+\!1)_{n-1} (n\!+\!\lambda\!+\!1)_{n-1}} \int_X \sigma^{\lambda+n-1} \tilde{L}_{n-1}(-n\!-\!\lambda\!-\!1)(\chi \E) dvol_g. Hence the residue formula (<ref>) yields \begin{align} \Res_{\lambda=-n} \left(\int_X \sigma^\lambda \E dvol_g \right) &= \frac{1}{(-n\!+\!1)\dots(-1) (n\!-\!1)!} \int_M \iota^* \tilde{L}_{n-1}(-1)(\chi \E) dvol_h \\ & = (-1)^{n-1} \frac{1}{(n\!-\!1)! (n\!-\!1)!} \int_M \iota^* \frac{1}{\|\NV\|} \tilde{L}_{n-1}(-1)(\E) dvol_h. \end{align} Thus, we find \int_M v_n dvol_h = (-1)^{n-1} \frac{1}{n! (n-1)!} \int_M \iota^ \frac{1}{\|\NV\|} \tilde{L}_{n-1}(-1)(\E) dvol_h. This proves the formula for the anomaly using $\dot{L}(0)(1) = (n-1) \rho - s \J$ (see (<ref>)) and the fact that $\grad(\sigma)$ in adapted coordinates equals $\|\NV\|^2 \partial_s$ (see (<ref>)). The formulas for the coefficients $c_k$ in Theorem <ref> and Theorem <ref> are special cases of <cit.> (for $\tau=1$). Similarly, the formulas for $\LA$ are special cases of <cit.>. But note that in <cit.> there are no local coefficients $v_k$ (defined in adapted coordinates). The present proofs differ from those in <cit.>. Whereas the above proofs rest on the conjugation formula and a Bernstein-Sato-type argument, the latter rest on a certain distributional calculus (see also Remark <ref>). Since the expansion in Theorem <ref> can be written in the form \sum_{k=0}^{n-1} \frac{1}{n-k} \left\langle \sigma^(\delta^{(k)}),dvol_g \right\rangle \varepsilon^{-n+k} - (-1)^{n-1}\left\langle \sigma^(\delta^{(n)}),dvol_g \right\rangle \log \varepsilon + V + o(1) using the formula \begin{equation}\label{BR} \left\langle \sigma^(\delta^{(k)}),dvol_g \right\rangle = (-1)^k k! \int_M v_k dvol_h \end{equation} for the currents $\sigma^(\delta^{(k)})$, this proves the equivalence of Theorem <ref> and <cit.> combined with <cit.> (for $\tau =1$). We continue with a proof of the relation (<ref>). First, we extend $g$ and $\sigma$ smoothly to a sufficiently small neighborhood $\tilde{X}$ of $M$ as in the discussion after Corollary <ref>. Then $\sigma^(\delta^{(k)})$ is defined as a continuous functional on smooth volume forms on $\tilde{X}$ with compact support.[Since $\sigma^(\delta)$ has compact support, we can pair it with any smooth volume form.] Next, we note the commutation rule \begin{equation}\label{B-1} \sigma^(u') = \mathfrak{X}_\sigma (\sigma^(u)), \; u \in C^\infty(\R) \end{equation} \mathfrak{X}(\tilde{X}) \ni \mathfrak{X}_\sigma \st \NV/\|\NV\|^2, \; \NV = \grad_g(\sigma). \begin{equation}\label{B-2} \sigma^(u^{(k)}) = \mathfrak{X}^k_\sigma (\sigma^(u)) \end{equation} for $k \in \N$. Now any $\varphi \in C^\infty(\tilde{X})$ defines a current on $\tilde{X}$ by \langle \varphi, \psi dvol_g \rangle = \int_{\tilde{X}} \varphi \psi dvol_g, \; \psi \in C_0^\infty(\tilde{X}). An extension of (<ref>) to $u=\delta$ yields an analogous relation for currents; it suffices to approximate $\delta$ by test functions in the weak topology. The relation (<ref>) for $u=\delta$ implies \begin{align} \left \langle \sigma^(\delta^{(k)}), \psi dvol_g \right \rangle & = \left\langle \mathfrak{X}_\sigma^k (\sigma^(\delta)), \psi dvol_g \right \rangle \\ & = \left \langle \sigma^(\delta), (\mathfrak{X}_\sigma^)^k (\psi) dvol_g \right \rangle, \end{align} where the adjoint operator $\mathfrak{X}^_\sigma$ is defined by \int_{\tilde{X}} \mathfrak{X}_\sigma(\varphi) \psi dvol_g = \int_{\tilde{X}} \varphi \mathfrak{X}_\sigma^(\psi) dvol_g, \; \psi \in C_c^\infty(\tilde{X}). Now we calculate (using adapted coordinates and partial integration) \begin{align} \int_{\tilde{X}} \mathfrak{X}^k_\sigma (\varphi) \psi dvol_g & = \int_{I \times M} \eta^(\mathfrak{X}_\sigma^k(\varphi)) \eta^(\psi) v ds dvol_h \\ & = \int_{I \times M} \partial_s^k (\eta^(\varphi)) \eta^(\psi) v ds dvol_h \qquad \mbox{(by \eqref{intertwine})} \\ & = (-1)^k \int_{I \times M} \eta^(\varphi) v^{-1} \partial_s^k(\eta^(\psi) v) v ds dvol_h \qquad \mbox{(by partial integration)} \\ & = (-1)^k \int_X \varphi \eta_ (v^{-1} \partial_s^k (v \eta^(\psi))) dvol_g. \end{align} This shows that $\mathfrak{X}_\sigma^* (\psi) = \eta_* (v^{-1} \partial_s^k (v \eta^(\psi)))$. Next, we observe that \left \langle \sigma^(\delta), \psi dvol_g \right\rangle = \int_M \iota^* \frac{\psi}{\|\NV\|} dvol_h. This formula is an extension of <cit.> (for a flat background).[With appropriate interpretation, this identity coincides with <cit.>. The current reference to Hörmander replaces the reference for this result which has been used in <cit.>. This reference actually only utilizes formal arguments for domains in flat space $\R^n$.] $\iota^(v) = v_0 = \frac{1}{\|\NV\|}$, these results imply \begin{align} \left\langle \sigma^(\delta^{(k)}),\psi dvol_g \right\rangle & = (-1)^k \int_M \iota^ \partial_s^k (v \eta^(\psi)) dvol_h. \end{align} In particular, for $\psi = 1$ we find (<ref>). By (<ref>), the anomaly $\LA$ is proportional to \int_M \iota^* \frac{1}{\sqrt{\SC}} \tilde{L}_{n-1}(-1) (\E) dvol_h with $\E = \dot{L}(0)(1) \SC^{-1} + (d \SC^{-1},d\sigma)_g$ (see (<ref>)). In the singular Yamabe case, this integral reduces to \int_M \iota^* L_{n-1}(-1) \dot{L}(0)(1) dvol_h = \int_M \iota^* \dot{L}_n(0)(1) dvol_h = - \int_M \QC_n dvol_h. This motivates Gover and Waldron <cit.> to regard the function \begin{equation}\label{tilde-Qn} \tilde{\QC}_n(g,\sigma) \st \iota^* \frac{1}{\sqrt{\SC}} \tilde{L}_{n-1}(g,\sigma;-1) (\E) \end{equation} as a generalized critical extrinsic $Q$-curvature. We show that this quantity shares basic properties with $\QC_n$. In this context, we consider conformal changes $(\hat{g},\hat{\sigma}) = (e^{2\varphi} g, e^\varphi \sigma)$ and we let $L = L(g,\sigma)$ and $\hat{L} = L(\hat{g},\hat{\sigma})$. Similarly $\hat{\cdot}$ also denotes other functionals of $(g,\sigma)$ for such conformal changes. Now differentiating the conformal transformation law e^{-(\lambda-1) \varphi} \circ \hat{L}(\lambda) = L(\lambda) \circ e^{-\lambda \varphi} at $\lambda = 0$, gives e^\varphi \dot{\hat{L}}(0)(1) = \dot{L}(0)(1) - L(0)(\varphi) using $L(0)(1)=0$. Hence e^\varphi \hat{\E} = \E - \SC^{-1} L(0)(\varphi) + \sigma (d\SC^{-1},d\varphi), and the conformal covariance of $\tilde{L}(\lambda)$ implies \begin{align} e^{n \iota^(\varphi)} \hat{\tilde{\QC}}_n & = \iota^* \frac{1}{\sqrt{\SC}} e^{n \varphi} \hat{\tilde{L}}_{n-1}(-1) (\hat{\E}) = \iota^* \frac{1}{\sqrt{\SC}}\tilde{L}_{n-1}(-1) (e^\varphi \hat{\E}) \\ & = \iota^* \frac{1}{\sqrt{\SC}} \tilde{L}_{n-1}(-1) (\E - \SC^{-1} L(0)(\varphi) + \sigma (d\SC^{-1},d\varphi)) \\ & = \tilde{\QC}_n - \iota^* \frac{1}{\sqrt{\SC}} \tilde{L}_{n-1}(-1) (\SC^{-1} L(0)(\varphi) - \sigma (d\SC^{-1},d\varphi)). \end{align} This proves the conformal transformation law \begin{equation}\label{CTL-tilde} e^{n \iota^(\varphi)} \hat{\tilde{\QC}}_n = \tilde{\QC}_n - \iota^* \tilde{\PO}_n(\varphi) \end{equation} \tilde{\PO}_n \st \frac{1}{\sqrt{\SC}} \tilde{L}_{n-1} (-1) ( \SC^{-1} L(0)(\cdot) - \sigma (d\SC^{-1},d\cdot)). Note that the latter operator again is conformally covariant: $e^{n\varphi} \tilde{\PO}_n(\hat{g},\hat{\sigma}) = \tilde{\PO}_n(g,\sigma)$. This immediately follows from the conformal covariance of $L$ and the invariance of $\SC$. The following result generalizes Corollary <ref> and clarifies the content of <cit.>. For closed $M^n$, the integral \int_M \tilde{\QC}_n (g,\sigma) dvol_h is invariant under conformal changes of the pair $(g,\sigma)$. We give two proofs. The first proof uses an argument of <cit.>. Since the integral of $\tilde{\QC}_n$ is proportional to the anomaly $\LA$ in Theorem <ref>, it suffices to prove that $\LA(\hat{g},\hat{\sigma}) = \LA(g,\sigma)$. We recall that $\hat{\sigma}^{-2} \hat{g} = \sigma^{-2} g$ and prove that the expansion of the difference \int_{\hat{\sigma} > \varepsilon} dvol_{\sigma^{-2} g} - \int_{\sigma > \varepsilon} dvol_{\sigma^{-2} g} does not contain a $\log \varepsilon$ term. Note that $\{\hat{\sigma} > \varepsilon \} = \{\sigma > \varepsilon e^{-\varphi} \}$. Now, using adapted coordinates, we find that the above difference equals \begin{align} & \int_{\varepsilon}^{\varepsilon e^{-\varphi}} s^{-n-1} \left( \int_M v(s) dvol_h \right) ds \\ & = \sum_{k=0}^{n-1} \varepsilon^{-n+k} \int_M \frac{v_k}{n-k} \left(1-e^{(n-k)\varphi}\right) dvol_h - \int_M \varphi v_n dvol_h + o(1). \end{align} This expansion does not contain a $\log \varepsilon$ term. The second proof rests on the conformal transformation law (<ref>). It shows that it suffices to prove that $\int_M \iota^* \tilde{\PO}_n(\varphi) dvol_h = 0$ for all $\varphi \in C^\infty(X)$. Now the generalization[With appropriate interpretations as in Remark <ref>, the relation (<ref>) follows from <cit.>.] \begin{equation}\label{extension} \int_M \iota^* \frac{1}{\|\NV\|} \tilde{L}_{n-1}(-1)(\psi) dvol_h \sim \int_M \iota^* \partial_s^{n-1}(v \psi) dvol_h \end{equation} \begin{align} \langle \sigma^(\delta), L_{n-1}(-1) (\psi) dvol_g \rangle \sim \int_M \iota^* \partial_s^{n-1} (v \psi) dvol_h \end{align} (see (<ref>)) shows that the assertion is equivalent to \begin{align}\label{van2} \int_M \iota^ \partial_s^{n-1} \left( v \left[\SC^{-1} L(0)(\varphi) - s (d\SC^{-1},d\varphi)_g \right] \right) dvol_h= 0. \end{align} We recall that $L(0)(\varphi) = (n-1) a \partial_s (\varphi) - s \Delta_g(\varphi)$ with $a = \|\NV\|^2$ and calculate \begin{align} & \iota^ \partial_s^{n-1} \left( v f L(0)(\varphi) \right) \\ & = (n-1) \iota^* \partial_s^{n-1} (v f a \varphi') - (n-1) \iota^* \partial_s^{n-2} (v f \Delta_g (\varphi)) \\ & = (n-1) \iota^* \partial_s^{n-2} ( v' f a \varphi' + v f' a \varphi' + v f a' \varphi' + v f a \varphi'') \\ & - (n-1) \iota^* \partial_s^{n-2}\left(v f \left (a \varphi'' + a \frac{1}{2} \tr (h_s^{-1} h_s') \varphi' + \frac{1}{2} a' \varphi' - \frac{1}{2} (d \log a,d\varphi)_{h_s} + \Delta_{h_s}(\varphi) \right)\right) \end{align} using (<ref>). Simplification of that result gives (n-1) \iota^ \partial_s^{n-2} \left( v a f' \varphi' + v f \frac{1}{2} (d \log a, d \varphi)_{h_s} - v f \Delta_{h_s} (\varphi)\right). Here we used that $v = a^{-\frac{1}{2}} dvol_{h_s}/dvol_h = a^{-\frac{1}{2}} \mathring{v}$ implies \frac{\mathring{v}'}{\mathring{v}} = \frac{1}{2} \tr (h_s^{-1} h_s') = \frac{v'}{v} + \frac{1}{2} \frac{a'}{a} (see (<ref>)). Thus, we obtain \begin{align} & \int_M \iota^ \partial_s^{n-1} \left( v f L(0)(\varphi) \right) dvol_h \\ & = (n-1) \iota^* \partial_s^{n-2} \left( \int_M v \left(a f' \varphi' + f \frac{1}{2} (d \log a, d \varphi)_{h_s} - f \Delta_{h_s}(\varphi)\right) dvol_h \right). \end{align} Now partial integration on $M$ yields \begin{align} \int_M v f \Delta_{h_s}(\varphi) dvol_h & = \int_M a^{-\frac{1}{2}} f \Delta_{h_s}(\varphi) dvol_{h_s} = - \int_M (d (a^{-\frac{1}{2}} f, d\varphi)_{h_s} dvol_{h_s} \\ & = - \int_M v a^{\frac{1}{2}} (d (a^{-\frac{1}{2}} f), d\varphi)_{h_s} dvol_h \\ & = - \int_M v (d f,d\varphi)_{h_s} dvol_h + \frac{1}{2} \int_M v f (d \log a,d\varphi)_{h_s} dvol_h \end{align} using $a^{\frac{1}{2}} (d (a^{-\frac{1}{2}} f), d\varphi)_{h_s} = - \frac{1}{2} f (d \log a, d\varphi)_{h_s} + (d f,d\varphi)_{h_s}$. Differentiating this relation by $s$ implies \begin{align} \int_M \iota^* \partial_s^{n-1} \left( v f L(0)(\varphi) \right) dvol_h & = (n-1) \iota^* \partial_s^{n-2} \left( \int_M v a f' \varphi' + v (df,d\varphi)_{h_s} dvol_h \right) \\ & = (n-1) \iota^* \partial_s^{n-2} \left( \int_M v (df,d\varphi)_g dvol_h \right). \end{align} For $f = \SC^{-1}$, this identity implies the vanishing of (<ref>). Note that similar arguments show that L(-n) ( f \sigma^(\delta^{(n-1)})) = L(0)(f) \sigma^(\delta^{(n-1)}) for $f \in C^\infty(X)$ (see <cit.>). We omit the details. A calculation shows that in adapted coordinates \begin{equation}\label{TQ} \tilde{\QC}_2(g,\sigma) = \iota^ (\|\NV\|^{-5} ( \rho^2 - \rho \SC' - 2 (\SC')^2) + \|\NV\|^{-3} (\J - \rho' + \SC'')). \end{equation} In particular, if $\iota^* \SC'$ and $\iota^* \SC''$ vanish (singular Yamabe case), then $\QC_2 = \iota^* (\rho^2 - \rho' + \J)$. By $\iota^* (\rho) = -H$ and $\iota^* (\rho') = \Rho_{00} + \|\lo\|^2$ (Lemma <ref>), we get $\QC_2 = - \Rho_{00} - \|\lo\|^2 + \iota^(\J)$. Now the Gauss equation $\iota^ \J = \J^h + \Rho_{00} + \frac{1}{2} \|\lo\|^2 - H^2$ implies \QC_2 = \J^h - \frac{1}{2} \|\lo\|^2. This is the known formula for the critical extrinsic $Q$-curvature in dimension $n=2$ (Example <ref>). If $\sigma= d_M$ is the distance function, then $\|\NV\|=1$ and geodesic normal coordinates are adapted coordinates. Now $\SC = 1 + 2 s \rho$ implies $\iota^(\SC') = 2 \iota^(\rho)$ and $\iota^(\SC'') = 4 \iota^(\rho')$. Thus, we get \tilde{\QC}_2 = \iota^* (\J + 3 \rho' - 9 \rho^2). But $\iota^(\rho) = - \frac{2}{3} H$ and $3 \iota^(\rho') = - \iota^* \partial_s (\Delta_g (s) + s \J) = - \iota^* (2 H' + \J) = \iota^* (\Ric_{00} - \J) + \|L\|^2 = \Rho_{00} + \|L\|^2$ (using $2 H' = -\|L\|^2 - \Ric_{00}$ - see (<ref>)) show that $\tilde{\QC}_2 = \Ric_{00} + \|\lo\|^2 - 2 H^2$. This confirms the formula $-2 \LA = \int_M u_2 dvol_h$ (with $u_2$ as in (<ref>)). Finally, we use similar arguments as above to establish analogous formulas for the integrated renormalized volume coefficients $w_k$, which are defined in terms of geodesic normal coordinates. Let $d_M$ be the distance function of $M$. Let $M$ be closed. Assume that $\sigma$ satisfies $\SCY$. Then \int_M w_k dvol_h = (-1)^k \frac{(n\!-\!1\!-\!k)!}{(n\!-\!1)! k!} \int_M \iota^* L_k(g,\sigma;-n\!+\!k) \left( \left(\frac{d_M}{\sigma}\right)^{n-k}\right) dvol_h for $0 \le k \le n-1$ and \int_M w_n dvol_h = \frac{(-1)^{n-1} }{(n\!-\!1)! n!} \int_M \iota^* \dot{L}_n(g,\sigma;0)(1) dvol_h. We compare two different calculations of the residues of the family \lambda \mapsto \int_X \sigma^\lambda \psi dvol_g, \; \Re(\lambda) > -1 for appropriate test functions $\psi$. On the one hand, arguments as above using the functional equation (<ref>) prove the relation \begin{equation}\label{res-cal-1} \Res_{\lambda=-k-1} \left( \int_X \sigma^\lambda \psi dvol_g \right) = \frac{1}{k!(-n\!+\!1)_k} \int_M \iota^* L_k(g,\sigma;-n\!+\!k)(\psi) dvol_h. \end{equation} On the other hand, we use geodesic normal coordinates and asymptotic expansions of the resulting integrand. By $\kappa^(d_M)=r$ and \kappa^ (dvol_g) = \left( \frac{\kappa^(\sigma)}{r}\right)^{n+1} w(r) dr dvol_h (see (<ref>)), we obtain \begin{align} \int_X \sigma^\lambda \psi dvol_g & = \int_X d_M^\lambda \left(\frac{\sigma}{d_M}\right)^\lambda \psi dvol_g \\ & = \int_{[0,\varepsilon)} \int_M r^\lambda \left( \frac{\kappa^(\sigma)}{r}\right)^{\lambda+n+1} \kappa^(\psi) w(r) dr dvol_h \end{align} for test functions $\psi$ with appropriate support. The classical formula (<ref>) implies that \Res_{\lambda=-k-1} \left( \int_X \sigma^\lambda \psi dvol_g \right) = \frac{1}{k!} \int_M \left( \left(\frac{\kappa^(\sigma)}{r}\right)^{-k+n} \kappa^(\psi) w(r)\right)^{(k)}(0) dvol_h. For the test function $\psi_k = (d_M/\sigma)^{n-k} \chi$ (with an appropriate cut-off function $\chi$), the latter result yields \begin{equation}\label{res-cal-2} \Res_{\lambda=-k-1} \left( \int_X \sigma^\lambda \psi_k dvol_g \right) = \int_M w_k dvol_h. \end{equation} Now comparing (<ref>) and (<ref>) proves the first assertion. The assertion in the critical case follows from $\int_M w_n dvol_h = \int_M v_n dvol_h$ (see (<ref>)) and Theorem <ref>. The first part of Theorem <ref> is a special case of <cit.>. § HOLOGRAPHIC FORMULÆ FOR EXTRINSIC $Q$-CURVATURES We work in adapted coordinates. In particular, $\J=\J^g$ is identified with $\eta^(\J)$. The following result is a local version of Theorem <ref>. Let $n$ be even and assume that $\sigma$ satisfies $\SCY$. Then \begin{equation}\label{Q-holo-form} \QC_n (g) = (-1)^n (n\!-\!1)!^2 \sum_{k=0}^{n-1} \frac{1}{n\!-\!1\!-\!2k} \T_k^(g;0) \left( (n\!-\!1)(n\!-\!k) v_{n-k} + 2k (v \J)_{n-k-2} \right). \end{equation} For $k=n-1$, the second term on the right-hand side is defined as $0$. The assumption that $n$ is even guarantees that the fractions on the right-hand side are well-defined. For odd $n$, we refer to Conjecture <ref>. Assume that $g_+ = r^{-2}(dr^2 + h_r)$ is an even Poincaré-Einstein metric in normal form relative to $h$ and let $g = r^2 g_+ = dr^2 + h_r$. Then \J^g= - \frac{1}{2r} \tr (h_r^{-1} h'_r) = - \frac{1}{r} \frac{v'}{v}(r) by <cit.> or Lemma <ref>, and the second term on the right-hand side of (<ref>) yields $-2k (n-k) v_{n-k}$. Therefore, the sum simplifies to \sum_{k=0}^{n-1} (n-k) \T_k^(0) (v_{n-k}) and the assertion reduces to the main result of <cit.> by noting that it only contains contributions for even $k$. Note that (<ref>) can be written in the form \QC_n = (-1)^{n} n! (n\!-\!1)! v_n + \sum_{k=1}^{n-1} \T_k^(0)(\cdots). Since $\T_k(0)(1) = 0$ for $k \ge 1$, integration of this identity (for closed $M$) reproduces Theorem <ref>, and the holographic formula provides a formula for the lower-order terms. There is a generalization of Theorem <ref> to subcritical $Q$-curvatures. Let $n$ be even and $\N \ni N < n$. Assume that $\sigma$ satisfies $\SCY$. Then \begin{align}\label{Q-holo-form-gen} \QC_N(g) & = (-1)^{N} (N\!-\!1)!^2 \sum_{k=0}^{N-1} \frac{1}{(2N\!-\!n\!-\!1\!-\!2k)} \notag \\ & \times \T_k^\left(g;\frac{n-N}{2}\right) \left( (N\!-\!1)(N\!-\!k) v_{N-k} + (n\!-\!N\!+\!2k) (v \J)_{N-k-2} \right). \end{align} For $k=N-1$, the second term on the right-hand side is defined as $0$. Again, the assumption that $n$ is even guarantees that the fractions on the right-hand side are well-defined. For odd $n$, we refer to Conjecture <ref>. In the even Poincaré-Einstein case, the formula is non-trivial only for even $N$. In that case, the solution operators in (<ref>) act on (N\!-\!1)(N\!-\!k) v_{N-k} - (n\!-\!N\!+\!2k) (N\!-\!k) v_{N-k} = (2N\!-\!n\!-\!1\!-\!2k)(N\!-\!k) v_{N-k} and the sum simplifies to \sum_{k=0}^{N-1} \T_k^\left(g;\frac{n-N}{2}\right) (N-k) v_{N-k}. Here only terms with even $k$ contribute. Thus the formula reduces to the main result of <cit.>. The following conjecture implies that for odd $n$ the respective terms with the singular fractions in the sums (<ref>) and (<ref>) do not contribute. For odd $n\ge 3$, it holds \begin{equation}\label{van-id} \tfrac{n+1}{2} v_{\frac{n+1}{2}} + (v \J)_{\frac{n-3}{2}} = 0. \end{equation} Moreover, the holographic formulas (<ref>) and (<ref>) are valid also for odd $n$. For $n=3$ and $n=5$, the respective relations (<ref>) read \begin{align} 2v_2 + \J_0 & = 0, \\ 3 v_3 + \J'_0 + v_1 \J_0 & = 0. \end{align} For details, we refer to the discussion in Examples <ref>–<ref>. These identities are consequences of the relation (<ref>). More generally, the relation (<ref>) implies that any coefficient $v_N$ can be written as a linear combination of products of derivatives of $\rho$ and $\J$ at $s=0$. The calculations of the resulting formulas for odd $n \le 11$ confirm the relation (<ref>) in these special cases. For the discussion of the holographic formula for $\QC_3$ in general dimensions, we refer to Example <ref>. Now we present the proof of Theorem <ref>. We evaluate the quantity $\dot{\D}_n^{res}(0)(1)$ using the factorization formula \D_n^{res}(\lambda) = \D_{n-1}^{res}(\lambda-1) L(\lambda) (Corollary <ref>). In view of $L(0)(1)=0$, it follows that \begin{equation}\label{start-crit} \dot{\D}_n^{res}(0)(1) = \D_{n-1}^{res}(-1) \dot{L}(0)(1). \end{equation} Now we apply the representation formula \D_{n-1}^{res}(-1) = (-1)^{n-1} (n\!-\!1)!^2 \sum_{j=0}^{n-1} \frac{1}{(n\!-\!1\!-\!j)!} \left[ \T_j^(0) v_0 + \cdots + \T_0^(0) v_j\right] \iota^ \partial_s^{n-1-j} (Theorem <ref>). We find \dot{\D}_n^{res}(0)(1) = (-1)^{n-1} (n-1)!^2 \sum_{j=0}^{n-1} \frac{1}{(n\!-\!1\!-\!j)!} \left[ \T_j^(0) v_0 + \cdots + \T_0^(0) v_j\right] \iota^* \partial_s^{n-1-j} (\dot{L}(0)(1)). In the latter sum, the operator $\T_k^(0)$ acts on the sum \left(v_0 \frac{1}{(n\!-\!1\!-\!k)!} \iota^ \partial_s^{n-1-k} + \cdots + v_{n-1-k} \iota^* \right)(\dot{L}(0)(1)). But this quantity equals the $(n\!-\!1\!-\!k)$'th Taylor coefficient $(v \dot{L}(0)(1))_{n-1-k}$ of $v \dot{L}(0)(1)$. Now the \begin{equation}\label{reduction} \iota^* \partial_s^k (v \dot{L}(0)(1)) = -\frac{n\!-\!1}{(n\!-\!1\!-\!2k)} \iota^* \partial_s^{k+1}(v) - \frac{2k(n\!-\!1\!-\!k)}{(n\!-\!1\!-\!2k)} \iota^* \partial_s^{k-1}(v\J) \end{equation} for $0 \le k \le n-1$ (which extends (<ref>))[For $k=0$, the second term on the right-hand side is defined as $0$.] yields \begin{align} (v \dot{L}(0)(1))_{n-1-k} & = \frac{1}{(n\!-\!1\!-\!k)!} \iota^ \partial_s^{n-1-k} (v \dot{L}(0)(1)) \\ & = \frac{n\!-\!1}{(n\!-\!1\!-\!2k)(n\!-\!1\!-\!k)!} \iota^* \partial_s^{n-k} (v) + \frac{2k (n\!-\!1\!-\!k)}{(n\!-\!1\!-\!2k)(n\!-\!1\!-\!k)!} \iota^* \partial_s^{n-2-k}(v\J) \\ & = \frac{(n\!-\!1)(n\!-\!k)}{(n\!-\!1\!-\!2k)} v_{n-k} + \frac{2k}{(n\!-\!1\!-\!2k)} (v\J)_{n-2-k} \end{align} for $0 \le k \le n-1$. This implies the claim. It only remains to prove the identity (<ref>). Note that $\dot{L}(0)(1) = (n-1)\rho-s\J$. Using the definition of $\rho$, we obtain \begin{equation}\label{d-delta} \iota^ \partial_s^k (v\dot{L}(0)(1)) = - \frac{n-1}{n+1} \left( \iota^* \partial_s^k(v \Delta (s)) + \frac{2kn}{n-1} \iota^* \partial_s^{k-1}(v\J)\right). \end{equation} Now the relation (<ref>) implies \begin{align} \iota^ \partial_s^k(v \Delta (s)) & = \iota^* \partial_s^{k+1}(v \|\NV\|^2) \\ & = \iota^* \partial_s^{k+1}(v) - 2 (k+1) \iota^* \partial_s^k(v \rho) \\ & = \iota^* \partial_s^{k+1}(v) + \frac{2(k+1)}{n+1} \iota^* \partial_s^k(v \Delta(s) + s v\J) \end{align} for $0 \le k \le n-1$ using $\|\NV\|^2 = 1-2s\rho + O(s^{n+1})$. We rewrite this identity in the form \frac{n-2k-1}{n+1} \iota^ \partial_s^k(v \Delta (s)) = \iota^* \partial_s^{k+1}(v) + \frac{2k(k+1)}{n+1} \iota^* \partial_s^{k-1}(v \J). By substituting this result into (<ref>), we obtain (<ref>). Note that the above arguments extend those in the proof of Theorem <ref>. Finally, we sketch a proof of Theorem <ref>. For $N < n$, we have \D_N^{res} \left(\frac{-n+N}{2}\right) (1) = \left(\frac{n-N}{2}\right) \QC_N by (<ref>). Hence \begin{align} \QC_N & = \D_N^{res}\left(\frac{-n+N}{2}\right) (1) \left(\frac{n-N}{2}\right)^{-1} \\ & = \D_{N-1}^{res}\left(\frac{-n+N}{2}-1\right) L\left(\frac{-n+N}{2}\right) (1) \left(\frac{n-N}{2}\right)^{-1} & \mbox{(by Corollary \ref{factor})} \\ & = \D_{N-1}^{res}\left(\frac{-n+N}{2}-1\right) (-(N-1) \rho + s \J) & \mbox{(by \eqref{aa})}. \end{align} This formula generalizes (<ref>). Now we proceed as in the proof of Theorem <ref>. In particular, Theorem <ref> and the above identity imply \begin{align} \QC_N & = (-1)^{N} (N\!-\!1)!^2 \\ & \times \sum_{j=0}^{N-1} \frac{1}{j!} \left[ \T_{N-1-j}^\left(\frac{n\!-\!N}{2}\right) v_0 + \cdots + \T_0^\left(\frac{n\!-\!N}{2}\right) v_j \right] \iota^ \partial_s^j ((N\!-\!1)\rho-s\J). \end{align} But in the latter double sum the operator $\T_k^\left(\frac{n-N}{2}\right)$ acts on the $(N-1-k)$'th Taylor coefficient of $((N-1)\rho-s\J)v$. A calculation using the definition of $\rho$ yields the extension \begin{equation}\label{reduction-2} \iota^* \partial_s^k (v ((N\!-\!1)\rho - s\J)) = - \frac{N\!-\!1}{(n\!-\!1\!-\!2k)} \iota^* \partial_s^{k+1}(v) - \frac{k(n\!-\!2k\!+\!N\!-\!2)}{(n\!-\!1\!-\!2k)} \iota^* \partial_s^{k-1}(v\J) \end{equation} of (<ref>) for $k=0,\dots,N-1$.[For $k=0$, the second term on the right-hand side is defined as $0$.] Hence \begin{equation} (v ((N\!-\!1)\rho-s\J))_{N-1-k} = - \frac{(N\!-\!1)(N\!-\!k)}{(n\!-\!2N\!+\!2k\!+\!1)} v_{N-k} - \frac{n\!-\!N\!+\!2k}{(n\!-\!2N\!+\!2k\!+\!1)} (v\J)_{N-2-k} \end{equation} for $k=0,\dots,N-1$. This implies the assertion. $\square$ We finish this section with an application to the singular Yamabe obstruction $\B_{n}$ (see Section <ref>). We recall that our discussion of extrinsic conformal Laplacians $\PO_N$ and extrinsic $Q$-curvatures $\QC_N$ is restricted to the range $N \le n$. Already the first super-critical $\QC$-curvature $\QC_{n+1}$ is not well-defined. Calculations in low-order cases point to the origin of its non-existence: $\QC_N$ has a pole in $n=N-1$. The following result interprets its residue. Let $n$ be even and $N \ge 3$. Then it holds \begin{equation}\label{QB-F} \Res_{n=N-1} (\QC_N) = (-1)^{n-1} n! (n\!+\!1)! \frac{n}{2} \B_n. \end{equation} Assume that $N \le n$. Theorem <ref> and Theorem <ref> imply that \QC_N = (-1)^N (N\!-\!1)! N! \frac{N\!-\!1}{2N\!-\!n\!-\!1} v_N + \cdots, where the hidden terms are regular at $n=N-1$. However, the term $v_N$ has a simple pole at $n=N-1$. More precisely, it follows from Corollary <ref> that v_N = -(n\!+\!1\!-\!2N) \frac{1}{N!} \partial^{N-1}_s (\rho)\|_0 + \cdots, where the hidden terms are regular at $n=N-1$, and Proposition <ref> explains the origin of the pole of $\partial^{N-1}_s (\rho)\|_0$. But a comparison of Proposition <ref> and Theorem <ref> shows that[See also <cit.>.] \begin{equation}\label{B-rho} \B_{n} = \frac{2}{(n+1)!} \Res_{n=N-1} (\partial_s^{N-1}(\rho))\|_0. \end{equation} The result follows from these facts. For a detailed discussion of the relation between $\B_2$ and $\QC_3$, we refer to Section <ref>. This proof rests on the key relation between $Q$ and $v$ (holographic formula)! Some possible additions for later versions: 1. the above proof requires some improvements: we should emphasize that we do a continuation in dimension argument. 2. There is no difference between taking partials in $s$ or partials in $N$: lower-order derivatives of $\rho$ are not singular. 3. the arguments also give the conformal invariance of $\B_n$: this requires that the pole of $\PO_{n+1}$ is only in the constant term. Combining Theorem <ref> with $\Res_{n=N-1} (\PO_N) \sim \Res_{n=N-1} (\QC_N)$ allows us to derive the conformal invariance $e^{(n+1) \iota^(\varphi)} \hat{\B}_{n} = \B_{n}$ of the obstruction from the conformal covariance of $\PO_N$. The above observations resemble the result that the residues \Res_{n=4}(P_6) = -16 (\delta (\B d) - (\B,\Rho)) \Res_{n=6}(P_8) = -48 (\delta(\OB_6 d) - (\OB_6,\Rho)) of super-critical GJMS-operators $P_6$ and $P_8$ are conformally covariant. Here $\B$ is the Bach tensor, and $\OB_6$ is the Fefferman-Graham obstruction tensor in dimension $n=6$. For more details, we refer to <cit.>. § COMMENTS ON FURTHER DEVELOPMENTS We recall that $\PO_n \iota^ = \D_n^{res}(0)$. For even $n$, this critical extrinsic conformal Laplacian is elliptic. For odd $n$, the leading part of this operator is determined by $\lo$. The lower-order terms are not known in general. For $n=3$, the explicit formula in Proposition <ref> shows that $\PO_3$ vanishes iff $\lo=0$, i.e., iff $M$ is totally umbilic. The vanishing of $\PO_n$ is a conformally invariant condition. Now if $\PO_n(g) = 0$, we define \begin{equation} \PO_n'(g) = \dot{\D}_n^{res}(g;0) \quad \mbox {and} \quad \QC_n'(g) = \frac{1}{2} \ddot{\D}_n^{res}(g;0). \end{equation} Then $\PO_n': C^\infty(X) \to C^\infty(M)$ is a conformally covariant operator e^{n \iota^(\varphi)} \PO_n'(\hat{g}) = \PO_n'(g), \; \varphi \in C^\infty(X) such that the pair (\PO_n', \QC_n') satisfies the fundamental identity \begin{equation}\label{FI-prime} e^{n \iota^(\varphi)} \QC_n'(\hat{g}) = \QC_n'(g) - \PO_n'(g)(\varphi) + \iota^(\varphi) \PO_n'(g)(1) \end{equation} for all $\varphi \in C^\infty(X)$. This identity follows by twice differentiation of the conformal transformation law of the critical residue family $\D_n^{res}(\lambda)$ at $\lambda=0$. Note that $\PO_n'(g)(1) = - \QC_n(g)$. It is interesting to analyze this construction further. First of all, it is a question of independent interest to characterize the vanishing of $\PO_n(g)$ for odd $n$. Is it true that $\QC_n(g)(1) = 0$? The operator $\PO_n$ may be regarded as a boundary operator for a conformally covariant boundary value problem for the critical GJMS-operator $P_{n+1}$ on $X$. For $n=3$, such boundary value problems were recently analyzed in <cit.>. A conformally covariant boundary operator of third order together with a $Q$-curvature like scalar curvature quantity was discovered in <cit.> in connection with the study of Polyakov formulas on four-dimensional manifolds with boundary. Later it was studied from various perspectives. For details, we refer to <cit.> and the references in these works. It would be interesting to develop an extension of the present theory in higher codimension situations again using solutions of singular Yamabe problems. We briefly describe some aspects of a special case of such a theory. Let $S^m \hookrightarrow S^n$ be an equatorial subsphere of $S^n$, $m \le n-1$. It is well-known that the complement of $S^m$ in $S^n$ with the round metric $g$ is conformally equivalent to the product $\mathbb{H}^{m+1} \times S^{n-m-1}$ with the respective hyperbolic and round metric of constant scalar curvature $-(m+1)(m+2)$ and $(n-m-1)(n-m)$ on the factors. The conformal factor $\sigma \in C^\infty(S^n\setminus S^m)$ can be defined in terms of a Knapp-Stein intertwining operator on $S^n$ applied to the delta-distribution of $S^m$ <cit.>. More explicitly, assume that $S^m$ is defined by the equations $x_{m+2} = \cdots = x_{n+1}=0$. Then $\sigma$ is the restriction of $(\sum_{j=m+2}^{n+1} x_j^2)^{1/2}$ to $S^n$. A stereographic projection yields an isometry of $\sigma^{-2} g$ and (x_{m+1}^2 + \cdots + x_{n}^2)^{-1} \sum_{i=1}^n dx_i^2 = r^{-2} \sum_{i=1}^{m+1} dx_i^2 + g_{S^{n-m-1}}, where $r^2 = \sum_{i=m+1}^{n} x_i^2$. The above $\sigma$ is a generalization of the height function $\He$ in Section <ref> (the case $m=n-1$). Now, for any eigenfunction $\tau$ of the Laplacian on $S^{n-m-1}$ (spherical harmonics), there is an intertwining operator D_{N,\tau}(\lambda): C^\infty(S^n) \to C^\infty(S^{m}) for the subgroup $SO(1,m+1) \subset SO(1,n+1)$ leaving $S^m$ invariant (symmetry breaking operator). These families can be constructed in terms of the residues of the family \lambda \mapsto \int_{S^n} \sigma^\lambda u \psi dvol_{S^n}, where $u$ is an eigenfunction of $\Delta_{\sigma^{-2}g}$ on the complement of $S^m$ which is compatible with $\tau$. The conformal factor $\sigma$ is a solution of the singular Yamabe problem on the complement of $S^m$. In fact, the scalar curvature of $\sigma^{-2}g$ is $-(n+1)(2m-n+2)$. It is negative iff $m > (n-2)/2$, i.e., iff the dimension of the hyperbolic space exceeds the dimension of the sphere. This is a special case of <cit.>, where it is proved that if $M \subset S^n$ is a smooth submanifold of dimension $m$, a solution of the singular Yamabe problem with negative scalar curvature exists iff $m > (n-2)/2$. The analogous result for $S^n$ replaced by an arbitrary $X$ is in <cit.>. For a description of the asymptotic expansions of solutions of the singular Yamabe problem in the negative case, we refer to <cit.>. Related representation theoretical aspects of the case $S^m \hookrightarrow S^n$ are studied in <cit.>. Here the spectral decomposition of spherical principal series representations of $O(1,n+1)$ under restriction to $O(1,m+1) \times O(n-m)$ is made fully explicit. The conformal invariance of the integral \int_M \QC_n(g) dvol_h for closed $M$ (see (<ref>)) generalizes the conformal invariance of the total Branson \int_M Q_n(h) dvol_h. The classification of scalar Riemannian curvature quantities of a manifold $(M,h)$ which - like $Q_n$ - upon integration give rise to a conformal invariant has been the subject of the Deser-Schwimmer conjecture <cit.>. S. Alexakis has achieved this classification in a series of works (see <cit.> and its references). The present context suggests asking for an analogous classification of scalar Riemannian invariants of a manifold $(X,g)$ which, upon integration over closed submanifolds, yield conformal invariants of $g$. For first results in that direction, we refer to <cit.>. For related results around ${\bf Q}_4$, we refer to <cit.> and <cit.>. § CALCULATIONS AND FURTHER RESULTS theoremsubsection equationsubsection In the present section, we collect formulas and computational details used in the main body of the text. In addition, we illustrate various aspects of the general theory by low-order examples. This often gives additional insight into the situation's nature and complexity. The results may also serve as material for future research. Moreover, we add a few further results. For the reader's convenience, we start with an outline of the content of this section. In Section <ref>, we recall basic forms of the hypersurface Gauss equations, recall the transformation laws of the second fundamental form and of some derived constructions under conformal changes of the metric, derive a basic formula for the Laplacian, which plays a key role in the proof of the conjugation formula and clarify the relation between high-order iterated normal derivatives $\nabla_\NV^k$ and their analogs in adapted coordinates. In Section <ref>, we determine the first three terms in the expansion of a general metric $g$ in geodesic normal and adapted coordinates. These results are fundamental for the later calculations. We also illustrate the results in several model cases with constant curvature and vanishing trace-free part of $L$. These model cases may serve as valuable test examples of identities of the general theory. In Section <ref>, we first determine the first four terms in the asymptotic expansion of solutions of the singular Yamabe problem in geodesic normal coordinates. Then, we use these results to calculate the obstructions $\B_2$ and $\B_3$ in these terms. Finally, we prove a formula for the obstructions in terms of a formal residue of the super-critical coefficient $\sigma_{(n+2)}$. The explicit form of the first few coefficients $\sigma_{(k)}$ enables us to confirm this formula in low-order cases. In Section <ref>, we derive explicit formulas for the first three renormalized volume coefficients $v_k$ (in adapted coordinates). In Section <ref>, we derive explicit formulas for the first two normal derivatives of $\rho$. The discussion illustrates the efficiency of the recursive relation for the Taylor coefficients of $\rho$ expressed in Proposition <ref>. We further use these results in Section <ref> for a detailed discussion of $\QC_2$ and $\QC_3$. In Section <ref>, we show that the well-known formula for the obstruction $\B_2$ naturally follows from Theorem <ref>. Moreover, in Section <ref>, we evaluate the special case $n=3$ of Theorem <ref> for a flat background metric. We find that the formula coincides with the one derived in Section <ref> as well with a formula of Gover and Waldron. In addition, we verify that in the conformally flat case, the result fits with the formula for $\B_3$ established in [3]. In Section <ref>, we illustrate the role of the obstruction in the variational formula for the singular Yamabe energy <cit.>, <cit.> in low-order cases. In <cit.> and [3], the authors developed a new variational calculus. In contrast to these references, here we only use classical style arguments as in <cit.>, for instance. In Section <ref>, we provide direct proofs of the conformal covariance of $\PO_3$ and the fundamental conformal transformation law of $\QC_3$. Section <ref> is devoted to a derivation of explicit formulas for the first two solution operators $\T_1(\lambda)$ and $\T_2(\lambda)$ and the resulting first two residue families $\D_1^{res}(\lambda)$ and $\D_2^{res}(\lambda)$. In addition, we determine the leading term of $\PO_3$ through the leading term of the third solution operator $\T_3(\lambda)$. In the last section, we describe low-order renormalized volume coefficients $w_k$ in terms of Laplace-Robin operators. These results imply low-order cases of Theorem <ref>. Throughout this section, we apply some additional conventions. We use indices $i,j$ for tensorial objects on $M$ and $0$ for $\NV$ when viewed as a normal vector of $M$. In particular, $h_{ij}$ are the components of the metric $h$ on the boundary and $\Rho_{00} = \Rho_X (\NV,\NV)$ is the restriction to $M$ of the Schouten tensor of $g$ for the normal vector $\NV$. Similar conventions are used in adapted coordinates. Sometimes, it will be convenient to distinguish curvature quantities of $g$ and $h$ not by superscripts but by adding a bar to those of $g$ and leaving those of $h$ unbared. Then $\bar{R}$, $\overline{\Ric}$ and $\overline{\scal}$ are the curvature tensor, the Ricci tensor and the scalar curvature of $g$, respectively. For instance, $\Ric^g(\NV,\cdot)$ and $\overline{\Ric}_0$ are the same $1$-forms on $M$. As before, we often use the same notation for a function on $X$ and its pull-back by $\kappa$ or $\eta$ without mentioning it. A prime denotes derivatives in the variable $r$ and $s$. The restriction of a function $f(s)$ to $s=0$ is also denoted by $f_0$. This notation often replaces $\iota^ (f)$. For instance, we write $\rho'_0$ for the restriction $\iota^(\partial_s (\rho))$ of $\partial_s(\rho)$ to $s=0$. If confusion is excluded, we sometimes even omit the symbols indicating restriction. §.§ Some basic identities §.§.§ Gauss equations Let $M^n$ be a hypersurface in $(X^{n+1},g)$ with the induced metric $h=\iota^(g)$. Then it holds \begin{align} \Ric^g_{ij} - \Ric^h_{ij} & = R^g_{0ij0} + L^2_{ij} - n H L_{ij}, \label{GRicci} \\ \scal^g - \scal^h & = 2 \Ric^g_{00} + \|L\|^2 - n^2 H^2 \label{G1} \end{align} on $M$. In the bar-notation, (<ref>) reads \iota^* \overline{\scal} - \scal = 2 \overline{\Ric}_{00} +\|L\|^2 - n^2 H^2. The following result follows from the Gauss equation (<ref>). For $n \ge 2$, it holds \begin{equation}\label{G2} \iota^* \J^g - \J^h = \Rho^g_{00} + \frac{1}{2(n\!-\!1)} \|\lo\|^2 - \frac{n}{2} H^2. \end{equation} The Gauss equation (<ref>) is equivalent to 2n \iota^* \J^g - 2(n\!-\!1) \J^h = 2((n\!-\!1) \Rho^g_{00} + \J^g) + \|L\|^2 - n^2 H^2. 2(n\!-\!1) \iota^* \J^g - 2(n\!-\!1) \J^h = 2(n\!-\!1)\Rho^g_{00} + \|\lo\|^2 - n(n-1) H^2. This implies the assertion. $\JF$ Fialkow tensor Next, let \begin{equation}\label{Fialkow-tensor} \JF \st \iota^* (\Rho^g) - \Rho^h + H \lo + \frac{1}{2} H^2 h. \end{equation} $\JF$ is a conformally invariant symmetric bilinear form, i.e., it holds $\hat{\JF} = \JF$ (<cit.>.[In <cit.>, the tensor $\JF$ is called the Fialkow tensor following <cit.> (in turn being inspired by <cit.>). For more details on the relation to Fialkow's classical work, we refer to <cit.>.] Moreover, it satisfies the fundamental relation \begin{align}\label{FW-relation} (n\!-\!2) \JF_{ij} & = (n\!-\!2) \left(\iota^* \Rho^g_{ij} - \Rho^h_{ij} + H \lo_{ij} + \frac{1}{2} H^2 h_{ij} \right) \notag \\ & \stackrel{!}{=} W_{0ij0} + \lo^2_{ij} - \frac{\|\lo\|^2}{2(n\!-\!1)} h_{ij} \end{align} (<cit.>), where $W$ is the Weyl tensor of $g$. We recall that $W$ Weyl tensor \begin{equation}\label{RW-deco} R = W - \Rho \owedge g, \end{equation} where the Kulkarni-Nomizu product of the bilinear forms $b_1$ and $b_2$ is defined by \begin{align} & (b_1 \owedge b_2) (X,Y,Z,W) \\ & \st b_1 (X,Z) b_2 (Y,W) - b_1 (Y,Z) b_2(X,W) + b_1(Y,W) b_2(X,Z) - b_1(X,W) b_2(Y.Z). \end{align} $\owedge$ Kulkarni-Nomizu product §.§.§ Conformal change and the second fundamental form Let $N$ be a fixed unit normal field of $M$ defining $L$. Let $\hat{L}$ denote the second fundamental form of $M$ with respect to the metric $\hat{g} = e^{2\varphi} g$. Then it holds e^{-\varphi} \hat{L} = L + \nabla_N(\varphi) h. As a consequence, we find e^{\varphi} \hat{H} = H + \nabla_n(\varphi). Both relations combine into the conformal invariance property \begin{equation}\label{CTL-L} e^{-\varphi} \loh = \lo \end{equation} of the trace-free part $\lo = L - H h$ of $L$. It follows that $\loh^2 = \lo^2$, where $(\lo^2) _{ij} \st h^{ab} \lo_{ia} \lo_{jb}$. For $\|L\|^2 = \tr_h (L^2) = h^{ij} (L^2)_{ij} = h^{ij} h^{ab} L_{ia} L_{jb}$, we find $e^{2\varphi} \|\loh\|^2 = \|\lo\|^2$.[Of course, the norm on the left-hand side is taken with respect to $\hat{h}$.] \begin{equation}\label{tf-square} (\loh^2)_\circ = (\lo^2)_\circ. \end{equation} §.§.§ Some formulas for the Laplacian Here we discuss some useful formulas for the Laplacian. In particular, we prove the identity Let $\sigma$ be a defining function of $M$ and $\partial_0 = \NV = \grad_g(\sigma)$. We assume that $\|\partial_0\|_g =1$ on $M$. Then it holds \begin{equation}\label{LBM} \Delta_g (u) = \partial_0^2 (u) + \Delta_h (u) + n H \partial_0 (u) - \langle du, \nabla_{\partial_0}(\partial_0) \rangle \end{equation} on $M$. Let $\left \{\partial_i \right \}$ be an orthonormal basis of the tangent spaces of the level surfaces $\sigma^{-1}(c)$ (for small $c$) and let $\partial_0 = \grad_g(\sigma)$. By assumption, these form an orthonormal basis on $M$. Let $\left\{ dx^i \right\}$ together with $\left\{dx^0\right\}$ be the dual basis. We calculate $\iota^* \Delta_g (f)$ using $\Delta = \tr (\nabla^2)$. First of all, we have $\nabla (u) = \partial_i (u) dx^i + \partial_0 (u) dx^0$. Hence[As usual, we sum over repeated indices.] \begin{align} \nabla^2(u) & = \partial_{ij}(u) dx^i \otimes dx^j + \partial_i(u) \nabla_{\partial_j}(dx^i) \otimes dx^j \\ & + \partial_{i0}(u) dx^i \otimes dx^0 + \partial_i(u) \nabla_{\partial_0} (dx^i) \otimes dx^0 \\ & + \partial_{0j}(u) dx^0 \otimes dx^j + \partial_0(u) \nabla_{\partial_j}(dx^0) \otimes dx^j \\ & + \partial_{00}(u) dx^0 \otimes dx^0 + \partial_0(u) \nabla_{\partial_0}(dx^0) \otimes dx^0. \end{align} Now taking traces gives \begin{align} \tr (\nabla^2)(u) & = \sum_{i=0}^n \partial_{ii}(u) + \partial_i(u) \langle \nabla_{\partial_j}(dx^i), \partial_j \rangle + \partial_i(u) \langle \nabla_{\partial_0} (dx^i), \partial_0 \rangle + \partial_0(u) \langle \nabla_{\partial_j}(dx^0), \partial_j \rangle \\ & + \partial_0^2(u) + \partial_0(u) \langle \nabla_{\partial_0}(dx^0), \partial_0 \rangle \end{align} on $M$. Thus, we obtain \begin{align} \tr (\nabla^2)(u) & = \sum_{i=0}^n \partial_{ii}(u) + \partial_i(u) \langle \nabla_{\partial_j}(dx^i), \partial_j \rangle - \partial_i(u) \langle dx^i, \nabla_{\partial_0} (\partial_0) \rangle - \partial_0(u) \langle dx^0, \nabla_{\partial_j}(\partial_j) \rangle \\ & + \partial_0^2(u) - \partial_0(u) \langle dx^0, \nabla_{\partial_0}(\partial_0) \rangle. \end{align} But since $\nabla^g_{\partial_i}(\partial_j) = \nabla^h_{\partial_i}(\partial_j)$ on $M$, the last display simplifies to \begin{align} \Delta_g (u) & = \partial_0^2(u) + \Delta_h (u) - \partial_0(u) \langle dx^0, \nabla_{\partial_j}(\partial_j) \rangle \\ & - \partial_i(u) \langle dx^i, \nabla_{\partial_0} (\partial_0) \rangle - \partial_0(u) \langle dx^0, \nabla_{\partial_0}(\partial_0) \rangle \end{align} on $M$. By $L (\partial_i,\partial_j) = - \langle \nabla_{\partial_i} (\partial_j), dx^0 \rangle $, we obtain \Delta_g (u) = \partial_0^2(u) + \Delta_h (u) + n H \partial_0(u) - \langle du,\nabla_{\partial_0}(\partial_0) \rangle on $M$. This completes the proof. Without the assumption that $\|\partial_0\|=1$ on $M$, an extension of the above arguments shows that \begin{equation} \Delta_g (u) = \frac{1}{\|\partial_0\|^2} \partial_0^2 (u) + \Delta_h (u) + n H \frac{1}{\|\partial_0\|} \partial_0 (u) - \frac{1}{\|\partial_0\|^2} \langle du, \nabla_{\partial_0}(\partial_0) \rangle \end{equation} on $M$. Similar arguments prove (<ref>). In the situation of Lemma <ref>, assume that $\sigma$ is the distance function of $M$. Then $\nabla_\NV (\NV) = 0$ and we recover the well-known formula \begin{equation} \Delta_g = \nabla_{\NV}^2 + \Delta_h + n H \nabla_{\NV} \end{equation} on $M$. The following result reproves <cit.>. If $\sigma$ satisfies $\SCY$, then \begin{equation}\label{LR} \Delta_g = \nabla_\NV^2 + \Delta_h + (n-1) H \nabla_\NV \end{equation} on $M$. By Lemma <ref>, it suffices to prove that $\nabla_\NV(\NV) = H \NV$ on $M$. For any $X \in \mathfrak{X}(X)$, we calculate \langle \nabla_\NV(\NV),X\rangle = \Hess_g(\sigma)(\NV,X) = \Hess_g(\sigma)(X,\NV) = \langle \nabla_X(\NV),\NV \rangle = 1/2 \langle d (\|\NV\|^2),X \rangle. Now the assertion follows from $\SCY$ using $\rho = -H$ on $M$ (Lemma <ref>). §.§.§ Iterated normal derivatives The identity \iota^* \partial_s^k \circ \eta^* = \iota^* (\|\NV\|^{-2} \nabla_\NV)^k (see (<ref>)) relates iterated normal derivatives $\iota^* \partial_s^k \circ \eta^$ with respect to $s$ to iterated weighted gradients $\iota^ (\|\NV\|^{-2} \nabla_\NV)^k$ of $\sigma$. Moreover, if $\SC(g,\sigma)=1$, i.e., if $\|\NV\|^2 = 1- 2 \sigma \rho$, any iterated weighted gradient $\iota^* (\|\NV\|^{-2} \nabla_\NV)^k$ can be written as a composition with $\iota^$ of a linear combination of iterated gradients $\nabla^j_\NV$ for $j \le k$ and polynomials in the curvature quantities $\iota^ \nabla_\NV^j(\rho) \in C^\infty(M)$ for $j \le k-1$. This follows by an easy induction using $\nabla_\NV (\sigma) = \|\NV\|^2$. In particular, we obtain the following low-order formulas. If $\SC(g,\sigma)=1$, then it holds \begin{equation} \iota^ \partial_s \eta^* = \iota^* \nabla_\NV \quad \mbox{and} \quad \iota^* \partial_s^2 \eta^* = \iota^* (\nabla_\NV^2 + 2 \rho \nabla_\NV). \end{equation} We calculate \iota^ \partial_s^2 \eta^* = \iota^* (\|\NV\|^{-2} \nabla_\NV)^2 = \iota^* \nabla_\NV (1 + 2 \sigma \rho) \nabla_\NV = \iota^* (\nabla_\NV^2 + 2 \nabla_\NV(\sigma) \rho \nabla_\NV) using $\|\NV\|^2 = 1-2\sigma \rho$. Now $\iota^* \nabla_\NV (\sigma) = \iota^* (\|\NV\|^2) = 1$ implies the second identity. These formulas can easily be inverted to express iterated normal gradients in terms of iterated normal derivatives with respect to $s$. In particular, we obtain the following identities. If $\SC(g,\sigma)=1$, then it holds \begin{equation} \iota^ \nabla_\NV = \iota^* \partial_s \eta^* \quad \mbox{and} \quad \iota^* \nabla_\NV^2 = (\iota^* \partial_s^2 - 2 \rho_0 \iota^* \partial_s ) \eta^. \end{equation} The above discussion obviously generalizes to the case that $\sigma$ only satisfies the condition $\SCY$ with a non-trivial remainder. §.§ Expansions of the metric. Model cases We start by discussing the normal forms of a given metric $g$ in geodesic normal coordinates and in adapted coordinates (as defined in Section <ref>). The formulas for these normal forms contain respective families $h_r$ and $h_s$ of metrics on $M$. In order to simplify notation, we shall use the same notation for the coefficients of their Taylor series in $r$ and $s$. It always will be clear from the context which coefficients are meant. We derive formulas for the first few Taylor coefficients of $h_r$ and $h_s$. A series of geometrically intuitive examples follow the discussion. $h_{(k)}$ coefficients of $h_r$ or $h_s$ We expand the family $h_r$ in the normal form $dr^2 + h_r$ of $g$ in geodesic normal coordinates as h_r = h + h_{(1)} r + h_{(2)} r^2 + \cdots. The coefficients $h_{(k)}$ can be expressed in terms of the curvature of the metric $g$, its covariant derivatives, and the second fundamental form $L$. The expansion starts \begin{equation}\label{h-geodesic} (h_r)_{ij} = h_{ij} + 2L_{ij} r + ((L^2)_{ij} - R_{0ij0}) r^2 + (h_{(3)})_{ij} r^3 + \cdots \end{equation} \begin{equation}\label{h-cubic} 3 (h_{(3)})_{ij} = - \nabla_{\partial_r} (R)_{0ij0} - 2 L_i^k R_{0jk0} - 2 L_j^k R_{0ik0}. \end{equation} Here $(L^2)_{ij} = L_{ik} L^k_j = L_{ik} L_{js} h^{ks}$. Next, assume that $\sigma$ satisfies $\SCY$. We expand the family $h_s$ in the normal form (<ref>) as h_s = h + h_{(1)} s + h_{(2)} s^2 + \cdots. The coefficients $h_{(k)}$ can be expressed in terms of the curvature of the metric $g$, its covariant derivatives, and the second fundamental form $L$. The expansion starts with \begin{equation}\label{h-adapted} (h_s)_{ij} = h_{ij} + 2 L_{ij} s + ((L \lo)_{ij} - R_{0ij0}) s^2 + (h_{(3)})_{ij} s^3 + \cdots \end{equation} \begin{align}\label{h-adapted-cubic} 3 (h_{(3)})_{ij} & = - \nabla_{\partial_s} (R)_{0ij0} - 2 \lo_i^k R_{0jk0} - 2 \lo_j^k R_{0ik0} \notag \\ & + \Hess_{ij}(H) - H R_{0ij0} - 3 H (L \lo)_{ij} + 2 L_{ij} \rho_0'. \end{align} Here $(L \lo)_{ij} = L_{ik} \lo_{js} h^{ks}$ and $\rho_0' = \Rho_{00} + \|\lo\|^2/(n-1)$. The notation $\rho_0'$ will be justified in Lemma <ref>. We recall the standard formulas R^s_{ijk} = \Gamma_{jk}^l \Gamma_{il}^s - \Gamma_{ik}^l \Gamma_{jl}^s + \partial_i (\Gamma_{jk}^s) - \partial_j (\Gamma_{ik}^s), \quad R_{ijkl} = R_{ijk}^s g_{sl} and $\Gamma_{ij}^k$ Christoffel symbol \Gamma_{ij}^m = \frac{1}{2} g^{km} (\partial_i (g_{jk}) + \partial_j (g_{ik}) - \partial_k (g_{ij})) for the components of the curvature tensor of a metric $g$.[As usual, we sum over repeated indices and denote derivatives by a lower index.] For the metric $g = dr^2 + h_r$, we find $\Gamma_{ij}^0 = -\frac{1}{2} \partial_r(h_r)_{ij}$. Hence $L_{ij} = - \Gamma_{ij}^0 = \frac{1}{2} (h_{(1)})_{ij}$ on $M$. This proves $h_{(1)} = 2L$. In order to verify the formula for $h_{(2)}$ in (<ref>), we calculate the components $R_{0ij0}$ of the curvature tensor of the metric $dr^2 + h_r$. We decompose $R_{0jk}^0$ as R_{0jk}^0 = \left( \Gamma_{jk}^l \Gamma_{0l}^0 - \Gamma_{0k}^l \Gamma_{jl}^0 + \partial_r (\Gamma_{jk}^0) - \partial_j (\Gamma_{0k}^0) \right) + \Gamma_{jk}^0 \Gamma_{00}^0 - \Gamma_{0k}^0 \Gamma_{j0}^0, where the summations run only over tangential indices. Now, for the metric $g = dr^2 + h_r$, we find the Christoffel symbols \Gamma_{ij}^0 = - \frac{1}{2} g_{ij}', \quad \Gamma_{0j}^0 = 0, \quad \Gamma_{00}^0 = 0, \quad \Gamma_{0k}^l = \frac{1}{2} g^{rl} g_{kr}', where $^\prime$ denotes $\partial_r$. It follows that \begin{equation}\label{curv-geo} R_{0jk}^0 = \frac{1}{4} g^{rl} g_{kr}' g_{jl}' - \frac{1}{2} g_{jk}''. \end{equation} We evaluate this formula at $r=0$. Using $g_{jk}' = 2 L_{jk}$, we find R_{0jk0} = R_{0jk}^0 = h^{rl} L_{kr} L_{jl} - (h_{(2)})_{jk} = (L^2)_{jk} - (h_{(2)})_{jk}. This implies the formula for $h_{(2)}$ in (<ref>). Next, we prove the formula for the cubic term. First, we note that \nabla_{\partial_r}(R)_{0jk0} = \partial_r (R_{0jk0}) - R(\partial_r,\nabla_{\partial_r}(\partial_j),\partial_k,\partial_r) - R(\partial_r,\partial_j,\nabla_{\partial_r}(\partial_k),\partial_r) using $\nabla_{\partial_r}(\partial_r) = 0$. Now (<ref>) implies \begin{align} \partial_r (R_{0jk0}) = \partial_r (R_{0jk}^0 g_{00}) = \partial_r (R_{0jk}^0) = \frac{1}{4} (g^{rl})' g_{kr}' g_{jl}' + \frac{1}{4} g^{rl} g_{kr}'' g_{jl}' + \frac{1}{4} g^{rl} g_{kr}' g_{jl}''- \frac{1}{2} g_{jk}'''. \end{align} Evaluation of this formula for $r=0$ yields \begin{align} \partial_r (R_{0jk0}) & = - 2 L^{rl} L_{kr} L_{jl} + h^{rl} (h_{(2)})_{kr} L_{jl} + h^{rl} L_{kr} (h_{(2)})_{jl} - 3 (h_{(3)})_{jk} \\ & = - L_j^r R_{0kr0} - L_k^l R_{0jl0} - 3 (h_{(3)})_{jk}. \end{align} The two remaining terms in the formula for $\partial_r (R_{0jk0})$ for $r=0$ are - R_{0jl0} L_k^l - R_{0lk0} L_j^l. Thus, we find \nabla_{\partial_r}(R)_{0jk0} = -2 L_j^r R_{0kr0} - 2 L_k^l R_{0jl0} - 3 (h_{(3)})_{jk}. This proves (<ref>). Similarly, the formula for $h_s$ in (<ref>) and (<ref>) follows from a calculation of the Christoffel symbols and the components $R_{0ij0} = R_{0ij}^0 g_{00}$ of the curvature tensor of the metric \eta^(g) = \eta^(\|\NV\|^2)^{-1} ds^2 + h_s = a^{-1} ds^2 + h_s with $a = \eta^(\|\NV\|^2)$. In order to simplify the notation, we shall write $g$ instead of $\eta^(g)$ and $\rho$ instead of $\eta^(\rho)$. For the above metric, we find the Christoffel symbols[As usual the prime means derivative in $s$.] \Gamma_{ij}^0 = - \frac{1}{2} g^{00} g_{ij}', \quad \Gamma_{0j}^0 = \frac{1}{2} g^{00} \partial_j (g_{00}), \quad \Gamma_{00}^0 = \frac{1}{2} g^{00} g_{00}', \quad \Gamma_{0k}^l = \frac{1}{2} g^{rl} g_{kr}'. Here $g^{00} = a = 1- 2s\rho$ (by assumption) and $g_{00} = a^{-1} = 1 + 2s\rho + \cdots$. In particular, $g^{00}=1$, $g_{00}' = -2H$ and $(g^{00})' = 2H$ on $M$. Here we used that $\rho = -H$ on $M$ (Lemma <ref>). Hence $L_{ij} = - \Gamma_{ij}^0 = \frac{1}{2} (h_{(1)})_{ij}$. The components $R_{0jk0}$ of the curvature tensor are given by R_{0jk0} = R_{0jk}^0 g_{00} \begin{align}\label{R-adapted} R_{0jk}^0 & = \frac{1}{2} \Gamma_{jk}^l g^{00} \partial_l (g_{00}) + \frac{1}{4} g^{00} g^{rl} g_{kr}' g_{jl}' - \frac{1}{2} ((g^{00})' g_{jk}' + g^{00} g_{jk}'') \notag\\ & - \frac{1}{2} ( \partial_j (g^{00}) \partial_k(g_{00}) + g^{00} \partial^2_{kj}(g_{00})) \notag \\ & - \frac{1}{4} (g^{00})^2 g_{jk}' g_{00}' - \frac{1}{4} (g^{00})^2 \partial_j(g_{00}) \partial_k(g_{00}). \end{align} Evaluation of this formula for $s=0$ gives R_{0jk0} = h^{rl} L_{kr} L_{jl} - (h_{(2)})_{jk} - 2H L_{jk} + H L_{jk} = (L^2)_{jk} - H L_{jk} - (h_{(2)})_{jk}. But $L^2 - H L = L (L - H h) = L \lo$. Hence $R_{0jk0} = (L \lo)_{jk} - (h_{(2)})_{jk}$. This implies the formula for $h_{(2)}$ in (<ref>). Finally, we prove the formula for the cubic term. Here we utilize the expansions g_{00} = (1-2 \rho s)^{-1} = 1 + 2\rho s + 4 \rho^2 s^2 + \cdots = 1 + 2\rho_0 s + (2\rho_0'+4\rho_0^2) s^2 + \cdots g^{00} = 1 -2\rho s = 1- 2\rho_0 s - 2\rho_0' s^2 + \cdots. We proceed as above and calculate $\nabla_{\partial_s} (R)_{0jk0}$ for $s=0$. In the present case, it holds \begin{align} \nabla_{\partial_s}(R)_{0jk0} & = \partial_s (R_{0jk0}) - R(\partial_s,\nabla_{\partial_s}(\partial_j),\partial_k,\partial_s) - R(\partial_s,\partial_j,\nabla_{\partial_s}(\partial_k),\partial_s) \\ & - R(\nabla_{\partial_s}(\partial_s),\partial_j,\partial_k,\partial_s) - R(\partial_s,\partial_j,\partial_k,\nabla_{\partial_s}(\partial_s)); \end{align} note that $\nabla_{\partial_s}(\partial_s)$ does not vanish in general. First, we calculate the term $\partial_s (R_{0jk0})$ for $s=0$ using (<ref>). We \begin{align} \partial_s (R_{0jk0}) & = \partial_s (R_{0jk}^0) - 2 H R_{0jk}^0 \\ & = -\Gamma_{jk}^l \partial_l(H) + \frac{1}{2} H h^{rl} g_{kr}' g_{jl}' + \frac{1}{4} (g^{rl})' g_{kr}' g_{jl}' + \frac{1}{4} h^{rl} g_{kr}'' g_{jl}' + \frac{1}{4} h^{rl} g_{kr}' g_{jl}'' \\ & + 4 \rho_0' L_{jk} - 2 H g_{jk}'' - \frac{1}{2} g_{jk}''' + \partial^2_{kj}(H) \\ & + \frac{1}{2} H g_{jk}'' - \rho_0' g_{jk}' - 2 H R_{0jk0} \end{align} using the expansions of $g_{00}$ and $g^{00}$. We further simplify that sum by using the known formula for the first derivative of $h_s$ and $\Hess_{jk} = \partial^2_{jk} - \Gamma_{jk}^l \partial_l$. Then \begin{align} \partial_s (R_{0jk0}) & = \Hess_{jk}(H) + 2 H L_k^l L_{jl} - 2 L^{rl} L_{kr} L_{jl} + L_j^r (h_{(2)})_{kr} + L_k^l (h_{(2)})_{jl} \\ & + 4 L_{jk} \rho_0' - 4 H (h_{(2)})_{jk} - 3 (h_{(3)})_{jk} + H (h_{(2)})_{jk} - 2 L_{jk} \rho_0' \\ & = \Hess_{jk}(H) + 2 H (L^2)_{jk} - 2 (L^3)_{jk} + L_j^r (h_{(2)})_{kr} + L_k^l (h_{(2)})_{jl} \\ & -3 H (h_{(2)})_{jk} + 2 L_{jk} \rho_0' - 3 (h_{(3)})_{jk} - 2 H R_{0jk0} \end{align} on $M$. The four remaining contributions to $\nabla_{\partial_s}(R)_{0jk0}$ for $s=0$ equal \begin{align} & - \Gamma_{0j}^l R_{0lk0} - \Gamma_{0k}^l R_{0jl0} - \Gamma_{00}^0 R_{0jk0} - \Gamma_{00}^0 R_{0jk0} \\ & = - \frac{1}{2} g^{rl} g_{jr}' R_{0lk0} - \frac{1}{2} g^{rl} g_{kr}' R_{0jl0} - g^{00} g_{00}' R_{0jk0} \\ & = - L_j^l R_{0lk0} - L_k^l R_{0jl0} + 2 H R_{0jk0} \\ & = -\lo_j^l R_{0lk0} - \lo_k^l R_{0jl0}. \end{align} \begin{align} \nabla_{\partial_s}(R)_{0jk0} & = \Hess_{jk}(H) + 2 H (L^2)_{jk} - 2 (L^3)_{jk} \\ & + L_j^r ((L \lo)_{kr} - R_{0kr0}) + L_k^l ((L \lo)_{jl} - R_{0jl0}) - 3 H ((L \lo)_{jk} - R_{0jk0}) \\ & - \lo_j^l R_{0lk0} - \lo_k^l R_{0jl0} + 2 L_{jk} \rho_0' - 3 (h_{(3)})_{jk} - 2 H R_{0jk0} \\ & = \Hess_{jk}(H) - L_j^r R_{0kr0} - L_k^l R_{0jl0} - 3 H (L \lo)_{jk} + 3 H R_{0jk0} \\ & -\lo_j^l R_{0lk0} - \lo_k^l R_{0jl0} + 2 L_{jk} \rho_0' - 3 (h_{(3)}) _{jk} - 2 H R_{0jk0} \\ & = \Hess_{jk}(H) - 2 \lo_j^r R_{0kr0} - 2 \lo_k^r R_{0jr0} - 3 H (L \lo)_{jk} - H R_{0jk0} \\ & + 2 L_{jk} \rho_0' - 3 (h_{(3)}) _{jk} \end{align} by the formula for $h_{(2)}$. This proves (<ref>). Formula (<ref>) was given in <cit.> and the formula (<ref>) for the cubic coefficient was displayed in <cit.>.[In our conventions, $R$ and $L$ have opposite signs as in <cit.>.] The proof of Proposition <ref> shows that $h_{(2)}$ depends only on $\rho_0$. Similarly, $h_{(3)}$ only depends on $\rho_0$ and $\rho_0'$. More generally, it is easy to see that $h_{(k)}$ only depends on $\partial_s^j(\rho)\|_0$ for $j \le k-2$. Formula (<ref>) for the cubic term $h_{(3)}$ of the expansion of $g$ in adapted coordinates yields \begin{equation}\label{h12-trace-n} \tr (h_{(1)}) = 2 n H, \quad \tr(h_{(2)}) = \|\lo\|^2 - \Ric_{00} \end{equation} \begin{equation}\label{h3-trace-n} 3 \tr (h_{(3)}) = - \nabla_{\partial_s}(\Ric)_{00} - 4 \lo^{ik} R_{0ik0} + \Delta H - H \Ric_{00} - 3 H\|\lo\|^2 + 2 n H \rho_0'. \end{equation} If $g_+ = r^{-2} g = r^{-2} (dr^2 + h_r)$ is a Poincaré-Einstein metric in normal form relative to $h = h_0$, then $h_{(1)}=0$ and $h_{(2)} = -\Rho^h$. Comparing this result with (<ref>), implies that $R^g_{0ij0} = \Rho^h_{ij}$. Hence $\Ric^g_{00} = \J^h$. We illustrate the above results in some simple model cases. Let $S^n \subset \R^{n+1}$ be defined by $\|x\|=1$. Let $g_0$ be the Euclidean metric on $\R^{n+1}$. Then the function $\sigma = (1-\|x\|^2)/2$ yields the Poincaré metric \sigma^{-2} g_0 = \frac{4}{(1-\|x\|^2)^2} \sum_{i=1}^{n+1} dx_i^2 on the unit ball $\|x\|<1$. Now $\NV = \grad(\sigma) = - \sum_i x_i \partial_i$. Hence the normalized gradient field $\mathfrak{X} = \NV/\|\NV\|^2$ is given by $\mathfrak{X} = -\frac{1}{\|x\|^2} \sum_i x_i \partial_i$. It follows that the map \eta: (-1/2,1/2) \times S^n \ni (s,x) \mapsto \sqrt{1-2s} x \in \R^{n+1} defines the adapted coordinates. Note that $\eta^(\sigma)=s$. Hence we obtain \eta^(g_0) = \frac{1}{1-2s} ds^2 + (1-2s) h, where the round metric $h$ on $S^n$ is induced by $g$. In particular, $h_s = h - 2sh$, i.e., $h_{(1)} = -2h$ and $h_{(2)}=0$ The coefficient $h_{(1)}$ is to be interpreted as $2L$ being defined by the unit normal field $-\partial_r$. The vanishing of the quadratic and the cubic term is confirmed by the general formulas. Note also that $\rho = 1$. It follows that $v(s)=(1-2s)^{\frac{n-1}{2}}$ and $\mathring{v}(s) = (1-2s)^{\frac{n}{2}}$. These results confirm the relation (<ref>) using $a=1-2s$. Moreover, the identity (<ref>) in Conjecture <ref> is trivially satisfied. Let $S^n \subset S^{n+1}$ be an equatorial subsphere. Let $g$ be the round metric on $S^{n+1}$. Then the height-function $\sigma = \He \in C^\infty(S^{n+1})$ defines the metric $\He^{-2} g$ on both connected components of the complement of the zero locus of $\He$. It is isometric to the Poincaré-metric on the unit ball (see Section <ref>). The map \eta: (-1,1) \times S^n \ni (s,x) \mapsto (\sqrt{1-s^2}x,s) \in S^{n+1} defines the adapted coordinates. Note that $\eta^(\sigma)=s$. Hence we obtain \eta^(g) = \frac{1}{1-s^2} ds^2 + (1-s^2) h, where the round metric $h$ on $S^n$ is induced by $g$. In particular, $h_{(1)} = 0$ and $h_{(2)}= -h$. The coefficient $h_{(1)}$ vanishes since $L=0$ and the coefficient $h_{(2)}$ is to be interpreted as $-R_{0ij0}$. The vanishing of the cubic term is confirmed by the general formula. Note that $\He$ is an eigenfunction of the Laplacian on $S^{n+1}$ with eigenvalue $-(n+1)$ and $\J = \frac{n+1}{2}$. Hence $\rho=\frac{1}{2} \He$ and $\eta^(\rho) = \frac{1}{2} s$. Finally, we have $v(s)=(1-s^2)^{\frac{n-1}{2}}$ and $\mathring{v}(s) = (1-s^2)^{\frac{n}{2}}$. These results confirm the relation (<ref>) using $a = 1-s^2$. Moreover, the identity (<ref>) in Conjecture <ref> reduces to the trivial relation \binom{\frac{n-1}{2}}{\frac{n+1}{4}} (-1)^{\frac{n+1}{4}} + \binom{\frac{n-1}{2}}{\frac{n-3}{4}} (-1)^{\frac{n-3}{4}} = 0. In the above two examples, either the curvature of the background metric or the second fundamental form vanishes. We finish this section with the discussion of a model case with non-trivial curvature and non-trivial second fundamental form. Let $\mathbb{H}^{n+1}$ be the upper half-space with the hyperbolic metric $g_+ = r^{-2} (dr^2 + dx^2)$. For $c>0$, we let $X = \{r \ge c\}$ with boundary $M = \{ r=c \}$ and background metric $g=g_+$. The metric $g$ restricts to $h = c^{-2} dx^2$ on $M$. The defining function $\sigma = 1 - c/r$ is smooth up to the boundary $M$. It solves the singular Yamabe problem since $\sigma^{-2} g_+ = (r-c)^{-2} (dr^2 + dx^2)$ has scalar curvature $-n(n+1)$. The map \eta: (0,1) \times \R^n \ni (s,x) \mapsto \left(\frac{c}{1-s},x\right) \in X defines the adapted coordinates. In fact, $\eta^(\sigma) = s$. We obtain the normal form \eta^(g) = \frac{1}{(1-s)^2} ds^2 + \frac{(1-s)^2}{c^2} dx^2 of $g$ in adapted coordinates. In particular, $h_{(1)} = -2h$, $h_{(2)} = h$ and $h_{(3)} = 0$. These results fit with the general formulas in Proposition <ref> since $L=-h$ and $R_{0ij0} = -h_{ij}$. The vanishing of $h_{(3)}$ follows using $\lo = 0$, $H = -1$ and $\Rho_{00} = - 1/2$. A calculation using the formula $\Delta_{g_+} = r^2 \partial_r^2 - (n-1) r \partial_r + \Delta_{\R^n}$ for the Laplacian of $g_+$ yields \rho = \frac{1}{2} + \frac{c}{2r}. Hence $\eta^(\rho) = 1- s/2$. Note that $\NV = \grad_g(\sigma) = -c \partial_r$, $\|\NV\|^2 = r^{-2} c^2$ and $\eta^(\|\NV\|^2) = (1-s)^2 \stackrel{!}{=} 1- 2s\eta^(\rho)$. Finally, we have $v(s) = (1-s)^{n-1}$ and $\mathring{v}(s) = (1-s)^n$. By $\J = - \frac{n+1}{2}$, the identity (<ref>) in Conjecture <ref> reduces to the trivial relation \binom{n-1}{\frac{n+1}{2}} (-1)^{\frac{n+1}{2}} - \binom{n-1}{\frac{n-3}{2}} (-1)^{\frac{n-3}{2}} = 0. Finally, these results confirm the relation (<ref>) using $a=(1-s)^2$. §.§ Approximate solutions of the singular Yamabe problem. The residue formula In the present section, we determine the first few terms in the expansion $\sigma_F = r + \sigma_{(2)} r^2 + \dots$ of a solution of the equation \begin{equation}\label{start-sol} \SC(g,\sigma_F) = 1+O(r^{n+1}). \end{equation} We also describe the obstruction $\B_n$ in terms of a formal residue of the supercritical term $\sigma_{n+2}$. By (<ref>), equation (<ref>) takes the form \begin{align} & \partial_r(\sigma_F)^2 + h_r^{ij} \partial_i (\sigma_F) \partial_j (\sigma_F) \notag \\ & - \frac{2}{n+1} \sigma_F \left (\partial_r^2 (\sigma_F) + \frac{1}{2} \tr (h_r^{-1} h_r') \partial_r (\sigma_F) + \Delta_{h_r} (\sigma_F) + \bar{\J} \sigma_F \right) = 1+ O(r^{n+1}). \end{align} Here and in the following, we use the bar notation for curvature quantities of $g$. We shall formulate the results in terms of the volume coefficients $u_k$ of the metric $g$ in geodesic normal coordinates (see (<ref>)) using \begin{equation}\label{volume-geodesic} \frac{u'(r)}{u(r)} = \frac{1}{2} \tr (h_r^{-1} h_r'). \end{equation} The following results describe the first three Taylor coefficients of a solution $\sigma_F$ of the singular Yamabe problem in general dimensions. $\sigma_{(2)}$, $\sigma_{(3)}$, $\sigma_{(4)}$ It holds \begin{align} \sigma_{(2)} & = \frac{1}{2n} u_1, \\ \sigma_{(3)} & = \frac{2}{3(n-1)} u_2 - \frac{1}{3n} u_1^2 + \frac{1}{3(n-1)} \bar{\J} \end{align} \begin{align}\label{sigma4-g} \sigma_{(4)} & = \frac{3}{4(n-2)} u_3 - \frac{9n^2-20n+7}{12n(n-1)(n-2)} u_1 u_2 + \frac{6n^2-11n+1}{24n^2(n-2)} u_1^3 \notag \\ & + \frac{2n-1}{6n(n-1)(n-2)} u_1 \bar{\J} + \frac{1}{4(n-2)} \bar{\J}' + \frac{1}{4(n-2)} \Delta (\sigma_{(2)}). \end{align} Alternatively, the above formulas can be derived from the description of the solution $\sigma$ in <cit.>. In fact, these formulas describe expansions of $\sigma$ into power series of any defining function with coefficients that live on the background space $X$. The calculation then requires expanding these coefficients into power series of the distance function. We omit the details. The volume coefficients $u_j$ may be expressed in terms of the Taylor coefficients of $h_r$. Such relations follow from (<ref>) by Taylor expansion in $r$ and resolving the resulting relations for $u_j$. We find \begin{align} u_1 & = \frac{1}{2} \tr (h_{(1)}), \\ u_2 & = \frac{1}{8} (\tr (h_{(1)})^2 + 4 \tr (h_{(2)}) - 2 \tr (h_{(1)}^2)) \end{align} \begin{align} u_3 & = \frac{1}{48} (\tr (h_{(1)})^3 + 12 \tr (h_{(1)}) \tr (h_{(2)}) + 24 \tr (h_{(3)}) - 6 \tr(h_{(1)}) \tr (h_{(1)}^2) \\ & - 24 \tr(h_{(1)} h_{(2)}) + 8 \tr (h_{(1)}^3)). \end{align} These formulas are valid in general dimensions. The expressions for $h_{(j)}$ (for $j \le 3$) in Proposition <ref> imply \begin{align} u_1 & = n H, \\ 2 u_2 & = -\overline{\Ric}_{00} - \|L\|^2 + n^2 H^2 = \overline{\Ric}_{00} + \scal - \overline{\scal} \end{align} (by the Gauss equation) or equivalently \begin{equation}\label{u2} 2 u_2 = -\overline{\Ric}_{00} - \|\lo\|^2 + n(n-1) H^2. \end{equation} Moreover, we find \begin{equation}\label{u3} 6 u_3 = - \bar{\nabla}_0 (\overline{\Ric})_{00} + 2 (L,\bar{\G}) - 3 n H \overline{\Ric}_{00} + 2 \tr(L^3) - 3n H \|L\|^2 + n^3 H^3, \end{equation} where $\bar{\G}_{ij} \st \bar{R}_{0ij0}$, or equivalently $\bar{\G}$ \begin{align} 6 u_3 & = -\bar{\nabla}_0 (\overline{\Ric})_{00} + 2 (\lo,\bar{\G}) - (3n-2) H \overline{\Ric}_{00} \\ & + 2 \tr (\lo^3) - 3(n-2) H \|\lo\|^2 + n(n-1)(n-2) H^3. \end{align} For these formulas for $u_1,u_2,u_3$, see also <cit.>. Note that \begin{equation}\label{det-2} - \|L\|^2 + n^2 H^2 = \sum_{ij} \begin{vmatrix} L_{ii} & L_{ij} \\ L_{ji} & L_{jj} \end{vmatrix} = 2 \sigma_2(L) \end{equation} and $\sigma_k(L)$ elementary symmetric polynomial \begin{equation}\label{det-3} 2 \tr(L^3) - 3n H \|L\|^2 + n^3 H^3 = \sum_{ijk} \begin{vmatrix} L_{ii} & L_{ij} & L_{ik} \\ L_{ji} & L_{jj} & L_{jk} \\ L_{ki} & L_{kj} & L_{kk} \end{vmatrix} = 6 \sigma_3(L) \end{equation} in an orthonormal basis. Here $\sigma_k(L)$ denotes the $k$-th elementary symmetric polynomial in the eigenvalues of the shape operator. The identities (<ref>) and (<ref>) are special cases of Newton's identities relating elementary symmetric polynomials to power series. In these terms, we have \begin{align} 2 u_2 & = -\overline{\Ric}_{00} + 2 \sigma_2(L) \\ 6 u_3 & = -\bar{\nabla}_0 (\overline{\Ric})_{00} + 2 (\lo,\bar{\G}) - (3n-2) H \overline{\Ric}_{00} + 6 \sigma_3(L). \end{align} Note also that versions of the Gauss identity express the quantities $\sigma_2(L)$ and $\sigma_3(L)$ in terms of the curvatures of $g$ and of the induced metric on $M$ <cit.>. Finally, we note that (<ref>) vanishes in $n=2$ - this identity is equivalent to $\tr(\lo^3) = 0$ in $n=2$. The above results imply It holds $2 \sigma_{(2)} = H$ and \begin{equation}\label{sigma3} 6 \sigma_{(3)} = -2 \bar{\Rho}_{00} - \frac{2}{n-1} \|\lo\|^2 = 2( \J - \iota^ \bar{\J}) - \frac{1}{n-1} \|\lo\|^2 - n H^2 \end{equation} for $n \ge 2$. The second equality follows by combining the first equality with (<ref>). The results of Lemma <ref> are contained in <cit.>. See also <cit.>.[These references use a different convention for $L$ and $H$.] The above formulas for $v_j$ are equivalent to the corresponding formulas in <cit.>. We also refer to <cit.> for the corresponding results in higher codimensions. However, the methods of proof in these references are different. For $n=1$, the above results easily imply \begin{equation}\label{B1=0} \B_1 = (r^{-2} (\SC(S_2)-1))\|_0 = -2u_2 - \bar{\J}_0 = \overline{\Ric}_{00} - \bar{\J}_0 = 0 \end{equation} using $\lo=0$ (see also Remark <ref>). This result is well-known <cit.>. It also is of interest to explicate the above formulas for flat backgrounds. In fact, it follows from the identity \begin{equation}\label{volume-flat} u(r) = \det (\id + r L) \end{equation} (see <cit.>) for a flat background that the formulas for $\sigma_{(k)}$ (for $k \le 4$) can be expressed in terms of $L$. Newton's identities imply \begin{align} u_1 & = n H, \\ u_2 & = \frac{1}{2} (n (n-1) H^2 - \|\lo\|^2), \\ u_3 & = \frac{1}{6} ( H^3 n(n-1)(n-2) - 3 (n-2) H \|\lo\|^2 + 2 \tr(\lo^3)) \end{align} and a direct calculation yields the following result. The expansion of a solution of the Yamabe problem for $M^n \hookrightarrow \R^{n+1}$ has the form \sigma_F = r + \frac{r^2}{2} H - \frac{r^3}{3(n-1)} \|\lo\|^2 + r^4 \sigma_{(4)} + \cdots with the coefficient \begin{align} \sigma_{(4)} & = \frac{1}{24(n-2)} \left(6 \tr(\lo^3) + \frac{7n-11}{n-1} H \|\lo\|^2 + 3 \Delta(H)\right). \end{align} Note that the coefficients in formula (<ref>) for $\sigma_{(4)}$ have a simple pole in $n=2$. The following result calculates the formal residue at $n=2$. It holds \res_{n=2}(\sigma_{(4)}) = \frac{3}{4} u_3 + \frac{1}{32} u_1^3 - \frac{1}{8} u_1 u_2 + \frac{1}{4} u_1 \bar{\J} + \frac{1}{4} \bar{\J}' + \frac{1}{4} \Delta (\sigma_{(2)}). For a flat background, we obtain \res_{n=2}(\sigma_{(4)}) = \frac{1}{8} (H \|\lo\|^2 + \Delta (H)). Note that for a flat background it holds $u_3 = 0$ in $n=2$. We also recall that $\tr(\lo^3) = 0$ for $n=2$. Alternatively, one can use formula <cit.> for $\sigma_{(4)}$ to confirm this residue formula for general backgrounds. Corollary <ref> and the following result are special cases of the residue formula in Lemma <ref>. It holds \res_{n=2}(\sigma_{(4)}) = - \frac{3}{8} \B_2. We recall that the obstruction $\B_2$ is defined by \B_2 = (r^{-3} (\SC(S_3)-1))\|_0, where $S_3= r + \sigma_{(2)} r^2 + \sigma_{(3)} r^3$. A calculation yields \begin{equation}\label{B2-new-g} \B_2 = - 2 u_3 - \frac{1}{12} u_1^3 + \frac{1}{3} u_1 u_2 - \frac{2}{3} u_1\bar{\J} - \frac{2}{3} \bar{\J}' - \frac{2}{3} \Delta (\sigma_{(2)}). \end{equation} We omit the details. We recall that the term $u_3$ vanishes in $n=2$ in the flat case but not in the curved case. In Section <ref>, we shall derive another formula for $\B_2$ from Theorem <ref>. Although the equivalence of both formulas is non-trivial, we leave the check of consistency to the reader. Similarly, we may either directly evaluate the definition $\B_3 = (r^{-4} (\SC(S_4)-1))\|_0$ or use the residue formula (<ref>) to derive a formula for $\B_3$ from the residue of $\sigma_{(5)}$ at $n=3$. The results read as follows. It holds \begin{align}\label{B3-start} \B_3 & = - 2u_4 + \frac{1}{2} u_1 u_3 + \frac{1}{3} u_2^2 - \frac{7}{18} u_1^2 u_2 + \frac{2}{27} u_1^4 - \frac{1}{3} \bar{\J} u_2 - \frac{5}{12} \bar{\J}' u_1 - \frac{1}{4} \bar{\J}'' \notag \\ & - \frac{1}{2} \Delta (\sigma_{(3)}) - \frac{1}{3} u_1 \Delta(\sigma_{(2)}) - \frac{1}{2} \Delta' (\sigma_{(2)}) + \|d\sigma_{(2)}\|^2, \end{align} \begin{equation} 6 \sigma_{(2)} = u_1 \quad \mbox{and} \quad 9 \sigma_{(3)} = 3 u_2 - u_1^2 + \frac{3}{2} \bar{\J} \end{equation} and $\Delta' = (d/dt)\|_0 (\Delta_{h+2sL})$ (see (<ref>)). For a flat background, it holds $\bar{\J}=0$, the term $u_4$ vanishes, and we obtain \begin{equation}\label{B3-flat-algo} 12 \B_3 = \Delta (\|\lo\|^2) + 6 H \tr (\lo^3) + \|\lo\|^4 + 3 \|dH\|^2 - 6 H \Delta(H) - 3 \Delta' (H). \end{equation} The above formula for $\B_3$ also follows from the following result through the residue formula \begin{equation}\label{res-5} \res_{n=3}(\sigma_{(5)}) = - \frac{2}{5} \B_3. \end{equation} The evaluation of that formula rests on the following result for the coefficient $\sigma_{(5)}$. In general dimensions, it holds \begin{align} \sigma_{(5)} & = - \frac{n+1}{10(n-3)} \|d\sigma_{(2)}\|^2 \\ & + \frac{1}{5(n-3)} \Delta'(\sigma_{(2)}) + \frac{1}{5(n-3)} \Delta(\sigma_{(3)}) + \frac{3n-1}{20(n-3)(n-2)n} \Delta(\sigma_{(2)}) u_1 \\ & + \frac{1}{10(n-3)} \bar{\J}'' + \frac{n-1}{4(n-3)(n-2)n} \bar{\J}' u_1 + \frac{2(3n-5)}{15(n-3)(n-1)^2} \bar{\J} u_2 \\ & - \frac{4n-3}{20 (n-2)(n-1)n} \bar{\J} u_1^2 + \frac{1}{30(n-1)^2} \bar{\J}^2 \\ & + \frac{48n^4-247n^3+387n^2-179n+3}{60(n-3)(n-2)(n-1)n^2} u_1^2 u_2 - \frac{2(3n^2-11n+10)}{15(n-3)(n-1)^2} u_2^2 \\ & - \frac{24n^4-110n^3+133n^2-24n-3}{120 (n-3)(n-2)n^3} u_1^4 - \frac{16n^2-53n+27}{20(n-3)(n-2)n} u_1 u_3 + \frac{4}{5(n-3)} u_4. \end{align} We omit the details of the proof. An alternative formula for $\sigma_{(5)}$ is given in <cit.>. Here the same comments as after Lemma <ref> apply. Lemma <ref> implies \res_{n=2}(\sigma_{(5)}) = - \frac{1}{2} u_1 \res_{n=2} (\sigma_{(4)}) = \frac{3}{4} H \B_2. This relation extends the identities \begin{align} \res_{n=1}(\sigma_{(4)}) & = - \frac{1}{2} u_1 \res_{n=1}(\sigma_{(3)}), \\ \res_{n=1}(\sigma_{(3)}) & = - \frac{1}{2} u_1 \res_{n=1}(\sigma_{(2)}) = 0. \end{align} More generally, we conjecture the factorization identities \res_{n=k-3}(\sigma_{(k)}) = - \frac{1}{2} u_1 \res_{n=k-3} (\sigma_{(k-1)}) for $k \ge 4$. Combining (<ref>) with the variation formula $\Delta'(u) = -2 (\Hess(u),L) - 3 (du,dH)$ (see (<ref>)) shows that 12 \B_3 = \Delta (\|\lo\|^2) + 12 \|dH\|^2 + 6 (\Hess(H),\lo) + \|\lo\|^4 + 6 H \tr(\lo^3). In Section <ref>, we shall alternatively derive that result from Theorem <ref>. A direct proof that, for conformally flat backgrounds, Lemma <ref> is equivalent to Lemma <ref> will be given in a separate work. Finally, we establish the residue formula for the singular Yamabe obstruction in full generality. It holds \begin{equation}\label{res-f} \res_{n=k-2}(\sigma_{(k)}) =-\frac{k-1}{2k} \B_{k-2}, \; k \ge 4. \end{equation} Let $S_k$ be the Taylor polynomial $r + r^2 \sigma_{(2)} + \cdots + r^k \sigma_{(k)}$ of degree $k$. Then a calculation shows that \begin{equation}\label{S-deco} \SC(S_k) = \SC(S_{k-1}) + r^{k-1} \frac{2k (n-k+2)}{n+1} \sigma_{(k)} + O(r^k). \end{equation} Assume that $S_{k-1}$ is the $(k-1)$-th approximate solution of the singular Yamabe problem, i.e., $\SC(S_{k-1}) = 1+ O(r^{k-1})$. Then $S_k$ is the $k$-th approximate solution if in the expansion \SC(S_k) = 1 + r^{k-1} \left (\frac{2k (n-k+2)}{n+1} \sigma_{(k)} + \cdots \right) + O(r^k) with an unknown coefficient $\sigma_{(k)}$ the coefficient of $r^{k-1}$ vanishes, i.e., if $\SC(S_k) = 1 + O(r^k)$. This can be solved for $\sigma_{(k)}$ if $k =2,3,\dots,n+1$.[This algorithm yields Lemma <ref>.] In the case $k=n+2$, we only have \SC(S_{n+1}) = 1 + O(r^{n+1}), and the restriction of the latter remainder is the obstruction $\B_n$. Now (<ref>) implies \frac{k-1}{2k} (r^{-k+1} (\SC(S_{k-1})-1)\|_0 + \res_{n=k-2} (\sigma_{(k)}) = \frac{k-1}{2k} (r^{-k+1}(\SC(S_k)-1)\|_0 if $S_{k-1}$ is the $(k-1)$-th approximate solution of the singular Yamabe problem. Then the left-hand side is well-defined and the right-hand side vanishes. This implies the assertion. §.§ Renormalized volume coefficients (adapted coordinates) Here we consider the first three coefficients in the expansion of $v(s)$ as defined in (<ref>) in adapted coordinates. As usual we identify $\rho$ with $\eta^(\rho)$. We also recall that derivatives with respect to $s$ are denoted by a prime. We first derive formulas for $v_1$ and $v_2$ from Proposition <ref>. Note that a^{-1/2} = (1-2s\rho)^{-1/2} = 1 + s \rho_0 + s^2 (\frac{3}{2} \rho_0^2 + \rho'_0) + \cdots. Hence the factorization $v(s) = a^{-1/2} \mathring{v}(s)$ and the identities \begin{equation} \mathring{v}_1 = \frac{1}{2} \tr (h_{(1)}) \quad \mbox{and} \quad \mathring{v}_2 = \frac{1}{8} \tr (h_{(1)})^2 + \frac{1}{2} \tr (h_{(2)}) - \frac{1}{4} \tr (h_{(1)}^2)) \end{equation} show that the expansion of $v(s)$ starts with \begin{align} & 1 + \frac{1}{2} (2 \rho_0 + \tr (h_{(1)})) s \\ & + \frac{1}{2} \left(\tr (h_{(2)}) - \frac{1}{2} \|h_{(1)}\|^2 + \frac{1}{4} \tr(h_{(1)})^2 + \rho_0 \tr (h_{(1)}) + 3 \rho_0^2 + 2 \rho'_0 \right) s^2 + \cdots. \end{align} Now (<ref>) and $\rho_0 = -H$ (Lemma <ref>) imply that the linear coefficient equals $(n-1)H$. Hence $v_1=(n-1)H$. This result fits with the formula <cit.> for its integral. Moreover, using Lemma <ref>, we \frac{1}{2} \left( \|\lo\|^2 - \overline{\Ric}_{00} - 2 \|L\|^2 + n^2 H^2 - 2n H^2 + 3 H^2 + 2 \bar{\Rho}_{00} + 2 \frac{\|\lo\|^2}{n-1}\right) for the quadratic coefficient. By $\overline{\Ric}_{00} = (n-1) \bar{\Rho}_{00} + \bar{\J}_0$ and $\|L\|^2 =\|\lo\|^2 + n H^2$, the latter sum simplifies to \begin{equation}\label{v2} v_2 = \frac{1}{2} \left(-\frac{n-3}{n-1} \|\lo\|^2 - (n-3) \bar{\Rho}_{00} + (n-1)(n-3) H^2 - \bar{\J}_0 \right). \end{equation} In particular, (<ref>) shows that $2 v_2 = - \bar{\J}_0$ for $n=3$. This proves the relation (<ref>) in Conjecture <ref> for $n=3$. In the following examples, we demonstrate how the identity (<ref>) can be used to calculate the renormalized volume coefficients $v_k$ (for $k=1,2,3$). This alternative method does not require the calculation of composition of $L$-operators and does not use explicit formulas for the Taylor coefficients of $h_s$. The restriction of (<ref>) to $s=0$ implies $v_1=-(n-1)\rho_0$. Now the fact $\rho_0 = \iota^ \rho = -H$ (Lemma <ref>) implies $v_1 = (n-1)H$. We restrict the derivative of (<ref>) in $s$ to $s=0$. Then 2v_2 - v_1^2 = -(n\!-\!3) \rho'_0 - \bar{\J}_0 - 2(n\!-\!1) \rho^2_0. Now we combine this result with the value of $v_1$ (Example <ref>) and Lemma <ref> to conclude that \begin{align}\label{v2n} -2v_2 & = \bar{\J}_0 + (n\!-\!3) \left(\bar{\Rho}_{00} + \frac{1}{n\!-\!1} \|\lo\|^2 \right) - (n\!-\!3)(n\!-\!1) H^2 \notag \\ & = \bar{\J}_0 + (n\!-\!3) \rho_0' - (n\!-\!3)(n\!-\!1) H^2. \end{align} This result fits with (<ref>) and with the formula <cit.> for its integral. Using (<ref>), we finally obtain \begin{equation}\label{v2n-2} -2v_2 = \J + (n\!-\!2) \bar{\Rho}_{00} + \frac{2n\!-\!5}{2(n\!-\!1)} \|\lo\|^2 - (n\!-\!2)(n\!-\!3/2) H^2. \end{equation} In particular, for $n=2$, we have \begin{equation}\label{v2-2} -2v_2 = \J - \frac{1}{2} \|\lo\|^2. \end{equation} For closed $M$, the total integral of this quantity is a conformal invariant (by Gauss-Bonnet). Note that $2v_2=\QC_2$ (by Example <ref>) which confirms Theorem <ref> for $n=2$. The equality of the coefficients of $s^2$ in (<ref>) yields the identity 3v_3 - 3 v_1 v_2 + v_1^3 = - \frac{n\!-\!5}{2} \rho''_0 - \bar{\J'}_0 - 4(n\!-\!2) \rho'_0 \rho_0 - 2 \rho_0 \bar{\J}_0 - 4(n\!-\!1) \rho_0^3. Hence, using $\rho_0 = -H$, we obtain 3v_3 = 3 v_1 v_2 - v_1^3 - \frac{n\!-\!5}{2} \rho''_0 - \bar{\J}'_0 + 4(n\!-\!2) \rho'_0 H + 2 H \bar{\J}_0 + 4(n\!-\!1) H^3. Combining this formula with the formulas for $v_1$ and $v_2$ in Example <ref> and Example <ref> gives 6 v_3 = (n\!-\!5)(n\!-\!3)(n\!-\!1) H^3 - (n\!-\!5)(3n\!-\!5) H \rho'_0 - (3n\!-\!7) H \bar{\J}_0 - (n\!-\!5) \rho''_0 - 2 \bar{\J}'_0. In particular, this formula implies 6 v_3 = - 8 H \bar{\J}_0 - 2 \bar{\J}'_0. if $n=5$. This proves the relation (<ref>) in Conjecture <ref> for $n=5$. Finally, we note that $\iota^* \nabla_{\NV}^2(\rho)$ corresponds to $\rho_0'' - 2 \rho_0 \rho_0'$ (see Example <ref>). Hence we can rewrite the latter formula as \begin{align}\label{v3-inter} 6 v_3 & = (n\!-\!5)(n\!-\!3)(n\!-\!1) H^3 - (3n\!-\!7) H \iota^* \bar{\J} \notag \\ & - (n\!-\!5)(3n\!-\!7) H \iota^* \nabla_\NV (\rho)- (n\!-\!5) \iota^* \nabla_\NV^2(\rho) - 2 \iota^* \nabla_\NV(\bar{\J}). \end{align} In particular, for $n=3$ we get 6 v_3 = - 2 H \iota^* \bar{\J} + 4 H \iota^* \nabla_\NV (\rho) + 2 \iota^* \nabla_\NV^2(\rho) - 2 \iota^* \nabla_\NV( \bar{\J} ). This quantity actually equals a multiple of $\QC_3$, up to a divergence term; for a discussion of $\QC_3$ we refer to Example <ref>. By <cit.>, the result (<ref>) implies that[There seems to be a misprint in the contribution of the term $H \iota^* \J$.] -\iota^* L(-n+1)L(-n+2)L(-n+3)(1) = 6 (n-1)(n-2)(n-3) v_3, up to a divergence term. Therefore, the result confirms the formula for $c_3$ in Theorem <ref>. In order to express the sum in (<ref>) in terms of standard curvature terms, it remains to determine $\iota^* \nabla_\NV^k (\rho)$ for $k=1,2$. Explicit formulas for these terms will be derived in Section <ref> (Lemma <ref>, Lemma <ref>). The case $k=2$ was first treated in <cit.> (see Remark <ref>). It is only here where we need the full information of Proposition Of course, the renormalized volume coefficients $v_k$, which are defined in terms of adapted coordinates, are to be distinguished from the renormalized volume coefficients $w_k$, which are defined in terms of geodesic normal coordinates <cit.>. By (<ref>), the latter ones are defined by the w(r) = (1 + \sigma_{(2)} r + \sigma_{(3)} r^2 + \cdots)^{-(n+1)} u(r). In particular, we find $w_1 = -(n+1) \sigma_{(2)} + u_1 = \frac{n-1}{2} H$. The above relation implies w_2 = 6 \sigma_{(2)}^2 - 3 \sigma_{(3)} - 3 \sigma_{(2)} u_1 + u_2 for $n=2$, and its evaluation yields 2 w_2 = - \J + \frac{1}{2} \|\lo\|^2. Note that $w_1 = v_1$ (for $n=1$) and $w_2 = v_2$ (for $n=2$). In general, the coefficients $w_n$ and $v_n$ differ by a non-trivial total divergence. In particular, we find w_3 = -20 \sigma_{(2)}^3 + 20 \sigma_{(2)} \sigma_{(3)} - 4 \sigma_{(4)} + 10 \sigma_{(2)}^2 u_1 - 4 \sigma_{(3)} u_1 - 4 \sigma_{(2)} u_2 + u_3 for $n=3$, and an evaluation yields \begin{align} 6 v_3 & = 6 w_3 + \Delta(H). \end{align} Explicit formulas for $w_1, w_2$ (in general dimensions) were derived in <cit.>. In Section <ref>, these coefficients will be described in terms of $L$-operators. the last relation follows from the later formula for v_3 and a calculation of w_3 (which we only did in a manuscript) §.§ Low-order Taylor coefficients of $\rho$ Assuming that $\sigma$ satisfies the condition $\SCY$, we derive formulas for the first few Taylor coefficients of $\rho$ in the variable $s$. We first use Lemma <ref> to derive formulas for the restrictions of $\rho$ and $\nabla_\NV(\rho)$ to $M$. The following result reproves part of <cit.>. Let $n \ge 2$. Then $\iota^* \rho = - H$ and \begin{equation}\label{rho-1} \iota^* \nabla_\NV(\rho) = \Rho_{00} + \frac{\|\lo\|^2}{n-1}. \end{equation} We calculate in geodesic normal coordinates. We expand the defining relation \begin{equation}\label{rho-def} \Delta_g (\sigma) = -(n\!+\!1) \rho - \sigma \J \end{equation} of $\rho$ into a power series of $r$. The Laplacian takes the form $\partial_r^2 + \frac{1}{2} \tr (h_r^{-1} h'_r) \partial_r + \Delta_{h_r}$. Hence the restriction of (<ref>) to $r=0$ implies $2 \sigma_{(2)} + n H = -(n+1)\rho_0$, and Lemma <ref> yields the first assertion. Next, we restrict the derivative of (<ref>) in $r$ to $r=0$. Then 6 \sigma_{(3)} + \frac{1}{2} \partial_r (\tr(h_r^{-1} h'_r))\|_0 + \tr(h_r^{-1} h'_r)\|_0 \sigma_{(2)} = -(n\!+\!1) \partial_r(\rho)\|_0 - \J_0. But (<ref>) implies the identities \tr(h_r^{-1} h'_r)\|_0 = 2 \tr (L) \quad \mbox{and} \quad \partial_r (\tr(h_r^{-1} h'_r))\|_0 = - 2\|L\|^2 - 2 \Ric_{00}. Hence the above relation transforms into 6 \sigma_{(3)} - \|L\|^2 - \Ric_{00} + 2 \tr(L) \sigma_{(2)} = -(n\!+\!1) \partial_r(\rho)\|_0 - \J_0. We combine this result with (<ref>) for $\Ric_{00}$, (<ref>) for $\sigma_{(3)}$ and $\|L\|^2 = \|\lo\|^2 + n H^2$ to obtain \partial_r(\rho)\|_0 = \iota^* (\J^g) - \J^h + \frac{\|\lo\|^2}{2(n\!-\!1)} + \frac{n}{2} H^2. Now (<ref>) implies the assertion. The equation (<ref>) justifies the notation $\rho_0'$ in Proposition <ref>. Next, we provide an alternative proof of Lemma <ref>. This illustrates the efficiency of the differential equation for $\rho$ (Lemma <ref>), the resulting recursive formula in Proposition <ref> and the formula for the obstruction (Theorem <ref>) in low-order cases. First of all, the restriction of (<ref>) to $s=0$ yields $n \rho(0) + \frac{1}{2} \tr (h_{(1)}) = 0$. Using $h_{(1)} = 2L$ by (<ref>), we get $\rho(0) = -H$. This reproves the first part of Lemma <ref>. Next, the formula (<ref>) for $k=1$ yields (n-1) \rho'_0 - 2 \rho_0 \left(\frac{\mathring{v}'}{\mathring{v}}\right)\|_0 + \partial_s \left(\frac{\mathring{v}'}{\mathring{v}} \right)\|_0 + \J_0 = 0 or, equivalently, \begin{equation}\label{rho-prime} (n-1) \rho_0' - \rho_0 \tr (h_{(1)}) + \tr (h_{(2)}) - \frac{1}{2} \tr (h_{(1)}^2) + \J_0 = 0. \end{equation} By (<ref>) and $\tr(L^2) = \tr (\mathring{L}^2) + n H^2$, this gives \begin{align} 0 & = (n-1) \rho_0' + 2n H^2 + \tr (\mathring{L}^2) - \Ric_{00} - 2 \tr (L^2) + \J_0 \\ & = (n-1) \rho_0' - \Ric_{00} - \|\mathring{L}\|^2 + \J_0 \\ & = (n-1) \rho_0' - (n-1) \Rho_{00}- \|\mathring{L}\|^2 . \end{align} This reproves the second part of Lemma <ref>. For $n=1$, Theorem <ref> states that \B_1 = - \partial_s \left( \frac{\mathring{v}'}{\mathring{v}}\right)\|_0 + 2 \rho_0 \left( \frac{\mathring{v}'}{\mathring{v}}\right)\|_0 - \J_0 or, equivalently, \B_1 = -\tr (h_{(2)}) + \frac{1}{2} \tr (h_{(1)}^2) + \rho_0 \tr (h_{(1)}) - \J_0. By $\lo = 0$, $\tr(h_{(1)}) = 2H$, $\tr(h_{(1)}^2) = 4 H^2$, $\tr(h_{(2)}) = -\Ric_{00}$ and $\rho_0 = -H$, this formula simplifies to \B_1 = \Ric_{00} - \J_0 = \Ric_{00} - K_0 = 0 using $\Ric = K g$, $K$ being the Gauss curvature of $g$. We continue discussing the second-order derivative of $\rho$ in general dimensions $n \ge 3$. The arguments are parallel to the discussion in Section <ref>. Proposition <ref> for $k=2$ gives \begin{equation} -(n\!-\!2) \rho_0'' = \partial_s^2 \left( \frac{\mathring{v}'}{\mathring{v}}\right)\|_0 - 4 \rho_0 \partial_s \left( \frac{\mathring{v}'}{\mathring{v}}\right)\|_0 - 4 \rho_0' \left( \frac{\mathring{v}'}{\mathring{v}}\right)\|_0 +2 \J'_0. \end{equation} This formula is equivalent to \begin{align}\label{rho-second} & -(n\!-\!2) \rho_0'' \notag \\ & = \tr ( 3 h_{(3)} - 3 h_{(1)} h_{(2)} + h_{(1)}^3) - 4 \rho_0 \tr (h_{(2)}) + 2 \rho_0 \tr (h_{(1)}^2) - 2 \rho_0' \tr (h_{(1)}) + 2 \J_0'. \end{align} In order to make that formula explicit, we use Proposition <ref> and in particular Corollary <ref>. We obtain \begin{align}\label{rho-second-sum} & -(n\!-\!2) \rho_0'' = -\nabla_{\partial_s}(\Ric)_{00} + 2 \partial_s(\J) - 4 \lo^{ij} R_{0ij0} + \Delta H - H \Ric_{00} - 3 H\|\lo\|^2 - 2n H \rho_0' \notag \\ & - 6 \tr (L^2 \lo) + 6 L^{ij} R_{0ij0} + 8 \tr (L^3) + 4 H \|\lo\|^2 - 4 H \Ric_{00} - 8 H \|L\|^2. \end{align} Now, by the obvious relation -\nabla_{\partial_s}(\Ric)_{00} + 2 \partial_s(\J) = - \nabla_{\partial_s} (G)_{00} - (n-2) \partial_s(\J) for the Einstein tensor $G = \Ric - \frac{1}{2} \scal g$ and the identity (<ref>), we get \begin{align} -(n\!-\!2) \rho_0'' & = \delta^h (\Ric(\partial_s,\cdot)) - \lo^{ij} \Ric_{ij} + (n\!+\!1) H \Ric_{00} - H \scal - (n\!-\!2) \J_0' - 2n H \rho_0' \notag \\ & - 4 \lo^{ij} R_{0ij0} + \Delta H - H \Ric_{00} - 3 H\|\lo\|^2 \notag \\ & - 6 \tr (L^2 \lo) + 6 L^{ij} R_{0ij0} + 8 \tr (L^3) + 4 H \|\lo\|^2 - 4 H \Ric_{00} - 8 H \|L\|^2 . \end{align} The decomposition (<ref>) shows that \begin{align} L^{ij} R_{0ij0} & = L^{ij} (W_{0ij0} + \Rho_{ij} + \Rho_{00} g_{ij}) \\ & = L^{ij} W_{0ij0} + L^{ij} \Rho_{ij} + n H \Rho_{00} \\ & = L^{ij} W_{0ij0} + \lo^{ij} \Rho_{ij} + H (\J - \Rho_{00}) + n H \Rho_{00} \\ & = L^{ij} W_{0ij0} + \lo^{ij} \Rho_{ij} + H \J + (n-1) H \Rho_{00} \\ & = \lo^{ij} W_{0ij0} + \lo^{ij} \Rho_{ij} + H \Ric_{00}. \end{align} Similarly, we find \lo^{ij} R_{0ij0} = \lo^{ij} W_{0ij0} + \lo^{ij} \Rho_{ij}. The latter two results and the formula (<ref>) for $\rho_0'$ imply \begin{align} -(n\!-\!2) \rho_0'' & = \delta^h (\Ric(\partial_s,\cdot)) - \lo^{ij} \Ric_{ij} + (n\!+\!1) H \Ric_{00} - H \scal - (n\!-\!2) \J_0' \notag \\ & - 4 \lo^{ij} W_{0ij0} - 4 \lo^{ij} \Rho_{ij} + \Delta H - H \Ric_{00} - 3 H\|\lo\|^2 \notag \\ & - 6 \tr (L^2 \lo) + 6 \lo^{ij} W_{0ij0} + 6 \lo^{ij} \Rho_{ij} + 6 H \Ric_{00} \notag \\ & + 8 \tr (L^3) + 4 H \|\lo\|^2 - 4 H \Ric_{00} - 8 H \|L\|^2 - 2n H \left(\Rho_{00} + \frac{\|\lo\|^2}{n\!-\!1} \right). \end{align} Simplification gives \begin{align} -(n\!-\!2) \rho_0'' & = \delta^h (\Ric(\partial_s,\cdot)) + \Delta H + 2 \lo^{ij} W_{0ij0} + 2 \lo^{ij} \Rho_{ij} - \lo^{ij} \Ric_{ij} \notag \\ & + (n\!+\!2) H \Ric_{00} - 2n H \Rho_{00} - H \scal \notag \\ & + H\|\lo\|^2 - 8 H \|L\|^2 - 6 \tr (L^2 \lo) + 8 \tr (L^3) - \frac{2n}{n\!-\!1} H \|\lo\|^2 - (n\!-\!2) \J_0'. \end{align} Further simplification using \|L\|^2 = \|\lo\|^2 + n H^2, \quad \tr (L^2 \lo) = 2 H \|\lo\|^2 \quad \mbox{and} \quad \tr (L^3) = \tr (\lo^3) +3 H \|\lo\|^2 + n H^3 \begin{align}\label{rho2-a} -(n\!-\!2) \rho_0'' & = \delta^h (\Ric(\partial_s,\cdot)) + \Delta H + 2 (\lo^{ij} W_{0ij0} + \tr (\lo^3)) - (n\!-\!3) \lo^{ij} \Rho^g_{ij} \notag \\ & + \frac{(n\!-\!2)(n\!+\!1)}{n\!-\!1} H \Ric_{00} - \frac{n\!-\!2}{n\!-\!1} H \scal + \frac{3n\!-\!5}{n\!-\!1} H \|\lo\|^2 - (n\!-\!2) \J_0' . \end{align} Next, we apply the basic identity (<ref>). It implies that \begin{equation}\label{F-L-trace} \lo^{ij} W_{0ij0} + \tr (\lo^3) \stackrel{!}{=} (n-2) \lo^{ij} \JF_{ij}. \end{equation} Now, combining the formula (<ref>) for $\Delta H$ and (<ref>) with (<ref>), yields \begin{align} -(n\!-\!2) \rho_0'' & = \frac{n\!-\!2}{n\!-\!1} \delta^h (\Ric(\partial_s,\cdot)) + \frac{1}{n\!-\!1} \delta \delta (\lo) + 2 (n\!-\!2) \lo^{ij} \JF_{ij} - (n\!-\!3) \lo^{ij} \Rho^g_{ij} \\ & + \frac{(n\!-\!2)(n\!+\!1)}{n\!-\!1} H \Ric_{00} - \frac{n\!-\!2}{n\!-\!1} H \scal + \frac{3n\!-\!5}{n\!-\!1} H \|\lo\|^2 - (n\!-\!2) \J_0'. \end{align} \begin{align} \rho_0'' & = - \frac{1}{(n\!-\!1)(n\!-\!2)} \delta \delta (\lo) - \frac{1}{n\!-\!1} \delta^h (\Ric(\partial_s,\cdot)) - 2 \lo^{ij} \JF_{ij} + \frac{n\!-\!3}{n\!-\!2} \lo^{ij} \Rho^g_{ij} \\ & - \frac{n\!+\!1}{n\!-\!1} H \Ric_{00} + \frac{1}{n\!-\!1} H \scal - \frac{3n\!-\!5}{(n\!-\!1)(n\!-\!2)} H \|\lo\|^2 + \J_0' . \end{align} Finally, we use the defining relation $\iota^* \Rho^g = \JF + \Rho^h - H \lo - 1/2 h H^2$ of $\JF$ to replace in this identity the Schouten tensor $\Rho^g$ by the Schouten tensor $\Rho^h$. Thus, we have proved the following result. For $n \ge 3$, it holds \begin{align}\label{rho-2-ex} \rho_0'' & = - \frac{1}{(n\!-\!1)(n\!-\!2)} \delta \delta (\lo) - \frac{1}{n\!-\!1} \delta^h (\Ric^g(\partial_s,\cdot)) - \frac{n\!-\!1}{n\!-\!2} (\lo,\JF) + \frac{n\!-\!3}{n\!-\!2} (\lo,\Rho^h) \notag \\ & - \frac{n\!+\!1}{n\!-\!1} H \Ric_{00} + \frac{1}{n\!-\!1} H \scal - \frac{n\!+\!1}{n\!-\!1} H \|\lo\|^2 + \J_0'. \end{align} In Section <ref>, we shall connect this result with the holographic formula for $\QC_3$. Lemma <ref> is equivalent to <cit.> and <cit.>, up to the sign of the term $(\lo,\Rho^h)$. In fact, the quoted result calculates the restriction of $\nabla_{\NV}^2(\rho)$ to $M$. By Example <ref>, it corresponds to $\rho''_0+ 2 H \rho_0'$. But (<ref>) implies \begin{align} \rho''_0+ 2 H\rho_0' & = (\cdots) -\frac{n\!+\!1}{n\!-\!1} H \Ric_{00} + \frac{1}{n\!-\!1} H \scal - \frac{n\!+\!1}{n\!-\!1} H \|\lo\|^2 + \J_0' \\ & + 2 H \left(\frac{1}{n\!-\!1} \Ric_{00} - \frac{1}{n\!-\!1} \J_0 + \frac{\|\lo\|^2}{n\!-\!1}\right) + \J_0' \\ & = (\cdots) - H \Ric_{00} + 2 H \J_0 - H\|\lo\|^2 + \J_0' \\ & = (\cdots) - H ((n\!-\!1) \Rho_{00} + \|\lo\|^2) + H \J_0 + \J_0', \end{align} where $(\cdots)$ indicates the four terms in the first line of (<ref>). This proves the claim. The present alternative proof rests on the recursive formula in Proposition <ref> and the explicit formula for $h_{(3)}$ in Proposition <ref>. The proof of Proposition <ref> shows that the calculation of the coefficient $h_{(2)}$ involves $h_{(1)}$ and $\rho_0$. Once $h_{(2)}$ has been determined, we calculated $\rho_0'$ using (<ref>). Similarly, the calculation of $h_{(3)}$ in the proof of Lemma <ref> involves the lower-order coefficients of $h_s$ and $\rho_0$, $\rho_0'$. Once $h_{(3)}$ has been determined, the recursive formula (<ref>) yields $\rho_0''$. A continuation of that iterative process yields the higher-order coefficients, at least in principle. We finish this section with some comments concerning the functions $\rho$ and the singular Yamabe obstructions in Examples <ref>–<ref>. In Example <ref>, it holds $\mathring{v}'/\mathring{v} = - n/(1-2s)$. Hence the differential equation (<ref>) reduces to - s \rho' + n \rho - n(1-2s \rho)/(1-2s) = 0. One readily checks that $\rho=1$ is the unique solution with initial value $\rho(0) = 1$. The vanishing of the obstruction is reproduced by the formula (<ref>). Indeed, we obtain (n+1)! \B_n = 2n \partial_s^n (1/(1-2s))\|_0 - 4 n^2 \partial_s^{n-1} (1/(1-2s))\|_0 = 2n 2^n n! - 4 n^2 2^{n-1} (n-1)! = 0. Similarly, in Example <ref>, it holds $\mathring{v}'/\mathring{v} = - ns/(1-s^2)$ and the differential equation (<ref>) reduces to - s\rho' + n\rho - n s (1-2s\rho)/(1-s^2) + \tfrac{n+1}{2} s = 0. One easily checks that $\rho = \frac{1}{2} s$ is the unique solution with the initial value $\rho(0)=0$. Again, the vanishing of the obstruction is reproduced by (n+1)! \B_n = 2 \partial_s^n(ns/(1-s^2))\|_0 - 4 \binom{n}{2} \partial_s^{n-2}(ns/(1-s^2))\|_0 = 2n n! - 4n \binom{n}{2} (n-2)! = 0 for odd $n$ and trivially for even $n$. §.§ The obstruction $\B_2$ for general backgrounds In the present section, we prove the equivalence of both formulas for the singular Yamabe obstruction $\B_2$ displayed in (<ref>) and derive the second of these formulas from Theorem <ref>. The Codazzi-Mainardi identity states that \begin{equation}\label{CME} \nabla^h_Y (L)(X,Z) - \nabla^h_X(L)(Y,Z) = R^g (X,Y,Z,N) \end{equation} for $X,Y,Z \in \mathfrak{X}(M)$ if $L(X,Y) = - h(\nabla^g_X(Y),N)$ for some unit normal vector field $N$ <cit.>. Now, we decompose the curvature tensor $R$ as \begin{align}\label{KN} & R(X,Y,Z,W) = W(X,Y,Z,W) \\ & + \Rho(Y,Z) g (X,W) - \Rho(X,Z) g(Y,W) - \Rho(Y,W)g(X,Z) + \Rho(X,W)g(Y,Z), \notag \end{align} where $W$ is the trace-free Weyl tensor (see (<ref>)), and take traces in (<ref>) in the arguments $X,Z$. Then n \langle dH, Y \rangle - \langle \delta^h (L), Y \rangle = -(n-1) \Rho(N,Y). \delta^h (\lo) - (n-1) dH = (n-1) \Rho(N,\cdot) [Lemma 6.25.2]J1). Now, taking a further divergence, yields \begin{equation}\label{ddL} \delta^h \delta^h (\lo) - (n-1) \Delta H = (n-1) \delta^h (\Rho(N,\cdot)). \end{equation} For $n=2$, this proves the equivalence of both formulas in (<ref>). We continue by showing that the second formula for $\B_2$ in (<ref>) is a special case of Theorem <ref>. First, we observe that the formula - 3 \B_2 = \partial_s^2 \left( \frac{\mathring{v}'}{\mathring{v}} \right)\|_0 - 4 \rho_0 \partial_s \left (\frac{\mathring{v}'}{\mathring{v}} \right)\|_0 - 4 \rho_0' \left(\frac{\mathring{v}'}{\mathring{v}}\right)\|_0 + 2 \J_0' in Theorem <ref> is equivalent to -3 \B_2 = \tr ( 3 h_{(3)} - 3 h_{(1)} h_{(2)} + h_{(1)}^3) - 4 \rho_0 \tr (h_{(2)}) + 2 \rho_0 \tr (h_{(1)}^2) - 2 \rho_0' \tr (h_{(1)}) + 2 \J_0'. In order to make that sum explicit, we use the results in Proposition <ref>. By Corollary <ref> for $n=2$ and $\rho_0 = - H$, we obtain \begin{align}\label{B2-sum} -3 \B_2 & = -\nabla^g_{\partial_s}(\Ric)_{00} + 2 \partial_s(\J) - 4 \lo^{ik} R_{0ik0} + \Delta H - H \Ric_{00} - 3 H\|\lo\|^2 \\ & - 6 \tr (L^2 \lo) + 6 L^{ij} R_{0ij0} + 8 \tr (L^3) + 4 H \|\lo\|^2 - 4 H \Ric_{00} - 8 H \|L\|^2 - 4 H \rho_0'. \notag \end{align} Now we prove the identity \begin{equation}\label{Einstein-00} \nabla^g_{\partial_s}(G)_{00} = -\delta^h (\Ric(\partial_s,\cdot)) + \lo^{ij} \Ric_{ij} - (n+1) H \Ric_{00} + H \scal \end{equation} for the Einstein tensor $G \st \Ric - \frac{1}{2} \scal g$. Note that $G= \Ric - 2 \J g$ in dimension $n=3$. It is well-known that the second Bianchi identity implies the relation $2 \delta^g (\Ric) = d\scal$. Hence \begin{align} \nabla^g_{\partial_s}(\Ric)(\partial_s,\partial_s) & = \delta^g(\Ric)(\partial_s) - g^{ij} \nabla^g_{\partial_i}(\Ric)(\partial_j,\partial_s) \\ & = \frac{1}{2} \langle d \scal,\partial_s \rangle \\ & - g^{ij} \partial_i (\Ric(\partial_j,\partial_s)) + g^{ij} \Ric(\nabla^g_{\partial_i}(\partial_j),\partial_s) + g^{ij} \Ric(\partial_j,\nabla^g_{\partial_i}(\partial_s)) \\ & = \frac{1}{2} \langle d \scal,\partial_s \rangle \\ & - h^{ij} \partial_i (\Ric(\partial_j,\partial_s)) + h^{ij} \Ric (\nabla^h_{\partial_i}(\partial_j) - L_{ij} \partial_s,\partial_s) + h^{ij} \Ric(\partial_j,\nabla^g_{\partial_i}(\partial_s)) \\ & = \frac{1}{2} \langle d \scal,\partial_s \rangle - \delta^h (\Ric(\partial_s,\cdot)) - n H \Ric_{00} + h^{ij} \Ric(\partial_j,\nabla^g_{\partial_i}(\partial_s)) \end{align} on $M$. Therefore, using $\nabla_{\partial_i}^g(\partial_s) = L_i^k \partial_k$, we obtain \begin{align} \nabla^g_{\partial_s}(G)_{00} & = -\delta^h (\Ric(\partial_s,\cdot)) - n H \Ric_{00} + h^{ij} L_i^k \Ric_{jk} \\ & = -\delta^h (\Ric(\partial_s,\cdot)) - n H\Ric_{00} + L^{ij} \Ric_{ij}. \end{align} This implies (<ref>). Next, we observe that the decomposition (<ref>) yields \begin{align} L^{ij} R_{0ij0} & = L^{ij} (W_{0ij0} + \Rho_{ij} + \Rho_{00} g_{ij}) \\ & = L^{ij} \Rho_{ij} + 2 H \Rho_{00} \\ & = \lo^{ij} \Rho_{ij} + H (\J - \Rho_{00}) + 2 H \Rho_{00} = \lo^{ij} \Rho_{ij} + H \J + H \Rho_{00} \\ & = \lo^{ij} \Rho_{ij} + H \Ric_{00} \end{align} since the Weyl tensor $W$ vanishes in dimension $3$. Similarly, we find \begin{equation} \lo^{ij} R_{0ij0} = \lo^{ij} \Rho_{ij}. \end{equation} These results and the formula (<ref>) for $\rho_0'$ show that the sum (<ref>) simplifies to (recall that $n=2$) \begin{align} & \delta^h (\Ric(\partial_s,\cdot)) - \lo^{ij} \Ric_{ij} + 3 H \Ric_{00} - H \scal - 4 \lo^{ij} \Rho_{ij} + \Delta H - H \Ric_{00} - 3 H \|\lo\|^2 \\ & - 6 \tr (L^2 \lo) + 6 \lo^{ij} \Rho_{ij} + 6 H \Ric_{00} + 8 \tr(L^3) + 4 H \|\lo\|^2 - 4 H \Ric_{00} - 8 H \|L\|^2 \\ & - 4H \Ric_{00} + H \scal - 4 H \|\lo\|^2 \\ & = \delta^h (\Ric(\partial_s,\cdot)) + \Delta H + \lo^{ij} \Ric_{ij}, \end{align} up to terms that are at least quadratic in $L$. In order to deal with these terms, we note that \|L\|^2 = \|\lo\|^2 + 2 H^2, \quad \tr (L^2 \lo) = 2 H \|\lo\|^2 \quad \mbox{and} \quad \tr (L^3) = 3 H \|\lo\|^2 + 2 H^3 using $\tr(\lo^3)=0$. It follows that the remaining terms are $(- 3 - 12 + 24 + 4 - 8 - 4) H \|\lo\|^2 = H \|\lo\|^2$. Summarizing we obtain -3 \B_2 = \delta^h (\Ric(\partial_s,\cdot)) + \Delta H + \lo^{ij} \Ric_{ij}+ H \|\lo\|^2. This proves the second formula for $\B_2$ in (<ref>). The identity (<ref>) is equivalent to <cit.> or <cit.>. In addition to the derivation of the above formula for the singular Yamabe obstruction $\B_2$, in these references, this crucial identity is used to prove a formula for the second normal derivative of $\rho$ and a formula for $\QC_3$. The calculations lead to the same results as here. Our discussion of the second normal derivative of $\rho$ and the explicit formula for $\QC_3$ is contained in Sections <ref> and <ref>. It uses the identity (<ref>). §.§ The obstruction $\B_3$ for conformally flat backgrounds We first evaluate Theorem <ref> for the obstruction $\B_3$ of a three-manifold $M \hookrightarrow \R^4$. The resulting formula is equivalent to a formula in <cit.> (see Lemma <ref>). This will imply a formula for $\B_3$ in a conformally flat background. The Yamabe obstruction of a hypersurface $(M^3,h)$ in the four-dimensional flat space $\R^4$ is given by the formula \begin{equation}\label{B3-final} 12 \B_3 = \Delta (\|\lo\|^2) + 12 \|dH\|^2 + 6 (\lo,\Hess (H)) - 2 \|\lo\|^4 + 6 \tr (\lo^4) + 6 H \tr (\lo^3) \end{equation} or, equivalently, \begin{equation}\label{B3-final2} 12 \B_3 = \Delta (\|\lo\|^2) + 12 \|dH\|^2 + 6 (\lo,\Hess (H)) + \|\lo\|^4 + 6 H \tr (\lo^3). \end{equation} Here the Hessian, the Laplacian, scalar products, and traces are taken with respect to the metric $h$ on $M$. Newton's identity 24 \sigma_4 (L) = \tr(L)^4 - 6 \tr(L)^2 \|L\|^2 + 3 \|L\|^4 + 8 \tr(L) \tr(L^3) - 6 \tr(L^4) for the elementary symmetric polynomial $\sigma_4(L)$ of the eigenvalues of $L$ (or rather of the shape operator) implies \begin{align} 24 \sigma_4(L) & = n(n-1)(n-2)(n-3) H^4 - 6 (n-2)(n-3) H \|\lo\|^2 + 8(n-3) H \tr(\lo^3) \\ & + 3 (\|\lo\|^4 - 2 \tr(\lo^4)). \end{align} Since $\sigma_4(L) = 0$ in dimension $n=3$, we get $\|\lo\|^4 = 2 \tr(\lo^4)$ for $n=3$. Corollary <ref> implies the equivalence of (<ref>) and (<ref>). We derive formula (<ref>) as a consequence of the identity \begin{equation}\label{3-flat} 12\B_3 = - \partial_s^3 \left( \frac{\mathring{v}'}{\mathring{v}}\right)\|_0 + 6 \rho_0 \partial_s^2 \left( \frac{\mathring{v}'}{\mathring{v}}\right)\|_0 + 12 \rho_0' \partial_s \left( \frac{\mathring{v}'}{\mathring{v}}\right)\|_0 + 6 \rho_0'' \left( \frac{\mathring{v}'}{\mathring{v}}\right)\|_0 - 3 \partial_s^2(\bar{\J})\|_0 \end{equation} in Theorem <ref>. Note that, for a flat background, the last term vanishes. By the relations \begin{align} \left( \frac{\mathring{v}'}{\mathring{v}}\right)\|_0 & = \frac{1}{2} \tr (h_{(1)}), \\ \partial_s \left( \frac{\mathring{v}'}{\mathring{v}}\right)\|_0 & = \frac{1}{2} \tr (2 h_{(2)} - h_{(1)}^2), \\ \partial_s^2 \left( \frac{\mathring{v}'}{\mathring{v}}\right)\|_0 & = 2! \frac{1}{2} \tr (3 h_{(3)} - 3 h_{(1)} h_{(2)} + h_{(1)}^3), \\ \partial_s^3 \left( \frac{\mathring{v}'}{\mathring{v}}\right)\|_0 & = 3! \frac{1}{2} \tr (4 h_{(4)} - 4 h_{(1)} h_{(3)} - 2 h_{(2)}^2 + 4 h_{(1)}^2 h_{(2)} - h_{(1)}^4), \end{align} the identity (<ref>) is equivalent to \begin{align}\label{B3-flat} 12 \B_3 & = - \tr( 12 h_{(4)} - 12 h_{(1)} h_{(3)} - 6 h_{(2)}^2 + 12 h_{(1)}^2 h_{(2)} - 3 h_{(1)}^4) \notag \\ & + 6 \rho_0 \tr (3 h_{(3)} - 3 h_{(1)} h_{(2)} + h_{(1)}^3) \notag \\ & + 6 \rho_0' \tr (2 h_{(2)} - h_{(1)}^2) \notag \\ & + 3 \rho_0'' \tr (h_{(1)}). \end{align} In order to evaluate that sum, we use the formulas for $h_{(k)}$ ($k \le 3$) in Proposition <ref> and the formulas for the first two normal derivatives of $\rho$ in Section <ref>. In addition, it remains to determine the coefficient $\tr(h_{(4)})$. The following result even provides a closed formula for the Taylor coefficient $h_{(4)}$ of $h_s$ (for a flat background). Let $n=3$. Then \begin{align}\label{h4-ex} 12 h_{(4)} & = - 9 H \Hess (H) + L \Hess (H) + \Hess (H) L - 6 dH \otimes dH - \gamma \notag \\ & - \Hess (\|\lo\|^2) + 4 \lo^2 \|\lo\|^2 + 15 H^2 ({(\lo^2)}_\circ + H \lo) - H \lo \|\lo\|^2 + 3 L \rho_0'', \end{align} \begin{equation}\label{gamma-form} \gamma_{jk} \st 2 h^{lm} (\nabla_j(L)_{km} + \nabla_k(L)_{jm} - \nabla_m(L)_{jk}) \partial_l(H) \end{equation} \begin{equation}\label{rho-pp} \rho_0'' = - \Delta (H) - 2 \tr (\lo^3) - 2 H \|\lo\|^2. \end{equation} We use the same notation as in the proof of Proposition <ref>. We expand the curvature components $R_{0jk}^0$ of the metric \eta^(g) = a^{-1} ds^2 + h + h_{(1)} s + h_{(2)} s^2 + h_{(3)} s^3 + h_{(4)} s^4 + \cdots into power series of $s$. We recall that $a = \eta^(\|\NV\|^2)$. In order to simplify the notation, we write $g$ for the metric $\eta^(g)$ and identify $\eta^(\rho)$ with $\rho$. By assumption, it holds $a = 1- 2s \rho$. As usual, the $0$-components refer to $\partial_s$. Since $g$ is flat, the components $R_{0jk}^0$ vanish. Now, for the above metric, we find the Christoffel symbols \Gamma_{ij}^0 = - \frac{1}{2} g^{00} g_{ij}', \quad \Gamma_{0j}^0 = \frac{1}{2} g^{00} \partial_j (g_{00}), \quad \Gamma_{00}^0 = \frac{1}{2} g^{00} g_{00}', \quad \Gamma_{0k}^l = \frac{1}{2} g^{rl} g_{kr}', where $'$ denotes the derivative in $s$. We recall that $g^{00} = a = 1- 2s \rho$ (by assumption) and $g_{00} = a^{-1} = 1 + 2s\rho + \cdots$. Hence \begin{align}\label{R-adapted-2} 0 \stackrel{!}{=} R_{0jk}^0 & = \frac{1}{2} \Gamma_{jk}^l g^{00} (g_{00})_l + \frac{1}{4} g^{00} g^{rl} g_{kr}' g_{jl}' - \frac{1}{2} ((g^{00})' g_{jk}' + g^{00} g_{jk}'') \notag \\ & - \frac{1}{2} ((g^{00})_j (g_{00})_k + g^{00} (g_{00})_{kj}) - \frac{1}{4} (g^{00})^2 g_{jk}' g_{00}' - \frac{1}{4} (g^{00})^2 (g_{00})_j (g_{00})_k. \end{align} Now we display the Taylor expansions of all $6$ terms in (<ref>) up to order $s^2$. Using these results, it is easy to see that the coefficients of $s$ reproduce the result of the earlier calculation of $h_{(3)}$ in the proof of Proposition <ref> (Remark <ref>). First, we observe that the Christoffel symbols $\Gamma_{jk}^l$ for $g$ restrict to the Christoffel symbols $\Gamma_{jk}^l$ for $h$. Moreover, we recall the general variation formula \delta (\Gamma_{jk}^l) = \frac{1}{2} g^{lm} (\nabla_j (\delta(g))_{km} + \nabla_k (\delta(g))_{jm} - \nabla_{m}(\delta(g))_{jk}). Let $(\Gamma_{jk}^l)' = \delta (\Gamma_{jk}^l)$ denote the variation of the Christoffel symbols for the variation $g = h + 2 L s + \cdots$. By $g_{00} = 1 + 2 s \rho + \cdots$ and $\rho_0 = - H$, it follows that the first term in (<ref>) contributes by \begin{align}\label{term0} s^2 \frac{1}{2} (\Gamma_{jk}^l)' (-2 \partial_l(H)) & = - s^2 (\Gamma_{jk}^l)' \partial_l(H) \notag \\ & = - s^2 h^{lm} (\nabla_j(L)_{km} + \nabla_k(L)_{jm} - \nabla_{m}(L)_{jk}) \partial_l(H) \notag \\ & = - s^2 \frac{1}{2} \gamma_{jk}. \end{align} The remaining contributions of the first term in (<ref>) match with the contributions by the second part of the fourth term to \begin{align}\label{term1} -\frac{1}{2} s [\Hess (2\rho_0)] - \frac{1}{2} s^2 \left[\Hess (2\rho_0' + 4 \rho_0^2) - 2 \rho_0 \Hess (2\rho_0) \right] + \cdots \end{align} with $\Hess$ defined with respect to $h$. Next, the second term contributes by $1/4$ times[Here we use the fact that $[h_{(1)},h_{(2)}] = 0$ for a flat background.] \begin{align}\label{term2} h_{(1)}^2 & + s \big[4h_{(1)} h_{(2)} - h_{(1)}^3 - 2 h_{(1)}^2 \rho_0\big] \notag \\ & + s^2 \big[3 h_{(1)} h_{(3)} + 3 h_{(3)} h_{(1)} + 4 h_{(2)}^2 - 5 h_{(1)}^2 h_{(2)} + h_{(1)}^4 \notag \\ & + 2 h_{(1)}^3 \rho_0 - 4 h_{(1)} h_{(2)} \rho_0 - 4 h_{(2)} h_{(1)} \rho_0 - 2 h_{(1)}^2 \rho_0'\big] + \cdots. \end{align} Finally, we find * Term three contributes by $-1/2$ times \begin{align}\label{term3} 2 h_{(2)} - 2 h_{(1)} \rho_0 & + s \big[6h_{(3)} - 8h_{(2)} \rho_0 - 4 h_{(1)} \rho_0'\big] \notag\\ & + s^2 \big[12 h_{(4)} - 18 h_{(3)}\rho_0 - 12 h_{(2)} \rho_0' - 3 h_{(1)} \rho_0''\big] + \cdots \end{align} * The first part of term four contributes by $-1/2$ times \begin{equation}\label{term4} s^2 (-4) \partial_j(\rho_0) \partial_k(\rho_0) \end{equation} * Term five contributes by $-1/4$ times \begin{equation}\label{term5} 2 h_{(1)} \rho_0 + s \big[4 h_{(2)} \rho_0 + 4 h_{(1)} \rho_0' \big] + s^2 \left[ 6 h_{(3)} \rho_0 + 8 h_{(2)} \rho_0' + 3 h_{(1)} \rho_0'' \right] + \cdots \end{equation} * Term six contributes by $-1/4$ times \begin{equation}\label{term6} s^2 4 \partial_j(\rho_0) \partial_k(\rho_0) \end{equation} Summarizing the coefficients of $s^2$ in (<ref>)-(<ref>) we obtain \begin{align}\label{h4-ev} 0 = & - \frac{1}{2} (\gamma + \Hess (2\rho_0' + 4 \rho_0^2) - 4 \rho_0 \Hess (\rho_0)) \notag \\ & + \frac{1}{4} \big[3 h_{(1)} h_{(3)} + 3 h_{(3)} h_{(1)} + 4 h_{(2)}^2 - 5 h_{(1)}^2 h_{(2)} + h_{(1)}^4 + 2 h_{(1)}^3 \rho_0 - 8 h_{(1)} h_{(2)} \rho_0 - 2 h_{(1)}^2 \rho_0'\big] \notag\\ & - \frac{1}{2} \big[12 h_{(4)} - 18 h_{(3)}\rho_0 - 12 h_{(2)} \rho_0' - 3 h_{(1)} \rho_0'' \big] \notag\\ & - \frac{1}{4} \left[ 6 h_{(3)} \rho_0 + 8 h_{(2)} \rho_0' + 3 h_{(1)} \rho_0'' \right] \notag\\ & + 2 d \rho_0 \otimes d \rho_0 \notag\\ & - d \rho_0 \otimes d\rho_0. \end{align} Note that the third and the fourth line can be summarized to -\frac{1}{4} (24 h_{(4)} - 30 h_{(3)} \rho_0 - 16 h_{(2)} \rho_0' - 3 h_{(1)} \rho_0''). Therefore, we find the formula \begin{align}\label{h4-gen} 24 h_{(4)} & = - 4 \Hess (\rho_0') - 8 \Hess (\rho_0^2) + 8 \rho_0 \Hess(\rho_0) - 2 \gamma \notag \\ & + \big[3 h_{(1)} h_{(3)} + 3 h_{(3)} h_{(1)} + 4 h_{(2)}^2 - 5 h_{(1)}^2 h_{(2)} + h_{(1)}^4 + 2 h_{(1)}^3 \rho_0 - 8 h_{(1)} h_{(2)} \rho_0 - 2 h_{(1)}^2 \rho_0'\big] \notag \\ & +30 h_{(3)} \rho_0 + 16 h_{(2)} \rho_0' + 3 h_{(1)} \rho_0'' \notag \\ & + 4 d \rho_0 \otimes d \rho_0. \end{align} Now we apply the known formulas for $h_{(j)}$ ($j \le 3$) (Proposition <ref>) and $\rho_0,\rho_0',\rho_0''$ (Lemma <ref>, Lemma <ref>) to make that sum fully explicit. First, we prove the remarkable simplification \begin{align}\label{rem-sim} & 3 h_{(1)} h_{(3)} + 3 h_{(3)} h_{(1)} + 4 h_{(2)}^2 - 5 h_{(1)}^2 h_{(2)} + h_{(1)}^4 + 2 h_{(1)}^3 \rho_0 - 8 h_{(1)} h_{(2)} \rho_0 - 2 h_{(1)}^2 \rho_0' \notag \\ & = 2 L \Hess(H) + 2 \Hess(H) L. \end{align} In fact, we calculate \begin{align} & 3 h_{(1)} h_{(3)} + 3 h_{(3)} h_{(1)} \\ & = 2 L \Hess (H) + 2 \Hess(H) L - 6 H L^2 \lo + 2 L^2 \|\lo\|^2 - 6 H L \lo L + 2 L^2 \|\lo\|^2 \end{align} \begin{align} 2 h_{(1)}^3 \rho_0 - 8h_{(1)} h_{(2)} \rho_0 - 2 h_{(1)}^2 \rho_0' = -16 H L^3 + 16 H L^2 \lo - 4 L^2 \|\lo\|^2. \end{align} The sum of these two results gives 2 L \Hess (H) + 2 \Hess(H) L + 4 H (\lo^3 + 2 H \lo^2 + H^2 \lo) - 16 H (\lo^3 + 3 H \lo^2 + 3 H^2 \lo + H^3 \id). Moreover, we get \begin{align} & 4 h_{(2)}^2 - 5 h_{(1)}^2 h_{(2)} + h_{(1)}^4 \\ & = 4 \lo^4 + 8 H \lo^3 + 4 H^2 \lo^2 - 20 \lo^4-60 H \lo^3 - 60 H^2 \lo^2 - 20 H^3 \lo \\ & + 16 \lo^4 + 64 H \lo^3 + 96 H^2 \lo^2 + 64 H^3 \lo + 16 H^4 \id \\ & = 12 H \lo^3 + 40 H^2 \lo^2 + 44 H^3 \lo + 16 H^4 \id. \end{align} Summing these identities proves (<ref>). The above results imply \begin{align} 12 h_{(4)} & = - \Hess (\|\lo\|^2) - 4 \Hess (H^2) + 4 H \Hess (H) + 2 d H \otimes dH - \gamma \\ & + L \Hess(H) + \Hess(H) L + \alpha \\ & = - \Hess (\|\lo\|^2) + L \Hess(H) + \Hess(H) L - 8 H \Hess (H) - 6 dH \otimes dH - \gamma + \alpha, \end{align} \begin{align} & \alpha \st 15 h_{(3)} \rho_0 + 8 h_{(2)} \rho_0' + 3/2 h_{(1)} \rho_0'' \\ & = - 5 H \Hess (H) + 15 H^2 L \lo - 5 H L \|\lo\|^2 + 4 L \lo \|\lo\|^2 + 3 L \rho_0'' \\ & = - 5 H \Hess (H) + 15 H^2 \lo^2 + 15 H^3 \lo - 5 H \lo \|\lo\|^2 - 5 H^2 \|\lo\|^2 \id + 4 \lo^2 \|\lo\|^2 + 4 H \lo \|\lo\|^2 \\ & + 3 L \rho_0'' \\ & = -5 H \Hess (H) + 4 \lo^2 \|\lo\|^2 + 5 H^2 (3 \lo^2 - \|\lo\|^2 \id) - H \lo \|\lo\|^2 + 15 H^3 \lo + 3 L \rho_0''. \end{align} Summarizing the last two results implies the first assertion. The formula for $\rho_0''$ in $n=3$ is a direct consequence of the formula for $\rho_0''$ for general $n$ (Lemma <ref>) using $(\lo,\JF) = \tr (\lo^3)$ and $\delta \delta (\lo) = 2 \Delta (H)$ (by Codazzi-Mainardi). Lemma <ref> implies 12 \tr (h_{(4)}) = - \Delta (\|\lo\|^2) - 9 H \Delta (H) + 2 (L,\Hess(H)) - 12 \|dH\|^2 + 4 \|\lo\|^4 + 9 H \rho_0''. It only remains to prove that \begin{equation}\label{trace-gamma} \tr (\gamma) = 6 (dH,dH). \end{equation} \begin{align} \tr (\gamma) & = 2 h^{jk} h^{lm} (\nabla_j(L)_{km} + \nabla_k(L)_{jm} - \nabla_m(L)_{jk}) \partial_l(H) \\ & = 2 h^{lm} (\delta(L)_m + \delta (L)_m - \nabla_m (\tr(L))) \partial_l(H) \\ & = 6 h^{lm} \partial_m(H) \partial_l(H) \\ & = 6 (dH,dH) \end{align} by $\delta(L) = 3 dH$ (Codazzi-Mainardi). The proof is complete. We proceed with the evaluation of (<ref>). \begin{align} & \tr(12 h_{(1)} h_{(3)} + 6 h_{(2)}^2 - 12 h_{(1)}^2 h_{(2)} + 3 h_{(1)}^4) \\ & = 8 (L,\Hess(H)) + 6 \tr (\lo^4) + 8 \|\lo\|^4 + 36 H \tr (\lo^3) + 126 H^2 \|\lo\|^2 + 144 H^4. \end{align} By the known formulas for the coefficients $h_{(k)}$ for $k \le 3$, we find \begin{align} & \tr(12 h_{(1)} h_{(3)} + 6 h_{(2)}^2 - 12 h_{(1)}^2 h_{(2)} + 3 h_{(1)}^4) \\ & = 8 \tr (L \Hess (H)) - 24 H \tr (L^2 \lo) + 8 \tr(L^2) \|\lo\|^2 + 6 \tr (L^2 \lo^2) - 48 \tr (L^3 \lo) + 48 \tr (L^4). \end{align} The latter sum equals the sum of $8 \tr (L \Hess (H))$ and \begin{align} & -24 H \tr (\lo^3 + 2H \lo^2) + 8 \tr(\lo^2 + 2H \lo + H^2 \id) \|\lo\|^2 + 6 \tr( (\lo^2 + 2H \lo + H^2) \lo^2) \\ & - 48 \tr (\lo^4 + 3 H \lo^3 + 3 H^2 \lo^2) + 48 \tr(\lo^4 + 4 H \lo^3 + 6 H^2 \lo^2 + H^4 \id). \end{align} The result follows by simplification. The following result evaluates the lower-order terms in (<ref>). \begin{align} & 6 \rho_0 \tr (3 h_{(3)} - 3 h_{(1)} h_{(2)} + h_{(1)}^3) + 6 \rho_0' \tr (2 h_{(2)} - h_{(1)}^2) + 3 \rho_0'' \tr (h_{(1)}) \notag \\ & = - 6 H \Delta (H) - 6 \|\lo\|^4 - 12 H \tr (\lo^3) - 108 H^2 \|\lo\|^2 - 144 H^4 + 18 H \rho_0''. \end{align} By $3 \tr (h_{(3)}) = \Delta (H)$, the sum equals \begin{align} & -6 H (\Delta (H) - 6 \tr (L^2 \lo) + 8 \tr (L^3)) + 6 \|\lo\|^2 \tr (L \lo - 2 L^2) + 18 H \rho_0'' \\ & = -6 H \Delta (H) + 36 H \tr (L^2 \lo) - 48 H \tr (L^3) - 6 \|\lo\|^4 - 36 H^2 \|\lo\|^2 + 18 H \rho_0'' \\ & = -6 H \Delta (H) + 36 H (\tr(\lo^3) + 2H \|\lo\|^2) - 48 ( H\tr(\lo^3) + 3H^2 \|\lo\|^2 + 3 H^4) \\ & - 6 \|\lo\|^4 -36 H^2 \|\lo\|^2 + 18 H \rho_0''. \end{align} Simplification completes the proof. Now we summarize the above results. We obtain \begin{align} 12 \B_3 & = \Delta (\|\lo\|^2) + 9 H \Delta (H) - 2 (L,\Hess(H)) + 12 \|dH\|^2 - 4 \|\lo\|^4 - 9 H \rho_0'' \\ & + 8 (L,\Hess(H)) + 6 \tr (\lo^4) + 8 \|\lo\|^4 + 36 H \tr (\lo^3) + 126 H^2 \|\lo\|^2 + 144 H^4 \\ & - 6 H \Delta (H) - 6 \|\lo\|^4 - 12 H \tr (\lo^3) - 108 H^2 \|\lo\|^2 - 144 H^4 + 18 H \rho_0'' \\ & = \Delta (\|\lo\|^2) + 12 \|dH\|^2 + 6 (\lo,\Hess(H)) - 2 \|\lo\|^4 + 6 \tr (\lo^4) \\ & + 9 H \Delta (H) + 18 H^2 \|\lo\|^2 + 24 H \tr (\lo^3) + 9 H \rho_0''. \end{align} The relation $\rho_0'' = - \Delta (H) - 2 \tr (\lo^3) - 2 H \|\lo\|^2$ implies the assertion. This completes the proof of (<ref>). The linear terms in the expansions (<ref>)-(<ref>) of Christoffel symbols show that \begin{align} 0 & = \Hess (H) + \frac{1}{4} (2 h_{(1)} h_{(2)} + 2 h_{(2)} h_{(1)} - h_{(1)}^3 - 2 \rho_0 h_{(1)}^2) \\ & - \frac{1}{2} ( 6h_{(3)} - 8 \rho_0 h_{(2)} - 4 \rho_0' h_{(1)}) \\ & - \frac{1}{4} (4 \rho_0 h_{(2)} + 4 \rho_0' h_{(1)}). \end{align} \begin{align} 0 & = \Hess(H) + h_{(1)} h_{(2)} - \frac{1}{4} h_{(1)}^3 + \frac{1}{2} H h_{(1)}^2 - 3 h_{(3)} - 3 H h_{(2)} + \rho_0' h_{(1)} \\ & = \Hess(H) - 3 h_{(3)} - 3 H L \lo + 2 \rho_0' L \end{align} using the formulas for $h_{(1)}$ and $h_{(2)}$ in Proposition <ref>. This reproduces the formula (<ref>) for $h_{(3)}$ for a flat background. We round up this section with a discussion of the relation of the formula for $\B_3$ in Proposition <ref> to alternative formulas in the literature. In <cit.> and <cit.>, it is stated that for a conformally flat background $\B_3$ equals $\BB_3$, where $\BB_3$ \begin{equation}\label{GGHW-B3} 6 \BB_3 \st \|\nabla \lo\|^2 + 2 (\lo,\Delta (\lo)) + 3/2 (\delta(\lo),\delta(\lo)) - 2 \J \|\lo\|^2 + \|\lo\|^4 \end{equation} with $\J = \J^h$. In the flat case, using $\delta(\lo) = 2 dH$ (Codazzi-Mainardi) and the Gauss identity \J = \frac{3}{2} H^2 - \frac{1}{4} \|\lo\|^2, this formula reads \begin{equation}\label{GW-B3} 6 \BB_3 = \|\nabla \lo\|^2 + 2 (\lo, \Delta (\lo)) + 6 \|dH\|^2 - 3H^2 \|\lo\|^2 + 3/2 \|\lo\|^4. \end{equation} For $n=3$, it holds \begin{equation}\label{Id-1} \frac{1}{2} \Delta (\|\lo\|^2) = 3 (\lo,\Hess(H)) + \|\nabla \lo\|^2 + 3 H \tr(L^3) - \|L\|^4 \end{equation} \begin{equation}\label{Id-2} \delta \delta (\lo^2) = 4 (\lo,\Hess(H)) + \|\nabla \lo\|^2 + 2 \|dH\|^2 + 3H \tr(L^3) - \|L\|^4. \end{equation} Hence we have the difference formula \begin{equation}\label{basic-div} \frac{1}{2} \Delta (\|\lo\|^2) - \delta \delta (\lo^2) = - (\lo,\Hess(H)) - 2 \|dH\|^2. \end{equation} As a consequence, we obtain $\BB_3 = \B_3$. On the one hand, we apply the identity (<ref>) to calculate \begin{align} 12 \B_3 & = 6 (\lo,\Hess(H)) + 2 \|\nabla \lo\|^2 + 6 H \tr(L^3) - 2 \|L\|^4 \\ & + 12 \|dH\|^2 + 6 (\lo,\Hess(H)) - 2 \|\lo\|^4 + 6 \tr (\lo^4) + 6 H \tr (\lo^3) \\ & = 12 (\lo,\Hess(H)) + 2 \|\nabla L\|^2 + 6 \|dH\|^2 \\ & + 6 H \tr(L^3) + 6 H \tr(\lo^3) - 2 \|L\|^4 - 2 \|\lo\|^4 + 6 \tr(\lo^4) \end{align} using the relation $\|\nabla L\|^2 = \|\nabla \lo\|^2 + 3 \|dH\|^2$. Hence 6 \B_3 = 6 (\lo,\Hess(H)) + \|\nabla L\|^2 + 3 \|dH\|^2 + 3 H \tr(L^3) + 3 H \tr(\lo^3) - \|L\|^4 - \|\lo\|^4 + 3 \tr(\lo^4). On the other hand, we use the identity $\Delta (\|\lo\|^2) = 2 (\lo,\Delta(\lo)) + 2 \|\nabla \lo\|^2$ and (<ref>) to find \begin{align} 6 \BB_3 & = \Delta (\|\lo\|^2) - \|\nabla \lo\|^2 + 6 \|dH\|^2- 3 H^2 \|\lo\|^2 + 3/2 \|\lo\|^4 \\ & = 6 (\lo,\Hess(H)) + \|\nabla L\|^2 + 3 \|dH\|^2 + 6 H \tr (L^3) - 3 H^2 \|\lo\|^2 -2 \|L\|^4 + 3/2 \|\lo\|^4. \end{align} Hence the difference $6 (\BB_3 - \B_3)$ equals \begin{align} & 3 H \tr(L^3) - 3 H \tr(\lo^3) - \|L\|^4 + \|\lo\|^4 + 3/2 \|\lo\|^4 - 3 H^2 \|\lo\|^2 - 3 \tr(\lo^4) \\ & = 3 H (3 H \tr (\lo^2) + 3 H^3) - 6H^2\|\lo\|^2 - 9H^4 - 3 H^2 \|\lo\|^2 + 3 (1/2 \|\lo\|^4 - \tr (\lo^4)) \\ & = 3 ((1/2 \|\lo\|^4 - \tr (\lo^4)) \\ & = 0 \end{align} by Corollary <ref>. This proves the assertion. Finally, we show that the formula for $\B_3$ established in Proposition <ref> implies For a conformally flat background, it holds \begin{equation}\label{B3-GW-0} 6 \B_3 = 3 \delta \delta ((\lo^2)_\circ) + 3 (\Rho,(\lo^2)_\circ) + \|\lo\|^4. \end{equation} Note that both $\B_3$ and the right-hand side of (<ref>) are conformally invariant. In fact, the operator $T: b \mapsto \delta \delta (b) + (\Rho,b)$ acting on trace-free symmetric bilinear forms $b$ on $M^3$ is well-known to be conformally invariant in the sense that $e^{4\varphi} \hat{T} (b) = T (b)$. In (<ref>), the operator $T$ acts on the trace-free part $(\lo^2)_\circ$ of $\lo^2$. The conformal invariance $\hat{\lo}^2 = \lo^2$ implies the conformal invariance of the trace-free part of $\lo^2$ (Section <ref>). This shows the claimed conformal invariance. In particular, the right-hand side of (<ref>) has the same conformal transformation law as $\B_3$. In more explicit terms, formula (<ref>) reads \begin{align}\label{B3-GW} 6 \B_3 & = 3 \delta \delta (\lo^2) - \Delta (\|\lo\|^2) + 3 (\Rho,\lo^2) - \|\lo\|^2 \J + \|\lo\|^4, \end{align} and it suffices to verify (<ref>) in the flat case. For a flat background, the formulas (<ref>) and (<ref>) are equivalent. The assertion is equivalent to the vanishing of the sum \begin{align} & 6 \delta \delta (\lo^2) - 2 \Delta (\|\lo\|^2) + 6 (\Rho,\lo^2) - 2 \|\lo\|^2 \J + 2 \|\lo\|^4 \\ & - \Delta (\|\lo\|^2) - 12 \|dH\|^2 - 6 (\lo,\Hess (H)) - \|\lo\|^4 - 6 H \tr (\lo^3). \end{align} But the identity \begin{equation}\label{Fial} \JF = \iota^* \bar{\Rho} - \Rho + H \lo + \frac{1}{2} H^2 h \stackrel{!}{=} \lo^2 - \frac{1}{4} \|\lo\|^2 h \end{equation} for the Fialkov tensor $\JF$ (see (<ref>)) and the Gauss identity $\J = 3/2 H^2 - 1/4 \|\lo\|^2$ (see (<ref>)) imply \begin{equation}\label{JP} 6(\lo^2,\Rho) - 2 \|\lo\|^2 \J = 6 H \tr(\lo^3) - \|\lo\|^4. \end{equation} Hence the above sum simplifies to 6 \delta \delta (\lo^2) - 3 \Delta (\|\lo\|^2) - 12 \|dH\|^2 - 6 (\lo,\Hess (H)). By (<ref>), this sum vanishes. The proof is complete. Alternatively, one may derive formula (<ref>) for conformally flat backgrounds by direct evaluation of the definition of the obstruction $\B_3$. For details, we refer to <cit.>. §.§ Variational aspects Here we relate the obstructions $\B_2$ (for general backgrounds) and $\B_3$ (for conformally flat backgrounds) to singular Yamabe energy functionals. The discussion illustrates the general results of <cit.> and connects with the classical literature. We first consider the classical situation of a variation of the Willmore functional. For a closed surface $f: M^2 \hookrightarrow \R^3$, we consider a normal variation $f_t: M^2 \hookrightarrow \R^3$ of $f$: f_t (x) = f (x) + t u(x) N_0, where $u \in C^\infty(M)$ and $N_0$ is the unit normal of $M$. The variation field of $f_t$ is $u N_0$. We set $M_t = f_t (M)$ and let $\W_2$ Willmore functional \begin{equation}\label{W2-flat} \W_2(f) \st \int_{f(M)} \|\lo\|^2 dvol_h, \end{equation} where the metric $h$ is induced by the Euclidean metric on $\R^3$. We often identify $M$ with $f(M)$. Set \var (\W_2)[u] \st (d/dt)\|_0 (\W_2(M_t)). In order to calculate that variation, we recall the well-known variation formulas \begin{align} \var (h)[u] & = 2 u L, \\ \var (L)[u] & = -\Hess(u) + u L^2, \\ 2 \var (H)[u] & = -\Delta (u) - u \|L\|^2 \end{align} \var(dvol_h)[u] = 2 u H dvol_h, where $L^2$, $\|L\|^2$ and $\Delta (u)$ are defined by the metric $h$ on $M$. First, we note that \var( \|\lo\|^2)[u] = 2 (\lo ,\var(\lo)[u]) - 4 u (L,\lo^2); the second term comes from raising $2$ indices: $\|\lo\|^2 = \tr (\lo^2) = h^{ia} h^{jb} \lo_{ij}\lo_{kb}$. \begin{align} \var (\|\lo\|^2)[u] & = 2 (\lo,\var(L)[u] - H \var (h)[u]) - 4 u \tr(\lo^3 + H \lo^2) \\ & = - 2 ((\lo,\Hess(u) - u L^2) + 2 H u (\lo,L)) + 4 u H \|\lo\|^2 \end{align} using $\tr(\lo^3)=0$. Now partial integration gives \begin{align} \var(\W_2)[u] & = - \int_M u \left[ 2 \delta \delta (\lo) -2 (\lo,L^2) + 4 H (\lo,L) + 4 H \|\lo\|^2 - 2 H \|\lo\|^2 \right] dvol_h; \end{align} the last term comes from the variation of the volume. Simplification yields \begin{equation}\label{W2} \var (\W_2)[u] = -\int_M u ( 2 \Delta (H) + 2 H \|\lo\|^2) dvol_h \end{equation} using $\delta \delta (\lo) = \Delta(H)$ and again $\tr (\lo^3)=0$. This proves the classical result that in a flat background, the Euler-Lagrange equation of the Willmore functional $\W_2$ is \Delta(H) + H \|\lo\|^2 = 0. By $\|\lo\|^2 = 2 (H^2- K)$, where $K$ is the Gauss curvature, the Euler-Lagrange equation of the Willmore functional $\W_2$ reads \Delta (H) + 2 H(H^2-K) = 0. This equation is known as the Willmore equation. It was already mentioned in <cit.> and Schadow (1922). We refer to <cit.> for more details. The variation formula (<ref>) implies the special case \begin{equation}\label{AB-2} \var(\A_2)[u] = \frac{3}{2} \int_{M^2} u \B_2 dvol_h \end{equation} of the variation formula \begin{equation}\label{var-form-1} \var(\A_n)[u] = (n+2)(n-1) \int_M u \LO_n dvol_h = \frac{(n+1)(n-1)}{2} \int_{M^n} u \B_n dvol_h \end{equation} for $\A_n = \int_M v_n dvol_h$. The first equality in (<ref>) was proved in <cit.>. The variational formula in terms of $\B_n$ was established in <cit.> by different arguments. For the second equality, we refer to (<ref>). In fact, (<ref>) shows that \A_2 = \int_{M^2} v_2 dvol = \int_{M^2} \left(-\frac{1}{2} \J + \frac{1}{4} \|\lo\|^2 \right) dvol_h. \var(\A_2)[u] = \frac{1}{4} \var(\W_2)[u] = -\frac{1}{2} \int_M u (\Delta (H) + H \|\lo\|^2) dvol_h by Gauss-Bonnet. On the other hand, we have \B_2 = -\frac{1}{3} (\Delta (H) + H\|\lo\|^2). This implies (<ref>). These results generalize as follows to closed surfaces $M^2 \hookrightarrow (X^3,g)$ in general backgrounds. For the following discussion, we also refer to <cit.>. We consider normal variations with a variation field of the form $u N_0$ with a unit normal field $N_0$. The following formula is well-known (see <cit.>, <cit.>, <cit.>). It can be proved by calculation in geodesic normal coordinates. For $u=1$, it plays a central role in <cit.>. It holds \begin{equation}\label{VL} \var(L_{ij})[u] = -\Hess_{ij}(u) + u (L^2)_{ij} - u \bar{R}_{0ij0}. \end{equation} \begin{align} \var(\|\lo\|^2)[u] & = 2 (\lo,\var(L)[u] - H \var(h)[u]) - 4 u \tr (\lo^3 + H \lo^2) \\ & = 2 (\lo, - \Hess (u) + u L^2) - 2 u \lo^{ij} R_{0ij0} - 4 H u (\lo,L) - 4 H \|\lo\|^2. \end{align} Now simplification and partial integration gives \var(\W_2)[u] = - \int_M u (2 \delta \delta (\lo) + 2 H \|\lo\|^2 + 2 \lo^{ij} R_{0ij0}) dvol_h. Since the Weyl tensor vanishes in dimension $3$, we have $\lo^{ij} R_{0ij0} = \lo^{ij} (\Rho_{ij} + \Rho_{00} h_{ij}) = \lo^{ij} \Rho_{ij}$. Thus the integrand is given by letting the operator \begin{equation}\label{op-2} b \mapsto \delta \delta (b) + (\iota^(\Rho), b) + H (\lo,b) \end{equation} act on $b = \lo$. The above operator is well-known to be conformally covariant on trace-free symmetric bilinear forms (see <cit.>). This implies the conformal invariance of the integrand. The above calculation shows that the Euler-Lagrange equation of the Willmore functional $\W_2$ is \begin{equation}\label{EL} \delta \delta (\lo) + H \|\lo\|^2 + \lo^{ij} \bar{R}_{0ij0} = 0. \end{equation} By Codazzi-Mainardi, $\delta \delta (\lo)$ equals $\Delta (H) + \delta (\bar{\Rho}_{0})$. Thus we obtain \Delta (H) + \delta (\overline{\Ric}_{0}) + H \|\lo\|^2 + \lo^{ij} \bar{R}_{0ij0} = 0. We also observe that the left-hand side of (<ref>) coincides with \delta \delta (\lo) + H \|\lo\|^2 + (\lo,\iota^ (\bar{\Rho})) = \delta \delta (\lo) + H \|\lo\|^2 + (\lo,\iota^* (\overline{\Ric})) since $\lo^{ij} \bar{R}_{0ij0} = \lo^{ij} \overline{\Ric}_{ij}$. In fact, since the Weyl tensor vanishes in dimension $3$, it holds \bar{R}_{0ij0} = \bar{\Rho}_{ij} + \bar{\Rho}_{00} h_{ij} and we obtain \lo^{ij} \bar{R}_{0ij0} = \lo^{ij} \bar{\Rho}_{ij} = \lo^{ij} \overline{\Ric}_{ij}. It follows that the variation of the Willmore functional $\W_2$ yields the Yamabe obstruction $\B_2$: \var (\A_2)[u] = - \frac{1}{4} \var (\W_2) [u] = -\frac{3}{2} \int u \B_2 dvol_h confirming (<ref>) for general backgrounds. $\W_3$ Willmore functional We continue with an analogous discussion of the variation of \begin{equation}\label{W3} \W_3 \st \int_{M^3} \tr(\lo^3) dvol_h \end{equation} for variations of a closed three-manifold $M^3 \hookrightarrow X^4$ in a conformally flat background $(X,g)$. We first determine the variation of $\W_3$ for a general background metric $g$. Here we use the variation \begin{align}\label{var-form} \var (h)[u] & = 2 u L, \notag \\ \var (L)[u] & = - \Hess(u) + u L^2 - u \bar{R}_{0 \cdot \cdot 0}, \notag \\ 3 \var (H)[u] & = - \Delta (u) - u \|L\|^2 - u \overline{\Ric}_{00} \end{align} \var(dvol_h)[u] = 3 u H dvol_h. Note that the operator $\Delta (u) + u \|L\|^2 + u \overline{\Ric}_{00}$ is the Jacobi operator appearing in the second variation formula for the area of minimal surfaces <cit.>. First, we observe that \var(\tr (\lo^3))[u] = 3 (\lo^2,\var(\lo)[u]) - 6 u (L,\lo^3); the second term comes from raising $3$ indices: $\tr (\lo^3) = h^{ia} h^{jb} h^{kc} \lo_{ij}\lo_{kb} \lo_{ca}$. \begin{align} & \var (\tr (\lo^3))[u] \\ & = 3 (\lo^2,\var(\lo)[u]) - 6 u (L,\lo^3) \\ & = 3 (\lo^2,\var (L)[u] - \var (H)[u] h - H \var(h)[u]) - 6 u (L,\lo^3) \\ & = 3 (\lo^2,-\Hess(u) + u L^2) - 3 u (\lo^2,\bar{R}_{0 \cdot\cdot 0}) + (\Delta(u) + u \|L\|^2 + u \overline{\Ric}_{00}) \|\lo\|^2 \\ & - 6 u H (\lo^2,L) - 6 u (L,\lo^3). \end{align} Now partial integration and simplification yields \begin{align}\label{var-W3-g} & \var\big(\int_M \tr(\lo^3) dvol_h\big)[u] \notag \\ & = \int_M u \big[ - 3 \delta \delta (\lo^2) + \Delta (\|\lo\|^2) - 3 \tr (\lo^4) + \|\lo\|^4 - 3 H \tr(\lo^3) \big] dvol_h \notag \\ & + \int_M u ( - 3 (\lo^2,\bar{R}_{0 \cdot \cdot 0}) + \|\lo^2\| \overline{\Ric}_{00}) dvol_h. \end{align} Now, for a conformally flat background, we reformulate this variation formula in a conformally invariant way. The following result is also covered in <cit.> using a different method. Assume that $n=3$ and that the Weyl tensor of $(X,g)$ vanishes. Then \begin{equation}\label{var-W3-CI} \var \left( \int_M \tr(\lo^3) dvol_h \right)[u] = - 3 \int_M u \left(\delta \delta ((\lo^2)_\circ) + (\Rho, (\lo^2)_\circ) + \frac{1}{3} \|\lo\|^4\right) dvol_h. \end{equation} By (<ref>), the claim is equivalent to the identity \begin{align} & \delta \delta ((\lo^2)_\circ) + (\Rho, (\lo^2)_\circ) + \frac{1}{3} \|\lo\|^4 \\ & = \delta \delta (\lo^2) - \frac{1}{3} \Delta (\|\lo\|^2) + \tr (\lo^4) - \frac{1}{3}\|\lo\|^4 + H \tr(\lo^3) + (\lo^2,\bar{R}_{0 \cdot \cdot 0}) - \frac{1}{3} \|\lo^2\| \overline{\Ric}_{00}. \end{align} Note that \delta \delta ((\lo^2)_\circ) = \delta \delta (\lo^2) - \frac{1}{3} \Delta (\|\lo\|^2) \begin{align} (\Rho, (\lo^2)_\circ) = (\Rho,\lo^2) - \frac{1}{3} (\Rho,\|\lo\|^2 h) = (\Ric,\lo^2) - \J \|\lo\|^2 - \frac{1}{3} \J \|\lo\|^2 = (\Ric,\lo^2) - \frac{4}{3} \J \|\lo\|^2. \end{align} Moreover, it holds \begin{align} (\lo^2,\bar{R}_{0 \cdot \cdot 0}) - \frac{1}{3} \|\lo^2\| \overline{\Ric}_{00} & = (\lo^2,\bar{\Rho}) + \bar{\Rho}_{00} \|\lo\|^2 - \frac{1}{3} \|\lo^2\| \overline{\Ric}_{00} \\ & = (\lo^2,\bar{\Rho}) + \frac{1}{3} \bar{\Rho}_{00} \|\lo\|^2 - \frac{1}{3} \bar{\J} \|\lo\|^2 \end{align} (by the vanishing of the Weyl tensor). Hence the claim reduces to \begin{align} & (\Ric,\lo^2) - \frac{4}{3} \J \|\lo\|^2 + \frac{1}{3} \|\lo\|^4 \\ & = \tr (\lo^4) - \frac{1}{3}\|\lo\|^4 + H \tr(\lo^3) + (\bar{\Rho},\lo^2) + \frac{1}{3} \bar{\Rho}_{00} \|\lo\|^2 - \frac{1}{3} \bar{\J} \|\lo\|^2 . \end{align} Now the Gauss equations \begin{align} \Ric_{ij} = \overline{\Ric}_{ij} - \bar{R}_{0ij0} + 3 H L_{ij} - (L^2)_{ij} \quad \mbox{and} \quad \J - \overline{\J} = - \bar{\Rho}_{00} - \frac{1}{4} \|\lo\|^2 + \frac{3}{2} H^2 \end{align} (see (<ref>) and (<ref>)) imply \begin{align} (\Ric,\lo^2) - \frac{4}{3} \J \|\lo\|^2 & = (\overline{\Ric},\lo^2) - (\bar{\Rho},\lo^2) - \bar{\Rho}_{00} \|\lo\|^2 + 3 H (L,\lo^2) - (L^2,\lo^2) \\ & - \frac{4}{3} \left(\bar{\J} - \bar{\Rho}_{00} - \frac{1}{4} \|\lo\|^2 + \frac{3}{2} H^2\right) \|\lo\|^2 \\ & = 2 (\bar{\Rho},\lo^2) + \bar{\J} \|\lo\|^2 - (\bar{\Rho},\lo^2) - \bar{\Rho}_{00} \|\lo\|^2 + \frac{4}{3} \bar{\Rho}_{00} \|\lo\|^2 - \frac{4}{3} \bar{\J} \|\lo\|^2 \\ & + 3 H(L,\lo^2) - (L^2,\lo^2) + \frac{1}{3} \|\lo\|^4 - 2 H^2 \|\lo\|^2 \\ & = (\bar{\Rho},\lo^2) + \frac{1}{3} \bar{\Rho}_{00} \|\lo\|^2 - \frac{1}{3} \bar{\J} \|\lo\|^2 \\ & + 3 H(L,\lo^2) - (L^2,\lo^2) + \frac{1}{3} \|\lo\|^4 - 2 H^2 \|\lo\|^2. \end{align} Hence it suffices to prove that 3 H(L,\lo^2) - (L^2,\lo^2) + \frac{2}{3} \|\lo\|^4 - 2 H^2 \|\lo\|^2 = \tr (\lo^4) - \frac{1}{3} \|\lo\|^4 + H \tr (\lo^3). By simplification, this identity is equivalent to \frac{2}{3}\|\lo\|^4 - \tr (\lo^4) = \tr (\lo^4) - \frac{1}{3} \|\lo\|^4, i.e., $\|\lo\|^4 = 2 \tr (\lo^4)$ (see Corollary <ref>). The proof is complete. In terms of \begin{equation}\label{A3} \A_3 = \int_{M^3} v_3 dvol_h = -\frac{2}{3} \int_{M^3} \tr (\lo^3) dvol_h = -\frac{2}{3} \W_3 \end{equation} we obtain \var (\A_3)[u] = 4 \int_{M^3} u \B_3 dvol_h. Combine (<ref>) with Lemma <ref>. This is a special case of (<ref>). Note that the second equality in (<ref>) follows by combining (<ref>) with (<ref>) and (<ref>). Another proof of Corollary <ref> (even for general backgrounds) has been given in [3] using a method that rests on a certain distributional calculus. Finally, the above classical style arguments have been extended to the general case in <cit.>. §.§ Low-order extrinsic $Q$-curvatures Here we discuss the low-order extrinsic $Q$-curvatures $\QC_2$ and $\QC_3$ from the perspective of their holographic formulas. We consider $\QC_2$ in general dimensions. The holographic formula (<ref>) states that -\QC_2 = \frac{1}{n-3} (2v_2 + (n-2) (v\J)_0) + \frac{1}{n-1} \T_1^\left(\frac{n}{2}-1\right) (v_1) for even $n$. Using $v_1 = (n-1)H$ and the formula (<ref>) for $v_2$ as well as $\T_1(\lambda)=-\lambda H$ (Lemma <ref>), we obtain - \QC_2 = \J_0 - \rho_0' + \frac{n}{2}H^2 = \J_0 - \Rho_{00} - \frac{\|\lo\|^2}{n-1} + \frac{n}{2} H^2. In particular, we see that the holographic formula makes sense for all $n \ge 2$. By the hypersurface Gauss identity (<ref>), the above formula simplifies to $\QC_2$ extrinsic $Q$-curvature of order $2$ \begin{equation}\label{Q2-gen} -\QC_2 = \J^h - \frac{\|\lo\|^2}{2(n-1)}. \end{equation} Note that this result fits with the formula $\PO_2$ second-order conformal Laplacian \begin{equation}\label{PO2} \PO_2 = \Delta_h - \left(\frac{n}{2}-1\right)\left (\J^h - \frac{\|\lo\|^2}{2(n-1)}\right) \end{equation} <cit.>. Independently, the latter formula for $\PO_2$ will be derived below from the solution operator $\T_2(\lambda)$ (Lemma <ref> and the discussion following it). We also recall that $\QC_2 = - Q_2$ in the Poincaré-Einstein case. Finally, we note that the formula for $\QC_2$ is singular for $n=1$ and $\Res_{n=1}(\QC_2) = \frac{1}{2} \|\lo\|^2 = 0$. The holographic formula (<ref>) for $\QC_3$ states that \begin{align} \tfrac{1}{4} \QC_3 & = \tfrac{1}{n-5} (6 v_3 + (n\!-\!3) (v\J)_1) + \tfrac{1}{n-3} \T_1^(\tfrac{n-3}{2}) (4 v_2 + (n\!-\!1) (v\J)_0) + \tfrac{1}{n-1} \T_2^(\tfrac{n-3}{2})(2v_1) \end{align} for even $n$. As a byproduct of the following discussion, we will see that the fractions in that formula do not prevent its validity in odd dimensions. First, we note that the above formula is equivalent to \begin{align} \frac{1}{4} \QC_3 & = \frac{1}{n\!-\!5} (6 v_3 + (n\!-\!3) \J'_0 + (n\!-\!3) v_1 \J_0) \\ & + \frac{1}{n\!-\!3} \T_1^\left(\frac{n-3}{2}\right) (4v_2 + (n\!-\!1) \J_0) + \frac{1}{n\!-\!1} \T_2^\left(\frac{n-3}{2}\right) (2v_1). \end{align} Now the formulas \begin{align} v_1 & = (n\!-\!1) H \\ 2 v_2 & = (n\!-\!3)(n\!-\!1) H^2 - \iota^* (\J) - (n\!-\!3) \iota^* \nabla_\NV(\rho) & \mbox{(by \eqref{v2n})} \\ 6 v_3 & = (n\!-\!5)(n\!-\!3)(n\!-\!1) H^3 - (3n\!-\!7) H \iota^* (\J) - (n\!-\!5)(3n\!-\!7) H \iota^* \nabla_\NV(\rho) \\ & - (n\!-\!5) \iota^* \nabla_\NV^2(\rho) - 2 \iota^* \nabla_\NV(\J) & \mbox{(by \eqref{v3-inter})} \end{align} show that the fractions are reduced. Indeed, it holds \begin{align} 6 v_3 + (n\!-\!3) \J'_0 + (n\!-\!3) v_1 \J_0 & = 0 \qquad \mbox{if $n=5$}, \\ 4 v_2 + (n\!-\!1) \J_0 & = 0 \qquad \mbox{if $n=3$} \end{align} (these are the relations mentioned in Remark <ref>). Then we obtain \begin{align}\label{Q3-ex-holo} \tfrac{1}{4} \QC_3 & = (n\!-\!3)(n\!-\!1) H^3 - (3n\!-\!7) H \iota^ \nabla_\NV(\rho) + \iota^* \nabla_\NV(\J) - \iota^* \nabla_\NV^2(\rho) + (n\!-\!2) H \iota^* (\J) \notag \\ & + \T_1^\left(\frac{n\!-\!3}{2}\right) (\iota^ (\J) - 2 \iota^* \nabla_\NV(\rho) + 2(n\!-\!1)H^2) + \T_2^\left(\frac{n\!-\!3}{2}\right) (2H). \end{align} A calculation using Lemma <ref>, Lemmas <ref>–<ref> and the hypersurface Gauss identity (<ref>) shows that \begin{equation}\label{Q3-rho} \frac{1}{4} \QC_3 = \Delta H + H \iota^ \J + \iota^* \nabla_\NV (\J) - (n\!-\!1) H \iota^* \nabla_\NV(\rho) - \iota^* \nabla_\NV^2(\rho). \end{equation} In particular, for $n=3$, we find \begin{align} \frac{1}{4} \QC_3 = \Delta H + H\iota^ (\J) + \iota^* \nabla_\NV(\J) - 2 H \iota^* \nabla_\NV(\rho) -\iota^* \nabla_\NV^2(\rho). \end{align} By comparison with Example <ref>, we see that 12 v_3 = - \QC_3 + 4\Delta H and consequently \begin{equation}\label{vQ3} 12 \int_{M^3} v_3 dvol_h = - \int_{M^3} \QC_3 dvol_h. \end{equation} This confirms Theorem <ref> for $n=3$. Formula (<ref>) also follows from the last display in the proof of <cit.> or <cit.> which evaluates the composition of three Laplace-Robin operators. We continue with the discussion of the holographic formula for $\QC_3$ in general dimensions $n \ge 3$. In fact, we prove that an evaluation of (<ref>) shows that the explicit formula (<ref>) for $\rho_0''$ is equivalent to a simple formula for $\QC_3$. Assume that $n \ge 3$. Then the formula (<ref>) for $\rho_0''$ is equivalent to $\QC_3$ extrinsic $Q$-curvature of order $3$ \begin{align}\label{Q3F} \frac{1}{4} \QC_3 & = \frac{1}{n\!-\!2} (\delta \delta (\lo) - (n\!-\!3) (\lo, \Rho^h) + (n\!-\!1) (\lo,\JF)) \\ & = \frac{1}{n\!-\!2} (\delta \delta (\lo) - 2(n\!-\!2) (\lo,\Rho^h) + (n\!-\!1) (\lo,\Rho^g) + (n\!-\!1) H \|\lo\|^2). \notag \end{align} Note that (<ref>) fits with the explicit formula for $\PO_3$ in Proposition <ref>. The proof of Proposition <ref> will show that the displayed formula for $\QC_3$ follows by combining the holographic formula for $\QC_3$ (in the form (<ref>)) with the formulas for the first two normal derivatives of $\rho$. Note also that in dimension $n=3$, the above formula reads $\QC_3$ critical extrinsic $Q$-curvature of order $3$ \begin{equation}\label{Q3-ex} \QC_3 = 4 \delta \delta (\lo) + 8 (\lo,\JF). \end{equation} It immediately follows from that expression that the integral $\int_{M^3} \QC_3 dvol_h$ is conformally invariant as a functional of $g$. It suffices to prove the equivalence of (<ref>) and the first identity. For this, we make explicit the equality of (<ref>) and (<ref>). In terms of adapted coordinates, it states the equality \begin{equation} \Delta H + H \J_0 + \J_0' - (n\!-\!1) H \rho_0' - \rho_0'' - 2 H \rho_0' = \frac{1}{n\!-\!2} \delta \delta (\lo) - \frac{n\!-\!3}{n\!-\!2} \lo^{ij} \Rho^h_{ij} + \frac{n\!-\!1}{n\!-\!2} \lo^{ij} \JF_{ij}. \end{equation} Here we used that $\iota^ \nabla_\NV^2(\rho)$ corresponds to $\rho_0'' - 2 \rho_0 \rho_0' = \rho_0'' + 2 H \rho_0'$ (see Example <ref>). By the identity (<ref>) for $\Delta H$, this relation is equivalent to \begin{equation} \rho_0'' = \frac{1}{n\!-\!1} \delta \delta (\lo) -\frac{1}{n\!-\!2} \delta \delta (\lo) + \frac{n\!-\!3}{n\!-\!2} \lo^{ij} \Rho^h_{ij} - \frac{n\!-\!1}{n\!-\!2} \lo^{ij} \JF_{ij} + H \J_0 - (n+1) H \rho_0' + \J_0'. \end{equation} H \J_0 - (n\!+\!1) H \rho_0' = - \frac{n\!+\!1}{n\!-\!1} H \Ric_{00} + \frac{1}{n\!-\!1} H \scal - \frac{n\!+\!1}{n\!-\!1} H \|\lo\|^2 shows that this expression for $\rho_0''$ coincides with the one given in (<ref>). The proof is complete. The above formula for $\QC_3$ is singular for $n=2$. But the second formula in (<ref>) shows that \begin{equation}\label{QB2} \Res_{n=2} (\QC_3) = 4 (\delta \delta (\lo) + (\lo, \Rho^g) + H \|\lo\|^2), \end{equation} up to the term $-8(n-2) (\lo,\Rho^h) = -8(\lo,\Ric^h)$ (which vanishes in dimension $n=2$ by $\Ric^h = K h$, $K$ being the Gauss curvature). The right-hand side of (<ref>) is proportional to the obstruction $\B_2$. From that perspective, its conformal invariance follows from the conformal covariance of $\PO_3$. The above argument to derive $\B_2$ from the constant term of $\PO_3$ is due to <cit.>. For the general relation between the singular Yamabe obstruction and the super-critical $Q$-curvature $Q_{n+1}$, we refer to Theorem <ref>. §.§ The pair $(\PO_3,\QC_3)$ The following explicit formula for $\PO_3$ was first proven in <cit.> by evaluation of the relevant composition of three Laplace-Robin operators.[See also the identical <cit.>.] $\PO_3$ extrinsic conformal Laplacian of order $3$ For $n \ge 3$, the operator \PO_3 = 8 \delta (\lo d) + \frac{n\!-\!3}{2} \frac{4}{n\!-\!2} (\delta \delta (\lo) - (n\!-\!3) (\lo,\Rho^h) + (n\!-\!1) (\lo,\JF)) is conformally covariant: e^{\frac{n+3}{2} \varphi} \circ \PO_3(\hat{g}) = \PO_3(g) \circ e^{\frac{n-3}{2}\varphi}, \; \varphi \in C^\infty(X). The formula for the leading term of $\PO_3$ will be derived in Lemma <ref>. We first calculate \begin{align} & e^{\frac{n+3}{2} \varphi} \hat{\delta} (\hat{\lo} d)(e^{-\frac{n-3}{2} \varphi} u) = \delta (e^{\frac{n-3}{2}\varphi} \lo d)(e^{-\frac{n-3}{2}\varphi} u) + \frac{n\!-\!3}{2} e^{\frac{n-3}{2}\varphi} i_{\grad(\varphi)}(\lo d)(e^{-\frac{n-3}{2} \varphi} u) \\ & = \delta (\lo d)(u) - \frac{n\!-\!3}{2} \delta (\lo d\varphi \cdot u) + \frac{n\!-\!3}{2} i_{\grad(\varphi)}(\lo du) - \left(\frac{n\!-\!3}{2}\right)^2 (d\varphi,\lo d\varphi) u \\ & = \delta (\lo d)(u) - \frac{n\!-\!3}{2} \delta (\lo d\varphi) u - \left(\frac{n\!-\!3}{2}\right)^2 (d\varphi,\lo d\varphi) u \end{align} using $\hat{\lo} = e^{-\varphi} \lo$ for the endomorphism $\lo$ corresponding to the trace-free second fundamental form and the transformation law $e^{(a+2)\varphi} \hat{\delta} e^{-a\varphi} = \delta + (n\!-\!2\!-\!a) i_{\grad(\varphi)}$ on $\Omega^1(M)$. Next, the term with the Schouten tensor yields (\lo,\Rho^h) - (\lo,\Hess(\varphi)) + (\lo,d\varphi \otimes \varphi). Finally, it holds \begin{align} & e^{3\varphi} \hat{\delta} \hat{\delta} (\hat{\lo}) = \delta(e^\varphi \hat{\delta} (e^\varphi \lo)) + (n\!-\!3) i_{\grad(\varphi)} e^{\varphi} \hat{\delta} (e^\varphi \lo) \\ & = \delta \delta (\lo) + (n\!-\!1) \delta (\lo d\varphi) + (n\!-\!3) i_{\grad(\varphi)} \delta (\lo) + (n\!-\!3)(n\!-\!1) i_{\grad(\varphi)} i_{\grad(\varphi)} (\lo) \end{align} using $\hat{\lo} = e^{\varphi} \lo$ and the transformation law e^{(a+2)\varphi} \hat{\delta} (e^{-a\varphi} b) = \delta (b) + (n\!-\!2\!-\!a) i_{\grad(\varphi)}(b) - \tr(b) d\varphi on symmetric bilinear forms $b$. Now simplification proves the claim. The operator $\frac{n-2}{4} \PO_3$ differs from the tractor calculus operator D_M^A L_{AB} D_X^B by the contribution of $(\lo,\JF)$ <cit.>. Here $L_{AB}$ is a tractor calculus version of the second fundamental form. This identification again implies its conformal covariance. Proposition <ref> yields an explicit formula for $\QC_3$ (see (<ref>)). As a direct cross-check of that formula, we note that in dimension $n=3$ \begin{align} e^{3\varphi} \hat{\QC}_3 & = 4 e^{3\varphi} \hat{\delta} \hat{\delta} (\hat{\lo}) + 8 e^{3\varphi} (\hat{\lo}, \hat{\JF})_{\hat{h}} \\ & = 4 \delta e^\varphi \hat{\delta} (e^{\varphi} \lo) +8 (\lo,\JF)_h \\ & = 4 (\delta \delta (\lo) + 2 \delta (\lo d)(\varphi)) + 8 (\lo,\JF)_h \\ & = \QC_3 + \PO_3 (\varphi). \end{align} This also confirms Theorem <ref> for $n=3$. Similar arguments provide an elementary proof of the conformal invariance of the obstruction $\B_2$. §.§ Low-order solution operators Under the assumption $\SCY$, we make the low-order solution operators $\T_1(\lambda)$ and $\T_2(\lambda)$ explicit in general dimensions. For that purpose, we use the ansatz $u = s^\lambda f + s^{\lambda+1} \T_1(\lambda) f + s^{\lambda+2} \T_2(\lambda) f + \dots$ for an approximate solution of the equation $-\Delta_{\sigma^{-2}g}(u)=\lambda(n-\lambda)u$ in adapted coordinates. $\T_1(\lambda) = - \lambda H$. By the assumption $\SCY$, the coefficients in (<ref>) expand as $a = 1-2s\rho + O(s^{n+1}) = 1 + 2s H + \cdots$ (using $\iota^* (\rho) = -H$) and $\tr (h_s^{-1} h'_s) = 2 \tr(L) + \cdots$. Now we let the operator (<ref>) act on $u$. The expansion of the result starts with s^2 \partial_s^2 (s^\lambda) f - (n-1) s\partial_s (s^\lambda) f \stackrel{!}{=} -\lambda(n-\lambda) s^\lambda f. The next term in the expansion reads \begin{align} & s^2 \partial_s^2(s^{\lambda+1}) \T_1(\lambda)(f)+ 2 s^3 H \partial_s^2(s^\lambda) f + s^2 \tr(L) \partial_s(s^\lambda) f \\ & - (n-1) s \partial_s(s^{\lambda+1}) \T_1(\lambda)(f) - 2(n-1) s^2 H \partial_s(s^\lambda) f + H s^2 \partial_s(s^\lambda) f \\ & = (\lambda\!-\!n\!+\!1)(\lambda\!+\!1) s^{\lambda+1} \T_1(\lambda)(f) + (2\lambda\!-\!n\!+\!1) \lambda H s^{\lambda+1} f. \end{align} This result equals $-\lambda(n-\lambda) s^{\lambda+1} \T_1(\lambda)(f)$ iff $\T_1(\lambda)(f)=-\lambda H$. It is worth emphasizing that $\T_1(\lambda)$ is regular in $\lambda$. In fact, this property does not hold for general asymptotically hyperbolic metrics <cit.>. As a consequence of Lemma <ref>, we find \begin{align}\label{D1-exp} \D_1^{res}(\lambda) & = (2\lambda\!+\!n\!-\!1)(\iota^* \partial_s + (v_1 + \T_1^(\lambda\!+\!n\!-\!1)) \iota^) \notag \\ & = (2\lambda\!+\!n\!-\!1)(\iota^* \partial_s -\lambda H \iota^). \end{align} This formula obviously confirms Theorem <ref> for the operator $\iota^ L(\lambda)$. The formula for $\T_2(\lambda)$ is a bit more complicated. Assume that $n \ge 2$. Then \begin{align} \T_2(\lambda) & = \frac{-\Delta_h + \lambda \J^h}{2(2\lambda\!-\!n\!+\!2)} + \frac{\lambda}{2} \Rho_{00} - \frac{\lambda}{2(2\lambda\!-\!n\!+\!2)} \left(\frac{n\!-\!1\!-\!2\lambda}{n\!-\!1} - \frac{1}{2(n\!-\!1)} \right) \|\lo\|^2 \\ & + \frac{\lambda}{2(2\lambda\!-\!n\!+\!2)} \left((\lambda\!+\!1)(2\lambda\!-\!n\!+\!3) - \frac{n}{2}\right) H^2. \end{align} By $\SCY$, the coefficients in (<ref>) expand as a = 1-2s\rho + \cdots = 1 + 2s H - 2s^2 \left(\Rho_{00} + \frac{\|\lo\|^2}{n-1}\right) + \cdots, up to a remainder in $O(s^{n+1})$, and \tr (h_s^{-1} h'_s) = 2 \tr(L) - s (2\Ric_{00} + 2 \|\lo\|^2 + 4n H^2) + \cdots. Hence the equality of the coefficients of $s^{\lambda+2}$ in the expansion of the eigenequation $\Delta_{s^{-2}\eta^( g)} u = -\lambda(n-\lambda) u$ yields the \begin{align} & 2(2\lambda\!-\!n\!+\!2) \T_2(\lambda)(f) + (\lambda\!+\!1)(2\lambda\!-\!n\!+\!3) H \T_1(\lambda)(f) \\ & = \left[-2\lambda(\lambda\!-\!1) \rho_0' - \lambda (\Ric_{00} + \|\lo\|^2) + 2(n\!-\!1) \lambda \rho_0' -2\lambda \rho_0' + \Delta \right](f) = 0. \end{align} This condition for $\T_2(\lambda)$ simplifies to \begin{align} & 2(2\lambda\!-\!n\!+\!2) \T_2(\lambda)(f) - \lambda(\lambda\!+\!1)(2\lambda\!-\!n\!+\!3) H^2 f \\ & = \lambda \left[ -2(n\!-\!1\!-\!\lambda) \rho_0' + \Ric_{00} + \|\lo\|^2 \right] f - \Delta f \\ & = \lambda \left [-2(n\!-\!1\!-\!\lambda)\left(\Rho_{00} + \frac{\|\lo\|^2}{n-1}\right) + ((n\!-\!1) \Rho_{00} + \J_0) + \|\lo\|^2 \right] f - \Delta f \\ & = -\lambda \left((n\!-\!1\!-\!2\lambda) \Rho_{00} + \frac{n\!-\!1\!-\!2\lambda}{n\!-\!1} \|\lo\|^2 - \J_0 \right) f - \Delta f \end{align} using Lemma <ref>. Finally, (<ref>) implies \begin{align} & 2(2\lambda\!-\!n\!+\!2) \T_2(\lambda)(f) \\ & = - \lambda \left((n\!-\!2\!-\!2\lambda) \Rho_{00} - \J^h + \left(\frac{n\!-\!1\!-\!2\lambda}{n\!-\!1} - \frac{1}{2(n\!-\!1)} \right) \|\lo\|^2 + \frac{n}{2} H^2 \right) f \\ & + \lambda(\lambda\!+\!1)(2\lambda\!-\!n\!+\!3) H^2 f - \Delta f. \end{align} This completes the proof. In particular, Lemma <ref> implies -4 \Res_{\frac{n}{2}-1}(\T_2(\lambda)) = \Delta_h - \left( \frac{n}{2}-1\right) \left(\J^h - \frac{ \|\lo\|^2}{2(n-1)} \right) = \PO_2(g). This is a special case of Theorem <ref>. It follows that -\QC_2(g) = \J^h - \frac{1}{2(n-1)} \|\lo\|^2. For $n=2$, Lemma <ref> implies \T_2(0)(1) = \frac{1}{4} \J^h - \frac{1}{8} \|\lo\|^2. For the function $\QC_2(\lambda)$ defined in (<ref>), it follows that $\QC_2(0)= - \J^h + \frac{1}{2} \|\lo\|^2 \stackrel{!}{=} \QC_2$. This confirms (<ref>) for $n=2$. The above results yield an explicit formula for $\D_2^{res}(\lambda)$. \begin{align}\label{D2-final} \D_2^{res}(\lambda) & = (2\lambda\!+\!n\!-\!3 ) \Big[(2\lambda\!+\!n\!-\!2) \iota^ \partial_s^2 - 2 (2\lambda\!+\!n\!-\!2)(\lambda\!-\!1) H \iota^* \partial_s \notag \\ & - \Big[ \Delta_h + \lambda \J^h - \lambda(2\lambda\!+\!n\!-\!2) \Rho_{00} - \lambda \left (\frac{2\lambda\!+\!n\!-\!2}{2(n\!-\!1)} + \frac{2\lambda\!+\!n\!-\!1}{2(n\!-\!1)} \right) \|\lo\|^2 \notag \\ & -(2\lambda\!+\!n\!-\!2)(\lambda\!-\!1/2) \lambda H^2\Big] \iota^* \Big]. \end{align} We omit the details of the calculation. This result fits with the formula for $\iota^* L(\lambda-1) L(\lambda)$ in <cit.>, \D_2^{res}(\lambda) \circ \eta^* = \iota^* L(\lambda-1) L(\lambda). In order to see this, it only remains to express the normal derivatives in the variable $s$ by iterated gradients. But $\iota^* \nabla_\NV$ corresponds to $\iota^\partial_s$ and $\iota^ \nabla_\NV^2$ corresponds to $\iota^* (\partial_s^2 + 2 H \partial_s)$ (see Example <ref>). Note that the prefactor in (<ref>) implies that $\D_2^{res}(-\frac{n-3}{2}) = 0$. The conformally covariant term in brackets (for $\lambda=-(n-3)/2$ and up to the contribution by $\|\lo\|^2$) has been used in <cit.> as a boundary operator associated to the Paneitz operator on $X$. For general $\lambda$, it appears in <cit.> (up to the term containing $\|\lo\|^2$). Concerning the classification of such boundary operators, we refer to <cit.>. Lemma <ref> immediately shows that $\D_2^{res}(-\frac{n}{2}+1)= \PO_2(g)\iota^$. In addition, we find the remarkable identity \begin{equation}\label{factor-big} \D_2^{res}\left(\frac{-n+1}{2}\right) \circ \eta^ = 2 \iota^* P_2(g), \end{equation} where $P_2(g)$ is the Yamabe operator of $g$. In fact, we calculate \begin{align} & \D_2^{res}\left(\frac{-n+1}{2}\right) \\ & = 2 \left(\iota^ \partial_s^2 + (n\!+\!1) H \iota^* \partial_s + \Delta_h \iota^* - \frac{n\!-\!1}{2} \left[\J^h + \Rho_{00} - \frac{n}{2} H^2 + \frac{1}{2(n\!-\!1)} \|\lo\|^2 \right] \iota^* \right) \\ & = 2 \left(\iota^* \partial_s^2 + (n\!+\!1) H \iota^* \partial_s + \Delta_h \iota^* - \frac{n\!-\!1}{2} \iota^* \J^g \right) \end{align} using (<ref>). Hence \begin{align} \D_2^{res}\left(\frac{-n\!+\!1}{2}\right) \circ \eta^* & = 2 \left(\iota^* \nabla_\NV^2 + (n\!-\!1) H \iota^* \nabla_\NV + \Delta_h \iota^* - \frac{n\!-\!1}{2} \iota^* \J^g \right) \\ & = 2 \iota^\left(\Delta_g - \frac{n\!-\!1}{2}\J^g \right) = 2 \iota^ P_2(g) \end{align} using $\iota^ \nabla_\NV^2 = (\iota^* \partial_s^2 + 2 H \iota^* \partial_s) \circ \eta^$ (Example <ref>) and the identity (<ref>). For Poincaré-Einstein metrics, the relation (<ref>) is one of the identities in a second set of so-called factorization identities <cit.>. It is an open problem whether the higher-order residue families in the general case continue to satisfy such identities. Finally, we note that Lemma <ref> implies that \begin{equation}\label{Q2-van} \QC^{res}_2(0) \st \D_2^{res}(0)(1) =0. \end{equation} This is a special case of the following conjecture. $\QC_N^{res}(0) \st \D_N^{res}(0)(1) \stackrel{!}{=} 0$ for $N \ge 1$. This vanishing result is well-known for residue families of even order in the Poincaré-Einstein case A conformally covariant second-order family of differential operators which interpolates between the GJMS operators $\iota^ P_2(g)$ and $P_2(h) \iota^$ (with $h = \iota^ (g))$ was given in <cit.>. This result suggests restating the above formula for $\D_2^{res}(\lambda)$ in the perhaps more enlightening \begin{align}\label{L2-nice} \iota^* L(\lambda-1) L(\lambda) & = (2\lambda\!+\!n\!-\!3) \Big[ (2\lambda\!+\!n\!-\!2) \iota^* P_2(g) - (2\lambda\!+\!n\!-\!1) \PO_2(g) \iota^* \notag \\ & -(2\lambda\!+\!n\!-\!1)(2\lambda\!+\!n\!-\!2) H \iota^* \nabla_\NV \notag \\ & + 2 \left(\lambda\!+\!\frac{n\!-\!1}{2}\right) \left(\lambda\!+\!\frac{n\!-\!2}{2}\right) (\QC_2(g) + \iota^* Q_2(g) + \lambda H^2) \iota^* \Big], \end{align} P_2(g) = \Delta_g - \frac{n\!-\!1}{2} \J^g = \Delta_g - \frac{n\!-\!1}{2} Q_2(g) is the Yamabe operator of $g$ and \PO_2(g) = \Delta_h - \left(\frac{n}{2}-1\right)\left (\J^h - \frac{\|\lo\|^2}{2(n-1)}\right) = \Delta_h + \left(\frac{n}{2}-1\right) \QC_2(g) is the extrinsic Yamabe operator on $M$ (see (<ref>)).[Note that we use the conventions $\PO_2(g)(1) = \frac{n-2}{2} \QC_2(g)$ on $M$ and $P_2(g)(1) = -\frac{n-1}{2} Q_2(g)$ on $X$.] In order to prove that formula, it is enough to relate the cubic polynomial in square brackets to the corresponding cubic polynomial in (<ref>). It is easy to relate the terms with derivatives. Moreover, the coincidence of the zeroth order terms follows from the Gauss identity - we omit the calculation. (<ref>) immediately shows that $\D_2^{res}(g;-\frac{n}{2}+1) = \PO_2(g) \iota^$ and \begin{equation} \D_2^{res}\left(g;\frac{-n+1}{2}\right) \circ \eta^* = 2 \iota^* P_2(g). \end{equation} The formula (<ref>) leads to a simple formula for the $Q$-curvature polynomial \QC_2^{res}(g;\lambda) \st \D_2^{res}(g;\lambda)(1). It holds \begin{align}\label{Q2-simple} & \QC_2^{res}(g;\lambda) = (2\lambda\!+\!n\!-\!3) \lambda \notag \\ & \times \left[ (2\lambda\!+\!n\!-\!2) \iota^ Q_2(g) + (2\lambda\!+\!n\!-\!1) \QC_2(g) + 2\left(\lambda\!+\!\frac{n\!-\!1}{2}\right) \left(\lambda\!+\!\frac{n\!-\!2}{2}\right) H^2 \right]. \end{align} This result clearly implies (<ref>). It suggests considering the quadratic polynomial in brackets as the actual interesting object. In the critical dimension $n=2$, we also find $\dot{\D}_2^{res}(0)(1) = \dot{\QC}_2^{res}(0) = -\QC_2$. One should compare Lemma <ref> with <cit.>. Note that in the Poincaré-Einstein case (i.e., if $g = r^2 g_+$), it holds $\iota^* Q_2(g) = \iota^* \J^g = \J^h = Q_2(h)$, $\QC_2(g) = - Q_2(h)$ (Remark <ref>) and $H=0$. Hence (<ref>) reduces to (2\lambda\!+\!n\!-\!3) \lambda \left[(2\lambda\!+\!n\!-\!2) Q_2(h) - (2\lambda\!+\!n\!-\!1) \QC_2(h) \right] = (2\lambda\!+\!n\!-\!3) \lambda Q_2(h) (see also Remark <ref>). It is an open problem whether for $N \ge 3$ the $Q$-curvature polynomial $\QC_N^{res}(\lambda) \st \D_N^{res}(\lambda)(1)$ similarly can be reduced to a lower degree polynomial and whether these polynomials admit a recursive description as in the Poincaré-Einstein case Next, we determine the leading part of $\PO_3$ from the leading part of the solution operator $\T_3(\lambda)$. $\PO_3$ is an operator of second-order. Let $\LT(\PO_3)$ denote its terms that contain one or two derivatives. The same notation will be used for other second-order operators. It holds \LT( \Res_{\lambda=\frac{n-3}{2}}(\T_3(\lambda)) = \frac{1}{3} \delta (\lo d). \LT(\PO_3) = 8 \delta (\lo d). The leading part of the solution operator $\T_3(\lambda)$ is determined by the equation \begin{align} & 3(2\lambda\!-\!n\!+\!3) \LT(\T_3(\lambda)) + 2 (\lambda\!+\!1)(\lambda\!+\!2) H \LT(\T_2(\lambda)) + (\lambda+2) \tr(L) \LT(\T_2(\lambda)) \\ & + (\lambda+2) H \LT(\T_2(\lambda)) - 2 (n\!-\!1)(\lambda+2) H \LT(\T_2(\lambda)) - (dH,d\cdot)_h \\ & + \LT(\Delta \T_1(\lambda)) + \Delta_h' = 0, \end{align} where $\Delta_{h_s} = \Delta_h + s \Delta_h' + \frac{s^2}{2!} \Delta_h'' + \cdots$. Simplification yields \begin{align} & -3(2\lambda\!-\!n\!+\!3) \LT(\T_3(\lambda)) \\ & = (\lambda\!+\!2)(2\lambda\!-\!n\!+\!5) H \LT(\T_2(\lambda)) - (dH,d\cdot)_h + \LT(\Delta \T_1(\lambda)) + \Delta'. \end{align} But the first variation $\Delta'$ of the Laplacian with respect to the variation $h_s$ of $h$ is given by (<ref>). Using $\LT( \T_2(\frac{n-3}{2}))= \frac{1}{2} \Delta$ (Lemma <ref>), it follows that $$-6 \LT( \Res_{\lambda=\frac{n-3}{2}}(\T_3(\lambda))$$ \begin{align} & \frac{n\!+\!1}{2} H \Delta - (dH,d\cdot)_h - \frac{n\!-\!3}{2} \LT(\Delta \circ H) -2H \Delta + (n\!-\!2) (dH,d\cdot)_h - 2 \delta (\lo d) \\ & = -(dH,d\cdot)_h - (n\!-\!3) (dH,d\cdot)_h + (n\!-\!2) (dH,d\cdot)_h - 2 \delta (\lo d) \\ & = - 2 \delta (\lo d). \end{align} This completes the proof. Similar arguments prove It holds \LT( \Res_{\lambda=\frac{n-5}{2}}(\T_5(\lambda)) = \frac{1}{30} (\Delta \delta (\lo d) + \delta(\lo ) \Delta). \LT(\PO_5) = 192 (\Delta \delta (\lo d) + \delta(\lo d) \Delta). These results are special cases of Proposition <ref>. §.§ Low order cases of Theorem <ref> The following result illustrates Theorem <ref> for $k=1,2$. Assume that $\sigma$ satisfies $\SCY$. Then \begin{equation} \iota^ L(-n\!+\!1) \left( \left( \frac{r}{\sigma} \right)^{n-1} \right) = - \frac{(n\!-\!1)^2}{2} H \end{equation} for $n \ge 1$ and \begin{align} & \iota^* L(-n\!+\!1) L(-n\!+\!2) \left( \left( \frac{r}{\sigma} \right)^{n-2} \right) \\ & = \frac{(n\!-\!1)(n\!-\!2)}{6} \left( -\frac{n\!-\!5}{n\!-\!1} \|\lo\|^2 - 2 (n\!-\!2) \iota^* \J^g + 2 (n\!-\!5) \J^h + \frac{(n\!-\!2)(n\!-\!3)}{2} H^2 \right) \end{align} for $n \ge 2$. Here $L(\lambda)$ is short for $L(g,\sigma;\lambda)$. By <cit.>, it holds $w_1 = \frac{n-1}{2} H$ and the right-hand side of the second identity equals $2(n\!-\!1)(n\!-\!2) w_2$. Hence w_1 = - \frac{1}{n-1} \iota^ L(-n\!+\!1)\left( \left( \frac{r}{\sigma} \right)^{n-1} \right) for $n \ge 2$ and = \frac{1}{2(n\!-\!1)(n\!-\!2)} \iota^* L(-n\!+\!1) L(-n\!+\!2) \left( \left(\frac{r}{\sigma} \right)^{n-2} \right) for $n \ge 3$. We expand in geodesic normal coordinates. First, we note that \left( \frac{r}{\sigma} \right)^{n-1} = 1 - (n\!-\!1) \sigma_{(2)} r + \cdots and recall that $\sigma_{(2)} = \frac{1}{2} H$. Now we calculate \begin{align} \iota^ L(-n\!+\!1) (1) = -(n\!-\!1)^2 H \quad \mbox{and} \quad \iota^* L(-n\!+\!1) (H r) = - (n\!-\!1) H \end{align} using $\rho_0 = -H$. This proves the first identity. Similarly, we find \left( \frac{r}{\sigma} \right)^{n-2} = 1 - (n\!-\!2) \sigma_{(2)} r + \left(-(n\!-\!2) \sigma_{(3)} + \binom{n\!-\!1}{2} \sigma_{(2)}^2\right) r^2 + \cdots. Now $\grad_g(\sigma) = \partial_r + H r \partial_r + \cdots$ and we obtain \begin{align} & \iota^* L(-n\!+\!1)L(-n\!+\!2)(1) \\ & = -(n\!-\!1) \iota^* (\partial_r + (n\!-\!1) H) ((-n\!+\!3) (\partial_r + H r \partial_r - (n\!-\!2) \rho) - r (\Delta_g - (n\!-\!2) \J^g))(1) \\ & = -(n\!-\!1)(n\!-\!2) ((n\!-\!3) \partial_r (\rho)\|_0 + \J^g\|_0 - (n\!-\!1)(n\!-\!3) H^2). \end{align} Note that this result coincides with $2(n\!-\!1)(n\!-\!2)v_2$, where $v_2$ is as in (<ref>). 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# A Closer Look at Temporal Sentence Grounding in Videos: Dataset and Metric Yitian Yuan<EMAIL_ADDRESS>Tsinghua University , Xiaohan Lan <EMAIL_ADDRESS>Tsinghua University , Xin Wang <EMAIL_ADDRESS>Tsinghua University & Pengcheng Laboratory , Long Chen<EMAIL_ADDRESS>Columbia University , Zhi Wang <EMAIL_ADDRESS>Tsinghua University and Wenwu Zhu∗ <EMAIL_ADDRESS>Tsinghua University & Pengcheng Laboratory (2021) ###### Abstract. Temporal Sentence Grounding in Videos (TSGV), i.e., grounding a natural language sentence which indicates complex human activities in a long and untrimmed video sequence, has received unprecedented attentions over the last few years. Although each newly proposed method plausibly can achieve better performance than previous ones, current TSGV models still tend to capture the moment annotation biases and fail to take full advantage of multi-modal inputs. Even more incredibly, several extremely simple baselines without training can also achieve state-of-the-art performance. In this paper, we take a closer look at the existing evaluation protocols for TSGV, and find that both the prevailing dataset splits and evaluation metrics are the devils to cause unreliable benchmarking. To this end, we propose to re-organize two widely-used TSGV benchmarks (ActivityNet Captions and Charades-STA). Specifically, we deliberately make the ground-truth moment distribution _different_ in the training and test splits, i.e., out-of-distribution (OOD) testing. Meanwhile, we introduce a new evaluation metric “dR@$n$,IoU@$m$” to calibrate the basic IoU scores by penalizing on the bias-influenced moment predictions and alleviate the inflating evaluations caused by the dataset annotation biases such as overlong ground-truth moments. Under our new evaluation protocol, we conduct extensive experiments and ablation studies on eight state-of-the-art TSGV methods. All the results demonstrate that the re- organized dataset splits and new metric can better monitor the progress in TSGV. Our reorganized datsets are available at https://github.com/yytzsy/grounding_changing_distribution. temporal sentence grounding in videos, dataset bias, evaluation metric, dataset re-splitting, out-of-distribution testing ††journalyear: 2021††copyright: acmcopyright††conference: Proceedings of the 2nd International Workshop on Human-centric Multimedia Analysis; October 20, 2021; Virtual Event, China††booktitle: Proceedings of the 2nd International Workshop on Human-centric Multimedia Analysis (HUMA ’21), October 20, 2021, Virtual Event, China††price: 15.00††doi: 10.1145/3475723.3484247††isbn: 978-1-4503-8671-5/21/10††ccs: Computing methodologies Artificial intelligence††ccs: Computing methodologies Natural language processing††ccs: Computing methodologies Computer vision ## 1\. Introduction Figure 1. (a): TSGV aims to localize a moment with the start timestamp (21.3s) and end timestamp (30.7s). (b): The performance comparisons of some SOTA TSGV models with Bias-based baseline (orange bar) on Charades-STA with evaluation metric R<EMAIL_ADDRESS>(c): The performance comparisons of some SOTA TSGV models with PredictAll baseline (orange bar) on ActivityNet Captions with evaluation metric R<EMAIL_ADDRESS> Detecting human activities of interest from untrimmed videos is a prominent and fundamental problem in video scene understanding. Early video action localization works (Shou et al., 2016; Wang et al., 2016) mainly focus on detecting activities belonging to some predefined categories (Shou et al., 2016; Wang et al., 2016), which extremely restrict their flexibility and can hardly cover various human activities in life. For this purpose, a more challenging but meaningful task which extends the limited categories to open natural language descriptions was proposed (Anne Hendricks et al., 2017; Gao et al., 2017; Hendricks et al., 2018), dubbed as Temporal Sentence Grounding in Videos (TSGV). As the example shown in Figure 1 (a), given a natural language query and an untrimmed video, TSGV needs to identify the start and end timestamps of one segment (i.e., moment) in the video, which semantically corresponds to the language query. Due to its profound significance, the TSGV task has received unprecedented attentions over the last few years — a surge of datasets (Anne Hendricks et al., 2017; Gao et al., 2017; Krishna et al., 2017; Regneri et al., 2013) and methods (Chen et al., 2018, 2020; Duan et al., 2018; Gao et al., 2019; Ge et al., 2019; Hahn et al., 2019; He et al., 2019; Jiang et al., 2019; Liu et al., 2018a, b; Lu et al., 2019; Wang et al., 2019; Wu et al., 2020; Xu et al., 2019; Yuan et al., 2019a, b; Zeng et al., 2020; Zhang et al., 2019a, 2020, b) have been developed. Although each newly proposed method can plausibly achieve better performance and make progress over previous ones, a recent study (Otani et al., 2020) shows that even today’s state-of-the-art (SOTA) TSGV models still fail to make full use of multi-modal inputs, i.e., they over-rely on the ground-truth moment annotation biases in current benchmarks, and lack of sufficient understanding of the multi-modal inputs. Specifically, taking one prevailing benchmark Charades-STA (Gao et al., 2017) as an example, suppose that there is a Bias-based baseline model which only makes predictions by sampling a moment from the frequency statistics of the ground-truth moment annotations in the training set. As illustrated in Figure 1 (b), this naive Bias-based model unexpected surpasses several SOTA deep models, i.e., Charades-STA has obvious moment location annotation biases. Therefore, _we argue that current TSGV datasets with heavily biased annotations cannot accurately monitor the progress in TSGV research._ Meanwhile, another characteristic of the ground-truth moment annotations in current TSGV benchmarks is that they usually have relatively long temporal durations. For example, 40% queries in the ActivityNet Captions dataset refer to a moment occupying over 30% temporal ranges of the whole input video. These overlong ground-truth moments incidentally make the current evaluation metrics unreliable. Specifically, the most prevalent evaluation metric for today’s TSGV is “R@$n$,IoU@$m$”, i.e., the percentage of testing samples which have at least one of the top-$n$ results with IoU larger than $m$. Due to the difficulty of the TSGV task, almost all published TSGV works tend to use a _small_ IoU threshold $m$ (i.e., 0.3) for evaluation, especially for challenging datasets (i.e., ActivityNet Captions (Krishna et al., 2017)). However, _we argue that the metric R@ $n$,IoU@$m$ with small $m$ is unreliable for the datasets with overlong ground-truth moment annotations_. For example, a small IoU threshold can be easily achieved by a long duration moment prediction. As an extreme case, a simple baseline which always directly takes the whole input video as the prediction (cf. the PredictAll baseline in Figure 1 (c)) can still achieve a SOTA performance under this metric. In this paper, to help disentangle the effects of ground-truth moment annotation biases, we propose to resplit the widely-used TSGV benchmarks (i.e., Charades-STA and ActivityNet Captions) by changing their ground-truth moment annotation distributions, obtaining two new evaluation benchmarks: Charades-CD (Charades-STA under Changing Distributions) and ActivityNet-CD. These new splits are created by re-organizing all splits (the training, validation and test sets) of original datasets, and the ground-truth moment distributions are deliberately designed _different_ in the training and test splits, i.e., out-of-distribution (OOD) testing. To better evaluate models’ generalization ability and compare the performance between the OOD samples and the independent and identically distributed (IID) samples, we also maintain a test split with IID samples, denoted as test-iid set (vs. test-ood set). Meanwhile, we propose a more reliable evaluation metric — dR@$n$, IoU@$m$ — for small threshold $m$. This metric calibrates the basic IoU scores with the temporal location discrepancy between the predicted and ground-truth moments, which is expected to reduce the influence of moment durations and restraint the inflating evaluations caused by overlong ground-truth moments in the datasets. To demonstrate the difficulty of our new splits and monitor the progress in TSGV, we evaluate the performance of eight representative SOTA models on our new evaluation protocol. Our key finding is that the performance of most tested models drops significantly when evaluated on the OOD samples (i.e., the test-ood set) compared to the IID samples (i.e., the test-iid set). This finding provides further evidences that existing methods only fit the moment annotation biases, and fail to bridge the semantic gaps between the video contents and natural language queries. Meanwhile, the proposed metric (dR@$n$,Iou@$m$) can effectively reduce the inflating performance caused by the annotation biases when the IoU threshold $m$ is small. In summary, we make three contributions in this paper: * • We propose new splits of two prevailing TSGV datasets, which are able to disentangle the effects of annotation biases. * • We propose a new metric: dR@$n$,IoU@$m$, which is more reliable than the existing metrics, especially when IoU threshold is small. * • We conduct extensive studies with several SOTA models. Consistent performance gaps between IID and OOD samples have proven that our new evaluation protocol can better monitor the progress in TSGV. ## 2\. Related Works ### 2.1. Temporal Sentence Grounding in Videos In this section, we coarsely group existing TSGV methods into four categories: _Two-Stage Methods._ Early TSGV methods typically solve this problem in a two- stage fashion: They first extract numerous video segment candidates by temporal sliding windows, and then either match the query sentence with these candidates (Anne Hendricks et al., 2017) or fuse query and video segment features to regress the final position, e.g., CTRL (Gao et al., 2017), ACL-K (Ge et al., 2019), SLTA (Jiang et al., 2019), ACRN (Liu et al., 2018a), ROLE (Liu et al., 2018b), VAL (Song and Han, 2018) and BPNet (Xiao et al., 2021). To speed up the sliding window processing, Xu et.al. (Xu et al., 2019) proposed QSPN, which injects text features early to generate segment candidates, and helps to eliminate the unlikely segment candidates and increases the grounding accuracy. _End-to-End Methods._ Besides the two-stage framework, some other TSGV works seek to solve the grounding problem in an end-to-end manner (Chen et al., 2018; Yuan et al., 2019a, b; Zeng et al., 2020; Zhang et al., 2019a, 2020, b). Chen et.al. (Chen et al., 2018) proposed TGN, which sequentially scores a set of temporal candidates ended at each frame and generates the final grounding result in one single pass. Similarly, ABLR model also processes video sequences via LSTMs (Yuan et al., 2019b), where the start and end timestamps of the predicted segments are regressed from the attention weights yielded by the multi-pass interaction between videos and queries. There are also some works leveraging temporal convolutional networks to solve the TSGV problem. Zhang et.al. (Zhang et al., 2019a) presented MAN, which assigns candidate segment representations aligned with language semantics over different temporal locations and scales in hierarchical temporal convolutional feature maps. Yuan et.al. (Yuan et al., 2019a) introduced the SCDM, where query semantic is used to control the feature normalization between different temporal convolutional layers, making the query-related video activities tightly compose together. Both MAN and SCDM only consider 1D temporal feature maps, while 2D-TAN (Zhang et al., 2020) models the temporal relations between video segments by a 2D map. In the 2D map, 2D-TAN encodes the adjacent temporal relation, and learns discriminative features for matching video segments with queries. _RL-based Methods._ Some recent models also regard the TSGV task as a sequence decision making problem, and resort to Reinforcement Learning (RL) algorithms. Specifically, Wang et.al. (Wang et al., 2019) introduced a semantic matching RL (SM-RL) model by extracting semantic concepts of videos and fusing them with global context features. Then, video contents are selectively observed and associated with the given sentence in a matching-based manner. Hahn et.al. (Hahn et al., 2019) presented TripNet, which uses RL to efficiently localize relevant activity clips in long videos, by learning how to intelligently hop around the video. Wu et.al. (Wu et al., 2020) formulated a tree-structured policy based progressive RL (TSP-PRL) model to sequentially regulate the predicated temporal boundaries by an iterative refinement process. _Weakly Supervised Methods._ Since the ground-truth annotations for the TSGV task are manually consuming, some works start to extend this problem to a weakly supervised scenario where the ground-truth segments are unavailable in the training stage (Duan et al., 2018; Gao et al., 2019; Mithun et al., 2019; Song et al., 2020; Tan et al., 2019). Mithun et.al. (Mithun et al., 2019) utilized a latent alignment between video frames and sentence descriptions with Text-Guided Attention (TGA), and TGA was used during the test stage to retrieve relevant moments. Duan et.al. (Duan et al., 2018) took the TSGV task as an intermediate step for dense video captioning, and then they established a cycle system and leveraged the captioning loss to train the whole model. Song et.al. (Song et al., 2020) presented a multi-level attentional reconstruction network, which leverages both intra- and inter-proposal interactions to learn a language-driven attention map, and can directly rank the candidate proposals at the inference stage. ### 2.2. Biases in TSGV Datasets A recent work (Otani et al., 2020) also discusses the dataset bias problem in current TSGV benchmarks. The main contribution of (Otani et al., 2020) is to find and analyse the moment annotation biases in previal benchmarks and perform human studies to demonstrate the disagreement among different annotators. In contrast, we propose to re-split these datasets to reduce these ground-truth moment annotation biases, and introduce a new metric to alleviate the inflating performance of SOTA models, i.e., we go one step further to build a more reasonable and reliable evaluation protocol. Meanwhile, we reproduce and analyse eight different SOTA methods from four different categories on both original and new evaluation protocols. ## 3\. Dataset and Metric Analysis ### 3.1. Dataset Analysis So far, there are four available TSGV datasets in our communities: DiDeMo (Anne Hendricks et al., 2017), TACoS (Regneri et al., 2013), Charades-STA (Gao et al., 2017) and ActivityNet Captions (Krishna et al., 2017). Since both DiDeMo and TACoS have some inherent and obvious disadvantages (e.g., For DiDeMo, the unit interval of annotations is five seconds; for TACoS, the visual scene is restricted in kitchen), dataset Charades-STA and ActivityNet Captions gradually become the mainstream benchmarks for TSGV evaluation (Chen et al., 2018; Hahn et al., 2019; Xu et al., 2019; Yuan et al., 2019a; Zeng et al., 2020; Zhang et al., 2020). The details about these two datasets are as follows: Charades-STA. It is built upon the Charades (Sigurdsson et al., 2016) dataset. The average length of videos in Charades is 30 seconds, and each video is annotated with multiple descriptions, action labels, action intervals, and classes of interacted objects. Gao et.al. (Gao et al., 2017) extended the Charades dataset to the TSGV task by assigning the temporal intervals to text descriptions and matching the common key words in the interval action labels and texts. In the official split (Gao et al., 2017), there are 5,338 videos and 12,408 query-moment pairs in the training set, and 1,334 videos and 3,720 query-moment pairs in the test set (cf. Table 1). ActivityNet Captions. It is originally developed for the dense video captioning task (Krishna et al., 2017). Since the official test set is withheld, previous TSGV works (Yuan et al., 2019a, b) merge the two available validation subsets “val1” and “val2” as the test set. In summary, there are 10,009 videos and 37,421 query-moment pairs in the training set, and 4,917 videos and 34,536 query-moment pairs in the test set. (cf. Table 1). Figure 2. The ground-truth moment annotation distributions of all query-moment pairs in Charades-STA and ActivityNet Captions. The deeper the color, the larger density in distributions. To examine the ground-truth moment annotation distributions of these two datasets, we normalize the start and end timestamps of all annotated moments in both the training and test sets, and use Gaussian kernel density estimation to fit the joint distribution of these normalized start and end timestamps. As shown in Figure 2, for both two datasets, the moment annotation distributions are almost identical in the training and test sets. For Charades-STA, most of the moments start at the beginning of the videos and end at around $20\%-40\%$ of the length of the videos. The moment annotation distributions present a strip with relatively uniform width, which indicates that the length of moment in Charades-STA roughly concentrates within a certain range. For ActivityNet Captions, the distributions are significantly different from those of Charades-STA, which concentrates in three local areas, i.e., the three corners. All these areas show that a considerable number of ground-truth moments start at the beginning of the video or end at the ending of the video, even exactly the same as the whole video (the left top area). This may be due to that the dataset ActivityNet Captions is originally annotated for dense video captioning, and the captions (queries) are always annotated based on the whole video. Therefore, we can observe that the ground-truth moment annotations in both benchmarks consists of strong biases. In other word, by fitting these moment annotation biases, a simple baseline can also achieve a state-of-the-art performance (cf. Figure 1). ### 3.2. Evaluation Metric Analysis To evaluate the temporal grounding accuracy, almost all existing TSGV works adopt the “R@$n$,IoU@$m$” as a standard evaluation metric. Specifically, for each query $q_{i}$, it first calculates the Intersection-over-Union (IoU) between the predicted moment and its ground-truth, and this metric is formally defined as: (1) $\text{R@$n$,IoU@$m$}=\frac{1}{N_{q}}\sum_{i}r(n,m,q_{i}),$ where $r(n,m,q_{i})=1$ if there is at least one of top-$n$ predicted moments of query $q_{i}$ having an IoU larger than threshold $m$, otherwise it equals to 0. $N_{q}$ is the total number of all queries. Most of previous TSGV methods (Chen et al., 2018; Liu et al., 2018b; Xu et al., 2019; Yuan et al., 2019b; Zhang et al., 2020) always report their scores on some small IoU thresholds like $m\in\\{0.1,0.3,0.5\\}$. However, as shown in Figure 3 (b), for dataset ActivityNet Captions, a substantial proportion of ground-truth moments have relatively long durations. Statistically, 40%, 20%, and 10% of sentence queries refer to a moment occupying over 30%, 50%, and 70% of the length of the whole video, respectively. Such annotation biases can obviously increase the chance of correct predictions under small IoU thresholds. Taking an extreme case as example, if the ground-truth moment is the whole video, any predictions with duration longer than 0.3 can achieve R<EMAIL_ADDRESS>Thus, metric R@$n$,IoU@$m$ with small $m$ is unreliable for current biased annotated datasets. Figure 3. The histogram of the normalized ground-truth moment durations in Charades-STA and ActivityNet Captions. Figure 4. (a) and (b) illustrate the ground-truth moment annotation distributions of each split in Charades-CD and ActivityNet-CD, respectively. (c) presents the moment annotation distributions of the query-indicated moments which contain action cook in the training and test-ood sets of Charades-CD. The deeper the color, the larger the density in the distribution. Figure 5. The frequency distributions of the top-30 actions in the query-moment pairs of different splits. The longer the bar, the more frequently the action appears. ## 4\. Proposed Evaluation Protocol ### 4.1. Dataset Re-splitting To accurately monitor the research progress in TSGV and reduce the influence of moment annotation biases, we propose to re-organize the two datasets (i.e., Charades-STA and ActivityNet Captions) by deliberately assigning different moment annotation distributions in each split. Particularly, each dataset is re-splitted into four sets: _training_ , _validation (val)_ , _test-iid_ , and _test-ood_. All samples in the training, val, and test-iid sets satisfy the independent and identical distribution, and the samples in test-ood set are out-of-distribution. The performance gap between the test-iid set and test-ood set can effectively reflect the generalization ability of the models. We name the two new re-organized datasets as Charades-CD and ActivityNet-CD. Dataset Aggregation and Splitting. For each dataset, we merge the training and test sets by aggregating all the query-moment pairs, i.e., Charades-STA has 12,408 + 3,720 = 16,128 pairs overall and ActivityNet Captions has 37,421 + 34,536 = 71,957 pairs in total (cf. Table 1). We first regard each query- moment pair as a data sample. Then, we use the Gaussian kernel density estimation as mentioned in Section 3.1 (cf. Figure 2) to fit the moment annotation distribution among these data samples. In the fitted distribution, each moment has a density value based on its temporal location in the video. We rank all the moments (as well as their paired queries) based on their density values in a descending order, and take the lower 20% data samples as the preliminary test-ood set, i.e., the temporal locations distribution of the preliminary test-ood set is furthest _different_ from the distribution of the whole dataset. The remaining 80% data sample are divided into the preliminary training set. Conflicting Video Elimination. Since each video is associated with multiple sentence queries, another concern is that we need to make sure that there is no video overlap between the training and test sets. Thus, after obtaining the preliminary test-ood set, we check whether the videos of these samples also appear in the preliminary training set. If it is the case, we move all samples (i.e., query-moment pairs) referring to the same video into the split with most of samples. In addition, to avoid the inflating performance of overlong predictions in ActivityNet-CD (cf. the PredictAll baseline in Figure 1), we leave all samples with ground-truth moment occupying over 50% of the length of the whole video into the training set. After eliminating all conflicting videos, we obtain the final test-ood set, which consists of around 20% query-moment pairs of the whole dataset. Then, we randomly divide the remaining samples (based on videos) into three splits: the training, val, and test-iid sets, which consist of around 70%, 5%, and 5% data samples, respectively. The statistics of the new proposed splits are reported in Table 1. Dataset \| Split \| # Videos \| # Pairs ---\|---\|---\|--- Charades-STA \| training \| 5,338 \| 12,408 test \| 1,334 \| 3,720 ActivityNet Captions \| training \| 10,009 \| 37,421 test \| 4,917 \| 34,536 Charades-CD \| training \| 4,564 \| 11,071 val \| 333 \| 859 test-iid \| 333 \| 823 test-ood \| 1,442 \| 3,375 ActivityNet-CD \| training \| 10,984 \| 51,415 val \| 746 \| 3,521 test-iid \| 746 \| 3,443 test-ood \| 2,450 \| 13,578 Table 1. The detailed statistics of the number of videos and query-moment pairs in different datasets and splits. ### 4.2. Charades-CD and ActivityNet-CD Moment Annotation Distributions. The ground-truth moment annotation distributions of Charades-CD and ActivityNet-CD are illustrated in Figure 4. From the Figure 4, we can observe that the moment annotation distributions of the test-ood set are significantly different from those of the other three sets (i.e., training, val, and test-iid sets). Compared with the moment annotation distributions of original test split (cf. Figure 2), the proposed test-ood split has several improvements: 1) For Charades-CD, the distributions of the start timestamps of the moments are more diverse (vs. concentrating on the beginning of the videos). 2) For ActivityNet-CD, more moments locate in relatively central areas of the videos, i.e., models will not perform well by over relying on the annotation biases. Action Distributions. We also investigate the action distributions of the original and re-organized datasets. Specifically, for each dataset, we extract the verbs from all sentence queries and count the frequency of each verb. Since the verb frequencies satisfy a long-tail distribution, we select the top-30 frequent verbs, which cover 92.7% of all action types in Charades-CD and 52.9% for ActivityNet-CD, respectively. The statistical results are illustrated in Figure 5. From this figure, we can observe that the new test- ood sets on both two datasets still have similar action distributions with the training set and original splits, which shows the OOD of moment annotations comes from each verb type. As shown in Figure 4 (c), the moment annotation distribution of the new training and test-ood set are totally different for the verb _cook_. ### 4.3. Proposed Evaluation Metric As discussed in Section 3.2, the most prevailing evaluation metric — R@$n$,IoU@$m$ — is unreliable under small threshold $m$. To alleviate this issue, as shown in Figure 6, we propose to calibrate the $r(n,m,q_{i})$ value by considering the “temporal distance” between the predicted and ground-truth moments. Specifically, we propose a new metric discounted-R@$n$,IoU@$m$ (dR@$n$,IoU@$m$): (2) $\text{dR@$n$,IoU@$m$}=\frac{1}{N_{q}}\sum_{i}r(n,m,q_{i})\cdot\alpha^{s}_{i}\cdot\alpha^{e}_{i},$ where $\alpha^{}_{i}=1-\text{abs}(p_{i}^{}-g_{i}^{})$, and $\text{abs}(p_{i}^{}-g_{i}^{})$ is the absolute distance between the boundaries of predicted and ground-truth moments. Both $p_{i}^{}$ and $g_{i}^{}$ are normalized to the range (0, 1) by dividing the whole video length. When the predicted and ground-truth moments are very close to each other, the discount ratio $\alpha^{}_{i}$ will be close to 1, i.e., the new metric can degrade to R@$n$,IoU@$m$ with exactly accurate predictions. Otherwise, even the IoU threshold condition is met, the score $r(n,m,q_{i})$ will still be discounted by $\alpha^{}_{i}$, which helps to alleviate the inflating recall scores under small IoU thresholds. With the proposed dR@$n$,IoU@$m$ metric, those speculation methods which over-rely on moments annotation biases (e.g., long moments annotations in ActivityNet Captions) will not perform well. Figure 6. An illustration of the proposed dR@$n$,IoU@$m$ metric. Figure 7. Performances (%) of SOTA TSGV methods on the test set of original splits (Charades-STA and ActivityNet Captions) and test sets (test-iid and test-ood) of proposed splits (Charades-CD and ActivityNet-CD). We use metric R<EMAIL_ADDRESS>in all cases. ## 5\. Experiments ### 5.1. Benchmarking the SOTA TSGV Methods To demonstrate the difficulty of the new proposed splits (i.e., Charades-CD and ActivityNet-CD), we compare the performance of two simple baselines and eight representative state-of-the-art methods on both the original and proposed splits. Specifically, we can categorize these methods into the following groups: • _Bias-based Method_ : It uses the Gaussian kernel density estimation to fit the moment annotation distribution, and randomly samples several locations based on the fitted distribution as the final moment predictions. * • _PredictAll Method_ : It directly predicts the whole video as the final moment predictions. * • _Two-Stage Methods_ : Cross-modal Temporal Regression Localizer (CTRL) (Gao et al., 2017), and Attentive Cross-modal Retrieval Network (ACRN) (Liu et al., 2018a). * • _End-to-End Methods_ : Attention-Based Location Regression (ABLR) (Yuan et al., 2019b), Semantic Conditioned Dynamic Modulation (SCDM) (Yuan et al., 2019a), 2D Temporal Adjacent Network (2D-TAN) (Zhang et al., 2020), and Dense Regression Network (DRN) (Zeng et al., 2020). * • _RL-based Method_ : Tree-Structured Policy based Progressive Reinforcement Learning (TSP-PRL) (Wu et al., 2020). * • _Weakly-supervised Method_ : Weakly-Supervised Sentence Localizer (WSSL) (Duan et al., 2018). For all these SOTA methods, we use the public available official implementations to get the temporal grounding results. The results of the proposed test-iid and test-ood sets on two datasets come from the same model finetuned on the val set. For more fair comparisons, we have unified the feature representations of the videos and sentence queries. To cater for most of TSGV methods, we use I3D feature (Carreira and Zisserman, 2017) for the videos in dataset Charades-STA (Charades-CD), and C3D feature (Tran et al., 2015) for the videos in dataset ActivityNet Captions (Activity-CD). Each word in the query sentences is encoded by a pretrained GloVe (Pennington et al., 2014) word embedding. Method \| Split \| Charades-CD \| ActivityNet-CD ---\|---\|---\|--- m=0.1 \| m=0.3 \| m=0.5 \| m=0.7 \| m=0.9 \| m=0.1 \| m=0.3 \| m=0.5 \| m=0.7 \| m=0.9 Bias-based \| test-iid \| 31.42 \| 26.25 \| 16.87 \| 9.34 \| 2.70 \| 36.15 \| 29.31 \| 19.81 \| 12.27 \| 7.68 test-ood \| 14.75 \| 9.30 \| 5.04 \| 2.21 \| 0.55 \| 21.89 \| 9.21 \| 0.26 \| 0.11 \| 0.03 PredictAll \| test-iid \| 31.04 \| 10.93 \| 0.00 \| 0.00 \| 0.00 \| 36.43 \| 29.62 \| 20.05 \| 12.45 \| 7.83 test-ood \| 37.43 \| 27.13 \| 0.06 \| 0.00 \| 0.00 \| 21.87 \| 9.01 \| 0.00 \| 0.00 \| 0.00 CTRL (Gao et al., 2017) \| test-iid \| 50.61 \| 42.65 \| 29.80 \| 11.86 \| 1.41 \| 27.34 \| 19.42 \| 11.27 \| 4.29 \| 0.25 test-ood \| 52.80 \| 44.97 \| 30.73 \| 11.97 \| 1.12 \| 26.23 \| 15.68 \| 7.89 \| 2.53 \| 0.20 ACRN (Liu et al., 2018a) \| test-iid \| 53.22 \| 47.50 \| 31.77 \| 12.93 \| 0.71 \| 27.69 \| 20.06 \| 11.57 \| 4.41 \| 0.75 test-ood \| 53.36 \| 44.69 \| 30.03 \| 11.89 \| 1.38 \| 27.03 \| 16.06 \| 7.58 \| 2.48 \| 0.17 ABLR (Yuan et al., 2019b) \| test-iid \| 59.26 \| 52.26 \| 41.13 \| 23.50 \| 3.66 \| 55.62 \| 46.86 \| 35.45 \| 20.57 \| 6.32 test-ood \| 54.09 \| 44.62 \| 31.57 \| 11.38 \| 1.39 \| 46.88 \| 33.45 \| 20.88 \| 10.03 \| 2.31 SCDM (Yuan et al., 2019a) \| test-iid \| 62.47 \| 58.14 \| 47.36 \| 30.79 \| 6.62 \| 55.15 \| 46.44 \| 35.15 \| 22.04 \| 6.07 test-ood \| 59.08 \| 52.38 \| 41.60 \| 22.22 \| 3.81 \| 45.08 \| 31.56 \| 19.14 \| 9.31 \| 1.94 2D-TAN (Zhang et al., 2020) \| test-iid \| 59.80 \| 53.71 \| 43.46 \| 24.99 \| 6.95 \| 57.11 \| 49.18 \| 39.63 \| 27.36 \| 9.00 test-ood \| 50.87 \| 43.45 \| 30.77 \| 11.75 \| 1.92 \| 44.37 \| 30.86 \| 18.38 \| 9.11 \| 2.05 DRN (Zeng et al., 2020) \| test-iid \| 57.03 \| 51.35 \| 41.91 \| 26.74 \| 6.46 \| 56.96 \| 48.92 \| 39.27 \| 25.71 \| 6.81 test-ood \| 49.17 \| 40.45 \| 30.43 \| 15.91 \| 3.13 \| 47.50 \| 36.86 \| 25.15 \| 14.33 \| 3.76 TSP-PRL (Wu et al., 2020) \| test-iid \| 54.60 \| 46.44 \| 35.43 \| 17.01 \| 3.57 \| 53.98 \| 44.93 \| 33.93 \| 19.50 \| 4.79 test-ood \| 42.21 \| 31.93 \| 19.37 \| 6.20 \| 1.16 \| 44.23 \| 29.61 \| 16.63 \| 7.43 \| 1.46 WSSL (Duan et al., 2018) \| test-iid \| 45.90 \| 34.99 \| 14.06 \| 4.27 \| 0.00 \| 36.67 \| 26.06 \| 17.20 \| 6.16 \| 1.24 test-ood \| 49.92 \| 35.86 \| 23.67 \| 8.27 \| 0.06 \| 30.71 \| 17.00 \| 7.17 \| 1.82 \| 0.17 Table 2. Performances (%) of SOTA TSGV methods on the Charades-CD and ActivityNet-CD datasets with metric dR@$1$,IoU@$m$. ### 5.2. Performance Comparisons on the Original and Proposed Data Splits We report the performance of all mentioned TSGV methods with metric R<EMAIL_ADDRESS>in Figure 7. From Figure 7, we can observe that almost all methods have a significant performance gap between the test-iid and test-ood sets, i.e., these methods always over-rely on the moment annotation biases, and fail to generalize to the OOD testing. Meanwhile, the performance results on the original test set and the proposed test-iid set are relatively close, which shows that the moment distribution of the test-iid set is similar to the majority of the whole dataset. We provide more detailed experimental result analyses in the following: Baseline Methods. After changing the moment annotation distributions in different splits, the Bias-based method cannot take advantage of the annotation biases and its performance degrades from 13.6% on the test-iid set to 0.1% on the test-ood set of ActivityNet-CD. For the PredictAll method, since all the ground-truth moments in Charades-CD are less than 50% range of the whole videos, naively predicting the whole video as the grounding results will inevitably cause the R<EMAIL_ADDRESS>scores to 0.0 on this dataset. Since the ground-truth moments in ActivityNet-CD are much longer, the PredictAll method achieves high results at 11.9% and 13.8% on the original test set and new test-iid set, respectively. However, in the test-ood set where the longer segments are excluded, the PredictAll method also degrades its performance to 0.0. Two-Stage Methods. We find that the two-stage methods (i.e., CTRL and ACRN) are less sensitive to the domain gaps between the test-iid and test-ood sets. This is due to that they use a sliding-window strategy to retrieve video moment candidates, and compare these moment candidates with each query sentence individually. In this manner, all moment candidates are independent to the overall video contents, and the moment annotation distributions have less influence on the model performance. We can also observe that the performance of these two methods on the test-ood set of ActivityNet-CD presents a more obvious drop compared to the performance on test-iid set. In contrast, the performance on the test-iid and test-ood sets of Charades-CD are competing. The main reason is that the ground-truth moments in the test-ood set of Charades-CD always occupy a longer range over the whole videos (cf. Figure 4 (a), which makes the sliding windows have more chance to hit the ground-truth moments. In summary, although CTRL and ACRN are less sensitive to the moment annotation biases, their grounding performances are still far behind other types of SOTA methods, e.g., SCDM and DRN. End-to-End Methods. As for the end-to-end methods (i.e., ABLR, SCDM, 2D-TAN and DRN), we can observe that their performances all drop significantly on the test-ood set compared to the test-iid set on both two datasets. These methods all have considered the whole video contexts and temporal information. The initial intention for this design is that numerous queries often contain some words referring to temporal orders and locations such as “before”, “after”, “begin” and “end”, or they want to encode the important temporal relations between video moments. Unfortunately, although our test-ood split does not break any video temporal relations, their performance on the OOD testing still drop significantly. This demonstrates that current methods do not play their advantages and fail to utilize the video temporal relation or vision-language interaction for TSGV. RL-based Method. The RL-based method TSP-PRL also suffers from obvious performance drops on the test-ood set compared to the test-iid set. Actually, TSP-PRL adopts IoU between the predicted and ground-truth moment as the training reward in the RL framework. In this case, the temporal annotations directly affect the model learning, and the changes of moment annotation distributions will inevitably influence the model performance. Weakly-supervised Method. The results of the weakly-supervised method WSSL is _thought-provoking_ : it achieves better performance on test-ood set compared to test-iid set in Charades-CD, but results of these splits in ActivityNet-CD are exactly the reverse. After carefully checking the predicted moment results, we find that the normalized (start, end) moment predictions on both two datasets converge on several certain predictions (i.e., (0, 1), (0, 0.5), (0.5, 1)). These results indicate that the WSSL method does not learn to align the video and sentence semantics at all. Instead, it only speculatively guesses several possible locations. ### 5.3. Performance Evaluation with dR@$n$,IoU@$m$ We report the performance of all mentioned TSGV methods with our proposed metric dR@$1$,IoU@$m$ in Table 2. The trend of performance drop on the test- ood set compared to the test-iid set in Table 2 is similar to that in Figure 7. These results verify again that current TSGV methods suffer from severe temporal annotation biases in the datasets, and fail to generalize to the OOD testing. Meanwhile, by comparing Table 2 and Figure 7, we can observe that the dR@$1$,IoU@$m$ values are smaller than the R@$1$,IoU@$m$ values. For example, the SCDM model achieves score 32.5% in R<EMAIL_ADDRESS>while score 30.8% in dR<EMAIL_ADDRESS>on the test-iid set of Charades-CD. Such phenomenon is adhere to our definition of dR@$1$,IoU@$m$. For more clearer illustration, we further compare the dR@$1$ and R@$1$ scores under different IoUs of some SOTA methods in Figure 8. When the IoU threshold is small, dR@$1$ is much lower than R@$1$, and the gap between them gradually decreases with the increase of IoU threshold. Interestingly, we find that the naive Bias-based baseline achieves even better results than SCDM and 2D-TAN methods in the R<EMAIL_ADDRESS>metric, while reversely in the dR<EMAIL_ADDRESS>metric. These results indicate that recall values under small IoU thresholds are unreliable and overrated: although some moment predictions meet the IoU requirement, they still have a great discrepancy to the ground-truth moments. Instead, our proposed dR@$n$,IoU@$m$ metric can alleviate this problem since it can discount the recall value based on the temporal distance between the predicted and ground- truth moment temporal locations. When the prediction meets the larger IoU requirements, the discount will be smaller, i.e., the dR@$n$,IoU@$m$ values and R@$n$,IoU@$m$ values will be closer to each other. Therefore, our predicted dR@$n$,IoU@$m$ metric is more stable on different IoU thresholds, and it can suppress some inflating results (such as Bias-based or PredictAll baselines) caused by the moment annotation biases in the datasets. Meanwhile, these results further reveal that it is more reliable to report the grounding accuracy on large IoUs. Figure 8. Performance (%) comparisons of SOTA TSGV methods between original metric (R@$1$,IoU@$m$) and proposed metric (dR@$1$,IoU@$m$). All results come from the test set of ActivityNet Captions. ## 6\. Conclusion In this paper, we take a closer look at the existing evaluation protocol of the Temporal Sentence Grounding in Videos (TSGV) task, and we find that both the prevailing dataset splits and evaluation metric are the devils to cause the unreliable benchmarking: the datasets have obvious moment annotation biases and the metric is prone to overrating the model performance. To solve these problems, we propose to re-split the current Charades-STA and ActivityNet Captions datasets by making the ground-truth moment annotation distributions different in the training and test set. Meanwhile, we propose a new evaluation metric to alleviate the inflating evaluations caused by dataset annotation biases such as overlong ground-truth moments. The proposed data splits and metric serve as a promising test-bed to monitor the progress in TSGV. We also thoroughly evaluate eight state-of-the-art TSGV methods with the new evaluation protocol, opening the door for future research. ## 7\. 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Checking Robustness Between Weak Transactional Consistency ModelsThis work is supported in part by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 678177). Sidi Mohamed Beillahi() Ahmed Bouajjani Constantin Enea S.M. Beillahi, A. Bouajjani, and C. Enea. Université de Paris, IRIF, CNRS, Paris, France, Concurrent accesses to databases are typically encapsulated in transactions in order to enable isolation from other concurrent computations and resilience to failures. Modern databases provide transactions with various semantics corresponding to different trade-offs between consistency and availability. Since a weaker consistency model provides better performance, an important issue is investigating the weakest level of consistency needed by a given program (to satisfy its specification). As a way of dealing with this issue, we investigate the problem of checking whether a given program has the same set of behaviors when replacing a consistency model with a weaker one. This property known as robustness generally implies that any specification of the program is preserved when weakening the consistency. We focus on the robustness problem for consistency models which are weaker than standard serializability, namely, causal consistency, prefix consistency, and snapshot isolation. We show that checking robustness between these models is polynomial time reducible to a state reachability problem under serializability. We use this reduction to also derive a pragmatic proof technique based on Lipton's reduction theory that allows to prove programs robust. We have applied our techniques to several challenging applications drawn from the literature of distributed systems and databases. § INTRODUCTION Concurrent accesses to databases are typically encapsulated in transactions in order to enable isolation from other concurrent computations and resilience to failures. Modern databases provide transactions with various semantics corresponding to different tradeoffs between consistency and availability. The strongest consistency level is achieved with serializable transactions [41] whose outcome in concurrent executions is the same as if the transactions were executed atomically in some order. Since serializability () carries a significant penalty on availability, modern databases often provide weaker consistency models, e.g., causal consistency () [37], prefix consistency () [21, 24], and snapshot isolation () [11]. Causal consistency requires that if a transaction $\tr_1$ “affects” another transaction $\tr_2$, e.g., $\tr_1$ executes before $\tr_2$ in the same session or $\tr_2$ reads a value written by $\tr_1$, then the updates in these two transactions are observed by any other transaction in this order. Concurrent transactions, which are not causally related to each other, can be observed in different orders, leading to behaviors that are not possible under . Prefix consistency requires that there is a total commit order between all the transactions such that each transaction observes all the updates in a prefix of this sequence ( is stronger than ). Two transactions can observe the same prefix, which leads to behaviors that are not admitted by . Snapshot isolation further requires that two different transactions observe different prefixes if they both write to a common variable. Since a weaker consistency model provides better performance, an important issue is identifying the weakest level of consistency needed by a program (to satisfy its specification). One way to tackle this issue is checking whether a program $P$ designed under a consistency model $S$ has the same behaviors when run under a weaker consistency model $W$. This property of a program is generally known as robustness against substituting $S$ with $W$. It implies that any specification of $P$ is preserved when weakening the consistency model (from $S$ to $W$). Preserving any specification is convenient since specifications are rarely present in practice. The problem of checking robustness for a given program has been investigated in several recent works, but only when the stronger model ($S$) is , e.g., [9, 10, 18, 25, 12, 39], or sequential consistency in the non-transactional case, e.g. [35, 14, 28]. there is a large class of specifications that can be implemented even in the presence of “anomalies”, i.e., behaviors which are not admitted under (see [45] for a discussion). In this context, an important question is whether a certain implementation (program) is robust against substituting a weak consistency model, e.g., , with a weaker one, e.g., . In this paper, we consider the sequence of increasingly strong consistency models mentioned above, , , and , and investigate the problem of checking robustness for a given program against weakening the consistency model to one in this range. We study the asymptotic complexity of this problem and propose effective techniques for establishing robustness based on abstraction. There are two important cases to consider: robustness against substituting with and with , respectively. Robustness against substituting with can be obtained as the conjunction of these two cases. In the first case ( vs ), checking robustness for a program $P$ is reduced to a reachability (assertion checking) problem in a composition of $P$ under with a monitor that checks whether a behavior is an “anomaly”, i.e., admitted by $P$ under , but not under . This approach raises two non-trivial challenges: (1) defining a monitor for detecting vs anomalies that uses a minimal amount of auxiliary memory (to remember past events), and (2) determining the complexity of checking if the composition of $P$ with the monitor reaches a specific control location[We assume that the monitor goes to an error location when detecting an anomaly.] under the (weaker) model . Interestingly enough, we address these two challenges by studying the relationship between these two weak consistency models, and , and serializability. The construction of the monitor is based on the fact that the vs anomalies can be defined as roughly, the difference between the vs and vs anomalies (investigated in previous work [12]), and we show that the reachability problem under can be reduced to a reachability problem under . These results lead to a polynomial-time reduction of this robustness problem (for arbitrary programs) to a reachability problem under , which is important from a practical point of view since the semantics (as opposed to the or semantics) can be encoded easily in existing verification tools (using locks to guard the isolation of transactions). These results also enable a precise characterization of the complexity class of this problem. Checking robustness against substituting with is reduced to the problem of checking robustness against substituting with . The latter has been shown to be polynomial-time reducible to reachability under in [10]. This surprising result relies on the reduction from reachability to reachability mentioned above. This reduction shows that a given program $P$ reaches a certain control location under iff a transformed program $P'$, where essentially, each transaction is split in two parts, one part containing all the reads, and one part containing all the writes, reaches the same control location under . Since this reduction preserves the structure of the program, vs anomalies of a program $P$ correspond to vs anomalies of the transformed program $P'$. Beyond enabling these reductions, the characterization of classes of anomalies or the reduction from the semantics to the semantics are also important for a better understanding of these weak consistency models and the differences between them. We believe that these results can find applications beyond robustness checking, e.g., verifying conformance to given specifications. As a more pragmatic approach for establishing robustness, which avoids a non-reachability proof under , we have introduced a proof methodology that builds on Lipton's reduction theory [38] and the concept of commutativity dependency graph introduced in [9], which represents mover type dependencies between the transactions in a program. We give sufficient conditions for robustness in all the cases mentioned above, which characterize the commutativity dependency graph associated to a given program. We tested the applicability of these verification techniques on a benchmark containing seven challenging applications extracted from previous work [29, 33, 18]. These techniques are precise enough for proving or disproving the robustness of all these applications, for all combinations of the consistency models. keywords = assume, select, return, stepnumber=1,numberblanklines=false,mathescape=true Process 1 CreateEvent(v, e1, 3): [ Tickets[v][e1] := 3 ] [ r := $\sum\limits_{\mbox{e}}$Tickets[v][e] ] Process 2 CreateEvent(v, e2, 3): [ Tickets[v][e2] := 3 ] [ r := $\sum\limits_{\mbox{e}}$Tickets[v][e] ] [->,>=stealth',shorten >=1pt,auto,node distance=4cm, semithick, transform shape,every text node part/.style=align=left] [shape=rectangle ,draw=none,font=] at (-4,0) (m) CreateEvent(v,e1,3); [shape=rectangle ,draw=none,font=, label=right://r=3] at (-4.3,-2) (n) CountTickets(v); [shape=rectangle ,draw=none,font=, ] at (0,0) (p)CreateEvent(v,e2,3); [shape=rectangle ,draw=none,font=, label=right://r=3] at (0,-2) (l) CountTickets(v); [ every edge/.style=draw=black,very thick] [->] (m.212) edge[left] node $\hbo$ (n.119); [->] (m.230) edge[right,dashed] node $\po$ (n.80); [->] (n) edge[below] node $\hbo$ (p); [->] (p.240) edge[left] node $\hbo$ (l.119); [->] (p.280) edge[right,dashed] node $\po$ (l.80); [->] (l) edge[above] node $\hbo$ (m); A trace of FusionTicket. Process 1 Register(u, p1): [ r := RegisteredUsers[u] assume r == 0 RegisteredUsers[u] := 1 Password[u] := p1 ] Process 2 Register(u, p2): [ r := RegisteredUsers[u] assume r == 0 RegisteredUsers[u] := 1 Password[u] := p2 ] [->,>=stealth',shorten >=1pt,auto,node distance=4cm, semithick, transform shape,every text node part/.style=align=left] [shape=rectangle ,draw=none,font=] at (-3.5,0) (m) Register(u,p1); [shape=rectangle ,draw=none,font=] at (0,0) (p)Register(u,p2); [ every edge/.style=draw=black,very thick] [->] (m) edge[bend right,above] node $\hbo$ (p); [->] (p) edge[bend right,above] node $\hbo$ (m); A and trace of Twitter. Process 1 RegisterRd(u, p1): [ r := RegisteredUsers[u] assume r == 0 ] RegisterWr(u, p1): [ RegisteredUsers[u] := 1 Password[u] := p1 ] Process 2 RegisterRd(u, p2): [ r := RegisteredUsers[u] assume r == 0 ] RegisterWr(u, p2): [ RegisteredUsers[u] := 1 Password[u] := p2 ] Transformed Twitter. [->,>=stealth',shorten >=1pt,auto,node distance=4cm, semithick, transform shape,every text node part/.style=align=left] [shape=rectangle ,draw=none,font=] at (-4,0) (m) RegisterRd(u,p1); [shape=rectangle ,draw=none,font=] at (-4,-2) (n) RegisterWr(u,p1); [shape=rectangle ,draw=none,font=] at (0,0) (p)RegisterRd(u,p2); [shape=rectangle ,draw=none,font=] at (0,-2) (l)RegisterWr(u,p2); [ every edge/.style=draw=black,very thick] [->] (m) edge[above] node $\hbo$ (l); [->] (m.235) edge[left] node $\hbo$ (n.120); [->] (m.275) edge[right,dashed] node $\po$ (n.80); [->] (n.360) edge[above] node $\hbo$ (l.180); [->] (p) edge[below] node $\hbo$ (n); [->] (p.235) edge[left] node $\hbo$ (l.120); [->] (p.275) edge[right,dashed] node $\po$ (l.80); A and trace of transformed Twitter. Process 1 [ assume time < TIMEOUT Bets[1] := 2 ] Process 2 [ assume time < TIMEOUT Bets[2] := 3 ] Process 3 [Bets' := Bets n := Bets'.Length assume time > TIMEOUT n > 0 select i s.t. Bets'[i] $\neq$ $\bot$ return := Bets'[i] ] [->,>=stealth',shorten >=1pt,auto,node distance=4cm, semithick, transform shape,every text node part/.style=align=left] [shape=rectangle ,draw=none,font=] at (-6,0) (m) PlaceBet(1,2); [shape=rectangle ,draw=none,font=] at (-3,0) (p) PlaceBet(2,3); [shape=rectangle ,draw=none,font=, label=right:// return=2] at (0.5,0) (q)SettleBet(); [ every edge/.style=draw=black,very thick] [->] (m) edge[bend right,above] node $\hbo$ (q); [->] (q) edge[above] node $\hbo$ (p); A and trace of Betting. [shape=rectangle ,draw=none,font=] (A) at (0,0) [] PlaceBet(1,2); [shape=rectangle ,draw=none,font=] (B) at (2.5,0) [] SettleBet(); [shape=rectangle ,draw=none,font=] (C) at (5,0) [] PlaceBet(2,3); [ every edge/.style=draw=black,very thick] [->] (A) edge [bend left] node [above,font=] (B); [->] (B) edge[bend left] node [above,font=] (A); [->] (C) edge [bend right] node [above,font=] (B); [->] (B) edge[bend right] node [above,font=] (C); Commutativity dependency graph of Betting. Transactional programs and traces under different consistency models. § OVERVIEW We give an overview of the robustness problems investigated in this paper, discussing first the case vs. , and then vs . We end with an example that illustrates the robustness checking technique based on commutativity arguments. Robustness vs . We illustrate the robustness against substituting with using the FusionTicket and the Twitter programs in Figure <ref> and Figure <ref>, respectively. FusionTicket manages tickets for a number of events, each event being associated with a venue. Its state consists of a two-dimensional map that stores the number of tickets for an event in a given venue ($r$ is a local variable, and the assignment in is interpreted as a read of the shared state). The program has two processes and each process contains two transactions. The first transaction creates an event $\mbox{e}$ in a venue $\mbox{v}$ with a number of tickets $\mbox{n}$, and the second transaction computes the total number of tickets for all the events in a venue $\mbox{v}$. A possible candidate for a specification of this program is that the values computed in are monotonically increasing since each such value is computed after creating a new event. Twitter provides a transaction for registering a new user with a given username and password, which is executed by two parallel processes. Its state contains two maps that record whether a given username has been registered (0 and 1 stand for non-registered and registered, respectively) and the password for a given username. Each transaction first checks whether a given username is free (see the statement). The intended specification is that the user must be registered with the given password when the registration transaction succeeds. A program is robust against substituting with if its set of behaviors under the two models coincide. We model behaviors of a given program as traces, which record standard control-flow and data-flow dependencies between transactions, e.g., the order between transactions in the same session and whether a transaction reads the value written by another (read-from). The transitive closure of the union of all these dependency relations is called happens-before. Figure <ref> pictures a trace of FusionTicket where the concrete values which are read in a transaction are written under comments. In this trace, each process registers a different event but in the same venue and with the same number of tickets, and it ignores the event created by the other process when computing the sum of tickets in the venue. Figure <ref> pictures a trace of FusionTicket under , which is a witness that FusionTicket is not robust against substituting with . This trace is also a violation of the intended specification since the number of tickets is not increasing (the sum of tickets is $3$ in both processes). The happens-before dependencies (pictured with $\hbo$ labeled edges) include the program-order $\po$ (the order between transactions in the same process), and read-write dependencies, since an instance of $\mbox{CountTickets(v)}$ does not observe the value written by the $\mbox{CreateEvent}$ transaction in the other process (the latter overwrites some value that the former reads). This trace is allowed under because the transaction $\mbox{CreateEvent(v, e1, 3)}$ executes concurrently with the transaction $\mbox{CountTickets(v)}$ in the other process, and similarly for $\mbox{CreateEvent(v, e2, 3)}$. However, it is not allowed under since it is impossible to define a total commit order between $\mbox{CreateEvent(v, e1, 3)}$ and $\mbox{CreateEvent(v, e2, 3)}$ that justifies the reads of both $\mbox{CountTickets(v)}$ transactions (these reads should correspond to the updates in a prefix of this order). For instance, assuming that $\mbox{CreateEvent(v, e1, 3)}$ commits before $\mbox{CreateEvent(v, e2, 3)}$, $\mbox{CountTickets(v)}$ in the second process must observe the effect of $\mbox{CreateEvent(v, e1, 3)}$ as well since it observes the effect of $\mbox{CreateEvent(v, e2, 3)}$. However, this contradicts the fact that $\mbox{CountTickets(v)}$ computes the sum of tickets as being $3$. On the other hand, Twitter is robust against substituting with . For instance, Figure <ref> pictures a trace of Twitter under , where the in both transactions pass. In this trace, the transactions $\mbox{Register(u,p1)}$ and $\mbox{Register(u,p2)}$ execute concurrently and are unaware of each other's writes (they are not causally related). The $\hbo$ dependencies include write-write dependencies since both transactions write on the same location (we consider the transaction in Process 2 to be the last one writing to the map), and read-write dependencies since each transaction reads that is written by the other. This trace is also allowed under since the commit order can be defined such that $\mbox{Register(u,p1)}$ is ordered before $\mbox{Register(u,p2)}$, and then both transactions read from the initial state (the empty prefix). Note that this trace has a cyclic happens-before which means that it is not allowed under serializability. Checking robustness vs . We reduce the problem of checking robustness against substituting with to the robustness problem against substituting with (the latter reduces to a reachability problem under [10]). This reduction relies on a syntactic program transformation that rewrites behaviors of a given program $P$ to $\serc{}$ behaviors of another program $P'$. The program $P'$ is obtained by splitting each transaction $\atr$ of $P$ into two transactions: the first transaction performs all the reads in $\atr$ and the second performs all the writes in $\atr$ (the two are related by program order). Figure <ref> shows this transformation applied on Twitter. The trace in Figure <ref> is a serializable execution of the transformed Twitter which is “observationally” equivalent to the trace in Figure <ref> of the original Twitter, i.e., each read of the shared state returns the same value and the writes on the shared state are applied in the same order (the acyclicity of the happens-before shows that this is a serializable trace). The transformed FusionTicket coincides with the original version because it contains no transaction that both reads and writes on the shared state. We show that behaviors and behaviors of the original and transformed program, respectively, are related by a bijection. In particular, we show that any vs. robustness violation of the original program manifests as a vs. robustness violation of the transformed program, and vice-versa. For instance, the trace of the original Twitter in Figure <ref> corresponds to the trace of the transformed Twitter in Figure <ref>, and the acyclicity of the latter (the fact that it is admitted by ) implies that the former is admitted by the original Twitter under . On the other hand, the trace in Figure <ref> is also a of the transformed FusionTicket and its cyclicity implies that it is not admitted by FusionTicket under , and thus, it represents a robustness violation. Robustness vs . We illustrate the robustness against substituting with using Twitter and the Betting program in Figure <ref>. Twitter is not robust against substituting with , the trace in Figure <ref> being a witness violation. This trace is also a violation of the intended specification since one of the users registers a password that is overwritten in a concurrent transaction. This trace is not possible under because $\mbox{Register(u,p1)}$ and $\mbox{Register(u,p2)}$ observe the same prefix of the commit order (i.e., an empty prefix), but they write to a common memory location $\mbox{Password[u]}$ which is not allowed under . On the other hand, the Betting program in Figure <ref>, which manages a set of bets, is robust against substituting with . The first two processes execute one transaction that places a bet of a value $\mbox{v}$ with a unique bet identifier $\mbox{id}$, assuming that the bet expiration time is not yet reached (bets are recorded in the map ). The third process contains a single transaction that settles the betting assuming that the bet expiration time was reached and at least one bet has been placed. This transaction starts by taking a snapshot of the map into a local variable , and then selects a random non-null value (different from $\bot$) in the map to correspond to the winning bet. The intended specification of this program is that the winning bet corresponds to a genuine bet that was placed. Figure <ref> pictures a trace of Betting where $\mbox{SettleBet}$ observes only the bet of the first process $\mbox{PlaceBet(1,2)}$. The $\hbo$ dependency towards the second process denotes a read-write dependency ($\mbox{SettleBet}$ reads a cell of the map which is overwritten by the second process). This trace is allowed under because no two transactions write to the same location. Checking robustness vs . We reduce robustness against substituting with to a reachability problem under . This reduction is based on a characterization of happens-before cycles[Traces with an acyclic happens-before are not robustness violations because they are admitted under serializability, which implies that they are admitted under the weaker model as well.] that are possible under but not , and the transformation described above that allows to simulate the semantics of a program on top of . The former is used to define an instrumentation (monitor) for the transformed program that reaches an error state iff the original program is not robust. Therefore, we show that the happens-before cycles in traces that are not admitted by must contain a transaction that (1) overwrites a value written by another transaction in the cycle and (2) reads a value overwritten by another transaction in the cycle. For instance, the trace of Twitter in Figure <ref> is not allowed under because $\mbox{Register(u,p2)}$ overwrites a value written by $\mbox{Register(u,p1)}$ (the password) and reads a value overwritten by $\mbox{Register(u,p1)}$ (checking whether the username $u$ is registered). The trace of Betting in Figure <ref> is allowed under because its happens-before is acyclic. Checking robustness using commutativity arguments. Based on the reductions above, we propose an approximated method for proving robustness based on the concept of mover in Lipton's reduction theory [38]. A transaction is a left (resp., right) mover if it commutes to the left (resp., right) of another transaction (by a different process) while preserving the computation. We use the notion of mover to characterize the data-flow dependencies in the happens-before. Roughly, there exists a data-flow dependency between two transactions in some execution if one doesn't commute to the left/right of the other one. We define a commutativity dependency graph which summarizes the happens-before dependencies in all executions of a transformed program (obtained by splitting the transactions of the original program as explained above), and derive a proof method for robustness which inspects paths in this graph. Two transactions $\atr_1$ and $\atr_2$ are linked by a directed edge iff $\atr_1$ cannot move to the right of $\atr_2$ (or $\atr_2$ cannot move to the left of $\atr_1$), or if they are related by the program order. Moreover, two transactions $\atr_1$ and $\atr_2$ are linked by an undirected edge iff they are the result of splitting the same transaction. A program is robust against substituting with if roughly, its commutativity dependency graph does not contain a simple cycle of directed edges with two distinct transactions $\atr_1$ and $\atr_2$, such that $\atr_1$ does not commute left because of another transaction $\atr_3$ in the cycle that reads a variable that $\atr_1$ writes to, and $\atr_2$ does not commute right because of another transaction $\atr_4$ in the cycle ($\atr_3$ and $\atr_4$ can coincide) that writes to a variable that $\atr_2$ either reads from or writes to[The transactions $\atr_1$, $\atr_2$, $\atr_3$, and $\atr_4$ correspond to $\atr_1$, $\atr_i$, $\atr_n$, and $\atr_{i+1}$, respectively, in Theorem <ref>.]. For instance, Figure <ref> shows the commutativity dependency graph of the transformed Betting program, which coincides with the original Betting because $\mbox{PlaceBet(1,2)}$ and $\mbox{PlaceBet(2,3)}$ are write-only transactions and $\mbox{SettleBet()}$ is a read-only transaction. Both simple cycles in Figure <ref> contain just two transactions and therefore do not meet the criterion above which requires at least 3 transactions. Therefore, Betting is robust against substituting with . A program is robust against substituting with , if roughly, its commutativity dependency graph does not contain a simple cycle with two successive transactions $\atr_1$ and $\atr_2$ that are linked by an undirected edge, such that $\atr_1$ does not commute left because of another transaction $\atr_3$ in the cycle that writes to a variable that $\atr_1$ writes to, and $\atr_2$ does not commute right because of another transaction $\atr_4$ in the cycle ($\atr_3$ and $\atr_4$ can coincide) that writes to a variable that $\atr_2$ reads from[The transactions $\atr_1$, $\atr_2$, $\atr_3$, and $\atr_4$ correspond to $\atr_1$, $\atr_2$, $\atr_n$, and $\atr_3$, respectively, in Theorem <ref>.]. Betting is also robust against substituting with for the same reason (simple cycles of size 2). § CONSISTENCY MODELS <prog> ::= program <process>$^{}$ <process> ::= process <pid> regs <reg>$^{}$ <txn>$^{}$ <txn> ::= begin <read>$^{}$ <test>$^{}$ <write>$^{}$ commit <read> ::= <label>":" <reg> ":=" <var>";" goto <label>";" <test> ::= <label>":" assume <bexpr>";" goto <label>";" <write> ::= <label>":" <var> ":=" <reg-expr>";" goto <label>";" The syntax of our programming language. $a^{}$ indicates zero or more occurrences of $a$. $\langle pid\rangle$, $\langle reg\rangle$, $\langle label \rangle$, and $\langle var\rangle$ represent a process identifier, a register, a label, and a shared variable, respectively. $\langle reg\text{-}expr \rangle$ is an expression over registers while $\langle bexpr \rangle$ is a Boolean expression over registers, or the non-deterministic choice $$. We present our results in the context of the simple programming language, defined in Figure <ref>, where a program is a parallel composition of processes distinguished using a set of identifiers $\mathbb{P}$. A process is a sequence of transactions and each transaction is a sequence of labeled instructions. A transaction starts with a begin instruction and finishes with a commit instruction. Instructions include assignments to a process-local register from a set $\mathbb{R}$ or to a shared variable from a set $\mathbb{V}$, or an assume. The assignments use values from a data domain $\mathbb{D}$. An assignment to a register $\langle reg\rangle := \langle var\rangle$ is called a read of the shared-variable $\langle var\rangle$ and an assignment to a shared variable $\langle var\rangle := \langle reg\rangle$ is called a write to the shared-variable $\langle var\rangle$. The assume $\langle bexpr\rangle$ blocks the process if the Boolean expression $\langle bexpr\rangle$ over registers is false. It can be used to model conditionals. The goto statement transfers the control to the program location (instruction) specified by a given label. Since multiple instructions can have the same label, goto statements can be used to mimic imperative constructs like loops and conditionals inside transactions. We assume w.l.o.g. that every transaction is written as a sequence of reads or assume statements followed by a sequence of writes (a single goto statement from the sequence of read/assume instructions transfers the control to the sequence of writes). In the context of the consistency models we study in this paper, every program can be equivalently rewritten as a set of transactions of this form. To simplify the technical exposition, programs contain a bounded number of processes and each process executes a bounded number of transactions. A transaction may execute an unbounded number of instructions but these instructions concern a bounded number of variables, which makes it impossible to model SQL (select/update) queries that may access tables with a statically unknown number of rows. Our results can be extended beyond these restrictions as explained in Remark <ref> and Remark <ref>. We describe the semantics of a program under four consistency models, i.e., causal consistency[We consider a variation known as causal convergence [19, 15]] (), prefix consistency (), snapshot isolation (), and serializability (). In the semantics of a program under , shared variables are replicated across each process, each process maintaining its own local valuation of these variables. During the execution of a transaction in a process, its writes are stored in a transaction log that can be accessed only by the process executing the transaction and that is broadcasted to all the other processes at the end of the transaction. To read a shared variable $\anaddr$, a process $\apr$ first accesses its transaction log and takes the last written value on $\anaddr$, if any, and then its own valuation of the shared variable, if $\anaddr$ was not written during the current transaction. Transaction logs are delivered to every process in an order consistent with the causal relation between transactions, i.e., the transitive closure of the union of the program order (the order in which transactions are executed by a process), and the read-from relation (a transaction $\atr_1$ reads-from a transaction $\atr_2$ iff $\atr_1$ reads a value that was written by $\atr_2$). When a process receives a transaction log, it immediately applies it on its shared-variable valuation. In the semantics of a program under and , shared variables are stored in a central memory and each process keeps a local valuation of these variables. When a process starts a new transaction, it fetches a consistent snapshot of the shared variables from the central memory and stores it in its local valuation of these variables. During the execution of a transaction in a process, writes to shared variables are stored in the local valuation of these variables, and in a transaction log. To read a shared variable, a process takes its own valuation of the shared variable. A process commits a transaction by applying the updates in the transaction log on the central memory in an atomic way (to make them visible to all processes). Under , when a process applies the writes in a transaction log on the central memory, it must ensure that there were no concurrent writes that occurred after the last fetch from the central memory to a shared variable that was written during the current transaction. Otherwise, the transaction is aborted and its effects discarded. In the semantics of a program under , we adopt a simple operational model where we keep a single shared-variable valuation in a central memory (accessed by all processes) with the standard interpretation of read and write statements. Transactions execute serially, one after another. We use a standard model of executions of a program called trace. A trace represents the order between transactions in the same process, and the data-flow in an execution using standard happens-before relations between transactions. We assume that each transaction in a program is identified uniquely using a transaction identifier from a set $\mathbb{T}$. Also, $\amap: \mathbb{T} \rightarrow 2^{\mathbb{S}}$ is a mapping that associates each transaction in $\mathbb{T}$ with a sequence of read and write events from the set \begin{align} \mathbb{S} = \{\readact(\atr,\anaddr,\aval), \writeact(\atr,\anaddr,\aval): \atr\in \mathbb{T}, \anaddr\in \mathbb{V}, \aval\in \mathbb{D}\} \end{align} where $\readact(\atr,\anaddr,\aval)$ is a read of $\anaddr$ returning $\aval$, and $\writeact(\atr,\anaddr,\aval)$ is a write of $\aval$ to $\anaddr$. A trace is a tuple $\atrace = (\rho,\amap,\tor,\po,\rfo,\sto,\cfo)$ where $\rho\subseteq \mathbb{T}$ is a set of transaction identifiers, and * $\tor$ is a mapping giving the order between events in each transaction, i.e., it associates each transaction $\atr$ in $\rho$ with a total order $\tor(\atr)$ on $\amap(\atr) \times \amap(\atr)$. * $\po$ is the program order relation, a strict partial order on $\rho \times \rho$ that orders every two transactions issued by the same process. * $\rfo$ is the read-from relation between distinct transactions $(\atr1, \atr2) \in \rho \times \rho$ representing the fact that $\atr2$ reads a value written by $\atr1$. * $\sto$ is the store order relation on $\rho \times \rho$ between distinct transactions that write to the same shared variable. * $\cfo$ is the conflict order relation between distinct transactions, defined by $\cfo = \rfo^{-1};\sto$ ($;$ denotes the sequential composition of two relations). For simplicity, for a trace $\atrace = (\rho,\amap,\tor,\po,\rfo,\sto,\cfo)$, we write $t\in \atrace$ instead of $t\in\rho$. We also assume that each trace contains a fictitious transaction that writes the initial values of all shared variables, and which is ordered before any other transaction in program order. Also, $\tracesconf_{\textsf{X}}(\aprog)$ is the set of traces representing executions of program $\aprog$ under a consistency model $\textsf{X}$. For each $\textsf{X}\in \{\ccc{},\pcc{},\sic{},\serc{}\}$, the set of traces $\tracesconf_{\textsf{X}}(\aprog)$ can be described using the set of properties in Table <ref>. A trace $\atrace$ is possible under causal consistency iff there exist two relations $\viso$ a partial order (causal order) and $\arbo$ a total order (arbitration order) that includes $\viso$, such that the properties $\axpoco$, $\axcoarb$, and $\axretval$ hold [26, 15]. $\axpoco$ guarantees that the program order and the read-from relation are included in the causal order, and $\axcoarb$ guarantees that the causal order and the store order are included in the arbitration order. $\axretval$ guarantees that a read returns the value written by the last write in the last transaction that contains a write to the same variable and that is ordered by $\viso$ before the read's transaction. We use $\axcc$ to denote the conjunction of these three properties. A trace $\atrace$ is possible under prefix consistency iff there exist a causal order $\viso$ and an arbitration order $\arbo$ such that $\axcc$ holds and the property $\axprefix$ holds as well [26]. $\axprefix$ guarantees that every transaction observes a prefix of transactions that are ordered by $\arbo$ before it. We use $\axpc$ to denote the conjunction of $\axcc$ and $\axprefix$. A trace $\atrace$ is possible under snapshot isolation iff there exist a causal order $\viso$ and an arbitration order $\arbo$ such that $\axpc$ holds and the property $\axconflict$ holds [26]. $\axconflict$ guarantees that if two transactions write to the same variable then one of them must observe the other. We use $\axsi$ to denote the conjunction of $\axpc$ and $\axconflict$. A trace $\atrace$ is serializable iff there exist a causal order $\viso$ and an arbitration order $\arbo$ such that the property $\axser$ holds which implies that the two relations $\viso$ and $\arbo$ coincide. Note that for any given program $\aprog$, $\tracesconf_{\serc{}}(\aprog)\subseteq \tracesconf_{\sic{}}(\aprog)\subseteq \tracesconf_{\pcc{}}(\aprog)\subseteq \tracesconf_{\ccc{}}(\aprog)$. Also, the four consistency models we consider disallow anomalies such as dirty and phantom reads. $\axpoco$ $\viso_{0}^{+} \subseteq \viso$ $\axcoarb$ $\arbo_{0}^+ \subseteq \arbo$ $\axcc$ $\axretval \wedge \axpoco \wedge \axcoarb$ $\axprefix$ $\arbo ; \viso \subseteq \viso$ $\axpc$ $ \axprefix \wedge \axcc$ $\axconflict$ $\sto \subseteq \viso$ $\axsi$ $\axconflict \wedge \axpc$ $\axser$ $\axretval \wedge \axpoco \wedge \axcoarb \wedge \viso = \arbo $ $\viso_{0} = \po \cup \rfo$ and $\arbo_{0} = \po \cup \rfo \cup \sto$ $\axretval$ = $\forall\ t\in \atrace.\ \forall\ \readact(\atr,\anaddr,\aval) \in \amap(\atr)$ we have that * there exist a transaction $\atr_0 = Max_{\arbo}(\{\atr' \in \atrace\ \|\ (\atr',\atr) \in \viso \wedge \exists\ \writeact(\atr',\anaddr,\cdot)\in\amap(\atr')\})$ and an event $\writeact(\atr_0,\anaddr,\aval) = Max_{\tor(\atr_0)}(\{\writeact(\atr_0,\anaddr,\cdot) \in \amap(\atr_0)\})$. Declarative definitions of consistency models. For an order relation $\leq$, $a = Max_{\leq}(A)$ iff $a \in A \wedge \forall\ b \in A.\ b \leq a$. For a given trace $\atrace=(\rho,\amap,\tor, \po, \rfo, \sto, \cfo)$, the happens before order is the transitive closure of the union of all the relations in the trace, i.e., $\hbo = (\po \cup \rfo \cup \sto \cup \cfo)^{+}$. A classic result states that a trace $\atrace$ is serializable iff $\hbo$ is acyclic [2, 46]. Note that $\hbo$ is acyclic implies that $\sto$ is a total order between transactions that write to the same variable, and $(\po \cup \rfo)^{+}$ and $(\po \cup \rfo \cup \sto)^{+}$ are acyclic. §.§ Robustness In this work, we investigate the problem of checking whether a program $\aprog$ under a semantics $\textsf{Y} \in \{\pcc{},\ \sic{}\}$ produces the same set of traces as under a weaker semantics $\textsf{X} \in \{\ccc{},\ \pcc{}\}$. When this holds, we say that $\aprog$ is robust against $\textsf{X}$ relative to $\textsf{Y}$. A program $\aprog$ is called robust against a semantics $\textsf{X} \in \{\ccc{},\ \pcc{},\ \sic{}\}$ relative to a semantics $\textsf{Y} \in \{\pcc{},\ \sic{},\ \serc{}\}$ such that $\textsf{Y}$ is stronger than $\textsf{X}$ iff $\tracesconf_{\textsf{X}}(\aprog)=\tracesconf_{\textsf{Y}}(\aprog)$. If $\aprog$ is not robust against $\textsf{X}$ relative to $\textsf{Y}$ then there must exist a trace $\atrace \in \tracesconf_{\textsf{X}}(\aprog) \setminus \tracesconf_{\textsf{Y}}(\aprog)$. We say that $\atrace$ is a robustness violation trace. [->,>=stealth',shorten >=1pt,auto,node distance=4cm, semithick, transform shape,every text node part/.style=align=left] [shape=rectangle ,draw=none,font=, label=left:$\atr_1$] at (-2.5,0) (m) $[x := 1]$; [shape=rectangle ,draw=none,font=, label=left:$\atr_2$] at (-2.5,-2) (n) $[r1 := y]\ //0$; [shape=rectangle ,draw=none,font=, label=right:$\atr_3$] at (0,0) (p)$[y := 1]$; [shape=rectangle ,draw=none,font=, label=right:$\atr_4$] at (0,-2) (l) $[r2 := x]\ //0$; [ every edge/.style=draw=black,very thick] [->] (m) edge[left] node $\po$ (n); [->] (n) edge[left] node $\cfo$ (p); [->] (p) edge[right] node $\po$ (l); [->] (l) edge[right] node $\cfo$ (m); Store Buffering ($\mathsf{SB}$). [->,>=stealth',shorten >=1pt,auto,node distance=4cm, semithick, transform shape,every text node part/.style=align=left] [shape=rectangle ,draw=none,font=, label=left:$\atr_1$] at (-3,0) (m) $[r1 := x\ \ //0$ $\ x := r1 + 1]$; [shape=rectangle ,draw=none,font=, label=right:$\atr_2$] at (0,0) (p)$[r2 := x\ \ //0$ $\ x := r2 + 1]$; [ every edge/.style=draw=black,very thick] [->] (m) edge[bend right,above] node $\sto$ (p); [->] (p) edge[bend right,above] node $\cfo$ (m); Lost Update ($\mathsf{LU}$). [->,>=stealth',shorten >=1pt,auto,node distance=4cm, semithick, transform shape,every text node part/.style=align=left] [shape=rectangle ,draw=none,font=, label=left:$\atr_1$] at (-3,0) (m) $[r1 := x\ \ //0$ $\ y := 1]$; [shape=rectangle ,draw=none,font=, label=right:$\atr_2$] at (0,0) (p)$[r2 := y\ \ //0$ $\ x := 1]$; [ every edge/.style=draw=black,very thick] [->] (m) edge[bend right,above] node $\cfo$ (p); [->] (p) edge[bend right,above] node $\cfo$ (m); Write Skew (WS). [->,>=stealth',shorten >=1pt,auto,node distance=4cm, semithick, transform shape,every text node part/.style=align=left] [shape=rectangle ,draw=none,font=, label=left:$\atr_1$] at (-2.5,0) (m) $[x := 1]$; [shape=rectangle ,draw=none,font=, label=left:$\atr_2$] at (-2.5,-2) (n) $[y := 1]$; [shape=rectangle ,draw=none,font=, label=right:$\atr_3$] at (0,0) (p)$[r1 := y]\ //1$; [shape=rectangle ,draw=none,font=, label=right:$\atr_4$] at (0,-2) (l) $[r2 := x]\ //1$; [ every edge/.style=draw=black,very thick] [->] (m) edge[left] node $\po$ (n); [->] (n) edge[left] node $\rfo$ (p); [->] (p) edge[right] node $\po$ (l); [->] (m) edge[right] node $\rfo$ (l); Message Passing (MP). Litmus programs We illustrate the notion of robustness on the programs in Figure <ref>, which are commonly used in the literature. In all programs, transactions of the same process are aligned vertically and ordered from top to bottom. Each read instruction is commented with the value it reads in some execution. The store buffering ($\mathsf{SB}$) program in Figure <ref> contains four transactions that are issued by two distinct processes. We emphasize an execution where $\atr_2$ reads $0$ from $y$ and $\atr_4$ reads $0$ from $x$. This execution is allowed under since the two writes by $\atr_1$ and $\atr_3$ are not causally dependent. Thus, $\atr_2$ and $\atr_4$ are executed without seeing the writes from $\atr_3$ and $\atr_1$, respectively. However, this execution is not feasible under (which implies that it is not feasible under both and ). In particular, we cannot have neither $(\atr_1,\atr_3) \in \arbo$ nor $(\atr_3,\atr_1) \in \arbo$ which contradicts the fact that $\arbo$ is total order. For example, if $(\atr_1,\atr_3) \in \arbo$, then $(\atr_1,\atr_4) \in \viso$ (since $\arbo;\viso \subset \viso$) which contradicts the fact that $\atr_4$ does not see $\atr_1$. Similarly, $(\atr_3,\atr_1) \in \arbo$ implies that $(\atr_3,\atr_2) \in \viso$ which contradicts the fact that $\atr_2$ does not see $\atr_3$. Thus, $\mathsf{SB}$ is not robust against relative to . The lost update ($\mathsf{LU}$) program in Figure <ref> has two transactions that are issued by two distinct processes. We highlight an execution where both transactions read $0$ from $x$. This execution is allowed under since both transactions are not causally dependent and can be executed in parallel by the two processes. However, it is not allowed under since both transactions write to a common variable (i.e., $x$). Thus, they cannot be executed in parallel and one of them must see the write of the other. Thus, $\mathsf{SB}$ is not robust against relative to . The write skew ($\mathsf{WS}$) program in Figure <ref> has two transactions that are issued by two distinct processes. We highlight an execution where $\atr_1$ reads $0$ from $x$ and $\atr_2$ reads $0$ from $y$. This execution is allowed under since both transactions are not causally dependent, do not write to a common variable, and can be executed in parallel by the two processes. However, this execution is not allowed under since one of the two transactions must see the write of the other. Thus, $\mathsf{WS}$ is not robust against relative to . The message passing ($\mathsf{MP}$) program in Figure <ref> has four transactions issued by two processes. Because $\atr_1$ and $\atr_2$ are causally dependent, under any semantics $\textsf{X} \in \{\ccc{},\ \pcc{},\ \sic{},\ \serc{}\}$ we only have three possible executions of $\mathsf{MP}$, which correspond to either $\atr_3$ and $\atr_4$ not observing the writes of $\atr_1$ and $\atr_2$, or $\atr_3$ and $\atr_4$ observe the writes of both $\atr_1$ and $\atr_2$, or $\atr_4$ observes the write of $\atr_1$ (we highlight the values read in the second case in Figure <ref>). Therefore, the executions of this program under the four consistency models coincide. Thus, $\mathsf{MP}$ is robust against relative to any other model. § ROBUSTNESS AGAINST RELATIVE TO We show that checking robustness against relative to can be reduced to checking robustness against relative to . The crux of this reduction is a program transformation that allows to simulate the semantics of a program $\aprog$ using the semantics of a program $\aprog_\pcinstr$. Checking robustness against relative to can be reduced in polynomial time to reachability under [10]. Given a program $\aprog$ with a set of transactions $\trsaprog{\aprog}$, we define a program $\aprog_\pcinstr$ that every transaction $\atr\in \trsaprog{\aprog}$ is split into a transaction $\atrrd{\atr}$ that contains all the read/assume statements in $\atr$ (in the same order) and another transaction $\atrwr{\atr}$ that contains all the write statements in $\atr$ (in the same order). In the following, we establish the following result: A program $\aprog$ is robust against relative to iff $\aprog_\pcinstr$ is robust against relative to . Intuitively, under , processes can execute concurrent transactions that fetch the same consistent snapshot of the shared variables from the central memory and subsequently commit their writes. Decoupling the read part of a transaction from the write part allows to simulate such behaviors even under . The proof of this theorem relies on several intermediate results concerning the relationship between traces of $\aprog$ and $\aprog_\pcinstr$. Let $\atrace= (\rho, \po, \rfo, \sto, \cfo) \in \tracesconf_{\textsf{X}}(\aprog)$ be a trace of a program $\aprog$ under a semantics $\textsf{X}$. We define the trace $\atrace_\pcinstr= (\rho_\pcinstr, \po_\pcinstr, \rfo_\pcinstr, \sto_\pcinstr, \cfo_\pcinstr)$ where every transaction $\atr \in \atrace$ is split into two transactions $\atrrd{\atr}\in \atrace_\pcinstr$ and $\atrwr{\atr} \in \atrace_\pcinstr$, and the dependency relations are straightforward adaptations, i.e., * $\po_\pcinstr$ is the smallest transitive relation that includes $(\atrrd{\atr},\atrwr{\atr})$ for every $\atr$, and $(\atrwr{\atr},\atrrd{\atr'})$ if $(\atr,\atr')\in \po$, $(\atrwr{\atr'},\atrrd{\atr}) \in \rfo_\pcinstr$, $(\atrwr{\atr'},\atrwr{\atr}) \in \sto_\pcinstr$, and $(\atrrd{\atr'},\atrwr{\atr}) \in \cfo_\pcinstr$ if $(\atr',\atr) \in \rfo$, $(\atr',\atr) \in \sto$, and $(\atr',\atr) \in \cfo$, respectively. [->,>=stealth',shorten >=1pt,auto,node distance=4cm, semithick, transform shape,every text node part/.style=align=left] [shape=rectangle ,draw=none,font=, label=left:$\atrrd{\atr_1}$] at (-3,0) (m) $[r1 = x]\ //0$; [shape=rectangle ,draw=none,font=, label=left:$\atrwr{\atr_1}$] at (-3,-1.5) (m1) $[x = r1 + 1]$; [shape=rectangle ,draw=none,font=, label=right:$\atrrd{\atr_2}$] at (0,0) (p)$[r2 = x]\ //0$; [shape=rectangle ,draw=none,font=, label=right:$\atrwr{\atr_2}$] at (0,-1.5) (p1)$[x = r2 + 1]$; [ every edge/.style=draw=black,very thick] [->] (m1) edge[below] node $\sto$ (p1); [->] (m) edge[left] node $\po$ (m1); [->] (m) edge[below] node $\cfo$ (p1); [->] (p) edge[above] node $\cfo$ (m1); [->] (p) edge[left] node $\po$ (p1); A trace of the transformed LU program ($\mathsf{LU}_{\pcinstr}$). For instance, Figure <ref> pictures the trace $\atrace_\pcinstr$ for the $\mathsf{LU}$ trace $\atrace$ given in Figure <ref>. For traces $\atrace$ of programs that contain singleton transactions, e.g., $\mathsf{SB}$ in Figure <ref>, $\atrace_\pcinstr$ coincides with $\atrace$. Conversely, for a given trace $\atrace_\pcinstr= (\rho_\pcinstr, \po_\pcinstr, \rfo_\pcinstr, \sto_\pcinstr, \cfo_\pcinstr) \in \tracesconf_{\textsf{X}}(\aprog_\pcinstr)$ of a program $\aprog_\pcinstr$ under a semantics $\textsf{X}$, we define the trace $\atrace= (\rho, \po, \rfo, \sto, \cfo)$ where every two components $\atrrd{\atr}$ and $\atrwr{\atr}$ are merged into a transaction $\atr \in \atrace$. The dependency relations are defined in a straightforward way, e.g., if $(\atrwr{\atr'},\atrwr{\atr}) \in \sto_\pcinstr$ then $(\atr',\atr) \in \sto$. The following lemma shows that for any semantics $\textsf{X} \in \{\ccc,\ \pcc{},\ \sic{}\}$, if $\atrace \in \tracesconf_{\textsf{X}}(\aprog)$ for a program $\aprog$, then $\atrace_\pcinstr$ is a valid trace of $\aprog_\pcinstr$ under $\textsf{X}$, i.e., $\atrace_\pcinstr \in \tracesconf_{\textsf{X}}(\aprog_\pcinstr)$. Intuitively, this lemma shows that splitting transactions in a trace and defining dependency relations appropriately cannot introduce cycles in these relations and preserves the validity of the different consistency axioms. The proof of this lemma relies on constructing a causal order $\viso_\pcinstr$ and an arbitration order $\arbo_\pcinstr$ for the trace $\atrace_\pcinstr$ starting from the analogous relations in $\atrace$. In the case of $\ccc$, these are the smallest transitive relations such that: * $\po_\pcinstr\subseteq \viso_\pcinstr\subseteq \arbo_\pcinstr$, and * if $(\atr_{1},\atr_{2}) \in \viso$ then $(\atrwr{\atr_{1}},\atrrd{\atr_{2}}) \in \viso_\pcinstr$, and if $(\atr_{1},\atr_{2}) \in \arbo$ then $(\atrwr{\atr_{1}},\atrrd{\atr_{2}}) \in \arbo_\pcinstr$. For and , $\viso_\pcinstr$ must additionally satisfy: if $(\atr_{1},\atr_{2}) \in \arbo$, then $(\atrwr{\atr_{1}},\atrwr{\atr_{2}}) \in \viso_\pcinstr$. This is required in order to satisfy the axiom $\axprefix$, i.e., $\arbo_\pcinstr;\viso_\pcinstr \subset \viso_\pcinstr$, when $(\atrwr{\atr_{1}},\atrrd{\atr_{2}}) \in \arbo_\pcinstr$ and $(\atrrd{\atr_{2}},\atrwr{\atr_{2}}) \in \viso_\pcinstr$. This construction ensures that $\viso_\pcinstr$ is a partial order and $\arbo_\pcinstr$ is a total order because $\viso$ is a partial order and $\arbo$ is a total order. Also, based on the above rules, we have that: if $(\atrwr{\atr_{1}},\atrrd{\atr_{2}}) \in \viso_\pcinstr$ then $(\atr_{1},\atr_{2}) \in \viso$, and similarly, if $(\atrwr{\atr_{1}},\atrrd{\atr_{2}}) \in \arbo_\pcinstr$ then $(\atr_{1},\atr_{2}) \in \arbo$. If $\atrace \in \tracesconf_{\textsf{X}}(\aprog)$, then $\atrace_\pcinstr \in \tracesconf_{\textsf{X}}(\aprog_\pcinstr)$. We start with the case $\textsf{X} = \ccc$. We first show that $\atrace_\pcinstr$ satisfies $\axpoco$ and $\axcoarb$. For $\axpoco$, let $\atr_{1}' \in \{\atrrd{\atr_{1}},\atrwr{\atr_{1}}\}$ and $\atr_{2}' \in \{\atrrd{\atr_{2}},\atrwr{\atr_{2}}\}$, such that $(\atr_{1}',\atr_{2}') \in (\po_\pcinstr\cup\rfo_\pcinstr)^{+}$. By the definition of $\viso_\pcinstr$, we have that either $(\atr_{1}'=\atrrd{\atr_{1}},\atr_{2}'=\atrwr{\atr_{2}}) \in \po_\pcinstr$ and $\atr_1 = \atr_2$ or $(\atr_{1},\atr_2) \in (\po\cup\rfo)^{+}$, which implies that $(\atr_{1},\atr_2) \in \viso$. In both cases we obtain that $(\atr_{1}',\atr_{2}') \in \viso_\pcinstr$. The axiom $\axpoco$ can be proved in a similar way. Next, we show that $\atrace_\pcinstr$ satisfies the property $\axretval$. Let $\atr$ be a transaction in $\atrace$ that contains a read event $\readact(\atr,\anaddr,\aval)$. Let $\atr_0$ be the transaction in $\atrace$ such that $$\atr_0 = Max_{\arbo}(\{\atr' \in \atrace\ \|\ (\atr',\atr) \in \viso \wedge \exists\ \writeact(\atr',\anaddr,\cdot)\in\amap(\atr')\}).$$ The read value $\aval$ must have been written by $\atr_0$ since $\atrace$ satisfies $\axretval$. Thus, the read $\readact(\atr,\anaddr,\aval)$ in $\atrrd{\atr}$ of $\atrace_\pcinstr$ must return the value written by $\atrwr{\atr_0}$. From the definitions of $\viso_\pcinstr$ and $\arbo_\pcinstr$, we get $$\atrwr{\atr_1}\in\{\atrwr{\atr'} \in \atrace_\pcinstr\ \|\ (\atrwr{\atr'},\atrrd{\atr}) \in \viso_\pcinstr \wedge \exists\ \writeact(\atrwr{\atr'},\anaddr,\cdot)\in\amap(\atrwr{\atr'})\}$$ $$\atr_1 \in \{\atr' \in \atrace\ \|\ (\atr',\atr) \in \viso \wedge \exists\ \writeact(\atr',\anaddr,\cdot)\in\amap(\atr')\}$$ because $(\atrwr{\atr_{1}},\atrrd{\atr_{2}}) \in \viso_\pcinstr$ implies $(\atr_{1},\atr_{2}) \in \viso$. Since $(\atrwr{\atr_{1}},\atrwr{\atr_{2}}) \in \arbo_\pcinstr$ implies $(\atr_{1},\atr_{2}) \in \arbo$, we also obtain that $$\atrwr{\atr_0} = Max_{\arbo_\pcinstr}(\{\atrwr{\atr'} \in \atrace_\pcinstr\ \|\ (\atrwr{\atr'},\atrrd{\atr}) \in \viso_\pcinstr \wedge \exists\ \writeact(\atrwr{\atr'},\anaddr,\cdot)\in\amap(\atrwr{\atr'})\})$$ and since the read $\readact(\atr,\anaddr,\aval)$ in $\atrrd{\atr}$ of $\atrace_\pcinstr$ returns the value written by $\atrwr{\atr_0}$, $\atrace_\pcinstr$ satisfies $\axretval$. For the case $\textsf{X} = \pcc$, we show that $\atrace_\pcinstr$ satisfies the property $\axprefix$ (the other axioms are proved as in the case of $\ccc$). Suppose we have $(\atr_{1}',\atr_{2}') \in \arbo_\pcinstr$ and $(\atr_{2}',\atr_{3}') \in \viso_\pcinstr$ where $\atr_{1}' \in \{\atrrd{\atr_{1}},\atrwr{\atr_{1}}\}$, $\atr_{2}' \in \{\atrrd{\atr_{2}},\atrwr{\atr_{2}}\}$, and $\atr_{3}' \in \{\atrrd{\atr_{3}},\atrwr{\atr_{3}}\}$. The are five cases to be discussed: * $(\atr_{1}'=\atrrd{\atr_{1}},\atr_{2}'=\atrwr{\atr_{2}}) \in \po_\pcinstr$ and $\atr_1 = \atr_2$ and $(\atr_{2},\atr_{3}) \in \viso$, * $(\atr_{1},\atr_{2}) \in \arbo$ and $(\atr_{2},\atr_{3}) \in \viso$, * $(\atr_{1},\atr_{2}) \in \arbo$ and $(\atr_{2}'=\atrrd{\atr_{2}},\atr_{3}'=\atrwr{\atr_{3}}) \in \po_\pcinstr$ and $\atr_2 = \atr_3$, * $(\atr_{1}'=\atrrd{\atr_{1}},\atr_{2}'=\atrwr{\atr_{2}}) \in \po_\pcinstr$ and $\atr_1 = \atr_2$ and $(\atr_{2},\atr_{3}) \in \arbo$ and $\atr_{3}'= \atrwr{\atr_{3}}$, * $(\atr_{1},\atr_{2}) \in \arbo$ and $(\atr_{2},\atr_{3}) \in \arbo$ and $\atr_{3}'= \atrwr{\atr_{3}}$. Cases (a) and (b) imply that $(\atr_{1},\atr_{3}) \in \viso$ since $\arbo;\viso \subset \arbo$, which implies that $(\atr_{1}',\atr_{3}') \in \viso_\pcinstr$. Cases (c), (d), and (e) imply that $(\atr_{1},\atr_{3}) \in \arbo$ and $\atr_{3}'= \atrwr{\atr_{3}}$ then we get that $(\atrwr{\atr_{1}},\atrwr{\atr_{3}}) \in \viso_\pcinstr$ and $\atr_{3}'= \atrwr{\atr_{3}}$ which means that $(\atr_{1}',\atr_{3}') \in \viso_\pcinstr$. Note that the rule $(\atrwr{\atr_{1}},\atrwr{\atr_{2}}) \in \viso_\pcinstr$ if $(\atr_{1},\atr_{2}) \in \arbo$ cannot change the fact that $$\atrwr{\atr_1}\in\{\atrwr{\atr'} \in \atrace_\pcinstr\ \|\ (\atrwr{\atr'},\atrrd{\atr}) \in \viso_\pcinstr \wedge \exists\ \writeact(\atrwr{\atr'},\anaddr,\cdot)\in\amap(\atrwr{\atr'})\}$$ $$\atr_1 \in \{\atr' \in \atrace\ \|\ (\atr',\atr) \in \viso \wedge \exists\ \writeact(\atr',\anaddr,\cdot)\in\amap(\atr')\}$$ Thus, the proof of $\axretval$ follows as in the previous case. For the case $\textsf{X} = \sic$, we show that $\atrace_\pcinstr$ satisfies $\axconflict$. If $(\atrwr{\atr_{1}},\atrwr{\atr_{2}}) \in \sto_\pcinstr$, then $(\atr_{1},\atr_{2}) \in \sto \subset \viso$, which implies that $(\atrwr{\atr_{1}},\atrrd{\atr_{2}}) \in \viso_\pcinstr$. Therefore, $(\atrwr{\atr_{1}},\atrwr{\atr_{2}}) \in \viso_\pcinstr$, which concludes the proof. The axiom $\axretval$ can be proved as in the previous cases. Before presenting a strengthening of Lemma <ref> when $\textsf{X}$ is , we give an important characterization of traces. This characterization is stated in terms of acyclicity properties. $\atrace$ is a trace under iff $\arbo_{0}^{+}$ and $\viso_{0}^{+};\cfo$ are acyclic ($\arbo_{0}$ and $\viso_{0}$ are defined in Table <ref>). ($\Rightarrow$) Let $\atrace$ be a trace under . From $\axpoco$ and $\axcoarb$ we get that $\arbo_{0}^{+} \subset \arbo$, and $\arbo_{0}^{+}$ is acyclic because $\arbo$ is total order. Assume by contradiction that $\viso_{0}^{+};\cfo$ is cyclic which implies that $\viso;\cfo$ is cyclic since $\viso_{0}^{+} \subset \viso$, which means that there exist $\atr_1$ and $\atr_2$ such that $(\atr_1, \atr_2) \in \viso$ and $(\atr_2, \atr_1) \in \cfo$. $(\atr_2, \atr_1) \in \cfo$ implies that there exists $\atr_3$ such that $(\atr_3, \atr_1) \in \sto$ and $(\atr_3, \atr_2) \in \rfo$. Based on the definition of $\axretval$, $\atr_3$ has two possible instances: * $\atr_3$ corresponds to the "fictional" transaction that wrote the initial values which cannot be the case when $(\atr_1, \atr_2) \in \viso$ and $\atr_1$ writes to the same variable that $\atr_2$ reads from, * $\atr_3$ is the last transaction that occurs before $\atr_2$ that writes the value read by $\atr_2$, which means that $(\atr_1,\atr_3) \in \arbo$ which contradicts the fact that $(\atr_3, \atr_1) \in \sto$ since $\sto \subset \arbo$. ($\Leftarrow$) Let $\atrace$ be a trace such that $\arbo_{0}^{+}$ and $\viso_{0}^{+};\cfo$ are acyclic. Then, we define the relations $\viso$ and $\arbo$ such that $\viso = \viso_{0}^{+}$ and $\arbo$ is any total order that includes $\arbo_{0}^{+}$. Then, we obtain that $(\viso \cup \sto)^{+} \subset \arbo$ and $\viso;\cfo$ is acyclic. Thus, $\atrace$ satisfies the properties $\axpoco$ and $\axcoarb$. Next, we will show that $\atrace$ satisfies $\axretval$. Let $\atr$ be a transaction in $\atrace$ that contains a read event $\readact(\atr,\anaddr,\aval)$. Let $\atr_0$ be transaction in $\atrace$ such that $$\atr_0 = Max_{\arbo}(\{\atr' \in \atrace\ \|\ (\atr',\atr) \in \viso \wedge \exists\ \writeact(\atr',\anaddr,\cdot)\in\amap(\atr')\})$$ then the read must return a value written by $\atr_0$. Assume by contradiction that there exists some other transaction $\atr_1 \neq \atr_0$ such that $(\atr_1,\atr) \in \rfo$. Then, we get that $(\atr_1,\atr_0) \in \arbo$ and both write to $\anaddr$, therefore, $(\atr_1,\atr_0) \in \sto$ since $\sto \subset \arbo$. Combining $(\atr_1,\atr) \in \rfo$ and $(\atr_1,\atr_0) \in \sto$ we obtain $(\atr,\atr_0) \in \cfo$ and since $(\atr_0,\atr) \in \viso$ then we obtain that $(\atr,\atr) \in \viso;\cfo$ which contradicts the fact that $\viso;\cfo$ is acyclic. Therefore, the read value was written by $\atr_0$ and $\atrace$ satisfies $\axretval$. Next we show that a trace $\atrace$ of a program $\aprog$ is iff the corresponding trace $\atrace_\pcinstr$ of $\aprog_\pcinstr$ is as well. This result is based on the observation that cycles in $\arbo_{0}^{+}$ or $\viso_{0}^{+};\cfo$ cannot be broken by splitting transactions. A trace $\atrace$ of $\aprog$ is iff the corresponding trace $\atrace_\pcinstr$ of $\aprog_\pcinstr$ is . The only-if direction follows from Lemma <ref>. For the if direction: consider a trace $\atrace_\pcinstr$ which is . We prove by contradiction that $\atrace$ must be as well. Assume that $\atrace$ is not then it must contain a cycle in either $\arbo_{0}^{+}$ or $\viso_{0}^{+};\cfo$ (based on Lemma <ref>). In the rest of the proof when we mention a cycle we implicitly refer to a cycle in either $\arbo_{0}^{+}$ or $\viso_{0}^{+};\cfo$. Splitting every transaction $\atr \in \atrace$ in a trace to a pair of transactions $\atrrd{\atr}$ and $\atrwr{\atr}$ that occur in this order might not maintain a cycle of $\atrace$. However, we prove that this is not possible and our splitting conserves the cycle. Assume we have a vertex $\atr$ as part of the cycle. We show that $\atr$ can be split into two transactions $\atrrd{\atr}$ and $\atrwr{\atr}$ while maintaining the cycle. Note that $\atr$ is part of a cycle iff either * $(\atr_{1},\atr) \in \arbo_{0}$ and $(\atr,\atr_{2})\in \arbo_{0}$ or * $(\atr_{1},\atr) \in \viso_{0}$ and $(\atr,\atr_{2})\in \viso_{0}$ or * $(\atr_{1},\atr) \in \viso_{0}$ and $(\atr,\atr_{2})\in \cfo$ or * $(\atr_{1},\atr) \in \cfo$ and $(\atr,\atr_{2})\in \viso_{0}$ where $\atr_{1}$ and $\atr_{2}$ might refer to the same transaction. Thus, by splitting $\atr$ to $\atrrd{\atr}$ and $\atrwr{\atr}$, the above four cases imply that: * if $(\atr_{1},\atr) \in \viso_{0}$ and $(\atr,\atr_{2})\in \arbo_{0}$ then $(\atr_{1}',\atrrd{\atr}) \in (\po_\pcinstr \cup \rfo_\pcinstr)$ and $(\atrwr{\atr},\atr_{2}')\in (\po_\pcinstr \cup \rfo_\pcinstr \cup \sto_\pcinstr)$ where $\atr_{1}' \in \{\atrrd{\atr_{1}},\atrwr{\atr_{1}}\}$ and $\atr_{2}' \in \{\atrrd{\atr_{2}},\atrwr{\atr_{2}}\}$. This maintains the vertices $\atr_{1}'$ and $\atr_{2}'$ connected in the cycle formed by the dependency relations of $\atrace_\pcinstr$ since $(\atrrd{\atr},\atrwr{\atr}) \in \po_\pcinstr$; * if $(\atr_{1},\atr) \in \sto$ and $(\atr,\atr_{2})\in \arbo_{0}$ then $(\atr_{1}',\atrwr{\atr}) \in \sto_\pcinstr$ and $(\atrwr{\atr},\atr_{2}')\in (\po_\pcinstr \cup \rfo_\pcinstr \cup \sto_\pcinstr)$ which maintains the vertices $\atr_{1}'$ and $\atr_{2}'$ connected in the cycle formed by the dependency relations of $\atrace_\pcinstr$; * $(\atr_{1},\atr) \in \viso_{0}$ and $(\atr_{2},\atr) \in \cfo$ then $(\atr_{1}',\atrrd{\atr}) \in (\po_\pcinstr \cup \rfo_\pcinstr)$ and $(\atrrd{\atr},\atr_{2}')\in \cfo_\pcinstr$ maintains the vertices $\atr_{1}'$ and $\atr_{2}'$ connected in the cycle formed by the dependency relations of $\atrace_\pcinstr$; * $(\atr_{1},\atr) \in \cfo$ and $(\atr_{2},\atr) \in \viso_{0}$ then $(\atr_{1}',\atrwr{\atr}) \in \cfo_\pcinstr$ and $(\atrwr{\atr},\atr_{2}')\in (\po_\pcinstr \cup \rfo_\pcinstr)$ which maintains the vertices $\atr_{1}'$ and $\atr_{2}'$ connected in the cycle formed by the dependency relations of $\atrace_\pcinstr$ as well. Therefore, doing the splitting creates a cycle in either $(\po_\pcinstr \cup \rfo_\pcinstr \cup \sto_\pcinstr)^{+}$ or $(\po_\pcinstr \cup \rfo_\pcinstr)^{+};\cfo_\pcinstr$ which implies that $\atrace_\pcinstr$ is not , a contradiction. The following lemma shows that a trace $\atrace$ is iff the corresponding trace $\atrace_\pcinstr$ is . The if direction in the proof is based on constructing a causal order $\viso$ and an arbitration order $\arbo$ for the trace $\atrace$ from the arbitration order $\arbo_\pcinstr$ in $\atrace_\pcinstr$ (since $\atrace_\pcinstr$ is a trace under serializability $\viso_\pcinstr$ and $\arbo_\pcinstr$ coincide). These are the smallest transitive relations such that: * if $(\atrwr{\atr_{1}},\atrrd{\atr_{2}}) \in \arbo_\pcinstr$ then $(\atr_{1},\atr_{2}) \in \viso$, * if $(\atrwr{\atr_{1}},\atrwr{\atr_{2}}) \in \arbo_\pcinstr$ then $(\atr_{1},\atr_{2}) \in \arbo$[If $\atrwr{\atr_{1}}$ is empty ($\atr_1$ is read-only), then we set $(\atr_{1},\atr_{2}) \in \arbo$ if $(\atrrd{\atr_{1}},\atrwr{\atr_{2}}) \in \viso_\pcinstr$. If $\atrwr{\atr_{2}}$ is empty, then $(\atr_{1},\atr_{2}) \in \arbo$ if $(\atrwr{\atr_{1}},\atrrd{\atr_{2}}) \in \viso_\pcinstr$. If both $\atrwr{\atr_{1}}$ and $\atrwr{\atr_{2}}$ are empty, then $(\atr_{1},\atr_{2}) \in \arbo$ if $(\atrrd{\atr_{1}},\atrrd{\atr_{2}}) \in \viso_\pcinstr$.]. The only-if direction is based on the fact that any cycle in the dependency relations of $\atrace$ that is admitted under (characterized in Lemma <ref>) is “broken” by splitting transactions. Also, splitting transactions cannot introduce new cycles that do not originate in $\atrace$. A trace $\atrace$ is iff $\atrace_\pcinstr$ is ($\Leftarrow$) Assume that $\atrace_\pcinstr$ is . We will show that $\atrace$ is . Notice that if $(\atr_{1},\atr_{2}) \in \viso_{0}^{+}$ then $(\atrwr{\atr_{1}},\atrrd{\atr_{2}}) \in \arbo_\pcinstr$ which implies that $(\atr_{1},\atr_{2}) \in \viso$. Similarly, if $(\atr_{1},\atr_{2}) \in \arbo_{0}^{+}$ then $(\atrwr{\atr_{1}},\atrwr{\atr_{2}}) \in \arbo_\pcinstr$ or $(\atrwr{\atr_{1}},\atrrd{\atr_{2}}) \in \arbo_\pcinstr$ which implies that $(\atrwr{\atr_{1}},\atrwr{\atr_{2}}) \in \arbo_\pcinstr$ which in both cases implies that $(\atr_{1},\atr_{2}) \in \arbo$. Thus, $\atrace$ satisfies the properties $\axpoco$ and $\axcoarb$. Now assume that $(\atr_{1},\atr_{2})\in \arbo$ and $(\atr_{2},\atr_{3})\in \viso$. We show that $(\atr_{1},\atr_{3})\in \viso$. The assumption implies that $(\atrwr{\atr_{1}},\atrwr{\atr_{2}}) \in \arbo_\pcinstr$ and $(\atrwr{\atr_{2}},\atrrd{\atr_{3}}) \in \arbo_\pcinstr$, which means that $(\atrwr{\atr_{1}},\atrrd{\atr_{3}}) \in \arbo_\pcinstr$. Therefore, $(\atr_{1},\atr_{3}) \in \viso$ and $\atrace$ satisfies the property $\axconflict$. Concerning $\axretval$, let $\atr$ be a transaction in $\atrace$ that contains a read event $\readact(\atr,\anaddr,\aval)$. Let $\atr_0$ be transaction in $\atrace$ such that $$\atr_0 = Max_{\arbo}(\{\atr' \in \atrace\ \|\ (\atr',\atr) \in \viso \wedge \exists\ \writeact(\atr',\anaddr,\cdot)\in\amap(\atr')\}).$$ We show that the read must return a value written by $\atr_0$. The definitions of $\viso$ and $\arbo$ imply that $$\atrwr{\atr_1}\in\{\atrwr{\atr'} \in \atrace_\pcinstr\ \|\ (\atrwr{\atr'},\atrrd{\atr}) \in \arbo_\pcinstr \wedge \exists\ \writeact(\atrwr{\atr'},\anaddr,\cdot)\in\amap(\atrwr{\atr'})\}$$ $$\atr_1 \in \{\atr' \in \atrace\ \|\ (\atr',\atr) \in \viso \wedge \exists\ \writeact(\atr',\anaddr,\cdot)\in\amap(\atr')\}$$ because $(\atrwr{\atr_{1}},\atrrd{\atr_{2}}) \in \arbo_\pcinstr$ implies $(\atr_{1},\atr_{2}) \in \viso$. Then, we obtain that $$\atrwr{\atr_0} = Max_{\arbo_\pcinstr}(\{\atrwr{\atr'} \in \atrace_\pcinstr\ \|\ (\atrwr{\atr'},\atrrd{\atr}) \in \arbo_\pcinstr \wedge \exists\ \writeact(\atrwr{\atr'},\anaddr,\cdot)\in\amap(\atrwr{\atr'})\})$$ and since $\atrace_\pcinstr$ is we know that the read must return the value written by $\atrwr{\atr_0}$. Thus, the read returns the value written by $\atr_0$, which implies that $\atrace$ satisfies $\axretval$ holds. Therefore, $\atrace$ is . ($\Rightarrow$) Assume that $\atrace$ is . We show that $\atrace_\pcinstr$ is . Since $\atrace_\pcinstr$ is the result of splitting transactions, a cycle in its dependency relations can only originate from a cycle in $\atrace$. Therefore, it is sufficient to show that any happens-before cycle in $\atrace$ is broken in $\atrace_\pcinstr$. From Lemma <ref>, we have that $\atrace$ either does not admit a happens-before cycle or any (simple) happens-before cycle in $\atrace$ must have either two successive $\cfo$ dependencies or a $\sto$ dependency followed by a $\cfo$ dependency. If $\atrace$ does not admit a happens-before cycle then it is , and $\atrace_\pcinstr$ is trivially (since splitting transactions cannot introduce new cycles). [shape=rectangle ,draw=none,font=] (A0) at (0,0) [] $\atr_1$ ; [shape=rectangle ,draw=none,font=] (A1) at (2,0) [] $\atr_2$; [shape=rectangle ,draw=none,font=] (B1) at (4,0) [] $\atr_3$; [shape=rectangle ,draw=none,font=] (B2) at (5,0) [] $\Longrightarrow$; [shape=rectangle ,draw=none,font=] (C1) at (6,0) [] $\atrrd{\atr_{1}}$; [shape=rectangle ,draw=none,font=] (D1) at (8,0) [] $\atrwr{\atr_{1}}$; [shape=rectangle ,draw=none,font=] (D2) at (10,0) [] $\atrrd{\atr_{2}}$; [shape=rectangle ,draw=none,font=] (D0) at (12,0) [] $\atrwr{\atr_{2}}$; [shape=rectangle ,draw=none,font=] (E0) at (14,0) [] $\atrrd{\atr_{3}}$ ; [shape=rectangle ,draw=none,font=] (E1) at (16,0) [] $\atrwr{\atr_{3}}$; [ every edge/.style=draw=black,very thick] [->] (A0) edge[] node [above,font=] $\sto \cup \cfo$ (A1); [->] (A1) edge[] node [above,font=] $\cfo$ (B1); [->] (B1) edge[bend left] node [above,font=] $\hbo$ (A0); [->] (C1) edge[] node [above,font=] $\po_\pcinstr$ (D1); [->] (D1) edge[bend left] node [below,font=] $\sto_\pcinstr$ (D0); [->] (C1) edge[bend right] node [above,font=] $\cfo_\pcinstr$ (D0); [->] (D2) edge[] node [above,font=] $\po_\pcinstr$ (D0); [->] (D2) edge[bend right] node [above,font=] $\cfo_\pcinstr$ (E1); [->] (E0) edge[] node [above,font=] $\po_\pcinstr$ (E1); Otherwise, if $\atrace$ admits a happens-before cycle like above, then $\atrace$ must contain three transactions $\atr_{1}$, $\atr_{2}$, and $\atr_{3}$ such that $(\atr_{1},\atr_{2}) \in \sto \cup \cfo$, $(\atr_{2},\atr_{3}) \in \cfo$, and $(\atr_{3},\atr_{1}) \in \hbo$ (like in the picture above). Then, by splitting transactions we obtain that $(\atrwr{\atr_{1}},\atrwr{\atr_{2}}) \in \sto_\pcinstr$ or $(\atrrd{\atr_{1}},\atrwr{\atr_{2}}) \in \cfo_\pcinstr$, and $(\atrrd{\atr_{2}},\atrwr{\atr_{3}}) \in \cfo_\pcinstr$. Since, we have $(\atrrd{\atr_{2}},\atrwr{\atr_{2}}) \in \po_\pcinstr$ (and not $(\atrwr{\atr_{2}},\atrrd{\atr_{2}}) \in \po_\pcinstr$), this cannot lead to a cycle in $\atrace_\pcinstr$, which concludes the proof that $\atrace_\pcinstr$ is The lemmas above are used to prove Theorem <ref> as follows: Proof of Theorem <ref>: For the if direction, assume by contradiction that $\aprog$ is not robust against relative to . Then, there must exist a trace $\atrace \in \tracesconf_{\ccc{}}(\aprog) \setminus \tracesconf_{\pcc{}}(\aprog)$. Lemmas <ref> and <ref> imply that the corresponding trace $\atrace_\pcinstr$ of $\aprog_\pcinstr$ is and not . Thus, $\aprog_\pcinstr$ is not robust against relative to . The only-if direction is proved similarly. Robustness against relative to has been shown to be reducible in polynomial time to the reachability problem under [10]. Given a program $\aprog$ and a control location $\ell$, the reachability problem under asks whether there exists an execution of $\aprog$ under that reaches $\ell$. Therefore, as a corollary of Theorem <ref>, we obtain the following: Checking robustness against relative to is reducible to the reachability problem under in polynomial time. In the following we discuss the complexity of this problem in the case of finite-state programs (bounded data domain). The upper bound follows from Corollary <ref> and standard results about the complexity of the reachability problem under sequential consistency, which extend to , with a bounded [34] or parametric number of processes [44]. For the lower bound, given an instance $(\aprog,\ell)$ of the reachability problem under sequential consistency, we construct a program $\aprog'$ where each statement $s$ of $\aprog$ is executed in a different transaction that guards[That is, the transaction is of the form [lock; $s$; unlock]] the execution of $s$ using a global lock (the lock can be implemented in our programming language as usual, e.g., using a busy wait loop for locking), and where reaching the location $\ell$ enables the execution of a “gadget” that corresponds to the $\mathsf{SB}$ program in Figure <ref>. Executing each statement under a global lock ensures that every execution of $\aprog'$ under $\ccc$ is serializable, and faithfully represents an execution of $\aprog$ under sequential consistency. Moreover, $\aprog$ reaches $\ell$ iff $\aprog'$ contains a robustness violation, which is due to the $\mathsf{SB}$ execution. Checking robustness of a program with a fixed number of variables and bounded data domain against relative to is PSPACE-complete when the number of processes is bounded and EXPSPACE-complete, otherwise. § ROBUSTNESS AGAINST RELATIVE TO In this section, we show that checking robustness against relative to can be reduced in polynomial time to a reachability problem under the semantics. We reuse the program transformation from the previous section that allows to simulate behaviors on top of , and additionally, we provide a characterization of traces that distinguish the semantics from . We use this characterization to define an instrumentation (monitor) that is able to detect if a program under admits such traces. We show that the happens-before cycles in a robustness violation (against relative to ) must contain a $\sto$ dependency followed by a $\cfo$ dependency, and they should not contain two successive $\cfo$ dependencies. This follows from the fact that every happens-before cycle in a trace must contain either two successive $\cfo$ dependencies, or a $\sto$ dependency followed by a $\cfo$ dependency. Otherwise, the happens-before cycle would imply a cycle in the arbitration order. Then, any trace under where all its simple happens-before cycles contain two successive $\cfo$ dependencies is possible under . For instance, the trace of the non-robust $\mathsf{LU}$ execution in Figure <ref> contains $\sto$ dependency followed by a $\cfo$ dependency and does not contain two successive $\cfo$ dependencies which is disallowed , while the trace of the robust $\mathsf{WS}$ execution in Figure <ref> contains two successive $\cfo$ dependencies. As a first step, we prove the following theorem characterizing traces that are allowed under both and . A program $\aprog$ is robust against relative to iff every happens-before cycle in a trace of $\aprog$ under contains two successive $\cfo$ dependencies. Before giving the proof of the above theorem, we state several intermediate results that characterize cycles in or traces. First, we show that every trace in which all simple happens-before cycles contain two successive $\cfo$ is also a trace. If a trace $\atrace$ is and all happens-before cycles in $\atrace$ contain two successive $\cfo$ dependencies, then $\atrace$ is . Let $\arbo_{1}$ be a total order that includes $\arbo_{0}^{+}$ and $\arbo_{0}^{+};\cfo;\arbo_{0}^{}$ ($\arbo_{0}^$ is the reflexive closure of $\arbo_{0}$). This is well defined because there exists no cycle between tuples in these two relations. Indeed, if $(\atr_{1},\atr_{2}) \in \arbo_{0}^{+}$ and there exist $\atr_{3}$ and $\atr_{4}$ such that $(\atr_{2}, \atr_{3}) \in \arbo_{0}^{+}$, $(\atr_{3}, \atr_{4}) \in \cfo$, and $(\atr_{4}, \atr_{1}) \in \arbo_{0}^{}$, then we have a cycle in $\arbo_{0}^{+};\cfo$ that does not contain two successive $\cfo$ dependencies, which contradicts the hypothesis. Also, for every pair of transactions $(\atr_{1},\atr_{2})$ there cannot exist $\atr_{3}$ and $\atr_{4}$ such that $$(\atr_{2}, \atr_{3}) \in \arbo_{0}^{+},\ (\atr_{3}, \atr_{4}) \in \cfo\mbox{ and }(\atr_{4}, \atr_{1}) \in \arbo_{0}^{}$$ $\atr_{3}'$ and $\atr_{4}'$ such that $$(\atr_{1}, \atr_{3}') \in \arbo_{0}^{+},\ (\atr_{3}', \atr_{4}') \in \cfo\mbox{ and }(\atr_{4}', \atr_{2}) \in \arbo_{0}^{}$$ This will imply a cycle in $\arbo_{0}^{+};\cfo;\arbo_{0}^{+};\cfo$ which again contradicts the hypothesis. Also, let $\viso_{1}$ be the smallest transitive relation that includes $\arbo_{0}^{+}$ and $\arbo_{1};\arbo_{0}^{+}$. We show that $\viso_{1}$ and $\arbo_{1}$ are causal and arbitration orders of $\atrace$ that satisfy all the axioms of . $\axpoco$ and $\axcoarb$ hold trivially. Since $\sto \subseteq \viso_{1}$, $\axconflict$ holds as well. $\axpc$ holds because $\arbo_{1} ; \viso_{1} = \arbo_{1};(\arbo_{0}^{+} \cup \arbo_{1};\arbo_{0}^{+})^+ = \arbo_{1};\arbo_{0}^{+} \subset \viso_{1}$. The axiom $\axretval$ is equivalent to the acyclicity of $\viso_{1};\cfo$ when $\axpoco$ and $\axcoarb$ hold. Assume by contradiction that $\viso_1;\cfo$ is cyclic. From the definition of $\viso_1$ and the fact that $\arbo_{1}$ is total order we obtain that either: $\arbo_{0}^{+};\cfo$ is cyclic, which implies that there exists a happens-before cycle that does not contain two successive $\cfo$, which contradicts the hypothesis, or * $\arbo_{1};\arbo_{0}^{+};\cfo$ is cyclic, which implies that there exist $\atr_{1}$, $\atr_{2}$, and $\atr_{3}$ such that $(\atr_{2}, \atr_{3}) \in \arbo_{0}^{+}$, $(\atr_{3}, \atr_{1}) \in \cfo$ and $(\atr_{1},\atr_{2}) \in \arbo_{1}$. This contradicts the fact that $(\atr_{2}, \atr_{3}) \in \arbo_{0}^{+}$ and $(\atr_{3}, \atr_{1}) \in \cfo$ implies $(\atr_{2},\atr_{1}) \in \arbo_{1}$. Therefore, $\atrace$ satisfies $\axretval$ for $\viso_1$ and $\arbo_1$, which concludes the proof. The proof of Theorem <ref> also relies on the following lemma that characterizes happens-before cycles permissible under . [22, 12] If a trace $\atrace$ is , then all its happens-before cycles must contain two successive $\cfo$ dependencies. Proof of Theorem <ref>: For the only-if direction, if $\aprog$ is robust against relative to then every trace $\atrace$ of $\aprog$ under is as well. Therefore, by Lemma <ref>, all cycles in $\atrace$ contain two successive $\cfo$ which concludes the proof of this direction. For the reverse, let $\atrace$ be a trace of $\aprog$ under such that all its happens-before cycles contain two successive $\cfo$. Then, by Lemma <ref>, we have that $\atrace$ is . Thus, every trace $\atrace$ of $\aprog$ under is . Next, we present an important lemma that characterizes happens before cycles possible under the semantics. This is a strengthening of a result in [12] which shows that all happens before cycles under must have two successive dependencies in $\{\cfo,\sto\}$ and at least one $\cfo$. We show that the two successive dependencies cannot be $\cfo$ followed $\sto$, or two successive $\sto$. If a trace $\atrace$ is then all happens-before cycles in $\atrace$ must contain either two successive $\cfo$ dependencies or a $\sto$ dependency followed by a $\cfo$ dependency. It was shown in [12] that all happens-before cycles under must contain two successive dependencies in $\{\cfo,\sto\}$ and at least one $\cfo$. Assume by contradiction that there exists a cycle with $\cfo$ dependency followed by $\sto$ dependency or two successive $\sto$ dependencies. This cycle must contain at least one additional dependency. Otherwise, the cycle would also have a $\sto$ dependency followed by a $\cfo$ dependency, or it would imply a cycle in $\sto$, which is not possible (since $\sto \subset \arbo$ and $\arbo$ is a total order). Then, we get that the dependency just before $\cfo$ is either $\po$ or $\rfo$ (i.e., $\viso_0$) since we cannot have $\cfo$ or $\sto$ followed by $\cfo$. Also, the relation after $\sto$ is either $\po$ or $\rfo$ or $\sto$ (i.e., $\arbo_0$) since we cannot have $\sto$ followed by $\cfo$. Thus, the cycle has the following shape: [shape=rectangle ,draw=none,font=] (A0) at (0,0) [] $\atr_1$ ; [shape=rectangle ,draw=none,font=] (A1) at (1.3,0) [] $\atr_2$; [shape=rectangle ,draw=none,font=] (B1) at (2.6,0) [] $\atr_3$; [shape=rectangle ,draw=none,font=] (B2) at (3.9,0) [] $\atr_4$; [shape=rectangle ,draw=none,font=] (C0) at (4.5,0) [] $\cdots$ ; [shape=rectangle ,draw=none,font=] (C1) at (5.1,0) [] $\atr_i$; [shape=rectangle ,draw=none,font=] (D1) at (6.4,0) [] $\atr_{i+1}$; [shape=rectangle ,draw=none,font=] (D2) at (7.9,0) [] $\atr_{i+2}$; [shape=rectangle ,draw=none,font=] (D0) at (9.4,0) [] $\atr_{i+3}$; [shape=rectangle ,draw=none,font=] (E0) at (10.2,0) [] $\cdots$ ; [shape=rectangle ,draw=none,font=] (E1) at (11,0) [] $\atr_{n-4}$; [shape=rectangle ,draw=none,font=] (F1) at (12.5,0) [] $\atr_{n-3}$; [shape=rectangle ,draw=none,font=] (F2) at (14,0) [] $\atr_{n-2}$; [shape=rectangle ,draw=none,font=] (F3) at (15.5,0) [] $\atr_{n-1}$; [shape=rectangle ,draw=none,font=] (F4) at (17,0) [] $\atr_{n}$; [ every edge/.style=draw=black,very thick] [->] (A0) edge[] node [above,font=] $\cfo$ (A1); [->] (A1) edge[] node [above,font=] $\sto$ (B1); [->] (B1) edge[] node [above,font=] $\arbo_0$ (B2); [->] (C1) edge[] node [above,font=] $\viso_0$ (D1); [->] (D1) edge[] node [above,font=] $\cfo$ (D2); [->] (D2) edge[] node [above,font=] $\sto$ (D0); [->] (E1) edge[] node [above,font=] $\viso_0$ (F1); [->] (F1) edge[] node [above,font=] $\cfo$ (F2); [->] (F2) edge[] node [above,font=] $\sto$ (F3); [->] (F3) edge[] node [above,font=] $\arbo_0$ (F4); [->] (F4) edge[bend left=11] node [above,font=] $\viso_0$ (A0); Since $\viso_0;\cfo\subseteq \arbo$ is a consequence of the axioms [26], we get that $(\atr_{n}, \atr_2) \in \arbo$, $(\atr_{i}, \atr_{i+2}) \in \arbo$ and $(\atr_{n-4}, \atr_{n-2}) \in \arbo$, which allows to “short-circuit” the cycle. Using the fact that $\sto \subset \arbo$, $\viso_0 \subset \arbo$, and $\arbo_0 \subset \arbo$, and applying the short-circuiting process multiple times, we obtain a cycle in the arbitration order $\arbo$ which contradicts the fact that $\arbo$ is a total order. Combining the results of Theorem <ref> and Lemmas <ref> and <ref>, we obtain the following characterization of traces which violate robustness against relative to . A program $\aprog$ is not robust against relative to iff there exists a trace $\atrace_\pcinstr$ of $\aprog_\pcinstr$ under such that the trace $\atrace$ obtained by merging[This transformation has been defined at the beginning of Section <ref>.] read and write transactions in $\atrace_\pcinstr$ contains a happens-before cycle that does not contain two successive $\cfo$ dependencies, and it contains a $\sto$ dependency followed by a $\cfo$ dependency. The results above enable a reduction from checking robustness against relative to to a reachability problem under the semantics. For a program $\aprog$, we define an instrumentation denoted by $\sem{\aprog}$, such that $\aprog$ is not robust against relative to iff $\sem{\aprog}$ violates an assertion under . The instrumentation consists in rewriting every transaction of $\aprog$ as shown in Figure <ref>. [->,>=stealth',shorten >=1pt,auto,node distance=4cm, semithick, transform shape,every text node part/.style=align=left] [shape=rectangle ,draw=none,font=] at (0,0) (m) $\alpha$; [shape=rectangle ,draw=none,font=] at (1.5,0) (n) $\atr_{\instr}$; [shape=rectangle ,draw=none,font=] at (3,0) (n1) $\beta$; [shape=rectangle ,draw=none,font=] at (4.5,0) (n2) $\atr_{0}$; [shape=rectangle ,draw=none,font=] at (6,0) (n3) $\gamma$; [shape=rectangle ,draw=none,font=] at (7.5,0) (n4) $\atr$; [ every edge/.style=draw=black,very thick] [->] (n) edge[bend left=17] node $\cfo$ (n2); [->] (n2) edge[bend left=17] node $\hbo$ (n4); [->] (n4) edge[bend left=23, above] node $\sto$ (n); Execution simulating a violation to robustness against relative to . The instrumentation $\sem{\aprog}$ running under $\serc$ simulates the semantics of $\aprog$ using the same idea of decoupling the execution of the read part of a transaction from the write part. It violates an assertion when it simulates a trace containing a happens-before cycle as in Theorem <ref>. The execution corresponding to this trace has the shape given in Figure <ref>, where $\atr_{\instr}$ is the transaction that occurs between the $\sto$ and the $\cfo$ dependencies, and every transaction executed after $\atr_{\instr}$ (this can be a full transaction in $\aprog$, or only the read or write part of a transaction in $\aprog$) is related by a happens-before path to $\atr_{\instr}$ (otherwise, the execution of this transaction can be reordered to occur before $\atr_{\instr}$). A transaction in $\aprog$ can have its read part included in $\alpha$ and the write part included in $\beta$ or $\gamma$. Also, $\beta$ and $\gamma$ may contain transactions in $\aprog$ that executed only their read part. It is possible that $\atr_{0} = \atr$, $\beta=\gamma=\epsilon$, and $\alpha = \epsilon$ (the $\mathsf{LU}$ program shown in Figure <ref> is an example where this can happen). The instrumentation uses auxiliary variables to track happens-before dependencies, which are explained below. \begin{figure}[!ht] \footnotesize \begin{minipage}{\linewidth} Transaction ``\plog{begin} $\langle$read$\rangle^{}$ $\langle$test$\rangle^{}$ $\langle$write$\rangle^{}$ \plog{commit}'' is rewritten to: \end{minipage} \begin{minipage}{0.655\linewidth} \begin{lstlisting} if ( !done$_\#$ ) if ( ) begin <read>$^{}$ <test>$^{}$ commit $\label{ln:part11}$ if ( !done$_\#$ ) begin <write>$^{}$ commit $\label{ln:part12}$ $\mathcal{I}$(begin) ($\mathcal{I}$(<write>))$^{}$ $\mathcal{I}$(commit)$\label{ln:part31}$ begin ($\mathcal{I}_\#$(<read>))$^{}$ <test>$^{}$ ($\mathcal{I}_\#$(<write>))$^{}$ $\mathcal{I}_\#$(commit)$\label{ln:part21}$ assume false; else if ( ) rdSet' := $\emptyset$; wrSet' := $\emptyset$; $\mathcal{I}$(begin) ($\mathcal{I}$(<read>))$^{}$ <test>$^{}$ $\mathcal{I}$(commit)$\label{ln:part32}$ $\mathcal{I}$(begin) ($\mathcal{I}$(<write>))$^{}$ $\mathcal{I}$(commit)$\label{ln:part33}$ \end{lstlisting} % $\mathcal{I}$(begin) ($\mathcal{I}$(<read>))$^{}$ <test>$^{}$ ($\mathcal{I}$(<write>))$^{}$ $\mathcal{I}$(commit) \end{minipage}\hfill \begin{minipage}{0.305\linewidth} % \lstset{numbers=none} $\mathcal{I}_\#$( r := x ): \begin{lstlisting}[xleftmargin=2mm,firstnumber=16] r := x; $\label{ln:delay1}$ hbR['x'] := 0; rdSet := rdSet $\cup$ { 'x' }; \end{lstlisting} $\mathcal{I}_\#$( x := e ): \begin{lstlisting}[xleftmargin=2mm,firstnumber=19] if ( varW == $\bot$ and * ) varW := 'x'; \end{lstlisting} $\mathcal{I}_\#$( commit ): \begin{lstlisting}[xleftmargin=2mm,firstnumber=21] assume ( varW != $\bot$ ) done$_\#$ := true $\label{ln:delay2}$ \end{lstlisting} \end{minipage} \vspace{1mm} \begin{minipage}{0.5\linewidth} $\mathcal{I}$( begin ): \begin{lstlisting}[xleftmargin=3mm,firstnumber=23] hb := $\bot$ if ( hbP != $\bot$ and hbP < 2 ) hb := 0; else if ( hbP = 2 ) hb := 2; \end{lstlisting} $\mathcal{I}$( commit ): \begin{lstlisting}[xleftmargin=3mm,firstnumber=29] assume ( hb != $\bot$ ) $\label{ln:assume}$ assert ( hb == 2 or varW $\not\in$ wrSet' ); $\label{ln:assert}$ if ( hbP == $\bot$ or hbP > hb ) $\label{ln:hbupdates1}$ hbP = hb; for each 'x' $\in$ wrSet' if ( hbW['x'] == $\bot$ or hbW['x'] > hb ) hbW['x'] = hb; for each 'x' $\in$ rdSet' if ( hbR['x'] == $\bot$ or hbR['x'] > hb ) hbR['x'] = hb; $\label{ln:hbupdates2}$ rdSet := rdSet $\cup$ rdSet'; $\label{ln:rdSetUpdate}$ wrSet := wrSet $\cup$ wrSet'; $\label{ln:wrSetUpdate}$ \end{lstlisting} \end{minipage} \hfill \begin{minipage}{0.45\linewidth} $\mathcal{I}$( r := x ): \begin{lstlisting}[xleftmargin=3mm,firstnumber=42] r := x; rdSet' := rdSet' $\cup$ { 'x' }; if ( 'x' $\in$ wrSet ) $\label{ln:rdHBCont}$ if ( hbW['x'] != 2 ) hb := 0 $\label{ln:rdHBConthb1}$ else if ( hb == $\bot$ ) hb := hbW['x'] $\label{ln:rdHBConthb2}$ \end{lstlisting} $\mathcal{I}$( x := e ): \begin{lstlisting}[xleftmargin=3mm,firstnumber=49] x := e; wrSet' := wrSet' $\cup$ { 'x' }; if ( 'x' $\in$ wrSet ) $\label{ln:wrHBCont1}$ if ( hbW['x'] != 2 ) hb := 0 else if ( hb == $\bot$ ) hb := hbW['x'] if ( 'x' $\in$ rdSet ) $\label{ln:wrHBCont2}$ if ( hb = $\bot$ or hb > hbR['x'] + 1 ) hb := min(hbR['x'] + 1,2) \end{lstlisting} \end{minipage} \vspace{-10pt} \normalsize \caption{A program instrumentation for checking robustness against \pcc{} relative to \sic{}. The auxiliary variables used by the instrumentation are shared variables, except for \texttt{hbP}, \texttt{rdSet'}, and \texttt{wrSet'}, which are process-local variables, and they are initially set to $\bot$. This instrumentation uses program constructs which can be defined as syntactic sugar from the syntax presented in Section~\ref{sec:consistency}, e.g., if-then-else statements (outside transactions).} \label{Figure:Instr} \vspace{-10pt} \end{figure} % \scriptsize % \begin{minipage}{0.30\linewidth} % \begin{eqnarray} % %%%%%%%%%%% % &&\semidlerpcinstr{\thetransitionnotogo{\atr}{[\rds;\ \wrs]}} = \notag\\ % &&\thetransitionnotogo{\atrrd{\atr}}{[\rds]}\notag\\ % &&\thetransitionnotogo{\atrwr{\atr}}{[\wrs]}\notag\\[4mm] % %%%%%%%%%%% % &&\semidlermove{\thetransitionnotogo{\atr}{[\rds;\ \wrs]}} = \notag\\ % &&\thetransitionnolabel{\thecondition{ a_{\wrflag} = \perp \land\ \apr_{\wrflag} = \perp}}\notag\\ % &&\thetransitionnolabel{\lit{Foreach}\ \ensuremath{\rdO}\in\ensuremath{\rds}}\notag\\ % &&\thetransitionnolabel{\ \ \ \sem{\rdO}_{\instr}}\notag\\ % &&\thetransitionnolabel{\lit{Foreach}\ \ensuremath{\wrO}\in\ensuremath{\wrs}}\notag\\ % &&\thetransitionnolabel{\ \ \ \sem{\wrO}_{\instr}}\notag\\ % &&\thetransitionnolabel{\thecondition{ a_{\wrflag} \neq \perp \land\ \apr_{\wrflag} = \apr}}\notag\\[4mm] % %%%%%%%%%%% % %%%%%%%%%%% % &&\semidlermove{\thetransitionnotogo{\rdO}{\theload{\areg}{\anaddr}}} =\notag\\ % &&\thetransitionnolabel{\theload{\areg}{\anaddr'}}\notag\\ % &&\thetransitionnolabel{\theassign{\rdaddr{\anaddr}}{0}}\notag\\ % &&\thetransitionnolabel{\theassign{\rdSet}{\rdSet \cup\ \{\anaddr\}}}\notag\\[4mm] % %%%%%%%%%% % &&\semidlermove{\thetransitionnotogo{\wrO}{\thestore{\anaddr}{e}}} = \notag\\ % &&\thetransitionnolabel{\thestore{\anaddr'}{e}}\notag\\ % &&\thetransitionnolabel{\theifcondition{ a_{\wrflag} = \perp \land\ \apr_{\wrflag} = \perp \land\ }}\notag\\ % &&\thetransitionnolabel{\ \ \ \theassign{a_{\wrflag}}{\anaddr}}\notag\\ % &&\thetransitionnolabel{\ \ \ \theassign{\apr_{\wrflag}}{\apr}}\notag % %%%%%%%%%%% % %%%%%%%%%%% % \end{eqnarray} % \end{minipage}\hfill % \begin{minipage}{0.30\linewidth} % \begin{eqnarray} % %%%%%%%%%%% % %%%%%%%%%%% % &&\semidler{\thetransitionnotogo{\atr}{[\rds;\ \wrs]}} = \notag\\ % &&\thetransitionnolabel{\thecondition{ a_{\wrflag} \neq \perp \land\ \apr \neq \apr_{\wrflag}}}\notag\\ % &&\thetransitionnolabel{\theifcondition{\rwMap{\apr} \neq \perp \land\ \rwMap{\apr} < 2 }}\notag\\ % &&\thetransitionnolabel{\ \ \ \theassign{\rw}{0}}\notag\\ % &&\thetransitionnolabel{\theelseifcondition{\rwMap{\apr} = 2}}\notag\\ % &&\thetransitionnolabel{\ \ \ \theassign{\rw}{2}}\notag\\ % &&\thetransitionnolabel{\lit{Foreach}\ \ensuremath{\rdO}\in\ensuremath{\rds}}\notag\\ % &&\thetransitionnolabel{\ \ \ \sem{\rdO}_{\textsf{H}}}\notag\\ % &&\thetransitionnolabel{\lit{Foreach}\ \ensuremath{\wrO}\in\ensuremath{\wrs}}\notag\\ % &&\thetransitionnolabel{\ \ \ \sem{\wrO}_{\textsf{H}}}\notag\\ % &&\thetransitionnolabel{\thecondition{\rw \neq \perp}}\notag\\ % &&\thetransitionnolabel{\theifcondition{\rw \neq 2 \land\ (\{a_{\wrflag}\} \cap\ \wrSet') \neq \perp}}\notag\\ % &&\thetransitionnolabel{\ \ \ \lit{assert}\ \myfalse\lit{;}}\notag\\ % &&\thetransitionnolabel{\theifcondition{\rwMap{\apr} = \perp \lor\ \rwMap{\apr} > \rw}}\notag\\ % &&\thetransitionnolabel{\ \ \ \theassign{\rwMap{\apr}}{\rw}}\notag\\ % &&\thetransitionnolabel{\lit{Foreach}\ \ensuremath{\anaddr}\in\ensuremath{\wrSet'}}\notag\\ % &&\thetransitionnolabel{\ \ \ \theifcondition{\wraddr{\anaddr} = \perp \lor\ \wraddr{\anaddr} > \rw}}\notag\\ % &&\thetransitionnolabel{\ \ \ \ \ \ \theassign{\wraddr{\anaddr}}{\rw}}\notag\\ % &&\thetransitionnolabel{\lit{Foreach}\ \ensuremath{\anaddr}\in\ensuremath{\rdSet'}}\notag\\ % &&\thetransitionnolabel{\ \ \ \theifcondition{\rdaddr{\anaddr} = \perp \lor\ \rdaddr{\anaddr} > \rw}}\notag\\ % &&\thetransitionnolabel{\ \ \ \ \ \ \theassign{\rdaddr{\anaddr}}{\rw}}\notag\\ % &&\thetransitionnolabel{\theassign{\wrSet}{\wrSet \cup\ \wrSet'}}\notag\\ % &&\thetransitionnolabel{\theassign{\rdSet}{\rdSet \cup\ \rdSet'}}\notag % %%%%%%%%%%% % %%%%%%%%%%% % \end{eqnarray} % \end{minipage}\hfill % \begin{minipage}{0.30\linewidth} % \begin{eqnarray} % %%%%%%%%%%% % &&\semidler{\thetransitionnotogo{\rdO}{\theload{\areg}{\anaddr}}} = \notag\\ % &&\thetransitionnolabel{\theload{\areg}{\anaddr}}\notag\\ % &&\thetransitionnolabel{\theassign{\rdSet'}{\rdSet' \cup\ \{\anaddr\}}}\notag\\ % &&\thetransitionnolabel{\theifcondition{(\wrSet \cap\ \{\anaddr\}) \neq \perp}}\notag\\ % &&\thetransitionnolabel{\ \ \ \theifcondition{\wraddr{\anaddr} \neq 2}}\notag\\ % &&\thetransitionnolabel{\ \ \ \ \ \ \theassign{\rw}{0}}\notag\\ % &&\thetransitionnolabel{\ \ \ \theelseifcondition{\rw = \perp}}\notag\\ % &&\thetransitionnolabel{\ \ \ \ \ \ \theassign{\rw}{\wraddr{\anaddr}}}\notag\\[4mm] % %%%%%%%%%%% % &&\semidler{\thetransitionnotogo{\wrO}{\thestore{\anaddr}{e}}} = \notag\\ % &&\thetransitionnolabel{\thestore{\anaddr}{e}}\notag\\ % &&\thetransitionnolabel{\theassign{\wrSet'}{\wrSet' \cup\ \{\anaddr\}}}\notag\\ % &&\thetransitionnolabel{\theifcondition{(\wrSet \cap\ \{\anaddr\}) \neq \perp}}\notag\\ % &&\thetransitionnolabel{\ \ \ \theifcondition{\wraddr{\anaddr} \neq 2}}\notag\\ % &&\thetransitionnolabel{\ \ \ \ \ \ \theassign{\rw}{0}}\notag\\ % &&\thetransitionnolabel{\ \ \ \theelseifcondition{\rw = \perp}}\notag\\ % &&\thetransitionnolabel{\ \ \ \ \ \ \theassign{\rw}{\wraddr{\anaddr}}}\notag\\ % &&\thetransitionnolabel{\theifcondition{(\rdSet \cap\ \{\anaddr\}) \neq \perp}}\notag\\ % &&\thetransitionnolabel{\ \ \ \theifcondition{\rw = \perp \vee\ \rw > \rdaddr{\anaddr} + 1}}\notag\\ % &&\thetransitionnolabel{\ \ \ \ \ \ \theassign{\rw}{\minOf{\rdaddr{\anaddr} + 1}{2}}}\notag % %%%%%%%%%%% % %%%%%%%%%%% % \end{eqnarray} % \end{minipage} % \normalsize % \caption{Program Instrumentation Rules. $min(a,b)$ is the smallest of $a$ and $b$.} % \label{Figure:Instr} % \end{figure} The instrumentation executes (incomplete) transactions without affecting the auxiliary variables (without tracking happens-before dependencies) (lines~\ref{ln:part11} and \ref{ln:part12}) until a non-deterministically chosen point in time when it declares the current transaction as the candidate for $_$ (line~\ref{ln:part21}). Only one candidate for $_$ can be chosen during the execution. This transaction executes only its reads and it chooses non-deterministically a variable that it could write as a witness for the $$ dependency (see lines~\ref{ln:delay1}-\ref{ln:delay2}). The name of this variable is stored in a global variable \texttt{varW} (see the definition of $ℐ_#$( x := e )). %The id of the process executing the current transaction is also recorded, in the variable \texttt{pW} (see the definition of $\mathcal{I}_\#$( x := e )). The writes are \emph{not} applied on the shared memory. Intuitively, $_$ should be thought as a transaction whose writes are delayed for later, after transaction $$ in Figure~\ref{fig:violationInstr} executed. The instrumentation checks that $_$ and $$ can be connected by some happens-before path that includes the $$ and $$ dependencies, and that does not contain two consecutive $$ dependencies. If it is the case, it violates an assertion at the commit point of $$. Since the write part of $_$ is intuitively delayed to execute after $$, the process executing $_$ is disabled all along the execution (see the \texttt{assume false}). After choosing the candidate for $_$, the instrumentation uses the auxiliary variables for tracking happens-before dependencies. Therefore, \texttt{rdSet} and \texttt{wrSet} record variables read and written, respectively, by transactions that are connected by a happens-before path to $_$ (in a trace of $$). This is ensured by the assume at line~\ref{ln:assume}. During the execution, the variables read or written by a transaction\footnote{These are stored in the local variables \texttt{rdSet'} and \texttt{wrSet'} while the transaction is running.} that writes a variable in \texttt{rdSet} (see line~\ref{ln:wrHBCont2}), or reads or writes a variable in \texttt{wrSet} (see lines~\ref{ln:rdHBCont} and~\ref{ln:wrHBCont1}), will be added to these sets (see lines~\ref{ln:rdSetUpdate} and~\ref{ln:wrSetUpdate}). %The \texttt{assume} at line~\ref{ln:assume} will pass anytime such a conflicting access happens. Since the variables that $_$ writes in $$ are not recorded in \texttt{wrSet}, these happens-before paths must necessarily start with a $$ dependency (from $_$). When the assertion fails (line~\ref{ln:assert}), the condition \texttt{varW} $∈$ \texttt{wrSet}' ensures that the current transaction has a $$ dependency towards the write part of $_$ (the current transaction plays the role of $$ in Figure~\ref{fig:violationInstr}). %A subtle point of the instrumentation is that after $\atr_{\instr}$, transactions can execute in their entirety (as in $\aprog$) or in two phases, first their read part and then the write part (to simulate the \pcc{} semantics). The two scenarios lead to different effects on the auxiliary variables tracking happens-before dependencies. The reads of a transaction that executes in two phases are not recorded in \texttt{rdSet} if none of them is conflicting with a write in a previous transaction, as opposed to the case when the transaction executes in its entirety (this transaction could be connected by happens-before to $\atr_{\instr}$ because of its writes). This is complete because a transaction $\atr'$ that must execute in two phases is related by $\cfo$ to another transaction $\atr''$ which is in happens before itself (more precisely before its write part). Then, either the happens-before from $\atr''$ to $\atr'$ is $\sto$ and this scenario would be captured when $\atr'$ is chosen as a candidate for $\atr_{\instr}$, or it is $\cfo$ and the path from $\atr_{\instr}$ to $\atr'$ contains two consecutive The rest of the instrumentation checks that there exists a happens-before path from $_$ to $$ that does not include two consecutive $$ dependencies, called a \sic{}$_$ path. This check is based on the auxiliary variables whose name is prefixed by \texttt{hb} and which take values in the domain ${,0,1,2}$ ($$ represents the initial value). \begin{itemize}[topsep=3pt] \item \texttt{hbR['x']} (resp., \texttt{hbW['x']}) is 0 iff there exists a transaction $\atr'$ that reads \texttt{x} (resp., writes to \texttt{x}), such that there exists a \sic{}$_{\neg}$ path from $\atr_{\instr}$ to $\atr'$ that ends with a dependency which is \emph{not} $\cfo$, \item \texttt{hbR['x']} (resp., \texttt{hbW['x']}) is 1 iff there exists a transaction $\atr'$ that reads \texttt{x} (resp., writes to \texttt{x}) that is connected to $\atr_{\instr}$ by a \sic{}$_{\neg}$ path, and \emph{every} \sic{}$_{\neg}$ path from $\atr_{\instr}$ to a transaction $\atr''$ that reads \texttt{x} (resp., writes to \texttt{x}) ends with an $\cfo$ dependency, \item \texttt{hbR['x']} (resp., \texttt{hbW['x']}) is 2 iff there exists no \sic{}$_{\neg}$ path from $\atr_{\instr}$ to a transaction $\atr'$ that reads \texttt{x} (resp., writes to \texttt{x}). \end{itemize} The local variable \texttt{hbP} has the same interpretation, except that $'$ and $”$ are instantiated over transactions in the same process (that already executed) instead of transactions that read or write a certain variable. Similarly, the variable \texttt{hb} is a particular case where $'$ and $”$ are instantiated to the current transaction. The violation of the assertion at line~\ref{ln:assert} implies that \texttt{hb} is 0 or 1, which means that there exists a \sic{}$_$ path from $_$ to $$. During each transaction that executes after $_$, the variable \texttt{hb} characterizing happens-before paths that end in this transaction is updated every time a new happens-before dependency is witnessed (using the values of the other variables). For instance, when witnessing a $$ dependency (line~\ref{ln:rdHBCont}), if there exists a \sic{}$_$ path to a transaction that writes to \texttt{x}, then the path that continues with the $$ dependency towards the current transaction is also a \sic{}$_$ path, and the last dependency of this path is not $$. Therefore, \texttt{hb} is set to 0 (see line~\ref{ln:rdHBConthb1}). Otherwise, if every path to a transaction that writes to \texttt{x} is not a \sic{}$_$ path, then every path that continues to the current transaction (by taking the $$ dependency) remains a non \sic{}$_$ path, and \texttt{hb} is set to the value of \texttt{hbW[`x`]}, which is 2 in this case (see line~\ref{ln:rdHBConthb2}). Before ending a transaction, the value of \texttt{hb} can be used to modify the \texttt{hbR}, \texttt{hbW}, and \texttt{hbP} variables, but only if those variables contain bigger values (see lines~\ref{ln:hbupdates1}--\ref{ln:hbupdates2}). %TODO I STOPPED HERE %Every transaction in $\aprog_\siinstr$ is constructed through the rewriting of the corresponding transaction %from $\aprog_\pcinstr$ where we use auxiliary flags to store the accessed locations and build the happens %before relation between $\atr_{\instr}$ and $\atr$. %The main ideas of the instrumentation consists executing the transactions of the transformed program as in %$\semidlerpcinstr{\atr}$ of Figure \ref{Figure:Instr} which constitute the elements of the sequence $\alpha$ in $\atrace_{\instr}$. Until reaching the transaction $\atr_{\instr}$ which is instrumented as in $\semidlermove{\atr_{\instr}}$ of Figure \ref{Figure:Instr} where we uses a copy of the variables in the original program denoted $\anaddr'$. Then, the instrumented transaction $\semidlermove{\atr_{\instr}}$ will write only to $\anaddr'$ and read only from $\anaddr'$. The writes made by all the other transactions that occur in $\alpha$ are applied to both $\anaddr'$ and $\anaddr$. Also, all of these transactions cannot read from $\anaddr'$. %We use the flag $a_{\wrflag}$ to store the name of variable that generated the dependency $(\atr,\atr_{\instr}) \in \sto$. %Also, we use the flag $\apr_{\wrflag}$ to store the identifier of the process that executed $\atr_{\instr}$. %Afterward, the transactions in $\beta \cdot \atr_{0} \cdot \gamma$ contribute to the happens-before relation between $\atr_{\instr}$ and $\atr$. The instrumentation only simulate traces such that $\beta \cdot \atr_{0} \cdot \gamma$ doesn't contain any transaction from the process that executed $\atr_{\instr}$, which is sound and complete. %Then, we for every transaction $\atr_1$ in $\beta \cdot \atr_{0} \cdot \gamma$ either we apply directly $\semidler{\atr_1}$ of Figure \ref{Figure:Instr} on $\atr_1$ or $\semidler{\semidlerpcinstr{\atr_1}}$ on the two transactions resulting from the splitting of $\atr_1$. %Note that in the instrumentation $\semidler{\atr_1}$ one of the sets $\ensuremath{\wrs}$ or $\ensuremath{\rds}$ can be empty. %To establish a happens-before relation between $\atr_{\instr}$ and $\atr$ in $\atrace_{\instr}$, we look whether a transaction can extend a happens-before started by $\atr_{\instr}$ (elements of $\beta \cdot \atr_{0} \cdot \gamma$). In order for a transaction $\atr_1$ to extend a happens-before relation, it has to satisfy one of the following conditions: % \begin{itemize}[topsep=0pt] % \item the transaction is from a process that has already another transaction in the happens-before. Thus, we ensure the continuity of the happens-before relation through $\po$ relation. % \item the transaction is reading from a variable that was written to by a previous transaction in the happens-before. Hence, we ensure the continuity of the happens-before relation through $\rfo$ relation. % \item the transaction writes to a variable that was written to by a previous transaction in the happens-before. Hence, we ensure the continuity of the happens-before relation through $\sto$ relation. % \item the transaction writes to a variable that was read by a previous transaction in the happens-before. Hence, we ensure the continuity of the happens-before relation through $\cfo$ relation. %Furthermore, as per Theorem \ref{corol:pcsi}, we must ensure that in the happens-before relation between $\atr_{\instr}$ and $\atr$ we don't have two successive $\cfo$ relations. %Therefore, we use two variables $\wrSet$ and $\rdSet$ to track the variables read and written by transactions in $\beta \cdot \atr_{0} \cdot \gamma$. %Also, for each variable, we introduce two flags $\wraddr{\anaddr}$ and $\rdaddr{\anaddr}$ for each variable $\anaddr$ to say that % \item if $\wraddr{\anaddr} = 0$ (resp., $\rdaddr{\anaddr} = 0$) there exists a transaction $\atr_1$ in the happens-before that writes to (resp., reads from) $\anaddr$ such that the happens-before relation between $\atr_{\instr}$ and $\atr_1$ doesn't contain two successive $\cfo$ relations and the last dependency relation in this happens-before is not $\cfo$. % \item if $\wraddr{\anaddr} = 1$ (resp., $\rdaddr{\anaddr} = 1$) there exists a transaction $\atr_1$ in the happens-before that writes to (resp., reads from) $\anaddr$ such that the happens-before relation between $\atr_{\instr}$ and $\atr_1$ doesn't contain two successive $\cfo$ relations, however, the last dependency relation in this happens-before is $\cfo$. % \item if $\wraddr{\anaddr} = 2$ (resp., $\rdaddr{\anaddr} = 2$) there exists a transaction $\atr_1$ in the happens-before that writes to (resp., reads from) $\anaddr$ such that the happens-before relation between $\atr_{\instr}$ and $\atr_1$ contains two successive $\cfo$ relations. %Similarly, we introduce a local flag $\rwMap{\apr}$ for each process $\apr$ to say that % \item if $\rwMap{\apr} = 0$, there exists a transaction $\atr_1$ from $\apr$ in the happens-before such that the happens-before relation between $\atr_{\instr}$ and $\atr_1$ doesn't contain two successive $\cfo$ relations and the last dependency relation in this happens-before is not $\cfo$. % \item if $\rwMap{\apr} = 1$, every transaction $\atr_1$ from $\apr$ in the happens-before where the happens-before relation between $\atr_{\instr}$ and $\atr_1$ doesn't contain two successive $\cfo$ relations, we have that the last dependency relation in this happens-before is $\cfo$. % \item if $\rwMap{\apr} = 2$, for every transaction $\atr_1$ from $\apr$ in the happens-before, we have that the happens-before relation between $\atr_{\instr}$ and $\atr_1$ contains two successive $\cfo$ relations. %Each transaction $\atr_1$ which is trying to join the happens-before is equipped with a local flag $\rw$ %that is initialized to null ($\perp$) at the start of $\atr_1$ and must contain a value that's different than null %when reaching the end of the transaction (meaning that $\atr_1$ satisfied one of the four conditions required in order to join the happens-before) such that % \item $\rw = 0$ means that there exists a happens-before relation between $\atr_{\instr}$ and $\atr_1$ that doesn't contain two successive $\cfo$ relations and the last dependency relation in this happens-before is not $\cfo$. % \item $\rw = 1$ means that in every happens-before relation between $\atr_{\instr}$ and $\atr_1$ that doesn't contain two successive $\cfo$ relations we have that the last dependency relation in this happens-before is $\cfo$. % \item $\rw = 2$ means that every happens-before relation between $\atr_{\instr}$ and $\atr_1$ contains two successive $\cfo$ relations. %Otherwise, the execution is blocked. %Also, note that at the start of the execution, all flags are initialized to null. %In general, whether a transaction is splitted and executed without instrumentation $\semidlerpcinstr{\atr}$, or instrumented as in $\semidlermove{\atr}$, or instrumented as in $\semidler{\atr}$, or splitted and each of the resulting transactions is instrumented as in $\semidler{\semidlerpcinstr{\atr}}$ is set non-deterministically and can vary from execution to execution. %In the instrumentation $\semidler{\atr_1}$ of a transaction $\atr_1$ in the happens before, we check whether we reached an error state if $\atr_1$ writes to the variable that was stored in $a_{\wrflag}$ and $\rw \neq 2$. %The final instrumentation of a given program $\aprog$, denoted by $\aprog_\siinstr$, is obtained by replacing each transaction $\atr$ with the concatenation of the four possible instrumentations, i.e., %\atr_\siinstr ::= \semidlerpcinstr{\atr} \hspace{0.17cm} \semidlermove{\atr} \hspace{0.17cm} \semidler{\atr} \hspace{0.17cm} \semidler{\semidlerpcinstr{\atr}} The correctness of the instrumentation is stated in the following theorem. \begin{theorem}\label{them:RobPcSiInstr} A program $\aprog$ is robust against \pcc{} relative to \sic{} iff the instrumentation in Figure~\ref{Figure:Instr} does not violate an assertion when executed under \serc{}. \end{theorem} %We give the proof of Theorem \ref{them:RobPcSiInstr} in the supplementary materials. Theorem~\ref{them:RobPcSiInstr} implies the following complexity result for finite-state programs. The lower bound is proved similarly to the case \ccc{} vs \pcc{}. %Similar to before, we obtain an upper bound of the robustness problem against \pcc{} relative to \sic{} based on the reachability problem under \serc{}. For the lower bound, we construct a program $\aprog'$ from $\aprog$ in the same manner as before just instead of the $\mathsf{SB}$ program, we use the $\mathsf{LU}$ program. Then, we obtain that $\aprog$ reaches a state $\astate$ iff $\aprog'$ is not robust against \pcc{} relative to \sic{}. \begin{corollary}\label{corol:SIRobcomplexity} Checking robustness of a program with a fixed number of variables and bounded data domain against \pcc{} relative to \sic{} is PSPACE-complete when the number of processes is bounded and EXPSPACE-complete, otherwise. % Checking robustness of a program with a fixed number of variables and bounded data domain against \pcc{} relative to \sic{} is PSPACE-complete when the number of processes is fixed and EXPSPACE-complete, otherwise. \end{corollary} Checking robustness against \ccc{} relative to \sic{} can be also shown to be reducible (in polynomial time) to a reachability problem under \serc{} by combining the results of checking robustness against \ccc{} relative to \pcc{} and \pcc{} relative to \sic{}. \begin{theorem} \label{them:RobCcSi} A program $\aprog$ is robust against \ccc{} relative to \sic{} iff $\aprog$ is robust against \ccc{} relative to \pcc{} and $\aprog$ is robust against \pcc{} relative to \sic{}. \end{theorem} \begin{remark} \label{rem:robustness} Our reductions of robustness checking to reachability apply to an extension of our programming language where the number of processes is unbounded and each process can execute an arbitrary number of times a statically known set of transactions. This holds because the instrumentation in Figure~\ref{Figure:Instr} and the one in~[10] (for the case \ccc{} vs. \serc{}) consist in adding a set of instructions that manipulate a fixed set of process-local or shared variables, which do not store process or transaction identifiers. These reductions extend also to SQL queries that access unbounded size tables. Rows in a table can be interpreted as memory locations (identified by primary keys in unbounded domains, e.g., integers), and SQL queries can be interpreted as instructions that read/write a set of locations in one shot. These possibly unbounded sets of locations can be represented symbolically using the conditions in the SQL queries (e.g., the condition in the WHERE part of a SELECT). The instrumentation in Figure 6 needs to be adapted so that read and write sets are updated by adding sets of locations for a given instruction (represented symbolically as mentioned above). \end{remark} % \begin{multicols}{2} % \begin{algorithmic}[1] % %\REQUIRE $n \geq 0 \vee x \neq 0$ % %\ENSURE $y = x^n$ % \STATE $y \Leftarrow 1$ % %\IF{$\hb{} = \perp$} % %\STATE $X \Leftarrow 1 / x$ % %\STATE $N \Leftarrow -n$ % %\ELSE % %\STATE $X \Leftarrow x$ % %\STATE $N \Leftarrow n$ % %\ENDIF % %\WHILE{$N \neq 0$} % %\IF{$N$ is even} % %\STATE $X \Leftarrow X \times X$ % %\STATE $N \Leftarrow N / 2$ % %\ELSE[$N$ is odd] % %\STATE $y \Leftarrow y \times X$ % %\STATE $N \Leftarrow N - 1$ % %\ENDIF % %\ENDWHILE % \end{algorithmic} % \end{multicols} % \caption{x} % \label{alg1} %!TEX root = draft.tex \vspace{-15pt} \section{Proving Robustness Using Commutativity Dependency Graphs} \label{sec:commutativitygraph} We describe an approximated technique for proving robustness, which leverages the concept of left/right mover in Lipton's reduction theory~[38]. This technique reasons on the \emph{commutativity dependency graph}~[9] associated to the transformation $_$ of an input program $$ that allows to simulate the \pcc{} semantics under serializability (we use a slight variation of the original definition of this class of graphs). We characterize robustness against \ccc{} relative to \pcc{} and \pcc{} relative to \sic{} in terms of certain properties that (simple) cycles in this graph must satisfy. We recall the concept of movers and the definition of commutativity dependency graphs. Given a program $$ and a trace $= _1·…·_n ∈_()$ of $$ under serializability, we say that $_i ∈$ \emph{moves right (resp., left)} in $$ if $_1·…·_i-1·_i+1·_i·_i+2·…·_n$ (resp., $_1·…·_i-2·_i·_i-1·_i+1·…·_n$) is also a valid execution of $$, $_i$ and $_i+1$ (resp., $_i-1$) are executed by distinct processes, and both traces reach the same end state. A transaction $∈$ is not a right (resp., left) mover iff there exists a trace $∈_()$ such that $∈$ and $$ doesn't move right (resp., left) in $$. Thus, when a transaction $$ is \emph{not} a right mover then there must exist another transaction $' ∈$ which caused $$ to not be permutable to the right (while preserving the end state). Since $$ and $'$ do not commute, then this must be because of either a write-read, write-write, or a read-write dependency relation between the two transactions. We say that $$ is not a right mover because of $'$ and a dependency relation that is either write-read, write-write, or read-write. Notice that when $$ is not a right mover because of $'$ then $'$ is not a left mover because of $$. We define $$ as a binary relation between transactions such that $(,') ∈$ when $$ is \emph{not} a right mover because of $'$ and a write-read dependency ($t'$ reads some value written by $t$). We define the relations $$ and $$ corresponding to write-write and read-write dependencies in a similar way. %We denote $(\atr,\atr') \in \msto$ when $\atr$ is not a right mover because $\atr'$ and write-write dependency. Similarly, we denote $(\atr,\atr') \in \mcfo$ when $\atr$ is not a right mover because $\atr'$ and a read-write dependency between the two transactions. We call $$, $$, and $$, \emph{non-mover} relations. The \emph{commutativity dependency graph} of a program $$ is a graph where vertices represent transactions in $$. Two vertices are linked by a program order edge if the two transactions are executed by the same process. The other edges in this graph represent the ``non-mover'' relations $$, $$, and $$. Two vertices that represent the two components $$ and $$ of the same transaction $$ (already linked by $$ edge) are also linked by an undirected edge labeled by $$ (same-transaction relation). %To model an arbitrary instantiation of the same transaction $\atr$, we take a symbolic integer $m > 0$ and add a sequence of $m$ vertices representing $m$ instantiations of $\atr$ in the commutativity graph. We relate every two successive vertices by a program order edge. Also, similar to before for every vertex in the sequence, we draw the incoming and outgoing edges using the ``non-mover'' relations $\mrfo$, $\msto$, and $\mcfo$. \begin{wrapfigure}{r}{0.53\textwidth} \vspace{-25pt} \lstset{basicstyle=\ttfamily\scriptsize} \centering \resizebox{!}{2.2cm}{ \begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=4cm, semithick, transform shape,every text node part/.style={align=left}] \node[shape=rectangle ,draw=none,font=\large, label={left:$\atrwr{\atr 1}$}] at (-3.2,0) (m) {$[x = 1]$}; \node[shape=rectangle ,draw=none,font=\large, label={left:$\atrwr{\atr 2}$}] at (-3.2,-2) (m1) {$[y = 1]$}; \node[shape=rectangle ,draw=none,font=\large, label={right:$\atrrd{\atr 3}$}] at (0,0) (p){$[r1 = y]$}; \node[shape=rectangle ,draw=none,font=\large, label={right:$\atrrd{\atr 4}$}] at (0,-2) (p1){$[r2 = x]$}; \begin{scope}[ every edge/.style={draw=black,very thick}] \path[->] (m) edge[left] node {$\po$} (m1); \path[->] (p1) edge[bend left, above] node[xshift=2.9mm,yshift=-1.3mm] {$\mcfo$} (m); \path[->] (m) edge[bend left, below] node {$\mrfo$} (p1); \path[->] (m1) edge[bend left, below] node[xshift=2.5mm,yshift=0.5mm] {$\mrfo$} (p); \path[->] (p) edge[bend left, above] node[xshift=-1.2mm,yshift=0.3mm] {$\mcfo$} (m1); \path[->] (p) edge[right] node {$\po$} (p1); \end{scope} \end{tikzpicture}} \vspace{-3pt} \caption{The commutativity dependency graph of the $\mathsf{MP}_{\pcinstr}$ program.} \label{fig:litmus4CDG} \vspace{-15pt} \end{wrapfigure} Our results about the robustness of a program $$ are stated over a slight variation of the commutativity dependency graph of $_$ (where a transaction is either read-only or write-only). This graph contains additional undirected edges that link every pair of transactions $$ and $$ of $_$ that were originally components of the same transaction $$ in $$. Given such a commutativity dependency graph, the robustness of $$ is implied by the absence of cycles of specific shapes. These cycles can be seen as an abstraction of potential robustness violations for the respective semantics (see Theorem~\ref{them:MovRobCcPc} and Theorem~\ref{them:MovRobPcSi}). % we ask whether this graph has cycles of some specific shapes based the characterization of robustness violation traces for the respective semantics. %The corresponding program is robust when the graph doesn't have these types of cycles Figure \ref{fig:litmus4CDG} pictures the commutativity dependency graph for the $𝖬𝖯$ program. Since every transaction in $𝖬𝖯$ is singleton, the two programs $𝖬𝖯$ and $𝖬𝖯_$ coincide. Using the characterization of robustness violations against \ccc{} relative to \serc{} from~[10] and the reduction in Theorem~\ref{them:RobCcPc}, we obtain the following result concerning the robustness against \ccc{} relative to \pcc{}. \begin{theorem} \label{them:MovRobCcPc} Given a program $\aprog$, if the commutativity dependency graph of the program $\aprog_\pcinstr$ does not contain a simple cycle formed by $\atr_1$ $\cdots$ $\atr_i$ $\cdots$ $\atr_n$ such that: \begin{itemize}[topsep=3pt] \item $(\atr_n,\atr_1) \in \mcfo$; \item $(\atr_j, \atr_{j+1}) \in (\po \cup \rfo)^{}$, for $j \in [1,i-1]$; \item $(\atr_i,\atr_{i+1}) \in (\mcfo \cup \msto)$; \item $(\atr_j,\atr_{j+1}) \in (\mcfo \cup \msto \cup \mrfo \cup \po)$, for $j \in [i+1,n-1]$. \end{itemize} then $\aprog$ is robust against \ccc{} relative to \pcc{}. \end{theorem} \begin{comment} \begin{proof} It is enough to show: if $\aprog$ is not robust against \ccc{} relative to \pcc{} then we have a simple cycle in the commutativity dependency graph of $\aprog_\pcinstr$ of the form above. Assume $\aprog$ is not robust against \ccc{} relative to \pcc{}. Then, from Theorem \ref{them:RobCcPc}, we obtain $\aprog_\pcinstr$ is not robust against \ccc{} relative to \serc{}. Also it was shown in [10] that if a program is not robust then there must exist a robustness violation trace (\ccc{} relative to \serc{}) $\atrace_\pcinstr$ of the shape $\atrace_\pcinstr = \alpha \cdot \atr_1 \cdot \beta \cdot \atr_i \cdot \atr_{i+1} \cdot \gamma \cdot \atr_n$ where $(\atr_1,\atr_i) \in (\po \cup \rfo)^{+}$, $(\atr_{i},\atr_{i+1}) \in (\sto \cup \cfo)$, $(\atr_{i+1},\atr_n) \in \hbo$, and $(\atr_n,\atr_1) \in \cfo$. Note that since transactions in the trace $\atrace_\pcinstr$ can either be read-only or write-only. Then, $(\atr_{i},\atr_{i+1}) \in (\sto \cup \cfo)$ and $(\atr_{n},\atr_1) \in \cfo$ imply that $\atr_1$ and $\atr_{i+1}$ must be a write-only transactions and $\atr_{n}$ must be a read-only transaction. Note that we may have $\beta = \gamma = \epsilon$ as the case for the trace of the $\mathsf{SB}$ program given in Figure \ref{fig:litmus1}. %In the trace $\atrace_\pcinstr$, we let $\atr_1$ to be $\atr_1$, $\atr_i$ to be $\atr_2$, $\atr_3$ to be $\atr_{i+1}$, and $\atr_4$ to be $\atr_n$ of Theorem \ref{them:MovRobCcPc}. We consider first the general case when $\atr_1 \not\equiv \atr_2$. The other case can be proved in the same way. Consider the prefix $\atrace_{p}$ of $\atrace_\pcinstr$: $\atrace_{p} = \alpha \cdot \atr_1 \cdot \beta \cdot \atr_{i}$ where $(\atr_1,\atr_{i}) \in (\po \cup \rfo)^{+}$ which is a \serc{} trace of $\aprog_{\pcinstr}$. Then, we have a sequence of transactions from $\atr_1$ to $\atr_{i}$ that are related by either $\po$ or $\rfo$. In the case two transactions are only related by $\rfo$, then the first transaction is not a right mover because of the second transaction reads from a write in the first transaction. Thus, we can relate the two transactions using the relation $\mrfo$ in the commutativity dependency graph. Similarly consider the following trace $\atrace_{s}$ extracted from $\atrace_\pcinstr$: $\atrace _{s} = \alpha \cdot \atr_{i+1} \cdot \gamma \cdot \atr_n$ where $(\atr_{i+1},\atr_n) \in \hbo$ which is a \serc{} trace of $\aprog_{\pcinstr}$. Similar to before, we have a sequence of transactions from $\atr_{i+1}$ to $\atr_n$ that are related by either $\po$, $\rfo$, $\sto$, or $\cfo$. For any two transactions that are related only by either $\rfo$, $\sto$, or $\cfo$, this implies that the first transaction is not a right mover because of the second transaction and a write-read, write-write, or read-write dependency between the two, respectively. Thus, we can relate the two transactions using either $\mrfo$, $\msto$, or $\mcfo$, respectively. Now consider the following trace $\atrace_{1}$ extracted from $\atrace_\pcinstr$: $\atrace_{1} = \alpha \cdot \atr_1 \cdot \beta \cdot \atr_{i} \cdot \atr_{i+1}$ where $(\atr_{i},\atr_{i+1}) \in (\sto \cup \cfo)$ is a \serc{} trace of $\aprog_{\pcinstr}$. Because $\atr_{i}$ and $\atr_{i+1}$ are related by either $\sto$ or $\cfo$, then $\atr_{i}$ is not a right mover because of $\atr_{i+1}$ and a write-write or read-write dependency between the two, respectively. Thus, we can relate the two transactions using either $\msto$ or $\mcfo$, respectively. Finally, consider the following trace $\atrace_{2}$ extracted from $\atrace_\pcinstr$: $\atrace_{2} = \alpha \cdot \atr_{i+1} \cdot \gamma \cdot \atr_n \cdot \atr_1$ where $(\atr_n,\atr_1) \in \cfo$ is a \serc{} trace of $\aprog_{\pcinstr}$. Because $\atr_{n}$ and $\atr_{1}$ are related by $\cfo$, then $\atr_{n}$ is not a right mover because of $\atr_{1}$ and a read-write dependency between the two. Thus, we can relate the two transactions using $\mcfo$. \end{proof} \end{comment} Next we give the characterization of commutativity dependency graphs required for proving robustness against \pcc{} relative to \sic{}. \begin{theorem} \label{them:MovRobPcSi} Given a program $\aprog$, if the commutativity dependency graph of the program $\aprog_\pcinstr$ does not contain a simple cycle formed by $\atr_1$ $\cdots$ $\atr_n$ such that: \begin{itemize}[topsep=3pt] \item $(\atr_n,\atr_1) \in \msto$, $(\atr_1,\atr_2) \in \sametro$, and $(\atr_2,\atr_3) \in \mcfo$; \item $(\atr_j,\atr_{j+1}) \in (\mcfo \cup \msto \cup \mrfo \cup \po \cup \sametro)^{}$, for $j \in [3,n-1]$; \item $\forall\ j \in [2,n-2].$ \begin{itemize} \item $\mbox{if }(\atr_j,\atr_{j+1}) \in \mcfo\mbox{ then }(\atr_{j+1},\atr_{j+2}) \in (\mrfo \cup \po \cup \msto)$; \item $\mbox{if }(\atr_{j+1},\atr_{j+2}) \in \mcfo\mbox{ then }(\atr_{j},\atr_{j+1}) \in (\mrfo \cup \po)$. \end{itemize} \item $\forall\ j \in [3,n-3]. \mbox{ if }(\atr_{j+1},\atr_{j+2}) \in \sametro\mbox{ and }(\atr_{j+2},\atr_{j+3}) \in \mcfo \mbox{ then }(\atr_{j},\atr_{j+1}) \in \msto$. \end{itemize} then $\aprog$ is robust against \pcc{} relative to \sic{}. \end{theorem} \begin{comment} \begin{proof} Similar to before it is enough to show: if $\aprog$ is not robust against \pcc{} relative to \sic{} then we have a simple cycle in the commutativity dependency graph of $\aprog_\pcinstr$ of the form above. Assume $\aprog$ is not robust against \pcc{} relative to \sic{}. Then, from Theorem \ref{them:RobPcSiInstr}, we obtain that if $\sem{\aprog}$ reaches an error state under \serc{} then we will have the following trace $\atrace$ under \serc: $\atrace = \alpha \cdot \atrrd{\atr_{\instr}} \cdot \atr_3 \cdot \beta \cdot \atr_n \cdot \atrwr{\atr_{\instr}}$\footnote{For simplicity, we assume here that after reaching the error state we execute the writes of $\atr_{\instr}$, i.e., $\atrwr{\atr_{\instr}}$} where $(\atrrd{\atr_{\instr}},\atr_3) \in \cfo$, $(\atr_3,\atr_n) \in \hbo$, $(\atr_n,\atrwr{\atr_{\instr}}) \in \sto$, and we don't have two successive $\cfo$ in the happens before between $\atr_3$ and $\atr_n$. In $\atrace$, $\atrwr{\atr_{\instr}}$ (resp., $\atrrd{\atr_{\instr}}$) represents $\atr_1$ (resp., $\atr_2$) in the theorem statement. Note that we may have $\alpha = \beta = \epsilon$ as is the case of the transformed $\mathsf{LU}$ program given in Figure \ref{fig:litmus2Instr}. The construction of the cycle in the commutativity dependency graph follows the same procedure taken in the proof of Theorem \ref{them:MovRobCcPc}. The only difference is that for every two transactions of $\atrace$ that are part of the happens before between $\atr_3$ and $\atr_n$, if the two are not connected by either $\po$, $\rfo$, $\sto$, or $\cfo$ then they must be the reads and writes of the same original transaction in $\aprog$. In this case, in the commutativity dependency graph we have the two transactions related by $\sametro$. %Notice that we abused notation by using the two components $\atrrd{\atr 1}$ and $\atrwr{\atr_1}$ of $\atr_1$ in $\atrace$ to denote that $\atr 1$ writes were not immediately written to the shared variables. \end{proof} \end{comment} In Figure \ref{fig:litmus4CDG}, we have three simple cycles in the graph: \begin{itemize}[topsep=3pt] \item $(\atrwr{\atr 1}, \atrrd{\atr 4}) \in \mrfo$ and $(\atrrd{\atr 4}, \atrwr{\atr 1}) \in \mcfo$, \item $(\atrwr{\atr 2}, \atrrd{\atr 3}) \in \mrfo$ and $(\atrrd{\atr 3}, \atrwr{\atr 2}) \in \mcfo$, \item $(\atrwr{\atr 1}, \atrwr{\atr 2}) \in \po$, $(\atrwr{\atr 2}, \atrrd{\atr 3}) \in \mrfo$, $(\atrrd{\atr 3}, \atrrd{\atr 4}) \in \po$, and $(\atrrd{\atr 4}, \atrwr{\atr 1}) \in \mcfo$. \end{itemize} Notice that none of the cycles satisfies the properties in Theorems \ref{them:MovRobCcPc} and \ref{them:MovRobPcSi}. Therefore, $𝖬𝖯$ is robust against \ccc{} relative to \pcc{} and against \pcc{} relative to \sic{}. \begin{remark} \label{rem:comgraph} For programs that contain an unbounded number of processes, an unbounded number of instantiations of a fixed number of process ``templates'', or unbounded loops with bodies that contain entire transactions, a sound robustness check consists in applying Theorem~\ref{them:MovRobCcPc} and Theorem~\ref{them:MovRobPcSi} to (bounded) programs that contain two copies of each process template, and where each loop is unfolded exactly two times. This holds because the mover relations are ``static'', they do not depend on the context in which the transactions execute, and each cycle requiring more than two process instances or more than two loop iterations can be short-circuited to a cycle that exists also in the bounded program. Every outgoing edge from a third instance/iteration can also be taken from the second instance/iteration. Two copies/iterations are necessary in order to discover cycles between instances of the same transaction (the cycles in Theorem~\ref{them:MovRobCcPc} and Theorem~\ref{them:MovRobPcSi} are simple and cannot contain the same transaction twice). These results extend easily to SQL queries as well because the notion of mover is independent of particular classes of programs or instructions. %Note that the notion of mover used in commutativity dependency graphs is quite generic and independent of particular classes of programs or instructions. For instance, a mover check between two transactions that access sets of locations defined using symbolic expressions corresponds to checking whether the conjunction of the two symbolic expressions is satisfiable. \end{remark} \input{experiments.tex} %!TEX root = draft.tex \vspace{-5pt} \section{Related Work} \label{sec:related} \vspace{-5pt} The consistency models in this paper were studied in several recent works~[20, 19, 24, 42, 15, 43, 13]. Most of them focused on their operational and axiomatic formalizations. % of programs semantics under the weak consistency models both operationally and declaratively. The formal definitions we use in this paper are based on those given in~[24, 15]. Biswas and Enea~[13] shows that checking whether an execution is \ccc{} is polynomial time while checking whether it is \pcc{} or \sic{} is NP-complete. %In this paper we tackle trace-based robustness problem. In the literature, the decidability and complexity of trace-based The robustness problem we study in this paper has been investigated in the context of weak memory models, but only relative to sequential consistency, against Release/Aquire (RA), TSO and Power~[35, 16, 14, 28]. Checking robustness against \ccc{} and \sic{} relative to \serc{} has been investigated in~[9, 10]. %All of these work study the robustness between a weak consistency model (e.g., RA, TSO, or \ccc{}) and the strong semantics model, i.e., sequential consistency and serialisability. %On the other hand, In this work, we study the robustness problem between two weak consistency models, which poses different non-trivial challenges. In particular, previous work proposed reductions to reachability under sequential consistency (or \serc{}) that relied on a concept of minimal robustness violations (w.r.t. an operational semantics), which does not apply in our case. % an important technique that was commonly used in previous work which consists of using borderline violations %cannot be applied here. Since, these violations defined such that removing the last action in the violation results in serialisable execution which is not valid in our case. %However, we have reduced the robustness against \ccc{} relative to \pcc{} to the robustness against \ccc{} relative to \serc{} which allowed us to use the results that were proved in [12, 10]. %Note that our reduction from \pcc{} to \serc{} is similar in spirit to the one shown in [43], however, %in our case execution traces include the store order dependency relation in order to construct the happens before relation. The relationship between \pcc{} and \serc{} is similar in spirit to the one given by Biswas and Enea~[13] in the context of checking whether an execution is \pcc{}. However, that relationship was proven in the context of a ``weaker'' notion of trace (containing only program order and read-from), and it does not extend to our notion of trace. For instance, that result does not imply preserving $$ dependencies which is crucial in our case. Some works describe various over- or under-approximate analyses for checking robustness relative to \serc{}. The works in~[12, 17, 18, 25, 39] propose static analysis techniques based on computing an abstraction of the set of computations, which is used for proving robustness. In particular, [18, 39] encode program executions under the weak consistency model using FOL formulas to describe the dependency relations between actions in the executions. These approaches may return false alarms due to the abstractions they consider in their encoding. Note that in this paper, we prove a strengthening of the results of [12] with regard to the shape of happens before cycles allowed under \pcc{}. An alternative to {\em trace-based} robustness, is {\em state-based} robustness which requires that a program is robust if the sets of reachable states under two semantics coincide. While state-robustness is the necessary and sufficient concept for preserving state-invariants, its verification, which amounts in computing the set of reachable states under the weak semantics models is in general a hard problem. The decidability and the complexity of this problem has been investigated in the context of relaxed memory models such as TSO and Power, and it has been shown that it is either decidable but highly complex (non-primitive recursive), or undecidable [5, 6]. Automatic procedures for approximate reachability/invariant checking have been proposed using either abstractions or bounded analyses, e.g., [7, 4, 27, 1]. Proof methods have also been developed for verifying invariants in the context of weakly consistent models such as [36, 31, 40, 3]. 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We first show that $\atrace_\pcinstr$ satisfies $\axpoco$ and $\axcoarb$. For $\axpoco$, let $\atr_{1}' \in \{\atrrd{\atr_{1}},\atrwr{\atr_{1}}\}$ and $\atr_{2}' \in \{\atrrd{\atr_{2}},\atrwr{\atr_{2}}\}$, such that $(\atr_{1}',\atr_{2}') \in (\po_\pcinstr\cup\rfo_\pcinstr)^{+}$. By the definition of $\viso_\pcinstr$, we have that either $(\atr_{1}'=\atrrd{\atr_{1}},\atr_{2}'=\atrwr{\atr_{2}}) \in \po_\pcinstr$ and $\atr_1 = \atr_2$ or $(\atr_{1},\atr_2) \in (\po\cup\rfo)^{+}$, which implies that $(\atr_{1},\atr_2) \in \viso$. In both cases we obtain that $(\atr_{1}',\atr_{2}') \in \viso_\pcinstr$. The axiom $\axpoco$ can be proved in a similar way. % \item $\axcoarb$: the same proof steps as in $\axpoco$. % \end{itemize} Next, we show that $\atrace_\pcinstr$ satisfies the property $\axretval$. Let $\atr$ be a transaction in $\atrace$ that contains a read event $\readact(\atr,\anaddr,\aval)$. Let $\atr_0$ be the transaction in $\atrace$ such that $$\atr_0 = Max_{\arbo}(\{\atr' \in \atrace\ \|\ (\atr',\atr) \in \viso \wedge \exists\ \writeact(\atr',\anaddr,\cdot)\in\amap(\atr')\}).$$ The read value $\aval$ must have been written by $\atr_0$ since $\atrace$ satisfies $\axretval$. Thus, the read $\readact(\atr,\anaddr,\aval)$ in $\atrrd{\atr}$ of $\atrace_\pcinstr$ must return the value written by $\atrwr{\atr_0}$. From the definitions of $\viso_\pcinstr$ and $\arbo_\pcinstr$, we get $$\atrwr{\atr_1}\in\{\atrwr{\atr'} \in \atrace_\pcinstr\ \|\ (\atrwr{\atr'},\atrrd{\atr}) \in \viso_\pcinstr \wedge \exists\ \writeact(\atrwr{\atr'},\anaddr,\cdot)\in\amap(\atrwr{\atr'})\}$$ $$\atr_1 \in \{\atr' \in \atrace\ \|\ (\atr',\atr) \in \viso \wedge \exists\ \writeact(\atr',\anaddr,\cdot)\in\amap(\atr')\}$$ because $(\atrwr{\atr_{1}},\atrrd{\atr_{2}}) \in \viso_\pcinstr$ implies $(\atr_{1},\atr_{2}) \in \viso$. Since $(\atrwr{\atr_{1}},\atrwr{\atr_{2}}) \in \arbo_\pcinstr$ implies $(\atr_{1},\atr_{2}) \in \arbo$, we also obtain that $$\atrwr{\atr_0} = Max_{\arbo_\pcinstr}(\{\atrwr{\atr'} \in \atrace_\pcinstr\ \|\ (\atrwr{\atr'},\atrrd{\atr}) \in \viso_\pcinstr \wedge \exists\ \writeact(\atrwr{\atr'},\anaddr,\cdot)\in\amap(\atrwr{\atr'})\})$$ and since the read $\readact(\atr,\anaddr,\aval)$ in $\atrrd{\atr}$ of $\atrace_\pcinstr$ returns the value written by $\atrwr{\atr_0}$, $\atrace_\pcinstr$ satisfies $\axretval$. For the case $\textsf{X} = \pcc$, we show that $\atrace_\pcinstr$ satisfies the property $\axprefix$ (the other axioms are proved as in the case of $\ccc$). % In the previous case we already showed that $\axpoco$ and $\axcoarb$ hold. % Now we will show that $\atrace_\pcinstr$ satisfies the property $\axprefix$. Suppose we have $(\atr_{1}',\atr_{2}') \in \arbo_\pcinstr$ and $(\atr_{2}',\atr_{3}') \in \viso_\pcinstr$ where $\atr_{1}' \in \{\atrrd{\atr_{1}},\atrwr{\atr_{1}}\}$, $\atr_{2}' \in \{\atrrd{\atr_{2}},\atrwr{\atr_{2}}\}$, and $\atr_{3}' \in \{\atrrd{\atr_{3}},\atrwr{\atr_{3}}\}$. The are five cases to be discussed: \begin{enumerate} %[label=\alph] \item $(\atr_{1}'=\atrrd{\atr_{1}},\atr_{2}'=\atrwr{\atr_{2}}) \in \po_\pcinstr$ and $\atr_1 = \atr_2$ and $(\atr_{2},\atr_{3}) \in \viso$, \item $(\atr_{1},\atr_{2}) \in \arbo$ and $(\atr_{2},\atr_{3}) \in \viso$, \item $(\atr_{1},\atr_{2}) \in \arbo$ and $(\atr_{2}'=\atrrd{\atr_{2}},\atr_{3}'=\atrwr{\atr_{3}}) \in \po_\pcinstr$ and $\atr_2 = \atr_3$, \item $(\atr_{1}'=\atrrd{\atr_{1}},\atr_{2}'=\atrwr{\atr_{2}}) \in \po_\pcinstr$ and $\atr_1 = \atr_2$ and $(\atr_{2},\atr_{3}) \in \arbo$ and $\atr_{3}'= \atrwr{\atr_{3}}$, \item $(\atr_{1},\atr_{2}) \in \arbo$ and $(\atr_{2},\atr_{3}) \in \arbo$ and $\atr_{3}'= \atrwr{\atr_{3}}$. \end{enumerate} Cases (a) and (b) imply that $(\atr_{1},\atr_{3}) \in \viso$ since $\arbo;\viso \subset \arbo$, which implies that $(\atr_{1}',\atr_{3}') \in \viso_\pcinstr$. Cases (c), (d), and (e) imply that $(\atr_{1},\atr_{3}) \in \arbo$ and $\atr_{3}'= \atrwr{\atr_{3}}$ then we get that $(\atrwr{\atr_{1}},\atrwr{\atr_{3}}) \in \viso_\pcinstr$ and $\atr_{3}'= \atrwr{\atr_{3}}$ which means that $(\atr_{1}',\atr_{3}') \in \viso_\pcinstr$. Note that the rule $(\atrwr{\atr_{1}},\atrwr{\atr_{2}}) \in \viso_\pcinstr$ if $(\atr_{1},\atr_{2}) \in \arbo$ cannot change the fact that $$\atrwr{\atr_1}\in\{\atrwr{\atr'} \in \atrace_\pcinstr\ \|\ (\atrwr{\atr'},\atrrd{\atr}) \in \viso_\pcinstr \wedge \exists\ \writeact(\atrwr{\atr'},\anaddr,\cdot)\in\amap(\atrwr{\atr'})\}$$ iff $$\atr_1 \in \{\atr' \in \atrace\ \|\ (\atr',\atr) \in \viso \wedge \exists\ \writeact(\atr',\anaddr,\cdot)\in\amap(\atr')\}$$ Thus, the proof of $\axretval$ follows as in the previous case. For the case $\textsf{X} = \sic$, we show that $\atrace_\pcinstr$ satisfies $\axconflict$. If $(\atrwr{\atr_{1}},\atrwr{\atr_{2}}) \in \sto_\pcinstr$, then $(\atr_{1},\atr_{2}) \in \sto \subset \viso$, which implies that $(\atrwr{\atr_{1}},\atrrd{\atr_{2}}) \in \viso_\pcinstr$. Therefore, $(\atrwr{\atr_{1}},\atrwr{\atr_{2}}) \in \viso_\pcinstr$, which concludes the proof. The axiom $\axretval$ can be proved as in the previous cases. %Thus, the property $\axconflict$ holds. Also, we prove in the same way as the previous two cases that $\atrace_\pcinstr$ satisfies the property $\axretval$. \end{proof} \begin{proof}[Proof of Lemma \ref{lem:cycles}] ($\Rightarrow$) Let $\atrace$ be a trace under \ccc{}. From $\axpoco$ and $\axcoarb$ we get that $\arbo_{0}^{+} \subset \arbo$, and $\arbo_{0}^{+}$ is acyclic because $\arbo$ is total order. Assume by contradiction that $\viso_{0}^{+};\cfo$ is cyclic which implies that $\viso;\cfo$ is cyclic since $\viso_{0}^{+} \subset \viso$, which means that there exist $\atr_1$ and $\atr_2$ such that $(\atr_1, \atr_2) \in \viso$ and $(\atr_2, \atr_1) \in \cfo$. $(\atr_2, \atr_1) \in \cfo$ implies that there exists $\atr_3$ such that $(\atr_3, \atr_1) \in \sto$ and $(\atr_3, \atr_2) \in \rfo$. Based on the definition of $\axretval$, $\atr_3$ has two possible instances: \begin{itemize} \item $\atr_3$ corresponds to the "fictional" transaction that wrote the initial values which cannot be the case when $(\atr_1, \atr_2) \in \viso$ and $\atr_1$ writes to the same variable that $\atr_2$ reads from, \item $\atr_3$ is the last transaction that occurs before $\atr_2$ that writes the value read by $\atr_2$, which means that $(\atr_1,\atr_3) \in \arbo$ which contradicts the fact that $(\atr_3, \atr_1) \in \sto$ since $\sto \subset \arbo$. \end{itemize} ($\Leftarrow$) Let $\atrace$ be a trace such that $\arbo_{0}^{+}$ and $\viso_{0}^{+};\cfo$ are acyclic. Then, we define the relations $\viso$ and $\arbo$ such that $\viso = \viso_{0}^{+}$ and $\arbo$ is any total order that includes $\arbo_{0}^{+}$. Then, we obtain that $(\viso \cup \sto)^{+} \subset \arbo$ and $\viso;\cfo$ is acyclic. Thus, $\atrace$ satisfies the properties $\axpoco$ and $\axcoarb$. Next, we will show that $\atrace$ satisfies $\axretval$. Let $\atr$ be a transaction in $\atrace$ that contains a read event $\readact(\atr,\anaddr,\aval)$. Let $\atr_0$ be transaction in $\atrace$ such that $$\atr_0 = Max_{\arbo}(\{\atr' \in \atrace\ \|\ (\atr',\atr) \in \viso \wedge \exists\ \writeact(\atr',\anaddr,\cdot)\in\amap(\atr')\})$$ then the read must return a value written by $\atr_0$. Assume by contradiction that there exists some other transaction $\atr_1 \neq \atr_0$ such that $(\atr_1,\atr) \in \rfo$. Then, we get that $(\atr_1,\atr_0) \in \arbo$ and both write to $\anaddr$, therefore, $(\atr_1,\atr_0) \in \sto$ since $\sto \subset \arbo$. Combining $(\atr_1,\atr) \in \rfo$ and $(\atr_1,\atr_0) \in \sto$ we obtain $(\atr,\atr_0) \in \cfo$ and since $(\atr_0,\atr) \in \viso$ then we obtain that $(\atr,\atr) \in \viso;\cfo$ which contradicts the fact that $\viso;\cfo$ is acyclic. Therefore, the read value was written by $\atr_0$ and $\atrace$ satisfies $\axretval$. %%(2): The only-if direction: similar to (1) since $\ccc{} \subset \pcc$, we have that $\arbo_{0}^{+}$ is acyclic. We assume by contradiction that $\arbo_{0}^{+}?;\viso_{0};\cfo$ is cyclic. The property $\axprefix$ implies that $\sto;\viso \subset \viso$ since $\sto \subset \arbo$ and since $\viso_{0}^{+} \subset \viso$ then $\arbo_{0}^{+}?;\viso_{0} \subset \viso$. Thus, $\viso;\cfo$ is cyclic which results in a contradiction as in (1). %%(2): The if direction: we define the relations $\viso$ and $\arbo$ such that $\arbo_{0}^{+}\subset \arbo$ and $\viso = \arbo ; \viso_{0} \cup \viso_{0}$ which imply that $\viso_{0}^{+} \subset \viso$ and $\arbo;\viso \subset \viso$. Thus, $\atrace$ satisfies the properties $\axpoco$, $\axcoarb$, and $\axprefix$. Using the same proof steps as in (1) we can show that $\atrace$ satisfies $\axretval$. %%(3): The only-if direction: similar to above we have that $\arbo_{0}^{+}$ is acyclic. We assume by contradiction that $\arbo_{0}^{+};\cfo$ is cyclic. Combining the properties $\axconflict$ and $\axpoco$ implies that $\arbo_{0}^{+} \subset \viso$. Thus, $\viso;\cfo$ is cyclic which results in a contradiction as in (1). %%(3): The if direction: we can define the relations $\viso$ and $\arbo$ such that $\viso = \arbo_{0}^{+}$ and $\arbo_{0}^{+}\subset \arbo$ which imply the properties $\axpoco$, $\axcoarb$, and $\axconflict$. Using the same proof steps as in (1) we can show that $\atrace$ satisfies $\axretval$. \end{proof} \begin{proof}[Proof of Lemma~\ref{lem:CcCc}] The only-if direction follows from Lemma \ref{lem:Transform}. For the if direction: consider a trace $\atrace_\pcinstr$ which is \ccc{}. We prove by contradiction that $\atrace$ must be \ccc{} as well. Assume that $\atrace$ is not \ccc{} then it must contain a cycle in either $\arbo_{0}^{+}$ or $\viso_{0}^{+};\cfo$ (based on Lemma \ref{lem:cycles}). In the rest of the proof when we mention a cycle we implicitly refer to a cycle in either $\arbo_{0}^{+}$ or $\viso_{0}^{+};\cfo$. Splitting every transaction $\atr \in \atrace$ in a trace to a pair of transactions $\atrrd{\atr}$ and $\atrwr{\atr}$ that occur in this order might not maintain a cycle of $\atrace$. However, we prove that this is not possible and our splitting conserves the cycle. Assume we have a vertex $\atr$ as part of the cycle. We show that $\atr$ can be split into two transactions $\atrrd{\atr}$ and $\atrwr{\atr}$ while maintaining the cycle. Note that $\atr$ is part of a cycle iff either \begin{enumerate} \item $(\atr_{1},\atr) \in \arbo_{0}$ and $(\atr,\atr_{2})\in \arbo_{0}$ or \item $(\atr_{1},\atr) \in \viso_{0}$ and $(\atr,\atr_{2})\in \viso_{0}$ or \item $(\atr_{1},\atr) \in \viso_{0}$ and $(\atr,\atr_{2})\in \cfo$ or \item $(\atr_{1},\atr) \in \cfo$ and $(\atr,\atr_{2})\in \viso_{0}$ \end{enumerate} where $\atr_{1}$ and $\atr_{2}$ might refer to the same transaction. Thus, by splitting $\atr$ to $\atrrd{\atr}$ and $\atrwr{\atr}$, the above four cases imply that: \begin{enumerate} \item if $(\atr_{1},\atr) \in \viso_{0}$ and $(\atr,\atr_{2})\in \arbo_{0}$ then $(\atr_{1}',\atrrd{\atr}) \in (\po_\pcinstr \cup \rfo_\pcinstr)$ and $(\atrwr{\atr},\atr_{2}')\in (\po_\pcinstr \cup \rfo_\pcinstr \cup \sto_\pcinstr)$ where $\atr_{1}' \in \{\atrrd{\atr_{1}},\atrwr{\atr_{1}}\}$ and $\atr_{2}' \in \{\atrrd{\atr_{2}},\atrwr{\atr_{2}}\}$. This maintains the vertices $\atr_{1}'$ and $\atr_{2}'$ connected in the cycle formed by the dependency relations of $\atrace_\pcinstr$ since $(\atrrd{\atr},\atrwr{\atr}) \in \po_\pcinstr$; \item if $(\atr_{1},\atr) \in \sto$ and $(\atr,\atr_{2})\in \arbo_{0}$ then $(\atr_{1}',\atrwr{\atr}) \in \sto_\pcinstr$ and $(\atrwr{\atr},\atr_{2}')\in (\po_\pcinstr \cup \rfo_\pcinstr \cup \sto_\pcinstr)$ which maintains the vertices $\atr_{1}'$ and $\atr_{2}'$ connected in the cycle formed by the dependency relations of $\atrace_\pcinstr$; \item $(\atr_{1},\atr) \in \viso_{0}$ and $(\atr_{2},\atr) \in \cfo$ then $(\atr_{1}',\atrrd{\atr}) \in (\po_\pcinstr \cup \rfo_\pcinstr)$ and $(\atrrd{\atr},\atr_{2}')\in \cfo_\pcinstr$ maintains the vertices $\atr_{1}'$ and $\atr_{2}'$ connected in the cycle formed by the dependency relations of $\atrace_\pcinstr$; \item $(\atr_{1},\atr) \in \cfo$ and $(\atr_{2},\atr) \in \viso_{0}$ then $(\atr_{1}',\atrwr{\atr}) \in \cfo_\pcinstr$ and $(\atrwr{\atr},\atr_{2}')\in (\po_\pcinstr \cup \rfo_\pcinstr)$ which maintains the vertices $\atr_{1}'$ and $\atr_{2}'$ connected in the cycle formed by the dependency relations of $\atrace_\pcinstr$ as well. \end{enumerate} Therefore, doing the splitting creates a cycle in either $(\po_\pcinstr \cup \rfo_\pcinstr \cup \sto_\pcinstr)^{+}$ or $(\po_\pcinstr \cup \rfo_\pcinstr)^{+};\cfo_\pcinstr$ which implies that $\atrace_\pcinstr$ is not \ccc{}, a contradiction. \end{proof} \begin{proof}[Proof of Lemma \ref{lem:PcSer}] ($\Leftarrow$) Assume that $\atrace_\pcinstr$ is \serc{}. We will show that $\atrace$ is \pcc{}. Notice that if $(\atr_{1},\atr_{2}) \in \viso_{0}^{+}$ then $(\atrwr{\atr_{1}},\atrrd{\atr_{2}}) \in \arbo_\pcinstr$ which implies that $(\atr_{1},\atr_{2}) \in \viso$. Similarly, if $(\atr_{1},\atr_{2}) \in \arbo_{0}^{+}$ then $(\atrwr{\atr_{1}},\atrwr{\atr_{2}}) \in \arbo_\pcinstr$ or $(\atrwr{\atr_{1}},\atrrd{\atr_{2}}) \in \arbo_\pcinstr$ which implies that $(\atrwr{\atr_{1}},\atrwr{\atr_{2}}) \in \arbo_\pcinstr$ which in both cases implies that $(\atr_{1},\atr_{2}) \in \arbo$. Thus, $\atrace$ satisfies the properties $\axpoco$ and $\axcoarb$. Now assume that $(\atr_{1},\atr_{2})\in \arbo$ and $(\atr_{2},\atr_{3})\in \viso$. We show that $(\atr_{1},\atr_{3})\in \viso$. The assumption implies that $(\atrwr{\atr_{1}},\atrwr{\atr_{2}}) \in \arbo_\pcinstr$ and $(\atrwr{\atr_{2}},\atrrd{\atr_{3}}) \in \arbo_\pcinstr$, which means that $(\atrwr{\atr_{1}},\atrrd{\atr_{3}}) \in \arbo_\pcinstr$. Therefore, $(\atr_{1},\atr_{3}) \in \viso$ and $\atrace$ satisfies the property $\axconflict$. Concerning $\axretval$, let $\atr$ be a transaction in $\atrace$ that contains a read event $\readact(\atr,\anaddr,\aval)$. Let $\atr_0$ be transaction in $\atrace$ such that $$\atr_0 = Max_{\arbo}(\{\atr' \in \atrace\ \|\ (\atr',\atr) \in \viso \wedge \exists\ \writeact(\atr',\anaddr,\cdot)\in\amap(\atr')\}).$$ We show that the read must return a value written by $\atr_0$. The definitions of $\viso$ and $\arbo$ imply that $$\atrwr{\atr_1}\in\{\atrwr{\atr'} \in \atrace_\pcinstr\ \|\ (\atrwr{\atr'},\atrrd{\atr}) \in \arbo_\pcinstr \wedge \exists\ \writeact(\atrwr{\atr'},\anaddr,\cdot)\in\amap(\atrwr{\atr'})\}$$ $$\atr_1 \in \{\atr' \in \atrace\ \|\ (\atr',\atr) \in \viso \wedge \exists\ \writeact(\atr',\anaddr,\cdot)\in\amap(\atr')\}$$ because $(\atrwr{\atr_{1}},\atrrd{\atr_{2}}) \in \arbo_\pcinstr$ implies $(\atr_{1},\atr_{2}) \in \viso$. Then, we obtain that $$\atrwr{\atr_0} = Max_{\arbo_\pcinstr}(\{\atrwr{\atr'} \in \atrace_\pcinstr\ \|\ (\atrwr{\atr'},\atrrd{\atr}) \in \arbo_\pcinstr \wedge \exists\ \writeact(\atrwr{\atr'},\anaddr,\cdot)\in\amap(\atrwr{\atr'})\})$$ and since $\atrace_\pcinstr$ is \serc{} we know that the read must return the value written by $\atrwr{\atr_0}$. Thus, the read returns the value written by $\atr_0$, which implies that $\atrace$ satisfies $\axretval$ holds. Therefore, $\atrace$ is \pcc{}. ($\Rightarrow$) Assume that $\atrace$ is \pcc{}. We show that $\atrace_\pcinstr$ is \serc{}. Since $\atrace_\pcinstr$ is the result of splitting transactions, a cycle in its dependency relations can only originate from a cycle in $\atrace$. Therefore, it is sufficient to show that any happens-before cycle in $\atrace$ is broken in $\atrace_\pcinstr$. From Lemma \ref{lem:pccycles}, we have that $\atrace$ either does not admit a happens-before cycle or any (simple) happens-before cycle in $\atrace$ must have either two successive $\cfo$ dependencies or a $\sto$ dependency followed by a $\cfo$ dependency. If $\atrace$ does not admit a happens-before cycle then it is \serc{}, and $\atrace_\pcinstr$ is trivially \serc{} (since splitting transactions cannot introduce new cycles). \scalebox{0.67} \begin{tikzpicture} \node[shape=rectangle ,draw=none,font=\large] (A0) at (0,0) [] {$\atr_1$ }; \node[shape=rectangle ,draw=none,font=\large] (A1) at (2,0) [] {$\atr_2$}; \node[shape=rectangle ,draw=none,font=\large] (B1) at (4,0) [] {$\atr_3$}; \node[shape=rectangle ,draw=none,font=\large] (B2) at (5,0) [] {$\Longrightarrow$}; %\node[shape=rectangle ,draw=none,font=\large] (C0) at (8,0) [] {$\cdots$ }; \node[shape=rectangle ,draw=none,font=\large] (C1) at (6,0) [] {$\atrrd{\atr_{1}}$}; \node[shape=rectangle ,draw=none,font=\large] (D1) at (8,0) [] {$\atrwr{\atr_{1}}$}; \node[shape=rectangle ,draw=none,font=\large] (D2) at (10,0) [] {$\atrrd{\atr_{2}}$}; \node[shape=rectangle ,draw=none,font=\large] (D0) at (12,0) [] {$\atrwr{\atr_{2}}$}; \node[shape=rectangle ,draw=none,font=\large] (E0) at (14,0) [] {$\atrrd{\atr_{3}}$ }; \node[shape=rectangle ,draw=none,font=\large] (E1) at (16,0) [] {$\atrwr{\atr_{3}}$}; \begin{scope}[ every edge/.style={draw=black,very thick}] \path [->] (A0) edge[] node [above,font=\small] {$\sto \cup \cfo$} (A1); \path [->] (A1) edge[] node [above,font=\small] {$\cfo$} (B1); \path [->] (B1) edge[bend left] node [above,font=\small] {$\hbo$} (A0); \path [->] (C1) edge[] node [above,font=\small] {$\po_\pcinstr$} (D1); \path [->] (D1) edge[bend left] node [below,font=\small] {$\sto_\pcinstr$} (D0); \path [->] (C1) edge[bend right] node [above,font=\small] {$\cfo_\pcinstr$} (D0); \path [->] (D2) edge[] node [above,font=\small] {$\po_\pcinstr$} (D0); \path [->] (D2) edge[bend right] node [above,font=\small] {$\cfo_\pcinstr$} (E1); \path [->] (E0) edge[] node [above,font=\small] {$\po_\pcinstr$} (E1); \end{scope} \end{tikzpicture}} Otherwise, if $\atrace$ admits a happens-before cycle like above, then $\atrace$ must contain three transactions $\atr_{1}$, $\atr_{2}$, and $\atr_{3}$ such that $(\atr_{1},\atr_{2}) \in \sto \cup \cfo$, $(\atr_{2},\atr_{3}) \in \cfo$, and $(\atr_{3},\atr_{1}) \in \hbo$ (like in the picture above). Then, by splitting transactions we obtain that $(\atrwr{\atr_{1}},\atrwr{\atr_{2}}) \in \sto_\pcinstr$ or $(\atrrd{\atr_{1}},\atrwr{\atr_{2}}) \in \cfo_\pcinstr$, and $(\atrrd{\atr_{2}},\atrwr{\atr_{3}}) \in \cfo_\pcinstr$. Since, we have $(\atrrd{\atr_{2}},\atrwr{\atr_{2}}) \in \po_\pcinstr$ (and not $(\atrwr{\atr_{2}},\atrrd{\atr_{2}}) \in \po_\pcinstr$), this cannot lead to a cycle in $\atrace_\pcinstr$, which concludes the proof that $\atrace_\pcinstr$ is \serc{} %Thus, in all cases $\atrace_\pcinstr$ doesn't contain cycles. Therefore, $\atrace_\pcinstr$ is \serc{}. \end{proof} %!TEX root = draft.tex \section{Proofs for Section~\ref{sec:robustness}} \label{sec:robustnessProofs} \begin{proof}[Proof of Lemma~\ref{lem:pcsicycles}] %Since $\atrace$ is \pcc{}, there exists a causal order $\viso$ and an arbitration order $\arbo$, such that $\viso\subseteq \arbo$ and $\axpc$ hold. Let $\arbo_{1}$ be a total order that includes $\arbo_{0}^{+}$ and $\arbo_{0}^{+};\cfo;\arbo_{0}^{}$ ($\arbo_{0}^$ is the reflexive closure of $\arbo_{0}$). This is well defined because there exists no cycle between tuples in these two relations. Indeed, if $(\atr_{1},\atr_{2}) \in \arbo_{0}^{+}$ and there exist $\atr_{3}$ and $\atr_{4}$ such that $(\atr_{2}, \atr_{3}) \in \arbo_{0}^{+}$, $(\atr_{3}, \atr_{4}) \in \cfo$, and $(\atr_{4}, \atr_{1}) \in \arbo_{0}^{}$, then we have a cycle in $\arbo_{0}^{+};\cfo$ that does not contain two successive $\cfo$ dependencies, which contradicts the hypothesis. Also, for every pair of transactions $(\atr_{1},\atr_{2})$ there cannot exist $\atr_{3}$ and $\atr_{4}$ such that $$(\atr_{2}, \atr_{3}) \in \arbo_{0}^{+},\ (\atr_{3}, \atr_{4}) \in \cfo\mbox{ and }(\atr_{4}, \atr_{1}) \in \arbo_{0}^{}$$ and $\atr_{3}'$ and $\atr_{4}'$ such that $$(\atr_{1}, \atr_{3}') \in \arbo_{0}^{+},\ (\atr_{3}', \atr_{4}') \in \cfo\mbox{ and }(\atr_{4}', \atr_{2}) \in \arbo_{0}^{}$$ This will imply a cycle in $\arbo_{0}^{+};\cfo;\arbo_{0}^{+};\cfo$ which again contradicts the hypothesis. %if $(\atr_{1},\atr_{2}) \not\in \arbo_{0}^{+}$ and there exist $\atr_{3}$ and $\atr_{4}$ such that $(\atr_{2}, \atr_{3}) \in \arbo_{0}^{+}$, $(\atr_{3}, \atr_{4}) \in \cfo$, and $(\atr_{4}, \atr_{1}) \in \arbo_{0}^{}$ ($\arbo_{0}^*$ denotes the reflexive closure of $\arbo_{0}$), then $(\atr_{2},\atr_{1}) \in \arbo_{1}$. % and $\arbo_{1}$ is transitive\footnote{Note that .}. Also, let $\viso_{1}$ be the smallest transitive relation that includes $\arbo_{0}^{+}$ and $\arbo_{1};\arbo_{0}^{+}$. We show that $\viso_{1}$ and $\arbo_{1}$ are causal and arbitration orders of $\atrace$ that satisfy all the axioms of \sic{}. $\axpoco$ and $\axcoarb$ hold trivially. Since $\sto \subseteq \viso_{1}$, $\axconflict$ holds as well. %We show first that $\arbo_{1}$ is a well defined total order. Note that because $\axcoarb$, $\arbo_{0}^{+} \subset \arbo$ is acyclic. % the fact that all happens-before cycles in $\atrace$ contain two successive $\cfo$. Thus, $\arbo_{1}$ is a well defined total order. %Also, since $\arbo_{0}^+ \subset \arbo_{1}$ then $\axcoarb$ holds. %Similar to above $\viso_{1}$ is a well defined partial order. %Note that $\viso_{1} \subset \arbo_{1}$ as well. Also, since $\viso_{0}^+ \subset \viso_{1}$ then $\axpoco$ holds. $\axpc$ holds because $\arbo_{1} ; \viso_{1} = \arbo_{1};(\arbo_{0}^{+} \cup \arbo_{1};\arbo_{0}^{+})^+ = \arbo_{1};\arbo_{0}^{+} \subset \viso_{1}$. The axiom $\axretval$ is equivalent to the acyclicity of $\viso_{1};\cfo$ when $\axpoco$ and $\axcoarb$ hold. Assume by contradiction that $\viso_1;\cfo$ is cyclic. From the definition of $\viso_1$ and the fact that $\arbo_{1}$ is total order we obtain that either: %$\arbo_{0}^{+};\cfo$ or $\arbo_{1};\arbo_{0}^{+};\cfo$ (since ) is cyclic: \begin{itemize} \item $\arbo_{0}^{+};\cfo$ is cyclic, which implies that there exists a happens-before cycle that does not contain two successive $\cfo$, which contradicts the hypothesis, or \item $\arbo_{1};\arbo_{0}^{+};\cfo$ is cyclic, which implies that there exist $\atr_{1}$, $\atr_{2}$, and $\atr_{3}$ such that $(\atr_{2}, \atr_{3}) \in \arbo_{0}^{+}$, $(\atr_{3}, \atr_{1}) \in \cfo$ and $(\atr_{1},\atr_{2}) \in \arbo_{1}$. This contradicts the fact that $(\atr_{2}, \atr_{3}) \in \arbo_{0}^{+}$ and $(\atr_{3}, \atr_{1}) \in \cfo$ implies $(\atr_{2},\atr_{1}) \in \arbo_{1}$. %by definition if $(\atr_{1},\atr_{2}) \not\in \arbo_{0}^{+}$, otherwise, we get $\arbo_{0}^{+};\arbo_{0}^{+};\cfo = \arbo_{0}^{+};\cfo$ is cyclic which leads to the previous case. \end{itemize} Therefore, $\atrace$ satisfies $\axretval$ for $\viso_1$ and $\arbo_1$, which concludes the proof. \end{proof} \medskip Next, we present an important lemma that characterizes happens before cycles possible under the \pcc{} semantics. This is a strengthening of a result in~[12] which shows that all happens before cycles under \pcc{} must have two successive dependencies in ${,}$ and at least one $$. We show that the two successive dependencies cannot be $$ followed $$, or two successive $$. \begin{proof}[Proof of Lemma~\ref{lem:pccycles}] It was shown in [12] that all happens-before cycles under \pcc{} must contain two successive dependencies in $\{\cfo,\sto\}$ and at least one $\cfo$. Assume by contradiction that there exists a cycle with $\cfo$ dependency followed by $\sto$ dependency or two successive $\sto$ dependencies. This cycle must contain at least one additional dependency. Otherwise, the cycle would also have a $\sto$ dependency followed by a $\cfo$ dependency, or it would imply a cycle in $\sto$, which is not possible (since $\sto \subset \arbo$ and $\arbo$ is a total order). % it does this implies a cycle in $\sto \subset \arbo$ which is a contradiction. Then, we get that the dependency just before $\cfo$ is either $\po$ or $\rfo$ (i.e., $\viso_0$) since we cannot have $\cfo$ or $\sto$ followed by $\cfo$. Also, the relation after $\sto$ is either $\po$ or $\rfo$ or $\sto$ (i.e., $\arbo_0$) since we cannot have $\sto$ followed by $\cfo$. Thus, the cycle has the following shape: \medskip \scalebox{0.65} \begin{tikzpicture} \node[shape=rectangle ,draw=none,font=\large] (A0) at (0,0) [] {$\atr_1$ }; \node[shape=rectangle ,draw=none,font=\large] (A1) at (1.3,0) [] {$\atr_2$}; \node[shape=rectangle ,draw=none,font=\large] (B1) at (2.6,0) [] {$\atr_3$}; \node[shape=rectangle ,draw=none,font=\large] (B2) at (3.9,0) [] {$\atr_4$}; \node[shape=rectangle ,draw=none,font=\large] (C0) at (4.5,0) [] {$\cdots$ }; \node[shape=rectangle ,draw=none,font=\large] (C1) at (5.1,0) [] {$\atr_i$}; \node[shape=rectangle ,draw=none,font=\large] (D1) at (6.4,0) [] {$\atr_{i+1}$}; \node[shape=rectangle ,draw=none,font=\large] (D2) at (7.9,0) [] {$\atr_{i+2}$}; \node[shape=rectangle ,draw=none,font=\large] (D0) at (9.4,0) [] {$\atr_{i+3}$}; \node[shape=rectangle ,draw=none,font=\large] (E0) at (10.2,0) [] {$\cdots$ }; \node[shape=rectangle ,draw=none,font=\large] (E1) at (11,0) [] {$\atr_{n-4}$}; \node[shape=rectangle ,draw=none,font=\large] (F1) at (12.5,0) [] {$\atr_{n-3}$}; \node[shape=rectangle ,draw=none,font=\large] (F2) at (14,0) [] {$\atr_{n-2}$}; \node[shape=rectangle ,draw=none,font=\large] (F3) at (15.5,0) [] {$\atr_{n-1}$}; \node[shape=rectangle ,draw=none,font=\large] (F4) at (17,0) [] {$\atr_{n}$}; \begin{scope}[ every edge/.style={draw=black,very thick}] \path [->] (A0) edge[] node [above,font=\small] {$\cfo$} (A1); \path [->] (A1) edge[] node [above,font=\small] {$\sto$} (B1); \path [->] (B1) edge[] node [above,font=\small] {$\arbo_0$} (B2); \path [->] (C1) edge[] node [above,font=\small] {$\viso_0$} (D1); \path [->] (D1) edge[] node [above,font=\small] {$\cfo$} (D2); \path [->] (D2) edge[] node [above,font=\small] {$\sto$} (D0); \path [->] (E1) edge[] node [above,font=\small] {$\viso_0$} (F1); \path [->] (F1) edge[] node [above,font=\small] {$\cfo$} (F2); \path [->] (F2) edge[] node [above,font=\small] {$\sto$} (F3); \path [->] (F3) edge[] node [above,font=\small] {$\arbo_0$} (F4); \path [->] (F4) edge[bend left=11] node [above,font=\small] {$\viso_0$} (A0); \end{scope} \end{tikzpicture}} \medskip Since $\viso_0;\cfo\subseteq \arbo$ is a consequence of the \pcc{} axioms~[26], we get that $(\atr_{n}, \atr_2) \in \arbo$, $(\atr_{i}, \atr_{i+2}) \in \arbo$ and $(\atr_{n-4}, \atr_{n-2}) \in \arbo$, which allows to ``short-circuit'' the cycle. Using the fact that $\sto \subset \arbo$, $\viso_0 \subset \arbo$, and $\arbo_0 \subset \arbo$, and applying the short-circuiting process multiple times, we obtain a cycle in the arbitration order $\arbo$ which contradicts the fact that $\arbo$ is a total order. \end{proof} \begin{proof}[Proof of Theorem~\ref{them:RobCcSi}] For the only-if direction: assume that $\aprog$ is robust against \ccc{} relative to \sic{}. Then, the set of traces of $\aprog$ under the two consistency models coincide. Since the set of traces under \sic{} is subset of the one under \pcc{}, then the set of traces under \ccc{} is subset of the one under \pcc{}. This implies that $\aprog$ is robust against \ccc{} relative to \pcc{}. % because the set of traces under \pcc{} is a subset of the set of executions under \ccc{}. Thus, we obtain that the set of traces of $\aprog$ under the three consistency models coincide. Therefore, $\aprog$ is robust against \pcc{} relative to \sic{} as well. For the if direction: assume that $\aprog$ is robust against \ccc{} relative to \pcc{} and $\aprog$ is robust against \pcc{} relative to \sic{}. Then, the set of traces of $\aprog$ under the three consistency models coincide. Thus, we obtain that $\aprog$ is robust against \ccc{} relative to \sic{}. \end{proof} %!TEX root = draft.tex \section{Proofs for Section~\ref{sec:commutativitygraph}} \label{sec:commutativitygraphProofs} \begin{proof}[Proof of Theorem~\ref{them:MovRobCcPc}] It is enough to show: if $\aprog$ is not robust against \ccc{} relative to \pcc{} then we have a simple cycle in the commutativity dependency graph of $\aprog_\pcinstr$ of the form above. Assume $\aprog$ is not robust against \ccc{} relative to \pcc{}. Then, from Theorem \ref{them:RobCcPc}, we obtain $\aprog_\pcinstr$ is not robust against \ccc{} relative to \serc{}. Also it was shown in [10] that if a program is not robust then there must exist a robustness violation trace (\ccc{} relative to \serc{}) $\atrace_\pcinstr$ of the shape $\atrace_\pcinstr = \alpha \cdot \atr_1 \cdot \beta \cdot \atr_i \cdot \atr_{i+1} \cdot \gamma \cdot \atr_n$ where $(\atr_1,\atr_i) \in (\po \cup \rfo)^{+}$, $(\atr_{i},\atr_{i+1}) \in (\sto \cup \cfo)$, $(\atr_{i+1},\atr_n) \in \hbo$, and $(\atr_n,\atr_1) \in \cfo$. Note that since transactions in the trace $\atrace_\pcinstr$ can either be read-only or write-only. Then, $(\atr_{i},\atr_{i+1}) \in (\sto \cup \cfo)$ and $(\atr_{n},\atr_1) \in \cfo$ imply that $\atr_1$ and $\atr_{i+1}$ must be a write-only transactions and $\atr_{n}$ must be a read-only transaction. Note that we may have $\beta = \gamma = \epsilon$ as the case for the trace of the $\mathsf{SB}$ program given in Figure \ref{fig:litmus1}. %In the trace $\atrace_\pcinstr$, we let $\atr_1$ to be $\atr_1$, $\atr_i$ to be $\atr_2$, $\atr_3$ to be $\atr_{i+1}$, and $\atr_4$ to be $\atr_n$ of Theorem \ref{them:MovRobCcPc}. We consider first the general case when $\atr_1 \not\equiv \atr_2$. The other case can be proved in the same way. Consider the prefix $\atrace_{p}$ of $\atrace_\pcinstr$: $\atrace_{p} = \alpha \cdot \atr_1 \cdot \beta \cdot \atr_{i}$ where $(\atr_1,\atr_{i}) \in (\po \cup \rfo)^{+}$ which is a \serc{} trace of $\aprog_{\pcinstr}$. Then, we have a sequence of transactions from $\atr_1$ to $\atr_{i}$ that are related by either $\po$ or $\rfo$. In the case two transactions are only related by $\rfo$, then the first transaction is not a right mover because of the second transaction reads from a write in the first transaction. Thus, we can relate the two transactions using the relation $\mrfo$ in the commutativity dependency graph. Similarly consider the following trace $\atrace_{s}$ extracted from $\atrace_\pcinstr$: $\atrace _{s} = \alpha \cdot \atr_{i+1} \cdot \gamma \cdot \atr_n$ where $(\atr_{i+1},\atr_n) \in \hbo$ which is a \serc{} trace of $\aprog_{\pcinstr}$. Similar to before, we have a sequence of transactions from $\atr_{i+1}$ to $\atr_n$ that are related by either $\po$, $\rfo$, $\sto$, or $\cfo$. For any two transactions that are related only by either $\rfo$, $\sto$, or $\cfo$, this implies that the first transaction is not a right mover because of the second transaction and a write-read, write-write, or read-write dependency between the two, respectively. Thus, we can relate the two transactions using either $\mrfo$, $\msto$, or $\mcfo$, respectively. Now consider the following trace $\atrace_{1}$ extracted from $\atrace_\pcinstr$: $\atrace_{1} = \alpha \cdot \atr_1 \cdot \beta \cdot \atr_{i} \cdot \atr_{i+1}$ where $(\atr_{i},\atr_{i+1}) \in (\sto \cup \cfo)$ is a \serc{} trace of $\aprog_{\pcinstr}$. Because $\atr_{i}$ and $\atr_{i+1}$ are related by either $\sto$ or $\cfo$, then $\atr_{i}$ is not a right mover because of $\atr_{i+1}$ and a write-write or read-write dependency between the two, respectively. Thus, we can relate the two transactions using either $\msto$ or $\mcfo$, respectively. Finally, consider the following trace $\atrace_{2}$ extracted from $\atrace_\pcinstr$: $\atrace_{2} = \alpha \cdot \atr_{i+1} \cdot \gamma \cdot \atr_n \cdot \atr_1$ where $(\atr_n,\atr_1) \in \cfo$ is a \serc{} trace of $\aprog_{\pcinstr}$. Because $\atr_{n}$ and $\atr_{1}$ are related by $\cfo$, then $\atr_{n}$ is not a right mover because of $\atr_{1}$ and a read-write dependency between the two. Thus, we can relate the two transactions using $\mcfo$. \end{proof} \begin{proof}[Proof of Theorem~\ref{them:MovRobPcSi}] Similar to before it is enough to show: if $\aprog$ is not robust against \pcc{} relative to \sic{} then we have a simple cycle in the commutativity dependency graph of $\aprog_\pcinstr$ of the form above. Assume $\aprog$ is not robust against \pcc{} relative to \sic{}. Then, from Theorem \ref{them:RobPcSiInstr}, we obtain that if $\sem{\aprog}$ reaches an error state under \serc{} then we will have the following trace $\atrace$ under \serc: $\atrace = \alpha \cdot \atrrd{\atr_{\instr}} \cdot \atr_3 \cdot \beta \cdot \atr_n \cdot \atrwr{\atr_{\instr}}$\footnote{For simplicity, we assume here that after reaching the error state we execute the writes of $\atr_{\instr}$, i.e., $\atrwr{\atr_{\instr}}$.} where $(\atrrd{\atr_{\instr}},\atr_3) \in \cfo$, $(\atr_3,\atr_n) \in \hbo$, $(\atr_n,\atrwr{\atr_{\instr}}) \in \sto$, and we don't have two successive $\cfo$ in the happens before between $\atr_3$ and $\atr_n$. In $\atrace$, $\atrwr{\atr_{\instr}}$ (resp., $\atrrd{\atr_{\instr}}$) represents $\atr_1$ (resp., $\atr_2$) in the theorem statement. Note that we may have $\alpha = \beta = \epsilon$ as is the case of the transformed $\mathsf{LU}$ program given in Figure \ref{fig:litmus2Instr}. The construction of the cycle in the commutativity dependency graph follows the same procedure taken in the proof of Theorem \ref{them:MovRobCcPc}. The only difference is that for every two transactions of $\atrace$ that are part of the happens before between $\atr_3$ and $\atr_n$, if the two are not connected by either $\po$, $\rfo$, $\sto$, or $\cfo$ then they must be the reads and writes of the same original transaction in $\aprog$. In this case, in the commutativity dependency graph we have the two transactions related by $\sametro$. %Notice that we abused notation by using the two components $\atrrd{\atr 1}$ and $\atrwr{\atr_1}$ of $\atr_1$ in $\atrace$ to denote that $\atr 1$ writes were not immediately written to the shared variables. \end{proof} \end{document}
# A quantum Boltzmann equation for strongly correlated electrons Antonio Picano Department of Physics, University of Erlangen-Nürnberg, 91058 Erlangen, Germany Jiajun Li Department of Physics, University of Erlangen- Nürnberg, 91058 Erlangen, Germany Martin Eckstein Department of Physics, University of Erlangen-Nürnberg, 91058 Erlangen, Germany ###### Abstract Collective orders and photo-induced phase transitions in quantum matter can evolve on timescales which are orders of magnitude slower than the femtosecond processes related to electronic motion in the solid. Quantum Boltzmann equations can potentially resolve this separation of timescales, but are often constructed within a perturbative framework. Here we derive a quantum Boltzmann equation which only assumes a separation of timescales (taken into account through the gradient approximation for convolutions in time), but is based on a non-perturbative scattering integral, and makes no assumption on the spectral function such as the quasiparticle approximation. In particular, a scattering integral corresponding to non-equilibrium dynamical mean-field theory is evaluated in terms of an Anderson impurity model in a non- equilibrium steady state with prescribed distribution functions. This opens the possibility to investigate dynamical processes in correlated solids with quantum impurity solvers designed for the study of non-equilibrium steady states. ## I Introduction One of the biggest challenges in the theoretical description of quantum many- particle systems is to predict their non-equilibrium dynamics at long times after a perturbation. This would be essential for the understanding of non- equilibrium phenomena in complex solids,Basov _et al._ (2017); Giannetti _et al._ (2016) including photo-induced metal-insulator transitions and hidden phases with spin, orbital, charge, or superconducting order.Ichikawa _et al._ (2011); Fausti _et al._ (2011); Beaud _et al._ (2014); Wegkamp _et al._ (2014); Stojchevska _et al._ (2014); Mor _et al._ (2017); Budden _et al._ (2020) The evolution of the electronic structure in these situations is often intertwined with the dynamics of the crystal lattice, collective orders, or slow electronic variables such as non-thermal band occupations, which is orders of magnitude slower than intrinsic electronic processes such as the electron tunnelling between atoms. Moreover, a large timescale separation becomes apparent in the thermalization of pre-thermal states,Polkovnikov _et al._ (2011); Berges _et al._ (2004); Moeckel and Kehrein (2008) where approximate conservation laws provide a dynamical constraint.Kollar _et al._ (2011); Langen _et al._ (2016) A major goal is therefore to devise an approach that can explore the dynamics on the slower timescale, while still taking into account accurately the fast degrees of freedom. Within the Keldysh formalism, non-equilibrium quantum many-particle systems can be described in terms of time and frequency- dependent spectral functions $A_{\bm{k}}(\omega,t)$ and distribution functions $F_{\bm{k}}(\omega,t)$. (For simplicity, spin and orbital indices in addition to momentum $\bm{k}$ are not shown here.) A large separation can become evident between the variation of the functions with time $t$, and the intrinsic timescales related to the linewidth of relevant spectral features. If these timescales are well separated, one can cast the full many-body dynamics into a differential equation known as the quantum Boltzmann equation (QBE).Kadanoff and Baym (1962); Kamenev (2011) In abstract form, the QBE defines a scattering contribution to the evolution of the distribution functions, $\displaystyle\big{[}\partial_{t}F_{\bm{k}}(\omega,t)\big{]}_{scatt}=I[F,A],$ (1) where the so-called scattering integral $I$ depends on the spectrum and distribution function at the same time. The full time-dependence is determined by additional contributions from the coherent single-particle propagation, and a separate equation for the evolution of the spectrum in terms of the distribution function. While the applicability of the QBE is in principle only controlled by the time-scale separation, the formalism is used predominantly for semiconductors or Fermi liquids with well-defined quasiparticles,Haug (1962) where one can make use of two additional and in practice rather important simplifications: (i) The quasiparticle approximation assumes the spectra $A_{\bm{k}}(\omega,t)$ to be sharply peaked at energies $\omega=\epsilon_{\bm{k}}$, and therefore allows to evaluate the QBE on-shell, essentially trading the frequency- dependent distribution function for quasiparticle occupations $n_{\bm{k}}(t)$. Moreover, (ii), the scattering kernel is often evaluated in a perturbative manner. In strongly correlated systems, both of these approximations are challenged. For example, doped Mott insulators show strange metallic behaviors without well-defined Fermi liquid quasiparticles in a wide parameter regime Georges _et al._ (2013); Deng _et al._ (2013), and similar behavior is observed in photo-doped Mott insulators.Eckstein and Werner (2013); Sayyad and Eckstein (2016); Dasari _et al._ (2020); Petersen _et al._ (2017); Sahota _et al._ (2019) Furthermore, the electronic structure in correlated systems depends strongly on the non-equilibrium distribution, as most clearly demonstrated through the possibility of photo-induced metal insulator transitions. For that reason, the dynamics of correlated systems has been mostly discussed within the formally exact non-equilibrium Green’s function (NEGF) techniques. In the NEGF formalism, the dynamics is described in terms of two-time Green’s functions $G_{\bm{k}}(t,t^{\prime})$, which are related to spectra and occupation functions through a Fourier transform with respect to relative time $t-t^{\prime}$. A two-time self-energy acts as a memory kernel in a non- Markovian propagation of the Green’s functions, the so-called Kadanoff-Baym equation. NEGF techniques can be combined with different diagrammatic approximations,Golež _et al._ (2016); Babadi _et al._ (2015); Rameau _et al._ (2016); Schlünzen _et al._ (2017) including in particular dynamical mean-field theory (DMFT),Aoki _et al._ (2014); Georges _et al._ (1996) and they do not rely on a quasiparticle approximation for the spectrum. On the other hand, they also do not make use of the time-scale separation, and therefore imply a high numerical cost: The effort scales like $\mathcal{O}(t_{\text{max}}^{3})$ with the simulation time $t_{\text{max}}$, as compared to $\mathcal{O}(t_{\text{max}})$ for the QBE. For weakly interacting systems, the perturbatively controlled generalized Kadanoff-Baym Ansatz (GKBA) Lipavský _et al._ (1986) has recently been set up to reach $\mathcal{O}(t_{\text{max}})$ scaling of the computational effort.Schlünzen _et al._ (2020) For strongly correlated systems, a systematic truncationSchüler _et al._ (2018) or compact compressionKaye and Golež (2020) of the memory kernel in the Kadanoff-Baym equations provide interesting perspectives, but so far the investigation of many fundamental questions has remained out of reach because of the $\mathcal{O}(t_{\text{max}}^{3})$ scaling. It would therefore be desirable to formulate a QBE which incorporates the simplifications due to the time-scale separation, but does not rely on quasiparticle or perturbative approximations. For example, if the equilibrium state of the system is described well by means of DMFT, the steady state fixed point of the QBE should be identical to this DMFT solution. A previous work has successfully employed a QBE without the quasiparticle approximation for a Mott insulator,Wais _et al._ (2018, 2020) assuming a rigid density of states and a renormalized second-order scattering integral. Here we show how such a scattering integral can be obtained from an auxiliary non-equilibrium steady state formalism. This allows to consistently combine the QBE with non- perturbative methods which have been developed to study true non-equilibrium steady states within DMFT.Joura _et al._ (2008); Li _et al._ (2015); Titvinidze _et al._ (2018); Matthies _et al._ (2018); Scarlatella _et al._ (2020); Li _et al._ (2020); Panas _et al._ (2019) The paper is organized as follows: In section II, we present the formulation of a non-perturbative QBE which is consistent with non-equilibrium DMFT. In Sec. III we compare its solution to a non-equilibrium DMFT simulation for the thermalization in a correlated metal. Section IV gives a conclusion and outlook. ## II Quantum Boltzmann equation ### II.1 General setting We will derive the QBE for a generic model, $\displaystyle H=\sum_{\bm{k},a,b}h_{\bm{k},ab}(t)c^{\dagger}_{\bm{k},a}c_{\bm{k},b}+H_{\text{int}},$ (2) where $c_{\bm{k},a}$ ($c_{\bm{k},a}^{\dagger}$) denotes the annihilation (creation) operator for a fermion with spin and orbital indices $a$ and momentum $\bm{k}$, and $H_{\text{int}}$ is an arbitrary two-particle interaction, and $h_{\bm{k},ab}(t)$ incorporates all single-particle terms. We assume that the system of interest is initially prepared in thermal equilibrium at temperature $T$, and driven out of equilibrium for times $t>0$ by external fields and a coupling to external heat and/or particle reservoirs. The description of this situation within many-body theory is based on contour- ordered Green’s functions, $\displaystyle G_{\bm{k},ab}(t,t^{\prime})=-i\langle T_{\mathcal{C}}c_{{\bm{k}},a}(t)c_{{\bm{k}},b}^{\dagger}(t^{\prime})\rangle,$ (3) with time arguments $t$ and $t^{\prime}$ on the Keldysh contour $\mathcal{C}$ that runs from 0 to time $t_{\text{max}}$ (the largest time of interest) on the real time axis, back to 0, and finally to $-i\beta$ along the imaginary time axis. (For an introduction to the Keldysh formalism and the notation, see, e.g., Ref. Aoki _et al._ , 2014.) Spin and orbital indices will be no longer shown in the following for simplicity; all Green’s functions, self- energies, dispersion functions $h_{\bm{k}}$ are matrices in these indices. From the contour-ordered function (3), one derives real and imaginary time Green’s functions, of which the retarded, lesser, and greater components are most important in the following. The retarded Green’s function (with real time arguments) $\displaystyle G^{R}_{\bm{k}}(t,t^{\prime})=-i\theta(t-t^{\prime})\langle[c_{\bm{k}}(t),c^{\dagger}_{\bm{k}}(t^{\prime})]_{+}\rangle,$ (4) is related to the spectral function of the system, while the occupied and unoccupied density of states are extracted from the lesser Green’s function, $\displaystyle G^{<}_{\bm{k}}(t,t^{\prime})=+i\langle c^{\dagger}_{\bm{k}}(t^{\prime})c_{\bm{k}}(t)\rangle.$ (5) $\displaystyle G^{>}_{\bm{k}}(t,t^{\prime})=-i\langle c_{\bm{k}}(t)c^{\dagger}_{\bm{k}}(t^{\prime})\rangle,$ (6) so that $\displaystyle G^{R}_{\bm{k}}(t,t^{\prime})=\theta(t-t^{\prime})[G^{>}_{\bm{k}}(t,t^{\prime})-G^{<}_{\bm{k}}(t,t^{\prime})].$ (7) In equilibrium, or in any time-translationally invariant state, all two-time correlation functions depend only on the relative time $t-t^{\prime}$. By taking the Fourier transform of $G^{R}$ with respect to this time difference, one obtains the spectral function $A$: $\displaystyle A_{\bm{k}}(\omega)=-\frac{1}{\pi}\imaginary G^{R}_{\bm{k}}(\omega+i0),$ (8) which is related to the lesser and greater Green’s functions through a fluctuation-dissipation theorem $\displaystyle G^{<}_{\bm{k}}(\omega)$ $\displaystyle=2\pi iA_{\bm{k}}(\omega)f_{\beta}(\omega),$ (9) $\displaystyle G^{>}_{\bm{k}}(\omega)$ $\displaystyle=-2\pi iA_{\bm{k}}(\omega)[1-f_{\beta}(\omega)],$ (10) where $f_{\beta}(\omega)$ is the Fermi distribution function, $f_{\beta}(\omega)=1/(e^{\beta\omega}+1)$. In a non-equilibrium steady-state, one can thus define the distribution function as the ratio $\displaystyle F_{\bm{k}}(\omega)=\frac{G^{<}_{\bm{k}}(\omega)}{2\pi iA_{\bm{k}}(\omega)}.$ (11) This is an energy distribution function, which is defined even in the absence of well-defined quasi-particles. The QBE provides an equation of motion for its time-dependent generalization, as introduced in the following. ### II.2 The QBE For every two-time quantity $X(t,t^{\prime})$ one can introduce the Wigner transform, $\displaystyle X(\omega,t)=\int ds\,e^{i\omega s}\,X(t+s/2,t-s/2),$ (12) where $t$ is the average time and $s=t-t^{\prime}$ is the relative time. In particular, this can be used to define a time-dependent spectrum and occupation function $F_{\bm{k}}(\omega,t)$ in analogy to Eqs. (7), (8), and (11), $\displaystyle A_{\bm{k}}(\omega,t)$ $\displaystyle=-\frac{1}{\pi}\imaginary G^{R}_{\bm{k}}(\omega+i0,t),$ (13) $\displaystyle=[G^{>}_{\bm{k}}(\omega,t)-G^{<}_{\bm{k}}(\omega,t)]/(-2\pi i),$ (14) $\displaystyle F_{\bm{k}}(\omega,t)$ $\displaystyle=G^{<}_{\bm{k}}(\omega,t)/[2\pi iA_{\bm{k}}(\omega,t)],$ (15) where $G^{R,<,>}_{\bm{k}}(\omega,t)$ are given by the Wigner transform. While Eqs. (13) to (15) always provide a valid mathematical definition, the functions gain a physical significance in particular in the limit in which there is a well-defined separation of timescales. Let us assume that there are scales $\delta\omega$ and $\delta t$ on which $G(\omega,t)$ varies in frequency and time, such that $\displaystyle\Big{\|}\frac{\partial_{\omega}G_{\bm{k}}(\omega,t)}{G_{\bm{k}}(\omega,t)}\Big{\|}<1/\delta\omega,\,\,\,\,\Big{\|}\frac{\partial_{t}G_{\bm{k}}(\omega,t)}{G_{\bm{k}}(\omega,t)}\Big{\|}<1/\delta t,$ (16) for lesser, greater, or retarded component. The scale $\delta\omega$ measures the relevant internal energy differences in the system, such as the linewidth or relevant spectral features, and $\delta t$ sets the scale for the time- evolution, with $\delta t\to\infty$ in a steady state. The QBE will be derived in the limit where these timescale are well separated, $\displaystyle\delta t\gg 1/\delta\omega.$ (17) This is also the limit in which the spectral and occupation functions gain their usual meaning in terms of a density of states: One can always approximate $G(\omega,t)$ by the average $\displaystyle G(\omega,t)\\!\approx\\!\\!\int\frac{dt^{\prime}d\omega^{\prime}}{\pi\Omega\tau}e^{-(\frac{t^{\prime}}{\tau})^{2}-(\frac{\omega^{\prime}}{\Omega})^{2}}G(\omega+\omega^{\prime},t+t^{\prime})$ (18) over a time interval $\tau\ll\delta t$ and a frequency interval $\Omega\ll\delta\omega$ on which the function varies weakly. With a sufficiently large time-scale separation (17), it is possible to choose $\Omega=1/\tau$ without violating the conditions $\tau\ll\delta t$ and $\Omega\ll\delta\omega$. With this, the average (18), with $G$ replaced by $-iG^{<}$, is the expression for the time-resolved photoemission spectrum Freericks _et al._ (2009); Eckstein and Kollar (2008) computed with a Gaussian probe pulse of duration $\tau$, and therefore has a well-defined interpretation in terms of an occupied density of states. In addition, this implies that the expression is real and positive, which can be proven by casting Eq. (18) in the form of a complete square using a Lehmann representation for the Green’s function. In the same way, $iG^{>}(\omega,t)$ can be interpreted as the unoccupied density of states (electron addition spectrum), and the spectral function $A(\omega,t)=[G^{>}(\omega,t)-G^{<}(\omega,t)]/(-2\pi i)$ has the usual meaning of a single-particle density of states in the many-body system. The QBE provides an equation of motion for the spectral and occupation functions (13) and (15) in the limit of well separated times.Kamenev (2011) Most importantly, the limit (17) allows for the simplification of the convolution $[A\ast B](t,t^{\prime})=\int d\bar{t}A(t,\bar{t})B(\bar{t},t^{\prime})$ of two real-time functions $A$ and $B$. In mathematical terms, the Wigner transform of the convolution is given by the Moyal product $\displaystyle[A\ast B](\omega,t)=e^{\frac{i}{2}[\partial_{t}^{A}\partial_{\omega}^{B}-\partial_{t}^{B}\partial_{\omega}^{A}]}A(\omega,t)B(\omega,t).$ (19) If Eqs. (16) and (17) hold for $A$ and $B$, the Moyal product can be simplified by considering only the leading term $\displaystyle[A\ast B](\omega,t)\approx A(\omega,t)B(\omega,t),$ (20) because $\|\partial_{t}A\>\partial_{\omega}B\|\ll\|AB\|$. This is the so-called gradient approximation. In a time-evolving state, Eq. (9) is generalized to the ansatz $\displaystyle G_{\bm{k}}^{<}(t,t^{\prime})=[F_{\bm{k}}\ast G_{\bm{k}}^{A}](t,t^{\prime})-[G_{\bm{k}}^{R}\ast F_{\bm{k}}](t,t^{\prime}),$ (21) where $F_{\bm{k}}(t,t^{\prime})$ depends on two-times, and $G_{\bm{k}}^{A}(t,t^{\prime})=G_{\bm{k}}^{R}(t^{\prime},t)^{\dagger}$ is the advanced Green’s function. By applying the gradient approximation (20) to this ansatz, we obtain the factorization $\displaystyle G^{<}_{\bm{k}}(\omega,t)=2\pi iA_{\bm{k}}(\omega,t)F_{\bm{k}}(\omega,t),$ (22) equivalent to Eq. (15), using $G^{A}_{\bm{k}}(\omega,t)=G^{R}_{\bm{k}}(\omega,t)^{\dagger}$. In order to derive the QBE for the evolution of the distribution function $F_{\bm{k}}$, one can consider the equations of motion for the Green’s function. For a non-interacting system with Green’s function $\mathcal{G}_{\bm{k}}(t,t^{\prime})=-i\langle\mathcal{T}_{\mathcal{C}}c_{\bm{k}}(t)c_{\bm{k}}^{\dagger}(t^{\prime})\rangle$ this is written as $\displaystyle\\{\mathcal{G}^{-1}_{\bm{k}}\ast\mathcal{G}_{{\bm{k}}}\\}(t,t^{\prime})=\delta_{\mathcal{C}}(t,t^{\prime}),$ (23) $\displaystyle\mathcal{G}^{-1}_{\bm{k}}(t,t^{\prime})=[i\partial_{t}+\mu- h_{\bm{k}}(t)]\delta_{\mathcal{C}}(t,t^{\prime}),$ (24) where $\delta_{\mathcal{C}}(t,t^{\prime})$ represents the delta-function on the Keldysh contour and $\mu$ is the chemical potential of the system. In the following, it will be convenient to also include the Hartree and Fock self- energy into the dispersion $h_{\bm{k}}(t)$. To include correlations we take into account the contour-ordered self-energy $\Sigma(t,t^{\prime})$ and obtain the interacting Green’s function $G$ via the Dyson equation $\displaystyle\\{[(\mathcal{G}_{\bm{k}})^{-1}-\Sigma_{\bm{k}}]\ast G_{\bm{k}}\\}(t,t^{\prime})=\delta_{\mathcal{C}}(t,t^{\prime})$ (25) on the Keldysh contour. From the Dyson equation for the lesser component, [$(\mathcal{G}^{R}_{\bm{k}})^{-1}-\Sigma^{R}_{\bm{k}}]\ast G_{\bm{k}}^{<}=\Sigma^{<}_{\bm{k}}\ast G_{\bm{k}}^{A}$, and the ansatz (21), we get $\displaystyle(\mathcal{G}^{R}_{\bm{k}})^{-1}\ast F_{\bm{k}}-$ $\displaystyle F_{\bm{k}}\ast(\mathcal{G}^{A}_{\bm{k}})^{-1}=$ $\displaystyle=\Sigma^{<}_{\bm{k}}+\Sigma^{R}_{\bm{k}}\ast F_{\bm{k}}-F_{\bm{k}}\ast\Sigma^{A}_{\bm{k}}.$ (26) (Real-time time arguments are shown only where otherwise ambiguous.) We thus obtain the equation of motion for $F_{\bm{k}}(t,t^{\prime})$: $\displaystyle i(\partial_{t}+\partial_{t^{\prime}})F_{\bm{k}}(t,t^{\prime})=$ $\displaystyle h_{\bm{k}}(t)F_{\bm{k}}(t,t^{\prime})-F_{\bm{k}}(t,t^{\prime})h_{\bm{k}}(t^{\prime})$ $\displaystyle+\Sigma^{<}_{\bm{k}}+\Sigma^{R}_{\bm{k}}\ast F_{\bm{k}}-F_{\bm{k}}\ast\Sigma^{A}_{\bm{k}}.$ (27) Equation (II.2) is still exact. To obtain the QBE, we then use the gradient approximation (20) to rewrite Eq. (II.2) as $\displaystyle\partial_{t}F_{\bm{k}}(\omega,t)$ $\displaystyle=-i[h_{\bm{k}}(t),F_{\bm{k}}(\omega,t)]+I_{\bm{k}}(\omega,t),$ (28) $\displaystyle I_{\bm{k}}(\omega,t)$ $\displaystyle=-i\big{[}\Sigma_{\bm{k}}^{R}(\omega,t)F_{\bm{k}}(\omega,t)-F_{\bm{k}}(\omega,t)\Sigma_{\bm{k}}^{A}(\omega,t)$ $\displaystyle\,\,\,\,\,\,\,+\Sigma^{<}_{\bm{k}}(\omega,t)\big{]},$ (29) where $I_{\bm{k}}(\omega,t)$ is the scattering integral. This equation is completed by the Dyson equation for the retarded Green’s function to leading order in the gradient approximation, $\displaystyle G^{R}_{\bm{k}}(\omega,t)$ $\displaystyle=[\omega+i0+\mu- h_{\bm{k}}(t)-\Sigma_{\bm{k}}^{R}(\omega,t)]^{-1}.$ (30) This set of equations must be combined with a given expression for the self- energy. For example, a simple perturbative expression would be a second-order diagram in terms of a two-particle density-density interaction $v_{\bm{q}},$ $\displaystyle\Sigma_{\bm{k}}(t,t^{\prime})=\sum_{\bm{k}^{\prime},\bm{q}}v_{\bm{q}}^{2}G_{\bm{k}^{\prime}+\bm{q}}(t,t^{\prime})G_{\bm{k}^{\prime}}(t^{\prime},t)G_{\bm{k}^{\prime}-\bm{q}}(t,t^{\prime}).$ (31) Such an analytic perturbative expression for $\Sigma$ can then be evaluated in the gradient approximation, thus closing the equation. In the following, we discuss a strategy to incorporate a non-perturbative self-energy approximation like DMFT into the QBE formalism, in which an explicit analytical expression for $\Sigma$ is not given. ### II.3 Non-perturbative evaluation of the scattering integral In general, the self-energy includes contributions from the interaction, and a possible coupling to a noninteracting environment, which can be used to represent thermal and particle reservoirs Tsuji _et al._ (2009); Büttiker (1985); Aoki _et al._ (2014). In the following, we write $\Sigma=\Sigma_{\text{int}}+\Gamma$, where $\Sigma_{\text{int}}$ is the interaction contribution, and $\Gamma$ represents the noninteracting reservoirs. Evaluating the interaction self-energy is the main challenge. We assume that the interaction self-energy $\Sigma_{\text{int}}(t,t^{\prime})=\hat{\Sigma}_{\bm{k},t,t^{\prime}}^{\text{skel}}[G]$ is a functional of the full Green’s function $G$, as obtained in particular as the so-called skeleton expansion through derivatives of the Luttinger-Ward functionalLuttinger and Ward (1960) for any conserving approximation.Baym and Kadanoff (1961) Also DMFT and its extensions can be cast in this language.Georges _et al._ (1996) A simple perturbative example would be the second-order diagram Eq. (31). Let us now imagine a system which has the same interaction but general non-interacting reservoirs so that the system resides in a non-equilibrium steady state (NESS) with steady state spectrum $\bar{A}_{\bm{k}}(\omega)$, and the steady state distribution $\bar{F}_{\bm{k}}(\omega)$. Evaluation of the full skeleton functional $\hat{\Sigma}_{\bm{k},t,t^{\prime}}^{\text{skel}}[G]$ at the translationally invariant Green’s function $\bar{G}[\bar{A},\bar{F}\big{]}$ defines a non- equilibrium steady-state functional through the Wigner transform (12) $\displaystyle\hat{\Sigma}^{\text{ness- skel}}_{\bm{k},\omega}\big{[}\bar{A},\bar{F}\big{]}=\int ds\,e^{i\omega s}\,\hat{\Sigma}_{\bm{k},s/2,-s/2}^{\text{skel}}[\bar{G}].$ (32) This skeleton functional is universal in the sense that it parametrically depends only on the interaction,Potthoff (2003) but not on the single-particle part of the Hamiltonian, and hence the functional (32) is independent of the choice of the reservoirs. In order to write the equations below in a more compact form, we note that the self-consistent evaluation of the functional (32), together with the steady state Dyson equation for the retarded function $\displaystyle\bar{A}_{\bm{k}}(\omega)$ $\displaystyle=-\frac{1}{\pi}\text{Im}\frac{1}{\omega^{+}+\mu-\bar{h}_{\bm{k}}-\bar{\Gamma}^{R}_{\bm{k}}(\omega)-\bar{\Sigma}^{R}_{\text{int},\bm{k}}(\omega)}$ (33) and given $\bar{h}_{\bm{k}}$ and $\bar{\Gamma}^{R}_{\bm{k}}(\omega)$, implicitly defines a steady-state functional of the self-energy and the spectral function in terms of the distribution function only, which we will denote by $\displaystyle\hat{\Sigma}_{\bm{k},\omega}^{\text{ness}}\big{[}\bar{F};\bar{h}_{\bm{k}},\bar{\Gamma}^{R}_{\bm{k}}\big{]},\,\,\,\,\hat{A}_{\bm{k},\omega}^{\text{ness}}\big{[}\bar{F};\bar{h}_{\bm{k}},\bar{\Gamma}^{R}_{\bm{k}}\big{]}.$ (34) Back to the QBE, at each order of a diagrammatic expression, the two-time self-energy $\Sigma_{\text{int}}(t,t^{\prime})$ can be written as a sum of convolutions and products of the full Green’s function $G$. In each of these terms, one can consistently use the leading order of the gradient approximation, in combination with the factorization (22). This procedure would be the same as evaluating $\hat{\Sigma}_{\text{int},\bm{k},t,t^{\prime}}^{\text{skel}}[\bar{G}]$ with a time-translationally invariant function $\bar{G}$ with spectral function $\bar{A}_{\bm{k}}(\omega)=A_{\bm{k}}(\omega,t)$ and distribution function $\bar{F}_{\bm{k}}(\omega)=F_{\bm{k}}(\omega,t)$. Hence the self-energy in the gradient approximation amounts to evaluating the NESS functional (32) $\displaystyle\Sigma_{\text{int},\bm{k}}(\omega,t)=\hat{\Sigma}^{\text{skel- ness}}_{\bm{k},\omega}\big{[}A(\cdot,t),F(\cdot,t)\big{]}.$ (35) Here the notation $X(\cdot,t)$ of the functional arguments $X=A,F$ indicates that the latter are considered as function of all their arguments except for $t$, which is considered as a fixed parameter. With Eq. (34), the QBE is now formally written as $\displaystyle\partial_{t}F_{\bm{k}}(\omega,t)$ $\displaystyle=-i[h_{\bm{k}}(t),F_{\bm{k}}(\omega,t)]+I_{\bm{k},\omega}[F(\cdot,t)],$ (36) $\displaystyle I_{\bm{k},\omega}[F(\cdot,t)]$ $\displaystyle=-i\big{[}\Sigma_{\bm{k}}^{R}(\omega,t)F_{\bm{k}}(\omega,t)-F_{\bm{k}}(\omega,t)\Sigma_{\bm{k}}^{A}(\omega,t)$ $\displaystyle\,\,\,\,\,\,\,+\Sigma^{<}_{\bm{k}}(\omega,t)\big{]},$ (37) where in the second line $\Sigma=\Sigma_{\text{int}}+\Gamma$, with $\displaystyle\Sigma_{\text{int},\bm{k}}(\omega,t)$ $\displaystyle=\hat{\Sigma}_{\bm{k},\omega}^{\text{ness}}\big{[}F(\cdot,t);h_{\bm{k}}(t),\Gamma^{R}_{\bm{k}}(\cdot,t)\big{]}.$ (38) In addition, the spectral function is given by $\displaystyle A_{\bm{k}}(\omega,t)$ $\displaystyle=\hat{A}_{\bm{k},\omega}^{\text{ness}}\big{[}F(\cdot,t);h_{\bm{k}}(t),\Gamma^{R}_{\bm{k}}(\cdot,t)\big{]}.$ (39) Physically, the last equation (39) means that we allow the electronic distribution function to instantaneously influence the electronic structure of the material. We will therefore refer to Eq. (39) as the instantaneous response approximation. Equations (36) to (39) now provide a closed set of time-dependent equations. This implicit scheme allows a non-perturbative evaluation of the QBE, provided that an efficient numerical description of a NESS is available: To evaluate $\hat{\Sigma}_{\bm{k},\omega}^{\text{ness}}\big{[}F(\cdot,t),...\big{]}$ and $A_{\bm{k}}(\omega,t)=\hat{A}_{\bm{k},\omega}^{\text{ness}}\big{[}F(\cdot,t),...\big{]}$ for a given distribution function $\bar{F}$, we choose an auxiliary steady state system with reservoir self-energy $\bar{\Gamma}^{R}_{\bm{k}}(\omega)=\Gamma^{R}_{\bm{k}}(\omega,t)$, while the bath occupation function, and hence $\bar{\Gamma}^{<}_{\bm{k}}(\omega)$ is treated as a free parameter. The latter is chosen such that the solution $\bar{F}_{\bm{k}}(\omega)$ gives the prescribed $F_{\bm{k}}(\omega,t)$, after which the outcomes $\bar{A}_{\bm{k}}(\omega)$ and $\bar{\Sigma}_{\text{int},\bm{k}}(\omega)$ are used to evaluate (38) and (39). In particular, within non-equilibrium DMFT, where only local self-energies need to be evaluated in a quantum impurity model, several promising non- perturbative techniques are available that can directly target such non- equilibrium states (see discussion in Sec. IV). Once Eqs. (38) and (39) can be evaluated for a given $F$, the QBE Eq. (36) can be solved as any differential equation. (In the implementation below, we use a simple Runge-Kutta algorithm.) In the following two sections, we will adapt the general formalism to the non- equilibrium DMFT framework. Before that, we conclude this section with a side remark: It is known even in equilibrium that the self-consistent solution of the Dyson equation with a skeleton self energy functional can have multiple unphysical solutions.Kozik _et al._ (2015) However, a possible multi- valuedness of the functional (34) will not be a problem here. The functions $A_{\bm{k}}(\omega,t)$, $F_{\bm{k}}(\omega,t)$, and $\Sigma_{\bm{k}}(\omega,t)$ evolve continuously as a function of time, so that even if unphysical steady-state solutions exist for a given distribution function, the physical solution is always selected by the requirement of continuity and the initial condition. On the other hand, if the system would evolve as a function of time into a branching point where multiple solutions of Eq. (34) meet, this would hint at a rather unconventional dynamical behavior. For example, in equilibrium it is known that the multi-valuedness of self-consistent perturbation theory is related to vertex singularities Schäfer _et al._ (2013), and in the Hubbard model these vertex singularities apparently fall together with the dynamical critical point found in Ref. Eckstein _et al._ , 2009. ### II.4 Scattering integral in DMFT In the following, we adapt the general QBE framework to non-equilibrium DMFT. Within DMFT, one maps the lattice model (2) onto an effective single-site impurity model. The impurity site has the same interaction as a site in the lattice, and its coupling to the environment is described by the so-called hybridization function $\Delta(t,t^{\prime})$, which is self-consistently determined such that the local ($\bm{k}$-averaged) lattice Green’s function $\displaystyle G_{\text{loc}}(t,t^{\prime})=\sum_{\bm{k}}G_{\bm{k}}(t,t^{\prime})$ (40) coincides with the impurity Green’s function. The key approximation of DMFT is that the lattice self-energy is local in space (independent of $\bm{k}$), and one requires the local lattice self-energy to be identical to the impurity self energy. In detail, the impurity model is defined by an action $\displaystyle\mathcal{S}=-i\int_{\mathcal{C}}dt\,H_{loc}(t)-i\int_{\mathcal{C}}dtdt^{\prime}\sum_{\sigma}c_{\sigma}^{\dagger}(t)\Delta(t,t^{\prime})c_{\sigma}(t^{\prime}),$ (41) in terms of the self-consistent hybridization function. The non-interacting Green’s function $\mathcal{G}$ is determined by the Dyson equation $\displaystyle\mathcal{G}^{-1}(t,t^{\prime})=[i\partial_{t}+\mu-h(t)]\delta_{\mathcal{C}}(t,t^{\prime})-\Delta(t,t^{\prime}),$ (42) where $h(t)$ is the single particle Hamiltonian in the impurity model. The interacting impurity Green’s function is given by $\displaystyle G_{\text{imp}}^{-1}=\mathcal{G}^{-1}-\Sigma_{\text{imp}},$ (43) and the self-consistency requires $\displaystyle G_{\text{imp}}=G_{\text{loc}},\,\,\,\,\Sigma_{\text{imp}}=\Sigma.$ (44) The self-consistent impurity model provides an implicit way to evaluate a non- perturbative expression $\hat{\Sigma}_{\text{int}}[G_{\text{loc}}]$ for a local self-energy in terms of a local Green’s functions. Along the line of the previous section, we can therefore use an impurity model in a NESS to construct the steady state functional (38) for the local self-energy. An impurity model in the steady state simply implies that the hybridization function itself is translationally invariant in time, and specified through its retarded and lesser components, $\Delta^{R}(\omega)$ and $\Delta^{<}(\omega)$. The evaluation of the functionals (38) and (39) within DMFT, for a given distribution function $\bar{F}_{\bm{k}}(\omega)$, depends on the type of impurity solver. Below we exemplify this for an impurity solver which determines the self energy from an expansion in terms of the noninteracting impurity Green’s function $\bar{\mathcal{G}}$, (such as weak-coupling Keldysh quantum Monte Carlo or iterated perturbation theory): * 1) Start with some guess for $\bar{\Sigma}_{\text{int}}^{R}(\omega)$ and $\bar{\Sigma}_{\text{int}}^{<}(\omega)$, and calculate the $\bm{k}$-dependent lattice Green’s functions [Eq. (30) with $\bm{k}$-independent self-energy] $\displaystyle\bar{G}_{\bm{k}}^{R}(\omega)=[\omega+\mu-\bar{h}_{\bm{k}}-\bar{\Gamma}_{\bm{k}}^{R}(\omega)-\bar{\Sigma}_{\text{int}}^{R}(\omega)]^{-1}.$ (45) and the spectrum $\bar{A}_{\bm{k}}(\omega)=-\frac{1}{\pi}\text{Im}G_{\bm{k}}^{R}(\omega+i0)$. * 2) Determine the lesser Green’s function from the given distribution function, $\displaystyle\bar{G}_{\bm{k}}^{<}(\omega)=2\pi i\bar{F}_{\bm{k}}(\omega)\bar{A}_{\bm{k}}(\omega).$ (46) * 3) Calculate the local lattice Green’s functions. $\displaystyle\bar{G}_{\text{loc}}^{R,<}(\omega)$ $\displaystyle=\sum_{\bm{k}}\bar{G}_{\bm{k}}^{R,<}(\omega,t).$ (47) * 4) Express the noninteracting Green’s function $\mathcal{G}$ of the impurity model in terms of $\Sigma_{\text{imp}}$ of $G_{\text{imp}}$ using the Dyson equation for the impurity model [Eqs. (42) and (43)] in the steady state. For example, this can be written as $\displaystyle\mathcal{G}^{R}(\omega)=[G_{\text{imp}}^{R}(\omega)^{-1}+\Sigma_{\text{imp}}^{R}(\omega)]^{-1},$ (48) $\displaystyle\Delta^{<}(\omega)=G^{R}_{\text{imp}}(\omega)^{-1}G^{<}_{\text{imp}}(\omega)G^{A}_{\text{imp}}(\omega)^{-1}-\Sigma^{<}_{\text{imp}}(\omega)$ $\displaystyle\mathcal{G}^{<}(\omega)=\mathcal{G}^{R}(\omega)\Delta^{<}\mathcal{G}^{A}(\omega),$ (49) Solve these equations for $\mathcal{G}(\omega)$ using the DMFT self- consistency for the lattice and impurity quantities, $\Sigma_{\text{imp}}(\omega)=\bar{\Sigma}_{\text{int}}(\omega)$ and $G_{\text{imp}}(\omega)=\bar{G}_{\text{loc}}(\omega)$. * 5) Calculate a new $\Sigma_{\text{imp}}$ by using an expansion in $\mathcal{G}^{R}(\omega)$. * 6) Set $\bar{\Sigma}^{R,<}_{\text{int}}(\omega)=\Sigma^{R,<}_{\text{imp}}(\omega)$, and iterate Step 2) to 5) until convergence. This iteration is basically a steady-state non-equilibrium DMFT simulation where the distribution function of the system is prescribed and the distribution of the reservoirs is determined, in contrast to conventional steady-state DMFT where the distribution function of the system of the system is determined by reservoirs with a given distribution function. ## III Comparison to the full DMFT simulation ### III.1 Model As a first test case for the methodology, we study the particle-hole symmetric single-band Hubbard model $\displaystyle\hat{H}=-t_{h}\sum_{\langle i,j\rangle,\sigma}c^{\dagger}_{i\sigma}c_{j\sigma}+U\sum_{j}\big{(}\hat{n}_{j\uparrow}-\tfrac{1}{2}\big{)}\big{(}\hat{n}_{j\downarrow}-\tfrac{1}{2}\big{)}.$ (50) Here $c_{j,\sigma}$ denotes the annihilation operator for a Fermion with spin $\sigma\in\\{\uparrow,\downarrow\\}$ at lattice site $j$, $\hat{n}_{j\sigma}=c^{\dagger}_{j\sigma}c_{j\sigma}$ is the particle number operator, $t_{h}$ the hopping matrix element between nearest neighbour sites, and $U$ the on-site interaction strength. The actual simulations assume a semi-elliptic local density of states $D(\epsilon)=\sqrt{4-\epsilon^{2}}/(2\pi)$ for the noninteracting model with bandwidth $4$, corresponding to a Bethe lattice with hopping $t_{h}=1$. The latter sets the unit of energy, and its inverse defines the unit of time ($\hbar=1$). The system is studied in the metallic regime, where $U$ is smaller than the bandwidth. Initially, the system is in equilibrium with a inverse temperature $\beta$. Within a short time interval, we then create a non-thermal population of electrons and holes similar to a photo-excited population (the precise protocol is given below). This non-thermal population will then relax under the influence of the electron-electron interaction and the coupling to a phonon bath, and we compare a simulation of this relaxation dynamics within the full non-equilibrium DMFT simulation and the QBE. For the excitation, we shortly couple a fermionic reservoir with density of states $\displaystyle A_{\text{bath}}(\omega)=A(\omega-2.5)+A(\omega+2.5)$ (51) consisting of two smooth bands with bandwidth $W_{\text{bath}}=6$ around the energies $\omega=\pm 2.5$; we choose $A(\omega)=\frac{1}{\pi}\cos[2](\pi\omega/W_{\text{bath}})$ in the interval $[-W_{\text{bath}}/2,W_{\text{bath}}/2]$, see dashed line at the bottom of Fig 1c for $A_{bath}(\omega)$. Choosing a population inversion in this reservoir will lead to a rapid transfer of electrons from the system into the negative energy part of the reservoir, and of electrons from the positive energy part of the bath to the system, thus generating an electron transfer similar to a photo-excitation process. The bath adds a local contribution $\Gamma(t,t^{\prime})$ to the self-energy (as obtained by integrating out the bath), $\displaystyle\Gamma(t,t^{\prime})=V(t)G_{\text{bath}}(t,t^{\prime})V(t^{\prime})^{},$ (52) where $V(t)$ is the time-profile of the coupling, and $G_{\text{bath}}(t,t^{\prime})$ is the bath Green’s function, $\displaystyle G_{\text{bath}}^{R}(t,t^{\prime})$ $\displaystyle=-i\theta(t-t^{\prime})\int d\omega\,e^{-i\omega(t-t^{\prime})}A_{\text{bath}}(\omega),$ (53) $\displaystyle G_{\text{bath}}^{<}(t,t^{\prime})$ $\displaystyle=i\int d\omega\,e^{-i\omega(t-t^{\prime})}f_{\text{bath}}(\omega)A_{\text{bath}}(\omega).$ (54) The bath occupation $f_{\text{bath}}(\omega)=f_{-\beta}(\omega)$ is taken to be, during the whole time-evolution of the system, a negative temperature Fermi-Dirac distribution (population inversion) , and the switching profile $V(t)=0.75\sin[2](\pi/5(t-t_{0}))$ is centred around an early time $t_{0}=27.5$ with a duration of just five inverse hoppings. In general, the QBE is expected to describe the evolution of the system only on timescales much longer than the inverse hopping, so that these details of the excitation protocol are not important for the present study. The coupling to the bosonic bath is included via a local electron-phonon self- energy $\Sigma_{\text{ph}}$. In order for the bosons to act as heath bath, we need to neglect the back-action of the electrons on the phonons, and we take $\Sigma_{\text{ph}}$ to be the simple first-order diagram of a local electron- phonon interaction, $\displaystyle\Sigma_{\text{ph}}(t,t^{\prime})$ $\displaystyle=g^{2}G(t,t^{\prime})D_{\text{ph}}(t,t^{\prime}),$ (55) where $G$ is the fully interacting local electron Green’s function of the system, $g$ measures the electron-phonon coupling strength, and $D_{\text{ph}}$ is the propagator for free bosons with an Ohmic density of states $\frac{\omega}{4\omega_{\text{ph}}^{2}}\exp(-\omega/\omega_{\text{ph}})$ with exponential cutoff $\omega_{\text{ph}}=0.2$. The occupation function of bosons is kept in equilibrium with inverse temperature $\beta$. The temperature of the heat bath is the same as the initial one of the system in equilibrium, such that the system will eventually thermalize back to its initial temperature long after the excitation. ### III.2 Full DMFT solution For the semi-elliptic density of states, the DMFT self-consistency can be formulated in closed form, and the hybridization of the impurity model is simply given by Georges _et al._ (1996); Aoki _et al._ (2014) $\displaystyle\Delta(t,t^{\prime})=G(t,t^{\prime})+\Gamma(t,t^{\prime})$ (56) in terms of the local Green’s function $G$. With the non-interacting Green’s function of the impurity model [Eq. (42)], the Dyson equation for the impurity model reads $\displaystyle G^{-1}(t,t^{\prime})=\mathcal{G}^{-1}(t,t^{\prime})-\Sigma_{\text{int}}(t,t^{\prime}).$ (57) Here $\displaystyle\Sigma_{\text{int}}(t,t^{\prime})=\Sigma_{U}(t,t^{\prime})+\Sigma_{\text{ph}}(t,t^{\prime})$ (58) is the interaction self-energy due to the electron phonon interaction and the Hubbard interaction. The latter is determined using the iterated perturbation theory (IPT) impurity solver, i.e., a second-order expansion in terms of $\mathcal{G}$, $\displaystyle\Sigma_{U}(t,t^{\prime})=U^{2}\mathcal{G}(t,t^{\prime})\mathcal{G}(t,t^{\prime})\mathcal{G}(t^{\prime},t).$ (59) In addition, the local energy $h(t)$ in Eq. (42) is the Hartree self-energy, $h(t)=Un_{\sigma}(t)$ with the density $n_{\sigma}(t)$ per spin. In the present case we study a half-filled system, so that $\mu=U/2$ and $\mu+h(t)=0$. The self-consistent solution of the system of Eq. (56) to (59) together with the excitation and phonon self energies Eq. (55) and Eq. (52) determines the time evolution of the physical system. The equations are solved on the Keldysh contour using the NESSi simulation package.Schüler _et al._ (2020) For the comparison with the QBE, the local spectral function and distribution function are then extracted from the Wigner transform of the local Green’s function $\displaystyle A(\omega,t)$ $\displaystyle=-\frac{1}{\pi}\text{Im}G^{R}(\omega+i0,t),$ (60) $\displaystyle F(\omega,t)$ $\displaystyle=\frac{G^{<}(\omega,t)}{2\pi iA(\omega,t)}.$ (61) Furthermore, we compute the total energy as: $\displaystyle E_{\text{DMFT}}=-2i(\DeltaG)^{<}(t,t)-i(\Sigma_{\text{int}}G)^{<}(t,t)$ (62) The first and second term represent the kinetic and interaction energy, respectively, with a factor two in the kinetic energy for the summation over spin components. ### III.3 QBE formulation For the present model, for which a closed set of equations is given in terms of local (momentum-averaged) quantities, the QBE can be derived directly for the local quantities. Instead of deriving Eq. (28) and (29) from the lattice Dyson equation (25), one can perform an analogous argument directly for the Dyson equation of the DMFT impurity model [Eq. (57)]. This leads to a local QBE $\displaystyle\partial_{t}F(\omega,t)=$ $\displaystyle\,I[F(\cdot)],$ (63) $\displaystyle I[F(\cdot)]=$ $\displaystyle-i\big{(}\Sigma^{<}(\omega,t)+[\Sigma^{R}(\omega,t)+\Delta^{R}(\omega,t)]F(\omega,t)+$ $\displaystyle-F(\omega,t)[\Sigma^{A}(\omega,t)+\Delta^{A}(\omega,t)]\big{)},$ (64) where again $\Sigma=\Gamma+\Sigma_{\text{int}}$, and $\displaystyle\Sigma_{\text{int}}(\omega,t)=$ $\displaystyle\Sigma_{\omega}^{\text{ness}}[F(\cdot,t)],\,\,A(\omega,t)=A_{\omega}^{\text{ness}}[F(\cdot,t)],$ (65) $\Sigma(\omega,t)$ and the spectrum $A(\omega,t)$ are understood in terms of an auxiliary steady state impurity model with given prescribed distribution function $\bar{F}(\omega)=F(\omega,t)$. The evaluation of these functionals is again done iteratively: 1. 1) Start from a guess for $\bar{\Sigma}_{\text{int}}(\omega)$. Solve the steady state variant of Eq. (57) for $\bar{G}^{R}(\omega)$, $\displaystyle\bar{G}^{R}(\omega)=[\omega+\mu-\bar{h}-\Delta^{R}(\omega)-\bar{\Sigma}_{\text{int}}^{R}(\omega)]^{-1}.$ (66) and determine $\bar{A}(\omega)=-\frac{1}{\pi}\bar{G}^{R}(\omega+i0)$. 2. 2) Determine the lesser Green’s function from the given distribution function, $\bar{G}^{<}(\omega)=2\pi i\bar{F}(\omega)\bar{A}(\omega)$. 3. 3) Use the self-consistency Eq. (56) to fix the hybridization function of the effective steady state impurity model, $\Delta(\omega)=\bar{G}(\omega)+\Gamma(\omega)$. 4. 4) Solve the impurity model. With IPT as an impurity solver, we first determine $\mathcal{G}(\omega)$ from $\Delta(\omega)$, $\displaystyle\mathcal{G}^{R}(\omega)=[\omega+\mu-h(t)-\Delta^{R}(\omega)]^{-1},$ (67) $\displaystyle\mathcal{G}^{<}(\omega)=\mathcal{G}^{R}(\omega)\Delta^{<}(\omega)\mathcal{G}^{A}(\omega),$ (68) transform to real time, evaluate Eq. (59), and transform back to frequency space to obtain $\Sigma_{U}^{R,<}(\omega)$. Similarly, $\Sigma_{\text{ph}}^{R,<}(\omega)$ is evaluated. 5. 5) Set $\bar{\Sigma}_{\text{int}}(\omega)=\Sigma_{U}(\omega)+\Sigma_{\text{ph}}(\omega)$, and iterate step 2) to 5) until convergence. The iteration serves as a way to evaluate $\Sigma^{\text{ness}}[F(\cdot,t)]$. The differential equation (63) is then solved using a Runge-Kutta algorithm. In addition to the spectral and distribution functions, we then compute the total energy $\displaystyle E_{\text{QBE}}=-2i[\Delta(\omega)G(\omega)]^{<}-i[\Sigma_{\text{int}}(\omega)G(\omega)]^{<}$ (69) in order to compare with the full solution (62). ### III.4 Results and Discussion Figure 1: a) Energy $-E_{\text{DMFT}}$ obtained from the full DMFT solution (interaction $U=3$, initial inverse temperature $\beta=20$, electron phonon coupling $g^{2}=0.5$). Coloured dots indicate the energies obtained from the auxiliary steady state $A_{\omega}^{\text{ness}}[F]$ [Eq. (65)], for different initial times $t_{0}$ at which the distribution functions $F(\omega,t_{0})$ is taken from the DMFT solution and copied in the auxiliary steady state problem. b) Distribution functions $F(\omega,t_{0})$ obtained from the full DMFT solution, at times $t_{0}$ corresponding to the dots in a), and copied in the auxiliary steady state problem. c) Dashed lines show the spectrum $A(\omega,t_{0})$ at various initial times, obtained from the full DMFT solution. Solid lines show the spectra obtained from the auxiliary steady state $A_{\omega}^{\text{ness}}[F]$ [Eq. (65)], evaluated with the distribution functions $F(\omega,t_{0})$ in b) taken from the DMFT solution. The dotted line at the bottom of c) shows the (rescaled) spectral function $A_{\text{bath}}(\omega)$ [Eq. (51)] for the excitation bath (the shaded orange area shows the occupied density of states for the bath), and the shaded area in a) the time window over which this bath is coupled to the system. In this subsection, we compare the QBE description with the full solution of the KB equations for the setting introduced in Sec. III.1. Figure 1a) shows the evolution of the energy in the full DMFT solution, which increases during the short excitation window, and subsequently relaxes back to the initial state due to electron thermalization and the electron-phonon interaction. Figure 1b) and c) then show the spectra and distribution functions at some points in time. In the initial and final state the spectrum has a central peak, representing a band of renormalized quasiparticles, which coexists with two Hubbard bands around $\omega=\pm U/2$. In equilibrium, with increasing $T$, the quasiparticle peak would be replaced by a dip in the spectral function, indicating that the high-temperature state is a bad-metal without coherent quasiparticles. After the excitation, the distribution function is highly non-thermal, and the quasiparticle band is strongly suppressed. With time, $F(\omega,t)$ approaches back the shape of an approximate Fermi distribution (electron thermalization), and simultaneously the effective temperature of this distribution relaxes back to the initial $1/\beta$. Together with this evolution of the distribution function, the quasiparticle peak in the spectrum is reformed. Before computing the time evolution generated by the QBE, we can independently evaluate the quality of the auxiliary steady-state representation of the spectra at each given time, i.e., the accuracy of the functional $A_{\omega}^{\text{ness}}[F]$, Eq. (65): We take the distribution function $F(\omega,t_{0})$ from the full solution at a given time $t_{0}$, evaluate $A_{\omega}^{\text{ness}}[\bar{F}]$ with $\bar{F}(\omega)=F(\omega,t_{0})$ as described below Eq. (65) to compute a steady state spectrum $\bar{A}(\omega)$, and compare the result with the full solution $A(\omega,t_{0})$. In Fig. 1c, dashed lines correspond to the DMFT solution $A(\omega,t_{0})$, while solid lines show the corresponding $\bar{A}(\omega)$. The comparison is perfect, even for relatively early times. Only for times immediately after the ultrafast excitation ($t=30$), where the gradient approximation is not supposed to work, can one observe a failure of the auxiliary steady state representation. We can therefore affirm that the density of states can be very accurately obtained as a steady state functional of the distribution function, even in the correlated metallic regime. For smaller values of $U$, the agreement is as good (not shown here). Furthermore, not only the density of states can be very accurately obtained as a steady state functional of the distribution function, but the whole Green’s function and self-energy: The energy values represented by coloured dots in Fig. 1a), calculated with Eq. (69), exactly match the ones of the full DMFT code at the same time, calculated with Eq. (62). Figure 2: Time-evolution of the total energy for $U=1$ (a), $U=2$ (b), and $U=3$ (c) (initial inverse temperature $\beta=20$, electron phonon coupling $g^{2}=0.5$). The black dashed lines show the energy $-E_{\text{DMFT}}$ obtained from the full DMFT evolution, solid lines show the energy $-E_{\text{QBE}}$ obtained from the QBE. The QBE is started at different times $t_{0}$ (indicated by the dots at the beginning of the dashed lines), taking the distribution function $F_{\text{DMFT}}(\omega,t_{0})$ as an initial state for a solution of the QBE at times $t>t_{0}$. In passing, we note that a non-equilibrium spectral function $A(\omega,t)$ defined by the Wigner transform (12) is real (hermitian) by construction, but not necessarily positive, while a steady-state fermionic spectral function is always positive. Moreover, for numerical reasons, for short times the integral in the Wigner transform (12) is truncated, possibly leading to small artefacts. In practice, the relation $F(\omega,t)=\bar{F}(\omega)$ will therefore not be enforced exactly, but as a best fit. It should be noted, however, that the positivity of $A(\omega,t)$ and $F(\omega,t)$ is indeed satisfied wherever the gradient approximation is accurate, as discussed in connection with Eq. (18). In particular, as one can see from Fig. 1b), the distribution functions are already positive in the relevant time interval for the present case. Next, we compare the relaxation dynamics of the system in the two descriptions. For this, we simply take the distribution function $F(\omega,t_{0})$ at a given time $t_{0}$ from the full DMFT solution as an initial state for a solution of the QBE for $t>t_{0}$. The time-evolution of the energy is shown in Fig. 2 for three different values of $U$, and different starting times $t_{0}$ of the QBE simulation. For small values of $U$ ($U=1$ and $U=2$ in Fig. 2a) and b), respectively), the energy relaxation rate obtained from the QBE is almost identical to the one from full DMFT. For $U=3$ (Fig. 2c), one can observe a difference in the magnitude of the time-constants related to the relaxation of the total energy in the two approaches. In particular, the QBE presents an artificially faster relaxation with respect to the full DMFT solution. This indicates that the gradient approximation is less justifies for $U=3$, which could be related to the existence of a more narrow quasiparticle band. As the starting point $t_{0}$ of the Boltzmann code shifts forward in time, the difference between the time evolution of the energies becomes less pronounced. If one decreases the coupling $g^{2}$ with the phonon bath (not shown), the relaxation dynamics of the system is slowed down, the gradient approximation is more justified, and the difference in the energy relaxation rate in the two approaches is less pronounced. Figure 3: Distribution function (upper panels) and spectral function (lower panels) obtained from the full DMFT solution (left panels) and the QBE (right panels) one at $U=3$. The QBE takes the DMFT distribution function $F(\omega,t_{0})$ at time $t_{0}=32$ as initial state for the evolution at $t>t_{0}$ (initial inverse temperature $\beta=20$, electron phonon coupling $g^{2}=0.5$). Although the relaxation rate for the energy in the QBE seems to be overestimated for larger values of $U$, Fig. 3 shows that the spectra and distribution functions obtained from the full DMFT and the QBE follow the same qualitative behavior, i.e., a relaxation of $F(\omega,t)$ to a Fermi function together with an evolution of the temperature in this Fermi function towards the initial temperature. ## IV Conclusion In conclusion, we developed a kinetic equation which works without the need to assume the existence of quasiparticles with well-defined dispersion, $\epsilon_{\bm{k}}$ and, above all, evaluates the scattering integral in a non-perturbative manner. In particular, a scattering integral which is consistent with DMFT is obtained by extracting self-energies from a quantum impurity model in an auxiliary non-equilibrium steady state. Most importantly, this guaranties that the final state of the evolution is a proper description of the fully interacting state of the correlated electron system, which makes the present formalism unique with respect to conventional quantum kinetic approaches based on perturbative scattering integrals or certain assumptions on the spectral function, such as assuming a rigid density of states or the quasiparticle approximation. While for full non-equilibrium Green’s function simulations the numerical effort for the propagation over a time interval $t_{\text{max}}$ scales with $\mathcal{O}(t_{\text{max}}^{3})$, and the required memory scales with $\mathcal{O}(t_{\text{max}}^{2})$, in the QBE the numerical effort is linear with $t_{\text{max}}$ and the memory required is independent of $t_{\text{max}}$. We have tested the framework on the relaxation of the electronic state in a correlated metal after a population transfer that simulates a photo- excitation. One assumption of the QBE, i.e., that the spectra at the correlated system can be obtained from an auxiliary steady state, is found to be satisfied with remarkable accuracy. Moreover, the relaxation dynamics for both spectral functions and distribution functions within the full non- equilibrium DMFT simulation and the QBE are consistent. Quantitatively, the gradient approximation underlying the QBE leads to a slight overestimation of the relaxation rate. Whether this can be corrected by higher order expansions of the gradient approximation is left for future investigations. The success of the QBE approach for the present setting motivates an application to different models. In particular this includes symmetry-broken states where interesting long-time phenomena have been observed,Picano and Eckstein (2020) and the evolution of the Mott phase, where already a QBE with an ad-hoc scattering integral has shown relative success.Wais _et al._ (2018) Possible applications of the formalism include the evolution of the density of states in correlated systems, in particular multi-orbital systems where a pronounced effect of the redistribution of weight has already been discussed using quasiparticle kinetic equations.He and Millis (2016) In this context, the method can be combined with GW Wegkamp _et al._ (2014) or DMFT+GWGolež _et al._ (2019), which have demonstrated again a pronounced dependence of the spectra on the distribution. Finally, another interesting perspective of the approach is that there are several promising numerical approaches to study non-equilibrium steady states within DMFT. This includes variants of the strong-coupling expansion Scarlatella2019; Li and Eckstein (2020), matrix product states,Schwarz _et al._ (2018) auxiliary master equations,Arrigoni _et al._ (2013) or Quantum Monte Carlo.Profumo _et al._ (2015); Bertrand _et al._ (2019) The QBE formalism would allow these non-perturbative techniques to access not only true steady states, but also non-equilibrium states of correlated electrons on the picosecond timescale relevant for photo-induced phase transition and collective orders. ###### Acknowledgements. We acknowledge Philipp Werner for useful discussions, and Nagamalleswararao Dasari for discussions as well as his contribution to the implementation of the Ohmic bath. This work was supported by the ERC Starting Grant No. 716648. The calculations have been done at the RRZE of the University Erlangen- Nuremberg. ## References Basov _et al._ (2017) D. N. Basov, R. D. Averitt, and D. 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# A Decentralized Analysis of Multiparty Protocols Bas van den Heuvel and Jorge A. Pérez ###### Abstract Protocols provide the unifying glue in concurrent and distributed software today; verifying that message-passing programs conform to such governing protocols is important but difficult. Static approaches based on multiparty session types (MPST) use protocols as types to avoid protocol violations and deadlocks in programs. An elusive problem for MPST is to ensure _both_ protocol conformance _and_ deadlock freedom for implementations with interleaved and delegated protocols. We propose a decentralized analysis of multiparty protocols, specified as global types and implemented as interacting processes in an asynchronous $\pi$-calculus. Our solution rests upon two novel notions: _router processes_ and _relative types_. While router processes use the global type to enable the composition of participant implementations in arbitrary process networks, relative types extract from the global type the intended interactions and dependencies between _pairs_ of participants. In our analysis, processes are typed using APCP, a type system that ensures protocol conformance and deadlock freedom with respect to _binary_ protocols, developed in prior work. Our decentralized, router-based analysis enables the sound and complete transference of protocol conformance and deadlock freedom from APCP to multiparty protocols. ###### Contents 1. 1 Introduction 2. 2 APCP: Asynchronous Processes, Deadlock Free by Typing 3. 3 Global Types and Relative Projection 1. 3.1 Relative Types 2. 3.2 Relative Projection and Well-Formedness 4. 4 Analyzing Global Types using Routers 1. 4.1 Synthesis of Routers 2. 4.2 Types for the Router’s Channels 1. 4.2.1 The Channels between Routers and Implementations 2. 4.2.2 The Channels between Pairs of Routers 3. 4.3 Networks of Routed Implementations 1. 4.3.1 The Typability of Routers 2. 4.3.2 Transference of Results (Operational Correspondence) 4. 4.4 Routers Strictly Generalize Centralized Orchestrators 1. 4.4.1 Synthesis of Orchestrators 2. 4.4.2 Orchestrators and Centralized Compositions of Routers are Behaviorally Equivalent 5. 5 Routers in Action 1. 5.1 Delegation and Interleaving 2. 5.2 Another Example of Delegation 3. 5.3 The Authorization Protocol in Action 6. 6 Related Work 7. 7 Conclusion 8. A Comparing Merge-based Well-formedness and Relative Well-formedness 1. A.1 Relative Well-Formed, Not Merge Well-Formed 2. A.2 Merge Well-Formed, Not Relative Well-Formed ## 1 Introduction This paper presents a new approach to the analysis of the _protocols_ that pervade concurrent and distributed software. Such protocols provide an essential unifying glue between communicating programs; ensuring that communicating programs implement protocols correctly, avoiding protocol violations and deadlocks, is an important but difficult problem. Here, we study _multiparty session types (MPST)_ [36], an approach to correctness in message-passing programs that uses governing multiparty protocols as types in program verification. As a motivating example, let us consider a _recursive authorization protocol_ , adapted from an example by Scalas and Yoshida [48]. It involves three participants: a Client, a Server, and an Authorization service. Intuitively, the protocol proceeds as follows. The Server requests the Client either to _login_ or to _quit_ the protocol. In the case of login, the Client sends a password to the Authorization service, which then may authorize the login with the Server; subsequently, the protocol can be performed again: this is useful when, e.g., clients must renew their authorization privileges after some time. In the case of quit, the protocol ends. MPST use _global types_ to specify multiparty protocols. The authorization protocol just described can be specified by the following global type between Client (‘$c$’), Server (‘$s$’), and Authorization service (‘$a$’): $G_{\mathsf{auth}}=\mu X\mathbin{.}s\mathbin{\twoheadrightarrow}c\left\\{\begin{array}[]{l}\mathsf{login}\mathbin{.}c\mathbin{\twoheadrightarrow}a\big{\\{}\mathsf{passwd}\langle\mathsf{str}\rangle\mathbin{.}a\mathbin{\twoheadrightarrow}s\\{\mathsf{auth}\langle\mathsf{bool}\rangle\mathbin{.}X\\}\big{\\}},\\\ \mathsf{quit}\mathbin{.}c\mathbin{\twoheadrightarrow}a\\{\mathsf{quit}\mathbin{.}\bullet\\}\end{array}\right\\}$ (1) After declaring a recursion on the variable $X$ (‘$\mu X$’), the global type $G_{\mathsf{auth}}$ stipulates that $s$ sends to $c$ (‘$s\mathbin{\twoheadrightarrow}c$’) a label login or quit. The rest of the protocol depends on this choice by $s$. In the login-branch, $c$ sends to $a$ a label passwd along with a string value (‘$\langle\mathsf{str}\rangle$’) and $a$ sends to $s$ a label auth and a boolean value, after which the protocol loops to the beginning of the recursion (‘$X$’). In the quit-branch, $c$ sends to $a$ a label quit after which the protocol ends (‘$\bullet$’). In MPST, participants are implemented as distributed processes that communicate asynchronously. Each process must correctly implement its corresponding portion of the protocol; these individual guarantees ensure that the interactions between processes conform to the given global type. Correctness follows from _protocol fidelity_ (processes interact as stipulated by the protocol), _communication safety_ (no errors or mismatches in messages), and _deadlock freedom_ (processes never get stuck waiting for each other). Ensuring that implementations satisfy these properties is a challenging problem, which is further compounded by two common and convenient features in interacting processes: _delegation_ and _interleaving_. We motivate them in the context of our example: * • _Delegation,_ or higher-order channel passing, can effectively express that the Client transparently reconfigures its involvement by asking another participant (say, a Password Manager) to act on its behalf; * • _Interleaving_ arises when a single process implements more than one role, as in, e.g., an implementation of both the Server and the Authorization service in a sequential process. Note that while delegation is explicitly specified in a global type, interleaving arises in its implementation as interacting processes, not in its specification. MPST have been widely studied from foundational and applied angles [7, 18, 37, 47, 5, 6, 48, 19, 39, 42]. The original theory by Honda _et al._ [35] defines a behavioral type system [38, 3] for a $\pi$-calculus, which exploits linearity to ensure protocol fidelity and communication safety; most derived works retain this approach and target the same properties. Deadlock freedom is hard to ensure by typing when implementations feature delegation and interleaving. In simple scenarios without interleaving and/or delegation, deadlock freedom is easy, as it concerns a single-threaded protocol. In contrast, deadlock freedom for processes running multiple, interleaved protocols (possibly delegated) is a much harder problem, addressed only by some advanced type systems [7, 44, 21]. In this paper, we tackle the problem of ensuring that networks of interacting processes correctly implement a given global type in a deadlock free manner, while supporting delegation and interleaving. Our approach is informed by the differences between _orchestration_ and _choreography_ , two salient approaches to the coordination and organization of interacting processes in service-oriented paradigms [45]: * • In _orchestration_ -based approaches, processes interact through a coordinator process which ensures that they all follow the protocol as intended. Quoting Van der Aalst, in an orchestration “the conductor tells everybody in the orchestra what to do and makes sure they all play in sync” [52]. * • In _choreography_ -based approaches, processes interact directly following the protocol without external coordination. Again quoting Van der Aalst, in a choreography “dancers dance following a global scenario without a single point of control” [52]. Specification and analysis techniques based on MPST fall under the choreography-based approach. The global type provides the protocol’s specification; based on the global type, implementations for each participant interact directly with each other, without an external coordinator. As we will see, the contrast between orchestration and choreography is relevant here because it induces a different _network topology_ for interacting processes. In an orchestration, the resulting process network is _centralized_ : all processes must connect to a central orchestrator process. In a choreography, the process network is _decentralized_ , as processes can directly connect to each other. ##### Contributions We develop a new decentralized analysis of multiparty protocols. * • Here ‘analysis’ refers to (i) ways of specifying such protocols as interacting processes _and_ (ii) techniques to verify that those processes satisfy the intended correctness properties. * • Also, aligned with the above discussion, ‘decentralized’ refers to the intended network topology for processes, which does not rely on an external coordinator. Our decentralized analysis of global types enforces protocol fidelity, communication safety, and deadlock freedom for process implementations, while uniformly supporting delegation, interleaving, and asynchronous communication. $P$router$Q$router$R$router $\begin{array}[]{c}\text{medium}\\\ \text{or}\\\ \text{arbiter}\end{array}$ $P$$Q$$R$ Figure 1: Given processes $P$, $Q$, and $R$ implementing the roles of $c$, $s$, and $a$, respectively, protocol $G_{\mathsf{auth}}$ can be realized as a choreography of routed implementations (our approach, left) and as an orchestration of implementations, with a medium or arbiter process (previous works, right). The _key idea_ of our analysis is to exploit global types to generate _router processes_ (simply _routers_) that enable participant implementations to communicate directly. There is a router per participant; it is intended to serve as a “wrapper” for an individual participant’s implementation. The composition of an implementation with its corresponding router is called a _routed implementation_. Collections of routed implementations can then be connected in arbitrary _process networks_ that correctly realize the multiparty protocol, subsuming centralized and decentralized topologies. Routers are _synthesized_ from global types, and do not change the behavior of the participant implementations they wrap; they merely ensure that networks of routed implementations correctly behave as described by the given multiparty protocol. Returning to Van der Aalst’s analogies quoted above, we may say that in our setting participant implementations are analogous to skilled but barefoot dancers, and that routers provide them with the appropriate shoes to dance without a central point of control. To make this analogy a bit more concrete, Figure 1 (left) illustrates the decentralized process network formed by routed implementations of the participants of $G_{\mathsf{auth}}$: once wrapped by an appropriate router, implementations $P$, $Q$, and $R$ can be composed directly in a decentralized process network. A central technical challenge in our approach is to ensure that compositions of routed implementations conform to their global type. The channels that enable the arbitrary composition of routed implementations need to be typed in accordance with the given multiparty protocol. Unfortunately, the usual notion of projection in MPST, which obtains a single participant’s perspective from a global type, does not suffice: we need a local perspective that is relative to the _two participants_ that the connected routed implementations represent. To this end, we introduce a new notion, _relative projection_ , which isolates the exchanges of the global type that relate to pairs of participants. In the case of $G_{\mathsf{auth}}$, for instance, we need three relative types, describing the protocol for $a$ and $c$, for $a$ and $s$, and for $c$ and $s$. A derived challenge is that when projecting a global type onto a pair of participants, it is possible to encounter _non-local choices_ : choices by other participants that affect the protocol between the two participants involved in the projection. To handle this, relative projection explicitly records non-local choices in the form of _dependencies_ , which inform the projection’s participants that they need to coordinate on the results of the non-local choices. To summarize, our decentralized analysis of global types relies on three intertwined novel notions: * • Routers that wrap participant implementations in order to enable their composition in arbitrary network topologies, whilst guaranteeing that the resulting process networks correctly follow the given global type in a deadlock free manner. * • Relative Types that type the channels between routed implementations, obtained by means of a new notion of projection of global types onto pairs of participants. * • Relative Projection and Dependencies that make it explicit in relative types that participants need to coordinate on non-local choices. The key ingredients of our decentralized analysis for $G_{\mathsf{auth}}$ are jointly depicted in Figure 2. With respect to prior analyses of multiparty protocols, a distinguishing feature of our work is its natural support of decentralized process networks, as expected in a choreography-based approach. Caires and Pérez [12] connect participant implementations through a central coordinator, called _medium process_. This medium process is generated from a global type, and intervenes in all exchanges to ensure that the participant implementations follow the multiparty protocol. The composition of the medium with the participant implementations can then be analyzed using a type system for binary sessions. In a similar vein, Carbone _et al._ [16] define a type system in which they use global types to validate choreographies of participant implementations. Their analysis of protocol implementations—in particular, deadlock freedom—relies on encodings into another type system where participant implementations connect to a central coordinator, called the _arbiter process_. Similar to mediums, arbiters are generated from the global type to ensure that participant implementations follow the protocol as intended. Both these approaches are clear examples of orchestration, and thus do not support decentralized network topologies. To highlight the differences between our decentralized analysis and prior approaches, compare the choreography of routed implementations in Figure 1 (left) with an implementation of $G_{\mathsf{auth}}$ in the style of Caires and Pérez and of Carbone _et al._ , given in Figure 1 (right). These prior works rely on orchestration because the type systems they use for verifying process implementations restrict connections between processes: they only admit a form of process composition that makes it impossible to simultaneously connect three or more participant implementations [24]. In this paper, we overcome this obstacle by relying on APCP (Asynchronous Priority-based Classical Processes) [51], a type system that allows for more general forms of process composition. By using annotations on types, APCP prevents _circular dependencies_ , i.e., cyclically connected processes that are stuck waiting for each other. This is how our approach supports networks of routed participants in both centralized and decentralized topologies, thus subsuming choreography and orchestration approaches. $G_{\mathsf{auth}}$$L_{c}$local projection (§ 4.2.1)$R_{cs},R_{ca}$relative projection (§ 3.2)$P$type check in APCP (§ 2)$\mathcal{R}_{c}$router synthesis (§ 4.1)$G_{\mathsf{auth}}$$P\mathbin{\|}\mathcal{R}_{c}$clients$Q\mathbin{\|}\mathcal{R}_{s}$serverl$R\mathbin{\|}\mathcal{R}_{a}$authorization servicesnetwork of routed implementations of $G_{\mathsf{auth}}$ (Def. 25)routed implementation of $c$ (Def. 24) Figure 2: Decentralized analysis of $G_{\mathsf{auth}}$ into a network of routed implementations. The definition of $G_{\mathsf{auth}}$ contains message types. Focusing on the client $c$ (on the left), $L_{c}$ denotes a session type, whereas $R_{cs}$ and $R_{ca}$ are relative types with respect to the server and the authorization service, respectively. ##### Outline This paper is structured as follows. Next, Section 2 recalls APCP as introduced by Van den Heuvel and Pérez [51] and summarizes the correctness properties for asynchronous processes derived from typing. The following three sections develop and illustrate our contributions: * • Section 3 introduces relative types and relative projection, and defines well- formed global types, a class of global types that includes protocols with non- local choices. * • Section 4 introduces the synthesis of routers. A main result is their typability in APCP (Theorem 11). We establish deadlock freedom for networks of routed implementations (Theorem 18), which we transfer to multiparty protocols via an operational correspondence result (Theorems 19 and 23). Moreover, we show that our approach strictly generalizes prior analyses based on centralized topologies (Theorem 27). * • Section 5 demonstrates our contributions in action, with a full development of the routed implementations for $G_{\mathsf{auth}}$, and an example of the flexible support for _delegation_ and _interleaving_ enabled by our router- based approach and APCP. We discuss further related works in § 6 and conclude the paper in § 7. We use colors to improve readability. ## 2 APCP: Asynchronous Processes, Deadlock Free by Typing We recall APCP as defined by Van den Heuvel and Pérez [51]. APCP is a type system for asynchronous $\pi$-calculus processes (with non-blocking outputs) [34, 9], with support for recursion and cyclic connections. In this type system, channel endpoints are assigned linear types that represent two-party (binary) _session types_ [33]. Well-typed APCP processes preserve typing (Theorem 2) and are deadlock free (Theorem 5). At its basis, APCP combines Dardha and Gay’s Priority-based Classical Processes (PCP) [22] with DeYoung _et al._ ’s continuation-passing semantics for asynchrony [27], and adds recursion, inspired by the work of Toninho _et al._ [49]. We refer the interested reader to the work by Van den Heuvel and Pérez [51] for a motivation of design choices and proofs of results. ##### Process Syntax Process syntax: $\displaystyle P,Q::=$ $\displaystyle\leavevmode\nobreak\ x[y,z]$ output $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \;\mbox{\large{$\mid$}}\;\leavevmode\nobreak\ \leavevmode\nobreak\ x(y,z)\mathbin{.}P$ input $\mid$ $\displaystyle\leavevmode\nobreak\ x[z]\triangleleft i$ selection $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \;\mbox{\large{$\mid$}}\;\leavevmode\nobreak\ \leavevmode\nobreak\ x(z)\triangleright\\{i{:}\leavevmode\nobreak\ P\\}_{i\in I}$ branching $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \;\mbox{\large{$\mid$}}\;\leavevmode\nobreak\ \leavevmode\nobreak\ (\bm{\nu}xy)P$ restriction $\mid$ $\displaystyle\leavevmode\nobreak\ P\mathbin{\|}Q$ parallel $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \;\mbox{\large{$\mid$}}\;\leavevmode\nobreak\ \leavevmode\nobreak\ \bm{0}$ inaction $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \;\mbox{\large{$\mid$}}\;\leavevmode\nobreak\ \leavevmode\nobreak\ x\mathbin{\leftrightarrow}y$ forwarder $\mid$ $\displaystyle\leavevmode\nobreak\ \mu X(\tilde{z})\mathbin{.}P$ recursive loop $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \;\mbox{\large{$\mid$}}\;\leavevmode\nobreak\ \leavevmode\nobreak\ X{\langle\tilde{z}\rangle}$ recursive call . Structural congruence: $\displaystyle P\equiv_{\alpha}P^{\prime}\implies{}$ $\displaystyle P$ $\displaystyle\equiv P^{\prime}$ $\displaystyle x\mathbin{\leftrightarrow}y$ $\displaystyle\equiv y\mathbin{\leftrightarrow}x$ $\displaystyle P\mathbin{\|}Q$ $\displaystyle\equiv Q\mathbin{\|}P$ $\displaystyle(\bm{\nu}xy)x\mathbin{\leftrightarrow}y$ $\displaystyle\equiv\bm{0}$ $\displaystyle P\mathbin{\|}\bm{0}$ $\displaystyle\equiv P$ $\displaystyle P\mathbin{\|}(Q\mathbin{\|}R)$ $\displaystyle\equiv(P\mathbin{\|}Q)\mathbin{\|}R$ $\displaystyle x,y\notin\mathrm{fn}(P)\implies{}$ $\displaystyle P\mathbin{\|}(\bm{\nu}xy)Q$ $\displaystyle\equiv(\bm{\nu}xy)(P\mathbin{\|}Q)$ $\displaystyle(\bm{\nu}xy)\bm{0}$ $\displaystyle\equiv\bm{0}$ $\displaystyle\|\tilde{z}\|=\|\tilde{y}\|\implies{}$ $\displaystyle\mu X(\tilde{z})\mathbin{.}P$ $\displaystyle\equiv P\big{\\{}(\mu X(\tilde{y})\mathbin{.}P\\{\tilde{y}/\tilde{z}\\})/X{\langle\tilde{y}\rangle}\big{\\}}$ $\displaystyle(\bm{\nu}xy)P$ $\displaystyle\equiv(\bm{\nu}yx)P$ $\displaystyle(\bm{\nu}xy)(\bm{\nu}zw)P$ $\displaystyle\equiv(\bm{\nu}zw)(\bm{\nu}xy)P$ . Reduction: $\displaystyle\beta_{\text{Id}}$ $\displaystyle z,y\neq x\implies{}$ $\displaystyle(\bm{\nu}yz)(x\mathbin{\leftrightarrow}y\mathbin{\|}P)$ $\displaystyle\longrightarrow P\\{x/z\\}$ $\displaystyle\beta_{\mathbin{\otimes}\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}}$ $\displaystyle(\bm{\nu}xy)(x[a,b]\mathbin{\|}y(v,z)\mathbin{.}P)$ $\displaystyle\longrightarrow P\\{a/v,b/z\\}$ $\displaystyle\beta_{{\oplus}\&}$ $\displaystyle j\in I\implies{}$ $\displaystyle(\bm{\nu}xy)(x[b]\triangleleft j\mathbin{\|}y(z)\triangleright\\{i{:}\leavevmode\nobreak\ P_{i}\\}_{i\in I})$ $\displaystyle\longrightarrow P_{j}\\{b/z\\}$ $\displaystyle\kappa_{\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}}$ $\displaystyle x\notin\tilde{v},\tilde{w}\implies{}$ $\displaystyle(\bm{\nu}\tilde{v}\tilde{w})(x(y,z)\mathbin{.}P\mathbin{\|}Q)$ $\displaystyle\longrightarrow x(y,z)\mathbin{.}(\bm{\nu}\tilde{v}\tilde{w})(P\mathbin{\|}Q)$ $\displaystyle\kappa_{\&}$ $\displaystyle x\notin\tilde{v},\tilde{w}\implies{}$ $\displaystyle(\bm{\nu}\tilde{v}\tilde{w})(x(z)\triangleright\\{i{:}\leavevmode\nobreak\ P_{i}\\}_{i\in I}\mathbin{\|}Q)$ $\displaystyle\longrightarrow x(z)\triangleright\\{i{:}\leavevmode\nobreak\ (\bm{\nu}\tilde{v}\tilde{w})(P_{i}\mathbin{\|}Q)\\}_{i\in I}$ $(P\equiv P^{\prime})\wedge(P^{\prime}\longrightarrow Q^{\prime})\wedge(Q^{\prime}\equiv Q)$ $\shortrightarrow_{\equiv}$ $P\longrightarrow Q$ $\raisebox{9.0pt}{}P\longrightarrow Q$ $\shortrightarrow_{\nu}$ $(\bm{\nu}xy)P\longrightarrow(\bm{\nu}xy)Q$ $\raisebox{9.0pt}{}P\longrightarrow Q$ $\shortrightarrow_{\mathbin{\|}}$ $P\mathbin{\|}R\longrightarrow Q\mathbin{\|}R$ Figure 3: Definition of the process language of APCP. We write $x,y,z,\ldots$ to denote (channel) _endpoints_ (also known as _names_), and write $\tilde{x},\tilde{y},\tilde{z},\ldots$ to denote sequences of endpoints. Also, we write $i,j,k,\ldots$ to denote _labels_ for choices and $I,J,K,\ldots$ to denote sets of labels. We write $X,Y,\ldots$ to denote _recursion variables_ , and $P,Q,\ldots$ to denote processes. Figure 3 (top) gives the syntax of processes, which communicate asynchronously by following a continuation-passing style. The output action ‘$x[y,z]$’ denotes the sending of endpoints $y$ and $z$ along $x$: while the former is the message, the latter is the protocol’s continuation; both $y$ and $z$ are free. The input prefix ‘$x(y,z)\mathbin{.}P$’ blocks until a message $y$ and a continuation endpoint $z$ are received on $x$, binding $y$ and $z$ in $P$. The selection action ‘$x[z]\mathbin{\triangleleft}i$’ sends a label $i$ and a continuation endpoint $z$ along $x$. The branching prefix ‘$x(z)\mathbin{\triangleright}\\{i{:}\leavevmode\nobreak\ P_{i}\\}_{i\in I}$’ blocks until it receives a label $i\in I$ and a continuation endpoint $z$ on $x$, binding $z$ in each $P_{i}$. Restriction ‘$(\bm{\nu}xy)P$’ binds $x$ and $y$ in $P$, thus declaring them as the two endpoints of the same channel and enabling communication, as in Vasconcelos [53]. The process ‘$\mkern-1.0muP\mathbin{\|}Q\mkern 1.0mu$’ denotes the parallel composition of $P$ and $Q$. The process ‘$\bm{0}$’ denotes inaction. The forwarder process ‘$x\mathbin{\leftrightarrow}y$’ is a primitive copycat process that links together $x$ and $y$. The prefix ‘$\mu X(\tilde{z})\mathbin{.}P$’ defines a recursive loop, where $\mu$ binds any free occurrences of $X$ in $P$ and the endpoints $\tilde{z}$ form a context for $P$. The recursive call ‘$X{\langle\tilde{z}\rangle}$’ loops to its corresponding $\mu X$, providing the endpoints $\tilde{z}$ as context. We only consider contractive recursion, disallowing processes with subexpressions of the form ‘$\mu X_{1}(\tilde{z})\ldots\mu X_{n}(\tilde{z})\mathbin{.}X_{1}{\langle\tilde{z}\rangle}$’. Endpoints and recursion variables are free unless they are bound somewhere. We write ‘$\mathrm{fn}(P)$’ and ‘$\mathrm{frv}(P)$’ for the sets of free names and free recursion variables of $P$, respectively. Also, we write ‘$P\\{x/y\\}$’ to denote the capture-avoiding substitution of the free occurrences of $y$ in $P$ for $x$. The notation ‘$P\big{\\{}(\mu X(\tilde{y})\mathbin{.}Q)/X{\langle\tilde{y}\rangle}\big{\\}}$’ denotes the substitution of occurrences of recursive calls ‘$X{\langle\tilde{y}\rangle}$’ for any sequence of names $\tilde{y}$ in $P$ with the recursive loop ‘$\mu X(\tilde{y})\mathbin{.}Q$’, which we call _unfolding_ recursion. We write sequences of substitutions ‘$P\\{x_{1}/y_{1}\\}\ldots\\{x_{n}/y_{n}\\}$’ as ‘$P\\{x_{1}/y_{1},\ldots,x_{n}/y_{n}\\}$’. In an output ‘$x[y,z]$’, both $y$ and $z$ are free, as mentioned above; they can be bound to a continuation process using parallel composition and restriction, as in, e.g., $(\bm{\nu}ya)(\bm{\nu}zb)(x[y,z]\mathbin{\|}P_{a,b})$. The same applies to selection ‘$x[z]\mathbin{\triangleleft}i$’. We introduce useful notations that elide the restrictions and continuation endpoints: ###### Notation 1 (Derivable Actions and Prefixes). We use the following syntactic sugar: $\displaystyle\overline{x}[y]\cdot P$ $\displaystyle:=(\bm{\nu}ya)(\bm{\nu}zb)(x[a,b]\mathbin{\|}P\\{z/x\\})$ $\displaystyle\overline{x}\mathbin{\triangleleft}\ell\cdot P$ $\displaystyle:=(\bm{\nu}zb)(x[b]\mathbin{\triangleleft}\ell\mathbin{\|}P\\{z/x\\})$ $\displaystyle x(y)\mathbin{.}P$ $\displaystyle:=x(y,z)\mathbin{.}P\\{z/x\\}$ $\displaystyle x\mathbin{\triangleright}\\{i{:}\leavevmode\nobreak\ P_{i}\\}_{i\in I}$ $\displaystyle:=x(z)\mathbin{\triangleright}\\{i{:}\leavevmode\nobreak\ P_{i}\\{z/x\\}\\}_{i\in I}$ Note the use of ‘${}\cdot{}$’ instead of ‘${}\mathbin{.}{}$’ in output and selection actions to stress that they are non-blocking. ##### Operational Semantics We define a reduction relation for processes ($P\longrightarrow Q$) that formalizes how complementary actions on connected endpoints may synchronize. As usual for $\pi$-calculi, reduction relies on _structural congruence_ ($P\equiv Q$), which equates the behavior of processes with minor syntactic differences; it is the smallest congruence relation satisfying the axioms in Figure 3 (center). Structural congruence defines the following properties of our process language. Processes are equivalent up to $\alpha$-equivalence. Parallel composition is associative and commutative, with unit ‘$\bm{0}$’. The forwarder process is symmetric, and equivalent to inaction if both endpoints are bound together through restriction. A parallel process may be moved into or out of a restriction as long as the bound channels do not appear free in the moved process: this is _scope inclusion_ and _scope extrusion_ , respectively. Restrictions on inactive processes may be dropped, and the order of endpoints in restrictions and of consecutive restrictions does not matter. Finally, a recursive loop is equivalent to its unfolding, replacing any recursive calls with copies of the recursive loop, where the call’s endpoints are pairwise substituted for the contextual endpoints of the loop. We define the reduction relation by the axioms and closure rules in Figure 3 (bottom). Axioms labeled ‘$\beta$’ are _synchronizations_ and those labeled ‘$\kappa$’ are _commuting conversions_ , which allow pulling prefixes on free channels out of restrictions; they are not necessary for deadlock freedom, but they are usually presented in Curry-Howard interpretations of linear logic as session types [14, 54, 22, 27]. Rule $\beta_{\text{Id}}$ implements the forwarder as a substitution. Rule $\beta_{\mathbin{\otimes}\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}}$ synchronizes an output and an input on connected endpoints and substitutes the message and continuation endpoint. Rule $\beta_{{\oplus}\&}$ synchronizes a selection and a branch: the received label determines the continuation process, substituting the continuation endpoint appropriately. Rule $\kappa_{\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}}$ (resp. $\kappa_{\&}$) pulls an input (resp. a branching) prefix on free channels out of enclosing restrictions. Rules $\rightarrow_{\equiv}$, $\rightarrow_{\nu}$, and $\rightarrow_{\mathbin{\|}}$ close reduction under structural congruence, restriction, and parallel composition, respectively. ###### Notation 2 (Reductions). We write ‘$\longrightarrow_{\beta}$’ for reductions derived from $\beta$-axioms, and ‘$\longrightarrow^{\ast}$’ for the reflexive, transitive closure of ‘$\longrightarrow$’. Also, we write ‘$P\longrightarrow^{\star}Q$’ if $P\longrightarrow^{\ast}Q$ in a finite number of steps, and ‘$P{\centernot\longrightarrow}^{\ast}Q$’ for the non-existence of a series of reductions from $P$ to $Q$. ##### Session Types Session Type Endpoint Behavior $A\mathbin{\otimes}^{\mathsf{o}}B$ output an endpoint of type $A$, then behave as $B$ $A\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{\mathsf{o}}B$ input an endpoint of type $A$, then behave as $B$ ${{\oplus}}^{\mathsf{o}}\\{i{:}\leavevmode\nobreak\ A_{i}\\}_{i\in I}$ select a label $i\in I$, then behave as $A_{i}$ $\&^{\mathsf{o}}\\{i{:}\leavevmode\nobreak\ A_{i}\\}_{i\in I}$ receive a choice for a label $i\in I$, then behave as $A_{i}$ $\bullet$ closed session; no behavior Table 1: Session types and their associated endpoint behaviors (cf. Definition 1). The type system assigns session types to channel endpoints. We present session types as linear logic propositions following, e.g., Wadler [54], Caires and Pfenning [13], and Dardha and Gay [22]. We extend these propositions with recursion and _priority_ annotations on connectives. Intuitively, actions typed with lower priority should be performed before those with higher priority. We write $\mathsf{o},\kappa,\pi,\rho,\ldots$ to denote priorities, and ‘$\omega$’ to denote the ultimate priority that is greater than all other priorities and cannot be increased further. That is, $\forall t\in\mathbb{N}.\leavevmode\nobreak\ \omega>t$ and $\forall t\in\mathbb{N}.\leavevmode\nobreak\ \omega+t=\omega$. ###### Definition 1 (Session Types). The following grammar defines the syntax of _session types_ $A,B$. Let $\mathsf{o}\in\mathbb{N}\cup\\{\omega\\}$. $\displaystyle A,B$ $\displaystyle::=A\mathbin{\otimes}^{\mathsf{o}}B\;\mbox{\large{$\mid$}}\;A\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{\mathsf{o}}B\;\mbox{\large{$\mid$}}\;{{\oplus}}^{\mathsf{o}}\\{i:A\\}_{i\in I}\;\mbox{\large{$\mid$}}\;\&^{\mathsf{o}}\\{i:A\\}_{i\in I}\;\mbox{\large{$\mid$}}\;\bullet\;\mbox{\large{$\mid$}}\;\mu X\mathbin{.}A\;\mbox{\large{$\mid$}}\;X$ Table 1 gives _session types_ and the behavior that is expected of an endpoint with each type (recursive types entail a communication behavior only after unfolding). Note that ‘$\bullet$’ does not require a priority, as closed endpoints do not exhibit behavior and thus are non-blocking. We define ‘$\bullet$’ as a single, self-dual type for closed endpoints (cf. Caires [11] and Atkey _et al._ [4]). Type ‘$\mu X\mathbin{.}A$’ denotes a recursive type, in which $A$ may contain occurrences of the recursion variable ‘$X$’. As customary, ‘$\mu$’ is a binder: it induces the standard notions of $\alpha$-equivalence, substitution (denoted ‘$A\\{B/X\\}$’), and free recursion variables (denoted ‘$\mathrm{frv}(A)$’). We work with tail-recursive, contractive types, disallowing types of the form ‘$\mu X_{1}\ldots\mu X_{n}\mathbin{.}X_{1}$’. We postpone the formalization of the unfolding of recursive types, as it requires additional definitions to ensure consistency of priorities in types. _Duality_ , the cornerstone of session types and linear logic, ensures that the two endpoints of a channel have matching actions. Furthermore, dual types must have matching priority annotations. The following inductive definition of duality suffices for our tail-recursive types (cf. Gay _et al._ [31]). ###### Definition 2 (Duality). The _dual_ of session type $A$, denoted ‘$\overline{A}$’, is defined inductively as follows: $\displaystyle\overline{A\mathbin{\otimes}^{\mathsf{o}}B}$ $\displaystyle:=\overline{A}\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{\mathsf{o}}\overline{B}$ $\displaystyle\overline{{{\oplus}}^{\mathsf{o}}\\{i:A_{i}\\}_{i\in I}}$ $\displaystyle:=\&^{\mathsf{o}}\\{i:\overline{A_{i}}\\}_{i\in I}$ $\displaystyle\overline{\bullet}$ $\displaystyle:=\bullet$ $\displaystyle\overline{\mu X\mathbin{.}A}$ $\displaystyle:=\mu X\mathbin{.}\overline{A}$ $\displaystyle\overline{A\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{\mathsf{o}}B}$ $\displaystyle:=\overline{A}\mathbin{\otimes}^{\mathsf{o}}\overline{B}$ $\displaystyle\overline{\&^{\mathsf{o}}\\{i:A_{i}\\}_{i\in I}}$ $\displaystyle:={{\oplus}}^{\mathsf{o}}\\{i:\overline{A_{i}}\\}_{i\in I}$ $\displaystyle\overline{X}$ $\displaystyle:=X$ The priority of a type is determined by the priority of the type’s outermost connective: ###### Definition 3 (Priorities). For session type $A$, ‘$\mathsf{pr}(A)$’ denotes its _priority_ : $\displaystyle\mathsf{pr}(A\mathbin{\otimes}^{\mathsf{o}}B):=\mathsf{pr}(A\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{\mathsf{o}}B)$ $\displaystyle:=\mathsf{o}$ $\displaystyle\mathsf{pr}(\mu X\mathbin{.}A)$ $\displaystyle:=\mathsf{pr}(A)$ $\displaystyle\mathsf{pr}({\oplus}^{\mathsf{o}}\\{i{:}\leavevmode\nobreak\ A_{i}\\}_{i\in I}):=\mathsf{pr}(\&^{\mathsf{o}}\\{i{:}\leavevmode\nobreak\ A_{i}\\}_{i\in I})$ $\displaystyle:=\mathsf{o}$ $\displaystyle\mathsf{pr}(\bullet):=\mathsf{pr}(X)$ $\displaystyle:=\omega$ The priority of ‘$\bullet$’ and ‘$X$’ is $\omega$: they denote “final”, non- blocking actions of protocols. Although ‘$\mathbin{\otimes}$’ and ‘${\oplus}$’ also denote non-blocking actions, their priority is not constant: duality ensures that the priority for ‘$\mathbin{\otimes}$’ (resp. ‘${\oplus}$’) matches the priority of a corresponding ‘$\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}$’ (resp. ‘$\&$’), which denotes a blocking action. Having defined the priority of types, we now turn to formalizing the unfolding of recursive types. Recall the intuition that actions typed with lower priority should be performed before those with higher priority. Based on this rationale, we observe that unfolding should increase the priorities of the unfolded type. This is because the actions related to the unfolded recursion should be performed _after_ the prefix. The following definition _lifts_ priorities in types: ###### Definition 4 (Lift). For proposition $A$ and $t\in\mathbb{N}$, we define ‘$\mkern 2.0mu{\uparrow^{t}}A\mkern-3.0mu$’ as the _lift_ operation: $\displaystyle{\uparrow^{t}}(A\mathbin{\otimes}^{\mathsf{o}}B)$ $\displaystyle:=({\uparrow^{t}}A)\mathbin{\otimes}^{\mathsf{o}+t}({\uparrow^{t}}B)$ $\displaystyle{\uparrow^{t}}({\oplus}^{\mathsf{o}}\\{i{:}\leavevmode\nobreak\ A_{i}\\}_{i\in I})$ $\displaystyle:={\oplus}^{\mathsf{o}+t}\\{i{:}\leavevmode\nobreak\ {\uparrow^{t}}A_{i}\\}_{i\in I}$ $\displaystyle{\uparrow^{t}}\bullet$ $\displaystyle:=\bullet$ $\displaystyle{\uparrow^{t}}(A\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{\mathsf{o}}B)$ $\displaystyle:=({\uparrow^{t}}A)\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{\mathsf{o}+t}({\uparrow^{t}}B)$ $\displaystyle{\uparrow^{t}}(\&^{\mathsf{o}}\\{i{:}\leavevmode\nobreak\ A_{i}\\}_{i\in I})$ $\displaystyle:=\&^{\mathsf{o}+t}\\{i{:}\leavevmode\nobreak\ {\uparrow^{t}}A_{i}\\}_{i\in I}$ $\displaystyle{\uparrow^{t}}(\mu X\mathbin{.}A)$ $\displaystyle:=\mu X\mathbin{.}{\uparrow^{t}}(A)$ $\displaystyle{\uparrow^{t}}X$ $\displaystyle:=X$ ###### Definition 5. The _unfolding_ of ‘$\mu X\mathbin{.}A$’ is ‘$A\\{\mu X\mathbin{.}({\uparrow^{t}}A)/X\\}$’, denoted ‘$\mathrm{unfold}^{t}(\mu X\mathbin{.}A)$’, where $t\in\mathbb{N}$. When unfolding $\mu X\mathbin{.}A$ as $\mathrm{unfold}^{t}(\mu X\mathbin{.}A)$, the “lifter” $t$ will depend on the highest priority of the types appearing in a typing context. The highest priority of a type is defined as follows: ###### Definition 6 (Highest Priority). For session type $A$, ‘$\max_{\mathsf{pr}}(A)$’ denotes its _highest priority_ : $\displaystyle\max_{\mathsf{pr}}(A\mathbin{\otimes}^{\mathsf{o}}B):=\max_{\mathsf{pr}}(A\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{\mathsf{o}}B)$ $\displaystyle:=\max(\max_{\mathsf{pr}}(A),\max_{\mathsf{pr}}(B),\mathsf{o})$ $\displaystyle\max_{\mathsf{pr}}(\mu X\mathbin{.}A)$ $\displaystyle:=\max_{\mathsf{pr}}(A)$ $\displaystyle\max_{\mathsf{pr}}({\oplus}^{\mathsf{o}}\\{i:A_{i}\\}_{i\in I}):=\max_{\mathsf{pr}}(\&^{\mathsf{o}}\\{i:A_{i}\\}_{i\in I})$ $\displaystyle:=\max(\max_{i\in I}(\max_{\mathsf{pr}}(A_{i})),\mathsf{o})$ $\displaystyle\max_{\mathsf{pr}}(\bullet):=\max_{\mathsf{pr}}(X)$ $\displaystyle:=0$ Notice how, in contrast to Definition 3, the highest priority of ‘$\bullet$’ and ‘$X$’ is 0: this is because they do not contribute to the increase in priority needed for unfolding recursive types. ##### Type Checking The typing (or, type checking) rules of APCP enforce that channel endpoints implement their ascribed session types, while ensuring that actions with lower priority are performed before those with higher priority (cf. Dardha and Gay [22]). They enforce the following laws: 1. 1. an action with priority $\mathsf{o}$ must be prefixed only by inputs and branches with priority strictly smaller than $\mathsf{o}$—this law only applies to inputs and branches, because outputs and selections are not prefixes; 2. 2. dual actions leading to synchronizations must have equal priorities (cf. Def. 1). Judgments are of the form ‘$P\vdash\Omega;\Gamma$’: * • $P$ is a process; * • $\Gamma$ is a context that assigns types to channels (‘$x{:}\leavevmode\nobreak\ A$’); * • $\Omega$ is a context that assigns tuples of types to recursion variables (‘$X{:}\leavevmode\nobreak\ (A,B,\ldots)$’). A judgment ‘$P\vdash\Omega;\Gamma$’ then means that $P$ can be typed in accordance with the type assignments for names recorded in $\Gamma$ and the recursion variables in $\Omega$. Intuitively, the recursive context $\Omega$ ensures that the context endpoints concur between recursive definitions and calls. Both contexts $\Gamma$ and $\Omega$ obey _exchange_ : assignments may be silently reordered. $\Gamma$ is _linear_ , disallowing _weakening_ (i.e., all assignments must be used) and _contraction_ (i.e., assignments may not be duplicated). $\Omega$ allows weakening and contraction, because a recursive definition may be called _zero or more_ times. The empty context is written ‘$\emptyset$’. We write ‘${\uparrow^{t}}\Gamma$’ to denote the component-wise extension of lift (Definition 4) to typing contexts. Also, we write ‘$\mathsf{pr}(\Gamma)$’ to denote the least priority of all types in $\Gamma$ (Definition 3). An assignment ‘$\tilde{z}{:}\leavevmode\nobreak\ \tilde{A}$’ means ‘$z_{1}{:}\leavevmode\nobreak\ A_{1},\ldots,z_{k}{:}\leavevmode\nobreak\ A_{k}$’. Empty $\bm{0}\vdash\Omega;\emptyset$ $P\vdash\Omega;\Gamma$ $\bullet$ $P\vdash\Omega;\Gamma,x{:}\leavevmode\nobreak\ \bullet$ Id $x\mathbin{\leftrightarrow}y\vdash\Omega;x{:}\leavevmode\nobreak\ \overline{A},y{:}\leavevmode\nobreak\ A$ $P\vdash\Omega;\Gamma$ $Q\vdash\Omega;\Delta$ Mix $P\mathbin{\|}Q\vdash\Omega;\Gamma,\Delta$ $P\vdash\Omega;\Gamma,x{:}\leavevmode\nobreak\ A,y{:}\leavevmode\nobreak\ \overline{A}$ Cycle $(\bm{\nu}xy)P\vdash\Omega;\Gamma$ $\mathbin{\otimes}$ $x[y,z]\vdash\Omega;x{:}\leavevmode\nobreak\ A\mathbin{\otimes}^{\mathsf{o}}B,y{:}\leavevmode\nobreak\ \overline{A},z{:}\leavevmode\nobreak\ \overline{B}$ $P\vdash\Omega;\Gamma,y{:}\leavevmode\nobreak\ A,z{:}\leavevmode\nobreak\ B$ $\mathsf{o}<\mathsf{pr}(\Gamma)$ $\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}$ $x(y,z)\mathbin{.}P\vdash\Omega;\Gamma,x{:}\leavevmode\nobreak\ A\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{\mathsf{o}}B$ $j\in I\vphantom{P_{i}\mathsf{pr}(\Gamma)}$ ${\oplus}$ $x[z]\triangleleft j\vdash\Omega;x{:}\leavevmode\nobreak\ {{\oplus}}^{\mathsf{o}}\\{i:A_{i}\\}_{i\in I},z{:}\leavevmode\nobreak\ \overline{A_{j}}\vphantom{j}$ $\forall i\in I.\leavevmode\nobreak\ P_{i}\vdash\Omega;\Gamma,z{:}\leavevmode\nobreak\ A_{i}$ $\mathsf{o}<\mathsf{pr}(\Gamma)$ $\&$ $x(z)\triangleright\\{i:P_{i}\\}_{i\in I}\vdash\Omega;\Gamma,x{:}\leavevmode\nobreak\ \&^{\mathsf{o}}\\{i:A_{i}\\}_{i\in I}$ $t\in\mathbb{N}>\max_{\mathsf{pr}}(\tilde{A})$ $P\vdash\Omega,X{:}\leavevmode\nobreak\ {\tilde{A}};\tilde{z}{:}\leavevmode\nobreak\ \tilde{U}$ where each $U_{i}=\mathrm{unfold}^{t}(\mu X\mathbin{.}A_{i})$ $\forall A_{i}\in\tilde{A}.\leavevmode\nobreak\ A_{i}\neq X$ Rec $\mu X(\tilde{z})\mathbin{.}P\vdash\Omega;\tilde{z}{:}\leavevmode\nobreak\ \widetilde{\mu X\mathbin{.}A}$ Var $X{\langle\tilde{z}\rangle}\vdash\Omega,X{:}\leavevmode\nobreak\ \tilde{A};\tilde{z}{:}\leavevmode\nobreak\ \widetilde{\mu X\mathbin{.}A}$ . $P\vdash\Omega;\Gamma,y{:}\leavevmode\nobreak\ A,x{:}\leavevmode\nobreak\ B$ $\mathbin{\otimes}^{\star}$ $\overline{x}[y]\cdot P\vdash\Omega;\Gamma,x{:}\leavevmode\nobreak\ A\mathbin{\otimes}^{\mathsf{o}}B$ $P\vdash\Omega;\Gamma,x{:}\leavevmode\nobreak\ A_{j}$ $j\in I$ ${\oplus}^{\star}$ $\overline{x}\triangleleft j\cdot P\vdash\Omega;\Gamma,x{:}\leavevmode\nobreak\ {{\oplus}}^{\mathsf{o}}\\{i:A_{i}\\}_{i\in I}$ $P\vdash\Omega;\Gamma$ $t\in\mathbb{N}$ Lift $P\vdash\Omega;{\uparrow^{t}}\Gamma$ Figure 4: The typing rules of APCP (top) and admissible rules (bottom). Figure 4 (top) gives the typing rules. In typing rules, we often write ‘$\Gamma,x{:}\leavevmode\nobreak\ A$’ (or similarly for $\Omega$) to denote _disjoint union_ , i.e. $x\notin\mathsf{dom}(\Gamma)$. Some type-preserving transformations of typing derivations correspond to process reductions (cf. Theorem 2). Other such transformations correspond to structural congruences (cf. Figure 3 (middle)); we sometimes use this explicitly in typing derivations in the form of a rule ‘$\equiv$’. If $P\equiv Q$ and $P\vdash\Omega;\Gamma$ and $Q\vdash\Omega;\Gamma^{\prime}$ where $\Gamma$ and $\Gamma^{\prime}$ are equal up to the unfolding of recursive types, then we say that $P\vdash\Omega;\Gamma^{\prime}$ and $Q\vdash\Omega;\Gamma$; in the context of a typing derivation, we equate recursive types and their unfoldings. We describe the typing rules from a _bottom-up_ perspective. Axiom ‘Empty’ types an inactive process with no endpoints. Rule ‘$\bullet$’ silently removes a closed endpoint to the typing context. Axiom ‘Id’ types forwarding between endpoints of dual type. Rule ‘Mix’ types the parallel composition of two processes that do not share assignments on the same endpoints. Rule ‘Cycle’ types a restriction, where the two restricted endpoints must be of dual type. Note that a single application of ‘Mix’ followed by ‘Cycle’ coincides with the usual rule ‘Cut’ in type systems based on linear logic [14, 54]. Axiom ‘$\mathbin{\otimes}$’ types an output action; this rule does not have premises to provide a continuation process, leaving the free endpoints to be bound to a continuation process using ‘Mix’ and ‘Cycle’. Similarly, axiom ‘${\oplus}$’ types an unbounded selection action. Priority checks are confined to rules ‘$\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}$’ and ‘$\&$’, which type an input and a branching prefix, respectively. In both cases, the used endpoint’s priority must be lower than the priorities of the other types in the continuation’s typing context. Rule ‘Rec’ types a recursive definition by introducing a recursion variable to the recursion context whose tuple of types concurs with the contents of the recursive types in the typing context, where contractivity is guaranteed by requiring that the eliminated recursion variable may not appear unguarded in each of the context’s types. At the same time, the recursive types in the context are unfolded, and their priorities are lifted by a common value, denoted $t$ in the rule, that must be greater than the highest priority appearing in the original types (cf. Definition 6). Using a “common lifter”, i.e., lifting the priorities of all types by the same amount is crucial: it maintains the relation between the priorities of the types in the context. Axiom ‘Var’ types a recursive call on a variable in the recursive context. The rule requires that all the types in the context are recursive on the recursion variable called, and that the types inside the recursive definitions concur with the respective types assigned to the recursion varialbe in the recursive context. As mentioned before, the types associated to the introduced and consequently eliminated recursion variable is crucial in ensuring that a recursion is called with endpoints of the same type as required by its definition. The binding of output and selection actions to continuation processes (1) is derivable in APCP. The corresponding typing rules in Figure 4 (bottom) are admissible using ‘Mix’ and ‘Cycle’ (cf. [51]). Figure 4 (bottom) also includes an admissible rule ‘Lift’ that lifts a process’ priorities. The following result assures that, given a type, we can construct a process with an endpoint typable with the given type: ###### Proposition 1. Given a type $A$, there exists a $P$ such that $P\vdash\Omega;x{:}\leavevmode\nobreak\ A$. ###### Proof. We inductively define a function ‘$\mathrm{char}^{x}(A)$’ that, given a type $A$ and an endpoint $x$, constructs a process that performs the behavior described by $A$: $\displaystyle\mathrm{char}^{x}(A\mathbin{\otimes}^{\mathsf{o}}B)$ $\displaystyle:=\overline{x}[y]\cdot(\mathrm{char}^{y}(A)\mathbin{\|}\mathrm{char}^{x}(B))$ $\displaystyle\mathrm{char}^{x}({\oplus}^{\mathsf{o}}\\{i:A_{i}\\}_{i\in I})$ $\displaystyle:=\overline{x}\mathbin{\triangleleft}j\cdot\mathrm{char}^{x}(A_{j})\quad\text{[any $j\in I$]}$ $\displaystyle\mathrm{char}^{x}(A\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{\mathsf{o}}B)$ $\displaystyle:=x(y)\mathbin{.}(\mathrm{char}^{y}(A)\mathbin{\|}\mathrm{char}^{x}(B))$ $\displaystyle\mathrm{char}^{x}(\&^{\mathsf{o}}\\{i:A_{i}\\}_{i\in I})$ $\displaystyle:=x\mathbin{\triangleright}\\{i:\mathrm{char}^{x}(A_{i})\\}_{i\in I}$ $\displaystyle\mathrm{char}^{x}(\bullet)$ $\displaystyle:=\bm{0}\qquad\qquad\mathrm{char}^{x}(\mu X\mathbin{.}A):=\mu X(x)\mathbin{.}\mathrm{char}^{x}(A)\qquad\qquad\mathrm{char}^{x}(X):=X{\langle x\rangle}$ For finite types, we have: $\mathrm{char}^{x}(A)\vdash\emptyset;x{:}\leavevmode\nobreak\ A$. For simplicity, we omit details about recursive types, which require unfolding. For closed, recursive types, we have: $\mathrm{char}^{x}(\mu X\mathbin{.}A)\vdash\emptyset;x{:}\leavevmode\nobreak\ \mu X\mathbin{.}A$. ∎ ##### Type Preservation Well-typed processes satisfy protocol fidelity, communication safety, and deadlock freedom. The first two properties follow directly from _type preservation_ (also known as _subject reduction_), which ensures that reduction preserves typing. In contrast to Caires and Pfenning [14] and Wadler [54], where type preservation corresponds to the elimination of (top-level) applications of rule Cut, in APCP it corresponds to the more general elimination of (top-level) applications of rule Cycle. ###### Theorem 2 (Type Preservation [51]). If $P\vdash\Omega;\Gamma$ and $P\longrightarrow Q$, then $Q\vdash\Omega;{\uparrow^{t}}\Gamma$ for $t\in\mathbb{N}$. ##### Deadlock Freedom The deadlock freedom result for APCP adapts that for PCP [22]. As mentioned before, binding asynchronous outputs and selections to continuations involves additional, low-level uses of Cycle, which we cannot eliminate through process reduction. Therefore, top-level deadlock freedom holds for _live processes_ (Theorem 4). A process is live if it is equivalent to a restriction on _active names_ that perform unguarded actions. This way, e.g., in ‘$x[y,z]$’ the name $x$ is active, but $y$ and $z$ are not. ###### Definition 7 (Active Names). The _set of active names_ of $P$, denoted ‘$\mathrm{an}(P)\mkern-2.0mu$’, contains the (free) names that are used for unguarded actions (output, input, selection, branching): $\displaystyle\mathrm{an}(x[y,z])$ $\displaystyle:=\\{x\\}$ $\displaystyle\mathrm{an}(x(y,z)\mathbin{.}P)$ $\displaystyle:=\\{x\\}$ $\displaystyle\mathrm{an}(\bm{0})$ $\displaystyle:=\emptyset$ $\displaystyle\mathrm{an}(x[z]\mathbin{\triangleleft}j)$ $\displaystyle:=\\{x\\}$ $\displaystyle\mathrm{an}(x(z)\mathbin{\triangleright}\\{i{:}\leavevmode\nobreak\ P_{i}\\}_{i\in I})$ $\displaystyle:=\\{x\\}$ $\displaystyle\mathrm{an}(x\mathbin{\leftrightarrow}y)$ $\displaystyle:=\\{x,y\\}$ $\displaystyle\mathrm{an}(P\mathbin{\|}Q)$ $\displaystyle:=\mathrm{an}(P)\cup\mathrm{an}(Q)$ $\displaystyle\mathrm{an}(\mu X(\tilde{x})\mathbin{.}P)$ $\displaystyle:=\mathrm{an}(P)$ $\displaystyle\mathrm{an}((\bm{\nu}xy)P)$ $\displaystyle:=\mathrm{an}(P)\setminus\\{x,y\\}$ $\displaystyle\mathrm{an}(X{\langle\tilde{x}\rangle})$ $\displaystyle:=\emptyset$ ###### Definition 8 (Live Process). A process $P$ is _live_ , denoted ‘$\mkern 2.0mu\mathrm{live}(P)\mkern-3.0mu$’, if there are names $x,y$ and process $P^{\prime}$ such that $P\equiv(\bm{\nu}xy)P^{\prime}$ with $x,y\in\mathrm{an}(P^{\prime})$. We additionally need to account for recursion: as recursive definitions do not entail reductions, we must fully unfold them before eliminating Cycles. ###### Lemma 3 (Unfolding). If $P\vdash\Omega;\Gamma$, then there is a process $P^{\star}$ such that $P^{\star}\equiv P$ and $P^{\star}$ is not of the form ‘$\mu X(\tilde{z});P^{\prime}\mkern-2.0mu$’ and $P^{\star}\vdash\Omega;\Gamma$. Deadlock freedom, given next, states that typable processes that are live can reduce. It follows from an analysis of the priorities in the typing of the process, which makes it possible to find a pair of non-blocked, parallel, dual actions on connected endpoints, such that a communication can occur. The analysis also considers the possibility that a blocking action is on an endpoint which is not connected (i.e., the endpoint is free), in which case a commuting conversion can be performed. Confer the full proof by Van den Heuvel and Pérez [51, Theorem 5] for more details. ###### Theorem 4 (Deadlock Freedom). If $P\vdash\emptyset;\Gamma$ and $\mathrm{live}(P)$, then there is process $Q$ such that $P\longrightarrow Q$. We now state the deadlock freedom result formalized by Van den Heuvel and Pérez [51]. Following, e.g., Caires and Pfenning [14] and Dardha and Gay [22], it concerns processes typable under empty contexts. This way, the reduction guaranteed by Theorem 4 corresponds to a synchronization ($\beta$-rule), rather than a commuting conversion ($\kappa$-rule). ###### Theorem 5 (Deadlock Freedom for Processes Typable under Empty Contexts [51]). If $P\vdash\emptyset;\emptyset$, then either $P\equiv\bm{0}$ or $P\longrightarrow_{\beta}Q\mkern 2.0mu$ for some $Q$. ##### Fairness Processes typable under empty contexts are not only deadlock free, they are _fair_ : for each endpoint in the process, we can eventually observe a reduction involving that endpoint. To formalize this property, we define _labeled reductions_ , which expose details about a communication: ###### Definition 9 (Labeled Reductions). Consider the labels $\displaystyle\alpha::=x\mathbin{\leftrightarrow}y\leavevmode\nobreak\ \;\mbox{\large{$\mid$}}\;\leavevmode\nobreak\ x\rangle y{:}a\leavevmode\nobreak\ \;\mbox{\large{$\mid$}}\;\leavevmode\nobreak\ x\rangle y{:}\ell\qquad\qquad\text{(forwarding, output/input, selection/branching)}$ where each label has subjects $x$ and $y$. The _labeled reduction_ ‘$\mkern 1.0muP\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>is defined by the following rules: $\displaystyle(\bm{\nu}yz)(x\mathbin{\leftrightarrow}y\mathbin{\|}P)\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x\mathbin{\leftrightarrow}y}}}}P\\{x/z\\}\qquad(\bm{\nu}xy)(x[a,b]\mathbin{\|}y(v,z)\mathbin{.}P)\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x\rangle y{:}a}}}}P\\{a/v,b/z\\}$ $\displaystyle(\bm{\nu}xy)(x[b]\mathbin{\triangleleft}j\mathbin{\|}y(z)\mathbin{\triangleright}\\{i{:}\leavevmode\nobreak\ P_{i}\\}_{i\in I})\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}x\rangle y{:}j}}}}P_{j}\\{b/z\\}\quad\text{(if $j\in I$)}$ $(P\equiv P^{\prime})\wedge(P^{\prime}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha}}}}Q^{\prime})\wedge(Q^{\prime}\equiv Q)$ $P\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha}}}}Q$ $P\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha}}}}Q$ $(\bm{\nu}xy)P\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha}}}}(\bm{\nu}xy)Q$ $P\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha}}}}Q$ $P\mathbin{\|}R\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha}}}}Q\mathbin{\|}R$ ###### Proposition 6. For any $P$ and $P^{\prime}$, $P\longrightarrow_{\beta}P^{\prime}$ if and only if there exists a label $\alpha$ such that $P\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha}}}}P^{\prime}$. ###### Proof. Immediate by definition, for each $\beta$-reduction in Figure 3 (bottom) corresponds to a labeled reduction, and vice versa. ∎ Our fairness result states that processes typable under empty contexts have at least one finite reduction sequence (‘$\longrightarrow^{\star}$’) that enables a labeled reduction involving a _pending_ endpoint—an endpoint that occurs as the subject of an action, and is not bound by input or branching (see below). Clearly, the typed process may have other reduction sequences, not necessarily finite. ###### Definition 10 (Pending Names). Given a process $P$, we define the set of _pending names_ of $P$, denoted ‘$\mkern 1.0mu\mathrm{pn}(P)\mkern-3.0mu$’, as follows: $\displaystyle\mathrm{pn}(x[y,z])$ $\displaystyle:=\\{x\\}$ $\displaystyle\mathrm{pn}(x(y,z).P)$ $\displaystyle:=\\{x\\}\cup(\mathrm{pn}(P)\setminus\\{y,z\\})$ $\displaystyle\mathrm{pn}(\bm{0})$ $\displaystyle:=\emptyset$ $\displaystyle\mathrm{pn}(x[z]\mathbin{\triangleleft}j)$ $\displaystyle:=\\{x\\}$ $\displaystyle\mathrm{pn}(x(z)\mathbin{\triangleright}\\{i:P_{i}\\}_{i\in I})$ $\displaystyle:=\\{x\\}\cup({\mathchoice{\textstyle}{}{}{}\bigcup}_{i\in I}\mathrm{pn}(P_{i})\setminus\\{z\\})$ $\displaystyle\mathrm{pn}(x\mathbin{\leftrightarrow}y)$ $\displaystyle:=\\{x,y\\}$ $\displaystyle\mathrm{pn}(P\mathbin{\|}Q)$ $\displaystyle:=\mathrm{pn}(P)\cup\mathrm{pn}(Q)$ $\displaystyle\mathrm{pn}(\mu X(\tilde{x})\mathbin{.}P)$ $\displaystyle:=\mathrm{pn}(P)$ $\displaystyle\mathrm{pn}((\bm{\nu}xy)P)$ $\displaystyle:=\mathrm{pn}(P)$ $\displaystyle\mathrm{pn}(X{\langle\tilde{x}\rangle})$ $\displaystyle:=\emptyset$ ###### Theorem 7 (Fairness). Suppose given a process $P\vdash\emptyset;\emptyset$. Then, for every $x\in\mathrm{pn}(P)$ there exists a process $P^{\prime}$ such that $P\longrightarrow^{\star}P^{\prime}$ and $P^{\prime}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha}}}}\,Q$, for some process $Q$ and label $\alpha$ with subject $x$. ###### Proof. Take any $x\in\mathrm{pn}(P)$. Because $P$ is typable under empty contexts, $x$ is bound to some $y\in\mathrm{pn}(P)$ by restriction. By typing, in $P$ there is exactly one action on $x$ and one action on $y$ (they may also appear in forwarder processes). Following the restrictions on priorities in the typing of $x$ and $y$ in $P$, the actions on $x$ and $y$ cannot appear sequentially in $P$ (cf. the proof by Van den Heuvel and Pérez [51] for details on this reasoning). By typability, the action on $y$ is dual to the action on $x$. We apply induction on the number of inputs, branches, and recursive definitions in $P$ blocking the actions on $x$ and $y$, denoted $n$ and $m$, respectively. Because $P$ is typable under empty contexts, the blocking inputs and branches that are on names in $\mathrm{pn}(P)$ also have to be bound to pending names by restriction. The actions on these connected names may also be prefixed by inputs, branches, and recursive definitions, so we may need to unblock those actions as well. Since there can only be a finite number of names in any given process, we also apply induction on the number of prefixes blocking these connected actions. * • If $n=0$ and $m=0$, then the actions on $x$ and $y$ occur at the top-level; because they do not appear sequentially, the communication between $x$ and $y$ can take place immediately. Hence, $P\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha}}}}Q$ where $x$ and $y$ are the subjects of $\alpha$. This proves the thesis, with $P^{\prime}=P$. * • If $n>0$ or $m>0$, the analysis depends on the foremost prefix of the actions on $x$ and $y$. If the foremost prefix of either action is a recursive definition (‘$\mu X(\tilde{y})$’), we unfold the recursion. Because a corresponding recursive call (‘$X{\langle\tilde{z}\rangle}$’) cannot occur as a prefix, the effect of unfolding either (i) triggers actions that occur in parallel to those on $x$ and $y$, or (ii) the actions on $x$ or $y$ prefix the unfolded recursive call. In either case, the number of prefixes decreases, and the thesis follows from the IH. Otherwise, if neither foremost prefix is a recursive definition, then the foremost prefixes must be actions on names in $\mathrm{pn}(P)$. Consider the action that is typable with the least priority. W.l.o.g. assume that this is the foremost prefix of $x$. Suppose this action is on some endpoint $w$ connected to another endpoint $z\in\mathrm{pn}(P)$ by restriction. By typability, the priority of $w$ is less than that of $x$ and all of the prefixes in between. This means that the number of prefixes blocking the action on $z$ strictly decreases. Hence, by the IH, $P\longrightarrow^{\star}P^{\prime\prime}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha^{\prime}}}}}\,Q^{\prime}$ in a finite number of steps, where $w$ and $z$ are the subjects of $\alpha^{\prime}$. The communication between $w$ and $z$ can be performed, and $n$ decreases. By Type Preservation (Theorem 2), $Q^{\prime}\vdash\emptyset;\emptyset$. The thesis then follows from the IH: $P\longrightarrow^{\star}P^{\prime\prime}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha^{\prime}}}}}\,Q^{\prime}\longrightarrow^{\star}P^{\prime}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha}}}}\,Q$ in finite steps, where $x$ and $y$ are the subjects of $\alpha$. ∎ ### Examples To illustrate APCP processes and their session types, we give implementations of the three participants in $G_{\mathsf{auth}}$ in Section 1. ###### Example 1. Processes $P$, $Q$, and $R$ are typed implementations for participants $c$, $s$, and $a$, respectively, where each process uses a single channel to perform the actions described by $G_{\mathsf{auth}}$. $\displaystyle P$ $\displaystyle:=\mu X({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}})\mathbin{.}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}\mathbin{\triangleright}\left\\{\begin{array}[]{ll}\mathsf{login}{:}&{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}(u)\mathbin{.}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}}\mathbin{\triangleleft}\mathsf{passwd}\cdot\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}}[\bm{logmein345}]\cdot X{\langle{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}\rangle},\\\ \mathsf{quit}{:}&{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}(w)\mathbin{.}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}}\mathbin{\triangleleft}\mathsf{quit}\cdot\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}}[z]\cdot\bm{0}\end{array}\right\\}$ $\displaystyle\vdash{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}{:}\leavevmode\nobreak\ \mu X\mathbin{.}\&^{2}\left\\{\begin{array}[]{ll}\mathsf{login}{:}&\bullet\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{3}{\oplus}^{4}\\{\mathsf{passwd}{:}\leavevmode\nobreak\ \bullet\mathbin{\otimes}^{5}X\\},\\\ \mathsf{quit}{:}&\bullet\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{3}{\oplus}^{4}\\{\mathsf{quit}{:}\leavevmode\nobreak\ \bullet\mathbin{\otimes}^{5}\bullet\\}\end{array}\right\\}$ $\displaystyle Q$ $\displaystyle:=\mu X({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}})\mathbin{.}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}}\mathbin{\triangleleft}\mathsf{login}\cdot\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}}[u]\cdot{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}\mathbin{\triangleright}\\{\mathsf{auth}{:}\leavevmode\nobreak\ {{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}(v)\mathbin{.}X{\langle{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}\rangle}\\}$ $\displaystyle\vdash{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}{:}\leavevmode\nobreak\ \mu X\mathbin{.}{\oplus}^{0}\\{\mathsf{login}{:}\leavevmode\nobreak\ \bullet\mathbin{\otimes}^{1}\&^{10}\\{\mathsf{auth}{:}\leavevmode\nobreak\ \bullet\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{11}X\\},\mathsf{quit}{:}\leavevmode\nobreak\ \bullet\mathbin{\otimes}^{1}\bullet\\}$ $\displaystyle R$ $\displaystyle:=\mu X({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}})\mathbin{.}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}\mathbin{\triangleright}\left\\{\begin{array}[]{ll}\mathsf{login}{:}&{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}\mathbin{\triangleright}\\{\mathsf{passwd}{:}\leavevmode\nobreak\ {{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}(u)\mathbin{.}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}}\mathbin{\triangleleft}\mathsf{auth}\cdot\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}}[v]\cdot X{\langle{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}\rangle}\\},\\\ \mathsf{quit}{:}&{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}\mathbin{\triangleright}\\{\mathsf{quit}{:}\leavevmode\nobreak\ {{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}(w)\mathbin{.}\bm{0}\\}\end{array}\right\\}$ $\displaystyle\vdash{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}{:}\leavevmode\nobreak\ \mu X\mathbin{.}\&^{2}\left\\{\begin{array}[]{ll}\mathsf{login}{:}&\&^{6}\\{\mathsf{passwd}{:}\leavevmode\nobreak\ \bullet\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{7}{\oplus}^{8}\\{\mathsf{auth}{:}\leavevmode\nobreak\ \bullet\mathbin{\otimes}^{9}X\\}\\},\\\ \mathsf{quit}{:}&\&^{6}\\{\mathsf{quit}{:}\leavevmode\nobreak\ \bullet\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{7}\bullet\\}\end{array}\right\\}$ Process $P$ is a specific implementation for $c$, where we use ‘$\bm{logmein345}$’ to denote a closed channel endpoint representing the password string “logmein345”. Similarly, $Q$ is a specific implementation for $s$ that continuously chooses the login branch. Note that the processes above cannot be directly connected to each other to implement $G_{\mathsf{auth}}$. Our goal is to enable the composition of (typed) implementations such as $P$, $Q$, and $R$ in a correct and deadlock free manner. We shall proceed as follows. After setting up the routers that enable the composition of these processes according to $G_{\mathsf{auth}}$ (Section 4), we will return to this example in Section 5. At that point, it will become clear that the priorities in the types of $P$, $Q$, and $R$ were chosen to ensure the correct composition with their respective routers. ## 3 Global Types and Relative Projection We analyze multiparty protocols specified as _global types_. We consider a standard syntax, with session delegation and recursion, subsuming the one given in the seminal paper by Honda _et al._ [36]. In the following, we write $p,q,r,s,\ldots$ to denote _(protocol) participants_. ###### Definition 11 (Types). _Global types_ $G$ and _message types_ $S,T$ are defined as: $\displaystyle G$ $\displaystyle::=p\mathbin{\twoheadrightarrow}q\\{i\langle S\rangle\mathbin{.}G\\}_{i\in I}\;\mbox{\large{$\mid$}}\;\mu X\mathbin{.}G\;\mbox{\large{$\mid$}}\;X\;\mbox{\large{$\mid$}}\;\bullet\;\mbox{\large{$\mid$}}\;\mathsf{skip}\mathbin{.}G$ $\displaystyle S,T$ $\displaystyle::={!}T\mathbin{.}S\;\mbox{\large{$\mid$}}\;{?}T\mathbin{.}S\;\mbox{\large{$\mid$}}\;{{\oplus}}\\{i{:}\leavevmode\nobreak\ S\\}_{i\in I}\;\mbox{\large{$\mid$}}\;\&\\{i{:}\leavevmode\nobreak\ S\\}_{i\in I}\;\mbox{\large{$\mid$}}\;\bullet$ We include basic types (e.g., unit, bool, int), which are all syntactic sugar for $\bullet$. The type ‘$p\mathbin{\twoheadrightarrow}q\\{i\langle S_{i}\rangle\mathbin{.}G_{i}\\}_{i\in I}$’ specifies a direct exchange from participant $p$ to participant $q$, which precedes protocol $G_{i}$: $p$ chooses a label $i\in I$ and sends it to $q$ along with a message of type $S_{i}$. Message exchange is _asynchronous_ : the protocol can continue as $G_{i}$ before the message has been received by $q$. The type ‘$\mu X\mathbin{.}G$’ defines a recursive protocol: whenever a path of exchanges in $G$ reaches the recursion variable $X$, the protocol continues as ‘$\mu X\mathbin{.}G$’. The type ‘$\bullet$’ denotes the completed protocol. For technical convenience, we introduce the construct ‘$\mathsf{skip}\mathbin{.}G$’, which denotes an unobservable step that precedes $G$. Recursive definitions bind recursion variables, so recursion variables not bound by a recursive definition are free. We write ‘$\mathrm{frv}(G)$’ to denote the set of free recursion variables of $G$, and say $G$ is _closed_ if $\mathrm{frv}(G)=\emptyset$. Recursion in global types is tail-recursive and _contractive_ (i.e. they contain no subexpressions of the form ‘$\mu X_{1}\ldots\mu X_{n}\mathbin{.}X_{1}$’). As for the session types in Section 2, we define the unfolding of a recursive global type by substituting copies of the recursive definition for recursive calls, i.e. ‘$\mu X\mathbin{.}G$’ unfolds to ‘$G\\{\mu X\mathbin{.}G/X\\}$’. In approaches based on MPST, the grammar of global types specifies multiparty protocols but does not ensure their correct implementability; such guarantees are given in terms of _well-formedness_ , defined as projectability onto all participants (cf. § 3.2). Message types $S,T$ define binary protocols, not to be confused with the types in § 2. Type ‘${!}T\mathbin{.}S$’ (resp. ‘${?}T\mathbin{.}S$’) denotes the output (resp. input) of a message of type $T$ followed by the continuation $S$. Type ‘${{\oplus}}\\{i{:}\leavevmode\nobreak\ S_{i}\\}_{i\in I}$’ denotes _selection_ : the output of choice for a label $i\in I$ followed by the continuation $S_{i}$. Type ‘$\&\\{i{:}\leavevmode\nobreak\ S_{i}\\}_{i\in I}$’ denotes _branching_ : the input of a label $i\in I$ followed by the continuation $S_{i}$. Type ‘$\bullet$’ denotes the end of the protocol. Note that, due to the tail-recursiveness of session and global types, there are no recursive message types. It is useful to obtain the set of participants of a global type: ###### Definition 12 (Participants). We define the _set of participants_ of global type $G$, denoted ‘$\mkern 1.0mu\mathsf{prt}(G)\mkern-2.0mu$’: $\displaystyle\mathsf{prt}(p\mathbin{\twoheadrightarrow}q\\{i\langle S_{i}\rangle\mathbin{.}G_{i}\\}_{i\in I})$ $\displaystyle:=\\{p,q\\}\cup({\mathchoice{\textstyle}{}{}{}\bigcup}_{i\in I}\leavevmode\nobreak\ \mathsf{prt}(G_{i}))$ $\displaystyle\mathsf{prt}(\mathsf{skip}\mathbin{.}G)$ $\displaystyle:=\mathsf{prt}(G)$ $\displaystyle\mathsf{prt}(\bullet)$ $\displaystyle:=\emptyset$ $\displaystyle\mathsf{prt}(\mu X\mathbin{.}G)$ $\displaystyle:=\mathsf{prt}(G)$ $\displaystyle\mathsf{prt}(X)$ $\displaystyle:=\emptyset$ ### 3.1 Relative Types While a global type such as $G_{\mathsf{auth}}$ (1) describes a protocol from a vantage point, we introduce _relative types_ that describe the interactions between _pairs_ of participants. This way, relative types capture the peer-to- peer nature of multiparty protocols. We develop projection from global types onto relative types (cf. § 3.2) and use it to establish a new class of _well- formed_ global types. A choice between participants in a global type is _non-local_ if it influences future exchanges between other participants. Our approach uses _dependencies_ to expose these non-local choices in the relative types of these other participants. Relative types express interactions between two participants. Because we obtain a relative type through projection of a global type, we know which participants are involved. Therefore, a relative type only mentions the sender of each exchange; we implicitly know that the recipient is the other participant. ###### Definition 13 (Relative Types). _Relative types_ $R$ are defined as follows, where the $S_{i}$ are message types (cf. Def. 11): $R::=p\\{i\langle S_{i}\rangle\mathbin{.}R\\}_{i\in I}\;\mbox{\large{$\mid$}}\;p{?}r\\{i\mathbin{.}R\\}_{i\in I}\;\mbox{\large{$\mid$}}\;p{!}r\\{i\mathbin{.}R\\}_{i\in I}\;\mbox{\large{$\mid$}}\;\mu X\mathbin{.}R\;\mbox{\large{$\mid$}}\;X\;\mbox{\large{$\mid$}}\;\bullet\;\mbox{\large{$\mid$}}\;\mathsf{skip}\mathbin{.}R$ We detail the syntax above, given participants $p$ and $q$. * • Type ‘$p\\{i\langle S_{i}\rangle\mathbin{.}R_{i}\\}_{i\in I}$’ specifies that $p$ must choose a label $i\in I$ and send it to $q$ along with a message of type $S_{i}$ after which the protocol continues with $R_{i}$. * • Given an $r$ which is _not_ involved in the relative type (i.e., $p\neq r,q\neq r$), type ‘$p{?}r\\{i\mathbin{.}R_{i}\\}_{i\in I}$’ expresses a dependency: a non-local choice between $p$ and $r$ which influences the protocol between $p$ and $q$. Here, the dependency indicates that after $p$ receives from $r$ the chosen label, $p$ must forward it to $q$, determining the protocol between $p$ and $q$. * • Similarly, type ‘$p{!}r\\{i\mathbin{.}R_{i}\\}_{i\in I}$’ expresses a dependency, which indicates that after $p$ sends to $r$ the chosen label, $p$ must forward it to $q$. * • Types ‘$\mu X\mathbin{.}R$’ and ‘$X$’ define recursion, just as their global counterparts. * • The type ‘$\bullet$’ specifies the end of the protocol between $p$ and $q$. * • The type ‘$\mathsf{skip}\mathbin{.}R$’ denotes an unobservable step that precedes $R$. ###### Definition 14 (Participants of Relative Types). We define the _set of participants_ of relative type $R$, denoted ‘$\mkern 1.0mu\mathsf{prt}(R)\mkern-2.0mu$’: $\displaystyle\mathsf{prt}(p\\{i\langle S_{i}\rangle\mathbin{.}R_{i}\\}_{i\in I})$ $\displaystyle:=\\{p\\}\cup({\mathchoice{\textstyle}{}{}{}\bigcup}_{i\in I}\leavevmode\nobreak\ \mathsf{prt}(R_{i}))$ $\displaystyle\mathsf{prt}(\mathsf{skip}\mathbin{.}R)$ $\displaystyle:=\mathsf{prt}(R)$ $\displaystyle\mathsf{prt}(\bullet)$ $\displaystyle:=\emptyset$ $\displaystyle\mathsf{prt}(p{?}r\\{i\mathbin{.}R_{i}\\}_{i\in I})$ $\displaystyle:=\\{p\\}\cup({\mathchoice{\textstyle}{}{}{}\bigcup}_{i\in I}\leavevmode\nobreak\ \mathsf{prt}(R_{i}))$ $\displaystyle\mathsf{prt}(\mu X\mathbin{.}R)$ $\displaystyle:=\mathsf{prt}(R)$ $\displaystyle\mathsf{prt}(X)$ $\displaystyle:=\emptyset$ $\displaystyle\mathsf{prt}(p{!}r\\{i\mathbin{.}R_{i}\\}_{i\in I})$ $\displaystyle:=\\{p\\}\cup({\mathchoice{\textstyle}{}{}{}\bigcup}_{i\in I}\leavevmode\nobreak\ \mathsf{prt}(R_{i}))$ We introduce some useful notation: ###### Notation 3. * • We write ‘$\mkern 1.0mup\mathbin{\twoheadrightarrow}q{:}i\langle S\rangle\mathbin{.}G\mkern-3.0mu$’ for a global type with a single branch ‘$\mkern 1.0mup\mathbin{\twoheadrightarrow}q\\{i\langle S\rangle\mathbin{.}G\\}\mkern-3.0mu$ (and similarly for exchanges and dependencies in relative types). * • We omit ‘unit’ message types from global and relative types, writing ‘$\mkern 1.0mui\mathbin{.}G\mkern-2.0mu$’ for ‘$\mkern 1.0mui\langle\mathsf{unit}\rangle\mathbin{.}G\mkern-2.0mu$’. * • Given $k>1$, we write ‘$\mkern 1.0mu\mathsf{skip}^{k}\mkern-3.0mu$’ for a sequence of $k$ $\mathsf{skip}$s. ### 3.2 Relative Projection and Well-Formedness $\displaystyle\mathrm{ddep}((p,q),s\mathbin{\twoheadrightarrow}r\\{i\langle S_{i}\rangle\mathbin{.}G_{i}\\}_{i\in I}):=\begin{cases}\mathsf{skip}\mathbin{.}(G_{i^{\prime}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q))\leavevmode\nobreak\ \text{[any $i^{\prime}\in I$]}&\text{if $\forall i,j.$}\\\ &\text{$G_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)=G_{j}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)$}\\\ p{!}r\\{i\mathbin{.}(G_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q))\\}_{i\in I}&\text{if $p=s$}\\\ q{!}r\\{i\mathbin{.}(G_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q))\\}_{i\in I}&\text{if $q=s$}\\\ p{?}s\\{i\mathbin{.}(G_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q))\\}_{i\in I}&\text{if $p=r$}\\\ q{?}s\\{i\mathbin{.}(G_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q))\\}_{i\in I}&\text{if $q=r$}\end{cases}$ . $\displaystyle(s\mathbin{\twoheadrightarrow}r\\{i\langle S_{i}\rangle\mathbin{.}G_{i}\\}_{i\in I})\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)$ $\displaystyle:=\begin{cases}p\\{i\langle S_{i}\rangle\mathbin{.}(G_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q))\\}_{i\in I}&\text{if $p=s$ and $q=r$}\\\ q\\{i\langle S_{i}\rangle\mathbin{.}(G_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q))\\}_{i\in I}&\text{if $q=s$ and $p=r$}\\\ \mathrm{ddep}((p,q),s\mathbin{\twoheadrightarrow}r\\{i\langle S_{i}\rangle\mathbin{.}G_{i}\\}_{i\in I})&\text{otherwise}\end{cases}$ $\displaystyle(\mu X\mathbin{.}G)\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)$ $\displaystyle:=\begin{cases}\mu X\mathbin{.}(G\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q))&\text{if $G\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)$ defined and contractive on $X$}\\\ \bullet&\text{otherwise}\end{cases}$ $\displaystyle X\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q):=X\qquad\bullet\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q):=\bullet\qquad(\mathsf{skip}\mathbin{.}G)\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q):=\mathsf{skip}\mathbin{.}(G\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q))$ Above, ‘$\mathsf{skip}^{\ast}$’ denotes a sequence of zero or more $\mathsf{skip}$. Figure 5: Dependency Detection (top), and Relative Projection (bottom, cf. Definition 16). When a side-condition does not hold, either is undefined. We define _relative projection_ for global types. We want relative projection to fail when it would return a non-contractive recursive type. To this end, we define a notion of contractiveness on relative types: ###### Definition 15 (Contractive Relative Types). Given a relative type $R$ and a recursion variable $X$, we say _$R$ is contractive on $X$_ if either of the following holds: * • $R$ contains an exchange, or * • $R$ ends in a recursive call on a variable other than $X$. Relative projection then relies on the contractiveness of relative types. It also relies on an auxiliary function to determine if a dependency message is needed and possible. ###### Definition 16 (Relative Projection). Given a global type $G$, we define its relative projection onto a pair of participants $p$ and $q$, denoted ‘ $G\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)$’, by induction on the structure of $G$ as given in Figure 5 (bottom), using the auxiliary function $\mathrm{ddep}$ (cf. Figure 5, top). We discuss how Definition 16 projects global types onto a pair of participants $(p,q)$, as per Figure 5 (bottom). The most interesting case is the projection of a direct exchange ‘$s\mathbin{\twoheadrightarrow}r\\{i\langle S_{i}\rangle\mathbin{.}G_{i}\\}_{i\in I}$’. When the exchange involves both $p$ and $q$, the projection yields an exchange between $p$ and $q$ with the appropriate sender. Otherwise, the projection relies on the function ‘$\mathrm{ddep}$’ in Figure 5 (top), which determines whether the exchange is a non-local choice for $p$ and $q$ and yields an appropriate projection accordingly: * • If the projections of all branches are equal, the exchange is not a non-local choice and $\mathrm{ddep}$ yields the unobservable step ‘$\mathsf{skip}$’ followed by the projection of any branch. * • If there are branches with different projections, the exchange is a non-local choice, so $\mathrm{ddep}$ yields a dependency if possible. If $p$ or $q$ is involved in the exchange, $\mathrm{ddep}$ yields an appropriate dependency (e.g., ‘$p{!}r$’ if $p$ is the sender, or ‘$q{?}s$’ if $q$ is the recipient). If neither $p$ nor $q$ are involved, then $\mathrm{ddep}$ cannot yield a dependency and projection is thus undefined. The projection of ‘$\mu X\mathbin{.}G^{\prime}$’ considers the projection of the body ‘$G^{\prime}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)$’ to see whether $p$ and $q$ interact in $G^{\prime}$. If $G^{\prime}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)$ is a (possibly empty) sequence of $\mathsf{skip}$s followed by $\bullet$ or $X$, then $p$ and $q$ do not interact and the projection yields $\bullet$. Otherwise, $p$ and $q$ do interact and projection preserves the recursive definition. Note that Definition 15 (contractiveness) is key here: e.g., $G^{\prime}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)=\mathsf{skip}\mathbin{.}\mu Y\mathbin{.}\mathsf{skip}\mathbin{.}X$ is not contractive on $X$, so $(\mu X\mathbin{.}G^{\prime})\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)=\bullet$. The projection of a recursive call ‘$X$’ is simply ‘$X$’. The projection of ‘$G_{1}\mathbin{\|}G_{2}$’ is standard [35]: it ensures that $G_{1}$ and $G_{2}$ do not share participants and only continues with either global type if both $p$ and $q$ are participants. The projections of ‘$\bullet$’ and ‘$\mathsf{skip}$’ are homomorphic. ###### Example 2 (Projections of $G_{\mathsf{auth}}$). To demonstrate relative projection, let us consider again $G_{\mathsf{auth}}$: $\displaystyle G_{\mathsf{auth}}=\mu X\mathbin{.}s\mathbin{\twoheadrightarrow}c\left\\{\begin{array}[]{@{}l@{}}\mathsf{login}\mathbin{.}c\mathbin{\twoheadrightarrow}a{:}\mathsf{passwd}\langle\mathsf{str}\rangle\mathbin{.}a\mathbin{\twoheadrightarrow}s{:}\mathsf{auth}\langle\mathsf{bool}\rangle\mathbin{.}X,\\\ \mathsf{quit}\mathbin{.}c\mathbin{\twoheadrightarrow}a{:}\mathsf{quit}\mathbin{.}\bullet\end{array}\right\\}$ The relative projection onto $(s,c)$ is straightforward, as there are no non- local choices to consider: $\displaystyle G_{\mathsf{auth}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(s,c)=\mu X\mathbin{.}s\left\\{\begin{array}[]{@{}l@{}}\mathsf{login}\mathbin{.}\mathsf{skip}^{2}\mathbin{.}X,\\\ \mathsf{quit}\mathbin{.}\mathsf{skip}\mathbin{.}\bullet\end{array}\right\\}$ However, compare the projection of the initial login branch onto $(s,a)$ and $(c,a)$ with the projection of the quit branch: they are different. Therefore, the initial exchange between $s$ and $c$ is a non-local choice in the protocols relative to $(s,a)$ and $(c,a)$. Since $s$ is involved in this exchange, the non-local choice is detected by ‘$\mkern 1.0mu\mathrm{ddep}\mkern-3.0mu$’: $\displaystyle\mathrm{ddep}((s,a),s\mathbin{\twoheadrightarrow}c\\{\mathsf{login}\ldots,\quad\mathsf{quit}\ldots\\})=s{!}c\\{\mathsf{login}\ldots,\quad\mathsf{quit}\ldots\\}$ Hence, this non-local choice can be included in the relative projection onto $(s,a)$ as a dependency: $\displaystyle G_{\mathsf{auth}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(s,a)=\mu X\mathbin{.}s{!}c\left\\{\begin{array}[]{@{}l@{}}\mathsf{login}\mathbin{.}\mathsf{skip}\mathbin{.}a{:}\mathsf{auth}\langle\mathsf{bool}\rangle\mathbin{.}X,\\\ \mathsf{quit}\mathbin{.}\mathsf{skip}\mathbin{.}\bullet\end{array}\right\\}$ Similarly, $c$ is involved in the initial exchange, so the non-local choice can also be included in the relative projection onto $(c,a)$ as a dependency: $\displaystyle G_{\mathsf{auth}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(c,a)=\mu X\mathbin{.}c{?}s\left\\{\begin{array}[]{@{}l@{}}\mathsf{login}\mathbin{.}c{:}\mathsf{passwd}\langle\mathsf{str}\rangle\mathbin{.}\mathsf{skip}\mathbin{.}X,\\\ \mathsf{quit}\mathbin{.}c{:}\mathsf{quit}\langle\mathsf{unit}\rangle\mathbin{.}\bullet\end{array}\right\\}$ Since relative types are relative to pairs of participants, the input order of participants for projection does not matter: ###### Proposition 8. Suppose a global type $G$ and distinct participants $p,q\in\mathsf{prt}(G)$. * • If $G\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)$ is defined, then $G\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)=G\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(q,p)$ and $\mathsf{prt}(G\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q))\subseteq\\{p,q\\}$; * • $G\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)$ is undefined if and only if $G\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(q,p)$ is undefined. ##### Well-formed Global Types We may now define _well-formedness_ for global types. Unlike usual MPST approaches, our definition relies exclusively on (relative) projection (Def. 16), and does not appeal to external notions such as merge and subtyping [37, 55]. ###### Definition 17 (Relative Well-Formedness). A global type $G$ is _relative well-formed_ if, for every distinct $p,q\in\mathsf{prt}(G)$, the projection $G\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)$ is defined. The following contrasts our new notion of relative well-formedness with notions of well-formedness based on the usual notion of local types [35, 26]. ###### Example 3. Consider the following global type involving participants $p,q,r,s$: $G_{3}:=p\mathbin{\twoheadrightarrow}q\left\\{\begin{array}[]{l}1\langle S_{a}\rangle\mathbin{.}p\mathbin{\twoheadrightarrow}r{:}1\langle S_{b}\rangle\mathbin{.}p\mathbin{\twoheadrightarrow}s{:}1\langle S_{c}\rangle\mathbin{.}q\mathbin{\twoheadrightarrow}r{:}1\langle S_{d}\rangle\mathbin{.}q\mathbin{\twoheadrightarrow}s{:}1\langle S_{e}\rangle\mathbin{.}\bullet,\\\ 2\langle S_{f}\rangle\mathbin{.}r\mathbin{\twoheadrightarrow}p{:}2\langle S_{g}\rangle\mathbin{.}s\mathbin{\twoheadrightarrow}p{:}2\langle S_{h}\rangle\mathbin{.}r\mathbin{\twoheadrightarrow}q{:}2\langle S_{i}\rangle\mathbin{.}s\mathbin{\twoheadrightarrow}q{:}2\langle S_{j}\rangle\mathbin{.}\bullet\end{array}\right\\}$ The initial exchange between $p$ and $q$ is a non-local choice influencing the protocols between other pairs of participants. Well-formedness as in [35, 26] forbids non-local choices. In contrast, $G_{3}$ is relative well-formed: $p$ and $q$ must both forward the selected label to both $r$ and $s$. The dependencies in the following relative projections express precisely this: $\displaystyle G_{3}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,r)$ $\displaystyle=p{!}q\\{1\mathbin{.}p{:}1\langle S_{b}\rangle\mathbin{.}\mathsf{skip}^{3}\mathbin{.}\bullet,\quad 2\mathbin{.}r{:}2\langle S_{g}\rangle\mathbin{.}\mathsf{skip}^{3}\mathbin{.}\bullet\\}$ $\displaystyle G_{3}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,s)$ $\displaystyle=p{!}q\\{1\mathbin{.}\mathsf{skip}\mathbin{.}p{:}1\langle S_{c}\rangle\mathbin{.}\mathsf{skip}^{2}\mathbin{.}\bullet,\quad 2\mathbin{.}\mathsf{skip}\mathbin{.}s{:}2\langle S_{h}\rangle\mathbin{.}\mathsf{skip}^{2}\mathbin{.}\bullet\\}$ $\displaystyle G_{3}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(q,r)$ $\displaystyle=q{?}p\\{1\mathbin{.}\mathsf{skip}^{2}\mathbin{.}q{:}1\langle S_{d}\rangle\mathbin{.}\mathsf{skip}\mathbin{.}\bullet,\quad 2\mathbin{.}\mathsf{skip}^{2}\mathbin{.}r{:}2\langle S_{i}\rangle\mathbin{.}\mathsf{skip}\mathbin{.}\bullet\\}$ $\displaystyle G_{3}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(q,s)$ $\displaystyle=q{?}p\\{1\mathbin{.}\mathsf{skip}^{3}\mathbin{.}q{:}1\langle S_{e}\rangle\mathbin{.}\bullet,\quad 2\mathbin{.}\mathsf{skip}^{3}\mathbin{.}s{:}2\langle S_{j}\rangle\mathbin{.}\bullet\\}$ Dependencies in relative types follow the non-local choices in the given global type: by implementing such choices, dependencies ensure correct projectability. They induce additional messages, but in our view this is an acceptable price to pay for an expressive notion of well-formedness based only on projection. It is easy to see that in a global type with $n$ participants, the number of messages per communication is $\mathcal{O}(n)$—an upper-bound following from the worst-case scenario in which both sender and recipient have to forward a label to $n-2$ participants due to dependencies, as in the example above. However, in practice, sender and recipient will rarely both have to forward labels, let alone both to all participants. ## 4 Analyzing Global Types using Routers $P$${\llbracket G_{\mathsf{auth}}\rrbracket}_{c}^{\\{s,a\\}}$${}\in\mathrm{ri}(G_{\mathsf{auth}},\\{c\\})$${\llbracket G_{\mathsf{auth}}\rrbracket}_{s}^{\\{a,s\\}}$$Q$${}\in\mathrm{ri}(G_{\mathsf{auth}},\\{s\\})$${\llbracket G_{\mathsf{auth}}\rrbracket}_{a}^{\\{s,c\\}}$$R$${}\in\mathrm{ri}(G_{\mathsf{auth}},\\{a\\})$ $P$${\llbracket G_{\mathsf{auth}}\rrbracket}_{c}^{\\{s,a\\}}$${}\in\mathrm{ri}(G_{\mathsf{auth}},\\{c\\})$${\llbracket G_{\mathsf{auth}}\rrbracket}_{s}^{\\{a,s\\}}$${\llbracket G_{\mathsf{auth}}\rrbracket}_{a}^{\\{s,c\\}}$$S$${}\in\mathrm{ri}(G_{\mathsf{auth}},\\{s,a\\})$ Figure 6: Two different networks of routed implementations for $G_{\mathsf{auth}}$ (1), without interleaving (left) and with interleaving (right). For participants $p$ and $\tilde{q}$, Definition 19 gives the router process $\mkern 1.0mu{\llbracket G\rrbracket}_{p}^{\tilde{q}}\mkern-3.0mu$ and Definition 24 gives the set $\mathrm{ri}(G,\tilde{q})$. Lines indicate channels and boxes are local compositions of processes. In this section, we develop our decentralized analysis of multiparty protocols (§ 3) using relative types (§ 3.1) and APCP (§ 2). The intended setup is as follows. Each participant’s role in a global type $G$ is implemented by a process, which is connected to a _router_ : a process that orchestrates the participant’s interactions in $G$. The resulting _routed implementations_ (Def. 24) can then directly connect to each other to form a decentralized _network of routed implementations_ that implements $G$. This way we realize the scenario sketched in Figure 1 (left), which is featured in more detail in Figure 6 (left). Key in our analysis is the _synthesis_ of a participant’s router from a global type (§ 4.1). To assert well-typedness—and thus deadlock freedom—of networks of routed implementations (Theorem 11), we extract binary session types from the global type and its associated relative types (§ 4.2): * • from the global type we extract types for channels between implementations and routers; * • from the relative types we extract types for channels between pairs of routers. After defining routers and showing their typability, we set up networks of routed implementations of global types (§ 4.3). To enable the transference of deadlock freedom APCP to multiparty protocols, we then establish an operational correspondence between global types and networks of routed implementations (Theorems 19 and 23). Finally, to show that our routed approach strictly generalizes the prior centralized analyses [12, 16], we define an orchestrated analysis of global types and show that it is behaviorally equivalent to a centralized composition of routers (§ 4.4). In the following section (§ 5), we will show routers in action. ### 4.1 Synthesis of Routers We synthesize routers by decomposing each exchange in the global type into four sub-steps, which we motivate by considering the initial exchange from $s$ to $c$ in $G_{\mathsf{auth}}$ 1: $s\mathbin{\twoheadrightarrow}c\\{\mathsf{login}\ldots,\quad\mathsf{quit}\ldots\\}$. As explained in Example 2, this exchange induces a dependency in the relative projections of $G_{\mathsf{auth}}$ onto $(s,a)$ and $(c,a)$. We decompose this initial exchange as follows, where $P$, $Q$, and $R$ are the implementations of $c$, $s$, and $a$, respectively (given in Example 1) and $\mathcal{R}_{x}$ stands for the router of each $x\in\\{s,c,a\\}$. Below, multiple actions in one step happen concurrently: 1. 1. $Q$ sends $\ell\in\\{\mathsf{login},\mathsf{quit}\\}$ to $\mathcal{R}_{s}$. 2. 2. $\mathcal{R}_{s}$ sends $\ell$ to $\mathcal{R}_{c}$ (recipient) and $\mathcal{R}_{a}$ (output dependency). $Q$ sends unit value $v$ to $\mathcal{R}_{s}$. 3. 3. $\mathcal{R}_{c}$ sends $\ell$ to $P$ and $\mathcal{R}_{a}$ (input dependency). $\mathcal{R}_{s}$ forwards $v$ to $\mathcal{R}_{c}$. 4. 4. $\mathcal{R}_{c}$ forwards $v$ to $P$. $\mathcal{R}_{a}$ sends $\ell$ to $R$. In Section 4.2, we follow this decomposition to assign to each consecutive step a consecutive priority: this ensures the consistency of priority checks required to establish the deadlock freedom of networks of routed implementations. We define router synthesis by means of an algorithm that returns a _router process_ for a given global type and participant. More precisely: given $G$, a participant $p$, and $\tilde{q}=\mathsf{prt}(G)\setminus\\{p\\}$, the algorithm generates a process, denoted ‘${\llbracket G\rrbracket}_{p}^{\tilde{q}}$’, which connects with a process implementing $p$’s role in $G$ on channel ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$; we shall write such channels in pink. This router for $p$ connects with the routers of the other participants in $G$ ($q_{i}\in\tilde{q}$) on channels ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q_{1}}},\ldots,{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q_{n}}}$; we shall write such channels in purple. (This convention explains the colors of the lines in Figure 6.) The router synthesis algorithm relies on relative projection to detect non- local choices; this way, the router can synchronize with the participant’s implementation and with other routers appropriately. To this end, we define the predicate ‘$\mathrm{hdep}$’, which is true for an exchange and a pair of participants if the exchange induces a dependency for either participant. Recall that relative projection produces a ‘$\mathsf{skip}$’ when an exchange is not non-local (cf. Figure 5). Thus, ‘$\mathrm{hdep}$’ only holds true if relative projection does not produce a ‘$\mathsf{skip}$’. ###### Definition 18. The predicate ‘$\mkern 2.0mu\mathrm{hdep}(q,p,G)\mkern-3.0mu$’ is true if and only if * • $G=s\mathbin{\twoheadrightarrow}r\\{i\langle S_{i}\rangle\mathbin{.}G_{i}\\}_{i\in I}$ and $q\notin\\{s,r\\}$ and $p\in\\{s,r\\}$, and * • $\mathrm{ddep}((p,q),G)\neq\mathsf{skip}\mathbin{.}R$ for all relative types $R$, where $\mathrm{ddep}$ is as in Fig. 5 (top). ###### Example 4. Consider the global type $G_{\mathsf{h}}:=p\mathbin{\twoheadrightarrow}q\\{\mathsf{a}\mathbin{.}p\mathbin{\twoheadrightarrow}r{:}\mathsf{a}\mathbin{.}\bullet,\quad\mathsf{b}\mathbin{.}r\mathbin{\twoheadrightarrow}p{:}\mathsf{b}\mathbin{.}\bullet\\}$. We have that $\mathrm{hdep}(q,p,G_{\mathsf{h}})$ is false because the initial exchange in $G_{\mathsf{h}}$ is not a dependency for $p$ and $q$, but $\mathrm{hdep}(r,p,G_{\mathsf{h}})$ is true because the initial exchange in $G_{\mathsf{h}}$ is indeed a dependency for $p$ and $r$. 1 def _${\llbracket G\rrbracket}_{p}^{\tilde{q}}$_ as 2 switch _$G$_ do 3 case _$s\mathbin{\twoheadrightarrow}r\\{i\langle S_{i}\rangle\mathbin{.}G_{i}\\}_{i\in I}$_ do 4 $\mathsf{deps}:=\\{q\in\tilde{q}\mid\mathrm{hdep}(q,p,G)\\}$ 5 6 if _$p=s$_ then return ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}\mathbin{\triangleright}\big{\\{}i{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}}\mathbin{\triangleleft}i\cdot{(\overline{{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}}\mathbin{\triangleleft}i)}_{q\in\mathsf{deps}}\cdot{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}(v)\mathbin{.}\overline{{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}}[w]\cdot(v\mathbin{\leftrightarrow}w\mathbin{\|}{\llbracket G_{i}\rrbracket}_{p}^{\tilde{q}})\big{\\}}_{i\in I}$ 7 8 else if _$p=r$_ then return ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}\mathbin{\triangleright}\big{\\{}i{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}i\cdot{(\overline{{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}}\mathbin{\triangleleft}i)}_{q\in\mathsf{deps}}\cdot{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}(v)\mathbin{.}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}[w]\cdot(v\mathbin{\leftrightarrow}w\mathbin{\|}{\llbracket G_{i}\rrbracket}_{p}^{\tilde{q}})\big{\\}}_{i\in I}$ 9 10 else if _$p\notin\\{s,r\\}$_ then 11 $\mathsf{depon}_{s}:=(s\in\tilde{q}\wedge\mathrm{hdep}(p,s,G))$ 12 $\mathsf{depon}_{r}:=(r\in\tilde{q}\wedge\mathrm{hdep}(p,r,G))$ 13 14 if _$\mathsf{depon}_{s}$ and $\neg\mathsf{depon}_{r}$_ then return ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}\mathbin{\triangleright}\big{\\{}i{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}i\cdot{\llbracket G_{i}\rrbracket}_{p}^{\tilde{q}}\big{\\}}_{i\in I}$ 15 16 else if _$\mathsf{depon}_{r}$ and $\neg\mathsf{depon}_{s}$_ then return ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}\mathbin{\triangleright}\big{\\{}i{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}i\cdot{\llbracket G_{i}\rrbracket}_{p}^{\tilde{q}}\big{\\}}_{i\in I}$ 17 18 else if _$\mathsf{depon}_{s}$ and $\mathsf{depon}_{r}$_ then return ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}\mathbin{\triangleright}\big{\\{}i{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}i\cdot{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}\mathbin{\triangleleft}\\{i{:}\leavevmode\nobreak\ {\llbracket G_{i}\rrbracket}_{p}^{\tilde{q}}\\}\big{\\}}_{i\in I}$ 19 20 else return ${\llbracket G_{j}\rrbracket}_{p}^{\tilde{q}}$ for any $j\in I$ 21 22 23 24 case _$\mu X\mathbin{.}G^{\prime}$_ do 25 $\tilde{q}^{\prime}:=\\{q\in\tilde{q}\mid G\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)\neq\bullet\\}$ 26 if _$\tilde{q}^{\prime}\neq\emptyset$_ then return $\mu X({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}},{({{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}})}_{q\in\tilde{q}^{\prime}})\mathbin{.}{\llbracket G^{\prime}\rrbracket}_{p}^{\tilde{q}^{\prime}}$ 27 else return $\bm{0}$ 28 29 30 case _$X$_ do return $X{\langle{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}},{({{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}})}_{q\in\tilde{q}}\rangle}$ 31 32 case _$\mathsf{skip}\mathbin{.}G^{\prime}$_ do return ${\llbracket G^{\prime}\rrbracket}_{p}^{\tilde{q}}$ 33 34 case _$\bullet$_ do return $\bm{0}$ 35 Algorithm 1 Synthesis of Router Processes (Def. 19). ###### Definition 19 (Router Synthesis). Given a global type $G$, a participant $p$, and participants $\tilde{q}$, Algorithm 1 defines the synthesis of a _router process_ , denoted ‘$\mkern 1.0mu{\llbracket G\rrbracket}_{p}^{\tilde{q}}\mkern-3.0mu$’, that interfaces the interactions of $p\mkern-2.0mu$ with the other protocol participants according to $G$. We often write ‘$\mathcal{R}_{p}$’ for ‘${\llbracket G\rrbracket}_{p}^{\mathsf{prt}(G)\setminus\\{p\\}}$’ when $G$ is clear from the context. Algorithm 1 distinguishes six cases depending on the syntax of $G$ (Def. 11). The key case is ‘${s\mathbin{\twoheadrightarrow}r\\{i\langle U_{i}\rangle\mathbin{.}G_{i}\\}_{i\in I}}$’ (algorithm 1). First, the algorithm computes a set $\mathsf{deps}$ of participants that depend on the exchange using $\mathrm{hdep}$ (cf. Def. 18). Then, the algorithm considers the three possibilities for $p$: 1. 1. If $p=s$ then $p$ is the sender (algorithm 1): the algorithm returns a process that receives a label $i\in I$ over ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$; sends $i$ over ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}$ and over ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}$ for every $q\in\mathsf{deps}$; receives a channel $v$ over ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$; forwards $v$ as $w$ over ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}$; and continues as ‘${\llbracket G_{i}\rrbracket}_{p}^{\tilde{q}}$’. 2. 2. If $p=r$ then $p$ is the recipient (algorithm 1): the algorithm returns a process that receives a label $i\in I$ over ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}$; sends $i$ over ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ and over ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}$ for every $q\in\mathsf{deps}$; receives a channel $v$ over ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}$; forwards $v$ as $w$ over ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$; and continues as ‘${\llbracket G_{i}\rrbracket}_{p}^{\tilde{q}}$’. 3. 3. Otherwise, if $p$ is not involved (algorithm 1), we use ‘$\mathrm{hdep}$’ to determine whether $p$ depends on an output from $s$, an input from $r$, or on both (algorithms 1 and 1). If $p$ only depends on the output from $s$, the algorithm returns a process that receives a label $i\in I$ over ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}$; sends $i$ over ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$; and continues as ‘${\llbracket G_{i}\rrbracket}_{p}^{\tilde{q}}$’ (algorithm 1). If $p$ only depends on an input from $r$, the returned process is similar; the only difference is that $i$ is received over ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}$ (algorithm 1). When $p$ depends on _both_ the output from $s$ and on the input from $r$ (algorithm 1), the algorithm returns a process that receives a label $i\in I$ over ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}$; sends $i$ over ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$; receives the label $i$ over ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}$; and continues as ‘${\llbracket G_{i}\rrbracket}_{p}^{\tilde{q}}$’. If there are no dependencies, the returned process is ‘${\llbracket G_{j}\rrbracket}_{p}^{\tilde{q}}$’, for arbitrary $j\in I$ (algorithm 1). In case ‘$\mu X\mathbin{.}G^{\prime}$’ (algorithm 1), the algorithm stores in ‘$\tilde{q}^{\prime}$’ those $q\in\tilde{q}$ that interact with $p$ in $G^{\prime}$ (i.e. $\mu X\mathbin{.}G^{\prime}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)\neq\bullet$). Then, if $\tilde{q}^{\prime}$ is non-empty (algorithm 1), the algorithm returns a recursive definition with as context the channels ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}$ for $q\in\tilde{q}^{\prime}$ and ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$. Otherwise, the algorithm returns ‘$\bm{0}$’ (algorithm 1). In case ‘$X$’ (algorithm 1), the algorithm returns a recursive call with as context the channels ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}$ for $q\in\tilde{q}$ and ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$. In case ‘$\mathsf{skip}\mathbin{.}G^{\prime}$’ (algorithm 1), it continues with ‘$G^{\prime}$’ immediately. Finally, in case ‘$\bullet$’ (algorithm 1), the algorithm returns ‘$\bm{0}$’. Considering the number of steps required to return a process, the complexity of Algorithm 1 is linear in the size of the given global type (defined as the sum of the number of communications over all branches). ### 4.2 Types for the Router’s Channels Here, we obtain session types (cf. Def. 1) for (i) the channels between routers and implementations (§ 4.2.1) and for (ii) the channels between pairs of routers (§ 4.2.2). While the former are extracted from global types, the latter are extracted from relative types. #### 4.2.1 The Channels between Routers and Implementations We begin with the session types for the channels between routers and implementations (given in pink), which we extract directly from the global type. A participant’s implementation performs on this channel precisely those actions that the participant must perform as per the global type. Hence, we define this extraction as a form of _local projection_ of the global type onto a _single participant_. The resulting session type may used as a guidance for specifying a participant implementation, which can then connect to the router’s dually typed channel endpoint. Below, $\mathsf{o}\in\mathbb{N}$ is arbitrary: $\displaystyle{{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}\bullet{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}$ $\displaystyle:=\bullet$ $\displaystyle{{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}{!}T\mathbin{.}S{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}$ $\displaystyle:={{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}T{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}\mathbin{\otimes}^{\mathsf{o}}{{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}$ $\displaystyle{{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}{{\oplus}}\\{i{:}\leavevmode\nobreak\ S_{i}\\}_{i\in I}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}$ $\displaystyle:={\oplus}^{\mathsf{o}}\\{i{:}\leavevmode\nobreak\ {{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}\\}_{i\in I}$ $\displaystyle{{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}{?}T\mathbin{.}S{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}$ $\displaystyle:={{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}T{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{\mathsf{o}}{{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}$ $\displaystyle{{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}\&\\{i{:}\leavevmode\nobreak\ S_{i}\\}_{i\in I}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}$ $\displaystyle:=\&^{\mathsf{o}}\\{i{:}\leavevmode\nobreak\ {{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}\\}_{i\in I}$ . If $G=s\mathbin{\twoheadrightarrow}r\\{i\langle S_{i}\rangle\mathbin{.}G_{i}\\}_{i\in I}$, $G\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}}p:=\begin{cases}{{\oplus}}^{\mathsf{o}}\\{i{:}\leavevmode\nobreak\ {{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}\mathbin{\otimes}^{\mathsf{o}+1}(G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}p)\\}_{i\in I}&\text{if $p=s$}\\\ \&^{\mathsf{o}+2}\\{i{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}}\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{\mathsf{o}+3}(G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}p)\\}_{i\in I}&\text{if $p=r$}\\\ \&^{\mathsf{o}+2}\\{i{:}\leavevmode\nobreak\ (G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}p)\\}_{i\in I}&\text{if $p\notin\\{s,r\\}$ and $\mathrm{hdep}(p,s,G)$}\\\ \&^{\mathsf{o}+3}\\{i{:}\leavevmode\nobreak\ (G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}p)\\}_{i\in I}&\text{if $p\notin\\{s,r\\}$ and $\neg\mathrm{hdep}(p,s,G)$ and $\mathrm{hdep}(p,r,G)$}\\\ G_{i^{\prime}}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}p\leavevmode\nobreak\ \text{[any $i^{\prime}\in I$]}&\text{otherwise}\end{cases}$ Otherwise, $\displaystyle\bullet\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}}p:=\bullet\qquad(\mathsf{skip}\mathbin{.}G^{\prime})\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}}p:=G^{\prime}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}p\qquad X\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}}p:=X$ $\displaystyle(\mu X\mathbin{.}G^{\prime})\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}}p$ $\displaystyle:=\begin{cases}\mu X\mathbin{.}(G^{\prime}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}}p)&\text{if $G^{\prime}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}}p$ defined and contractive on $X$}\\\ \bullet&\text{otherwise}\end{cases}$ Figure 7: Extracting Session Types from Message Types (top), and Local Projection: Extracting Session Types from a Global Type (bottom, cf. Definition 22). Global types contain message types (Def. 11), so we must first define how we extract session types from message types. This is a straightforward definition, which leaves priorities unspecified: they do not matter for the typability of routers, which forward messages between implementations and other routers. Note that one must still specify these priorities when type- checking implementations, making sure they concur between sender and recipient. ###### Definition 20 (From Message Types to Session Types). We define the extraction of a session type from message type $S$, denoted ‘$\mkern 1.0mu{{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S\mkern 1.0mu{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}\mkern-3.0mu$’, by induction on the structure of $S$ as in Figure 7 (top). We now define local projection. To deal with non-local choices, local projection incorporates dependencies by relying on the dependency detection of relative projection (cf. Def. 16). Also similar to relative projection, local projection relies on a notion of contractiveness for session types. ###### Definition 21 (Contractive Session Types). Given a session type $A$ and a recursion variable $X$, we say _$A$ is contractive on $X$_ if either of the following holds: * • $A$ contains a connective in $\\{\mathbin{\otimes},\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}},{\oplus},\&\\}$, or * • $A$ is a recursive call on a variable other than $X$. ###### Definition 22 (Local Projection: From Global Types to Session Types). We define the local projection of global type $G$ onto participant $p$ with priority $\mathsf{o}$, denoted ‘$\mkern 1.0muG\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}}p\mkern-2.0mu$’, by induction on the structure of $G$ as in Figure 7 (bottom), relying on message type extraction (Def. 20) and the predicate ‘$\mkern 1.0mu\mathrm{hdep}\mkern-3.0mu$’ (Def. 18). We consider the local projection of an exchange in a global type onto a participant $p$ with priority $\mathsf{o}$. The priorities in local projection reflect the four sub-steps into which we decompose exchanges in global types (cf. Section 4.1). There are three possibilities, depending on the involvement of $p$ in the exchange: 1. 1. If $p$ is the sender, local projection specifies a choice (${\oplus}$) between the exchange’s labels at priority $\mathsf{o}$ and an output ($\mathbin{\otimes}$) of the associated message type at priority $\mathsf{o}+1$, followed by the projection of the chosen branch at priority $\mathsf{o}+4$. 2. 2. If $p$ is the recipient, local projection specifies a branch ($\&$) on the exchange’s labels at priority $\mathsf{o}+2$ and an input ($\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}$) of the associated message type at priority $\mathsf{o}+3$, followed by the projection of the chosen branch at priority $\mathsf{o}+4$. 3. 3. If $p$ is neither sender nor recipient, local projection uses the predicate ‘$\mathrm{hdep}$’ (Def. 18) to detect a dependency on the sender’s output or the recipient’s input. If there is a dependency on the output, local projection specifies a branch on the exchange’s labels at priority $\mathsf{o}+2$. If there is a dependency on the input, local projection specifies a branch at priority $\mathsf{o}+3$. Otherwise, when there is no dependency at all, local projection simply continues with the projection of any branch at priority $\mathsf{o}+4$. Projection only preserves recursive definitions if they contain actual behavior (i.e. the projection of the recursive loop is contractive, cf. Definition 21). The projections of ‘$\bullet$’ and recursion variables are homomorphic. The projection of ‘$\mathsf{skip}$’ simply projects the skip’s continuation, at priority $\mathsf{o}+4$ to keep the priority aligned with the priorities of the other types of the router. #### 4.2.2 The Channels between Pairs of Routers $\displaystyle{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0mus\\{i\langle S_{i}\rangle\mathbin{.}R_{i}\\}_{i\in I}{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}}$ $\displaystyle:=\begin{cases}{{\oplus}}^{\mathsf{o}+1}\left\\{i{:}\leavevmode\nobreak\ {{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}\mathbin{\otimes}^{\mathsf{o}+2}{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muR_{i}{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}+4}\right\\}_{i\in I}&\text{if $p=s$}\\\\[6.0pt] \&^{\mathsf{o}+1}\left\\{i{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}}\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{\mathsf{o}+2}{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muR_{i}{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}+4}\right\\}_{i\in I}&\text{if $q=s$}\end{cases}$ $\displaystyle{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0mur{?}s\\{i\mathbin{.}R_{i}\\}_{i\in I}{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}}$ $\displaystyle:=\begin{cases}{{\oplus}}^{\mathsf{o}+2}\left\\{i{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muR_{i}{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}+4}\right\\}_{i\in I}&\text{if $p=r$}\\\\[6.0pt] \&^{\mathsf{o}+2}\left\\{i{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muR_{i}{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}+4}\right\\}_{i\in I}&\text{if $q=r$}\end{cases}$ $\displaystyle{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0mus{!}r\\{i\mathbin{.}R_{i}\\}_{i\in I}{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}}$ $\displaystyle:=\begin{cases}{{\oplus}}^{\mathsf{o}+1}\left\\{i{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muR_{i}{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}+4}\right\\}_{i\in I}&\text{if $p=s$}\\\\[6.0pt] \&^{\mathsf{o}+1}\left\\{i{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muR_{i}{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}+4}\right\\}_{i\in I}&\text{if $q=s$}\end{cases}$ $\displaystyle{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0mu\bullet{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}}:=\bullet\qquad{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0mu\mathsf{skip}\mathbin{.}R{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}}:={{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muR{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}+4}\qquad{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0mu\mu X\mathbin{.}R{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}}:=\mu X\mathbin{.}{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muR{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}}\qquad{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muX{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}}:=X$ Figure 8: Extracting Session Types from Relative Types (cf. Definition 23). For the channels between pairs of routers (given in purple), we extract session types from relative types (Def. 13). Considering a relative type that describes the protocol between $p$ and $q$, this entails decomposing it into a type for $p$ and a dual type for $q$. ###### Definition 23 (From Relative Types to Session Types). We define the extraction of a session type from relative type $R$ between $p$ and $q$ at $p$’s perspective with priority $\mathsf{o}$, denoted ‘$\mkern 1.0mu{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muR{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}}\mkern-3.0mu$’, by induction on the structure of $R$ as in Figure 8. Here, extraction is _directional_ : in ‘${{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muR{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}}$’, the annotation ‘$p\rangle q$’ says that the session type describes the perspective of $p$’s router with respect to $q$’s. Messages with sender $p$ are decomposed into selection (${\oplus}$) at priority $\mathsf{o}+1$ followed by output ($\mathbin{\otimes}$) at priority $\mathsf{o}+2$. Dependencies on messages recieved by $p$ become selection types (${\oplus}$) at priority $\mathsf{o}+1$, and dependencies on messages sent by $p$ become selection types (${\oplus}$) at priority $\mathsf{o}+2$. Messages from $q$ and dependencies on $q$ yield dual types. Extraction from ‘$\bullet$’ and recursion is homomorphic, and extraction from ‘$\mathsf{skip}$’ simply extracts from the skip’s continuation at priority $\mathsf{o}+4$. This way, the channel endpoint of $p$’s router that connects to $q$’s router will be typed ‘${{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}}$’, i.e. the session type extracted from the relative projection of $G$ onto $p,q$ at $p$’s perspective. Similarly, the endpoint of this channel at $q$’s router will have the type ‘${{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{q{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}p}^{\mathsf{o}}$’, i.e. the same relative projection but at $q$’s perspective. Clearly, these session types must be dual. ###### Theorem 9. Given a relative well-formed global type $G$ and $p,q\in\mathsf{prt}(G)$, ${{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}}=\overline{{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{q{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}p}^{\mathsf{o}}}.$ ###### Proof. By construction from Definition 16 and Definition 23. ∎ ### 4.3 Networks of Routed Implementations Having defined routers and types for their channels, we now turn to defining _networks of routed implementations_ , i.e., process networks of routers and implementations that correctly represent a given multiparty protocol. Then, we appeal to the types obtained in § 4.2 to establish the typability of routers (Theorem 11). Finally, we show that all networks of routed implementations of well-formed global types are deadlock free (Theorem 18), and that networks of routed implementations behave as depicted by the global types from which they are generated (Theorems 19 and 23). We begin by defining routed implementations, which connect implementations of subsets of protocol participants with routers: ###### Definition 24 (Routed Implementations). Given a closed, relative well-formed global type $G$, for participants $\tilde{p}\subseteq\mathsf{prt}(G)$, the _set of routed implementations_ of $\tilde{p}$ in $G$ is defined as follows (cf. Def. 22 for local projection ‘$\mkern 1.0mu\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}\mkern-3.0mu$’ and Def. 19 for router synthesis ‘$\mkern 1.0mu{\llbracket\ldots\rrbracket}\mkern-3.0mu$’): $\mathrm{ri}(G,\tilde{p}):=\left\\{(\bm{\nu}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}})_{p\in\tilde{p}}\,(Q\mathbin{\|}{\mathchoice{\textstyle}{}{}{}\prod}_{p\in\tilde{p}}\mathcal{R}_{p})\mathrel{}\middle\|\mathrel{}\begin{array}[]{l}Q\vdash\emptyset;\Gamma,{({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}{:}\leavevmode\nobreak\ G\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{0}p)}_{p\in\tilde{p}}\\\ {}\wedge\forall p\in\tilde{p}.\leavevmode\nobreak\ \mathcal{R}_{p}={\llbracket G\rrbracket}_{p}^{\mathsf{prt}(G)\setminus\\{p\\}}\end{array}\right\\}$ We write $\mathcal{N}_{\tilde{p}},\mathcal{N}^{\prime}_{\tilde{p}},\ldots$ to denote elements of $\mathrm{ri}(G,\tilde{p})$. Thus, the composition of a collection of routers and an implementation $Q$ is a routed implementation as long as $Q$ can be typed in a context that includes the corresponding projected types. Note that the parameter $\tilde{p}$ indicates the presence of _interleaving_ : when $\tilde{p}$ is a singleton, the set $\mathrm{ri}(G,\tilde{p})$ contains processes in which there is a single router and the implementation $Q$ is single-threaded (non-interleaved); more interestingly, when $\tilde{p}$ includes two or more participants, the set $\mathrm{ri}(G,\tilde{p})$ consists of processes in which the implementation $Q$ interleaves the roles of the multiple participants in $\tilde{p}$. A network of routed implementations of a global type, or simply a _network_ , is then the composition of any combination of routed implementations that together account for all the protocol’s participants. Hence, we define sets of networks, quantified over all possible combinations of sets of participants and their respective routed implementations. The definition relies on _complete partitions_ of the participants of a global type, i.e., a split of $\mathsf{prt}(G)$ into non-empty, disjoint subsets whose union yields $\mathsf{prt}(G)$. ###### Definition 25 (Networks). Suppose given a closed, relative well-formed global type $G$. Let $\mathbb{P}_{G}$ be the set of all complete partitions of $\mathsf{prt}(G)$ with elements $\pi,\pi^{\prime},\ldots$. The _set of networks_ of $G$ is defined as $\mathrm{net}(G):=\big{\\{}(\bm{\nu}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}})_{p,q\in\mathsf{prt}(G)}({\mathchoice{\textstyle}{}{}{}\prod}_{\tilde{p}\in\pi}\mathcal{N}_{\tilde{p}})\leavevmode\nobreak\ \big{\|}\leavevmode\nobreak\ \pi\in\mathbb{P}_{G}\wedge\forall\tilde{p}\in\pi.\leavevmode\nobreak\ \mathcal{N}_{\tilde{p}}\in\mathrm{ri}(G,\tilde{p})\big{\\}}.$ We write $\mathcal{N},\mathcal{N}^{\prime},\ldots$ to denote elements of $\mathrm{net}(G)$. ###### Example 5. Figure 6 depicts two networks in $\mathrm{net}(G_{\mathsf{auth}})$ related to different partitions of $\mathsf{prt}(G_{\mathsf{auth}})$, namely $\big{\\{}\\{a\\},\\{s\\},\\{c\\}\big{\\}}$ (non-interleaved) on the left and $\big{\\{}\\{a,s\\},\\{c\\}\big{\\}}$ (interleaved) on the right. Because a network $\mathcal{N}$ may not be typable under the empty typing context, we have the following definition to “complete” networks. ###### Definition 26 (Completable Networks). Suppose given a network $\mathcal{N}$ such that $\mathcal{N}\vdash\emptyset;\Gamma$. We say that $\mathcal{N}$ is _completable_ if (i) $\Gamma$ is empty or (ii) there exist $\tilde{v},\tilde{w}$ such that $(\bm{\nu}\tilde{v}\tilde{w})\mathcal{N}\vdash\emptyset;\emptyset$. When $\mathcal{N}$ is completable, we write ‘$\mkern 1.0mu\mathcal{N}^{\circlearrowright}\mkern-3.0mu$’ to stand for $\mathcal{N}$ (if $\mathcal{N}\vdash\emptyset;\emptyset$) or $(\bm{\nu}\tilde{v}\tilde{w})\mathcal{N}$ (otherwise). ###### Proposition 10. For any closed, relative well-formed global type $G$, there exists at least one completable network $\mathcal{N}\in\mathrm{net}(G)$. ###### Proof. To construct a completable network in $\mathrm{net}(G)$, we construct a routed implementation (Def. 24) for every $p\in\mathsf{prt}(G)$. Given a $p\in\mathsf{prt}(G)$, by Proposition 1, there exists $Q\vdash\emptyset;{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}{:}\leavevmode\nobreak\ G\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{0}p$. Composing each such characteristic implementation process with routers, and then composing the routed implementations, we obtain a network $\mathcal{N}\in\mathrm{net}(G)$, where $\mathcal{N}\vdash\emptyset;\emptyset$. Hence, $\mathcal{N}$ is completable. ∎ $P$implementation${\llbracket G_{\mathsf{auth}}\rrbracket}_{c}^{\\{s,a\\}}$router${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{:}\leavevmode\nobreak\ G_{\mathsf{auth}}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}}c$${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{:}\leavevmode\nobreak\ \overline{G_{\mathsf{auth}}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}}c}$${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{\mathsf{auth}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(c,s){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{c{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}s}^{\mathsf{o}}$${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{\mathsf{auth}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(c,a){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{c{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}a}^{\mathsf{o}}$Definition 22Definition 19Definition 16Definition 23 Figure 9: Overview of Theorem 11, with the definitions and notations for synthesizing and typing routers, using participant $c$ of $G_{\mathsf{auth}}$ implemented as $P$ (cf. Example 1). Lines indicate channels and boxes indicate processes. #### 4.3.1 The Typability of Routers We wish to establish that the networks of a global type are deadlock free. This result, formalized by Theorem 18 (Theorem 18), hinges on the typability of routers, which we address next. Figure 9 gives an overview of the definitions and notations involved in this theorem’s statement. ###### Theorem 11. Suppose given a closed, relative well-formed global type $G$, and a $p\in\mathsf{prt}(G)$. Then, ${\llbracket G\rrbracket}_{p}^{\mathsf{prt}(G)\setminus\\{p\\}}\vdash\emptyset;\leavevmode\nobreak\ {{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}{:}\leavevmode\nobreak\ \overline{G\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{0}p},\leavevmode\nobreak\ {\big{(}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{0}\big{)}}_{q\in\mathsf{prt}(G)\setminus\\{p\\}}.$ This result is a corollary of Theorem 16 (Theorem 16), which we show next. We give a full proof on Section 4.3.1, after the proof of Theorem 16. ##### Alarm Processes We focus on networks of routed implementations—compositions of synthesized routers and well-typed processes. However, in order to establish the typability of routers we must account for an edge case that goes beyond these assumptions, namely when a routed implementation is connected to some undesirable implementation, not synthesized by Algorithm 1. Consider the following example: ###### Example 6. Consider again the global type $G_{\mathsf{auth}}$, which, for the purpose of this example, we write as follows: $\displaystyle G_{\mathsf{auth}}=s\mathbin{\twoheadrightarrow}c\left\\{\begin{array}[]{@{}l@{}}\mathsf{login}{:}\leavevmode\nobreak\ G_{\mathsf{login}},\\\ \mathsf{quit}{:}\leavevmode\nobreak\ G_{\mathsf{quit}}\end{array}\right\\}$ As established in Example 2, the initial exchange between $s$ and $c$ determines a dependency for the interactions of $a$ with both $s$ and $c$. Therefore, the implementation of $a$ needs to receive the choice between login and quit from the implementations of both $s$ and $c$. An undesirable implementation for $c$, without a router, could be for instance as follows: $\displaystyle R^{\prime}:={{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}\mathbin{\triangleright}\left\\{\begin{array}[]{@{}l@{}}\mathsf{login}{:}\leavevmode\nobreak\ {{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}\mathbin{\triangleleft}\mathsf{quit}\cdot\ldots,\\\ \mathsf{quit}{:}\leavevmode\nobreak\ {{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}\mathbin{\triangleleft}\mathsf{quit}\cdot\ldots\end{array}\right\\}$ Notice how $R^{\prime}$ always sends to $a$ the label quit, even if the choice made by $s$ (and sent to $c$) is login. Now, if $s$ chooses login, the router of $a$ is in limbo: on the one hand, it expects $s$ to behave as specified in $G_{\mathsf{login}}$; on the other hand, it expects $c$ to behave as specified in $G_{\mathsf{quit}}$. Clearly, the router of $a$ is in an inconsistent state due to $c$’s implementation. Because routers always forward the chosen label correctly, this kind of undesirable behavior never occurs in the networks of Definition 25—we state this formally in § 4.3.2 (Theorem 17). Still, in order to prove that our routers are well-typed, we must accommodate the possibility that a router ends up in an undesirable state due to inconsistent forwarding. For this, we extend APCP with an _alarm process_ that signals an inconsistency on a given set of channel endpoints. ###### Definition 27 (Alarm Process). Given channel endpoints $\tilde{x}=x_{1},\ldots,x_{n}$, we write ‘$\mkern 2.0mu{\mathchoice{\textstyle}{}{}{}\mathsf{alarm}({\tilde{x}})}\mkern-3.0mu$’ to denote an inconsistent state on those endpoints. In a way, ${\mathchoice{\textstyle}{}{}{}\mathsf{alarm}({\tilde{x}})}$ is closer to an observable action (a “barb”) than to an actual process term: ${\mathchoice{\textstyle}{}{}{}\mathsf{alarm}({\tilde{x}})}$ does not have reductions, and no process from Figure 3 (top) can reduce to ${\mathchoice{\textstyle}{}{}{}\mathsf{alarm}({\tilde{x}})}$. We assume that ${\mathchoice{\textstyle}{}{}{}\mathsf{alarm}({\tilde{x}})}$ does not occur in participant implementations (cf. $Q$ in Definition 24); we treat it as a process solely for the purpose of refining the router synthesis algorithm (Algorithm 1) with the possibility of inconsistent forwarding. The refinement concerns the process on algorithm 1: $\displaystyle{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}\mathbin{\triangleright}\big{\\{}i{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}i\cdot{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}\mathbin{\triangleleft}\\{i{:}\leavevmode\nobreak\ {\llbracket G_{i}\rrbracket}_{p}^{\tilde{q}}\\}\big{\\}}_{i\in I}$ We extend it with additional branches, as follows: $\displaystyle{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}\mathbin{\triangleright}\big{\\{}i{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}i\cdot{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}\mathbin{\triangleleft}\left(\begin{array}[]{@{}l@{}}\\{i{:}\leavevmode\nobreak\ {\llbracket G_{i}\rrbracket}_{p}^{\tilde{q}}\\}\\\\[5.0pt] \leavevmode\nobreak\ \underline{\cup\leavevmode\nobreak\ \\{i^{\prime}{:}\leavevmode\nobreak\ {\mathchoice{\textstyle}{}{}{}\mathsf{alarm}({{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}},{({{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}})}_{q\in\tilde{q}}})}\\}_{i^{\prime}\in I\setminus\\{i\\}}}\nobreak\leavevmode\nobreak\leavevmode\end{array}\right)\big{\\}}_{i\in I}$ (4) This new process for algorithm 1 captures the kind of inconsistency illustrated by Example 6, which occurs when a label $i\in I$ is received over ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}$ after which a label $i^{\prime}\in I\setminus\\{i\\}$ is received over ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}$. We account for this case by using the underlined alarm processes. Routers are then made of processes as in Figure 3 (top), selectively extended with alarms as just described. Because ${\mathchoice{\textstyle}{}{}{}\mathsf{alarm}({\tilde{x}})}$ merely acts as an observable that signals undesirable behavior, we find it convenient to type it using the following axiom: Alarm ${\mathchoice{\textstyle}{}{}{}\mathsf{alarm}({x_{1},\ldots,x_{n}})}\vdash\Omega;x_{1}{:}\leavevmode\nobreak\ A_{1},\ldots,x_{n}{:}\leavevmode\nobreak\ A_{n}$ where the recursive context $\Omega$ and types $A_{1},\ldots,A_{n}$ are arbitrary. ##### Context-based Typability Considering the refinement of Algorithm 1 with alarm processes, we prove Theorem 16 on Theorem 16, from which Theorem 11 follows as a corollary. It relies on some additional auxiliary definitions and results. To type the router for a participant at any point in the protocol, we need the definition of the entire protocol. It is not enough to only consider the current (partial) protocol at such points: we need information about bound recursion variables in order to perform unfolding in types. To this end, we define _global contexts_ , that allow us to look at part of a protocol while retaining definitions that concern the entire protocol. ###### Definition 28 (Global Contexts). _Global contexts_ $\mathcal{C}$ are given by the following grammar: $\displaystyle C::=p\mathbin{\twoheadrightarrow}q\left(\begin{array}[]{@{}l@{}}\\{i\langle S\rangle\mathbin{.}G\\}_{i\in I}\\\ {}\cup\\{i^{\prime}\langle S\rangle\mathbin{.}C\\}_{i^{\prime}\notin I}\end{array}\right)\;\mbox{\large{$\mid$}}\;\mathsf{skip}\mathbin{.}C\;\mbox{\large{$\mid$}}\;\mu X\mathbin{.}C\;\mbox{\large{$\mid$}}\;[]$ We often simply write ‘context’ when it is clear that we are referring to a global context. Given a context $C$ and a global type $G$, we write ‘$\mkern 1.0muC[G]\mkern-3.0mu$’ to denote the global type obtained by replacing the hole ‘$\mkern 1.0mu[]\mkern-3.0mu$’ in $C$ with $G$. If $G=C[G_{s}]$ for some context $C$ and global type $G_{s}$, then we write ‘$\mkern 1.0muG_{s}\leq_{C}G\mkern-3.0mu$’. As mentioned before, a context captures information about the recursion variables that are bound at any given point in a global type. Our goal is to obtain a _context-based_ typability result for routers. The order in which recursive variables are bound is important to correctly unfold types: ###### Example 7. Consider the following global type with three nested recursive definitions: $\displaystyle G_{\mathsf{rec}}=\mu X\mathbin{.}a\mathbin{\twoheadrightarrow}b:1\mathbin{.}\mu Y\mathbin{.}a\mathbin{\twoheadrightarrow}b:2\mathbin{.}\mu Z\mathbin{.}a\mathbin{\twoheadrightarrow}b\\{\mathsf{x}:X,\quad\mathsf{y}:Y,\quad\mathsf{z}:Z\\}$ To type the router for, e.g., $a$ at the final exchange between $a$ and $b$, we need to be aware of the unfolding of recursion. The recursion on $X$, $Y$, and $Z$ have all to be unfolded, and the recursion on $Z$ must include first the unfolding of $X$ and then the unfolding of $Y$, which must in turn include the prior unfolding of $X$. To account for nested recursions, the following definition gives the bound variables of a context exactly in the order in which they appear: ###### Definition 29 (Recursion Binders of Contexts). Given a global context $C$, the _sequence of recursion binders to the hole_ of $C$, denoted ‘$\mkern 1.0mu\mathrm{ctxbind}(C)\mkern-3.0mu$’, is defined as follows: $\displaystyle\mathrm{ctxbind}(\mu X\mathbin{.}C):=(X,\mathrm{ctxbind}(C))\qquad\mathrm{ctxbind}(\mathsf{skip}\mathbin{.}C):=\mathrm{ctxbind}(C)\qquad\mathrm{ctxbind}([]):=()$ $\displaystyle\mathrm{ctxbind}(p\mathbin{\twoheadrightarrow}q\left(\begin{array}[]{@{}l@{}}\\{i\langle S_{i}\rangle\mathbin{.}G_{i}\\}_{i\in I}\\\ {}\cup\\{i^{\prime}\langle S_{i^{\prime}}\rangle\mathbin{.}C\\}_{i^{\prime}\notin I}\end{array}\right))$ $\displaystyle:=\mathrm{ctxbind}(C)$ Given $G_{s}\leq_{C}G$, the sequence of recursion binders of $G_{s}$, denoted ‘$\mkern 1.0mu\mathrm{subbind}(G_{s},G)\mkern-3.0mu$’, is defined as $\mathrm{ctxbind}(C)$. The following retrieves the body of a recursive definition from a global context, informing us on how to unfold types: ###### Definition 30 (Recursion Extraction). The function ‘$\mkern 1.0mu\mathrm{recdef}(X,G)\mkern-3.0mu$’ extracts the recursive definition on $X$ from $G$, i.e. $\mathrm{recdef}(X,G)=G^{\prime}$ if $\mu X\mathbin{.}G^{\prime}\leq_{C}G$ for some context $C$. Also, ‘$\mkern 1.0mu\mathrm{recCtx}(X,G)\mkern-3.0mu$’ extracts the context of the recursive definition on $X$ in $G$, i.e. $\mathrm{recCtx}(X,G)=C$ if $\mu X\mathbin{.}\mathrm{recdef}(X,G)\leq_{C}G$. When unfolding bound recursion variables, we need the priorities of the unfolded types. The following definition gives a priority that is expected at the hole in a context, as well as the priority expected at any recursive definition in a global type: ###### Definition 31 (Absolute Priorities of Contexts). Given a context $C$ and $\mathsf{o}\in\mathbb{N}$, we define $\mathrm{ctxpri}^{\mathsf{o}}(C)$ as follows: $\displaystyle\mathrm{ctxpri}^{\mathsf{o}}([]):=\mathsf{o}\qquad\mathrm{ctxpri}^{\mathsf{o}}(\mathsf{skip}\mathbin{.}C):=\mathrm{ctxpri}^{\mathsf{o}+4}(C)\qquad\mathrm{ctxpri}^{\mathsf{o}}(\mu X\mathbin{.}C):=\mathrm{ctxpri}^{\mathsf{o}}(C)$ $\displaystyle\mathrm{ctxpri}^{\mathsf{o}}(p\mathbin{\twoheadrightarrow}q\left(\begin{array}[]{@{}l@{}}\\{i\langle S_{i}\rangle\mathbin{.}G_{i}\\}_{i\in I}\\\ {}\cup\\{i^{\prime}\langle S_{i^{\prime}}\mathbin{.}C\\}_{i^{\prime}\notin I}\end{array}\right))$ $\displaystyle:=\mathrm{ctxpri}^{\mathsf{o}+4}(C)$ Then, the _absolute priority_ of $C$, denoted ‘$\mkern 1.0mu\mathrm{ctxpri}(C)\mkern-3.0mu$’, is defined as $\mathrm{ctxpri}^{0}(C)$. The absolute priority of $X$ in $G$, denoted ‘$\mkern 1.0mu\mathrm{varpri}(X,G)\mkern-3.0mu$’, is defined as $\mathrm{ctxpri}(C)$ for some context $C$ such that $\mu X\mathbin{.}\mathrm{recdef}(X,G)\leq_{C}G$. To avoid non-contractive recursive types, relative projection (cf. Figure 5) closes a type when the participants do not interact inside a recursive definition. Hence, when typing a router for a recursive definition, we must determine which pairs of participants are “active” at any given point in a protocol, and close the connections with the “inactive” participants. ###### Example 8. Consider the following global type, where a client (‘$c$’) requests two independent, infinite Fibonacci sequences (‘$f_{1}$’ and ‘$f_{2}$’): $\displaystyle G_{\mathsf{fib}}=c\mathbin{\twoheadrightarrow}f_{1}:\mathsf{init}\langle\mathsf{int}\times\mathsf{int}\rangle\mathbin{.}c\mathbin{\twoheadrightarrow}f_{2}:\mathsf{init}\langle\mathsf{int}\times\mathsf{int}\rangle\mathbin{.}\underbrace{\mu X\mathbin{.}f_{1}\mathbin{\twoheadrightarrow}c:\mathsf{next}\langle\mathsf{int}\rangle\mathbin{.}f_{2}\mathbin{\twoheadrightarrow}c:\mathsf{next}\langle\mathsf{int}\rangle\mathbin{.}X}_{G^{\prime}_{\mathsf{fib}}}$ Participants $f_{1}$ and $f_{2}$ do not interact with each other in the body of the recursion, as formalized by their relative projection: $\displaystyle\mathrm{recdef}(X,G_{\mathsf{fib}})\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(f_{1},f_{2})=\mathsf{skip}\mathbin{.}\mathsf{skip}\mathbin{.}X$ Hence, $G^{\prime}_{\mathsf{fib}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(f_{1},f_{2})=\bullet$, and $f_{1}$ and $f_{2}$ do not form an active pair of participants for the recursion in $G_{\mathsf{fib}}$. Therefore, $f_{1}$’s router closes its connection with $f_{2}$’s router at the start of the recursion on $X$, and vice versa. The following definition uses relative projection to determine the pairs of active participants at the hole of a context, as well as at any recursive definition in a global type. We consider pairs of participants $(p,q)$ and $(q,p)$ to be equivalent. ###### Definition 32 (Active Participants). Suppose given a relative well-formed global type $G$. The following mutually defined functions compute sets of _pairs of active participants_ for recursive definitions and contexts, denoted ‘$\mkern 1.0mu\mathrm{recactive}(X,G)\mkern-3.0mu$’ and ‘$\mkern 1.0mu\mathrm{active}(C,G)\mkern-3.0mu$’, respectively. $\displaystyle\mathrm{recactive}(X,G)$ $\displaystyle:=\\{(p,q)\in\mathrm{active}(\mathrm{recCtx}(X,G),G)\mid(\mu X\mathbin{.}\mathrm{recdef}(X,G))\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)\neq\bullet\\}$ $\displaystyle\mathrm{active}(C,G)$ $\displaystyle:=\begin{cases}\mathrm{recactive}(Y,G)&\text{if $\mathrm{ctxbind}(C)=(\tilde{X},Y)$}\\\ \mathsf{prt}(G)^{2}&\text{otherwise}\end{cases}$ The interdependency between ‘$\mkern 1.0mu\mathrm{recactive}(X,G)\mkern-3.0mu$’ and ‘$\mkern 1.0mu\mathrm{active}(C,G)\mkern-3.0mu$’ is well-defined: the former function considers the active participants of a context, which contains less recursive definitions. When typing a router for a given protocol, we have to keep track of assignments in the recursive context at any point in the protocol. The following two lemmas ensure that the active participants of recursive definitions are consistent with the active participants of their bodies. ###### Lemma 12. Suppose given a closed, relative well-formed global type $G$, and a global type $G_{s}$ and context $C$ such that $G_{s}\leq_{C}G$. For any $Z\in\mathrm{ctxbind}(C)$, $\mathrm{active}(C,G)\subseteq\mathrm{recactive}(Z,G)$. ###### Proof. Take any $Z\in\mathrm{ctxbind}(C)$. Then $\mathrm{ctxbind}(C)=(\tilde{X},Y)$. By definition, ${\mathrm{active}(C,G)=\mathrm{recactive}(Y,G)}$. If $Y=Z$, the thesis is proven. Otherwise, by definition, ${\mathrm{recactive}(Y,G)\subseteq\mathrm{active}(\mathrm{recCtx}(Y,G),G)}$. Since the recursive definition on $Z$ appears in $\mathrm{recCtx}(Y,G)$, it follows by induction on the size of $\tilde{X}$ that $\mathrm{active}(\mathrm{recCtx}(Y,G),G)\subseteq\mathrm{recactive}(Z,G)$. This proves the thesis. ∎ The following lemma ensures that when typing a recursive call, the endpoints given as context for the recursive call concur with the endpoints in the recursive context: ###### Lemma 13. Suppose given a closed, relative well-formed global type $G$, a recursion variable $Z$, and a context $C$ such that $Z\leq_{C}G$. Then, $\mathrm{active}(C,G)=\mathrm{recactive}(Z,G)$. ###### Proof. Because $G=C[Z]$ and $G$ is closed (i.e. $\mathrm{frv}(G)=\emptyset$), there is a recursive definition on $Z$ in $G$. Hence, $\mathrm{ctxbind}(C)\neq\emptyset$, i.e. $\mathrm{ctxbind}(C)=(\tilde{X},Y)$ and $\mathrm{active}(C,G)=\mathrm{recactive}(Y,G)$. If $Y=Z$, the thesis is proven. Otherwise, the recursive definition on $Y$ in $G$ appears somewhere inside the recursive definition on $Z$. Suppose, for contradiction, that $\mathrm{active}(C,G)\neq\mathrm{recactive}(Z,G)$. There are two cases: there exists $(p,q)\in{\mathsf{prt}(G)}^{2}$ s.t. (i) $(p,q)\in\mathrm{active}(C,G)$ and $(p,q)\notin\mathrm{recactive}(Z,G)$, or (ii) $(p,q)\in\mathrm{recactive}(Z,G)$ and $(p,q)\notin\mathrm{active}(C,G)$. Case (i) contradicts Lemma 12. In case (ii), $(\mu Z\mathbin{.}\mathrm{recdef}(Z,G))\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)\neq\bullet$ and $(\mu Y\mathbin{.}\mathrm{recdef}(Y,G))\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)=\bullet$. The recursive call on $Z$ in $G$ appears somewhere inside the recursive definition on $Y$, and hence $\mathrm{recdef}(Y,G)\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)$ contains the recursive call on $Z$. This means that $\mathrm{recdef}(Y,G)\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)$ is contractive on $Y$ (Def. 15), and hence $(\mu Y\mathbin{.}\mathrm{recdef}(Y,G))\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)\neq\bullet$, contradicting the assumption. ∎ Our typability result for routers relies on relative and local projection. Hence, we need to guarantee that all the projections we need at any given point of a protocol are defined. The following result shows a form of compositionality for relative and local projection, guaranteeing the definedness of projections for all active participants of a given context: ###### Proposition 14. Suppose given a closed, relative well-formed global type $G$, and a global type $G_{s}$ such that $G_{s}\leq_{C}G$. Then, for every $(p,q)\in\mathrm{active}(C,G)$, the relative projection $G_{s}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)$ is defined. Also, for every $p\in\\{p\in\mathsf{prt}(G)\mid\exists q\in\mathsf{prt}(G).\leavevmode\nobreak\ (p,q)\in\mathrm{active}(C,G)\\}$, the local projection $G_{s}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}}p$ is defined for any priority $\mathsf{o}$. ###### Proof. Suppose that, for contradiction, $G_{s}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)$ is undefined. We show by induction on the structure of $C$ that this means that $G\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)$ is undefined, contradicting the relative well-formedness of $G$. * • Hole: $C=[]$. We have $G_{s}=G$, and the thesis follows immediately. * • Exchange: $C=r\mathbin{\twoheadrightarrow}s\left(\begin{array}[]{@{}l@{}}\\{i\langle S_{i}\rangle\mathbin{.}G_{i}\\}_{i\in I}\\\ {}\cup\\{i^{\prime}\langle S_{i^{\prime}}\rangle\mathbin{.}C^{\prime}\\}_{i^{\prime}\notin I}\end{array}\right)$. By the IH, $C^{\prime}[G_{s}]\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)$ is undefined. Since the relative projection of an exchange relies on the relative projection of each of the exchange’s branches, $G\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)$ is undefined. * • Skip: $C=\mathsf{skip}\mathbin{.}C^{\prime}$. By the IH, $C^{\prime}[G_{s}]\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)$ is undefined. Since the relative projection of a skip relies on the relative projection of the skip’s continuation, $G\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)$ is undefined. * • Recursive definition: $C=\mu X\mathbin{.}C^{\prime}$. It follows from Lemma 12 that $\mathrm{active}(C,G)\subseteq\mathrm{recactive}(X,G)$. Hence, $(p,q)\in\mathrm{recactive}(X,G)$, and thus $(\mu X\mathbin{.}\mathrm{recdef}(X,G))\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)=(\mu X\mathbin{.}C^{\prime}[G_{s}])\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)\neq\bullet$, which means that $C^{\prime}[G_{s}]\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)$ is defined. This contradicts the IH. The proof for the definedness of local projection is analogous. ∎ Recall Example 7, where nested recursive definitions in a protocol require nested unfolding of recursive types. The following definition gives us a concise way of writing such nested (or _deep_) unfoldings: ###### Definition 33 (Deep Unfolding). Suppose given a sequence of tuples $\tilde{U}$, with each tuple consisting of a recursion variable $X_{i}$, a lift $t_{i}\in\mathbb{N}$, and a type $B_{i}$. The _deep unfolding_ of the type $A$ with $\tilde{U}$, denoted ‘$\mkern 1.0mu\mathrm{deepUnfold}(A,\tilde{U})\mkern-3.0mu$’, is the type defined as follows: $\displaystyle\mathrm{deepUnfold}(A,())$ $\displaystyle:=A$ $\displaystyle\mathrm{deepUnfold}(A,(\tilde{U},(X,t,B)))$ $\displaystyle:=\mathrm{deepUnfold}(A,\tilde{U})\\{\big{(}\mu X\mathbin{.}({\uparrow^{t}}\mathrm{deepUnfold}(B,\tilde{U}))\big{)}/X\\}$ When typing a router’s recursive call, the types of the router’s endpoints are unfoldings of the types in the recursive context. However, because of the deep unfolding in types, this is far from obvious. The following result connects a particular form of deep unfolding with regular unfolding (cf. Definition 5). ###### Proposition 15. Suppose given a type $A$ and a sequence of tuples $\tilde{U}$ consisting of a recursion variable, a lift, and a substitution type. Then, $\displaystyle\mathrm{deepUnfold}(A,(\tilde{U},(X,t,A)))$ $\displaystyle=\mathrm{unfold}^{t}(\mu X\mathbin{.}\mathrm{deepUnfold}(A,\tilde{U})).$ ###### Proof. By Definition 33: $\displaystyle\mathrm{deepUnfold}(A,(\tilde{U},(X,t,A)))$ $\displaystyle=\mathrm{deepUnfold}(A,\tilde{U})\\{\big{(}\mu X\mathbin{.}({\uparrow^{t}}\mathrm{deepUnfold}(A,\tilde{U}))\big{)}/X\\}$ $\displaystyle=\mathrm{unfold}^{t}(\mu X\mathbin{.}\mathrm{deepUnfold}(A,\tilde{U}))\qed$ Armed with these definitions and results, we can finally state our context- based typability result for routers: ###### Theorem 16. Suppose given a closed, relative well-formed global type $G$. Also, suppose given a global type $G_{s}$ such that $G_{s}\leq_{C}G$, and a $p\in\mathsf{prt}(G)$ for which there is a $q\in\mathsf{prt}(G)$ such that $(p,q)\in\mathrm{active}(C,G)$. Consider: * • the participants with whom $p$ interacts in $G_{s}$: $\tilde{q}=\\{q\in\mathsf{prt}(G)\mid(p,q)\in\mathrm{active}(C,G)\\}$, * • the absolute priority of $G_{s}$: $\mathsf{o}_{C}=\mathrm{ctxpri}(C)$, * • the sequence of bound recursion variables of $G_{s}$: $\widetilde{X_{C}}=\mathrm{ctxbind}(C)$, * • for every $X\in\widetilde{X_{C}}$: * – the body of the recursive definition on $X$ in $G$: $G_{X}=\mathrm{recdef}(X,G)$, * – the participants with whom $p$ interacts in $G_{X}$: $\tilde{q}_{X}=\\{q\in\mathsf{prt}(G)\mid(p,q)\in\mathrm{recactive}(X,G)\\}$, * – the absolute priority of $G_{X}$: $\mathsf{o}_{X}=\mathrm{varpri}(X,G)$, * – the sequence of bound recursion variables of $G_{X}$ excluding $X$: $\widetilde{Y_{X}}=\mathrm{subbind}(\mu X\mathbin{.}G_{X},G)$, * – the type required for ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ for a recursive call on $X$: $\displaystyle A_{X,p}=\mathrm{deepUnfold}(\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}p},{(Y,t_{Y},\overline{G_{Y}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{Y}}p})}_{Y\in\widetilde{Y_{X}}}),$ * – the type required for ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}$ for a recursive call on $X$: $\displaystyle B_{X,q}=\mathrm{deepUnfold}({{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{X}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{X}},{(Y,t_{Y},{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{Y}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{Y}})}_{Y\in\widetilde{Y_{X}}}),$ * – the minimum lift for typing a recursive definition on $X$: $t_{X}=\max_{\mathsf{pr}}\left(A_{X},{(B_{X,q})}_{q\in\tilde{q}_{X}}\right)+1$, * • the type expected for ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ for $p$’s router for $G_{s}$: $\displaystyle D_{p}=\mathrm{deepUnfold}(\overline{G_{s}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}p},{(X,t_{X},\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}p})}_{X\in\widetilde{X_{C}}}),$ * • the type expected for ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}$ for $p$’s router for $G_{s}$: $\displaystyle E_{q}=\mathrm{deepUnfold}({{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{s}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}},{(X,t_{X},{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{X}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{X}})}_{X\in\widetilde{X_{C}}}).$ Then, we have: $\displaystyle{\llbracket G_{s}\rrbracket}_{p}^{\tilde{q}}\vdash{\Big{(}X{:}\leavevmode\nobreak\ \big{(}A_{X},{(B_{X,q})}_{q\in\tilde{q}_{X}}\big{)}\Big{)}}_{X\in\widetilde{X_{C}}};\leavevmode\nobreak\ {{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}{:}\leavevmode\nobreak\ D_{p},\leavevmode\nobreak\ {({{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}{:}\leavevmode\nobreak\ E_{q})}_{q\in\tilde{q}}$ ###### Proof. We apply induction on the structure of $G_{s}$, with six cases as in Algorithm 1. We only detail the cases of exchange and recursion. Axiom Alarm is used in only one sub-case (case 3(c), cf. Figure 11 below). * • _Exchange_ : $G_{s}=s\mathbin{\twoheadrightarrow}r\\{i\langle S_{i}\rangle\mathbin{.}G_{i}\\}_{i\in I}$ (algorithm 1). In this case, we add connectives to the types obtained from the IH. Since we do not introduce any recursion variables to these types, the substitutions in the types from the IH are not affected. Hence, we can omit these substitutions from the types. Also, for each $i\in I$, we have $\mathrm{frv}(G_{i})\subseteq\mathrm{frv}(G_{s})$, i.e. the recursive context remains untouched in this derivation, so we also omit the recursive context. Let $\mathsf{deps}:=\\{q\in\tilde{q}\mid\mathrm{hdep}(q,p,G_{s})\\}$ (as on algorithm 1). There are three cases depending on the involvement of $p$. 1. 1. If $p=s$, then $p$ is the sender (algorithm 1). Let us consider the relative projections onto $p$ and the participants in $\tilde{q}$. For the recipient $r$, $G_{s}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,r)=p\\{i\mathbin{.}(G_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,r))\\}_{i\in I}.$ (5) For each $q\in\mathsf{deps}$, by Definition 18, $\mathrm{ddep}((q,p),G)\neq\mathsf{skip}\mathbin{.}R$ for some $R$. That is, since $p$ is the sender of the exchange, for each $q\in\mathsf{deps}$, by the definitions in Figure 5, $G_{s}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)=p{!}r\\{i\mathbin{.}(G_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q))\\}_{i\in I}.$ (6) On the other hand, for each $q\in\tilde{q}\setminus\mathsf{deps}\setminus\\{r\\}$, $G_{s}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)=\mathsf{skip}\mathbin{.}(G_{i^{\prime}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q))$ (7) for any $i^{\prime}\in I$, because for each $i,j\in I$, $G_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)=G_{j}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q).$ (8) Let us take stock of the types we expect for each of the router’s channels. For ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ we expect $\displaystyle\overline{G_{s}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}p}$ $\displaystyle=\overline{{{\oplus}}^{\mathsf{o}_{C}}\\{i{:}\leavevmode\nobreak\ {{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}\mathbin{\otimes}^{\mathsf{o}_{C}+1}(G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}+4}p)\\}_{i\in I}}$ $\displaystyle=\&^{\mathsf{o}_{C}}\\{i{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}}\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{\mathsf{o}_{C}+1}\overline{(G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}+4}p)}\\}_{i\in I}.$ (9) For ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}$ we expect $\displaystyle{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{s}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}r}^{\mathsf{o}_{C}}$ $\displaystyle={{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0mup\\{i\mathbin{.}(G_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,r))\\}_{i\in I}{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}r}^{\mathsf{o}_{C}}$ $\displaystyle={{\oplus}}^{\mathsf{o}_{C}+1}\\{i{:}\leavevmode\nobreak\ {{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}\mathbin{\otimes}^{\mathsf{o}_{C}+2}{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,r){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}r}^{\mathsf{o}_{C}+4}\\}_{i\in I}.$ (10) For each $q\in\mathsf{deps}$, for ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}$ we expect $\displaystyle{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{s}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}}$ $\displaystyle={{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0mup{!}r\\{i.\leavevmode\nobreak\ (G_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q))\\}_{i\in I}{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}}$ $\displaystyle={{\oplus}}^{\mathsf{o}_{C}+1}\\{i{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}+4}\\}_{i\in I}.$ (11) For each $q\in\tilde{q}\setminus\mathsf{deps}\setminus\\{r\\}$, for ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}$ we expect $\displaystyle{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{s}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}}$ $\displaystyle={{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0mu\mathsf{skip}\mathbin{.}(G_{i^{\prime}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}}$ $\displaystyle={{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i^{\prime}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}+4}\leavevmode\nobreak\ \text{for any $i^{\prime}\in I$.}$ (12) Let us now consider the process returned by Algorithm 1, with each prefix marked with a number: ${\llbracket G_{s}\rrbracket}_{p}^{\tilde{q}}=\underbrace{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}\mathbin{\triangleright}\big{\\{}i{:}\big{.}}_{1}\leavevmode\nobreak\ \underbrace{\overline{{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}}\mathbin{\triangleleft}i}_{2_{i}}\cdot\underbrace{{(\overline{{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}}\mathbin{\triangleleft}i)}_{q\in\mathsf{deps}}}_{3_{i}}\cdot\underbrace{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}(v)}_{4_{i}}\mathbin{.}\underbrace{\overline{{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}}[w]}_{5_{i}}\cdot(v\mathbin{\leftrightarrow}w\mathbin{\|}{\llbracket G_{i}\rrbracket}_{p}^{\tilde{q}})\big{.}\big{\\}}_{i\in I}$ For each $i^{\prime}\in I$, let $C_{i^{\prime}}:=C[s\mathbin{\twoheadrightarrow}r(\\{i\langle S_{i}\rangle\mathbin{.}G_{i}\\}_{i\in I\setminus\\{i^{\prime}\\}}\cup\\{i^{\prime}\langle S_{i^{\prime}}\rangle\mathbin{.}[]\\})]$. Clearly, $G_{i^{\prime}}\leq_{C_{i^{\prime}}}G$. Also, because we are not adding recursion binders, the current value of $\tilde{q}$ is appropriate for the IH. With this context $C_{i^{\prime}}$ and $\tilde{q}$, we apply the IH to obtain the typing of ${\llbracket G_{i^{\prime}}\rrbracket}_{p}^{\tilde{q}}$, where priorities start at $\mathrm{ctxpri}(C_{i^{\prime}})=\mathrm{ctxpri}(C)+4=\mathsf{o}_{C}+4$ (cf. Def. 31). Following these typings, Figure 10 gives the typing of ${\llbracket G_{s}\rrbracket}_{p}^{\tilde{q}}$, referring to parts of the process by the number marking its foremost prefix above. Clearly, the priorities in the derivation of Figure 10 meet all requirements. The order of the applications of ${\oplus}^{\star}$ for each $q\in\mathsf{deps}$ does not matter, since the selection actions are asynchronous. Id $\forall i\in I.\leavevmode\nobreak\ v\mathbin{\leftrightarrow}w\vdash v{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}},w{:}\leavevmode\nobreak\ {{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}$ $\forall i\in I.\leavevmode\nobreak\ {\llbracket G_{i}\rrbracket}_{p}^{\tilde{q}}\vdash{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}{:}\leavevmode\nobreak\ \overline{(G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}+4}p)},{\big{(}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}+4}\big{)}}_{q\in\tilde{q}}$ Mix $\forall i\in I.\leavevmode\nobreak\ v\mathbin{\leftrightarrow}w\mathbin{\|}{\llbracket G_{i}\rrbracket}_{p}^{\tilde{q}}\vdash\begin{array}[t]{@{}lr@{}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}{:}\leavevmode\nobreak\ \overline{(G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}+4}p)},v{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}},w{:}\leavevmode\nobreak\ {{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}},\\\ {\big{(}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}+4}\big{)}}_{q\in\tilde{q}}\end{array}$ $\mathbin{\otimes}^{\star}$ $\forall i\in I.\leavevmode\nobreak\ 5_{i}\vdash\begin{array}[t]{@{}lr@{}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}{:}\leavevmode\nobreak\ \overline{(G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}+4}p)},v{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}},\\\ {{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}{:}\leavevmode\nobreak\ {{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}\mathbin{\otimes}^{\mathsf{o}_{C}+2}{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,r){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}r}^{\mathsf{o}_{C}+4},\\\ {\big{(}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}+4}\big{)}}_{q\in\tilde{q}\setminus\\{r\\}}\end{array}$ $\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}$ $\forall i\in I.\leavevmode\nobreak\ 4_{i}\vdash\begin{array}[t]{@{}lr@{}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}}\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{\mathsf{o}_{C}+1}\overline{(G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}+4}p)},\\\ {{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}{:}\leavevmode\nobreak\ {{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}\mathbin{\otimes}^{\mathsf{o}_{C}+2}{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,r){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}r}^{\mathsf{o}_{C}+4},\\\ {\big{(}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}+4}\big{)}}_{q\in\tilde{q}\setminus\\{r\\}}\end{array}$ $\forall q\in\mathsf{deps}.\leavevmode\nobreak\ {\oplus}^{\star}$ $\forall i\in I.\leavevmode\nobreak\ 3_{i}\vdash\begin{array}[t]{@{}lr@{}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}}\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{\mathsf{o}_{C}+1}\overline{(G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}+4}p)},\\\ {{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}{:}\leavevmode\nobreak\ {{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}\mathbin{\otimes}^{\mathsf{o}_{C}+2}{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,r){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}r}^{\mathsf{o}_{C}+4},\\\ {\big{(}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}{:}\leavevmode\nobreak\ {{\oplus}}^{\mathsf{o}_{C}+1}\\{i{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}+4}\\}_{i\in I}\big{)}}_{q\in\mathsf{deps}},\\\ {\big{(}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}+4}\big{)}}_{q\in\tilde{q}\setminus\mathsf{deps}}\end{array}$ ${\oplus}^{\star}$ $\forall i\in I.\leavevmode\nobreak\ 2_{i}\vdash\begin{array}[t]{@{}lr@{}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}}\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{\mathsf{o}_{C}+1}\overline{(G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}+4}p)},\\\ {{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}{:}\leavevmode\nobreak\ {{\oplus}}^{\mathsf{o}_{C}+1}\\{i{:}\leavevmode\nobreak\ {{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}\mathbin{\otimes}^{\mathsf{o}_{C}+2}{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,r){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}r}^{\mathsf{o}_{C}+4}\\}_{i\in I},\\\ {\big{(}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}{:}\leavevmode\nobreak\ {{\oplus}}^{\mathsf{o}_{C}+1}\\{i{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}+4}\\}_{i\in I}\big{)}}_{q\in\mathsf{deps}},\\\ {\big{(}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}+4}\big{)}}_{q\in\tilde{q}\setminus\mathsf{deps}}&\text{(cf.\ \eqref{eq:outSkipSame})}\end{array}$ $\&$ ${\llbracket G_{s}\rrbracket}_{p}^{\tilde{q}}=1\vdash\begin{array}[t]{@{}lr@{}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}{:}\leavevmode\nobreak\ \&^{\mathsf{o}_{C}}\\{i{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}}\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{\mathsf{o}_{C}+1}\overline{(G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}+4}p)}\\}_{i\in I},&\text{(cf.\ \eqref{eq:outCiType})}\\\ {{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}{:}\leavevmode\nobreak\ {{\oplus}}^{\mathsf{o}_{C}+1}\\{i{:}\leavevmode\nobreak\ {{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}\mathbin{\otimes}^{\mathsf{o}_{C}+2}{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,r){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}r}^{\mathsf{o}_{C}+4}\\}_{i\in I},&\text{(cf.\ \eqref{eq:outCrtRecvType})}\\\ {\big{(}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}{:}\leavevmode\nobreak\ {{\oplus}}^{\mathsf{o}_{C}+1}\\{i{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}+4}\\}_{i\in I}\big{)}}_{q\in\mathsf{deps}},&\text{(cf.\ \eqref{eq:outCrtDepsType})}\\\ {\big{(}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i^{\prime}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}+4}\big{)}}_{q\in\tilde{q}\setminus\mathsf{deps}}&\text{(cf.\ \eqref{eq:outCrtSkipType})}\end{array}$ Figure 10: Typing derivation used in the proof of Theorem 11. 2. 2. If $p=r$, then $p$ is the recipient (algorithm 1). This case is analogous to the previous one. 3. 3. If $p\notin\\{r,s\\}$ (algorithm 1), then further analysis depends on whether the exchange is a dependency for $p$. Let $\displaystyle\mathsf{depon}_{s}$ $\displaystyle:=(s\in\tilde{q}\wedge\mathrm{hdep}(p,s,G))$ (as on algorithm 1), and $\displaystyle\mathsf{depon}_{r}$ $\displaystyle:=(r\in\tilde{q}\wedge\mathrm{hdep}(p,r,G))$ (as on algorithm 1). To see what the truths of $\mathsf{depon}_{s}$ and $\mathsf{depon}_{r}$ mean, we follow Definition 18 and the definitions in Figure 5. $\displaystyle G_{s}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,s)$ $\displaystyle=\begin{cases}s{!}r\\{i\mathbin{.}(G_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,s))\\}_{i\in I}&\text{if $\mathsf{depon}_{s}$ is true}\\\ \mathsf{skip}\mathbin{.}(G_{i^{\prime}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,s))\leavevmode\nobreak\ \text{for any $i^{\prime}\in I$}&\text{otherwise}\end{cases}$ (13) $\displaystyle G_{s}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,r)$ $\displaystyle=\begin{cases}r{?}s\\{i\mathbin{.}(G_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,r))\\}_{i\in I}&\text{if $\mathsf{depon}_{r}$ is true}\\\ \mathsf{skip}\mathbin{.}(G_{i^{\prime}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,r))\leavevmode\nobreak\ \text{for any $i^{\prime}\in I$}&\text{otherwise}\end{cases}$ (14) Let us also consider the relative projections onto $p$ and the participants in $\tilde{q}$ besides $r$ and $s$, which follow by the relative well-formedness of $G_{s}$. For each $q\in\tilde{q}\setminus\\{r,s\\}$, $G_{s}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)=\mathsf{skip}\mathbin{.}(G_{i^{\prime}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q))$ (15) for any $i^{\prime}\in I$. The rest of the analysis depends on the truth of $\mathsf{depon}_{s}$ and $\mathsf{depon}_{r}$. There are four cases. 1. (a) If $\mathsf{depon}_{s}$ is true and $\mathsf{depon}_{r}$ is false (algorithm 1), let us take stock of the types we expect for each of the router’s channels. For ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ we expect $\displaystyle\overline{G_{s}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}p}$ $\displaystyle=\overline{\&^{\mathsf{o}_{C}+2}\\{i{:}\leavevmode\nobreak\ (G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}+4}p)\\}_{i\in I}}$ $\displaystyle={{\oplus}}^{\mathsf{o}_{C}+2}\\{i{:}\leavevmode\nobreak\ \overline{(G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}+4}p)}\\}_{i\in I}.$ (16) For ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}$ we expect $\displaystyle{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{s}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,s){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}s}^{\mathsf{o}_{C}}$ $\displaystyle={{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0mus{!}r\\{i\mathbin{.}(G_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,s))\\}_{i\in I}{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}s}^{\mathsf{o}_{C}}$ (cf. (13)) $\displaystyle=\&^{\mathsf{o}_{C}+1}\\{i{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,s){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}s}^{\mathsf{o}_{C}+4}\\}_{i\in I}.$ (17) For each $q\in\tilde{q}\setminus\\{s\\}$, for ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}$ we expect $\displaystyle{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{s}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}}$ $\displaystyle={{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0mu\mathsf{skip}\mathbin{.}(G_{i^{\prime}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}}$ (cf. (14) and (15)) $\displaystyle={{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i^{\prime}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}+4}\leavevmode\nobreak\ \text{for any $i^{\prime}\in I$}.$ (18) Similar to case (1), we apply the IH to obtain the typing of ${\llbracket G_{i}\rrbracket}_{p}^{\tilde{q}}$ for each $i\in I$, starting at priority $\mathsf{o}_{C}+4$. We derive the typing of ${\llbracket G_{s}\rrbracket}_{p}^{\tilde{q}}$: $\forall i\in I.\leavevmode\nobreak\ {\llbracket G_{i}\rrbracket}_{p}^{\tilde{q}}\vdash{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}{:}\leavevmode\nobreak\ \overline{G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}+4}p},{\big{(}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}+4}\big{)}}_{q\in\tilde{q}}$ ${\oplus}^{\star}$ $\forall i\in I.\leavevmode\nobreak\ \overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}i\cdot{\llbracket G_{i}\rrbracket}_{p}^{\tilde{q}}\vdash{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}{:}\leavevmode\nobreak\ {{\oplus}}^{\mathsf{o}_{C}+2}\\{i{:}\leavevmode\nobreak\ \overline{G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}+4}p}\\}_{i\in I},{\big{(}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}+4}\big{)}}_{q\in\tilde{q}}$ $\&$ ${\llbracket G_{s}\rrbracket}_{p}^{\tilde{q}}={{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}\mathbin{\triangleright}\\{i{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}i\cdot{\llbracket G_{i}\rrbracket}_{p}^{\tilde{q}}\\}_{i\in I}\vdash\begin{array}[t]{@{}lr@{}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}{:}\leavevmode\nobreak\ {{\oplus}}^{\mathsf{o}_{C}+2}\\{i{:}\leavevmode\nobreak\ \overline{G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}+4}p}\\}_{i\in I},&\text{(cf.\ \eqref{eq:rDepSCiType})}\\\ {{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}{:}\leavevmode\nobreak\ \&^{\mathsf{o}_{C}+1}\\{i{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}+4}\\}_{i\in I},&\text{(cf.\ \eqref{eq:rDepSCrtSType})}\\\ {\big{(}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i}^{\prime}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}+4}\big{)}}_{q\in\tilde{q}}&\text{(cf.\ \eqref{eq:rDepSCrtOtherType})}\end{array}$ 2. (b) The case where $\mathsf{depon}_{s}$ is false and $\mathsf{depon}_{r}$ is true (algorithm 1) is analogous to the previous one. 3. (c) If both $\mathsf{depon}_{s}$ and $\mathsf{depon}_{r}$ are true (algorithm 1 and (4)), let us once again take stock of the types we expect for each of the router’s channels. For ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ we expect $\displaystyle\overline{G_{s}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}p}$ $\displaystyle=\overline{\&^{\mathsf{o}_{C}+2}\\{i{:}\leavevmode\nobreak\ (G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}+4}p)\\}_{i\in I}}$ $\displaystyle={{\oplus}}^{\mathsf{o}_{C}+2}\\{i{:}\leavevmode\nobreak\ \overline{(G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}+4}p)}\\}_{i\in I}$ (19) For ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}$ we expect $\displaystyle{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{s}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,s){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}s}^{\mathsf{o}_{C}}$ $\displaystyle={{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0mus{!}r\\{i\mathbin{.}(G_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,s))\\}_{i\in I}{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}s}^{\mathsf{o}_{C}}$ (cf. (13)) $\displaystyle=\&^{\mathsf{o}_{C}+1}\\{i{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,s){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}s}^{\mathsf{o}_{C}+4}\\}_{i\in I}$ (20) For ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}$ we expect $\displaystyle{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{s}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,r){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}r}^{\mathsf{o}_{C}}$ $\displaystyle={{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0mur{?}s\\{i\mathbin{.}(G_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,r))\\}_{i\in I}{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}r}^{\mathsf{o}_{C}}$ (cf. (14)) $\displaystyle=\&^{\mathsf{o}_{C}+2}\\{i{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,r){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}r}^{\mathsf{o}_{C}+4}\\}_{i\in I}$ (21) For each $q\in\tilde{q}\setminus\\{s,r\\}$, for ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}$ we expect $\displaystyle{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{s}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}}$ $\displaystyle={{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0mu\mathsf{skip}\mathbin{.}(G_{i^{\prime}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}}$ (cf. (15)) $\displaystyle={{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i^{\prime}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}+4}\leavevmode\nobreak\ \text{for any $i^{\prime}\in I$}$ (22) It is clear from (20) and (21) that the router will receive label $i\in I$ first on ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}$ and then $i^{\prime}\in I$ on ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}$. We rely on alarm processes (Definition 27) to handle the case $i^{\prime}\neq i$. Similar to case (1), we apply the IH to obtain the typing of ${\llbracket G_{i}\rrbracket}_{p}^{\tilde{q}}$ for each $i\in I$, starting at priority $\mathsf{o}_{C}+4$. Figure 11 gives the typing of ${\llbracket G_{s}\rrbracket}_{p}^{\tilde{q}}$. $\begin{array}[b]{@{}l@{}}\forall i\in I.\\\ {\llbracket G_{i}\rrbracket}_{p}^{\tilde{q}}\vdash\begin{array}[t]{@{}l@{}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}{:}\overline{G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}+4}p},\\\ {\big{(}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}{:}{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}+4}\big{)}}_{q\in\tilde{q}}\end{array}\end{array}$ Alarm $\begin{array}[b]{@{}l@{}}\forall i\in I.\\\ \forall i^{\prime}\in I\setminus\\{i\\}.\leavevmode\nobreak\ {\mathchoice{\textstyle}{}{}{}\mathsf{alarm}({\mathsf{chs}})}\vdash\begin{array}[t]{@{}l@{}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}{:}\leavevmode\nobreak\ \overline{G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}+4}p},\\\ {{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,s){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}s}^{\mathsf{o}_{C}+4},\\\ {{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i^{\prime}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,r){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}r}^{\mathsf{o}_{C}+4},\\\ {\big{(}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}+4}\big{)}}_{q\in\tilde{q}\setminus\\{s,r\\}}\end{array}\end{array}$ $\&$ $\forall i\in I.\leavevmode\nobreak\ {{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}\mathbin{\triangleright}\\{i{:}\leavevmode\nobreak\ {\llbracket G_{i}\rrbracket}_{p}^{\tilde{q}}\\}\cup\\{i^{\prime}{:}\leavevmode\nobreak\ {\mathchoice{\textstyle}{}{}{}\mathsf{alarm}({\mathsf{chs}})}\\}_{i^{\prime}\in I\setminus\\{i\\}}\vdash\begin{array}[t]{@{}l@{}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}{:}\leavevmode\nobreak\ \overline{G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}+4}p},\\\ {{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,s){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}s}^{\mathsf{o}_{C}+4},\\\ {{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}{:}\leavevmode\nobreak\ \&^{\mathsf{o}_{C}+2}\\{i{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i^{\prime}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,r){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}r}^{\mathsf{o}_{C}+4}\\}_{i\in I},\\\ {\big{(}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}+4}\big{)}}_{q\in\tilde{q}\setminus\\{s,r\\}}\end{array}$ ${\oplus}^{\star}$ $\forall i\in I.\leavevmode\nobreak\ \overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}i\cdot{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}\mathbin{\triangleright}\\{i{:}\leavevmode\nobreak\ {\llbracket G_{i}\rrbracket}_{p}^{\tilde{q}}\\}\cup\\{i^{\prime}{:}\leavevmode\nobreak\ {\mathchoice{\textstyle}{}{}{}\mathsf{alarm}({\mathsf{chs}})}\\}_{i^{\prime}\in I\setminus\\{i\\}}\vdash\begin{array}[t]{@{}l@{}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}{:}\leavevmode\nobreak\ {{\oplus}}^{\mathsf{o}_{C}+2}\\{i{:}\leavevmode\nobreak\ \overline{G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}+4}p}\\}_{i\in I},\\\ {{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,s){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}s}^{\mathsf{o}_{C}+4},\\\ {{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}{:}\leavevmode\nobreak\ \&^{\mathsf{o}_{C}+2}\\{i{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i^{\prime}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,r){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}r}^{\mathsf{o}_{C}+4}\\}_{i\in I},\\\ {\big{(}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}+4}\big{)}}_{q\in\tilde{q}\setminus\\{s,r\\}}\end{array}$ $\&$ $\underbrace{{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}\mathbin{\triangleright}\\{i{:}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}i\cdot{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}\mathbin{\triangleright}\\{i{:}{\llbracket G_{i}\rrbracket}_{p}^{\tilde{q}}\\}{\cup}\\{i^{\prime}{:}{\mathchoice{\textstyle}{}{}{}\mathsf{alarm}({\mathsf{chs}})}\\}_{i^{\prime}\in I\setminus\\{i\\}}\\}_{i\in I}}_{{\llbracket G_{s}\rrbracket}_{p}^{\tilde{q}}}\vdash\begin{array}[t]{@{}lr@{}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}{:}\leavevmode\nobreak\ {{\oplus}}^{\mathsf{o}_{C}+2}\\{i{:}\leavevmode\nobreak\ \overline{G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}+4}p}\\}_{i\in I},&\text{(cf.\ \eqref{eq:rDepCiType})}\\\ {{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}{:}\leavevmode\nobreak\ \&^{\mathsf{o}_{C}+1}\\{i{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,s){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}s}^{\mathsf{o}_{C}+4}\\}_{i\in I},&\text{(cf.\ \eqref{eq:rDepCrtSType})}\\\ {{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}{:}\leavevmode\nobreak\ \&^{\mathsf{o}_{C}+2}\\{i{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i^{\prime}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,r){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}r}^{\mathsf{o}_{C}+4}\\}_{i\in I},&\text{(cf.\ \eqref{eq:rDepCrtRType})}\\\ {\big{(}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i^{\prime}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}+4}\big{)}}_{q\in\tilde{q}\setminus\\{s,r\\}}&\text{(cf.\ \eqref{eq:rDepCrtOtherType})}\end{array}$ Figure 11: Typing derivation used in the proof of Theorem 11, where $\mathsf{chs}=\\{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}\\}\cup\\{{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}\mid q\in\tilde{q}\\}$. 4. (d) If both $\mathsf{depon}_{s}$ and $\mathsf{depon}_{r}$ are false, let us again take stock of the types we expect for each of the router’s channels. For ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ we expect $\displaystyle\overline{G_{s}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}p}$ $\displaystyle=\overline{G_{i^{\prime}}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}+4}p}\leavevmode\nobreak\ \text{for any $i^{\prime}\in I$}.$ For each $q\in\tilde{q}$, for ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}$ we expect $\displaystyle{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{s}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}}$ $\displaystyle={{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0mu\mathsf{skip}\mathbin{.}(G_{i^{\prime}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}}$ (cf. (13), (14) and (15)) $\displaystyle={{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{i^{\prime}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}+4}\leavevmode\nobreak\ \text{for any $i^{\prime}\in I$.}$ Similar to case (1), we apply the IH to obtain the typing of ${\llbracket G_{i^{\prime}}\rrbracket}_{p}^{\tilde{q}}$, starting at priority $\mathsf{o}_{C}+4$. This directly proves the thesis. * • _Recursive definition_ : $G_{s}=\mu Z\mathbin{.}G^{\prime}$ (algorithm 1). Let $\displaystyle\tilde{q}^{\prime}:=\\{q\in\tilde{q}\mid G_{s}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)\neq\bullet\\}$ (23) (as on algorithm 1). We consider the relative projections onto $p$ and the participants in $\tilde{q}$. For each $q\in\tilde{q}^{\prime}$, we know $G_{s}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)\neq\bullet$, while for each $q\in\tilde{q}\setminus\tilde{q}^{\prime}$, we know $G_{s}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)=\bullet$. More precisely, by Definition 16, for each $q\in\tilde{q}^{\prime}$, $\displaystyle G_{s}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)$ $\displaystyle=(\mu Z\mathbin{.}G^{\prime})\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)=\mu Z\mathbin{.}(G^{\prime}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)).$ (24) and thus $\displaystyle G^{\prime}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)$ $\displaystyle\neq\mathsf{skip}^{\ast}\mathbin{.}\bullet\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ G^{\prime}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)\neq\mathsf{skip}^{\ast}\mathbin{.}Z.$ For each $q\in\tilde{q}\setminus\tilde{q}^{\prime}$, $\displaystyle G_{s}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)$ $\displaystyle=(\mu Z\mathbin{.}G^{\prime})\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)=\bullet,$ (25) and thus $\displaystyle G^{\prime}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)$ $\displaystyle=\mathsf{skip}^{\ast}\mathbin{.}\bullet\leavevmode\nobreak\ \text{or}\leavevmode\nobreak\ G^{\prime}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)=\mathsf{skip}^{\ast}\mathbin{.}Z.$ Further analysis depends on whether $\tilde{q}^{\prime}=\emptyset$ or not. We thus examine two cases: * – If $\tilde{q}^{\prime}=\emptyset$ (algorithm 1), let us consider the local projection $G_{s}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}p$. We prove that $G_{s}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}p=\bullet$. Suppose, for contradiction, that $G_{s}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}p\neq\bullet$. Then, by the definitions in Figure 7, $G^{\prime}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}p\neq X$ and $G^{\prime}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}p\neq\bullet$. That is, $G^{\prime}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}p$ contains communication actions or some recursion variable other than $Z$. However, communication actions in $G^{\prime}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}p$ originate from exchanges in $G^{\prime}$, either involving $p$ and some $q\in\tilde{q}$, or as a dependency on an exchange involving some $q\in\tilde{q}$. Moreover, recursion variables in $G^{\prime}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}p$ originate from recursion variables in $G^{\prime}$. But this would mean that for this $q$, $G^{\prime}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)$ contains interactions or recursion variables, contradicting (25). Therefore, it cannot be the case that $G_{s}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}p\neq\bullet$. Let us take stock of the types we expect for each of the router’s channels. For now, we omit the substitutions in the types. For ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ we expect $\displaystyle\overline{G_{s}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}p}$ $\displaystyle=\overline{\bullet}=\bullet.$ For each $q\in\tilde{q}$, for ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}$ we expect $\displaystyle{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{s}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}}$ $\displaystyle={{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0mu\bullet{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}}=\bullet.$ (cf. (25)) Because all expected types are $\bullet$, the substitutions do not affect the types, so we can omit them altogether. First we apply Empty, giving us an arbitrary recursive context, and thus the recursive context we need. Then, we apply $\bullet$ for ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ and for ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}$ for each $q\in\tilde{q}$, and obtain the typing of ${\llbracket G_{s}\rrbracket}_{p}^{\tilde{q}}$ (omitting the recursive context): ${\llbracket G_{s}\rrbracket}_{p}^{\tilde{q}}=\bm{0}\vdash{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}{:}\leavevmode\nobreak\ \bullet,{({{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}{:}\leavevmode\nobreak\ \bullet)}_{q\in\tilde{q}}$ * – If $\tilde{q}^{\prime}\neq\emptyset$ (algorithm 1), then, following similar reasoning as in the previous case, ${G_{s}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}p=\mu Z\mathbin{.}(G^{\prime}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}p)}$. We take stock of the types we expect for each of the router’s channels. Note that, because of the recursive definition on $Z$ in $G_{s}$, there cannot be another recursive definition in the context $C$ capturing the recursion variable $Z$. Therefore, by Definition 29, $Z\notin\widetilde{X_{C}}$. For ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ we expect $\displaystyle\mathrm{deepUnfold}(\overline{G_{s}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}p},\ldots)$ $\displaystyle{}=\mathrm{deepUnfold}(\overline{\mu Z\mathbin{.}(G^{\prime}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}p)},\ldots)$ $\displaystyle{}=\mathrm{deepUnfold}(\mu Z\mathbin{.}\overline{G^{\prime}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}p},\ldots)$ $\displaystyle{}=\mu Z\mathbin{.}\mathrm{deepUnfold}(\overline{G^{\prime}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}p},{(X,t_{X},\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}p})}_{X\in\widetilde{X_{C}}}).$ (26) For each $q\in\tilde{q}^{\prime}$, for ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}$ we expect $\displaystyle\mathrm{deepUnfold}({{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{s}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}},\ldots)$ $\displaystyle{}=\mathrm{deepUnfold}({{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0mu\mu Z\mathbin{.}(G^{\prime}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q)){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}},\ldots)$ $\displaystyle{}=\mathrm{deepUnfold}(\mu Z\mathbin{.}{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG^{\prime}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}},\ldots)$ (cf. (24)) $\displaystyle=\mu Z\mathbin{.}\mathrm{deepUnfold}({{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG^{\prime}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}},{(X,t_{X},{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{X}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{X}})}_{X\in\widetilde{X_{C}}}).$ (27) For each $q\in\tilde{q}\setminus\tilde{q}^{\prime}$, for ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}$ we expect $\displaystyle\mathrm{deepUnfold}({{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{s}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}},\ldots)$ $\displaystyle{}=\mathrm{deepUnfold}({{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0mu\bullet{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}},\ldots)$ $\displaystyle{}=\mathrm{deepUnfold}(\bullet,\ldots)$ (cf. (25)) $\displaystyle{}=\bullet.$ (28) We also need an assignment in the recursive context for every $X\in\widetilde{X_{C}}$, but not for $Z$. Let $C^{\prime}=C[\mu Z\mathbin{.}[]]$. Clearly, $G^{\prime}\leq_{C^{\prime}}G$. Let us first establish some facts about the recursion binders, priorities, and active participants related to $C^{\prime}$, $G^{\prime}$, and $Z$: * * $\widetilde{X_{C^{\prime}}}=\mathrm{ctxbind}(C^{\prime})=(\mathrm{ctxbind}(C),Z)=(\widetilde{X_{C}},Z)$ (cf. Def. 29). * * $G_{Z}=\mathrm{recdef}(Z,G)=G^{\prime}$, as proven by the context $C^{\prime}$ (cf. Def. 30). * * $\widetilde{Y_{Z}}=\mathrm{subbind}(\mu Z\mathbin{.}G_{Z},G)=\mathrm{ctxbind}(C)=\widetilde{X_{C}}$. * * $\mathsf{o}_{C^{\prime}}=\mathrm{ctxpri}(C^{\prime})=\mathrm{ctxpri}(C)=\mathsf{o}_{C}$, and $\mathsf{o}_{Z}=\mathrm{varpri}(Z,G)=\mathrm{ctxpri}(C)=\mathsf{o}_{C}$, and hence $\mathsf{o}_{C^{\prime}}=\mathsf{o}_{Z}$ (cf. Def. 31). * * $\tilde{q}_{Z}=\tilde{q}^{\prime}$ (cf. Def. 32 and (23)). Because $\widetilde{X_{C^{\prime}}}=(\widetilde{X_{C}},Z)$ and $\tilde{q}^{\prime}=\tilde{q}_{Z}$, $\tilde{q}^{\prime}$ is appropriate for the IH. We apply the IH on $C^{\prime}$, $G^{\prime}$, and $\tilde{q}^{\prime}$ to obtain a typing for ${\llbracket G^{\prime}\rrbracket}_{p}^{\tilde{q}^{\prime}}$, where we immediately make use of the facts established above. We give the assignment to $Z$ in the recursive context separate from those for the recursion variables in $\widetilde{X_{C}}$. Also, by Proposition 15, we can write the final unfolding on $Z$ in the types separately. For example, the type for ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ is $\displaystyle\mathrm{deepUnfold}(\overline{G^{\prime}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C^{\prime}}}p},{(X,t_{X},\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}p})}_{X\in\widetilde{X_{C^{\prime}}}})$ $\displaystyle=\mathrm{deepUnfold}(\overline{G^{\prime}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}p},{(X,t_{X},\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}p})}_{X\in(\widetilde{X_{C}},Z)})$ $\displaystyle=\mathrm{deepUnfold}(\overline{G^{\prime}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}p},\big{(}{(X,t_{X},\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}p})}_{X\in\widetilde{X_{C}}},(Z,t_{Z},\overline{G_{Z}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{Z}}p})\big{)})$ $\displaystyle=\mathrm{deepUnfold}(\overline{G^{\prime}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}p},\big{(}{(X,t_{X},\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}p})}_{X\in\widetilde{X_{C}}},(Z,t_{Z},\overline{G^{\prime}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}p})\big{)})$ $\displaystyle=\mathrm{unfold}^{t_{Z}}\big{(}\mu Z\mathbin{.}\mathrm{deepUnfold}(\overline{G^{\prime}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}p},{(X,t_{X},\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}p})}_{X\in\widetilde{X_{C}}})\big{)}.$ The resulting typing is as follows: $\displaystyle{\llbracket G^{\prime}\rrbracket}_{p}^{\tilde{q}^{\prime}}\vdash\begin{array}[t]{@{}l@{}}{\left(X{:}\leavevmode\nobreak\ \left(\begin{array}[]{@{}l@{}}\mathrm{deepUnfold}(\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}p},{(Y,t_{Y},\overline{G_{Y}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{Y}}p})}_{Y\in\widetilde{Y_{X}}}),\\\\[4.0pt] {\Big{(}\mathrm{deepUnfold}({{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{X}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{X}},{(Y,t_{Y},{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{Y}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{Y}})}_{Y\in\widetilde{Y_{X}}})\Big{)}}_{q\in\tilde{q}_{X}}\end{array}\right)\right)}_{X\in\widetilde{X_{C}}},\\\\[16.0pt] Z{:}\leavevmode\nobreak\ \left(\begin{array}[]{@{}l@{}}\mathrm{deepUnfold}(\overline{G^{\prime}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}p},{(X,t_{X},\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}p})}_{X\in\widetilde{X_{C}}}),\\\\[4.0pt] {\Big{(}\mathrm{deepUnfold}({{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG^{\prime}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}},{(X,t_{X},{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{X}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{X}})}_{X\in\widetilde{X_{C}}})\Big{)}}_{q\in\tilde{q}^{\prime}}\end{array}\right);\\\\[16.0pt] {{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}{:}\leavevmode\nobreak\ \mathrm{unfold}^{t_{Z}}\big{(}\mu Z\mathbin{.}\mathrm{deepUnfold}(\overline{G^{\prime}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}p},{(X,t_{X},\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}p})}_{X\in\widetilde{X_{C}}})\big{)},\\\\[6.0pt] {\Big{(}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}{:}\leavevmode\nobreak\ \mathrm{unfold}^{t_{Z}}\big{(}\mu Z\mathbin{.}\mathrm{deepUnfold}({{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG^{\prime}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}},{(X,t_{X},{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{X}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{X}})}_{X\in\widetilde{X_{C}}})\big{)}\Big{)}}_{q\in\tilde{q}^{\prime}}\end{array}$ By assumption, we have $\displaystyle t_{Z}$ $\displaystyle=\max_{\mathsf{pr}}\left(\begin{array}[]{@{}l@{}}\mathrm{deepUnfold}(\overline{G^{\prime}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}p},{(X,t_{X},\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}p})}_{X\in\widetilde{X_{C}}})\\\\[4.0pt] {\Big{(}\mathrm{deepUnfold}({{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG^{\prime}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}},{(X,t_{X},{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{X}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{X}})}_{X\in\widetilde{X_{C}}})\Big{)}}_{q\in\tilde{q}^{\prime}}\end{array}\right)+1,$ so $t_{Z}$ is clearly greater than the maximum priority appearing in the types before unfolding. Hence, we can apply Rec to eliminate $Z$ from the recursive context, and to fold the types, giving the typing of ${\llbracket G_{s}\rrbracket}_{p}^{\tilde{q}}=\mu Z({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}},{({{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}})}_{q\in\tilde{q}^{\prime}})\mathbin{.}{\llbracket G^{\prime}\rrbracket}_{p}^{\tilde{q}^{\prime}}$: $\displaystyle{\llbracket G_{s}\rrbracket}_{p}^{\tilde{q}}\vdash\begin{array}[t]{@{}l@{}}{\left(X{:}\leavevmode\nobreak\ \left(\begin{array}[]{@{}l@{}}\mathrm{deepUnfold}(\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}p},{(Y,t_{Y},\overline{G_{Y}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{Y}}p})}_{Y\in\widetilde{Y_{X}}}),\\\\[4.0pt] {\Big{(}\mathrm{deepUnfold}({{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{X}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{X}},{(Y,t_{Y},{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{Y}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{Y}})}_{Y\in\widetilde{Y_{X}}})\Big{)}}_{q\in\tilde{q}_{X}}\end{array}\right)\right)}_{X\in\widetilde{X_{C}}};\\\\[16.0pt] {{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}{:}\leavevmode\nobreak\ \mu Z\mathbin{.}\mathrm{deepUnfold}(\overline{G^{\prime}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}p},{(X,t_{X},\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}p})}_{X\in\widetilde{X_{C}}}),\\\\[6.0pt] {\Big{(}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}{:}\leavevmode\nobreak\ \mu Z\mathbin{.}\mathrm{deepUnfold}({{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG^{\prime}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}},{(X,t_{X},{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{X}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{X}})}_{X\in\widetilde{X_{C}}})\Big{)}}_{q\in\tilde{q}^{\prime}}\end{array}$ In this typing, the type for ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ concurs with (26), and, for every $q\in\tilde{q}^{\prime}$, the type for ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}$ concurs with (27). For every $q\in\tilde{q}\setminus\tilde{q}^{\prime}$, we can add the type for ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}$ in (28) by applying $\bullet$. This proves the thesis. * • _Recursive call_ : $G_{s}=Z$ (algorithm 1). Clearly, because $G$ is closed (i.e. $\mathrm{frv}(G)=\emptyset$), $Z\in\widetilde{X_{C}}$. More precisely, $\widetilde{X_{C}}=(\tilde{X}_{1},Z,\tilde{X}_{2})$. Note that the recursive definitions on the variables in $\tilde{X}_{1}$ appear in $G$ after the recursive definitions on the variables in $(Z,\tilde{X}_{2})$. Because the unfoldings of $(Z,\tilde{X}_{2})$ occur before the unfoldings of $\tilde{X}_{1}$, the recursive definitions on the variables in $\tilde{X}_{1}$ are renamed in order to avoid capturing these variables when performing the unfoldings of $(Z,\tilde{X}_{2})$. So, after the unfoldings of $(Z,\tilde{X}_{2})$, there are no recursive calls on the variables in $\tilde{X}_{1}$ anymore, so the unfoldings on $\tilde{X}_{1}$ do not have any effect on the types. Also, note that $\tilde{X}_{2}=\widetilde{Y_{Z}}$ (cf. Def. 29). Let us take stock of the types we expect for our router’s channels. For ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ we expect $\displaystyle\mathrm{deepUnfold}(\overline{G_{s}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}p},\ldots)$ $\displaystyle{}=\mathrm{deepUnfold}(\overline{Z\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}p},\ldots)$ $\displaystyle{}=\mathrm{deepUnfold}(\overline{Z},\ldots)$ $\displaystyle{}=\mathrm{deepUnfold}(Z,{(X,t_{X},\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}p})}_{X\in(\tilde{X}_{1},Z,\widetilde{Y_{Z}})})$ $\displaystyle{}=\mathrm{deepUnfold}(Z,{(X,t_{X},\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}p})}_{X\in(Z,\widetilde{Y_{Z}})})$ $\displaystyle{}=\mu Z\mathbin{.}({\uparrow^{t_{Z}}}\mathrm{deepUnfold}(\overline{G_{Z}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{Z}}p},{(X,t_{X},\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}p})}_{X\in\widetilde{Y_{Z}}}))$ (29) For each $q\in\tilde{q}$, for ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}$ we expect $\displaystyle\mathrm{deepUnfold}({{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{s}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}},\ldots)$ $\displaystyle{}=\mathrm{deepUnfold}({{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muZ{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}},\ldots)$ $\displaystyle{}=\mathrm{deepUnfold}(Z,{(X,t_{X},{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{X}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{X}})}_{X\in(\tilde{X}_{1},Z,\widetilde{Y_{Z}})})$ $\displaystyle{}=\mathrm{deepUnfold}(Z,{(X,t_{X},{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{X}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{X}})}_{X\in(Z,\widetilde{Y_{Z}})})$ $\displaystyle{}=\mu Z\mathbin{.}({\uparrow^{t_{Z}}}\mathrm{deepUnfold}({{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{Z}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{Z}},{(X,t_{X},{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{X}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{X}})}_{X\in\widetilde{Y_{Z}}}))$ (30) Also, we need an assignment in the recursive context for every $X\in\widetilde{X_{C}}$. By Lemma 13, $\tilde{q}=\tilde{q}_{Z}$. Hence, for $Z$, the assignment should be as follows: $\displaystyle Z{:}\leavevmode\nobreak\ \left(\begin{array}[]{@{}l@{}}\mathrm{deepUnfold}(\overline{G_{Z}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{Z}}p},{(X,t_{X},\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}p})}_{X\in\widetilde{Y_{Z}}}),\\\ {\Big{(}\mathrm{deepUnfold}({{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{Z}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{Z}},{(X,t_{X},{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{X}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{X}})}_{X\in\widetilde{Y_{Z}}})\Big{)}}_{q\in\tilde{q}}\end{array}\right)$ (33) We apply Var to obtain the typing of ${\llbracket G_{s}\rrbracket}_{\tilde{q}}^{p}$, where we make us the rule’s allowance for an arbitrary recursive context up to the assignment to $Z$. Var is applicable, because the types are recursive definitions on $Z$, concurring with the types assigned to $Z$, and lifted by a common lifter $t_{Z}$. Var $\begin{array}[]{@{}l@{}}{\llbracket G_{s}\rrbracket}_{\tilde{q}}^{p}=X{\langle{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}},{({{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}})}_{q\in\tilde{q}}\rangle}\\\\[6.0pt] {}\vdash\begin{array}[t]{@{}l@{}}{\left(X{:}\leavevmode\nobreak\ \left(\begin{array}[]{@{}l@{}}\mathrm{deepUnfold}(\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}p},{(Y,t_{Y},\overline{G_{Y}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{Y}}p})}_{Y\in\widetilde{Y_{X}}}),\\\\[4.0pt] {\Big{(}\mathrm{deepUnfold}({{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{X}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{X}},{(Y,t_{Y},{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{Y}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{Y}})}_{Y\in\widetilde{Y_{X}}})\Big{)}}_{q\in\tilde{q}_{X}}\end{array}\right)\right)}_{X\in\widetilde{X_{C}}\setminus(Z)},\\\\[8.0pt] Z{:}\leavevmode\nobreak\ \left(\begin{array}[]{@{}l@{}}\mathrm{deepUnfold}(\overline{G_{Z}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{Z}}p},{(X,t_{X},\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}p})}_{X\in\widetilde{Y_{Z}}}),\\\ {\Big{(}\mathrm{deepUnfold}({{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{Z}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{Z}},{(X,t_{X},{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{X}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{X}})}_{X\in\widetilde{Y_{Z}}})\Big{)}}_{q\in\tilde{q}}\end{array}\right);\\\\[8.0pt] {{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}{:}\leavevmode\nobreak\ \mu Z\mathbin{.}({\uparrow^{t_{Z}}}\mathrm{deepUnfold}(\overline{G_{Z}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{Z}}p},{(X,t_{X},\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}p})}_{X\in\widetilde{Y_{Z}}})),\\\\[6.0pt] {\left({{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}{:}\leavevmode\nobreak\ \mu Z\mathbin{.}({\uparrow^{t_{Z}}}\mathrm{deepUnfold}({{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{Z}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{Z}},{(X,t_{X},{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{X}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{X}})}_{X\in\widetilde{Y_{Z}}}))\right)}_{q\in\tilde{q}}\end{array}\end{array}$ In this typing, the type of ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ concurs with the expected type in (29), the types of ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}$ for each $q\in\tilde{q}$ concur with the expected types in (30), and the assignment to $Z$ in the recursive context concurs with (33). This proves the thesis. ∎ Now, we can prove Theorem 11 as a corollary of Theorem 16: ###### Proof of Theorem 11 on Theorem 11. We have been given a closed, relative well-formed global type $G$, and a participant $p\in\mathsf{prt}(G)$. Let $C:=[]$ and $G_{s}:=G$. Clearly, $G_{s}\leq_{C}G$. By Definition 32, ${\mathrm{active}(C,G)={\mathsf{prt}(G)}^{2}}$. For $p$ to be a participant of $G$, there must be an exchange involving $p$ and some other participant $q$, i.e. there exists a $q\in\mathsf{prt}(G)$ such that $(p,q)\in\mathrm{active}(C,G)$. Moreover, $\tilde{q}$ as defined in Theorem 16 is $\\{q\in\mathsf{prt}(G)\mid(p,q)\in\mathrm{active}(C,G)\\}=q\in\mathsf{prt}(G)\setminus\\{p\\}$. Hence, Theorem 16 allows us to find a typing for ${\llbracket G\rrbracket}_{p}^{\mathsf{prt}(G)\setminus\\{p\\}}$. Let us consider the precise values of the ingredients of Theorem 16 in our application: 1. 1. $\mathsf{o}_{C}=\mathrm{ctxpri}(C)=0$, 2. 2. $\widetilde{X_{C}}=\mathrm{ctxbind}(C)=()$, 3. 3. $\begin{array}[t]{@{}rll@{}}D_{p}&=\mathrm{deepUnfold}(\overline{G_{s}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}p},{(X,t_{X},\overline{G_{x}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}p})}_{X\in\widetilde{X_{C}}})\\\ &=\overline{G\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{0}p}&\text{(cf.\ \lx@cref{creftypecap~refnum}{d:deepUnfold}),}\end{array}$ 4. 4. $\begin{array}[t]{@{}rll@{}}E_{q}&=\mathrm{deepUnfold}({{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{s}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{C}},{(X,t_{X},{{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{X}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{\mathsf{o}_{X}})}_{X\in\widetilde{X_{C}}})\\\ &={{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{0}&\text{(cf.\ \lx@cref{creftypecap~refnum}{d:deepUnfold}).}\end{array}$ Finally, the result of Theorem 16 is as follows: $\displaystyle{\llbracket G_{s}\rrbracket}_{p}^{\tilde{q}}\vdash{\Big{(}X{:}\leavevmode\nobreak\ \big{(}A_{X},{(B_{X,q})}_{q\in\tilde{q}_{X}}\big{)}\Big{)}}_{X\in\widetilde{X_{C}}};\leavevmode\nobreak\ {{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}{:}\leavevmode\nobreak\ D_{p},\leavevmode\nobreak\ {({{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}{:}\leavevmode\nobreak\ E_{q})}_{q\in\tilde{q}}$ Applying (1)–(4) above, we get the following: $\displaystyle{\llbracket G\rrbracket}_{p}^{\mathsf{prt}(G)\setminus\\{p\\}}\vdash\emptyset;\leavevmode\nobreak\ {{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}{:}\leavevmode\nobreak\ \overline{G\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{0}p},\leavevmode\nobreak\ {\big{(}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}{:}\leavevmode\nobreak\ {{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{p{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}q}^{0}\big{)}}_{q\in\mathsf{prt}(G)\setminus\\{p\\}}$ This coincides exactly with the result of Theorem 11. ∎ #### 4.3.2 Transference of Results (Operational Correspondence) Given a global type $G$, we now formalize the transference of correctness properties such as deadlock freedom from ‘$\mathrm{net}(G)$’ (cf. Definition 25) to ‘$G$’. Here, we define an _operational correspondence_ between networks and global types, in both directions. That is, we show that a network performs interactions between implementations and routers and between pairs of routers if and only if that communication step is stipulated in the corresponding global type (Theorems 19 and 23). Before formalizing the operational correspondence, we show that networks of routed implementations never reduce to alarm processes. To be precise, because alarm processes only can occur in routers (not in implementations), we show that none of the routers of a network reduces to an alarm process, formalized using evaluation contexts: ###### Definition 34 (Evaluation Context). We define an _evaluation context_ as a process with a single hole ‘$\mkern 1.0mu[\,]\mkern-3.0mu$’, not prefixed by input or branching: $\displaystyle E::=(\bm{\nu}xy)\,E\;\mbox{\large{$\mid$}}\;P\mathbin{\|}E\;\mbox{\large{$\mid$}}\;\mu X(\tilde{z})\mathbin{.}E\;\mbox{\large{$\mid$}}\;[\,]$ Given an evaluation context $E$, we write ‘$\mkern 1.0muE[P]\mkern-3.0mu$’ to denote the process obtained by replacing the hole in $E$ with $P$. ###### Theorem 17. Given a relative well-formed global type $G$ and a network of routed implementations $\mathcal{N}\in\mathrm{net}(G)$, then $\mathcal{N}{\centernot\longrightarrow}^{\ast}E[{\mathchoice{\textstyle}{}{}{}\mathsf{alarm}({\tilde{x}})}],$ for any evaluation context $E$ and set of endpoints $\tilde{x}$. ###### Proof. By definition (Definition 25), $\mathcal{N}$ consists only of routers (Definition 19) and well-typed processes not containing the alarm process (cf. the assumption below Definition 27). Suppose, for contradiction, that there are $E\in\mathcal{E}$ and $\tilde{x}$ such that $\mathcal{N}\longrightarrow^{\ast}E[{\mathchoice{\textstyle}{}{}{}\mathsf{alarm}({\tilde{x}})}]$. Since only routers can contain the alarm process, there is a router $\mathcal{R}_{p}$ in $\mathcal{N}$ for participant $p\in\mathsf{prt}(G)$ that reduces to the alarm process. Since it is the only possibility for a router synthesized by Algorithm 1 to contain the alarm process, it must contain the process in (4). This process is synthesized on algorithm 1 of Algorithm 1, so there is an exchange in $G$ with sender $s\in\mathsf{prt}(G)\setminus\\{p\\}$ and recipient $r\in\mathsf{prt}(G)\setminus\\{p\\}$ that is a dependency for the interactions of $p$ with both $s$ and $r$. For this exchange, the router $\mathcal{R}_{s}$ for $s$ contains the process returned on algorithm 1 of Algorithm 1, and the router $\mathcal{R}_{r}$ for $r$ contains the process returned on algorithm 1. Suppose $s$ has a choice between the labels in $I$, and the implementation of $s$ chooses $i\in I$. Then, $\mathcal{R}_{s}$ sends $i$ to $\mathcal{R}_{r}$ and $\mathcal{R}_{p}$. Now, for $\mathcal{R}_{p}$ to reduce to the alarm process, it has to receive from $\mathcal{R}_{r}$ a label $i^{\prime}\in I\setminus\\{i\\}$. However, this contradicts algorithm 1 of Algorithm 1, which clearly defines $\mathcal{R}_{r}$ to send $i$ to $\mathcal{R}_{p}$. Hence, $\mathcal{N}{\centernot\longrightarrow}^{\ast}E[{\mathchoice{\textstyle}{}{}{}\mathsf{alarm}({\tilde{x}})}]$. ∎ It follows from this and the typability of routers (Theorem 11) that networks of routed implementations are deadlock free: ###### Theorem 18. For relative well-formed global type $G$, every $\mathcal{N}\in\mathrm{net}(G)$ is deadlock free. ###### Proof. By the typability of routers (Theorem 11) and the duality of the types of router channels (Theorem 9), $\mathcal{N}\vdash\emptyset;\emptyset$. Hence, by Theorem 5, $\mathcal{N}$ is deadlock free, and by Theorem 17, $\mathcal{N}$ never reduces to the alarm process. ∎ To formalize our operational correspondence result, we apply the labeled reductions for processes ‘$\mkern 1.0muP\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha}}}}Q$’ (cf. Definition 9) and define a labeled transition system (LTS) for global types. ###### Definition 35 (LTS for Global Types). We define the relation ‘$\mkern 1.0muG\xrightarrow{\vspace{-.8ex}\alpha}G^{\prime}\mkern-2.0mu$’, with labels ‘$\mkern 1.0mu\beta\mkern-3.0mu$’ of the form ‘$\mkern 1.0mup\rangle q{:}\ell\langle S\rangle\mkern-3.0mu$’ (sender, recipient, label, and message type), by the following rules: $\raisebox{8.0pt}{}j\in I$ $\mkern-6.0mup\mkern 1.0mu{\mathbin{\twoheadrightarrow}}\mkern 1.0muq\\{i\langle S_{i}\rangle\mathbin{.}G_{i}\\}_{i\in I}\mkern 1.0mu{\xrightarrow{\vspace{-.8ex}p\rangle q{:}j\langle S_{j}\rangle}}\mkern 1.0muG_{j}\mkern-6.0mu$ $G\xrightarrow{\vspace{-.8ex}\alpha}G^{\prime}$ $\mkern-6.0mu\raisebox{12.0pt}{}\mathsf{skip}\mathbin{.}G\mkern 1.0mu{\xrightarrow{\vspace{-.8ex}\alpha}}\mkern 1.0muG^{\prime}\mkern-6.0mu$ $G\\{\mu X\mathbin{.}G/X\\}\xrightarrow{\vspace{-.8ex}\alpha}G^{\prime}$ $\mu X\mathbin{.}G\xrightarrow{\vspace{-.8ex}\alpha}G^{\prime}$ Intuitively, operational correspondence states: 1. 1. every transition of a global type is mimicked by a precise sequence of labeled reductions originating from an associated completable network (_completeness_ ; Theorem 19), and 2. 2. for every labeled reduction originated in a completable network there is a corresponding global type transition (_soundness_ ; Theorem 23). We write ‘$\rho_{1}\rho_{2}$’ for the composition of relations ‘$\rho_{1}$’ and ‘$\rho_{2}$’. Recall that the notation ‘$\longrightarrow^{\star}$’ stands for finite sequences of reductions, as defined in 2. ###### Theorem 19 (Operational Correspondence: Completeness). Suppose given a relative well-formed global type $G$. Also, suppose given $p,q\in\mathsf{prt}(G)$ and a set of labels $J$ such that $j\in J$ if and only if $G\xrightarrow{\vspace{-.8ex}p\rangle q:j\langle S_{j}\rangle}G_{j}$ for some $S_{j}$. Then, 1. 1. for any completable $\mathcal{N}\in\mathrm{net}(G)$, there exists a $j^{\prime}\in J$ such that $\mathcal{N}^{\circlearrowright}\longrightarrow^{\star}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}\rangle{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}:j^{\prime}}}}}\,\mathcal{N}_{0}$; 2. 2. for any $j^{\prime}\in J$, there exists a completable $\mathcal{N}\in\mathrm{net}(G)$ such that $\mathcal{N}^{\circlearrowright}\longrightarrow^{\star}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}\rangle{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}:j^{\prime}}}}}\,\mathcal{N}_{0}$; 3. 3. for any completable $\mathcal{N}\in\mathrm{net}(G)$ and any $j^{\prime}\in J$, if $\mathcal{N}^{\circlearrowright}\longrightarrow^{\star}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}\rangle{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}:j^{\prime}}}}}\,\mathcal{N}_{0}$, then there exists a completable $\mathcal{N}_{j^{\prime}}\in\mathrm{net}(G_{j^{\prime}})$ such that, $\displaystyle\mathcal{N}_{0}\,\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}\rangle{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}{:}j^{\prime}}}}}\longrightarrow^{\star}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}\rangle{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}{:}j^{\prime}}}}}\longrightarrow^{\star}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}\rangle{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}{:}v}}}}\,\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}\rangle{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}{:}w}}}}\,\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}v\mathbin{\leftrightarrow}w}}}}\longrightarrow^{\star}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}\rangle{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}{:}w}}}}\,\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}v\mathbin{\leftrightarrow}w}}}}\,\mathcal{N}_{j^{\prime}}^{\circlearrowright}.$ ###### Proof. By the labelled transitions of global types (Def. 35) and relative well- formedness, $G$ is a sequence of $\mathsf{skip}$s followed by an exchange from $p$ to $q$ over the labels in $J$. Since the $\mathsf{skip}$s do not influence the behavior of routers, let us assume simply that $\displaystyle G=p\mathbin{\twoheadrightarrow}q\\{j\langle S_{j}\rangle\mathbin{.}G_{j}\\}_{j\in J}.$ We prove each Subitem separately. 1. (a) Take any completable $\mathcal{N}\in\mathrm{net}(G)$. By definition (Def. 26), $\mathcal{N}^{\circlearrowright}\vdash\emptyset;\emptyset$. By the construction of networks of routed implementations (Def. 25), ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}\in\mathrm{bn}(\mathcal{N}^{\circlearrowright})$, and ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}$ is connected to ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$. Also by construction, the type of ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}$ in the typing derivation of $\mathcal{N}^{\circlearrowright}$ is $\displaystyle G\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{0}p={\oplus}^{0}\\{j:{{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{j}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}\mathbin{\otimes}^{1}(G_{j}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{4}p)\\})_{j\in J}.$ By the well-typedness of $\mathcal{N}^{\circlearrowright}$, we can infer the kind of action that is defined on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}$: a selection, or a forwarder. By induction on the number of connected forwarders (which is finite by the finiteness of process terms), eventually a forwarder has to be connected to a selection. So, after reducing the forwarders, we have a selection on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}$, of some $j^{\prime}\in J$. Hence, by Fairness (Theorem 7), after a finite number of steps, we can observe a communication of the label $j^{\prime}$ from ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}$ to ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$. This proves the thesis: $\mathcal{N}^{\circlearrowright}\longrightarrow^{\star}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}\rangle{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}:j^{\prime}}}}}\,\mathcal{N}_{0}$. 2. (b) Following the proof of the existence of completable networks (Proposition 10), we can generate an implementation process for all of $G$’s participants from local projections (cf. Proposition 1). Take any $j^{\prime}\in J$. For the implementation process of $p$, we specifically generate an implementation process that sends the label $j^{\prime}$. These implementation processes allow us to construct $\mathcal{N}$, which by construction is in $\mathrm{net}(G)$ and is completable. Following the reasoning as in Subitem (a), ${\mathcal{N}^{\circlearrowright}\longrightarrow^{\star}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}\rangle{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}:j^{\prime}}}}}\,\mathcal{N}_{0}}$. 3. (c) By definition (Def. 26), $\mathcal{N}^{\circlearrowright}\vdash\emptyset;\emptyset$. Hence, by Fairness (Theorem 7), for any of the pending names of $\mathcal{N}^{\circlearrowright}$, we can observe a communication after a finite number of steps. By construction (Def. 25), the endpoints that we are required to observe by thesis are bound in $\mathcal{N}^{\circlearrowright}$. From the shape of $G$, the definition of routed implementations (Def. 24), and the typability of routers (Theorem 11), we know the types of all the required endpoints in $\mathcal{N}^{\circlearrowright}$. We can deduce the required labeled reductions following the reasoning as in Subitem (a). Let us summarize the origin of each of the network’s steps: 1. 1. $\mathcal{N}^{\circlearrowright}\longrightarrow^{\star}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}\rangle{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}:j^{\prime}}}}}\,\mathcal{N}_{0}$: The implementation of $p$ selects label $j^{\prime}$ with $p$’s router. 2. 2. $\mathcal{N}_{0}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}\rangle{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}:j^{\prime}}}}}\mathcal{N}_{1}$: The router of $p$ forwards $j^{\prime}$ to $q$’s router. 3. 3. $\mathcal{N}_{1}\longrightarrow^{\star}\mathcal{N}_{2}$: The router of $p$ forwards $j^{\prime}$ to the routers of the participant that depend on the output by $p$, and these routers forward $j^{\prime}$ to their respective implementations. 4. 4. $\mathcal{N}_{2}\,\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}\rangle{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}:j^{\prime}}}}}\longrightarrow^{\star}\mathcal{N}_{3}$: The router of $q$ forwards $j^{\prime}$ to $q$’s implementation, and to the routers of the participants that depend on the input by $q$, and these routers forward $j^{\prime}$ to their respective implementation (if they have not done so already for the output dependency on $p$). 5. 5. $\mathcal{N}_{3}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}\rangle{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}:v}}}}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}\rangle{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}:w}}}}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}v\mathbin{\leftrightarrow}w}}}}\mathcal{N}_{4,v}$: The implementation of $p$ sends an endpoint $v$ to $p$’s router, which sends a fresh endpoint $w$ to $q$’s router, and $v$ is forwarded to $w$. 6. 6. $\mathcal{N}_{4,v}\longrightarrow^{\star}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}\rangle{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}:w}}}}\,\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}v\mathbin{\leftrightarrow}w}}}}\,\mathcal{N}_{j^{\prime}}^{\circlearrowright}$: The router of $q$ sends a fresh endpoint $w$ to $q$’s implementations, and $v$ is forwarded to $w$. In $\mathcal{N}_{j^{\prime}}^{\circlearrowright}$, all routers have transitioned to routers for $G_{j^{\prime}}$. Moreover, by Type Preservation (Theorem 2), $\mathcal{N}_{j^{\prime}}^{\circlearrowright}\vdash\emptyset;\emptyset$. By isolating restrictions on endpoints that belong only to implementation processes, we can find $\mathcal{N}_{j^{\prime}}\in\mathrm{net}(G_{j^{\prime}})$ such that $\mathcal{N}_{j^{\prime}}^{\circlearrowright}$ is its completion. This proves the thesis. Note that $G$ can also contain recursive definitions before the initial exchange; this case can be dealt with by unfolding. ∎ Our soundness result, given below as Theorem 23, will capture the notion that after any sequence of reductions from the network of a global type $G$, a network of another global type $G^{\prime}$ can be reached. Crucially, $G^{\prime}$ can be reached from $G$ through a series of transitions. Networks are inherently concurrent, whereas global types are built out of sequential compositions; as a result, the network could have enabled (asynchronous) actions that correspond to exchanges that are not immediately enabled in the global type. For example, consider the global types $G=a\mathbin{\twoheadrightarrow}b\big{\\{}1\langle S_{1}\rangle.c\mathbin{\twoheadrightarrow}d\\{1\langle S^{\prime}\rangle.\mathsf{end}\\},2\langle S_{2}\rangle.c\mathbin{\twoheadrightarrow}d\\{1\langle S^{\prime}\rangle.\mathsf{end}\\}\big{\\}}$ and $G^{\prime}=a\mathbin{\twoheadrightarrow}b\big{\\{}1\langle S\rangle.b\mathbin{\twoheadrightarrow}c\\{1\langle S^{\prime}\rangle.\mathsf{end}\\}\big{\\}}$. Clearly, the initial exchange in $G$ between $a$ and $b$ is not a dependency for the following exchange between $c$ and $d$. The routers of $c$ and $d$ synthesized from $G$ thus start with their exchange, without awaiting the initial exchange between $a$ and $b$ to complete. Hence, in a network of $G$, both exchanges in $G$ may be enabled simultaneously. We further refer to exchanges that may be simultaneously enabled in networks as _independent (global) exchanges_. While all exchanges appearing in $G$ are independent, the two exchanges in $G^{\prime}$ are not. In the proof of soundness, we may encounter in a network reductions related to independent exchanges, so we have to be able to identify the independent exchanges in the global type to which the network belongs. Lemma 21 states that independent exchanges related to observed reductions in a network of a global type $G$ can be reached from $G$ after any sequence of transitions in a finite number of steps. The proof of this lemma relies on Lemma 20, which ensures that if a participant does not depend on a certain exchange, then the routers synthesized at each of the branches of the exchange are equal. ###### Lemma 20. Suppose given a relative well-formed global type $G=s\mathbin{\twoheadrightarrow}r\\{i\langle S_{i}\rangle.G_{i}\\}_{i\in I}$, and take any ${p\in\mathsf{prt}(G)\setminus\\{s,r\\}}$ and $\tilde{q}\subseteq\mathsf{prt}(G)\setminus\\{p\\}$. If neither $\mathrm{hdep}(p,s,G)$ nor $\mathrm{hdep}(p,r,G)$ holds, then ${{\llbracket G_{i}\rrbracket}_{p}^{\tilde{q}}={\llbracket G_{j}\rrbracket}_{p}^{\tilde{q}}}$ for every $i,j\in I$. ###### Proof. The analysis proceeds by cases on the structure of $G$. As a representative case we consider ${G=s\mathbin{\twoheadrightarrow}r\\{1\langle S_{1}\rangle.G_{1},2\langle S_{2}\rangle.G_{2}\\}}$. Towards a contradiction, we assume ${\llbracket G_{1}\rrbracket}_{p}^{\tilde{q}}\neq{\llbracket G_{2}\rrbracket}_{p}^{\tilde{q}}$. There are many cases where Algorithm 1 generates differents routers for $p$ at $G_{1}$ and at $G_{2}$. We discuss the interesting case where ${\llbracket G_{1}\rrbracket}_{p}^{\tilde{q}}={{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}\mathbin{\triangleright}\ldots$ (algorithm 1) and ${\llbracket G_{2}\rrbracket}_{p}^{\tilde{q}}={{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q_{2}}}\mathbin{\triangleright}\ldots$ (algorithm 1). Then $G_{1}=p\mathbin{\twoheadrightarrow}q_{1}\\{\ldots\\}$ and $G_{2}=q_{2}\mathbin{\twoheadrightarrow}p\\{\ldots\\}$. We have $G_{1}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q_{1})=p\\{\ldots\\}$ and $G_{2}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q_{1})=\mathsf{skip}\ldots$ or $G_{2}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q_{1})=p{?}q_{2}\\{\ldots\\}$ (w.l.o.g., assume the former). Since $G$ is relative well-formed, the projection $G\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q_{1})$ must exist. Hence, since $p\notin\\{s,r\\}$ and $G_{1}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q_{1})\neq G_{2}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q_{1})$, it must be the case that $q_{1}\in\\{s,r\\}$—w.l.o.g., assume $q_{1}=s$. Then $G\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(p,q_{1})=q_{1}{!}r\\{1.p\\{\ldots\\},2.\mathsf{skip}\ldots\\}$, and thus $\mathrm{hdep}(p,q_{1},G)=\mathrm{hdep}(p,s,G)$ is true. This contradicts the assumption that $\mathrm{hdep}(p,s,G)$ is false. ∎ ###### Lemma 21. Suppose given a relative well-formed global type $G$ and a completable $\mathcal{N}\in\mathrm{net}(G)$ such that $\mathcal{N}^{\circlearrowright}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}\rangle{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}:\ell}}}}$, for some $c\in\mathsf{prt}(G)$. For every $G^{\prime}$ and $\beta_{1},\ldots,\beta_{n}$ ($n\geq 0$) such that $G\xrightarrow{\vspace{-.8ex}\beta_{1}}\ldots\xrightarrow{\vspace{-.8ex}\beta_{n}}G^{\prime}$ where $c$ is not involved in any $\beta_{k}$ (with $G=G^{\prime}$ if $n=0$), there exist $G^{\prime\prime}$, $d\in\mathsf{prt}(G)$, and $\beta^{\prime}_{1},\ldots,\beta^{\prime}_{m}$ ($m\geq 0$) such that ${G^{\prime}\xrightarrow{\vspace{-.8ex}\beta^{\prime}_{1}}\ldots\xrightarrow{\vspace{-.8ex}\beta^{\prime}_{m}}G^{\prime\prime}=c\mathbin{\twoheadrightarrow}d\\{i\langle S_{i}\rangle.G_{i}\\}_{i\in I}}$ where $c$ is not involved in any $\beta^{\prime}_{k}$ (with $G^{\prime\prime}=c\mathbin{\twoheadrightarrow}d\\{i\langle S_{i}\rangle.G_{i}\\}_{i\in I}$ if $m=0$). ###### Proof. By induction on $n$ (IH1). We first observe that the behavior on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}$ in $\mathcal{N}^{\circlearrowright}$ can only arise from the router generated for $c$ at $G$, following Algorithm 1 (algorithm 1) after finitely many passes through lines 1 (no dependency) and 1 (skip); for simplicity, assume only algorithm 1 applies. * • Case $n=0$. Let $x\geq 0$ denote the number of passes through algorithm 1 to generate the router for $c$ at $G$. We apply induction on $x$ (IH2): * – Case $x=0$. The router for $c$ at $G$ is generated through algorithm 1, so $G=c\mathbin{\twoheadrightarrow}d\\{i\langle S_{i}\rangle.G_{i}\\}_{i\in I}$, proving the thesis. * – Case $x=x^{\prime}+1$. Then $G=a\mathbin{\twoheadrightarrow}b\\{i\langle S_{i}\rangle.G_{i}\\}_{i\in I}$ and algorithm 1 returns the router for $c$ at $G_{j}$ for any $j\in I$. We have $G\xrightarrow{\vspace{-.8ex}a\rangle b:j\langle S_{j}\rangle}G_{j}$. Given the same implementation process for $c$ as in $\mathcal{N}$, we can construct a completable $\mathcal{M}\in\mathrm{net}(G_{j})$ such that $\mathcal{M}^{\circlearrowright}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}\rangle{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}:\ell}}}}$. Hence, the thesis follows from IH2. * • Case $n=n^{\prime}+1$. By assumption, $G\xrightarrow{\vspace{-.8ex}\beta_{1}}G^{\prime}_{1}$ where $c$ is not the sender or recipient in $\beta_{1}$. Hence, $G=a\mathbin{\twoheadrightarrow}b\\{i.\langle S_{i}\rangle.G_{i}\\}_{i\in I}$ where $G^{\prime}_{1}=G_{j}$ for some $j\in I$. The router for $c$ at $G$ is thus generated through algorithm 1 of Algorithm 1. It follows from Lemma 20 that this router is equal to the router for $c$ at $G^{\prime}_{1}$, but with one less pass through algorithm 1. Given the same implementation process for $c$ as in $\mathcal{N}$, we can construct a completable $\mathcal{M}\in\mathrm{net}(G^{\prime}_{1})$ such that $\mathcal{M}^{\circlearrowright}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}\rangle{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}:\ell}}}}$. Hence, the thesis follows from IH1. ∎ The proof of soundness relies on Proposition 22: if different reductions are enabled for a given process, then they do not exclude each other. That is, the same process is reached no matter the order in which those reductions are executed. We refer to simultaneously enabled reductions as _independent reductions_. ###### Proposition 22 (Independent Reductions). Suppose given a process $P\vdash\Omega;\Gamma$ and reduction labels $\alpha$ and $\alpha^{\prime}_{1},\ldots,\alpha^{\prime}_{n}$ ($n\geq 1$) where $\alpha\notin\\{\alpha^{\prime}_{1},\ldots,\alpha^{\prime}_{n}\\}$ (cf. Definition 9). If $P\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha}}}}$ and $P\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha^{\prime}_{1}}}}}\ldots\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha^{\prime}_{n}}}}}$, then there exists a process $Q$ such that $P\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha}}}}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha^{\prime}_{1}}}}}\ldots\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha^{\prime}_{n}}}}}Q$ and $P\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha^{\prime}_{1}}}}}\ldots\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha^{\prime}_{n}}}}}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha}}}}\leavevmode\nobreak\ Q$. ###### Proof. By induction on $n$: * • $n=1$. By assumption, $P\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha}}}}$ and $P\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha^{\prime}_{1}}}}}$. The proof proceeds by considering all possible combinations of shapes for $\alpha$ and $\alpha^{\prime}_{1}$ (forwarder, output/input, and selection/branching). Consider the case where ${\alpha=x\rangle y:a}$ and $\alpha^{\prime}_{1}=w\rangle z:b$. Because $P$ is well-typed, we infer that there are evaluation contexts $E_{1}$ and $E_{2}$ such that $P\equiv E_{1}[(\bm{\nu}xy)(x[a,c]\mathbin{\|}y(a,c).P_{1})]\equiv E_{2}[(\bm{\nu}wz)(w[b,d]\mathbin{\|}z(b,d).P_{2}])$ (Definition 34). Since the reductions labeled $\alpha$ and $\alpha^{\prime}_{1}$ are both enabled in $P$, it cannot be the case that $x,y\in\mathrm{fn}(P_{2})$ and $w,z\in\mathrm{fn}(P_{1})$. Hence, there exists an evaluation context $E_{3}$ such that $P\equiv E_{3}[(\bm{\nu}xy)(x[a,c]\mathbin{\|}y(a,c).P_{1})\mathbin{\|}(\bm{\nu}wz)(w[b,d]\mathbin{\|}z(b,d).P_{2}])$. Then $P\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha}}}}Q_{1}\equiv E_{3}[P_{1}\mathbin{\|}(\bm{\nu}wz)(w[b,d]\mathbin{\|}z(b,d).P_{2}])$ and $P\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha^{\prime}_{1}}}}}Q_{2}\equiv E_{3}[(\bm{\nu}xy)(x[a,c]\mathbin{\|}y(a,c).P_{1})\mathbin{\|}P_{2}]$. Let $Q=E_{3}[P_{1}\mathbin{\|}P_{2}]$; then $Q_{1}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha^{\prime}_{1}}}}}Q$ and $Q_{2}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha}}}}Q$. Hence, $P\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha}}}}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha^{\prime}_{1}}}}}Q$ and $P\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha^{\prime}_{1}}}}}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha}}}}Q$. All other cases proceed similarly. Note that when one of the reductions (say, $\alpha$) has a selection/branching label, such a reduction would discard some branches and thus possible behaviors. This is not an issue for establishing the thesis, because typability guarantees that the sub-process that enables the $\alpha^{\prime}$-labeled reduction does not appear under the to-be- discarded branches. Hence, the execution of $\alpha$ will not jeopardize the $\alpha^{\prime}$-labeled reduction. * • $n=n^{\prime}+1$ for $n^{\prime}\geq 1$. By the IH, ${P\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha}}}}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha^{\prime}_{1}}}}}\ldots\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha^{\prime}_{n^{\prime}}}}}}Q^{\prime}_{1}}$ and $P\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha^{\prime}_{1}}}}}\ldots\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha^{\prime}_{n^{\prime}}}}}}P^{\prime}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha}}}}Q^{\prime}_{1}$. By assumption, $P^{\prime}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha^{\prime}_{n}}}}}Q^{\prime}_{2}$. Since $P$ is well-typed, by Theorem 2 (Subject Reduction), $P^{\prime}$ is well-typed. Since $P^{\prime}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha}}}}Q^{\prime}_{1}$ and $P^{\prime}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha^{\prime}_{n}}}}}Q^{\prime}_{2}$, we can follow the same argumentation as in the base case to show that $Q^{\prime}_{1}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha^{\prime}_{n}}}}}Q$ and $Q^{\prime}_{2}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha}}}}Q$. Hence, $P\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha}}}}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha^{\prime}_{1}}}}}\ldots\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha^{\prime}_{n^{\prime}}}}}}Q^{\prime}_{1}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha^{\prime}_{n}}}}}Q$ and $P\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha^{\prime}_{1}}}}}\ldots\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha^{\prime}_{n^{\prime}}}}}}P^{\prime}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha^{\prime}_{n}}}}}Q^{\prime}_{2}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha}}}}Q$. ∎ To understand the proof of soundness and the rôle of independent reductions therein, consider the following example. We first introduce some notation which we also use in the proof of soundness: given an ordered sequence of reduction labels $A=(\alpha_{1},\ldots,\alpha_{k})$, we write $P\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}A}}}}Q$ to denote $P\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha_{1}}}}}\ldots\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha_{k}}}}}Q$. ###### Example 9. The recursive global type $G=\mu X.a\mathbin{\twoheadrightarrow}b:1\langle S\rangle.c\mathbin{\twoheadrightarrow}d:1\langle S\rangle.X$ features two independent exchanges. Consider a network $\mathcal{N}\in\mathrm{net}(G)$. Let $A$ denote the sequence of labeled reductions necessary to complete the exchange in $G$ between $a$ and $b$, and $C$ similarly for the exchange between $c$ and $d$. Assuming that communication with routers is not blocked by implementation processes, we have $\mathcal{N}^{\circlearrowright}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}A}}}}$ and $\mathcal{N}^{\circlearrowright}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}C}}}}$, because the exchanges are independent. Now, suppose that from $\mathcal{N}^{\circlearrowright}$ we observe $m$ times the sequence of $C$ reductions: $\mathcal{N}^{\circlearrowright}\underbrace{\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}C}}}}\ldots\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}C}}}}}_{\text{$m$ times}}N^{\prime}$. We see that $N^{\prime}$ is not a network of a global type reachable from $G$: there are still $m$ exchanges between $a$ and $b$ pending. Still, we can exhibit a series of transitions from $G$ that includes $m$ times the exchange between $c$ and $d$: $G\underbrace{\xrightarrow{\vspace{-.8ex}a\rangle b:1\langle S\rangle}\xrightarrow{\vspace{-.8ex}c\rangle d:1\langle S\rangle}\ldots\xrightarrow{\vspace{-.8ex}a\rangle b:1\langle S\rangle}\xrightarrow{\vspace{-.8ex}c\rangle d:1\langle S\rangle}}_{\text{$m$ times}}G$ Following these transitions, we can exhibit a corresponding sequence of reductions from $\mathcal{N}^{\circlearrowright}$ that includes $m$ times the sequence $C$ and ends up in another network $\mathcal{M}\in\mathrm{net}(G)$: $\mathcal{N}^{\circlearrowright}\underbrace{\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}A}}}}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}C}}}}\ldots\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}A}}}}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}C}}}}}_{\text{$m$ times}}\mathcal{M}^{\circlearrowright}$ At this point it is crucial that from $\mathcal{N}^{\circlearrowright}$ the sequences of reductions $A$ and $C$ can be performed independently. Hence, by Proposition 22, $\mathcal{N}^{\circlearrowright}\underbrace{\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}C}}}}\ldots\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}C}}}}}_{\text{$m$ times}}N^{\prime}\underbrace{\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}A}}}}\ldots\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}A}}}}}_{\text{$m$ times}}\mathcal{M}^{\circlearrowright}$. In the proof of soundness, whenever we assure that certain reductions are independent, we refer to those assurances as _independence facts_ (IFacts). Also, in the proof we consider labeled reductions, and distinguish between _protocol_ and _implementation_ reductions: the former are reductions with labels that indicate any interaction with a router, and the latter are any other reductions (which, by the definition of networks, can only occur within participant implementation processes). By a slight abuse of notation, given ordered sequences of reduction labels $A$ and $A^{\prime}$, we write $A^{\prime}\subseteq A$ to denote that $A^{\prime}$ is a subsequence of $A$, where the labels in $A^{\prime}$ appear in the same order in $A$ but not necessarily in sequence (and similarly for $A^{\prime}\subset A$). With $A\setminus A^{\prime}$ we denote the sequence obtained from $A$ by removing all the labels in $A^{\prime}$, and $A\cup A^{\prime}$ denotes the sequence obtained by adding the labels from $A^{\prime}$ to the end of $A$. ###### Theorem 23 (Operational Correspondence: Soundness). Suppose given a relative well-formed global type $G$ and a completable $\mathcal{N}\in\mathrm{net}(G)$. For every ordered sequence of $k\geq 0$ reduction labels $A=(\alpha_{1},\ldots,\alpha_{k})$ and $N^{\prime}$ such that $\mathcal{N}^{\circlearrowright}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}A}}}}N^{\prime}$, there exist $G^{\prime}$ and $\beta_{1},\ldots,\beta_{n}$ (with $n\geq 0$) such that (i) $G\xrightarrow{\vspace{-.8ex}\beta_{1}}\ldots\xrightarrow{\vspace{-.8ex}\beta_{n}}G^{\prime}$ and (ii) $N^{\prime}\longrightarrow^{\ast}\mathcal{M}^{\circlearrowright}$, with $\mathcal{M}\in\mathrm{net}(G^{\prime})$. ###### Proof. By induction on the structure of $G$; we detail the interesting cases of labeled exchanges with implicitly unfolded recursive definitions. We exhibit transitions $G\xrightarrow{\vspace{-.8ex}\beta_{1}}\ldots\xrightarrow{\vspace{-.8ex}\beta_{n}}G^{\prime}$ and establish a corresponding sequence of reductions $\mathcal{N}^{\circlearrowright}\longrightarrow^{\ast}\mathcal{M}^{\circlearrowright}$ that includes all the labels in $A$, with $\mathcal{M}\in\mathrm{net}(G^{\prime})$. During this step, we assure the independence between the observed reductions $A$ and the reductions we establish (IFacts). Using these independence assurances, we show that also $N^{\prime}\longrightarrow^{\ast}\mathcal{M}^{\circlearrowright}$. We apply induction on the size of $A$ (IH1) to show the existence of (i) $G^{\prime}$ and $\beta_{1},\ldots,\beta_{n}$ such that (i) $G\xrightarrow{\vspace{-.8ex}\beta_{1}}\ldots\xrightarrow{\vspace{-.8ex}\beta_{n}}G^{\prime}$ and (ii) $\mathcal{N}^{\circlearrowright}\longrightarrow^{\ast}\mathcal{M}^{\circlearrowright}$ including all reductions in $A$, with $\mathcal{M}\in\mathrm{net}(G^{\prime})$: * • Base case: then $A$ is empty, and the thesis holds trivially, with $G^{\prime}=G$ and $\mathcal{M}=\mathcal{N}$. * • Inductive case: then $A$ is non-empty. By the definition of networks (Definition 25), we know that reductions starting at $\mathcal{N}^{\circlearrowright}$ are protocol reductions related to an independent exchange in $G$, or implementation reductions. Every protocol reduction in $A$ is related to some exchange in $G$, and so we can group sequences of protocol reductions related to the same exchange. By construction, every such sequence of protocol reductions $A_{\ast}\subseteq A$ starts with an implementation sending a label to a router, i.e., with a label of the form $\alpha_{\ast}={{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}\rangle{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}:\ell$. For each such $\alpha_{\ast}$, the router in $\mathcal{N}$ of the sender $c$ has been synthesized from $G$ in a finite number of inductive steps. We take the $\alpha_{\ast}$ that originates from the router synthesized in the least number of steps. This gives us the $A_{\ast}$ starting with $\alpha_{\ast}$ that relates to an exchange in $G$ which is not prefixed by exchanges relating to any of the other $A^{\prime}_{\ast}\subseteq A\setminus A_{\ast}$. Networks are well-typed by definition. None of the reductions in $A_{\ast}$ are blocked by protocol reductions appearing earlier in $A$ (IFact 1): they originate from exchanges in $G$ appearing after the exchange related to $A_{\ast}$, and the priorities in their related types are thus higher than those in the types related to $A_{\ast}$, i.e., blocking by input or branching would contradict the well-typedness of $\mathcal{N}^{\circlearrowright}$. However, it may be that some implementation reductions $A_{+}\subseteq A\setminus A_{\ast}$ do block the reductions in $A_{\ast}$; they are also not blocked by any prior protocol reductions due to priorities (IFact 2). Hence, from $\mathcal{N}^{\circlearrowright}$ we can perform the implementation reductions in $A_{+}$. By Subject Reduction (Theorem 2), this results in another completed network $\mathcal{N}_{0}^{\circlearrowright}$ of $G$. This establishes the reduction sequence $\mathcal{N}^{\circlearrowright}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}A_{+}}}}}\mathcal{N}_{0}^{\circlearrowright}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}A_{\ast}}}}}$. By Lemma 21, there are $m\geq 0$ transitions $G\xrightarrow{\vspace{-.8ex}\beta_{1}}G_{1}\ldots\xrightarrow{\vspace{-.8ex}\beta_{m}}G_{m}$ where the initial prefix of $G_{m}$ corresponds to the labeled choice by the implementation of $c$: $G_{m}=c\mathbin{\twoheadrightarrow}d\\{i\langle S_{i}\rangle.G^{\prime}_{i}\\}_{i\in I}$, with $\ell\in I$. Additionally, $G_{m}$ contains exchanges related to every sequence of protocol reductions in $A\setminus A_{+}\setminus A_{\ast}$: all these sequences start with a selection from implementation to router, and thus the involved participants do not depend on any of the exchanges between $G$ and $G_{m}$, such that Lemma 20 applies. To establish a sequence of reductions from $\mathcal{N}_{0}^{\circlearrowright}$ to the completion of a network $\mathcal{N}_{m}\in\mathrm{net}(G_{m})$, we apply induction on $m$ (IH2): * – The base case where $m=0$ is trivial, with $G_{m}=G$ and thus $\mathcal{N}_{m}^{\circlearrowright}=\mathcal{N}_{0}^{\circlearrowright}$. * – In the inductive case, following the same approach as in the proof of completeness (Theorem 19), we reduce $\mathcal{N}_{0}^{\circlearrowright}\longrightarrow^{\ast}\mathcal{N}_{1}^{\circlearrowright}$ such that $\mathcal{N}_{1}\in\mathrm{net}(G_{1})$. Then, by IH2, $\mathcal{N}_{1}^{\circlearrowright}\longrightarrow^{\ast}\mathcal{N}_{m}^{\circlearrowright}$ where $\mathcal{N}_{m}\in\mathrm{net}(G_{m})$. Note that these reductions may require implementation reductions to unblock protocol reductions, and these implementation reductions may appear in $A$. None of the reductions from $\mathcal{N}_{0}^{\circlearrowright}$ to $\mathcal{N}_{m}^{\circlearrowright}$ can be blocked by any of the other protocol reductions in $A$, following again from priorities in types; hence, the leftover reductions in $A$ are independent from these reductions (IFact 3). Additionally, the sequence of protocol reductions $A_{\ast}$ was already enabled from $\mathcal{N}_{0}^{\circlearrowright}$, so those reductions are also independent (IFact 4). We know that $\mathcal{N}_{m}^{\circlearrowright}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}\rangle{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}:\ell}}}}$ and $G_{m}\xrightarrow{\vspace{-.8ex}c\rangle d:\ell\langle S_{\ell}\rangle}G^{\prime}_{\ell}$. From $\mathcal{N}_{m}^{\circlearrowright}$, we again follow the proof of completeness to show that $\mathcal{N}_{m}^{\circlearrowright}\longrightarrow^{\ast}\mathcal{M}_{\ell}^{\circlearrowright}$, where $\mathcal{M}_{\ell}\in\mathrm{net}(G^{\prime}_{\ell})$. Given the definition of routers, it must be that all the reductions in $A_{\ast}$ appear in this sequence of reductions. Let $A^{\prime}\subset A$ denote the leftover reductions from $A$ (i.e., $A$ except all reductions that occurred between $\mathcal{N}^{\circlearrowright}$ and $\mathcal{M}_{\ell}^{\circlearrowright}$, including $A_{\ast}$ and $A_{+}$). By IFacts 1–4, $\mathcal{M}_{\ell}^{\circlearrowright}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}A^{\prime}}}}}M^{\prime}$. Then by IH1, there exist $G^{\prime}$ and $\beta_{m+2},\ldots,\beta_{n}$ (with $n\geq m+1$) such that (i) $G^{\prime}_{\ell}\xrightarrow{\vspace{-.8ex}\beta_{m+2}}\ldots\xrightarrow{\vspace{-.8ex}\beta_{n}}G^{\prime}$ and (ii) $\mathcal{M}_{\ell}^{\circlearrowright}\longrightarrow^{\ast}\mathcal{M}^{\circlearrowright}$ including all reductions in $A^{\prime}$, with $\mathcal{M}\in\mathrm{net}(G^{\prime})$. Let $\beta_{m+1}=c\rangle d:\ell\langle S_{\ell}\rangle$. We have shown the existence of $G^{\prime}$ and $\beta_{1},\ldots,\beta_{n}$ such that (i) $G\xrightarrow{\vspace{-.8ex}\beta_{1}}\ldots\xrightarrow{\vspace{-.8ex}\beta_{m}}G_{m}\xrightarrow{\vspace{-.8ex}\beta_{m+1}}G^{\prime}_{\ell}\xrightarrow{\vspace{-.8ex}\beta_{m+2}}\ldots\xrightarrow{\vspace{-.8ex}\beta_{n}}G^{\prime}$ and (ii) $\mathcal{N}^{\circlearrowright}\longrightarrow^{\ast}\mathcal{N}_{m}^{\circlearrowright}\longrightarrow^{\ast}\mathcal{M}_{\ell}^{\circlearrowright}\longrightarrow^{\ast}\mathcal{M}^{\circlearrowright}$ including all reductions in $A$, with $\mathcal{M}\in\mathrm{net}(G^{\prime})$. We are left to show that from $\mathcal{N}^{\circlearrowright}\longrightarrow^{\ast}\mathcal{M}^{\circlearrowright}$ and $\mathcal{N}^{\circlearrowright}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}A}}}}N^{\prime}$, we can conclude that $N^{\prime}\longrightarrow^{\ast}\mathcal{M}^{\circlearrowright}$. We apply induction on the size of $A$ (IH3), using IFacts 1–4 and Proposition 22: * • Base case: Then $A$ is empty, there is nothing to do, and the thesis is proven. * • Inductive case: Then $A=A^{\prime}\cup(\alpha^{\prime})$. By IH3, $\mathcal{N}^{\circlearrowright}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}A^{\prime}}}}}N^{\prime\prime}\longrightarrow^{\ast}N^{\prime\prime\prime}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha^{\prime}}}}}\longrightarrow^{\ast}\mathcal{M}^{\circlearrowright}$. Moreover, by assumption, $N^{\prime\prime}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha^{\prime}}}}}N^{\prime}$. IFacts 1–4 show that the $\alpha^{\prime}$-labeled reduction is independent from the reductions between $N^{\prime\prime}$ and $N^{\prime\prime\prime}$. Hence, by Proposition 22, we have $\mathcal{N}^{\circlearrowright}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}A^{\prime}}}}}N^{\prime\prime}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha^{\prime}}}}}N^{\prime}\longrightarrow^{\ast}\mathcal{M}^{\circlearrowright}$. That is, $\mathcal{N}^{\circlearrowright}\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}A}}}}N^{\prime}\longrightarrow^{\ast}\mathcal{M}^{\circlearrowright}$, proving the thesis. ∎ In the light of Theorem 23, let us revisit Example 9: ###### Example 10 (Revisiting Example 9). Recall the global type $G=\mu X.a\mathbin{\twoheadrightarrow}b:1\langle S\rangle.c\mathbin{\twoheadrightarrow}d:1\langle S\rangle.X$ from Example 9, with two independent exchanges. We take some $\mathcal{N}\in\mathrm{net}(G)$ such that $\mathcal{N}^{\circlearrowright}\underbrace{\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}C}}}}\ldots\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}C}}}}}_{\text{$m$ times}}N^{\prime}$, where $C$ denotes the sequence of reduction labels corresponding to the exchange between $c$ and $d$. By Theorem 23, there indeed are $G^{\prime}$ and $\beta_{1},\ldots,\beta_{n}$ such that $G\xrightarrow{\vspace{-.8ex}\beta_{1}}\ldots\xrightarrow{\vspace{-.8ex}\beta_{n}}G^{\prime}$ and $N^{\prime}\longrightarrow^{\ast}\mathcal{M}^{\circlearrowright}$, with $\mathcal{M}\in\mathrm{net}(G^{\prime})$. To be precise, following Theorem 23, indeed $G\underbrace{\xrightarrow{\vspace{-.8ex}a\rangle b:1\langle S\rangle}\xrightarrow{\vspace{-.8ex}c\rangle d:1\langle S\rangle}\ldots\xrightarrow{\vspace{-.8ex}a\rangle b:1\langle S\rangle}\xrightarrow{\vspace{-.8ex}c\rangle d:1\langle S\rangle}}_{\text{$m$ times}}G\quad\text{ and }\quad\mathcal{N}^{\circlearrowright}\underbrace{\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}C}}}}\ldots\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}C}}}}}_{\text{$m$ times}}N^{\prime}\underbrace{\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}A}}}}\ldots\mathbin{{\color[rgb]{0.7578125,0.38671875,0.89453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.7578125,0.38671875,0.89453125}\xrightharpoondown{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}A}}}}}_{\text{$m$ times}}\mathcal{M}^{\circlearrowright}$ where $A$ is the sequence of reduction labels corresponding to the exchange between $a$ and $b$ and ${\mathcal{M}\in\mathrm{net}(G)}$. ### 4.4 Routers Strictly Generalize Centralized Orchestrators Unlike our decentralized analysis, previous analyses of global types using binary session types rely on centralized orchestrators (called mediums [12] or arbiters [16]). Here, we show that our approach strictly generalizes these centralized approaches. Readers interested in our decentralized approach in action may safely skip this section and go directly to Section 5. We introduce an algorithm that synthesizes an orchestrator—a single process that orchestrates the interactions between a protocol’s participants (§ 4.4.1). We show that the composition of this orchestrator with a context of participant implementations is behaviorally equivalent to the specific case in which routed implementations are organized in a _centralized composition_ (Theorem 27 in § 4.4.2). #### 4.4.1 Synthesis of Orchestrators 1 def _${\mathsf{O}}_{\tilde{q}}[G]$_ as 2 switch _$G$_ do 3 case _$s\mathbin{\twoheadrightarrow}r\\{i\langle S_{i}\rangle\mathbin{.}G_{i}\\}_{i\in I}$_ do 4 $\mathsf{deps}:=\\{q\in\tilde{q}\mid\mathrm{hdep}(q,s,G)\vee\mathrm{hdep}(q,r,G)\\}$ 5 return ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}\triangleright\\{i{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}r}}}\triangleleft i\cdot\underline{{(\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}}\triangleleft i)}_{q\in\mathsf{deps}}}\cdot{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}(v)\mathbin{.}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}r}}}[w]\cdot(v\mathbin{\leftrightarrow}w\mathbin{\|}{\mathsf{O}}_{\tilde{q}}[G_{i}])\\}_{i\in I}$ 6 7 case _$\mu X\mathbin{.}G^{\prime}$_ do 8 $\tilde{q}^{\prime}:=\\{q\in\tilde{q}\mid G\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{0}q\neq\bullet\\}$ 9 if _$\tilde{q}^{\prime}\neq\emptyset$_ then return $\mu X({({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}})}_{q\in\tilde{q}^{\prime}})\mathbin{.}{\mathsf{O}}_{\tilde{q}^{\prime}}[G^{\prime}]$ 10 else return $\bm{0}$ 11 12 13 case _$X$_ do return $X{\langle{({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}})}_{q\in\tilde{q}}\rangle}$ 14 15 case _$\mathsf{skip}\mathbin{.}G^{\prime}$_ do return ${\mathsf{O}}_{\tilde{q}}[G^{\prime}]$ 16 17 case _$\mathsf{end}$_ do return $\bm{0}$ 18 Algorithm 2 Synthesis of Orchestrator Processes (Def. 36). We define the synthesis of an orchestrator from a global type. The orchestrator of $G$ will have a channel endpoint ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p_{i}}}$ for connecting to the process implementation of every $p_{i}\in\mathsf{prt}(G)$. ###### Definition 36 (Orchestrator). Given a global type $G$ and participants $\tilde{q}$, Algorithm 2 defines the synthesis of an _orchestrator process_ , denoted ‘$\mkern 1.0mu{\mathsf{O}}_{\tilde{q}}[G]\mkern-3.0mu$’, that orchestrates interactions according to $G$. Algorithm 2 follows a similar structure as the router synthesis algorithm (Algorithm 1). The input parameter ‘$\tilde{q}$’ keeps track of active participants, making sure recursions are well-defined; it should be initialized as ‘$\mathsf{prt}(G)$’. We briefly discuss how the orchestrator process is generated. The interesting case is an exchange ‘$p\mathbin{\twoheadrightarrow}q\\{i\langle U_{i}\rangle\mathbin{.}G_{i}\\}_{i\in I}$’ (algorithm 2), where the algorithm combines the several cases of the router’s algorithm (that depend on the involvement of the router’s participant). First, the sets of participants ‘$\mathsf{deps}$’ that depend on the sender and on the recipient are computed (algorithm 2) using the auxiliary predicate ‘$\mathrm{hdep}$’ (cf. Def. 18). Then, the algorithm returns a process (algorithm 2) that receives a label $i\in I$ over ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}$; forwards it over ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}r}}$ and over ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}$ for all $q\in\mathsf{deps}$; receives a channel over ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}$; forwards it over ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}r}}$; and continues as ‘${\mathsf{O}}_{\tilde{q}}[G_{i}]$’. The synthesis of a recursive definition ‘$\mu X\mathbin{.}G^{\prime}$’ (algorithm 2) requires care, as the set of active participants $\tilde{q}$ may change. In order to decide which $q\in\tilde{q}$ are active in $G^{\prime}$, the algorithm computes the local projection of $G$ onto each $q\in\tilde{q}$ to determine the orchestrator’s future behavior on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}$, creating a new set $\tilde{q}^{\prime}$ with those $q\in\tilde{q}$ for which the projection is different from ‘$\bullet$’ (algorithm 2). Then, the algorithm returns a recursive process with as context the channel endpoints ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}$ for $q\in\tilde{q}^{\prime}$, with ‘${\mathsf{O}}_{\tilde{q}^{\prime}}[G^{\prime}]$’ as the body. The synthesis of a recursive call ‘$X$’ (algorithm 2) yields a recursive call with as context the channels ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}$ for $q\in\tilde{q}$. Finally, for ‘$\mathsf{skip}\mathbin{.}G^{\prime}$’ (algorithm 2) the algorithm returns the orchestrator for $G^{\prime}$, and for ‘$\bullet$’ (algorithm 2) the algorithm returns ‘$\bm{0}$’. There is a minor difference between the orchestrators synthesized by Algorithm 2 and the mediums defined by Caires and Pérez [12]. The difference is in the underlined portion in algorithm 2, which denotes explicit messages (obtained via dependency detection) needed to deal with non-local choices. The mediums by Caires and Pérez do not include such communications, as their typability is based on local types, which rely on a merge operation at projection time. The explicit actions in algorithm 2 make the orchestrator compatible with participant implementations that connect with routers. Aside from these actions, our concept of orchestrator is essentially the same as that of the mediums by Caires and Pérez. Crucially, orchestrators can be typed using local projection (cf. Def. 22) similar to the typing of routers using relative projection (cf. Theorem 11). This result follows by construction: ###### Theorem 24. Given a closed, relative well-formed global type $G$, $\displaystyle{\mathsf{O}}_{\mathsf{prt}(G)}[G]\vdash\emptyset;{({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}{:}\leavevmode\nobreak\ \overline{(G\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{0}p)})}_{p\in\mathsf{prt}(G)}.$ ###### Proof. We prove a more general statement. Suppose given a closed, relative well- formed global type $G$. Also, suppose given a global type $G_{s}\leq_{C}G$. Consider: * • the participants that are active in $G_{s}$: $\tilde{q}=\\{q\in\mathsf{prt}(G)\mid\exists p\in\mathsf{prt}(G).\leavevmode\nobreak\ (p,q)\in\mathrm{active}(C,G)\\}$, * • the absolute priority of $G_{s}$: $\mathsf{o}_{C}=\mathrm{ctxpri}(C)$, * • the sequence of bound recursion variables of $G_{s}$: $\widetilde{X_{C}}=\mathrm{ctxbind}(C)$, * • for every $X\in\widetilde{X_{C}}$: * – the body of the recursive definition on $X$ in $G$: $G_{X}=\mathrm{recdef}(X,G)$, * – the participants that are active in $G_{X}$: $\tilde{q}_{X}=\\{q\in\mathsf{prt}(G)\mid\exists p\in\mathsf{prt}(G).\leavevmode\nobreak\ (p,q)\in\mathrm{recactive}(X,G)\\}$, * – the absolute priority of $G_{X}$: $\mathsf{o}_{X}=\mathrm{varpri}(X,G)$, * – the sequence of bound recursion variables of $G_{X}$ excluding $X$: $\widetilde{Y_{X}}=\mathrm{subbind}(\mu X\mathbin{.}G_{X},G)$, * – the type required for ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}$ for a recursive call on $X$: $\displaystyle A_{X,q}=\mathrm{deepUnfold}(\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}q},{(Y,t_{Y},\overline{G_{Y}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{Y}}q})}_{Y\in\widetilde{Y_{X}}})$ * – the minimum lift for typing a recursive definition on $X$: $t_{X}=\max_{\mathsf{pr}}\left({(A_{X,q})}_{q\in\tilde{q}_{X}}\right)+1$, * • the type expected for ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}$ for the orchestrator for $G_{s}$: $\displaystyle D_{q}=\mathrm{deepUnfold}(\overline{G_{s}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}q},{(X,t_{X},\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}q})}_{X\in\widetilde{X_{C}}}).$ Then, we have: $\displaystyle{\mathsf{O}}_{\tilde{q}}[G_{s}]\vdash{\left(X{:}\leavevmode\nobreak\ {\big{(}A_{X,q}\big{)}}_{q\in\tilde{q}_{X}}\right)}_{X\in\widetilde{X_{C}}};\leavevmode\nobreak\ {\left({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}{:}\leavevmode\nobreak\ D_{q}\right)}_{q\in\tilde{q}}$ Similar to how Theorem 11 follows from Theorem 16, the thesis follows as a corollary from this more general statement (cf. the proof of Theorem 11 on Section 4.3.1). We apply induction on the structure of $G_{s}$, with six cases as in Algorithm 2. We only detail the cases of exchange and recursion. * • _Exchange_ : $G_{s}=s\mathbin{\twoheadrightarrow}r\\{i\langle S_{i}\rangle\mathbin{.}G_{i}\\}_{i\in I}$ (algorithm 2). Following similar reasoning as in the case for exchange in the proof of Theorem 16, we can omit the unfoldings on types, as well as the recursive context. Let $\mathsf{deps}_{s}:=\\{q\in\tilde{q}\mid\mathrm{hdep}(q,s,G_{s})\\}$ and $\mathsf{deps}_{r}:=\\{q\in\tilde{q}\setminus\mathsf{deps}_{s}\mid\mathrm{hdep}(q,r,G_{s})\\}$. Note that $\mathsf{deps}_{s}\cup\mathsf{deps}_{r}$ coincides with $\mathsf{deps}$ as defined on algorithm 2 and that $s,r\notin\mathsf{deps}_{s}\cup\mathsf{deps}_{r}$. Let us take stock of the types we expect for each of the orchestrator’s channels. For ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}$ we expect $\displaystyle\overline{G_{s}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}}s}$ $\displaystyle=\overline{{{\oplus}}^{\mathsf{o}}\\{i{:}\leavevmode\nobreak\ {{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}\mathbin{\otimes}^{\mathsf{o}+1}(G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}s)\\}_{i\in I}}$ $\displaystyle=\&^{\mathsf{o}}\\{i{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}}\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{\mathsf{o}+1}\overline{(G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}s)}\\}_{i\in I}.$ (34) For ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}r}}$ we expect $\displaystyle\overline{G_{s}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}}r}$ $\displaystyle=\overline{{\&}^{\mathsf{o}+2}\\{i{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}}\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{\mathsf{o}+3}(G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}r)\\}_{i\in I}}$ $\displaystyle={{\oplus}}^{\mathsf{o}+2}\\{i{:}\leavevmode\nobreak\ {{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}\mathbin{\otimes}^{\mathsf{o}+3}\overline{(G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}r)}\\}_{i\in I}.$ (35) For each $q\in\mathsf{deps}_{s}$, for ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}$ we expect $\displaystyle\overline{G_{s}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}}q}$ $\displaystyle=\overline{\&^{\mathsf{o}+2}\\{i{:}\leavevmode\nobreak\ (G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}q)\\}_{i\in I}}$ $\displaystyle={{\oplus}}^{\mathsf{o}+2}\\{i{:}\leavevmode\nobreak\ \overline{(G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}q)}\\}_{i\in I}.$ (36) For each $q\in\mathsf{deps}_{r}$, for ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}$ we expect $\displaystyle\overline{G_{s}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}}q}$ $\displaystyle=\overline{\&^{\mathsf{o}+3}\\{i{:}\leavevmode\nobreak\ (G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}q)\\}_{i\in I}}$ $\displaystyle={{\oplus}}^{\mathsf{o}+3}\\{i{:}\leavevmode\nobreak\ \overline{(G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}q)}\\}_{i\in I}.$ (37) For each $q\in\tilde{q}\setminus\mathsf{deps}_{s}\setminus\mathsf{deps}_{r}\setminus\\{s,r\\}$, for ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}$ we expect $\displaystyle\overline{G_{s}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}}q}$ $\displaystyle=\overline{G_{i^{\prime}}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}q}\leavevmode\nobreak\ \text{for any $i^{\prime}\in I$}.$ (38) Let us now consider the process returned by Algorithm 1, with each prefix marked with a number. ${\mathsf{O}}_{\tilde{q}}[G]=\underbrace{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}\triangleright\\{i{:}}_{1}\leavevmode\nobreak\ \underbrace{\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}r}}}\triangleleft i}_{2_{i}}\cdot\underbrace{{(\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}}\triangleleft i)}_{q\in\mathsf{deps}}}_{3_{i}}\cdot\underbrace{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}(v)}_{4_{i}}\mathbin{.}\underbrace{\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}r}}}[w]}_{5_{i}}\cdot(v\mathbin{\leftrightarrow}w\mathbin{\|}{\mathsf{O}}_{\tilde{q}}[G_{i}])\\}_{i\in I}$ For each $i^{\prime}\in I$, let $C_{i^{\prime}}:=C[s\mathbin{\twoheadrightarrow}r(\\{i\langle S_{i}\rangle\mathbin{.}G_{i}\\}_{i\in I\setminus\\{i^{\prime}\\}}\cup\\{i^{\prime}\langle S_{i^{\prime}}\rangle\mathbin{.}[]\\})]$. Clearly, $G_{i^{\prime}}\leq_{C_{i^{\prime}}}G$. Also, because we are not adding recursion binders, the current value of $\tilde{q}$ is appropriate for the IH. With $C_{i^{\prime}}$ and $\tilde{q}$, we apply the IH to obtain the typing of ${\mathsf{O}}_{\tilde{q}}[G_{i^{\prime}}]$, where priorities start at $\mathrm{ctxpri}(C_{i^{\prime}})=\mathrm{ctxpri}(C)+4$ (cf. Def. 31). Following these typings, Figure 12 gives the typing of ${\mathsf{O}}_{\tilde{q}}[G_{s}]$, referring to parts of the process by the number marking its foremost prefix above. Clearly, the priorities in the derivation in Figure 12 meet all requirements. The order of the applications of ${\oplus}^{\star}$ for each $q\in\mathsf{deps}_{s}\cup\mathsf{deps}_{r}$ does not matter, since the selection actions are asynchronous. Id $\forall i\in I.\leavevmode\nobreak\ v\mathbin{\leftrightarrow}w\vdash\begin{array}[t]{@{}l@{}}v{:}\leavevmode\nobreak\ \overline{S_{i}},w{:}\leavevmode\nobreak\ S_{i}\end{array}$ $\forall i\in I.\leavevmode\nobreak\ {\mathsf{O}}_{\tilde{q}}[G_{i}]\vdash\begin{array}[t]{@{}l@{}}{({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}{:}\leavevmode\nobreak\ \overline{G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}q})}_{q\in\tilde{q}}\end{array}$ Mix $\forall i\in I.\leavevmode\nobreak\ v\mathbin{\leftrightarrow}w\mathbin{\|}{\mathsf{O}}_{\tilde{q}}[G_{i}]\vdash\begin{array}[t]{@{}l@{}}v{:}\leavevmode\nobreak\ \overline{S_{i}},w{:}\leavevmode\nobreak\ S_{i},\\\ {({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}{:}\leavevmode\nobreak\ \overline{G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}}q)}_{q\in\tilde{q}}\end{array}$ $\mathbin{\otimes}^{\star}$ $\forall i\in I.\leavevmode\nobreak\ 5_{i}\vdash\begin{array}[t]{@{}l@{}}v{:}\leavevmode\nobreak\ \overline{S_{i}},\\\ {{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}r}}{:}\leavevmode\nobreak\ {{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}\mathbin{\otimes}^{\mathsf{o}+3}\overline{(G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}r)},\\\ {({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}{:}\leavevmode\nobreak\ \overline{G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}q})}_{q\in\tilde{q}\setminus\\{r\\}}\end{array}$ $\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}$ $\forall i\in I.\leavevmode\nobreak\ 4_{i}\vdash\begin{array}[t]{@{}l@{}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}}\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{\mathsf{o}+1}\overline{(G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}s)},\\\ {{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}r}}{:}\leavevmode\nobreak\ {{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}\mathbin{\otimes}^{\mathsf{o}+3}\overline{(G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}r)},\\\ {({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}{:}\leavevmode\nobreak\ \overline{G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}q})}_{q\in\tilde{q}\setminus\\{s,r\\}}\end{array}$ $\forall q\in\mathsf{deps}_{s}\cup\mathsf{deps}_{r}.\leavevmode\nobreak\ {\oplus}^{\star}$ $\forall i\in I.\leavevmode\nobreak\ 3_{i}\vdash\begin{array}[t]{@{}l@{}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}}\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{\mathsf{o}+1}\overline{(G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}s)},\\\ {{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}r}}{:}\leavevmode\nobreak\ {{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}\mathbin{\otimes}^{\mathsf{o}+3}\overline{(G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}r)},\\\ {({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}{:}\leavevmode\nobreak\ {{\oplus}}^{\mathsf{o}+2}\\{i{:}\leavevmode\nobreak\ \overline{(G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}q)}\\}_{i\in I})}_{q\in\mathsf{deps}_{s}}\\\ {({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}{:}\leavevmode\nobreak\ {{\oplus}}^{\mathsf{o}+3}\\{i{:}\leavevmode\nobreak\ \overline{(G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}q)}\\}_{i\in I})}_{q\in\mathsf{deps}_{r}}\\\ {({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}{:}\leavevmode\nobreak\ \overline{G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}q})}_{q\in\tilde{q}\setminus\mathsf{deps}_{s}\setminus\mathsf{deps}_{r}\setminus\\{s,r\\}}\end{array}$ ${\oplus}^{\star}$ $\forall i\in I.\leavevmode\nobreak\ 2_{i}\vdash\begin{array}[t]{@{}l@{}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}}\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{\mathsf{o}+1}\overline{(G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}s)},\\\ {{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}r}}{:}\leavevmode\nobreak\ {{\oplus}}^{\mathsf{o}+2}\\{i{:}\leavevmode\nobreak\ {{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}\mathbin{\otimes}^{\mathsf{o}+3}\overline{(G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}r)}\\}_{i\in I},\\\ {({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}{:}\leavevmode\nobreak\ {{\oplus}}^{\mathsf{o}+2}\\{i{:}\leavevmode\nobreak\ \overline{(G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}q)}\\}_{i\in I})}_{q\in\mathsf{deps}_{s}}\\\ {({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}{:}\leavevmode\nobreak\ {{\oplus}}^{\mathsf{o}+3}\\{i{:}\leavevmode\nobreak\ \overline{(G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}q)}\\}_{i\in I})}_{q\in\mathsf{deps}_{r}}\\\ {({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}{:}\leavevmode\nobreak\ \overline{G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}q})}_{q\in\tilde{q}\setminus\mathsf{deps}_{s}\setminus\mathsf{deps}_{r}\setminus\\{s,r\\}}\end{array}$ $\&$ ${\mathsf{O}}_{\tilde{q}}[G_{s}]=1\vdash\begin{array}[t]{@{}lr@{}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}{:}\leavevmode\nobreak\ \&^{\mathsf{o}}\\{i{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}}\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{\mathsf{o}+1}\overline{(G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}s)}\\}_{i\in I},&\text{(cf.\ \eqref{eq:mSType})}\\\ {{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}r}}{:}\leavevmode\nobreak\ {{\oplus}}^{\mathsf{o}+2}\\{i{:}\leavevmode\nobreak\ {{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S_{i}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}\mathbin{\otimes}^{\mathsf{o}+3}\overline{(G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}r)}\\}_{i\in I},&\text{(cf.\ \eqref{eq:mRType})}\\\ {({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}{:}\leavevmode\nobreak\ {{\oplus}}^{\mathsf{o}+2}\\{i{:}\leavevmode\nobreak\ \overline{(G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}q)}\\}_{i\in I})}_{q\in\mathsf{deps}_{s}}&\text{(cf.\ \eqref{eq:mDepSType})}\\\ {({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}{:}\leavevmode\nobreak\ {{\oplus}}^{\mathsf{o}+3}\\{i{:}\leavevmode\nobreak\ \overline{(G_{i}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}q)}\\}_{i\in I})}_{q\in\mathsf{deps}_{r}}&\text{(cf.\ \eqref{eq:mDepRType})}\\\ {({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}{:}\leavevmode\nobreak\ \overline{G_{i^{\prime}}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}+4}q})}_{q\in\tilde{q}\setminus\mathsf{deps}_{s}\setminus\mathsf{deps}_{r}\setminus\\{s,r\\}}&\text{(cf.\ \eqref{eq:mDepOtherType})}\end{array}$ Figure 12: Typing derivation used in the proof of Theorem 24. * • _Recursive definition_ : $G_{s}=\mu Z\mathbin{.}G^{\prime}$ (algorithm 2). Let $\displaystyle\tilde{q}^{\prime}:=\\{q\in\tilde{q}\mid G_{s}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}}q\neq\bullet\\}$ (39) (as on algorithm 2). The analysis depends on whether $\tilde{q}^{\prime}=\emptyset$ or not. * – If $\tilde{q}^{\prime}=\emptyset$ (algorithm 2), let us take stock of the types expected for each of the orchestrator’s channels. For now, we omit the substitutions in the types. For each $q\in\tilde{q}$, for ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}$ we expect $\displaystyle\overline{G_{s}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}q}$ $\displaystyle=\bullet.$ (40) Because all expected types are $\bullet$, the substitutions do not affect the types, so we can omit them altogether. First we apply Empty, giving us an arbitrary recursive context, thus the recursive context we need. Then, we apply $\bullet$ for ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}$ for each $q\in\tilde{q}$ (cf. (40)), and obtain the typing of ${\mathsf{O}}_{\tilde{q}}[G_{s}]$ (omitting the recursive context): ${\mathsf{O}}_{\tilde{q}}[G_{s}]=\bm{0}\vdash{({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}{:}\leavevmode\nobreak\ \bullet)}_{q\in\tilde{q}}.$ * – If $\tilde{q}^{\prime}\neq\emptyset$ (algorithm 2), let us take stock of the types expected for each of the orchestrator’s channels. Note that, because of the recursive definition on $Z$ in $G_{s}$, there cannot be another recursive definition in the context $C$ capturing the recursion variable $Z$. Therefore, by Definition 29, $Z\notin\widetilde{X_{C}}$. For each $q\in\tilde{q}^{\prime}$, for ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}$ we expect $\displaystyle\mathrm{deepUnfold}(\overline{G_{s}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}q},\ldots)$ $\displaystyle=\mathrm{deepUnfold}(\overline{\mu Z\mathbin{.}(G^{\prime}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}q)},\ldots)$ $\displaystyle=\mathrm{deepUnfold}(\mu Z\mathbin{.}\overline{G^{\prime}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}q},\ldots)$ $\displaystyle=\mu Z\mathbin{.}\mathrm{deepUnfold}(\overline{G^{\prime}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}q},{(X,t_{X},\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}q})}_{X\in\widetilde{X_{C}}}).$ (41) For each $q\in\tilde{q}\setminus\tilde{q}^{\prime}$, for ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}$ we expect $\displaystyle\mathrm{deepUnfold}(\overline{G_{s}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}q},\ldots)$ $\displaystyle=\mathrm{deepUnfold}(\bullet,\ldots)=\bullet.$ (42) We also need an assignment in the recursive context for every $X\in\widetilde{X_{C}}$, but not for $Z$. Let $C^{\prime}=C[\mu Z\mathbin{.}[]]$. Clearly, $G^{\prime}\leq_{C^{\prime}}G$. Let us establish some facts about the recursion binders, priorities, and active participants related to $C^{\prime}$, $G^{\prime}$, and $Z$: * * $\widetilde{X_{C^{\prime}}}=\mathrm{ctxbind}(C^{\prime})=(\mathrm{ctxbind}(C),Z)=(\widetilde{X_{C}},Z)$ (cf. Def. 29). * * $G_{Z}=\mathrm{recdef}(Z,G)=G^{\prime}$, as proven by the context $C^{\prime}$ (cf. Def. 30). * * $\widetilde{Y_{Z}}=\mathrm{subbind}(\mu Z\mathbin{.}G_{Z},G)=\mathrm{ctxbind}(C)=\widetilde{X_{C}}$. * * $\mathsf{o}_{C^{\prime}}=\mathrm{ctxpri}(C^{\prime})=\mathrm{ctxpri}(C)=\mathsf{o}_{C}$, and $\mathsf{o}_{Z}=\mathrm{varpri}(Z,G)=\mathrm{ctxpri}^{C}(=)\mathsf{o}_{C}$, and hence $\mathsf{o}_{C^{\prime}}=\mathsf{o}_{Z}$ (cf. Def. 31). * * $\tilde{q}_{Z}=\tilde{q}^{\prime}$ (cf. Def. 32 and (39)). Because $\widetilde{X_{C^{\prime}}}=(\widetilde{X_{C}},Z)$ and $\tilde{q}^{\prime}=\tilde{q}_{Z}$, $\tilde{q}^{\prime}$ is appropriate for the IH. We apply the IH on $C^{\prime}$, $G^{\prime}$, and $\tilde{q}^{\prime}$ to obtain a typing for ${\mathsf{O}}_{\tilde{q}^{\prime}}[G^{\prime}]$, where we immediately make use of the facts established above. We given the assignment to $Z$ in the recursive context separate from those for the recursion variables in $\widetilde{X_{C}}$. Also, by Proposition 15, we can write the final unfolding on $Z$ in the types separately. $\displaystyle{\mathsf{O}}_{\tilde{q}^{\prime}}[G^{\prime}]\vdash\begin{array}[t]{@{}l@{}}{\left(X{:}\leavevmode\nobreak\ {\big{(}\mathrm{deepUnfold}(\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}q},{(Y,t_{Y},\overline{G_{Y}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{Y}}q})}_{Y\in\widetilde{Y_{X}}})\big{)}}_{q\in\tilde{q}_{X}}\right)}_{X\in\widetilde{X_{C}}},\\\\[6.0pt] Z{:}\leavevmode\nobreak\ {\big{(}\mathrm{deepUnfold}(\overline{G^{\prime}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}q},{(X,t_{X},\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}q})}_{X\in\widetilde{X_{C}}})\big{)}}_{q\in\tilde{q}^{\prime}};\\\\[6.0pt] {\left({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}{:}\leavevmode\nobreak\ \mathrm{unfold}^{t_{Z}}(\mu Z\mathbin{.}\mathrm{deepUnfold}(\overline{G^{\prime}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}q},{(X,t_{X},\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}q})}_{X\in\widetilde{X_{C}}}))\right)}_{q\in\tilde{q}^{\prime}}\end{array}$ By assumption, we have $\displaystyle t_{Z}$ $\displaystyle=\max_{\mathsf{pr}}{\left(\mathrm{deepUnfold}(\overline{G^{\prime}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}q},{(X,t_{X},\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}q})}_{X\in\widetilde{X_{C}}})\right)}_{q\in\tilde{q}^{\prime}}+1,$ so $t_{Z}$ is clearly bigger than the maximum priority appearing in the types before unfolding. Hence, we can apply Rec to eliminate $Z$ from the recursive context, and to fold the types, giving the typing of ${\mathsf{O}}_{\tilde{q}}[G_{s}]=\mu Z({({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}})}_{q\in\tilde{q}^{\prime}})\mathbin{.}{\mathsf{O}}_{\tilde{q}^{\prime}}[G^{\prime}]$: $\displaystyle{\mathsf{O}}_{\tilde{q}}[G_{s}]\vdash\begin{array}[t]{@{}l@{}}{\left(X{:}\leavevmode\nobreak\ {\big{(}\mathrm{deepUnfold}(\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}q},{(Y,t_{Y},\overline{G_{Y}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{Y}}q})}_{Y\in\widetilde{Y_{X}}})\big{)}}_{q\in\tilde{q}_{X}}\right)}_{X\in\widetilde{X_{C}}};\\\\[6.0pt] {\left({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}{:}\leavevmode\nobreak\ \mu Z\mathbin{.}\mathrm{deepUnfold}(\overline{G^{\prime}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}q},{(X,t_{X},\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}q})}_{X\in\widetilde{X_{C}}})\right)}_{q\in\tilde{q}^{\prime}}\end{array}$ In this typing, the type for ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}$ for every $q\in\tilde{q}^{\prime}$ concurs with (41). For every $q\in\tilde{q}\setminus\tilde{q}^{\prime}$, we can add the type for ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}$ in (42) by applying $\bullet$. This proves the thesis. * • _Recursive call_ : $G_{s}=Z$ (algorithm 2). Following similar reasoning as in the case of recursive call in the proof of Theorem 16, let us take stock of the types we expect for our orchestrator’s channels. For each $q\in\tilde{q}$, for ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}$ we expect $\displaystyle\mathrm{deepUnfold}(\overline{G_{s}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}q},\ldots)$ $\displaystyle=\mathrm{deepUnfold}(\overline{Z\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{C}}q},\ldots)$ $\displaystyle=\mathrm{deepUnfold}(\overline{Z},\ldots)$ $\displaystyle=\mathrm{deepUnfold}(Z,{(X,t_{X},\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}q})}_{X\in(\tilde{X}_{1},Z,\widetilde{Y_{Z}})})$ $\displaystyle=\mathrm{deepUnfold}(Z,{(X,t_{X},\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}q})}_{X\in(Z,\widetilde{Y_{Z}})})$ $\displaystyle=\mu Z\mathbin{.}({\uparrow^{t_{Z}}}\mathrm{deepUnfold}(\overline{G_{Z}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{Z}}q},{(X,t_{X},\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}q})}_{X\in\widetilde{Y_{Z}}}))$ (43) Also, we need an assignment in the recursive context for every $X\in\widetilde{X_{C}}$. By Lemma 13, $\tilde{q}=\tilde{q}_{Z}$. Hence, for $Z$, the assignment should be as follows: $\displaystyle Z{:}\leavevmode\nobreak\ {\left(\mathrm{deepUnfold}(\overline{G_{Z}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{Z}}q},{(X,t_{X},\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}q)}_{X\in\widetilde{Y_{Z}}}})\right)}_{q\in\tilde{q}}$ (44) We apply Var to obtain the typing of ${\mathsf{O}}_{\tilde{q}}[G_{s}]$, where we make us the rule’s allowance for an arbitrary recursive context up to the assignment to $Z$. Var is applicable, because the types are recursive definitions on $Z$, concurring with the types assigned to $Z$, and lifted by a common lifter $t_{Z}$. Var ${\mathsf{O}}_{\tilde{q}}[G_{s}]=X{\langle{({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}})}_{q\in\tilde{q}}\rangle}\vdash\begin{array}[t]{@{}l@{}}{\left(X{:}\leavevmode\nobreak\ {\big{(}\mathrm{deepUnfold}(\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}q},{(Y,t_{Y},\overline{G_{Y}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{Y}}q})}_{Y\in\widetilde{Y_{X}}})\big{)}}_{q\in\tilde{q}_{X}}\right)}_{X\in\widetilde{X_{C}}\setminus(Z)},\\\ Z{:}\leavevmode\nobreak\ {\left(\mathrm{deepUnfold}(\overline{G_{Z}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{Z}}q},{(X,t_{X},\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}q)}_{X\in\widetilde{Y_{Z}}}})\right)}_{q\in\tilde{q}};\\\ {\big{(}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}{:}\leavevmode\nobreak\ \mu Z\mathbin{.}({\uparrow^{t_{Z}}}\mathrm{deepUnfold}(\overline{G_{Z}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{Z}}q},{(X,t_{X},\overline{G_{X}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}_{X}}q})}_{X\in\widetilde{Y_{Z}}}))\big{)}}_{q\in\tilde{q}}\end{array}$ In this typing, the types of ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}$ for each $q\in\tilde{q}$ concur with the expected types in (43), and the assignment to $Z$ in the recursive context concurs with (44). This proves the thesis. ∎ #### 4.4.2 Orchestrators and Centralized Compositions of Routers are Behaviorally Equivalent First, we formalize what we mean with a centralized composition of routers, which we call a _hub of routers_. A hub of routers is just a specific composition of routers, formalized as the centralized composition of the routers of all a global type’s participants synthesized from the global type: ###### Definition 37 (Hub of a Global Type). Given global type $G$, we define the _hub of routers_ of $G$ as follows: $\mathcal{H}_{G}:=(\bm{\nu}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}})_{p,q\in\mathsf{prt}(G)}\big{(}{\mathchoice{\textstyle}{}{}{}\prod}_{p\in\mathsf{prt}(G)}\mathcal{R}_{p}\big{)}$ Hubs of routers can be typed using local projection (cf. Def. 22), identical to the typing of orchestrators (cf. Theorem 24): ###### Theorem 25. For relative well-formed global type $G$ and priority $\mathsf{o}$, $\mathcal{H}_{G}\vdash\emptyset;{({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}{:}\leavevmode\nobreak\ \overline{(G\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}}p)})}_{p\in\mathsf{prt}(G)}.$ ###### Proof. By the typability of routers (Theorem 11) and the duality of the types of the endpoints connecting pairs of routers (Theorem 9). ∎ out $x[a,b]\xrightarrow{\vspace{-.8ex}x[a,b]}\bm{0}$ $P\xrightarrow{\vspace{-.8ex}x[a,b]}P^{\prime}$ out-open $(\bm{\nu}ya)(\bm{\nu}zb)P\xrightarrow{\vspace{-.8ex}(\bm{\nu}ya)(\bm{\nu}zb)x[a,b]}P^{\prime}$ in $x(v,w)\mathbin{.}P\xrightarrow{\vspace{-.8ex}x(v,w)}P$ $P\xrightarrow{\vspace{-.8ex}(\bm{\nu}ya)(\bm{\nu}zb)x[a,b]}P^{\prime}$ $Q\xrightarrow{\vspace{-.8ex}x(v,w)}Q^{\prime}$ out-close $P\mathbin{\|}Q\xrightarrow{\vspace{-.8ex}\tau}(\bm{\nu}yv)(\bm{\nu}zw)(P^{\prime}\mathbin{\|}Q^{\prime})$ sel $x[b]\triangleleft j\xrightarrow{\vspace{-.8ex}x[b]\triangleleft j}\bm{0}$ $P\xrightarrow{\vspace{-.8ex}x[b]\triangleleft j}P^{\prime}$ sel-open $(\bm{\nu}zb)P\xrightarrow{\vspace{-.8ex}(\bm{\nu}zb)x[b]\triangleleft j}P^{\prime}$ $j\in I\raisebox{16.79158pt}{}$ bra $x(w)\triangleright\\{i:P_{i}\\}_{i\in I}\xrightarrow{\vspace{-.8ex}x(w)\triangleright j}P_{j}$ $P\xrightarrow{\vspace{-.8ex}(\bm{\nu}zb)x[b]\triangleleft j}P^{\prime}$ $Q\xrightarrow{\vspace{-.8ex}x(w)\triangleright j}Q^{\prime}$ sel-close $P\mathbin{\|}Q\xrightarrow{\vspace{-.8ex}\tau}(\bm{\nu}zw)(P^{\prime}\mathbin{\|}Q^{\prime})$ $P\xrightarrow{\vspace{-.8ex}\alpha}Q$ $\mathrm{bn}(\alpha)\cap\mathrm{fn}(R)=\emptyset$ par-L $P\mathbin{\|}R\xrightarrow{\vspace{-.8ex}\alpha}Q\mathbin{\|}R$ $P\xrightarrow{\vspace{-.8ex}\alpha}Q$ $\mathrm{bn}(\alpha)\cap\mathrm{fn}(R)=\emptyset$ par-R $R\mathbin{\|}P\xrightarrow{\vspace{-.8ex}\alpha}R\mathbin{\|}Q$ id $(\bm{\nu}yz)(x\mathbin{\leftrightarrow}y\mathbin{\|}P)\xrightarrow{\vspace{-.8ex}\tau}P\\{x/z\\}$ $P\xrightarrow{\vspace{-.8ex}\alpha}Q$ $\\{y,y^{\prime}\\}\cap\mathrm{fn}(\alpha)=\emptyset$ res $(\bm{\nu}yy^{\prime})P\xrightarrow{\vspace{-.8ex}\alpha}(\bm{\nu}yy^{\prime})Q$ Figure 13: Labeled transition system for APCP (cf. Definition 38). In order to state the behavioral equivalence of orchestrators and hubs of routers, we first define the specific behavioral equivalence we desire. To this end, we first define a labeled transition system (LTS) for APCP: ###### Definition 38 (LTS for APCP). We define the labels $\alpha$ for transitions for processes as follows: $\displaystyle\alpha::=$ $\displaystyle\leavevmode\nobreak\ \tau$ communication $\mid$ $\displaystyle\leavevmode\nobreak\ x[a,b]$ output $\displaystyle\qquad\;\mbox{\large{$\mid$}}\;(\bm{\nu}ya)(\bm{\nu}zb)x[a,b]$ bound output $\mid$ $\displaystyle\leavevmode\nobreak\ x[b]\triangleleft j$ selection $\displaystyle\qquad\;\mbox{\large{$\mid$}}\;(\bm{\nu}zb)x[b]\triangleleft j$ bound selection $\mid$ $\displaystyle\leavevmode\nobreak\ x(v,w)$ input $\displaystyle\qquad\;\mbox{\large{$\mid$}}\;x(w)\triangleright j$ branch The relation _labeled transition_ ($P\xrightarrow{\vspace{-.8ex}\alpha}Q$) is then defined by the rules in Figure 13. ###### Proposition 26. $P\longrightarrow_{\beta}Q$ if and only if $P\xrightarrow{\vspace{-.8ex}\tau}Q$. As customary, we write ‘$\Rightarrow$’ for the reflexive, transitive closure of $\xrightarrow{\vspace{-.8ex}\tau}$, and we write ‘$\xRightarrow{\alpha}$’ for $\Rightarrow\xrightarrow{\vspace{-.8ex}\alpha}\Rightarrow$ if $\alpha\neq\tau$ and for $\Rightarrow$ otherwise. We can now define the behavioral equivalence we desire: ###### Definition 39 (Weak bisimilarity). A binary relation $\mathbb{B}$ on processes is a _weak bisimulation_ if whenever $(P,Q)\in\mathbb{B}$, * • $P\xrightarrow{\vspace{-.8ex}\alpha}P^{\prime}$ implies that there is $Q^{\prime}$ such that $Q\xRightarrow{\alpha}Q^{\prime}$ and $(P^{\prime},Q^{\prime})\in\mathbb{B}$, and * • $Q\xrightarrow{\vspace{-.8ex}\alpha}Q^{\prime}$ implies that there is $P^{\prime}$ such that $P\xRightarrow{\alpha}P^{\prime}$ and $(P^{\prime},Q^{\prime})\in\mathbb{B}$. Two processes $P$ and $Q$ are _weakly bisimilar_ if there exists a weak bisimulation $\mathbb{B}$ such that $(P,Q)\in\mathbb{B}$. Our equivalence result shall relate the behavior of an orchestrator and a hub on a single but arbitrary channel. More specifically, our result will demonstrate that both settings exhibit the same actions on a channel endpoint connect to a particular participant’s implementation. In order to isolate such a channel, we place the orchestrator and hub of routers in an evaluation context consisting of restrictions and parallel compositions with arbitrary processes, such that it connects all but one of the orchestrator’s or hub’s channels. For example, given a global type $G$ and implementations $P_{q}\vdash\emptyset;{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}{:}\leavevmode\nobreak\ G\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{0}q$ for every participant $q\in\mathsf{prt}(G)\setminus\\{p\\}$, we could use the following evaluation context: $\displaystyle E:=(\bm{\nu}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}})_{q\in\mathsf{prt}(G)\setminus\\{p\\}}\big{(}{{\mathchoice{\textstyle}{}{}{}\prod}}_{q\in\mathsf{prt}(G)\setminus\\{p\\}}P_{q}\mathbin{\|}[\,]\big{)}$ Replacing the hole in this evaluation context with the orchestrator or hub of routers of $G$ leaves one channel free: the channel ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ for the implementation of $p$. Now, we can observe the behavior of these two processes on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$. In what follows we write $\mathcal{O}_{G}^{\tilde{q}}$ instead of ${\mathsf{O}}_{\tilde{q}}[G]$. When we appeal to router and orchestrator synthesis, we often omit the parameter $\tilde{q}$. That is, we write ${\llbracket G\rrbracket}_{p}$ instead of ${\llbracket G\rrbracket}_{p}^{\tilde{q}}$, and $\mathcal{O}_{G}$ instead of $\mathcal{O}_{G}^{\tilde{q}}$. ###### Theorem 27. Suppose given a relative well-formed global type $G$. Let $\mathcal{H}_{G}$ be the hub of routers of $G$ (Def. 37) and take the orchestrator $\mathcal{O}_{G}^{\mathsf{prt}(G)}$ of $G$ (Def. 36). Let $p\in\mathsf{prt}(G)$, and let $E$ be an evaluation context such that $E[\mathcal{H}_{G}]\vdash\emptyset;{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}{:}\leavevmode\nobreak\ \overline{(G\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{\mathsf{o}}p)}$. Then, $E[\mathcal{H}_{G}]$ and $E[\mathcal{O}_{G}]$ are weakly bisimilar (Def. 39). We first give an intuition for the proof of Theorem 27 and its ingredients, after which we give the proof using these ingredients; then, we detail the ingredients. The proof is by coinduction, i.e., by exhibiting a weak bisimulation $\mathbb{B}$ that contains the pair $(E[\mathcal{H}_{G}],E[\mathcal{O}_{G}])$. To construct $\mathbb{B}$ and prove that it is a weak bisimulation we require the following: * • We define a function that, given a global type $G$ and a _starting relation_ $\mathbb{B}_{0}$, computes a corresponding _candidate relation_. This function is denoted $\mathbb{B}(G,\mathbb{B}_{0})$ (Def. 40). * • Suppose $G\xrightarrow{\vspace{-.8ex}\beta_{1}}\ldots\xrightarrow{\vspace{-.8ex}\beta_{k}}G^{\prime}$, with $k\geq 0$. Given some starting relation $\mathbb{B}_{0}$, we want to show that the relation obtained from $\mathbb{B}(G^{\prime},\mathbb{B}_{0})$ is a weak bisimulation, for which we need to assert that $\mathbb{B}_{0}$ is an appropriate starting relation. To this end, we define a function that computes a _consistent_ starting relation for a bisimulation relation, given a pair $(P,Q)$ of processes and a participant $p$ of $G$. This function is denoted $\langle G\xrightarrow{\vspace{-.8ex}\beta_{1}}\ldots\xrightarrow{\vspace{-.8ex}\beta_{k}}G^{\prime},(P,Q),p\rangle$ (Def. 41). * • The property that processes in such a consistent starting relation follow a pattern of specific labeled transitions, passing through a context containing the router of $p$ or the orchestrator (Lemma 28). * • The property that the relation obtained from $\mathbb{B}(G^{\prime},\mathbb{B}_{0},p)$ is a weak bisimulation, given the consistent starting relation $\mathbb{B}_{0}=\langle G\xrightarrow{\vspace{-.8ex}\beta_{1}}\ldots\xrightarrow{\vspace{-.8ex}\beta_{k}}G^{\prime},(E[\mathcal{H}_{G}],E[\mathcal{O}_{G}]),p\rangle$ (Lemma 29). Theorem 27 follows from these definitions and results: ###### Proof of Theorem 27. Let $\mathbb{B}=\mathbb{B}(G,\mathbb{B}_{0})$, where $\mathbb{B}_{0}=\langle G,(E[\mathcal{H}_{G}],E[\mathcal{O}_{G}]),p\rangle$. By Lemma 29, $\mathbb{B}$ is a weak bisimulation. Because $(E[\mathcal{H}_{G}],E[\mathcal{O}_{G}])\in\mathbb{B}_{0}\subseteq\mathbb{B}$, it then follows that $E[\mathcal{H}_{G}]$ and $E[\mathcal{O}_{G}]$ are weakly bisimilar. ∎ We setup some notations: ###### Notation 4. We adopt the following notational conventions. * • We write $\mathsf{Proc}$ to denote the set of all typable APCP processes. * • In the LTS for APCP (Def. 38), we simplify labels: we write an overlined variant for output and selection (e.g., for $(\bm{\nu}ab){{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}[a]\mathbin{\triangleleft}\ell$ we write $\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}\ell$), and omit continuation channels for input and branching (e.g., for ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}(a)\mathbin{\triangleright}\ell$ we write ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}\mathbin{\triangleright}\ell$). * • Also, we write $P\xRightarrow{\alpha_{1}\ldots\alpha_{n}}Q$ rather than $P\xRightarrow{\alpha_{1}}P_{1}\xRightarrow{\alpha_{2}}P_{2}\ldots\xRightarrow{\alpha_{n}}Q$. * • We write $\tilde{\alpha}$ to denote a sequence of labels, e.g., if $\tilde{\alpha}=\alpha_{1}\ldots\alpha_{n}$ then ${\xRightarrow{\tilde{\alpha}}}={\xRightarrow{\alpha_{1}\ldots\alpha_{n}}}$. If $\tilde{\alpha}=\epsilon$ (empty sequence), then ${\xRightarrow{\tilde{\alpha}}}={\Rightarrow}$. The following function defines a relation on processes, which we will use as the weak bisimulation between $E[\mathcal{H}_{G}]$ and $E[\mathcal{O}_{G}]$: ###### Definition 40 (Candidate Relation). Let $G$ be a global type and let $p$ be a participant of $G$. Also, let $\mathbb{B}_{0}\subseteq\mathsf{Proc}\times\mathsf{Proc}$ denote a relation on processes. We define a _candidate relation_ for a weak bisimulation of the hub and orchestrator of $G$ observed on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ starting at $\mathbb{B}_{0}$, by abuse of notation denoted $\mathbb{B}(G,\mathbb{B}_{0},p)$. The definition is inductive on the structure of $G$: * • $G=\bullet$. Then $\mathbb{B}(G,\mathbb{B}_{0},p)=\mathbb{B}_{0}$. * • $G=s\mathbin{\twoheadrightarrow}r\\{i\langle S_{i}\rangle\mathbin{.}G_{i}\\}_{i\in I}$. We distinguish four cases, depending on the involvement of $p$: * – $p=s$. For every $i\in I$, let $\displaystyle\mathbb{B}_{1}^{i}$ $\displaystyle=\\{(P_{1},Q_{1})\mid\exists(P_{0},Q_{0})\in\mathbb{B}_{0}\leavevmode\nobreak\ \text{s.t.}\leavevmode\nobreak\ P_{0}\xrightarrow{\vspace{-.8ex}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}\mathbin{\triangleright}i}\Rightarrow P_{1}\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ Q_{0}\xrightarrow{\vspace{-.8ex}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}\mathbin{\triangleright}i}\Rightarrow Q_{1}\\};$ $\displaystyle\mathbb{B}_{2}^{i}$ $\displaystyle=\\{(P_{2},Q_{2})\mid\exists(P_{1},Q_{1})\in\mathbb{B}_{1}^{i}\leavevmode\nobreak\ \text{s.t.}\leavevmode\nobreak\ P_{1}\xrightarrow{\vspace{-.8ex}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}(y)}\Rightarrow P_{2}\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ Q_{1}\xrightarrow{\vspace{-.8ex}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}(y)}\Rightarrow Q_{2}\\}$ Then $\mathbb{B}(G,\mathbb{B}_{0},p)=\mathbb{B}_{0}\cup{\mathchoice{\textstyle}{}{}{}\bigcup}_{i\in I}(\mathbb{B}_{1}^{i}\cup\mathbb{B}(G_{i},\mathbb{B}_{2}^{i},p)).$ * – $p=r$. For every $i\in I$, let $\displaystyle\mathbb{B}_{1}^{i}$ $\displaystyle=\\{(P_{1},Q_{1})\mid\exists(P_{0},Q_{0})\in\mathbb{B}_{0}\leavevmode\nobreak\ \text{s.t.}\leavevmode\nobreak\ P_{0}\xrightarrow{\vspace{-.8ex}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}i}\Rightarrow P_{1}\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ Q_{0}\xrightarrow{\vspace{-.8ex}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}i}\Rightarrow Q_{1}\\};$ $\displaystyle\mathbb{B}_{2}^{i}$ $\displaystyle=\\{(P_{2},Q_{2})\mid\exists(P_{1},Q_{1})\in\mathbb{B}_{1}\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ y\leavevmode\nobreak\ \text{s.t.}\leavevmode\nobreak\ P_{1}\xrightarrow{\vspace{-.8ex}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}[y]}\Rightarrow P_{2}\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ Q_{1}\xrightarrow{\vspace{-.8ex}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}[y]}\Rightarrow Q_{2}\\}.$ Then $\mathbb{B}(G,\mathbb{B}_{0},p)=\mathbb{B}_{0}\cup{\mathchoice{\textstyle}{}{}{}\bigcup}_{i\in I}(\mathbb{B}_{1}^{i}\cup\mathbb{B}(G_{i},\mathbb{B}_{2}^{i},p)).$ * – $p\notin\\{s,r\\}$ and $\mathrm{hdep}(p,s,G)$ or $\mathrm{hdep}(p,r,G)$. For every $i\in I$, let $\mathbb{B}_{1}^{i}=\\{(P_{1},Q_{1})\mid\exists(P_{0},Q_{0})\in\mathbb{B}_{0}\leavevmode\nobreak\ \text{s.t.}\leavevmode\nobreak\ P_{0}\xrightarrow{\vspace{-.8ex}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}i}\Rightarrow P_{1}\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ Q_{0}\xrightarrow{\vspace{-.8ex}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}i}\Rightarrow Q_{1}\\}$ Then $\mathbb{B}(G,\mathbb{B}_{0},p)=\mathbb{B}_{0}\cup{\mathchoice{\textstyle}{}{}{}\bigcup}_{i\in I}\mathbb{B}(G_{i},\mathbb{B}_{1}^{i},p).$ * – $p\notin\\{s,r\\}$ and neither $\mathrm{hdep}(p,s,G)$ nor $\mathrm{hdep}(p,r,G)$. Then $\mathbb{B}(G,\mathbb{B}_{0},p)=\mathbb{B}(G_{j},\mathbb{B}_{0},p)$ for any $j\in I$. * • $G=\mu X\mathbin{.}G^{\prime}$. Then $\mathbb{B}(G,\mathbb{B}_{0},p)=\mathbb{B}(G^{\prime}\\{\mu X\mathbin{.}G^{\prime}/X\\},\mathbb{B}_{0},p)$. * • $G=\mathsf{skip}\mathbin{.}G^{\prime}$. Then $\mathbb{B}(G,\mathbb{B}_{0},p)=\mathbb{B}(G^{\prime},\mathbb{B}_{0},p)$. The function $\mathbb{B}(G,\mathbb{B}_{0},p)$ constructs a relation between processes by following labeled transitions on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ that concur with the expected behavior of $p$’s router and the orchestrator depending on the shape of $G$. For example, for $G=s\mathbin{\twoheadrightarrow}p\\{i\langle S_{i}\rangle\mathbin{.}G_{i}\\}_{i\in I}$, for each $i\in I$, the function constructs $\mathbb{B}_{1}^{i}$ containing the processes reachable from $\mathbb{B}_{0}$ through a transition labeled $\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}i$ (selection of the label chosen by $s$), and $\mathbb{B}_{2}^{i}$ containing the processes reachable from $\mathbb{B}_{0}$ through a transition labeled $\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}[y]$ (output of the endpoint sent by $s$); the resulting relation then consists of $\mathbb{B}_{0}$ and, for each $i\in I$, $\mathbb{B}_{1}^{i}$ and $\mathbb{B}(G_{i},\mathbb{B}_{2}^{i},p)$ (i.e., the candidate relation for $G_{i}$ starting with $B_{2}^{i}$). Since we are interested in a _weak_ bisimulation, the $\tau$-transitions of one process do not need to be simulated by related processes. Hence, e.g., if $(P,Q)\in\mathbb{B}_{0}$ and $P\xrightarrow{\vspace{-.8ex}\tau}P^{\prime}$ and $Q\xrightarrow{\vspace{-.8ex}\tau}Q^{\prime}$, then $\\{(P,Q),(P^{\prime},Q),(P,Q^{\prime}),(P^{\prime},Q^{\prime})\\}\subseteq\mathbb{B}(G,\mathbb{B}_{0},p)$. This way, we only _synchronize_ related processes when they can both take the same labeled transition. We intend to show that, if $G\xrightarrow{\vspace{-.8ex}\beta_{1}}\ldots\xrightarrow{\vspace{-.8ex}\beta_{k}}G^{\prime}$, the function $\mathbb{B}(G^{\prime},\mathbb{B}_{0},p)$ constructs a weak bisimulation. However, for this to hold, the starting relation $\mathbb{B}_{0}$ cannot be arbitrary: the pairs of processes in $\mathbb{B}_{0}$ have to be reachable from $E[\mathcal{H}_{G}]$ and $E[\mathcal{O}_{G}]$ through labeled transitions that concur with the transitions from $G$ to $G^{\prime}$. Moreover, the processes must have “passed through” evaluation contexts containing the router for $p$ at $G^{\prime}$ and the orchestrator at $G^{\prime}$. The following defines a _consistent starting relation_ , parametric on $k$, that satisfies these requirements. Note that for constructing the relation $\mathbb{B}$, we only need the following definition for $k=0$. However, in the proof that $\mathbb{B}$ is a weak bisimulation we need to generalize it to $k\geq 0$ to assure that the starting relation of coinductive steps is consistent. ###### Definition 41 (Consistent Starting Relation). Let $G\xrightarrow{\vspace{-.8ex}\beta_{1}}\ldots\xrightarrow{\vspace{-.8ex}\beta_{k}}G^{\prime}$ (with $k\geq 0$) be a sequence of labeled transitions from $G$ to $G^{\prime}$ including the intermediate global types (cf. Definition 35) and let $p$ be a participant of $G$. Also, let $(P,Q)$ be a pair of initial processes. We define the _consistent starting relation_ for observing the hub and orchestrator of $G^{\prime}$ on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ starting at $(P,Q)$ after the transitions from $G$ to $G^{\prime}$, denoted $\langle G\xrightarrow{\vspace{-.8ex}\beta_{1}}\ldots\xrightarrow{\vspace{-.8ex}\beta_{k}}G^{\prime},(P,Q),p\rangle$. The definition is inductive on the number $k$ of transitions: * • $k=0$. Then $\langle G,(P,Q),p\rangle=\\{(P^{\prime},Q^{\prime})\mid P\Rightarrow P^{\prime}\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ Q\Rightarrow Q^{\prime}\\}$. * • $k=k^{\prime}+1$. Then $\displaystyle\langle G\xrightarrow{\vspace{-.8ex}\beta_{1}}\ldots\xrightarrow{\vspace{-.8ex}\beta_{k^{\prime}}}G_{k^{\prime}}\xrightarrow{\vspace{-.8ex}\beta_{k}}G_{k},(P,Q),p\rangle={}$ $\displaystyle\quad\\{(P_{k},Q_{k})\mid\begin{array}[t]{@{}l@{}}\exists(P_{k^{\prime}},Q_{k^{\prime}})\in\langle G\xrightarrow{\vspace{-.8ex}\beta_{1}}\ldots\xrightarrow{\vspace{-.8ex}\beta_{k^{\prime}}}G_{k^{\prime}},(P,Q),p\rangle\\\ \text{s.t.}\leavevmode\nobreak\ (\begin{array}[t]{@{}l@{}l@{}}&\leavevmode\nobreak\ (\exists C\leavevmode\nobreak\ \text{s.t.}\leavevmode\nobreak\ P_{k^{\prime}}\xRightarrow{\tilde{\alpha}}C[{\llbracket G_{k}\rrbracket}_{p}]\Rightarrow P_{k})\\\ \text{and}&\leavevmode\nobreak\ (\exists D\leavevmode\nobreak\ \text{s.t.}\leavevmode\nobreak\ Q_{k^{\prime}}\xRightarrow{\tilde{\alpha}}D[\mathcal{O}_{G_{k}}]\Rightarrow Q_{k}))\\},\end{array}\end{array}$ where $\tilde{\alpha}$ depends on $\beta_{k}=s\rangle r:j\langle S_{j}\rangle$ and $G_{k^{\prime}}$ (in unfolded form if $G_{k^{\prime}}=\mu X\mathbin{.}G^{\prime}_{k^{\prime}}$): * – If $p=s$, then $\tilde{\alpha}={{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}\mathbin{\triangleright}j\,{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}(y)$. * – If $p=r$, then $\tilde{\alpha}=\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}j\,\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}[y]$. * – If $p\notin\\{s,r\\}$ and $\mathrm{hdep}(p,s,G_{k})$ or $\mathrm{hdep}(p,r,G_{k})$, then $\tilde{\alpha}=\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}j$. * – If $p\notin\\{s,r\\}$ and neither $\mathrm{hdep}(p,s,G_{k})$ nor $\mathrm{hdep}(p,r,G_{k})$, then $\tilde{\alpha}=\epsilon$. ###### Lemma 28. Let $G$ be a relative well-formed global type such that $G\xrightarrow{\vspace{-.8ex}\beta_{1}}\ldots\xrightarrow{\vspace{-.8ex}\beta_{k}}G^{\prime}$ for $k\geq 0$ and let $p$ be a participant of $G$. Also, let $E$ be an evaluation context such that $\mathrm{fn}(E)=\\{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}\\}$. Then there exists $\tilde{\alpha}$ such that, for every $(P,Q)\in\langle G\xrightarrow{\vspace{-.8ex}\beta_{1}}\ldots\xrightarrow{\vspace{-.8ex}\beta_{k}}G^{\prime},(E[\mathcal{H}_{G}],E[\mathcal{O}_{G}]),p\rangle$, * • $E[\mathcal{H}_{G}]\xRightarrow{\tilde{\alpha}}C\big{[}{\llbracket G^{\prime}\rrbracket}_{p}\big{]}\Rightarrow P$ where $C$ is an evaluation context without an output or selection on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$; and * • $E[\mathcal{O}_{G}]\xRightarrow{\tilde{\alpha}}D\big{[}\mathcal{O}_{G^{\prime}}\big{]}\Rightarrow Q$ where $D$ is an evaluation context without an output or selection on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$. ###### Proof. By induction on $k$. In the base case ($k=0$), we have $G=G^{\prime}$, so $E[\mathcal{H}_{G}]=C[{\llbracket G^{\prime}\rrbracket}_{p}]\Rightarrow P$ and $E[\mathcal{O}]=D[\mathcal{O}_{G^{\prime}}]\Rightarrow Q$. For the inductive case ($k=k^{\prime}+1$), we detail the representative case where $G\xrightarrow{\vspace{-.8ex}\beta_{1}}\ldots\xrightarrow{\vspace{-.8ex}\beta_{k^{\prime}}}G_{k^{\prime}}=p\mathbin{\twoheadrightarrow}s\\{i\langle S_{i}\rangle\mathbin{.}G^{\prime}_{i}\\}_{i\in I}\xrightarrow{\vspace{-.8ex}p\rangle s:i^{\prime}\langle S_{i^{\prime}}\rangle}G^{\prime}$ for some $i^{\prime}\in I$. By the IH, for every $(P_{k^{\prime}},Q_{k^{\prime}})\in\langle G\xrightarrow{\vspace{-.8ex}\beta_{1}}\ldots\xrightarrow{\vspace{-.8ex}\beta_{k^{\prime}}}G_{k^{\prime}},(E[\mathcal{H}_{G}],E[\mathcal{O}_{G}]),p\rangle$, there exists $\tilde{\alpha}^{\prime}$ such that $E[\mathcal{H}_{G}]\xRightarrow{\tilde{\alpha}^{\prime}}C^{\prime}[{\llbracket G_{k^{\prime}}\rrbracket}^{p}]\Rightarrow P_{k^{\prime}}$ and $E[\mathcal{O}_{G}]\xRightarrow{\tilde{\alpha}^{\prime}}D^{\prime}[\mathcal{O}_{G_{k^{\prime}}}]\Rightarrow Q_{k^{\prime}}$ where $C^{\prime}$ and $D^{\prime}$ are without output or selection on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$. Take any $(P,Q)\in\langle G\xrightarrow{\vspace{-.8ex}\beta_{1}}\ldots\xrightarrow{\vspace{-.8ex}\beta_{k^{\prime}}}G_{k^{\prime}}\xrightarrow{\vspace{-.8ex}s\rangle p:i^{\prime}\langle S_{i^{\prime}}\rangle}G^{\prime},(E[\mathcal{H}_{G}],E[\mathcal{O}_{G}]),p\rangle$. By definition, there exists $(P_{k^{\prime}},Q_{k^{\prime}})\in\langle G\xrightarrow{\vspace{-.8ex}\beta_{1}}\ldots\xrightarrow{\vspace{-.8ex}\beta_{k^{\prime}}}G_{k^{\prime}},(E[\mathcal{H}_{G}],E[\mathcal{O}_{G}]),p\rangle$ such that $P_{k^{\prime}}\xRightarrow{\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}i^{\prime}\,\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}[y]}C[{\llbracket G^{\prime}\rrbracket}^{p}]\Rightarrow P\text{ and }Q_{k^{\prime}}\xRightarrow{\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}i^{\prime}\,\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}[y]}D[\mathcal{O}_{G^{\prime}}]\Rightarrow Q$ where there are no outputs or selection on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ in $C$ and $D$. Let $\tilde{\alpha}=\tilde{\alpha}^{\prime}\,\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}\mathbin{\triangleleft}i^{\prime}}\,\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}[y]$. Then ${E[\mathcal{H}_{G}]\xRightarrow{\tilde{\alpha}}C[{\llbracket G^{\prime}\rrbracket}_{p}]\Rightarrow P}$ and $E[\mathcal{O}_{G}]\xRightarrow{\tilde{\alpha}}D[\mathcal{O}_{G^{\prime}}]\Rightarrow Q$. ∎ ###### Lemma 29. Let $G$ be a relative well-formed global type such that $G\xrightarrow{\vspace{-.8ex}\beta_{1}}\ldots\xrightarrow{\vspace{-.8ex}\beta_{k}}G^{\prime}$ (with $k\geq 0$) and let $p$ be a participant of $G$. Also, let $E$ be an evaluation context such that $\mathrm{fn}(E)=\\{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}\\}$. Then the relation $\mathbb{B}(G^{\prime},\mathbb{B}_{0})$, with $\mathbb{B}_{0}=\langle G\xrightarrow{\vspace{-.8ex}\beta_{1}}\ldots\xrightarrow{\vspace{-.8ex}\beta_{k}}G^{\prime},(E[\mathcal{H}_{G}],E[\mathcal{O}_{G}]),p\rangle$, is a weak bisimulation (cf. Definition 39). ###### Proof. By coinduction on the structure of $G^{\prime}$; there are four cases (communication, recursion, $\mathsf{skip}$, and $\bullet$). We only detail the interesting case of communication, which is the only case which involves transitions with labels other than $\tau$. There are four subcases depending on the involvement of $p$ in the communication ($p$ is sender, $p$ is recipient, $p$ depends on the communication, or $p$ does not depend on the communication). In each subcase, the proof follows the same pattern, so as a representative case, we detail when $p$ is the recipient of the communication, i.e., $G^{\prime}=s\mathbin{\twoheadrightarrow}p\\{i\langle S_{i}\rangle\mathbin{.}G^{\prime}_{i}\\}_{i\in I}$. Recall $\displaystyle{\llbracket G^{\prime}\rrbracket}_{p}$ $\displaystyle={{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}\mathbin{\triangleright}\big{\\{}i{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}i\cdot{(\overline{{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}}\mathbin{\triangleleft}i)}_{q\in\mathsf{deps}}\cdot{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}(v)\mathbin{.}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}[w]\cdot(v\mathbin{\leftrightarrow}w\mathbin{\|}{\llbracket G^{\prime}_{i}\rrbracket}_{p})\big{\\}}_{i\in I},$ (Algorithm 1 algorithm 1) $\displaystyle{\llbracket G^{\prime}\rrbracket}_{s}$ $\displaystyle={{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}\mathbin{\triangleright}\big{\\{}i{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}}\mathbin{\triangleleft}i\cdot{(\overline{{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}}\mathbin{\triangleleft}i)}_{q\in\mathsf{deps}}\cdot{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}(v)\mathbin{.}\overline{{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}}[w]\cdot(v\mathbin{\leftrightarrow}w\mathbin{\|}{\llbracket G^{\prime}_{i}\rrbracket}_{s})\big{\\}}_{i\in I},$ (Algorithm 1 algorithm 1) $\displaystyle\mathcal{O}_{G^{\prime}}$ $\displaystyle={{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}\triangleright\\{i{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\triangleleft i\cdot{(\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}}\triangleleft i)}_{q\in\mathsf{deps}}\cdot{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}(v)\mathbin{.}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}[w]\cdot(v\mathbin{\leftrightarrow}w\mathbin{\|}\mathcal{O}_{G^{\prime}_{i}})\\}_{i\in I}.$ (Algorithm 2 algorithm 2) Let $\mathbb{B}=\mathbb{B}(G^{\prime},\mathbb{B}_{0})$. We have $\mathbb{B}=\mathbb{B}_{0}\cup{\mathchoice{\textstyle}{}{}{}\bigcup}_{i\in I}(\mathbb{B}_{1}^{i}\cup\mathbb{B}(G^{\prime}_{i},\mathbb{B}_{2}^{i}))$ with $\mathbb{B}_{1}^{i}$ and $\mathbb{B}_{2}^{i}$ as defined above. Take any $(P,Q)\in\mathbb{B}$; we distinguish cases depending on the subset of $\mathbb{B}$ to which $(P,Q)$ belongs: * • $(P,Q)\in\mathbb{B}_{0}$. By Lemma 28, we have $E[\mathcal{H}_{G}]\xRightarrow{\tilde{\alpha}}C[{\llbracket G^{\prime}\rrbracket}_{p}]\Rightarrow P$ and $E[\mathcal{O}]\xRightarrow{\tilde{\alpha}}D[\mathcal{O}_{G^{\prime}}]\Rightarrow Q$, where $C$ and $D$ do not contain an output or selection on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$. Suppose $P\xrightarrow{\vspace{-.8ex}\alpha}P^{\prime}$; we need to exhibit a matching weak transition from $Q$. By assumption, there are no outputs or selections on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ in $C$ and $D$. Since there are no outputs or selections on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ in $C$, by definition of ${\llbracket G^{\prime}\rrbracket}_{p}$, we need only consider two cases for $\alpha$: * – $\alpha=\tau$. We have $Q\Rightarrow Q$, so $Q\xRightarrow{\tau}Q$. Since $C[{\llbracket G^{\prime}\rrbracket}_{p}]\Rightarrow P^{\prime}$ and $D[\mathcal{O}_{G^{\prime}}]\Rightarrow Q$, we have $(P^{\prime},Q)\in\mathbb{B}_{0}\subseteq\mathbb{B}$. * – $\alpha=\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}j$ for some $j\in I$. To enable this transition, which originates from $p$’s router, somewhere in the $\tau$-transitions between $C[{\llbracket G^{\prime}\rrbracket}_{p}]$ and $P$ the label $j$ was received on ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}$, sent by the router of $s$ on ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}$. For this to happen, the label $j$ was received on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}$, sent from the context on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}$. Since $\mathcal{H}_{G}$ and $\mathcal{O}$ are embedded in the same context, the communication of $j$ between ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}$ and ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}$ can also take place after a number of $\tau$-transitions from $D[\mathcal{O}_{G^{\prime}}]$, after which the selection of $j$ on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ becomes enabled. Hence, since there are no outputs or selection on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ in $D$, we have $Q\Rightarrow Q_{0}\xrightarrow{\vspace{-.8ex}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}j}Q^{\prime}$. We have $D[\mathcal{O}_{G^{\prime}}]\Rightarrow Q_{0}$, so $(P,Q_{0})\in\mathbb{B}_{0}$. Since $P\xrightarrow{\vspace{-.8ex}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}j}\Rightarrow P^{\prime}$ and $Q_{0}\xrightarrow{\vspace{-.8ex}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}j}\Rightarrow Q^{\prime}$, we have $(P^{\prime},Q^{\prime})\in\mathbb{B}_{1}^{j}\subseteq\mathbb{B}^{\prime}$. Now suppose $Q\xrightarrow{\vspace{-.8ex}\alpha}Q^{\prime}$; we need to exhibit a matching weak transition from $P$. Again, we need only consider two cases for $\alpha$: * – $\alpha=\tau$. Analogous to the similar case above. * – $\alpha=\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}j$ for some $j\in I$. To enable this transition, which originates from the orchestrator, somewhere in the $\tau$-transitions between $D[\mathcal{O}_{G^{\prime}}]$ and $Q$ the label $j$ was received on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}$, sent from the context on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}$. Hence, this communication can also take place after a number of transitions from $E[\mathcal{H}_{G}]$, where the label is received by the router of $s$. After this, from $C[{\llbracket G^{\prime}\rrbracket}_{p}]$, the router of $s$ forwards $j$ to $p$’s router (communication between ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}$ and ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}$), enabling the selection of $j$ on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ in $p$’s router. Hence, since there are no outputs or selections in $C$, we have $P\Rightarrow P_{0}\xrightarrow{\vspace{-.8ex}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}j}P^{\prime}$. We have $C[{\llbracket G^{\prime}\rrbracket}_{p}]\Rightarrow P_{0}$, so $(P_{0},Q)\in\mathbb{B}_{0}$. Since $P_{0}\xrightarrow{\vspace{-.8ex}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}j}\Rightarrow P^{\prime}$ and $Q\xrightarrow{\vspace{-.8ex}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}j}\Rightarrow Q^{\prime}$, we have $(P^{\prime},Q^{\prime})\in\mathbb{B}_{1}^{j}\subseteq\mathbb{B}$. * • $(P,Q)\in\mathbb{B}_{1}^{j}$ for some $j\in I$. We have $E[\mathcal{H}_{G}]\xRightarrow{\tilde{\alpha}}C[{\llbracket G^{\prime}\rrbracket}_{p}]\Rightarrow P_{0}\xrightarrow{\vspace{-.8ex}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}j}\Rightarrow P$ and ${E[\mathcal{O}_{G}]\xRightarrow{\tilde{\alpha}}D[\mathcal{O}_{G^{\prime}}]\Rightarrow Q_{0}\xrightarrow{\vspace{-.8ex}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}j}\Rightarrow Q}$ where $(P_{0},Q_{0})\in\mathbb{B}_{0}$. Since we have already observed the selection of $j$ on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ from both the hub and the orchestrator, we know that the routers of $p$ and $s$ are in branch $j$, and similarly the orchestrator is in branch $j$. Suppose $P\xrightarrow{\vspace{-.8ex}\alpha}P^{\prime}$. To exhibit a matching weak transition from $Q$ we only need to consider two cases for $\alpha$: * – $\alpha=\tau$. We have $Q\xRightarrow{\tau}Q$, and $P_{0}\xrightarrow{\vspace{-.8ex}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}j}\Rightarrow P^{\prime}$ and $Q_{0}\xrightarrow{\vspace{-.8ex}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}j}\Rightarrow Q$, so $(P^{\prime},Q)\in\mathbb{B}_{1}^{j}\subseteq\mathbb{B}$. * – $\alpha=\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}[y]$ for some $y$. The observed output of some $y$ on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ must originate from $p$’s router. This output is only enabled after receiving some $v$ over ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}$, which must be sent by the router of $s$ over ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}$. The output by the router of $s$ is only enabled after receiving some $v$ over ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}$, sent by the context over ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}$. Since the hub and the orchestrator are embedded in the same context, the communication of $v$ from ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}$ to ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}$ can also occur (or has already occurred) for the orchestrator. After this, the output of $y$ over ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ is enabled in the orchestrator, i.e., $Q\Rightarrow Q_{1}\xrightarrow{\vspace{-.8ex}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}[y]}Q^{\prime}$. We have $Q_{0}\xrightarrow{\vspace{-.8ex}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}j}\Rightarrow Q_{1}$, so $(P,Q_{1})\in\mathbb{B}_{1}^{j}$. Since $P\xrightarrow{\vspace{-.8ex}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}[y]}\Rightarrow P^{\prime}$ and $Q_{1}\xrightarrow{\vspace{-.8ex}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}[y]}\Rightarrow Q^{\prime}$, we have $(P^{\prime},Q^{\prime})\in\mathbb{B}_{2}^{j}$. By definition, $\mathbb{B}_{2}^{j}\subseteq\mathbb{B}(G^{\prime}_{j},B_{2}^{j})\subseteq\mathbb{B}$, so $(P^{\prime},Q^{\prime})\in\mathbb{B}$. Now suppose $Q\xrightarrow{\vspace{-.8ex}\alpha}Q^{\prime}$. To exhibit a matching weak transition from $P$ we only need to consider two cases for $\alpha$: * – $\alpha=\tau$. Analogous to the similar case above. * – $\alpha=\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}[y]$ for some $y$. The observed output of some $y$ on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ must originate from the orchestrator. This output is only enabled after receiving some $v$ over ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}$, sent by the context of ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}$. Since the hub and the orchestrator are embeded in the same context, the communication of $v$ from ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}$ to ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}$ can also occur (or has already occurred) for the router of $s$. After this, the router of $s$ sends another channel $v^{\prime}$ over ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}$, received by $p$’s router on ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}$. This enables the output of $y$ on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ by $p$’s router, i.e., $P\Rightarrow P_{1}\xrightarrow{\vspace{-.8ex}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}[y]}P^{\prime}$. We have $P_{0}\xrightarrow{\vspace{-.8ex}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}j}\Rightarrow P_{1}$, so $(P_{1},Q)\in\mathbb{B}_{1}^{j}$. Since $P\xrightarrow{\vspace{-.8ex}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}[y]}\Rightarrow P^{\prime}$ and $Q_{1}\xrightarrow{\vspace{-.8ex}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}[y]}\Rightarrow Q^{\prime}$, we have $(P^{\prime},Q^{\prime})\in\mathbb{B}_{2}^{j}$. As above, this implies that $(P^{\prime},Q^{\prime})\in\mathbb{B}$. * • For some $j\in I$, $(P,Q)\in\mathbb{B}(G^{\prime}_{j},\mathbb{B}_{2}^{j})$. The thesis follows from proving that $\mathbb{B}(G^{\prime}_{j},\mathbb{B}_{2}^{j})$ is a weak bisimulation. For this, we want to appeal to the coinduction hypothesis, so we have to show that $\mathbb{B}_{2}^{j}=\langle G\xrightarrow{\vspace{-.8ex}\beta_{1}}\ldots\xrightarrow{\vspace{-.8ex}\beta_{k}}G^{\prime}\xrightarrow{\vspace{-.8ex}s\rangle p:j\langle S_{j}\rangle}G^{\prime}_{j},(E[\mathcal{H}_{G}],E[\mathcal{O}]),p\rangle$. We prove that $(P_{2},Q_{2})\in\mathbb{B}_{2}^{j}$ if and only if $(P_{2},Q_{2})\in\langle G\xrightarrow{\vspace{-.8ex}\beta_{1}}\ldots\xrightarrow{\vspace{-.8ex}\beta_{k}}G^{\prime}\xrightarrow{\vspace{-.8ex}s\rangle p:j\langle S_{j}\rangle}G^{\prime}_{j},(E[\mathcal{H}_{G}],E[\mathcal{O}]),p\rangle$, i.e., we prove both directions of the bi-implication: * – Take any $(P_{2},Q_{2})\in\mathbb{B}_{2}^{j}$. We have $E[\mathcal{H}_{G}]\xRightarrow{\tilde{\alpha}}C[{\llbracket G^{\prime}\rrbracket}_{p}]\Rightarrow P_{0}\xrightarrow{\vspace{-.8ex}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}j}\Rightarrow P_{1}\xrightarrow{\vspace{-.8ex}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}[y]}\Rightarrow P_{2}$ and $E[\mathcal{O}]\xRightarrow{\tilde{\alpha}}D[\mathcal{O}_{G^{\prime}}]\Rightarrow Q_{0}\xrightarrow{\vspace{-.8ex}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}j}\Rightarrow Q_{1}\xrightarrow{\vspace{-.8ex}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}[y]}\Rightarrow Q_{2}$, where $(P_{0},Q_{0})\in\mathbb{B}_{0}$ and $(P_{1},P_{1})\in\mathbb{B}_{1}^{j}$. By definition, somewhere during the transitions from $C[{\llbracket G^{\prime}\rrbracket}_{p}]$ to $P_{1}$, we find $C^{\prime}[{\llbracket G^{\prime}_{j}\rrbracket}_{p}]$, which may then further reduce by $\tau$-transitions towards $P_{2}$. As soon as we do find $C^{\prime}[{\llbracket G^{\prime}_{j}\rrbracket}_{p}]$, the output on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ is available, and the selection on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ has already occurred or is still available. Because they are asynchronous actions, we can observe the selection and output on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ as soon as they are available, before further reducing $p$’s router. Hence, we can observe $C[{\llbracket G^{\prime}\rrbracket}_{p}]\Rightarrow\xrightarrow{\vspace{-.8ex}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}j}\Rightarrow\xrightarrow{\vspace{-.8ex}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}[y]}\Rightarrow C^{\prime\prime}[{\llbracket G^{\prime}_{j}\rrbracket}_{p}]\Rightarrow P_{2}$, i.e., ${E[\mathcal{H}_{G}]\xRightarrow{\tilde{\alpha}}C[{\llbracket G^{\prime}\rrbracket}_{p}]\xRightarrow{{\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}j}\,{\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}[y]}}C^{\prime\prime}[{\llbracket G^{\prime}_{j}\rrbracket}_{p}]\Rightarrow P_{2}}.$ By definition, ${\llbracket G^{\prime}_{j}\rrbracket}_{p}$ has no output or selection on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ available, so there are no outputs or selections on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ in $C^{\prime\prime}$. By a similar argument, we can observe $D[\mathcal{O}_{G^{\prime}}]\Rightarrow\xrightarrow{\vspace{-.8ex}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}j}\Rightarrow\xrightarrow{\vspace{-.8ex}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}[y]}\Rightarrow D^{\prime\prime}[\mathcal{O}_{G^{\prime}_{j}}]\Rightarrow Q_{2}$, i.e., $E[\mathcal{O}]\xRightarrow{\tilde{\alpha}}D[\mathcal{O}_{G^{\prime}}]\xRightarrow{{\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}j}\,{\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}[y]}}D^{\prime\prime}[\mathcal{O}_{G^{\prime}_{j}}]\Rightarrow Q_{2}$. Also in this case, there are no outputs or selections on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ in $D^{\prime\prime}$. By assumption and definition, $(C^{\prime\prime}[{\llbracket G^{\prime}\rrbracket}_{p}],D^{\prime\prime}[\mathcal{O}_{G^{\prime}}])\in\mathbb{B}_{0}=\langle G\xrightarrow{\vspace{-.8ex}\beta_{1}}\ldots\xrightarrow{\vspace{-.8ex}\beta_{k}}G^{\prime},(E[\mathcal{H}_{G}],E[\mathcal{O}]),p\rangle.$ Hence, by definition, $(P_{2},Q_{2})\in\langle G\xrightarrow{\vspace{-.8ex}\beta_{1}}\ldots\xrightarrow{\vspace{-.8ex}\beta_{k}}G^{\prime}\xrightarrow{\vspace{-.8ex}s\rangle p:j\langle S_{j}\rangle}G^{\prime}_{j},(E[\mathcal{H}_{G}],E[\mathcal{O}]),p\rangle$. * – Take any $(P,Q)\in\langle G\xrightarrow{\vspace{-.8ex}\beta_{1}}\ldots\xrightarrow{\vspace{-.8ex}\beta_{k}}G^{\prime}\xrightarrow{\vspace{-.8ex}s\rangle p:j\langle S_{j}\rangle}G^{\prime}_{j},(E[\mathcal{H}_{G}],E[\mathcal{O}]),p\rangle$. By definition, there are $(P^{\prime},Q^{\prime})\in\langle G\xrightarrow{\vspace{-.8ex}\beta_{1}}\ldots\xrightarrow{\vspace{-.8ex}\beta_{k}}G^{\prime},(E[\mathcal{H}_{G}],E[\mathcal{O}]),p\rangle$ such that $P^{\prime}\xRightarrow{\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}j\,\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}[y]}C[{\llbracket G^{\prime}\rrbracket}_{p}]\Rightarrow P$ and $Q^{\prime}\xRightarrow{\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}\mathbin{\triangleleft}j\,\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}}[y]}D[\mathcal{O}_{G^{\prime}}]\Rightarrow Q$. Since, $\mathbb{B}_{0}=\langle G\xrightarrow{\vspace{-.8ex}\beta_{1}}\ldots\xrightarrow{\vspace{-.8ex}\beta_{k}}G^{\prime},(E[\mathcal{H}_{G}],E[\mathcal{O}]),p\rangle$, by definition $(P,Q)\in\mathbb{B}_{2}^{j}$. ∎ ## 5 Routers in Action We demonstrate our router-based analysis of global types by means of several examples. First, in § 5.1 and § 5.2 we consider two simple protocols: they illustrate the different components of our approach, and our support for delegation and interleaving. Then in § 5.3 we revisit the authorization protocol $G_{\mathsf{auth}}$ from Section 1 to illustrate how our analysis supports also more complex protocols featuring also non-local choices and recursion. ### 5.1 Delegation and Interleaving We illustrate our analysis by considering a global type with delegation and interleaving, based on an example by Toninho and Yoshida [50, Ex. 6.9]. Consider the global type: $\displaystyle G_{\mathsf{intrl}}:=p\mathbin{\twoheadrightarrow}q{:}1\langle{!}\mathsf{int}\mathbin{.}\bullet\rangle\mathbin{.}r\mathbin{\twoheadrightarrow}t{:}2\langle\mathsf{int}\rangle\mathbin{.}p\mathbin{\twoheadrightarrow}q{:}3\mathbin{.}\bullet$ Following Toninho and Yoshida [50], we define implementations of the roles of the four participants ($p,q,r,t$) of $G_{\mathsf{intrl}}$ using three processes ($P_{1}$, $P_{2}$, and $P_{3}$): $P_{2}$ and $P_{3}$ implement the roles of $q$ and $r$, respectively, and $P_{1}$ interleaves the roles of $p$ and $t$ by sending a channel $s$ to $q$ and receiving an int value $v$ from $r$, which it should forward to $q$ over $s$. $\displaystyle P_{1}$ $\displaystyle:=\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}}\mathbin{\triangleleft}1\cdot\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}}[s]\cdot({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}t}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}\mathbin{\triangleright}\\{2{:}\leavevmode\nobreak\ {{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}t}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}(v)\mathbin{.}\overline{s}[w]\cdot v\mathbin{\leftrightarrow}w\\}\mathbin{\|}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}}\mathbin{\triangleleft}3\cdot\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}}[z]\cdot\bm{0})$ $\displaystyle\vdash{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}{:}\leavevmode\nobreak\ {\oplus}^{0}\big{\\{}1{:}\leavevmode\nobreak\ {{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}{!}\mathsf{int}\mathbin{.}\bullet{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}\mathbin{\otimes}^{1}{\oplus}^{8}\\{3{:}\leavevmode\nobreak\ \bullet\mathbin{\otimes}^{9}\bullet\\}\big{\\}},\leavevmode\nobreak\ {{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}t}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}{:}\leavevmode\nobreak\ \&^{6}\\{2{:}\leavevmode\nobreak\ \bullet\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{7}\bullet\\}$ $\displaystyle P_{2}$ $\displaystyle:={{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}\mathbin{\triangleright}\\{1{:}\leavevmode\nobreak\ {{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}(y)\mathbin{.}y(x)\mathbin{.}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}\mathbin{\triangleright}\\{3{:}\leavevmode\nobreak\ {{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}(u)\mathbin{.}\bm{0}\\}\\}\vdash{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}{:}\leavevmode\nobreak\ \&^{2}\big{\\{}1{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}{!}\mathsf{int}\mathbin{.}\bullet{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}}\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{3}\&^{10}\\{3{:}\leavevmode\nobreak\ \bullet\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{11}\bullet\\}\big{\\}}$ $\displaystyle P_{3}$ $\displaystyle:=\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}r}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}}\mathbin{\triangleleft}2\cdot\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}r}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}}[\bm{33}]\cdot\bm{0}\vdash{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}r}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}{:}\leavevmode\nobreak\ {\oplus}^{4}\\{2{:}\leavevmode\nobreak\ \bullet\mathbin{\otimes}^{5}\bullet\\}$ where ‘$\bm{33}$’ denotes a closed channel endpoint representing the number “$33$”. To prove that $P_{1}$, $P_{2}$, and $P_{3}$ correctly implement $G_{\mathsf{intrl}}$, we compose them with the routers synthesized from $G_{\mathsf{intrl}}$. For example, the routers for $p$ and $t$, to which $P_{1}$ will connect, are as follows (omitting curly braces for branches on a single label): $\displaystyle\mathcal{R}_{p}$ $\displaystyle={{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}\mathbin{\triangleright}1\mathbin{.}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}\mathbin{\triangleleft}1\cdot{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}(s)\mathbin{.}\overline{{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}}[s^{\prime}]\cdot(s\mathbin{\leftrightarrow}s^{\prime}\mathbin{\|}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}\mathbin{\triangleright}3\mathbin{.}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}\mathbin{\triangleleft}3\cdot{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}(z)\mathbin{.}\overline{{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}}[z^{\prime}]\cdot(z\mathbin{\leftrightarrow}z^{\prime}\mathbin{\|}\bm{0}))$ $\displaystyle\mathcal{R}_{t}$ $\displaystyle={{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}t}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}\mathbin{\triangleright}2\mathbin{.}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}t}}\mathbin{\triangleleft}2\cdot{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}t}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}(v)\mathbin{.}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}t}}}[v^{\prime}]\cdot(v\mathbin{\leftrightarrow}v^{\prime}\mathbin{\|}\bm{0})$ We assign values to the priorities in ${{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}{!}\mathsf{int}\mathbin{.}\bullet{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}=\bullet\mathbin{\otimes}^{\mathsf{o}}\bullet$ to ensure that $P_{1}$ and $P_{2}$ are well-typed; assigning $\mathsf{o}=8$ works, because the output on $s$ in $P_{1}$ occurs after the input on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}t}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}$ (which has priority 6–7) and the input on $y$ in $P_{2}$ occurs before the second input on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}$ (which has priority 10–11). The types assigned to ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}$ and ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}t}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}$ in $P_{1}$ coincide with $(G_{\mathsf{intrl}}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{0}p)$ and $(G_{\mathsf{intrl}}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{0}t)$, respectively (cf. Def. 22). Therefore, by Theorem 11, the process $P_{1}$ connect to the routers for $p$ and $t$ $(\bm{\nu}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}})(\bm{\nu}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}t}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu t}})(P_{1}\mathbin{\|}\mathcal{R}_{p}\mathbin{\|}\mathcal{R}_{t})$ is well- typed. Similarly, $(\bm{\nu}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}})(P_{2}\mathbin{\|}\mathcal{R}_{q})$ and $(\bm{\nu}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}r}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}r}})(P_{3}\mathbin{\|}\mathcal{R}_{r})$ are well-typed. The composition of these routed implementations results in the following network: $\displaystyle N_{\mathsf{intrl}}:=\begin{array}[]{c}(\bm{\nu}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}})(\bm{\nu}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}})\\\ (\bm{\nu}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}t}}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}t}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}})(\bm{\nu}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}})\\\ (\bm{\nu}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}t}}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}t}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}})(\bm{\nu}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}t}}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}t}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}r}})\end{array}\left(\begin{array}[]{l}\phantom{{}\mathbin{\|}{}}(\bm{\nu}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}})(\bm{\nu}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}t}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}t}})(P_{1}\mathbin{\|}\mathcal{R}_{p}\mathbin{\|}\mathcal{R}_{t})\\\ {}\mathbin{\|}(\bm{\nu}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}q}})(P_{2}\mathbin{\|}\mathcal{R}_{q})\\\ {}\mathbin{\|}(\bm{\nu}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}r}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}r}})(P_{3}\mathbin{\|}\mathcal{R}_{r})\end{array}\right)$ We have $N_{\mathsf{intrl}}\in\mathrm{net}(G_{\mathsf{intrl}})$ (cf. Def. 25), so, by Theorem 18, $N_{\mathsf{intrl}}$ is deadlock free and, by Theorem 19 and Theorem 23, it correctly implements $G_{\mathsf{intrl}}$. ### 5.2 Another Example of Delegation Here, we further demonstrate our support for interleaving, showing how a participant can delegate the rest of its interactions in a protocol. The following global type formalizes a protocol in which a Client ($c$) asks an online Password Manager ($p$) to login with a Server ($s$): $\displaystyle G_{\mathsf{deleg}}:=c\mathbin{\twoheadrightarrow}p{:}\mathsf{login}\langle S\rangle\mathbin{.}G^{\prime}_{\mathsf{deleg}}$ where $\displaystyle S$ $\displaystyle:={!}({?}\mathsf{bool}\mathbin{.}\bullet)\mathbin{.}S^{\prime}$ $\displaystyle S^{\prime}$ $\displaystyle:=\&\\{\mathsf{passwd}{:}\leavevmode\nobreak\ {?}\mathsf{str}\mathbin{.}{\oplus}\\{\mathsf{auth}{:}\leavevmode\nobreak\ {!}\mathsf{bool}\mathbin{.}\bullet\\}\\}$ $\displaystyle G^{\prime}_{\mathsf{deleg}}$ $\displaystyle:=c\mathbin{\twoheadrightarrow}s{:}\mathsf{passwd}\langle\mathsf{str}\rangle\mathbin{.}s\mathbin{\twoheadrightarrow}c{:}\mathsf{auth}\langle\mathsf{bool}\rangle\mathbin{.}\bullet$ Here $S^{\prime}$ expresses the type of $\mathcal{R}_{c}$’s channel endpoint ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}$. This means that we can give implementations for $c$ and $p$ such that $c$ can send its channel endpoint ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}$ to $p$, after which $p$ logs in with $s$ in $c$’s place, forwarding the authorization boolean received from $s$ to $c$. Giving such implementations is relatively straightforward, demonstrating the flexibility of our global types and analysis using APCP and routers. Using local projection, we can compute a type for $c$’s implementation to safely connect with its router $\displaystyle G_{\mathsf{deleg}}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{0}c={\oplus}^{0}\\{\mathsf{login}{:}\leavevmode\nobreak\ {{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}\mathbin{\otimes}^{1}(G^{\prime}_{\mathsf{deleg}}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{4}c)\\}$ where $\displaystyle{{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}$ $\displaystyle=(\bullet\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{\mathsf{o}}\bullet)\mathbin{\otimes}^{\kappa}{{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S^{\prime}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}$ $\displaystyle{{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S^{\prime}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}$ $\displaystyle=\&^{\pi}\\{\mathsf{passwd}{:}\leavevmode\nobreak\ \bullet\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{\rho}{\oplus}^{\delta}\\{\mathsf{auth}{:}\leavevmode\nobreak\ \bullet\mathbin{\otimes}^{\phi}\bullet\\}\\}$ $\displaystyle G^{\prime}_{\mathsf{deleg}}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{4}c$ $\displaystyle={\oplus}^{4}\\{\mathsf{passwd}{:}\leavevmode\nobreak\ \bullet\mathbin{\otimes}^{5}\&^{10}\\{\mathsf{auth}{:}\leavevmode\nobreak\ \bullet\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{11}\bullet\\}\\}$ Notice how $\overline{{{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S^{\prime}{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}}=G^{\prime}_{\mathsf{deleg}}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{4}c$, given the assignments $\pi=4,\rho=5,\delta=10,\phi=11$. We can use these types to guide the design of a process implementation for $c$. Consider the process: $\displaystyle Q:=\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}}\mathbin{\triangleleft}\mathsf{login}\cdot\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}}[u]\cdot\overline{u}[v]\cdot(u\mathbin{\leftrightarrow}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}\mathbin{\|}v(a)\mathbin{.}\bm{0})\vdash\emptyset;{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}{:}\leavevmode\nobreak\ G_{\mathsf{deleg}}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{0}c$ This implementation is interesting: after the first exchange in $G_{\mathsf{deleg}}$—sending a fresh channel $u$ (to $p$)—$c$ sends another fresh channel $v$ over $u$; then, $c$ delegates the rest of its exchanges in $G^{\prime}_{\mathsf{deleg}}$ by forwarding all traffic on ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}$ over $u$; in the meantime, $c$ awaits an authorization boolean over $v$. Again, using local projection, we can compute a type for $p$’s implementation to connect with its router: $\displaystyle G_{\mathsf{deleg}}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{0}p=\&^{2}\\{\mathsf{login}{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}}\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{3}\bullet\\}$ We can then use it to type the following implementation for $p$: $\displaystyle P:={{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}\mathbin{\triangleright}\left\\{\begin{array}[]{rl}\mathsf{login}{:}&{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}})\mathbin{.}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}(v)\\\ &{}\mathbin{.}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}}\mathbin{\triangleleft}\mathsf{passwd}\cdot\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}}[\bm{pwd123}]\\\ &{}\cdot{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}\mathbin{\triangleleft}\\{\mathsf{auth}{:}\leavevmode\nobreak\ {{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}(a)\mathbin{.}\overline{v}[a^{\prime}]\cdot a\mathbin{\leftrightarrow}a^{\prime}\\}\end{array}\right\\}\vdash\emptyset;{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}{:}\leavevmode\nobreak\ G_{\mathsf{deleg}}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{0}p$ In this implementation, $p$ receives a channel ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}$ (from $c$) over which it first receives a channel $v$. Then, it behaves over ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}$ according to $c$’s role in $G^{\prime}_{\mathsf{deleg}}$. Finally, $p$ forwards the authorization boolean received from $s$ over $v$, effectively sending the boolean to $c$. Given an implementation for $s$, say $S\vdash\emptyset;{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}{:}\leavevmode\nobreak\ G_{\mathsf{deleg}}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{0}s$, what remains is to assign values to the remaining priorities in ${{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\llparenthesis}S{\color[rgb]{0.7421875,0.2890625,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7421875,0.2890625,0}\rrparenthesis}}$: assigning $\mathsf{o}=12,\kappa=4$ works. Now, we can compose the implementations $P$, $Q$ and $S$ with their respective routers and then compose these routed implementations together to form a deadlock free network of $G_{\mathsf{deleg}}$. This way, e.g., the router for $c$ is as follows (again, omitting curly braces for branches on a single label): $\displaystyle\mathcal{R}_{c}$ $\displaystyle={{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}\mathbin{\triangleright}\mathsf{login}\mathbin{.}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}\mathbin{\triangleleft}\mathsf{login}\cdot{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}(u)\mathbin{.}\overline{{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}}[u^{\prime}]\cdot($ $\displaystyle\phantom{{}={}}\quad u\mathbin{\leftrightarrow}u^{\prime}\mathbin{\|}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}\mathbin{\triangleright}\mathsf{passwd}\mathbin{.}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}\mathbin{\triangleleft}\mathsf{passwd}\cdot{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}(v)\mathbin{.}\overline{{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}}[v^{\prime}]\cdot($ $\displaystyle\phantom{{}={}}\qquad v\mathbin{\leftrightarrow}v^{\prime}\mathbin{\|}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}\mathbin{\triangleright}\mathsf{auth}\mathbin{.}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}\mathbin{\triangleleft}\mathsf{auth}\cdot{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}(w)\mathbin{.}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}}[w^{\prime}]\cdot(w\mathbin{\leftrightarrow}w^{\prime}\mathbin{\|}\bm{0})))$ Interestingly, the router is agnostic of the fact that the endpoint $u$ it receives over ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}$ is in fact the opposite endpoint of the channel formed by ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}$. ### 5.3 The Authorization Protocol in Action Let us repeat $G_{\mathsf{auth}}$ from Section 1: $\displaystyle G_{\mathsf{auth}}=\mu X\mathbin{.}s\mathbin{\twoheadrightarrow}c\left\\{\begin{array}[]{@{}l@{}}\mathsf{login}\mathbin{.}c\mathbin{\twoheadrightarrow}a{:}\mathsf{passwd}\langle\mathsf{str}\rangle\mathbin{.}a\mathbin{\twoheadrightarrow}s{:}\mathsf{auth}\langle\mathsf{bool}\rangle\mathbin{.}X,\\\ \mathsf{quit}\mathbin{.}c\mathbin{\twoheadrightarrow}a{:}\mathsf{quit}\mathbin{.}\bullet\end{array}\right\\}$ The relative projections of $G_{\mathsf{auth}}$ are as follows: $\displaystyle G_{\mathsf{auth}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(s,a)$ $\displaystyle=\mu X\mathbin{.}s{!}c\left\\{\begin{array}[]{@{}l@{}}\mathsf{login}\mathbin{.}\mathsf{skip}\mathbin{.}a{:}\mathsf{auth}\langle\mathsf{bool}\rangle\mathbin{.}X,\\\ \mathsf{quit}\mathbin{.}\mathsf{skip}\mathbin{.}\bullet\end{array}\right\\}$ $\displaystyle G_{\mathsf{auth}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(c,a)$ $\displaystyle=\mu X\mathbin{.}c{?}s\left\\{\begin{array}[]{@{}l@{}}\mathsf{login}\mathbin{.}c{:}\mathsf{passwd}\langle\mathsf{str}\rangle\mathbin{.}\mathsf{skip}\mathbin{.}X,\\\ \mathsf{quit}\mathbin{.}c{:}\mathsf{quit}\mathbin{.}\bullet\end{array}\right\\}$ $\displaystyle G_{\mathsf{auth}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(s,c)$ $\displaystyle=\mu X\mathbin{.}s\left\\{\begin{array}[]{@{}l@{}}\mathsf{login}\mathbin{.}\mathsf{skip}^{2}\mathbin{.}X,\\\ \mathsf{quit}\mathbin{.}\mathsf{skip}\mathbin{.}\bullet\end{array}\right\\}$ $\displaystyle\mathcal{R}_{c}$ $\displaystyle=\mu X({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}},{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}},{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}})\mathbin{.}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}\mathbin{\triangleright}\left\\{\begin{array}[]{ll}\mathsf{login}{:}&\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}}\mathbin{\triangleleft}\mathsf{login}\cdot\overline{{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}}\mathbin{\triangleleft}\mathsf{login}\cdot{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}(u)\mathbin{.}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}}[u^{\prime}]\\\ &{}\cdot(u\mathbin{\leftrightarrow}u^{\prime}\mathbin{\|}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}\mathbin{\triangleright}\left\\{\begin{array}[]{ll}\mathsf{passwd}{:}&\overline{{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}}\mathbin{\triangleleft}\mathsf{passwd}\cdot{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}(v)\mathbin{.}\overline{{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}}[v^{\prime}]\\\ &{}\cdot(v\mathbin{\leftrightarrow}v^{\prime}\mathbin{\|}X{\langle{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}},{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}},{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}\rangle})\end{array}\right\\}),\\\ \mathsf{quit}{:}&\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}}\mathbin{\triangleleft}\mathsf{quit}\cdot\overline{{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}}\mathbin{\triangleleft}\mathsf{quit}\cdot{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}(w)\mathbin{.}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}}[w^{\prime}]\\\ &{}\cdot(w\mathbin{\leftrightarrow}w^{\prime}\mathbin{\|}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}\mathbin{\triangleright}\left\\{\begin{array}[]{ll}\mathsf{quit}{:}&\overline{{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}}\mathbin{\triangleleft}\mathsf{quit}\cdot{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}(z)\mathbin{.}\overline{{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}}[z^{\prime}]\\\ &\cdot(z\mathbin{\leftrightarrow}z^{\prime}\mathbin{\|}\bm{0})\end{array}\right\\})\end{array}\right\\}$ $\displaystyle\vdash\begin{array}[t]{l}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}{:}\leavevmode\nobreak\ \mu X\mathbin{.}{\oplus}^{2}\left\\{\begin{array}[]{ll}\mathsf{login}{:}&\bullet\mathbin{\otimes}^{3}\&^{4}\\{\mathsf{passwd}{:}\leavevmode\nobreak\ \bullet\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{5}X\\},\\\ \mathsf{quit}{:}&\bullet\mathbin{\otimes}^{3}\&^{4}\\{\mathsf{quit}{:}\leavevmode\nobreak\ \bullet\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{5}\bullet\\}\end{array}\right\\}=\overline{(G_{\mathsf{auth}}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{0}c)},\\\ {{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}{:}\leavevmode\nobreak\ \mu X\mathbin{.}\&^{1}\\{\mathsf{login}{:}\leavevmode\nobreak\ \bullet\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{2}X,\mathsf{quit}{:}\leavevmode\nobreak\ \bullet\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{2}\bullet\\}={{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{\mathsf{auth}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(c,s){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{c{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}s}^{0},\\\ {{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}{:}\leavevmode\nobreak\ \mu X\mathbin{.}{\oplus}^{2}\left\\{\begin{array}[]{ll}\mathsf{login}{:}&{\oplus}^{5}\\{\mathsf{passwd}{:}\leavevmode\nobreak\ \bullet\mathbin{\otimes}^{6}X\\},\\\ \mathsf{quit}{:}&{\oplus}^{5}\\{\mathsf{quit}{:}\leavevmode\nobreak\ \bullet\mathbin{\otimes}^{6}\bullet\\}\end{array}\right\\}={{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{\mathsf{auth}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(c,a){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{c{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}a}^{0}\end{array}$ $\displaystyle\mathcal{R}_{s}$ $\displaystyle=\mu X({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}},{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}},{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}})\mathbin{.}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}\mathbin{\triangleright}\left\\{\begin{array}[]{ll}\mathsf{login}{:}&\overline{{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}}\mathbin{\triangleleft}\mathsf{login}\cdot\overline{{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}}\mathbin{\triangleleft}\mathsf{login}\cdot{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}(u)\mathbin{.}\overline{{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}}[u^{\prime}]\\\ &{}\cdot(u\mathbin{\leftrightarrow}u^{\prime}\mathbin{\|}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}\mathbin{\triangleright}\left\\{\begin{array}[]{rl}\mathsf{auth}{:}&\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}}\mathbin{\triangleleft}\mathsf{auth}\cdot{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}(v)\mathbin{.}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}}[v^{\prime}]\\\ &{}\cdot(v\mathbin{\leftrightarrow}v^{\prime}\mathbin{\|}X{\langle{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}},{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}},{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}\rangle})\end{array}\right\\}),\\\ \mathsf{quit}{:}&\overline{{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}}\mathbin{\triangleleft}\mathsf{quit}\cdot\overline{{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}}\mathbin{\triangleleft}\mathsf{quit}\cdot{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}(v)\mathbin{.}\overline{{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}}[v^{\prime}]\\\ &\cdot(v\mathbin{\leftrightarrow}v^{\prime}\mathbin{\|}\bm{0})\end{array}\right\\}$ $\displaystyle\vdash\begin{array}[t]{l}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}{:}\leavevmode\nobreak\ \mu X\mathbin{.}\&^{0}\\{\mathsf{login}{:}\leavevmode\nobreak\ \bullet\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{1}{\oplus}^{10}\\{\mathsf{auth}{:}\leavevmode\nobreak\ \bullet\mathbin{\otimes}^{11}X\\},\mathsf{quit}{:}\leavevmode\nobreak\ \bullet\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{1}\bullet\\}=\overline{(G_{\mathsf{auth}}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{0}s)},\\\ {{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}{:}\leavevmode\nobreak\ \mu X\mathbin{.}{\oplus}^{1}\\{\mathsf{login}{:}\leavevmode\nobreak\ \bullet\mathbin{\otimes}^{2}X,\mathsf{quit}{:}\leavevmode\nobreak\ \bullet\mathbin{\otimes}^{2}\bullet\\}={{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{\mathsf{auth}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(s,c){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{s{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}c}^{0},\\\ {{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}{:}\leavevmode\nobreak\ \mu X\mathbin{.}{\oplus}^{1}\\{\mathsf{login}{:}\leavevmode\nobreak\ \&^{9}\\{\mathsf{auth}{:}\leavevmode\nobreak\ \bullet\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{10}X\\},\mathsf{quit}{:}\leavevmode\nobreak\ \bullet\\}={{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{\mathsf{auth}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(s,a){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{s{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}a}^{0}\end{array}$ $\displaystyle\mathcal{R}_{a}$ $\displaystyle=\mu X({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}},{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}},{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}})\mathbin{.}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}\mathbin{\triangleright}\left\\{\begin{array}[]{l}\mathsf{login}{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}}\mathbin{\triangleleft}\mathsf{login}\\\ {}\cdot{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}\mathbin{\triangleright}\left\\{\begin{array}[]{l}\mathsf{login}{:}\\\ {{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}\mathbin{\triangleright}\left\\{\begin{array}[]{l}\mathsf{passwd}{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}}\mathbin{\triangleleft}\mathsf{passwd}\cdot{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}(u)\mathbin{.}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}}[u^{\prime}]\\\ {}\cdot\left(\begin{array}[]{l}u\mathbin{\leftrightarrow}u^{\prime}\\\ {}\mathbin{\|}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}\mathbin{\triangleright}\left\\{\begin{array}[]{l}\mathsf{auth}{:}\\\ \overline{{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}}\mathbin{\triangleleft}\mathsf{auth}\cdot{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}(v)\mathbin{.}\overline{{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}}[v^{\prime}]\\\ {}\cdot(v\mathbin{\leftrightarrow}v^{\prime}\mathbin{\|}X{\langle{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}},{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}},{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}\rangle})\end{array}\right\\}\end{array}\right)\end{array}\right\\},\\\ \mathsf{quit}{:}\leavevmode\nobreak\ {\mathchoice{\textstyle}{}{}{}\mathsf{alarm}({{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}},{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}},{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}})}\end{array}\right\\},\\\ \mathsf{quit}{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}}\mathbin{\triangleleft}\mathsf{quit}\\\ {}\cdot{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}\mathbin{\triangleright}\left\\{\begin{array}[]{l}\mathsf{login}{:}\leavevmode\nobreak\ {\mathchoice{\textstyle}{}{}{}\mathsf{alarm}({{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}},{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}},{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}})},\\\ \mathsf{quit}{:}\leavevmode\nobreak\ {{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}\mathbin{\triangleright}\\{\mathsf{quit}{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}}\mathbin{\triangleleft}\mathsf{quit}\cdot{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}(w)\mathbin{.}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}}[w^{\prime}]\cdot(w\mathbin{\leftrightarrow}w^{\prime}\mathbin{\|}\bm{0})\\},\end{array}\right\\}\end{array}\right\\}$ $\displaystyle\vdash\begin{array}[t]{l}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}{:}\leavevmode\nobreak\ \mu X\mathbin{.}{\oplus}^{2}\left\\{\begin{array}[]{ll}\mathsf{login}{:}&{\oplus}^{6}\\{\mathsf{passwd}{:}\leavevmode\nobreak\ \bullet\mathbin{\otimes}^{7}\&^{8}\\{\mathsf{auth}{:}\leavevmode\nobreak\ \bullet\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{9}X\\}\\},\\\ \mathsf{quit}{:}&{\oplus}^{6}\\{\mathsf{quit}{:}\leavevmode\nobreak\ \bullet\mathbin{\otimes}^{7}\bullet\\}\end{array}\right\\}=\overline{(G_{\mathsf{auth}}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{0}a)},\\\ {{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}{:}\leavevmode\nobreak\ \mu X\mathbin{.}\&^{2}\left\\{\begin{array}[]{ll}\mathsf{login}{:}&\&^{5}\\{\mathsf{passwd}{:}\leavevmode\nobreak\ \bullet\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{6}X\\},\\\ \mathsf{quit}{:}&\&^{5}\\{\mathsf{quit}{:}\leavevmode\nobreak\ \bullet\mathbin{\mathchoice{\rotatebox[origin={c}]{180.0}{$\displaystyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\textstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptstyle{\&}$}}{\rotatebox[origin={c}]{180.0}{$\scriptscriptstyle{\&}$}}}^{6}\bullet\\}\end{array}\right\\}={{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{\mathsf{auth}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(a,c){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{a{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}c}^{0},\\\ {{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}{:}\leavevmode\nobreak\ \mu X\mathbin{.}\&^{1}\\{\mathsf{login}{:}\leavevmode\nobreak\ {\oplus}^{9}\\{\mathsf{auth}{:}\leavevmode\nobreak\ \bullet\mathbin{\otimes}^{10}X\\},\mathsf{quit}{:}\leavevmode\nobreak\ \bullet\\}={{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\llparenthesis}\mkern-1.0muG_{\mathsf{auth}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(a,s){\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rrparenthesis}}_{a{\color[rgb]{0,0.56640625,0.61328125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.56640625,0.61328125}\rangle}s}^{0}\end{array}$ Figure 14: Routers synthesized from $G_{\mathsf{auth}}$. The typed routers synthesized from $G_{\mathsf{auth}}$ are given in Figure 14. Let us explain the behavior of $\mathcal{R}_{a}$, the router of $a$. $\mathcal{R}_{a}$ is a recursive process on recursion variable $X$, using the endpoint for the implementation ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}$ and the endpoint for the other routers ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}$ and ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}$ as context. The initial message in $G_{\mathsf{auth}}$ from $s$ to $c$ is a dependency for $a$’s interactions with both $s$ and $c$. Therefore, the router first branches on the first dependency with $s$: a label received over ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}$ (login or quit). Let us detail the login branch. Here, the router sends login over ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}$. Then, the router branches on the second dependency with $c$: a label received over ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}$ (again, login or quit). * • In the second login branch, the router receives the label passwd over ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}$, which it then sends over ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}$. The router then receives an endpoint (the password) over ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}$, which it forwards over ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}$. Finally, the router receives the label auth over ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}$, which it sends over ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}$. Then, the router receives an endpoint (the authorization result) over ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}$, which it forwards over ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}$. The router then recurses to the beginning of the loop on the recursion variable $X$, passing the endpoints ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}},{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}},{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}$ as recursive context. * • In the quit branch, the router is in an inconsistent state, because it has received a label over ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}$ which does not concur with the label received over ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}$. Hence, the router signals an alarm on its endpoints ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}},{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}},{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}$. Notice how the typing of the routers in Figure 14 follows Theorem 11: for each $p\in\\{c,s,a\\}$, the endpoint ${{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}p}}$ is typed with local projection (Def. 22), and for each $q\in\\{c,s,a\\}\setminus\\{p\\}$ the endpoint ${{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}p}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}q}}$ is typed with relative projection (Defs. 16 and 23). $\displaystyle{\mathsf{O}}_{\\{c,s,a\\}}[G_{\mathsf{auth}}]$ $\displaystyle\hskip 10.00002pt=\begin{array}[t]{l}\mu X({{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}},{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}},{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}})\\\ {}\mathbin{.}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}\mathbin{\triangleright}\left\\{\begin{array}[]{ll}\mathsf{login}{:}&\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}}\mathbin{\triangleleft}\mathsf{login}\cdot\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}}\mathbin{\triangleleft}\mathsf{login}\cdot{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}(u)\mathbin{.}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}}[u^{\prime}]\\\ &{}\cdot(u\mathbin{\leftrightarrow}u^{\prime}\mathbin{\|}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}\mathbin{\triangleright}\left\\{\begin{array}[]{ll}\mathsf{auth}{:}&\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}}\mathbin{\triangleleft}\mathsf{passwd}\cdot{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}(v)\mathbin{.}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}}[v^{\prime}]\\\ &{}\cdot(v\mathbin{\leftrightarrow}v^{\prime}\mathbin{\|}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}\mathbin{\triangleright}\left\\{\begin{array}[]{ll}\mathsf{auth}{:}&\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}}\mathbin{\triangleleft}\mathsf{auth}\cdot{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}(w)\mathbin{.}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}}[w^{\prime}]\\\ &{}\cdot(w\mathbin{\leftrightarrow}w^{\prime}\mathbin{\|}X{\langle{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}},{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}},{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}\rangle})\end{array}\right\\})\end{array}\right\\}),\\\ \\\ \mathsf{quit}{:}&\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}}\mathbin{\triangleleft}\mathsf{quit}\cdot\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}}\mathbin{\triangleleft}\mathsf{quit}\cdot{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}(z)\mathbin{.}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}}[z^{\prime}]\\\ &{}\cdot(z\mathbin{\leftrightarrow}z^{\prime}\mathbin{\|}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}\mathbin{\triangleright}\\{\mathsf{quit}{:}\leavevmode\nobreak\ \overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}}\mathbin{\triangleleft}\mathsf{quit}\cdot{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}(y)\mathbin{.}\overline{{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}}[y^{\prime}]\cdot(y\mathbin{\leftrightarrow}y^{\prime}\mathbin{\|}\bm{0})\\})\end{array}\right\\}\end{array}$ $\displaystyle\hskip 10.00002pt\vdash{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}{:}\leavevmode\nobreak\ \overline{G_{\mathsf{auth}}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{0}c},{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}{:}\leavevmode\nobreak\ \overline{G_{\mathsf{auth}}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{0}s},{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}{:}\leavevmode\nobreak\ \overline{G_{\mathsf{auth}}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{0}a}$ Figure 15: Orchestrator synthesized from $G_{\mathsf{auth}}$ (cf. Def. 36). Consider again the participant implementations given in Example 1: $P$ implements the role of $c$, $Q$ the role of $s$, and $R$ the role of $a$. Notice that the types of the channels of these processes coincide with relative projections: $\displaystyle P$ $\displaystyle\vdash\emptyset;{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}{:}\leavevmode\nobreak\ G_{\mathsf{auth}}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{0}c$ $\displaystyle Q$ $\displaystyle\vdash\emptyset;{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}{:}\leavevmode\nobreak\ G_{\mathsf{auth}}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{0}s$ $\displaystyle R$ $\displaystyle\vdash\emptyset;{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}{:}\leavevmode\nobreak\ G_{\mathsf{auth}}\mathbin{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\downharpoonright}}^{0}a$ Let us explore how to compose these implementations with their respective routers. The order of composition determines the network topology. Decentralized By first composing each router with their respective implementation, and then composing the resulting routed implementations, we obtain a decentralized topology: $N_{\mathsf{auth}}^{\mathsf{decentralized}}:=\begin{array}[]{r}(\bm{\nu}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}})\\\ (\bm{\nu}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}})\\\ (\bm{\nu}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}})\end{array}\left(\begin{array}[]{l}\phantom{{}\mathbin{\|}{}}(\bm{\nu}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}})\\\ {}\mathbin{\|}(\bm{\nu}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}})\\\ {}\mathbin{\|}(\bm{\nu}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}})\end{array}\begin{array}[]{l}(\mathcal{R}_{c}\mathbin{\|}P)\\\ (\mathcal{R}_{s}\mathbin{\|}Q)\\\ (\mathcal{R}_{a}\mathbin{\|}R)\end{array}\right)$ This composition is in fact a network of routed implementations of $G$ (cf. Def. 25), so Theorems 19, 23 and 18 apply: we have $N_{\mathsf{auth}}^{\mathsf{decentralized}}\in\mathrm{net}(G_{\mathsf{auth}})$, so $N_{\mathsf{auth}}^{\mathsf{decentralized}}$ behaves as specified by $G_{\mathsf{auth}}$ and is deadlock free. Centralized By first composing the routers, and then composing the connected routers with each implementation, we obtain a centralized topology: $N_{\mathsf{auth}}^{\mathsf{centralized}}:=\begin{array}[]{c}(\bm{\nu}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}})\\\ (\bm{\nu}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}})\\\ (\bm{\nu}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}})\end{array}\left(\begin{array}[]{l}\begin{array}[]{c}(\bm{\nu}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}})\\\ (\bm{\nu}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}c}})\\\ (\bm{\nu}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}a}}_{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}s}})\end{array}\left(\begin{array}[]{l}\phantom{{}\mathbin{\|}{}}\mathcal{R}_{c}\\\ {}\mathbin{\|}\mathcal{R}_{s}\\\ {}\mathbin{\|}\mathcal{R}_{a}\end{array}\right)\begin{array}[]{l}{}\mathbin{\|}P\\\ {}\mathbin{\|}Q\\\ {}\mathbin{\|}R\end{array}\end{array}\right)$ Note that the composition of routers is a _hub of routers_ (Def. 37). Consider the composition of $P$, $Q$ and $R$ with the orchestrator of $G_{\mathsf{auth}}$ (given in Figure 15): $N_{\mathsf{auth}}^{\mathsf{orchestrator}}:=\begin{array}[]{c}(\bm{\nu}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}c}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}})\\\ (\bm{\nu}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}s}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}})\\\ (\bm{\nu}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}a}}_{{\color[rgb]{0.84765625,0,0.4140625}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0,0.4140625}\mu}})\end{array}\left({\mathsf{O}}_{\\{c,s,a\\}}[G_{\mathsf{auth}}]\begin{array}[]{l}{}\mathbin{\|}P\\\ {}\mathbin{\|}Q\\\ {}\mathbin{\|}R\end{array}\right)$ By Theorem 27, the hub of routers and the orchestrator of $G_{\mathsf{auth}}$ are weakly bisimilar (Def. 39). Hence, $N_{\mathsf{auth}}^{\mathsf{centralized}}$ and $N_{\mathsf{auth}}^{\mathsf{orchestrator}}$ behave the same. Since each of $N_{\mathsf{auth}}^{\mathsf{top}}$ with $\mathsf{top}\in\\{\mathsf{decentralized},\mathsf{centralized},\mathsf{orchestrator}\\}$ is typable in empty contexts, by Theorem 18, each of these compositions is deadlock free. Moreover, $N_{\mathsf{auth}}^{\mathsf{decentralized}}$ and $N_{\mathsf{auth}}^{\mathsf{centralized}}$ are structurally congruent, so, by Theorems 19 and 23, they behave as prescribed by $G_{\mathsf{auth}}$. Finally, by Theorem 27, $N_{\mathsf{auth}}^{\mathsf{centralized}}$ and $N_{\mathsf{auth}}^{\mathsf{orchestrator}}$ are bisimilar, and so $N_{\mathsf{auth}}^{\mathsf{orchestrator}}$ also behaves as prescribed by $G_{\mathsf{auth}}$. ## 6 Related Work ###### Types for Deadlock Freedom Our decentralized analysis of global types is related to type systems that ensure deadlock freedom for multiparty sessions with delegation and interleaving [7, 44, 21]. Unlike these works, we rely on a type system for _binary_ sessions which is simple and enables an expressive analysis of global types. Coppo _et al._ [7, 20, 21] give type systems for multiparty protocols, with asynchrony and support for interleaved sessions by tracking of mutual dependencies between them; as per Toninho and Yoshida [50], our example in Section 5.1 is typable in APCP but untypable in their system. Padovani _et al._ [44] develop a type system that enforces liveness properties for multiparty sessions, defined on top of a $\pi$-calculus with labeled communication. Rather than global types, their type structure follows approaches based on _conversation types_ [15]. Toninho and Yoshida [50] analyze binary sessions, leveraging on deadlock freedom results for multiparty sessions to extend Wadler’s CLL [54] with cyclic networks. Their process language is synchronous and uses replication rather than recursion. We note that their Examples 6.8 and 6.9 can be typed in APCP (cf. § 5.1); a detailed comparison between their extended CLL and APCP is interesting future work. ###### MPST and Binary Analyses of Global Types There are many works on MPST and their integration into programming languages; see [38, 3] for surveys. Triggered by flawed proofs of type safety and limitations of usual theories, Scalas and Yoshida [48] define a meta-framework of multiparty protocols based on local types, without global types and projection. Their work has been a source of inspiration for our developments; we address similar issues by adopting relative types, instead of cutting ties with global types. As already mentioned, Caires and Pérez [12] and Carbone _et al._ [16] reduce the analysis of global types to binary session type systems based on intuitionistic and classical linear logic, respectively. Our routers strictly generalize the centralized mediums of Caires and Pérez (cf. § 4.4). We substantially improve over the expressivity of the decentralized approach of Carbone _et al._ based on coherence, but reliant on encodings into centralized arbiters; for instance, their approach does not support the example from Toninho and Yoshida [50] we discuss in § 5.1. Also, Caires and Pérez support neither recursive global types nor asynchronous communication, and neither do Carbone _et al._. Scalas _et al._ [47] leverage on an encoding of binary session types into _linear types_ [23, 41] to reduce multiparty sessions to processes typable with linear types, with applications in Scala programming. Their analysis is decentralized but covers processes with synchronous communication only; also, their deadlock freedom result is limited with respect to ours: it does not support interleaving, such as in the example in § 5.1. ###### Monitoring through MPST Our work and the works discussed so far all consider the verification of implementations of multiparty protocols through static type checking. Bocchi _et al._ [8] use a _dynamic_ approach: communication between implementations is enacted by _monitors_ , which are derived from the global type to prevent protocol violations. In their approach, Bocchi _et al._ rely on the traditional workflow for MPST: projection onto binary session types based on the merge operation. Interestingly, Bocchi _et al._ ’s semantics relies on _routing_ , which is similar in spirit, but not in details, to our routers: their routing approach abstracts away from the actual network structure, while our routers enable the concrete realization of a decentralized network structure. We also note that Bocchi _et al._ ’s monitors, based on finite state machines, live on the level of semantics, while our routers, $\pi$-calculus processes, live on the same level as implementations. The theory by Bocchi _et al._ has resulted in the development of tools for a practical application of monitoring in Python [25], including an extension to real-time systems [43]. ###### Other Approaches to Multiparty Protocols In a broader context, Message Sequence Charts (MSCs) provide graphical specifications of multiparty protocols. Alur _et al._ [2] and Abdallah _et al._ [1] study the decidability of model-checking properties such as implementability of MSC Graphs and High-level MSCs (HMSCs) as Communicating FSMs (CFSMs). Genest _et al._ [32] study the synthesis of implementations of HMSCs as CFSMs; as we do, they use extra synchronization messages in some cases. We follow an entirely different research strand: our analysis is type- based and targets well-formed global types that are implementable by design. We note that the decidability of key notions for MPST (such as well-formedness and typability) has been addressed in [36]. Collaboration diagrams are another visual model for communicating processes (see, e.g. [10]). Salaün _et al._ [46] encode collaboration diagrams into the LOTOS process algebra [28] to enable model-checking [30], realizability checks for synchronous and asynchronous communication, and synthesis of participant implementations. Their implementation synthesis is reminiscent of our router synthesis, and also adds extra synchronization messages to realize otherwise unrealizable protocols with non-local choices. ## 7 Conclusion We have developed a new analysis of multiparty protocols specified as global types. One distinguishing feature of our analysis is that it accounts for multiparty protocols implemented by arbitrary process networks, which can be centralized (as in orchestration-based approaches) but also decentralized (as in choreography-based approaches). Another salient feature is that we can ensure both protocol conformance (protocol fidelity, communication safety) and deadlock freedom, which is notoriously hard to establish for protocols/implementations involving delegation and interleaving. To this end, we have considered asynchronous process implementations in APCP, the typed process language that we introduced in [51]. Our analysis enables the transference of correctness properties from APCP to multiparty protocols. We have illustrated these features using the authorization protocol $G_{\mathsf{auth}}$ adapted from Scalas and Yoshida [48] as a running example; additional examples further justify how our approach improves over previous analyses (cf. Section 5). Our analysis of multiparty protocols rests upon three key innovations: _routers_ , which enable global type analysis as decentralized networks; _relative types_ that capture protocols between pairs of participants; _relative projection_ , which admits global types with non-local choices. In our opinion, these notions are interesting on their own. In particular, relative types shed new light on more expressive protocol specifications than usual MPST, which are tied to notions of local types and merge/subtyping. There are several interesting avenues for future work. Comparing relative and merge-based well-formedness would continue the tread of new projections of global types (cf. App. A for initial findings). We would also like to develop a type system based on relative types, integrating the logic of routers into a static type checking that ensures deadlock freedom for processes. Finally, we are interested in developing practical tool support based on our findings. For this latter point, following [40], we would like to first formalize a theory of runtime monitoring based on routers, which can already be seen as an elementary form of _choreographed monitoring_ (cf. [29]). ##### Acknowledgments We are grateful to the anonymous reviewers for their constructive feedback and suggestions, which were enormously helpful to improve the presentation. 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Before we recall the definition of merge-based well-formedness, we define the projection of global types to local types. Local types express one particular participant’s perspective of a global protocol. Although $\mathsf{skip}$ is not part of standard definitions of local types, we include it to enable a fair comparison with relative types. ###### Definition 42 (Local types). _Local types_ $L$ are defined as follows, where the $S_{i}$ are the message types from Def. 11: $L::={?}p\\{i\langle S\rangle\mathbin{.}L\\}_{i\in I}\;\mbox{\large{$\mid$}}\;{!}p\\{i\langle S\rangle\mathbin{.}L\\}_{i\in I}\;\mbox{\large{$\mid$}}\;\mu X\mathbin{.}L\;\mbox{\large{$\mid$}}\;X\;\mbox{\large{$\mid$}}\;\bullet\;\mbox{\large{$\mid$}}\;\mathsf{skip}\mathbin{.}L$ The local types ${?}p\\{i\langle S_{i}\rangle\mathbin{.}L_{i}\\}_{i\in I}$ and ${!}p\\{i\langle S_{i}\rangle\mathbin{.}L_{i}\\}_{i\in I}$ represent receiving a choice from $p$ and sending a choice to $p$, respectively. All of $\bullet$, $\mu X\mathbin{.}L$, $X$, and $\mathsf{skip}$ are just as before. Instead of external dependencies, the projection onto local types relies on an operation on local types called _merge_. Intuitively, merge allows combining overlapping but not necessarily identical receiving constructs. This is one main difference with respect to our relative projection. ###### Definition 43 (Merge of Local Types). For local types $L_{1}$ and $L_{2}$, we define $L_{1}\sqcup L_{2}$ as the _merge_ of $L_{1}$ and $L_{2}$: $\displaystyle\mathsf{skip}\mathbin{.}L_{1}\sqcup\mathsf{skip}\mathbin{.}L_{2}$ $\displaystyle:=L_{1}\sqcup L_{2}$ $\displaystyle\bullet\sqcup\bullet$ $\displaystyle:=\bullet$ $\displaystyle\mu X\mathbin{.}L_{1}\sqcup\mu X\mathbin{.}L_{2}$ $\displaystyle:=\mu X\mathbin{.}(L_{1}\sqcup L_{2})$ $\displaystyle X\sqcup X$ $\displaystyle:=X$ $\displaystyle{!}p\\{i\langle S_{i}\rangle\mathbin{.}L_{i}\\}_{i\in I}\sqcup{!}p\\{i\langle S_{i}\rangle\mathbin{.}L_{i}\\}_{i\in I}$ $\displaystyle:={!}p\\{i\langle S_{i}\rangle\mathbin{.}L_{i}\\}_{i\in I}$ $\displaystyle{?}p\\{i\langle S_{i}\rangle\mathbin{.}L_{i}\\}_{i\in I}\sqcup{?}p\\{j\langle S^{\prime}_{j}\rangle\mathbin{.}L^{\prime}_{j}\\}_{j\in J}$ $\displaystyle:={?}p\left(\begin{array}[]{l}\phantom{{}\cup{}}\\{i\langle S_{i}\rangle\mathbin{.}L_{i}\\}_{i\in I\setminus J}\\\ {}\cup\\{j\langle S^{\prime}_{j}\rangle\mathbin{.}L^{\prime}_{j}\\}_{j\in J\setminus I}\\\ {}\cup\\{k\langle S_{k}\sqcup S^{\prime}_{k}\rangle\mathbin{.}(L_{k}\sqcup L^{\prime}_{k})\\}_{k\in I\cap J}\end{array}\right)$ The merge between message types $S_{1}\sqcup S_{2}$ corresponds to the identity function. If the local types do not match the above definition, their merge is undefined. We can now define local projection based on merge: ###### Definition 44 (Merge-based Local Projection). For global type $G$ and participant $p$, we define $G\mathbin{\upharpoonright}p$ as the _merge-based local projection_ of $G$ under $p$: $\displaystyle\bullet\mathbin{\upharpoonright}p$ $\displaystyle:=\bullet$ $\displaystyle(\mathsf{skip}\mathbin{.}G)\mathbin{\upharpoonright}p$ $\displaystyle:=\mathsf{skip}\mathbin{.}(G\mathbin{\upharpoonright}p)$ $\displaystyle X\mathbin{\upharpoonright}p$ $\displaystyle:=X$ $\displaystyle(\mu X\mathbin{.}G)\mathbin{\upharpoonright}p$ $\displaystyle:=\mathrlap{\begin{cases}\bullet&\text{if $G\mathbin{\upharpoonright}p=\mathsf{skip}^{\ast}\mathbin{.}\bullet$ or $G\mathbin{\upharpoonright}p=\mathsf{skip}^{\ast}\mathbin{.}X$}\\\ \mu X\mathbin{.}(G\mathbin{\upharpoonright}p)&\text{otherwise}\end{cases}}$ $\displaystyle(r\mathbin{\twoheadrightarrow}s\\{i\langle U_{i}\rangle\mathbin{.}G_{i}\\}_{i\in I})\mathbin{\upharpoonright}p$ $\displaystyle:=\mathrlap{\begin{cases}{?}r\\{i\langle U_{i}\rangle\mathbin{.}(G_{i}\mathbin{\upharpoonright}p)\\}_{i\in I}&\text{if $p=s$}\\\ {!}s\\{i\langle U_{i}\rangle\mathbin{.}(G_{i}\mathbin{\upharpoonright}p)\\}_{i\in I}&\text{if $p=r$}\\\ \mathsf{skip}\mathbin{.}(\sqcup_{i\in I}(G_{i}\mathbin{\upharpoonright}p))&\text{otherwise}\end{cases}}$ $\displaystyle(G_{1}\mathbin{\|}G_{2})\mathbin{\upharpoonright}p$ $\displaystyle:=\mathrlap{\begin{cases}G_{1}\mathbin{\upharpoonright}p&\text{if $p\in\mathsf{prt}(G_{1})$ and $p\notin\mathsf{prt}(G_{2})$}\\\ G_{2}\mathbin{\upharpoonright}p&\text{if $p\in\mathsf{prt}(G_{2})$ and $p\notin\mathsf{prt}(G_{1})$}\\\ \bullet&\text{if $p\notin\mathsf{prt}(G_{1})\cup\mathsf{prt}(G_{2})$}\end{cases}}$ ###### Definition 45 (Merge Well-Formedness). A global type $G$ is _merge well-formed_ if, for every $p\in\mathsf{prt}(G)$, the merge-based local projection $G\mathbin{\upharpoonright}p$ is defined. The classes of relative and merge-based well-formed global types overlap: there are protocols that can be expressed using dependencies in relative types, as well as using merge in local types. Interestingly, the classes are _incomparable_ : some relative well-formed global types are not merge-based well-formed, and vice versa. We now explore these differences. ### A.1 Relative Well-Formed, Not Merge Well-Formed The merge of local types with outgoing messages of different labels is undefined. Therefore, if a global type has communications, e.g., from $s$ to $a$ with different labels across branches of a prior communication between $b$ and $a$, the global type is not merge well-formed. In contrast, such global types can be relative well-formed, because the prior communication may induce a dependency. Similarly, global types with communications with different participants across branches of a prior communication are never merge well- formed, but may be relative well-formed. The following example demonstrates a global type with messages of different labels across branches of a prior communication: ###### Example. We give an adaptation of the two-buyer-seller protocol in which Seller ($s$) tells Alice ($a$) to pay or not, depending on whether Bob ($b$) tells $a$ to buy or not. $\displaystyle G_{\mathsf{rwf}}:=b\mathbin{\twoheadrightarrow}a\left\\{\begin{array}[]{@{}l@{}}\mathsf{ok}\mathbin{.}s\mathbin{\twoheadrightarrow}a{:}\mathsf{pay}\langle\mathsf{int}\rangle\mathbin{.}\bullet,\\\ \mathsf{cancel}\mathbin{.}s\mathbin{\twoheadrightarrow}a{:}\mathsf{cancel}\mathbin{.}\bullet\end{array}\right\\}$ This protocol is relative well-formed, as the relative projections under every combination of participants are defined. Notice how there is a dependency in the relative projection under $s$ and $a$: $\displaystyle G_{\mathsf{rwf}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(s,a)=a{?}b\left\\{\begin{array}[]{@{}l@{}}\mathsf{ok}\mathbin{.}s{:}\mathsf{pay}\langle\mathsf{int}\rangle\mathbin{.}\bullet,\\\ \mathsf{cancel}\mathbin{.}s{:}\mathsf{cancel}\mathbin{.}\bullet\end{array}\right\\}$ However, we do not have merge well-formedness: the merge-based local projection under $s$ is not defined: $\displaystyle G_{\mathsf{rwf}}\mathbin{\upharpoonright}s=\mathsf{skip}\mathbin{.}({!}a{:}\mathsf{pay}\langle\mathsf{int}\rangle\mathbin{.}\bullet\sqcup{!}a{:}\mathsf{cancel}\mathbin{.}\bullet)$ ### A.2 Merge Well-Formed, Not Relative Well-Formed For a communication between, e.g., $a$ and $b$ to induce a dependency for subsequent communications between other participants, at least one of $a$ and $b$ must be involved. Therefore, global types where communications with participants other than $a$ and $b$ have different labels across branches of a prior communication between $a$ and $b$ are never relative well-formed. In contrast, merge can combine the reception of different labels, so such global types may be merge well-formed—as long as the sender is aware of which branch has been taken before. The following example demonstrates such a situation, and explains how such global types can be modified to be relative well-formed: ###### Example. Consider a variant of the two-buyer-seller protocol in which Seller ($s$) invokes a new participant, Mail-service ($m$), to deliver the requested product. In the following global type, Bob ($b$) tells Alice ($a$) of its decision to buy or not, after which $b$ sends the same choice to $s$, who then either invokes $m$ to deliver the product or not: $\displaystyle G_{\mathsf{mwf}}:=b\mathbin{\twoheadrightarrow}a\left\\{\begin{array}[]{@{}l@{}}\mathsf{ok}\mathbin{.}b\mathbin{\twoheadrightarrow}s{:}\mathsf{ok}\mathbin{.}s\mathbin{\twoheadrightarrow}m{:}\mathsf{deliver}\langle\mathsf{str}\rangle\mathbin{.}\bullet,\\\ \mathsf{quit}\mathbin{.}b\mathbin{\twoheadrightarrow}s{:}\mathsf{quit}\mathbin{.}s\mathbin{\twoheadrightarrow}m{:}\mathsf{quit}\mathbin{.}\bullet\end{array}\right\\}$ $G_{\mathsf{mwf}}$ is merge well-formed: the merge-based local projections under all participants are defined. Notice how the two different messages from $s$ are merged in the merge-based local projection under $m$: $\displaystyle G_{\mathsf{mwf}}\mathbin{\upharpoonright}m=\mathsf{skip}^{2}\mathbin{.}{?}s\left\\{\begin{array}[]{@{}l@{}}\mathsf{deliver}\langle\mathsf{str}\rangle\mathbin{.}\bullet,\\\ \mathsf{quit}\mathbin{.}\bullet\end{array}\right\\}$ $G_{\mathsf{mwf}}$ is not relative well-formed: the relative projection under $s$ and $m$ is not defined. The initial exchange between $b$ and $a$ cannot induce a dependency, since neither of $s$ and $m$ is involved. Hence, the relative projections of both branches must be identical, but they are not: $\displaystyle\mathsf{skip}\mathbin{.}s{:}\mathsf{deliver}\langle\mathsf{str}\rangle\mathbin{.}\bullet\neq\mathsf{skip}\mathbin{.}s{:}\mathsf{quit}\mathbin{.}\bullet$ We recover relative well-formedness by modifying $G_{\mathsf{mwf}}$: we give $s$ the same options to send to $m$ in both branches of the initial communication: $\displaystyle G^{\prime}_{\mathsf{mwf}}:=b\mathbin{\twoheadrightarrow}a\left\\{\begin{array}[]{@{}l@{}}\mathsf{ok}\mathbin{.}b\mathbin{\twoheadrightarrow}s{:}\mathsf{ok}\mathbin{.}s\mathbin{\twoheadrightarrow}m\\{\mathsf{deliver}\langle\mathsf{str}\rangle\mathbin{.}\bullet,\quad\mathsf{quit}\mathbin{.}\bullet\\},\\\ \mathsf{quit}\mathbin{.}b\mathbin{\twoheadrightarrow}s{:}\mathsf{quit}\mathbin{.}s\mathbin{\twoheadrightarrow}m\\{\mathsf{deliver}\langle\mathsf{str}\rangle\mathbin{.}\bullet,\quad\mathsf{quit}\mathbin{.}\bullet\\}\end{array}\right\\}$ The new protocol is still merge well-formed, but it is now relative well- formed too; the relative projection under $s$ and $m$ is defined: $\displaystyle G^{\prime}_{\mathsf{mwf}}\mathbin{{\color[rgb]{0.09765625,0,0.54296875}\definecolor[named]{pgfstrokecolor}{rgb}{0.09765625,0,0.54296875}\upharpoonright}}(s,m)=\mathsf{skip}^{2}\mathbin{.}s\left\\{\begin{array}[]{@{}l@{}}\mathsf{deliver}\langle\mathsf{address}\rangle\mathbin{.}\bullet,\\\ \mathsf{quit}\mathbin{.}\bullet\end{array}\right\\}$ This modification may not be ideal, though, because $s$ can quit the protocol even if $b$ has ok’ed the transaction, and that $s$ can still invoke a delivery even if $b$ has quit the transaction.
# Blast waves in a paraxial fluid of light Murad Abuzarli Tom Bienaimé Elisabeth Giacobino Alberto Bramati Quentin Glorieux<EMAIL_ADDRESS>ALaboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-Université PSL, Collège de France - Paris, France ###### Abstract We study experimentally blast wave dynamics on a weakly interacting fluid of light. The fluid density and velocity are measured in 1D and 2D geometries. Using a state equation arising from the analogy between optical propagation in the paraxial approximation and the hydrodynamic Euler’s equation, we access the fluid hydrostatic and dynamic pressure. In the 2D configuration, we observe a negative differential hydrostatic pressure after the fast expansion of a localized over-density, which is a typical signature of a blast wave for compressible gases. Our experimental results are compared to the Friedlander waveform hydrodynamical model[1]. Velocity measurements are presented in 1D and 2D configurations and compared to the local speed of sound, to identify supersonic region of the fluid. Our findings show an unprecedented control over hydrodynamic quantities in a paraxial fluid of light. ††preprint: APS/123-QED ## Introduction In classical hydrodynamics, a blast wave is characterized by an increased pressure and flow resulting from the rapid release of energy from a concentrated source [2]. The particular characteristics of a blast wave is that it is followed by a wind of negative pressure, which induces an attractive force back towards the origin of the shock. Typical blast waves occur after the detonation of trinitrotoluene [3, 4], nuclear fission [5], break of a pressurized container [6] or heating caused by a focused pulsed laser [7]. The sudden release of energy causes a rapid expansion, which in a three dimensional space is analogous to a spherical piston [8] and produces a compression wave in the ambient gas. For a fast enough piston, the compression wave develops into a shock wave which is characterized by the rapid increase of all the physical properties of the gas, namely, the hydrostatic pressure, density and particle velocity [9]. In 1946, Friedlander predicted that immediately after the shock front the physical properties at a given position in space decay exponentially [1, 10]. In this model, for 3-dimensional and 2-dimensional spaces the hydrostatic pressure and the density are expected to fall below the values of the ambient atmosphere leading to a blast wind [2]. Shock waves have been studied in several contexts in physics, including acoustics, plasma physics, ultra-cold atomic gases [11, 12, 13] and non-linear optics [14, 15, 16, 17, 6]. In optics, the hydrodynamics interpretation relies on the Madelung transforms which identify the light intensity to the fluid density and the phase gradient to the fluid velocity[18]. Recently several works have studied analytically shock wave formation in one and two dimensions [19, 20]. Optical systems allow for repeatable experiments and precise control of the experimental parameters. For example dispersive superfluid-like shock waves have been observed [14], as well as generation of solitons [16], shocks in non-local media [21, 15], shocks in disordered media [17], analogue dam break [6] and Riemann waves [22]. However, an experimental study of blast waves has not been done in atomic gases nor in non-linear optics systems. In this work, we demonstrate the generation of a blast wave in a fluid of light. Interestingly, the prediction of a blast wind with negative pressure and density holds in two dimensional space but not in 1 dimension [23]. Optical analogue systems allow for an experimental validation of this prediction. In this letter, we study the formation of blast waves in a paraxial fluid of light. We measure the time evolution of analogue physical properties such as the hydrostatic pressure, the density, the particle velocity and the dynamic pressure at a fixed point for 1 and 2-dimensional systems. We report the observation of a negative hydrostatic differential pressure after a shock wave in 2-dimensional system and we show that the Friedlander waveform describes quantitatively our experimental results for all physical parameters. This paper is organized as follows. We first introduce the analogy between the propagation equation of a laser beam through non-linear medium (a warm atomic vapor) and the hydrodynamics equation and derive the relevant analogue physical properties. In the second section of this work, we describe our experimental setup and present our results on the density and hydrostatic pressure measurements. We highlight the striking differences between 1 and 2-dimensional systems. Finally, we study the time evolution of the velocity and dynamic pressure. ## Theoretical Model We describe the propagation of a linearly polarized monochromatic beam in a local Kerr medium. We separate the electric field’s fast oscillating carrier from the slowly varying (with respect to the laser wavelength) envelope : $E=\mathcal{E}(\mathbf{r},z)$e${}^{i(kz-k_{0}ct)}+$ complex conjugate. Under the paraxial approximation, the propagation equation for the envelope $\mathcal{E}$ is the Non-Linear Schrödinger Equation (NLSE) [18]: $i\frac{\partial\mathcal{E}}{\partial z}=\left(-\frac{1}{2k}\nabla^{2}_{\perp}+g{\mid}\mathcal{E}{\mid}^{2}-\frac{i\alpha}{2}\right)\mathcal{E},$ (1) where $k$ is the laser wavevector in the medium, $\alpha$ is the extinction coefficient accounting for losses due to absorption, and the $g$ parameter is linked to the intensity dependent refractive index variation $\Delta n$ via: $g{\mid}\mathcal{E}{\mid}^{2}=-k_{0}\Delta n$ (with $k_{0}$ the laser wavevector in vacuum). The NLSE is analogous to a 2D Gross-Pitaevskii equation describing the dynamics of a quantum fluid in the mean-field approximation. This analogy is possible by mapping the envelope $\mathcal{E}$ to the quantum fluid many-body wavefunction and the axial coordinate $z$ to an effective evolution time. The non-linear refractive index variation plays then the role of a repulsive photon-photon interaction, since all measurements in this work are done in the self-defocusing regime i.e. $\Delta n<0$ and therefore $g>0$. Diffraction acts as kinetic energy with the effective mass emerging from the paraxial approximation and given by the laser wavevector $k=8.10^{6}$ m-1. Using the Madelung transformation: $\mathcal{E}=\sqrt{\rho}\textrm{e}^{i\phi}$, $\mathbf{v}=\frac{c}{k}\nabla_{\perp}\phi$ one can derive from the NLSE hydrodynamic equations [19, 14], linking the fluid’s density $\rho$ with its velocity $\mathbf{v}$: $\displaystyle\frac{\partial\rho}{\partial z}+\nabla_{\perp}.\left(\rho\frac{\mathbf{v}}{c}\right)=-\alpha\rho$ (2) $\displaystyle\frac{\partial\mathbf{v}}{\partial z}+\frac{1}{2c}\nabla_{\perp}\mathbf{v}^{2}=-\nabla_{\perp}\left(\frac{cg\rho}{k}-\frac{c}{2k^{2}\sqrt{\rho}}\nabla^{2}_{\perp}\sqrt{\rho}\right).$ (3) Eq. (2) is the continuity equation with a loss term accounting for photon absorption. Eq. (3) is similar to the Euler equation without viscosity, in which the driving force stems from interaction and the so-called quantum pressure term due to diffraction. Establishing the formal analogy requires, however, defining an analogue pressure $P$ to be able to re-express the right- hand side of Eq. (3) as: $-1/\rho\cdot\nabla_{\perp}P$. This is possible for the first term stemming from interactions. Using the identity: $-\nabla_{\perp}\rho=-1/(2\rho)\nabla_{\perp}\rho^{2}$ one can define the so called bulk hydrostatic pressure $P$ as: $\displaystyle P=\frac{c^{2}}{2}\frac{\rho^{2}g}{k}=\frac{1}{2}\rho c_{s}^{2},$ (4) where the last equality comes from $c_{s}^{2}=c^{2}\cdot g\rho/k$. Eq. (4) is the state equation linking the fluid hydrostatic pressure $P$ to its density if one neglects the quantum pressure term. It is the consequence of the mean- field formulation of the interaction. It also implies that the fluid of light is compressible with the compressibility equal to: $k/(c^{2}\rho^{2}g)$. One then gets the analogue Euler equation: $\displaystyle\frac{\partial\mathbf{v}}{\partial(z/c)}+\frac{1}{2}\nabla_{\perp}\mathbf{v}^{2}=-\frac{1}{\rho}\nabla_{\perp}P,$ (5) with a pressure $P$ of dimension [density]$\times$[speed]2. As already mentioned, the fluid dynamics can be studied by accessing its state at different $z$ positions, however this is not recommend practically since imaging inside a non-linear medium is highly challenging task. Alternatively, one can instead re-scale the effective time by incorporating fluid interaction [24, 20]. Fluid interaction can then be varied experimentally and the fluid dynamics can be studied while imaging only the state at the medium output plane. Re-scaling the time is based on defining following quantities: $\displaystyle z_{NL}=\frac{1}{g\rho(0,L)},\ \ \textrm{non-linear axial length}$ (6) $\displaystyle\xi=\sqrt{\frac{z_{NL}}{k},}\ \ \textrm{transverse healing length}$ (7) $\displaystyle c_{s}=\frac{c}{k\xi},\ \ \textrm{speed of sound}$ (8) $\displaystyle\psi=\frac{\mathcal{E}}{\sqrt{\rho(0,L)}},$ (9) and substituting the time and space variables as: $\tau=~{}z/z_{NL}$, $\tilde{\mathbf{r}}=\mathbf{r}/\xi$, $\tilde{\nabla}_{\perp}=\xi\nabla_{\perp}$. $L$ is the non-linear medium length. The propagation equation then reads: $i\frac{\partial\psi}{\partial\tau}=\left(-\frac{1}{2}\tilde{\nabla}^{2}_{\perp}+{\mid}\psi{\mid}^{2}\right)\psi.$ (10) One can note that dynamics of $\psi$ is not anymore dissipative, due to the normalization with respect to the exponentially decaying density: $\rho(0,L)=\rho(0,0)\textrm{exp}(-\alpha L)$, measured at at the medium exit plane. This formulation is necessary to describe accurately the experimental results of this work probing temporal dynamics of a fluid of light by varying the strength of the optical non-linearity and not the imaged $z$ plane. The effective time $\tau=\|\Delta n(\mathbf{r}_{\perp}=0,L)\|k_{0}L$ equals to the maximal accumulated non-linear phase. Rewriting the Madelung transformation with the new variables, we obtain: $\psi=\sqrt{\tilde{\rho}}\textrm{e}^{i\phi}=\sqrt{\frac{\rho}{\rho(0,L)}}\textrm{e}^{i\phi},\ \ \ \tilde{\mathbf{v}}=\frac{\mathbf{v}}{c_{s}}=\tilde{\nabla}_{\perp}\phi.$ (11) One gets dimensionless Euler and the continuity equations: $\displaystyle\frac{\partial\tilde{\rho}}{\partial\tau}+\tilde{\nabla_{\perp}}.\left(\tilde{\rho}\tilde{\mathbf{v}}\right)=0$ (12) $\displaystyle\frac{\partial\tilde{\mathbf{v}}}{\partial\tau}+\frac{1}{2}\tilde{\nabla_{\perp}}\tilde{\mathbf{v}}^{2}=-\tilde{\nabla_{\perp}}\left(\tilde{\rho}-\frac{1}{2\sqrt{\tilde{\rho}}}\tilde{\nabla^{2}_{\perp}}\sqrt{\tilde{\rho}}\right),$ (13) where the link between Eq. (13) and the Euler equation is made by neglecting the quantum pressure and defining the the dimensionless hydrostatic pressure as: $\tilde{P}=\frac{1}{2}\tilde{\rho}^{2}.$ (14) Finally, the dynamic pressure is defined as a vector quantity by: $\tilde{P_{d}}=\frac{1}{2}\tilde{\rho}\tilde{\mathbf{v}}\|\tilde{\mathbf{v}}\|,$ (15) The dynamic pressure is the fluid kinetic energy flux and accounts for the amount of pressure due to fluid motion. The impact force on an obstacle hit by a shockwave is proportional to its dynamic pressure. Expressed in dimensionless units, the dynamic pressure gives the strength of the convection term normalized by the pressure due to the interactions in the Eq. (13). It can be computed directly from the density and velocity measurements. ## Shock waves and blast wind In this work, we study the dynamics of a fluid of light disturbed by a localized Gaussian over-density $\delta\rho(\mathbf{r},0)=~{}\rho_{1}~{}\textrm{exp}\left(-2\mathbf{r}^{2}/\omega_{1}^{2}\right)$. $\rho_{1}$ is of the same magnitude as the background fluid density $\rho_{0}$ and $\omega_{1}$ quantifies the perturbation width. We can write $\rho(\mathbf{r},L)=\rho_{0}+\delta\rho(\mathbf{r},L).$ Normalizing the total density by its maximal undisturbed value one gets: $\tilde{\rho}(\mathbf{r},\tau)=\rho(\mathbf{r},L)/\rho_{0}(0,L)$. Extending this definition to $\rho_{0}$ and $\rho_{1}$, we obtain $\tilde{\rho}_{0}$ bound between 0 and 1 and having a Gaussian shape, and $\tilde{\rho}_{1}$ expressing the perturbation strength with respect to the fluid background density. To take into account the Gaussian profile of the density $\rho_{0}$, we define the over-pressure from the pressure difference between the case with and without perturbation: $\delta\tilde{P}(\mathbf{r},\tau)=\tilde{P}(\mathbf{r},\tau)-\tilde{P}_{0}(\mathbf{r},\tau).$ (16) To evaluate the differential pressure $\Delta\tilde{P}(\tau)$, showing the instantaneous difference in pressure between the perturbation center and the external undisturbed area, we define: $\Delta\tilde{P}(\tau)=\tilde{P}(0,\tau)-P_{0}(r_{ext},\tau).$ (17) The differential pressure $\Delta\tilde{P}(\tau)$ is the most important quantity we study in this work and we expect major differences in the non- linear perturbation dynamics between the 1D and the 2D geometries. Finally, the fluid velocity can be measured experimentally. It requires a measurement of the beam wavefront which is realized using off-axis interferometry. Calculating numerically the gradient of the phase, we obtain the background fluid velocity $\mathbf{v}_{0}$ and the perturbation velocity $\mathbf{v}_{1}$ by analyzing the images without and with the perturbation, respectively. Several studies have been performed in both $\rho_{1}\ll~{}\rho_{0}$ and $\rho_{1}\gg\rho_{0}$ regimes, observing the Bogoliubov dispersion of the linearized waves created by the perturbation [25, 26, 27], and the shock waves [14, 20], respectively. In this work we investigate the case $\rho_{1}\sim\rho_{0}$ by analyzing the fluid density, velocity and pressure both in the 1D and 2D geometries. The NLSE is known to give rise to sound-like dispersion to the low amplitude waves, governed by the Bogoliubov theory. Here, a perturbation of the same order (or larger) than the background results in the sound velocity variation following the local density inside the perturbation. This is the prerequisite for observing shock waves, a special type of waves changing their shape during propagation towards a steepening profile. In hydrodynamics, shock waves are usually reported as a time evolution measurement of a physical quantity (pressure, density…) at a fixed point in space. After the passage of a the shock wave front, a blast wind (a negative differential pressure) should be observed in 2 and 3 dimensional space. A direct physical consequence of this wind in classical hydrodynamics is observed for example after an explosion inside an edifice: the presence of glass pieces within the building is the signature of the blast wind . In the next section we report the time evolution as well as the time snapshots (spatial map of a physical quantity at fixed time) typically not accessible in classical hydrodynamics experiments. ## Experimental setup In our experiment, we investigate the propagation of a near-resonance laser beam through a warm rubidium vapor cell, which induces effective photon-photon interactions [28]. Two configurations are studied: the 2D geometry with a radially symmetric dynamics and the 1D geometry with a background much larger along $x$ than along $y$ which allows for a 1D description of non-linear wave dynamics [20]. A tapered amplified diode laser is split into a background, a reference and a perturbation beams (see supplementary materials for details). The background beam is enlarged with a telescope up to a waist of 2.5$\pm$0.5 mm along $x$ and 0.8$\pm$0.1 mm along $y$ in the 1D geometry, and 1.8$\pm$0.3 mm along the radial coordinate in the 2D configuration. The reference beam (for interferometric phase measurement) is matched to the same dimensions. The perturbation beam is focused to get the waist of 0.12$\pm$0.03 mm in the middle of the cell (the corresponding Rayleigh range is 55 mm). The background and perturbation are recombined with a 90R:10T beam splitter such that 90 % of the background beam power is reflected towards the cell. The second arm of the BS is sent through a 200 $\mu m$ diameter pinhole into a photodiode to stabilize the interferometer. The control is realized by locking on local minimum acting on a piezoelectric mirror mount with a RedPitaya hardware run by the PyRPL software [29]. Cell temperature is 149(2)° C leading to an atomic density of 8.3$\pm$0.8$\times 10^{13}$ cm-3. The cell output is imaged with a $\times$4.2 magnifying 4-f setup onto a camera. Sets of 4 images (background only, background with reference, background with perturbation and finally background with both perturbation and reference) at different input powers $\mathcal{P}$ ranging from 50 to 600 mW and different laser detunings $\Delta$ from the 85Rb D2 line $F=3\rightarrow F^{\prime}$ transition are taken (see supplementary materials for details). The reference beam is superimposed with other beams with an angle of 30 milli-radians, giving rise to interferogramms with vertical fringes of average periodicity of 25$\pm$1 $\mu$m. Figure 1: Density data: The left column corresponds to the 1D configuration and the right column to the 2D case. a) and b) are over-density maps at time $\tau=31$, obtained by subtracting the images with no perturbation from the ones with perturbation in the 1D and 2D geometry, respectively. c), d) are density profiles without (blue) and with (red) perturbation in the 1D and 2D geometry, respectively. The profiles are shifted vertically (spacing of 2) for better visibility. ## Density The density is an important physical parameter needed to compute the static and hydrodynamic pressure. It is directly given by the intensity measurement. In figure 1 a) and b), we present the experimental maps of the over-density $\delta\tilde{\rho}$ at time $\tau=31$, after subtracting the background fluid, in the 1D and 2D geometries respectively. By changing the laser intensity and detuning, we can modify the effective time $\tau$. The associated time $\tau$ is calculated from the nonlinear index $\Delta n$ via the off-axis interferometric measurement for each experimental configuration ($\mathcal{P},\Delta$) (see supplementary materials for details). Fig. 1 a) and b) show the spatio-temporal over-density diagrams. We present the corresponding density profiles at different times in figure 1 c) and d). The 1D density data are averaged over the $y$ direction for ${\mid}y{\mid}<0.1$ mm and the 2D images are radially averaged to get the background fluid density (blue curves) and the total fluid density including the perturbation (red curves). These results show two important effects. In the 1D geometry, a clear steepening of the perturbation front and the development of dispersive shock waves can be seen as an oscillating pattern developing in beyond the shock front with effective time $\tau$. In the 2D geometry, interestingly, the steepening of the shock front is less pronounced. Moreover, a density much lower than the background density is observed in the center of the 2D profiles for long time $\tau>20$, which is not the case in 1D. This negative differential density has a direct consequence on the differential pressure calculated using Eq. (17). ## Static pressure Figure 2: Pressure analysis: a),c): 1D & 2D over-pressure profiles evaluated at different effective times $\tau$. Each following profile shifted vertically by 2 for better visibility. b),d) show the 1D and 2D spatio-temporal diagrams of the over-pressure evolution, respectively. The dotted black lines show the trajectory of expansion at the speed of sound according to the parabolic equation with the prefactor given by $k/L=107$ mm-1 in both geometries. The blue dotted lines show the same trajectories shifted horizontally by 250 $\mu m$ and 200 $\mu m$ in 1D and 2D cases, respectively. It corresponds to external undisturbed area used for the measurement of the differential pressure. Dashed green rectangles around $\tau=40$ show the presence of a second shock due to an increasing differential pressure. Figure 3: Differential pressure calculated from Eq. (17) for the 1D (circular dots) and the 2D cases (square dots). The uncertainty bars correspond to the statistical analysis of multiple images. The pressure is normalized as described in the main text. Blue line is the ambient pressure outside of the shock. Black dashed line is the Friedlander model for a blast wave described in Eq. (18) with $P_{s}=1$ and $t^{}=20$. To isolate the effect of the perturbation on the static pressure, we compute the over-pressure from images of the background with and without the bump taken at same effective times $\tau(\mathcal{P},\Delta)$, using Eq. (16) and (14). The over-pressure as a function of time $\tau$ is shown in Fig. (2) b) and d) and profiles averaged along $y$ in the 1D case and radially in the 2D case are presented in Fig. (2) a) and c) for various times. The trajectory of a density pulse spreading with no dispersion at the speed of sound can be expressed as follows: $r=c_{s}(\tau)\times(L/c)$. The coefficient can be calculated using the time dependence of the sound velocity: $c_{s}=c\sqrt{\tau/(kL)}$ obtained from Eqs. (6) and (8). It directly leads to $\tau=kr^{2}/L$ and knowing that: $L=75$ mm and $k=8\times 10^{3}$ mm-1, one gets: $\tau=107\times r^{2}$. The coefficient does not depend on the dimensionality of the system. In the pressure maps (Fig. (2) b) and d)), we have added a black dashed line following this trend: $\tau=107\times x^{2}$ (1D) and $\tau=107\times r^{2}$ (2D). As expected, this trajectory follows closely the shock front in the 1D geometry. The differential pressure is defined as the pressure difference between inside and outside of the shock as expressed in Eq. (17). The undisturbed pressure as function of time is evaluated along the same trend line $\tau=107\times(r_{ext}-r_{0})^{2}$, translated $r_{0}=250~{}\mu$m in 1D and $r_{0}=200~{}\mu$m in 2D, which corresponds to $\sim 1.5$ times the perturbation beam waist (blue dashed line). In 2D, the shock front expansion is slower than the calculated trajectory, as described in [20], and the blue dashed line can therefore still used to define the undisturbed pressure. The temporal evolution of the differential static pressure (at $x=0$) is presented in Fig. 3. 1D (red circles) and 2D (gray triangles) geometries are compared from $\tau=0$ to $\tau=45$. An important difference can be seen between the two geometries: in the 2D situation the differential pressure becomes negative at $\tau=20$ as it goes to zero in the 1D case. The observation of the negative pressure is the typical signature of a blast wind. This measurement reveals the dramatic impact of the geometry on blast wind in a fluid of light and exemplifies the analogy with classical hydrodynamics. To quantify this analogy, we use the Friedlander waveform model which is known to describe the dynamics of physical quantities in a free-field (i.e. in a open 3-dimensional space) blast wave [2]. In this model the differential pressure follows an exponential decay of the form: $\Delta\tilde{P}=P_{s}e^{-\tau/t^{}}(1-\tau/t^{}),$ (18) where $P_{s}$ and $t^{}$ are two parameters which corresponds respectively to the peak differential pressure immediately behind the shock and to the time when the differential pressure becomes negative. The period when the hydrostatic pressure is above the ambient value is known as the positive phase, and the period when the properties are below the ambient value is the negative phase. We use $P_{s}=1$ (since the differential pressure is normalized) and $t^{}=20$ and plot the corresponding model with a black dashed line in Fig. 3. An intriguing feature can also be seen in the 2D time evolution at $\tau=40$. Close to the minimum of the negative phase, a second peak of differential pressure is observed (the single point at $\tau=40$ Fig. 3 is the average of several realizations with errors bars indicating the standard deviation of the measurement) in our optical analogue which is reminiscent of the second shock observed in classical explosion. In classical blast wave dynamics, this second shock is believed to be a consequence of the expansion and subsequent implosion of the detonation products and source materials. Our results suggest that this second shock might be of more general nature than currently thought. ## Velocity Figure 4: Fluid velocities from the off-axis interferometry. a),b) Space-time evolution of the Mach number with respect to the background’s local speed of sound, in the 1D and 2D geometry, respectively. The dotted black line in a) shows the calculated trajectory of expansion at the speed of sound (see main text). c),d) show the background’s $\tilde{v}_{0}$ (blue) and total $\tilde{v}$ (red) Mach number profiles, at different times, for the 1D (x coordinate) and 2D geometry (radial coordinate), respectively. Each following profile shifted vertically (spacing of 1) for visibility. For blast waves, there are no simple thermodynamic relationships between the physical properties of the fluid at a fixed point [30]. This means that the temporal evolution of the static pressure measured at a fixed point is not sufficient to calculate the temporal evolution of the velocity or the dynamic pressure from that single measurement. To fully describe the physical properties of a fluid in a blast wave it is necessary to independently measure at least three of the physical properties, such as, the static pressure, the density and the fluid velocity or the dynamic pressure. In the last section of this work, we report the measurement of last two physical properties, which are vector quantities. The fluid velocity is calculated from its phase (see Eq. (11)) which is measured using off-axis interferometric imaging. The off-axis configuration consists in the tilted recombination of the signal beam with the reference beam on the camera plane. This results in the set of linear fringes evolving along the relative tilt direction and locally deformed (stretched or compressed) according to the beams relative curvature. Using a collimated Gaussian beam as the reference, the measured curvature is the one of the signal beam. The acquired interferogramm carries the information on the beam phase via its amplitude modulated term. This term shows spatial periodicity and in the Fourier space it translates to two peaks shifted by a distance proportional to the off-axis tilt angle, symmetric with respect to the origin. By numerically calculating the spatial spectrum and filtering one of these peaks, the inverse Fourier transform gives the beam complex envelope with a spatial resolution bound by the fringe wavelength. The measured phase is unwrapped and the contribution due to the relative tilt is removed by subtracting the phase ramp. The resulting phase is averaged and numerically differentiated to get the velocity map. Using this procedure, the off-axis interferograms of the background fluid and of the background fluid with the perturbation are analyzed to give access to $v_{0}(r,\tau)$ and $v(r,\tau)$, respectively. The difference of these quantities gives the perturbation velocity $v_{1}(r,\tau)$. The non-zero velocity $v_{0}$ of the background fluid arises from its finite size causing its expansion due to a non-zero pressure gradient. The knowledge of $v_{0}$ is essential to calculate the effective interaction $g$ and therefore the time $\tau$ and the sound velocity. Indeed, $\phi_{0}=\tau\tilde{\rho}_{0}$ can be accessed by integrating $v_{0}$ over the transverse coordinate and using the fact that $\phi(\tilde{r}\rightarrow\infty,\tau)\rightarrow 0$. Knowing $\tau$, the sound velocity is $c_{s}(\mathbf{r}_{\perp},\tau)=c\sqrt{\tau\tilde{\rho}_{0}(\mathbf{r}_{\perp},\tau)/(k_{0}L)}$. The velocity maps normalized by the local sound velocity (in Mach units) are presented in figure 4 a) and b) for the 1D and 2D configurations, respectively. Since velocity is a vector quantity, negative values correspond to a propagation along $-x$ direction. Figure 4 c) and d) show the corresponding profiles obtained for three specific times $\tau=2;\ 23$ and $45$. The maximal speed of sound at these times is 0.18, 0.62 and 0.86 percent of the speed of light in vacuum. Positive outward velocity, as well as zero velocity at the center is observed at all times both in the 1D and 2D cases. Whereas it is intuitively expected in the 1D geometry with the differential pressure never dropping to negative values, it also holds in the 2D case in which a negative phase for the differential pressure exists. A possible explanation lies in the fact that when the negative phase is reached for the differential pressure, the perturbation has already expanded enough such that the net resulting force is smaller due to a larger distance. It is also worth noting that the velocity is at least 2 times larger in the 1D geometry than in 2D, as seen by comparison of the y-axis scales in Figure 4 c) and d). Additionally, clear steepening of the velocity profiles is observed in the 1D case reaching a Mach number of 1 at the steepest position. Figure 5: Dynamic pressure analysis. a) and b) show the spatio-temporal evolution maps of the dynamic pressure profiles, for the 1D (the x component) and 2D geometry (the radial component), respectively. Below, the c) and d) panels show various superimposed dynamic pressure profiles at different times, in 1D and 2D geometry, respectively. ## Dynamic pressure Alternatively, we can measure the dynamic pressure to compute a third thermodynamic quantity: the total pressure. The dynamic pressure is also a vector quantity and can be obtained from a phase measurement similar to fluid velocity using Eq. (15). The dynamic pressure maps are presented in Figs. 5 a) and b). Once again Figs. 5 c) and d) show dynamic pressure profiles for three selected times. In 1D, the dynamic pressure forms a steep overpressure characteristic of the shock front which increases as function of time. In the 2D geometry, on the contrary the dynamic pressure reaches a plateau at the shock front without forming a steep overpressure peak. This behavior is in agreement with the velocity distributions presented previously. ## Conclusion Relying on detailed measurements of all thermodynamic quantities in a fluid of light blast wave, we have demonstrated for the first time the occurence of a blast wave in a fluid of light. We compare 1D and 2D geometry and report the observation of a negative phase during the blast only for the 2-dimensional case. The differential pressure in the 2D geometry is compared to the classical hydrodynamics of Friedlander blast-wave and we see a very good agreement with this model. Velocity maps and dynamic pressure are finally presented to complete the study. Our work opens the way to precise engineering of a fluid of light density and velocity distribution which will prove to be a valuable tool to design new experiments studying superfluid turbulence [31] or analogue gravity where an excitation of a fluid of light changes from a subsonic to a supersonic region. ###### Acknowledgements. 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Hughes, Absolute absorption on rubidium d lines: comparison between theory and experiment, Journal of Physics B: Atomic, Molecular and Optical Physics 41, 155004 (2008). * Weller _et al._ [2011] L. Weller, R. J. Bettles, P. Siddons, C. S. Adams, and I. G. Hughes, Absolute absorption on the rubidium d1line including resonant dipole–dipole interactions, Journal of Physics B: Atomic, Molecular and Optical Physics 44, 195006 (2011). Supplemental Materials: Blast waves in a paraxial fluid of light Experimental details The scheme of the experimental setup is shown on Figure S1. Toptica DLCpro 780 with TA was used for all measurements. The laser frequency was tuned around 780 nm and measured with a MogWave Multimeter LambdaMeter and calibrated with Saturable absorption spectroscopy (SAS). The laser beam was mode cleaned with a single mode fiber and then split into the Background, Bump and the Reference arms. The respective intensity ratio was fixed by the angles of the Half-Wave-Plates (HWP), placed before the Polarizing Beam Splitters (PBS), in agreement with experimental requirements: $\tilde{\rho}_{1}(\textbf{r}=0,\tau=0)\approx 3$ and minimal sufficient power into the reference beam to have noticeable fringe contrast. Since the Background and the Reference have the same polarization during recombination, their interference needs to be constructive at the cell input in order to create the desired input state for the fluid’s density. The beamsplitter’s unused arm’s power at the Background-Bump overlap area should then be minimal. This signal was measured with a 200 $\mu m$ diameter pinhole centered at the overlap area and a photodiode. The relative phase needs to be locked in order to minimize permanently this signal and make it insensitive to perturbations such as air currents. Therefore the photodiode signal was transformed into an error signal of a piezoelectric mirror mount controlling the relative phase. The error signal generation from the photodiode signal was realized with the PyRPL software running on a Red Pitaya FPGA [29]. The modulation frequency was around 2-3 kHz. Figure S1: Schematic visualization of the experimental setup. Diode laser frequency calibration was performed with Saturable Absorption Spectroscopy (SAS), and during the experiment the frequency measurement was performed with a MogWave Lambdameter. The laser was mode cleaned with a polarization maintaining single mode siber (PMSMF), before being split into the Background, Bump and the Reference. The Background-Bump interference arm complementary to the Rb vapor cell was cropped with a 200 $\mu m$ diameter pinhole (Ph) to measure the the power of the overlap area on a photodiode. This signal was minimized by controlling the relative phase via piezoelectric motion of a mirror mount to have permanently constructive interference on the vapor cell arm. The error signal was generated from the photodiode signal with the PyRPL lockbox software. Vapor Temperature Figure S2: Vapor’s transmission and its intensity dependent refractive index measurement. a) and b) show the maximal refractive index variation calculated from the off-axis interferograms of the backgroung beam with a reference, in 1D and 2D geometry, respectively. c) Background beam’s transmission spectrum with respect to 85Rb cooling transition measured at different input powers. Dashed line is the theory of a linear multilevel vapor at temperature 150 °C and 0.5 % the isotopic fraction of 87Rb inside the cell. Checking the ”Kerr” approximation: d) and e) show the variation of the refractive index with laser power at fixed laser detuning in both geometries. One of the useful knobs to control the light-matter interaction in hot vapor cells is the atomic density. The latter is directly linked to the vapor pressure via the ideal gas law (neglecting the atom-atom interactions). It equals the Rb vapor’s saturation pressure at thermal liquid-gas equilibrium and can be increased by several orders of magnitude when heating the cell from 50° C to 150° C. Keeping the vapor temperature constant during the experiment is therefore necessary to control the atomic susceptibility. In our experiment, several electric resistors were wound around the cell and connected in parallel to a DC power supply to heat up the cell. The vapor temperature was accessed by measuring the transmission spectrum around the Rb D2 line in the weak beam limit. The frequency calibration was performed via Saturable absorption spectroscopy, as shown on Figure S1. The experimental spectrum was fitted with the linear susceptibility model developped in [32] taking into account all hyperfine transitions of both isotopes and the collisional self-broadening due to resonant dipole-dipole interactions [33], with the atomic density and the number fraction of 87Rb isotope as free parameters. The temperature was measured before and after each experiment to prevent any temperature drift. Non-linear refractive index variation measurement The intensity dependent refractive index of our hot atomic vapor is the key parameter governing the fluid’s dynamics as it is linked to the effective evolution time: $\tau=\Delta nk_{0}L$ and its speed of sound: $c_{s}=c\sqrt{\Delta n}$. In this work it was measured using the off-axis interferometry which gives access to the transverse phase variations at the cell exit plane. The transverse phase profile of the Background beam is assumed to depend as follows on the beam’s intensity I(r): $\phi_{th}(\textbf{r},L)=k_{0}L\frac{n_{2}I(\textbf{r})}{1+I(\textbf{r})/I_{s}}+\phi_{0}$ (S1) Where $n_{2}$ is the Kerr index, $I_{s}$ the saturation intensity of the Kerr effect and $\phi_{0}$ a constant phase. The gradient of the phase, giving access to the fluid velocity, is numerically calculated and fitted with $\nabla\phi_{th}$ with $n_{2}$ and $I_{s}$ as free parameters. Figure S2 a) and b) show measured maximal variation of refractive index for different experimental configurations of the laser detuning $\Delta$ and power $P$. Each point corresponds to a processed image. c) Shows the transmission spectra through the cell for different input powers. No saturation of the absortpion can be evidenced. The black dashed line is the theoretical calculation of the linear susceptibility used for the measurement of the vapor’s temperature. Finally, d) and e) show the variation of the refractive index with intensity. The graphs show that the results of this work are obtained below the regime of the saturation of the Kerr effect. Background beam’s expansion Figure S3: Background fluid’s expansion. a) shows the Background’s expansion in the transverse y direction in the 1D geometry and b) shows the Background’s radial expansion in the 2D geometry. In the theoretical discussion developed in the main text and for the $\Delta n$ measurement it is assumed that the background fluid beam’s density is invariant with time. The experimental data to verify this hypothesis are shown in Figure S3. No expansion in the x direction of the 1D case was observed. The expansion is most pronounced in the transverse y direction of the 1D case. In the 2D case the background’s insignificant expansion is observed. Relevance of the Quantum Pressure Figure S4: Quantum pressure calculated from experimental density profiles. a) in the 1D geometry and b) in the 2D geometry. Same colormap is used for both graphs. As mentioned in the main text, the Quantum pressure was neglected in the theoretical description of the experimental data as we are interested in the fluid’s behavior in the long wavelength limit. This term is known to have a dispersive contribution to the shockwave profile which, upon steepening, becomes composed of an increased amount of various momentum components moving at different velocities. To evaluate the relevance of the Quantum Pressure in this work we calculated it from the experimental density profiles at different evolution times for both 1D and 2D geometry as: $\tilde{P}_{q}=\frac{1}{2\sqrt{\tilde{\rho}}}\tilde{\nabla}^{2}_{\perp}\sqrt{\tilde{\rho}}$ (S2) Depending on the dimensionality the Laplacian was calculated as: $\tilde{\nabla}^{2}_{\perp}=\xi^{2}\partial^{2}/\partial x^{2}$ in 1D or as: $\tilde{\nabla}^{2}_{\perp}=\xi^{2}[\partial^{2}/\partial r^{2}+(1/r)\times\partial/\partial r]$ in 2D using the radial symmetry. With this formulation the value of the dimensionless Quantum Pressure directly compares with the dimensionless density (stemming from interactions) in the right hand side of the Euler-like Madelung equation. The result is shown on Figure S4 a) for the 1D and b) for the 2D case. The Quantum Pressure seems to be most pronounced at the vicinity of the Shock front in 1D. In 2D it seems to decay with time. In both cases it does not exceed 0.1 for times $\tau>5$. This validates the theoretical approach chosen in work and consisting in neglecting the Quantum Pressure. For lower times the calculation seems inaccurate. This may be due to a large uncertainty on the healing length $\xi$ in this regime.
# Homotopy Methods for Eigenvector-Dependent Nonlinear Eigenvalue Problems ††thanks: The research was supported in part by the National Natural Science Foundation of China (11971092) Xuping Zhang School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning 116025, P. R. China (zhangxp@dlut.edu.cn). Haimei Huo School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning 116025, P. R. China (ab1234@mail.dlut.edu.cn). ###### Abstract Eigenvector-dependent nonlinear eigenvalue problems are considered which arise from the finite difference discretizations of the Gross-Pitaevskii equation. Existence and uniqueness of positive eigenvector for both one and two dimensional cases and existence of antisymmetric eigenvector for one dimensional case are proved. In order to compute eigenpairs corresponding to excited states as well as ground state, homotopies for both one and two dimensional problems are constructed respectively and the homotopy paths are proved to be regular and bounded. Numerical results are presented to verify the theories derived for both one and two dimensional problems. Key Words eigenvector-dependent nonlinear eigenvalue problem, Gross-Pitaevskii equation, homotopy continuation method Subject Classification(AMS):65H17, 65H20, 65N06, 65N25 ## 1 Introduction In this paper, we are concerned with the eigenvector-dependent nonlinear eigenvalue problems resulting from the finite difference discretizations of the Gross-Pitaevskii equation (GPE) describing Bose-Einstein condensates (BEC). BEC are clouds of ultracold alkali-metal atoms or molecules that occupy a single quantum state [1, 2]. The properties of a BEC at temperature $T$ much smaller than the critical condensation temperature $T_{c}$ are usually described by the nonlinear Schrödinger equation (NLS) for the macroscopic wave function known as the Gross-Pitaevskii equation $\begin{array}[]{lcl}{i\psi_{t}=-\frac{1}{2}\Delta\psi+V(x)\psi+\beta\ \|\psi\|^{2}\psi},&&{t>0,~{}x\in\Omega},\\\ {\psi(x,t)=0},&&{t\geq 0,~{}x\in\partial\Omega},\end{array}\\\ $ (1) where $\psi=\psi(x,t)$ is the macroscopic wave function of the BEC, $V(x)=\frac{1}{2}(x_{1}^{2}+x_{2}^{2}+\cdots+x_{N}^{2})$ is a typical trapping potential, $\Omega$ is a bounded domain in $\mathbb{R}^{N}$, $N\leq 3$, and $\beta$ positive or negative corresponds to the defocusing or focusing NLS. Two important invariants of GPE are the normalization of the wave function $\begin{array}[]{lcl}N(\psi)=\int_{\Omega}\|\psi(x,t)\|^{2}dx=1,&&t\geq 0,\end{array}$ (2) and the energy $\displaystyle E(\psi(x,t))=\int_{\Omega}\left[\frac{1}{2}\|\nabla\psi\|^{2}+V(x)\|\psi\|^{2}+\frac{\beta}{2}\|\psi\|^{4}\right]dx=E(\psi(x,0)).$ (3) To find stationary solution of (1), we substitute the formula $\psi(x,t)=e^{-i\lambda t}\phi(x)$ into (1) and (2) and obtain the time- independent Schrödinger equation with Dirichlet boundary condition and the normalized condition $\displaystyle\lambda\phi(x)$ $\displaystyle=-\frac{1}{2}\Delta\phi(x)+V(x)\phi(x)+\beta\phi^{3}(x),\quad{x\in\Omega},$ (4) $\displaystyle\phi(x)$ $\displaystyle=0,\quad x\in\partial\Omega,$ (5) $\displaystyle\int_{\Omega}\|\phi(x)\|^{2}dx$ $\displaystyle=1,$ (6) where $\lambda$ is the chemical potential of the condensate and $\phi(x)$ is a real function independent of $t$ [3]. (4)-(6) is a nonlinear eigenvalue problem. The eigenfunction corresponding to the minimum energy is called ground state and other eigenfunctions corresponding to larger energy are called excited states in the literature. There have been many theoretical studies as well as numerical studies for the time-independent Schrödinger equation. Bao and Cai [4] pointed out when $\beta>0$, the positive ground state is unique, and if $V(x)$ is radially symmetric in 2D, the positive ground state must be radially symmetric. Bao and Tang [1] proposed methods by directly minimizing the energy functional via finite element approximation to obtain the ground state and by continuation method to obtain excited states. Edwards and Burnett [5] presented a Runge- Kutta type method and employed it to solve the spherically symmetric time- independent GPE. Adhikari [6] used this approach to get the ground state solution of GPE in 2D with radial symmetry. Chang and Chien [7] and Chang, Chien and Jeng [8] investigated stationary state solutions of $(\ref{sec 1: transformed nonlinear problem})$ using numerical continuation method, where $\lambda$ was treated as a continuation parameter. The solution curves branching from the first few bifurcation points of $(\ref{sec 1: transformed nonlinear problem})$ were numerically traced using continuation method under the normalization condition $(\ref{sec 1:transformed constraint})$. Since nonlinearity rather than discretization method is our main concern and finite difference discretization will lead to a simpler nonlinear structure, finite difference discretization is adopted in this paper. The finite difference discretization of (4)-(6) is the following eigenvector-dependent nonlinear eigenvalue problem, $\begin{array}[]{c}D\varphi+\beta\varphi^{3}=\lambda\varphi,\\\ h\varphi^{\mathrm{T}}\varphi-1=0,\end{array}$ (7) where $D=\frac{1}{2}D_{1}+V$, $D_{1}$ is the coefficient matrix corresponding to $-\Delta$, $V$ is the diagonal matrix corresponding to the potential $V(x)$, $\lambda$ and $\varphi$ are the unknowns, and $h$ is a constant related to mesh size. $\varphi^{3}$ represents the vector with elements being the corresponding elements of $\varphi$ to the power 3. This convention will be used throughout this paper. With respect to the theoretical aspects of eigenvector-dependent nonlinear eigenvalue problem, [9] and [10] studied the following general nonlinear eigen-value problem $Ax+F(x)=\lambda x,$ (8) where $A$ is an $n\times n$ irreducible Stieltjes matrix, i.e, an irreducible symmetric positive definite matrix with off-diagonal entries nonpositive, $F(x)=(f_{1}(x_{1}),\ldots,f_{n}(x_{n}))^{\mathrm{T}}$ and $x=(x_{1},\cdots,x_{n})^{\mathrm{T}}$. The functions $f_{i}(x_{i})$ are assumed to have the property that $f_{i}(x_{i})>0$, when $x_{i}>0$, $i=1,\ldots,n$. It is shown that under certain conditions on $F(x)$, there exists a positive eigenvector $x(\lambda)$ if and only if $\lambda>\mu$, where $\mu$ is the smallest eigenvalue of $A$, and for every $\lambda>\mu$, the positive eigenvector is unique. Moreover, such a solution is a monotone increasing function of $\lambda$. The most popular numerical method to the eigenvector-dependent nonlinear eigenvalue problems is the self-consistent field (SCF) iteration, which is suitable for computing the ground state; for instance, see [11, 12] and the references therein. In [13], inverse iteration method was applied to solve eigenvector-dependent nonlinear eigenvalue problems. Most of the above papers concentrate on the ground state and the first excited state. As far as we know, there are only a few numerical works on other excited states, such as [1, 14, 15, 16]. The main purpose of this paper is to design algorithms for computing excited states of high energy. Homotopy method is one of the effective methods for solving eigenvalue problems. A great advantage of the homotopy method is that it is to a large degree parallel, in the sense that each eigenpath is traced independently of the others. There are several works on homotopy methods for linear eigenvalue problems. Remarkable numerical results have been obtained by using homotopy algorithm on eigenvalue problems of tridiagonal symmetric matrices [17, 18]. Solving eigenvalue problems of real nonsymmetric matrices with real homotopy was developed in [19, 20]. The homotopy method is also used to solve the generalized eigenvalue problem [21]. For eigenvalue-dependent nonlinear eigen- problems such as $\lambda$-matrix problems, a homotopy was given by Chu, Li and Sauer [22]. The major part of this paper is the construction of homotopy for computing many eigenpairs of the eigenvector-dependent nonlinear eigen-problem. Key issues encountered in constructing the homotopy are the selection of the homotopy parameter and that of an appropriate initial eigenvalue problem so that the homotopy paths determined by the homotopy equation are regular and the numerical work in following these paths is at reasonable cost. The parameter $\beta$ in the original problem seems to be a natural choice for the homotopy parameter. However, it seems difficult to prove that 0 is a regular value for such homotopy. In fact, 0 is probably not a regular value of the natural homotopy with parameter $\beta$. Instead, an artificial parameter $t$ is chosen as the homotopy parameter to connect a constructed initial eigenvalue problem and the target one. As for the selection of an initial eigenvalue problem, random matrix with certain sparse structure is designed, which guarantees that 0 is a regular value of the homopoty with probability one and which renders the initial problem and the target problem possess similar structures. The rest of this paper is organized as follows. In Section 2, the time- independent GPE Dirichlet problem (4)-(6) is discretized by finite difference method and existence of certain types of solution of the discretized problems is derived. In Section 3, homotopies for (7) are constructed with $\Omega\subset\mathbb{R}$ and $\Omega\subset\mathbb{R}^{2}$ respectively, and regularity and boundedness of the homotopy paths are proved. In Section 4, numerical results are presented to verify the theoretical results derived for $\Omega\subset\mathbb{R}$ and $\Omega\subset\mathbb{R}^{2}$ respectively. Conclusions are drawn in the last section. ## 2 Discretizations of the nonlinear eigenvalue problem ### 2.1 Finite difference discretizations For one dimensional problem (4)-(6) with $\Omega=[a,b]\subset\mathbb{R}$, the grid points are $x_{j}=a+jh$, $j=0,\ldots,n+1$, where $n\in\mathbb{N}^{+}$ and $h=\frac{b-a}{n+1}$ is the mesh size. The finite difference discretization of the differential equation and a simple quadrature of the normalization condition lead to the following system of algebraic equations, $\begin{array}[]{c}D\varphi+\beta\varphi^{3}-\lambda\varphi=0,\\\ \frac{1}{2}\left(\frac{1}{h}-\varphi^{\mathrm{T}}\varphi\right)=0,\end{array}$ (9) where $\varphi=(\varphi_{1},\cdots,\varphi_{n})^{\mathrm{T}}$, $\varphi_{j}$ are the approximations of $\phi(x_{j})$, $v_{j}=V(x_{j})$, $j=1,\ldots,n$, and $D=\frac{1}{2}D_{1}+V$ with $\displaystyle D_{1}=\frac{1}{h^{2}}\left(\begin{array}[]{cccc}2&-1&\\\ -1&2&\ddots&\\\ &\ddots&\ddots&-1\\\ &&-1&2\end{array}\right),\quad V=\left(\begin{array}[]{ccc}v_{1}&&\\\ &\ddots&\\\ &&v_{n}\end{array}\right).$ (17) The discretization of the normalized condition is rewritten so that the Jacobian matrix of the nonlinear mapping with respect to $(\varphi,\lambda)$ is symmetric, as will be seen below. For two dimensional problem (4)-(6) with $\Omega=[a,b]\times[c,d]$, the domain is divided into a $(m+1)\times(n+1)$ mesh with step size $h_{1}=\frac{b-a}{m+1}$ in $x$-direction, $h_{2}=\frac{d-c}{n+1}$ in $y$-direction. The grid points $(x_{i},y_{j})$ are $x_{i}=a+ih_{1}$, $i=0,\ldots,m+1$, and $y_{j}=c+jh_{2}$, $j=0,\ldots,n+1$. Using central difference, we get $\begin{array}[]{c}D\varphi+\beta\varphi^{3}-\lambda\varphi=0,\\\ \frac{1}{2}\left(\frac{1}{h_{1}h_{2}}-\varphi^{\mathrm{T}}\varphi\right)=0,\end{array}$ (18) where $\varphi=(\varphi_{11},\cdots,\varphi_{1n},\cdots,\varphi_{m1},\cdots,\varphi_{mn})^{\mathrm{T}}$, $\varphi_{ij}$ are the approximations of $\phi(x_{i},y_{j})$, $v_{ij}=V(x_{i},y_{j})$, $i=1,\ldots,m$, $j=1,\ldots,n$ and D is a block tridiagonal matrix $D=\frac{1}{2}\left(\begin{array}[]{cccc}D_{11}&D_{12}&\\\ D_{21}&D_{22}&\ddots&\\\ &\ddots&\ddots&D_{{m-1},m}\\\ &&D_{m,{m-1}}&D_{mm}\end{array}\right)+\left(\begin{array}[]{cccc}V_{1}&&\\\ &V_{2}&&\\\ &&\ddots&\\\ &&&V_{m}\end{array}\right),$ where $\displaystyle D_{ii}=\left(\begin{array}[]{cccc}\frac{2}{h_{1}^{2}}+\frac{2}{h_{2}^{2}}&-\frac{1}{h_{2}^{2}}&\\\ -\frac{1}{h_{2}^{2}}&\frac{2}{h_{1}^{2}}+\frac{2}{h_{2}^{2}}&\ddots&\\\ &\ddots&\ddots&-\frac{1}{h_{2}^{2}}\\\ &&-\frac{1}{h_{2}^{2}}&\frac{2}{h_{1}^{2}}+\frac{2}{h_{2}^{2}}\end{array}\right)\in\mathbb{R}^{n\times n},$ (23) $\displaystyle D_{i-1,i}=\left(\begin{array}[]{cccc}-\frac{1}{h_{1}^{2}}&&&\\\ &-\frac{1}{h_{1}^{2}}&&\\\ &&\ddots&\\\ &&&-\frac{1}{h_{1}^{2}}\end{array}\right)\in\mathbb{R}^{n\times n},\quad D_{{i-1},i}=D_{i,{i-1}},$ (28) $\displaystyle V_{i}=\left(\begin{array}[]{cccc}v_{i1}&&&\\\ &v_{i2}&&\\\ &&\ddots&\\\ &&&v_{in}\end{array}\right),\quad i=1,\ldots,m.$ (33) ###### Remark 2.1 For both one and two dimensional cases, D is an irreducible symmetric diagonal dominant matrix and the diagonal entries of D are all positive. Therefore D is positive definite. ### 2.2 Existence of certain types of solution In this subsection, we will study the existence of solution for the discretized nonlinear eigenvalue problem. From [4], we know for (4)-(6), when $\beta>0$, the positive ground state is unique, and if $V(x)$ is radially symmetric in 2D, the positive ground state must be radially symmetric. We will prove the existence of positive solution and the existence of antisymmetric solution for discretized nonlinear eigenvalue problem. For convenient reading, two underlying theorems from [9] are quoted as underlying lemmas. ###### Lemma 2.2 ([9]) Let A be an irreducible Stieltjes matrix and $\mu$ be the smallest positive eigenvalue of A. Let $\lambda>\mu$ and let $F(x)=\left(\begin{array}[]{c}f_{1}(x_{1})\\\ \vdots\\\ f_{n}(x_{n})\end{array}\right),$ (34) where for $i=1,\ldots,n$, $f_{i}(x):[0,\infty)\rightarrow[0,\infty)$ are $C^{1}$ functions satisfying the conditions: $\lim\limits_{t\rightarrow 0}\frac{f_{i}(t)}{t}=0,\qquad\lim\limits_{t\rightarrow\infty}\frac{f_{i}(t)}{t}=\infty.$ (35) Then $Ax+F(x)=\lambda x$ has a positive solution. If, in addition, for $i=1,\ldots,n$, $\frac{f_{i}(s)}{s}<\frac{f_{i}(t)}{t}$ (36) whenever $0<s<t$, then the solution is unique. ###### Lemma 2.3 ([9]) Let the conditions $(\ref{sec 2: limit conditon})$ and $(\ref{sec 2: inequality condition})$ of Lemma 2.2 be satisfied and let $x(\lambda)$ denote the unique positive eigenvector corresponding to $\lambda\in(\mu,\infty)$. Then: 1. (i) $x(\lambda_{1})<x(\lambda_{2})$, if $\mu<\lambda_{1}<\lambda_{2}<\infty$; 2. (ii) $x(\lambda)$ is continuous on $(\mu,\infty)$; 3. (iii) $\lim\limits_{\lambda\rightarrow\infty}{x_{i}(\lambda)}=\infty$, $i=1,\cdots,n$; 4. (iv) $\lim\limits_{\lambda\rightarrow\mu^{+}}{x_{i}(\lambda)}=0$, $i=1,\cdots,n$. ###### Remark 2.4 Lemma 2.3 indicates for any given normalization $r>0$, there exist a unique $\lambda>\mu$ and unique positive $x(\lambda)$ such that $\|\|x(\lambda)\|\|=r$. ###### Theorem 2.5 If $\beta>0$, there exist unique positive eigenvectors for problem $(\ref{sec 2: finite difference - one dimen})$ and $(\ref{sec 2: finite difference - two dimen})$ respectively. Proof. From Remark 2.1, we know $D$ in both $(\ref{sec 2: finite difference - one dimen})$ and $(\ref{sec 2: finite difference - two dimen})$ is an irreducible Stieltjes matrix. In addition it can be verified that $\beta\varphi^{3}$ in both $(\ref{sec 2: finite difference - one dimen})$ and $(\ref{sec 2: finite difference - two dimen})$ satisfies the conditions of $F(x)$ in Lemma 2.2. From Remark 2.4, the claim is proved. ###### Theorem 2.6 Let $\beta>0$ and $\Omega=[-a,a]$ for problem $(\ref{sec 2: finite difference - one dimen})$. 1. (i) When $n$ is odd and the grid points $x_{i}$, $i=1,\ldots,n$, satisfy $x_{1}=-x_{n},~{}x_{2}=-x_{n-1},~{}\cdots,~{}x_{\frac{n-1}{2}}=-x_{\frac{n+3}{2}},~{}x_{\frac{n+1}{2}}=0,$ there exists a unique solution $\varphi=(\varphi_{1},\varphi_{2},\cdots,\varphi_{n})^{\mathrm{T}}$ with $\varphi_{1}=-\varphi_{n}$, $\varphi_{2}=-\varphi_{n-1},\cdots$, $\varphi_{\frac{n-1}{2}}=-\varphi_{\frac{n+3}{2}}$, $\varphi_{\frac{n+1}{2}}=0$, $\varphi_{j}>0$, $j=1,\ldots,n$. 2. (ii) When $n$ is even and the grid points $x_{i}$, $i=1,\ldots,n$, satisfy $x_{1}=-x_{n},~{}x_{2}=-x_{n-1},~{}\cdots,~{}x_{\frac{n}{2}}=-x_{\frac{n}{2}+1},$ there exists a unique solution $\varphi=(\varphi_{1},\varphi_{2},\cdots,\varphi_{n})^{\mathrm{T}}$ with $\varphi_{1}=-\varphi_{n}$, $\varphi_{2}=-\varphi_{n-1}$, $\cdots$, $\varphi_{\frac{n}{2}}=-\varphi_{\frac{n}{2}+1}$, $\varphi_{j}>0$, $j=1,\ldots,n$. Proof. (i). When $n$ is odd, consider the following equations $\begin{array}[]{c}D_{2}\varphi+\beta\varphi^{3}=\lambda\varphi,\\\ \varphi^{\mathrm{T}}\varphi-\frac{1}{2h}=0,\end{array}$ (37) where $\displaystyle D_{2}=\left(\begin{array}[]{cccc}\frac{1}{h^{2}}+\frac{1}{2}x_{1}^{2}&-\frac{1}{2h^{2}}&&\\\ -\frac{1}{2h^{2}}&\frac{1}{h^{2}}+\frac{1}{2}x_{2}^{2}&\ddots&\\\ &\ddots&\ddots&-\frac{1}{2h^{2}}\\\ &&-\frac{1}{2h^{2}}&\frac{1}{h^{2}}+\frac{1}{2}x_{\frac{n-1}{2}}^{2}\end{array}\right)\in\mathbb{R}^{\frac{n-1}{2}\times\frac{n-1}{2}}.$ (42) Note that $D_{2}$ is an irreducible Stieltjes matrix and $\beta\varphi^{3}$ satisfies the conditions of $F(x)$ in Lemma 2.2. Therefore there exists a unique positive solution $\varphi=(\varphi_{1},\varphi_{2},\ldots,\varphi_{\frac{n-1}{2}})^{\mathrm{T}}$ for (37). Due to the relations $x_{1}=-x_{n},~{}x_{2}=-x_{n-1},~{}\cdots,~{}x_{\frac{n-1}{2}}=-x_{\frac{n+3}{2}},~{}x_{\frac{n+1}{2}}=0$, set $\varphi_{n}=-\varphi_{1}$, $\varphi_{n-1}=-\varphi_{2}$, $\cdots$, $\varphi_{\frac{n+3}{2}}=-\varphi_{\frac{n-1}{2}}$, $\varphi_{\frac{n+1}{2}}=0$. Then $\varphi=(\varphi_{1},\varphi_{2},\ldots,\varphi_{n})^{\mathrm{T}}$ is a solution of $(\ref{sec 2: finite difference - one dimen})$. (ii). When $n$ is even, consider the following equations $\begin{array}[]{c}D_{2}\varphi+\beta\varphi^{3}=\lambda\varphi,\\\ \varphi^{\mathrm{T}}\varphi-\frac{1}{2h}=0,\end{array}$ (43) where $\displaystyle D_{2}=\left(\begin{array}[]{cccc}\frac{1}{h^{2}}+\frac{1}{2}x_{1}^{2}&-\frac{1}{2h^{2}}&&\\\ -\frac{1}{2h^{2}}&\frac{1}{h^{2}}+\frac{1}{2}x_{2}^{2}&\ddots&\\\ &\ddots&\ddots&-\frac{1}{2h^{2}}\\\ &&-\frac{1}{2h^{2}}&\frac{3}{2h^{2}}+\frac{1}{2}x_{\frac{n}{2}}^{2}\end{array}\right)\in\mathbb{R}^{\frac{n}{2}\times\frac{n}{2}}.$ (48) Similarly there exists a unique positive solution $\varphi=(\varphi_{1},\varphi_{2},\ldots,\varphi_{\frac{n}{2}})^{\mathrm{T}}$ for (43). Due to the relations $x_{1}=-x_{n},~{}x_{2}=-x_{n-1},~{}\cdots,~{}x_{\frac{n}{2}}=-x_{\frac{n}{2}+1}$, set $\varphi_{n}=-\varphi_{1}$, $\varphi_{n-1}=-\varphi_{2}$, $\cdots$, $\varphi_{\frac{n}{2}+1}=-\varphi_{\frac{n}{2}}$. Then $\varphi=(\varphi_{1},\varphi_{2},\ldots,\varphi_{n})^{\mathrm{T}}$ is a solution of $(\ref{sec 2: finite difference - one dimen})$. ## 3 Homotopy methods In this section, in order to compute many eigenpairs, we construct homotopy equations for 1D discretized problem $(\ref{sec 2: finite difference - one dimen})$ and 2D discretized problem $(\ref{sec 2: finite difference - two dimen})$ respectively. We shall prove the regularity and boundedness of the homotopy paths. The regularity of homotopy paths can be usually obtained by random perturbations of appropriate parameters, so the most important feature of our construction is the choice of appropriate parameters. In addition, if the initial matrix can be chosen as close to the matrix $D$ as possible, then most of the homotopy paths are close to straight lines and will be easy to follow [19]. ### 3.1 One dimensional case For $(\ref{sec 2: finite difference - one dimen})$, a homotopy $H:\mathbb{R}^{n}\times\mathbb{R}\times[0,1]\rightarrow\mathbb{R}^{n}\times\mathbb{R}$ is defined as $H(\varphi,\lambda,t)=\left(\begin{array}[]{c}(1-t)A(K)\varphi+D\varphi+t\beta\varphi^{3}-\lambda\varphi\\\ \frac{1}{2}\left(\frac{1}{h}-\varphi^{\mathrm{T}}\varphi\right)\end{array}\right)=0,$ (49) where $A(K)=\mbox{diag}(a_{1},\cdots,a_{n})$ is a random diagonal matrix with $K=(a_{1},\cdots,a_{n})^{\mathrm{T}}\in\mathbb{R}^{n}$. At $t=0$, $H(\varphi,\lambda,0)$ corresponds to the linear eigenvalue problem $H(\varphi,\lambda,0)=\left(\begin{array}[]{c}A(K)\varphi+D\varphi-\lambda\varphi\\\ \frac{1}{2}\left(\frac{1}{h}-\varphi^{\mathrm{T}}\varphi\right)\end{array}\right)=0,$ (50) while at $t=1$, $H(\varphi,\lambda,1)=0$ corresponds to the problem (9). Assuming that the eigenpairs of $H(\varphi,\lambda,0)=0$ are $(\varphi^{(i)},\lambda_{i})$, $i=1,\ldots,n$, $\lambda_{1}\leq\cdots\leq\lambda_{n}$, we shall use these $n$ points $(\varphi^{(i)},\lambda_{i},0)$ as our initial points when tracing the homotopy curves leading to the desired solutions of $H(\varphi,\lambda,1)=0$. The choice of the initial matrix $A(K)+D$ provides some advantages. First, it makes sure that $\forall t\in[0,1)$, what we need to solve is a sparse nonlinear eigenvalue problem with constraint. Second, since $\forall K\in\mathbb{R}^{n}$, $\forall t\in[0,1)$ and $\varphi\in\mathbb{R}^{n}$, $(1-t)A(K)+D+t\beta\mbox{diag}(\varphi^{2})$ is a real symmetric matrix, the solution curves starting from the initial points at $t=0$ are real. Finally, since $A(K)+D$ is a tridiagonal matrix with all the subdiagonal and supdiagonal entries nonzero, all the eigenvalues of $A(K)+D$ are simple and the Jacobian matrix of $H$ at $(\varphi_{0},\lambda_{0},0)$ is nonsingular for $\forall(\varphi_{0},\lambda_{0})$ such that $H(\varphi_{0},\lambda_{0},0)=0$. Thus locally a unique curve around $(\varphi_{0},\lambda_{0},0)$ is guaranteed. The effectiveness of the homotopy is based on the following Parametrized Sard’s Theorem. ###### Theorem 3.1 (Parametrized Sard’s Theorem) Let $f:M\times P\subset\mathbb{R}^{m}\times\mathbb{R}^{q}\to\mathbb{R}^{n}$ be a $C^{k}$ mapping with $k>max(0,m-n)$, where $M$ and $P$ are open sets in $\mathbb{R}^{m}$ and $\mathbb{R}^{q}$ respectively. If $y$ is a regular value of f, then y is also a regular value of $f(\cdot,p)$ for almost all $p\in P$. In the rest of this paper, we will denote the $i$-th row of a matrix $M$ as $M(i,:)$ and the $j$-th column of $M$ as $M(:,j)$. If $I_{1}$ is a row index set, $M(I_{1},:)$ will be the submatrix formed by the $I_{1}$ rows of $M$. $M(i:end,:)$ will be the submatrix formed by the rows from the $i$-th row to the last. Similarly $M(:,J_{1})$ will be the submatrix formed by the $J_{1}$ columns of $M$. If $I_{1}$ and $I_{2}$ are two index sets, $[M(I_{1},:);M(I_{2},:)]$ denotes the submatrix formed by the $I_{1}$ rows and the $I_{2}$ rows of $M$. In Theorem 3.2, we prove regularity and boundedness of the homotopy paths determined by the homotopy equation (49). ###### Theorem 3.2 For the homotopy $H:\mathbb{R}^{n}\times\mathbb{R}\times[0,1)\rightarrow\mathbb{R}^{n}\times\mathbb{R}$ in (49), for almost all $K\in\mathbb{R}^{n}$, 1. (i) 0 is a regular value of $H$ and therefore the homotopy curves corresponding to different initial points do not intersect each other for $t\in[0,1)$; 2. (ii) Every homotopy path $(\varphi(s),\lambda(s),t(s))\subset H^{-1}(0)$ is bounded. Proof. (i) Define a mapping $\tilde{H}:\mathbb{R}^{n}\times\mathbb{R}\times[0,1]\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\times\mathbb{R}$ related to $H$ as follows $\tilde{H}(\varphi,\lambda,t,K)=\left(\begin{array}[]{c}(1-t)A(K)\varphi+D\varphi+t\beta\varphi^{3}-\lambda\varphi\\\ \frac{1}{2}\left(\frac{1}{h}-\varphi^{\mathrm{T}}\varphi\right)\end{array}\right)=0,$ (51) such that $H(\varphi,\lambda,t)=\tilde{H}(\varphi,\lambda,t,K)$. The Jacobian matrix of $\tilde{H}$, $\frac{\partial\tilde{H}}{\partial(\varphi,\lambda,t,K)}$, is $\left(\begin{array}[]{cccc}(1-t)A(K)+D+3t\beta\mbox{diag}(\varphi^{2})-\lambda I&-\varphi&\beta\varphi^{3}-A(K)\varphi&(1-t)\mbox{diag}(\varphi)\\\ -\varphi^{\mathrm{T}}&0&0&0\end{array}\right).$ Divide $\\{1,\cdots,n\\}$ into two parts $C^{0}$ and $C^{}$, where $C^{0}$ denotes the indices $i$ such that $\varphi_{i}=0$ and $C^{}$ denotes the indices $i$ such that $\varphi_{i}\neq 0$. Since $\forall i\in C^{0}$, both the columns $3t\beta\mbox{diag}(\varphi^{2})(:,i)$ and $t\beta\mbox{diag}(\varphi^{2})(:,i)$ are equal to zero, it holds that, $\displaystyle\left((1-t)A(K)+D+3t\beta\mbox{diag}(\varphi^{2})-\lambda I\right)(:,C^{0})$ $\displaystyle=$ $\displaystyle\left((1-t)A(K)+D+t\beta\mbox{diag}(\varphi^{2})-\lambda I\right)(:,C^{0}).$ (52) For the columns in $C^{}$, the diagonal matrix $\mbox{diag}(\varphi)$ is nonzero. By elementary column transformations, the Jacobian matrix $\frac{\partial\tilde{H}}{\partial(\varphi,\lambda,t,K)}$ is transformed to the following, $\left(\begin{array}[]{cccc}(1-t)A(K)+D+t\beta\mbox{diag}(\varphi^{2})-\lambda I&-\varphi&\beta\varphi^{3}-A(K)\varphi&(1-t)\mbox{diag}(\varphi)\\\ -\varphi^{\mathrm{T}}&0&0&0\end{array}\right).$ Define $\displaystyle F_{1}=\left(\begin{array}[]{cc}(1-t)A(K)+D+t\beta\mbox{diag}(\varphi^{2})-\lambda I&-\varphi\\\ -\varphi^{\mathrm{T}}&0\end{array}\right).$ (55) For $\forall K\in\mathbb{R}^{n}$, $\forall(\varphi,\lambda,t)\in\mathbb{R}^{n}\times R\times[0,1)$ satisfying $(\ref{sec3: mapping-related-to-homotopy-1D})$, since the subdiagonals and supdiagonals of the matrix $(1-t)A(K)+D+t\beta\mbox{diag}(\varphi^{2})$ are nonzero, $\lambda$ is a simple eigenvalue of $(1-t)A(K)+D+t\beta\mbox{diag}(\varphi^{2})$. Therefore $F_{1}$ is nonsingular and $\frac{\partial\tilde{H}}{\partial(\varphi,\lambda,t,K)}$ is row full rank. As a result, 0 is a regular value of the mapping $\tilde{H}$. From Theorem 3.1, for almost all $K\in\mathbb{R}^{n}$, 0 is a regular value of the restricted mapping $\tilde{H}(\cdot,\cdot,\cdot,K)$, i.e., $H$. (ii) From $(\ref{sec 3: homotopy one dimen})$, for fixed $K\in\mathbb{R}^{n}$, $\forall(\varphi,\lambda,t)\in\mathbb{R}^{n}\times\mathbb{R}\times[0,1)$ satisfying $H(\varphi,\lambda,t)$$=$$0$, $\\|\varphi\\|=1/\sqrt{h}$ and $\lambda=h\left((1-t)\varphi^{\mathrm{T}}A(K)\varphi+\varphi^{\mathrm{T}}D\varphi+t\beta\varphi^{\mathrm{T}}\varphi^{3}\right)$. Then $\displaystyle\|\lambda\|$ $\displaystyle=$ $\displaystyle h\|(1-t)\varphi^{\mathrm{T}}A(K)\varphi+\varphi^{\mathrm{T}}D\varphi+t\beta\varphi^{\mathrm{T}}\varphi^{3}\|$ $\displaystyle\leq$ $\displaystyle(1-t)\rho(A(K))+\rho(D)+\frac{t\beta}{h}$ $\displaystyle\leq$ $\displaystyle\rho(A(K))+\rho(D)+\frac{\beta}{h},$ where $\rho(M)$ denotes the spectral radius of $M$. ### 3.2 Two dimensional case #### 3.2.1 The homotopy with random tridiagonal matrix For two dimensional case, the homotopy $H:\mathbb{R}^{mn}\times\mathbb{R}\times[0,1]\rightarrow\mathbb{R}^{mn}\times\mathbb{R}$ is constructed as follows $H(\varphi,\lambda,t)=\left(\begin{array}[]{c}(1-t)A(K)\varphi+D\varphi+t\beta\varphi^{3}-\lambda\varphi\\\ \frac{1}{2}\left(\frac{1}{h_{1}h_{2}}-\varphi^{\mathrm{T}}\varphi\right)\end{array}\right)=0,$ (56) where $A(K)\in\mathbb{R}^{mn\times mn}$ is a random block diagonal matrix with tridiagonal blocks, namely, $A(K)=\left(\begin{array}[]{cccc}A_{1}&&&\\\ &A_{2}&&\\\ &&\ddots&\\\ &&&A_{m}\end{array}\right)\mbox{with}~{}A_{i}=\left(\begin{array}[]{ccccc}a_{11}^{(i)}&a_{12}^{(i)}&&&\\\ a_{12}^{(i)}&a_{22}^{(i)}&a_{23}^{(i)}&&\\\ &a_{23}^{(i)}&a_{33}^{(i)}&\ddots&\\\ &&\ddots&\ddots&a_{n-1,n}^{(i)}\\\ &&&a_{n-1,n}^{(i)}&a_{nn}^{(i)}\end{array}\right),~{}i=1,\ldots,m,$ and $K=\left(K_{1},K_{2},\ldots,K_{m}\right)^{\mathrm{T}}$ with $K_{i}=\left(a_{11}^{(i)},a_{12}^{(i)},a_{22}^{(i)},a_{23}^{(i)},\ldots,a_{n-1,n-1}^{(i)},a_{n-1,n}^{(i)},a_{nn}^{(i)}\right)$. $H(\varphi,\lambda,0)$ corresponds to the linear eigenvalue problem $H(\varphi,\lambda,0)=\left(\begin{array}[]{c}A(K)\varphi+D\varphi-\lambda\varphi\\\ \frac{1}{2}\left(\frac{1}{h_{1}h_{2}}-\varphi^{\mathrm{T}}\varphi\right)\end{array}\right)=0,$ while $H(\varphi,\lambda,1)$ corresponds to the problem $(\ref{sec 2: finite difference - two dimen})$. In order to show the effectiveness of this homotopy $H$ by the Parametrized Sard’s Theorem, define a mapping $\tilde{H}:\mathbb{R}^{mn}\times\mathbb{R}\times[0,1]\times\mathbb{R}^{(2nm-m)}\rightarrow\mathbb{R}^{mn}\times\mathbb{R}$ related to $H$ as follows $\tilde{H}(\varphi,\lambda,t,K)=\left(\begin{array}[]{c}(1-t)A(K)\varphi+D\varphi+t\beta\varphi^{3}-\lambda\varphi\\\ \frac{1}{2}\left(\frac{1}{h_{1}h_{2}}-\varphi^{\mathrm{T}}\varphi\right)\end{array}\right)=0,$ (57) such that $H(\varphi,\lambda,t)=\tilde{H}(\varphi,\lambda,t,K)$. The Jacobian matrix of $\tilde{H}$, $\frac{\partial\tilde{H}}{\partial(\varphi,\lambda,t,K)}$, is $\left(\begin{array}[]{cccc}(1-t)A(K)+D+3t\beta\mbox{diag}(\varphi^{2})-\lambda I&-\varphi&\beta\varphi^{3}-A(K)\varphi&(1-t)B\\\ -\varphi^{\mathrm{T}}&0&0&0\end{array}\right),$ (58) where $B=\frac{\partial(A(K)\varphi)}{\partial K}\in\mathbb{R}^{{(mn)}\times{(2nm-m)}}$. Denote $\varphi_{i}=\left(\varphi_{i1},\varphi_{i2},\ldots,\varphi_{in}\right)^{\mathrm{T}}$. It can be verified that $B=\left(\begin{array}[]{cccc}B_{1}&&&\\\ &B_{2}&&\\\ &&\ddots&\\\ &&&B_{m}\end{array}\right)$ (59) with $B_{i}=\frac{\partial(A_{i}\varphi_{i})}{\partial K_{i}}=\left(\begin{array}[]{ccccccccccc}\varphi_{i1}&\varphi_{i2}&&&&&&&&&\\\ &\varphi_{i1}&\varphi_{i2}&\varphi_{i3}&&&&&&&\\\ &&&\varphi_{i2}&\varphi_{i3}&\varphi_{i4}&&&&&\\\ &&&&&&\ddots&&&&\\\ &&&&&&&\varphi_{i,n-2}&\varphi_{i,n-1}&\varphi_{in}&\\\ &&&&&&&&&\varphi_{i,n-1}&\varphi_{in}\end{array}\right).$ (60) For example, when $m=2,n=3$, we have $B=\begin{pmatrix}\varphi_{11}&\varphi_{12}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}\\\ {0}&\varphi_{11}&\varphi_{12}&\varphi_{13}&{0}&{0}&{0}&{0}&{0}&{0}\\\ {0}&{0}&{0}&\varphi_{12}&\varphi_{13}&{0}&{0}&{0}&{0}&{0}\\\ {0}&{0}&{0}&{0}&{0}&\varphi_{21}&\varphi_{22}&{0}&{0}&{0}\\\ {0}&{0}&{0}&{0}&{0}&{0}&\varphi_{21}&\varphi_{22}&\varphi_{23}&{0}\\\ {0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&\varphi_{22}&\varphi_{23}\end{pmatrix}.$ Denote by $G_{a}$ all the inner grid points and by $R_{a}$ the ordering of the grid points in $G_{a}$, i.e., $\displaystyle G_{a}=\\{(i,j):i=1,\ldots,m,j=1,\ldots,n\\},$ (61) $\displaystyle R_{a}=\\{j+(i-1)n:(i,j)\in G_{a}\\},$ (62) and by $G^{0}$ the grid points with function value being zero, $\displaystyle G^{0}=\\{(i,j)\in G_{a}:\varphi_{ij}=0\\}.$ (63) Note that $G_{a}$ and $R_{a}$ have a one to one correspondence. Define such correspondence as a mapping $\Gamma:G_{a}\rightarrow R_{a}$, $\displaystyle\Gamma(i,j)=j+(i-1)n.$ (64) Denote by $R^{0}$ the indices of rows in which $B$ is zero, by $R^{}$ the indices of rows, in which $B$ is not zero, and $S_{i}^{0}$ and $S_{i}^{}$ with similar meanings for $B_{i}$, $\displaystyle R^{0}=\\{r:B(r,:)=0\\},\quad R^{}=\\{r:B(r,:)\neq 0\\},$ (65) $\displaystyle S^{0}_{i}=\\{r:B_{i}(r,:)=0\\},\quad S_{i}^{}=\\{r:B_{i}(r,:)\neq 0\\}.$ (66) It can be verified that $\displaystyle R^{0}=\bigcup_{i=1}^{m}S^{0}_{i},\quad R^{}=\bigcup_{i=1}^{m}S_{i}^{}.$ (67) ###### Lemma 3.3 Let $B$ be the matrix defined in (59). Then the nonzero rows of $B$ are linearly independent, i.e., the submatrix $B(R^{},:)$ is row full rank. Proof. Note that $B(R^{},:)=[B(S_{1}^{},:);\ldots;B(S_{m}^{},:)]$. Due to the block structure of $B$, it suffices to prove that for any $i$, $1\leq i\leq m$, $B_{i}(S_{i}^{},:)$ is row full rank if $S_{i}^{}$ is not empty. Now suppose that $S_{i}^{}$ is not empty, that is $\varphi_{i}\neq 0$. Let the nonzero components of $\varphi_{i}$ be $\varphi_{i,i_{1}},\varphi_{i,i_{2}},\ldots,\varphi_{i,i_{r}}$, with $i_{1}<i_{2}<\ldots<i_{r}$. Denote by $Z(\varphi_{i,i_{1}},\ldots,\varphi_{i,i_{k}})$ the rows of $B_{i}$ containing $\varphi_{i,i_{1}},\varphi_{i,i_{2}},\ldots,\varphi_{i,i_{k}}$, $1\leq k\leq r$. Then $B_{i}(S_{i}^{},:)=Z(\varphi_{i,i_{1}},\ldots,\varphi_{i,i_{r}})$. The claim will be proved by successively adding rows with nonzero component of $\varphi_{i}$ to $Z$. (1) Prove $Z(\varphi_{i,i_{1}})$ is row full rank. If $i_{1}=1$, $Z(\varphi_{i,i_{1}})$ is the first two rows of $B_{i}$ and it is row full rank. If $1<i_{1}<n$, $Z(\varphi_{i,i_{1}})$ is three successive rows of $B_{i}$ which contains the following submatrix involving $\varphi_{i,i_{1}}$, $\left(\begin{array}[]{ccc}\varphi_{i,i_{1}}&0&0\\\ &\varphi_{i,i_{1}}&\\\ 0&0&\varphi_{i,i_{1}}\end{array}\right)$ where $$ stands for an element which may be zero or nonzero. Therefore $Z(\varphi_{i,i_{1}})$ is row full rank. If $i_{1}=n$, $Z(\varphi_{i,i_{1}})$ is the last two rows of $B_{i}$ and is row full rank too. (2) Prove that when $Z(\varphi_{i,i_{1}},\ldots,\varphi_{i,i_{k}})$ is row full rank, $Z(\varphi_{i,i_{1}},\ldots,\varphi_{i,i_{k+1}})$ is also row full rank, $1\leq k<r$. If $i_{k}=n-1$, then $i_{k+1}=n$ and $Z(\varphi_{i,i_{1}},\ldots,\varphi_{i,i_{k+1}})=Z(\varphi_{i,i_{1}},\ldots,\varphi_{i,i_{k}})$. Next suppose $i_{k}<n-1$. If $i_{k+1}=i_{k}+1$, i.e., $\varphi_{i,i_{k}}$ and $\varphi_{i,i_{k+1}}$ are successive, then $Z(\varphi_{i,i_{1}},\ldots,\varphi_{i,i_{k+1}})$ consists of $Z(\varphi_{i,i_{1}},\ldots,\varphi_{i,i_{k}})$ and a new row with $\varphi_{i,i_{k+1}}$ as its first nonzero element and with $\varphi_{i,i_{k+1}}$ located in a column different from those of $\varphi_{i,i_{1}},\ldots,\varphi_{i,i_{k}}$, the connecting submatrix illustrated in the following, $\left(\begin{array}[]{cccccc}&\varphi_{i,i_{k}}&\varphi_{i,i_{k+1}}&0&0&0\\\ 0&0&\varphi_{i,i_{k}}&\varphi_{i,i_{k+1}}&&0\\\ 0&0&0&0&\varphi_{i,i_{k+1}}&\end{array}\right)$ Thus $Z(\varphi_{i,i_{1}},\ldots,\varphi_{i,i_{k+1}})$ is row full rank. If $i_{k+1}=i_{k}+2<n$, then $Z(\varphi_{i,i_{1}},\ldots,\varphi_{i,i_{k+1}})$ consists of $Z(\varphi_{i,i_{1}},\ldots,\varphi_{i,i_{k}})$ and two new rows and similarly these two new rows are linearly independent with $Z(\varphi_{i,i_{1}},\ldots,\varphi_{i,i_{k}})$. If $i_{k+1}\geq i_{k}+3$ and $i_{k+1}<n$, then $Z(\varphi_{i,i_{1}},\ldots,\varphi_{i,i_{k+1}})$ consists of $Z(\varphi_{i,i_{1}},\ldots,\varphi_{i,i_{k}})$ and three new rows. If $i_{k+1}=n$, $i_{k+1}=i_{k}+2$, then $Z(\varphi_{i,i_{1}},\ldots,\varphi_{i,i_{k+1}})$ consists of $Z(\varphi_{i,i_{1}},\ldots,\varphi_{i,i_{k}})$ and a new row. If $i_{k+1}=n$, $i_{k+1}=i_{k}+s$ and $s\geq 3$, then $Z(\varphi_{i,i_{1}},\ldots,\varphi_{i,i_{k+1}})$ consists of $Z(\varphi_{i,i_{1}},\ldots,\varphi_{i,i_{k}})$ and two new rows. All the latter cases can be similarly proved. In the following, first we prove there exists a zero measure set $U_{1}\subset\mathbb{R}^{(2nm-m)}$, such that if $K\in\mathbb{R}^{(2nm-m)}\setminus U_{1}$, the eigenvalues of $A(K)+D$ are simple. Then we prove that 0 is a regular value of $H(\varphi,\lambda,t)$ for almost all $K\in\mathbb{R}^{(2nm-m)}\setminus(U_{1}\cup U_{2})$, where $U_{2}=(\mathbb{R}^{+})^{(2nm-m)}$. The removal of $U_{2}$ is to make the elements in the subdiagonal and supdiagonal of the matrix $(1-t)A(K)+D$ negative for $t\in[0,1)$. ###### Lemma 3.4 The eigenvalues of $A(K)+D$ are simple for $K$ almost everywhere in $\mathbb{R}^{(2nm-m)}$ except on a subset of real codimension 1. Proof. Let $f(\lambda)=det(A(K)+D-\lambda I)$. The polynomial $f(\lambda)$ has no multiple roots if and only if its discriminant $R(K)$ is nonzero [22]. It is obvious that $R(K)$ is not identically zero. Furthermore, since $R(K)$ is a polynomial in the elements of vector $K$, it can vanish only on a hypersurface of real codimension 1. The hypersurface is $U_{1}=\\{K\|R(K)=0\\}.$ In the following Lemma 3.5, we prove that for any matrix $F$ consisting of several submatrices by row, if every submatrix of $F$ is row full rank and the index sets of nonzero columns do not intersect for any two submatrices, then $F$ is row full rank. ###### Lemma 3.5 Let $F\in\mathbb{R}^{p\times q}$ be a matrix. Suppose $\\{1,\ldots,p\\}=\bigcup_{i=1}^{s}I_{i}$, with $I_{i}\bigcap I_{j}=\emptyset$, if $i\neq j$. Denote $E_{i}=\\{k\in\\{1,\ldots,q\\}:F(I_{i},k)\neq 0\\}.$ Suppose that for any $1\leq i\leq s$, $F(I_{i},:)$ is row full rank and $E_{i}\bigcap E_{j}=\emptyset$, if $i\neq j$. Then $F$ is row full rank. Proof. Since for any $1\leq i\leq s$, $F(I_{i},:)$ is row full rank, there exist a column index set $J_{i}\subset E_{i}$ such that $F(I_{i},J_{i})$ is a nonsingular submatrix. Correspondingly, the matrix $[F(I_{1},:);F(I_{2},:);\ldots;F(I_{s},:)]$ has a nonsingular submatrix as follows, $\left(\begin{array}[]{cccc}F(I_{1},J_{1})&&&\\\ &F(I_{2},J_{2})&&\\\ &&\ddots&\\\ &&&F(I_{s},J_{s})\end{array}\right).$ As a result, the matrix $[F(I_{1},:);F(I_{2},:);\ldots;F(I_{s},:)]$ is row full rank and so is $F$. To prove that $0$ is a regular value of $\tilde{H}(\varphi,\lambda,t,K):\mathbb{R}^{mn}\times R\times[0,1)\times\mathbb{R}^{(2nm-m)}\setminus(U_{1}\cup U_{2})\to\mathbb{R}^{mn+1}$, we need to prove $\forall(\varphi,\lambda,t,K)\in\mathbb{R}^{mn}\times R\times[0,1)\times\mathbb{R}^{(2nm-m)}\setminus(U_{1}\cup U_{2})$ satisfying $\tilde{H}(\varphi,\lambda,t,K)=0$, the Jacobian matrix of $\tilde{H}(\varphi,\lambda,t,K)$ is row full rank. $\forall(\varphi,\lambda,t,K)$ satisfying $\tilde{H}(\varphi,\lambda,t,K)=0$, for $B$ defined in (59), we have $B(R^{0},:)=0$. Correspondingly for the Jacobian matrix defined in (58), we have $\displaystyle((1-t)A+D+3t\beta\mbox{diag}(\varphi)^{2}-\lambda I)(R^{0},:)=((1-t)A+D-\lambda I)(R^{0},:).$ Through row permutations, the Jacobian matrix of $\tilde{H}(\varphi,\lambda,t,K)$, $\frac{\partial\tilde{H}}{\partial(\varphi,\lambda,t,K)}$, can be rewritten as $\left(\begin{array}[]{cccc}((1-t)A+D+3t\beta\mbox{diag}(\varphi)^{2}-\lambda I)(R^{},:)&-\varphi(R^{})&(\beta\varphi^{3}-A\varphi)(R^{})&(1-t)B(R^{},:)\\\ ((1-t)A+D-\lambda I)(R^{0},:)&-\varphi(R^{0})&(\beta\varphi^{3}-A\varphi)(R^{0})&0\\\ -\varphi^{\mathrm{T}}&0&0&0\end{array}\right).$ $\forall(\varphi,\lambda,t,K)$ satisfying $\tilde{H}(\varphi,\lambda,t,K)=0$, from Lemma 3.3, we know $B(R^{},:)$ is row full rank. Therefore, if we can prove $\left(\begin{array}[]{c}((1-t)A+D-\lambda I)(R^{0},:)\\\ -\varphi^{\mathrm{T}}\end{array}\right)$ (68) is row full rank, then $\frac{\partial\tilde{H}}{\partial(\varphi,\lambda,t,K)}$ is row full rank. From $\tilde{H}=0$, we have $((1-t)A+D-\lambda I)(R^{0},:)\varphi=0$, i.e., $\varphi$ is orthogonal to the rows of $((1-t)A+D-\lambda I)(R^{0},:)$. Therefore, if we can prove $((1-t)A+D-\lambda I)(R^{0},:)$ is row full rank, $\frac{\partial\tilde{H}}{\partial(\varphi,\lambda,t,K)}$ is row full rank. Now the problem is turned into proving that $((1-t)A+D-\lambda I)(R^{0},:)$ is row full rank. For easy exposition, some concepts concerning the topology of the grid points with zero function value are introduced. In addition, $(i,j)$ will be considered as grid point in the rest of this section, representing $(x_{i},y_{j})$. ###### Definition 3.6 In the grid $G_{a}$, a zero valued node $(i,j)$ is a grid point with $\varphi_{i,j}=0$. ###### Definition 3.7 Two zero valued nodes $(i_{1},j_{1})$ and $(i_{p},j_{p})$ are said to be zero valued connected, if there exist a sequence of zero valued nodes, $\displaystyle(i_{1},j_{1}),(i_{2},j_{2}),\ldots,(i_{p},j_{p}),$ (69) such that for any two successive nodes $(i_{k},j_{k})$ and $(i_{k+1},j_{k+1})$ of the sequence, the distance of these two nodes is 1 in the sense that $\displaystyle\|i_{k}-i_{k+1}\|+\|j_{k}-j_{k+1}\|=1.$ (70) ###### Definition 3.8 A set $S$ consisting of zero valued nodes is called a zero valued connected set if any two nodes of $S$ are zero valued connected. ###### Definition 3.9 A set $S$ consisting of zero valued nodes is called a zero valued connected component if $S$ is connected and S is the largest zero connected set containing $S$. From (56), the discretization of the differential equation in a stencil is written explicitly, $\displaystyle\alpha_{1}^{(ij)}\varphi_{ij}+\alpha_{2}^{(ij)}\varphi_{i-1,j}+\alpha_{3}^{(ij)}\varphi_{i+1,j}+\alpha_{4}^{(ij)}\varphi_{i,j+1}+\alpha_{5}^{(ij)}\varphi_{i,j-1}=0,$ (71) $\displaystyle\varphi_{0j}=\varphi_{m+1,j}=0,$ (72) $\displaystyle\varphi_{i0}=\varphi_{i,n+1}=0,$ (73) $\displaystyle\varphi^{\mathrm{T}}\varphi-\frac{1}{h_{1}h_{2}}=0,$ (74) where $\alpha_{1}^{(ij)}=(1-t)a_{jj}^{(i)}+\frac{1}{h_{1}^{2}}+\frac{1}{h_{2}^{2}}+v_{ij}+t\beta\varphi_{ij}^{2}-\lambda$, $\alpha_{2}^{(ij)}=\alpha_{3}^{(ij)}=(1-t)a_{jj+1}^{(i)}-\frac{1}{2h_{2}^{2}}$, $\alpha_{4}^{(ij)}=\alpha_{5}^{(ij)}=-\frac{1}{2h_{1}^{2}}$, and $i=1,\ldots,m$, $j=1,\ldots,n$. When $K\in R^{(2nm-m)}\setminus(U_{1}\bigcup U_{2})$, the sign relationships among the components of $\varphi$ are given in the following remark. ###### Remark 3.10 Let $(i,j)$ be an inner zero valued node. Assume that all except two of its neighbouring points are known to be zero valued nodes. If one of the rest two points is a zero valued node, then so is the other; If one of the rest two points is not a zero valued node, then neither is the other. For any set $G$ of zero valued nodes, denote by $R_{G}$ the set of indices of rows corresponding to $G$, in which the matrix $B$ is zero, i.e., $\displaystyle R_{G}=\\{r:B(r,:)=0,r=j+(i-1)n,(i,j)\in G\\}.$ (75) Note that if a zero valued node $(i,j)\in G$ is such that $B(s,:)=0$ with $s=j+(i-1)n$, then the neighbouring inner grid points in $y$-direction should be zero valued nodes, that is, both $(i,j-1)$ and $(i,j+1)$ are zero valued nodes if $1<j<n$, or $(i,j+1)$ is a zero valued node if $j=1$, or $(i,j-1)$ is a zero valued node if $j=n$. If $(i,j)\in G$ and the upper point $(i,j+1)$ or the lower point $(i,j-1)$ is not a zero valued node, then the corresponding row index of $s=\Gamma(i,j)$ will not be in $R_{G}$. Therefore $\Gamma^{-1}(R_{G})\subset G$. Note that $R_{G}$ may be empty even if $G$ is not empty. Let $M=(1-t)A+D-\lambda I$. Denote by $C_{G}$ the indices of columns corresponding to the zero valued connected set $G$, in which $M(R_{G},:)$ is not zero, i.e., $\displaystyle C_{G}=\\{c\in R_{a}:M(R_{G},c)\neq 0\\}.$ (76) If $R_{G}=\emptyset$, define $C_{G}=\emptyset$. For any $s\in R_{G}$, let $(i,j)=\Gamma^{-1}(s)$. Then the $\Gamma$ images of $(i,j)$ and its neighbouring inner grid points are possibly included in $C_{G}$. Denote by $g_{i}$ the $i$-th column of grid points in $x$-direction, i.e., $\displaystyle g_{i}=\\{(i,j):1\leq j\leq n\\},$ (77) and by $O_{1}$ ($O_{m}$) the ordering of all the inner grid points except the first (last) column in $y$-direction, $\displaystyle O_{1}=\\{r=j+(i-1)n:(i,j)\in G_{a}\backslash g_{1}\\},$ (78) $\displaystyle O_{m}=\\{r=j+(i-1)n:(i,j)\in G_{a}\backslash g_{m}\\}.$ (79) ###### Lemma 3.11 Both $M(O_{1},:)$ and $M(O_{m},:)$ are row full rank. Proof. It is obvious that $M(O_{1},:)$ has the following form: $\left(\begin{array}[]{ccccc}&\ldots&\ldots&\ldots&\ldots\\\ &&\ldots&\ldots&\ldots\\\ &&\ddots&\ldots&\ldots\\\ &&&&\ldots\end{array}\right),$ where $$ represents nonzero element. Therefore $M(O_{1},:)$ is row full rank. $M(O_{m},:)$ has the following form: $\left(\begin{array}[]{ccccc}\ldots&&&&\\\ \ldots&\ldots&&&\\\ \ldots&\ldots&\ldots&\ddots&\\\ \ldots&\ldots&\ldots&\ldots&\end{array}\right).$ Therefore $M(O_{m},:)$ is row full rank. Note that if $G^{0}\bigcap g_{1}=\emptyset$, then $R^{0}\subset O_{1}$, from Lemma 3.11, $M(R^{0},:)$ is row full rank. In the following, we consider the case $G^{0}\bigcap g_{1}\not=\emptyset$. The set $G^{0}\bigcap g_{1}$ can have its own zero valued connected components. ###### Lemma 3.12 For $m\geq n\geq 6$, suppose that there is a zero valued connected component $\gamma$ of $G^{0}\bigcap g_{1}$ with $s$ points, $s\geq 2$. Then 1. (i) $s<n$; 2. (ii) If $\gamma$ contains the point $(1,1)$ or contains the point $(1,n)$, then the zero valued connected component $G$ of $G^{0}$ containing $\gamma$ will be the set of zero valued nodes starting from $\gamma$ and ending on column $s$ with $s+1-i$ zero valued nodes on column $i$, $1\leq i\leq s$; 3. (iii) Suppose that $\gamma$ is located in the inner part of $g_{1}$, i.e., the grid point $(1,j)$ of $\gamma$ is such that $2\leq j\leq n-1$. If $s$ is even, then the zero valued connected component $G$ of $G^{0}$ containing $\gamma$ will be the set of zero valued nodes starting from $\gamma$ and ending on column $s/2$ with $s+2-2i$ zero valued nodes on column $i$, $1\leq i\leq s/2$; 4. (iv) Suppose that $\gamma$ is located in the inner part of $g_{1}$. If $s$ is odd, then there is a zero valued connected set $G$ of $G^{0}$ containing $\gamma$ and arriving at a single zero valued node on column $(s+1)/2$ with $s+2-2i$ zero valued nodes on column $i$, $1\leq i\leq(s+1)/2$; 5. (v) For the cases (2), (3), (4), if $s\geq 3$, the index set of nonzero columns of $M$ corresponding to $G$ satisfies $C_{G}=\Gamma(G)$. Proof. 1. (i) If $s=n$, we have $\varphi_{11}=\varphi_{12}=\ldots=\varphi_{1n}=0$. From the sign relationships among the components of $\varphi$ as stated in Remark 3.10, the grid points in the column 2 are all zero valued nodes, namely, $\varphi_{21}=\varphi_{22}=\ldots=\varphi_{2n}=0$. By induction, all the grid points are zero valued nodes, namely, $\varphi_{11}=\ldots=\varphi_{1n}=\ldots=\varphi_{mn}=0$, a contradiction with the condition that $\varphi^{T}\varphi\neq 0$. Therefore $s<n$. 2. (ii) If $\gamma$ starts from the point $(1,1)$, from the sign relationships among the components of $\varphi$, the zero valued nodes connecting $\gamma$ on the column 2 of grid points are $(2,j)$, $1\leq j\leq s-1$. By induction, it can be seen that the zero valued connected component $G$ of $G^{0}$ containing $\gamma$ will end on column $s$. The zero valued nodes connecting $\gamma$ are illustrated in Fig 1 and form a zero valued component, where black dot represents zero and white dot represents nonzero. Similarly, the case $\gamma$ ends at the point $(1,n)$ can be proved. Figure 1: Illustration of a flag$(1,1)$$(1,2)$$(1,s-1)$$(1,s)$$(1,s+1)$$(1,n)$$(2,1)$$(s-1,1)$$(s,1)$$(s+1,1)$$(m,1)$ 3. (iii) Assume that $\gamma$ starts from the point $(1,j_{1})$ and ends at the point $(1,j_{2})$ with $j_{1}\geq 2$ and $j_{2}\leq n-1$. From the sign relationships among the components of $\varphi$, the zero valued nodes connecting $\gamma$ on the column 2 of grid points are $(2,j)$, $3\leq j\leq s-2$. The zero valued nodes connecting $\gamma$ are illustrated in Fig 2. The zero valued nodes connecting $\gamma$ will end on column $s/2$ with two zero valued nodes and form a zero valued component $G$. Figure 2: Illustration of a flag$(1,1)$$(1,t-1)$$(1,t)$$(1,t+1)$$(1,t+\frac{s}{2}-2)$$(1,t+\frac{s}{2}-1)$$(1,t+\frac{s}{2})$$(1,t+\frac{s}{2}+1)$$(1,t+s-2)$$(1,t+s-1)$$(1,t+s)$$(1,n)$$(2,1)$$(\frac{s}{2}-1,1)$$(\frac{s}{2},1)$$(m,1)$ 4. (iv) Similar to the case (3), the zero valued nodes connecting $\gamma$ are illustrated in Fig 3. However, since $s$ is odd, the zero valued nodes connecting $\gamma$ arrives at one single point on column $(s+1)/2$. All these zero valued nodes connecting $\gamma$ from column 1 to column $(s+1)/2$ form a zero valued connected set $G$. It may be or may not be a zero valued connected component. Figure 3: Illustration of a flag$(1,10)$$(1,t-1)$$(1,t)$$(1,t+1)$$(1,t+\frac{s-3}{2})$$(1,t+\frac{s-1}{2})$$(1,t+\frac{s+1}{2})$$(1,t+s-2)$$(1,t+s-1)$$(1,t+s)$$(1,n)$$(2,1)$$(\frac{s-1}{2},1)$$(\frac{s+1}{2},1)$$(m,1)$ 5. (v) From the figures for cases (2), (3), (4), it can be seen that $\Gamma^{-1}(R_{G})$ is the set of zero valued nodes by shrinking $G$ one layer from its nonzero boundary. $\Gamma^{-1}(C_{G})$ is the set of zero valued nodes by extending $\Gamma^{-1}(R_{G})$ one layer towards the nonzero boundary, which is $G$ itself. ###### Definition 3.13 Let $\gamma$ be a zero valued connected component of $G^{0}\bigcap g_{1}$ with $s$ points, $s\geq 2$. The zero valued connected set $G$ mentioned in (2), (3), (4) in Lemma 3.12 is called a flag of $\gamma$, denoted as $F_{\gamma}$. ###### Lemma 3.14 Let $\gamma$ be a zero valued connected component of $G^{0}\bigcap g_{1}$ of $s$ points with $s>2$ an odd number and $F_{\gamma}$ be its flag. Let $G$ be another set of zero valued nodes. Let $I_{1}=R_{F_{\gamma}}$ and $I_{2}=R_{G}$ be the row index sets as defined in (75) such that $M(I_{2},:)$ is row full rank. Denote $\displaystyle J_{1}=\\{j\in R_{a}:M(I_{1},j)\neq 0\\},\quad J_{2}=\\{j\in R_{a}:M(I_{2},j)\neq 0\\},$ (80) $\displaystyle J_{12}=J_{1}\bigcap J_{2}.$ (81) Suppose that $J_{12}$ has only one element $k$ and that both the column $M(I_{1},k)$ and the column $M(I_{2},k)$ have only one nonzero element, denoted as $M(s_{1},k)\neq 0$ and $M(s_{2},k)\neq 0$ respectively. Let $(i_{1},j_{1})$ and $(i_{2},j_{2})$ be the grid point corresponding to $s_{1}$ and $s_{2}$ respectively. Suppose $i_{2}=i_{1}+2$ and $j_{1}=j_{2}$. Then $[M(I_{1},:);M(I_{2},:)]$ is row full rank. Proof. Note that since $J_{12}$ is not empty, the zero valued connected component $\gamma$ corresponds to the case (4) of Lemma 3.12. Geometrically, for that $k$, the two grid points $(i_{1},j_{1})$ and $(i_{2},j_{2})$ corresponding to $M(s_{1},k)\neq 0$ and $M(s_{2},k)\neq 0$ lie in the same row of grid points with distance 2 in the sense that $\|i_{2}-i_{1}\|+\|j_{2}-j_{1}\|=2$, as illustrated locally in Figure 4. Figure 4: Local illustration of connection with a flag$(i_{1},j_{1})$$(i_{2},j_{2})$ Note that by Lemma 3.11, $M(I_{1},:)$ is row full rank. By row permutations and column permutations, $[M(I_{1},:);M(I_{2},:)]$ is transformed to the following form, denoted as $W$, $\begin{array}[]{l@{\hspace{-5pt}}llll}\begin{array}[]{@{\hspace{10pt}}l@{\hspace{4pt}}l@{\hspace{2pt}}l@{\hspace{18pt}}l}\hskip 10.0pt\lx@intercol\overbrace{\hphantom{\begin{array}[]{ccccccc}\ast&\ast&\ast&\ddots&\ddots&\ddots&\ast\end{array}}}^{\displaystyle s}\hfil\hskip 4.0&\overbrace{\hphantom{\begin{array}[]{lcccc}\ast&\ddots&\ddots&\ast&\ast\end{array}}}^{\displaystyle s-2}\hfil\hskip 2.0&\overbrace{\hphantom{\begin{array}[]{lcc}\ast&\ddots&\ast\end{array}}}^{\displaystyle s-4}\hfil\hskip 18.0&\overbrace{\hphantom{\begin{array}[]{lcc}\ast&\ast&\ast\end{array}}}^{\displaystyle 3}\par\end{array}&\\\ \left(\begin{array}[]{ccccccc\|ccccc\|ccc\|c\|ccc\|c\|cc}\ast&\ast&\ast&&&&&\ast&&&&&&&&&&&&&&\\\ &\ast&\ast&\ast&&&&&\ast&&&&&&&&&&&&&\\\ &&\ast&\ast&\ast&&&&&\ast&&&&&&&&&&&&\\\ &&&\ddots&\ddots&\ddots&&&&&\ddots&&&&&&&&&&&\\\ &&&&\ast&\ast&\ast&&&&&\ast&&&&&&&&&\\\ &&\ast&&&&&\ast&\ast&\ast&&&\ast&&&&&&&&&\\\ &&&\ddots&&&&&\ddots&\ddots&\ddots&&&\ddots&&&&&&&&\\\ &&&&\ast&&&&&&\ast&\ast&&&\ast&&&&&&&\\\ &&&&&&&&&\ddots&&&&\ddots&&\ddots&&&&&&\\\ &&&&&&&&&&&&&&&&\ast&&&&&\\\ &&&&&&&&&&&&&&&&&\ast&&&&\\\ &&&&&&&&&&&&&&&&&&\ast&&&\\\ &&&&&&&&&&&&&&&\cdots&\ast&\ast&\ast&\ast&&\\\ \hline\cr&&&&&&&&&&&&&&&&&&&\ast&\ast&\cdots\\\ &&&&&&&&&&&&&&&&&&&&\ast&\\\ &&&&&&&&&&&&&&&&&&&&&\ddots\end{array}\right)&\begin{array}[]{l}\vspace{35pt}\left.\vphantom{\begin{array}[]{c}\ast\\\ \ast\\\ \ast\\\ \ddots\\\ \ast\\\ \ast\\\ \ddots\\\ \ast\\\ \ddots\\\ \ast\\\ \ast\\\ \ast\\\ \ast\end{array}}\right\\}\frac{(s-1)^{2}}{4}\end{array}\end{array}.$ Note that the diagonal entries of the submatrix $W(1:\frac{(s-1)^{2}}{4},s+1:\frac{(s+1)^{2}}{4})$ are all nonzero. Therefore, by elementary matrix transformations on the first $\frac{(s+1)^{2}}{4}$ columns, the above matrix $W$ is transformed to the following, denoted as $\overline{W}$, $\begin{array}[]{l@{\hspace{-5pt}}llll}\begin{array}[]{@{\hspace{1pt}}l@{\hspace{2pt}}l@{\hspace{2pt}}l@{\hspace{15pt}}l}\hskip 1.0pt\lx@intercol\overbrace{\hphantom{\begin{array}[]{ccccccc}\ast&\ast&\ast&\ddots&\ddots&\ddots&\ast\end{array}}}^{\displaystyle s}\hfil\hskip 2.0&\overbrace{\hphantom{\begin{array}[]{lcccc}\ast&\ddots&\ddots&\ast&\ast\end{array}}}^{\displaystyle s-2}\hfil\hskip 2.0&\overbrace{\hphantom{\begin{array}[]{lcc}\ast&\ddots&\ast\end{array}}}^{\displaystyle s-4}\hfil\hskip 15.0&\overbrace{\hphantom{\begin{array}[]{lcc}\ast&\ast&\ast\end{array}}}^{\displaystyle 3}\par\end{array}&\\\ \left(\begin{array}[]{ccccccc\|ccccc\|ccc\|c\|ccc\|c\|cc}&&&&&&&\ast&&&&&&&&&&&&&&\\\ &&&&&&&&\ast&&&&&&&&&&&&&\\\ &&&&&&&&&\ast&&&&&&&&&&&&\\\ &&&&&&&&&&\ddots&&&&&&&&&&&\\\ &&&&&&&&&&&\ast&&&&&&&&&\\\ &&&&&&&&&&&&\ast&&&&&&&&&\\\ &&&&&&&&&&&&&\ddots&&&&&&&&\\\ &&&&&&&&&&&&&&\ast&&&&&&&\\\ &&&&&&&&&&&&&&&\ddots&&&&&&\\\ &&&&&&&&&&&&&&&&\ast&&&&&\\\ &&&&&&&&&&&&&&&&&\ast&&&&\\\ &&&&&&&&&&&&&&&&&&\ast&&&\\\ &&&&&&&&&&&&&&&&&&&\ast&&\\\ \hline\cr\times&\ast&\ast&\cdots&\ast&\ast&\ast&\ast&\ast&\cdots&\ast&\ast&\ast&\cdots&\ast&\cdots&\ast&\ast&\ast&\ast&\ast&\cdots\\\ &&&&&&&&&&&&&&&&&&&&\ast&\\\ &&&&&&&&&&&&&&&&&&&&&\ddots\end{array}\right)&\begin{array}[]{l}\vspace{35pt}\left.\vphantom{\begin{array}[]{c}\ast\\\ \ast\\\ \ast\\\ \ddots\\\ \ast\\\ \ast\\\ \ddots\\\ \ast\\\ \ddots\\\ \ast\\\ \ast\\\ \ast\\\ \ast\end{array}}\right\\}\frac{(s-1)^{2}}{4}\end{array}\end{array}.$ By elementary matrix transformations on the first $\frac{(s-1)^{2}}{4}+1$ rows, the above matrix $\overline{W}$ is further transformed to the following, denoted as $\overline{\overline{W}}$, $\begin{array}[]{l@{\hspace{-5pt}}llll}\begin{array}[]{@{\hspace{0.1pt}}l@{\hspace{1pt}}l@{\hspace{1pt}}l@{\hspace{9pt}}l}\hskip 0.1pt\lx@intercol\overbrace{\hphantom{\begin{array}[]{ccccccc}\ast&\ast&\ast&\ddots&\ddots&\ddots&\ast\end{array}}}^{\displaystyle s}\hfil\hskip 1.0&\overbrace{\hphantom{\begin{array}[]{lcccc}\ast&\ddots&\ddots&\ast&\ast\end{array}}}^{\displaystyle s-2}\hfil\hskip 1.0&\overbrace{\hphantom{\begin{array}[]{lcc}\ast&\ddots&\ast\end{array}}}^{\displaystyle s-4}\hfil\hskip 9.0&\overbrace{\hphantom{\begin{array}[]{lcc}\ast&\ast&\ast\end{array}}}^{\displaystyle 3}\par\end{array}&\\\ \left(\begin{array}[]{ccccccc\|ccccc\|ccc\|c\|ccc\|c\|cc}&&&&&&&\ast&&&&&&&&&&&&&&\\\ &&&&&&&&\ast&&&&&&&&&&&&&\\\ &&&&&&&&&\ast&&&&&&&&&&&&\\\ &&&&&&&&&&\ddots&&&&&&&&&&&\\\ &&&&&&&&&&&\ast&&&&&&&&&\\\ &&&&&&&&&&&&\ast&&&&&&&&&\\\ &&&&&&&&&&&&&\ddots&&&&&&&&\\\ &&&&&&&&&&&&&&\ast&&&&&&&\\\ &&&&&&&&&&&&&&&\ddots&&&&&&\\\ &&&&&&&&&&&&&&&&\ast&&&&&\\\ &&&&&&&&&&&&&&&&&\ast&&&&\\\ &&&&&&&&&&&&&&&&&&\ast&&&\\\ &&&&&&&&&&&&&&&&&&&\ast&&\\\ \hline\cr\times&\ast&\ast&\cdots&\ast&\ast&\ast&&&&&&&&&&&&&&\ast&\cdots\\\ &&&&&&&&&&&&&&&&&&&&\ast&\\\ &&&&&&&&&&&&&&&&&&&&&\ddots\end{array}\right)&\begin{array}[]{l}\vspace{35pt}\left.\vphantom{\begin{array}[]{c}\ast\\\ \ast\\\ \ast\\\ \ddots\\\ \ast\\\ \ast\\\ \ddots\\\ \ast\\\ \ddots\\\ \ast\\\ \ast\\\ \ast\\\ \ast\end{array}}\right\\}\frac{(s-1)^{2}}{4}\end{array}\end{array}.$ Note that in the row $\frac{(s-1)^{2}}{4}+1$, at least one element $\overline{\overline{W}}(\frac{(s-1)^{2}}{4}+1,1)$, denoted as ${}^{\prime}\times^{\prime}$, is not zero. Therefore, the lower submatrix $\overline{\overline{W}}(\frac{(s-1)^{2}}{4}+1:end,:)$ is still row full rank. Now Lemma 3.5 can be applied to conclude that $\overline{\overline{W}}$ is row full rank. Therefore $[M(I_{1},:);M(I_{2},:)]$ is row full rank. ###### Theorem 3.15 For the homotopy $H:\mathbb{R}^{mn}\times\mathbb{R}\times[0,1)\rightarrow\mathbb{R}^{mn}\times\mathbb{R}$ in (56), $\forall n\in\mathbb{N}^{+}$, $m\geq n\geq 6$, for almost all $K\in\mathbb{R}^{(2nm-m)}\setminus(U_{1}\cup U_{2})$, 1. (i) 0 is a regular value of $H$ defined in $(\ref{sec3:eqn-homotopy-3diagonal})$ and therefore the homotopy paths corresponding to different initial points do not intersect each other for $t\in[0,1)$; 2. (ii) Every homotopy path $(\varphi(s),\lambda(s),t(s))\subset H^{-1}(0)$ is bounded. Proof. (i). It suffices to prove that $\forall(\varphi,\lambda,t,K)\in\mathbb{R}^{mn}\times\mathbb{R}\times[0,1)\times\mathbb{R}^{(2nm-m)}\setminus(U_{1}\cup U_{2})$ satisfying $\tilde{H}(\varphi,\lambda,t,K)=0$, $((1-t)A+D-\lambda I)(R^{0},:)$ is row full rank, or in short hand notation $M(R^{0},:)$ is row full rank. Note that $R^{0}$ corresponds to the zero valued nodes $G^{0}$. If $G^{0}\bigcap g_{1}=\emptyset$, then $R^{0}\subset O_{1}$. By Lemma 3.11 $M(R^{0},:)$ is row full rank. Next $G^{0}\bigcap g_{1}\neq\emptyset$ is assumed. If the zero valued connected components of the set $G^{0}\bigcap g_{1}$ are all single point sets, $R^{0}$ is a subset of $O_{1}$, and again by Lemma 3.11, $M(R^{0},:)$ is row full rank. Assume that there are $q$ zero valued connected components of the set $G^{0}\bigcap g_{1}$, each of which has more than one point. These connected components of the set $G^{0}\bigcap g_{1}$ are denoted as $\gamma_{i}$, with corresponding flags $G_{i}=F_{\gamma_{i}}$, $i=1,\cdots,q$. Denote $\displaystyle\overline{G}=G_{a}\setminus(\bigcup_{i=1}^{q}G_{i}),$ (82) $\displaystyle\overline{R}=R_{\overline{G}},\quad\overline{C}=C_{\overline{G}},\quad R_{i}=R_{G_{i}},\quad C_{i}=C_{G_{i}},\quad\quad\forall 1\leq i\leq q.$ (83) By Lemma 3.12, it can be verified that $\displaystyle R^{0}=\overline{R}\bigcup\left(\bigcup_{i=1}^{q}R_{i}\right),$ (84) $\displaystyle R_{i}\bigcap\overline{R}=\emptyset,\quad\forall 1\leq i\leq q,$ (85) $\displaystyle R_{i}\bigcap R_{j}=\emptyset,\quad C_{i}\bigcap C_{j}=\emptyset,\quad\forall i\neq j.$ (86) Note that if a zero valued connected component $\gamma$ of $G^{0}\bigcap g_{1}$ has only two nodes, then the flag of $\gamma$ is the $\gamma$ itself, and $R_{\gamma}=\emptyset$. Assume that all the zero valued connected components $\gamma_{i}$ have more than two nodes. Since $\displaystyle\overline{R}\subset O_{1},\quad R_{i}\subset O_{m},\quad\forall 1\leq i\leq q,$ (87) by Lemma 3.11, all of $M(\overline{R},:)$ and $M(R_{i},:)$, $1\leq i\leq q$, are row full rank. It is possible that for some $\gamma_{i}$, $C_{i}\bigcap\overline{C}\neq\emptyset$. If so, $\gamma_{i}$ is the case described in case $(4)$ in Lemma 3.12, $C_{i}\bigcap\overline{C}$ has only one element and the corresponding grid points are illustrated in Figure 5. Figure 5: Local illustration of a connection$G_{i}$$\overline{G}$ Without loss of generality, suppose that all the $\gamma_{i}$ satisfying such property are the first $p$ $\gamma_{i}$, i.e., $\displaystyle C_{i}\bigcap\overline{C}\neq\emptyset,\quad 1\leq i\leq p,$ (88) $\displaystyle C_{i}\bigcap\overline{C}=\emptyset,\quad p+1\leq i\leq q.$ (89) Denote $\displaystyle G$ $\displaystyle=\overline{G}\bigcup\left(\bigcup_{i=p+1}^{q}G_{i}\right),$ (90) $\displaystyle Z(G_{k},\ldots,G_{1},G)$ $\displaystyle=[M(R_{k},:);\ldots;M(R_{1},:);M(R_{G},:)],\quad\forall 1\leq k\leq p.$ (91) Then $M(R^{0},:)=Z(G_{p},\ldots,G_{1},G)$. The claim that $M(R^{0},:)$ is row full rank will be proved by recursion. Firstly, prove $Z(G_{1},G)$ is row full rank. Take $I_{1}=R_{1}$ and $I_{2}=R_{G}$. The conditions of Lemma 3.14 are satisfied. Thus $[M(R_{1},:);M(R_{G},:)]$ is row full rank. Secondly, prove $Z(G_{k+1},\ldots,G_{1},G)$ is row full rank if $Z(G_{k},\ldots,G_{1},G)$ is, $\forall 1\leq k<p$. Take $I_{1}=R_{k+1}$ and $I_{2}=R_{k}\cup\cdots\cup R_{1}\cup R_{G}$. Since $C_{k+1}\cap C_{i}=\emptyset$, $1\leq i\leq k$, and $C_{k+1}\bigcap C_{G}$ has only one element, $I_{1}$ and $I_{2}$ satisfy the conditions of Lemma 3.14. Thus $[M(R_{k},:);\ldots;M(R_{1},:);M(R_{G},:)]$ is row full rank. (ii). Similar to the one dimensional case, that $H^{-1}(0)$ is bounded can be proved also. #### 3.2.2 The homotopy with random pentadiagonal matrix A homotopy with random pentadiagonal matrix is also possible. Specifically, for $\tilde{H}$ defined in (57), replacing $A(K)$ with $\overline{A}(\overline{K})$, where $\overline{A}(\overline{K})\in\mathbb{R}^{mn\times mn}$ is a random pentadiagonal matrix with the same sparse structure as $D$, namely, $\overline{A}(\overline{K})=\left(\begin{array}[]{cccc}A_{1}&\overline{A}_{1}&&\\\ \overline{A}_{1}&A_{2}&\overline{A}_{2}&\\\ &&\ddots&\overline{A}_{m-1}\\\ &&\overline{A}_{m-1}&A_{m}\end{array}\right),$ where $A_{i}=\left(\begin{array}[]{ccccc}a_{11}^{(i)}&a_{12}^{(i)}&&&\\\ a_{12}^{(i)}&a_{22}^{(i)}&a_{23}^{(i)}&&\\\ &a_{23}^{(i)}&a_{33}^{(i)}&\ddots&\\\ &&\ddots&\ddots&a_{n-1,n}^{(i)}\\\ &&&a_{n-1,n}^{(i)}&a_{nn}^{(i)}\end{array}\right),\quad\overline{A}_{i}=\left(\begin{array}[]{ccccc}b_{1}^{(i)}&&&&\\\ &b_{2}^{(i)}&&&\\\ &&b_{3}^{(i)}&&\\\ &&&\ddots&\\\ &&&&b_{n}^{(i)}\end{array}\right),$ $\displaystyle\overline{K}=\left(K_{1},K_{2},\ldots,K_{m},\overline{K}_{1},\ldots,\overline{K}_{m-1}\right)^{\mathrm{T}},$ $\displaystyle K_{i}=\left(a_{11}^{(i)},a_{12}^{(i)},a_{22}^{(i)},a_{23}^{(i)},\ldots,a_{n-1,n-1}^{(i)},a_{n-1,n}^{(i)},a_{nn}^{(i)}\right),~{}i=1,\ldots,m,$ $\displaystyle\overline{K}_{i}=\left(b_{1}^{(i)},b_{2}^{(i)},\ldots,b_{n}^{(i)}\right),~{}i=1,\ldots,m-1.$ The Jacobian matrix of $\tilde{H}(\varphi,\lambda,t,\overline{K})$, $\frac{\partial\tilde{H}}{\partial(\varphi,\lambda,t,\overline{K})}$, is : $\left(\begin{array}[]{cccc}(1-t)\overline{A}(\overline{K})+D+3t\beta\mbox{diag}(\varphi^{2})-\lambda I&-\varphi&\beta\varphi^{3}-\overline{A}(\overline{K})\varphi&(1-t)\overline{B}\\\ -\varphi^{\mathrm{T}}&0&0&0\end{array}\right),$ where $\overline{B}=\frac{\partial(\overline{A}(\overline{K})\varphi)}{\partial\overline{K}}\in\mathbb{R}^{{(mn)}\times{(3nm- m-n)}}$. Recall $\varphi_{i}=\left(\varphi_{i1},\varphi_{i2},\ldots,\varphi_{in}\right)^{\mathrm{T}}$. It can be verified that $\displaystyle\overline{B}$ $\displaystyle=\left(\begin{array}[]{ccccc\|cccc}B_{1}&&&&&\overline{B}_{2}&&&\\\ &B_{2}&&&&\overline{B}_{1}&\overline{B}_{3}&&\\\ &&\ddots&&&&\overline{B}_{2}&\overline{B}_{4}&\\\ &&&\ddots&&&&\ddots&\ddots\\\ &&&&B_{m}&&&&\overline{B}_{m-1}\end{array}\right)$ (97) $\displaystyle=\left(\begin{array}[]{ccccc\|cccc}&&&&&\overline{B}_{2}&&&\\\ &&&&&\overline{B}_{1}&\overline{B}_{3}&&\\\ &&B&&&&\overline{B}_{2}&\overline{B}_{4}&\\\ &&&&&&&\ddots&\ddots\\\ &&&&&&&&\overline{B}_{m-1}\end{array}\right),$ (103) with $\displaystyle\overline{B}_{i}=\left(\begin{array}[]{cccc}\varphi_{i1}&&&\\\ &\varphi_{i2}&&\\\ &&\ddots&\\\ &&&\varphi_{in}\end{array}\right).$ (108) Note that the left part $B$ of $\overline{B}$ is nothing but the matrix in (59). ###### Lemma 3.16 The eigenvalues of $\overline{A}(\overline{K})+D$ are simple for $\overline{K}$ almost everywhere except on a subset of real codimension 1. Proof. Similar to the proof of Lemma 3.4. For our discussion, $\overline{U}_{2}=(\mathbb{R}^{+})^{(3nm-m-n)}$ is removed. To prove that $\forall(\varphi,\lambda,t,\overline{K})\in\mathbb{R}^{mn}\times R\times[0,1)\times\mathbb{R}^{(3nm- m-n)}\setminus(\overline{U}_{1}\cup\overline{U}_{2})$ satisfying $\tilde{H}(\varphi,\lambda,t,\overline{K})=0$, the Jacobian matrix of $\tilde{H}(\varphi,\lambda,t,\overline{K})$ in (3.2.2) is row full rank, it suffices to prove that the following submatrix of the Jacobian matrix in (3.2.2) $\left(\begin{array}[]{cccc}(1-t)\overline{A}(\overline{K})+D+3t\beta\mbox{diag}(\varphi^{2})-\lambda I&-\varphi&\beta\varphi^{3}-\overline{A}(\overline{K})\varphi&(1-t)B\\\ -\varphi^{\mathrm{T}}&0&0&0\end{array}\right)$ (109) is row full rank. With the notations $R^{0}$ and $R^{}$ as in (65) and similar arguments as in Subsection 3.2.1, it can be proved that $((1-t)\overline{A}+D-\lambda I))(R^{0},:)$ is row full rank. Therefore the matrix (109) is row full rank. That is $0$ is a regular value of the homotopy $H$ with random pentadiagonal matrix for almost all $\overline{K}\in\mathbb{R}^{(3nm-m-n)}\setminus(U_{1}\cup U_{2})$ with $m\geq n\geq 6$. ## 4 Algorithm and numerical results ### 4.1 Algorithm Thanks to Theorems 3.2 and 3.15, since 0 is a regular value of the homotopies constructed, the homotopy paths determined by the homotopy equations (49) and (56) have no bifurcation points with probability one. Therefore the usual path following algorithm, i.e., the predictor-corrector method as in [23, 24], can be adapted to trace the homotopy paths of the homotopy equations (49) and (56). The adapted algorithm is stated in Algorithm 1. For notation convenience, in Algorithm 1, we denote the iterate at step $k$ as $x_{k}=(\varphi^{(k)},\lambda_{k})$, $k=1,\ldots$. Initialization: Set $(x_{0},t_{0})=(x_{0},0)$, $k=0$, the minimum step size $ds_{min}$, the initial step size $ds$. Compute the tangent vector $(\dot{x}_{0},\dot{t}_{0})$ such that $\dot{t}_{0}>0$ and record the orientation $ori$ ${H_{x}\dot{x}_{0}+H_{t}\dot{t}_{0}=0},\qquad ori=\mbox{sign}\left(\left\|\begin{array}[]{cc}H_{x}&H_{t}\\\ \dot{x}_{0}&\dot{t}_{0}\end{array}\right\|\right).$ (110) while _$t_{k} <1$_ do Predictor: $(\bar{x}_{k+1},\bar{t}_{k+1})=(x_{k},t_{k})+ds(\bar{x}_{k},\bar{t}_{k})$; if _$\bar{t}_{k+1} >1$_ then change ds such that $\bar{t}_{k+1}=1$; end if Corrector: if _$\bar{t}_{k+1}=1$_ then $(v,\tau)=(0,1)$; else $(v,\tau)=(\dot{x}_{k},\dot{t}_{k})$; end if Employ Newton method to solve the following nonlinear equations: $\begin{array}[]{ccc}\left(\begin{array}[]{c}H(x,t)\\\ v^{T}(x-\bar{x}_{k+1})+\tau(t-\bar{t}_{k+1})\end{array}\right)&=&0.\end{array}$ (111) Judgement: if _the above iteration converges to $(x_{k+1},t_{k+1})$_ then compute the tangent vector $(\dot{x}_{k+1},\dot{t}_{k+1})$ satisfying ${H_{x}\dot{x}_{k+1}+H_{t}\dot{t}_{k+1}=0},\qquad\mbox{sign}\left(\left\|\begin{array}[]{cc}H_{x}&H_{t}\\\ \dot{x}_{k+1}&\dot{t}_{k+1}\end{array}\right\|\right)=ori$ (112) else $ds=\frac{1}{2}ds$; go to Predictor; end if Compute angle $\theta$ of $(\bar{x}_{k},\bar{t}_{k})$ and $(\bar{x}_{k+1},\bar{t}_{k+1})$; if _$\theta >18^{0}$_ then $ds=\frac{1}{2}ds,$ go to Predictor; end if Accept iterates: $(x_{k},t_{k})=(x_{k+1},t_{k+1})$,$(\dot{x}_{k},\dot{t}_{k})=(\dot{x}_{k+1},\dot{t}_{k+1})$; if _$\theta <6^{0}$_ then $ds=2ds$; end if if _$ds <ds_{min}$_ then stop the algorithm; end if end while Algorithm 1 Predictor-Corrector method ### 4.2 Numerical results The first numerical example is the 1D discretized problem (9) with $\beta=20$, $V(x)=\frac{1}{2}x^{2}$, $\Omega=[-2,2]$ and $n=999$. Some eigenvectors, i.e., approximate eigenfunctions are plotted in Figures 6(a) to 6(i). The eigenvector in Figure 6(a) is the unique positive solution of (9) as stated in Theorem 2.5, which corresponds to the unique positive ground state, as proved in [4] for the continuous nonlinear eigenvalue problem (4)-(6) when $\beta>0$. The approximate eigenfunction in Figure 6(b) is antisymmetric as described in Theorem 2.6 and is an approximate first excited state. Others are approximate excited states corresponding to higher energy. The order preserving property of the eigenvalue curves as stated in [17] was observed, that is, if $\lambda(0)$ is the $k$th smallest eigenvalue of the initial problem, then $\lambda(t)$ is the $k$th smallest eigenvalue of the intermediate problem for each $t\in[0,1)$. However, we are not able to prove such property for the eigenvector-dependent nonlinear eigen-problem yet. (a) $\lambda=6.76$ (b) $\lambda=8.39$ (c) $\lambda=10.35$ (d) $\lambda=12.73$ (e) $\lambda=15.62$ (f) $\lambda=19.08$ (g) $\lambda=23.13$ (h) $\lambda=27.79$ (i) $\lambda=33.05$ The second numerical example is the 2D discretized problem $(\ref{sec 2: finite difference - two dimen})$ with $\beta=20$, $V(x)=\frac{1}{2}(x_{1}^{2}+x_{2}^{2})$, $\Omega=[0,1]\times[0,1]$ and $m=n=29$. Some approximate eigenfunctions are collected in Figures 6(j) to 6(u). The approximate eigenfunction in Figure 6(j) corresponds to the unique positive ground state. Others are approximate excited states corresponding to higher energy. The order preserving property of the eigenvalue curves is also observed for the 2D case. (j) $\lambda=43.36$ (k) $\lambda=60.89$ (l) $\lambda=65.31$ (m) $\lambda=78.81$ (n) $\lambda=86.28$ (o) $\lambda=94.06$ (p) $\lambda=104.4$ (q) $\lambda=108.82$ (r) $\lambda=120.47$ (s) $\lambda=129.67$ (t) $\lambda=133.81$ (u) $\lambda=138.68$ ## 5 Conclusion Solutions to the discretized problem with the finite difference disretization for the GPE inherit certain properties of the solutions to the continuous problem, such as the existence and uniqueness of positive eigenvector (eigenfunction). The designed homotopy continuation methods are suitable for computing eigenpairs corresponding to excited states of high energy as well as the ground state and the first excited state. In order to make sure that the homotopy paths are regular and that the path following is efficient, artificial homotopy parameter and random matrices with certain structures in the homotopies seem indispensable. ## References [1] W. Bao and W. 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# Spin dynamics and unconventional Coulomb phase in Nd2Zr2O7 M. Léger Institut Néel, CNRS and Université Grenoble Alpes, 38000 Grenoble, France Laboratoire Léon Brillouin, Université Paris-Saclay, CNRS, CEA, CE- Saclay, F-91191 Gif-sur-Yvette, France E. Lhotel<EMAIL_ADDRESS>Institut Néel, CNRS and Université Grenoble Alpes, 38000 Grenoble, France M. Ciomaga Hatnean Department of Physics, University of Warwick, Coventry, CV4 7AL, United Kingdom J. Ollivier Institut Laue Langevin, F-38042 Grenoble, France A. R. Wildes Institut Laue Langevin, F-38042 Grenoble, France S. Raymond Université Grenoble Alpes, CEA, IRIG, MEM, MDN, 38000 Grenoble, France E. Ressouche Université Grenoble Alpes, CEA, IRIG, MEM, MDN, 38000 Grenoble, France G. Balakrishnan Department of Physics, University of Warwick, Coventry, CV4 7AL, United Kingdom S. Petit<EMAIL_ADDRESS>Laboratoire Léon Brillouin, Université Paris-Saclay, CNRS, CEA, CE-Saclay, F-91191 Gif-sur-Yvette, France ###### Abstract We investigate the temperature dependence of the spin dynamics in the pyrochlore magnet Nd2Zr2O7 by neutron scattering experiments. At low temperature, this material undergoes a transition towards an “all in - all out” antiferromagnetic phase and the spin dynamics encompass a dispersion-less mode, characterized by a dynamical spin ice structure factor. Unexpectedly, this mode is found to survive above $T_{\rm N}\approx 300$ mK. Concomitantly, elastic correlations of the spin ice type develop. These are the signatures of a peculiar correlated paramagnetic phase which can be considered as a new example of Coulomb phase. Our observations near $T_{\rm N}$ do not reproduce the signatures expected for a Higgs transition, but show reminiscent features of the “all in - all out” order superimposed on a Coulomb phase. Geometrical frustration is well known to be one of the key ingredients leading to unconventional states of matter, especially in magnetism [1, 2]. Among them, spin ice and more generally Coulomb phases [3] have attracted significant interest. These can be considered as an original state of matter formed by disordered degenerate configurations where local degrees of freedom remain strongly constrained at the local scale by an organizing principle. In the case of spin ice, these degrees of freedom are Ising spins, sitting on the sites of a pyrochlore lattice formed of corner sharing tetrahedra and aligned along the axes which connect the corners of the tetrahedra to their center. The organizing principle, the “ice rule”, states that each tetrahedron should have two spins pointing in and two out, in close analogy with the rule which controls the hydrogen position in water ice [4]. Importantly, the idea that this local constraint can be considered as the conservation law of an “emergent” magnetic flux (${\bf\nabla}\cdot{\bf B}=0$) was quickly imposed [5, 6, 7]. Quantum fluctuations can cause this flux to change with time, giving rise to an emergent electric field, and eventually to an emergent quantum electromagnetism [8, 9, 10]. This quantum spin ice state hosts spinon (monopole in the spin ice language [11]) and photon like excitations. Despite much work, however, experimental evidence for this enigmatic physics remains elusive, with the possible exception of Pr2Hf2O7 [12]. Indeed, the conditions for the realisation of this so-called quantum spin ice state are drastic: transverse terms have to be sizable in the Hamiltonian to enable fluctuations out of the local Ising axes, but should remain small enough to prevent the stabilization of classical phases, called Higgs phases, characterized by ordered components perpendicular to these axes [13, 11, 14]. The pyrochlore material Nd2Zr2O7 offers the opportunity to approach this issue. Recent studies suggest that below 1 K this compound hosts a correlated state, which could be a remarkable novel example of Coulomb phase [15, 16]. This phase would be described by a “two in – two out” rule as in spin ice, but built on a pseudospin component different from the conventional $\langle 111\rangle$ Ising one. The “all in – all out” (AIAO) ordering previously observed below $T_{\rm N}\approx 300$ mK [17, 18] would then correspond to the pseudospin ordering in directions perpendicular to the components responsible for the “high temperature” Coulomb phase. It was proposed that a Higgs mechanism may account for this transition [16]. Such a process is invoked in $U(1)$ quantum spin liquids when the deconfined spinon excitations undergo a Bose-Einstein condensation, resulting in a Higgs phase along with a gapped photon excitation [19, 13, 20]. In this letter, we show that the paramagnetic phase of Nd2Zr2O7 does carry elastic spin ice-like correlations, and thus confirm the proposed Coulomb phase picture above $T_{\rm N}$. We present a detailed study of the spin dynamics as a function of temperature and explore the nature of this Coulomb phase above and close to the transition. The spin excitations of Nd2Zr2O7 deep in the AIAO phase include a peculiar spectrum with a flat band at the energy $E_{0}\approx$ 70 $\mu$eV characterized by a spin ice-like ${\bf Q}$-dependence [15, 21, 22]. Using neutron scattering experiments, we report the temperature dependence of the gap $E_{0}$, and reveal that this gap persists above $T_{\rm N}$. This result is robust, and withstands a small substitution at the Zr site. The spectra recorded above $T_{\rm N}$ do not show the spinon continuum expected in the Higgs scenario. Instead, we observe dispersive features reminiscent of the AIAO ordered phase superimposed on the Coulomb phase signal. This coexistence suggests that a strong exchange competition is at work in this temperature range, emphasizing the originality of the Coulomb phase above the transition. The single crystal samples used in this work are the same as in our previous studies (labeled #1 [17, 15, 21] and #2 [21]). In addition, results on a single crystal of Nd2(Zr1-xTix)2O7, with $x=2.4$ % (Sample #3) (See supplementary material [23]) are presented, not in order to analyze the role of disorder but to illustrate the robustness of the results. Magnetic properties were measured in very low temperature SQUID magnetometers developed at the Institut Néel [29]. The composition and magnetic structure at low temperature were determined using the D23 (CEA-CRG@ILL) neutron diffractometer [23]. Polarized neutron scattering experiments were carried out at D7 (ILL) on Sample #1. Inelastic neutron scattering (INS) experiments were carried out on the IN5 (ILL) time of flight spectrometer on all samples and on the triple axis spectrometer IN12 (CEA-CRG@ILL) for Sample #1. The INS data have been analyzed using the cefwave software developed at LLB. The XYZ Hamiltonian proposed to describe the properties of Nd based pyrochlores due to the peculiar dipolar-octupolar character of the Nd3+ion [30], writes: ${\cal H}=\sum_{\langle i,j\rangle}\left[{\sf J}_{x}\tau^{x}_{i}\tau^{x}_{j}+{\sf J}_{y}\tau^{y}_{i}\tau^{y}_{j}+{\sf J}_{z}\tau^{z}_{i}\tau^{z}_{j}+{\sf J}_{xz}(\tau^{x}_{i}\tau^{z}_{j}+\tau^{z}_{i}\tau^{x}_{j})\right]$ (1) In this Hamiltonian, $\tau_{i}$ is not the actual spin, but a pseudospin which resides on the rare-earth sites of the pyrochlore lattice. Its $z$ component relates to the usual magnetic moment and is directed along the local $\langle 111\rangle$ directions of the tetrahedra of the pyrochlore lattice. This Hamiltonian can be rewritten by rotating the ${\bf x}$ and ${\bf z}$ axes in the $({\bf x},{\bf z})$ plane by an angle $\theta$. In this $({\bf\tilde{x}},{\bf\tilde{z}})$ rotated frame, the relevant parameters of the Hamiltonian ${\cal H}$ are labeled $\tilde{{\sf J}}_{x,y,z}$, leading to [30, 31]: $\displaystyle{\cal H}_{\rm XYZ}=$ $\displaystyle\sum_{\langle i,j\rangle}\left[{\tilde{\sf J}_{x}}\tilde{\tau}^{\tilde{x}}_{i}\tilde{\tau}^{\tilde{x}}_{j}+{\tilde{\sf J}_{y}}\tilde{\tau}^{\tilde{y}}_{i}\tilde{\tau}^{\tilde{y}}_{j}+{\tilde{\sf J}_{z}}\tilde{\tau}^{\tilde{z}}_{i}\tilde{\tau}^{\tilde{z}}_{j}\right]$ (2) $\displaystyle{\rm with}\quad\tan(2\theta)=\frac{2{\sf J}_{xz}}{{\sf J}_{x}-{\sf J}_{z}}$ With time and maturation of the subject, the estimated parameters for Nd2Zr2O7 have evolved. Determinations of the $\tilde{\sf J}_{i}$ parameters are based on the spin wave spectra measured at very low temperature in zero field [15, 31, 22] or applied field [21], while the angle $\theta$ is deduced from the Curie-Weiss temperature [31] and/or the ordered AIAO magnetic moment [21, 22]. The sets of reported parameters are summarized in Table 1, where we have added the parameters refined here for the Nd2(Zr1-xTix)2O7 sample (Sample #3) [23] and have revisited the ones of Samples #1 and #2. From these values, two interesting features stand out, which remain unexplained to date and should be further explored to ascertain their relevance: (i) the larger the Néel temperature, the larger the ordered moment along z is. (ii) very similar $\tilde{\sf J}_{i}$ parameters are obtained for the various samples, despite differences with regard to the amount of impurities or to the ordering parameters. The ${\sf J}$ parameters lead to an ordered AIAO ground state, where the pseudospins point along the (local) direction ${\bf\tilde{z}}$, turned around the ${\bf z}$-axis towards the ${\bf x}$-axis by the angle $\theta$ [31]. As shown by INS experiments, peculiar excitations are associated with this ground state. They manifest as an inelastic spin ice like flat mode at an energy $E_{0}\approx$ 70 $\mu$eV, above which spin wave branches disperse (See Figure 4a for Sample #1) [15]. This excitation spectrum is understood in the framework of the dynamic fragmentation [31, 32] as the sum of a dynamic divergence-free contribution, giving rise to the flat mode at $E_{0}$ and of a dynamic curl-free contribution, which takes the form of the dispersing branches. These spin waves correspond to the propagation of magnetically charged excitations and have a spectral weight made of half-moons in reciprocal space [15, 33]. Sample / Ref. \| $m_{\rm ord}~{}(\mu_{\rm B})$ \| $T_{\rm N}$ (mK) \| Hamiltonian parameters (K) \| $\theta$ (rad) ---\|---\|---\|---\|--- \| \| \| $\tilde{{\sf J}}_{x}$ \| $\tilde{{\sf J}}_{y}$ \| $\tilde{{\sf J}}_{z}$ \| $\tilde{{\sf J}}_{x}/\|\tilde{{\sf J}}_{z}\|$ \| #1 \| $0.8\pm 0.05$ \| 285 \| 1.18 \| -0.03 \| -0.53 \| 2.20 \| 1.23 #2 \| $1.1\pm 0.1$ \| 340 \| 1.0 \| 0.066 \| -0.5 \| 2.0 \| 1.09 #3 \| $1.19\pm 0.03$ \| 375 \| 0.97 \| 0.21 \| -0.53 \| 1.83 \| 1.08 [22] \| 1.26 \| 400 \| 1.05 \| 0.16 \| -0.53 \| 1.98 \| 0.98 [31] \| 1.4 \| - \| 1.2 \| 0.0 \| -0.55 \| 2.18 \| 0.83 Table 1: Ordered moment $m_{\rm ord}$ along ${\bf z}$, transition temperature $T_{\rm N}$ and Hamiltonian parametrization reported in different studies. $\tilde{{\sf J}}_{i}$ parameters for Sample #1 and from Ref. 31 were obtained from fits of the INS data reported in Ref. 15 and, for Sample #2 in Ref. 21. $m_{\rm ord}$ from Ref. 31 is a calculated value. The total Nd3+ magnetic moment is estimated to $\approx 2.4~{}\mu_{\rm B}$ [17, 18]. Figure 1: (a-b) Magnetic instantaneous correlations in Sample #1 as a function of temperature. The 10 K dataset has been subtracted as a background reference. Measurements in (a) were symmetrized. (c) “Spin ice” moment $m_{1}$ and AIAO ordered moment $m_{2}$ along ${\bf z}$ as a function of temperature [23]. Lines are guides to the eye. Instantaneous spin-spin correlations $S({\bf Q})$ were measured in Sample #1 as a function of temperature between 60 mK and 1 K through polarized neutron scattering experiments and are displayed in Figure 1 [23]. These measurements integrate over the neutron energy loss up to 3.5 meV, and thus contain both elastic and inelastic signals. At 1 K, a spin ice pattern can barely be observed, revealing the onset of a Coulomb phase. Upon cooling, the spin ice pattern becomes clearly visible below 600 mK. At 450 mK, the magnetic moment $m_{1}$ responsible for the spin ice-like diffuse scattering is estimated to $2.05\pm 0.3~{}\mu_{\rm B}$ [23], to be compared to the 2.4 $\mu_{\rm B}$ full Nd moment [17, 18]. In addition to this signal, below 800 mK, magnetic diffuse scattering spots appear around $(220)$, $(113)$ and symmetry related positions. Intensity on these positions increases with cooling until they transform into Bragg peaks below $T_{\rm N}$ (285 mK in this sample) characteristic of the AIAO phase. At low temperature, the corresponding ordered magnetic moment is $m_{2}=m_{\rm ord}=0.8\pm 0.05~{}\mu_{\rm B}$ (from diffraction measurements) and the magnetic contribution to the spin ice like diffuse scattering amounts to $m_{1}=2\pm 0.3~{}\mu_{\rm B}$ [23] (see Figure 1c). The moment embedded in the spin ice correlations is thus at maximum around $T_{\rm N}$ and slightly decreases at lower temperature. The diffuse scattering observed in the vicinity of the Bragg peak positions above $T_{\rm N}$ might arise from AIAO diffuse scattering just above the ordering transition, but could also be a signature of deconfined excitations, as proposed in Ref. 16. Figure 2: Spectral function $S(E)$ at different temperatures [23] measured at a wavelength $\lambda=$ 8.5 Å, hence an energy resolution of 20 $\mu$eV: (a) in Sample #1, integrated around ${\bf Q}=(0.8~{}0.8~{}0.8)$. The grey and red lines correspond to the fitted incoherent elastic $I_{c}$ and inelastic $S_{0}$ contributions respectively. (b) and (c): integrated over the measured ${\bf Q}$ range in Samples #2 (b) and #3 (c). To determine the spectral profile contained in those magnetic correlations, and especially the elastic or inelastic nature of the spin ice correlations associated to $m_{1}$, INS measurements have been carried out on the three aforementioned samples (see Table 1) as a function of temperature. To highlight the possible presence of an inelastic flat mode, the ${\bf Q}$-integrated spectral function $S(E)=\int d{\bf Q}S({\bf Q},E)$ was computed. As this quantity is akin to a density of states, it enhances the contribution of the flat modes contained in the spectrum. Figure 2 displays $S(E)$ at different temperatures. As previously shown [15], the inelastic flat band is clearly seen at low temperature. It is still visible at finite energy close to $T_{\rm N}$ (320 mK for Sample #2 and 315 mK for Sample #3) and above $T_{\rm N}$ (340 mK for Sample #1), yet broadens significantly upon warming. At the highest temperatures, the signal looks almost quasielastic. To obtain a quantitative insight into the temperature evolution of the mode, data were fitted for the three samples (as shown in Figure 2a for Sample #1) to the following model [23]: $S(E)=b+I_{c}(E)+F(E,T)\times\left[S_{0}(E)+S_{1}(E)\right]$ (3) $b$ is a flat background, $I_{c}(E)$ is a Gaussian function centered at zero energy to account for the elastic incoherent scattering. $F(E,T)=1+n(E,T)$ is the detailed balance factor ($n$ is the Bose-Einstein distribution). $S_{0}(E)$ and $S_{1}(E)$ are two Lorentzian profiles, centered on the energy $E_{0,1}$ and of intensity $I_{0,1}$, which represent respectively the flat band and the dispersive mode typical of the spin wave spectrum in Nd2Zr2O7. The determined positions $E_{0}$ and intensities $I_{0}$ are shown in Figure 3 as a function of the temperature normalized to $T_{\rm N}$ for the three samples. As anticipated from Figure 2, with increasing temperature, the band at $E_{0}$ softens and broadens while its intensity decreases. Nevertheless, $E_{0}$ is non-zero at $T_{\rm N}$ and a persistent dynamical behaviour is observed in all samples at and above $T_{\rm N}$, up to about $2T_{\rm N}$. Finally, the width of the features above the flat mode makes it hard to extract quantitative information from $S_{1}$. However, close examination of $S({\bf Q},E)$ measured for Sample #1 above $T_{\rm N}$ at 340 mK (see Figure 4) shows that, in all investigated directions, besides a strong quasielastic contribution (the inelastic mode being hardly discernible due to the energy resolution and the color scale), weak features are present close to the position of the low temperature dispersions. These spin wave fingerprints, highlighted by arrows on Figure 4(b) and which manifest as a broad signal in ${\bf Q}$-cuts (Figure 4(c-d)), are not compatible with the excitation spectrum expected in the presence of monopole creation and hopping [16]. Several striking features emerge from these measurements. INS experiments reveal that the intensity $I_{0}$ of the inelastic spin ice mode decreases when increasing temperature. Since D7 polarized experiments show that the full spin ice correlations, elastic and inelastic, are strongest around $T_{\rm N}$, the spin ice pattern observed above $T_{\rm N}$ must contain a new spin ice contribution, likely elastic, and different from the inelastic mode at $E_{0}$. This is confirmed by magnetization measurements, which point to ferromagnetic-like correlations, as expected for spin ice [23]. This elastic signal could not be directly identified in the elastic line of the IN5 data [23] certainly due to background issues, but we should stress that the D7 polarization analysis is definitely the most appropriate way to remove properly nuclear contributions and visualize small magnetic contributions. These results thus point to the coexistence of two spin ice-like contributions, an elastic and an inelastic one with different origins, and different temperature dependences. Figure 3: (a) $E_{0}$ and (b) $I_{0}$ as a function of reduced temperature $T/T_{\rm N}$ obtained from measurements on IN5 (dots - see Figure 2) and IN12 (triangles), together with results from Monte-Carlo (MC) calculations from Ref. 16 (red dots) [23]. Lines are guides to the eye. The large $I_{0}$ experimental value when $E_{0}=0$ is the signature of the persistent quasielastic contribution above $T_{\rm N}$. These two contributions can be understood as the manifestation of the strong competition at play between the pseudospin components of Nd. The negative value of $\tilde{\sf J}_{z}$ (see Table 1) promotes an AIAO phase built on ${\tilde{\tau}}^{\tilde{z}}$ while the positive $\tilde{\sf J}_{x}$ favors a Coulomb phase, similar to a spin ice phase, but built on ${\tilde{\tau}}^{\tilde{x}}$. For $\tilde{\sf J}_{x}/\|\tilde{\sf J}_{z}\|\approx 2$, the value determined for Nd2Zr2O7, the former is stabilized at low temperature and the latter at finite temperature, due to the large entropy associated to the Coulomb phase. In these two regimes, spin ice contributions are expected, an elastic one in the Coulomb phase at “high” temperature, and an inelastic one in the AIAO ordered phase (accompanied by dispersive excitations). Remarkably, the observable $\tau_{z}$, which corresponds to the magnetic dipolar moment along the local $\langle 111\rangle$ axes, is a combination of the ${\tilde{\tau}}^{\tilde{x}}$ and ${\tilde{\tau}}^{\tilde{z}}$ components of the pseudospin. It thus holds the two competing contributions (AIAO and Coulomb), which contrasts with the conventional spin ice case where the $z$ component carries elastic spin ice correlations only. Figure 4: INS spectra of Sample #1 along several high symmetry directions at 60 mK (a) and 340 mK (b), measured on IN5 with $\lambda$=6 Å. Red arrows highlight the dispersive modes and their fingerprints above $T_{\rm N}$. (c-d) Constant ${\bf Q}$-cuts at these two temperatures, integrated (c) along $(hh0)$ and (d) along $(hh2)$. The present results shed light on the manner in which the system evolves from the “high” temperature Coulomb phase to the low temperature AIAO ordered phase. At high temperature, around 1 K, the elastic spin ice signal characteristic of the ${\tilde{\tau}}^{\tilde{x}}$ Coulomb phase appears first. Upon cooling, the inelastic spin ice contribution along with dispersive spin wave branches emerge above $T_{\rm N}$ and coexist with the elastic one. They can naturally be considered as excitations stemming from the short-range AIAO correlations of the ${\tilde{\tau}}^{\tilde{z}}$ component observed below 800 mK (see Figure 1). The system enters the long-range AIAO ordered state at a temperature $T_{\rm N}\approx 300$ mK. It corresponds to about $\|\tilde{\sf J}_{x}\|/4$, thus to a temperature scale far above the one obtained theoretically for the stabilization of the quantum regime of spin ice, which is estimated to a few percents of the characteristic exchange interaction [34, 35]. This indicates that the Coulomb phase remains in its thermal regime down to $T_{\rm N}$. Surprisingly, the ordering temperature is larger than semi-classical Monte- Carlo calculations predictions [16]. At $T_{\rm N}$, the excitation spectrum is gapped, with the coexistence between the elastic spin ice component and the inelastic spectrum typical of AIAO ordering. The lack of a spinon continuum which would condense at $T_{\rm N}$ seems to preclude a transition driven by a Higgs mechanism. Deeper in the AIAO phase, the inelastic component - together with the Bragg peaks - develops at the expense of the elastic component. The weak maximum of the spin ice $m_{1}$ moment around $T_{\rm N}$ can thus be interpreted as due to the rise of the inelastic spin ice mode along with the persistence of the elastic contribution of the Coulomb phase. The coexistence of the elastic and inelastic signals is consistent with MC calculations [16], even if, close to $T_{\rm N}$, the two modes are less distinguishable in the experiments than in the calculations due to the strong broadening of the inelastic mode. Although some distribution is observed between the samples, the measured temperature dependence of the inelastic spin ice mode, described by the energy $E_{0}(T)$ and intensity $I_{0}(T)$, is also consistent with calculations [23], despite a slightly stronger inelastic component in experiments above $T_{\rm N}$ (see Figure 3). In summary, we find that with increasing temperature, the now well-established flat spin ice band characteristic of the AIAO ground state in Nd2Zr2O7, softens while its intensity decreases. The energy of this mode remains however finite at and above $T_{\rm N}$ and becomes overdamped with increasing the temperature further. At the same time, a new elastic spin ice component appears. The nature of the correlated phase above $T_{\rm N}$ is thus highly unconventional with the coexistence of an (elastic) Coulomb phase and fragmented excitations, resulting from the competition between the different terms of the Hamiltonian. Our observations support a picture where the AIAO ordering arises from a thermal spin ice phase, a scenario which is well accounted for by semi-classical MC calculations from Ref. 16, and is different from the proposed Higgs transition. When increasing the ratio $\tilde{\sf J}_{x}/\|\tilde{\sf J}_{z}\|$, reentrant behaviors are predicted [16] while the system approaches a quantum spin liquid ground state [31]. 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Spin dynamics and unconventional Coulomb phase in Nd2Zr2O7 Supplementary Material ## I Single crystal growth Single crystals of Nd2(Zr1-xTix)2O7 ($x=0$ and 0.025) were grown by the floating zone method, using a four-mirror xenon arc lamp optical image furnace [24, 25]. A summary of the conditions used for each crystal growth is given in Table S1. Crystal \| Sample label \| Lattice parameter \| Growth rate \| Growth atmosphere, \| Feed / seed ---\|---\|---\|---\|---\|--- \| Å \| (mm/h) \| pressure \| rotation rate (rpm) Nd2Zr2O7 \| Sample #1 \| $10.66\pm 0.02$ \| 12.5 \| Air, ambient \| 15 / 30 Nd2Zr2O7 \| Sample #2 \| $10.66\pm 0.04$ \| 15 \| Air, ambient \| 20 / 25 Nd2(Zr1-xTix)2O7 \| Sample #3 \| $10.65\pm 0.02$ \| 10 \| Air, ambient \| 15 / 5 Table S1: Summary of the samples with their crystal growth conditions. The lattice parameters were obtained at 6 K on neutron diffractometers (Sample #1 and #3) and triple axis spectrometers (all samples). Two different pure Nd2Zr2O7 samples had to be used in inelastic neutron scattering experiments, because the first one broke when warming up the dilution fridge after an experiment. ## II Characterization of the Ti substituted sample (Sample #3) Figure S1: (a) Refinement of the crystal neutron structure factors at 6 K, giving a refined Ti content equal to 2.4 %. (b) Measured intensity on the magnetic peaks (220), (113), (351) and (260), and symmetry related peaks at 60 mK, obtained from the difference with the 6 K data, and compared to the refined intensity. We have studied a substituted sample, in which a small content of Zr is replaced by Ti, slightly shrinking the structure. The nominal composition of the studied sample is 2.5 % of Ti atoms. As shown below, this substitution only slightly affects the magnetic properties of the magnetic Nd3+ sublattice and the low temperature properties are qualitatively the same. The value of the Ti content was refined by neutron diffraction, thanks to the significant contrast between Zr and Ti. A series of Bragg peak intensities was collected at 6 K on the single crystal neutron diffractometer D23 (CEA CRG- ILL). The data are in agreement with the pyrochlore structure ($Fd{\bar{3}}m$ space group), with a lattice parameter of 10.65 Å and the 48f oxygen atoms at the position $x_{\rm 48f}=0.336$. The Ti content is found to be 2.4 %. The Fullprof refinement [26] of the structure factor is shown on Fig. S1(a). The Néel temperature was determined from very low temperature magnetization measurements, and found to be $T_{\rm N}=375$ mK. The magnetic contribution raises below $T_{\rm N}$ in neutron diffraction measurements on top of the crystalline peaks. The Fullprof refinement below $T_{\rm N}$ confirms the same “all in - all out” (AIAO) magnetic structure as in the pure sample (Fig. S1(b)). At 60 mK, the refined ordered Nd3+ magnetic moment is $1.19\pm 0.03~{}\mu_{\rm B}$. ## III Measurements and analysis of polarized neutron scattering experiments Figure S2: D7 data and analysis. Panels (a,b,c) display respectively the raw data at 60 mK, the symmetrized and noise filtered data (10 K data have been subtracted). Panel (d) shows the spin ice magnetic scattering function calculated for a lattice containing 5488 spins. Panel (e) shows the same calculation assuming random $\pm 1$ Ising spins. It serves as a background reference, and is subtracted from (d), just as the 10 K data are subtracted from the low temperature data. In polarized neutron experiments carried out at D7 (ILL, France), we used the $P_{z}$ polarization mode, ${\bf z_{Q}}$ being the axis normal to the scattering plane and parallel to $[1\bar{1}0]$. We measured $N+I^{(z)}$ in the NSF channel, and $I^{(y)}$ in the SF channel. Here we use conventional notations: $N$ is the crystalline structure factor, while $I^{(y)}$ and $I^{(z)}$ denote the spin-spin correlation functions between spin components parallel to ${\bf y_{Q}}$ and ${\bf z_{Q}}$ respectively. The ${\bf y_{Q}}$ axis lies within the scattering plane, perpendicular both to ${\bf Q}$ and ${\bf z_{Q}}$. We used a wavelength $\lambda=4.85$ Å. The sample was rotated by steps of 1 degree, and 2 positions of the detector bank have been combined. Standard corrections (vanadium and quartz) have been processed. Finally, in order to eliminate any background contribution, the data recorded at 10 K have been subtracted from the data taken at lower temperatures. In Figure 1 of the main article, the data have been symmetrized (for the three lowest temperatures) while the noise was reduced by a mean filtering. This image processing based treatment tends to reduce the variation between one pixel and the next. The idea of mean filtering is to replace each pixel value with the average value of its neighbors, including itself. This has the effect of eliminating pixel values which are unrepresentative of their surroundings. For the sake of illustration, Figure S2(a-c) shows the different steps of this processing for the 60 mK data. Unfortunately, it was not possible to determine intensities in absolute units from the D7 measurements. To determine the magnetic moment responsible for the spin ice-like diffuse scattering, we had to proceed in an alternative manner. To this end, we carried out a series of calculations, assuming a “theoretical sample crystal” consisting of Ising spins (of length unity) located at the rare earth sites of a pyrochlore lattice of size $L$. We have considered $n$ spin ice configurations generated on this pyrochlore lattice (via a Monte- Carlo algorithm) and computed the average structure factor from the obtained magnetic moment ${\bf m}_{i,a=x,y,z}$ at each site $i$. The total magnetic neutron intensity, proportional to the spin-spin correlation, is calculated as: $I^{(y)}+I^{(z)}=\sum_{i,j}\sum_{a,b=x,y,z}m_{i,a}\left(\delta_{ab}-\frac{{\bf Q}_{a}{\bf Q}_{b}}{{\bf Q}^{2}}\right)m_{j,b}~{}e^{i{\bf Q}.({\bf R}_{i}-{\bf R}_{j})}$ and the intensity in the SF $P_{z}$ mode, corresponding to the data, is given by $\displaystyle I^{(y)}=\sum_{i,j}{\bf m}_{i}.{\bf y}_{\bf Q}\ {\bf m}_{j}.{\bf y}_{\bf Q}\ e^{i{\bf Q}.({\bf R}_{i}-{\bf R}_{j})}$. Noteworthy, the Monte-Carlo sampling was checked using the analytical method proposed by C.L. Henley [6]. The spin-spin correlation function (per unit cell) is written as: $I^{(y)}=4~{}t_{0}^{2}~{}\sum_{a=x,y,z}M^{T}_{a}\left[I-E(E^{+}E)^{-1}E^{+}\right]M_{a}$ $E$ is a 2-column matrix and $M_{a=x,y,z}$ is a collection of 1-column vectors containing the coordinates $a=x,y,z$ of the four magnetic moments ${\bf m}_{i}$ belonging to a given tetrahedron: $\displaystyle E$ $\displaystyle=\begin{pmatrix}e^{-i\pi{\bf Q.u_{1}}}&e^{i\pi{\bf Q.u_{1}}}\\\ e^{-i\pi{\bf Q.u_{2}}}&e^{i\pi{\bf Q.u_{2}}}\\\ e^{-i\pi{\bf Q.u_{3}}}&e^{i\pi{\bf Q.u_{3}}}\\\ e^{-i\pi{\bf Q.u_{4}}}&e^{i\pi{\bf Q.u_{4}}}\\\ \end{pmatrix}$ $\displaystyle M_{a}$ $\displaystyle=\begin{pmatrix}M_{1,a}\\\ M_{2,a}\\\ M_{3,a}\\\ M_{4,a}\end{pmatrix}\quad{\rm where}\quad M_{i,a}=m_{i,a}-\frac{{\bf m}_{i}.{\bf Q}}{Q^{2}}Q_{a}-\left(\sum_{b}\left(m_{i,b}-\frac{{\bf m}_{i}.{\bf Q}}{Q^{2}}Q_{b}\right).z_{{\bf Q},b}\right)z_{{\bf Q},a}\quad{\rm for}\quad{i=1,2,3,4}$ The four moments are defined as ${\bf m}_{1}=(c,c,c)$, ${\bf m}_{2}=(-c,-c,c)$, ${\bf m}_{3}=(-c,c,-c)$, ${\bf m}_{4}=(c,-c,-c)$ with $c=1/\sqrt{3}$ and attached to a tetrahedron. The ${\bf u_{i}}$ are vectors pointing towards the four corners of a tetrahedron: ${\bf u_{1}}=(d,d,d)$, ${\bf u_{2}}=(-d,-d,d)$, ${\bf u_{3}}=(-d,d,-d)$, ${\bf u_{4}}=(d,-d,-d)$ and $d=1/4$. Proper normalization condition imposes $t_{0}^{2}=2$. In the same way, we computed the structure factor $I^{(y)}_{\rm rd}$ assuming that the Ising spins have purely random $\pm 1$ values. This quantity was used as a background reference, just as the 10 K measurement was used as described above. We eventually considered the case where the spins are arranged in an “all in – all out” (AIAO) ordering, leading to magnetic Bragg peaks with a structure factor denoted hereafter $I^{(y)}_{\rm AIAO}$. First we calculated the integrated intensity around $(11\bar{3})$ from the experimental data at different temperatures, yielding $I^{\rm exp}_{\rm AIAO}(T)$. On the other hand, the same quantity was determined from $I^{(y)}_{\rm AIAO}$, yielding $I^{\rm c}_{\rm AIAO}$. Since the actual value of the ordered AIAO moment at low temperature $m_{\rm AIAO}$ (called $m_{2}$ in the main text) is precisely known from diffraction measurements (D23), we introduced the normalization factor $c$: $c=\frac{I^{\rm exp}_{\rm AIAO}}{m_{\rm AIAO}^{2}~{}I^{\rm c}_{\rm AIAO}}$ In a second step, we have computed the experimental integrated intensity within a box delineating the arm along $(hh{\bar{h}})$, yielding $\Delta I^{\rm exp}_{\rm arm}$. To obtain an estimate of the moment $m_{\rm SI}$ (called $m_{1}$ in the main text) involved in the spin ice component, which reflects the evolution seen on the maps presented in Figure 1 of the main text, we proposed to compare $\Delta I^{\rm exp}_{\rm arm}$ to $\Delta I^{\rm c}_{\rm arm}=m_{\rm SI}^{2}(I^{(y)}-I^{(y)}_{\rm rd})$. The estimation is then made quantitative by looking for $m_{\rm SI}$ such that : $\Delta I^{\rm exp}_{\rm arm}(T)=c\times\Delta I^{\rm c}_{\rm arm}$ The obtained values are listed in Table S2. Other calculation methods have been tested and lead to similar results in terms of absolute values and evolution with temperature. Furthermore, using $m_{\rm AIAO}=0.8\pm 0.05~{}\mu_{\rm B}$, the accuracy on $m_{\rm SI}$ is estimated to $\pm 0.3~{}\mu_{\rm B}$. This analysis confirms that the spin ice pattern has a weak maximum close to $T_{\rm N}$ and persists up to 600 mK, i.e. far above $T_{\rm N}$. $T$ (mK) \| $I^{\rm exp}_{\rm arm}(T)$ \| $m_{\rm SI}$ ($\mu_{\rm B}$) ---\|---\|--- 60 \| 0.027 \| 1.97 235 \| 0.035 \| 2.25 450 \| 0.030 \| 2.05 600 \| 0.020 \| 1.70 800 \| 0.015 \| 1.46 1000 \| 0.0095 \| 1.16 Table S2: Values of the spin moment $m_{\rm SI}$ vs temperature, determined from the procedure described in the text. Note that the temperature of 235 mK was estimated from the amplitude of the magnetic Bragg peaks (the thermometer indicated 300 mK). For 450 mK, no precise determination of the sample temperature could be done, but, in the absence of magnetic Bragg peaks, the temperature was definitely above $T_{\rm N}$. ## IV Time of flight inelastic scattering measurements Inelastic neutron scattering experiments were carried out on the IN5 disk chopper time of flight spectrometer (ILL, France). A good compromise between flux, energy resolution and accessible ${\bf Q}$ space was obtained with a wave length $\lambda=6$ or 6.5 Å. However, to ensure a better energy resolution $\Delta E=20~{}\mu$eV, necessary to fully resolve the dynamic spin ice mode at $E_{0}$, experiments were also conducted with $\lambda=8.5$ Å. The data were processed with the Mantid [27] and horace [28] softwares, transforming the recorded time of flight, sample rotation and scattering angle into energy transfer and ${\bf Q}$-wave vectors. The offset of the sample rotation was determined based on the Bragg peak positions. In all the experiments, the sample was rotated in steps of 1 degree and the counting time was about 10 minutes per sample position. It should be noticed that a very long thermalization time was systematically necessary to cool down the sample to the lowest temperature. In addition, we realized that when warming up from the lowest temperature, the sample temperature was not necessarily the same as the temperature indicated by the thermometer. For this reason, when possible (depending on the ratio between the resolution and the temperature), we have refined the “true” temperature by fitting the negative energy part of the spectra. It leads to the temperatures indicated on Figures 2, 3 and 4 of the main text, which are quite different from the thermometer temperatures. These temperatures are summarized in Table S3. Sample \| Thermometer \| Estimated ---\|---\|--- \| temperature \| temperature Sample #1 \| 450 mK \| $341\pm 100$ mK Sample #2 \| 60 mK \| $323\pm 78$ mK \| 300 mK \| $313\pm 68$ mK \| 450 mK \| $444\pm 111$ mK Sample #3 \| 275 mK \| $242\pm 35$ mK \| 350 mK \| $317\pm 38$ mK Table S3: Estimated effective temperatures in the different experiments performed on IN5. Figure S3: Constant ${\bf Q}$-cuts at two temperatures, the base temperature of 60 mK (blue) and above $T_{\rm N}$ (red) and which clearly show the vestiges of spin waves. Constant ${\bf Q}$-cuts from the data have been performed at the base temperature (typically 60 mK) and above $T_{\rm N}$ (340 mK), to clearly show the persistence of the spin wave signal above $T_{\rm N}$. These cuts, displayed in Figure S3 are along $(1,1,\ell)$, $(2,2,\ell)$, $(h,h,0)$ and $(h,h,2)$. Two of them are reproduced in the main text. This residual spin wave signal above $T_{\rm N}$ is not expected in conventional three dimensional paramagnets, in the absence of magnetic frustration. It thus would not be observed in a standard “all in – all out” antiferromagnet, which is predicted to behave classically close to the antiferromagnetic transition. The persistence of the spin wave signal (and of “all in – all out” diffuse scattering) quite far above $T_{\rm N}$ in Nd2Zr2O7 thus points out the unconventional nature of the magnetism in this compound and is likely related to the strong competition at play with the Coulomb phase observed above $T_{\rm N}$. ## V Inelastic scattering measurements on a triple axis spectrometer The temperature dependence of the spin dynamics in Sample #1 was also investigated on the cold TAS spectrometer IN12 (ILL, France). Scans at specific ${\bf Q}$ positions (0.5 0.5 2), (0 0 2.5) and (1.8 1.8 0) have been performed at different temperatures ranging from 50 up to 800 mK. Those positions were chosen since they probe different regions with respect to the dispersion. (0.5 0.5 2) essentially probes the flat spin ice band, (1.8 1.8 0) is sensitive to the zone boundary dispersive spin wave mode and (0.5 0.5 2) is somehow intermediate. A final wave vector $k_{f}=1.05$ Å-1 was used (in combination with nitrogen cooled Be filter) to ensure the best energy resolution, $\Delta E=50~{}\mu$eV. A magnetic field was also applied along $[1\bar{1}0]$. After correction from the detailed balance factor, we computed the difference between data taken a given temperature $T$ and the 800 mK data. Where applicable, we subtracted the data obtained at the same temperature but under a 1 T magnetic field. We could then extract the energy and intensity of the inelastic mode in the same way as for TOF measurements. The temperatures below $T_{\rm N}$ were estimated from the intensity of the (220) magnetic Bragg peak. ## VI Spin dynamics in the Ti substituted sample (Sample #3) ### VI.1 Determination of the parameters Inelastic neutron scattering data carried out at IN5 (ILL) on a single crystal sample show little evolution compared to the pure sample. The inelastic flat spin ice mode is observed at $E_{0}\approx 70~{}\mu$eV, while the dispersing mode stemming from the pinch point positions unfolds towards the zone centers, for instance $(220)$ or $(113)$. This is illustrated in Figure S4, which shows the dispersion along several reciprocal directions at 45 mK. Figure S4: Top: INS data taken at IN5 at 45 mK on the Sample #3 along high symmetry directions. Black and white dots are the energies $E_{0}$ and $E_{1}$ respectively, fitted according to the procedure described in the main text (see also equation (S1). Bottom: Spin wave calculations performed with the parameters given in Table 1 (main text). To determine the parameters of the XYZ Hamiltonian ${\cal H}_{\rm XYZ}=\sum_{\langle i,j\rangle}\left[{\tilde{\sf J}_{x}}\tilde{\tau}^{\tilde{x}}_{i}\tilde{\tau}^{\tilde{x}}_{j}+{\tilde{\sf J}_{y}}\tilde{\tau}^{\tilde{y}}_{i}\tilde{\tau}^{\tilde{y}}_{j}+{\tilde{\sf J}_{z}}\tilde{\tau}^{\tilde{z}}_{i}\tilde{\tau}^{\tilde{z}}_{j}\right]$ (see also equation (2) of the main text), we use analytic calculations giving the energy of the spin ice band [31]: $E_{0}=\sqrt{(3\|\tilde{{\sf J}}_{z}\|-\tilde{{\sf J}}_{x})(3\|\tilde{{\sf J}}_{z}\|-\tilde{{\sf J}}_{y})}$ as well as the energy of the dispersive modes at some high symmetry ${\bf Q}$ vectors [22]: $\displaystyle{\bf Q}=(110),(112)$ $\displaystyle,~{}$ $\displaystyle\Delta_{2}=\sqrt{(3\|\tilde{{\sf J}}_{z}\|+\tilde{{\sf J}}_{x})(3\|\tilde{{\sf J}}_{z}\|+\tilde{{\sf J}}_{y})}$ $\displaystyle{\bf Q}=(220),(113)$ $\displaystyle,~{}$ $\displaystyle\Delta_{3}=3\sqrt{(\|\tilde{{\sf J}}_{z}\|+\tilde{{\sf J}}_{x})(\|\tilde{{\sf J}}_{z}\|+\tilde{{\sf J}}_{y})}$ Simulations have then been performed to reproduce the data with the cefwave software developed at LLB using the Hamiltonian (1): ${\cal H}=\sum_{\langle i,j\rangle}\left[{\sf J}_{x}\tau^{x}_{i}\tau^{x}_{j}+{\sf J}_{y}\tau^{y}_{i}\tau^{y}_{j}+{\sf J}_{z}\tau^{z}_{i}\tau^{z}_{j}+{\sf J}_{xz}(\tau^{x}_{i}\tau^{z}_{j}+\tau^{z}_{i}\tau^{x}_{j})\right]$ The ground state configuration is first determined by solving this Hamiltonian at the mean field level, where the expectation values $\langle\tau^{x,y,z}_{j}\rangle$ are determined in a self-consistent manner. Spin wave calculations are performed using a generalized susceptibility approach out of the obtained configurations. Finally, the neutron cross section is calculated from $\tau^{z}_{i}\tau^{z}_{j}$ correlations. Notably, the simulations performed with the parameters of Table 1 (main text) reproduce quite well the data, as shown in Figure S4. ### VI.2 Temperature dependence of the spin dynamics Inelastic data in Sample #3 were fitted in the whole measured ${\bf Q}$ space using the model described in the main text: $S({\bf Q},E)=b+I_{c}(E)+F(E,T)\times\left[S_{0}(E)+S_{1}(E)\right]$ (S1) $b$ is a flat background (wavelength dependent), $I_{c}$ is a Gaussian function centered at zero energy to represent the elastic incoherent scattering. $F(E,T)=(1+n(E))$ is the detailed balance factor, and $S_{0}$ and $S_{1}$ are two Lorentzian profiles which represent respectively the flat band and the dispersive mode typical of the spin wave spectrum in Nd2Zr2O7. Figure S5 shows the energy $E_{0}$ of the flat band at different temperatures in the form of a map over the sector probed by TOF measurements. Figure S6 displays the intensities $I_{0}$ (panel a, upper row) and $I_{1}$ (panel b, lower row). The map on the right of the same figure shows the energy $E_{1}$ of the dispersive spin wave mode. To check the overall consistency of the fitting procedure, dashed lines visualize the directions of the scans reported in Figure S4. Note that for Sample #1, the fit was carried out at selected ${\bf Q}$ values (${\bf Q}=(0.8\ 0.8\ 0.8)$, $(1.1\ 1.1\ 1.1)$, $(1/2\ 1/2\ 1/2)$, $(1/2\ 1/2\ 3/2)$ and $(3/4\ 3/4\ 3/2)$). Figure S5: Temperature dependence of the flat spin ice band at $E_{0}$ deduced from the fit in Sample #3, as described in the main text. The portion of $({\bf Q},E)$ space corresponds to the sector probed by TOF measurements with $\lambda$=8.5 Å. Figure S6: (a) Temperature dependence of the intensity $I_{0}$ of the flat spin ice band from the fit in Sample #3, as described in the main text. (b) shows the intensity $I_{1}$ of the dispersive mode, and (c) shows its energy $E_{1}$. Dashed lines correspond to the directions of the cuts shown in Figure S4. The portion of $({\bf Q},E)$ space corresponds to the sector probed by TOF measurements with $\lambda=8.5$ Å. Figure S7: Temperature dependence of the intensity of the incoherent scattering $I_{c}(E=0)$ in Sample #3. The portion of $({\bf Q},E)$ space corresponds to the sector probed by TOF measurements with $\lambda=8.5$ Å. Finally, aiming at identifying a possible elastic contribution with the spin ice structure factor, Figure S7 shows the temperature evolution of the intensity of the incoherent elastic contribution, i.e. the dominant contribution $I_{c}(E=0)$ in the spectrum, to which the 45 mK map was subtracted. Within experimental uncertainties, these maps are featureless and no spin ice pattern can be clearly distinguished. ## VII Analysis of classical dynamics results Figure S8: (a) From Ref. 16 (courtesy of J. Xu): Evolution of the gapped flat mode for several temperatures (0.05, 0.1, 0.125, 0.15, 0.165, 0.175, 0.18, 0.185, 0.2 K) simulated using semi-classical molecular dynamics averaging over Q from (0.1 0.1 0) to (0.9 0.9 0). (b-c) Temperature dependences fitted from (a) (see equation S2): (b) Temperature dependence of the intensities $I_{0}$ and $I^{\prime}_{0}$ of the flat spin ice band and of the elastic contributions respectively. (c) Temperature dependence of the energy $E_{0}$ of the flat spin ice band. The main text of the present work compares the measured temperature dependence of $E_{0}$ and $I_{0}$ in our three samples with Monte Carlo calculations reported in Ref. 16. These calculations use effective exchange parameters from Ref. 22, which are detailed in Table 1 of the main text. They give a Néel temperature of 0.18 K. From these calculations, as illustrated in the Figure 3 of Ref. 16, two contributions are obtained: a spin ice elastic contribution which projects onto the ${\bf z}$ axis with a factor $\sin^{2}\theta$, as well as spin waves features characteristic of the AIAO phase, with especially a flat spin ice band. To compare quantitatively these results with our data, those theoretical curves have been fitted to two modes, following: $I(E)=I_{0}~{}e^{-4\log 2(\frac{E-E_{0})}{\delta_{0}})^{2}}+\frac{I^{\prime}_{0}}{1+(\frac{E}{\delta^{\prime}_{0}})^{2}}$ (S2) The result of this fit is illustrated in Figures S8(b) and S8(c), which display respectively the intensity of the modes ($I_{0}$ and $I^{\prime}_{0}$) and the position $E_{0}$ of the flat spin ice band. Interestingly, this energy remains finite even above the calculated critical temperature $T_{\rm N}=0.18$ K. ## VIII Correlations in magnetization measurements Figure S9: Magnetization $M$ vs $H/T$ measured for Sample #1 with the field applied along: (a) [100], (b) [110] and (c) [111] at 1, 1.8 and 4.2 K. In a paramagnet, isothermal magnetization curves scale as a function of the variable $H/T$. The deviations to this scaling give insight into the nature of the correlations that develop in the system. Upon cooling, if the magnetization curve increases faster (slower) than the higher temperature curve, it is the signature of the development of ferromagnetic-like (antiferromagnetic-like) correlations. We have plotted the magnetization as a function of $H/T$ for Nd2Zr2O7, measured in Sample #1. As shown in Figure S9, the $M(H/T)$ curves rise above the 4.2 K curves upon cooling down to 1 K, which indicates the development of ferromagnetic correlations, consistent with the elastic spin ice picture inferred from our neutron scattering measurements. At 500 mK, the curves lie between the 1 and 4.2 K curves (not shown on the figure for clarity), showing the development of antiferromagnetic correlations compared to 1 K, but the persistence of global ferromagnetic correlations. These antiferromagnetic correlations will end in the “all in – all out” ordering at about 300 mK.
# Growth of Sobolev norms in linear Schrödinger equations as a dispersive phenomenon A. Maspero111 International School for Advanced Studies (SISSA), Via Bonomea 265, 34136, Trieste, Italy Email<EMAIL_ADDRESS> ###### Abstract In this paper we consider linear, time dependent Schrödinger equations of the form ${\rm i}\partial_{t}\psi=K_{0}\psi+V(t)\psi$, where $K_{0}$ is a strictly positive selfadjoint operator with discrete spectrum and constant spectral gaps, and $V(t)$ a time periodic potential. We give sufficient conditions on $V(t)$ ensuring that $K_{0}+V(t)$ generates unbounded orbits. The main condition is that the resonant average of $V(t)$, namely the average with respect to the flow of $K_{0}$, has a nonempty absolutely continuous spectrum and fulfills a Mourre estimate. These conditions are stable under perturbations. The proof combines pseudodifferential normal form with dispersive estimates in the form of local energy decay. We apply our abstract construction to the Harmonic oscillator on ${\mathbb{R}}$ and to the half-wave equation on ${\mathbb{T}}$; in each case, we provide large classes of potentials which are transporters. ## 1 Introduction We consider the abstract linear Schrödinger equation ${\rm i}\partial_{t}\psi=K_{0}\psi+V(t)\psi\ $ (1.1) on a scale of Hilbert spaces ${\mathcal{H}}^{r}$; here $V(t)$ is a time $2\pi$-periodic potential and $K_{0}$ a selfadjoint, strictly positive operator with compact resolvent, pure point spectrum and constant spectral gaps. We prove some abstract results ensuring, $\forall r>0$, the existence of solutions $\psi(t)$ whose ${\mathcal{H}}^{r}$-norms grow polynomially fast, $\\|\psi(t)\\|_{r}\geq C_{r}\,\left\langle t\right\rangle^{r},\quad\forall t\gg 1\ ,$ whereas their ${\mathcal{H}}^{0}$-norms are constant for all times, $\\|\psi(t)\\|_{0}=\\|\psi(0)\\|_{0}$ $\forall t$. Here $\left\langle t\right\rangle:=\sqrt{1+t^{2}}$. These solutions therefore exhibit weak turbulent behavior in the form of energy cascade towards high frequencies. We apply our abstract result to two models: the Harmonic oscillator on ${\mathbb{R}}$ and the half-wave equation on ${\mathbb{T}}$. In both cases we exhibit large classes of potentials $V(t)$, bounded, smooth and periodic in time, so that the Hamiltonian $K_{0}+V(t)$ generates unbounded orbits. The phenomenon is purely perturbative: for $V=0$ each norm of each solution is constant for all times. So the central question is the existence of potentials able to transport energy to high-frequencies; we formalize this notion in the following definition: ###### Definition 1.1. We shall say that $V(t)$ is a transporter if $\forall r>0$ there exists a solution $\psi(t)\in{\mathcal{H}}^{r}$ of (1.1) with unbounded growth of norm, i.e. $\limsup_{t\to\infty}\\|\psi(t)\\|_{r}=\infty.$ (1.2) If this happens for every nonzero solution, we shall say that $V(t)$ is a universal transporter. Starting with the pioneering work of Bourgain [9], in the last few years there has been some efforts to construct both transporters [17, 22, 44] and universal transporters [6, 34] for different types of Schrödinger equations. All these papers provide explicit examples of potentials, constructed ad hoc for the problem at hand. The novelty of our result is that we identify sufficient, explicit and robust (i.e. stable under perturbations) conditions ensuring $V(t)$ to be a transporter. Precisely, its resonant average $\displaystyle\left\langle V\right\rangle:=\frac{1}{2\pi}\int_{0}^{2\pi}e^{{\rm i}sK_{0}}\,V(s)\,e^{-{\rm i}sK_{0}}\,{\rm d}s\ $ (1.3) must have nontrivial absolutely continuous spectrum in an interval, and over this interval it has to fulfill a Mourre estimate – see (2.7) below (actually we also require that both $K_{0}$ and $V(t)$ belong to some abstract graded algebra of pseudodifferential operators, as in [5]). The crucial point is that these conditions imply dispersive estimates for $\left\langle V\right\rangle$ of the form $\\|K_{0}^{-k}e^{-{\rm i}t\left\langle V\right\rangle}P_{c}\phi\\|_{0}\lesssim\left\langle t\right\rangle^{-k}\\|K_{0}^{k}\phi\\|_{0}\ ,\quad\forall t\in{\mathbb{R}},$ (1.4) where $P_{c}$ is a projection on a subset of the absolutely continuous spectral space of $\left\langle V\right\rangle$. A consequence of (1.4) is that we obtain solutions of ${\rm i}\partial_{t}\phi=\left\langle V\right\rangle\phi$ with decaying negative Sobolev norms and so, by duality, growing positive Sobolev norms. The fact that Mourre estimates imply dispersive estimates as above has origin from the work of Sigal-Soffer in quantum scattering theory [41] and it has been extended by many authors (see e.g. [42, 23, 31, 30, 24, 2]). See also the recent results [13, 12, 20] where similar dispersive properties are studied for pseudodifferential operators of order 0 on compact manifolds of dimension greater equal $2$. To explain the connection between the dynamics of (1.1) and the dispersive properties of the flow of $\left\langle V\right\rangle$, let us briefly describe the main ideas of the proof. The first step is to put system (1.1) into its resonant pseudodifferential normal form. This is the resonant variant of the normal form developed in [5] for non-resonant systems (and essentially an abstract version of the normal form of Delort [17]); it allows, $\forall N\in{\mathbb{N}}$, to conjugate equation (1.1) to ${\rm i}\partial_{t}{\varphi}=\big{(}K_{0}+Z^{(N)}(t)+R^{(N)}(t)\big{)}{\varphi}$ (1.5) where $Z^{(N)}(t)$ is a time dependent operator fulfilling ${\rm i}\partial_{t}Z^{(N)}(t)=[K_{0},Z^{(N)}(t)],\qquad Z^{(N)}(0)=\left\langle V\right\rangle+\mbox{lower order terms}$ (1.6) whereas $R^{(N)}(t)$ is an $N$-smoothing operator (it maps ${\mathcal{H}}^{r}\to{\mathcal{H}}^{r+N}$ continuously $\forall r$). The difference with the non-resonant case of [5] is that, in that paper, $Z^{(N)}(t)$ commutes with $K_{0}$. This is not anymore true in the resonant case we deal with; however (1.6) implies that $e^{{\rm i}tK_{0}}Z^{(N)}(t)e^{-{\rm i}tK_{0}}$ is time independent and thus coincides with $Z^{(N)}(0)$. Thus, conjugating (1.5) with $e^{-{\rm i}tK_{0}}$, we arrive at the equation ${\rm i}\partial_{t}\phi=\big{(}\left\langle V\right\rangle+T+R(t)\big{)}\phi$ (1.7) where $T:=Z^{(N)}(0)-\left\langle V\right\rangle$ is a time independent selfadjoint compact operator and $R(t)$ is $N$-smoothing. Then we analyze the dynamics of the truncated equation ${\rm i}\partial_{t}\phi=\big{(}\left\langle V\right\rangle+T\big{)}\phi$ (1.8) and prove that it has solutions with decaying negative Sobolev norms and so, by duality, growing positive Sobolev norms. This is the core of the proof; after this step, it is not difficult to construct a solution of the complete equation (1.7) exhibiting energy cascade, exploiting that $R(t)$ is regularizing. So let us concentrate on (1.8). The goal is to prove a dispersive estimate of the form (1.4) with $\left\langle V\right\rangle$ replaced by $\left\langle V\right\rangle+T$. However the point is delicate because the absolutely continuous spectrum of $\left\langle V\right\rangle$ (which exists by assumption) could be completely destroyed by adding $T$; a celebrated theorem by Weyl-von Neumann ensures that any selfadjoint operator (in a separable Hilbert space) can be perturbed by a (arbitrary small) compact selfadjoint operator so that its spectrum becomes pure point (see e.g. [32, pag. 525]). This is exactly the situation we want to avoid, as pure point spectrum prevents dispersive estimates. To get around this, we exploit that Mourre estimates are stable under pseudodifferential perturbations. This allows us to prove that $\left\langle V\right\rangle+T$ fulfills Mourre estimates and thus a dispersive estimate as (1.4). We also stress that fulfilling a Mourre estimate seems to be a quite general condition, and in the applications we exhibit large classes of operators which are transporters. For example, for the half wave equation we prove that any operator of the form $\cos(mt)v(x)$ with $v\in C^{\infty}({\mathbb{T}},{\mathbb{R}})$ and $m\in{\mathbb{Z}}$ is a transporter provided the $m$-th Fourier coefficient of $v(x)$ is not zero. Finally, the conditions we identity to be transporters are robust: if a potential $V(t)$ fulfills them, so does $V(t)+W(t)$ for any sufficiently small pseudodifferential operator $W(t)$. This shows that weakly turbulent phenomena induced by certain transporters are stable under perturbations. Up to our knowledge, this “stability of instability” is new in the literature and we consider it one of the main novelty of the paper. We conclude the introduction by reviewing the known results about existence of transporters for linear time dependent Schrödinger equations. As we already mentioned, the first result is due to Bourgain [9], who constructed a transporter for the Schrödinger equation on the torus; in this case $V(t,x)$ is a bounded real analytic function. Delort [17] constructs a transporter for the harmonic oscillator on ${\mathbb{R}}$, which is a time $2\pi$-periodic pseudodifferential operator of order zero. In [6] we proved that $ax\sin(t)$, $a>0$, is a universal transporter for the harmonic oscillator on ${\mathbb{R}}$; in this case the potential is an unbounded operator. In [34] we constructed universal transporters for the abstract equation (1.1), and applied the result to the harmonic oscillator on ${\mathbb{R}}$, the half-wave equation on ${\mathbb{T}}$ and on a Zoll manifold; in all cases the universal transporters are time periodic pseudodifferential operators of order 0. Finally recently Faou-Raphael [22] constructed a transporter for the harmonic oscillator on ${\mathbb{R}}$ which is a time dependent function (and not a pseudodifferential operator), and Thomann [44] has constructed a transporter for the harmonic oscillator on the Bargman-Fock space. Finally we recall the long-time growth result [25] for the semiclassical anharmonic oscillator on ${\mathbb{R}}^{d}$. Acknowledgments: We thank Matteo Gallone for helpful discussions on spectral theory and Dario Bambusi and Didier Robert for useful suggestions during the preparation of this work. ## 2 The abstract result We start with a Hilbert space ${\mathcal{H}}$, endowed with the scalar product $\left\langle\cdot,\cdot\right\rangle$, and a reference operator $K_{0}$, which we assume to be selfadjoint, positive, namely such that $\langle\psi;K_{0}\psi\rangle\geq c_{K_{0}}\\|\psi\\|^{2}\ ,\quad\forall\psi\in D(K_{0}^{1/2})\ ,\quad c_{K_{0}}>0\ ,$ and with compact resolvent. We define as usual a scale of Hilbert spaces by ${\mathcal{H}}^{r}:=D(K_{0}^{r})$ (the domain of the operator $K_{0}^{r}$) if $r\geq 0$, and ${\mathcal{H}}^{r}=({\mathcal{H}}^{-r})^{\prime}$ (the dual space) if $r<0$. Finally we denote by ${\mathcal{H}}^{-\infty}=\bigcup_{r\in{\mathbb{R}}}{\mathcal{H}}^{r}$ and ${\mathcal{H}}^{+\infty}=\bigcap_{r\in{\mathbb{R}}}{\mathcal{H}}^{r}$. We endow ${\mathcal{H}}^{r}$ with the natural norm $\\|\psi\\|_{r}:=\\|(K_{0})^{r}\psi\\|_{0}$, where $\\|\cdot\\|_{0}$ is the norm of ${\mathcal{H}}^{0}\equiv{\mathcal{H}}$. Notice that for any $m\in{\mathbb{R}}$, ${\mathcal{H}}^{+\infty}$ is a dense linear subspace of ${\mathcal{H}}^{m}$ (this is a consequence of the spectral decomposition of $K_{0}$). ###### Remark 2.1. By the very definition of ${\mathcal{H}}^{r}$, the unperturbed flow $e^{-{\rm i}tK_{0}}$ preserves each norm, $\\|e^{-{\rm i}tK_{0}}\psi\\|_{r}=\\|\psi\\|_{r}$ $\,\forall t\in{\mathbb{R}}$. Consequently, every orbit of equation (1.1) with $V(t)=0$ is bounded. Following [5], we introduce now a graded algebra ${\mathcal{A}}$ of operators which mimic some fundamental properties of different classes of pseudodifferential operators. For $m\in{\mathbb{R}}$ let ${\mathcal{A}}_{m}$ be a linear subspace of $\bigcap_{s\in{\mathbb{R}}}{\mathcal{L}}({\mathcal{H}}^{s},{\mathcal{H}}^{s-m})$ and define ${\mathcal{A}}:=\bigcup_{m\in{\mathbb{R}}}{\mathcal{A}}_{m}$. We notice that the space $\bigcap_{s\in{\mathbb{R}}}{\mathcal{L}}({\mathcal{H}}^{s},{\mathcal{H}}^{s-m})$ is a Fréchet space equipped with the semi-norms: $\\|A\\|_{m,s}:=\\|A\\|_{{\mathcal{L}}({\mathcal{H}}^{s},{\mathcal{H}}^{s-m})}$. We shall need to control the smoothing properties of the operators in the scale $\\{{\mathcal{H}}^{r}\\}_{r\in{\mathbb{R}}}$. If $A\in{\mathcal{A}}_{m}$ then $A$ is more and more smoothing if $m\rightarrow-\infty$ and the opposite as $m\rightarrow+\infty$. We will say that $A$ is of order $m$ if $A\in{\mathcal{A}}_{m}$. ###### Definition 2.2. We say that $S\in{\mathcal{L}}({\mathcal{H}}^{+\infty},{\mathcal{H}}^{-\infty})$ is $N$-smoothing if $\forall\kappa\in{\mathbb{R}}$, it can be extended to an operator in ${\mathcal{L}}({\mathcal{H}}^{\kappa},{\mathcal{H}}^{\kappa+N})$. When this is true for every $N\geq 0$, we say that $S$ is a smoothing operator. The first set of assumptions concerns the properties of ${\mathcal{A}}_{m}$: Assumption I: Pseudodifferential algebra * (i) For each $m\in{\mathbb{R}}$, $K_{0}^{m}\in{\mathcal{A}}_{m}$; in particular $K_{0}$ is an operator of order one. * (ii) For each $m\in{\mathbb{R}}$, ${\mathcal{A}}_{m}$ is a Fréchet space for a family of filtering semi-norms $\\{\wp^{m}_{j}\\}_{j\geq 0}$ such that the embedding ${\mathcal{A}}_{m}\hookrightarrow\bigcap_{s\in{\mathbb{R}}}{\mathcal{L}}({\mathcal{H}}^{s},{\mathcal{H}}^{s-m})$ is continuous222A family of seminorms $\\{\wp^{m}_{j}\\}_{j\geq 0}$ is called filtering if for any $j_{1},j_{2}\geq 0$ there exist $k\geq 0$ and $c_{1},c_{2}>0$ such that the two inequalities $\wp^{m}_{j_{1}}(A)\leq c_{1}\wp^{m}_{k}(A)$ and $\wp^{m}_{j_{2}}(A)\leq c_{2}\wp^{m}_{k}(A)$ hold for any $A\in{\mathcal{A}}_{m}$.. If $m^{\prime}\leq m$ then ${\mathcal{A}}_{m^{\prime}}\subseteq{\mathcal{A}}_{m}$ with a continuous embedding. * (iii) ${\mathcal{A}}$ is a graded algebra, i.e. $\forall m,n\in{\mathbb{R}}$: if $A\in{\mathcal{A}}_{m}$ and $B\in{\mathcal{A}}_{n}$ then $AB\in{\mathcal{A}}_{m+n}$ and the map $(A,B)\mapsto AB$ is continuous from ${\mathcal{A}}_{m}\times{\mathcal{A}}_{n}$ into ${\mathcal{A}}_{m+n}$. * (iv) ${\mathcal{A}}$ is a graded Lie-algebra333This property will impose the choice of the semi-norms $\\{\wp^{m}_{j}\\}_{j\geq 1}$. We will see in the examples that the natural choice $(\\|\cdot\\|_{m,s})_{s\geq 0}$ has to be refined. : if $A\in{\mathcal{A}}_{m}$ and $B\in{\mathcal{A}}_{n}$ then the commutator $[A,B]\in{\mathcal{A}}_{m+n-1}$ and the map $(A,B)\mapsto[A,B]$ is continuous from ${\mathcal{A}}_{m}\times{\mathcal{A}}_{n}$ into ${\mathcal{A}}_{m+n-1}$. * (v) ${\mathcal{A}}$ is closed under perturbation by smoothing operators in the following sense: let $A$ be a linear map: ${\mathcal{H}}^{+\infty}\rightarrow{\mathcal{H}}^{-\infty}$. If there exists $m\in{\mathbb{R}}$ such that for every $N>0$ we have a decomposition $A=A^{(N)}+S^{(N)}$, with $A^{(N)}\in{\mathcal{A}}_{m}$ and $S^{(N)}$ is $N$-smoothing, then $A\in{\mathcal{A}}_{m}$. * (vi) If $A\in{\mathcal{A}}_{m}$ then also the adjoint operator $A^{}\in{\mathcal{A}}_{m}$. The duality here is defined by the scalar product $\langle\cdot,\cdot\rangle$ of ${\mathcal{H}}={\mathcal{H}}^{0}$. The adjoint $A^{}$ is defined by $\langle u,Av\rangle=\langle A^{}u,v\rangle$ for $u,v\in{\mathcal{H}}^{\infty}$ and extended by continuity. It is well known that classes of pseudodifferential operators satisfy these properties, provided one chooses for $K_{0}$ a suitable operator of the right order (see e.g. [28]). ###### Remark 2.3. One has that $\forall A\in{\mathcal{A}}_{m}$, $\forall B\in{\mathcal{A}}_{n}$ $\displaystyle\forall m,s\quad\exists N\ s.t.\ $ $\displaystyle\\|A\\|_{m,s}\leq C_{1}\,\wp^{m}_{N}(A)\ ,$ (2.1) $\displaystyle\forall m,n,j\quad\exists N\ s.t.\ $ $\displaystyle\wp^{m+n}_{j}(AB)\leq C_{2}\,\wp^{m}_{N}(A)\,\wp^{n}_{N}(B)\ ,$ (2.2) $\displaystyle\forall m,n,j\quad\exists N\ s.t.\ $ $\displaystyle\wp^{m+n-1}_{j}([A,B])\leq C_{3}\,\wp^{m}_{N}(A)\,\wp^{n}_{N}(B)\ ,$ (2.3) for some positive constants $C_{1}(s,m)$, $C_{2}(m,n,j)$, $C_{3}(m,n,j)$. ###### Remark 2.4. Any $A\in{\mathcal{A}}_{m}$ with $m<0$ is a compact operator on ${\mathcal{H}}$. Indeed write $A=AK_{0}^{-m}\,K_{0}^{m}$. Then $AK_{0}^{-m}\in{\mathcal{A}}_{0}$ is a bounded operator on ${\mathcal{H}}$ (Assumption I (i)–(iii)), whereas $K_{0}^{m}\equiv(K_{0}^{-1})^{-m}$ is compact on ${\mathcal{H}}$, as $K_{0}^{-1}$ is a compact operator by assumption. For $\Omega\subseteq{\mathbb{R}}^{d}$ and ${\mathcal{F}}$ a Fréchet space, we will denote by $C_{b}^{m}(\Omega,{\mathcal{F}})$ the space of $C^{m}$ maps $f:\Omega\ni x\mapsto f(x)\in{\mathcal{F}}$ such that, for every seminorm $\\|\cdot\\|_{j}$ of ${\mathcal{F}}$, one has $\sup_{x\in\Omega}\\|\partial_{x}^{\alpha}f(x)\\|_{j}<+\infty\ ,\quad\forall\alpha\in{\mathbb{N}}^{d}\ :\ \left\|\alpha\right\|\leq m\ .$ (2.4) If (2.4) is true $\forall m$, we say $f\in C^{\infty}_{b}(\Omega,{\mathcal{F}})$. Similarly we denote by $C^{\infty}({\mathbb{T}},{\mathcal{F}})$ the space of smooth maps from the torus ${\mathbb{T}}={\mathbb{R}}/(2\pi{\mathbb{Z}})$ to the Fréchet space ${\mathcal{F}}$. The second set of assumptions concerns the operator $K_{0}$, its spectral structure and an Egorov-like property, also well known for pseudo-differential operators. Assumption II: Properties of $K_{0}$ (i) The operator $K_{0}$ has purely discrete spectrum fulfilling ${\rm spec}(K_{0})\subseteq{\mathbb{N}}+\lambda$ (2.5) for some $\lambda\geq 0$. * (ii) For any $m\in{\mathbb{R}}$ and $A\in{\mathcal{A}}_{m}$, the map defined on ${\mathbb{R}}$ by $\tau\mapsto A(\tau):={\rm e}^{{\rm i}\tau K_{0}}\,A\,{\rm e}^{-{\rm i}\tau K_{0}}$ belongs to $C^{\infty}_{b}({\mathbb{R}},{\mathcal{A}}_{m})$ and one has $\forall j\quad\exists N\ s.t.\ \ \sup_{\tau\in{\mathbb{R}}}\wp^{m}_{j}(A(\tau))\leq C_{4}\,\wp^{m}_{N}(A)$ (2.6) for some positive constant $C_{4}(m,j)$. ###### Remark 2.5. Assumption II (i) guarantees that $e^{{\rm i}2\pi K_{0}}=e^{{\rm i}2\pi\lambda}$. As a consequence, for any operator $V$, the map $\tau\mapsto e^{{\rm i}\tau K_{0}}Ve^{-{\rm i}\tau K_{0}}$ is $2\pi$-periodic. We denote by $C^{\infty}_{c}({\mathbb{R}}^{d},{\mathbb{R}}_{\geq 0})$ the set of smooth functions with compact support from ${\mathbb{R}}^{d}$ to ${\mathbb{R}}_{\geq 0}$ (hence non-negative). Furthermore from now on, given two operators $\mathsf{A},\mathsf{B}\in{\mathcal{L}}({\mathcal{H}})$, we write $\mathsf{A}\leq\mathsf{B}$ with the meaning $\left\langle\mathsf{A}{\varphi},{\varphi}\right\rangle\leq\left\langle\mathsf{B}{\varphi},{\varphi}\right\rangle$ $\,\forall{\varphi}\in{\mathcal{H}}$. The last set of assumptions concerns the resonant average $\left\langle V\right\rangle$ of the potential $V(t)$ (see (1.3)) and its spectrum $\sigma(\left\langle V\right\rangle)$. Note that if $V(t)$ is selfadjoint $\forall t$, so is $\left\langle V\right\rangle$. Assumption III: Properties of the potential $V(t)$ The operator $V\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{0})$, $V(t)$ selfadjoint $\forall t$, and its resonant average $\left\langle V\right\rangle$ fulfills: * (i) There exists an interval $I_{0}\subset{\mathbb{R}}$ such that $\left\|\sigma(\left\langle V\right\rangle)\cap I_{0}\right\|>0$; here $\left\|\cdot\right\|$ denotes the Lebesgue measure. * (ii) Mourre estimate over $I_{0}$: there exist a selfadjoint operator $A\in{\mathcal{A}}_{1}$ and a function $g_{I_{0}}\in C_{c}^{\infty}({\mathbb{R}},{\mathbb{R}}_{\geq 0})$ with $g_{I_{0}}\equiv 1$ on $I_{0}$ such that $g_{I_{0}}(\left\langle V\right\rangle)\,{\rm i}[\left\langle V\right\rangle,A]\,g_{I_{0}}(\left\langle V\right\rangle)\geq\theta\,g_{I_{0}}(\left\langle V\right\rangle)^{2}+K$ (2.7) for some $\theta>0$ and $K$ a selfadjoint compact operator. The operator $g_{I_{0}}(\left\langle V\right\rangle)$ above is defined via functional calculus, see Appendix B. Following the literature, we shall say that $\left\langle V\right\rangle$ is conjugated to $A$ over $I_{0}$. ###### Remark 2.6. By Mourre theory [36] $\left\langle V\right\rangle$ has, in the interval $I_{0}$, a nontrivial absolutely continuous spectrum with finitely many eigenvalues of finite multiplicity and no singular continuous spectrum. In general one cannot exclude the existence of embedded eigenvalues in the absolutely continuous spectrum.444 For example consider $H\in{\mathcal{L}}(L^{2}({\mathbb{T}}))$ given by $(Hu)(x):=\cos(x)u(x)+\delta(1-\delta^{-1}\cos(x))\frac{1}{2\pi}\int_{\mathbb{T}}u(x)\big{(}1-\delta^{-1}\cos(x)\big{)}\,{\rm d}x\ ,\quad\delta\in\left(-\frac{1}{2},\frac{1}{2}\right)\setminus\\{0\\}\ .$ $H$ is selfadjoint, a 1-rank perturbation of the multiplication operator by $\cos(x)$, it has absolutely continuous spectrum in the interval $(-1,1)$, and $\delta$ is an embedded eigenvalue with eigenvector $u(x)\equiv 1$. Moreover $H$ is conjugated to $\sin(x)\frac{\partial_{x}}{{\rm i}}+\frac{\partial_{x}}{{\rm i}}\sin(x)$ over $[-\frac{1}{2},\frac{1}{2}]$. We are ready to state our main results. The first says that, under the set of assumptions above, $V(t)$ is a transporter in the sense of Definition 1.1: ###### Theorem 2.7. Assume that ${\mathcal{A}}$ is a graded algebra as in Assumption I, and that $K_{0}$ and $V(t)\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{0})$ satisfy Assumptions II and III. Then $V(t)$ is a transporter for the equation ${\rm i}\partial_{t}\psi=(K_{0}+V(t))\psi\ .$ (2.8) More precisely, for any $r>0$ there exist a solution $\psi(t)$ of (2.8) in ${\mathcal{H}}^{r}$ and constants $C,T>0$ such that $\\|\psi(t)\\|_{r}\geq C\left\langle t\right\rangle^{r},\quad\forall t\geq T\ .$ (2.9) We also prove a stronger result: namely not only $V(t)$ is a transporter, but also any operator sufficiently close to it (in the ${\mathcal{A}}_{0}$-topology). Here the precise statement: ###### Theorem 2.8. With the same assumptions of Theorem 2.7, there exist $\epsilon_{0}>0$ and ${\mathtt{M}}\in{\mathbb{N}}$ such that for any $W\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{0})$, $W(t)$ selfadjoint $\forall t$, fulfilling $\sup_{t\in{\mathbb{T}}}\,\wp^{0}_{\mathtt{M}}(W(t))\leq\epsilon_{0},$ (2.10) then $V(t)+W(t)$ is a transporter for the equation ${\rm i}\partial_{t}\psi=\big{(}K_{0}+V(t)+W(t)\big{)}\psi\ .$ (2.11) More precisely, for any $r>0$ there exist a solution $\psi(t)$ in ${\mathcal{H}}^{r}$ of (2.11) and constants $C,T>0$ such that $\\|\psi(t)\\|_{r}\geq C\left\langle t\right\rangle^{r},\quad\forall t\geq T\ .$ (2.12) Let us comment the above results. 1. 1. The growth of Sobolev norms of Theorem 2.7 is truly an energy cascade phenomenon; indeed the ${\mathcal{H}}^{0}$-norm of any solution of (2.8) is preserved for all times, $\\|\psi(t)\\|_{0}=\\|\psi(0)\\|_{0}$, $\,\forall t\in{\mathbb{R}}$. This is due to the selfadjointness of $K_{0}+V(t)$ (the same happens to solutions of (2.11)). 2. 2. Estimates (2.9), (2.12) provide optimal lower bounds for the speed of growth of the Sobolev norms. Indeed we proved [33] that, under the assumptions above555 in particular the fact that $[K_{0},V(t)]$ and $[K_{0},V(t)+W(t)]$ are uniformly (in $t$) bounded operators on the scale ${\mathcal{H}}^{r}$, any solution of (2.8) or (2.11) fulfills the upper bounds $\forall r>0\ \ \ \exists\,{\widetilde{C}}_{r}>0\colon\quad\\|\psi(t)\\|_{r}\leq{\widetilde{C}}_{r}\left\langle t\right\rangle^{r}\,\\|\psi(0)\\|_{r}.$ Thus, Theorems 2.7, 2.8 construct unbounded solutions with optimal growth. 3. 3. Theorem 2.8 proves robustness of certain type of transporters under small pseudodifferential perturbations. This shows a sort of “stability of instability”, which, up to our knowledge, is new in this context. 4. 4. Actually there are infinitely many distinct solutions undergoing growth of Sobolev norms. Their initial data are constructed in a unique way starting from functions belonging to the absolutely continuous spectral subspace of the operator $\left\langle V\right\rangle$. We describe such initial data in Corollary 3.16. 5. 5. Energy cascade is a resonant phenomenon; here it happens because $V(t)$ oscillates at frequency $\omega=1$ which resonates with the spectral gaps of $K_{0}$. In [5] we proved that if $V(t)\equiv\mathsf{V}(\omega t)$ is quasiperiodic in time with a frequency vector $\omega\in{\mathbb{R}}^{n}$ fulfilling the non-resonant condition $\exists\gamma,\tau>0\colon\quad\left\|\ell+\omega\cdot k\right\|\geq\frac{\gamma}{\left\langle k\right\rangle^{\tau}}\quad\forall\ell,k\in{\mathbb{Z}}\times{\mathbb{Z}}^{n}\setminus\\{0\\}$ (which is violated if $V(t)$ is $2\pi$-periodic) then the Sobolev norms grow at most as $\left\langle t\right\rangle^{\epsilon}$ $\forall\epsilon>0$. The $\left\langle t\right\rangle^{\epsilon}$-speed of growth is also known for systems with increasing [37, 33, 5] or shrinking [21, 35] spectral gaps and for Schrödinger equation on ${\mathbb{T}}^{d}$ with bounded [10, 16, 8] and even unbounded [7] potentials. 6. 6. In concrete models one can typically prove that if $V(t)$ is sufficiently small in size and oscillates in time with a strongly non resonant frequency $\omega$ (typically belonging to some Cantor set of large measure), then all solutions have uniformly in time bounded Sobolev norms. Therefore the stability/instability of the system depends only on the resonance property of the frequency $\omega$. We mention just the recent results [4, 6] which deal with the harmonic oscillator (as we consider it in the applications) and refer to those papers for a complete bibliography. 7. 7. The most delicate assumption to verify is (2.7). In the applications, one can try to construct an escape function for the principal symbol $\left\langle v\right\rangle$ of $\left\langle V\right\rangle$. This means to find a symbol $a(x,\xi)$ of order 1 such that the Poisson bracket $\\{\left\langle v\right\rangle,a\\}$ is strictly positive in some energy levels: $\exists c>0\colon\quad\\{\left\langle v\right\rangle,a\\}\geq c\qquad\mbox{ in }\\{(x,\xi):\ \ \left\|\left\langle v\right\rangle(x,\xi)-\lambda\right\|\leq\delta\\}\ .$ Then symbolic calculus and sharp Gårding inequality imply that (2.7) holds in the interval $I=(\lambda-\delta,\lambda+\delta)$; see [13] Section 6.2 for details. We finally note that the second theorem is stronger than the first one and implies it in the special case $W(t)\equiv 0$. However we think that the statement of Theorem 2.7 is clear and useful in the applications (see e.g. Section 4), so we decided to state it on its own. Having said so, in the sequel we shall only prove Theorem 2.8. ## 3 Proof of the abstract result As already mentioned, we shall only prove Theorem 2.8. The proof is divided in three steps; in the first one we put system (2.11) in its resonant pseudodifferential normal form. In the second one we analyze the dynamics of the effective Hamiltonian and prove the existence of solutions with decaying negative Sobolev norms. The final step is to construct a solution of the complete equation exhibiting growth of Sobolev norms. ### 3.1 Resonant pseudodifferential normal form The goal of this section is to put system (2.11) into its resonant pseudodifferential normal form up to an arbitrary $N$-smoothing operator. In this first step we shall only require Assumptions I and II. It is slightly more convenient to deal with the equation ${\rm i}\partial_{t}\psi=\big{(}K_{0}+\mathsf{V}(t)\big{)}\psi$ (3.1) and then to specify the result for $\mathsf{V}(t)=V(t)+W(t)$ as in (2.11). Given $\mathsf{V}\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{m})$, $m\in{\mathbb{R}}$, we define the averaged operator $\displaystyle{\widehat{\mathsf{V}}}(t):=\frac{1}{2\pi}\int_{0}^{2\pi}e^{{\rm i}sK_{0}}\,\mathsf{V}(t+s)\,e^{-{\rm i}sK_{0}}\,{\rm d}s\ .$ (3.2) We shall prove below that ${\widehat{\mathsf{V}}}(t)\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{m})$ (see Lemma 3.2). ###### Proposition 3.1 (Resonant pseudodifferential normal form). Consider equation (3.1) with $\mathsf{V}\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{0})$, $\mathsf{V}(t)$ selfadjoint $\forall t$. There exists a sequence $\\{X_{j}(t)\\}_{j\geq 1}$ of selfadjoint (time-dependent) operators in ${\mathcal{H}}$ with $X_{j}\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{1-j})$ and fulfilling $\forall r\in{\mathbb{R}},\ \exists c_{r,j},C_{r,j}>0\colon\qquad c_{r,j}\\|{\varphi}\\|_{r}\leq\\|e^{\pm{\rm i}X_{j}(t)}{\varphi}\\|_{r}\leq C_{r,j}\\|{\varphi}\\|_{r},\qquad\forall t\in{\mathbb{R}},$ (3.3) such that the following holds true. For any $N\geq 1$, the change of variables $\psi=e^{-{\rm i}X_{1}(t)}\cdots e^{-{\rm i}X_{N}(t)}{\varphi}$ (3.4) transforms (3.1) into the equation ${\rm i}\partial_{t}{\varphi}=\big{(}K_{0}+Z^{(N)}(t)+\mathsf{V}^{(N)}(t)\big{)}{\varphi}\ ;$ (3.5) here $\mathsf{V}^{(N)}\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{-N})$ whereas $Z^{(N)}\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{0})$, it is selfadjoint $\forall t$, it fulfills ${\rm i}\partial_{t}Z^{(N)}(t)=[K_{0},Z^{(N)}(t)]$ (3.6) and it has the expansion $Z^{(N)}(t)={\widehat{\mathsf{V}}}(t)+T^{(N)}(t),\qquad T^{(N)}\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{-1})\ .$ (3.7) Here ${\widehat{\mathsf{V}}}(t)$ is the averaged operator defined in (3.2). In order to prove the proposition we start with some preliminary results. The first regards the properties of ${\widehat{\mathsf{V}}}(t)$. ###### Lemma 3.2. Let $\mathsf{V}\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{m})$, $m\in{\mathbb{R}}$, $\mathsf{V}(t)$ selfadjoint $\forall t$. Then the following holds true. * (i) ${\widehat{\mathsf{V}}}\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{m})$, it is selfadjoint $\forall t$, it commutes with ${\rm i}\partial_{t}-K_{0}$, i.e. ${\rm i}\partial_{t}{\widehat{\mathsf{V}}}(t)=[K_{0},{\widehat{\mathsf{V}}}(t)]$ and $\forall j,\ell\geq 0\quad\exists\,M\in{\mathbb{N}},\ C>0\ \ {\rm s.t.}\ \ \sup_{t\in{\mathbb{T}}}\wp^{m}_{j}(\partial_{t}^{\ell}{\widehat{\mathsf{V}}}(t))\leq C\,\sup_{t\in{\mathbb{T}}}\ \wp^{m}_{M}(\mathsf{V}(t))\ .$ (3.8) * (ii) The resonant averaged operator $\left\langle\mathsf{V}\right\rangle$, defined in (1.3), belongs to ${\mathcal{A}}_{m}$, it is selfadjoint and $\forall j\geq 0\quad\exists\,M\in{\mathbb{N}},\ C>0\ \ {\rm s.t.}\ \ \wp^{m}_{j}(\left\langle\mathsf{V}\right\rangle)\leq C\,\sup_{t\in{\mathbb{T}}}\ \wp^{m}_{M}(\mathsf{V}(t))\ .$ (3.9) * (iii) One has the chain of identities ${\widehat{\mathsf{V}}}(0)=\left\langle\mathsf{V}\right\rangle=e^{{\rm i}tK_{0}}\,{\widehat{\mathsf{V}}}(t)\,e^{-{\rm i}tK_{0}}=\langle\,{\widehat{\mathsf{V}}}\,\rangle,\qquad\forall t\in{\mathbb{R}}\ .$ (3.10) ###### Proof. $(i)$ The properties ${\widehat{\mathsf{V}}}\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{m})$ and ${\widehat{\mathsf{V}}}(t)$ selfadjoint $\forall t$ follow from Assumption II and the fact that $\mathsf{V}(t)$ is $2\pi$-periodic in $t$ and selfadjoint $\forall t$. Let us prove it commutes with ${\rm i}\partial_{t}-K_{0}$. Using $\partial_{s}\left(e^{{\rm i}sK_{0}}\,\mathsf{V}(t+s)\,e^{-{\rm i}sK_{0}}\right)=e^{{\rm i}sK_{0}}\big{(}{\rm i}[K_{0},\mathsf{V}(t+s)]+\partial_{s}\mathsf{V}(t+s)\big{)}e^{-{\rm i}sK_{0}}$ we get $\displaystyle\partial_{t}{\widehat{\mathsf{V}}}(t)$ $\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}e^{{\rm i}sK_{0}}\,\partial_{t}\mathsf{V}(t+s)\,e^{-{\rm i}sK_{0}}\,{\rm d}s=\frac{1}{2\pi}\int_{0}^{2\pi}e^{{\rm i}sK_{0}}\,\partial_{s}\mathsf{V}(t+s)\,e^{-{\rm i}sK_{0}}\,{\rm d}s$ $\displaystyle=\frac{1}{2\pi{\rm i}}\int_{0}^{2\pi}e^{{\rm i}sK_{0}}\,[K_{0},\mathsf{V}(t+s)]\,e^{-{\rm i}sK_{0}}\,{\rm d}s={\rm i}^{-1}\,[K_{0},{\widehat{\mathsf{V}}}(t)]$ where in the second line we used the periodicity of $s\mapsto e^{{\rm i}sK_{0}}\,\mathsf{V}(t+s)\,e^{-{\rm i}sK_{0}}$ (see Remark 2.5) to remove the boundary terms. Estimate (3.8) for $\ell=0$ follows from Assumption II. For $\ell\geq 1$ we use induction: assume (3.8) is true up to a certain $\ell$; using $\partial_{t}^{\ell+1}{\widehat{\mathsf{V}}}(t)=-{\rm i}\partial_{t}^{\ell}[K_{0},{\widehat{\mathsf{V}}}(t)]=-{\rm i}[K_{0},\partial_{t}^{\ell}{\widehat{\mathsf{V}}}(t)]$, we get $\forall j\in{\mathbb{N}}$ $\wp_{j}^{m}(\partial_{t}^{\ell+1}{\widehat{\mathsf{V}}}(t))\leq\ \wp_{j}^{m}([K_{0},\partial_{t}^{\ell}{\widehat{\mathsf{V}}}(t)])\leq C\wp_{j_{1}}^{m}(\partial_{t}^{\ell}{\widehat{\mathsf{V}}}(t))\leq C\wp_{j_{2}}^{m}(\mathsf{V}(t))$ using also the inductive assumption. This proves (3.8). $(ii)$ It is clear that $\left\langle\mathsf{V}\right\rangle$ is time independent, selfadjoint and in ${\mathcal{A}}_{m}$ by Assumption II. Estimate (3.9) follows from Assumption II. $(iii)$ Clearly ${\widehat{\mathsf{V}}}(0)=\left\langle\mathsf{V}\right\rangle$. Then, as the map $\tau\mapsto e^{{\rm i}\tau K_{0}}\,\mathsf{V}(\tau)\,e^{-{\rm i}\tau K_{0}}$ is $2\pi$-periodic, one has $\forall t\in{\mathbb{R}}$ $e^{{\rm i}tK_{0}}\,{\widehat{\mathsf{V}}}(t)\,e^{-{\rm i}tK_{0}}=\frac{1}{2\pi}\int_{0}^{2\pi}e^{{\rm i}(t+s)K_{0}}\,\mathsf{V}(t+s)\,e^{-{\rm i}(s+t)K_{0}}\,{\rm d}s=\left\langle\mathsf{V}\right\rangle\ .$ Finally, exploiting this last identity, one has $\langle\,{\widehat{\mathsf{V}}}\,\rangle=\frac{1}{2\pi}\int_{0}^{2\pi}e^{{\rm i}tK_{0}}\,{\widehat{\mathsf{V}}}(t)\,e^{-{\rm i}tK_{0}}{\rm d}t=\frac{1}{2\pi}\int_{0}^{2\pi}\left\langle\mathsf{V}\right\rangle{\rm d}t=\left\langle\mathsf{V}\right\rangle$ which completes the proof of (3.10). ∎ The second preliminary result regards how to solve the homological equations which appear during the normal form procedure. More precisely we look for a time periodic operator $X(t)$ solving the homological equation $\partial_{t}X(t)+{\rm i}[K_{0},X(t)]=\mathsf{V}(t)-{\widehat{\mathsf{V}}}(t),$ (3.11) where ${\widehat{\mathsf{V}}}(t)$ is the averaged operator defined in (3.2). This is done in the next lemma. ###### Lemma 3.3. Let $\mathsf{V}\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{m})$, $m\in{\mathbb{R}}$, $\mathsf{V}(t)$ selfadjoint $\forall t$. The homological equation (3.11) has a solution $X\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{m})$ and $X(t)$ is selfadjoint $\forall t$. ###### Proof. We look for a solution of (3.11) using the method of variation of constants. In particular we take $X(t)=e^{-{\rm i}tK_{0}}\,Y(t)\,e^{{\rm i}tK_{0}}$ for some $Y\in C^{\infty}({\mathbb{R}},{\mathcal{A}}_{m})$ with $Y(0)=0$ to be determined. Then $X$ solves (3.11) provided $\partial_{t}Y(t)=e^{{\rm i}tK_{0}}\,(\mathsf{V}(t)-{\widehat{\mathsf{V}}}(t))\,e^{-{\rm i}tK_{0}}$, giving $Y(t)=\int_{0}^{t}e^{{\rm i}sK_{0}}\big{(}\mathsf{V}(s)-{\widehat{\mathsf{V}}}(s)\big{)}\,e^{-{\rm i}sK_{0}}\,{\rm d}s.$ By Lemma 3.2 and Assumption II, $Y\in C^{\infty}({\mathbb{R}},{\mathcal{A}}_{m})$ and it is selfadjoint $\forall t$. Therefore one gets $X(t)=\int_{0}^{t}e^{{\rm i}(s-t)K_{0}}\big{(}\mathsf{V}(s)-{\widehat{\mathsf{V}}}(s)\big{)}\,e^{-{\rm i}(s-t)K_{0}}\,{\rm d}s.$ Again $X\in C^{\infty}({\mathbb{R}},{\mathcal{A}}_{m})$ and it is selfadjoint $\forall t$. Finally (recall Remark 2.5) $\displaystyle X(t+2\pi)-X(t)$ $\displaystyle=\int_{t}^{t+2\pi}e^{{\rm i}(s-t)K_{0}}\,\big{(}\mathsf{V}(s)-{\widehat{\mathsf{V}}}(s)\big{)}\,e^{-{\rm i}(s-t)K_{0}}\,{\rm d}s$ $\displaystyle=e^{-{\rm i}tK_{0}}\int_{0}^{2\pi}e^{{\rm i}sK_{0}}\,\big{(}\mathsf{V}(s)-{\widehat{\mathsf{V}}}(s)\big{)}\,e^{-{\rm i}sK_{0}}\,{\rm d}s\ e^{{\rm i}tK_{0}}$ $\displaystyle=2\pi e^{-{\rm i}tK_{0}}\big{(}\left\langle\mathsf{V}\right\rangle-\langle\,{\widehat{\mathsf{V}}}\,\rangle\big{)}e^{{\rm i}tK_{0}}\stackrel{{\scriptstyle\eqref{av.W}}}{{=}}0$ which proves the periodicity of $t\mapsto X(t)$. ∎ We are ready to prove Proposition 3.1. During the proof we shall use some results proved in [5] about the flow generated by pseudodifferential operators; we collect them, for the reader’s convenience, in Appendix A. ###### Proof of Proposition 3.1. The proof is inductive on $N$. Let us start with $N=1$. We look for a change of variables of the form $\psi=e^{-{\rm i}X_{1}(t)}{\varphi}$ where $X_{1}(t)\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{0})$ is selfadjoint $\forall t$, to be determined. By Lemma A.1, ${\varphi}$ fulfills the Schrödinger equation ${\rm i}\partial_{t}{\varphi}=H^{+}(t){\varphi}$ with $\displaystyle H^{+}(t)$ $\displaystyle:=e^{{\rm i}X_{1}(t)}\,\big{(}K_{0}+\mathsf{V}(t)\big{)}\,e^{-{\rm i}X_{1}(t)}-\int_{0}^{1}e^{{\rm i}sX_{1}(t)}\,(\partial_{t}X_{1}(t))\,e^{-{\rm i}sX_{1}(t)}\ {\rm d}s\ .$ Then a commutator expansion, see Lemma A.2, gives $\displaystyle H^{+}(t)$ $\displaystyle=K_{0}+{\rm i}[X_{1}(t),K_{0}]+\mathsf{V}(t)-\partial_{t}X_{1}+\mathsf{V}^{(1)}(t)$ with $\mathsf{V}^{(1)}\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{-1})$, selfadjoint $\forall t$. By Lemma 3.3, we choose $X_{1}\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{0})$, selfadjoint $\forall t$, s.t. ${\rm i}[K_{0},X_{1}(t)]+\partial_{t}X_{1}(t)=\mathsf{V}(t)-{\widehat{\mathsf{V}}}(t)\ ,$ (3.12) where ${\widehat{\mathsf{V}}}(t)$ is the averaged operator (see (3.2)). With this choice we have $\displaystyle H^{+}(t)$ $\displaystyle=K_{0}+Z^{(1)}(t)+\mathsf{V}^{(1)}(t)\ ,\quad Z^{(1)}(t):={\widehat{\mathsf{V}}}(t)\ .$ (3.13) By Lemma 3.2, $Z^{(1)}\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{0})$, it is selfadjoint $\forall t$, it commutes with ${\rm i}\partial_{t}-K_{0}$. The map $e^{-{\rm i}X_{1}(t)}$ fulfills (3.3) thanks to Lemma A.3. This concludes the first step. The iterative step $N\to N+1$ is proved following the same lines, just adding the remark that $e^{{\rm i}X_{N+1}}Z^{(N)}e^{-{\rm i}X_{N+1}}-Z^{(N)}\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{-N-1})$, and solving the homological equation ${\rm i}[K_{0},X_{N+1}(t)]+\partial_{t}X_{N+1}(t)=\mathsf{V}^{(N)}(t)-{\widehat{\mathsf{V}^{(N)}}}(t)\ .$ (3.14) So one puts $Z^{(N+1)}:=Z^{(N)}+{\widehat{\mathsf{V}^{(N)}}}$. Note that ${\widehat{\mathsf{V}^{(N)}}}\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{-N})$, so $Z^{(N)}$ has an expansion in operators of decreasing order. ∎ It turns out that property (3.6) implies that $e^{{\rm i}tK_{0}}\,Z^{(N)}(t)\,e^{-{\rm i}tK_{0}}$ is time independent. A consequence of this fact is the following corollary. ###### Corollary 3.4. Consider equation (3.1) with $\mathsf{V}\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{0})$, $\mathsf{V}(t)$ selfadjoint $\forall t$. Fix $N\in{\mathbb{N}}$ arbitrary. There exists a change of coordinates ${\mathcal{U}}_{N}(t)$ unitary in ${\mathcal{H}}$ and fulfilling $\forall r\geq 0\quad\exists c_{r},C_{r}>0\colon\qquad c_{r}\\|{\varphi}\\|_{r}\leq\\|{\mathcal{U}}_{N}(t)^{\pm}{\varphi}\\|_{r}\leq C_{r}\\|{\varphi}\\|_{r},\qquad\forall t\in{\mathbb{R}},$ (3.15) such that $\psi(t)$ is a solution of (3.1) if and only if $\phi(t):={\mathcal{U}}_{N}(t)\psi(t)$ solves ${\rm i}\partial_{t}\phi=\big{(}\left\langle\mathsf{V}\right\rangle+T_{N}+R_{N}(t)\big{)}\phi\ ;$ (3.16) here $\left\langle\mathsf{V}\right\rangle$ is the resonant average of $\mathsf{V}$ (see (1.3)), $T_{N}\in{\mathcal{A}}_{-1}$ is time independent and selfadjoint and $R_{N}\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{-N})$. ###### Proof. Fix $N\in{\mathbb{N}}$ and apply Proposition 3.1 to conjugate equation (3.1) to the form (3.5) via the change of variables (3.4). Then we gauge away $K_{0}$ by the change of coordinates ${\varphi}=e^{-{\rm i}tK_{0}}\phi$, getting ${\rm i}\partial_{t}\phi=e^{{\rm i}tK_{0}}\,\big{(}Z^{(N)}(t)+\mathsf{V}^{(N)}(t)\big{)}\,e^{-{\rm i}tK_{0}}\,\phi.$ Define $\mathsf{H}_{N}:=e^{{\rm i}tK_{0}}\,Z^{(N)}(t)\,e^{-{\rm i}tK_{0}},\qquad R_{N}(t):=e^{{\rm i}tK_{0}}\,\mathsf{V}^{(N)}(t)\,e^{-{\rm i}tK_{0}}.$ The operator $R_{N}\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{-N})$ by Assumption II since $\mathsf{V}^{(N)}\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{-N})$. Let us now prove that $\mathsf{H}_{N}$ is time independent. We know by Lemma 3.1 that $Z^{(N)}(t)$ commutes with ${\rm i}\partial_{t}-K_{0}$; therefore $\partial_{t}\big{(}e^{{\rm i}tK_{0}}\,Z^{(N)}(t)\,e^{-{\rm i}tK_{0}}\big{)}=e^{{\rm i}tK_{0}}\,\big{(}{\rm i}[K_{0},Z^{(N)}(t)]+\partial_{t}Z^{(N)}(t)\big{)}\,e^{-{\rm i}tK_{0}}=0$ and we get $\mathsf{H}_{N}=e^{{\rm i}tK_{0}}\,Z^{(N)}(t)\,e^{-{\rm i}tK_{0}}\|_{t=0}=Z^{(N)}(0)\stackrel{{\scriptstyle\eqref{Z.exp}}}{{=}}{\widehat{\mathsf{V}}}(0)+T^{(N)}(0)\stackrel{{\scriptstyle\eqref{av.W}}}{{=}}\left\langle\mathsf{V}\right\rangle+T^{(N)}(0).$ So we put $T_{N}:=T^{(N)}(0)$; clearly it belongs to ${\mathcal{A}}_{-1}$, it is selfadjoint and time independent. Finally we put ${\mathcal{U}}_{N}(t):=e^{{\rm i}tK_{0}}\,e^{{\rm i}tX_{N}(t)}\,\cdots e^{{\rm i}tX_{1}(t)}$; estimate (3.15) follows from (3.3) and Remark 2.1. ∎ Coming back to the original equation (2.11), we apply Corollary 3.4 with $\mathsf{V}=V+W\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{0})$, getting the following result: ###### Corollary 3.5. With the same assumptions of Theorem 2.8, the following holds true. Fix $N\in{\mathbb{N}}$ arbitrary. There exists a change of coordinates ${\mathcal{U}}_{N}(t)$, unitary in ${\mathcal{H}}$ and fulfilling (3.15) such that $\psi(t)$ is a solution of (2.11) if and only if $\phi(t):={\mathcal{U}}_{N}(t)\psi(t)$ solves ${\rm i}\partial_{t}\phi=\big{(}\left\langle V\right\rangle+\left\langle W\right\rangle+T_{N}+R_{N}(t)\big{)}\phi$ (3.17) where $T_{N}\in{\mathcal{A}}_{-1}$ is selfadjoint and time independent whereas $R_{N}\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{-N})$. ### 3.2 Local energy decay estimates From now on we are going to assume also Assumption III. In the previous section we have conjugated the original equation (2.11) to the resonant equation (3.17). In this section we consider the effective equation obtained removing $R_{N}(t)$ from (3.17), namely ${\rm i}\partial_{t}{\varphi}=H_{N}{\varphi},\qquad H_{N}:=\left\langle V\right\rangle+\left\langle W\right\rangle+T_{N},$ (3.18) with $T_{N}\in{\mathcal{A}}_{-1}$ of Corollary 3.5. Note that $H_{N}$ is selfadjoint by Lemma 3.2 and Corollary 3.5. The goal is to construct a solution of (3.18) with polynomially in time growing Sobolev norms. Actually we will prove the following slightly stronger result, namely the existence of a solution with decaying negative Sobolev norms: ###### Proposition 3.6 (Decay of negative Sobolev norms). With the same assumptions of Theorem 2.8, consider the operator $H_{N}$ in (3.18). For any $k\in{\mathbb{N}}$, there exist a nontrivial solution ${\varphi}(t)\in{\mathcal{H}}^{k}$ of (3.18) and $\forall r\in[0,k]$ a constant $C_{r}>0$ such that $\\|{\varphi}(t)\\|_{{-r}}\leq C_{r}\left\langle t\right\rangle^{-r}\,\\|{\varphi}(0)\\|_{r}\ ,\qquad\forall t\in{\mathbb{R}}\ .$ (3.19) ###### Remark 3.7. As $H_{N}$ is selfadjoint, the conservation of the ${\mathcal{H}}^{0}$-norm and Cauchy-Schwartz inequality give $\\|{\varphi}(0)\\|_{0}^{2}=\\|{\varphi}(t)\\|_{0}^{2}\leq\\|{\varphi}(t)\\|_{r}\ \\|{\varphi}(t)\\|_{{-r}}\ ,\qquad\forall t\in{\mathbb{R}}\ ,$ so that (3.19) implies the growth of positive Sobolev norms: $\\|{\varphi}(t)\\|_{r}\geq\frac{1}{C_{r}}\frac{\\|{\varphi}(0)\\|_{0}^{2}}{\\|{\varphi}(0)\\|_{r}}\,\left\langle t\right\rangle^{r}\ ,\quad\forall t\in{\mathbb{R}}\ .$ The rest of the section is devoted to the proof of Proposition 3.6. As we shall see, it follows from a local energy decay estimate for the operator $H_{N}$, namely a dispersive estimate of the form $\\|\left\langle A\right\rangle^{-k}\,e^{-{\rm i}H_{N}t}\,g_{J}(H_{N})\,{\varphi}\\|_{0}\leq C_{k}\left\langle t\right\rangle^{-k}\\|\left\langle A\right\rangle^{k}g_{J}(H_{N}){\varphi}\\|_{0}\ ,\qquad\forall t\in{\mathbb{R}}$ (3.20) where $A\in{\mathcal{A}}_{1}$, $J\subset I_{0}$ is an interval and $g_{J}\in C^{\infty}_{c}({\mathbb{R}},{\mathbb{R}}_{\geq 0})$ with $g_{J}\equiv 1$ on $J$. ###### Remark 3.8. Actually estimate (3.20) show the existence of infinitely many solutions of (3.18) with decaying negative Sobolev norms. In particular this happens to any solution whose initial datum ${\varphi}(0)$ belongs to the (infinite dimensional) set ${\rm Ran}\,E_{J}(H_{N})$, where $E_{J}(H_{N})$ is the spectral projection of $H_{N}$ corresponding to the interval $J$. A possible approach (which we will follow here) to obtain such estimate is via Sigal-Soffer minimal velocity estimates [42, 23, 31, 30, 24, 2]. These estimates are based on Mourre theory, let us recall this last one. #### Mourre theory. Let $\mathsf{H}$ be a selfadjoint operator on the Hilbert space ${\mathcal{H}}$, and denote by $\sigma(\mathsf{H})$ its spectrum. We further denote by $\sigma_{d}(\mathsf{H})$ its discrete spectrum, $\sigma_{ess}(\mathsf{H})$ its essential spectrum, $\sigma_{pp}(\mathsf{H})$ its pure point spectrum, $\sigma_{ac}(\mathsf{H})$ its absolutely continuous spectrum and $\sigma_{sc}(\mathsf{H})$ its singular spectrum; see e.g. [38] pag. 236 and 231 for their definitions. Furthermore we denote by $E_{\Omega}(\mathsf{H})$ the spectral projection of $\mathsf{H}$ corresponding to the Borel set $\Omega$ and by $m_{\varphi}(\Omega):=\left\langle E_{\Omega}(\mathsf{H}){\varphi},{\varphi}\right\rangle$ the spectral measure associated to ${\varphi}\in{\mathcal{H}}$. Assume a selfadjoint operator $\mathsf{A}$ can be found such that $D(\mathsf{A})\cap{\mathcal{H}}$ is dense in ${\mathcal{H}}$. We put ${\rm ad}^{0}_{\mathsf{A}}(\mathsf{H}):=\mathsf{H},\qquad{\rm ad}_{\mathsf{A}}(\mathsf{H}):=[\mathsf{H},\mathsf{A}],\qquad{\rm ad}^{n}_{\mathsf{A}}(\mathsf{H}):=[{\rm ad}^{n-1}_{\mathsf{A}}(\mathsf{H}),\mathsf{A}],\quad\forall n\geq 2\ .$ (3.21) Consider the following properties: * (M1) For some ${\mathtt{N}}\geq 1$, the operators ${\rm ad}^{n}_{\mathsf{A}}(\mathsf{H})$ with $n=1,\ldots,{\mathtt{N}}$, can all be extended to bounded operators on ${\mathcal{H}}$. * (M2) Mourre estimate: there exists an open interval $I\subset{\mathbb{R}}$ with compact closure and a function $g_{I}\in C^{\infty}_{c}({\mathbb{R}},{\mathbb{R}}_{\geq 0})$ with $g_{I}\equiv 1$ on $I$ such that $g_{I}(\mathsf{H})\,{\rm i}[\mathsf{H},\mathsf{A}]\,g_{I}(\mathsf{H})\geq\theta g_{I}(\mathsf{H})^{2}+\mathsf{K}$ (3.22) for some $\theta>0$ and $\mathsf{K}$ a selfadjoint compact operator on ${\mathcal{H}}$. If the estimate (3.22) holds true with $\mathsf{K}=0$, we shall say that $\mathsf{H}$ fulfills a strict Mourre estimate. Mourre theorem [36] says the following: ###### Theorem 3.9 (Mourre). Assume conditions (M1) – (M2) with ${\mathtt{N}}=2$. In the interval $I$, the operator $\mathsf{H}$ can have only absolutely continuous spectrum and finitely many eigenvalues of finite multiplicity. If $\mathsf{K}=0$, there are no eigenvalues in the interval $I$, i.e. $\sigma(\mathsf{H})\cap I=\sigma_{ac}(\mathsf{H})\cap I$. ###### Remark 3.10. The version stated here of Mourre theorem is taken from [3, Lemma 5.6] and [14, Theorem 4.7 – 4.9], and it has slightly weaker assumptions compared to [36]. ###### Remark 3.11. Mourre theorem guarantees that $\sigma_{sc}(\mathsf{H})\cap I=\emptyset$ and, in case $\mathsf{K}=0$, $\sigma_{pp}(\mathsf{H})\cap I=\emptyset$. However it does not guarantee that $\sigma(\mathsf{H})\cap I\neq\emptyset$; in our case we shall verify this property explicitly. The key point is that if $H_{N}$ fulfills a strict Mourre estimate (namely with $\mathsf{K}=0$) then one can prove a local energy decay estimate like (3.20) for the Schrödinger flow of $H_{N}$. This is a quite general fact which follows exploiting minimal velocity estimates [30] and we prove it for completeness in Appendix C. So the next goal is to prove that $H_{N}$ satisfies a strict Mourre estimate over a certain interval $J\subset I_{0}$. During the proof we will use some standard results from functional calculus; we recall them in Appendix B. We shall also use the following lemma: ###### Lemma 3.12. Let $\mathsf{H}\in{\mathcal{L}}({\mathcal{H}})$ be selfadjoint. If $\lambda\in\sigma_{ac}(\mathsf{H})$, then $\forall\delta>0$ one has $\left\|[\lambda-\delta,\lambda+\delta]\cap\sigma(\mathsf{H})\right\|>0\ .$ ###### Proof. By contradiction, assume that $\exists\delta_{0}>0$ such that $\left\|[\lambda-\delta_{0},\lambda+\delta_{0}]\cap\sigma(\mathsf{H})\right\|=0$. As $\lambda\in\sigma_{ac}(\mathsf{H})$, there exists $f\in{\mathcal{H}}$ such that $E_{[\lambda-\delta_{0},\lambda+\delta_{0}]}(\mathsf{H})f\neq 0$ and the spectral measure $m_{f}=\left\langle E(\mathsf{H})f,f\right\rangle$ is absolutely continuous. Then $0=m_{f}([\lambda-\delta_{0},\lambda+\delta_{0}])=\left\langle E_{[\lambda-\delta_{0},\lambda+\delta_{0}]}(\mathsf{H})f,f\right\rangle=\\|E_{[\lambda-\delta_{0},\lambda+\delta_{0}]}(\mathsf{H})f\\|_{0}^{2}>0$ giving a contradiction. ∎ ###### Lemma 3.13. There exist $\epsilon_{0},{\mathtt{M}}>0$ such that, provided $W$ fulfills (2.10), the following holds true: * (i) There exists an interval $I\subset I_{0}$ such that $\left\|I\cap\sigma(H_{N})\right\|>0$. * (ii) $H_{N}$ fulfills a strict Mourre estimate over $I$: there exists a function $g_{I}\in C^{\infty}_{c}({\mathbb{R}},{\mathbb{R}}_{\geq 0})$ with ${\rm supp}\,g_{I}\subset I_{0}$, $g_{I}\equiv 1$ on $I$, and $\theta^{\prime}>0$ such that $g_{I}(H_{N})\,{\rm i}[H_{N},A]\,g_{I}(H_{N})\geq\theta^{\prime}g_{I}(H_{N})^{2}\ .$ (3.23) Here $I_{0}$ is the interval and $A$ is the operator of Assumption III. ###### Proof. During the proof we shall often use that for $\mathsf{A},\mathsf{B},\mathsf{C}\in{\mathcal{L}}({\mathcal{H}})$ and selfadjoints $\displaystyle\mathsf{A}\leq\mathsf{B}\ \ \Rightarrow\ \ \mathsf{C}\mathsf{A}\mathsf{C}\leq\mathsf{C}\mathsf{B}\mathsf{C},\qquad\\|\mathsf{A}\\|_{{\mathcal{L}}({\mathcal{H}})}\leq a\ \ \Rightarrow\ \ -a\leq\mathsf{A}\leq a\ .$ (3.24) To shorten notation, throughout the proof we shall put $H_{0}:=\left\langle V\right\rangle\ .$ We split the proof in several steps. Step 1: By Assumption III, $H_{0}$ fulfills a Mourre estimate over the interval $I_{0}$. The first step of the proof is to exhibit a subinterval $I_{1}\subset I_{0}$ containing only absolutely continuous spectrum of $H_{0}$, namely $\sigma(H_{0})\cap I_{1}=\sigma_{ac}(H_{0})\cap I_{1}\ ,\qquad\left\|\sigma(H_{0})\cap I_{1}\right\|>0\ ,$ (3.25) and over which $H_{0}$ fulfills a strict Mourre estimate: $\exists g_{I_{1}}\in C^{\infty}_{c}({\mathbb{R}},{\mathbb{R}}_{\geq 0})$, $g_{I_{1}}\equiv 1$ on $I_{1}$, ${\rm supp}\,g_{I_{1}}\subset I_{0}$, such that $g_{I_{1}}(H_{0})\,{\rm i}[H_{0},A]\,g_{I_{1}}(H_{0})\geq\frac{\theta}{2}g_{I_{1}}(H_{0})^{2}\ .$ (3.26) To prove this claim, first apply Mourre theorem to $H_{0}$ (note that (M1) and (M2) are verified $\forall{\mathtt{N}}\in{\mathbb{N}}$ by symbolic calculus and Assumption III), getting that $\sigma(H_{0})\cap I_{0}$ contains only finitely many eigenvalues with finite multiplicity and absolutely continuous spectrum. In particular $\|\overline{\sigma_{pp}(H_{0})}\cap I_{0}\|=0$ and by Assumption III $(i)$ it follows that $\|\sigma_{ac}(H_{0})\cap I_{0}\|=\|\sigma(H_{0})\cap I_{0}\|>0$. So we take $\lambda_{0}\in I_{0}\cap(\sigma_{ac}(H_{0})\setminus\sigma_{pp}(H_{0}))$ and a sufficiently small interval $I_{1}(\overline{\delta}):=(\lambda_{0}-\overline{\delta},\lambda_{0}+\overline{\delta})\subset I_{0}$, $\overline{\delta}>0$, which does not contain eigenvalues of $H_{0}$; this is possible as the eigenvalues of $H_{0}$ in $I_{0}$ are finite. Moreover by Lemma 3.12, $\left\|\sigma(H_{0})\cap I_{1}(\delta)\right\|>0$ for any $\delta>0$. Now take $\delta\in(0,\overline{\delta})$ and a function $g_{\delta}\in C^{\infty}_{c}({\mathbb{R}},{\mathbb{R}}_{\geq 0})$ with ${\rm supp}\,g_{\delta}\subset I_{1}(\delta)$ and $g_{\delta}=1$ on $I_{1}(\frac{\delta}{2})$. We claim that provided $\delta\in(0,\overline{\delta})$ is sufficiently small $\\|g_{\delta}(H_{0})Kg_{\delta}(H_{0})\\|_{{\mathcal{L}}({\mathcal{H}})}\leq\frac{\theta}{2}\ ,$ (3.27) where $\theta>0$ is the one of Assumption III. Indeed in $I_{1}(\overline{\delta})$ the spectrum of $H_{0}$ is absolutely continuous; this means that $\forall{\varphi}\in{\mathcal{H}}$, the vector ${\varphi}^{\prime}:=E_{I_{1}(\overline{\delta})}(H_{0}){\varphi}$ belongs to the absolutely continuous subspace of $H_{0}$, namely its spectral measure $m_{{\varphi}^{\prime}}$ is absolutely continuous w.r.t. the Lebesgue measure. Now, since for any ${\varphi}\in{\mathcal{H}}$ one has by functional calculus $g_{\delta}(H_{0})=g_{\delta}(H_{0})E_{I_{1}(\overline{\delta})}(H_{0})$, one has that $\\|g_{\delta}(H_{0}){\varphi}\\|_{0}^{2}=\\|g_{\delta}(H_{0})E_{I_{1}(\overline{\delta})}(H_{0}){\varphi}\\|_{0}^{2}=\int_{\mathbb{R}}g_{\delta}(\lambda)^{2}\,{\rm d}m_{{\varphi}^{\prime}}(\lambda)\to 0\ \ \ \mbox{ as }\delta\to 0$ by Lebesgue dominated convergence theorem. In particular $g_{\delta}(H_{0})\to 0$ strongly as $\delta\to 0$ and then, being $K$ compact, $g_{\delta}(H_{0})K\to 0$ uniformly as $\delta\to 0$ (see e.g. [1]). Therefore for $\delta\in(0,\overline{\delta})$ sufficiently small (3.27) holds true. Using the assumption (2.7), (3.27) and (3.24) we deduce that $\displaystyle g_{\delta}(H_{0})\,g_{I_{0}}(H_{0})\,{\rm i}[H_{0},A]\,g_{I_{0}}(H_{0})\,g_{\delta}(H_{0})$ $\displaystyle\geq\theta g_{\delta}(H_{0})\,g_{I_{0}}(H_{0})^{2}\,g_{\delta}(H_{0})-\frac{\theta}{2}\ ;$ next apply $g_{\frac{\delta}{2}}(H_{0})$ to the right and left of the previous inequality, use again (3.24) and the identity $g_{I_{0}}(H_{0})\,g_{\delta}(H_{0})\,g_{\frac{\delta}{2}}(H_{0})=g_{\frac{\delta}{2}}(H_{0})$ (which follows from $g_{I_{0}}\,g_{\delta}\,g_{\frac{\delta}{2}}=g_{\frac{\delta}{2}}$), to get the strict Mourre estimate $g_{I_{1}}(H_{0})\,{\rm i}[H_{0},A]\,g_{I_{1}}(H_{0})\geq\frac{\theta}{2}g_{I_{1}}(H_{0})^{2}$ (3.28) where $I_{1}:=I_{1}(\frac{\delta}{4})$ and $g_{I_{1}}:=g_{\frac{\delta}{2}}$ fulfills $g_{I_{1}}\equiv 1$ on $I_{1}$, ${\rm supp}\,g_{I_{1}}\subset I_{1}(\frac{\delta}{2})$. Clearly $I_{1}$ fulfills (3.25). Step 2: We shall prove that the selfadjoint operator $H_{\left\langle W\right\rangle}:=H_{0}+\left\langle W\right\rangle$ has a nontrivial spectrum in a subinterval $I_{2}\subseteq I_{1}$, and over this interval it fulfills the strict Mourre estimate $g_{I_{2}}\big{(}H_{\left\langle W\right\rangle}\big{)}\,{\rm i}[H_{\left\langle W\right\rangle},A]\,g_{I_{2}}\big{(}H_{\left\langle W\right\rangle}\big{)}\geq\frac{\theta}{4}g_{I_{2}}\big{(}H_{\left\langle W\right\rangle}\big{)}^{2}\ $ (3.29) for any $g_{I_{2}}\in C^{\infty}_{c}({\mathbb{R}},{\mathbb{R}}_{\geq 0})$ with ${\rm supp}\,g_{I_{2}}\subset I_{1}$, $g_{I_{2}}\equiv 1$ on $I_{2}$. To prove this, we exploit that $\left\langle W\right\rangle\in{\mathcal{A}}_{0}$ is a small bounded perturbation of $H_{0}$, fulfilling, by (2.1), (3.9) $\exists M_{0}\in{\mathbb{N}},C_{0}>0\colon\qquad\\|\left\langle W\right\rangle\\|_{{\mathcal{L}}({\mathcal{H}})}{\leq}\,C_{0}[W]_{M_{0}},$ (3.30) where we denoted $[W]_{M}:=\sup_{t\in{\mathbb{T}}}\,\wp^{0}_{M}(W(t))\ .$ First let us prove that $\sigma(H_{\left\langle W\right\rangle})\cap I_{1}\neq\emptyset$. Take again the same $\lambda_{0}\in\sigma(H_{0})\cap I_{1}$ as in the previous step. We claim that ${\rm dist}\big{(}\lambda_{0},\sigma(H_{\left\langle W\right\rangle})\big{)}\leq C_{0}\,[W]_{M_{0}}.$ (3.31) If $\lambda_{0}\in\sigma(H_{\left\langle W\right\rangle})$ this is trivial. So assume that $\lambda_{0}$ belongs to the resolvent set of $H_{\left\langle W\right\rangle}$. As $\lambda_{0}\in\sigma(H_{0})$, by Weyl criterion $\exists(f_{n})_{n\geq 1}\in{\mathcal{H}}$ with $\\|f_{n}\\|_{0}=1$ such that $\\|(H_{0}-\lambda_{0})f_{n}\\|_{0}\to 0$ as $n\to\infty$. Then $\forall n\geq 1$ $\displaystyle 1$ $\displaystyle=\\|f_{n}\\|_{0}=\\|(H_{\left\langle W\right\rangle}-\lambda_{0})^{-1}\,(H_{\left\langle W\right\rangle}-\lambda_{0})f_{n}\\|_{0}\leq\frac{1}{{\rm dist}\big{(}\lambda_{0},\sigma(H_{\left\langle W\right\rangle})\big{)}}\\|(H_{\left\langle W\right\rangle}-\lambda_{0})f_{n}\\|_{0}$ $\displaystyle\stackrel{{\scriptstyle\eqref{pm.s20}}}{{\leq}}\frac{1}{{\rm dist}\big{(}\lambda_{0},\sigma(H_{\left\langle W\right\rangle})\big{)}}\Big{(}\\|(H_{0}-\lambda_{0})f_{n}\\|_{0}+C_{0}[W]_{M_{0}}\Big{)}$ which proves (3.31) passing to the limit $n\to\infty$. Then, provided $[W]_{M_{0}}$ is sufficiently small, (3.31) implies that ${\rm dist}\big{(}\lambda_{0},\sigma(H_{\left\langle W\right\rangle})\big{)}<\delta/8$. From this we learn that (recall $I_{1}=(\lambda_{0}-\frac{\delta}{4},\lambda_{0}+\frac{\delta}{4})$) $\sigma(H_{\left\langle W\right\rangle})\cap I_{1}\neq\emptyset\ .$ (3.32) Next we prove the Mourre estimate (3.29); we shall work perturbatively from (3.26). First $\displaystyle g_{I_{1}}(H_{0})\,{\rm i}[H_{\left\langle W\right\rangle},A]\,g_{I_{1}}(H_{0})=g_{I_{1}}(H_{0})\,{\rm i}[H_{0},A]\,g_{I_{1}}(H_{0})+g_{I_{1}}(H_{0})\,{\rm i}[\left\langle W\right\rangle,A]\,g_{I_{1}}(H_{0});$ we bound the first term in the right hand side above from below using (3.26). Concerning the second term, we use $\displaystyle\exists\,M_{1}\in{\mathbb{N}},\,C_{1}>0\colon\quad\\|{\rm i}[\left\langle W\right\rangle,A]\\|_{{\mathcal{L}}({\mathcal{H}})}\leq C_{1}[W]_{M_{1}}\ $ (3.33) (by (2.1), (2.3), (3.9)) and the inequalities (3.24) to bound it from above getting $g_{I_{1}}(H_{0})\,{\rm i}[\left\langle W\right\rangle,A]\,g_{I_{1}}(H_{0})\geq- C_{1}\,[W]_{M_{1}}\,g_{I_{1}}(H_{0})^{2}\ .$ Therefore we find $\displaystyle g_{I_{1}}(H_{0})\,{\rm i}[H_{\left\langle W\right\rangle},A]\,g_{I_{1}}(H_{0})\geq\left(\frac{\theta}{2}-C_{1}[W]_{M_{1}}\right)\,g_{I_{1}}(H_{0})^{2}\ .$ (3.34) Take now an open interval $I_{2}\subset I_{1}$ such that $\sigma(H_{\left\langle W\right\rangle})\cap{I}_{2}\neq\emptyset$ (it is possible by (3.32)); take also $g_{I_{2}}\in C^{\infty}_{c}({\mathbb{R}},{\mathbb{R}}_{\geq 0})$ with ${\rm supp}\,g_{I_{2}}\subseteq I_{1}$ and $g_{I_{2}}\equiv 1$ on $I_{2}$; remark that $g_{I_{1}}g_{I_{2}}=g_{I_{2}}$. Now we wish to replace $g_{I_{1}}(H_{0})$ by $g_{I_{2}}(H_{\left\langle W\right\rangle})$ in (3.34), thus getting the claimed estimate (3.29). So write $\displaystyle g_{I_{2}}(H_{\left\langle W\right\rangle})\,{\rm i}[H_{\left\langle W\right\rangle},A]$ $\displaystyle\,g_{I_{2}}(H_{\left\langle W\right\rangle})=g_{I_{2}}(H_{\left\langle W\right\rangle})\,g_{I_{1}}(H_{\left\langle W\right\rangle})\,{\rm i}[H_{\left\langle W\right\rangle},A]\,g_{I_{1}}(H_{\left\langle W\right\rangle})\,g_{I_{2}}(H_{\left\langle W\right\rangle})$ $\displaystyle=g_{I_{2}}(H_{\left\langle W\right\rangle})\,g_{I_{1}}(H_{0})\,{\rm i}[H_{\left\langle W\right\rangle},A]\,g_{I_{1}}(H_{0})\,g_{I_{2}}(H_{\left\langle W\right\rangle})$ (3.35) $\displaystyle\ +g_{I_{2}}(H_{\left\langle W\right\rangle})\,\Big{(}\big{(}g_{I_{1}}(H_{\left\langle W\right\rangle})-g_{I_{1}}(H_{0})\big{)}\,{\rm i}[H_{\left\langle W\right\rangle},A]\,g_{I_{1}}(H_{0})$ (3.36) $\displaystyle\ +g_{I_{1}}(H_{\left\langle W\right\rangle})\,{\rm i}[H_{\left\langle W\right\rangle},A]\,\big{(}g_{I_{1}}(H_{\left\langle W\right\rangle})-g_{I_{1}}(H_{0})\big{)}\Big{)}\,g_{I_{2}}(H_{\left\langle W\right\rangle})$ (3.37) Again we estimate (3.35) from below and the other lines from above. First $\displaystyle\eqref{pm6}$ $\displaystyle\stackrel{{\scriptstyle\eqref{pm5}}}{{\geq}}\left(\frac{\theta}{2}-C_{1}[W]_{M_{1}}\right)\,g_{I_{2}}(H_{\left\langle W\right\rangle})\,g_{I_{1}}(H_{0})^{2}\,g_{I_{2}}(H_{\left\langle W\right\rangle})\ .$ (3.38) We still have to bound from below $g_{I_{2}}(H_{\left\langle W\right\rangle})\,g_{I_{1}}(H_{0})^{2}\,g_{I_{2}}(H_{\left\langle W\right\rangle})$. To proceed we use that $g_{I_{1}}(H_{\left\langle W\right\rangle})-g_{I_{1}}(H_{0})$ is small in size, being bounded, via Lemma B.6 and (3.30), by $\displaystyle\\|g_{I_{1}}(H_{\left\langle W\right\rangle})-g_{I_{1}}(H_{0})\\|_{{\mathcal{L}}({\mathcal{H}})}\leq C\,[W]_{M_{0}}\,.$ (3.39) So write $\displaystyle g_{I_{2}}(H_{\left\langle W\right\rangle})\,g_{I_{1}}(H_{0})^{2}\,g_{I_{2}}(H_{\left\langle W\right\rangle})=g_{I_{2}}(H_{\left\langle W\right\rangle})\,g_{I_{1}}(H_{\left\langle W\right\rangle})^{2}\,g_{I_{2}}(H_{\left\langle W\right\rangle})$ (3.40) $\displaystyle\ \ +g_{I_{2}}(H_{\left\langle W\right\rangle})\,\Big{(}g_{I_{1}}(H_{\left\langle W\right\rangle})\,(g_{I_{1}}(H_{0})-g_{I_{1}}(H_{\left\langle W\right\rangle}))\,+(g_{I_{1}}(H_{0})-g_{I_{1}}(H_{\left\langle W\right\rangle}))\,g_{I_{1}}(H_{0})\Big{)}\,g_{I_{2}}(H_{\left\langle W\right\rangle}).$ Therefore, using $g_{I_{1}}g_{I_{2}}=g_{I_{2}}$, estimates (3.39) and (3.24), we deduce $g_{I_{2}}(H_{\left\langle W\right\rangle})\,g_{I_{1}}(H_{0})^{2}\,g_{I_{2}}(H_{\left\langle W\right\rangle})\geq\big{(}1-C[W]_{M_{0}}\big{)}\,g_{I_{2}}(H_{\left\langle W\right\rangle})^{2}\ .$ Thus we can finally estimate line (3.35) from below using (3.38) and the previous estimate, concluding $\eqref{pm6}\geq\left(\frac{\theta}{2}-C_{1}[W]_{M_{1}}\right)\big{(}1-C[W]_{M_{0}}\big{)}\,g_{I_{2}}(H_{\left\langle W\right\rangle})^{2}.$ (3.41) Next consider lines (3.36), (3.37). We use the bound (see (3.33)) $\\|[H_{\left\langle W\right\rangle},A]\\|_{{\mathcal{L}}({\mathcal{H}}^{0})}\leq C\big{(}1+[W]_{M_{1}}\big{)}\ ,$ and (3.39) to get $\eqref{pm7}+\eqref{pm8}\geq-C\,[W]_{M_{0}}\,(1+[W]_{M_{1}})\,g_{I_{2}}(H_{\left\langle W\right\rangle})^{2}.$ (3.42) Putting together (3.41) and (3.42) we finally find $g_{I_{2}}(H_{\left\langle W\right\rangle})\,{\rm i}[H_{\left\langle W\right\rangle},A]\,g_{I_{2}}(H_{\left\langle W\right\rangle})\geq\left(\frac{\theta}{2}-C([W]_{M_{1}}+[W]_{M_{0}}+[W]_{M_{0}}\,[W]_{M_{1}})\right)\,g_{I_{2}}(H_{\left\langle W\right\rangle})^{2}.$ Thus, provided (2.10) holds true for ${\mathtt{M}}$ sufficiently large and $\epsilon_{0}$ sufficiently small, the strict Mourre estimate (3.29) follows. Mourre theorem implies that the spectrum of $H_{\left\langle W\right\rangle}$ in $I_{2}$ is absolutely continuous and by (3.32) it is also nonempty; summarizing (use also Lemma 3.12) $\sigma(H_{\left\langle W\right\rangle})\cap I_{2}=\sigma_{ac}(H_{\left\langle W\right\rangle})\cap I_{2}\qquad\mbox{and}\qquad\left\|\sigma(H_{\left\langle W\right\rangle})\cap I_{2}\right\|>0\ .$ (3.43) Step 3: The last step is to consider the operator $H_{N}=H_{0}+\left\langle W\right\rangle+T_{N}=H_{\left\langle W\right\rangle}+T_{N}$, which, for the remaining part of the proof, we shall denote just by $H$. We shall constantly use that any pseudodifferential operator of strictly negative order is a compact operator on ${\mathcal{H}}$ (see Remark 2.4); in particular $T_{N}\in{\mathcal{A}}_{-1}$ is compact. We begin by proving that $\left\|\sigma(H)\cap I_{2}\right\|>0\ .$ (3.44) Indeed by Weyl theorem $\sigma_{ess}(H)=\sigma_{ess}(H_{\left\langle W\right\rangle})$ and therefore $\displaystyle\sigma(H)\cap I_{2}$ $\displaystyle\supset\sigma_{ess}(H)\cap I_{2}=\sigma_{ess}(H_{\left\langle W\right\rangle})\cap I_{2}=\sigma(H_{\left\langle W\right\rangle})\cap I_{2}\ ,$ since $\sigma_{d}(H_{\left\langle W\right\rangle})\cap I_{2}=\emptyset$ having $H_{\left\langle W\right\rangle}$ no eigenvalues in $I_{2}$. Then (3.44) follows by (3.43). Next we prove that $H$ fulfills a Mourre estimate over $I_{2}$, i.e. $g_{I_{2}}\big{(}H\big{)}\,{\rm i}[H,A]\,g_{I_{2}}\big{(}H\big{)}\geq\frac{\theta}{4}g_{I_{2}}\big{(}H\big{)}^{2}+K$ (3.45) with $K$ a compact operator. We work perturbatively from (3.29). Again first we compute $g_{I_{2}}\big{(}H_{\left\langle W\right\rangle}\big{)}\,{\rm i}[H,A]\,g_{I_{2}}\big{(}H_{\left\langle W\right\rangle}\big{)}=g_{I_{2}}\big{(}H_{\left\langle W\right\rangle}\big{)}\,{\rm i}[H_{\left\langle W\right\rangle},A]\,g_{I_{2}}\big{(}H_{\left\langle W\right\rangle}\big{)}+g_{I_{2}}\big{(}H_{\left\langle W\right\rangle}\big{)}\,{\rm i}[T_{N},A]\,g_{I_{2}}\big{(}H_{\left\langle W\right\rangle}\big{)}\ ;$ we estimate the first term in the r.h.s. above by (3.29), whereas the second term is a compact operator since $[T_{N},A]\in{\mathcal{A}}_{-1}$. We obtain $g_{I_{2}}\big{(}H_{\left\langle W\right\rangle}\big{)}\,{\rm i}[H,A]\,g_{I_{2}}\big{(}H_{\left\langle W\right\rangle}\big{)}\geq\frac{\theta}{4}g_{I_{2}}\big{(}H_{\left\langle W\right\rangle}\big{)}^{2}+K_{1}$ (3.46) with $K_{1}$ a compact operator. Now we must replace $g_{I_{2}}\big{(}H_{\left\langle W\right\rangle}\big{)}$ with $g_{I_{2}}(H)$. We write $\displaystyle g_{I_{2}}(H)\,{\rm i}[H,A]\,g_{I_{2}}(H)=g_{I_{2}}(H_{\left\langle W\right\rangle})\,{\rm i}[H,A]\,g_{I_{2}}(H_{\left\langle W\right\rangle})$ (3.47) $\displaystyle\quad+\big{(}g_{I_{2}}(H)-g_{I_{2}}(H_{\left\langle W\right\rangle})\big{)}\,{\rm i}[H,A]\,g_{I_{2}}(H_{\left\langle W\right\rangle})+g_{I_{2}}(H)\,{\rm i}[H,A]\,\big{(}g_{I_{2}}(H)-g_{I_{2}}(H_{\left\langle W\right\rangle})\big{)}$ (3.48) This time we use that $g_{I_{2}}(H)-g_{I_{2}}(H_{\left\langle W\right\rangle})$ is a compact operator, see Lemma B.6. Thus $\displaystyle\eqref{pm600}$ $\displaystyle\stackrel{{\scriptstyle\eqref{pm.12}}}{{\geq}}\frac{\theta}{4}\,g_{I_{2}}\big{(}H_{\left\langle W\right\rangle}\big{)}^{2}+K_{1}=\frac{\theta}{4}g_{I_{2}}(H)^{2}+K_{2}$ where $K_{1}$, $K_{2}$ are compact operators. Similarly, using that ${\rm i}[H,A]\in{\mathcal{A}}_{0}$ is a bounded operator, we deduce that (3.48) is a compact operator. Estimate (3.45) follows. In particular $H$ is conjugated to $A$ over the interval $I_{2}$ fulfilling (3.43). Proceeding as in Step 1, we produce a subinterval $I\subset I_{2}$ such that $\left\|I\cap\sigma(H)\right\|>0\ ,\qquad I\cap\sigma(H)=I\cap\sigma_{ac}(H)$ and over which $H$ fulfills the strict Mourre estimate (3.23). ∎ The previous result has proved the existence of an interval $I$ over which $H_{N}$ fulfills a strict Mourre estimate. This implies that $H_{N}$ fulfills dispersive estimates in the form of local energy decay. In the literature there are various variants of this result, thus in Appendix C we state and prove the one we apply here. ###### Corollary 3.14. Fix $k\in{\mathbb{N}}$. For any interval $J\subset I$, any function $g_{J}\in C^{\infty}_{c}({\mathbb{R}},{\mathbb{R}}_{\geq 0})$ with ${\rm supp}\,g_{J}\subset I$, $g_{J}\equiv 1$ on $J$, there exists a constant $C_{k}>0$ such that $\\|\left\langle A\right\rangle^{-k}\,e^{-{\rm i}H_{N}t}\,g_{J}(H_{N})\,{\varphi}\\|_{0}\leq C_{k}\left\langle t\right\rangle^{-k}\\|\left\langle A\right\rangle^{k}g_{J}(H_{N}){\varphi}\\|_{0}\ ,\quad\forall t\in{\mathbb{R}}\ ,\ \ \ \forall{\varphi}\in{\mathcal{H}}^{k}\ .$ (3.49) Moreover $J$ can be chosen so that $\left\|J\cap\sigma(H_{N})\right\|>0$ and $\sigma(H_{N})\cap J=\sigma_{ac}(H_{N})\cap J$. ###### Proof. Apply Theorem C.1, noting that condition (M1) at page (M1) is trivially satisfied $\forall n\in{\mathbb{N}}$ as ${\rm ad}^{n}_{A}(H_{N})\in{\mathcal{A}}_{0}\subset{\mathcal{L}}({\mathcal{H}})$, whereas the whole point of Lemma 3.13 was to verify (M2). This gives estimate (3.49). The right hand side is finite for ${\varphi}\in{\mathcal{H}}^{k}$ by Lemma 3.15 below, which ensures that $g_{J}(H_{N}){\varphi}\in{\mathcal{H}}^{k}$. Finally note that, since $\left\|I\cap\sigma(H_{N})\right\|>0$, it is certainly possible to choose $J\subset I$ so that $\left\|J\cap\sigma(H_{N})\right\|>0$; as $H_{N}$ fulfills a strict Mourre estimate over $I$, its spectrum in this interval is absolutely continuous, so the same is true in $J$. ∎ ###### Lemma 3.15. For any $k\in{\mathbb{N}}$, $g_{J}(H_{N})$ extends to a bounded operator ${\mathcal{H}}^{k}\to{\mathcal{H}}^{k}$. ###### Proof. As $K_{0}^{k}\,g_{J}(H_{N})\,K_{0}^{-k}=g_{J}(H_{N})-[g_{J}(H_{N}),K_{0}^{k}]K_{0}^{-k},$ it is clearly sufficient to show that $[g_{J}(H_{N}),K_{0}^{k}]K_{0}^{-k}$ is bounded on ${\mathcal{H}}$. The adjoint formula (B.6) gives $[g_{J}(H_{N}),K_{0}^{k}]K_{0}^{-k}=\sum_{j=1}^{k}c_{k,j}\,{\rm ad}^{j}_{K_{0}}(g_{J}(H_{N}))\,K_{0}^{-j}\ ;$ then it is enough to show that ${\rm ad}^{j}_{K_{0}}(g_{J}(H_{N}))\in{\mathcal{L}}({\mathcal{H}})$. As ${\rm ad}^{j}_{K_{0}}(H_{N})$ is a bounded operator $\forall j$ (symbolic calculus), the result is an immediate application of Lemma B.5. ∎ We finally prove Proposition 3.6. ###### Proof of Proposition 3.6. First we show that for any $k\in{\mathbb{N}}$, there exists $C_{2k}>0$ such that $\\|e^{-{\rm i}tH_{N}}g_{J}(H_{N}){\varphi}\\|_{-2k}\leq C_{2k}\,\left\langle t\right\rangle^{-2k}\,\\|g_{J}(H_{N}){\varphi}\\|_{2k},\qquad\forall t\in{\mathbb{R}},\ \ \ \forall{\varphi}\in{\mathcal{H}}^{2k}\ .$ (3.50) This follows from Corollary 3.14 with $k\leadsto 2k$. Indeed, as $A\in{\mathcal{A}}_{1}$, the operator $\left\langle A\right\rangle^{2k}=(1+A^{2})^{k}\in{\mathcal{A}}_{2k}$ and therefore, by symbolic calculus, $K_{0}^{-2k}\left\langle A\right\rangle^{2k}$ and $\left\langle A\right\rangle^{2k}K_{0}^{-2k}$ belong to ${\mathcal{A}}_{0}\subset{\mathcal{L}}({\mathcal{H}})$. Then $\displaystyle\\|e^{-{\rm i}tH_{N}}g_{J}(H_{N}){\varphi}\\|_{-2k}$ $\displaystyle\leq\\|K_{0}^{-2k}\,\left\langle A\right\rangle^{2k}\\|_{{\mathcal{L}}({\mathcal{H}})}\,\\|\left\langle A\right\rangle^{-2k}e^{-{\rm i}tH_{N}}g_{J}(H_{N}){\varphi}\\|_{0}$ $\displaystyle\leq C_{2k}\left\langle t\right\rangle^{-2k}\\|\left\langle A\right\rangle^{2k}g_{J}(H_{N}){\varphi}\\|_{0}$ $\displaystyle\leq C_{2k}\left\langle t\right\rangle^{-2k}\\|\left\langle A\right\rangle^{2k}K_{0}^{-2k}\\|_{{\mathcal{L}}({\mathcal{H}})}\\|g_{J}(H_{N}){\varphi}\\|_{2k}$ proving (3.50). Then linear interpolation with the equality $\\|e^{-{\rm i}tH_{N}}{\varphi}_{0}\\|_{0}=\\|{\varphi}_{0}\\|_{0}$ $\,\forall t$ gives $\forall r\in[0,2k]$ $\\|e^{-{\rm i}tH_{N}}g_{J}(H_{N}){\varphi}\\|_{-r}\leq C_{r}\left\langle t\right\rangle^{-r}\\|g_{J}(H_{N}){\varphi}\\|_{r}\ ,\quad\forall t\in{\mathbb{R}}\ ,\ \ \ \forall{\varphi}\in{\mathcal{H}}^{r}\ .$ Finally we must show that this estimate is not trivial, namely that $\exists{\varphi}\in{\mathcal{H}}^{k}$ so that $g_{J}(H_{N}){\varphi}\neq 0$. So take $J\subset I$ with $\left\|J\cap\sigma(H_{N})\right\|>0$ and $\sigma(H_{N})\cap J=\sigma_{ac}(H_{N})\cap J$, which is possible by Corollary 3.14. As $g_{J}(H_{N}){\mathcal{H}}\neq\\{0\\}$ and ${\mathcal{H}}^{k}$ is dense in ${\mathcal{H}}$, we have that $g_{J}(H_{N}){\mathcal{H}}^{k}\neq\\{0\\}$. Then it is enough to take $f\in{\mathcal{H}}^{k}$ so that $g_{J}(H_{N})f\neq 0$, and put ${\varphi}_{0}:=g_{J}(H_{N})f$ which, by Lemma 3.15, belongs to ${\mathcal{H}}^{k}$. Such initial datum fulfills the claim of Proposition 3.6. ∎ ### 3.3 Proof of Theorem 2.8 We are finally in position of proving Theorem 2.8. Recall that in Corollary 3.5 we have conjugated equation (2.11) to (3.17) with a change of variables bounded ${\mathcal{H}}^{r}\to{\mathcal{H}}^{r}$ uniformly in time, whereas in Proposition 3.6 we have constructed a solution of the effective equation ${\rm i}\partial_{t}\psi=H_{N}\psi$ with decaying negative Sobolev norms, therefore with growing positive Sobolev norms. The last step is to construct a solution of the full equation (3.17) with growing Sobolev norms. To achieve this, we exploit that the perturbation $R_{N}(t)$ is $N$-smoothing (Definition 2.2). So to proceed we fix the parameters. First fix $r>0$, then choose $N,k\in{\mathbb{N}}$ such that $N\geq 2r+2,\quad k\geq N-r.$ (3.51) Apply Corollary 3.5 with such $N$, producing the operators $T_{N}$, $R_{N}(t)$ and conjugating (2.11) to (3.17). By Proposition 3.6, $\exists\,{\varphi}_{0}\in{\mathcal{H}}^{k}$ such that ${\varphi}(t):=e^{-{\rm i}tH_{N}}{\varphi}_{0}$ fulfills $\forall{\tt r}\in[0,k]$: $\\|{\varphi}(t)\\|_{-{\tt r}}\leq C_{{\tt r},N}\left\langle t\right\rangle^{-{\tt r}}\,\\|{\varphi}_{0}\\|_{{\tt r}},\qquad\forall t\in{\mathbb{R}}\ .$ (3.52) We look for an exact solution $\phi(t)$ of (3.17) of the form $\phi(t)={\varphi}(t)+u(t)$, i.e. $u(t)$ has to satisfy ${\rm i}\partial_{t}u=\big{(}H_{N}+R_{N}(t)\big{)}u+R_{N}(t){\varphi}(t).$ (3.53) Denoting by $U_{N}(t,s)$ the linear propagator of $H_{N}+R_{N}(t)$, we choose $u(t):={\rm i}\int\limits_{t}^{+\infty}U_{N}(t,s)\,R_{N}(s)\,{\varphi}(s)\,{\rm d}s.$ (3.54) We estimate the ${\mathcal{H}}^{r}$ norm of $u(t)$. As $\sup_{t}\\|[H_{N}+R_{N}(t),\,K_{0}]\\|_{{\mathcal{L}}({\mathcal{H}}^{m})}<C_{m}<\infty\ ,\qquad\forall m\in{\mathbb{R}},$ Theorem 1.5 of [33] guarantees that the propagator $U_{N}(t,s)$ extends to a bounded operator ${\mathcal{H}}^{r}\to{\mathcal{H}}^{r}$ fulfilling666apply the theorem with $\tau=0$ and note that in that paper we defined $\\|\psi\\|_{r}\equiv\\|K_{0}^{r/2}\psi\\|_{0}$, therefore the estimate in that paper reads explicitly $\\|K_{0}^{r/2}U_{N}(t,s)\psi\\|_{0}\leq C_{r}\,\left\langle t-s\right\rangle^{r/2}\\|K_{0}^{r/2}\psi\\|_{0}$ $\forall r>0\ \ \ \exists\,C_{r}>0\colon\qquad\\|U_{N}(t,s)\\|_{{\mathcal{L}}({\mathcal{H}}^{r})}\leq C_{r}\,\left\langle t-s\right\rangle^{r},\quad\forall t,s\in{\mathbb{R}}\ .$ This estimate, the smoothing property $R_{N}(t)\colon{\mathcal{H}}^{r-N}\to{\mathcal{H}}^{r}$ and (3.52) with ${\tt r}:=N-r\in[0,k]$ give $\displaystyle\\|u(t)\\|_{r}$ $\displaystyle\leq C_{r}\int\limits_{t}^{+\infty}\left\langle t-s\right\rangle^{r}\\|R_{N}(s)\,{\varphi}(s)\\|_{r}\,{\rm d}s\leq C_{r}\int\limits_{t}^{+\infty}\left\langle t-s\right\rangle^{r}\,\\|{\varphi}(s)\\|_{-(N-r)}\,{\rm d}s$ $\displaystyle\leq C_{r,N}\,\\|{\varphi}_{0}\\|_{N-r}\int\limits_{t}^{+\infty}\left\langle t-s\right\rangle^{r}\,\frac{1}{\left\langle s\right\rangle^{N-r}}\,\,{\rm d}s\leq C_{r,N}\,\\|{\varphi}_{0}\\|_{k}\left\langle t\right\rangle^{-1}\ .$ In particular the ${\mathcal{H}}^{r}$ norm of $u(t)$ decreases to 0 as $t\to\infty$. Then $\phi(t)={\varphi}(t)+u(t)$ fulfills $\\|\phi(t)\\|_{r}\geq\\|{\varphi}(t)\\|_{r}-\\|u(t)\\|_{r}\geq c_{r}\frac{\\|{\varphi}_{0}\\|_{0}^{2}}{\\|{\varphi}_{0}\\|_{r}}\left\langle t\right\rangle^{r}-C_{r,N}\\|{\varphi}_{0}\\|_{k}\left\langle t\right\rangle^{-1}\geq C\left\langle t\right\rangle^{r}\ ,\quad\forall\|t\|\geq T\,,$ (3.55) where we used (3.52) with ${\tt r}=r$ and Remark 3.7. Finally we get a solution of the original equation (2.11) putting $\psi(t)={\mathcal{U}}_{N}(t)^{-1}\phi(t)$, recall Proposition 3.4. The operator ${\mathcal{U}}_{N}(t)$ fulfills (3.15), thus $\psi(t)$ has polynomially growing Sobolev norms as (2.12), concluding the proof of Theorem 2.8. We can also prove the existence of infinitely many solutions undergoing growth of Sobolev norms. ###### Corollary 3.16. There are infinitely many distinct solutions of equation (2.11) with growing Sobolev norms. ###### Proof. We fix $r>0$ and choose $N,k$ as in (3.51). From the previous proof, it follows that any initial data of the form $\psi(0):=({\rm Id}+{\mathcal{K}}_{0}){\varphi}\ ,\qquad{\mathcal{K}}_{t}{\varphi}:={\rm i}\int_{t}^{+\infty}U_{N}(t,s)R_{N}(s)e^{-{\rm i}sH_{N}}{\varphi}\,{\rm d}s,\,\qquad t\geq 0,$ with ${\varphi}\in{\rm Ran}\,g_{J}(H_{N})\cap{\mathcal{H}}^{k}$, gives rise to a solution with growing Sobolev norms (see also Remark 3.8). Here $J$ is the interval of Corollary 3.14. In particular, as $\left\|J\cap\sigma(H_{N})\right\|>0$ and $\sigma(H_{N})\cap J=\sigma_{ac}(H_{N})\cap J$, the set ${\rm Ran}\,g_{J}(H_{N})$ has infinite dimension. Let us prove that ${\rm Id}+{\mathcal{K}}_{0}$ is injective. Assume there are ${\varphi}_{1}\neq{\varphi}_{2}\in{\rm Ran}\,g_{J}(H_{N})\cap{\mathcal{H}}^{k}$ with $({\rm Id}+{\mathcal{K}}_{0}){\varphi}_{1}=({\rm Id}+{\mathcal{K}}_{0}){\varphi}_{2}$. Put $u_{j}(t):={\mathcal{K}}_{t}{\varphi}_{j}$, $j=1,2$; arguing as in the previous proof one has $\\|u_{j}(t)\\|_{r}\to 0$ as $t\to\infty$. Then ${\mathcal{U}}_{N}(t)^{-1}(e^{-{\rm i}tH_{N}}{\varphi}_{j}+u_{j}(t))$, $j=1,2$, both solve (2.11) and have the same initial datum, so they are the same solution $\psi(t)$ of equation (2.11). Then $\displaystyle\\|{\varphi}_{1}-{\varphi}_{2}\\|_{0}$ $\displaystyle=\\|e^{-{\rm i}tH_{N}}({\varphi}_{1}-{\varphi}_{2})\\|_{0}$ $\displaystyle\leq C_{r}\\|{\mathcal{U}}_{N}^{-1}(t)e^{-{\rm i}tH_{N}}({\varphi}_{1}-{\varphi}_{2})\\|_{r}\leq C_{r}\big{(}\\|u_{1}(t)\\|_{r}+\\|u_{2}(t)\\|_{r}\big{)}\to 0$ as $t\to\infty$. Hence ${\varphi}_{1}={\varphi}_{2}$. ∎ ## 4 Applications In the following section we apply Theorem 2.8 to the harmonic oscillator on ${\mathbb{R}}$ and the half-wave equation on ${\mathbb{T}}$. In both cases we construct transporters which are stable under small, time periodic, pseudodifferential perturbations. ### 4.1 Harmonic oscillator on ${\mathbb{R}}$ Consider the quantum harmonic oscillator $\displaystyle{\rm i}\partial_{t}\psi=\frac{1}{2}(-\partial_{x}^{2}+x^{2})\psi+V(t,x,D)\psi,\quad x\in{\mathbb{R}}.$ (4.1) Here $K_{0}:=\frac{1}{2}\left(-\partial_{x}^{2}+x^{2}\right)$ is the quantum Harmonic oscillator, the scale of Hilbert spaces is defined as usual by ${\mathcal{H}}^{r}={\rm Dom}\left(K_{0}^{r}\right)$, and the base space $({\mathcal{H}}^{0},\left\langle\cdot,\cdot\right\rangle)$ is $L^{2}({\mathbb{R}},{\mathbb{C}})$ with its standard scalar product. The perturbation $V$ is chosen as the Weyl quantization of a symbol belonging to the following class: ###### Definition 4.1. A function $f$ is a symbol of order $\rho\in{\mathbb{R}}$ if $f\in C^{\infty}({\mathbb{R}}_{x}\times{\mathbb{R}}_{\xi},{\mathbb{C}})$ and $\forall\alpha,\beta\in{\mathbb{N}}_{0}$, there exists $C_{\alpha,\beta}>0$ such that $\|\partial_{x}^{\alpha}\,\partial_{\xi}^{\beta}f(x,\xi)\|\leq C_{\alpha,\beta}\ (1+\|x\|^{2}+\|\xi\|^{2})^{\rho-\frac{\beta+\alpha}{2}}\ .$ We will write $f\in S^{\rho}_{{\rm har}}$. We endow $S^{\rho}_{\rm har}$ with the family of seminorms $\wp^{\rho}_{j}(f):=\sum_{\|\alpha\|+\|\beta\|\leq j}\ \ \sup_{(x,\xi)\in{\mathbb{R}}^{2}}\frac{\left\|\partial_{x}^{\alpha}\,\partial_{\xi}^{\beta}f(x,\xi)\right\|}{\left(1+\|x\|^{2}+\|\xi\|^{2}\right)^{\rho-\frac{\beta+\alpha}{2}}}\ ,\qquad j\in{\mathbb{N}}\cup\\{0\\}\ .$ Such seminorms turn $S^{\rho}_{\rm har}$ into a Fréchet space. If a symbol $f$ depends on additional parameters (e.g. it is time dependent), we ask that all the seminorms are uniform w.r.t. such parameters. To a symbol $f\in S^{\rho}_{\rm har}$ we associate the operator $f(x,D)$ by standard Weyl quantization $\Big{(}f(x,D)\psi\Big{)}(x):=\frac{1}{2\pi}\iint_{y,\xi\in{\mathbb{R}}}{\rm e}^{{\rm i}(x-y)\xi}\,f\left(\frac{x+y}{2},\xi\right)\,\psi(y)\,{\rm d}y{\rm d}\xi\ .$ ###### Definition 4.2. We say that $F\in{\mathcal{A}}_{\rho}$ if it is a pseudodifferential operator with symbol of class $S^{\rho}_{{\rm har}}$, i.e., if there exists $f\in S^{\rho}_{{\rm har}}$ and $S$ smoothing (in the sense of Definition 2.2) such that $F=f(x,D_{x})+S$. ###### Remark 4.3. With our numerology, the symbol of the harmonic oscillator $K_{0}$ is of order 1, $\frac{1}{2}({x^{2}+\xi^{2}})\in S^{1}_{{\rm har}}$, and not of order 2 as typically in the literature. As an application of the abstract theorems, we describe a class of operators which are transporters. This class, which we call smooth Töplitz operators, is easily described in terms of their matrix elements, which we now introduce. We denote by $\\{{\bf e}_{n}\\}_{n\in{\mathbb{N}}}$ the Hermite basis, formed by the (orthonormal) eigenvectors of the Harmonic oscillator $K_{0}$: $K_{0}{\bf e}_{n}=\left(n-\frac{1}{2}\right){\bf e}_{n},\quad\\|{\bf e}_{n}\\|_{0}=1,\quad n\in{\mathbb{N}}\ .$ (4.2) To each operator $\mathsf{H}\in{\mathcal{L}}({\mathcal{H}})$ we associate its matrix $(\mathsf{H}_{mn})_{m,n\in{\mathbb{N}}}$ with respect to the Hermite basis, whose elements are given by $\mathsf{H}_{mn}:=\left\langle\mathsf{H}\,{\bf e}_{n},{\bf e}_{m}\right\rangle\ ,\qquad\forall m,n\in{\mathbb{N}}\ .$ (4.3) ###### Remark 4.4. If $\mathsf{H}$ is selfadjoint, so is its matrix $(\mathsf{H}_{mn})_{m,n\in{\mathbb{N}}}$, in particular $\mathsf{H}_{mn}=\overline{\mathsf{H}_{nm}}$. ###### Definition 4.5 (Smooth Töplitz operators). A linear operator $\mathsf{H}\in{\mathcal{L}}({\mathcal{H}})$ is said a Töplitz operator if the entries of its matrix are constant along each diagonal, i.e. $\mathsf{H}_{m_{1}n_{1}}=\mathsf{H}_{m_{2}n_{2}},\quad\forall m_{1},n_{1},m_{2},n_{2}\in{\mathbb{N}}\colon\ \ \ m_{1}-n_{1}=m_{2}-n_{2}\ .$ (4.4) A Töplitz operator is said smooth if its matrix elements decay fast off diagonal, i.e. $\forall N>0$, $\exists C_{N}>0$ such that $\left\|\mathsf{H}_{mn}\right\|\leq\frac{C_{N}}{\left\langle m-n\right\rangle^{N}}\ ,\qquad\forall m,n\in{\mathbb{N}}\ .$ (4.5) ###### Example 4.6. The shift operators $S$ and its adjoint $S^{}$ are defined on the Hermite functions $\\{{\bf e}_{n}\\}_{n\geq 1}$ by $S{\bf e}_{n}={\bf e}_{n+1}\ ,\quad\forall n\in{\mathbb{N}}\ ,\qquad S^{}{\bf e}_{n}=\begin{cases}0&\mbox{ if }n=1\\\ {\bf e}_{n-1}&\mbox{ if }n\geq 2\end{cases}\ .$ (4.6) The action of $S$ (and of $S^{}$) is extended on all ${\mathcal{H}}$ by linearity, giving $S\psi=\sum_{n\geq 1}\psi_{n}{\bf e}_{n+1}$, where we defined $\psi_{n}:=\left\langle\psi,{\bf e}_{n}\right\rangle$ for $n\geq 1$. Their matrices are given by $(S_{mn})_{m,n\in{\mathbb{N}}}=\begin{pmatrix}0&&&\\\ 1&0&&\\\ &1&0&\\\ &&\ddots&\ddots\end{pmatrix},\qquad(S^{}_{mn})_{m,n\in{\mathbb{N}}}=\begin{pmatrix}0&1&&\\\ &0&1&\\\ &&0&1\\\ &&&\ddots\end{pmatrix},$ from which it is clear that both $S$ and $S^{}$ are smooth Töplitz operators. We prove in the following that any smooth Töplitz operator is actually a pseudodifferential operator in ${\mathcal{A}}_{0}$, see Lemma 4.10. As an application of the abstract theorems, we show that any smooth Töplitz operator becomes a transporter for the Harmonic oscillator once it is multiplied by an appropriate scalar time periodic function. ###### Theorem 4.7. Let $\mathsf{V}(x,D)$ be a selfadjoint and smooth Töplitz operator (see Definition 4.5). Take $m,n\in{\mathbb{N}}$, $m>n$, such that the matrix element $\mathsf{V}_{m-n}:=\left\langle\mathsf{V}(x,D)\,{\bf e}_{n},{\bf e}_{m}\right\rangle\neq 0\ .$ Then $V(t,x,D):=\cos((m-n)t)\,\mathsf{V}(x,D)$ (4.7) is a transporter for (4.1). More precisely, $\forall r\geq 0$ there exist a solution $\psi(t)\in{\mathcal{H}}^{r}$ of (4.1) and constants $C,T>0$ such that $\\|\psi(t)\\|_{r}\geq C\left\langle t\right\rangle^{r},\quad\forall t>T.$ The theorem follows applying Theorem 2.7. So we check that Assumptions I-III are fulfilled. Regarding Assumption I, it is the usual Weyl calculus for symbols in $S^{\rho}_{{\rm har}}$, see e.g. [40]. Concerning Assumption II, one has $\sigma(K_{0})=\\{n-\frac{1}{2}\\}_{n\in{\mathbb{N}}}$. Furthermore Egorov theorem for the Harmonic oscillator [27] states that the map $t\mapsto e^{-{\rm i}tK_{0}}\mathsf{A}e^{{\rm i}tK_{0}}\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{\rho})$ for any $\mathsf{A}\in{\mathcal{A}}_{\rho}$ (use also the periodicity of the flow of $K_{0}$). This can be seen e.g. by remarking that the symbol of $e^{-{\rm i}tK_{0}}\mathsf{A}e^{{\rm i}tK_{0}}$ is $a\circ\phi_{{\rm har}}^{t}$, where $a\in S^{\rho}_{\rm har}$ is the symbol of $\mathsf{A}$ and $\phi^{t}_{{\rm har}}$ is the time $t$ flow of the harmonic oscillator; explicitly $\displaystyle\left(a\circ\phi^{t}_{{\rm har}}\right)(x,\xi)=a(x\cos t+\xi\sin t,-x\sin t+\xi\cos t)\ .$ (4.8) Verification of Assumption III. First we show that smooth Töplitz operators belong to ${\mathcal{A}}_{0}$. We exploit Chodosh’s characterization [11], which we now recall. Define the discrete difference operator $\triangle$ on a function $M\colon{\mathbb{N}}\times{\mathbb{N}}\to{\mathbb{C}}$ by $(\triangle M)(m,n):=M(m+1,n+1)-M(m,n)\ ,$ and its powers $\triangle^{\gamma}$, $\gamma\in{\mathbb{N}}$, by $\triangle$ applied $\gamma$-times. ###### Definition 4.8 (Symbol matrix). A function $M\colon{\mathbb{N}}\times{\mathbb{N}}\to{\mathbb{C}}$ will be said to be a symbol matrix of order $\rho$ if for any $\gamma\in{\mathbb{N}}_{0}$, $N\in{\mathbb{N}}$, there exists $C_{\gamma,N}>0$ such that $\left\|(\triangle^{\gamma}M)(m,n)\right\|\leq C_{\gamma,N}\frac{(1+m+n)^{\rho-\|\gamma\|}}{\left\langle m-n\right\rangle^{N}}\ ,\quad\forall m,n\in{\mathbb{N}}\ .$ (4.9) The connection between pseudodifferential operators and symbol matrices is given by Chodosh’s characterization: ###### Theorem 4.9 ([11]). An operator $\mathsf{H}$ belongs to ${\mathcal{A}}_{\rho}$ if and only if its matrix $M^{(\mathsf{H})}(m,n):=\mathsf{H}_{mn}$ (as defined in (4.3)) is a symbol matrix of order $\rho$. As a direct consequence we have the following result: ###### Lemma 4.10. Any smooth Töplitz operator is a pseudodifferential operator in ${\mathcal{A}}_{0}$. ###### Proof. We use Theorem 4.9. Let $\mathsf{H}$ be smooth Töplitz and put $M^{(\mathsf{H})}(m,n):=\mathsf{H}_{mn}$. Then (4.9) holds with $\rho=\gamma=0$ by (4.5). By (4.4) one has $\triangle M^{(\mathsf{H})}=0$; so (4.9) holds also $\forall\gamma\geq 1$. ∎ In particular $V(t,x,D)=\cos((m-n)t)\mathsf{V}(x,D)$ belongs to $C^{\infty}({\mathbb{T}},{\mathcal{A}}_{0})$, which is the first required property of Assumption III. ###### Remark 4.11. The shift operators $S,S^{}$, defined in (4.6), belong to ${\mathcal{A}}_{0}$ being smooth Töplitz. Also their (integer) powers $S^{k}$, $S^{k}$, given for $k\in{\mathbb{N}}$ by $S^{k}{\bf e}_{n}={\bf e}_{n+k}\ ,\quad\forall n\in{\mathbb{N}}\ ,\qquad S^{k}{\bf e}_{n}=\begin{cases}0&\mbox{ if }n\leq k\\\ {\bf e}_{n-k}&\mbox{ if }n\geq k+1\end{cases}\ $ (4.10) are smooth Töplitz, so in ${\mathcal{A}}_{0}$. Next we compute the resonant average of $V(t,x,D)$. ###### Lemma 4.12. Let $V(t,x,D)$ as in (4.7). Its resonant average $\left\langle V\right\rangle$ (see (1.3)) is $\left\langle V\right\rangle=\frac{1}{2}\left(\mathsf{V}_{k}\,S^{k}+\overline{\mathsf{V}_{k}}\,S^{k}\right)\ ,\qquad k:=m-n\in{\mathbb{N}}\ ,$ (4.11) where $S\in{\mathcal{A}}_{0}$ is defined in (4.6) and $\mathsf{V}_{k}:=\mathsf{V}_{m-n}:=\left\langle\mathsf{V}\,{\bf e}_{n},{\bf e}_{m}\right\rangle\in{\mathbb{C}}$. ###### Proof. For $\ell\in{\mathbb{N}}$, denote by $\Pi_{\ell}{\varphi}:=\left\langle{\varphi},{\bf e}_{\ell}\right\rangle\,{\bf e}_{\ell}$ the projector on the Hermite function ${\bf e}_{\ell}$. Clearly $e^{{\rm i}sK_{0}}\,\Pi_{\ell}=\Pi_{\ell}\,e^{{\rm i}sK_{0}}=e^{{\rm i}s(\ell-\frac{1}{2})}\,\Pi_{\ell},\quad\forall\ell\in{\mathbb{N}}\ .$ From now on we simply write $\mathsf{V}\equiv\mathsf{V}(x,D)$. Using this identity and writing ${\rm Id}=\sum_{\ell\geq 1}\Pi_{\ell}$ we get $\displaystyle e^{{\rm i}sK_{0}}\,\mathsf{V}\,e^{-{\rm i}sK_{0}}$ $\displaystyle=\sum_{j,\ell\geq 1}e^{{\rm i}s(j-\ell)}\Pi_{j}\,\mathsf{V}\,\Pi_{\ell}=\sum_{j,\ell\geq 1}e^{{\rm i}s(j-\ell)}\,\left\langle\cdot,{\bf e}_{\ell}\right\rangle\,\left\langle\mathsf{V}{\bf e}_{\ell},{\bf e}_{j}\right\rangle\,{\bf e}_{j}\ .$ Now we compute, with $k:=m-n\in{\mathbb{N}}$, $\displaystyle\left\langle V\right\rangle$ $\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}\cos(ks)\,e^{{\rm i}sK_{0}}\,\mathsf{V}\,e^{-{\rm i}sK_{0}}{\rm d}s=\sum_{j,\ell\geq 1}\left\langle\mathsf{V}{\bf e}_{\ell},{\bf e}_{j}\right\rangle\,\left\langle\cdot,{\bf e}_{\ell}\right\rangle\,{\bf e}_{j}\,\frac{1}{2\pi}\int_{0}^{2\pi}\cos(ks)\,e^{{\rm i}s(j-\ell)}\,{\rm d}s$ $\displaystyle=\frac{1}{2}\sum_{\ell\geq 1}\left\langle\mathsf{V}{\bf e}_{\ell},{\bf e}_{\ell+k}\right\rangle\left\langle\cdot,{\bf e}_{\ell}\right\rangle\,{\bf e}_{\ell+k}+\frac{1}{2}\sum_{\ell\geq k+1}\left\langle\mathsf{V}{\bf e}_{\ell},{\bf e}_{\ell-k}\right\rangle\left\langle\cdot,{\bf e}_{\ell}\right\rangle\,{\bf e}_{\ell-k}=\frac{1}{2}\mathsf{V}_{k}\,S^{k}+\frac{1}{2}\overline{\mathsf{V}_{k}}\,S^{k}$ where in the last line we used $\mathsf{V}_{-k}=\left\langle\mathsf{V}{\bf e}_{\ell},{\bf e}_{\ell-k}\right\rangle=\overline{\left\langle\mathsf{V}{\bf e}_{\ell-k},{\bf e}_{\ell}\right\rangle}=\overline{\mathsf{V}_{k}}$ being $\mathsf{V}$ selfadjoint and smooth Töplitz (see Remark 4.4). ∎ Now define the selfadjoint operator $A:=\frac{\mathsf{V}_{k}}{{\rm i}}\,(K_{0}+\frac{1}{2})\,S^{k}-\frac{\overline{\mathsf{V}_{k}}}{{\rm i}}\,S^{k}\,(K_{0}+\frac{1}{2})-\frac{\overline{\mathsf{V}_{k}}}{{\rm i}}\,(K_{0}+\frac{1}{2})\,S^{k}+\frac{{\mathsf{V}_{k}}}{{\rm i}}\,S^{k}\,(K_{0}+\frac{1}{2})\ ,$ (4.12) which belongs to ${\mathcal{A}}_{1}$ by symbolic calculus as $K_{0}\in{\mathcal{A}}_{1}$ and $S,S^{}\in{\mathcal{A}}_{0}$ (see Remark 4.11). The next lemma verifies Assumption III. ###### Lemma 4.13. Assume that $\mathsf{V}_{k}\neq 0$. The following holds true: (i) The spectrum of the operator $H_{0}:=\left\langle V\right\rangle$ fulfills $\sigma(H_{0})\supseteq\left[-\|\mathsf{V}_{k}\|,\,\|\mathsf{V}_{k}\|\right]$. * (ii) For any interval $I_{0}\subset\left[-\|\mathsf{V}_{k}\|,\,\|\mathsf{V}_{k}\|\right]$, any $g_{I_{0}}\in C^{\infty}_{c}({\mathbb{R}},{\mathbb{R}}_{\geq 0})$ with $g_{I_{0}}\equiv 1$ over $I_{0}$ and ${\rm supp}\,g_{I_{0}}\subset\left[-\|\mathsf{V}_{k}\|,\,\|\mathsf{V}_{k}\|\right]$, there exist $\theta>0$ and $\mathsf{K}$ compact operator such that $g_{I_{0}}(H_{0})\,{\rm i}[H_{0},A]\,g_{I_{0}}(H_{0})\geq\theta\,g_{I_{0}}(H_{0})^{2}+\mathsf{K}\ .$ Here $A$ is defined in (4.12). ###### Proof. $(i)$ Let ${\mathtt{f}}(\rho):={\rm Re}(\mathsf{V}_{k}\,e^{-{\rm i}\rho k})$. We shall prove that ${\mathtt{f}}(\rho)\in\sigma(H_{0})$ $\forall\rho\in{\mathbb{R}}$, from which the claim follows. As $H_{0}$ is selfadjoint, it is enough to construct a Weyl sequence for ${\mathtt{f}}(\rho)$, i.e. a sequence $(\psi^{(n)})_{n\geq 1}$ with $\\|\psi^{(n)}\\|_{0}=1$ $\,\forall n$ and $\\|(H_{0}-{\mathtt{f}}(\rho))\psi^{(n)}\\|_{0}\to 0$ as $n\to\infty$. We put $\psi^{(n)}:=\frac{1}{\sqrt{n}}\sum_{\ell=1}^{n}e^{{\rm i}\rho\ell}{\bf e}_{\ell}\ .$ Then $\\|\psi^{(n)}\\|_{0}=1$ $\,\forall n$ and a direct computation shows that for $n>k$ $H_{0}\psi^{(n)}=\frac{1}{\sqrt{n}}\frac{\overline{\mathsf{V}}_{k}}{2}e^{{\rm i}\rho k}\sum_{m=1}^{k}e^{{\rm i}\rho m}\,{\bf e}_{m}+\frac{1}{\sqrt{n}}{\mathtt{f}}(\rho)\sum_{m=k+1}^{n-k}e^{{\rm i}\rho m}\,{\bf e}_{m}+\frac{1}{\sqrt{n}}\frac{\mathsf{V}_{k}}{2}e^{-{\rm i}\rho k}\sum_{m=n-k+1}^{n+k}e^{{\rm i}\rho m}\,{\bf e}_{m}\ .$ Thus one finds a constant $C_{k}>0$ such that $\\|(H_{0}-{\mathtt{f}}(\rho))\psi^{(n)}\\|_{0}\leq\frac{C_{k}}{\sqrt{n}}\to 0\quad\mbox{ as }n\to\infty\ ,$ proving that $\psi^{(n)}$ is a Weyl sequence; by Weyl criterium ${\mathtt{f}}(\rho)\in\sigma(H_{0})$. $(ii)$ First note that, by (4.2) and (4.6), one has $\forall k\in{\mathbb{N}}$ $\displaystyle[S^{k},K_{0}]=-kS^{k},\qquad[S^{k},K_{0}]=kS^{k},\qquad[S^{k},S^{k}]=\Pi_{\leq k}$ (4.13) $\displaystyle S^{k}S^{k}={\rm Id}-\Pi_{\leq k}\,\qquad S^{k}S^{k}={\rm Id}$ (4.14) where $\Pi_{\leq k}:=\sum_{\ell=1}^{k}\Pi_{\ell}$ is the projector on the Hermite modes with index $\leq k$. Using (4.13) a direct computation gives $\displaystyle{\rm i}[H_{0},A]$ $\displaystyle=k\big{(}2\|\mathsf{V}_{k}\|^{2}-\mathsf{V}_{k}^{2}S^{2k}-\overline{\mathsf{V}}^{2}_{k}S^{2k}-\|\mathsf{V}_{k}\|^{2}\Pi_{\leq k}\big{)}+2\|\mathsf{V}_{k}\|^{2}(K_{0}+\frac{1}{2})\Pi_{\leq k}$ $\displaystyle=4k\big{(}\|\mathsf{V}_{k}\|^{2}-H_{0}^{2}\big{)}+2\|\mathsf{V}_{k}\|^{2}(K_{0}+\frac{1}{2}-k)\Pi_{\leq k}\ .$ Clearly $\mathsf{K}:=2\|\mathsf{V}_{k}\|^{2}(K_{0}+\frac{1}{2}-k)\Pi_{\leq k}$ is compact, being finite rank. Next put $\tilde{f}(\lambda)=4k(\|\mathsf{V}_{k}\|^{2}-\lambda^{2})$ getting $\forall{\varphi}\in{\mathcal{H}}$ $\left\langle g_{I_{0}}(H_{0})\,{\rm i}[H_{0},A]\,g_{I_{0}}(H_{0}){\varphi},{\varphi}\right\rangle=\left\langle g_{I_{0}}(H_{0})\,\tilde{f}(H_{0})\,g_{I_{0}}(H_{0}){\varphi},{\varphi}\right\rangle+\left\langle g_{I_{0}}(H_{0})\,\mathsf{K}\,g_{I_{0}}(H_{0}){\varphi},{\varphi}\right\rangle\ .$ (4.15) Note that $\tilde{f}$ is strictly positive in the interior of $\left[-\|\mathsf{V}_{k}\|,\,\|\mathsf{V}_{k}\|\right]$; we put $\theta:=\inf\\{\tilde{f}(\lambda)\colon\lambda\in{\rm supp}\,g_{I_{0}}\\}>0\ .$ With this information we apply the spectral theorem and get $\displaystyle\left\langle g_{I_{0}}(H_{0})\,\tilde{f}(H_{0})\,g_{I_{0}}(H_{0}){\varphi},{\varphi}\right\rangle$ $\displaystyle=\int\limits_{\lambda\in\sigma(H_{0})}g_{I_{0}}(\lambda)^{2}\,\tilde{f}(\lambda)\,{\rm d}m_{\varphi}(\lambda)$ $\displaystyle\geq\theta\int\limits_{\lambda\in\sigma(H_{0})}g_{I_{0}}(\lambda)^{2}\,\,{\rm d}m_{\varphi}(\lambda)=\theta\\|g_{I_{0}}(H_{0}){\varphi}\\|_{0}^{2}\ .$ This estimate and (4.15) proves that $H_{0}$ fulfills a Mourre estimate over $I_{0}$. ∎ To conclude this section, we recall that in [34] it is proved that the pseudodifferential operator $\displaystyle V(t):=e^{-{\rm i}tK_{0}}\,(S+S^{})\,e^{{\rm i}tK_{0}}$ (4.16) is a universal transporter (see Definition 1.1). Using the abstract Theorem 2.8 we prove its stability under perturbations of class $C^{\infty}({\mathbb{T}},{\mathcal{A}}_{0})$: ###### Theorem 4.14. Consider equation (4.1) with $V(t)$ defined in (4.16). There exist $\epsilon_{0},{\mathtt{M}}>0$ such that $\forall W\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{0})$ with $\sup_{t}\wp^{0}_{\mathtt{M}}(W(t))\leq\epsilon_{0}$, the operator $V+\epsilon W$ is a transporter. More precisely $\forall r>0$ there exist a solution $\psi(t)\in{\mathcal{H}}^{r}$ of ${\rm i}\partial_{t}\psi=\big{(}\frac{-\partial_{x}^{2}+x^{2}}{2}+V(t)+W(t)\big{)}\psi$ and constants $C,T>0$ such that $\\|\psi(t)\\|_{r}\geq C\left\langle t\right\rangle^{r},\quad\forall t\geq T\ .$ ###### Proof. Again we verify Assumption III. It is clear that $V(t)\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{0})$ and that $\left\langle V\right\rangle=S+S^{}$, so it has the form (4.11) with $k=1$ and $\mathsf{V}_{1}=2$. Then Lemma 4.13 implies that $\left\langle V\right\rangle$ fulfills a Mourre estimate. ∎ ### 4.2 Half-wave equation on ${\mathbb{T}}$ The half-wave equation on ${\mathbb{T}}$ is given by ${\rm i}\partial_{t}\psi=\|D\|\psi+V(t,x,D)\psi\ ,\qquad x\in{\mathbb{T}}\ .$ (4.17) Here $\|D\|$ is the Fourier multiplier defined by $\|D\|\psi:=\sum_{j\in{\mathbb{Z}}}\|j\|\,\psi_{j}\,e^{{\rm i}jx}\ ,\qquad\psi_{j}:=\frac{1}{2\pi}\int_{\mathbb{T}}\psi(x)e^{-{\rm i}jx}{\rm d}x\ ,$ whereas $V(t,x,D)$ is a pseudodifferential operator of order 0. In this case $K_{0}:=\|D\|+1$, the scale of Hilbert spaces defined as ${\mathcal{H}}^{r}={\rm Dom}\left(K_{0}^{r}\right)$ coincides with standard Sobolev spaces on the torus $H^{r}({\mathbb{T}})$, and the base space $({\mathcal{H}}^{0},\left\langle\cdot,\cdot\right\rangle)$ is $L^{2}({\mathbb{T}},{\mathbb{C}})$ with its standard scalar product. In this setting we shall use pseudodifferential operators with periodic symbols, belonging to the following class: ###### Definition 4.15. A function $a(x,\xi)$ is a periodic symbol of order $\rho\in{\mathbb{R}}$ if $a\in C^{\infty}({\mathbb{T}}_{x}\times{\mathbb{R}}_{\xi},{\mathbb{C}})$ and for any $\alpha,\beta\in{\mathbb{N}}_{0}$, there exists a constant $C_{\alpha\beta}>0$ such that $\left\|\partial_{x}^{\alpha}\,\partial_{\xi}^{\beta}\,a(x,\xi)\right\|\leq C_{\alpha\beta}\,\left\langle\xi\right\rangle^{\rho-\beta},\quad\forall x\in{\mathbb{T}},\,\forall\xi\in{\mathbb{R}}\,.$ (4.18) We will write $a\in S^{\rho}_{{\rm per}}$. We also put $S^{-\infty}_{\rm per}:=\bigcap_{\rho\in{\mathbb{R}}}S^{\rho}_{\rm per}$ the class of smoothing symbols. We endow $S^{\rho}_{\rm per}$ with the family of seminorms $\wp^{\rho}_{j}(a):=\sum_{\|\alpha\|+\|\beta\|\leq j}\ \ \sup_{(x,\xi)\in{\mathbb{T}}\times{\mathbb{R}}}{\left\|\partial_{x}^{\alpha}\,\partial_{\xi}^{\beta}a(x,\xi)\right\|\,\left\langle\xi\right\rangle^{-\rho+\beta}}\ ,\qquad j\in{\mathbb{N}}_{0}\ .$ (4.19) Such seminorms turn $S^{\rho}_{\rm per}$ into a Fréchet space. If a symbol $a$ depends on additional parameters (e.g. it is time dependent), we ask that all the seminorms are uniform w.r.t. such parameters. To a symbol $a\in S^{\rho}_{\rm per}$ we associate its quantization $a(x,D)$ acting on a $2\pi$-periodic function $u(x)=\sum_{j\in{\mathbb{Z}}}u_{j}e^{{\rm i}jx}$ as $a(x,D)u:={\rm Op}{(a)}[u]:=\sum_{j\in{\mathbb{Z}}}\,a(x,j)\,u_{j}\,e^{{\rm i}jx}\,.$ (4.20) ###### Remark 4.16. Given a symbol $a(\xi)$ independent of $x$, then ${\rm Op}{(a)}$ is the Fourier multiplier operator $a(D)u=\sum_{j\in{\mathbb{Z}}}\,a(j)\,u_{j}\,e^{{\rm i}jx}$. If instead the symbol $a(x)$ is independent of $\xi$, then ${\rm Op}{(a)}$ is the multiplication operator ${\rm Op}{(a)}u=a(x)u$. ###### Definition 4.17. We say that $A\in{\mathcal{A}}_{\rho}$ if $A={\rm Op}(a)$ with $a\in S^{\rho}_{{\rm per}}$. ###### Example 4.18. The operator $\|D\|\in{\mathcal{A}}_{1}$ with symbol given by ${\mathtt{d}}(\xi):=\|\xi\|\chi(\xi)$ where $\chi$ is an even, positive smooth cut-off function satisfying $\chi(\xi)=0$ for $\|\xi\|\leq\frac{1}{5}$, $\chi(\xi)=1$ for $\|\xi\|\geq\frac{2}{5}$ and $\partial_{\xi}\chi(\xi)>0$ $\,\forall\xi\in(\frac{1}{5},\frac{2}{5})$. Also the Fourier projectors $\Pi_{\pm}$ and $\Pi_{0}$ defined by $\Pi_{+}u:=\sum_{j\geq 1}u_{j}\,e^{{\rm i}jx},\qquad\Pi_{-}u:=\sum_{j\leq-1}u_{j}\,e^{{\rm i}jx},\qquad\Pi_{0}u:=u_{0}$ (4.21) are pseudodifferential operators. In particular $\Pi_{\pm}={\rm Op}\left(\pi_{\pm}\right)\in{\mathcal{A}}_{0}$ and $\Pi_{0}={\rm Op}\left(\pi_{0}\right)\in{\mathcal{A}}_{-\infty}$, where $\pi_{\pm},\pi_{0}$ are a smooth partition of unity, $\pi_{+}(\xi)+\pi_{-}(\xi)+\pi_{0}(\xi)=1$ $\,\forall\xi$, fulfilling $\pi_{+}(\xi)=\begin{cases}1&\mbox{ if }\xi\geq\frac{4}{5}\\\ 0&\mbox{ if }\xi\leq\frac{3}{5}\end{cases}\ ,\quad\pi_{-}(\xi)=\begin{cases}1&\mbox{ if }\xi\leq-\frac{4}{5}\\\ 0&\mbox{ if }\xi\geq-\frac{3}{5}\end{cases}\ ,\quad\pi_{0}(\xi)=\begin{cases}1&\mbox{ if }\|\xi\|\leq\frac{3}{5}\\\ 0&\mbox{ if }\|\xi\|\geq\frac{1}{5}\end{cases}\ .$ (4.22) In this setting we prove that any multiplication operator, multiplied by an appropriate time periodic function, becomes a transporter. Here the result. ###### Theorem 4.19. Let $v\in C^{\infty}({\mathbb{T}},{\mathbb{R}})$. Choose $j\in{\mathbb{Z}}\setminus\\{0\\}$ such that the Fourier coefficient $v_{j}\neq 0$. Then the selfadjoint operator $V(t,x):=\cos(jt)\,v(x)$ (4.23) is a transporter. More precisely, $\forall r>0$ there exist a solution $\psi(t)\in{\mathcal{H}}^{r}$ of ${\rm i}\partial_{t}\psi=\big{(}\|D\|+V(t,x)\big{)}\psi$ and constants $C,T>0$ such that $\\|\psi(t)\\|_{r}\geq C\left\langle t\right\rangle^{r},\quad\forall t>T.$ The theorem follows from Theorem 2.7. So first we put ourselves in the setting of the abstract theorem and rewrite (4.17) as ${\rm i}\partial_{t}\psi=K_{0}\psi+{\widetilde{V}}(t,x)\psi,\qquad{\widetilde{V}}(t,x):=\cos(jt)v(x)-1\in{\mathcal{A}}_{0}\ .$ (4.24) Again we check Assumptions I-III. Regarding Assumption I, it is the usual pseudodifferential calculus for periodic symbols, see e.g. [39]. Verification of Assumption II. One has $\sigma(K_{0})=\\{n\\}_{n\in{\mathbb{N}}}$. To prove Assumption II $(ii)$ we use the identity $e^{-{\rm i}tK_{0}}\mathsf{A}\,e^{{\rm i}tK_{0}}=e^{-{\rm i}t\|D\|}\mathsf{A}\,e^{{\rm i}t\|D\|}$ and Egorov theorem for $\|D\|$, see e.g. [43, Theorem 4.3.6]. Actually we need also the following version of Egorov theorem. ###### Lemma 4.20. Let $a\in S^{\rho}_{\rm per}$, $\rho\in{\mathbb{R}}$. Then $e^{{\rm i}t\|D\|}\,{\rm Op}\left(a\right)\,e^{-{\rm i}t\|D\|}={\rm Op}\left(a(x+t,\xi)\right)\Pi_{+}+{\rm Op}\left(a(x-t,\xi)\right)\Pi_{-}+R(t)$ (4.25) where $\Pi_{\pm}$ are defined in (4.21) and $R(t)\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{\rho-1})$. If ${\rm Op}\left(a\right)$ is selfadjoint, so is $e^{{\rm i}t\|D\|}{\rm Op}\left(a\right)\,e^{-{\rm i}t\|D\|},$ $\,\forall t$. ###### Proof. The classical Egorov theorem for the half-Laplacian $\|D\|$ says that $e^{{\rm i}t\|D\|}\,{\rm Op}\left(a\right)\,e^{-{\rm i}t\|D\|}={\rm Op}\left(a\circ\phi_{\mathtt{d}}^{t}(x,\xi)\right)+{\widetilde{R}}(t)$ where $\phi_{\mathtt{d}}^{t}(x,\xi)$ is the time $t$ flow of the classical Hamiltonian ${\mathtt{d}}(\xi)=\|\xi\|\chi(\xi)$ (the symbol of $\|D\|$) and ${\widetilde{R}}(t)\in C^{\infty}({\mathbb{R}},{\mathcal{A}}_{\rho-1})$, see e.g. [43, Theorem 4.3.6]. We compute more explicitly $a\circ\phi_{\mathtt{d}}^{t}(x,\xi)$. The Hamiltonian equations of ${\mathtt{d}}(\xi)$ and its flow $\phi^{t}_{\mathtt{d}}$ are given by $\begin{cases}\dot{x}=\partial_{\xi}{\mathtt{d}}(\xi)={\mathtt{d}}^{\prime}(\xi)\\\ \dot{\xi}=-\partial_{x}{\mathtt{d}}(\xi)=0\end{cases}\ ,\qquad\phi^{t}_{\mathtt{d}}(x,\xi)=(x+t{\mathtt{d}}^{\prime}(\xi),\,\xi)\ .$ As ${\mathtt{d}}^{\prime}(\xi)=1$ for $\xi\geq\frac{2}{5}$ and ${\mathtt{d}}^{\prime}(\xi)=-1$ for $\xi\leq-\frac{2}{5}$, we write $(a\circ\phi_{\mathtt{d}}^{t})(x,\xi)=a(x+t,\xi)\,\pi_{+}(\xi)+a(x-t,\xi)\,\pi_{-}(\xi)+a(x+t{\mathtt{d}}^{\prime}(\xi),\xi)\,\pi_{0}(\xi)\ .$ As $\pi_{0}\in S^{-\infty}_{\rm per}$, the operator ${\rm Op}\left(a(x+t{\mathtt{d}}^{\prime}(\xi),\xi)\,\pi_{0}(\xi)\right)\in C^{\infty}({\mathbb{R}},{\mathcal{A}}_{-\infty})$. Moreover by symbolic calculus ${\rm Op}\left(a(x\pm t,\xi)\,\pi_{\pm}(\xi)\right)={\rm Op}\left(a(x\pm t,\xi)\right)\Pi_{\pm}+R_{\pm}(t),\quad R_{\pm}(t)\in C^{\infty}({\mathbb{R}},{\mathcal{A}}_{\rho-1})\ .$ Formula (4.25) follows with $R(t):={\widetilde{R}}(t)+R_{+}(t)+R_{-}(t)+{\rm Op}\left(a(x+t{\mathtt{d}}^{\prime}(\xi),\xi)\,\pi_{0}(\xi)\right)$. We claim that $R(t)$ is periodic in time. This follows by difference since both $e^{{\rm i}t\|D\|}\,{\rm Op}\left(a\right)\,e^{-{\rm i}t\|D\|}$ and ${\rm Op}\left(a(x\pm t,\xi)\right)\Pi_{\pm}$ are periodic in $t$ (recall that the symbol $a(x,\xi)$ is periodic in $x$). Finally as $e^{\pm{\rm i}t\|D\|}$ are unitary, the claim on the selfadjointness of $e^{{\rm i}t\|D\|}\,{\rm Op}\left(a\right)\,e^{-{\rm i}t\|D\|}$ follows. ∎ Verification of Assumption III. First we compute $\langle{\widetilde{V}}\rangle$. ###### Lemma 4.21. The resonant average $\langle{\widetilde{V}}\rangle\in{\mathcal{A}}_{0}$ of ${\widetilde{V}}$ (defined in (4.24)) is given by $\langle{\widetilde{V}}\rangle={\mathtt{v}}(x)-1+R,\qquad{\mathtt{v}}(x):={\rm Re}(v_{j}e^{{\rm i}jx})$ (4.26) and $R\in{\mathcal{A}}_{-1}$ is selfadjoint. ###### Proof. First remark that, as $e^{{\rm i}tK_{0}}=e^{{\rm i}t\|D\|}e^{{\rm i}t}$, $\displaystyle\langle{\widetilde{V}}\rangle$ $\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}e^{{\rm i}s\|D\|}\,{\widetilde{V}}(s)\,e^{-{\rm i}s\|D\|}{\rm d}s=\frac{1}{2\pi}\int_{0}^{2\pi}\cos(js)\,e^{{\rm i}s\|D\|}\,v(x)\,e^{-{\rm i}s\|D\|}\,{\rm d}s-1\ .$ (4.27) We compute $e^{{\rm i}s\|D\|}\,v(x)\,e^{-{\rm i}s\|D\|}$ with the aid of Lemma 4.20, getting $e^{{\rm i}s\|D\|}\,v(x)\,e^{-{\rm i}s\|D\|}=v(x+s)\,\Pi_{+}+v(x-s)\,\Pi_{-}+{\widetilde{R}}(s),$ (4.28) where ${\widetilde{R}}(s)\in C^{\infty}({\mathbb{T}},{\mathcal{A}}_{-1})$. Then, recalling that $v_{j}=\overline{v_{-j}}$ being $v(x)$ real valued, $\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}\cos(js)\,e^{{\rm i}s\|D\|}\,v(x)\,e^{-{\rm i}s\|D\|}\,{\rm d}s$ $\displaystyle\stackrel{{\scriptstyle\eqref{eg.per}}}{{=}}\sum_{\sigma=\pm}\frac{1}{2\pi}\int_{0}^{2\pi}\cos(js)\,v(x\sigma s)\,{\rm d}s\,\Pi_{\sigma}+\frac{1}{2\pi}\int_{0}^{2\pi}\cos(js){\widetilde{R}}(s){\rm d}s$ $\displaystyle={\rm Re}\left(v_{j}e^{{\rm i}jx}\right)\,(\Pi_{+}+\Pi_{-})+\frac{1}{2\pi}\int_{0}^{2\pi}\cos(js){\widetilde{R}}(s){\rm d}s$ $\displaystyle={\rm Re}\left(v_{j}e^{{\rm i}jx}\right)+R$ where $R:=\frac{1}{2\pi}\int_{0}^{2\pi}\cos(js){\widetilde{R}}(s){\rm d}s-{\rm Re}\left(v_{j}e^{{\rm i}jx}\right)\Pi_{0}\in{\mathcal{A}}_{-1}$. Together with (4.27), this proves (4.26). Finally $R$ is selfadjoint by difference, since both $\langle{\widetilde{V}}\rangle$ and ${\mathtt{v}}(x)-1$ are selfadjoint operators. ∎ Define the selfadjoint operator $\displaystyle A:={\mathtt{w}}(x)\,\frac{\partial_{x}}{{\rm i}}+\frac{\partial_{x}}{{\rm i}}\,{\mathtt{w}}(x)\ ,\qquad{\mathtt{w}}(x):={\rm Im}(v_{j}e^{{\rm i}jx})\ $ (4.29) belonging to ${\mathcal{A}}_{1}$. The next lemma verifies Assumption III. ###### Lemma 4.22. Assume that $v_{j}\neq 0$. The following holds true: * (i) The operator $H_{0}:=\langle{\widetilde{V}}\rangle$ has spectrum $\sigma(H_{0})\supseteq[-\|v_{j}\|-1,\,\|v_{j}\|-1]=:I$. * (ii) For any interval $I_{0}\subset I$, any $g_{I_{0}}\in C^{\infty}_{c}({\mathbb{R}},{\mathbb{R}}_{\geq 0})$ with $g_{I_{0}}\equiv 1$ over $I_{0}$ and ${\rm supp}\,g_{I_{0}}\subset I$, there exist $\theta>0$ and a compact operator $\mathsf{K}$ such that $g_{I_{0}}(H_{0})\,{\rm i}[H_{0},A]\,g_{I_{0}}(H_{0})\geq\theta\,g_{I_{0}}(H_{0})^{2}+\mathsf{K}\ .$ Here $A$ is defined in (4.29). ###### Proof. During the proof we shall use that any operator in ${\mathcal{A}}_{-1}$ is compact. Moreover we shall simply denote any compact operator by $\mathsf{K}$, which can change from line to line. $(i)$ By Lemma 4.21, $H_{0}$ is a compact perturbation of the multiplication operator by ${\mathtt{v}}(x)-1$, whose spectrum coincides with $I$. Then by Weyl’s theorem $\sigma(H_{0})\supseteq\sigma_{ess}(H_{0})=\sigma_{ess}({\mathtt{v}}(x)-1)=I\ .$ $(ii)$ First notice that, as ${\mathtt{v}}(x)={\rm Re}(v_{j}e^{{\rm i}jx})$ and ${\mathtt{w}}(x)={\rm Im}(v_{j}e^{{\rm i}jx})$, one has the identities ${\mathtt{v}}(x)^{2}+{\mathtt{w}}(x)^{2}=\|v_{j}\|^{2},\qquad{\mathtt{v}}^{\prime}(x)=-j\,{\mathtt{w}}(x)\ .$ (4.30) Next we compute $\displaystyle{\rm i}[H_{0},A]$ $\displaystyle={\rm i}[{\mathtt{v}}(x)-1+R,A]=-2{\mathtt{w}}(x)\,{\mathtt{v}}^{\prime}(x)+{\rm i}[R,A]$ $\displaystyle\stackrel{{\scriptstyle\eqref{vw}}}{{=}}2j\big{(}\|v_{j}\|^{2}-{\mathtt{v}}(x)^{2}\big{)}+\mathsf{K}=2j\big{(}\|v_{j}\|^{2}-(H_{0}+1-R)^{2}\big{)}+\mathsf{K}$ $\displaystyle=2j\big{(}\|v_{j}\|^{2}-(H_{0}+1)^{2}\big{)}+\mathsf{K}\ .$ Putting $f(\lambda):=2j\big{(}\|v_{j}\|^{2}-(\lambda+1)^{2}\big{)}$, we get $\forall{\varphi}\in{\mathcal{H}}$ $\displaystyle\left\langle g_{I_{0}}(H_{0})\,{\rm i}[H_{0},A]\,g_{I_{0}}(H_{0}){\varphi},{\varphi}\right\rangle=\left\langle g_{I_{0}}(H_{0})\,f(H_{0})\,g_{I_{0}}(H_{0}){\varphi},{\varphi}\right\rangle+\left\langle\mathsf{K}{\varphi},{\varphi}\right\rangle.$ (4.31) Now we notice that $f(\lambda)$ is positive in the interior of $I$; so we put $\theta:=\inf\left\\{f(\lambda)\colon\ \ \lambda\in{\rm supp}\,g_{I_{0}}\right\\}>0\ .$ With this information we apply the spectral theorem, getting, as in the previous section, $\displaystyle\left\langle g_{I_{0}}(H_{0})\,f(H_{0})\,g_{I_{0}}(H_{0}){\varphi},{\varphi}\right\rangle$ $\displaystyle\geq\theta\int\limits_{\lambda\in I}g_{I_{0}}(\lambda)^{2}\,\,{\rm d}m_{\varphi}(\lambda)=\theta\\|g_{I_{0}}(H_{0}){\varphi}\\|_{0}^{2}\ .$ (4.32) This together with (4.31) establishes the Mourre estimate over $I_{0}$. ∎ ## Appendix A Flows of pseudodifferential operators In this appendix we collect some known results about the flow generated by pseudodifferential operators belonging to the algebra ${\mathcal{A}}$. The setting is the same as [5] and we refer to that paper for the proofs. The first result describes how a Schrödinger equation is changed under a change of variables induced by the flow of a pseudodifferential operator, see Lemma 3.1 of [5]: ###### Lemma A.1. Let $H(t)$ be a time dependent selfadjoint operator, and $X(t)$ be a selfadjoint family of operators. Assume that $\psi(t)=e^{-{\rm i}X(t)}{\varphi}(t)$ then ${\rm i}\partial_{t}\psi=H(t)\psi\ \quad\iff\quad{\rm i}\partial_{t}{\varphi}=H^{+}(t){\varphi}$ (A.1) where $\displaystyle H^{+}(t):=e^{{\rm i}X(t)}\,H(t)\,e^{-{\rm i}X(t)}-\int_{0}^{1}e^{{\rm i}sX(t)}\,(\partial_{t}X(t))\,e^{-{\rm i}sX(t)}\ {\rm d}s\ .$ (A.2) The next property we shall need is the Lie expansion of $e^{{\rm i}X}\,A\,e^{-{\rm i}X}$ in operators of decreasing order, see Lemma 3.2 of [5]: ###### Lemma A.2. Let $X\in{\mathcal{A}}_{\rho}$ with $\rho<1$ be a symmetric operator. Let $A\in{\mathcal{A}}_{m}$ with $m\in{\mathbb{R}}$. Then $e^{{\rm i}\tau X}\,A\,e^{-{\rm i}\tau X}$ is selfadjoint and for any $M\geq 1$ we have777in [5] we have defined ${\rm ad}_{X}(A)={\rm i}[X,A]$ rather than (3.21); so we formulate the next result with the current notation $e^{{\rm i}\tau X}\,A\,e^{-{\rm i}\tau X}=\sum_{\ell=0}^{M}\frac{\tau^{\ell}}{{\rm i}^{\ell}\,\ell!}{\rm ad}_{X}^{\ell}(A)+R_{M}(\tau,X,A)\ ,\qquad\forall\tau\in{\mathbb{R}}\ ,$ (A.3) where $R_{M}(\tau,X,A)\in{\mathcal{A}}_{m-(M+1)(1-\rho)}$. In particular ${\rm ad}_{X}^{\ell}(A)\in{\mathcal{A}}_{m-\ell(1-\rho)}$ and $e^{{\rm i}\tau X}\,A\,e^{-{\rm i}\tau X}\in{\mathcal{A}}_{m}$, $\forall\tau\in{\mathbb{R}}$. The last result concerns boundedness properties of the operator $e^{-{\rm i}\tau X}$, see Lemma 3.3 of [5]: ###### Lemma A.3. Assume that $X(t)$ is a family of selfadjoint operators in ${\mathcal{A}}_{1}$ s.t. $\sup_{t\in{\mathbb{R}}}\wp^{1}_{j}(X(t))<\infty\ ,\quad\forall j\geq 1\ .$ (A.4) Then $e^{-{\rm i}\tau X(t)}$ extends to an operator in ${\mathcal{L}}({\mathcal{H}}^{r})$ $\,\forall r\in{\mathbb{R}}$, and moreover there exist $c_{r},C_{r}>0$ s.t. $c_{r}\\|\psi\\|_{r}\leq\\|e^{-{\rm i}\tau X(t)}\psi\\|_{r}\leq C_{r}\\|\psi\\|_{r}\ ,\qquad\forall t\in{\mathbb{R}}\ ,\quad\forall\tau\in[0,1]\ .$ (A.5) ## Appendix B Functional calculus In this section we collect some known results about functional calculus of selfadjoint operators which are used thorough the paper. We begin recalling Helffer-Sjöstrand formula [26], following the presentation of [15]. ###### Definition B.1. A function $f\in C^{\infty}({\mathbb{R}},{\mathbb{C}})$ will be said to belong to the class $S^{\rho}$, $\rho\in{\mathbb{R}}$, if $\forall m\in{\mathbb{N}}_{0}$, $\exists C_{m}>0$ such that $\left\|\frac{{\rm d}^{m}}{{\rm d}x^{m}}f(x)\right\|\leq C_{m}\left\langle x\right\rangle^{\rho-m},\quad\forall x\in{\mathbb{R}}\ .$ As usual we set the seminorms $\wp^{\rho}_{m}(f):=\sum_{0\leq j\leq m}\ \sup_{x\in{\mathbb{R}}}\left\|\frac{{\rm d}^{m}f(x)}{{\rm d}x^{m}}\right\|\,\left\langle x\right\rangle^{-\rho+m}\ ,\qquad m\in{\mathbb{N}}_{0}\ .$ Given $f\in S^{\rho}$, we define its almost analytic extension as follows: for any $N\in{\mathbb{N}}$, put ${\widetilde{f}}_{N}\colon{\mathbb{R}}^{2}\to{\mathbb{C}},\qquad{\widetilde{f}}_{N}(x,y):=\left(\sum_{\ell=0}^{N}f^{(\ell)}(x)\frac{({\rm i}y)^{\ell}}{\ell!}\right)\,\tau\left(\frac{y}{\left\langle x\right\rangle}\right)$ (B.1) where $\tau\in C^{\infty}({\mathbb{R}},{\mathbb{R}}_{\geq 0})$ is a cut-off function fulfilling $\tau(s)=1$ for $\|s\|\leq 1$ and $\tau(s)=0$ for $\|s\|\geq 2$. It is well known [15] that the choice of $N$ and of the cut-off function $\tau$ are by no means critical, and even other choices of $\widetilde{f}_{N}$ are possible (see e.g. [18]). The following properties are true [15]: let $f\in S^{\rho}$ with $\rho<0$, then $\displaystyle{\widetilde{f}}_{N}\|_{\mathbb{R}}=f,\qquad{\rm supp}\,{\widetilde{f}}_{N}\subset\left\\{x+{\rm i}y\ \colon\ \ \ x\in{\rm supp}\,f,\quad\|y\|\leq 2\left\langle x\right\rangle\right\\}\ ,$ (B.2) $\displaystyle\left\|\frac{\partial{\widetilde{f}}_{N}(x,y)}{\partial{\overline{z}}}\right\|\leq C_{N}\left\langle x\right\rangle^{\rho-N-1}\,\|y\|^{N},\qquad\frac{\partial{\widetilde{f}}_{N}}{\partial{\overline{z}}}:=\left(\frac{\partial{\widetilde{f}}_{N}}{\partial x}+{\rm i}\frac{\partial{\widetilde{f}}_{N}}{\partial y}\right)$ (B.3) $\displaystyle\int_{{\mathbb{R}}^{2}}\left\|\frac{\partial{\widetilde{f}}_{N}(z)}{\partial{\overline{z}}}\right\|\,\left\|{\rm Im}\,(z)\right\|^{-p-1}{\rm d}\overline{z}\wedge{\rm d}z\leq C_{N}\,\wp^{\rho}_{N+2}(f),\qquad\forall p=0,\ldots,N,$ (B.4) where $z=x+{\rm i}y$ and ${\rm d}\overline{z}\wedge{\rm d}z$ is the Lebesgue measure on ${\mathbb{C}}$. Given $\mathsf{H}$ a selfadjoint operator and $f\in S^{\rho}$, $\rho<0$, the Helffer-Sjöstrand formula defines $f(\mathsf{H})$ as $f(\mathsf{H}):=\frac{{\rm i}}{2\pi}\int_{{\mathbb{R}}^{2}}\frac{\partial{\widetilde{f}}_{N}(z)}{\partial{\overline{z}}}\,(z-\mathsf{H})^{-1}{\rm d}\overline{z}\wedge{\rm d}z=-\frac{1}{\pi}\int_{{\mathbb{R}}^{2}}\frac{\partial{\widetilde{f}}_{N}(z)}{\partial{\overline{z}}}\,(z-\mathsf{H})^{-1}{\rm d}x\,{\rm d}y\ .$ (B.5) ###### Theorem B.2 ([15]). Let $f\in S^{\rho}$, $g\in S^{\mu}$ with $\rho,\mu<0$ and $\mathsf{H}$ a selfadjoint operator. Then * (i) The operator $f(\mathsf{H})$ is independent of $N$ and of the cut-off function $\tau$. * (ii) The integral in (B.5) is norm convergent and $\\|f(\mathsf{H})\\|_{{\mathcal{L}}({\mathcal{H}})}\leq\\|f\\|_{L^{\infty}}$. * (iii) $f(\mathsf{H})\,g(\mathsf{H})=(fg)(\mathsf{H})$. * (iv) $\overline{f}(\mathsf{H})=f(\mathsf{H})^{}$. (v) If $f\in C^{\infty}_{c}$ has support disjoint from $\sigma(\mathsf{H})$, then $f(\mathsf{H})=0$. * (vi) If $z\notin{\mathbb{R}}$ and $f_{z}(x):=(z-x)^{-1}$ for all $x\in{\mathbb{R}}$, then $f_{z}\in S^{-1}$ and $f_{z}(\mathsf{H})=(z-\mathsf{H})^{-1}$. ###### Remark B.3. Given $f\in S^{\rho}$, $\rho<0$ and $\mathsf{H}$ selfadjoint, the operator $f(\mathsf{H})$ defined via Helffer-Sjöstrand formula coincides with the classical definition given by the spectral theorem, namely $f(\mathsf{H})=\int_{{\mathbb{R}}}f(\lambda)\,{\rm d}E(\lambda)$ where ${\rm d}E(\lambda)$ is the spectral resolution of $\mathsf{H}$. For a proof, see e.g. [19], Theorem 8.1. Next we recall expansion formulas for commutators. We start from the basic identities $\displaystyle{\rm ad}^{n}_{\mathsf{A}}(\mathsf{P}\mathsf{Q})$ $\displaystyle=\sum_{k=0}^{n}\binom{n}{k}\,{\rm ad}^{n-k}_{\mathsf{A}}(\mathsf{P})\,{\rm ad}^{k}_{\mathsf{A}}(\mathsf{Q}),\qquad[\mathsf{P},\mathsf{A}^{n}]=\sum_{j=1}^{n}c_{n,j}\,{\rm ad}^{j}_{\mathsf{A}}(\mathsf{P})\,\mathsf{A}^{n-j}\ .$ (B.6) For the next lemma see e.g. [18, Lemma C.3.1] or [29, Appendix B]. ###### Lemma B.4 (Commutator expansion formula). Let $k\in{\mathbb{N}}$ and $\mathsf{A},\mathsf{B}$ selfadjoint operators with $\\|{\rm ad}_{\mathsf{A}}^{j}(\mathsf{B})\\|_{{\mathcal{L}}({\mathcal{H}})}<\infty,\qquad\forall\,1\leq j\leq k\ .$ Let $f\in S^{\rho}$ with $\rho<0$, then one has the right and left commutator expansions $\displaystyle[\mathsf{B},f(\mathsf{A})]$ $\displaystyle=\sum_{j=1}^{k-1}\frac{1}{j!}\,f^{(j)}(\mathsf{A})\,{\rm ad}_{\mathsf{A}}^{j}(\mathsf{B})+R_{k}(f,\mathsf{A},\mathsf{B})$ (B.7) $\displaystyle=\sum_{j=1}^{k-1}\frac{(-1)^{j-1}}{j!}\,{\rm ad}_{\mathsf{A}}^{j}(\mathsf{B})\,f^{(j)}(\mathsf{A})\,+{\widetilde{R}}_{k}(f,\mathsf{A},\mathsf{B})$ (B.8) where the operators $R_{k},{\widetilde{R}}_{k}$ fulfill $\\|R_{k}(f,\mathsf{A},\mathsf{B})\\|_{{\mathcal{L}}({\mathcal{H}})},\quad\\|{\widetilde{R}}_{k}(f,\mathsf{A},\mathsf{B})\\|_{{\mathcal{L}}({\mathcal{H}})}\leq\,C_{N}\,\wp^{\rho}_{k+2}(f)\,\\|{\rm ad}_{\mathsf{A}}^{k}(\mathsf{B})\\|_{{\mathcal{L}}({\mathcal{H}})}\ .$ (B.9) ###### Lemma B.5. Let $k\in{\mathbb{N}}$ and $\mathsf{A},\mathsf{H}$ selfadjoint operators such that $\\|{\rm ad}_{\mathsf{A}}^{j}(\mathsf{H})\\|_{{\mathcal{L}}({\mathcal{H}})}<\infty,\qquad\forall\,1\leq j\leq k\ .$ (B.10) Let $g\in S^{\rho}$ with $\rho<0$. Then $\\|{\rm ad}_{\mathsf{A}}^{j}(g(\mathsf{H}))\\|_{{\mathcal{L}}({\mathcal{H}})}<\infty\,\quad\forall 1\leq j\leq k\ .$ ###### Proof. Take $N\geq k$ and use Helffer-Sjöstrand formula to write ${\rm ad}^{j}_{\mathsf{A}}(g(\mathsf{H}))=\frac{{\rm i}}{2\pi}\int_{{\mathbb{R}}^{2}}\frac{\partial{\widetilde{g}}_{N}(z)}{\partial\overline{z}}\,{\rm ad}^{j}_{\mathsf{A}}\big{(}(z-\mathsf{H})^{-1}\big{)}\,{\rm d}\overline{z}\wedge{\rm d}z.$ (B.11) As ${\rm ad}_{\mathsf{A}}\big{(}(z-\mathsf{H})^{-1}\big{)}=(z-\mathsf{H})^{-1}\,{\rm ad}_{\mathsf{A}}(\mathsf{H})\,(z-\mathsf{H})^{-1}$, by induction one gets for $j=1,\ldots,k$ ${\rm ad}_{\mathsf{A}}^{j}\big{(}(z-\mathsf{H})^{-1}\big{)}=\sum_{\ell=1}^{j}\sum_{k_{1}+\cdots+k_{\ell}=j\atop k_{1},\ldots,k_{\ell}\geq 1}c_{k_{1}\cdots k_{\ell}}^{\ell,j}\,(z-\mathsf{H})^{-1}\,{\rm ad}_{\mathsf{A}}^{k_{1}}(\mathsf{H})\,(z-\mathsf{H})^{-1}{\rm ad}_{\mathsf{A}}^{k_{2}}(\mathsf{H})\cdots(z-\mathsf{H})^{-1}\,{\rm ad}_{\mathsf{A}}^{k_{\ell}}(\mathsf{H})\,(z-\mathsf{H})^{-1}$ Using (B.10) and the estimate $\\|(z-\mathsf{H})^{-1}\\|_{{\mathcal{L}}({\mathcal{H}})}\leq{\left\|{\rm Im}\,(z)\right\|^{-1}}$, $\forall z\in{\mathbb{C}}\setminus{\mathbb{R}}$, one has for $j=1,\ldots,k$ $\\|{\rm ad}_{\mathsf{A}}^{j}\big{(}(z-\mathsf{H})^{-1}\big{)}\\|_{{\mathcal{L}}({\mathcal{H}})}\leq\sum_{\ell=1}^{j}C_{\ell}\,{\left\|{\rm Im}\,(z)\right\|}^{-\ell-1},\quad\forall z\in{\mathbb{C}}\setminus{\mathbb{R}}.$ Inserting this estimate into (B.11) and using (B.4) we bound for any $j=1,\ldots,k$ $\\|{\rm ad}^{j}_{\mathsf{A}}(g_{J}(\mathsf{H}))\\|_{{\mathcal{L}}({\mathcal{H}})}\lesssim\sum_{\ell=1}^{j}\int_{{\mathbb{R}}^{2}}\left\|\frac{\partial{\widetilde{g}}_{N}(z)}{\partial\overline{z}}\right\|\,\,{\left\|{\rm Im}\,(z)\right\|}^{-\ell-1}{\rm d}\overline{z}\wedge{\rm d}z\lesssim\wp^{\rho}_{N+2}(g)<\infty\ .$ ∎ ###### Lemma B.6. Let $g\in C^{\infty}_{c}({\mathbb{R}},{\mathbb{R}})$. Let $\mathsf{H},\mathsf{B}\in{\mathcal{L}}({\mathcal{H}})$ be selfadjoint. Then $\exists\,C>0$ such that $\\|g(\mathsf{H}+\mathsf{B})-g(\mathsf{H})\\|_{{\mathcal{L}}({\mathcal{H}})}\leq C\\|\mathsf{B}\\|_{{\mathcal{L}}({\mathcal{H}})}\ .$ (B.12) If $\mathsf{B}$ is compact on ${\mathcal{H}}$, so is $g(\mathsf{H}+\mathsf{B})-g(\mathsf{H})$. ###### Proof. Take $N\geq 1$. Using Helffer-Sjöstrand formula and the resolvent identity we obtain $g(\mathsf{H}+\mathsf{B})-g(\mathsf{H})=\frac{{\rm i}}{2\pi}\int_{{\mathbb{R}}^{2}}\frac{\partial{\widetilde{g}}_{N}(z)}{\partial\overline{z}}\,\big{(}z-(\mathsf{H}+\mathsf{B})\big{)}^{-1}\,\mathsf{B}\,(z-\mathsf{H})^{-1}\,{\rm d}\overline{z}\wedge{\rm d}z\ .$ Then use $\\|\big{(}z-(\mathsf{H}+\mathsf{B})\big{)}^{-1}\\|_{{\mathcal{L}}({\mathcal{H}})}$, $\\|(z-\mathsf{H})^{-1}\\|_{{\mathcal{L}}({\mathcal{H}})}\leq\left\|{\rm Im}(z)\right\|^{-1}$ for $z\in{\mathbb{C}}\setminus{\mathbb{R}}$ and (B.4). If $\mathsf{B}$ is compact then $\big{(}z-(\mathsf{H}+\mathsf{B})\big{)}^{-1}\,\mathsf{B}\,(z-\mathsf{H})^{-1}$ is a compact operator for any $z\in{\mathbb{C}}\setminus{\mathbb{R}}$. ∎ ## Appendix C Local energy decay estimates In this section we prove a local energy decay estimate starting from Mourre estimate. The result is essentially known but we could not find in the literature a statement exactly as the one we use in the paper, so we include here a proof, which follows closely the one of Lemma 4.1 of [24]. In this part we do not require pseudodifferential properties of the operators. We shall assume conditions (M1) and (M2) at page (M1). ###### Theorem C.1 (Local energy decay estimate). Fix $k\in{\mathbb{N}}$ and assume (M1)–(M2) with ${\mathtt{N}}\geq 4k+2$ and $\mathsf{K}=0$. Then for any interval $J\subset I$, any function $g_{J}\in C^{\infty}_{c}({\mathbb{R}},{\mathbb{R}}_{\geq 0})$ with ${\rm supp}\,g_{J}\subset I$, $g_{J}=1$ on $J$, there exists $C>0$ such that $\\|\left\langle\mathsf{A}\right\rangle^{-k}\,e^{-{\rm i}\mathsf{H}t}\,g_{J}(\mathsf{H})\,\psi\\|_{0}\leq C\left\langle t\right\rangle^{-k}\\|\left\langle\mathsf{A}\right\rangle^{k}\,g_{J}(\mathsf{H})\psi\\|_{0},\quad\forall t\in{\mathbb{R}},$ (C.1) for any $\psi$ such that the r.h.s. is finite. ###### Proof. Take $\chi(\xi):=\frac{1}{2}(1-\tanh\xi)$. Put $\eta(\xi):=\frac{1}{\sqrt{2}\cosh\xi}$ and note that $\chi^{\prime}=-\eta^{2},\qquad\left\|\eta^{(m)}(\xi)\right\|\leq C_{m}\,\eta(\xi),\quad\forall\xi\in{\mathbb{R}},\ \ \forall m\in{\mathbb{N}}.$ (C.2) Next we set for $a\in{\mathbb{R}}$, $s\geq 1$ and $\vartheta:=\frac{\theta}{2}$ (with $\theta$ of (M2) ) $\mathsf{A}_{t,s}:=\frac{1}{s}\big{(}\mathsf{A}-a-\vartheta t\big{)}$ and define via functional calculus the operators $\chi(\mathsf{A}_{t,s})$ and $\eta(\mathsf{A}_{t,s})$; both are bounded and selfadjoint on ${\mathcal{H}}$. To shorten the notation, from now on we write $\chi_{t,s}\equiv\chi(\mathsf{A}_{t,s})$, $\eta_{t,s}\equiv\eta(\mathsf{A}_{t,s})$, $g_{J}\equiv g_{J}(\mathsf{H})$ and $\psi_{t}:=e^{-{\rm i}t\mathsf{H}}\psi$. Note that $e^{-{\rm i}\mathsf{H}t}g_{J}(\mathsf{H})\psi=g_{J}(\mathsf{H})e^{-{\rm i}\mathsf{H}t}\psi\equiv g_{J}\psi_{t}$. The starting point of the proof is an energy estimate for the quantity $\\|(\chi_{t,s})^{\frac{1}{2}}\,g_{J}\psi_{t}\\|_{0}$. We have $\frac{d}{dt}\\|(\chi_{t,s})^{\frac{1}{2}}\,g_{J}\psi_{t}\\|_{0}^{2}=\frac{d}{dt}\left\langle\chi_{t,s}g_{J}\psi_{t},\,g_{J}\psi_{t}\right\rangle=\frac{\vartheta}{s}\\|\eta_{t,s}\,g_{J}\,\psi_{t}\\|_{0}^{2}+\left\langle{\rm i}[\mathsf{H},\chi_{t,s}]\,g_{J}\psi_{t},\,g_{J}\psi_{t}\right\rangle.$ (C.3) To evaluate the right hand side we shall use the commutator formulas in Lemma B.4, the identity ${\rm ad}^{j}_{\mathsf{A}_{t,s}}(\mathsf{H})=\frac{1}{s^{j}}{\rm ad}^{j}_{\mathsf{A}}(\mathsf{H})\ ,\quad\forall s\geq 1,\ \ \forall 1\leq j\leq{\mathtt{N}}$ (C.4) and the fact that all the operators ${\rm ad}^{j}_{\mathsf{A}}(\mathsf{H})$ are bounded $\forall 1\leq j\leq{\mathtt{N}}$ by (M1). The goal now is to estimate the second term in the right hand side of (C.3). For an arbitrary $f\in{\mathcal{H}}$ we write $\left\langle{\rm i}[\mathsf{H},\chi_{t,s}]f,f\right\rangle\stackrel{{\scriptstyle\eqref{r.exp},\eqref{ad.Ats}}}{{=}}-\frac{1}{s}\left\langle\eta_{t,s}^{2}{\rm i}[\mathsf{H},\mathsf{A}]f,f\right\rangle+\sum_{j=2}^{{\mathtt{N}}-1}\frac{1}{j!}\frac{1}{s^{j}}\left\langle\chi_{t,s}^{(j)}\,{\rm i}\,{\rm ad}_{\mathsf{A}}^{j}(\mathsf{H})f,f\right\rangle+\frac{1}{s^{\mathtt{N}}}\left\langle R_{\mathtt{N}}f,f\right\rangle$ (C.5) where $\chi_{t,s}^{(j)}:=\chi^{(j)}(\mathsf{A}_{t,s})$ and the remainder $R_{\mathtt{N}}$ fulfills the estimate (see (B.9)) $\\|R_{\mathtt{N}}\\|_{{\mathcal{L}}({\mathcal{H}})}\leq C_{\mathtt{N}}\,\\|{\rm ad}_{\mathsf{A}}^{\mathtt{N}}(\mathsf{H})\\|\leq C_{{\mathtt{N}}}.$ (C.6) Note that the constant $C_{\mathtt{N}}>0$ in the previous estimate is uniform in $a\in{\mathbb{R}}$. In the following we shall simply denote by $R_{\mathtt{N}}$ any bounded operator fulfilling an estimate like (C.6). Consider now the first term in the expansion (C.5) above. This time we use the left expansion (B.8) and write $\displaystyle\frac{1}{s}\left\langle\eta_{t,s}^{2}\,{\rm i}[\mathsf{H},\mathsf{A}]f,f\right\rangle$ $\displaystyle=\frac{1}{s}\left\langle\,{\rm i}[\mathsf{H},\mathsf{A}]\,\eta_{t,s}f,\,\eta_{t,s}f\right\rangle$ (C.7) $\displaystyle\quad+\sum_{j=2}^{{\mathtt{N}}-1}\frac{(-1)^{j-1}}{(j-1)!}\frac{1}{s^{j}}\langle\,{\rm i}\,{\rm ad}_{\mathsf{A}}^{j}(\mathsf{H})\,\eta_{t,s}^{(j-1)}f,\eta_{t,s}f\rangle+\frac{1}{s^{\mathtt{N}}}\left\langle R_{\mathtt{N}}f,f\right\rangle$ (C.8) where $R_{\mathtt{N}}$ is estimated as in (C.6). Consider now the second term in (C.5). From $\chi^{\prime}=-\eta^{2}$, we have by functional calculus $\chi^{(j)}(\mathsf{A}_{t,s})=\sum_{\ell=1}^{j}c_{\ell j}\,\eta^{(j-\ell)}(\mathsf{A}_{t,s})\,\eta^{(\ell)}(\mathsf{A}_{t,s})\,.$ Thus we get that $\displaystyle\frac{1}{s^{j}}\langle\chi_{t,s}^{(j)}\,{\rm i}\,{\rm ad}_{\mathsf{A}}^{j}(\mathsf{H})f,f\rangle$ $\displaystyle\stackrel{{\scriptstyle\eqref{l.exp}}}{{=}}\frac{1}{s^{j}}\sum_{\ell=1}^{j}c_{\ell j}\langle\,{\rm i}\,{\rm ad}_{\mathsf{A}}^{j}(\mathsf{H})\,\eta_{t,s}^{(\ell)}f,\eta_{t,s}^{(j-\ell)}f\rangle$ (C.9) $\displaystyle\quad+\sum_{\ell=1}^{j}\sum_{n=1}^{{\mathtt{N}}-j-1}\frac{c_{\ell jn}}{s^{j+n}}\langle\,{\rm i}\,{\rm ad}_{\mathsf{A}}^{j+n}(\mathsf{H})\,\eta_{t,s}^{(\ell+n)}f,\,\eta_{t,s}^{(j-\ell)}f\rangle+\frac{1}{s^{{\mathtt{N}}}}\left\langle R_{\mathtt{N}}f,f\right\rangle$ (C.10) By (C.5), (C.7), (C.9) we have found that $\left\langle{\rm i}[\mathsf{H},\chi_{t,s}]f,f\right\rangle$ is a sum of terms of the form $\displaystyle\left\langle{\rm i}[\mathsf{H},\chi_{t,s}]f,f\right\rangle=-\frac{1}{s}\left\langle\,{\rm i}[\mathsf{H},\mathsf{A}]\,\eta_{t,s}f,\,\eta_{t,s}f\right\rangle+\sum_{j=2}^{{\mathtt{N}}-1}\frac{1}{s^{j}}\sum_{n,\ell,m}\langle\,R_{n}\,\eta_{t,s}^{(\ell)}f,\eta_{t,s}^{(m)}f\rangle+\frac{1}{s^{\mathtt{N}}}\left\langle R_{\mathtt{N}}f,f\right\rangle$ where $R_{n},R_{\mathtt{N}}$ are bounded operators. Furthermore, from the second of (C.2) and the spectral theorem, we bound $\left\|\langle\,R_{n}\,\eta_{t,s}^{(\ell)}f,\eta_{t,s}^{(m)}f\rangle\right\|\leq C\,\\|\eta_{t,s}f\\|_{0}^{2}\ .$ (C.11) We thus obtain, for any $f\in{\mathcal{H}}$ and $s\geq 1$, the estimate $\left\langle{\rm i}[\mathsf{H},\chi_{t,s}]f,f\right\rangle\leq-\frac{1}{s}\left\langle\,{\rm i}[\mathsf{H},\mathsf{A}]\,\eta_{t,s}f,\,\eta_{t,s}f\right\rangle+\frac{C_{\mathtt{N}}}{s^{2}}\\|\eta_{t,s}f\\|_{0}^{2}+\frac{C_{\mathtt{N}}}{s^{\mathtt{N}}}\\|f\\|_{0}^{2}\ .$ (C.12) Now we evaluate such inequality at $f=g_{J}\psi_{t}$, getting $\left\langle{\rm i}[\mathsf{H},\chi_{t,s}]g_{J}\psi_{t},\,g_{J}\psi_{t}\right\rangle\leq-\frac{1}{s}\left\langle\,{\rm i}[\mathsf{H},\mathsf{A}]\,\eta_{t,s}g_{J}\psi_{t},\,\eta_{t,s}g_{J}\psi_{t}\right\rangle+\frac{C_{\mathtt{N}}}{s^{2}}\\|\eta_{t,s}g_{J}\psi_{t}\\|_{0}^{2}+\frac{C_{\mathtt{N}}}{s^{\mathtt{N}}}\\|g_{J}\psi_{t}\\|_{0}^{2}\ .$ (C.13) The next step is to prove that the first term in the right hand side above has a sign, up to higher order terms in $s^{-j}$. This is the point where the Mourre estimate (M2) comes into play. To see this, we analyze $\displaystyle\left\langle\,{\rm i}[\mathsf{H},\mathsf{A}]\,\eta_{t,s}\,g_{J}\psi_{t},\,\eta_{t,s}\,g_{J}\psi_{t}\right\rangle\equiv\left\langle\,{\rm i}[\mathsf{H},\mathsf{A}]\,\eta_{t,s}\,g_{I}\,g_{J}\psi_{t},\,\eta_{t,s}g_{I}\,g_{J}\psi_{t}\right\rangle$ (C.14) where we used that $g_{J}g_{I}=g_{J}$. Next we commute and expand in commutators $\eta_{t,s}g_{I}$: $\displaystyle\eta_{t,s}g_{I}=g_{I}\eta_{t,s}+[\eta_{t,s},g_{I}]\stackrel{{\scriptstyle\eqref{l.exp}}}{{=}}g_{I}\,\eta_{t,s}+\sum_{j=1}^{{\mathtt{N}}-2}\frac{c_{j}}{s^{j}}\,{\rm ad}_{\mathsf{A}}^{j}(g_{I}(\mathsf{H}))\,\eta_{t,s}^{(j)}+\frac{1}{s^{{\mathtt{N}}-1}}{\widetilde{R}}_{{\mathtt{N}}-1}\ ;$ (C.15) note that Lemma B.5 assures that the operators ${\rm ad}_{\mathsf{A}}^{j}(g_{I}(\mathsf{H}))$ are bounded $\forall j=1,\ldots,{\mathtt{N}}$, so is the operator ${\widetilde{R}}_{{\mathtt{N}}-1}$ which fulfills $\\|{\widetilde{R}}_{{\mathtt{N}}-1}\\|_{{\mathcal{L}}({\mathcal{H}})}\leq C_{\mathtt{N}}\,{\rm ad}_{\mathsf{A}}^{{\mathtt{N}}-1}(g_{I}(\mathsf{H}))<\infty\ .$ (C.16) Again in the following we shall denote by ${\widetilde{R}}_{{\mathtt{N}}-1}$ any operator fulfilling an estimate like (C.16). Inserting the expansion (C.15) into (C.14) one gets, with $w:=g_{J}\psi_{t}$, $\displaystyle\eqref{HAeta}=\left\langle{\rm i}[\mathsf{H},\mathsf{A}]\,g_{I}\,\eta_{t,s}w,\ g_{I}\,\eta_{t,s}w\right\rangle+\sum_{j=1}^{{\mathtt{N}}-2}\frac{c_{j}}{s^{j}}\sum_{n,\ell,m}\langle R_{n}\,\eta^{(\ell)}_{t,s}w,\ \eta^{(m)}_{t,s}w\rangle+\frac{1}{s^{{\mathtt{N}}-1}}\langle{\widetilde{R}}_{{\mathtt{N}}-1}w,w\rangle$ where each term of the form $\langle R_{n}\,\eta^{(\ell)}_{t,s}w,\ \eta^{(m)}_{t,s}w\rangle$ fulfills an estimate like (C.11). It is finally time to use the strict Mourre estimate: by assumption (M2) we have for $s\geq 1$ $\displaystyle\left\langle{\rm i}[\mathsf{H},\mathsf{A}]\,g_{I}\,\eta_{t,s}w,\ g_{I}\,\eta_{t,s}w\right\rangle$ $\displaystyle\geq\theta\\|g_{I}\,\eta_{t,s}w\\|_{0}^{2}\ .$ (C.17) Using again the expansion (C.15) and estimates (C.11), (C.16) we get therefore $\displaystyle\left\langle{\rm i}[\mathsf{H},\mathsf{A}]\,g_{I}\,\eta_{t,s}w,\ g_{I}\,\eta_{t,s}w\right\rangle$ $\displaystyle\geq\theta\\|\,\eta_{t,s}g_{I}w\\|_{0}^{2}-\frac{C_{\mathtt{N}}}{s}\\|\eta_{t,s}w\\|_{0}^{2}-\frac{C_{\mathtt{N}}}{s^{{\mathtt{N}}-1}}\\|w\\|_{0}^{2}.$ (C.18) This proves that the first term in the right hand side of (C.13) has a sign; we proceed from (C.13) and using inequality (C.18) (recall $w=g_{J}\psi_{t})$ we get $\left\langle{\rm i}[\mathsf{H},\chi_{t,s}]g_{J}\psi_{t},\,g_{J}\psi_{t}\right\rangle\leq-\frac{\theta}{s}\\|\eta_{t,s}\,g_{J}\psi_{t}\\|_{0}^{2}+\frac{C_{\mathtt{N}}}{s^{2}}\\|\eta_{t,s}\,g_{J}\psi_{t}\\|_{0}^{2}+\frac{C_{\mathtt{N}}}{s^{\mathtt{N}}}\\|\,g_{J}\psi_{t}\\|_{0}^{2}\ .$ (C.19) We come back to the estimate (C.3) of $\\|(\chi_{t,s})^{\frac{1}{2}}\,g_{J}\psi_{t}\\|_{0}$. We finally obtain, with $\vartheta=\frac{\theta}{2}$ and $s\geq 1$ sufficiently large, $\displaystyle\frac{d}{dt}\\|(\chi_{t,s})^{\frac{1}{2}}\,g_{J}\psi_{t}\\|_{0}^{2}$ $\displaystyle\stackrel{{\scriptstyle\eqref{min53}}}{{\leq}}\frac{1}{s}\left(\vartheta-\theta\right)\\|\eta_{t,s}\,g_{J}\psi_{t}\\|_{0}^{2}+\frac{C_{\mathtt{N}}}{s^{2}}\\|\eta_{t,s}\,g_{J}\psi_{t}\\|_{0}^{2}+\frac{C_{\mathtt{N}}}{s^{\mathtt{N}}}\\|\,g_{J}\psi_{t}\\|_{0}^{2}$ $\displaystyle\leq\frac{1}{s}\left(-\frac{\theta}{2}+\frac{C_{\mathtt{N}}}{s}\right)\\|\eta_{t,s}\,g_{J}\psi_{t}\\|_{0}^{2}+\frac{C_{\mathtt{N}}}{s^{\mathtt{N}}}\\|\,g_{J}\psi_{t}\\|_{0}^{2}$ So, for $s\geq 1$ sufficiently large, the first term in the right hand side above is negative and, using also that $e^{-{\rm i}t\mathsf{H}}$ is unitary and commutes with $g_{J}\equiv g_{J}(\mathsf{H})$, we get $\frac{d}{dt}\\|(\chi_{t,s})^{\frac{1}{2}}\,g_{J}\psi_{t}\\|_{0}^{2}\leq\frac{C_{\mathtt{N}}}{s^{\mathtt{N}}}\\|\,g_{J}\psi_{0}\\|_{0}^{2}\ .$ Integrating this inequality between $0$ and $t$ we find $\forall t>0$ $\\|\chi^{\frac{1}{2}}\Big{(}\frac{\mathsf{A}-a-\vartheta t}{s}\Big{)}\,g_{J}\psi_{t}\\|_{0}^{2}\leq\\|\chi^{\frac{1}{2}}\Big{(}\frac{\mathsf{A}-a}{s}\Big{)}\,g_{J}\psi\\|_{0}^{2}+\frac{C_{\mathtt{N}}\,t}{s^{\mathtt{N}}}\\|\,g_{J}\psi\\|_{0}^{2},$ uniformly for $a\in{\mathbb{R}}$ and $s\geq 1$ sufficiently large. We evaluate this inequality at $a=-\frac{\vartheta}{2}t$ and $s=\sqrt{t}$, obtaining for $t\geq 1$ sufficiently large, the minimal velocity estimate $\\|\chi^{\frac{1}{2}}\Big{(}\frac{\mathsf{A}-\frac{\vartheta}{2}t}{\sqrt{t}}\Big{)}\,g_{J}\psi_{t}\\|_{0}\leq\\|\chi^{\frac{1}{2}}\Big{(}\frac{\mathsf{A}+\frac{\vartheta}{2}t}{\sqrt{t}}\Big{)}\,g_{J}\psi\\|_{0}+C_{\mathtt{N}}\,t^{-\frac{{\mathtt{N}}}{4}+\frac{1}{2}}\\|\,g_{J}\psi\\|_{0}\ .$ (C.20) To conclude, take $k\in{\mathbb{N}}$ and consider $\\|\left\langle\mathsf{A}\right\rangle^{-k}\,g_{J}\,\psi_{t}\\|_{0}$. Clearly $\displaystyle\\|\left\langle\mathsf{A}\right\rangle^{-k}\,g_{J}\,\psi_{t}\\|_{0}$ $\displaystyle\leq\\|\chi^{\frac{1}{2}}\Big{(}\frac{\mathsf{A}-\frac{\vartheta}{2}t}{\sqrt{t}}\Big{)}\,\left\langle\mathsf{A}\right\rangle^{-k}\,g_{J}\,\psi_{t}\\|_{0}$ (C.21) $\displaystyle\quad+\\|\Big{(}1-\chi^{\frac{1}{2}}\Big{(}\frac{\mathsf{A}-\frac{\vartheta}{2}t}{\sqrt{t}}\Big{)}\,\Big{)}\,\left\langle\mathsf{A}\right\rangle^{-k}\,g_{J}\,\psi_{t}\\|_{0}$ (C.22) We estimate first (C.22). By Theorem B.2 (ii) we have $\\|\Big{(}1-\chi^{\frac{1}{2}}\Big{(}\frac{\mathsf{A}-\frac{\vartheta}{2}t}{\sqrt{t}}\Big{)}\,\Big{)}\,\left\langle\mathsf{A}\right\rangle^{-k}\\|_{{\mathcal{L}}({\mathcal{H}})}\leq\sup_{\lambda\in{\mathbb{R}}}\left\|\Big{(}1-\chi^{\frac{1}{2}}\Big{(}\frac{\lambda-\frac{\vartheta}{2}t}{\sqrt{t}}\Big{)}\Big{)}\,\left\langle\lambda\right\rangle^{-k}\right\|\leq C_{k}\left\langle t\right\rangle^{-k}.$ (C.23) To prove the last inequality, use that for $\lambda\geq\frac{\vartheta}{4}t$ then $\left\langle\lambda\right\rangle^{-k}\leq\left\langle t\right\rangle^{-k}$, whereas when $\lambda<\frac{\vartheta}{4}t$ then, being $\lambda\mapsto 1-\chi^{\frac{1}{2}}(\lambda)$ monotone increasing and exponentially decaying at $-\infty$, $1-\chi^{\frac{1}{2}}\Big{(}\frac{\lambda-\frac{\vartheta}{2}t}{\sqrt{t}}\Big{)}\leq 1-\chi^{\frac{1}{2}}\Big{(}-\frac{\vartheta}{4}\sqrt{t}\Big{)}\leq C\left(e^{-\frac{\vartheta}{4}\sqrt{t}}\right)^{\frac{1}{2}}\leq C_{k}\left\langle t\right\rangle^{-k}.$ Next we estimate (C.21) using the minimal velocity estimate. As $\left\langle\mathsf{A}\right\rangle^{-k}$ is a bounded operator, $\displaystyle\\|\chi^{\frac{1}{2}}\Big{(}\frac{\mathsf{A}-\frac{\vartheta}{2}t}{\sqrt{t}}\Big{)}\,\left\langle\mathsf{A}\right\rangle^{-k}\,g_{J}\,\psi_{t}\\|_{0}$ $\displaystyle\leq\\|\chi^{\frac{1}{2}}\Big{(}\frac{\mathsf{A}-\frac{\vartheta}{2}t}{\sqrt{t}}\Big{)}\,g_{J}\,\psi_{t}\\|_{0}$ $\displaystyle\stackrel{{\scriptstyle\eqref{min61}}}{{\leq}}\\|\chi^{\frac{1}{2}}\Big{(}\frac{\mathsf{A}+\frac{\vartheta}{2}t}{\sqrt{t}}\Big{)}\,g_{J}\psi\\|_{0}+C_{\mathtt{N}}\,t^{-\frac{{\mathtt{N}}}{4}+\frac{1}{2}}\\|\,g_{J}\psi\\|_{0}$ $\displaystyle\leq\\|\chi^{\frac{1}{2}}\Big{(}\frac{\mathsf{A}+\frac{\vartheta}{2}t}{\sqrt{t}}\Big{)}\,\left\langle\mathsf{A}\right\rangle^{-k}\\|_{{\mathcal{L}}({\mathcal{H}})}\,\\|\left\langle\mathsf{A}\right\rangle^{k}\,g_{J}\psi\\|_{0}+C_{\mathtt{N}}\,t^{-\frac{{\mathtt{N}}}{4}+\frac{1}{2}}\\|\,g_{J}\psi\\|_{0}$ Again we have $\\|\chi^{\frac{1}{2}}\Big{(}\frac{\mathsf{A}+\frac{\vartheta}{2}t}{\sqrt{t}}\Big{)}\,\left\langle\mathsf{A}\right\rangle^{-k}\\|_{{\mathcal{L}}({\mathcal{H}})}\leq\sup_{\lambda\in{\mathbb{R}}}\left\|\chi^{\frac{1}{2}}\Big{(}\frac{\lambda+\frac{\vartheta}{2}t}{\sqrt{t}}\Big{)}\,\left\langle\lambda\right\rangle^{-k}\right\|\leq C_{k}\left\langle t\right\rangle^{-k},$ (C.24) since for $\lambda\leq-\frac{\vartheta}{4}t$ one has $\left\langle\lambda\right\rangle^{-k}\leq C\left\langle t\right\rangle^{-k}$, whereas in case $\lambda>-\frac{\vartheta}{4}t$, as $\lambda\mapsto\chi^{\frac{1}{2}}(\lambda)$ is monotone decreasing exponentially fast at $+\infty$, one has $\chi^{\frac{1}{2}}\Big{(}\frac{\lambda+\frac{\vartheta}{2}t}{\sqrt{t}}\Big{)}\leq\chi^{\frac{1}{2}}\Big{(}\frac{\vartheta}{4}\sqrt{t}\Big{)}\leq C\left(e^{-\frac{\vartheta}{4}\sqrt{t}}\right)^{\frac{1}{2}}\leq C_{k}\left\langle t\right\rangle^{-k}.$ Altogether, from (C.21), (C.22) we have proved that for $t\geq 1$ sufficiently large, $\displaystyle\\|\left\langle\mathsf{A}\right\rangle^{-k}\,g_{J}\,\psi_{t}\\|_{0}$ $\displaystyle\leq C_{k}\left\langle t\right\rangle^{-k}\\|g_{J}\psi_{t}\\|_{0}+C_{k}\left\langle t\right\rangle^{-k}\\|\left\langle\mathsf{A}\right\rangle^{k}g_{J}\psi\\|_{0}+C_{\mathtt{N}}\,t^{-\frac{{\mathtt{N}}}{4}+\frac{1}{2}}\\|\,g_{J}\psi\\|_{0}$ $\displaystyle\leq C_{k}\left\langle t\right\rangle^{-k}\,\\|\left\langle\mathsf{A}\right\rangle^{k}g_{J}\psi\\|_{0}$ provided ${\mathtt{N}}=4k+2$. 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# An ensemble solver for segregated cardiovascular FSI Xue Li Department of Applied and Computational Mathematics and Statistics University of Notre Dame Notre Dame, IN 46556 <EMAIL_ADDRESS> &Daniele E. Schiavazzi Department of Applied and Computational Mathematics and Statistics University of Notre Dame Notre Dame, IN 46556 <EMAIL_ADDRESS> ###### Abstract Computational models are increasingly used for diagnosis and treatment of cardiovascular disease. To provide a quantitative hemodynamic understanding that can be effectively used in the clinic, it is crucial to quantify the variability in the outputs from these models due to multiple sources of uncertainty. To quantify this variability, the analyst invariably needs to generate a large collection of high-fidelity model solutions, typically requiring a substantial computational effort. In this paper, we show how an explicit-in-time _ensemble_ cardiovascular solver offers superior performance with respect to the embarrassingly parallel solution with implicit-in-time algorithms, typical of an inner-outer loop paradigm for non-intrusive uncertainty propagation. We discuss in detail the numerics and efficient distributed implementation of a segregated FSI cardiovascular solver on both CPU and GPU systems, and demonstrate its applicability to idealized and patient-specific cardiovascular models, analyzed under steady and pulsatile flow conditions. _K_ eywords Ensemble solvers $\cdot$ Uncertainty Quantification $\cdot$ Computational Hemodynamics $\cdot$ Explicit Time Integration $\cdot$ Biomechanics ## 1 Introduction In this study we focus on the development of efficient solvers for complex fluid-structure interaction (FSI) phenomena arising in cardiovascular (CV) hemodynamics. For this and many other applications, output variability is induced by uncertainty or ignorance in the input processes, e.g., material property distribution, physiologically sound boundary conditions or model anatomy, resulting from operator-dependent image volume segmentation. In this context, the new paradigm of Uncertainty Quantification (UQ) is rapidly becoming an integral part of the modeling exercise, and an indispensable tool to rigorously quantify confidence in the simulation outputs, enabling robust predictions of greater clinical impact. However, running a complete UQ study on a large scale cardiovascular model is typically associated with a substantial computational cost. Non-intrusive approaches for the solution of the forward problem in uncertainty quantification (also known as _uncertainty propagation_) typically consider the underlying deterministic solver as a black box (see Figure 1), requiring model solutions at various parameter realizations to be _independently_ (and possibly simultaneously) computed. The scalability of this paradigm is limited due to two main reasons. First, in the embarrassingly parallel solution of multiple instances of the same problem, a large number of operations is repeated. Second, the solution of large linear systems of equations from numerical integration in time with implicit schemes presents, in general, a less than ideal scalability when computed on large multi-core architectures. Figure 1: Schematic representation of the forward and inverse problems in UQ. To tackle these challenges, we propose an efficient computational approach for solving multiple instances of the same model (so-called model _ensemble_) at the same time, in a highly scalable fashion, enabled by CPU/GPU implementations of numerical integration schemes that rely heavily on distributed sparse matrix-vector products. We validate our approach by characterizing the effect of variability in vessel wall thickness and elastic modulus on the mechanical response of ideal and patient-specific cardiovascular models analyzed under steady and pulsatile flow conditions. A stochastic model for the spatial distribution of thickness and elastic modulus is provided, in this study, by approximating a Gaussian random field with Matérn covariance through the solution of a stochastic partial differential equation on a triangular finite element mesh of the vessel lumen [1]. Additionally, we follow a one-way coupled, segregated approach to fluid- structure interaction, where the wall stress is computed by an implicit Variational Multiscale fluid solver and passed to a structural, three-d.o.f.s shell model of the vascular walls [2]. Unique contributions of our approach are: 1. 1. We propose, for the first time, an ensemble solver in the context of cardiovascular hemodynamics for UQ, with the aim of drastically reduce the computational effort to perform campaigns of high-fidelity model solutions. 2. 2. Our approach paves the way to a fully explicit treatment of fluid structure interaction in cardiovascular modeling, whose potential has not yet been fully explored in the literature. In our opinion, this new paradigm will provide simpler approaches to simulating complex physiological dysfunctions (e.g. aortic dissection, involving large vessel deformations and contact/auto- contact phenomena, see [3]). 3. 3. The combination of explicit time integration schemes with the solution of model ensembles leads to an efficient distribution of computing load and memory usage in the GPU, enabling scalability in a way that is currently not possible with embarrassingly parallel model solutions and implicit solvers. We motivate the computational advantage of explicit-in-time ensemble solvers through a back-of-the-envelope argument. It is well known how explicit numerical integration schemes are only conditionally stable. For structural problems, the larger stable time step is determined through a CFL condition as $0.9\cdot\Delta t$ where $\Delta t$ is the amount of time required by an elastic wave to cross the smallest element in the mesh. A reasonable value of $\Delta t$ for CV modeling is $\Delta t=l_{e,\text{min}}/c$, where $l_{e,\text{min}}$ is the diameter of the circle/sphere inscribed in the smallest element in the mesh, $c=\sqrt{E/\rho}$ is the elastic wave speed, $E$ and $\rho$ the elastic modulus and density of the vascular tissue (assumed homogeneous and isotropic in the current argument). Typical values of $l_{e,\text{min}}=1.0\times 10^{-3}$ m, $E=0.7$ MPa and $\rho=1.06$ kg/m3 lead to a stable time step equal to approximately $0.9\cdot 1.2\times$10${}^{-6}\approx 1.0\times$10-6. In contrast, the typical time step adopted by implicit CV FSI solvers is one millisecond (an upper bound, typically much smaller). At every time step, these solvers require multiple linear systems to be solved with iterative methods, each consisting in several matrix-vector products per iteration. Thus, if we assume an average of 10 non- linear iterations consisting of 10 linear solver iterations each (with two matrix-vector multiplications per iteration), an explicit solver is only a factor of five more expensive. However, the cost of explicit methods can be further reduced through several means, for example, increasing the critical time step via mass scaling (see, e.g., [4]), selectively updating the stiffness, damping and mass matrices between successive time steps to avoid the repeated assembly of element matrices (in the linear regime), and by solving multiple realizations of the boundary conditions, material properties and geometry at the same time. Additionally, large matrix-vector operations needed by the explicit solution of model ensembles are particularly well suited for GPU computing. In summary, explicit time integration schemes have many advantages over implicit approaches in the simulation of phenomena occurring over small time intervals and, combined with the solution of model ensembles, bear substantial potential to boost the efficiency of solving CV models on modern GPU-based systems. Even though our work focuses on CV hemodynamics, the proposed solver paradigm is applicable to study fluid- structure interaction phenomena in other fields, but its efficiency is affected by the stiffness and mass properties in the selected application. Use of explicit structural solvers in cardiovascular flow problems is mainly related to their flexibility in modeling complex contact configurations. Studies involving coronary stent deployment following endoscopic balloon inflation are proposed, e.g., in [5, 6], and coupling with an implicit fluid solver is discussed in [7]. Implementation of structural explicit solvers on GPU are discussed in various studies in the literature. In [8] the authors describe in detail an application involving thin shells, while an overview on applications in biomechanics is discussed in [9]. Additionally, Ensemble methods for fluid problems have been recently proposed by [10, 11, 12, 13, 14] in the context of the Navier-Stokes equations with distinct initial conditions and forcing terms. This is based on the observation that solution of linear systems is responsible for a significant fraction of the overall running time for linearly implicit methods, and that is far more efficient to solve multiple times a system of equation with the same coefficient matrix and different right-hand-side than different systems altogether. Extensions have also been proposed to magnetohydrodynamics [15], natural convection problems [16] and parametrized flow problems [17, 18]. We note how, in our case, there is no approximation introduced in the formulation of the ensemble numerical scheme. In addition, no ensemble method appears to be available from the literature in the context of fluid-structure interaction problems. The basic methodology behind the proposed solver is discussed in Section 2, including the generation of random field material properties, the structural finite element formulation, the variational multiscale fluid solver, thier weak coupling and CPU/GPU implementation. Validation of the proposed approach is discussed in Section 3 with reference to an idealized model of the thoracic aorta and a patient-specific coronary model. Performance and scalability of the approach is discussed in Section 3.3 followed by conclusions and future work in Section 4. ## 2 Methodology ### 2.1 Random field material properties A homogeneous and isotropic vascular tissue with uncertain elastic modulus and thickness is assumed in this study, modeled through a Gaussian random field with Matérn covariance. A Gaussian marginal distribution appears to be the simplest idealized distribution compatible with the scarce experimental observations, while the choice of a Matérn covariance relates to its finite differentiability, which make this model more desirable than other kernels [19]. Let $\left\\|\cdot\right\\|$ denote the Euclidean distance in $\mathbb{R}^{d}$. The Matérn covariance between two points at distance $\\|\bm{h}\\|$ is $r(\\|\bm{h}\\|)=\frac{\sigma^{2}}{2^{\nu-1}\Gamma(\nu)}(\kappa\\|\bm{h}\\|)^{\nu}K_{\nu}(\kappa\\|\bm{h}\\|),\,\,\bm{h}\in\mathbb{R}^{d}$ (1) where $\Gamma(\cdot)$ is the gamma function, $K_{\nu}$ is the modified Bessel function of the second kind, $\sigma^{2}$ is the marginal variance, $\nu$ is a scaling parameter which determines the mean square differentiability of the underling process, and $\kappa$ is related to the correlation length $\rho=\sqrt{8\nu}/\kappa$, i.e., the distance which corresponds to a correlation of approximately $0.1$, for all $\nu$. It is known from the literature [20, 21] how Gaussian random fields with Matérn covariance can be obtained as solutions of a linear fractional stochastic partial differential equation (SPDE) of the form $(\kappa^{2}-\Delta)^{\alpha/2}\,x(\bm{s})=\mathcal{W}(\bm{s}),\,\,\bm{s}\in\mathbb{R}^{d},$ (2) where $\alpha=\nu+d/2$, $\kappa>0$, $\nu>0$, $\mathcal{W}(\bm{s})$ is a white noise spatial process, $\Delta=\Sigma_{i}\,\partial^{2}/\partial\,s_{i}^{2}$, and the marginal variance is $\sigma^{2}=\frac{\Gamma(\nu)}{\Gamma(\nu+d/2)(4\pi)^{d/2}\kappa^{2\nu}}.$ Since the lumen wall is modelled with a surface of triangular elements, we are interested in generating realizations from discretely indexed Gaussian random fields. This is achieved through an approximate stochastic weak solution of the SPDE (2), as discussed in [1]. We construct a discrete approximation of the solution $x(\bm{s})$ using a linear combination of basis functions, $\\{\psi_{k}\\},k=1,\dots,n$, and appropriate weights, $\\{w_{k}\\},k=1,\dots,n$, i.e., $x(\bm{s})=\sum_{k=1}^{n}\,\psi_{k}(\bm{s})\,w_{k}$. We then introduce an appropriate Sobolev space with inner product $\langle\cdot,\cdot\rangle$, a family of _test functions_ $\\{\varphi_{k}\\},k=1,\dots,n$, and derive a Galerkin functional for (2) of the form $\langle\varphi_{i},(\kappa^{2}-\Delta)^{\alpha/2}\,\psi_{j}\rangle\,w_{j}=\langle\varphi_{i},\mathcal{W}\rangle$ (3) We then choose $\varphi_{k}=(\kappa^{2}-\Delta)^{1/2}\,\psi_{k}$ for $\alpha=1$ and $\varphi_{k}=\psi_{k}$ for $\alpha=2$, leading to precision matrices $\bm{Q}_{\alpha}$ expressed as $\displaystyle\bm{Q}_{\alpha}=\varkappa^{2}\bm{C+G}$ $\displaystyle\text{for}\,\,\alpha=1$ (4) $\displaystyle\bm{Q}_{\alpha}=(\varkappa^{2}\bm{C+G})^{T}\bm{C}^{-1}(\varkappa^{2}\bm{C+G})$ $\displaystyle\text{for}\,\,\alpha=2$ $\displaystyle\bm{Q}_{\alpha}=(\varkappa^{2}\bm{C+G})^{T}\bm{C}^{-1}\bm{Q}_{\alpha-2}\bm{C}^{-1}(\varkappa^{2}\bm{C+G})$ $\displaystyle\text{for}\,\,\alpha>2,$ where, for $\alpha\geq 3$ a recursive Galerkin formulation is used, letting $\alpha=2$ on the left-hand side of equation (2) and replacing the right-hand side with a field generated by $\alpha-2$, assigning $\varphi_{k}=\psi_{k},\,k=1,\dots,n$. Note how the use of piecewise linear basis functions $\\{\psi_{k}\\},k=1,\dots,n$, lead to matrices $\bm{G}_{ij}=\langle\nabla\psi_{i},\nabla\psi_{j}\rangle\,\,\text{and}\,\,\bm{C}_{ij}=\langle\psi_{i},\psi_{j}\rangle,$ (5) that are _sparse_ , and often found in the finite element discretization of second order elliptic PDEs. However, the precision matrices $\bm{Q}_{\alpha}$ are, in general, not sparse as they contain the inverse $\bm{C}^{-1}$. Thus, by replacing the matrix $\bm{C}$ with the _lumped_ diagonal matrix $\widetilde{\bm{C}}$, sparsity is restored and $\bm{Q}_{\alpha}$ can be efficiently manipulated and decomposed through fast routines for sparse linear algebra available on a wide range of architectures. In addition, the introduction of $\widetilde{\bm{C}}$ leads to non zero terms on each row only for the _immediate neighbors_ $\mathcal{I}(s_{k})$ of a given node $k$ on the triangular surface mesh, since the basis function $\\{\psi_{k}\\}$ is supported only on the elements connected to node $k$. This reduces the Gaussian random field to a Gaussian Markov random field, for which $\begin{split}&\rho\left(x(s_{i}),x(s_{k})\,\|\,x(s_{j}),\,s_{j}\in\mathcal{I}(s_{i})\right)=\\\ &=\rho\left(x(s_{i})\,\|\,x(s_{j}),\,s_{j}\in\mathcal{I}(s_{i})\right),\end{split}$ (6) or, in other words, _given the value of $x$ on its neighbors_, at any node $k$ the random field $x(s_{k})$ is statistically independent from any other location. The approximation error introduced in (6) is, however, small and often negligible in applications [22]. The interested reader is referred to [1] for additional detail on the derivation of $\bm{Q}_{\alpha}$. Numerically generated realizations for various correlation lengths are shown in Figure 3 on an ideal cylindrical representation of the descending thoracic aorta, while Figure 3 shows the agreement of the generated field with the Matérn model. (a) $\rho=.95$ cm (b) $\rho=3.7$ cm (c) $\rho=7.2$ cm Figure 2: Random field generated on a cylindrical mesh for various correlation lengths. Figure 3: Comparison between spatial correlations from a numerically generated field $x(\bm{s})$ with precision matrix $\bm{Q}_{\alpha}$ and the exact Matérn model. ### 2.2 A segregated solver for fluid-structure interaction phenomena #### 2.2.1 Finite element model for the vessel wall We use a small strain, linear, 3 d.o.f. elastic thin shell which allows for a full compatibility between the fluid mesh discretized with tetrahedral elements and the solid walls. The in-plane stiffness of the shell is complemented with a transverse shear stiffness which provides stability under transverse loading [2]. Using a superscript $l$ and lower case $x,y,z$ to indicate quantities expressed in the local shell reference frame, we introduce a constitutive relation in Voigt notation expressed as $\bm{\sigma}^{l}=\bm{C}\cdot\bm{\varepsilon}^{l},\,\,\text{with}\,\,\bm{\sigma}^{l}=\begin{bmatrix}\sigma_{xx}\\\ \sigma_{yy}\\\ \tau_{xy}\\\ \tau_{xz}\\\ \tau_{yz}\end{bmatrix},\bm{\varepsilon}^{l}=\begin{bmatrix}\partial u_{x}/\partial x\\\ \partial u_{y}/\partial y\\\ \left(\partial u_{x}/\partial y+\partial u_{y}/\partial x\right)\\\ \partial u_{z}/\partial x\\\ \partial u_{z}/\partial y\end{bmatrix}.$ (7) We assume $\varepsilon^{l}_{zz}=0$, i.e., zero deformation through the thickness, disregarding the effect of both the normal pressure acting at the lumen surface and the Poisson effect due to the membrane deformations. Strains $\bm{\varepsilon}^{l}$ and nodal displacements $\bm{u}$ are related through the matrix $\bm{B}$ of shape function derivatives for a linear triangular element, i.e. $\bm{\varepsilon}^{l}=\bm{B}\,\bm{u}=\frac{1}{2\,A_{e}}\begin{bmatrix}y_{23}&0&0&y_{31}&0&0&y_{12}&0&0\\\ 0&x_{32}&0&0&x_{13}&0&0&x_{21}&0\\\ x_{32}&y_{23}&0&x_{13}&y_{31}&0&x_{21}&y_{12}&0\\\ 0&0&y_{23}&0&0&y_{31}&0&0&y_{12}\\\ 0&0&x_{32}&0&0&x_{13}&0&0&x_{21}\end{bmatrix}\begin{bmatrix}u_{x,1}\\\ u_{y,1}\\\ u_{z,1}\\\ u_{x,2}\\\ u_{y,2}\\\ u_{z,2}\\\ u_{x,3}\\\ u_{y,3}\\\ u_{z,3}\end{bmatrix}$ (8) where $x_{j},y_{j},\,j\in\\{1,2,3\\}$ are the local coordinates of the $j$-th element node, $x_{ij}=x_{i}-x_{j}$ (similarly for $y_{ij}$), $u_{x,j},u_{y,j},u_{z,j}$ are the local nodal displacements and $A_{e}$ is the triangular element area. The constitutive matrix is expressed as $\bm{C}=\frac{E}{(1-\nu^{2})}\begin{bmatrix}1&\nu&0&0&0\\\ \nu&1&0&0&0\\\ 0&0&0.5\,(1-\nu)&0&0\\\ 0&0&0&0.5\,k\,(1-\nu)&0\\\ 0&0&0&0&0.5\,k\,(1-\nu)\end{bmatrix}$ (9) where $E$ and $\nu$ are the Young’s modulus and Poisson’s ratio coefficient, respectively, and the shear factor $k$ accounts for a parabolic variation of transverse shear stress through the shell thickness (assumed as $5/6$ for a shell with rectangular cross section). Finally, the local element stiffness matrix $\bm{k}_{e}\in\mathbb{R}^{9\times 9}$ can be expressed as $\bm{k}_{e}=\int_{\Omega^{s}_{e}}\,\bm{B}^{T}\bm{C}\bm{B}\,\,\mathrm{d}\Omega^{s}_{e}=\sum_{i=1}^{n_{\text{gp}}}\,\bm{B}^{T}\bm{C}\bm{B}\,A_{e}\,\zeta_{i}\,w_{i},$ (10) where $n_{\text{gp}}$ is the total number of integration points and $\zeta_{i},w_{i},i\in\\{1,2,\dots,n_{\text{gp}}\\}$, are the element thickness and integration rule weights, respectively. In this study, we adopt a three- point Gauss integration rule to capture a linear variation for $E$ and $\zeta$ through each element, generated from a Gauss Markov random field with Matérn covariance $\bm{Q}_{\alpha}$, i.e., $E=E(x,y,\omega)$ and $\zeta=\zeta(x,y,\omega)$. Nodal vectors $\bm{E}\sim\mathcal{N}(\overline{\bm{E}},\bm{Q}^{-1}_{\alpha})$ and $\bm{\zeta}\sim\mathcal{N}(\overline{\bm{\zeta}},\bm{Q}^{-1}_{\alpha})$ are generated as $\bm{E}=\overline{\bm{E}}+(\bm{L}^{T})^{-1}\bm{z},\,\,\text{and}\,\,\bm{\zeta}=\overline{\bm{\zeta}}+(\bm{L}^{T})^{-1}\bm{z},$ (11) where $\bm{z}\sim\mathcal{N}(\bm{0},\bm{I}_{n})$ is a vector with standard Gaussian components and $\bm{L}$ is the sparse Cholesky factor of $\bm{Q}_{\alpha}$, i.e., $\bm{Q}_{\alpha}=\bm{L}\,\bm{L}^{T}$. The covariance matrix $\bm{Q}_{\alpha}$ is assembled in compressed sparse column (CSC) format and the Cholesky decomposition is computed using the _cholmod_ routine provided by the _scikit-sparse_ Python library. Finally, the product $(\bm{L}^{T})^{-1}\bm{z}$ is performed using the _solve_Lt_ routine, as the solution of a triangular system. #### 2.2.2 Variational multiscale finite element fluid solver The evolution of blood flow and pressure in the human cardiovascular system can be modeled using the Navier-Stokes equations. Even though many simplifying assumptions can be made to the equations to reduce the computational complexity, here we focus on high-fidelity models, i.e., models associated with large discretizations of a three-dimensional ($n_{\text{sd}}=3$) fluid domain $\Omega^{f}\subseteq\mathbb{R}^{n_{\text{sd}}}$. The boundary $\Gamma^{f}$ of $\Omega^{f}$ coincides with the mid-plane of the solid domain $\Omega^{s}$, and is partitioned into $\Gamma^{f}=\Gamma_{g}^{f}\cup\Gamma_{h}^{f}\cup\Gamma_{s}^{f}$ which correspond to the application of Dirichlet, Neumann boundary conditions and interaction with the solid, respectively. Consider also the vector fields $\bm{h}:\Gamma_{h}\times(0,T)\to\mathbb{R}^{n_{sd}}$, $\bm{g}:\Gamma_{g}\times(0,T)\to\mathbb{R}^{n_{sd}}$, $\bm{f}:\Omega\times(0,T)\to\mathbb{R}^{n_{sd}}$ and $\bm{v}^{0}:\Omega\to\mathbb{R}^{n_{sd}}$. We would like to solve the problem of finding $\bm{v}(\bm{X},t)$ and $p(\bm{X},t)$, $\forall\,\bm{X}\in\Omega$, $\forall\,t\in[0,T]$ such that $\begin{cases}\rho\,\dot{\bm{v}}+\rho\,\bm{v}\cdot\nabla\bm{v}=-\nabla p+\nabla\cdot\bm{\tau}+\bm{f}&(\bm{X},t)\in\Omega\times(0,T)\\\ \nabla\cdot\bm{v}=0&(\bm{X},t)\in\Omega\times(0,T),\\\ \end{cases}$ (12) subject to the boundary conditions $\begin{cases}\bm{v}=\bm{g}&(\bm{X},t)\in\Gamma_{g}\times(0,T)\\\ \bm{t}_{\bm{n}}=\bm{\sigma}\cdot\bm{n}=[-p\,\bm{I}+\bm{\tau}]\cdot\bm{n}=\bm{h}&(\bm{X},t)\in\Gamma_{h}\times(0,T)\\\ \bm{t}_{\bm{n}}=\bm{t}^{f}&(\bm{X},t)\in\Gamma_{s}\times(0,T)\\\ \bm{v}(\bm{X},0)=\bm{v}^{0}(\bm{X})&\bm{X}\in\Omega,\end{cases}$ (13) where $\bm{\tau}=\mu(\nabla\bm{v}+\nabla\bm{v}^{\,T})$ is the viscous stress tensor resulting from considering blood as a Newtonian fluid. Solution of (12) in weak form requires to define four approximation spaces, i.e., two trial spaces for the velocity $\bm{v}$ and pressure $p$ $\begin{split}\mathscr{S}_{k}^{h}&=\Big{\\{}\bm{v}\,\|\,\bm{v}(\cdot,t)\in\bm{H}^{1}(\Omega),\mathnormal{t}\in[0,\mathnormal{T}],\\\ &\bm{v}\,\|_{\bm{x}\in\Omega_{e}}\in\mathnormal{P}_{k}(\Omega_{e}),\,\bm{v}(\cdot,t)=\bm{g}\text{ on }\Gamma_{g}\Big{\\}},\\\ \mathscr{P}_{k}^{h}&=\left\\{p\,\|\,p(\cdot,t)\in L^{2}(\Omega),t\in[0,\mathnormal{T}],\,p\,\|_{\bm{x}\in\bar{\Omega}_{e}}\in\mathnormal{P}_{k}(\Omega_{e})\right\\},\end{split}$ and two test spaces for $\vec{w}$ and $q$ $\begin{split}\mathscr{W}_{k}^{h}=\Big{\\{}&\bm{w}\,\|\,\bm{w}(\cdot,t)\in\bm{H}^{1}(\Omega),\,\mathnormal{t}\in[0,\mathnormal{T}],\\\ &\bm{w}\,\|_{\bm{x}\in\Omega_{e}}\in\mathnormal{P}_{k}(\Omega_{e}),\,\bm{w}(\cdot,t)=\bm{0}\text{ on }\Gamma_{g}\Big{\\}},\,\,\mathscr{Q}_{k}^{h}=\mathscr{P}_{k}^{h}\end{split}$ where $\bm{H}^{1}(\Omega)$ is the Sobolev space of function triplets in $L^{2}(\Omega)$ with derivatives in $L^{2}(\Omega)$ and $P_{k}(\Omega)$ is the space of polynomials of order $k$ in $\Omega$. The above spaces are separated into large and a small scale contributions $\mathscr{S}_{k}^{h}=\overline{\mathscr{S}_{k}^{h}}\oplus\widetilde{\mathscr{S}_{k}^{h}}$, $\mathscr{W}_{k}^{h}=\overline{\mathscr{W}_{k}^{h}}\oplus\widetilde{\mathscr{W}_{k}^{h}}$ and $\mathscr{P}_{k}^{h}=\overline{\mathscr{P}_{k}^{h}}\oplus\widetilde{\mathscr{P}_{k}^{h}}$ with a corresponding decomposition of velocity and pressures as $\bm{v}=\overline{\bm{v}}+\widetilde{\bm{v}}$ and $p=\overline{p}+\widetilde{p}$, respectively. This decomposition is introduced in a weak form of the Navier-Stokes equations (12) and a _closure_ obtained by expressing the small scale variables $\widetilde{\bm{v}}$ and $\widetilde{p}$ in terms of their large scale counterparts $\overline{\bm{v}}$ and $\overline{p}$ using [23] $\begin{bmatrix}\widetilde{\bm{v}}\\\ \widetilde{\bm{p}}\end{bmatrix}=\begin{bmatrix}\tau_{M}\,\bm{R}_{M}(\overline{\bm{v}},\overline{p})\\\ \tau_{C}\,R_{C}(\overline{\bm{v}},\overline{p})\end{bmatrix},$ (14) where $\bm{R}_{M}$ and $R_{C}$ represent the momentum and continuity residual expressed as $\begin{cases}\bm{R}_{M}(\overline{\bm{v}},\overline{p})=\bm{R}_{M}=\rho\,\dot{\overline{\bm{v}}}+\rho\,\overline{\bm{v}}\cdot\nabla\overline{\bm{v}}+\nabla\overline{p}-\nabla\cdot\bm{\tau}-\bm{f},\\\ R_{C}(\overline{\bm{v}},\overline{p})=R_{C}=\nabla\cdot\overline{\bm{v}},\end{cases}$ (15) and the _stabilization_ coefficients are expressed as $\begin{split}\tau_{M}&=\left(\frac{4}{\Delta t^{2}}+\bm{u}\cdot\bm{G}\,\bm{u}+C_{I}\,\nu^{2}\,\bm{G}:\bm{G}\right)^{-1/2}\\\ \tau_{C}&=(\tau_{M}\,\bm{g}\cdot\bm{g})^{-1},\,\,G_{i,j}=\sum_{k=1}^{3}\,\dfrac{\partial\xi_{k}}{\partial x_{i}}\,\dfrac{\partial\xi_{k}}{\partial x_{j}},\,\,\,g_{i}=\sum_{k=1}^{3}\,\dfrac{\partial\xi_{k}}{\partial x_{i}},\end{split}$ (16) where $\partial\bm{\xi}/\partial\bm{x}$ is the inverse Jacobian of the element mapping between the parametric and physical domains. To simplify the notation, in what follows the large scale variables $\overline{\bm{v}}$ and $\overline{p}$ will be denoted simply by $\bm{v}$ and $p$ and the spaces $\overline{\mathscr{S}_{k}^{h}}$, $\overline{\mathscr{W}_{k}^{h}}$ and $\overline{\mathscr{P}_{k}^{h}}$ by $\mathscr{S}_{k}^{h}$, $\mathscr{W}_{k}^{h}$ and $\mathscr{P}_{k}^{h}$. A weak solution of the Navier-Stokes equations can now be determined by finding $\bm{v}\in\mathscr{S}_{h}^{k}$ and $p\in\mathscr{P}_{h}^{k}$ such that $\begin{split}&B(\bm{w},q\,;\,\bm{v},p)=B_{G}(\bm{w},q\,;\,\bm{v},p)+\\\ &+\sum_{e=1}^{n_{\text{el}}}\,\int_{\Omega_{e}}\left\\{(\bm{v}\cdot\nabla)\,\bm{w}\cdot(\tau_{M}\,\bm{R}_{M})+\nabla\cdot\bm{w}\,\tau_{C}\,R_{C}\right\\}\,\mathrm{d}\Omega_{e}+\\\ &+\sum_{e=1}^{n_{\text{el}}}\,\int_{\Omega_{e}}\left\\{\bm{w}\cdot\left[-\tau_{M}\,\bm{R}_{M}\cdot\nabla\bm{v}\right]+\left[\bm{R}_{M}\cdot\nabla\bm{w}\right]\cdot\left[\overline{\tau}\,\bm{R}_{M}\cdot\bm{v}\right]\right\\}\,\mathrm{d}\Omega_{e}+\\\ &+\sum_{e=1}^{n_{\text{el}}}\,\int_{\Omega_{e}}\nabla q\cdot\frac{\tau_{M}}{\rho}\,\bm{R}_{M}\,\mathrm{d}\Omega_{e}=0,\end{split}$ (17) for all $\vec{w}\in\vec{\mathscr{W}}_{h}^{k}$ and $q\in\mathscr{P}_{h}^{k}$, where the Galerkin functional $B_{G}$ is expressed as $\begin{split}&B_{G}(\bm{w},q\,;\,\bm{v},p)=\int_{\Omega}\bm{w}\cdot(\rho\,\dot{\bm{v}}+\rho\,\bm{v}\cdot\nabla\bm{v}-\bm{f})\,\,\mathrm{d}\Omega+\\\ &+\int_{\Omega}\left\\{\nabla\bm{w}\,:\,(-p\bm{I}+\bm{\tau})-\nabla q\cdot\bm{v}\right\\}\,\mathrm{d}\Omega+\\\ &+\int_{\Gamma_{h}}\,\left\\{-\bm{w}\cdot\bm{h}+q\,v_{n}\right\\}\,\mathrm{d}\Gamma+\int_{\Gamma_{s}}\,\left\\{-\bm{w}\cdot\bm{t}^{f}+q\,v_{n}\right\\}\,\mathrm{d}\Gamma+\\\ &+\int_{\Gamma_{g}}\,q\,v_{n}\,\mathrm{d}\Gamma.\end{split}$ (18) The discrete variables $\bm{w}^{h}$, $\bm{v}^{h}$, $q^{h}$ and $p^{h}$ are introduced in (17), leading to a non linear system of equations of the form $\begin{cases}\bm{N}_{M}(\dot{\bm{v}}^{h},\bm{v}^{h},p^{h})=0\\\ N_{C}(\dot{\bm{v}}^{h},\bm{v}^{h},p^{h})=0,\end{cases}$ (19) A predictor-multicorrector scheme [24] is used for time integration and, at each time step, the resulting non linear system (19) is solved through successive Newton iterations. Additional details can be found in [23, 25]. #### 2.2.3 Fluid-structure coupling Since this study focuses more on the development of an ensemble solver, we provide a one-way coupled, simplified treatment of the interaction between fluid and structure, leaving a more rigorous treatment to future work. Here we assume a fluid model with rigid walls and a structural model of the lumen governed by the three d.o.f. shell discussed in Section 2.2.1. The lumen deformation does not, in turn, affect the geometry of the fluid region. The elastic forces in the lumen wall are in equilibrium with the shear and pressure exerted by the fluid, or, in other words $\bm{t}^{s}=-\bm{t}^{f}$, where the wall stress $\bm{t}^{f}$ computed by the fluid solver is $\bm{t}^{f}=\bm{\sigma}^{f}\cdot\bm{n},\,\,\bm{\sigma}=2\mu\bm{\varepsilon}-p\bm{I},$ (20) $p$ is the main _nodal_ pressure unknown in the VMS fluid solver, and $\bm{\varepsilon}=\nabla^{s}\bm{u}$ is the symmetric part of the velocity gradient, which is constant on each P1 element in the fluid mesh. An example of shear forces computed by the VMS fluid solver for a coronary model is illustrated in Figure 4. At every node on the wall, the shear forces and normal vectors from adjacent elements are averaged and the nodal pressure added, leading to the three components of the nodal force that are passed to the structural solver. Finally, stress components in the circumferential, radial and axial directions are obtained by transforming the solid stress tensor to a local cylindrical coordinate system (see Figure 4). For an arbitrary Gauss point or lumen shell node $\bm{s}$, we identify the closest location on the vessel centerline. The tangent vector to the centerline at $\bm{s}$ is defined as the local _axial_ direction $z$, the normal at the Gauss point is the local _radial_ direction $r$, and the local _circumferential_ direction $\theta$ is obtained as the cross product between $r$ and $z$. Figure 4: Visualization of interface shear forces exchanged between the fluid and the structural solver. The local axis system used for stress post- processing is also shown in the close up. ### 2.3 Distributed explicit FSI solver on multiple CPUs The solution in time for the dynamics of the undamped non-linear structural system $\bm{M}\,\bm{\ddot{u}}+\bm{C}\,\bm{\dot{u}}+\bm{K}(\bm{u})\,\bm{u}=\bm{f}(t),$ (21) is computed using central differences in time, resulting in an update formula in the $[n\,\Delta t,(n+1)\,\Delta t]$ intervals expressed as $\left(\bm{\widetilde{M}}+\frac{\Delta t}{2}\bm{\widetilde{C}}\right)\,\bm{u}_{n+1}=\Delta t^{2}\,\bm{f}_{n}-\left(\Delta t^{2}\bm{K}-2\,\bm{M}\right)\,\bm{u}_{n}-\left(\bm{M}-\frac{\Delta t}{2}\bm{C}\right)\,\bm{u}_{n-1}$ (22) which is performed independently by an arbitrary number of mesh partitions. To do so efficiently, $\bm{\widetilde{M}},\bm{\widetilde{C}}$ in (22) are lumped mass matrix pre-assembled before the beginning of the time loop, containing the elemental contributions from all finite elements, even those belonging to separate mesh partitions. Given the limited amount of deformation typically observed in cardiovascular applications over a single heart cycle, the assumption of a fixed nodal mass over time is considered realistic. This assumption removes the need of communicating nodal masses during finite element assembly, improving scalability. In some cases, a viscous force $\bm{f}_{v}$ is added to the right-hand-side in order to damp the high- frequency oscillations as $\bm{f}_{v}=-c_{d}\,\dot{\bm{u}}_{n}$, through an appropriate damping coefficient $c_{d}$. Even though the geometry of the vessel lumen is updated at every step by adding $\bm{u}_{n+1}$, the displacement magnitudes over a typical heart cycle remain of the same order as the thickness of the vessel wall, hence in the linear regime. In this context, the stiffness do not significantly change and it is possible to save computational time by avoiding to assemble it at every iteration. In our code, we therefore provide the option to selectively update the stiffness matrix after a prescribed number of iterations. Synchronization of displacements for the nodes shared by multiple partitions is implemented using _Send_ , _Recv_ to the root CPU and broadcast back. The performance of the proposed ensemble solver was first tested on multiple CPUs. We use distributed sparse matrices in the Yale compressed sparse row (CSR) format [26] with dense coefficient entries of size $9\cdot n_{s}$, where $n_{s}$ is the number of material property realizations and a local element matrix of size 3$\times$3 results from selecting a three-d.o.f.s shell finite element. The code for the finite element assembly and sparse matrix-vector multiplication was developed in Cython+MPI+openMP and compared both with a C implementation and with the mkl_cspblas_dcsrgemv routine provided through the Intel MKL library, using single and multiple threads. We verified the satisfactory performance of our implementation under a wide range of mesh sizes, number of cores, and with/without multithreading. Encouraging speedups were obtained on multiple CPUs, as shown in Figure 5 and Figure 6. (a) (b) Figure 5: Performance of matrix-vector product kernel on CPU by our Cython+openMP implementation, GPU implementation and MKL library on multiple threads (a). Optimization of GPU matrix-vector kernel and CPU-GPU communication performance (b). Tests were performed using 1 CPU and 1 GPU on a cylindrical model associated with a sparse matrix having nnz = 160,587 and 1,000 material property realizations. (a) (b) Figure 6: Performance of explicit ensemble solver on multiple CPUs for a mesh with 5,074 elements, 2,565 nodes (a) and for a mesh with 15,136 elements and 7,628 nodes (b). (a) (b) Figure 7: Performance of explicit ensemble solver on multiple CPUs/GPUs for a mesh with 5,074 elements, 2,565 nodes (a) and for a mesh with 15,136 elements and 7,628 nodes (b). ### 2.4 Distributed explicit FSI solver on multiple GPUs We developed an hardware-independent openCL implementation of the solver running on multiple GPUs. We started with a naïve CSR-based parallel sparse matrix-vector product (SpMV), known as _CSR-Scalar_ , where each row of the sparse matrix is assigned to a separate thread [27]. This works well on CPUs, but causes uncoalesced, slow memory accesses on GPUs, since elements in each row occupy consecutive addresses in memory, but consecutive threads access elements on different rows. In addition, long rows lead to an unequal amount of work among the threads and some of them need to wait for others to finish. We then transitioned to a _CSR-Vector_ scheme [28], assigning a wavefront (or so-called _warp_ on NVIDIA architectures) to work on a single row of the matrix. This allows for access to consecutive memory locations in parallel, resulting in fast coalesced loads. However, CSR-Vector can lead to poor GPU occupancy for short rows due to unused execution resources. Improved performance can be achieved using _CSR-Stream_ [29] which statically fixes the number of nonzeros that will be processed by one wavefront and streams all of these values into the local scratchpad memory, effectively utilizing the GPU’s DRAM bandwidth and improving over CSR-Scalar. CSR-Stream also dynamically determines the number of rows on which each wavefront will operate, thus improving over CSR-Vector. While CSR-Stream substantially improves the performance of the spMV product kernel, the CPU-to-GPU data transfer still dominates the time step update, as shown in Figure 5(b). This problem can be mitigated by sending data to the GPU in smaller chunks, thus overlapping data transfer and kernel execution. In addition, data transfer can be minimized by an assembly-free approach. Even though this is a standard practice in explicit finite element codes, it is particularly effective on a GPU for three reasons. First, storage of a sparse global matrix would occupy a significant portion of the GPU memory, posing restrictions on the model size. Second, indexing operations to access entries in the global matrix would cause severe uncoalesced memory access, reducing significantly the degree of parallelism in the GPU. Third, for the most common sparse matrix storage schemes, indexing always include searching, which would be particularly slow on GPU. We compute local matrices directly in the GPU and take their product with a partition-based displacement vector, where only the displacements of shared nodes need to be synchronized through the root CPU. To compute element matrices, we buffer data to the GPU before the beginning of the time integration loop, in order to avoid any CPU to GPU data transfer due to element assembly. Buffered quantities include mesh geometry and material properties, particularly the product between the Young’s modulus and thickness at each Gauss point for all random field realizations. Note how the rest of the local stiffness matrix is constant for linear triangular elements. In addition, the left-hand-side lumped mass matrix resulting from the central difference scheme does not change throughout the time loop. We also leverage a mesh coloring algorithm, allowing working units to process different elements at the same time. During each time step, each computing group in the GPU works on one element, while the working units in the same group work on different realizations and therefore have access to coalesced GPU memory. Communication is only triggered by displacement synchronization. We use pinned host memory to speed up the data transfer between CPU and GPU. All displacements for each realization and the local matrices are stored in private memory since they do not need to be shared with other working units. Final GPU speed ups are illustrated in Figure 7. ### 2.5 A Python code-base The CVFES solver is developed in Python 3 with optimization in Cython [30] and element assembly and matrix product implementation on openCL [31] (though the Python pyOpenCL library [32]). The code leverages the VTK library [33] to read the solid, fluid mesh and boundary conditions. In this context the solver is fully compatible with the input files generated by the SimVascular software platform [34] and can be easily integrated with the SimVascular modeling workflow. Partitioning on multiple CPUs and GPUs is obtained for both solvers using parMETIS [35]. The code used to generate the results discussed in Section 3 is available through a public GitHub repository at https://github.com/desResLab/CVFES. ## 3 Results ### 3.1 Ideal cylindrical benchmark The first benchmark represents an ideal cylindrical lumen subject to aortic flow. The cylinder has a diameter equal to 4 cm and length of 30 cm, while two Matérn random fields for thickness and elastic modulus have been assigned as discussed in Section 2.1. Specifically, a mean $\mu=7.0\times 10^{6}$ Barye and a standard deviation $\sigma=7.0\times 10^{5}$ Barye have been assumed for the elastic modulus, whereas the thickness random field is characterized through a mean $\mu=0.4$ cm and a standard deviation $\sigma=0.04$ cm. Three values of the correlation length were considered, equal to 0.95 cm, 3.7 cm and 7.2 cm, respectively (see discussion in [36]). A uniform pressure of 13 mmHg was added to the pressure computed by the VMS fluid solver. We considered diastole as a natural (unstressed) state and applied the difference between a diastolic pressure of 80 mmHg and the mean brachial pressure in a healthy subject (i.e., with systolic pressure equal to 120 mmHg). This configuration is analyzed both under steady state and pulsatile flow conditions, under fully fixed structural boundary conditions at the two cylinder ends. #### 3.1.1 Steady state analysis The fluid solution is computed with the VMS fluid solver discussed in Section 2.2.2 using a parabolic velocity profile at the inflow corresponding to a $-66.59$ mL/s volumetric flow rate, zero-traction boundary condition at the outlet and a no-slip condition at the lumen wall. As expected, the fluid solution show a perfectly linear relative pressure profile along the cylinder center path and a uniform parabolic velocity profile from inlet to outlet, typical of viscous-dominated Poiseuille flow, as shown in Figure 8. The steady state pressure resulting from the fluid solver is applied as discussed in Section 2.2.3 and 100 thickness and elastic modulus realizations are solved simultaneously. The explicit structural simulation is run for $0.5$ seconds, until a steady state was observed. Three wall mesh densities (coarse, medium and fine) are finally considered, consisting of 5,074, 15,136 and 32,994 triangular shell elements, with explicit time steps set to $\Delta t=4.0\times 10^{-5}$ s, $\Delta t=4.0\times 10^{-5}$ s and $\Delta t=1.0\times 10^{-5}$ s, respectively and no viscous damping. Displacement magnitudes along the longitudinal $z$ axis (cylinder generator) are shown in the top row of Figure 9 for the finer mesh and various correlation lengths. The mean displacement one diameter away from the fully fixed ends (thick black line) is consistent with an homogeneous solution for a thick cylinder with average elastic modulus and thickness (blue dashed line). Displacements associated with single random field realizations are also shown, in Figure 9 (top row), using colors. As expected, the displacement wave length increases with the correlation length, and so does the displacement uncertainty quantified through the 5%-95% confidence interval (gray shaded area). Finally, the second row of Figure 9 shows how increasing the mesh density produces a limited difference in the 5%-95% confidence interval. Circumferential stress was found to be the most significant component, as expected, showing uncertainty increasing with the correlation length, similarly to what observed for the displacement magnitude. The in-plane and out-of-plane shear components ($\sigma_{\theta z}$ and $\sigma_{rz}$), though much smaller than circumferential and axial stresses, are essentially related to the material property non-homogeneity and reduce for an increasing correlation length. These stress components are zero both on average and by solving the model with average material properties. Thus, they can only be captured by explicitly modeling the spatial variability of material properties as in the proposed approach. (a) (b) Figure 8: Steady state pressure (a) and velocity (b) distributions from variational multi-scale fluid solver. Figure 9: Displacement magnitude along cylinder generator (longitudinal $z$ axis) for various correlation lengths (first row). 5%-95% confidence intervals for displacement magnitudes for three increasing mesh densities (i.e., coarse, medium and fine, second row). Figure 10: Ensemble means, 5%-95% confidence intervals and single realizations of longitudinal stress profiles for various mesh densities and random field correlation lengths. #### 3.1.2 Validation under pulsatile flow conditions A parabolic pulsatile inflow (Figure 11(a)) was applied to the same cylindrical geometry discussed in the previous section, while all the other boundary conditions (walls and outlets) were kept the same as in the steady case. The time step is set to $1.0\times 10^{-5}$ and the simulation run for two heart cycles and 100 material property realizations. To avoid the application of impulsive loads which can excite a broad range of frequencies, significantly affecting the undamped dynamics, a ramp was applied to the wall loads resulting from the fluid solver in the last heart cycle. The ramp follows a sine wave and is kept active for the first $0.2$ seconds of the simulation. No damping was applied to the simulation, i.e., $\bm{f}_{v}=0$. As expected, the resulting displacements follow a time profile similar to the inflow, and the 5%-95%confidence intervals increase with the correlation length of the underlying random field. Circumferential and axial stress components are the dominant stress components, which also increase with the correlation length, as observed for the steady state results. (a) (b) (c) Figure 11: Results for pulsatile flow on ideal cylindrical vessel. Inflow time history and spatial location for acquisition of displacement and stress outputs (a). Displacement time history at selected spatial location with thick lines representing ensemble averages, and thin lines used for 5%/95% percentiles (b). Circumferential and axial stress time history at selected spatial location (c). ### 3.2 Benchmark on patient-specific coronary model We also demonstrate the results of the proposed ensemble solver on a patient- specific model of the left anterior descending coronary branch. An ensemble of 100 model solutions were obtained using random field parameters for the elastic modulus equal to $\mu=1.15\times 10^{7}$ Barye and $\sigma=1.7\times 10^{6}$ Barye. For the thickness, we considered a mean equal to $\mu=0.075$ cm and a standard deviation of $\sigma=0.017$ cm. A slip-free boundary condition was applied at the outlets of the fluid domain, whereas fully fixed mechanical restraints (i.e., all three nodal translations) where applied along the edges at both the inlets and outlets. A uniform pressure has been superimposed to the lumen stress obtained from the fluid solver as discussed for the ideal aortic model in Section 3.1. #### 3.2.1 Steady state analysis A constant flow rate equal to $-0.28$ mL/s is applied at the model inlet with a parabolic profile, while a zero-traction boundary condition is applied at the outlets and a no-slip condition at the walls. The explicit time step is set to $2.5\times 10^{-6}$ and the model was run for $0.13$ seconds to reach the steady state. No additional viscous force $\bm{f}_{v}$ was considered. Figure 12 shows the displacements and stress for all three analyzed correlation lengths, averaged through a cross sectional slice of the branch of interest, and plotted for successive slices along the longitudinal $z$ axis. Displacement results confirm the absence of torsion, and a prevalent radial deformation mode, with rigid body motion evident from the axial displacements $d_{z}$, affected by the centerline path geometry. The stress results confirm the importance of the circumferential followed by the axial component. The circumferential stress reduces with the coronary branch radius as intuitively suggested by the Barlow (or Mariotte) formula for thin-walled cylinders. Similar cylindrical displacements, circumferential and axial stress are observed for all three correlation lengths. For the selected coronary branch, the smaller correlation length (0.95 cm) is approximately equal to twice the largest diameter, inducing minimal changes in local deformability. Figure 12: Displacement and stress profiles in left coronary artery LAD branch under steady flow, averaged over all material property realizations and cross- sectional slice. #### 3.2.2 Validation under pulsatile flow conditions The same geometry analyzed in the previous section is subject to a parabolic pulsatile inflow shown in Figure 13. A time step equal to $2.0\times 10^{-6}$ is selected, and the model is run for two complete heart cycles (1.6 seconds) and 100 material property realizations. Similar to the pulsatile cylindrical test case, a time ramp is applied during the first 0.2 s, to avoid impulsive loads produced by a non-zero wall stress at $t=0$, and a pressure of 13 mmHg is superimposed to the fluid wall stress. A viscous force $\bm{f}_{v}$ was applied, using a damping coefficient equal to $c_{d}=0.005$, which was found from various tests to remove the high-frequency oscillations without affecting the system dynamics. Results for the average displacements and stress are shown in Figure 13 at four successive locations over the center path of the LAD branch. Even for this case the circumferential and axial stress are the most relevant components and their time history is similar to the inflow and exhibit a maximum at diastole. Figure 13: Displacement and stress profiles in left coronary artery LAD branch under pulsatile flow, averaged over all material property realizations and cross-sectional slice. ### 3.3 Performance assessment In this section, we compare the performance obtained by running the proposed ensemble solver on three cylindrical models with an increasing number of elements, as shown in Table 1, Table 2 and Table 3. The explicit time step was set to $1.0\times 10^{-5}$ s for all models, and each run consisted of 1,000 time steps. The GPUs used for these tests are four _GeForce RTX 2080 Ti_ with 11GB of RAM equipped with 4352 NVIDIA CUDA Cores and connected to the server main board through PCI express ports. Two types of speedup are investigated, the first relates to solving the same number of material property realizations on an increasing number of processors, either CPUs or GPUs, and provides an idea of the effectiveness of the various optimizations presented in Section 2.3 and Section 2.4. This speedup (first number in parenthesis) is observed to increase with the model size and the number of random field realizations. The computation/communication tradeoff and the small mesh sizes selected for these tests also justify the negligible benefits of using 24 CPU cores instead of 12. The speedup achieved by our GPU implementation is instead very relevant, i.e., approximately three orders of magnitude with respect to a single CPU implementation. The second type of speedup quantifies the efficiency in the proposed ensemble solver, i.e., how much faster one can obtain the solution of multiple realizations by solving them at the same time, with respect to the solution of a single realization on the same hardware, multiplied by the total number of realizations. The computational savings of an ensemble solution are quantified between one and two orders of magnitude, confirming our initial claim. Table 1: Speedup - Model with 5074 elements, 2565 nodes - 1000 time steps with $\Delta t=1.0\times 10^{-5}$ Hardware \| 1 Smp (Spd) \| 10 Smp (Spd) \| 50 Smp (Spd) \| 100 Smp (Spd) \| 200 Smp (Spd) \| 500 Smp (Spd) \| ---\|---\|---\|---\|---\|---\|---\|--- 1 CPU \| 0:00:16 (1.0,1.0) \| 0:01:54 (1.43,1.0) \| 0:09:46 (1.39,1.0) \| 0:18:53 (1.44,1.0) \| 0:37:20 (1.46,1.0) \| 1:40:13 (1.36,1.0) \| 12 CPU \| 0:00:10 (1.0,1.50) \| 0:00:18 (5.92,6.20) \| 0:01:00 (8.92,9.62) \| 0:01:58 (9.21,9.59) \| 0:03:45 (9.65,9.94) \| 0:09:35 (9.46,10.45) \| 24 CPU \| 0:00:06 (1.0,2.35) \| 0:00:17 (3.93,6.45) \| 0:01:21 (4.27,7.22) \| 0:02:18 (5.03,8.21) \| 0:05:25 (4.26,6.87) \| 0:11:38 (4.98,8.61) \| 1 GPU \| 0:00:01 (1.0,9.42) \| 0:00:01 (10.20,67.19) \| 0:00:02 (29.82,201.96) \| 0:00:05 (32.75,214.30) \| 0:00:09 (37.84,244.67) \| 0:00:20 (42.31,293.77) \| 2 GPU \| 0:00:01 (1.0,10.56) \| 0:00:01 (9.17,67.69) \| 0:00:02 (36.27,275.28) \| 0:00:03 (44.01,322.76) \| 0:00:05 (55.86,404.83) \| 0:00:11 (66.74,519.27) \| 3 GPU \| 0:00:01 (1.0,9.49) \| 0:00:01 (9.53,63.25) \| 0:00:01 (44.86,305.92) \| 0:00:02 (59.66,393.12) \| 0:00:04 (75.29,490.21) \| 0:00:09 (91.75,641.43) \| 4 GPU \| 0:00:01 (1.0,9.34) \| 0:00:01 (9.70,63.35) \| 0:00:01 (46.12,309.52) \| 0:00:02 (58.67,380.35) \| 0:00:04 (75.09,481.08) \| 0:00:09 (90.21,620.49) \| () All time entries are in a _days:hours:minutes:seconds_ format. The speedup is indicated as $(x,y)$, where $x$ is the speedup with respect to the number of samples and $y$ the speedup by distributing the computation on multiple CPUs or GPUs. Table 2: Speedup - Model with 15136 elements, 7628 nodes - 1000 time steps with $\Delta t=1.0\times 10^{-5}$ Hardware \| 1 Smp (Spd) \| 10 Smp (Spd) \| 50 Smp (Spd) \| 100 Smp (Spd) \| 200 Smp (Spd) \| 500 Smp (Spd) \| ---\|---\|---\|---\|---\|---\|---\|--- 1 CPU \| 0:00:37 (1.0,1.0) \| 0:05:24 (1.16,1.0) \| 0:25:33 (1.23,1.0) \| 0:52:29 (1.20,1.0) \| 2:03:22 (1.02,1.0) \| 4:48:25 (1.09,1.0) \| 12 CPU \| 0:00:10 (1.0,3.59) \| 0:00:21 (4.83,14.89) \| 0:01:50 (4.74,13.82) \| 0:04:05 (4.29,12.83) \| 0:07:23 (4.74,16.69) \| 0:19:19 (4.54,14.92) \| 24 CPU \| 0:00:11 (1.0,3.32) \| 0:00:28 (4.02,11.48) \| 0:01:57 (4.83,13.01) \| 0:03:58 (4.77,13.21) \| 0:07:20 (5.16,16.79) \| 0:18:38 (5.08,15.47) \| 1 GPU \| 0:00:03 (1.0,9.72) \| 0:00:03 (10.16,84.84) \| 0:00:07 (26.56,209.58) \| 0:00:13 (27.89,225.99) \| 0:00:24 (31.22,297.26) \| 0:00:56 (34.28,305.28) \| 1 GPU \| 0:00:02 (1.0,15.01) \| 0:00:02 (10.11,130.45) \| 0:00:04 (29.17,355.60) \| 0:00:08 (30.94,387.40) \| 0:00:13 (37.20,547.27) \| 0:00:30 (40.80,561.29) \| 1 GPU \| 0:00:02 (1.0,17.02) \| 0:00:02 (9.47,138.46) \| 0:00:03 (29.37,405.87) \| 0:00:06 (33.57,476.46) \| 0:00:10 (40.38,673.50) \| 0:00:24 (44.55,694.70) \| 1 GPU \| 0:00:01 (1.0,20.01) \| 0:00:02 (8.21,141.15) \| 0:00:03 (28.94,470.08) \| 0:00:05 (35.15,586.47) \| 0:00:08 (42.02,823.93) \| 0:00:19 (48.60,891.06) \| () All time entries are in a _days:hours:minutes:seconds_ format. The speedup is indicated as $(x,y)$, where $x$ is the speedup with respect to the number of samples and $y$ the speedup by distributing the computation on multiple CPUs or GPUs. Table 3: Speedup - Model with 131552 elements, 65896 nodes - 1000 time steps with $\Delta t=1.0\times 10^{-5}$ Hardware \| 1 Smp (Spd) \| 10 Smp (Spd) \| 50 Smp (Spd) \| 100 Smp (Spd) \| 200 Smp (Spd) \| 500 Smp (Spd) \| ---\|---\|---\|---\|---\|---\|---\|--- 1 CPU \| 0:06:47 (1.0,1.0) \| 0:56:58 (1.19,1.0) \| 4:06:55 (1.38,1.0) \| 7:55:06 (1.43,1.0) \| - \| - \| 12 CPU \| 0:00:22 (1.0,18.13) \| 0:02:03 (1.83,27.74) \| 0:10:59 (1.71,22.48) \| 0:21:32 (1.74,22.06) \| 0:50:50 (1.46,1.0) \| - \| 24 CPU \| 0:00:23 (1.0,17.50) \| 0:02:57 (1.32,19.29) \| 0:13:34 (1.43,18.20) \| 0:26:51 (1.45,17.69) \| 0:54:27 (1.43,0.93) \| 2:47:47 (1.16,1.0) \| 1 GPU \| 0:00:29 (1.0,13.93) \| 0:00:30 (9.74,113.79) \| 0:01:00 (24.17,244.68) \| 0:02:00 (24.33,236.96) \| 0:03:33 (27.48,14.32) \| - \| 2 GPU \| 0:00:16 (1.0,25.45) \| 0:00:16 (9.95,212.32) \| 0:00:31 (25.24,466.69) \| 0:01:02 (25.54,454.35) \| 0:01:53 (28.31,26.95) \| 0:04:18 (30.99,38.94) \| 3 GPU \| 0:00:11 (1.0,37.06) \| 0:00:11 (9.68,300.82) \| 0:00:21 (25.24,679.48) \| 0:00:42 (25.69,665.52) \| 0:01:17 (28.57,39.60) \| 0:02:54 (31.48,57.60) \| 4 GPU \| 0:00:08 (1.0,46.39) \| 0:00:09 (9.52,370.13) \| 0:00:17 (25.33,853.57) \| 0:00:33 (26.47,858.07) \| 0:00:59 (29.47,51.13) \| 0:02:16 (32.24,73.82) \| () All time entries are in a _days:hours:minutes:seconds_ format. The speedup is indicated as $(x,y)$, where $x$ is the speedup with respect to the number of samples and $y$ the speedup by distributing the computation on multiple CPUs or GPUs. ## 4 Conclusions and future work Our study investigates the efficiency achievable by a ensemble cardiovascular solver on modern GPU architectures. The basic methodological approaches are discussed together with details on our efforts to optimize execution on such architectures, both algorithmically and in terms of host-to-device communication. In particular, computation of local element matrices is performed directly on the GPU, and working units are used to determine displacement increments for independent material property realizations. The main result from our analysis is the observation that explicit-in-time ensemble solvers based on matrix-vector product naturally achieve high scalability, due to their ability to generate large amount of computations and re-usable data patterns provided by independent realizations. This also suggests how ensemble cardiovascular solvers are _ideal_ to efficiently generate campaigns of high-fidelity model solutions that form a pre-requisite for uncertainty quantification studies, the state-of-the-art paradigm for the analysis of cardiovascular systems, due to their ability to quantify the effects of multiple sources of uncertainty on the simulation outputs. We have also validated our solver for the steady state and pulsatile solution of an idealized aortic flow and a patient-specific model of the left coronary artery, under vessel wall mechanical property uncertainty, modeled through a Gaussian random field approximation. Future work will be devoted to further improve the computational efficiency of the proposed approach and to transition from a segregated to a fully coupled Arbitrary Lagrangian-Eulerian fluid-structure interaction paradigm. In addition, the proposed approach only considered uncertainties due to vessel wall thickness and elastic modulus. We plan to support uncertainty also in the boundary conditions and model geometry. ## Acknowledgements This work was supported by a National Science Foundation award #1942662 _CAREER: Bayesian Inference Networks for Model Ensembles_ (PI Daniele E. Schiavazzi). This research used computational resources provided through the Center for Research Computing at the University of Notre Dame. We also acknowledge support from the open source SimVascular project at www.simvascular.org. ## Conflict of interest The authors declare that they have no conflict of interest. ## References [1] F. Lindgren, H. Rue, and J. Lindström. 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Present address: ]Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA Present address: ]Westlake University, 310024 Hangzhou, China Present address: ]Department of Quantum Matter Physics, University of Geneva, 24 Quai Ernest-Ansermet, 1211 Geneva 4, Switzerland # Quasi-isotropic orbital magnetoresistance in lightly doped SrTiO3 Clément Collignon [ JEIP, USR 3573 CNRS, Collège de France, PSL Research University, 11, place Marcelin Berthelot, 75231 Paris Cedex 05, France Laboratoire de Physique et d’Étude des Matériaux (ESPCI Paris - CNRS - Sorbonne Université), PSL Research University, 75005 Paris, France Yudai Awashima Department of Engineering Science, University of Electro- Communications, Chofu, Tokyo 182-8585, Japan Ravi Laboratoire de Physique et d’Étude des Matériaux (ESPCI Paris - CNRS - Sorbonne Université), PSL Research University, 75005 Paris, France Xiao Lin [ Laboratoire de Physique et d’Étude des Matériaux (ESPCI Paris - CNRS - Sorbonne Université), PSL Research University, 75005 Paris, France Carl Willem Rischau [ Laboratoire de Physique et d’Étude des Matériaux (ESPCI Paris - CNRS - Sorbonne Université), PSL Research University, 75005 Paris, France Anissa Acheche JEIP, USR 3573 CNRS, Collège de France, PSL Research University, 11, place Marcelin Berthelot, 75231 Paris Cedex 05, France Baptiste Vignolle Laboratoire National des Champs Magnétiques Intenses (LNCMI-EMFL), CNRS ,UGA, UPS, INSA, Grenoble/Toulouse, France Institut de Chimie de la Matière Condensée, Bordeaux, France Cyril Proust Laboratoire National des Champs Magnétiques Intenses (LNCMI-EMFL), CNRS ,UGA, UPS, INSA, Grenoble/Toulouse, France Yuki Fuseya Department of Engineering Science, University of Electro- Communications, Chofu, Tokyo 182-8585, Japan Institute for Advanced Science, University of Electro-Communications, Chofu, Tokyo 182-8585, Japan Kamran Behnia Laboratoire de Physique et d’Étude des Matériaux (ESPCI Paris - CNRS - Sorbonne Université), PSL Research University, 75005 Paris, France Benoit Fauqué<EMAIL_ADDRESS>JEIP, USR 3573 CNRS, Collège de France, PSL Research University, 11, place Marcelin Berthelot, 75231 Paris Cedex 05, France ###### Abstract A magnetic field parallel to an electrical current does not produce a Lorentz force on the charge carriers. Therefore, orbital longitudinal magnetoresistance is unexpected. Here we report on the observation of a large and non saturating magnetoresistance in lightly doped SrTiO3-x independent of the relative orientation of current and magnetic field. We show that this quasi-isotropic magnetoresistance can be explained if the carrier mobility along all orientations smoothly decreases with magnetic field. This anomalous regime is restricted to low concentrations when the dipolar correlation length is longer than the distance between carriers. We identify cyclotron motion of electrons in a potential landscape tailored by polar domains as the cradle of quasi-isotropic orbital magnetoresistance. The result emerges as a challenge to theory and may be a generic feature of lightly-doped quantum paralectric materials. ††preprint: APS/123-QED Magnetoresistance (MR), the change in electrical resistivity under the application of a magnetic field is an old topic in condensed matter physics Pippard (1989). It can be simply understood as a consequence of the Lorentz force exerted on mobile electrons by the magnetic field. This orbital magnetoresistance (which neglects the spin of electrons) is largest when the magnetic field is perpendicular to the electrical current. The transverse magnetoresistance (labelled TMR) is expected to increase quadratically with magnetic field at low fields and then saturate at high fields. The boundary between the two regimes is set by $\mu_{H}B\approx 1$ (where $\mu_{H}$ is the Hall mobility). When the field and the current are parallel, we are in presence of longitudinal magnetoresistance (labelled LMR), expected to be negligibly small due to the cancellation of the Lorentz force. However, this simple picture is known to fail in numerous cases. Non- saturating linear TMR has been observed in electronic systems ranging from potassium Pippard (1989), to doped silicon Delmo _et al._ (2009), 2D electron gas system Khouri _et al._ (2016), 3D doped semi-conductors and Dirac materials Xu _et al._ (1997); Hu _et al._ (2005); Kozlova _et al._ (2012); Schneider _et al._ (2014); Fauqué _et al._ (2013); Novak _et al._ (2015); Narayanan _et al._ (2015); Xiong _et al._ (2016), density wave materials Feng _et al._ (2019), or correlated materials Hayes _et al._ (2016). LMR, one order of magnitude smaller (than and with an opposite sign to) its transverse counterpart has been observed in silver chalcogenides Hu _et al._ (2005) and topological materials Wiedmann _et al._ (2016); dos Reis _et al._ (2016). The exact conditions for the emergence of a sizeable LMR is the subject of ongoing debate Pal and Maslov (2010); Burkov (2015); Goswami _et al._ (2015). Figure 1: Electrical transport properties of a lightly doped SrTiO3-x at low temperature : a) Fermi surface of the lower band of SrTiO3 according to Allen _et al._ (2013) b) Temperature dependence of the resistivity ($\rho$) of sample S7. Insert: $\rho$ vs $T^{2}$. c) Transverse ($j\perp B$) and longitudinal ($j//B$) magnetoresistance ($\frac{\Delta\rho}{\rho_{0}}$=$\frac{\rho(B)-\rho(B=0)}{\rho(B=0)}$) as function of the magnetic field up to 54 T, at $T=1.5$ K, for S6 ($n_{H}$(S${}_{6})=3.2\times 10^{17}$ cm-3). d) Normalised angular magneto- resistance of S1 ($n_{H}($S${}_{1})=6.5\times 10^{16}$ cm-3) in polar plot at $B=10$ T for two temperatures : $T=2$ K (in blue) and $T=10$ K (in red). $\theta=0^{\circ}$ and $90^{\circ}$ correspond respectively to ${\bf{j}}\perp{\bf{B}}$ and ${\bf{j}}//{\bf{B}}$) In this paper, we report on the case of lightly doped SrTiO3. Undoped strontium titanate is an incipient ferroelectric, dubbed quantum paraelectric Müller and Burkard (1979), which can be turned into a metal by non-covalent substitution or by removing oxygen Spinelli _et al._ (2010). This dilute metal Collignon _et al._ (2019) has attracted recent attention Zhou and Bernardi (2019); Kumar _et al._ (2020) due to the persistence of $T$-square resistivity in absence of Umklapp and interband scattering among electrons Lin _et al._ (2015) and the unexpected survival of metallicity at high temperatures Collignon _et al._ (2020). We will see below that its magnetoresistance is also remarkably non-trivial. In contrast with any other documented material, it shows a large and quasi-linear TMR, accompanied by a positive LMR of comparable amplitude. Intriguingly, the amplitude of magnetoresistance depends only on the amplitude of the magnetic field, independent of the mutual orientation of the current and the magnetic field. We will show that this unusual quasi-isotropic magnetoresistance is restricted to a range of doping where the inter-electron distance exceeds the typical size of a polar domain. The observation implies that this phenomenon is driven by the interplay between cyclotron orbits and the potential landscape shaped by polar domains and suggests that it may be generic to lightly doped quantum paraelectrics. Fig.1 presents our main result. When the carrier density in SrTiO3-δ is $n_{H}=3\times 10^{17}$ cm-3, there is a single Fermi pocket at the center of the Brillouin zone shown Fig.1a). This is what is expected by band calculations van der Marel _et al._ (2011) and found by quantum oscillations Uwe _et al._ (1985); Allen _et al._ (2013); Lin _et al._ (2014). Nevertheless, not only this dilute metal displays a $T^{2}$behavior (see Fig.1b)), but it also responds to magnetic field in a striking manner. Upon the application of a magnetic field of 54 T, there is a forty (twenty)-fold enhancement of resistance for the transverse (longitudinal) configuration (see Fig.1c)). In both cases, the evolution with field is quasi-linear and there is no sign of saturation even if the high field regime ($\mu_{H}B>>1$) is clearly attained. . Figure 2: Temperature dependence of the transverse and longitudinal magneto- resistance : a)-c) Field dependence of the resistivity for ${\bf{j}}\perp{\bf{B}}$ ($\rho_{\perp}$), $\frac{\rho_{xy}}{B}$ and the resistivity for ${\bf{j}}//{\bf{B}}$ ($\rho_{//}$) from $T=2$ to 14 K for S7 ($n_{H}($S${}_{7})=3.3\times 10^{17}$ cm-3) d)-f) same as a)-c) from $T=16$ to 60 K. g) Field dependence of the Hall mobility $\mu_{H}=\frac{\rho_{xy}}{B\rho_{\perp}}$ for $T=2$, 18 and 60 K. h) Longitudinal magnetoconductivity ($\frac{1}{\rho_{//}}$) compared with $\sigma_{//}=ne\mu_{H}(B)$ with the deduced $\mu_{H}(B)$ shown on g). A polar plot of the normalised angular magnetoresistance (AMR) at a fixed magnetic field for another sample (S1) with a slightly lower carrier density ($n_{H}($S${}_{1})=6.5\times 10^{16}$ cm-3) is shown on Fig.1d). The magnetic field rotates from the transverse ($\theta=0^{\circ}$) to the longitudinal ($\theta=90^{\circ}$) configuration at two different temperatures. While at $T=10$ K, the longitudinal magnetoresistance shrinks towards zero, at $T=2$ K the magnetoresistance is quasi-isotropic and the relative direction of the magnetic field and the current injection barely affects its amplitude. When a magnetic field is aligned along the $z$-axis, in presence of a single- component Fermi surface, the three components of the conductivity tensor have remarkably simple expressions: $\sigma_{zz}=\sigma_{//}=ne\mu_{H}$ (1) $\sigma_{xx}=\sigma_{\perp}=\frac{ne\mu_{H}}{1+\mu^{2}_{H}B^{2}}$ (2) $\sigma_{xy}=\mu_{H}B\frac{ne\mu_{H}}{1+\mu^{2}_{H}B^{2}}$ (3) Now, if $\mu_{H}$ remains constant as a function of magnetic field, one does not expect the longitudinal magnetoresistance, since $\rho_{//}=\sigma_{//}^{-1}$ would not depend on magnetic field. One would not even expect a transverse magnetoresistance, because the same is true for $\rho_{\perp}=\frac{\sigma_{\perp}}{\sigma_{\perp}^{2}+\sigma_{xy}^{2}}=\frac{1}{ne\mu_{H}}$. These equations hold in presence of a quadratic dispersion when the effective mass $m^{}$ and the Hall mobility, $\mu_{H}=e\tau/m^{}$ are well defined. The Fermi pocket associated with the lowest band in dilute metallic strontium titanate is not an ellipsoid. This can lead to a finite TMR and LMR Pippard (1989); Pal and Maslov (2010). However, as discussed in the supplement SM , the results computed using the specific geometry of the Fermi surface are well below the experimentally observed magnitudes at low temperature. As we will see below, to explain our result, one needs to assume a field-dependent $\mu_{H}$. Figure 3: Doping dependence of the transverse and the longitudinal magnetoresistance at $T=2$ K : a) and b) TMR for nH ranging from $6.5\times 10^{16}$ to $3.6\times 10^{19}$ cm-3. c) and d) same as a) and b) for the LMR. e) Field dependence of Hall mobility ($\mu_{H}$) deduced from the TMR and the Hall effect measurements in four low doped samples. f) Comparison of the longitudinal magnetoconductance ($\frac{1}{\rho_{//}}$) with $\sigma_{//}=ne\mu_{H}(B)$ with the deduced $\mu_{H}(B)$ shown on e). Fig.2 shows the evolution of the quasi-isotropic magnetoresistance with temperature. The amplitude of the TMR decreases with warming (see Fig.2a) and d)). The same is true for the LMR (see Fig.2c) and f)). On the other hand, the Hall coefficient is barely temperature-dependent (see Fig.2b) and e)). Upon warming, the longitudinal magnetoresistance decreases faster than its transverse counterpart and above 14 K it almost vanishes (Fig.2c) and d)). Above this temperature, a small TMR persists with an amplitude comparable with what the semi-classical theory expects (see supplement SM ). Fig.3 shows the evolution with doping. Increasing carrier concentration diminishes both TMR and LMR (see Fig.3a)-d)). As in the case of thermal evolution, the LMR decreases faster than the TMR. At low doping, the two configurations yield a similar amplitude. With increasing carrier concentration, the LMR becomes smaller than the TMR (see Fig.3 b) and d)). Therefore, the unusual regime of the magnetoresistance detected by the present study emerges only at low temperature (when resistivity is dominated by its elastic component) and at low carrier concentration. Remarkably, even in this unusual context, the three components of the conductivity tensor keep the links expected by Eq.(1-3). This is demonstrated in the final panels of Fig.2 and Fig.3. The Hall mobility at a given magnetic field can be extracted using $\mu_{H}=\frac{1}{B}\frac{\sigma_{xy}}{\sigma_{\perp}}=\frac{1}{B}\frac{\rho_{xy}}{\rho_{\perp}}$ (see Fig.2g)). The deduced $\mu_{H}(B)$ can then be compared with the field dependence of the longitudinal conductance $\sigma_{//}$ (see Fig.2h)). As seen in the figure, there is a satisfactory agreement. This is the case of all samples at low doping levels, as shown in Fig.3e) and f). Thus, assuming that mobility smoothly evolves with magnetic field, would explain both the quasi-linear non saturating TMR and the large finite LMR, which emerge at low doping. Fig.4a) shows the doping dependence of the LMR to TMR ratio ($\frac{\Delta\rho_{//}}{\Delta\rho_{\perp}}$ at $B=10$ T and $T=2.5$ K). Clearly, the finite LMR kicks in below a cut-off concentration and grows steadily with decreasing carrier density. The unusual magnetoresistance of lightly doped SrTiO3-δ is therefore restricted to carrier densities below a threshold of $3\times 10^{18}$ cm-3. As we will see below, a clue to the origin of this phenomenon is provided by this boundary. Figure 4: Size of polar domains vs. inter-electron distance in lightly doped SrTiO3 : a) Doping dependence of the ratio of the LMR and TMR at B=10T for T=2.5K (in blue closed circles). Insert : sketch of the cubic unit cell of SrTiO3 lattices in presence of an oxygen vacancy. b) Doping dependence of $\ell_{ee}$=$(n_{H})^{\frac{-1}{3}}$ (the inter-electron distance), of the screening length scale from charged impurities $r_{TF}=\sqrt{\frac{\pi a_{B}}{4k_{F}}}$ compares with the polar domain diameter, 2Rc=5.4nm at low temperature. c) Sketch of the SrTiO3 lattice in presence of two oxygen vacancies separated by a distance $\ell_{ee}$ and of the polar domains (in gray light) which form around each oxygen vacancies with a radius Rc. Below a critical doping where $\ell_{ee}$ becomes shorter than 2Rc non zero LMR appears. How can the mobility decrease with magnetic field along both orientations? Why does this decrease happens in a restricted window of doping? We will see below that a length scale specific to quantum paraelectrics plays a key role in finding answers to both of these questions. A field-dependent mobility has been previously invoked in other contexts Song _et al._ (2015); Fauqué _et al._ (2018). The time between collision events can become shorter in presence of magnetic field, because disorder is scanned differently at zero and finite magnetic fields. Compared to zero-field counterparts, charge carriers following a cyclotron orbit are more vulnerable to shallow scattering centers. Such a picture has been invoked to explain the linear TMR in 3D high mobility dilute semiconductors Song _et al._ (2015) and the sub-quadratic TMR in semi-metals Fauqué _et al._ (2018). There are three already identified relevant length scales to the problem. These are, i) the Thomas-Fermi screening radius $r_{TF}=\sqrt{\frac{\pi a_{B}}{4k_{F}}}$; ii) the magnetic length, $\ell_{B}=\sqrt{\frac{\hbar}{eB}}$; and iii) the Fermi wavelength, $\lambda_{F}=2\pi k_{F}^{-1}$. When disorder is smooth and $r_{TF}$ is longer than the cyclotron radius ($r_{c}=\ell_{B}^{2}k_{F}$), the magnetic field, by quenching the kinetic energy of electrons in the plane of cyclotron motion, would guide them along the minimum of the electrostatic potential fluctuations Song _et al._ (2015). This would lead to a decrease in mobility in the plane perpendicular to the magnetic field. The doping dependence of the Thomas-Fermi screening radius is shown in Fig.4b). Thanks to a Bohr radius as long as 600 nm in strontium titanate, $r_{TF}$ is remarkably long Behnia (2015) and easily exceeds the cyclotron radius in a field of the order of Tesla. Therefore, shallow extended disorder, screened at zero-field will become visible as the cyclotron radius shrinks. One can invoke this picture to explain the quasi-linear TMR. However, the finite LMR and the low-field TMR remain both unexplained, because only the plane perpendicular of the orientation to the magnetic field is concerned. In a polar crystal, defects, by distorting the lattice, generate electric dipoles. The typical length for correlation between such dipoles is set by $R_{c}$=$\frac{v_{s}}{\omega_{O}}$ (where $v_{s}$ and $\omega_{O}$ are the sound velocity and the frequency of the soft optical mode, respectively). In highly polarizable crystals, $\omega_{O}$ is small and $R_{c}$ can become remarkably long Vugmeister and Glinchuk (1990); Samara (2003). In the specific case of strontium titanate $v_{s}\simeq 7500$ m.s-1 Rehwald (1970), $\omega_{0}(300K)\simeq 11$ meV and $\omega_{0}(2$K$)\simeq 1.8$ meV Yamada and Shirane (1969), therefore, $R_{c}$ varies from 0.5 nm at 300 K to 2.7 nm at 2 K. As a consequence, defects can cooperate with other defects over long distances to generate mesoscopic dipoles. In the case of a co-valent substitution, such as Sr1-xCaxTiO3, a Ca atom can break the local inversion symmetry. It can cooperate with other Ca sites within a range of $R_{c}$ to choose the same orientation for dipole alignment. When the Ca density exceeds a threshold, these domains percolate and generate a ferroelectric ground states. Remarkably, this critical density ($x>0.002$ Bednorz and Müller (1984)) corresponds to replacement of 1 out of 500 Sr atoms by Ca, that is when their average distance falls below $\frac{R_{c}}{a}$ (here, $a=0.39$ nm is the lattice parameter and $\frac{R_{c}}{a}\approx 500^{-1/3}$). In the case of an oxygen vacancy, the donor, in addition to a local potential well, brings also a local dipole capable of cooperation with neighboring donors over long distances. Recent studies Rischau _et al._ (2017); Wang _et al._ (2019) have confirmed the survival of dipolar physics in presence of dilute metallicity and the generation of ripples by electric dipoles inside the shallow Fermi sea. Specifically, it was found that in Sr1-xCaxTiO3-δ, the ferroelectric-like alignment of dipoles is destroyed when there is more than one mobile electron per $7.9\pm 0.6$ Ca atoms, the Fermi sea is dense enough to impede the percolation between polar domains. This threshold corresponds to an inter- electron distance approximately twice ($7.9^{-1/3}\approx 2$) the inter-dipole distance. An oxygen vacancy (in addition to being a donor and an ionized point defect) generates an extended distortion of the size of $R_{c}$. This leads us to identify the origin of the doping window for the unusually isotropic magnetoresistance. If the inter-electron distance, ($\ell_{ee}$), which increases with decreasing carrier concentration, becomes significantly longer than $R_{c}$, then mobile carriers cannot adequately screen polar domains. Our data indicates that this is where the large quasi-isotropic magnetoresistance emerges. Fig.4b) shows the evolution of $\ell_{ee}=(n_{H})^{-1/3}$ with doping. One can see that the threshold of $3\times 10^{18}$ cm-3 corresponds to $\ell_{ee}=6.7$ nm. In other words, when the inter-electron distance becomes shorter than $2R_{c}$, the unusual magnetoresistance disappears, presumably because the Fermi sea is dense enough to impede the inhomogeneity generated by polar domains. This unusual MR is the largest at low temperature when resistivity is dominated by elastic scattering events. It vanishes with warming, when the inelastic $T^{2}$-term dominates over residual resistivity. This cross-over typically occurs around 15 K (see Fig.1b)). A possible solution to the mystery of the isotropic reduction of the mobility with magnetic field is offered by this length scale, which does not depend on the orientation of magnetic field. In presence of randomly oriented mesoscopic dipoles, the charge current does not align locally parallel to its macroscopic orientation. Instead, it will meander along a trajectory set by dipoles’ electric field. The disorder affecting the whirling electrons will reduce mobility along different orientations. Remarkably, the inhomogeneity brought by these polar domains do not impede the existence of a percolated Fermi sea and the observation of quantum oscillations in this range of carrier concentration. A solid explanation of this apparent paradox remains a a task for future theoretical investigations. In summary, we found a large and quasi-isotropic magnetoresistance in lightly doped strontium titanate. 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# Convergence Results of Forward-Backward Algorithms for Sum of Monotone Operators in Banach Spaces Yekini Shehu111Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, People’s Republic of China; Institute of Science and Technology (IST), Am Campus 1, 3400, Klosterneuburg, Vienna, Austria; e-mail: <EMAIL_ADDRESS> (January 22, 2021) ###### Abstract It is well known that many problems in image recovery, signal processing, and machine learning can be modeled as finding zeros of the sum of maximal monotone and Lipschitz continuous monotone operators. Many papers have studied forward-backward splitting methods for finding zeros of the sum of two monotone operators in Hilbert spaces. Most of the proposed splitting methods in the literature have been proposed for the sum of maximal monotone and inverse-strongly monotone operators in Hilbert spaces. In this paper, we consider splitting methods for finding zeros of the sum of maximal monotone operators and Lipschitz continuous monotone operators in Banach spaces. We obtain weak and strong convergence results for the zeros of the sum of maximal monotone and Lipschitz continuous monotone operators in Banach spaces. Many already studied problems in the literature can be considered as special cases of this paper. Keywords: inclusion problem; 2-uniformly convex Banach space; forward-backward algorithm; weak convergence; strong convergence. 2010 MSC classification: 47H05, 47J20, 47J25, 65K15, 90C25. ## 1 Introduction Let $E$ be a real Banach space with norm $\\|.\\|,$ we denote by $E^{}$ the dual of $E$ and $\langle f,x\rangle$ the value of $f\in E^{}$ at $x\in E.$ Let $B:E\rightarrow 2^{E^{}}$ be a maximal monotone operator and $A:E\rightarrow E^{}$ be a Lipschitz continuous monotone operator. We consider the following inclusion problem: find $x\in E$ such that $\displaystyle 0\in(A+B)x.$ (1) Throughout this paper, we denote the solution set of the inclusion problem (1) by $(A+B)^{-1}(0)$. The inclusion problem (1) contains, as special cases, convexly constrained linear inverse problem, split feasibility problem, convexly constrained minimization problem, fixed point problems, variational inequalities, Nash equilibrium problem in noncooperative games, and many more. See, for instance, [11, 15, 28, 33, 35, 36] and the references therein. A popular method for solving problem (1) in real Hilbert spaces, is the well- known forward–backward splitting method introduced by Passty [35] and Lions and Mercier [28]. The method is formulated as $\displaystyle x_{n+1}=(I+\lambda_{n}B)^{-1}(I-\lambda_{n}A)x_{n},\leavevmode\nobreak\ \leavevmode\nobreak\ \lambda_{n}>0,$ (2) under the condition that $Dom(B)\subset Dom(A)$. It was shown, see for example [11], that weak convergence of (2) requires quite restrictive assumptions on $A$ and $B$, such that the inverse of $A$ is strongly monotone or $B$ is Lipschitz continuous and monotone and the operator $A+B$ is strongly monotone on $Dom(B)$. Tseng in [48], weakened these assumptions and included an extra step per each step of (2) (called Tseng’s splitting algorithm) and obtained weak convergence result in real Hilbert spaces. Quite recently, Gibali and Thong [18] have obtained strong convergence result by modifying Tseng’s splitting algorithm in real Hilbert spaces. In this paper, we extend Tseng’s result [48] to a Banach space. We first prove the weak convergence of the sequence generated by our proposed method, assuming that the duality mapping is weakly sequentially continuous. This weak convergence is a generalization of Theorem 3.4 given in [48]. We next prove the strong convergence result for problem (1) under some mild assumptions and this extends Theorems 1 and 2 in [18] to Banach spaces. Finally, we apply our convergence results to the composite convex minimization problem in Banach spaces. ## 2 Preliminaries In this section, we define some concepts and state few basic results that we will use in our subsequent analysis. Let $S_{E}$ be the unit sphere of $E$, and $B_{E}$ the closed unit ball of $E$. Let $\rho_{E}:[0,\infty)\rightarrow[0,\infty)$ be the modulus of smoothness of $E$ defined by $\rho_{E}(t):=\sup\Big{\\{}\frac{1}{2}(\\|x+y\\|+\\|x-y\\|)-1:\,x\in S_{E},\,\\|y\\|\leq t\Big{\\}}.$ A Banach space $E$ is said to be $2$-uniformly smooth, if there exists a fixed constant $c>0$ such that $\rho_{E}(t)\leq ct^{2}$. The space $E$ is said to be smooth if $\displaystyle\lim_{t\rightarrow 0}\frac{\\|x+ty\\|-\\|x\\|}{t}$ (3) exists for all $x,y\in S_{E}$. The space $E$ is also said to be uniformly smooth if (3) converges uniformly in $x,y\in S_{E}$. It is well known that if $E$ is $2$-uniformly smooth, then $E$ is uniformly smooth. It is said to be strictly convex if $\\|(x+y)/2\\|<1$ whenever $x,y\in S_{E}$ and $x\neq y$. It is said to be uniformly convex if $\delta_{E}(\epsilon)>0$ for all $\epsilon\in(0,2]$, where $\delta_{E}$ is the modulus of convexity of $E$ defined by $\displaystyle\delta_{E}(\epsilon):=\inf\Big{\\{}1-\Big{\|}\Big{\|}\frac{x+y}{2}\Big{\|}\Big{\|}\mid x,y\in B_{E},\\|x-y\\|\geq\epsilon\Big{\\}}$ (4) for all $\epsilon\in[0,2]$. The space $E$ is said to be 2-uniformly convex if there exists $c>0$ such that $\delta_{E}(\epsilon)\geq c\epsilon^{2}$ for all $\epsilon\in[0,2]$. It is obvious that every 2-uniformly convex Banach space is uniformly convex. It is known that all Hilbert spaces are uniformly smooth and 2-uniformly convex. It is also known that all the Lebesgue spaces $L_{p}$ are uniformly smooth and 2-uniformly convex whenever $1<p\leq 2$ (see [7]). The normalized duality mapping of $E$ into $E^{}$ is defined by $Jx:=\\{x^{}\in E^{}\mid\langle x^{},x\rangle=\\|x^{}\\|^{2}=\\|x\\|^{2}\\}$ for all $x\in E$. The normalized duality mapping $J$ has the following properties (see, e.g., [47]): • if $E$ is reflexive and strictly convex with the strictly convex dual space $E^{}$, then $J$ is single-valued, one-to-one and onto mapping. In this case, we can define the single-valued mapping $J^{-1}:E^{}\rightarrow E$ and we have $J^{-1}=J_{}$, where $J_{}$ is the normalized duality mapping on $E^{}$; • if $E$ is uniformly smooth, then $J$ is uniformly norm-to-norm continuous on each bounded subset of $E.$ Let us recall from [1, 13] some examples for the normalized duality mapping $J$ in the uniformly convex and uniformly smooth Banach spaces $\ell_{p}$ and $L_{p},1<p<\infty$. * • For $\ell_{p}:Jx=\\|x\\|_{\ell_{p}}^{2-p}y\in\ell_{q}$, where $x=(x_{j})_{j\geq 1}$ and $y=(x_{j}\|x_{j}\|^{p-2})_{j\geq 1}$, $\frac{1}{p}+\frac{1}{q}=1$. * • For $L_{p}:Jx=\\|x\\|_{L_{p}}^{2-p}\|x\|^{p-2}x\in L_{q}$, $\frac{1}{p}+\frac{1}{q}=1$. Now, we recall some fundamental and useful results. ###### Lemma 2.1. The space $E$ is 2-uniformly convex if and only if there exists $\mu_{E}\geq 1$ such that $\displaystyle\frac{\\|x+y\\|^{2}+\\|x-y\\|^{2}}{2}\geq\\|x\\|^{2}+\\|\mu^{-1}_{E}y\\|^{2}$ (5) for all $x,y\in E$. The minimum value of the set of all $\mu_{E}\geq 1$ satisfying (5) for all $x,y\in E$ is denoted by $\mu$ and is called the 2-uniform convexity constant of $E$; see [5]. It is obvious that $\mu=1$ whenever $E$ is a Hilbert space. ###### Lemma 2.2 ([4]). Let $\displaystyle\frac{1}{p}+\frac{1}{q}=1,\leavevmode\nobreak\ \leavevmode\nobreak\ p,q>1$. The space $E$ is $q-$uniformly smooth if and only if its dual $E^{}$ is $p-$uniformly convex. ###### Lemma 2.3 ([51]). Let $E$ be a real Banach space. The following are equivalent: (1) $E$ is 2-uniformly smooth * (2) There exists a constant $\kappa>0$ such that $\forall\ x,y\in E$, $\\|x+y\\|^{2}\leq\\|x\\|^{2}+2\langle y,J(x)\rangle+2\kappa^{2}\\|y\\|^{2},$ where $\kappa$ is the 2-uniform smoothness constant. In Hilbert spaces, $\kappa=\frac{1}{\sqrt{2}}$. ###### Definition 2.4. Let $X\subseteq E$ be a nonempty subset. Then a mapping $A:X\to E^{}$ is called (a) strongly monotone with modulus $\gamma>0$ on $X$ if $\langle Ax-Ay,x-y\rangle\geq\gamma\\|x-y\\|^{2},\forall x,y\in X.$ In this case, we say that $A$ is $\gamma$-strongly monotone; * (b) monotone on $X$ if $\langle Ax-Ay,x-y\rangle\geq 0,\forall x,y\in X;$ * (c) Lipschitz continuous on $X$ if there exists a constant $L>0$ such that $\\|Ax-Ay\\|\leq L\\|x-y\\|$ for all $x,y\in X$. We give some examples of monotone operator in Banach spaces as given in [2]. ###### Example 2.5. Let $G\subset\mathbb{R}^{n}$ be a bounded measurable domain. Define the operator $A:L^{p}(G)\rightarrow L^{q}(G),\leavevmode\nobreak\ \leavevmode\nobreak\ \frac{1}{p}+\frac{1}{q}=1,\leavevmode\nobreak\ \leavevmode\nobreak\ p>1$, by the formula $Ay(x):=\varphi(x,\|y(x)\|^{p-1})\|y(x)\|^{p-2}y(x),\leavevmode\nobreak\ \leavevmode\nobreak\ x\in G,$ where the function $\varphi(x,s)$ is measurable as a function of $x$ for every $s\in[0,\infty)$ and continuous for almost all $x\in G$ as a function on $s,\|\varphi(x,s)\|\leq M$ for all $s\in[0,\infty)$ and for almost all $x\in G$. Observe that the operator $A$ really maps $L^{p}(G)$ to $L^{q}(G)$ because of the inequality $\|Ay\|\leq M\|y\|^{p-1}$. Then it can be shown that $A$ is a monotone map on $L^{p}(G)$. Let us consider another example from quantum mechanics. ###### Example 2.6. Define the operator $Au:=-a^{2}\triangle u+(g(x)+b)u(x)+u(x)\int_{\mathbb{R}^{3}}\frac{u^{2}(y)}{\|x-y\|}dy,$ where $\triangle:=\sum_{i=1}^{3}\frac{\partial^{2}}{\partial x_{i}^{2}}$ is the Laplacian in $\mathbb{R}^{3}$, $a$ and $b$ are constants, $g(x)=g_{0}(x)+g_{1}(x),\leavevmode\nobreak\ \leavevmode\nobreak\ g_{0}(x)\in L^{\infty}(\mathbb{R}^{3}),g_{1}(x)\in L^{2}(\mathbb{R}^{3})$. Let $A:=L+B$, where the operator $L$ is the linear part of $A$ (it is the Schrödinger operator) and $B$ is defined by the last term. It is known that $B$ is a monotone operator on $L^{2}(\mathbb{R}^{3})$ (see page 23 of [2]) and this implies that $A:L^{2}(\mathbb{R}^{3})\rightarrow L^{2}(\mathbb{R}^{3})$ is also a monotone operator. ###### Example 2.7. This example gives one of the perhaps most famous example of monotone operators, viz. the $p$-Laplacian $-{\rm div}(\|\nabla u\|^{p-2}\nabla u):W^{1}_{0}(L_{p}(\Omega))\rightarrow\Big{(}W^{1}_{0}(L_{p}(\Omega))\Big{)}^{}$, where $u:\Omega\rightarrow\mathbb{R}$ is a real function defined on a domain $\Omega\subset\mathbb{R}^{n}$. The $p$-Laplacian operator is a monotone operator for $1<p<\infty$ (in fact, it is strongly monotone for $p\geq 2$, and strictly monotone for $1<p<2$). The $p$-Laplacian operator is an extremely important model in many topical applications and certainly played an important role in the development of the theory of monotone operators. ###### Definition 2.8. A multi-valued operator $B:E\rightarrow 2^{E^{}}$ with graph $G(T)=\\{(x,x^{}):x^{}\in Tx\\}$ is said to be monotone if for any $x,y\in D(T),x^{}\in Tx$ and $y^{}\in Ty$ $\langle x-y,x^{}-y^{}\rangle\geq 0.$ A monotone operator $B$ is said to be maximal if $B=S$ whenever $S:E\rightarrow 2^{E^{}}$ is monotone and $G(B)\subset G(S)$. Let $E$ be a reflexive, strictly convex and smooth Banach space and let $B:E\rightarrow 2^{E^{}}$ be a maximal monotone operator. Then for each $r>0$ and $x\in E$, there corresponds a unique element $x_{r}\in E$ such that $Jx\in Jx_{r}+rBx_{r}.$ We define this unique element $x_{r}$, the resolvent of $B$, denoted by $J^{B}_{r}x$. In other words, $J_{r}^{B}=(J+rB)^{-1}J$ for all $r>0$. It is easy to show that $B^{-1}0=F(J^{B}_{r})$ for all $r>0$, where $F(J^{B}_{r})$ denotes the set of all fixed points of $J^{B}_{r}$. We can also define, for each $r>0$, the Yosida approximation of $B$ by $A_{r}=\frac{J-JJ^{B}_{r}}{r}$. For more details, see, for instance [6]. Suppose $E$ is a smooth Banach space. We introduce the functional studied in [1, 25, 38]: $\phi:E\times E\rightarrow\mathbb{R}$ defined by: $\displaystyle\phi(x,y):=\\|x\\|^{2}-2\langle x,Jy\rangle+\\|y\\|^{2}.$ (6) Clearly, $\phi(x,y)\geq(\\|x\\|-\\|y\\|)^{2}\geq 0.$ The following lemma gives some identities of functional $\phi$ defined in (6). ###### Lemma 2.9. (see [3] and [1]) Let $E$ be a real uniformly convex, smooth Banach space. Then, the following identities hold: (i) $\displaystyle\phi(x,y)=\phi(x,z)+\phi(z,y)+2\langle x-z,Jz-Jy\rangle,\ \forall x,y,z\in E.$ (ii) $\displaystyle\phi(x,y)+\phi(y,x)=2\langle x-y,Jx-Jy\rangle,\ \forall x,y\in E.$ Let $C\subseteq E$ be a nonempty, closed and convex subset of a real, uniformly convex Banach space $E$. Let us introduce the functional $V(x,y):E\times E^{}\rightarrow\mathbb{R}$ by the formula: $V(x,y):=\\|x\\|^{2}_{E}-2\langle x,y\rangle+\\|y\\|^{2}_{E^{}}.$ (7) Then, it is easy to see that $V(y,x)=\phi(y,J^{-1}x),\leavevmode\nobreak\ \leavevmode\nobreak\ \forall x\in E^{},y\in E.$ In the next lemma, we describe the property of the operator $V(.,.)$ defined in (7). ###### Lemma 2.10. ([1]) $V(x,x^{})+2\langle J^{-1}x^{}-x,y^{}\rangle\leq V(x,x^{}+y^{}),\leavevmode\nobreak\ \leavevmode\nobreak\ \forall x\in E,\leavevmode\nobreak\ \leavevmode\nobreak\ x^{},y^{}\in E^{}.$ The lemma that follows is stated and proven in [3, Lem. 2.2]. ###### Lemma 2.11. Suppose that $E$ is 2-uniformly convex Banach space. Then, there exists $\mu\geq 1$ such that $\frac{1}{\mu}\\|x-y\\|^{2}\leq\phi(x,y)\leavevmode\nobreak\ \leavevmode\nobreak\ \forall x,y\in E.$ The following lemma was given in [21]. ###### Lemma 2.12. Let $S$ be a nonempty, closed convex subset of a uniformly convex, smooth Banach space $E$. Let $\\{x_{n}\\}$ be a sequence in $E$. Suppose that, for all $u\in S$, $\phi(u,x_{n+1})\leq\phi(u,x_{n}),\leavevmode\nobreak\ \leavevmode\nobreak\ \forall n\geq 1.$ Then $\\{\Pi_{S}(x_{n})\\}$ is a Cauchy sequence. The following property of $\phi(.,.)$ was given in [1, Thm. 7.5] (see also [16, 17]). ###### Lemma 2.13. Let $E$ be a uniformly smooth Banach space which is also uniformly convex. If $\\|x\\|\leq c,\\|y\\|\leq c$, then $2L_{1}^{-1}c^{2}\delta_{E}\Big{(}\frac{\\|x-y\\|}{4c}\Big{)}\leq\phi(y,x)\leq 4L_{1}^{-1}c^{2}\rho_{E}\Big{(}\frac{4\\|x-y\\|}{c}\Big{)},$ where $L_{1}(1<L_{1}<3.18)$ is the Figiel’s constant. We next recall some existing results from the literature to facilitate our proof of strong convergence. The first is taken from [31]. ###### Lemma 2.14. Let $\\{a_{n}\\}$ be sequence of real numbers such that there exists a subsequence $\\{n_{i}\\}$ of $\\{n\\}$ such that $a_{n_{i}}<a_{{n_{i}}+1}$, for all $i\in\mathbb{N}$. Then there exists a nondecreasing sequence $\\{m_{k}\\}\subset\mathbb{N}$ such that $m_{k}\rightarrow\infty$ and the following properties are satisfied by all (sufficiently large) numbers $k\in\mathbb{N}$ $a_{m_{k}}\leq a_{{m_{k}}+1}\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm and}\leavevmode\nobreak\ \leavevmode\nobreak\ a_{k}\leq a_{{m_{k}}+1}.$ In fact, $m_{k}=\max\\{j\leq k:a_{j}<a_{j+1}\\}$. ###### Lemma 2.15. ([52]) Let $\\{a_{n}\\}$ be a sequence of nonnegative real numbers satisfying the following relation: $a_{n+1}\leq(1-\alpha_{n})a_{n}+\alpha_{n}\sigma_{n}+\gamma_{n},\leavevmode\nobreak\ \leavevmode\nobreak\ n\geq 1,$ where (a) $\\{\alpha_{n}\\}\subset[0,1],$ $\sum_{n=1}^{\infty}\alpha_{n}=\infty;$ * (b) $\limsup\sigma_{n}\leq 0$; * (c) $\gamma_{n}\geq 0\ (n\geq 1),$ $\sum_{n=1}^{\infty}\gamma_{n}<\infty.$ Then, $a_{n}\rightarrow 0$ as $n\rightarrow\infty$. The following lemma is needed in our proof to show that the weak limit point is a solution to the inclusion problem (1). ###### Lemma 2.16. ([6]) Let $B:E\to 2^{E^{}}$ be a maximal monotone mapping and $A:E\to E^{}$ be a Lipschitz continuous and monotone mapping. Then the mapping $A+B$ is a maximal monotone mapping. The following result gives an equivalence of fixed point problem and problem (1). ###### Lemma 2.17. Let $B:E\to 2^{E^{}}$ be a maximal monotone mapping and $A:E\to E^{}$ be a mapping. Define a mapping $T_{\lambda}x:=J_{\lambda}^{B}oJ^{-1}(J-\lambda A)(x),\leavevmode\nobreak\ \leavevmode\nobreak\ x\in E,\lambda>0.$ Then $F(T_{\lambda})=(A+B)^{-1}(0),$ where $F(T_{\lambda})$ denotes the set of all fixed points of $T_{\lambda}$. ###### Proof. Let $x\in F(T_{\lambda})$. Then $\displaystyle x\in F(T_{\lambda})$ $\displaystyle\Leftrightarrow$ $\displaystyle x=T_{\lambda}x=J_{\lambda}^{B}oJ^{-1}(J-\lambda A)(x)$ $\displaystyle\Leftrightarrow$ $\displaystyle x=(J+\lambda B)^{-1}JoJ^{-1}(Jx-\lambda Ax)$ $\displaystyle\Leftrightarrow$ $\displaystyle Jx-\lambda Ax\in Jx+\lambda Bx$ $\displaystyle\Leftrightarrow$ $\displaystyle 0\in\lambda(Ax+Bx)$ $\displaystyle\Leftrightarrow$ $\displaystyle 0\in Ax+Bx$ $\displaystyle\Leftrightarrow$ $\displaystyle x\in(A+B)^{-1}(0).$ ∎ We shall adopt the following notation in this paper: . $x_{n}\rightarrow x$ means that $x_{n}\rightarrow x$ strongly. . $x_{n}\rightharpoonup x$ means that $x_{n}\rightarrow x$ weakly. ## 3 Approximation Method In this section, we propose our method and state certain conditions under which we obtain the desired convergence for our proposed methods. First, we give the conditions governing the cost function and the sequence of parameters below. ###### Assumption 3.1. * (a) Let $E$ be a real 2-uniformly convex Banach space which is also uniformly smooth. * (b) Let $B:E\to 2^{E^{}}$ be a maximal monotone operator; $A:E\to E^{}$ a monotone and $L$-Lipschitz continuous. * (c) The solution set $(A+B)^{-1}(0)$ of the inclusion problem (1) is nonempty. Throughout this paper, we assume that the duality mapping $J$ and the resolvent $J_{\lambda_{n}}^{B}:=(J+\lambda_{n}B)^{-1}J$ of maximal monotone operator $B$ are easy to compute. ###### Assumption 3.2. Suppose the sequence $\\{\lambda_{n}\\}_{n=1}^{\infty}$ of step-sizes satisfies the following condition: $0<a\leq\lambda_{n}\leq b<\displaystyle\frac{1}{\sqrt{2\mu}\kappa L}$ where * $\mu$ is the 2-uniform convexity constant of $E$; * $\kappa$ is the 2-uniform smoothness constant of $E^{}$; $L$ is the Lipschitz constant of $A$. Assumption 3.2 is satisfied, e.g., for $\lambda_{n}=a+\frac{n}{n+1}\Big{(}\frac{1}{\sqrt{2\mu}\kappa L}-a\Big{)}$ for all $n\geq 1$. We now give our proposed method below. ###### Algorithm 3.3. Step 0: Let Assumptions 3.1 and 3.2 hold. Let $x_{1}\in E$ be a given starting point. Set $n:=1$. Step 1: Compute $y_{n}:=J_{\lambda_{n}}^{B}oJ^{-1}(Jx_{n}-\lambda_{n}Ax_{n})$. If $x_{n}-y_{n}=0$: STOP. Step 2: Compute $\displaystyle x_{n+1}=J^{-1}[Jy_{n}-\lambda_{n}(Ay_{n}-Ax_{n})].$ (8) Step 3: Set $n\leftarrow n+1$, and go to Step 1. We observe that in real Hilbert spaces, the duality mapping $J$ becomes the identity mapping and our Algorithm 3.3 reduces to the algorithm proposed by Tseng in [48]. Note that both sequences $\\{y_{n}\\}$ and $\\{x_{n}\\}$ are in $E$. Furthermore, by Lemma 2.17, we have that if $x_{n}=y_{n}$, then $x_{n}$ is a solution of problem (1). To the best of our knowledge, the proposed Algorithm 3.3 is the only known algorithm which can solve monotone inclusion problem (1) without the inverse- strongly monotonicity of $A$. We consider some various cases of Algorithm 3.3. * • When $A=0$ in Algorithm 3.3, then Algorithm 3.3 reduces to the methods proposed in [6, 20, 24, 26, 27, 28, 32, 35, 38, 39, 43]. In this case, the assumption that $E$ is 2-uniformly convex Banach space and uniformly smooth is not needed. In fact, the convergence can be obtained in reflexive Banach spaces in this case. However, we do not know if the convergence of Algorithm 3.3 can be obtained in a more general reflexive Banach space for problem (1). * • When $B=N_{C}$, the normal cone for closed and convex subset $C$ of $E$ ($N_{C}(x):=\\{x^{}\in E^{}:\langle y-x,x^{}\rangle\leq 0,\forall y\in C\\}$), then the inclusion problem (1) reduces to a variational inequality problem (i.e., find $x\in C:\langle Ax,y-x\rangle\geq 0,\leavevmode\nobreak\ \forall y\in C$). It is well known that $N_{C}=\partial\delta_{C}$, where $\delta_{C}$ is the indicator function of $C$ at $x$, defined by $\delta_{C}(x)=0$ if $x\in C$ and $\delta_{C}(x)=+\infty$ if $x\notin C$ and $\partial(.)$ is the subdifferential, defined by $\partial f(x):=\\{x^{}\in E^{}:f(y)\geq f(x)+\langle x^{},y-x\rangle,\leavevmode\nobreak\ \leavevmode\nobreak\ \forall y\in E\\}$ for a proper, lower semicontinuous convex functional $f$ on $E$. Using the theorem of Rockafellar in [40, 41], $N_{C}=\partial\delta_{C}$ is maximal monotone. Hence, $Jz\in J(J_{\lambda_{n}}^{B})+\lambda_{n}\partial\delta_{C}(J_{\lambda_{n}}^{B}),\leavevmode\nobreak\ \leavevmode\nobreak\ \forall z\in E.$ This implies that $\displaystyle 0\in\partial\delta_{C}(J_{\lambda_{n}}^{B})+\frac{1}{\lambda_{n}}J(J_{\lambda_{n}}^{B})-\frac{1}{\lambda_{n}}Jz=\partial\Big{(}\delta_{C}+\frac{1}{2\lambda_{n}}\\|.\\|^{2}-\frac{1}{\lambda_{n}}Jz\Big{)}J_{\lambda_{n}}^{B}.$ Therefore, $J_{\lambda_{n}}^{B}(z)={\rm argmin}_{y\in E}\Big{\\{}\delta_{C}(y)+\frac{1}{2\lambda_{n}}\\|y\\|^{2}-\frac{1}{\lambda_{n}}\langle y,Jz\rangle\Big{\\}}$ and $y_{n}$ in Algorithm 3.3 reduces to $y_{n}={\rm argmin}_{y\in E}\Big{\\{}\delta_{C}(y)+\frac{1}{2\lambda_{n}}\\|y\\|^{2}-\frac{1}{\lambda_{n}}\langle y,Jx_{n}-\lambda_{n}Ax_{n}\rangle\Big{\\}}.$ However, in implementing our proposed Algorithm 3.3, we assume that the resolvent $(J+\lambda_{n}B)^{-1}J$ is easy to compute and the duality mapping $J$ is easily computable as well. On the other hand, one has to obtain the Lipschitz constant, $L$, of the monotone mapping $A$ (or an estimate of it). In a case when the Lipschitz constant cannot be accurately estimated or overestimated, this might result in too small step-sizes $\lambda_{n}$. This is a drawback of our proposed Algorithm 3.3. One way to overcome this obstacle is to introduce linesearch in our Algorithm 3.3. This case will be considered in Algorithm 3.8. ### 3.1 Convergence Analysis In this Section, we give the convergence analysis of the proposed Algorithm 3.3. First, we establish the boundedness of the sequence of iterates generated by Algorithm 3.3. ###### Lemma 3.4. Let Assumptions 3.1 and 3.2 hold. Assume that $x^{}\in(A+B)^{-1}(0)$ and let the sequence $\\{x_{n}\\}_{n=1}^{\infty}$ be generated by Algorithm 3.3. Then $\\{x_{n}\\}$ is bounded. ###### Proof. By the Lyaponuv functional $\phi$, we have $\displaystyle\phi(x^{},x_{n+1})=$ $\displaystyle\phi(x^{},J^{-1}(Jy_{n}-\lambda_{n}(Ay_{n}-Ax_{n})))$ $\displaystyle=$ $\displaystyle\\|x^{}\\|^{2}-2\langle x^{},JJ^{-1}(Jy_{n}-\lambda_{n}(Ay_{n}-Ax_{n}))\rangle$ $\displaystyle+\\|J^{-1}(Jy_{n}-\lambda_{n}(Ay_{n}-Ax_{n}))\\|^{2}$ $\displaystyle=$ $\displaystyle\\|x^{}\\|^{2}-2\langle x^{},Jy_{n}-\lambda_{n}(Ay_{n}-Ax_{n})\rangle+\\|(Jy_{n}-\lambda_{n}(Ay_{n}-Ax_{n}))\\|^{2}$ $\displaystyle=$ $\displaystyle\\|x^{}\\|^{2}-2\langle x^{},Jy_{n}\rangle+2\lambda_{n}\langle x^{},Ay_{n}-Ax_{n}\rangle$ $\displaystyle+\\|Jy_{n}-\lambda_{n}(Ay_{n}-Ax_{n})\\|^{2}.$ (9) Using Lemma 2.2, we get that $E^{}$ is 2-uniformly smooth and so by Lemma 2.3, we get $\displaystyle\\|Jy_{n}-\lambda_{n}(Ay_{n}-Ax_{n})\\|^{2}\leq$ $\displaystyle\\|Jy_{n}\\|^{2}-2\lambda_{n}\langle Ay_{n}-Ax_{n},y_{n}\rangle$ $\displaystyle+2\kappa^{2}\\|\lambda_{n}(Ay_{n}-Ax_{n})\\|^{2}.$ (10) Substituting (10) into (9), we get $\displaystyle\phi(x^{},x_{n+1})\leq$ $\displaystyle\\|Jy_{n}\\|^{2}-2\lambda_{n}\langle Ay_{n}-Ax_{n},y_{n}\rangle+2\kappa^{2}\\|\lambda_{n}(Ay_{n}-Ax_{n})\\|^{2}$ $\displaystyle+\\|x^{}\\|^{2}-2\langle x^{},Jy_{n}\rangle+2\lambda_{n}\langle x^{},Ay_{n}-Ax_{n}\rangle$ $\displaystyle=$ $\displaystyle\\|x^{}\\|^{2}-2\langle x^{},Jy_{n}\rangle+\\|y_{n}\\|^{2}-2\lambda_{n}\langle Ay_{n}-Ax_{n},y_{n}-x^{}\rangle$ $\displaystyle+2\kappa^{2}\\|\lambda_{n}(Ay_{n}-Ax_{n})\\|^{2}$ $\displaystyle=$ $\displaystyle\phi(x^{},y_{n})-2\lambda_{n}\langle Ay_{n}-Ax_{n},y_{n}-x^{}\rangle+2\kappa^{2}\\|\lambda_{n}(Ay_{n}-Ax_{n})\\|^{2}.$ (11) Using Lemma 2.9 (i), we get $\displaystyle\phi(x^{},y_{n})=$ $\displaystyle\phi(x^{},x_{n})+\phi(x_{n},y_{n})+2\langle x^{}-x_{n},Jx_{n}-Jy_{n}\rangle$ $\displaystyle=$ $\displaystyle\phi(x^{},x_{n})+\phi(x_{n},y_{n})+2\langle x_{n}-x^{},Jy_{n}-Jx_{n}\rangle.$ (12) Putting (12) into (11), we get $\displaystyle\phi(x^{},x_{n+1})=$ $\displaystyle\phi(x^{},x_{n})+\phi(x_{n},y_{n})+2\langle x_{n}-x^{},Jy_{n}-Jx_{n}\rangle$ $\displaystyle-2\lambda_{n}\langle Ay_{n}-Ax_{n},y_{n}-x^{}\rangle+2\kappa^{2}\\|\lambda_{n}(Ay_{n}-Ax_{n})\\|^{2}$ $\displaystyle=$ $\displaystyle\phi(x^{},x_{n})+\phi(x_{n},y_{n})-2\langle y_{n}-x_{n},Jy_{n}-Jx_{n}\rangle+2\langle y_{n}-x^{},Jy_{n}-Jx_{n}\rangle$ $\displaystyle-2\lambda_{n}\langle Ay_{n}-Ax_{n},y_{n}-x^{}\rangle+2\kappa^{2}\\|\lambda_{n}(Ay_{n}-Ax_{n})\\|^{2}.$ (13) Using Lemma 2.9 (ii), we get $-\phi(y_{n},x_{n})+2\langle y_{n}-x_{n},Jy_{n}-Jx_{n}\rangle=\phi(x_{n},y_{n}).$ (14) Substituting (14) into (13), we have $\displaystyle\phi(x^{},x_{n+1})\leq$ $\displaystyle\phi(x^{},x_{n})-\phi(y_{n},x_{n})+2\langle y_{n}-x^{},Jy_{n}-Jx_{n}\rangle-2\lambda_{n}\langle Ay_{n}-Ax_{n},y_{n}-x^{}\rangle$ $\displaystyle+2\kappa^{2}\\|\lambda_{n}(Ay_{n}-Ax_{n})\\|^{2}$ $\displaystyle=$ $\displaystyle\phi(x^{},x_{n})-\phi(y_{n},x_{n})+2\kappa^{2}\\|\lambda_{n}(Ay_{n}-Ax_{n})\\|^{2}$ $\displaystyle-2\langle Jx_{n}-Jy_{n}-\lambda_{n}(Ax_{n}-Ay_{n}),y_{n}-x^{}\rangle.$ (15) Since $y_{n}=(J+\lambda_{n}B)^{-1}JoJ^{-1}(Jx_{n}-\lambda_{n}Ax_{n})$, we have $Jx_{n}-\lambda_{n}Ax_{n}\in(J+\lambda_{n}B)y_{n}$. Using the fact that $B$ is maximal monotone, then there exists $v_{n}\in By_{n}$ such that $Jx_{n}-\lambda_{n}Ax_{n}=Jy_{n}+\lambda_{n}v_{n}$. Therefore $v_{n}=\frac{1}{\lambda_{n}}(Jx_{n}-Jy_{n}-\lambda_{n}Ax_{n}).$ (16) On the other hand, we know that $0\in(Ax^{}+Bx^{})$ and $Ay_{n}+v_{n}\in(A+B)y_{n}$. Since $A+B$ is maximal monotone, we obtain $\displaystyle\langle Ay_{n}+v_{n},y_{n}-x^{}\rangle\geq 0.$ (17) Putting (16) into (17), we get $\displaystyle\langle Jx_{n}-Jy_{n}-\lambda_{n}(Ax_{n}-Ay_{n}),y_{n}-x^{}\rangle\geq 0.$ (18) Now, using (18) in (15), we get $\displaystyle\phi(x^{},x_{n+1})\leq$ $\displaystyle\phi(x^{},x_{n})-\phi(y_{n},x_{n})+2\kappa^{2}\\|\lambda_{n}(Ay_{n}-Ax_{n})\\|^{2}$ $\displaystyle\leq$ $\displaystyle\phi(x^{},x_{n})-\phi(y_{n},x_{n})+2\kappa^{2}\lambda_{n}^{2}L^{2}\mu\phi(y_{n},x_{n})$ $\displaystyle=$ $\displaystyle\phi(x^{},x_{n})-(1-2\kappa^{2}\lambda_{n}^{2}L^{2}\mu)\phi(y_{n},x_{n}).$ (19) Using Assumption 3.2, we get $\displaystyle\phi(x^{},x_{n+1})\leq\phi(x^{},x_{n}),$ (20) which shows that $\lim\phi(x^{},x_{n})$ exists and hence, $\\{\phi(x^{},x_{n})\\}$ is bounded. Therefore $\\{x_{n}\\}$ is bounded. ∎ ###### Definition 3.5. The duality mapping $J$ is weakly sequentially continuous if, for any sequence $\\{x_{n}\\}\subset E$ such that $x_{n}\rightharpoonup x$ as $n\rightarrow\infty$, then $Jx_{n}\rightharpoonup^{}Jx$ as $n\rightarrow\infty$. It is known that the normalized duality map on $\ell_{p}$ spaces, $1<p<\infty$, is weakly sequentially continuous. We now obtain the weak convergence result of Algorithm 3.3 in the next theorem. ###### Theorem 3.6. Let Assumptions 3.1 and 3.2 hold. Assume that $J$ is weakly sequentially continuous on $E$ and let the sequence $\\{x_{n}\\}_{n=1}^{\infty}$ be generated by Algorithm 3.3. Then $\\{x_{n}\\}$ converges weakly to $z\in(A+B)^{-1}(0)$. Moreover, $z:=\underset{n\rightarrow\infty}{\lim}\Pi_{(A+B)^{-1}(0)}(x_{n})$. ###### Proof. Let $x^{}\in(A+B)^{-1}(0)$. From (19), we have $\displaystyle 0$ $\displaystyle<$ $\displaystyle[1-2\kappa^{2}b^{2}L^{2}\mu]\phi(y_{n},x_{n})\leq[1-2\kappa^{2}\lambda_{n}^{2}L^{2}\mu]\phi(y_{n},x_{n})$ (21) $\displaystyle\leq$ $\displaystyle\phi(x^{},x_{n})-\phi(x^{},x_{n+1}).$ Since $\lim_{n\rightarrow\infty}\phi(x^{},x_{n})$ exists, we obtain from (21) that $\underset{n\rightarrow\infty}{\lim}\phi(y_{n},x_{n})=0.$ Applying Lemma 2.11, we get $\underset{n\rightarrow\infty}{\lim}\\|x_{n}-y_{n}\\|=0.$ Since $E$ is uniformly smooth, the duality mapping $J$ is uniformly norm-to- norm continuous on each bounded subset of $E$. Hence, we have $\underset{n\rightarrow\infty}{\lim}\\|Jx_{n}-Jy_{n}\\|=0.$ Since $\\{x_{n}\\}$ is bounded by Lemma 3.4, there exists a subsequence $\\{x_{n_{i}}\\}$ of $\\{x_{n}\\}$ and $z\in C$ such that $x_{n_{i}}\rightharpoonup z$. Since $\underset{n\rightarrow\infty}{\lim}\\|x_{n}-y_{n}\\|=0$, it follows that $x_{{n_{i}}+1}\rightharpoonup z$. We now show that $z\in(A+B)^{-1}(0)$. Suppose $(v,u)\in\textrm{Graph}(A+B)$. This implies that $Ju-Av\in Bv$. Furthermore, we obtain from $y_{n_{i}}=(J+\lambda_{n_{i}}B)^{-1}JoJ^{-1}(Jx_{n_{i}}-\lambda_{n_{i}}Ax_{n_{i}})$ that $(J-\lambda_{n_{i}}A)x_{n_{i}}\in(J+\lambda_{n_{i}}B)y_{n_{i}},$ and thus $\frac{1}{\lambda_{n_{i}}}(Jx_{n_{i}}-Jy_{n_{i}}-\lambda_{n_{i}}Ax_{n_{i}})\in By_{n_{i}}.$ Using the fact that $B$ is maximal monotone, we obtain $\langle v-y_{n_{i}},Ju- Av-\frac{1}{\lambda_{n_{i}}}(Jx_{n_{i}}-Jy_{n_{i}}-\lambda_{n_{i}}Ax_{n_{i}})\rangle\geq 0.$ Therefore, $\displaystyle\langle v-y_{n_{i}},Ju\rangle$ $\displaystyle\geq$ $\displaystyle\langle v-y_{n_{i}},Av+\frac{1}{\lambda_{n_{i}}}(Jx_{n_{i}}-Jy_{n_{i}}-\lambda_{n_{i}}Ax_{n_{i}})\rangle$ $\displaystyle=$ $\displaystyle\langle v-y_{n_{i}},Av- Ax_{n_{i}}\rangle+\langle v-y_{n_{i}},\frac{1}{\lambda_{n_{i}}}(Jx_{n_{i}}-Jy_{n_{i}})\rangle$ $\displaystyle=$ $\displaystyle\langle v-y_{n_{i}},Av- Ay_{n_{i}}\rangle+\langle v-y_{n_{i}},Ay_{n_{i}}-Ax_{n_{i}}\rangle$ $\displaystyle+\langle v-y_{n_{i}},\frac{1}{\lambda_{n_{i}}}(Jx_{n_{i}}-Jy_{n_{i}})\rangle$ $\displaystyle\geq$ $\displaystyle\langle v-y_{n_{i}},Ay_{n_{i}}-Ax_{n_{i}}\rangle+\langle v-y_{n_{i}},\frac{1}{\lambda_{n_{i}}}(Jx_{n_{i}}-Jy_{n_{i}})\rangle.$ By the fact that $\underset{n\rightarrow\infty}{\lim}\\|x_{n}-y_{n}\\|=0$ and $A$ is Lipschitz continuous, we obtain $\underset{n\rightarrow\infty}{\lim}\\|Ax_{n}-Ay_{n}\\|=0$. Consequently, we obtain that $\langle v-z,Ju\rangle\geq 0.$ By the maximal monotonicity of $A+B$, we have $0\in(A+B)z$. Hence, $z\in(A+B)^{-1}(0)$. Let $u_{n}:=\Pi_{(A+B)^{-1}(0)}(x_{n})$. By (20) and Lemma 2.12, we have that $\\{u_{n}\\}$ is a Cauchy sequence. Since $(A+B)^{-1}(0)$ is closed, we have that $\\{u_{n}\\}$ converges strongly to $w\in(A+B)^{-1}(0)$. By the uniform smoothness of $E$, we also have $\underset{n\rightarrow\infty}{\lim}\\|Ju_{n}-Jw\\|=0$. We then show that $z=w$. Using Lemma 2.10 (i), $u_{n}=\Pi_{(A+B)^{-1}(0)}(x_{n})$ and $z\in(A+B)^{-1}(0)$, we have $\langle z-u_{n},Ju_{n}-Jx_{n}\rangle\geq 0,\leavevmode\nobreak\ \leavevmode\nobreak\ \forall n\geq 1.$ Therefore, $\displaystyle\langle z-w,Jx_{n}-Ju_{n}\rangle$ $\displaystyle=$ $\displaystyle\langle z-u_{n},Jx_{n}-Ju_{n}\rangle+\langle u_{n}-w,Jx_{n}-Ju_{n}\rangle$ $\displaystyle\leq$ $\displaystyle\\|u_{n}-w\\|\\|Jx_{n}-Ju_{n}\\|\leq M\\|u_{n}-w\\|,\leavevmode\nobreak\ \leavevmode\nobreak\ \forall n\geq 1,$ where $M:=\underset{n\geq 1}{\sup}\\|Jx_{n}-Ju_{n}\\|$. Using $n=n_{i}$ in $\underset{n\rightarrow\infty}{\lim}\\|u_{n}-w\\|=0,\underset{n\rightarrow\infty}{\lim}\\|Ju_{n}-Jw\\|=0$ and the weakly sequential continuity of $J$, we obtain $\langle z-w,Jz-Jw\rangle\leq 0$ as $i\rightarrow\infty$. Therefore, $\langle z-w,Jz-Jw\rangle=0$. Since $E$ is strictly convex, we have $z=w$. Therefore, the sequence $\\{x_{n}\\}$ converges weakly to $z=\underset{n\rightarrow\infty}{\lim}\Pi_{(}A+B)^{-1}(0)(x_{n})$. This completes the proof. ∎ It is easy to see from Algorithm 3.3 above and Lemma 2.17 that $x_{n}=y_{n}$ if and only if $x_{n}\in(A+B)^{-1}(0)$. Also, we have already established that $\\|x_{n}-y_{n}\\|\rightarrow 0$ holds when $(A+B)^{-1}(0)\neq\emptyset$. Therefore, using the $\\|x_{n}-y_{n}\\|$ as a measure of convergence rate, we obtain the following non asymptotic rate of convergence of our proposed Algorithm 3.3. ###### Theorem 3.7. Let Assumptions 3.1 and 3.2 hold. Let the sequence $\\{x_{n}\\}_{n=1}^{\infty}$ be generated by Algorithm 3.3. Then $\min_{1\leq k\leq n}\\|x_{k}-y_{k}\\|=O(1/\sqrt{n})$. ###### Proof. We obtain from (19) that $\displaystyle\phi(x^{},x_{n+1})\leq\phi(x^{},x_{n})-(1-2\kappa^{2}\lambda_{n}^{2}L^{2}\mu)\phi(y_{n},x_{n}).$ Hence, we have from Lemma 2.11 that $\displaystyle\frac{1}{\mu}(1-2\kappa^{2}\lambda_{n}^{2}L^{2}\mu)\\|x_{n}-y_{n}\\|^{2}$ $\displaystyle\leq$ $\displaystyle(1-2\kappa^{2}\lambda_{n}^{2}L^{2}\mu)\phi(y_{n},x_{n})$ $\displaystyle\leq$ $\displaystyle\phi(x^{},x_{n})-\phi(x^{},x_{n+1}).$ By Assumption 3.2, we get $\displaystyle\sum_{k=1}^{n}\\|x_{k}-y_{k}\\|^{2}\leq\frac{\mu}{(1-2\kappa^{2}\lambda_{n}^{2}L^{2}\mu)}\phi(x^{},x_{1}).$ Therefore, $\min_{1\leq k\leq n}\\|x_{k}-y_{k}\\|^{2}\leq\frac{\mu}{n(1-2\kappa^{2}\lambda_{n}^{2}L^{2}\mu)}\phi(x^{},x_{1}).$ This implies that $\min_{1\leq k\leq n}\\|x_{k}-y_{k}\\|=O(1/\sqrt{n}).$ ∎ Next, we propose another iterative method such that the sequence of step-sizes does not depend on the Lipschitz constant of monotone operator $A$ in problem (1). ###### Algorithm 3.8. Step 0: Let Assumption 3.1 hold. Given $\gamma>0,l\in(0,1)$ and $\theta\in(0,\frac{1}{\sqrt{2\mu}\kappa})$. Let $x_{1}\in E$ be a given starting point. Set $n:=1$. Step 1: Compute $y_{n}:=J_{\lambda_{n}}^{B}J^{-1}(Jx_{n}-\lambda_{n}Ax_{n})$, where $\lambda_{n}$ is chosen to be the largest $\lambda\in\\{\gamma,\gamma l,\gamma l^{2},\ldots\\}$ satisfying $\displaystyle\lambda\\|Ax_{n}-Ay_{n}\\|\leq\theta\\|x_{n}-y_{n}\\|.$ (22) If $x_{n}-y_{n}=0$: STOP. Step 2: Compute $\displaystyle x_{n+1}=J^{-1}[Jy_{n}-\lambda_{n}(Ay_{n}-Ax_{n})].$ (23) Step 3: Set $n\leftarrow n+1$, and go to Step 1. Before we establish the weak convergence analysis of Algorithm 3.8, we first show that the line search rule given in (22) is well-defined in this lemma. ###### Lemma 3.9. The line search rule (22) in Algorithm 3.8 is well-defined and $\min\Big{\\{}\gamma,\frac{\theta l}{L}\Big{\\}}\leq\lambda_{n}\leq\gamma.$ ###### Proof. Using the Lipschitz continuity of $A$ on $E$, we obtain $\\|Ax_{n}-A(J_{\lambda_{n}}^{B}J^{-1}(Jx_{n}-\lambda_{n}Ax_{n}))\\|\leq L\\|x_{n}-J_{\lambda_{n}}^{B}J^{-1}(Jx_{n}-\lambda_{n}Ax_{n})\\|.$ This implies that $\frac{\theta}{L}\\|Ax_{n}-A(J_{\lambda_{n}}^{B}J^{-1}(Jx_{n}-\lambda_{n}Ax_{n}))\\|\leq\theta\\|x_{n}-J_{\lambda_{n}}^{B}J^{-1}(Jx_{n}-\lambda_{n}Ax_{n})\\|.$ Therefore, (22) holds whenever $\lambda_{n}\leq\frac{\theta}{L}$. Hence, $\lambda_{n}$ is well-defined. From the way $\lambda_{n}$ is chosen, we can clearly see that $\lambda_{n}\leq\gamma$. Now, suppose $\lambda_{n}=\gamma$, then (22) is satisfied and the lemma is proved. Suppose $\lambda_{n}<\gamma$. Then $\frac{\lambda_{n}}{l}$ violates (22) and we get $\displaystyle L\\|x_{n}-J_{\lambda_{n}}^{B}J^{-1}(Jx_{n}-\lambda_{n}Ax_{n})\\|$ $\displaystyle\geq$ $\displaystyle\\|Ax_{n}-A(J_{\lambda_{n}}^{B}J^{-1}(Jx_{n}-\lambda_{n}Ax_{n}))\\|$ $\displaystyle>$ $\displaystyle\frac{\theta}{\frac{\lambda_{n}}{l}}\\|x_{n}-J_{\lambda_{n}}^{B}J^{-1}(Jx_{n}-\lambda_{n}Ax_{n})\\|.$ This implies that $\lambda_{n}>\frac{\theta l}{L}$. This completes the proof. ∎ We now give a weak convergence result using Algorithm 3.8 in the next theorem. ###### Theorem 3.10. Let Assumptions 3.1. Assume that $J$ is weakly sequentially continuous on $E$ and let the sequence $\\{x_{n}\\}_{n=1}^{\infty}$ be generated by Algorithm 3.8. Then $\\{x_{n}\\}$ converges weakly to $z\in(A+B)^{-1}(0)$. Moreover, $z:=\underset{n\rightarrow\infty}{\lim}\Pi_{(A+B)^{-1}(0)}(x_{n})$. ###### Proof. Using the same line of arguments as in the proof of Lemma 3.4, we can obtain from (19) that $\displaystyle\phi(x^{},x_{n+1})$ $\displaystyle\leq$ $\displaystyle\phi(x^{},x_{n})-\phi(y_{n},x_{n})+2\kappa^{2}\\|\lambda_{n}(Ay_{n}-Ax_{n})\\|^{2}$ (24) $\displaystyle\leq$ $\displaystyle\phi(x^{},x_{n})-\phi(y_{n},x_{n})+2\kappa^{2}\theta^{2}\\|y_{n}-x_{n}\\|^{2}$ $\displaystyle\leq$ $\displaystyle\phi(x^{},x_{n})-\phi(y_{n},x_{n})+2\kappa^{2}\theta^{2}\mu\phi(y_{n},x_{n})$ $\displaystyle=$ $\displaystyle\phi(x^{},x_{n})-(1-2\kappa^{2}\theta^{2}\mu)\phi(y_{n},x_{n}).$ Since $\theta^{2}<\frac{1}{2\kappa^{2}\mu}$, we get $\displaystyle\phi(x^{},x_{n+1})\leq\phi(x^{},x_{n}),$ (25) which shows that $\lim\phi(x^{},x_{n})$ exists and hence, $\\{\phi(x^{},x_{n})\\}$ is bounded. Therefore $\\{x_{n}\\}$ is bounded. The rest of the proof follows by using the same arguments as in the proof of Theorem 3.6. The completes the proof. ∎ Finally, we give a modification of Algorithm 3.3 and consequently obtain the strong convergence analysis below. ###### Algorithm 3.11. Step 0: Let Assumptions 3.1 and 3.2 hold. Suppose that $\\{\alpha_{n}\\}$ is a real sequence in (0,1) and let $x_{1}\in E$ be a given starting point. Set $n:=1$. Step 1: Compute $y_{n}:=J_{\lambda_{n}}^{B}J^{-1}(Jx_{n}-\lambda_{n}Ax_{n})$. If $x_{n}-y_{n}=0$: STOP. Step 2: Compute $\displaystyle w_{n}=J^{-1}[Jy_{n}-\lambda_{n}(Ay_{n}-Ax_{n})]$ (26) and $\displaystyle x_{n+1}=J^{-1}[\alpha_{n}Jx_{1}+(1-\alpha_{n})Jw_{n}].$ (27) Step 3: Set $n\leftarrow n+1$, and go to Step 1. ###### Theorem 3.12. Let Assumptions 3.1 and 3.2 hold. Suppose that $\lim_{n\to\infty}\alpha_{n}=0$ and $\sum_{n=1}^{\infty}\alpha_{n}=\infty$. Let the sequence $\\{x_{n}\\}_{n=1}^{\infty}$ be generated by Algorithm 3.11. Then $\\{x_{n}\\}$ converges strongly to $z=\Pi_{(A+B)^{-1}(0)}(x_{1})$. ###### Proof. By Lemma 3.4, we have that $\\{x_{n}\\}$ is bounded. Furthermore, using Lemma 2.10 with (26) and (27), we have $\displaystyle\phi(z,x_{n+1})$ $\displaystyle=$ $\displaystyle\phi(z,J^{-1}(\alpha_{n}Jx_{1}+(1-\alpha_{n})Jw_{n}))$ (28) $\displaystyle=$ $\displaystyle V(z,\alpha_{n}Jx_{1}+(1-\alpha_{n})Jw_{n}))$ $\displaystyle\leq$ $\displaystyle V(z,\alpha_{n}Jx_{1}+(1-\alpha_{n})Jw_{n}-\alpha_{n}(Jx_{1}-Jz))$ $\displaystyle+2\alpha_{n}\langle Jx_{1}-Jz,x_{n+1}-z\rangle$ $\displaystyle=$ $\displaystyle V(z,\alpha_{n}Jz+(1-\alpha_{n})Jw_{n})+2\alpha_{n}\langle Jx_{1}-Jz,x_{n+1}-z\rangle$ $\displaystyle\leq$ $\displaystyle\alpha_{n}V(z,Jz)+(1-\alpha_{n})V(z,Jw_{n})+2\alpha_{n}\langle Jx_{1}-Jz,x_{n+1}-z\rangle$ $\displaystyle=$ $\displaystyle(1-\alpha_{n})V(z,Jw_{n})+2\alpha_{n}\langle Jx_{1}-Jz,x_{n+1}-z\rangle$ $\displaystyle\leq$ $\displaystyle(1-\alpha_{n})V(z,Jx_{n})+2\alpha_{n}\langle Jx_{1}-Jz,x_{n+1}-z\rangle$ $\displaystyle=$ $\displaystyle(1-\alpha_{n})\phi(z,x_{n})+2\alpha_{n}\langle Jx_{1}-Jz,x_{n+1}-z\rangle.$ Set $a_{n}:=\phi(x_{n},z)$ and divide the rest of the proof into two parts as follows. Case 1: Suppose that there exists $n_{0}\in\mathbb{N}$ such that $\\{\phi(z,x_{n})\\}_{n=n_{0}}^{\infty}$ is non-increasing. Then $\\{\phi(z,x_{n})\\}_{n=1}^{\infty}$ converges, and we therefore obtain $a_{n}-a_{n+1}\rightarrow 0,\leavevmode\nobreak\ \leavevmode\nobreak\ n\rightarrow\infty.$ (29) Using (20) in (27), we have $\displaystyle V(z,Jx_{n+1})$ $\displaystyle\leq$ $\displaystyle\alpha_{n}V(z,Jx_{1})+(1-\alpha_{n})V(z,Jw_{n})$ (30) $\displaystyle\leq$ $\displaystyle\alpha_{n}V(Jx_{1},z)+(1-\alpha_{n})V(Jx_{n},z)$ $\displaystyle-(1-\alpha_{n})[1-2\kappa^{2}\theta^{2}\mu]V(y_{n},Jx_{n}).$ This implies from (30) that $(1-\alpha_{n})[1-2\kappa^{2}\theta^{2}\mu]V(y_{n},Jx_{n})\leq V(Jx_{n},z)-V(Jx_{n+1},z)+\alpha_{n}M_{1},$ for some $M_{1}>0$. Thus, $(1-\alpha_{n})[1-2\kappa^{2}\theta^{2}\mu]\phi(y_{n},x_{n})\rightarrow 0,\leavevmode\nobreak\ \leavevmode\nobreak\ n\rightarrow\infty.$ Hence, $\phi(y_{n},x_{n})\rightarrow 0,\leavevmode\nobreak\ \leavevmode\nobreak\ n\rightarrow\infty.$ Consequently, $\\|x_{n}-y_{n}\\|\rightarrow 0,\leavevmode\nobreak\ \leavevmode\nobreak\ n\rightarrow\infty.$ By (26), we get $\displaystyle\\|Jw_{n}-Jy_{n}\\|$ $\displaystyle=$ $\displaystyle\lambda_{n}\\|Ay_{n}-Ax_{n}\\|$ $\displaystyle\leq$ $\displaystyle b\\|Ay_{n}-Ax_{n}\\|\rightarrow 0,\leavevmode\nobreak\ \leavevmode\nobreak\ n\rightarrow\infty.$ Therefore, $\\|w_{n}-y_{n}\\|\rightarrow 0,\leavevmode\nobreak\ \leavevmode\nobreak\ n\rightarrow\infty.$ Moreover, we obtain from (27) that $\displaystyle\\|Jx_{n+1}-Jw_{n}\\|$ $\displaystyle=$ $\displaystyle\alpha_{n}\\|Jx_{1}-Jw_{n}\\|\leq\alpha_{n}M_{2}\rightarrow 0,\leavevmode\nobreak\ \leavevmode\nobreak\ n\rightarrow\infty,$ (31) for some $M_{2}>0$. Since $J^{-1}$ is norm-to-norm uniformly continuous on bounded subsets of $E^{}$, we have that $\\|x_{n+1}-w_{n}\\|\rightarrow 0,\leavevmode\nobreak\ \leavevmode\nobreak\ n\rightarrow\infty.$ Now, $\\|x_{n+1}-x_{n}\\|\leq\\|x_{n+1}-w_{n}\\|+\\|w_{n}-y_{n}\\|+\\|y_{n}-x_{n}\\|\rightarrow 0,\leavevmode\nobreak\ \leavevmode\nobreak\ n\rightarrow\infty.$ Since $\\{x_{n}\\}$ is a bounded sunset of $E$, we can choose a subsequence $\\{x_{n_{k}}\\}$ of $\\{x_{n}\\}$ such that $x_{n_{k}}\rightharpoonup p\in E$ and $\displaystyle\limsup_{n\rightarrow\infty}\langle Jx_{1}-Jz,x_{n}-z\rangle\leq 2\lim_{k\rightarrow\infty}\langle Jx_{1}-Jz,x_{n_{k}}-z\rangle.$ Since $z=\Pi_{C}x_{1}$, we get $\displaystyle\limsup_{n\rightarrow\infty}\langle Jx_{1}-Jz,x_{n}-z\rangle$ $\displaystyle\leq$ $\displaystyle 2\lim_{k\rightarrow\infty}\langle Jx_{1}-Jz,x_{n_{k}}-z\rangle$ (32) $\displaystyle=$ $\displaystyle 2\langle Jx_{1}-Jz,p-z\rangle\leq 0.$ This implies that $\limsup_{n\rightarrow\infty}\langle Jx_{1}-Jz,x_{n}-z\rangle\leq 0.$ Using Lemma 2.15 and (32) in (28), we obtain $\lim_{n\rightarrow\infty}\phi(z,x_{n})=0.$ Thus, $x_{n}\rightarrow z$, $n\rightarrow\infty$. Case 2: Suppose that there exists a subsequence $\\{x_{n_{j}}\\}$ of $\\{x_{n}\\}$ such that $\phi(z,x_{m_{j}})<\phi(z,x_{{m_{j}}+1}),\leavevmode\nobreak\ \leavevmode\nobreak\ \forall j\in\mathbb{N}.$ From Lemma 2.14, there exists a nondecreasing sequence $\\{n_{k}\\}$ of $\mathbb{N}$ such that $\lim_{k\rightarrow\infty}\lim n_{k}=\infty$ and the following inequalities hold for all $k\in\mathbb{N}$: $\displaystyle\phi(z,x_{n_{k}})\leq\phi(z,x_{{n_{k}}+1})\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm and}\leavevmode\nobreak\ \leavevmode\nobreak\ \phi(z,x_{k})\leq\phi(z,x_{{n_{k}}+1}).$ (33) Observe that $\displaystyle\phi(z,x_{n_{k}})$ $\displaystyle\leq$ $\displaystyle\phi(z,x_{{n_{k}}+1})\leq\alpha_{n_{k}}\phi(z,x_{1})+(1-\alpha_{n_{k}})\phi(z,w_{n_{k}})$ $\displaystyle\leq$ $\displaystyle\alpha_{n_{k}}\phi(z,x_{1})+(1-\alpha_{n_{k}})\phi(z,x_{n_{k}}).$ Since $\lim_{n\rightarrow\infty}\alpha_{n}=0$, we get $\phi(z,x_{{n_{k}}+1})-\phi(z,x_{n_{k}})\rightarrow 0,\leavevmode\nobreak\ \leavevmode\nobreak\ k\rightarrow\infty.$ Since $\\{x_{n_{k}}\\}$ is bounded, there exists a subsequence of $\\{x_{n_{k}}\\}$ still denoted by $\\{x_{n_{k}}\\}$ which converges weakly to $p\in E$. Repeating the same arguments as in Case 1 above, we can show that $\\|x_{n_{k}}-y_{n_{k}}\\|\rightarrow 0,\leavevmode\nobreak\ \leavevmode\nobreak\ k\rightarrow\infty,\\|y_{n_{k}}-w_{n_{k}}\\|\rightarrow 0,\leavevmode\nobreak\ \leavevmode\nobreak\ k\rightarrow\infty\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm and}\leavevmode\nobreak\ \leavevmode\nobreak\ \\|x_{{n_{k}}+1}-x_{n_{k}}\\|\rightarrow 0,\leavevmode\nobreak\ \leavevmode\nobreak\ k\rightarrow\infty.$ Similarly, we can conclude that $\displaystyle\limsup_{k\rightarrow\infty}\langle x_{{n_{k}}+1}-z,Jx_{1}-Jz\rangle$ $\displaystyle=$ $\displaystyle\limsup_{k\rightarrow\infty}\langle x_{n_{k}}-z,Jx_{1}-Jz\rangle\leq 0.$ (34) It then follows from (28) and (33) that $\displaystyle\phi(z,x_{{n_{k}}+1})$ $\displaystyle\leq$ $\displaystyle(1-\alpha_{n_{k}})\phi(z,x_{n_{k}})+\alpha_{n_{k}}\langle x_{{n_{k}}+1}-z,Jx_{1}-Jz\rangle$ $\displaystyle\leq$ $\displaystyle(1-\alpha_{n_{k}})\phi(z,x_{{n_{k}}+1})+\alpha_{n_{k}}\langle x_{{n_{k}}+1}-z,Jx_{1}-Jz\rangle.$ Since $\alpha_{n_{k}}>0$, we get $\phi(z,x_{n_{k}})\leq\phi(z,x_{{n_{k}}+1})\leq\langle x_{{n_{k}}+1}-z,Jx_{1}-Jz\rangle.$ By (34), we have that $\limsup_{k\rightarrow\infty}\phi(z,x_{n_{k}})\leq\limsup_{k\rightarrow\infty}\langle x_{{n_{k}}+1}-z,Jx_{1}-Jz\rangle.$ Therefore, $x_{k}\rightarrow z,\leavevmode\nobreak\ \leavevmode\nobreak\ k\rightarrow\infty.$ This concludes the proof. ∎ ###### Remark 3.13. Our proposed Algorithms 3.3 and 3.11 are more applicable than the proposed methods in [10, 12, 23, 29, 30, 44, 45, 46, 42, 49] even in Hilbert spaces. The methods proposed in [12, 23, 29, 30, 44, 45, 46, 42, 49] are only applicable for solving problem (1) in the case when $B$ is maximal monotone and $A$ is inverse-strongly monotone (co-coercive) operator in real Hilbert spaces. Our Algorithms 3.3 and 3.11 are applicable for the case when $B$ is maximal monotone and $A$ is monotone operator even in 2-uniformy convex and uniformly smooth Banach spaces (e.g., $L_{p},1<p\leq 2$). Our results in this paper also complement the results of [14, 22]. ## 4 Application In this section, we apply our results to the minimization of composite objective function of the type $\displaystyle\min_{x\in E}f(x)+g(x),$ (35) where $f:E\rightarrow\mathbb{R}\cup\\{+\infty\\}$ is proper, convex and lower semi-continuous functional and $g:E\rightarrow\mathbb{R}$ is convex functional. Many optimization problems from image processing [9], statistical regression, machine learning (see, e.g., [50] and the references contained therein), etc can be adapted into the form of (35). In this setting, we assume that $g$ represents the ”smooth part” of the functional where $f$ is assumed to be non- smooth. Specifically, we assume that $g$ is G$\hat{a}$teaux-differentiable with derivative $\nabla g$ which is Lipschitz-continuous with constant $L$. Then by [37, thm. 3.13], we have $\langle\nabla g(x)-\nabla g(y),x-y\rangle\geq\frac{1}{L}\\|\nabla g(x)-\nabla g(y)\\|^{2},\leavevmode\nobreak\ \leavevmode\nobreak\ \forall x,y\in E.$ Therefore, $\nabla g$ is monotone and Lipschitz continuous with Lipschitz constant $L$. Observe that problem (35) is equivalent to find $\in E$ such that $\displaystyle 0\in\partial f(x)+\nabla g(x).$ (36) Then problem (36) is a special case of inclusion problem (1) with $A:=\nabla g$ and $B:=\partial f$. Next, we obtain the resolvent of $\partial f$. Let us fix $r>0$ and $z\in E$. Suppose $J_{r}^{\partial f}$ is the resolvent of $\partial f$. Then $Jz\in J(J_{r}^{\partial f})+r\partial f(J_{r}^{\partial f}).$ Hence we obtain $\displaystyle 0\in\partial f(J_{r}^{\partial f})+\frac{1}{r}J(J_{r}^{\partial f})-\frac{1}{r}Jz=\partial\Big{(}f+\frac{1}{2r}\\|.\\|^{2}-\frac{1}{r}Jz\Big{)}J_{r}^{\partial f}.$ Therefore, $J_{r}^{\partial f}(z)={\rm argmin}_{y\in E}\Big{\\{}f(y)+\frac{1}{2r}\\|y\\|^{2}-\frac{1}{r}\langle y,Jz\rangle\Big{\\}}.$ We can then write $y_{n}$ in Algorithm 3.3 as $y_{n}={\rm argmin}_{y\in E}\Big{\\{}f(y)+\frac{1}{2\lambda_{n}}\\|y\\|^{2}-\frac{1}{\lambda_{n}}\langle y,Jx_{n}-\lambda_{n}\nabla g(x_{n})\rangle\Big{\\}}.$ We obtain the following weak and strong convergence results for problem (35). ###### Theorem 4.1. Let $E$ be a real 2-uniformly convex Banach space which is also uniformly smooth and the solution set $S$ of problem (35) be nonempty. Suppose $\\{\lambda_{n}\\}_{n=1}^{\infty}$ satisfies the condition $0<a\leq\lambda_{n}\leq b<\displaystyle\frac{1}{\sqrt{2\mu}\kappa L}$. Assume that $J$ is weakly sequentially continuous on $E$ and let the sequence $\\{x_{n}\\}_{n=1}^{\infty}$ be generated by $\displaystyle\left\\{\begin{array}[]{llll}&x_{1}\in E,\\\ &y_{n}={\rm argmin}_{y\in E}\Big{\\{}f(y)+\frac{1}{2\lambda_{n}}\\|y\\|^{2}-\frac{1}{\lambda_{n}}\langle y,Jx_{n}-\lambda_{n}\nabla g(x_{n})\rangle\Big{\\}}\\\ &x_{n+1}=J^{-1}[Jy_{n}-\lambda_{n}(\nabla g(y_{n})-\nabla g(x_{n}))],\leavevmode\nobreak\ \leavevmode\nobreak\ n\geq 1.\end{array}\right.$ (40) Then $\\{x_{n}\\}$ converges weakly to $z\in S$. Moreover, $z:=\underset{n\rightarrow\infty}{\lim}\Pi_{S}(x_{n})$. ###### Theorem 4.2. Let $E$ be a real 2-uniformly convex Banach space which is also uniformly smooth and the solution set $S$ of problem (35) be nonempty. Suppose $\\{\lambda_{n}\\}_{n=1}^{\infty}$ satisfies the condition $0<a\leq\lambda_{n}\leq b<\displaystyle\frac{1}{\sqrt{2\mu}\kappa L}$. Suppose that $\\{\alpha_{n}\\}$ is a real sequence in (0,1) with $\lim_{n\to\infty}\alpha_{n}=0$ and $\sum_{n=1}^{\infty}\alpha_{n}=\infty$. Let the sequence $\\{x_{n}\\}_{n=1}^{\infty}$ be generated by $\displaystyle\left\\{\begin{array}[]{llll}&x_{1}\in E,\\\ &y_{n}={\rm argmin}_{y\in E}\Big{\\{}f(y)+\frac{1}{2\lambda_{n}}\\|y\\|^{2}-\frac{1}{\lambda_{n}}\langle y,Jx_{n}-\lambda_{n}\nabla g(x_{n})\rangle\Big{\\}}\\\ &w_{n}=J^{-1}[Jy_{n}-\lambda_{n}(\nabla g(y_{n})-\nabla g(x_{n}))],\\\ &x_{n+1}=J^{-1}[\alpha_{n}Jx_{1}+(1-\alpha_{n})Jw_{n}],\leavevmode\nobreak\ \leavevmode\nobreak\ n\geq 1.\end{array}\right.$ (45) Then $\\{x_{n}\\}$ converges strongly to $z=\Pi_{S}(x_{1})$. ###### Remark 4.3. • Our result in Theorems 4.1 and 4.2 complement the results of Bredies [9, 19]. Consequently, our results in Section 3.1 extend the results of Bredies [9, 19] to inclusion problem (1). In particular, we do not assume boundedness of $\\{x_{n}\\}$ (which was imposed on the results of [9, 19]) in our results. Therefore, our result improves on the results of [9, 19]. * • The minimization problem (35) in this section extends the problem studied in [8, 15, 34, 50] and other related papers from Hilbert spaces to Banach spaces. ## 5 Conclusion We study the Tseng-type algorithm for finding a solution to monotone inclusion problem involving a sum of maximal monotone and a Lipschitz continuous monotone mapping in 2-uniformly convex Banach space which is also uniformly smooth. We prove both weak and strong convergence of sequences of iterates to the solution of the inclusion problem under some appropriate conditions. Many results on monotone inclusion problems with single maximal monotone operator can be considered as special cases of the problem studied in this paper. As far as we know, this is the first time an inclusion problem involving sum of maximal monotone and Lipschitz continuous monotone operators will be studied in Banach spaces. Therefore, the results of this paper open up many forthcoming results regarding the inclusion problem studied in this paper. Our next project involves the following. * • The results in this paper exclude $L_{p}$ spaces with $p>2$. Therefore, extension of the results in this paper to a more general reflexive Banach space will be desired. * • How to effectively compute the duality mapping $J$ and the resolvent of maximal monotone mapping $B$ during implementations of our proposed algorithms will be considered further. * • The numerical implementations of problem (1) arising from signal processing, image reconstruction, etc will be studied; * • Other ways of implementation of the step-sizes $\lambda_{n}$ to give faster convergence of the proposed methods in this paper will be given. 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# A sparse grid discrete ordinate discontinuous Galerkin method for the radiative transfer equation Jianguo Huang*****Corresponding author<EMAIL_ADDRESS>Yue Yu <EMAIL_ADDRESS>School of Mathematical Sciences, and MOE-LSC, Shanghai Jiao Tong University Shanghai 200240, China (Version 0.0, June 12, 2019) ###### Abstract The radiative transfer equation is a fundamental equation in transport theory and applications, which is a 5-dimensional PDE in the stationary one-velocity case, leading to great difficulties in numerical simulation. To tackle this bottleneck, we first use the discrete ordinate technique to discretize the scattering term, an integral with respect to the angular variables, resulting in a semi-discrete hyperbolic system. Then, we make the spatial discretization by means of the discontinuous Galerkin (DG) method combined with the sparse grid method. The final linear system is solved by the block Gauss-Seidal iteration method. The computational complexity and error analysis are developed in detail, which show the new method is more efficient than the original discrete ordinate DG method. A series of numerical results are performed to validate the convergence behavior and effectiveness of the proposed method. ###### keywords: Radiative transfer equation, Sparse grid method , Discrete ordinate method , Discontinuous Galerkin method ## 1 Introduction Radiation transport is a physical process of energy transfer in the form of electromagnetic radiation which is affected by absorption, emission and scattering as it passes through the background materials. The radiative transfer equation (RTE) is an important mathematical model used to describe these interactions, finds applications in a wide variety of subjects, including neutron transport, heat transfer, optics, astrophysics, inertial confinement fusion, and high temperature flow systems, see for examples [27, 12, 16, 2, 20, 40]. The RTE can be viewed as a hyperbolic-type integro-differential equation. Even for the stationary monochromatic RTE, it is five-dimensional in the phase space, and hence cannot have a closed-form solution in general. Thus, the numerical solution of the equation is unavoidable and critical in applications. In history, the Monte-Carlo method is a typical approach for numerical simulation (cf. [11] and the references therein). The advantage is its simplicity and dimension-free convergence, and the weakness is its heavy computational cost and slow convergence. Until now, there have developed many other numerical methods as well. For the angular discretization, the typical methods include discrete ordinate methods (or $S_{N}$ methods) and spherical harmonic methods (or $P_{N}$ method); for the spatial discretization, the typical methods include finite difference methods, finite element methods and spectral methods. We refer to [31, 7, 6, 27, 12, 18, 30, 20] for details. Due to the flexibility and easy implementation, the discrete ordinate method is frequently used for angular discretization in practice. If the spatial domain is regular, this semi-discrete method is further discretized by the Chebyshev spectral method in [28, 17, 5] and the meshless discretization in [34, 29, 42]. In recent years, the positivity-preserving schemes are also developed very technically in [45, 15, 47]. For numerical solvers such as source iteration and multigrid algorithms, one can refer to [13, 1, 38, 36]. On the other hand, except the Monte-Carlo method, all the methods mentioned above solve the problems with reduced dimensions. In this paper, we intend to attack the problem in its original form with 3-spatial variables and 2-angular variables. In this case, most usual methods suffer from the so-called “the curse of dimension”, which indicates the low rate of convergence in terms of number of degrees of freedom due to the high dimensionality of the underlying problem. To the best of our knowledge, the sparse grid method, also called the sparse tensor product method, is an effective way to overcome the bottleneck. Historically, the idea of sparse grids can be traced back to Smoljak’s construction of multivariate quadrature formulas using combinations of tensor products of suitable one-dimensional formulas (cf. [39, 19]). More recently, the systematic and thorough studies on the method can be found in [46, 22, 23, 19]. In addition, several sparse grid methods are devised in [44, 21] for solving the RTE through conforming spatial discretization. However, according to the computational experience, it is preferable to use the discontinuous Galerkin (DG) method for spatial discretization for hyperbolic problems (cf. [9, 14, 10]), in order to capture non-smooth physical solutions. In [43], the sparse grid technique combined with the DG method has been developed for elliptic equations. This method is also applied to transport equations in [24, 25], but the scattering effect is not considered. The adaptive analogues of their methods are also given in [25, 41]. In this paper, we are intended to propose and analyze a sparse grid DG method to solve the RTE, following the ideas in [27] and [24]. Unlike the studies in [44, 21], the DG method will be used to carry out the spatial discretization. And different from [24], we will discuss in detail the efficient solution of the 5-dimensional RTE with scattering effect. Concretely speaking, the discrete ordinate technique is first applied to discretize the scattering term, an integral with respect to the angular variables, by simply picking several directions spanning the solid angle, resulting in a semi-discrete coupled hyperbolic system. In view of the hyperbolic nature of the semi- discrete system, the DG method is further employed for spatial discretization, yielding a fully discrete method. To overcome the curse of dimension, the sparse DG space is constructed by using the techniques in wavelet analysis to replace the original piecewise polynomial approximation space. We achieve the complexity analysis and error analysis of the method using some arguments in [27] and [24], which show the new approach can greatly reduce the spatial degrees of freedom while keeping almost the same accuracy up to multiplication of an $\log$ factor. For the resulting linear system, considering its block structure, we solve it using the block Gauss-Seidal iteration method. A series of numerical examples are reported to validate the accuracy and performance of the proposed method. Furthermore, we also extend the method to solve the RTE efficiently for some non-tensor product spatial domains in two dimensions. We end this section by introducing some notations and symbols frequently used in this paper. For a bounded Lipschitz domain $D$, the symbol $(\cdot,\cdot)_{D}$ denotes the $L^{2}$-inner product on $D$, $\\|\cdot\\|_{0,D}$ denotes the $L^{2}$-norm, and $\|\cdot\|_{s,D}$ is the $H^{s}(D)$-seminorm. For all integer $k\geq 0$, $\mathbb{P}_{k}(D)$ is the set of polynomials of degree $\leq k$ on $D$. The jumps and averages for scalar and vector-valued functions ($v,\boldsymbol{\tau}$, respectively) on an edge $e$ common to two elements $K_{1},K_{2}$ are defined by $[\\![v]\\!]=v_{1}\boldsymbol{n}_{1}+v_{2}\boldsymbol{n}_{2},\quad\\{\\!\\!\\{v\\}\\!\\!\\}=\frac{v_{1}+v_{2}}{2},$ $[\\![\boldsymbol{\tau}]\\!]=\boldsymbol{\tau}_{1}\cdot\boldsymbol{n}_{1}+\boldsymbol{\tau}_{2}\cdot\boldsymbol{n}_{2},\quad\\{\\!\\!\\{\boldsymbol{\tau}\\}\\!\\!\\}=\frac{\boldsymbol{\tau}_{1}+\boldsymbol{\tau}_{2}}{2},$ where $\boldsymbol{n}_{1},\boldsymbol{n}_{2}$ are the unit outward normals to $K_{1},K_{2}$, respectively. On a boundary edge or face, $[\\![v]\\!]=v\boldsymbol{n}$ and $\\{\\!\\!\\{\boldsymbol{\tau}\\}\\!\\!\\}=\boldsymbol{\tau}$. Moreover, for any two quantities $a$ and $b$, “$a\lesssim b$” indicates “$a\leq Cb$” with the hidden constant $C$ independent of the mesh size $h_{K}$, and “$a\eqsim b$” abbreviates “$a\lesssim b\lesssim a$”. ## 2 Radiative transfer equation The steady-state monoenergetic version of the radiative transfer equation is expressed as (cf. [27, 6]) $\boldsymbol{\omega}\cdot\nabla u(\boldsymbol{x},\boldsymbol{\omega})+\sigma_{t}(\boldsymbol{x})u(\boldsymbol{x},\boldsymbol{\omega})=\sigma_{s}(\boldsymbol{x})(Su)(\boldsymbol{x},\boldsymbol{\omega})+f(\boldsymbol{x},\boldsymbol{\omega}),\quad\boldsymbol{x}\in D,\boldsymbol{\omega}\in S^{2}.$ (2.1) Here, $D$ is a domain in $\mathbb{R}^{3}$ and $S^{2}$ denotes the unit sphere in $\mathbb{R}^{3}$, $u(\boldsymbol{x},\boldsymbol{\omega})$ is a function of three space variables $\boldsymbol{x}$ and two angular variables $\boldsymbol{\omega}$, $\sigma_{t}=\sigma_{a}+\sigma_{s}$ with $\sigma_{a}$ being the macroscopic absorption cross section, and $\sigma_{s}$ the macroscopic scattering cross section, and $f$ is a source function in $D$. We impose an inflow boundary value condition $u(\boldsymbol{x},\boldsymbol{\omega})=\alpha(\boldsymbol{x},\boldsymbol{\omega}),\quad(\boldsymbol{x},\boldsymbol{\omega})\in{\Gamma_{-}},$ (2.2) where $\Gamma_{-}$ is defined by $\Gamma_{-}=\\{(\boldsymbol{x},\boldsymbol{\omega}):\boldsymbol{x}\in\partial D,~{}\boldsymbol{\omega}\in S^{2},\quad\boldsymbol{n}(\boldsymbol{x})\cdot\boldsymbol{\omega}<0\\}.$ (2.3) The symbol $S$ on the right-hand side of (2.1) is an integral operator defined by $(Su)(\boldsymbol{x},\boldsymbol{\omega})=\int_{S^{2}}g(\boldsymbol{x},\boldsymbol{\omega}\cdot\hat{\boldsymbol{\omega}})u(\boldsymbol{x},\hat{\boldsymbol{\omega}}){\rm d}\sigma(\hat{\boldsymbol{\omega}})$ (2.4) with $g$ being a nonnegative normalized phase function $\int_{S^{2}}g(\boldsymbol{x},\boldsymbol{\omega}\cdot\hat{\boldsymbol{\omega}}){\rm d}\sigma(\hat{\boldsymbol{\omega}})=1,\quad\boldsymbol{x}\in D,\boldsymbol{\omega}\in S^{2}.$ In most applications, the function $g$ is assumed to be independent of $\boldsymbol{x}$. One well-known example considered in this paper is the Henyey-Greenstein phase function $g(t)=\frac{1-\eta^{2}}{4\pi(1+\eta^{2}-2\eta t)^{3/2}},~{}~{}t\in[-1,1],$ (2.5) where the parameter $\eta\in(-1,1)$ is the anisotropy factor for the scattering medium which measures the strength of forward peakedness of the phase function. Note that $\eta=0$ for isotropic scattering, $\eta>0$ for forward scattering, and $\eta<0$ for backward scattering. We assume that 1. $\sigma_{t},\sigma_{s}\in L^{\infty}(D)$, $\sigma_{s}\geq 0$ a.e. in $D$, $\sigma_{a}=\sigma_{t}-\sigma_{s}\geq c_{0}$ in $D$ for a constant $c_{0}>0$. * 2. $f(\boldsymbol{x},\boldsymbol{\omega})\in L^{2}(D\times S^{2})$ and is a continuous function with respect to $\boldsymbol{\omega}\in S^{2}$. Under these assumptions, the problem (2.1)-(2.2) has a unique solution $u\in H_{2}^{1}(D\times S^{2})$ (cf. [27]), where $H_{2}^{1}(D\times S^{2}):=\\{v\in L^{2}(D\times S^{2}):\boldsymbol{\omega}\cdot\nabla v\in L^{2}(D\times S^{2})\\}.$ ## 3 The sparse grid discrete-ordinate DG method for the RTE In this section, we first recall the construction of sparse discontinuous finite element spaces; One can refer to [43, 3, 4] and the references therein for details. Then, we will present in detail the sparse grid discrete-ordinate DG method for the RTE. ### 3.1 Construction of sparse DG spaces Let $\Omega=[0,1]$ and partition it into $2^{n}$ cells with uniform cell size $h=2^{-n}$. The resulting $n$-th level grid is denoted by ${\Omega_{n}}$ and the $j$-th cell is given by $I_{n}^{j}=(2^{-n}j,2^{-n}(j+1)],\quad j=0,1,\cdots,2^{n}-1.$ We define $V_{n}^{k}=\\{v:v\|_{I_{n}^{j}}\in\mathbb{P}_{k}(I_{n}^{j}),\quad j=0,1,\cdots,2^{n}-1\\}$ to be the piecewise polynomial space on ${\Omega_{n}}$. One can check that there exists the nested structure for different values of $n$: $V_{0}^{k}\subset V_{1}^{k}\subset\cdots\subset V_{n}^{k}\subset\cdots$. Denote $W_{n}^{k}$ to be the orthogonal complement of $V_{n-1}^{k}$ in $V_{n}^{k}$ with respect to the $L^{2}(\Omega)$ inner product, i.e., $V_{n-1}^{k}\oplus W_{n}^{k}=V_{n}^{k},\quad W_{n}^{k}\bot V_{n-1}^{k},\quad n\geq 1,$ where for simplicity set $W_{0}^{k}=V_{0}^{k}$. We then obtain an orthogonal decomposition of the DG space $V_{N}^{k}=\mathop{\bigoplus}\limits_{0\leq n\leq N}W_{n}^{k}.$ We proceed to review the construction in multi-dimensions. For $\Omega=[0,1]^{d}$, let $h_{m}=2^{-n_{m}}$ be the step size along $x_{m}$-direction. For simplicity, we use the notations of multi-indices in the following. Let $\boldsymbol{n}=(n_{1},n_{2},\cdots,n_{d})$. Then the cell size can be denoted by $h_{\boldsymbol{n}}=(2^{-n_{1}},2^{-n_{2}},\cdots,2^{-n_{d}})=2^{-\boldsymbol{n}}$ and the associated grid is written by $\Omega_{\boldsymbol{n}}$ whose $\boldsymbol{j}$-th cell is given by $I_{\boldsymbol{n}}^{\boldsymbol{j}}=I_{n_{1}}^{j_{1}}\times I_{n_{2}}^{j_{2}}\times\cdots\times I_{n_{d}}^{j_{d}},\quad\boldsymbol{j}=(j_{1},j_{2},\cdots,j_{d}),$ where $I_{n_{m}}^{j_{m}}=(2^{-n_{m}}j_{m},2^{-n_{m}}(j_{m}+1)],\quad j_{m}=0,1,\cdots,2^{n_{m}}-1$ is the element along $x_{m}$-axis. With multi-indices notation we have $\boldsymbol{0}\leq\boldsymbol{j}\leq 2^{\boldsymbol{n}}-\boldsymbol{1}$. Introduce a tensor-product piecewise polynomial space as $\boldsymbol{V}_{\boldsymbol{n}}^{k}=\\{\boldsymbol{v}:\boldsymbol{v}(\boldsymbol{x})\in\mathbb{Q}_{k}(I_{\boldsymbol{n}}^{\boldsymbol{j}}),\quad\boldsymbol{0}\leq\boldsymbol{j}\leq 2^{\boldsymbol{n}}-\boldsymbol{1}\\},$ where $\mathbb{Q}_{k}(I_{\boldsymbol{n}}^{\boldsymbol{j}})$ consists of polynomials of degree up to $k$ in each dimension on cell $I_{\boldsymbol{n}}^{\boldsymbol{j}}$. If we use an equal refinement of size $h:=h_{N}=2^{-N}$ in each coordinate direction, the grid and space will be denoted by $\Omega_{N}$ and $\boldsymbol{V}_{N}^{k}$, respectively. With the usual convention, we also use $\mathcal{T}_{h}$ and $V_{h}^{k}$ instead. It is obvious that $\boldsymbol{V}_{\boldsymbol{n}}^{k}=V_{n_{1}}^{k}\times V_{n_{2}}^{k}\times\cdots\times V_{n_{d}}^{k}.$ We similarly define the tensor-product multiwavelet space as $\boldsymbol{W}_{\boldsymbol{n}}^{k}=W_{n_{1}}^{k}\times W_{n_{2}}^{k}\times\cdots\times W_{n_{d}}^{k}.$ Observing the fact that $V_{n_{m}}^{k}=\mathop{\bigoplus}\limits_{0\leq j_{m}\leq n_{m}}W_{j_{m}}^{k},$ we have the following expansion $\boldsymbol{V}_{\boldsymbol{n}}^{k}=\mathop{\bigoplus}\limits_{\boldsymbol{0}\leq\boldsymbol{j}\leq\boldsymbol{n}}\boldsymbol{W}_{\boldsymbol{j}}^{k},\quad~{}\boldsymbol{V}_{N}^{k}=\mathop{\bigoplus}\limits_{\|\boldsymbol{j}\|_{\infty}\leq N}\boldsymbol{W}_{\boldsymbol{j}}^{k}.$ The sparse finite element approximation space on $\Omega_{N}$ is defined by the following truncated space $\widehat{\boldsymbol{V}}_{N}^{k}:=\mathop{\bigoplus}\limits_{\|\boldsymbol{n}\|_{1}\leq N}\boldsymbol{W}_{\boldsymbol{n}}^{k},\quad\|\boldsymbol{n}\|_{1}=n_{1}+n_{2}+\cdots+n_{d}.$ The number of degrees of freedom of sparse DG space is $\mathcal{O}(h^{-1}\|\log_{2}h\|^{d-1})$ with $h=2^{-N}$, which is significantly less than that of DG space with exponential dependence on $d$. ### 3.2 The sparse grid discrete-ordinate DG method For any continuous function $F(\boldsymbol{\omega})$ defined on the unit sphere $S^{2}$, we write the numerical quadrature to be used in the form $\int_{S^{2}}F(\boldsymbol{\omega}){\rm d}\sigma(\boldsymbol{\omega})\approx\sum\limits_{l=1}^{L}{w_{l}F(\boldsymbol{\omega}^{l})},\quad\boldsymbol{\omega}^{l}\in S^{2},~{}1\leq l\leq L.$ (3.1) The integral operator $S$ is then approximated by $(Su)(\boldsymbol{x},\boldsymbol{\omega})\approx(S_{d}u)(\boldsymbol{x},\boldsymbol{\omega}):=\sum\limits_{l=1}^{L}w_{l}g(\boldsymbol{x},\boldsymbol{\omega}\cdot\boldsymbol{\omega}^{l})u(\boldsymbol{x},\boldsymbol{\omega}^{l}).$ (3.2) Regarding the accuracy of the quadrature (3.1), we will write $n$ for the algebraic precision, i.e., the quadrature integrates exactly all spherical polynomials of total degree no more than $n$ and does not integrate exactly some spherical polynomial of total degree $n+1$. Then we have the following estimate (cf. [27]) $\Big{\|}\int_{S^{2}}F(\boldsymbol{\omega}){\rm d}\sigma(\boldsymbol{\omega})-\sum\limits_{l=1}^{L}w_{l}F(\boldsymbol{\omega}^{l})\Big{\|}\leq c_{s}n^{-s}\\|F\\|_{s,S^{2}},\quad F\in H^{s}({S^{2}}),\quad s>1,$ (3.3) where $c_{s}$ is a universal constant depending only on $s$. Associated with the numerical quadrature, we further define $m(\boldsymbol{x})=\mathop{\max}\limits_{1\leq i\leq L}\sum\limits_{l=1}^{L}w_{l}g(\boldsymbol{x},\boldsymbol{\omega}^{l}\cdot\boldsymbol{\omega}^{i})$ (3.4) and make the following assumption (cf. [27]): $\sigma_{t}-m\sigma_{s}\geq c_{0}^{\prime}~{}\mbox{in $D$ for some constant $c_{0}^{\prime}>0$}.$ (3.5) Using the quadrature (3.2), we can discretize (2.1) in angular direction to get $\boldsymbol{\omega}^{l}\cdot\nabla u^{l}+\sigma_{t}u^{l}=\sigma_{s}(\boldsymbol{x})\sum\limits_{i=1}^{L}w_{i}g(\boldsymbol{x},\boldsymbol{\omega}^{l}\cdot\boldsymbol{\omega}^{i})u^{i}+f^{l},\quad 1\leq l\leq L$ (3.6) with boundary value condition $u^{l}(\boldsymbol{x})=\alpha^{l}(\boldsymbol{x}),\quad(\boldsymbol{x},\boldsymbol{\omega}^{l})\in\Gamma_{-},~{}~{}1\leq l\leq L,$ (3.7) where $u^{l}=u^{l}(\boldsymbol{x})$ is the approximation to $u(\boldsymbol{x},\boldsymbol{\omega}^{l})$. The system (3.6) is a first-order hyperbolic problem in space, which will be further discretized by DG method. Let $\\{\mathcal{T}_{h}\\}_{h>0}$ be a regular family of triangulations of $D$. Assume that $\mathcal{E}_{h}$ consists of the set of all edges ($d=2$) or faces ($d=3$) in $\mathcal{T}_{h}$ and $\mathcal{E}_{h}^{0}$ the set of all interior edges or faces. By a direct manipulation, we obtain the following identity (cf. [10]): ###### Lemma 3.1. For $(\varphi,\boldsymbol{\tau})\in H^{s}(\mathcal{T}_{h})\times[H^{s}(\mathcal{T}_{h})]^{d}$, $s>1/2$, there holds $\sum\limits_{K\in\mathcal{T}_{h}}\int_{\partial K}\varphi\boldsymbol{\tau}\cdot\boldsymbol{n}{\rm d}s=\sum\limits_{e\in\mathcal{E}_{h}}\int_{e}\\{\\!\\!\\{\boldsymbol{\tau}\\}\\!\\!\\}\cdot[\\![\varphi]\\!]{\rm d}s+\sum\limits_{e\in\mathcal{E}_{h}^{0}}\int_{e}\\{\\!\\!\\{\varphi\\}\\!\\!\\}\cdot[\\![\boldsymbol{\tau}]\\!]{\rm d}s.$ (3.8) Further, if $u\in H^{s}(\omega_{e})$ and $s>1/2$, then we have the following weak continuity $\int_{e}[\\![u]\\!]v{\rm d}s=0,\quad v\in L^{2}(e),\quad e\in\mathcal{E}_{h}^{0},$ where $\omega_{e}$ is the set of elements sharing $e$ as an edge ($d=2$) or faces ($d=3$). We define a discontinuous finite element space by $V_{h}=\Big{\\{}v\in L^{2}(D):v\|_{K}\in\mathbb{P}_{k}(K),~{}~{}K\in\mathcal{T}_{h}\Big{\\}},$ (3.9) where $\mathbb{P}_{k}(K)$ denotes the set of all polynomials on $K$ with degree $\leq k$. Multiplying (3.6) by any $v_{h}\in V_{h}$, we obtain from the integration by parts that $\displaystyle\sum\limits_{K\in\mathcal{T}_{h}}\Big{[}\int_{K}(-u^{l}(\boldsymbol{\omega}^{l}\cdot\nabla v_{h})+\sigma_{t}u^{l}v_{h}){\rm d}x+\int_{\partial K}(\boldsymbol{\omega}^{l}\cdot\boldsymbol{n})u^{l}v_{h}{\rm d}s\Big{]}$ $\displaystyle\quad\quad=\int_{D}\sigma_{s}\sum\limits_{i=1}^{L}w_{i}g(\cdot,\boldsymbol{\omega}^{l}\cdot\boldsymbol{\omega}^{i})u^{i}v_{h}{\rm d}x+\int_{D}f^{l}v_{h}{\rm d}x,~{}~{}1\leq l\leq L.$ Taking $\boldsymbol{\tau}=\boldsymbol{\omega}^{l}u^{l}$ and $\varphi=v_{h}$ in (3.8), we immediately obtain the following system $a_{h}^{(l)}(u^{l},v_{h})+b_{h}^{(l)}(u^{l},v_{h})=(f^{l},v_{h})+\langle\alpha^{l},v_{h}\rangle^{(l)},\quad v_{h}\in V_{h},$ where $\displaystyle a_{h}^{(l)}(u^{l},v_{h})$ $\displaystyle=\sum\limits_{K\in\mathcal{T}_{h}}\int_{K}(-u^{l}(\boldsymbol{\omega}^{l}\cdot\nabla v_{h})+\sigma_{t}u^{l}v_{h}){\rm d}x$ $\displaystyle\quad-\int_{D}\sigma_{s}\sum\limits_{i=1}^{L}w_{i}g(\cdot,\boldsymbol{\omega}^{l}\cdot\boldsymbol{\omega}^{i})u^{i}v_{h}{\rm d}x,$ (3.10) $b_{h}^{(l)}(u^{l},v_{h})=\sum\limits_{e\not\subset\Gamma_{-}}\int_{e}\\{\\!\\!\\{\boldsymbol{\omega}^{l}u^{l}\\}\\!\\!\\}\cdot[\\![v_{h}]\\!]{\rm d}s,$ (3.11) $(f^{l},v_{h})=\int_{D}f^{l}v_{h}{\rm d}x,\quad\langle\alpha^{l},v_{h}\rangle^{(l)}=-\sum\limits_{e\subset\Gamma_{-}}\int_{e}\boldsymbol{\omega}^{l}\cdot\boldsymbol{n}\alpha^{l}v_{h}{\rm d}s.$ (3.12) Define $\boldsymbol{V}_{h}=(V_{h})^{L}$ and write a generic element as $\boldsymbol{v}_{h}:=\\{v_{h}^{l}\\}_{l=1}^{L}$. The global formulation can be expressed as $\sum\limits_{l=1}^{L}w_{l}(a_{h}^{(l)}(u^{l},v_{h}^{l})+b_{h}^{(l)}(u^{l},v_{h}^{l}))=\sum\limits_{l=1}^{L}w_{l}((f^{l},v_{h}^{l})+\langle\alpha^{l},v_{h}^{l}\rangle^{(l)}).$ Then the discrete-ordinate DG method is: Find $\boldsymbol{u}_{h}:=\\{u_{h}^{l}\\}\in\boldsymbol{V}_{h}$ such that $a_{h}(\boldsymbol{u}_{h},\boldsymbol{v}_{h})=F(\boldsymbol{v}_{h}),\quad\boldsymbol{v}_{h}\in\boldsymbol{V}_{h},$ (3.13) where $a_{h}(\boldsymbol{u}_{h},\boldsymbol{v}_{h})=\sum\limits_{l=1}^{L}w_{l}(a_{h}^{(l)}(u_{h}^{l},v_{h}^{l})+b_{h}^{(l)}(u_{h}^{l},v_{h}^{l})),$ $F(\boldsymbol{v}_{h})=\sum\limits_{l=1}^{L}w_{l}((f^{l},v_{h}^{l})+\langle\alpha^{l},v_{h}^{l}\rangle^{(l)}).$ It is preferable to add some stabilization terms in the DG scheme to penalize the jump of the solution across interior edges or faces of the triangulation. One approach introduced in [10] is to replace the average $\\{\\!\\!\\{\boldsymbol{\omega}^{l}u^{l}\\}\\!\\!\\}$ in (3.11) by $\\{\\!\\!\\{\boldsymbol{\omega}^{l}u_{h}^{l}\\}\\!\\!\\}+c_{e}^{l}[\\![u_{h}^{l}]\\!]$, where $c_{e}^{l}$ is a nonnegative function over $e$ satisfying $c_{e}^{l}=\theta_{0}\|\boldsymbol{\omega}^{l}\cdot\boldsymbol{n}\|$ with $\theta_{0}$ a constant independent of $e$ and $h$. The stabilized discrete- ordinate DG method is to find $\boldsymbol{u}_{h}:=\\{u_{h}^{l}\\}\in\boldsymbol{V}_{h}$ such that $a_{h}^{s}(\boldsymbol{u}_{h},\boldsymbol{v}_{h})=F(\boldsymbol{v}_{h}),\quad\boldsymbol{v}_{h}\in\boldsymbol{V}_{h},$ (3.14) where $a_{h}^{s}(\boldsymbol{u}_{h},\boldsymbol{v}_{h})=\sum\limits_{l=1}^{L}w_{l}(a_{h}^{(l)}(u_{h}^{l},v_{h}^{l})+b_{hs}^{(l)}(u_{h}^{l},v_{h}^{l}))$ (3.15) and $\displaystyle b_{hs}^{(l)}(u_{h}^{l},v_{h}^{l})$ $\displaystyle=b_{h}^{(l)}(u_{h}^{l},v_{h}^{l})+\sum\limits_{e\in\mathcal{E}_{h}^{0}}\int_{e}c_{e}^{l}[\\![u_{h}^{l}]\\!]\cdot[\\![v_{h}^{l}]\\!]{\rm d}s$ $\displaystyle=\sum\limits_{e\not\subset\Gamma_{-}}\int_{e}\\{\\!\\!\\{\boldsymbol{\omega}^{l}u_{h}^{l}\\}\\!\\!\\}\cdot[\\![v_{h}^{l}]\\!]{\rm d}s+\sum\limits_{e\in\mathcal{E}_{h}^{0}}\int_{e}c_{e}^{l}[\\![u_{h}^{l}]\\!]\cdot[\\![v_{h}^{l}]\\!]{\rm d}s.$ (3.16) ###### Remark 3.1. The sparse grid discrete-ordinate DG method is obtained by replacing the DG space $V_{h}$ in (3.9) with the sparse DG space $\widehat{V}_{h}^{k}:=\widehat{\boldsymbol{V}}_{N}^{k}\subset V_{h}$. ## 4 Error analysis ### 4.1 Error estimate of the sparse projection operator We define the broken $H^{s}$ Sobolev norm on $\Omega_{N}$ by $\\|v\\|_{H^{s}(\Omega_{N})}^{2}=\sum\limits_{\boldsymbol{0}\leq\boldsymbol{j}\leq 2^{\boldsymbol{N}-\boldsymbol{1}}-\boldsymbol{1}}\\|v\\|_{H^{s}(I_{\boldsymbol{N}}^{\boldsymbol{j}})}^{2}.$ For any nonnegative integer $m$ and the multi-index $\alpha=\\{i_{1},i_{2},\cdots,i_{r}\\}\subset\\{1,2,\cdots,d\\}$, define $\|v\|_{H^{m,\alpha}(\Omega)}=\Big{\\|}\Big{(}\frac{\partial^{m}}{\partial x_{i_{1}}^{m}}\cdots\frac{\partial^{m}}{\partial x_{i_{r}}^{m}}\Big{)}v\Big{\\|}_{L^{2}(\Omega)}$ and $\|v\|_{\mathcal{H}^{q+1}(\Omega)}=\mathop{\max}\limits_{1\leq r\leq d}\Big{(}\mathop{\max}\limits_{\alpha\in\\{1,2,\cdots,d\\},\|\alpha\|=r}\|v\|_{H^{q+1,\alpha}(\Omega)}\Big{)},$ which is the norm for the mixed derivative of $v$ of at most degree $q+1$ in each direction. In the following, we denote by $\boldsymbol{P}$ the sparse projection operator to be the $L^{2}$ projection onto $\widehat{\boldsymbol{V}}_{N}^{k}$. ###### Lemma 4.1. Let $\boldsymbol{P}$ be the sparse projector, $k\geq 1$, $N\geq 1$ and $d\geq 2$. Then for $v\in\mathcal{H}^{p+1}(\Omega)$ there hold $\|\boldsymbol{P}v-v\|_{L^{2}(\Omega_{N})}\lesssim\|\log_{2}h\|^{d}h^{k+1}\|v\|_{\mathcal{H}^{k+1}(\Omega)},$ $\|\boldsymbol{P}v-v\|_{H^{1}(\Omega_{N})}\lesssim h^{k}\|v\|_{\mathcal{H}^{k+1}(\Omega)},$ and $\Big{(}\sum\limits_{K\in\mathcal{T}_{h}}\\|\boldsymbol{P}v-v\\|_{0,\partial K}^{2}\Big{)}^{1/2}\lesssim\|\log_{2}h\|^{d}h^{k+1/2}\|v\|_{\mathcal{H}^{k+1}(\Omega)}.$ ###### Proof. It follows from [35, 43, 24] that for any $v\in\mathcal{H}^{p+1}(\Omega)$ and $1\leq q\leq\min\\{p,k\\}$, there holds $\|\boldsymbol{P}v-v\|_{H^{s}(\Omega_{N})}\lesssim\begin{cases}N^{d}2^{-N(q+1)}\|v\|_{\mathcal{H}^{q+1}(\Omega)},\quad&s=0,\\\ 2^{-Nq}\|v\|_{\mathcal{H}^{q+1}(\Omega)},\quad&s=1.\end{cases}$ Noting that $h=h_{N}=2^{-N}$, we have $\|\boldsymbol{P}v-v\|_{L^{2}(\Omega_{N})}\lesssim\|\log_{2}h\|^{d}h^{k+1}\|v\|_{\mathcal{H}^{k+1}(\Omega)}$ and $\|\boldsymbol{P}v-v\|_{H^{1}(\Omega_{N})}\lesssim h^{k}\|v\|_{\mathcal{H}^{k+1}(\Omega)}.$ Recalling the trace inequality (cf. [8]) $\\|\phi\\|_{0,\partial K}^{2}\lesssim h_{K}^{-1}\\|\phi\\|_{0,K}^{2}+h_{K}\|\phi\|_{1,K}^{2},$ (4.1) where $K\in\mathcal{T}_{h}$ with diameter $h_{K}$, we then have $\displaystyle\Big{(}\sum\limits_{K\in\mathcal{T}_{h}}\\|\boldsymbol{P}v-v\\|_{0,\partial K}^{2}\Big{)}^{1/2}$ $\displaystyle\lesssim\Big{(}\sum\limits_{K\in\mathcal{T}_{h}}h_{K}^{-1}\\|\boldsymbol{P}v-v\\|_{0,K}^{2}+h_{K}\|\boldsymbol{P}v-v\|_{1,K}^{2}\Big{)}^{1/2}$ $\displaystyle\lesssim(\|\log_{2}h\|^{2d}+1)^{1/2}h^{k+1/2}\|v\|_{\mathcal{H}^{k+1}(\Omega)}$ $\displaystyle\lesssim\|\log_{2}h\|^{d}h^{k+1/2}\|v\|_{\mathcal{H}^{k+1}(\Omega)}.$ This completes the proof. ∎ ### 4.2 Error analysis of the sparse grid discrete-ordinate DG method Using the similar arguments in [10, 27], one can deduce the following stability result whose proof is omitted for simplicity. ###### Lemma 4.2. Let $\|\\!\|\\!\|\boldsymbol{v}_{h}\|\\!\|\\!\|=\Big{[}\sum\limits_{l=1}^{L}w_{l}\Big{(}\sum\limits_{e\in\mathcal{E}_{h}}\int_{e}c_{e}^{l}\|[\\![v_{h}^{l}]\\!]\|^{2}{\rm d}s+\int_{D}(v_{h}^{l})^{2}{\rm d}x\Big{)}\Big{]}^{1/2}.$ Under the assumption (3.5), there holds $a_{h}^{s}(\boldsymbol{v}_{h},\boldsymbol{v}_{h})\gtrsim\|\\!\|\\!\|\boldsymbol{v}_{h}\|\\!\|\\!\|^{2},\quad\boldsymbol{v}_{h}\in\boldsymbol{V}_{h}.$ We denote the solutions of the original problem (2.1), the semi-discrete problem (3.6) and the stabilized discrete-ordinate DG method (3.14) by $\\{u(\boldsymbol{x},\boldsymbol{\omega}^{l})\\}$, $\boldsymbol{u}=\\{u^{l}(\boldsymbol{x})\\}$ and $\boldsymbol{u}_{h}=\\{u_{h}^{l}\\}$, respectively. The error is decomposed as $\\{u(\boldsymbol{x},\boldsymbol{\omega}^{l})\\}-\boldsymbol{u}_{h}=(\\{u(\boldsymbol{x},\boldsymbol{\omega}^{l})\\}-\\{u^{l}(\boldsymbol{x})\\})+(\\{u^{l}(\boldsymbol{x})\\}-\\{u_{h}^{l}(\boldsymbol{x})\\}),$ (4.2) and measured by $\\|u-u_{h}\\|_{h}=\Big{(}\sum\limits_{l=1}^{L}w_{l}\\|u(\cdot,\boldsymbol{\omega}^{l})-u_{h}^{l}\\|_{0,D}^{2}\Big{)}^{1/2}.$ (4.3) ###### Theorem 4.1. Let $n$ be the degree of precision of the numerical quadrature and $d\geq 2$. Then under the assumption (3.5), for the sparse grid discrete-ordinate DG method, we have $\displaystyle\\|u-u_{h}\\|_{h}$ $\displaystyle\lesssim c(\theta_{0})\|\log_{2}h\|^{d}h^{k+1/2}\Big{(}\sum\limits_{l=1}^{L}w_{l}\|u^{l}\|_{\mathcal{H}^{k+1}(\Omega)}^{2}\Big{)}^{1/2}$ $\displaystyle\quad+c(r^{\prime},g)n^{-r^{\prime}}\Big{(}\int_{D}\\|u(\boldsymbol{x},\cdot)\\|_{r^{\prime},S^{2}}^{2}{\rm d}x\Big{)}^{1/2},$ where, $h=2^{-N}$, $u^{l}$ is the solution of (3.6), $c(\theta_{0})=\theta_{0}^{-1/2}+\theta_{0}^{1/2}$ and $c(r^{\prime},g)$ is defined in (4.4). ###### Proof. For the first part in (4.2), let $\varepsilon^{l}(\boldsymbol{x}):=u(\boldsymbol{x},\boldsymbol{\omega}^{l})-u^{l}(\boldsymbol{x}),\quad 1\leq l\leq L.$ In view of the equation (4.19) in [27], one has $\sum\limits_{l=1}^{L}w_{l}\int_{D}(\varepsilon^{l})^{2}{\rm d}x\lesssim c(r^{\prime},g)^{2}n^{-2r^{\prime}}\int_{D}\\|u(\boldsymbol{x},\cdot)\\|_{r^{\prime},S^{2}}^{2}{\rm d}x,$ where $c(r^{\prime},g):=c(r^{\prime})\mathop{\sup}\limits_{\boldsymbol{x}\in D,\boldsymbol{\omega}\in S^{2}}\\|g(\boldsymbol{x},\boldsymbol{\omega}\cdot)\\|_{r^{\prime},S^{2}}$ (4.4) and $c(r^{\prime})$ is a positive constant depending only on $r^{\prime}$. For the second part, let $u^{l}-u_{h}^{l}=(u^{l}-P_{h}^{k}u^{l})-(u_{h}^{l}-P_{h}^{k}u^{l})=:\eta^{l}+\delta^{l},$ where $P_{h}^{k}$ is the $L^{2}$-projection onto the sparse DG space $\widehat{V}_{h}^{k}$ (cf. Remark 3.1). Similarly, we denote $\boldsymbol{\eta}=\\{\eta^{l}\\}$ and $\boldsymbol{\delta}=\\{\delta^{l}\\}$. When $\boldsymbol{u}_{h}$ is replaced by the exact solution $\boldsymbol{u}$ of the semi-discrete problem, the weak continuity in Lemma 3.1 yields $\int_{e}c_{e}^{l}[\\![u^{l}]\\!][\\![v_{h}^{l}]\\!]{\rm d}s=0,\quad e\in\mathcal{E}_{h}^{0}.$ From (3.2) we have $b_{hs}^{(l)}(u^{l},v_{h}^{l})=b_{h}^{(l)}(u^{l},v_{h}^{l})$, and hence $a_{h}^{s}(\boldsymbol{u},\boldsymbol{v}_{h})=a_{h}(\boldsymbol{u},\boldsymbol{v}_{h})$, which yields the following Galerkin orthogonality $a_{h}^{s}(\boldsymbol{u}-\boldsymbol{u}_{h},\boldsymbol{v}_{h})=a_{h}(\boldsymbol{u}-\boldsymbol{u}_{h},\boldsymbol{v}_{h})=0,\quad\boldsymbol{v}_{h}\in\boldsymbol{V}_{h}.$ (4.5) According to the stability estimate in Lemma 4.2, we have $\|\\!\|\\!\|\boldsymbol{\delta}\|\\!\|\\!\|^{2}\lesssim a_{h}^{s}(\boldsymbol{\delta},\boldsymbol{\delta})=a_{h}^{s}(\boldsymbol{u}_{h}-P_{h}^{k}\boldsymbol{u},\boldsymbol{\delta})=a_{h}^{s}(\boldsymbol{u}-P_{h}^{k}\boldsymbol{u},\boldsymbol{\delta})=a_{h}^{s}(\boldsymbol{\eta},\boldsymbol{\delta}).$ (4.6) We now estimate the right-hand side of (4.6). Let $a_{h}^{(l)}(u^{l},v_{h})={\rm I}_{1}^{l}(u^{l},v_{h})-{\rm I}_{2}^{l}(u^{l},v_{h}),$ where $\displaystyle{\rm I}_{1}^{l}(u^{l},v_{h})$ $\displaystyle=\sum\limits_{K\in\mathcal{T}_{h}}\int_{K}\left(-u^{l}(\boldsymbol{\omega}^{l}\cdot\nabla v_{h})+\sigma_{t}u^{l}v_{h}\right){\rm d}x,$ $\displaystyle{\rm I}_{2}^{l}(u^{l},v_{h})$ $\displaystyle=\int_{D}\sigma_{s}\sum\limits_{i=1}^{L}w_{i}g(\cdot,\boldsymbol{\omega}^{l}\cdot\boldsymbol{\omega}^{i})u^{i}v_{h}{\rm d}x.$ For the first term ${\rm I}_{1}^{l}$, noting that $\boldsymbol{\omega}^{l}\cdot\nabla\delta^{l}\|_{K}\in\mathbb{Q}_{k}(K)$, by the definition of the projector $P_{h}^{k}$, $\int_{K}\eta^{l}(\boldsymbol{\omega}^{l}\cdot\nabla\delta^{l}){\rm d}x=\int_{K}(u^{l}-P_{h}^{k}u^{l})(\boldsymbol{\omega}^{l}\cdot\nabla\delta^{l}){\rm d}x=0,$ which gives $\|{\rm I}_{1}^{l}(\eta^{l},\delta^{l})\|=\Big{\|}\sum\limits_{K\in\mathcal{T}_{h}}\int_{K}\left(-\eta^{l}(\boldsymbol{\omega}^{l}\cdot\nabla\delta^{l})+\sigma_{t}\eta^{l}\delta^{l}\right){\rm d}x\Big{\|}\lesssim\sum\limits_{K\in\mathcal{T}_{h}}\\|\eta^{l}\\|_{0,K}\\|\delta^{l}\\|_{0,K}.$ The Cauchy-Schwarz inequality yields $\Big{\|}\sum\limits_{l=1}^{L}w_{l}{\rm I}_{1}^{l}(\eta^{l},\delta^{l})\Big{\|}=\Big{(}\sum\limits_{l=1}^{L}w_{l}\\|\eta^{l}\\|_{0,D}^{2}\Big{)}^{1/2}\Big{(}\sum\limits_{l=1}^{L}w_{l}\\|\delta^{l}\\|_{0,D}^{2}\Big{)}^{1/2}.$ For the second one, using Lemma 4.3 in [27], we obtain $\displaystyle\Big{\|}\sum\limits_{l=1}^{L}{w_{l}{\rm I}_{2}^{l}(\eta^{l},\delta^{l})}\Big{\|}$ $\displaystyle\leq\Big{(}\sum\limits_{l=1}^{L}w_{l}\int_{D}m\sigma_{s}(\eta^{l})^{2}{\rm d}x\Big{)}^{1/2}\Big{(}\sum\limits_{l=1}^{L}w_{l}\int_{D}m\sigma_{s}(\delta^{l})^{2}{\rm d}x\Big{)}^{1/2}$ $\displaystyle\lesssim\Big{(}\sum\limits_{l=1}^{L}w_{l}\\|\eta^{l}\\|_{0,D}^{2}\Big{)}^{1/2}\Big{(}\sum\limits_{l=1}^{L}w_{l}\\|\delta^{l}\\|_{0,D}^{2}\Big{)}^{1/2},$ where $m=m(\boldsymbol{x})$ is given in (3.4). It remains to consider $b_{hs}^{(l)}(\eta^{l},\delta^{l})$. From (49) in [10] we have $\|b_{hs}^{(l)}(\eta^{l},\delta^{l})\|\leq\sum\limits_{e\in\mathcal{E}_{h}}\Big{(}\frac{1}{\theta_{0}}\\|c_{e}^{1/2}\\{\\!\\!\\{\eta^{l}\\}\\!\\!\\}\\|_{0,e}+\\|c_{e}^{1/2}[\\![\eta^{l}]\\!]\\|_{0,e}\Big{)}\\|c_{e}^{1/2}[\\![\delta^{l}]\\!]\\|_{0,e}.$ According to the choice of $c_{e}^{l}$, the Cauchy-Schwarz inequality gives $\Big{\|}\sum\limits_{l=1}^{L}{w_{l}b_{hs}^{(l)}(\eta^{l},\delta^{l})}\Big{\|}\lesssim c(\theta_{0})\Big{(}\sum\limits_{l=1}^{L}w_{l}\sum\limits_{K\in\mathcal{T}_{h}}\\|\eta^{l}\\|_{0,\partial K}^{2}\Big{)}^{1/2}\Big{(}\sum\limits_{l=1}^{L}w_{l}\sum\limits_{e\in\mathcal{E}_{h}}\\|c_{e}^{1/2}[\\![\delta^{l}]\\!]\\|_{0,e}^{2}\Big{)}^{1/2},$ where $c(\theta_{0})=\theta_{0}^{-1/2}+\theta_{0}^{1/2}$ is a constant, and hence $a_{h}^{s}(\boldsymbol{\eta},\boldsymbol{\delta})\lesssim c(\theta_{0})\Big{[}\sum\limits_{l=1}^{L}w_{l}\Big{(}\\|\eta^{l}\\|_{0,D}^{2}+\sum\limits_{K\in\mathcal{T}_{h}}\\|\eta^{l}\\|_{0,\partial K}^{2}\Big{)}\Big{]}^{1/2}\|\\!\|\\!\|\boldsymbol{\delta}\|\\!\|\\!\|.$ This combined with (4.6) yields $\|\\!\|\\!\|\boldsymbol{\delta}\|\\!\|\\!\|\lesssim c(\theta_{0})\Big{[}\sum\limits_{l=1}^{L}w_{l}\Big{(}\\|\eta^{l}\\|_{0,D}^{2}+\sum\limits_{K\in\mathcal{T}_{h}}\\|\eta^{l}\\|_{0,\partial K}^{2}\Big{)}\Big{]}^{1/2},$ and with the error estimates of the sparse projection in Lemma 4.1 leads to the desired result. ∎ ## 5 Numerical results In this section, we shall provide a series of numerical examples for solving the RTE (2.1)-(2.2) to illustrate the performance of the proposed sparse grid discrete coordinate DG method. ### 5.1 The linear system from the discrete problem The $S_{n}$ method has $n(n+2)$ directions with $n$ an even natural number. The discrete-ordinate sets satisfying the required moment equations to fourteen digits of accuracy have been given in [7]. Note that only the ordinates in the first octant are given there. The remaining ordinates can be obtained by using symmetry arguments. For example, $S_{2}$ data is given in Tab. 1. Tab. 1: Discrete-ordinate sets for $S_{2}$ ($\boldsymbol{\omega}=(s_{1},s_{2},s_{3})$) $s_{1}$ \| $s_{2}$ \| $s_{3}$ \| $w$ ---\|---\|---\|--- $\pm 0.5773502691896257$ \| $\pm 0.5773502691896257$ \| $\pm 0.5773502691896257$ \| 1.5707963267948966 Relabel the sparse bases by a single index $i=1,2,\cdots,M$ and denote them by $\varphi_{i}$, where $M=\dim\widehat{\boldsymbol{V}}_{N}^{k}$. The variational problem (3.14) can be written in matrix form $\boldsymbol{A}^{(l)}\hat{\boldsymbol{U}}^{l}-\sum\limits_{i=1}^{L}\boldsymbol{B}_{i}^{(l)}\hat{\boldsymbol{U}}^{i}+\boldsymbol{C}^{(l)}\hat{\boldsymbol{U}}^{l}=\boldsymbol{F}^{(l)},\quad 1\leq l\leq L,$ (5.1) where $\boldsymbol{A}^{(l)}=(a_{nm}^{(l)}),\quad\boldsymbol{B}_{i}^{(l)}=(b_{nmi}^{(l)}),\quad\boldsymbol{C}^{(l)}=(c_{nm}^{(l)}),$ $\hat{\boldsymbol{U}}^{l}=[\hat{u}_{1}^{l},\hat{u}_{2}^{l},\cdots,\hat{u}_{M}^{l}]^{T},\quad\boldsymbol{F}^{(l)}=[F_{1}^{(l)},F_{2}^{(l)},\cdots,F_{M}^{(l)}]^{T},$ and $a_{nm}^{(l)}=\sum\limits_{K\in\mathcal{T}_{h}}\int_{K}(-\varphi_{m}(\boldsymbol{\omega}^{l}\cdot\nabla\varphi_{n})+\sigma_{t}\varphi_{m}\varphi_{n}){\rm d}x,$ $b_{nmi}^{(l)}=w_{i}\int_{D}\sigma_{s}g(\cdot,\boldsymbol{\omega}^{l}\cdot{\boldsymbol{\omega}}^{i}){\varphi_{m}}\varphi_{n}{\rm d}x,$ $c_{nm}^{(l)}=\sum\limits_{e\not\subset\Gamma_{-}}\int_{e}\\{\\!\\!\\{\boldsymbol{\omega}^{l}\varphi_{m}\\}\\!\\!\\}\cdot[\\![\varphi_{n}]\\!]{\rm d}s+\sum\limits_{e\in\mathcal{E}_{h}^{0}}\int_{e}c_{e}^{l}[\\![\varphi_{m}]\\!]\cdot[\\![\varphi_{n}]\\!]{\rm d}s,$ $F_{n}^{(l)}=\int_{D}f^{l}\varphi_{n}{\rm d}x-\sum\limits_{e\subset\Gamma_{-}}\int_{e}\boldsymbol{\omega}^{l}\cdot\boldsymbol{n}\alpha^{l}\varphi_{n}{\rm d}s.$ The system (5.1) can be further rewritten in block matrix form $\boldsymbol{D}^{(l)}\hat{\boldsymbol{U}}=\boldsymbol{F}^{(l)},\quad 1\leq l\leq L,$ where $\boldsymbol{D}^{(l)}:=[-\boldsymbol{B}_{1}^{(l)},\cdots,-\boldsymbol{B}_{l-1}^{(l)},{\boldsymbol{A}^{(l)}}-\boldsymbol{B}_{l}^{(l)}+\boldsymbol{C}^{(l)},\cdots,-\boldsymbol{B}_{L}^{(l)}].$ The final linear system is $\boldsymbol{D}\hat{\boldsymbol{U}}=\boldsymbol{F},$ (5.2) where $\boldsymbol{D}=\left[\begin{array}[]{{20}{c}}\boldsymbol{D}^{(1)}\\\ \vdots\\\ \boldsymbol{D}^{(L)}\end{array}\right],\quad\hat{\boldsymbol{U}}=\left[\begin{array}[]{{20}{c}}\hat{\boldsymbol{U}}^{1}\\\ \vdots\\\ {\hat{\boldsymbol{U}}}^{L}\end{array}\right],\quad\boldsymbol{F}=\left[{\begin{array}[]{{20}{c}}\boldsymbol{F}^{(1)}\\\ \vdots\\\ \boldsymbol{F}^{(L)}\end{array}}\right].$ We solve (5.2) by using the block Gauss-Seidal iteration method. The accuracy is measured by the weighted relative error defined by $\\|u-u_{h}\\|_{rel}=\frac{\left(\sum\limits_{l=1}^{L}\omega_{l}\\|u(\cdot,\boldsymbol{\omega}^{l})-u_{h}^{l}\\|_{0,D}^{2}\right)^{1/2}}{\left(\sum\limits_{l=1}^{L}\omega_{l}\\|u(\cdot,\boldsymbol{\omega}^{l})\\|_{0,D}^{2}\right)^{1/2}},$ where $u_{h}^{l}=\sum\limits_{m=1}^{M}\hat{u}_{m}^{l}\varphi_{m}$ is the numerical solution. ### 5.2 Examples in three dimensions ###### Example 5.1. We take $\sigma_{t}=2$, $\sigma_{s}=1$ and $\eta=0$. The domain $D$ is a unit cube. With the right-hand side function $\displaystyle f(\boldsymbol{x},\boldsymbol{\omega})$ $\displaystyle=\pi s_{1}\cos(\pi x_{1})\sin(\pi x_{2})\sin(\pi x_{3})+\pi s_{2}\sin(\pi x_{1})\cos(\pi x_{2})\sin(\pi x_{3})$ $\displaystyle\quad+\pi s_{3}\sin(\pi x_{1})\sin(\pi x_{2})\cos(\pi x_{3})+\sin(\pi x_{1})\sin(\pi x_{2})\sin(\pi x_{3}),$ where $\boldsymbol{\omega}=(s_{1},s_{2},s_{3})$, the exact solution is $u(\boldsymbol{x},\boldsymbol{\omega})=\sin(\pi x_{1})\sin(\pi x_{2})\sin(\pi x_{3}).$ Fig. 1: Sparse pattern of the coefficient matrix for Example 5.1 ($N=3,k=2,n=2$) The sparse pattern for the coefficient matrix is shown in Fig. 1. The total number of the entries is $8200\times 8200={\text{67371264}}$ and the number of nonzero elements is ${\text{nz}}=239760$. Thus the sparsity ratio is 99.64%. (a) Exact (b) $\theta_{0}=10$ (c) $\theta_{0}=100$ (d) $\theta_{0}=500$ Fig. 2: The expansion coefficients for Example 5.1 with different stabilization parameters ($S_{2},k=2$) Tab. 2: Relative errors for Example 5.1: $k$ v.s. $S_{n}$ ($N=2$) $n$ \| 2 \| 4 \| 6 \| 8 \| 10 ---\|---\|---\|---\|---\|--- $k=1$ \| 1.7133e-01 \| 1.7329e-01 \| 1.7480e-01 \| 1.7491e-01 \| 1.7500e-01 $k=2$ \| 8.9453e-03 \| 8.3749e-03 \| 8.0365e-03 \| 8.0532e-03 \| 8.0755e-03 Tab. 3: Relative errors for Example 5.1: $N$ v.s. $S_{n}$ ($k=2$) $n$ \| 2 \| 4 \| 6 \| 8 \| 10 ---\|---\|---\|---\|---\|--- $N=2$ \| 8.9453e-03 \| 8.3749e-03 \| 8.0365e-03 \| 8.0532e-03 \| 8.0755e-03 $N=3$ \| 2.2512e-03 \| 2.1269e-03 \| 2.0150e-03 \| 2.0138e-03 \| 2.0136e-03 Fig. 2 displays the expansion coefficients which coincide in each angular direction since the true solution is independent of the angular variable $\boldsymbol{\omega}$. We observe a better result for bigger stabilization parameter $\theta_{0}$. In the following we always choose $\theta_{0}=10^{N+k}$ due to its good performance in different cases. For the isotropic case $\eta=0$, $S_{2}$ method is enough to resolve the solution accurately in angle as indicated by the numerical results in Tabs. 2 and 3. For the given example, the error is then dominated by the spatial problems. According to Theorem 4.1, the error bound is $\mathcal{O}(c(\theta_{0})\|\log_{2}h\|^{d}h^{k+1/2})$. With the choice for $\theta_{0}$ in this case, we have $c(\theta_{0})\|\log_{2}h\|^{d}h^{k+1/2}\approx\mathcal{O}(\|\log_{2}h\|^{d}h^{k})$ and the logarithmic factor implies a slightly lower order than $k$. From Tab. 4, we see that the convergence rates for $k=1,3$ are better than $\mathcal{O}(h^{k+1/2})$ and even the $(k+1)$-th order can be obtained for $k=3$. For $k=2,4$ the order is about $k$. Tab. 4: $L^{2}$ errors of $S_{2}$ method for Example 5.1 $N$ \| $k=1$ \| \| $k=2$ \| \| $k=3$ \| \| $k=4$ ---\|---\|---\|---\|---\|---\|---\|--- Err \| rate \| \| Err \| rate \| \| Err \| rate \| \| Err \| rate 1 \| 4.8695e-01 \| - \| \| 3.7626e-02 \| - \| \| 3.8603e-03 \| - \| \| 2.9324e-04 \| - 2 \| 1.7133e-01 \| 1.5070 \| \| 8.9453e-03 \| 2.0725 \| \| 2.1133e-04 \| 4.1911 \| \| 1.6406e-05 \| 4.1598 3 \| 5.6436e-02 \| 1.6021 \| \| 2.2512e-03 \| 1.9904 \| \| 1.2971e-05 \| 4.0261 \| \| 7.8260e-07 \| 4.3898 4 \| 1.6990e-02 \| 1.7319 \| \| 5.6295e-04 \| 1.9996 \| \| 8.2285e-07 \| 3.9785 \| \| - \| - ###### Example 5.2. We take $\sigma_{t}=3$ and $\sigma_{s}=1$. The domain $D$ is a unit cube. The true solution is taken as $u(\boldsymbol{x},\boldsymbol{\omega})=10\omega_{3}\sin(\pi x_{1})\sin(\pi x_{2})\sin(\pi x_{3}),$ from which we know after a direct manipulation that the right-hand side function is $\displaystyle f(\boldsymbol{x},\boldsymbol{s})=$ $\displaystyle 10(\sigma_{t}-\eta\sigma_{s})s_{3}\sin(\pi x_{1})\sin(\pi x_{2})\sin(\pi x_{3})$ $\displaystyle+10\pi s_{3}^{2}\sin(\pi x_{1})\sin(\pi x_{2})\cos(\pi x_{3})+10\pi s_{2}s_{3}\sin(\pi x_{1})\cos(\pi x_{2})\sin(\pi x_{3})$ $\displaystyle+10\pi s_{1}s_{3}\cos(\pi x_{1})\sin(\pi x_{2})\sin(\pi x_{3})$ where $\boldsymbol{\omega}=(s_{1},s_{2},s_{3})$. Tab. 5: Relative errors for Example 5.2 ($S_{2},\eta=0.1$) $N$ \| $k=1$ \| $k=2$ \| $k=3$ \| $k=4$ ---\|---\|---\|---\|--- 1 \| 2.2797e-01 \| 1.6584e-02 \| 1.7683e-03 \| 2.1850e-04 2 \| 8.2048e-02 \| 3.7848e-03 \| 2.0045e-04 \| 1.7282e-04 Tab. 6: Relative errors for Example 5.2 ($N=1,\eta=0.9$) $n$ \| $k=1$ \| $k=2$ \| $k=3$ \| $k=4$ ---\|---\|---\|---\|--- 2 \| 6.6685e-01 \| 6.6872e-01 \| 6.6969e-01 \| 2.0653e+01 4 \| 6.4693e-01 \| 1.4665e+01 \| 1.5884e+01 \| 1.6226e+01 6 \| 1.1411e+00 \| 1.2068e+00 \| 1.2539e+00 \| 1.2606e+00 8 \| 1.1511e-01 \| 1.3098e-01 \| 1.3206e-01 \| 1.3212e-01 10 \| 7.3980e-02 \| 7.8657e-02 \| 7.9128e-02 \| 7.9144e-02 12 \| 4.3524e-02 \| 3.2606e-02 \| 3.2684e-02 \| 3.2690e-02 We observe from Tab. 5 that $S_{2}$ method is accurate enough for the anisotropy factor close to isotropic cases. However, for strong forward scattering of $\eta=0.9$, it does not give a satisfactory result. We have to choose a larger $n$ to get an improved result, which, however, is not expected in real applications since $S_{n}$ method has $n(n+2)$ angular directions and hence $n(n+2)$ coupled spatial problems. In this case, some models have been developed to approximate the integral operator (cf. [26, 37, 48]). Another approach is to combine the sparse grid technique with the spherical harmonic method. Fig. 3: Sparse pattern of the coefficient matrix for Example 5.3 ($N=1,k=2,n=2$) ###### Example 5.3. This example is taken from the reference [32], where the Henyey-Greenstein function is replaced by the simplified approximate Mie (SAM): $g(t)=K_{S}(1+t)^{n_{p}},~{}~{}t\in[-1,1],$ where $n_{p}=\frac{2\eta}{1-\eta}$ is the anisotropic index and $K_{S}=\frac{1}{2\pi}\frac{n_{p}+1}{2^{n_{p}+1}}$ is the normalization factor. The geometric parameters and the true solution are the same as Example 5.2. For $S_{2}$ method with $N=1$ and $k=2$, the sparse pattern for the coefficient matrix is shown in Fig. 3. We also display the numerical and exact coefficients and $L^{2}$ projections at $z=0$ associated with the first angular direction in Fig. 4. We repeat the test for highly forward-peaked scattering with $\eta=0.9$. From Tab. 7 we observe a relatively smaller errors than that from Tab. 6, but the convergence behaviours are the same since the errors do not decrease significantly with the increase of $k$ and $n$. Fig. 4: Numerical and exact coefficients and $L^{2}$ projections for Example 5.3 ($N=1,k=2,n=2$) Tab. 7: Relative errors for Example 5.3 ($N=1,\eta=0.9$) $n$ \| $k=1$ \| $k=2$ \| $k=3$ \| $k=4$ ---\|---\|---\|---\|--- 2 \| 6.0818e-01 \| 7.2904e-01 \| 7.3901e-01 \| 7.3967e-01 4 \| 3.7295e-01 \| 4.0517e-02 \| 3.1238e-02 \| 3.1143e-02 6 \| 3.7622e-01 \| 3.7490e-02 \| 2.7350e-02 \| 2.7243e-02 8 \| 3.7823e-01 \| 2.8034e-02 \| 1.0114e-02 \| 9.7213e-03 10 \| 3.7872e-01 \| 2.6477e-02 \| 3.5129e-03 \| 2.0719e-03 12 \| 3.7875e-01 \| 2.6452e-02 \| 3.2775e-03 \| 1.6395e-03 ### 5.3 Flux distributions in two and three dimensions We now investigate the impact of the source term on the flux distributions. The isotropic photon flux is defined by $q(\boldsymbol{x})=\frac{1}{4\pi}\int_{S^{2}}u(\boldsymbol{x},\hat{\boldsymbol{\omega}}){\rm d}\sigma(\hat{\boldsymbol{\omega}}).$ For simplicity, vacuum boundary conditions are applied on all the boundaries. We always consider the isotropic scattering, and take $N=k=2$ for the spatial discretization. The examples in this subsection are taken from the reference [33]. ###### Example 5.4. This problem is defined on a unit cube with vacuum boundaries. The first 0.2 by 0.2 by 0.2 region $R$ contains a uniform isotropic source. For simplicity, we consider the following right-hand side function: $f(\boldsymbol{x},\boldsymbol{\omega})=f(\boldsymbol{x})=\begin{cases}1,\quad\boldsymbol{x}\in R=[0,0.2]^{3},\\\ 0,\quad\boldsymbol{x}\in D\backslash R.\end{cases}$ The entire box is of uniform composition with the following data: $\sigma_{t}=1$ and $\sigma_{s}=0.4$. For $z=0.1$ fixed, the contour plot of the flux distributions with varying orders of the discrete ordinates is displayed in Fig. 5. We can see clearly that the contour map shows rays emanating from the source. (a) $S_{2}$ (b) $S_{4}$ (c) $S_{6}$ Fig. 5: Contour map of the photon flux distributions for Example 5.4 We now perform the test on the problem in $(x,y)$-geometry. In this case, all coefficients, the boundary data and the solution of (2.1)-(2.2) are independent of the space variable $x_{3}=z$. ###### Example 5.5. This problem is defined on a unit square with vacuum boundaries. The first 0.2 by 0.2 region localized in the lower left corner contains a uniform isotropic source. The entire box is of uniform composition with the following data: $\sigma_{t}=1$ and $\sigma_{s}=0.4$. For this example, we only consider the $S_{4}$ method. The contour plot of the flux distributions is displayed in Fig. 6 (a). Numerical results of other cases are listed in Fig. 6 (b)-(f) by changing the positions or increasing the numbers of the isotropic sources. In all cases, we again observe the rays emanating from the sources. Fig. 6: Contour map of the photon flux distributions for Example 5.5 ### 5.4 Examples with complex spatial domains in two dimensions In the following, we extend the method to solve the RTE for some non-tensor product spatial domains in two dimensions. We always consider the isotropic scattering. Fig. 7: Initial subdivision of a $L$-shaped region for Example 5.6 ###### Example 5.6. The spatial domain $D$ is an $L$-shaped region in 2-D displayed in Fig. 7, consisting of three rectangles $R_{1}$, $R_{2}$ and $R_{2}$, where $R_{1}=\Big{\\{}(x_{1},x_{2}):0\leq x_{1}\leq 1,~{}~{}1\leq x_{2}\leq 2\Big{\\}},$ $R_{2}=\Big{\\{}(x_{1},x_{2}):0\leq x_{1}\leq 1,~{}~{}0\leq x_{2}\leq 1\Big{\\}},$ $R_{3}=\Big{\\{}(x_{1},x_{2}):1\leq x_{1}\leq 2,~{}~{}0\leq x_{2}\leq 1\Big{\\}}.$ The parameters are the same as Example 5.1 and the true solution is $u(\boldsymbol{x},\boldsymbol{\omega})=\sin(\pi x_{1})\sin(\pi x_{2}).$ (a) Exact solution (b) $N=2,k=1$ (c) $N=2,k=2$ Fig. 8: Exact and numerical solutions for Example 5.6 Let $R=R_{1}\cup R_{2}\cup R_{3}$ be the initial subdivision of the $L$-shaped region and $\boldsymbol{V}_{0}^{k}(R)$ denote the piecewise polynomial space on $R$. We have the following orthogonal decomposition $\boldsymbol{V}_{0}^{k}(R)=\boldsymbol{V}_{0}^{k}(R_{1})\oplus\boldsymbol{V}_{0}^{k}(R_{2})\oplus\boldsymbol{V}_{0}^{k}(R_{3}),$ where functions in $\boldsymbol{V}_{0}^{k}({R_{j}})~{}(j=1,2,3)$ are extended by zero to $\mathbb{R}^{2}$. For each $R_{j}$, one can regard it as $[0,1]^{2}$ and give the sparse representation by using an affine transformation. In Fig. 8, we display the numerical solutions for different $N$ and $k$ and the relative errors are given in Tab. 8. Tab. 8: Relative errors for Example 5.6 $N$ \| $k=1$ \| $k=2$ \| $k=3$ \| $k=4$ ---\|---\|---\|---\|--- 1 \| 2.2059e-01 \| 1.6769e-02 \| 1.7691e-03 \| 1.3197e-04 2 \| 6.1359e-02 \| 2.1758e-03 \| 1.1360e-04 \| 4.1841e-06 3 \| 1.7434e-02 \| 3.0707e-04 \| 7.2792e-06 \| 2.1799e-07 4 \| 4.8163e-03 \| 4.2519e-05 \| 4.6425e-07 \| - (a) Initial division (b) The final division $\mathcal{T}_{h}$ Fig. 9: Subdivision of the circular domain ###### Example 5.7. The spatial domain $D$ is a circular region displayed in Fig. 9. The parameters and the true solution are the same as the last example. To use the sparse grid method, we first plot a sufficiently large rectangle in the domain and approximate the boundary curve by a polygon as depicted in Fig. 9 (a). For simplicity, the boundary data corresponding to the polygon is obtained from the exact solution. For the general case, some approximation should be implemented, for example, the technique from the isoparametric finite elements. To avoid hanging nodes, the polygon approximation and the corresponding triangulation can be made consistent with the final partition of the rectangle, see Fig. 9 (b). We should note that the hanging nodes are allowed in our procedure since no interelement continuity is required. Denote the rectangle by $R$ and the other triangles by $T_{1},\cdots,T_{8}$, respectively. Let $\Omega=R\cup{T_{1}}\cup\cdots\cup{T_{8}}$. We then consider the initial DG space given by $\boldsymbol{V}_{0}^{k}(\Omega)=\boldsymbol{V}_{0}^{k}({R})\oplus\boldsymbol{V}_{0}^{k}({T_{1}})\oplus\cdots\oplus\boldsymbol{V}_{0}^{k}({T_{8}}).$ Tab. 9: Orthonormal bases on the reference triangle $\tau$ for $k\leq 2$ ($r:=\lambda_{1},s:=\lambda_{2}$) $k=0$ \| ---\|--- \| $\varphi_{1}=\sqrt{2}$ $k=1$ \| \| $\begin{gathered}\varphi_{2}=6r-2\hfill\\\ \varphi_{3}=2\sqrt{3}(2r+s-1)\hfill\\\ \end{gathered}$ $k=2$ \| \| $\begin{gathered}\varphi_{4}=\sqrt{6}(10r^{2}-8r+1)\hfill\\\ \varphi_{5}=3\sqrt{2}(5r-1)(r+2s-1)\hfill\\\ \varphi_{6}=\sqrt{30}(r^{2}+6rs-2r+6s^{2}-6s+1)\hfill\\\ \end{gathered}$ The orthonormal bases corresponding to $R$ has been given in the previous section, while the orthonormal bases on each $T_{i}$ can be obtained by using the Gram-Schmidt procedure. For any triangle $T$ with vertices $z_{i}={({x_{i}},{y_{i}})}$, $i=1,2,3$, any point $z={(x,y)}$ can be represented by the barycentric coordinates as $\begin{cases}x=x_{1}\lambda_{1}+x_{2}\lambda_{2}+x_{3}\lambda_{3},\\\ y=y_{1}\lambda_{1}+y_{2}\lambda_{2}+y_{3}\lambda_{3},\\\ 1=\lambda_{1}+\lambda_{2}+\lambda_{3}.\end{cases}$ Note that $\iint_{T}f(x,y)g(x,y){\rm d}x{\rm d}y=2\|T\|\int_{0}^{1}\int_{0}^{1-\lambda_{1}}\tilde{f}(\lambda_{1},\lambda_{2})\tilde{g}(\lambda_{1},\lambda_{2}){\rm d}\lambda_{2}{\rm d}\lambda_{1},$ where $\tilde{f}(\lambda_{1},\lambda_{2}):=f(x(\lambda_{1},\lambda_{2}),y(\lambda_{1},\lambda_{2})).$ We then define an inner product on the reference triangle $\tau$ by $(\tilde{f},\tilde{g})_{\tau}=\int_{0}^{1}\int_{0}^{1-\lambda_{1}}\tilde{f}(\lambda_{1},\lambda_{2})\tilde{g}(\lambda_{1},\lambda_{2}){\rm d}\lambda_{2}{\rm d}\lambda_{1}.$ Given the orthonormal bases on $\tau$ by $\\{\tilde{\varphi}_{i}\\}$, we then obtain the bases on $T$ given by $\psi_{i}(x,y)=\sqrt{\frac{1}{2\|T\|}}\varphi_{i}(x,y).$ For any function $f(x,y)$ defined on $T$, the projection coefficients are computed as $\displaystyle c_{i}$ $\displaystyle=\iint_{T}f(x,y)\psi_{i}(x,y){\rm d}x{\rm d}y=2\|T\|\int_{0}^{1}\int_{0}^{1-\lambda_{1}}\tilde{f}(\lambda_{1},\lambda_{2}){{\tilde{\psi}}_{i}}(\lambda_{1},\lambda_{2}){\rm d}\lambda_{2}{\rm d}\lambda_{1}$ $\displaystyle=\sqrt{2\|T\|}\int_{0}^{1}\int_{0}^{1-\lambda_{1}}\tilde{f}(\lambda_{1},\lambda_{2}){\tilde{\varphi}}_{i}(\lambda_{1},\lambda_{2}){\rm d}\lambda_{2}{\rm d}\lambda_{1}=:\sqrt{2\|T\|}{\tilde{c}}_{i}.$ The orthogonal bases on $\tau$ are obtained by using Gram-Schmidt procedure to the polynomial set $\\{1,\lambda_{1},\lambda_{2},\lambda_{1}^{2},\lambda_{1}\lambda_{2},\lambda_{2}^{2},\cdots\\}$, some of which are listed in Tab. 9. The relative error is defined by ${\rm Err}=\\|f-f_{h}\\|_{L^{2}(\mathcal{T}_{h})}/\\|f\\|_{L^{2}(\mathcal{T}_{h})}$ and given in Tab. 10. Tab. 10: Relative errors for Example 5.7 $(N=2)$ $k$ \| 0 \| 1 \| 2 \| 3 ---\|---\|---\|---\|--- Err \| 5.8510e-01 \| 6.1678e-02 \| 9.3273e-03 \| 5.5965e-04 Summarizing our main observations from the numerical results reported in all previous examples, we may conclude that 1. The sparse discrete ordinate DG method can greatly reduce the spatial degrees of freedom while keeping almost the same accuracy up to multiplication of an log factor. * 2. The proposed method is highly effective for problems away from strong forward scattering. To get an improved result, large discrete-ordinate sets are needed for highly forward-peaked case. * 3. 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# Lower bounds for the warping degree of a knot projection Atsushi Ohya Department of Computer Science and Engineering, University of Yamanashi, 4-4-37, Takeda, Kofu-shi, Yamanashi, 400-8510, Japan. Ayaka Shimizu Department of Mathematics, National Institute of Technology (KOSEN), Gunma College, 580 Toriba, Maebashi-shi, Gunma, 371-8530, Japan. Email: <EMAIL_ADDRESS><EMAIL_ADDRESS> ###### Abstract The warping degree of an oriented knot diagram is the minimal number of crossings which we meet as an under-crossing first when we travel along the diagram from a fixed point. The warping degree of a knot projection is the minimal value of the warping degree for all oriented alternating diagrams obtained from the knot projection. In this paper, we consider the maximal number of regions which share no crossings for a knot projection with a fixed crossing, and give lower bounds for the warping degree. ## 1 Introduction In this paper we assume that every knot diagram and knot projection has at least one crossing. A based knot diagram is a knot diagram which is given a base point on the diagram avoiding crossings. We denote by $D_{b}$ a based diagram $D$ with the base point $b$. The warping degree, $d(D_{b})$, of an oriented based knot diagram $D_{b}$ is the number of crossings such that we encounter the crossing as an under-crossing first when we travel along $D$ with the orientation starting at $b$. We call such a crossing a warping crossing point of $D_{b}$ (see Figure 1). Figure 1: The oriented based knot diagram $D_{b}$ has warping degree one. The crossing $p$ is the warping crossing point of $D_{b}$. The warping degree, $d(D)$, of an oriented knot diagram $D$ is the minimal value of $d(D_{b})$ for all base points $b$ of $D$ ([4]). A knot diagram is said to be monotone, or descending, if the warping degree is zero. Conversely, we can assume that the warping degree represents a complexity of a diagram in terms of how distant a knot diagram is from a monotone diagram. Note that a monotone knot diagram is a diagram of the trivial knot. A knot diagram is said to be alternating if we encounter an over-crossing and an under-crossing alternatively when traveling the diagram starting at any point on the diagram. Let $P$ be an unoriented knot projection. The warping degree of $P$ is defined to be the minimal value of the warping degree for all the oriented alternating diagrams obtained from $P$ by giving the orientation and crossing information. In Figure 2, all the reduced knot projections with warping degree one and two are shown ([6]). Further examples for warping degree three or four are listed in the table in Section 5 in this paper. As we may see, the warping degree of a knot projection shows somewhat complexity of a knot projection, like how “curly” a knot projection is, or how “quick” to back to a crossing when traveling the projection. Figure 2: All the reduced knot projections of warping degree one or two. The first two knot projections have warping degree one, and the others have two. The knot projections of warping degree two are determined in [6] by considering all possibilities of connections of the unavoidable parts. Further explorations in the same way for warping degree three or more would be difficult since there are too many kinds of unavoidable parts and too many possibilities of their connections. In this paper, we introduce the maximal independent region number, $\mathrm{IR}(P)$, of a knot projection in Section 2, and show the following inequality which is useful to estimate the warping degree. x ###### Theorem 1.1. The inequality $\mathrm{IR}(P)\leq d(P)\leq c(P)-\mathrm{IR}(P)-1$ holds for every reduced knot projection $P$, where $c(P)$ denotes the crossing number of $P$. x As mentioned in Sections 2 and 3, the value of $\mathrm{IR}(P)$ can be obtained without traveling along the knot projection, and also calculated just by solving simultaneous equations. Figure 3: Knot projections of warping degree three with 10, 11, 12 crossings. Since all the reduced knot projections with warping degree one and two are determined and we can find some knot projections with warping degree three (see Figure 3 and Section 5), we obtain the following table about the minimal value of the warping degree of reduced knot projections for each crossing number. $c$ \| 3 \| 4 \| 5 \| 6 \| 7 \| 8 \| 9 \| 10 \| 11 \| 12 ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- $d^{\text{min}}(c)$ \| 1 \| 1 \| 2 \| 2 \| 2 \| 2 \| 3 \| 3 \| 3 \| 3 Table 1: The crossing number $c$ and the minimal value of warping degree $d^{\text{min}}(c)$ for all reduced knot projections with $c$ crossings. Regions of a knot or link projection are independent if they share no crossings. We also give the following lower bound for the warping degree which would be helpful to extend the above table. x ###### Theorem 1.2. If all the connected link projections with $n$, $n+1$ or $n+2$ crossings have $m$ or more independent regions, then $d(P)\geq m-1$ holds for all reduced knot projections $P$ with $n$ or more crossings. x The rest of the paper is organized as follows: In Section 2, we define the maximal independent region number $\mathrm{IR}(P)$ and prove Theorem 1.1. In Section 3, we introduce the calculation for $\mathrm{IR}(P)$ by simultaneous equations. In Section 4, we estimate $\mathrm{IR}(P)$ and the warping degree $d(P)$ and prove Theorem 1.2. In Section 5, we list and compare the values of $\mathrm{IR}(P)$ and $d(P)$. ## 2 Independent region sets In this section, we define the maximal independent region number, and using it we estimate the warping degree of a knot projection. Throughout this section, we assume that every knot diagram and knot projection is reduced. We have the following lemma (cf. [6]). x ###### Lemma 2.1. Let $D$ be an oriented alternating knot diagram. Let $c$ be a crossing of $D$. Take a base point $b$ just before an over-crossing of $c$. If $D$ has a region $R$ which does not incident to $c$, then one of the crossings on the boundary of $R$ is a warping crossing point of $D_{b}$, and one of that is a non- warping crossing point of $D_{b}$. x ###### Proof. Let $e$ be the edge on the boundary of $R$ such that we meet it first from $b$. Then $e$ has an under-crossing and an over-crossing, that is, a warping crossing point and a non-warping crossing point. Figure 4: The edge $e$ on the boundary of $R$ which we meet first from the base point $b$ has one under-crossing and one over-crossing, and they are a warping crossing point and non-warping crossing point of $D_{b}$, respectively, regardless of the orientation. ∎ x Similarly, we have the following. x ###### Corollary 2.2. Let $D$ be an oriented alternating knot diagram. Let $c$ be a crossing of $D$. Take a base point $b$ just before an over-crossing of $c$. If $D$ has independent $n$ regions which are not incident to $c$, then the inequality $n\leq d(D)\leq c(D)-n-1$ holds. x ###### Proof. By Lemma 2.1, $D$ has at least $n$ warping crossing points of $D_{b}$. Also, $D$ has at least $n+1$ non-warping crossing points since the crossing $c$ is a non-warping crossing point, too. Therefore we have $n\leq d(D_{b})\leq c(D)-n-1$. By the location of the base point $b$, we have $d(D_{b})=d(D)$ ([8]). ∎ x By definition, we have the following corollary for knot projections. x ###### Corollary 2.3. Let $P$ be a knot projection, and $c$ a crossing of $P$. If $P$ has independent $n$ regions which are not incident to $c$, then the inequality $n\leq d(P)\leq c(P)-n-1$ holds. x The strong point is that we can estimate the warping degree without traveling the projection (see Figure 5). Figure 5: The warping degree can be determined for some knot projections from the inequality of Corollary 2.3, without traveling along the knot projection. The knot projection $P$ has $1\leq d(P)\leq 3-1-1$, and $d(P)=1$. We also obtain $d(Q)=2,d(R)=3$ from the inequalities. This would enable us to estimate the warping degree more combinatorically. We call the set of regions of a knot projection $P$ which are independent and are not incident to a crossing $c$ an independent region set for $P^{c}$. We call the crossing $c$ a base crossing. We define the maximal independent region number of $P^{c}$, $\mathrm{IR}(P^{c})$, to be the maximal cardinality of an independent region set for $P^{c}$. We define the maximal independent region number of $P$, $\mathrm{IR}(P)$, to be the maximal value of $\mathrm{IR}(P^{c})$ for all base crossings $c$. We prove Theorem 1.1. Proof of Theorem 1.1. It follows from Corollary 2.3. $\Box$ x ## 3 Independent region sets and region choice matrix In this section we explore how to find the independent region sets. A region choice matrix $M$, defined in [2], of a knot projection $P$ of $n$ crossings is the following $n\times(n+2)$ matrix. (The transposition is known as an incidence matrix defined in [3].) If a crossing $c_{i}$ is on the boundary of a region $R_{j}$, the $(i,j)$ component of $M$ is 1, and otherwise 0 (see Figure 6). Figure 6: A region choice matrix $M=(m_{ij})$, where $m_{ij}=1$ if $R_{j}$ is incident to $c_{i}$, and otherwise $m_{ij}=0$. Now we find out all of the independent region sets for $P^{c_{3}}$ for the knot projection $P$ and the crossing $c_{3}$ in Figure 6 by looking at its region choice matrix. Since the crossing $c_{3}$ is involved with the four regions $R_{1},R_{4},R_{5}$ and $R_{7}$, namely, the third row has 1 at the first, fourth, fifth and seventh column, we can not choose them as independent regions for $P^{c_{3}}$. Hence we choose the regions from the rest regions $R_{2},R_{3}$ and $R_{6}$. Namely, we choose columns from the second, third and sixth so that there are no components with the value two or more in the sum of the columns. Thus we obtain all the independent region sets for $P^{c_{3}}$, as $\\{R_{2}\\}$, $\\{R_{3}\\}$, $\\{R_{6}\\}$ and $\\{R_{2},R_{6}\\}$. More generally, we can find out all the independent region sets for a knot projection $P^{c}$ for all base crossings $c$ from the region choice matrix by solving the following simultaneous equations111 If it works on $\mathbb{Z}_{2}$, it is known that the simultaneous equations have solutions for any $b_{i}$’s and any region choice matrix of a knot projection ([7], [3]). Besides, if $x_{i}$’s are permitted to have the value for any integer, it is also known that the simultaneous equations have solutions for any region choice matrix of a knot projection even if $b_{i}$’s have the value for any integers ([2]). In this case, however, the equation has no solutions for some $b_{i}$’s. In Lemma 4.2, we will see that it definitely has solutions for some $b_{i}$’s. for $x_{i}\in\\{1,0\\}\ (i=1,2,\dots,7)$ $\displaystyle\left(\begin{array}[]{ccccccc}1&1&1&0&0&0&1\\\ 1&1&1&1&0&0&0\\\ 1&0&0&1&1&0&1\\\ 0&0&0&1&1&1&1\\\ 0&0&1&1&0&1&1\end{array}\right)\left(\begin{array}[]{c}x_{1}\\\ x_{2}\\\ x_{3}\\\ x_{4}\\\ x_{5}\\\ x_{6}\\\ x_{7}\end{array}\right)=\left(\begin{array}[]{c}b_{1}\\\ b_{2}\\\ b_{3}\\\ b_{4}\\\ b_{5}\end{array}\right),$ for all $b_{i}\in\\{1,0\\}\ (i=1,2,\dots,7)$, where $b_{k}\neq b_{l}$ for some $k$ and $l$; If $b_{i}=0$ for all $i$, it implies that no regions are chosen. If $b_{i}=1$ for all $i$, it means all the crossings are on the chosen regions, and we can not have a base crossing. ## 4 Estimation for the maximal independent region number As Theorem 1.1 implies, the warping degree is estimated by the maximal independent region numbers. In this section, we estimate the maximal independent region number itself, and prove Theorem 1.2. First, we have the following: x ###### Lemma 4.1. The inequality $\displaystyle\mathrm{IR}(P)\leq\frac{c(P)-1}{2}$ holds for every reduced knot projection $P$. x ###### Proof. By Theorem 1.1, we have $\mathrm{IR}(P)\leq c(P)-\mathrm{IR}(P)-1$, and then have $2\mathrm{IR}(P)\leq c(P)-1$. ∎ x Next, we give a routine lower bound for $\mathrm{IR}(P)$. x ###### Lemma 4.2. For any knot projection $P$ with $c(P)\geq 2$, we have $\mathrm{IR}(P)\geq 1$. x ###### Proof. For the case that $c(P)=2$, $P$ has two independent bigons. By taking a base crossing at a crossing which belongs to one of the two bigons, we can take an independent region at the other bigon. For the case that $c(P)\geq 3$, then the number of regions is $3+2=5$ or more by the Euler characteristic (see, for example, [2]). Take a base crossing $c$. Then either three or four regions are incident to $c$. This means there exists a region which is not incident to $c$. Hence $\mathrm{IR}(P)\geq 1$ holds. ∎ x To give further lower bounds for $\mathrm{IR}(P)$, we show the following lemma for link projections. x ###### Lemma 4.3. If all the connected link projections with $n$, $n+1$ or $n+2$ crossings have $m$ or more independent regions, then all the connected link projections with $n$ or more crossings have $m$ or more independent regions. x ###### Proof. Let $P$ be a link projection with $n+3$ crossings. If $P$ is reducible, splice it at a reducible crossing as shown in Figure 7. Figure 7: Splice $P$ at a reducible crossing. Then we obtain a link projection, $P^{\prime}$, with $n+2$ crossings. By assumption, $P^{\prime}$ has $m$ independent regions. Take the corresponding regions of $P$; If the region $R$ of $P^{\prime}$ created by the splice has been chosen, take one of the parts $R^{1}$ and $R^{2}$ as a corresponding region (see Figure 7). Thus, we obtain $m$ independent regions of $P$. If $P$ is reduced, it is shown in [1] that $P$ has a bigon or trigon. Splice it at a bigon or a trigon as shown in Figure 8. Figure 8: Splice $P$ at a bigon or a trigon and obtain a link projection $P^{\prime}$ or $P^{\prime\prime}$, respectively. Note that $P^{\prime}$ has $n+2$ and $P^{\prime\prime}$ has $n$ crossings. Then $P^{\prime}$ and $P^{\prime\prime}$ have $m$ independent regions. Similarly, take $m$ regions of $P$ properly from the corresponding regions, which are independent. ∎ x We have the following corollary. x ###### Corollary 4.4. If all the connected link projections with $n$, $n+1$ or $n+2$ crossings have $m$ or more independent regions, then $\mathrm{IR}(P)\geq m-1$ holds for all reduced knot projections $P$ with $n$ or more crossings. x ###### Proof. All reduced knot projections with $n$ or more crossings have $m$ independent regions by Lemma 4.3. Take a base crossing $c$ at the boundary of one of the $m$ regions. Then, the rest $m-1$ regions are independent regions for $P^{c}$. ∎ x We prove Theorem 1.2. Proof of Theorem 1.2. From Corollary 4.4 we have $\mathrm{IR}(P)\geq m-1$, and from Theorem 1.1 we have $d(P)\geq\mathrm{IR}(P)$. $\Box$ ## 5 Table of $\mathrm{IR}(P)$ and $d(P)$ For the knot projections $P$ of prime alternating knots with up to nine crossings which are obtained from the knot diagrams in Rolfsen’s knot table ([5]), the values of $\mathrm{IR}(P)$ and $d(P)$ ([8]) are listed below. The values of $\mathrm{IR}(P)$ were obtained by the calculation using the SAT solver. ## Acknowledgment The authors thank Yoshiro Yaguchi for helpful comments. ## References * [1] C. C. Adams, R. Shinjo and K. Tanaka, Complementary regions of knot and link diagrams, Ann. Comb. 15 (2011), 549–563. * [2] K. Ahara and M. Suzuki, An integral region choice problem on knot projection, J. Knot Theory Ramifications 21 (2012), 1250119 [20 pages]. * [3] Z. Cheng and H. Gao, On region crossing change and incidence matrix, Sci. China Math. 55 (2012), 1487–1495. * [4] A. Kawauchi, Lectures on knot theory (in Japanese), Kyoritsu shuppan Co. Ltd, 2007. * [5] D. Rolfsen, Knots and links, Publish or Perish, Inc. (1976). * [6] A. Shimizu, Prime alternating knots of minimal warping degree two, J. Knot Theory Ramifications 29 (2020), 2050060. * [7] A. Shimizu, Region crossing change is an unknotting operation, J. Math. Soc. Japan 66 (2014), 693–708. * [8] A. Shimizu, The warping degree of a knot diagram, J. Knot Theory Ramifications 19 (2010), 849–857.
Distance Estimation for BLE-based Contact Tracing – A Measurement Study Bernhard Etzlinger, Barbara Nußbaummüller, Philipp Peterseil, and Karin Anna Hummel Johannes Kepler University, Linz, Austria<EMAIL_ADDRESS> This work has been supported by the LCM K2 Center within the framework of the Austrian COMET-K2 program. Mobile contact tracing apps are – in principle – a perfect aid to condemn the human-to-human spread of an infectious disease such as COVID-19 due to the wide use of smartphones worldwide. Yet, the unknown accuracy of contact estimation by wireless technologies hinders the broader use. We address this challenge by conducting a measurement study with a custom testbed to show the capabilities and limitations of Bluetooth Low Energy (BLE) in different scenarios. Distance estimation is based on interpreting the signal pathloss with a basic linear and a logarithmic model. Further, we compare our results with accurate ultra-wideband (UWB) distance measurements. While the results indicate that distance estimation by BLE is not accurate enough, a contact detector can detect contacts below 2.5$\,$m with a true positive rate of 0.65 for the logarithmic and of 0.54 for the linear model. Further, the measurements reveal that multi-path signal propagation reduces the effect of body shielding and thus increases detection accuracy in indoor scenarios. Contact Tracing, Wireless Communications, BLE § INTRODUCTION Contact tracing aims at fighting human-to-human infection spreading by identifying – and isolating – persons who were in close contact to an infected person. Quarantine is one of the most effective measures to break infection chains [1]. To support time-intense manual contact tracing, mobile contact tracing apps have been recently introduced that estimate and capture contacts as a measure against COVID-19. Bluetooth Low Energy (BLE) is seen as the most promising wireless technology for contact tracing [2]. The spatial distance between two smartphones is estimated based on the received signal strength indicator (RSSI), which is known to be challenged by non line-of-sight (LOS) conditions such as shielding by human bodies and multi-path propagation in particular in indoor environments. It has been shown that BLE RSSI based distance estimation requires additional concepts to increase accuracy [3]. Among alternative technologies, ultra-wideband (UWB) is a well-known technology often used for location estimation. Ultra-wideband (UWB) based distance calculation uses time-of-flight (ToF) measures and may be leveraged for contact tracing. Yet, UWB is currently only available in a few flagship smartphones such as Samsung Galaxy Note 20 or Apple iPhone 11. Further, UWB distance estimation relies on the cooperative exchange of time information [4], which is a potential privacy threat and will consume additional energy. In our work, we will use UWB for comparison and consider it as ground-truth measurement system. The focus of this paper rests on characterizing BLE RSSI based distance estimation for contact tracing. We make the following contributions: * We introduce our measurement testbed consisting of a mobile app that allows to capture BLE and UWB-based distance estimates, the latter for comparison. BLE-based distance estimation makes use of a linear and a logarithmic pathloss model that interpret BLE RSSI values measured onboard of the smartphone. UWB distance readings are retrieved from connected external UWB modules. * We summarize the achieved accuracy of distance and exposure estimation in our measurement campaign in different scenarios consisting of 20'535 BLE logs. Our results confirm that using BLE RSSI for distance estimation is challenging and that the linear distance estimation model provides better distance estimation accuracy than the logarithmic estimation model. We further show that in order to enhance exposure detection, awareness of the smartphones carrying position is more beneficial than knowledge about the environment. * We study the effect of body shielding and multi-path propagation in an anechoic chamber (comparable to outdoor scenarios), LOS indoors, and in a corner scenario indoors with varying phone carrying positions. We find that BLE multi-path propagation can reduce body shielding effects. Finally, we compare BLE-based distance estimation with the more robust and accurate UWB ToF based estimation. § RELATED WORK BLE is widely used in mobile contact tracing apps currently promoted by health authorities worldwide. To unify and support contact tracing apps, the Google/Apple API for exposure notification based on BLE has been proposed for iOS[https://developer.apple.com/documentation/exposurenotification, accessed November 17th, 2020] and Android[https://developers.google.com/android/exposure-notifications/exposure-notifications-api, accessed November 17th, 2020] platforms. These APIs notify users about exposures to infected people according to commonly agreed thresholds (too close, too long) while preserving privacy to a high degree. The Google/Apple notification API has been widely adopted in contact tracing apps [2]. Yet, BLE-based distance estimation is error-prone. The study presented in [5] shows that staged contacts in a tram (less than 2 m, longer than 15 minutes) do not lead to expected notifications of the API when using the contact tracing app of Switzerland or Germany, and only 50% of exposures are detected by the Italian app. Thus, it is crucial to study and improve the effectiveness of BLE for contact tracing in real environments. The use of BLE for distance estimation and indoor localization has been thoroughly studied in the past, see, e.g. [6, 7, 8, 9]. These studies focus on communication of devices with no or little human interaction. As the contact tracing use case requires to capture realistic use scenarios of smartphones, those studies need to be extended. Major contact tracing data-sets and data repositories are provided by the MIT PACT project [10] and the DP-3T initiative [11]. Closest to the research presented in this paper is the study of BLE RSSI based distance estimation presented in [3], where several isolated effects such as body shielding or complex real-world situations are investigated. In our study, we will add a new dataset (joint effect of body shielding and multi-path), study model options to derive distance estimates based on BLE RSSI values, and extend the investigation by providing UWB-based measurements in addition to BLE RSSI logs. § TESTBED IMPLEMENTATION The measurement testbed comprises off-the-shelf smartphones running a mobile measurement app that has been developed for the purpose of evaluating contact tracing technologies. The app captures BLE RSSI values provided by the onboard BLE module and UWB distance measurements retrieved from an external module. Fig. <ref> visualizes the testbed and its use. §.§ Testbed Hardware and Software The testbed smartphones are of type Samsung Galaxy S7 and Samsung XCover 4s (Android version 8 and 9, Android BLE API). We do not make use of the Google/Apple API as the aim of the study is to investigate lower-level information provided by the BLE module. Via USB port, the smartphones connect to the UWB sensors [12]. The core component of the sensor is the Decawave DW1000 UWB transceiver chip, which is widely used for indoor localization. (a) (b) Measurement equipment: (a) smartphones connected to UWB sensors, (b) UWB sensor mounted on the left arm of a person, smartphone held in front of trunk. The measurement app is a native Android app that provides a user interface to control an experiment and to log BLE RSSI and UWB distance values. The app logs last recent scan values at configurable time intervals; 1 s is the default setting. Upon start, the app sends BLE advertisements based on the Generic Attribute Profile (GATT). When enabled by the user, a BLE scan is started resulting in an asynchronous return of newly scanned devices in BLE range. To restrict the devices reported in the scan, filtering by universally unique identifier (UUID) is employed. §.§ BLE RSSI Transformation BLE RSSI values are recorded on the smartphones as obtained through the getRSSI() Android function, which returns the received signal strength $P_{RX,i}$ in dBm, limited to the range of $[-127, 126]$ dBm. Note that in the practical implementation, the lowest recorded RSSI value was -105$\,$dBm. From the RSSI values, the pathloss (PL) is calculated on the receiving device using calibration values from the GAEN database[https://developers.google.com/android/exposure-notifications/files/en-calibration-2020-08-12.csv] as \begin{equation} \text{PL} = P_{\text{TX},j} - (P_{\text{RX},i} + \Delta_{\text{RX},i})\, , \label{eq:BLEcalibration} \end{equation} where $\Delta_{\text{RX},i}$ denotes the RX calibration value for scanning device $i$ and $P_{\text{TX},j}$ the calibrated TX power level of advertising device $j$. For the used Samsung XCover these are $P_{\text{TX}} = -24\,$dBm and $\Delta_{\text{RX}} = 5$ dB, and for the Samsung S7 $P_{\text{TX}} = -33\,$dBm and $\Delta_{\text{RX}} = 10$ dB. §.§ BLE Distance Estimation and Exposure Detection To detect exposures, we apply a distance based threshold detector. Thereby, we first obtain a model-based distance estimate $\hat{d}$ from a pathloss measurement. If the estimated distance is below the threshold distance, the data point is marked as positive, and otherwise negative. The model-based exposure estimation makes use of a linear pathloss model and a logarithmic pathloss model corresponding to known free space signal propagation properties. The linear pathloss model is chosen to describe a basic correlation of RSSI values with ground truth distance and has no physical interpretation. It is given by \begin{equation} \text{PL}_\text{lin} = \text{PL}_\text{0,lin} + k\, d\, , \label{eq:model_lin} \end{equation} where $\text{PL}_\text{0,lin}$ in dB is the pathloss at distance $d=0$ and $k$ is the slope in dB/m. The distance is then estimated by \begin{equation} \hat{d}_\text{lin} = \text{max}\Big( \frac{\text{PL} - \text{PL}_\text{0,lin}}{k}, 0 \Big)\, ,\label{eq:estimation_lin} \end{equation} where the maximum function $\text{max}(\cdot,0)$ avoids the estimation of negative distances. The log-distance pathloss model [6] is given by \begin{equation} \text{PL}_\text{ld} = \text{PL}_\text{0,ld} + 10\, \gamma \, \text{log}10\Big(\frac{d}{d_0}\Big) + X_g \, , \label{eq:model_ld} \end{equation} where $\text{PL}_\text{0,ld}$ in dB is the pathloss at reference distance $d_0$, $\gamma$ is the pathloss exponent and $X_g$ is the zero-mean Gaussian noise in dB. In this work, we have chosen $d_0 = 2\,$m as it reflects the COVID-19 distance of interest. The estimated distance is then given by \begin{equation} \hat{d}_\text{ld} = d_0 \, 10^{\frac{\text{PL} - \text{PL}_\text{0,ld}}{10\,\gamma}} \,. \label{eq:estimation_ld} \end{equation} §.§ UWB Distance Estimation The UWB modules retrieve a distance estimate by time-of-flight (ToF) calculation, obtained from timestamps that are recorded upon packet transmission and reception. As the clocks of the receiver and the transmitter are not synchronized, the ToF can only be estimated. The most common estimation approach double-sided two-way ranging (DS-TWR) [4], which requires TX and RX time stamps of three packet exchanges. We implement an extension to DS-TWR, known as cooperative synchronization and ranging [13]. The distance estimates are updated every 250$\,$ms. § MEASUREMENT STUDY In our measurement study, we aim at quantifying the accuracy of our proposed distance estimators (linear model, logarithmic model) based on BLE RSSI values. As BLE signal propagation is known to be effected by body shielding and multi-path propagation [14], we will in particular study these effects. §.§ Experiment Setup Two test persons are each equipped with a smartphone (Samsung xCover, Samsung Galaxy S7) and connected UWB sensor that is mounted on the left upper arm of the person. The experiments are recorded by the measurement app. The following properties are varied in our experiments: Carrying position: The two test persons are always carrying the smartphone at the same position, which is either (i) head – the smartphone is held at the left ear, (ii) trunk – the smartphone is held in front of the trunk, or (iii) pelvis – the smartphone is carried in the left front trouser's pocket. Distance $d$: The distance between the two persons is varied from 1$\,$m to 6$\,$m in steps of Environment: An anechoic chamber, a corridor, and a corner are selected, as depicted in Fig. <ref> (a)-(c). Orientation: The relative orientation between the persons can be 0$^{\circ}$, 90$^{\circ}$, 180$^{\circ}$, or 270$^{\circ}$ for head and pelvis carrying positions, and 0$^{\circ}$or 180$^{\circ}$ for trunk; see Fig. <ref> (d). Overall, the experiment consists of 180 combinations, each setting is measured with a duration of 3 min. The BLE pathloss of the whole experiment is visualized in Fig. <ref> and ranges from 22 dB to 70 dB. In addition, the fitted linear model in (<ref>), here referred to as 'lin', and the fitted logarithmic model in (<ref>), referred to as 'l-d', are depicted. The correlation coefficient of the linear approximation is $r = 0.51$, which is a weak correlation between distance and pathloss caused by the high variability of pathloss at each distance. The noise $X_g$ standard deviation is high with $\sigma = 8.48$ dB. Schematics of environments: (a) anechoic chamber, (b) corridor, (c) corner. Locations of Person 1 and Person 2 are depicted by blue and red circles, respectively, at six timesteps (at each location, each person is $d/2$ away from 'x', the start location), and (d) relative orientations. Heat map of BLE pathloss at six distances, grouped by three environments (corner, corridor, anechoic chamber). The model parameters of the linear model are: $\text{PL}_{0,\text{lin}}=\arrayij{mydatalin}{1}{1}\,$dB, $k=\arrayij{mydatalin}{2}{1}\,$dB/m with a correlation coefficient of $r = \arrayij{mydatalin}{3}{1}$. The parameters of the log-distance model are: $\text{PL}_{0,\text{ld}}=\arrayij{mydatald}{1}{1}\,$dB for reference distance $d_0=2.5\,$m, $\gamma=\arrayij{mydatald}{2}{1}$, $X_g$ noise standard deviation $\sigma = \arrayij{mydatald}{3}{1}\,$dB. BLE mean pathloss over actual distance (mean std. deviation is 3.90$\,$dB). The black dashed and dotted lines depict the linear and, respectively, the log-distance fit, per environment and carrying position. Distance estimation error for (a) BLE with known carrying position and unknown environment, (b) BLE with unknown carrying position and known environment, and (c) UWB. The boxplots depict the median, 0.25 and 0.75 quantile and the corresponding whiskers. §.§ Distance and Exposure Estimation Accuracy The distance estimation accuracy is described by the root-mean-square error (RMSE) of the distance estimate and the ground truth distance. The exposure detector is configured with a threshold of 2.5 m, meaning that an exposure is detected when the estimated distance is below this threshold. The detector is evaluated by the true positive rate $r_p$, the true negative rate $r_n$, the $F_1$ score and the Matthews correlation coefficient (MCC).[The MCC is in the interval $[-1,1]$, where values >0 indicate a performance better than a random guess.] Four scenarios are studied to assess how awareness of the environment and/or the carrying position influences estimation accuracy. For evaluation, the dataset is first divided along the known context settings (e.g., anechoic, corridor, corner), then each class is split equally into a training set and a test set. The training data is used to derive individual pathloss models for both model types (lin, l-d) considering censored pathloss measurements [15]. Comparing the case without context awareness (unknown/unknown) with the case of full awareness (known/known) in Tab. <ref>, the detection rates $r_p$ and $r_n$ increase; hereby the impact of knowing the carrying position is larger than the impact of knowing the environment. This effect is also described by the MCC. Note that the MCC is always above zero, which indicates that the estimate is always better than a random guess. The highest observed accuracy is MCC$=0.47$ and $F_1=0.7$. Accuracy of Exposure Estimation §.§ Effect of Body Shielding To isolate the effect of body shielding, we now discuss the BLE attenuation in the anechoic chamber as visualized in Fig. <ref>, left column. The phone's carrying position and orientation have a major influence on the pathloss. In LOS scenarios (head, 180$^{\circ}$ and 270$^{\circ}$; trunk, 0$^{\circ}$; pelvis, 0$^{\circ}$) the mean pathloss is always lower than in the other non-LOS scenarios, at the respective same distance. The spread of pathloss is high in all carrying positions, i.e., up to 7.11 dB (head, $d=1$ m), 11.69 dB (trunk, $d=1$ m), and 6.53 dB (pelvis, $d=1$ m). The spread due to body shielding is often higher than the distance-dependent increase of attenuation. This is confirmed by the weak linear correlation coefficients between pathloss and distance (lin model), which are $r=0.4$ (head), $r=0.5$ (trunk), and $r=0.36$ (pelvis). (The l-d model shows a similar behavior.) These results show that in particular in short ranges important for contact tracing (up to 2m), accurate distance estimation cannot be expected. The largest spread is found for head scenarios. For example, at a distance of 1 m, the pathloss is 42$\,$dB and 44$\,$dB under LOS conditions (180$^{\circ}$ and 270$^{\circ}$) and when fully blocked by the head, it is 57$\,$dB (90$^{\circ}$). It is worth noting that with increasing distance this difference is decreasing due to a mean pathloss saturation at approximately 64$\,$dB (BLE packets with higher pathloss are lost). §.§ Effect of Multi-path Propagation Multi-path propagation is usually thought to hinder precise distance measurements. However, comparing the pathloss in a multi-path propagation environment (corridor, corner) to the pathloss in the anechoic chamber, the effect of body shielding is less severe due to reflections from the walls, as expressed by the smaller spread of pathloss at a given distance (see Fig. <ref>). The linear correlation coefficients between distance and pathloss (lin model) reflect this effect as well: $0.35 \leq r \leq 0.56$ (anechoic), $0.46 \leq r \leq 0.6$ (corridor), and $0.77 \leq r \leq 0.83$ (corner). These results indicate that in environments with multi-path signal propagation, distance estimation based on BLE pathloss may be possible without knowing the carrying position and orientation, which is not the case for non multi-path environments such as outdoor environments. The highest correlation between distance and pathloss is observed in the corner scenario. At lower distances $d < 2$ m, a low pathloss below 42 dB is measured, which increases strongly with larger distances. The corner itself is a major cause of this behavior as at smaller distances LOS conditions are given. The corner obstructs the LOS path of the signal at larger distances. §.§ Distance Estimation Error Fig. <ref>(a) and (b) visualize the error statistics of BLE (lin and l-d models, individually fit to the scenario (head, trunk, pelvis and anechoic, corridor, corner)), and (c) the error statistics of UWB. For UWB, an overall distance RMSE of $0.9\,$m is observed, while BLE shows a RMSE of $3\,$m. This makes UWB a model technology w.r.t. its accuracy. Further, UWB may be used to collect ground truth measurements in realistic scenarios. UWB measurements yield precise distance estimates for all experiment settings that allow LOS propagation. The signal of the LOS path is correctly recognized by the UWB transceiver amongst other reflected paths. Notably, in the anechoic chamber either the signal is strong enough to propagate through particular body parts or no signal is received at all. Wall penetration of the UWB signal may lead to small error-prone distance estimates (corridor, corner). For example, see the negative distance errors in the corner environment. (Note that due to mounting the UWB sensors on the arm, position and orientation is not an issue.) Concerning BLE distance estimation errors, Fig. <ref>(a) and (b) show that the spread of the distance errors increases with distance, irrespective of the context. The l-d model also shows a larger spread of values than the lin model in most § CONCLUSIONS To assess mobile contact tracing technology, we introduced a flexible measurement smartphone app capable to capture BLE RSSI and UWB time-of-flight distance measurements. Our experimental results reveal that BLE-based estimation of distance is sensitive to carrying positions and distance estimation. Distance estimation is not accurate but exposure detection is feasible (distances below 2.5 m). Remarkably, multi-path propagation can reduce the effect of body shielding which may be leveraged in indoor environments where reflections from walls occur. 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# Evaluation Discrepancy Discovery: A Sentence Compression Case-study Yevgeniy Puzikov Ubiquitous Knowledge Processing Lab (UKP Lab), Department of Computer Science, Technical University of Darmstadt https://www.ukp.tu-darmstadt.de Research done during an internship at Bloomberg L.P., London, United Kingdom. ###### Abstract Reliable evaluation protocols are of utmost importance for reproducible NLP research. In this work, we show that sometimes neither metric nor conventional human evaluation is sufficient to draw conclusions about system performance. Using sentence compression as an example task, we demonstrate how a system can game a well-established dataset to achieve state-of-the-art results. In contrast with the results reported in previous work that _showed correlation_ between human judgements and metric scores, our manual analysis of state-of- the-art system outputs demonstrates that high metric scores _may only indicate a better fit to the data_ , but not better outputs, as perceived by humans. The prediction and error analysis files are publicly released. 111https://github.com/UKPLab/arxiv2021-evaluation-discrepancy-nsc ## 1 Introduction ### 1.1 Task description Sentence compression is a Natural Language Processing (NLP) task in which a system produces a concise summary of a given sentence, while preserving the grammaticality and the important content of the original input. Both abstractive [Cohn and Lapata, 2008, Rush et al., 2015] and extractive [Filippova and Altun, 2013, Filippova et al., 2015, Wang et al., 2017, Zhao et al., 2018] approaches have been proposed to tackle this problem. Most researchers have focused on the extractive methods, which treat this as a deletion-based task where each compression is a subsequence of tokens from its original sentence (Figure 1). Dickinson, who competed in triple jump at the 1936 Berlin Games, was also a bronze medalist in both the long jump and triple jump at the 1938 Empire Games. (a) Input sentence Dickinson competed in triple jump at the 1936 Berlin Games. (b) Reference compression Dickinson was a bronze medalist in the long jump and triple jump at the 1938 Empire Games. (c) Another possible compression Figure 1: Sentence compression example from the Google Dataset: an input sentence and a reference compression. The compression candidate at the bottom is also a valid one, but would score low, because the n-gram overlap with the reference is small. In the past few years several novel methods have been proposed to tackle the task of sentence compression. Most of these methods have been evaluated using the Google Dataset [Filippova and Altun, 2013] or its derivatives. Most authors present approaches that show better metric scores; a few of them also describe human evaluation experiments and show that the proposed methods outperform previous work. However, there has been a serious lack of analysis done on actual model predictions. In this work we show that metric scores obtained on the Google Dataset might be misleading; a closer look at model predictions reveals a considerable amount of noise, which renders the trained model predictions ungrammatical. Another problem is that valid system outputs which do not match the references are severely penalized. For example, a plausible compression for the introductory example we used above would be Dickinson was a bronze medalist in the long jump and triple jump at the 1938 Empire Games. However, with the established evaluation protocol, this compression would score very low because of the insignificant token overlap with the reference. We showed that evaluating a system on the Google Dataset is tricky, and even human evaluation done in the previous years could not detect the issues described in this chapter. To summarize, our contributions in this study are: * • We introduce a simple method of sentence compression that established new state-of-the-art results, as measured by common metrics. * • We design an experiment with a contrived system which achieved even higher scores, but produced less grammatical and less informative outputs. * • We show that this discrepancy may be attributed to the noise in the dataset. ## 2 Data Analysis In our experiments we use the Google Dataset introduced by ?) 222https://bit.ly/2ZvTK9z This dataset was constructed automatically by collecting English news articles from the Internet, treating the first sentence of each article as an uncompressed sentence and creating its extractive compression using a set of heuristics and the headline of the article. The dataset contains 200 000 training and 10 000 evaluation instances; the first 1 000 data points from the latter are commonly used as a test set and the remaining 9 000 as a development set. Exploratory data analysis showed that the distribution of the training data is highly skewed, which is not surprising though, given the nature of the data. (a) Sentence length (b) Reference length (c) Token length (d) Compression ratio (CR) values Figure 2: Data analysis of the Google Dataset: length distributions of sentences, ground-truth compressions and tokens, and the distribution of compression ratio values. The numbers to the right of each box denote median values. In order to remove outliers and fit the computation budget, we removed instances which contained sentences longer than 50 tokens and compressions longer than 17 tokens. We also removed examples with tokens longer than 15 characters, since those in most cases denoted website links. Finally, we excluded cases with a compression ratio of more than 0.85 — those rare cases in most cases were too long to qualify as compressions. Evaluation on the development and test sets was done without any data filtering. ## 3 BERT-based Sentence Compression Most modern deletion-based compression systems adopt either a tree-pruning, or a sequence labeling approach. The former uses syntactic information to navigate over a syntactic tree of a sentence and decide which parts of it to remove [Knight and Marcu, 2000, McDonald, 2006, Filippova and Altun, 2013]. With the advent of sequence-to-sequence models it became possible to skip the syntactic parsing step and solve the task directly, by processing a sentence one token at a time and making binary decisions as to whether to keep a token or delete it [Filippova et al., 2015, Wang et al., 2017, Zhao et al., 2018, Kamigaito and Okumura, 2020]. The advantages of such approaches include a lesser chance of introducing error propagation from incorrect parsing decisions, as well as higher training and inference speed. For a long time the space of sequence-to-sequence models has been dominated by different variants of Recurrent Neural Networks (RNN) [Rumelhart et al., 1986]. However, a more recent Transformer architecture [Vaswani et al., 2017] has shown very promising results in many NLP tasks. Given the success of Bidirectional Encoder Representations from Transformers (BERT) [Devlin et al., 2019], and the fact that there has been no empirical evaluation of its performance in sentence compression, we decided to fill this gap and find out how well BERT-based models would cope with the task. We used pretrained BERT-base-uncased model weights 333https://huggingface.co/bert-base-uncased provided by the HuggingFace library [Wolf et al., 2020], and implemented a simple BertUni model which encodes the source sentence $S=\\{w_{1},w_{2},\dots w_{n}\\}$ and produces a sequence of vectors $V=\\{v_{1},v_{2},\dots v_{n}\\},v_{i}\in\mathbb{R}^{h}$. Each vector is fed into a dense layer with a logistic function as a non-linear function to produce a score $s_{i}\in[0,1]$ (Figure 3). If $s_{i}\geq 0.5$, the model output is 1 (and 0, otherwise). Figure 3: Schematic of the BertUni architecture. A simple decision rule was used to make a binary prediction: $decision=\begin{cases}\text{1},&\text{if }\mathit{s}\geq 0.5,\\\ \text{0},&\text{otherwise}.\end{cases}$ ## 4 Experiments ### 4.1 Automatic Metric Evaluation For automatic evaluation of sentence compression systems, most researchers follow ?; ?) and use the following two metrics: * • F1-score: harmonic mean of the recall and precision in terms of tokens kept in the target and the generated compressions. * • Compression ratio (CR): the length of the compression divided over the original sentence length. The former metric shows how close the model outputs are to the references. The latter one is supposed to measure the compression effectiveness. To make our results comparable to previous work, in our experiments we followed the same convention. However, we would like to note that _measuring CR in sentence compression might be redundant_ for several reasons. The first reason comes from the fact that data-driven sentence compressors are likely to produce outputs with a compression ratio most commonly seen in the training data references. In other words, CR is less a property of a system and more a characteristic of the dataset. This is supported by the fact that most models reported in the literature have the same compression ratio (in the range of 0.38–0.43, see Table 1). Secondly, it is not even clear how to treat compression ratio values: is a CR of 0.4 better than a CR of 0.5? Intuitively, yes, because it means a more concise compression. However, the compression is really better only if it retained more valuable information from the source. On the other side, defining the notion of informativeness/importance in sentence compression (and document summarization, in general) is an open problem and currently is not measured automatically. This means that the CR metric is a very one-sided proxy, too crude to be used for real automatic evaluation without balancing it with some recall-oriented metric. To put the evaluation of our approach into better context, we compare it with the following systems. All systems predict a sequence of binary labels which decide which tokens to keep or remove from the input sentence. ##### LSTM. ?) use a three-layer uni-directional Long Short-term Memory (LSTM) network [Hochreiter and Schmidhuber, 1997] and pretrained word2vec [Mikolov et al., 2013] embeddings as input representations. For comparison, we use the results for the best configuration reported in the paper (LSTM-PAR-PRES). This system parses the input sentence into a dependency tree, encodes the tree structure and passes the aggregated feature representations to the decoder LSTM. Unlike our approach, this system relies on beam search at inference time. ##### BiLSTM. ?) build upon LSTM approach, but introduces several modifications. It employs a bi-directional LSTM encoder and enriches the feature representation with syntactic context. In addition, it uses Integer Linear Programming (ILP) methods to enforce explicit constraints on the syntactic structure and sentence length of the output. ##### Evaluator-LM. ?) uses a bi-directional RNN to encode the input sentence and predict a binary label for each input token. In addition to token embeddings, the network uses vector representations of part-of-speech (POS) tags and dependency relations. The system is trained using the REINFORCE algorithm [Williams, 1992], the reward signal comes from a pretrained syntax-based language model (LM). ##### SLAHAN. ?) propose a modular sequence-to-sequence model that consists of several components. The system encodes a sequence of tokens using a combination of pretrained embeddings (Glove [Pennington et al., 2014], ELMO [Peters et al., 2018], BERT) and parses the input into a dependency graph. Three attention modules are employed to encode the relations in the graph, their weighted sum is passed to a selective gate. The output of the latter forms an input to a LSTM decoder. Despite its simplicity, the proposed BERT-based approach achieved very competitive scores (Table 1). Model \| F1$\uparrow$ \| CR$\downarrow$ ---\|---\|--- Evaluator-LM \| 0.851 \| 0.39 BiLSTM \| 0.800 \| 0.43 LSTM \| 0.820 \| 0.38 SLAHAN \| 0.855 \| 0.407 BertUni \| 0.857 $\pm 0.002$ \| 0.413 $\pm 0.004$ BertUni (dev) \| 0.860 $\pm 0.001$ \| 0.418 $\pm 0.004$ Table 1: The performance of the BertUni model on the test portion of the Google Dataset, compared to recent approaches. The last row shows BertUni’s performance on the development set. Comparing single performance scores (and not score distributions) of neural approaches is meaningless, because training neural models is non-deterministic in many aspects and depends on random weight initialization, random shuffling of the training data for each epoch, applying random dropout masks [Reimers and Gurevych, 2017]. This makes it hard to compare the scores reported in previous works and our approach. To facilitate a fair comparison with future systems, we report the mean and standard deviation of the BertUni scores averaged across ten runs with different random seeds. In order to understand where BertUni fails and what we could potentially improve upon, we conducted manual error analysis of its predictions. ### 4.2 Error analysis The purpose of error analysis is to find weak spots of a system, from the point of view of human evaluation. In sentence compression, previous work typically analyzed system predictions of the first 200 sentences of the test set, using a 5-point Likert scale to assess annotators’ opinions of the compressions’ _readability_ and _informativeness_ [Filippova et al., 2015]. Since error analysis is used for further system improvement and test sets should be used only for final evaluation, we perform error analysis on the development set. In order to do that, we retrieved BertUni’s predictions on the 200 dev set sentences which received the lowest F1 scores and manually examined them. Note that those are not random samples; the reason why we chose worst predictions is because we know that the system performed poorly on them. As for the quality criteria, we had to make certain adjustments. ?) mention that readability _covers the grammatical correctness, comprehensibility and fluency of the output_ , while informativeness measures _the amount of important content preserved in the compression_. In our opinion, merging several criteria into one synthetic index is a bad idea, because annotators can’t easily decide on the exact facet of evaluation. Given that there already exists a problem of distinguishing fluency and grammaticality, adding both of them to assess readability seems to be a bad design decision. The problem is aggravated by the fact that readability as a text quality criterion is already used by NLP researchers for estimating the _text complexity_ from a reader’s point of view [Vajjala and Meurers, 2012, Štajner and Saggion, 2013, Venturi et al., 2015, De Clercq and Hoste, 2016]. This made us conclude that readability is another overloaded criterion. Instead, we chose _grammaticality_ as the first quality criterion. We manually analyzed BertUni predictions on the 200 aforementioned samples, trying to identify common error patterns. The results are presented below. ##### Grammaticality. Out of 200 compressions, 146 (73 %) were deemed to be grammatical. The errors in the remaining instances have been classified into several groups (marked with _G_ in Figure 4a). (a) BertUni errors (b) BertBi-TF errors (c) Ground-truth errors Figure 4: Number of errors made by the evaluated approaches on the 200 development set instances where BertUni achieved the lowest F1 scores, as well as errors found in ground-truth compressions. Error types marked with _G_ are _grammaticality_ flaws; the remaining ones are errors of _informativeness_. Most of them were cases where grammatical clauses miss linking words, are _stitched_ together, making the output ungrammatical, as in the following compressions: * • I ’m said It ’s not Kitty Pryde superhero is the leader of the X-Men . * • He first Postal Vote result can be announced before 10PM . Another large error category was _finish_ : the compression was grammatical until the last retained token, where the sentence ended abruptly, rendering the compression incomplete: * • Activision Blizzard has confirmed some new statistics for its games including . * • The South Sydney star had no case to . A few system outputs incorrectly started with a relative or demonstrative pronoun. This happened when the system failed to retain parts of main clause of the sentence (_rd-pron_): * • That shows young people rapping while flashing cash and a handgun in a public park . Finally, one output missed a verb which was essential for ensuring grammaticality (_verb-miss_): * • People giant waves crash against the railway line and buildings at Dawlish . ##### Informativeness. Out of 200 compressions, 105 (52.5 %) were deemed to be informative, the errors in the remaining instances have been classified into several groups (marked with _I_ in Figure 4a). Most of these erroneous cases were compressions which missed certain information that was needed for understanding the context (_info-miss_). For example: * • Dolly Bindra filed a case . * • Mount Hope became the third largest city . A smaller, but still a large group of compressions started with unresolved personal pronouns, which made it hard to understand the subject (_p-pron_): * • She hopes her album Britney Jean will inspire people . * • He should be allowed to work freely till proven guilty . In some cases, omitting the context caused a change in the meaning of the sentence (_mean-change_). For example: * • Reference: […] Aleksandar Vucic […] voiced hope that Germany will give even stronger support to Serbia […] * • System: Aleksandar spoke Germany will give stronger support to Serbia . A large number of both grammatical and informative compressions did not match references (_I2_). Interestingly enough, in some cases the system outputs were better then the references: * • Reference: We saw their two and raised to three. * • System: Newport beat Hartlepool 2 0 . * • Reference: Who joins for the remainder of the season subject . * • System: Watford have announced the signing of Lucas Neill . More examples of compression errors are provided in Section A.1. ## 5 Evaluation Discrepancy When assessing the sentence compressions, we needed to compare system outputs with references. Manual examination revealed that many references themselves were flawed. This, in turn, meant that noise is inherent to the Google Dataset, and metric-based improvements on this data are misleading. To corroborate this claim, we conducted two experiments: the first tested the capacity of a more accurate system to ignore the noise and output compressions of better quality. In the second, we verified whether the noise came from the ground-truth data and attempted to quantify it. At first, we decided to implement more complex models that could potentially achieve better scores. We attempted to improve the grammatical quality of BertUni compressions by using the history of model predictions for making more informed decisions. We impelented and tested models that use BERT-encoded lastly-retained tokens at each prediction step as an additional input to the model (prediction history), similar to n-gram language models. As a history, BertBi and BertTri used one and two previously predicted tokens, respectively. BertBiSS and BertTriSS were the same as BertBi and BertTri, but used scheduled sampling training scheme to mitigate the exposure bias issue [Bengio et al., 2015]. According to the metric evaluation results, none of the more complex models outperformed BertUni (Table 2). Model \| F1$\uparrow$ \| CR$\downarrow$ ---\|---\|--- BertUni \| 0.860 $\pm 0.001$ \| 0.418 $\pm 0.004$ BertBi \| 0.849 $\pm 0.001$ \| 0.423 $\pm 0.005$ BertBiSS \| 0.840 $\pm 0.003$ \| 0.370 $\pm 0.005$ BertTri \| 0.847 $\pm 0.002$ \| 0.423 $\pm 0.007$ BertTriSS \| 0.843 $\pm 0.003$ \| 0.382 $\pm 0.006$ BertBi-TF \| 0.901 \| 0.423 Table 2: BERT-based model variants’ performance on the development set (mean and standard deviation across ten random seed values). BertBi-TF was run only once, since it is a “cheating” model that is not meant to be used in production. We used an unrealistic scenario and artificially made it easier for the model to make correct predictions. We trained a BertBi-TF model which builds upon BertBi, but at prediction time for history instead of model predictions uses ground-truth labels 444We call this model BertBi-TF, since it builds upon BertBi, but uses teacher forcing (TF) both at training and prediction time.. The development set result of BertBi-TF was an F1 score of 0.901, a 4-point improvement over BertUni. We retrieved this model’s predictions for the same 200 dev set sentences used for the error analysis of BertUni outputs, and manually examined them. The usual evaluation practice is to draw samples randomly, in order to not give an advantage to any system and not to bias the evaluation. However, in this work we approached the problem from a system- development perspective and attempted to assess the comparative performance of the approaches in the _worst-case_ scenario. If such a comparison is biased, then only in favor of BertBi-TF, because the drawn samples were the worst ones for BertUni, not BertBi-TF. We view this as sanity step, a regression test to ensure that the newer version of the system performs at least as well as the baseline on the challenging cases. We assessed BertBi-TF outputs from the same aspects of _grammaticality_ and _informativeness_ , as described in Section 4.2. ##### Grammaticality. Out of 200 compressions, only 44 (22 %) were found to be grammatical; we classified the errors in the remaining instances into groups (marked with _G_ in Figure 4b). The first and most prevalent is the already mentioned _stitch_ group which comprises around 80 % of all grammatical errors: * • The program has received FBS college game 2014 season . * • Tskhinvali region with Russia . The remaining errors are faulty compression endings (_finish_): * • The fine has been described as a slap on the . * • P Chidambaram sought . ##### Informativeness. A similar situation was observed when assessing the compressions’ informativeness — only 41 (20.5 %) instances were considered as correct. The distribution of errors (marked with _I_ in Figure 4b) indicates that more than 80 % of cases miss information by omitting important words: * • Dickinson was a . * • Wynalda is mixing . A smaller fraction of errors was comprised by the cases with unresolved personal pronouns: * • He is an education . * • It would win 45 to 55 seats in Odisha . The remaining errors were the cases where the system compressions changed the semantics of the input: * • Sentence: 612 ABC Mornings intern Saskia Edwards hit the streets of Brisbane to find out what frustrates you about other people. * • System: Saskia frustrates people . More examples of BertBi-TF errors are provided in Section A.2. We counted the cases in which predictions of BertBi-TF had better or worse quality, compared to BertUni. In terms of informativeness, BertBi-TF improved 15 and worsened 78 instances; in terms of grammaticality, 115 instances were perceived as less grammatical, versus only 13 improved cases, which makes it clear that BertBi-TF makes many more mistakes than BertUni, despite the higher metric scores. In order to verify our findings, we examined the ground-truth compressions in more detail. Only 63 (31.5 %) of these compressions were both grammatical and informative. Figure 4c shows a visualization of the error type distribution. We provide examples of noisy ground-truth compressions in Section A.3. The abundant errors related to the use of pronouns in the compressions were predominantly caused by the fact that many instances contained ground-truth compressions with unresolved pronouns; cleaning the data would likely result in better outputs. The _stitch_ , _finish_ and _info_miss_ errors can be attributed to the fact that many references have missing information or artifacts remaining from the automatic procedure that was used to create these compressions [Filippova and Altun, 2013]. Resolving these issues may require more elaborate strategies, beyond simple text substitution. ## 6 Discussion In this study we advanced the state-of-the-art for the task of sentence compression, and achieved that by designing a simple, but effective sequence labeling system based on the Transformer neural network architecture. While the proposed approach achieved the highest scores reported in the research literature, the main message of the study is not a higher score — it is the idea that NLP system evaluation might need to go beyond simple comparison of metric scores with human judgements. We found that a higher-scoring system can produce worse-quality outputs. We further provided some empirical evidence that this issue is caused by the noise in the training data. We call this finding a _discrepancy discovery_ , because existent sentence compression work does not explain our results, based on the established evaluation practices. The research papers we analyzed present automatic and human evaluation statistics that seem to overlook the data quality issue. Of course, the approaches proposed so far could still produce high-quality sentence compressions, but the absence of error analysis plants a seed of doubt into the reader. In this work, we question not the reported results, but the principles of the conventional evaluation workflow. None of the examined research papers drew attention to the quality of the data, even though it is known that the dataset was constructed automatically, and therefore should contain noisy examples, which should affect the output quality of any data-driven system. Previous work also overlooked the use of the compression ratio which seems to be too simplistic to call it a metric that measures the compression effectiveness. Finally, the employed sentence compression evaluation protocols do not assume having multiple references. We did not go into much detail about this issue, but provided an illustration at the beginning of the paper (the Dickson example). The space of possible compressions in deletion-based sentence compression is bound by sentence length. But because the definition of importance is left out, the candidate space is very large. The existence of only one reference brings additional requirements for evaluation metrics to work, and commonly used n-gram overlap metrics clearly do not satisfy these requirements. ## 7 Conclusion The presented results show that system output analysis is indispensable when assessing the quality of NLP systems. 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Association for Computational Linguistics. ## Appendix A Error Examples This section contains error examples of BertUni and BertBi-TF models, as well as errors in the gold standard. We publicly share the prediction and error analysis files at https://github.com/UKPLab/arxiv2021-evaluation-discrepancy- nsc. ##### Grammaticality. Error types: * • _finish_ : incomplete sentence with an abrupt ending, caused by omitting the last token(s); * • _stitch_ : grammatical clauses with missing linking words, as if they are stitched together, which renders the sentence ungrammatical; * • _rd-pron_ : incorrect sentence start with a relative or demonstrative pronoun; * • _verb-miss_ : missing a verb which is essential for ensuring the grammaticality of a sentence. ##### Informativeness. Error types: * • _info-miss_ : missing certain information that is needed for understanding the context; * • _p-pron_ : starting with an unresolved personal pronoun, which makes it hard to understand the subject; * • _mean-change_ : omitting the context causing a change of the meaning of the sentence. We also provide examples of alternative compressions which focus on parts of the input sentence, which are different from those present in the reference compressions. They are marked with an _I2_ label and are listed here, together with erroneous cases, because due to mismatches with references they lower the metric scores of the evaluated approaches. ### A.1 BertUni #### Grammatical Errors Table 3: Manual error analysis results: examples of _ungrammatical_ outputs of BertUni. Error \| Sentence \| Compression ---\|---\|--- finish \| A young cow trying to grab a cooling drink from a river in Hampshire in the hot weather had to be rescued by firefighters when it got stuck in the mud. \| A cow trying to grab a drink had rescued . \| Japan and the US reaffirmed Monday during a meeting in Tokyo with visiting Under Secretary for Terrorism and Financial Intelligence David S Cohen their ongoing cooperation on sanctions against Iran, a Japanese government source said Tuesday. \| Japan and the US reaffirmed . \| 612 ABC Mornings intern Saskia Edwards hit the streets of Brisbane to find out what frustrates you about other people. \| Saskia Edwards hit . \| Google Helpouts, which was launched this week, is a service that allows you to pay for brief one-on-one webcam master classes with a range of experts in various fields. \| Google Helpouts is . rd-pron \| Some parents and others in Bessemer City are complaining about a YouTube video that shows young people rapping while flashing cash and a handgun in a public park. \| That shows young people rapping while flashing cash and a handgun in a public park . \| “In my district, already reeling from the shutdown of our largest private employer, the highest energy costs in the country, and reduced government revenues, this shutdown, if it continues any longer, can be the final nail in our economic coffin”, said Rep. Christensen. \| This shutdown can be the final nail in our economic coffin . stitch \| Our obsession with Sachin Tendulkar and records has made us lose perspective to such an extent that what should have been widely condemned is being conveniently ignored. \| Should have been condemned is . \| Direct Link Pisegna JR. Without will amoxicillin work for a uti this sweet little boy told his best friend that he loved him. \| boy told his he loved . \| Visitors will find a mixture of old and new at Silver Springs State Park, which opened Tuesday near Ocala. \| will find and at Silver Springs State Park opened . \| Twitter took the first step toward its IPO by filing with US regulators on Thursday, September 12, 2013. \| Twitter took by filing with US regulators . \| Arsenio Hall lost control of his brand new Porsche Cayenne S and crashed his car on Monday night in El Lay! \| Arsenio Hall lost his and crashed his car in El Lay verb-miss \| People watch giant waves crash against the already damaged railway line and buildings at Dawlish during storms in south west England February 8, 2014. \| People giant waves crash against the railway line and buildings at Dawlish . #### Informativeness Errors Table 4: Manual error analysis results: examples of _uninformative_ outputs of BertUni. We also show examples of alternative compressions (_I2_) which deviate from ground-truth compressions, but cannot be considered as errors. Error \| Sentence \| Compression ---\|---\|--- info-miss \| Buyers should beware, even though there was a safety recall on some GM cars those vehicles are still being sold on Craigslist. \| Buyers should beware . \| New results show some improvement in test scores and graduation rates for local students, but experts say there’s still more work to be done. \| There ’s still more work to be done . \| Counting of postal votes have already commenced while Elections Commissioner Mahinda Deshapriya stated that he first Postal Vote result can be announced before 10PM. \| He first Postal Vote result can be announced before 10PM . \| When President Obama was elected in 2008 for his first term, he made a presidential decision that he would not give up his blackberry. \| He would not give up his blackberry . mean-change \| On this week’s “Hostages” season 1, episode 13: “Fight or Flight,” Ellen reveals to Duncan that she will not kill the President but will help him get what he needs, as long as he gives her something in return. \| Ellen reveals she will not kill the President . \| Serbia’s First Deputy Prime Minister Aleksandar Vucic spoke in Germany with former German chancellor Helmut Kohl about Serbia’s path towards the EU and its economic recovery/ During the talks, Vucic highlighted the important role of German investors in Serbia and voiced hope that Germany will give even stronger support to Serbia in the realisation of its European goals. \| Aleksandar spoke Germany will give stronger support to Serbia . p-pron \| “Being a five-time champion, he knows how to handle pressure. Anand generally puts a lid on his emotions.” \| He knows how to handle pressure \| Chelsea manager Jose Mourinho has made light of the managerial instability under Roman Abramovich by admitting he’s trying to break his own record. \| He ’s trying to break his own record . \| Katy Perry wasn’t lying when she said she had some “beautiful news to share,” because she is now the new face of COVERGIRL. \| She had some beautiful news to share . I2 \| Provisur Technologies has entered into an agreement with Scanico in which Scanico has become Provisur’s global partner in commercial freezing technology. \| Reference: Scanico has become Provisur ’s global partner in commercial freezing technology . \| \| System: Provisur Technologies has entered into an agreement with Scanico . \| Davina McCall has undergone medical tests after fears she may be suffering from hypothermia after battling severe weather in a Sport Relief challenge. \| Reference: she may be suffering from hypothermia after battling in a Sport Relief challenge . \| \| System: Davina McCall has undergone medical tests . ### A.2 BertBi-TF #### Grammatical Errors Table 5: Manual error analysis results: examples of _ungrammatical_ outputs of BertBi-TF Error \| Sentence \| Compression ---\|---\|--- finish \| A UKIP candidate who is standing for election in Enfield Town has defended a tweet in which he said a black comedian “should emigrate to a black country.” \| UKIP candidate has defended a tweet he said a black comedian should \| Wynalda, who was introduced as the Silverbacks’ new manager on Tuesday, is mixing a bit of Europe with a bit of Mexico with a bit of Silicon Valley in an approach that will eliminate the head-coaching position. \| Wynalda is mixing . \| Cell C has announced new pre-paid and contract packages that offer unlimited calls to any network. \| Cell C has announced . \| Radnor police received a report Sept. 3 from a cadet at Valley Forge Military Academy that another cadet struck him in the face. \| Radnor police received a report another . \| Diego Forlan scored directly from a corner this weekend to help Internacional to a 3-2 win over Fluminense. \| Diego Forlan scored to help Internacional to . \| This 1930s-built four bedroom detached seafront home in Worthing is immaculately presented and has been expertly modernised. \| This seafront home is . stitch \| Buyers should beware, even though there was a safety recall on some GM cars those vehicles are still being sold on Craigslist. \| Buyers should beware are being sold on Craigslist . \| Davina McCall has undergone medical tests after fears she may be suffering from hypothermia after battling severe weather in a Sport Relief challenge. \| Davina McCall has undergone tests be suffering from hypothermia . \| New results show some improvement in test scores and graduation rates for local students, but experts say there’s still more work to be done. \| Show some improvement in test scores graduation rates students but there . \| POLICE are looking for witnesses after a car was hit by a van which failed to stop on Friday, January 7. \| POLICE a . \| Our obsession with Sachin Tendulkar and records has made us lose perspective to such an extent that what should have been widely condemned is being conveniently ignored. \| With Sachin Tendulkar has made lose perspective what . \| Watford have this evening announced the signing of experienced defender Lucas Neill, who joins for the remainder of the season subject to international clearance. \| Watford announced the signing Lucas season . . \| \| #### Informativeness Errors Table 6: Manual error analysis results: examples of _uninformative_ outputs of BertBi-TF. We also show examples of alternative compressions (_I2_) which deviate from ground-truth compressions, but cannot be considered as errors. Error \| Sentence \| Compression ---\|---\|--- info-miss \| “Former President Mandela is still in a critical condition in hospital but shows sustained improvement,” President Jacob Zuma said in a statement. \| Mandela is still in a critical condition shows improvement Jacob . \| This 1930s-built four bedroom detached seafront home in Worthing is immaculately presented and has been expertly modernised. \| This seafront home is . \| ASI’s additional director general BR Mani said he was hopeful of Nalanda making it to the list, claiming that Nalanda was an important centre of art and culture even before the university came into being. \| He Nalanda was . \| Tata Martino explained that after two months in Barcelona, he is starting to understand how things work at the club, assuring that there seems to be a need to create a weekly crisis regarding the team’s style of play. \| He crisis . mean-change \| Provisur Technologies has entered into an agreement with Scanico in which Scanico has become Provisur’s global partner in commercial freezing technology. \| Provisur has become Provisur ’s partner in commercial freezing technology . \| TV presenter Katie Piper shows off her baby daughter for the first time and said motherhood has given her “true fulfillment”. \| Piper shows off her baby daughter motherhood . \| AFL powerbrokers yesterday met Collingwood officials over convicted criminal Marley Williams, who faces a possible club ban or even a league suspension. \| Powerbrokers met ban suspension . \| British rapper Professor Green was arrested on suspicion of drink driving on Sunday after he called police to report a robbery. \| Professor Green was arrested on suspicion of report a robbery . p-pron \| “Being a five-time champion, he knows how to handle pressure. Anand generally puts a lid on his emotions.” \| He puts a lid on his emotions \| PersonalWeb® today announced that it has reached a confidential settlement in its lawsuit with NEXSAN®, to which Nexsan has licensed the Truename patents from PersonalWeb. \| It has reached patents PersonalWeb . \| Britney Spears has said she hopes her latest album Britney Jean will inspire people and she wants to ’project positive energy out into the world. \| She hopes her album Britney will inspire . I2 \| On this week’s “Hostages” season 1, episode 13: “Fight or Flight,” Ellen reveals to Duncan that she will not kill the President but will help him get what he needs, as long as he gives her something in return. \| Reference: On this week ’s Hostages season 1 episode 13 Fight or Flight Ellen reveals . \| \| System: Episode Fight or Flight Ellen reveals she will not kill the President . \| Blustery winds arrived in Gwinnett on Wednesday and brought with them lower temperatures that caused the National Weather Service to issue a freeze warning for the area. \| Reference: that caused the National Weather Service to issue a freeze warning . \| \| System: Blustery winds arrived issue a freeze warning . \| BJP and JD today welcomed the five-year jail term handed down to RJD chief Lalu Prasad in the fodder scam case, saying it would send out a message that the law will catch up with the corrupt, however influential they might be. \| Reference: The law will catch up with the corrupt influential . \| \| System: BJP law will catch up with the corrupt . ### A.3 Ground Truth #### Grammatical Errors Table 7: Manual error analysis results: examples of _grammatical errors_ in ground-truth compressions, sampled from 200 development set instances with lowest BertUni F1 scores). Error \| Sentence \| Compression ---\|---\|--- finish \| Police investigating the unexplained death of a man in Taupo say his van appears to have broken down. \| Police investigating the unexplained death say . \| Akkineni Nageswara Rao was one of the Indian cinema’s stalwarts, who will be remembered for his rich contribution. \| Akkineni Nageswara Rao was one . \| Mortgage fees are going up so where does Pa. \| Where does Pa . \| Coffee chain Starbucks has said guns are no longer “welcome” in its US cafes, although it has stopped short of an outright ban. \| Starbucks has said guns are . rd-pron \| Way back in May 2011, Google filed a patent application for eye tracking technology, which would allow it to charge advertisers on a ’pay per gaze’ basis. \| Which would allow it to charge advertisers on a pay per gaze basis . \| POLICE are looking for witnesses after a car was hit by a van which failed to stop on Friday, January 7. \| Which failed to stop . \| Tomorrow South Africa will celebrate the centenary of the Union Buildings in Pretoria which have recently been declared a national heritage site by the South African Heritage Resources Agency. \| Which have been declared a national heritage site . \| In a press release, Patrick said Goldstein will be replaced by Rachel Kaprielian, who is currently the state’s registrar of motor vehicles. \| Who is the state ’s registrar . stitch \| Iran wants to end the stand-off with global powers over its nuclear programme swiftly, but will not sacrifice its rights or interests for the sake of a solution, President Hassan Rouhani said on Friday. \| Iran wants but will not sacrifice its rights Hassan Rouhani said . \| Maggie Rose sheds her innocence in her brand new music video for “Looking Back Now.” \| Maggie Rose sheds for Looking Back Now \| The Muskingum University chapter of Omicron Delta Kappa has made a donation of more than $600 to the New Concord Food Pantry in an effort to give back to the community. \| The Muskingum University chapter of Omicron Delta Kappa has made in an effort to give back to the community . \| Dolly Bindra filed a case on an unknown person for having threatened her at gun point today in Oshiwara, Mumbai. \| Dolly Bindra filed for having threatened her at gun point in Oshiwara Mumbai . \| North Korean leader Kim Jong-un has met with the top military leaders and warned them of a grave situation and threatened a new nuclear test. \| Kim Jong-un has met and warned of a grave situation and threatened a nuclear test . #### Informativeness Errors Table 8: Manual error analysis results: examples of _informativeness_ errors in ground-truth compressions, sampled from 200 development set instances with lowest BertUni F1 scores). Error \| Sentence \| Compression ---\|---\|--- info-miss \| Some parents and others in Bessemer City are complaining about a YouTube video that shows young people rapping while flashing cash and a handgun in a public park. \| Some parents in Bessemer City are complaining about a video . \| Tata Martino explained that after two months in Barcelona, he is starting to understand how things work at the club, assuring that there seems to be a need to create a weekly crisis regarding the team’s style of play. \| There seems to be a need to create a weekly crisis . \| Nothing is ever left behind in a BREACHED performance as the loud rocking, heavy amp cranky band announce Toronto show dates since performing last October at Indie Week. \| Band announce Toronto show dates \| Prime Minister Kevin Rudd has missed the deadline for an August 24 election, with his deputy saying “people should just chill out” about the election date. \| People should chill out about the election date . p-pron \| Davina McCall has undergone medical tests after fears she may be suffering from hypothermia after battling severe weather in a Sport Relief challenge. \| She may be suffering from hypothermia after battling in a Sport Relief challenge . \| TV presenter Nick Knowles has been the recipient of some unexpected abuse as a result of an announcement that he will not be present at the birth of his child. \| He will not be present at the birth . \| England fast bowler James Anderson does not feel sorry for Australia and has said his team wants to win the Ashes 5-0. \| His team wants to win the Ashes 5 0 . \| If he decides to run for president, New Jersey Gov. Chris Christie will need to push back against the inevitable pressure that he will encounter to move to the right. \| He will encounter to move to the right . \| Armaan will be taken for a medical examination and post that he will be presented in the court today. \| He will be presented in the court .
# Repeated randomized algorithm for the Multicovering Problem Abbass Gorgi<EMAIL_ADDRESS>Mourad El Ouali<EMAIL_ADDRESS>Anand Srivastav<EMAIL_ADDRESS>Mohamed Hachimi <EMAIL_ADDRESS>Engineering Science Laboratory, University Ibn Zohr, Agadir, Morocco Department of Computer Science, Christian Albrechts University, Kiel, Germany ###### Abstract Let $\mathcal{H}=(V,\mathcal{E})$ be a hypergraph with maximum edge size $\ell$ and maximum degree $\Delta$. For given numbers $b_{v}\in\mathbb{N}_{\geq 2}$, $v\in V$, a set multicover in $\mathcal{H}$ is a set of edges $C\subseteq\mathcal{E}$ such that every vertex $v$ in $V$ belongs to at least $b_{v}$ edges in $C$. set multicover is the problem of finding a minimum-cardinality set multicover. Peleg, Schechtman and Wool conjectured that unless $\mathcal{P}=\mathcal{NP}$, for any fixed $\Delta$ and $b:=\min_{v\in V}b_{v}$, no polynomial-time approximation algorithm for the set multicover problem has an approximation ratio less than $\delta:=\Delta-b+1$. Hence, it’s a challenge to know whether the problem of set multicover is not approximable within a ratio of $\beta\delta$ with a constant $\beta<1$. This paper proposes a repeated randomized algorithm for the set multicover problem combined with an initial deterministic threshold step. Boosting success by repeated trials, our algorithm yields an approximation ratio of $\max\left\\{\frac{15}{16}\delta,\left(1-\frac{(b-1)\exp\left(\frac{3\delta+1}{8}\right)}{72\ell}\right)\delta\right\\}$. The crucial fact is not only that our result improves over the approximation ratio presented by Srivastav et al (Algorithmica 2016) for any $\delta\geq 13$, but it’s more general since we set no restriction on the parameter $\ell$. Furthermore, we prove that it is NP-hard to approximate the set multicover problem on $\Delta$-regular hypergraphs within a factor of $(\delta-1-\epsilon)$. Moreover we show that the integrality gap for the set multicover problem is at least $\frac{\ln_{2}(n+1)}{2b}$, which for constant $b$ is $\Omega(\ln n)$. ###### keywords: Integer linear programs, hypergraphs, approximation algorithms, randomized rounding, set cover and set multicover. ## 1 Introduction This work was intended as an attempt to solve approximately the set multicover problem. A nice formulation of this problem may be given by the notion of hypergraphs. A hypergraph is a pair $\mathcal{H}=(V,\mathcal{E})$, where $V$ is a finite set and $\mathcal{E}\subseteq 2^{V}$ is a family of some subsets of $V$. We call the elements of $V$ vertices and the elements of $\mathcal{E}$ (hyper-)edges. Further, let $n:=\|V\|$, $m:=\|{\cal E}\|$. W.l.o.g. let the vertices be enumerated as $v_{1},v_{2},\dots,v_{n}$ and the edges as $E_{1},E_{2},\dots,E_{m}$. As usually the degree of a vertex $v$ (notation $d(v)$) is the number of hyperedges it appears in. Let $\Delta:=\max_{v\in V}d(v)$ be the maximum degree. Furthermore, if the degree of every vertex is exactly $\Delta$, then ${\cal H}$ is called $\Delta$-regular. We define the number of vertices of a hyperedge as its size. If the size of all hyperedges is exactly $\ell$, i.e., $\forall E\in\mathcal{E},\,\|E\|=\ell$, then $\mathcal{H}$ is $\ell$-uniform. Let $\mathbf{b}:=(b_{1},b_{2},\dots,b_{n})\in\mathbb{N}_{\geq 2}^{n}$ be given. If a vertex $v_{i}$, $i\in[n]$, is contained in at least $b_{i}$ edges of some subset $C\subseteq\mathcal{E}$, we say that the vertex $v_{i}$ is fully covered by $b_{i}$ edges in $C$. A set multicover in $\mathcal{H}$ is a set of edges $C\subseteq\mathcal{E}$ such that every vertex $v_{i}$ in $V$ is fully covered by $b_{i}$ edges in $C$. The set multicover problem is the task of finding a set multicover of minimum cardinality. Related Work. The set cover problem $(b=1)$ is known to be NP-hard [14] and has been intensively explored for decades. Several deterministic approximation algorithms are exhibited for this problem [1, 10, 12, 16], all with approximation ratios $\Delta$. Furthermore, Johnson [13] and Lovász [17] gave a greedy algorithm with performance ratio $H(\ell)$, where $H(\ell)=\sum_{i=1}^{\ell}\frac{1}{i}$ is the harmonic number. Notice that $H(\ell)\leq 1+\ln(\ell)$. For hypergraphs with bounded $\ell$, Duh and Fürer [4] used the technique called semi-local optimization, improving $H(\ell)$ to $H(\ell)-\frac{1}{2}$. Unlike the set cover problem, the case $b\geq 2$ of the set multicover problem is less known. Let us give a summary of the known approximability results. In paper [21], Vazirani using primal-dual schema extended the result of Lovász [17] for $b\geq 1$. Later Fujito et al. [9] improved the algorithm of Vazirani and achieved an approximation ratio of $H(\ell)-\frac{1}{6}$ for $\ell$ bounded. Hall and Hochbaum [11] achieved by a greedy algorithm based on LP duality an approximation ratio of $\Delta$. By a deterministic threshold algorithm Peleg, Schechtman, and Wool in 1997 [19, 20] improved this result and gave an approximation ratio of $\delta$. They were also the first to propose an approximation algorithm for the set multicover problem with approximation ratio below $\delta$, namely a randomized rounding algorithm with performance ratio $(1-(\frac{c}{n})^{\frac{1}{\delta}})\cdot\delta$ for a small constant $c>0$. However, their ratio is depending on $n$, and asymptotically tends to $\delta$. Furthermore Peleg, Schechtman and Wool conjectured that for any fixed $\Delta$ and $b:=\min_{i\in[n]}b_{i}$ the problem cannot be approximated by a ratio smaller than $\delta:=\Delta-b+1$ unless $\mathcal{P}=\mathcal{NP}$. Hence it remained an open problem whether an approximation ratio of $\beta\delta$ with $\beta<1$ constant can be proved. A randomized algorithm of hybrid type was later given by Srivastav et al [7]. Their algorithm achieves for hypergraphs with $l\in\mathcal{O}\left(\max\\{(nb)^{\frac{1}{5}},n^{\frac{1}{4}}\\}\right)$ an approximation ratio of $\left(1-\frac{11(\Delta-b)}{72l}\right)\cdot\delta$ with constant probability. Concerning the algorithmic complexity, the set multicover problem has still not been investigated. In contrast to the set cover problem, it is known that the problem is hard to approximate to within $\Delta-1-\epsilon$, unless $\mathcal{P}=\mathcal{NP}$ [2], and to within $\Delta-\epsilon$ under the UGC [15] for any fixed $\epsilon>0$. Unless $\mathcal{P}=\mathcal{NP}$ there is no $(1-\epsilon)\ln n$ approximation [8]. This motivated us to study this aspect of the problem. Our Results. The main contribution of our paper is the combination of a deterministic threshold-based algorithm with repeated randomized rounding steps. The idea is to algorithmically discard instances that can be handled deterministically in favor of instances for which we obtain a constant-factor approximation less than $\delta$ using a repeated randomized strategy. Our hybrid randomized algorithm is designed as a cascade of a deterministic and a repeated randomized rounding step followed by greedy repair if the randomized solution is not feasible. First, the relaxed problem of the set multicover problem is solved. The successive actions depend on the cardinality of a set of hyperedges that will be defined according to the relaxed problem output. Our algorithm is an extension of an example given in [5, 6, 7, 10, 11, 20] for the vertex cover, partial vertex cover and set multicover problem in graphs and hypergraphs. The methods used in this paper rely on an application of an extension of the Chernoff-Hoeffding bound theorem for sums of independent random variables and are based on estimating the variance of the summed random variables for invoking the Chebychev-Cantelli inequality. Our algorithm yields a performance ratio of $\max\left\\{\frac{15}{16}\delta,\left(1-\frac{(b-1)\exp\left(\frac{3\delta+1}{8}\right)}{72\ell}\right)\delta\right\\}$. This ratio means a constant factor of less than $\delta$ for many settings of the parameters $\delta$, $b$, and $\ell$. It is asymptotically better than the former approximation ratios due to Peleg et al. and Srivastav et al. Furthermore, using a reduction of the set cover problem on $\Delta$-regular hypergraphs to the set multicover problem on $\Delta+b-1$-regular hypergraphs, we show that it is NP-hard to approximate the set multicover problem on $\Delta$-regular hypergraphs within a factor of $(\delta-1-\epsilon)$. Moreover, we show that the integrality gap for the natural LP formulation of the set multicover problem is at least $\frac{\ln_{2}(n+1)}{2b}$, which for constant $b$ is $\Omega(\ln n)$. Fundamental results and approximations for set multicover problem Hypergraph \| Approximation ratio ---\|--- - \| $H(\ell)$[21] bounded $\ell$ \| $H(\ell)-\frac{1}{6}$ [9] - \| $\delta$ [11, 20] - \| $(1-(\frac{c}{n})^{\frac{1}{\delta}})\cdot\delta$ where $c>0$ is a constant. [19] $l\in\mathcal{O}\left(\max\\{(nb)^{\frac{1}{5}},n^{\frac{1}{4}}\\}\right)$ \| $\left(1-\frac{11(\Delta-b)}{72\ell}\right)\cdot\delta$ [7] - \| $\max\left\\{\frac{15}{16}\delta,\left(1-\frac{(b-1)\exp\left(\frac{3\delta+1}{8}\right)}{72\ell}\right)\delta\right\\}$ (this paper) Outline of the paper. In Section 2, we give all the definitions and the tools needed for our analysis. In Section 3, we present a randomized algorithm of hybrid type and its analysis. In Section 4, we give a lower bound for the problem. In Section 5, we discuss the integrality gap of the LP formulation of the problem. ## 2 Definitions and preliminaries For the later analysis we will use the following extension of Chernoff- Hoeffding Bound inequality for a sum of independent random variables. It is often used if one only has a bound on the expectation: ###### Theorem 1 (see [3]) Let $X_{1},\ldots,X_{n}$ be independent $\\{0,1\\}$-random variables. Let $X=\sum_{i=1}^{n}X_{i}$ and suppose $\mathbb{E}(X)<\mu$. For every $0<\beta\leq 1$ we have $\Pr[X\geq(1+\beta)\mu]\leq\exp{\left(-\frac{\beta^{2}\mu}{3}\right)}.$ A further useful concentration theorem we will use is the Chebychev-Cantelli inequality: ###### Theorem 2 (see [18], page 64) Let $X$ be a non-negative random variable with finite mean $\mathbb{E}(X)$ and variance Var$(X)$. Then for any $a>0$ it holds that $\displaystyle\Pr(X\leq\mathbb{E}(X)-a)$ $\displaystyle\leq$ $\displaystyle\frac{{\rm Var}(X)}{{\rm Var}(X)+a^{2}}\cdot$ Our lower bound proof for the problem relies on extending the following theorem from the case of $b=1$ to the case of $b\geq 2$. ###### Theorem 3 (I. Dinur et al, 2005 [2]) For every integer $l\geq 3$ and every $\epsilon>0$, it is NP-hard to approximate the minimum vertex cover problem on $\ell$-uniform hypergraphs within a factor of $(\ell-1-\epsilon)$. A key notion of linear programming relaxations is the concept of Integrality Gap. ###### Definition 1 Let $\cal{I}$ be a set of instances, the Integrality Gap for minimization problems is defined as $\sup_{i\in\cal{I}}{\frac{{\rm Opt}(I)}{{\rm Opt}^{}(I)}}.$ ## 3 The multi-randomized rounding algorithm Let ${\cal H}=(V,{\cal E})$ be a hypergraph with maximum vertex degree $\Delta$ and maximum edge size $\ell$. An integer linear programming formulation of the set multicover problem is the following: $\displaystyle\min\sum_{j=1}^{m}x_{j},$ $\displaystyle\mbox{ILP}(\Delta,{\bf b}):\qquad$ $\displaystyle\sum_{j=1}^{m}a_{ij}x_{j}\geq b_{i}\quad\mbox{ for all }i\in[n],$ $\displaystyle x_{j}\in\\{0,1\\}\quad\mbox{ for all }j\in[m],$ where $A=(a_{ij})_{i\in[n],\,j\in[m]}\in\\{0,1\\}^{n\times m}$ is the vertex- edge incidence matrix of ${\cal H}$ and ${\bf b}=(b_{1},b_{2},\dots,b_{n})\in\mathbb{N}_{\geq 2}^{n}$ is the given integer vector. For every vertex $v$, we define $\Gamma(v):=\\{E\in\mathcal{E}\mathrel{\|}v\in E\\}$ the set of edges incident to $v$. The linear programming relaxation LP($\Delta,\,{\bf b}$) of ILP($\Delta,\,{\bf b}$) is given by relaxing the integrality constraints to $x_{j}\in[0,1]$ for all $j\in[m]$. Let $\mathrm{Opt}$ resp. ${\rm Opt}^{}$ be the value of an optimal solution to ILP($\Delta,\,{\bf b}$) resp. LP($\Delta,\,{\bf b}$). Let $(x^{\ast}_{1},\ldots,x^{\ast}_{m})$ be the optimal solution of the LP($\Delta,\,{\bf b}$). So ${\rm Opt}^{}=\sum_{j=1}^{m}x^{}_{j}$ and ${\rm Opt}^{}\leq\mathrm{Opt}$. The next lemma shows that the $b_{i}$ greatest values of the LP variables corresponding to the incident edges for any vertex $v_{i}$ are all greater than or equal to $\frac{1}{\delta}$. ###### Lemma 1 (see [20]) Let $b_{i},d,\Delta,n\in\mathbb{N}$ with $2\leqslant b_{i}\leqslant d-1\leqslant\Delta-1,i\in[n]$ . Let $x_{j}\in[0,1],j\in[d]$, such that $\displaystyle\sum_{j=1}^{d}x_{j}\geqslant b_{i}$. Then at least $b_{i}$ of the $x_{j}$ fulfill the inequality $x_{j}\geqslant\frac{1}{\delta}$. Our second lemma shows that the $b_{i}-1$ greatest values of the LP variables corresponding to the incident edges for any vertex $v_{i}$ are all greater than or equal to $\frac{2}{\delta+1}$ and with Lemma 1 we take the sum over the $b_{i}$ greatest values of the LP variables corresponding to the incident edges for any vertex $v_{i}$. ###### Lemma 2 Let $b_{i},d,\Delta,n\in\mathbb{N}$ with $2\leqslant b_{i}\leqslant d-1\leqslant\Delta-1,i\in[n]$ . Let $x_{j}\in[0,1],j\in[d]$, such that $\displaystyle\sum_{j=1}^{d}x_{j}\geqslant b_{i}$. Then at least $b_{i}-1$ of the $x_{j}$ fulfill the inequality $x_{j}\geqslant\frac{2}{\delta+1}$ and there exists an element $x_{j}$, distinct to all of them, that fulfills the inequality $x_{j}\geqslant\frac{1}{\delta}$ . ###### Proof 1 W.l.o.g. we suppose $x_{1}\geq x_{2}\geq\cdots\geq x_{b_{i}}\geq\cdots\geq x_{d}$. Hence $b_{i}-2\geq\displaystyle\sum_{j=1}^{b_{i}-2}x_{j}$ and $(d-b_{i}+2)x_{b_{i-1}}\geq\displaystyle\sum_{j=b_{i}-1}^{d}x_{j}$. Then $\displaystyle b_{i}-2+(\Delta-b+2)x_{b_{i}-1}$ $\displaystyle\geq$ $\displaystyle b_{i}-2+(\Delta-b_{i}+2)x_{b_{i}-1}$ $\displaystyle\geq$ $\displaystyle b_{i}-2+(d-b_{i}+2)x_{b_{i}-1}$ $\displaystyle\geq$ $\displaystyle\sum_{j=1}^{b_{i}-2}x_{j}+\sum_{j=b_{i}-1}^{d}x_{j}=\displaystyle\sum_{j=1}^{d}x_{j}$ $\displaystyle\geq$ $\displaystyle b_{i}$ So we have $x_{b_{i}-1}\geq\frac{2}{\delta+1}$. Since for all $j\in[b_{i}-1]\;,\ x_{j}\geq x_{b_{i}-1}$ then for all $j\in[b_{i}-1]\;,\;x_{j}\geq\frac{2}{\delta+1}$. Furthermore, by Lemma $1$ and the assumption on the orders of the variables $x_{j}$, for all $j\in[b_{i}]\;$ we have $x_{j}\geq\frac{1}{\delta}$ and particularly $x_{b_{i}}\geq\frac{1}{\delta}$. ### 3.1 The algorithm In this section we present an algorithm with conditioned randomized rounding based on the properties satisfied by two generated sets, $C_{1}$ and $C_{2}$. Input : A hypergraph $\mathcal{H}=(V,\,\mathcal{E})$ with maximum degree $\Delta$ and maximum hyperedge size $\ell$, numbers $b_{i}\in\mathbb{N}_{\geq 2}\text{ for }i\in[n]$, $b:=\min_{i\in[n]}b_{i}$, $\epsilon\in(0,1)$, a constant $k\in\mathbb{N}_{\geq 2}$ and $\delta=\Delta-b+1$. Output : A set multicover $C$ 1. 1. Initialize $C:=\emptyset$. Set $\lambda=\frac{\delta+1}{2}\;$, $\alpha=\frac{(b-1)\delta\epsilon^{k}}{6\ell}\times\exp\left(a_{k,\epsilon}\right)$ with $a_{k,\epsilon}=\frac{k(1-\epsilon)+(\delta-1)(1-\epsilon^{a})}{2}$ and $\lambda_{0}=(1-\epsilon)\delta$. 2. 2. Obtain an optimal solution $x^{}\in[0,1]^{m}$ by solving the LP($\Delta,\,{\bf b}$) relaxation. 3. 13. Set $C_{1}:=\\{E_{j}\in\mathcal{E}\mathrel{\|}x_{j}^{\ast}\geq\frac{1}{\lambda}\\}$, $\ C_{2}:=\\{E_{j}\in\mathcal{E}\mathrel{\|}\frac{1}{\lambda}>x_{j}^{\ast}\geq\frac{1}{\delta}\\}$ and $C_{3}:=\\{E_{j}\in\mathcal{E}\mathrel{\|}0<x_{j}^{\ast}<\frac{1}{\delta}\\}$. * 4. Take all edges of the set $C_{1}$ in the cover $C$. 25. if $\|C_{1}\|\geq\alpha\cdot\mathrm{Opt}^{}$ then return $C=C_{1}\cup C_{2}$. Else (Multi-randomized Rounding) 1. 3(a) For all edges $E_{j}\in C_{2}$ include the edge $E_{j}$ in the cover $C$, independently for all such $E_{j}$, with probability $\lambda_{0}x_{j}^{}$, $k$ times. 4 $\left(\text{ If, in any of these $k$ biased coin flips shows head, include the edge }E_{j}\text{ in the cover.}\right)$ * (b) For all edges $E_{j}\in C_{3}$ include the edge $E_{j}$ in the cover $C$, independently for all such $E_{j}$, with probability $(1-\epsilon^{k})\delta x_{j}^{}$. (c) (Repairing) Repair the cover $C$ (if necessary) as follows: Include arbitrary edges from $C_{2}$, incident to the vertices $v_{i}$ not fully covered, to $C$ until all vertices are fully covered. (d) Return the cover $C$. Algorithm 1 SET MULTICOVER In step $2$ we solve the linear programming relaxation LP($\Delta,\,{\bf b}$) in polynomial time, using some known polynomial-time procedure, e.g. the interior point method. Next we take into the cover all edges of the sets $C_{1}$ resp. $C_{2}$. Since the LP variable value $x^{}_{j}$ that corresponds to an edge $E_{j}$ from the set $C_{1}$ is greater than or equal to $\frac{2}{\delta+1}$ and the value $x^{}_{j}$ that corresponds to an edge $E_{j}$ from the set $C_{2}$ is less than $\frac{2}{\delta+1}$, we have $\|C_{1}\|+\|C_{2}\|=\|C\|\text{\quad and\quad}C_{1}\cap C_{2}=\emptyset$ (1) ### 3.2 Analysis of the algorithm Case $\mathbf{\|C_{1}\|\geq\alpha\cdot\mathrm{Opt}^{}}$. ###### Theorem 4 Let $\mathcal{H}$ be a hypergraph with maximum vertex degree $\Delta$ and maximum edge size $\ell$. Let $\alpha=\frac{(b-1)\delta\epsilon^{k}}{6\ell}\times\exp\left(a_{k,\epsilon}\right)$ with $a_{k,\epsilon}=\frac{k(1-\epsilon)+(\delta-1)(1-\epsilon^{a})}{2}$ as defined in Algorithm 1. If $\|C_{1}\|\geq\alpha\cdot\mathrm{Opt}^{}$ then Algorithm 1 returns a set multicover $C$ such that $\|C\|<\left(1-\frac{(b-1)\epsilon^{k}}{18\ell}\times\exp\left(a_{k,\epsilon}\right)\right)\delta\cdot\mathrm{Opt}^{}$ ###### Proof 2 The proof is straightforward, using the definitions of the sets $C_{1}$ and $C_{2}$. $\displaystyle\delta\mathrm{Opt}^{}=\sum_{j=1}^{m}\delta x^{}_{j}$ $\displaystyle\geq$ $\displaystyle\displaystyle\sum_{E_{j}\in C_{1}}\delta x^{}_{j}+\sum_{E_{j}\in C_{2}}\delta x^{}_{j}$ $\displaystyle\geq$ $\displaystyle\frac{2\delta}{\delta+1}\|C_{1}\|+\|C_{2}\|$ $\displaystyle\geq$ $\displaystyle\frac{2\delta}{\delta+1}\|C_{1}\|+\left(\|C\|-\|C_{1}\|\right)$ $\displaystyle\geq$ $\displaystyle\frac{\delta-1}{\delta+1}\|C_{1}\|+\|C\|$ $\displaystyle\overset{\delta\geq 2}{\geq}$ $\displaystyle\frac{1}{3}\|C_{1}\|+\|C\|$ $\displaystyle\geq$ $\displaystyle\frac{1}{3}\alpha\cdot\mathrm{Opt}^{}+\|C\|.$ Hence $\|C\|\leq\left(1-\frac{(b-1)\epsilon^{k}}{18\ell}\times\exp\left(a_{k,\epsilon}\right)\right)\delta\cdot\mathrm{Opt}^{}$ Case $\mathbf{\|C_{1}\|<\alpha\cdot\mathrm{Opt}^{}}$. Let $X_{1},\ldots,X_{m}$ be $\\{0,1\\}$-random variables defined as follows: $\displaystyle X_{j}=\begin{cases}1&\text{if the edge}\,E_{j}\,\text{was picked into the cover before repairing}\\\ 0&\text{otherwise}.\end{cases}$ Note that the $X_{1},\ldots,X_{m}$ are independent for a given $x^{}\in[0,1]^{m}$. For all $i\in[n]$ we define the $\\{0,1\\}$\- random variables $Y_{i}$ as follows: $\displaystyle Y_{i}=\begin{cases}1&\text{if the vertex}~{}v_{i}~{}\text{is fully covered before repairing}\\\ 0&\text{otherwise}.\end{cases}$ We denote by $X:=\sum_{j=1}^{m}X_{j}$ and $Y:=\sum_{i=1}^{n}Y_{i}$ the cardinality of the cover and the cardinality of the set of fully covered vertices before the step of repairing, respectively. At this step by Lemma 2, one more edge for each vertex is at most needed to be fully covered. The cover $C$ obtained by Algorithm 1 is bounded by $\left\lvert C\right\rvert\leq X+n-Y.$ (2) Our next lemma provides upper bounds on the expectation of the random variable $X$ and the expectation and variance of the random variable $Y$, which we will use to proof Theorem 5. This is a restriction of Lemma $4$ in [7] to the last case in Algorithm 1. ###### Lemma 3 Let $l$ and $\Delta$ be the maximum size of an edge and the maximum vertex degree, respectively. Let $\epsilon\in\left[\frac{\delta-1}{2\delta},\left(\frac{\delta-1}{2\delta}\right)^{\frac{1}{k}}\right]$, $a_{k,\epsilon}=\frac{k(1-\epsilon)+(\delta-1)(1-\epsilon^{a})}{2}$, $\lambda_{0}=(1-\epsilon)\delta$ and $\lambda=\frac{\delta+1}{2}$ as in Algorithm 1. We have $\mathrm{(i)}$ $\mathbb{E}(Y)\geq(1-\exp\left(-2a_{k,\epsilon}\right))n$. $\mathrm{(ii)}$ ${\rm Var}(Y)\leq 2n^{2}\exp\left(-2a_{k,\epsilon}\right)$. $\mathrm{(iii)}$ $\mathbb{E}(X)\leq(1-\epsilon^{k})\delta\mathrm{Opt}^{}$. $\mathrm{(iv)}$ $\dfrac{(b-1)n}{\alpha\ell}<\mathrm{Opt}^{}$. ###### Proof 3 (i) Let $i\in[n]$, $r=d(i)-b_{i}+1$. If $\left\lvert C_{1}\cap\Gamma(v_{i})\right\rvert\geq b_{i}$, then the vertex $v_{i}$ is fully covered and $\Pr(Y_{i}=0)=0$. Otherwise we get by Lemma 2 that $\left\lvert C_{1}\cap\Gamma(v_{i})\right\rvert=b_{i}-1$ and there exists at least one more edge from $C_{2}$ with $x_{j}\geq\frac{1}{\delta}$, so we have $\sum_{E_{j}\in\Gamma(v_{i})\cap C_{2}}x_{j}^{}\geq\frac{1}{\delta}$ and by the inequality constraints it holds that $\sum_{E_{j}\in\Gamma(v_{i})\cap\left(C_{2}\cup C_{3}\right)}x_{j}^{}\geq 1$. Therefore $\displaystyle\Pr(Y_{i}=0)$ $\displaystyle=$ $\displaystyle\left(\prod_{E_{j}\in\Gamma(v_{i})\cap C_{2}}(1-\lambda_{0}x_{j}^{})\right)^{k}\prod_{E_{j}\in\Gamma(v_{i})\cap C_{3}}\left(1-(1-\epsilon^{k})\delta x_{j}^{}\right)$ $\displaystyle=$ $\displaystyle\prod_{E_{j}\in\Gamma(v_{i})\cap C_{2}}\left(1-\lambda_{0}x_{j}^{}\right)^{k}\prod_{E_{j}\in\Gamma(v_{i})\cap C_{3}}\left(1-(1-\epsilon^{k})\delta x_{j}^{}\right)$ $\displaystyle\leq$ $\displaystyle\prod_{E_{j}\in\Gamma(v_{i})\cap C_{2}}\exp(-k\lambda_{0}x_{j}^{})\prod_{E_{j}\in\Gamma(v_{i})\cap C_{3}}\exp(-(1-\epsilon^{k})\delta x_{j}^{})$ $\displaystyle=$ $\displaystyle\exp\left(-k(1-\epsilon)\delta\sum_{E_{j}\in\Gamma(v_{i})\cap C_{2}}x_{j}^{}\right)\cdot\exp\left(-(1-\epsilon^{k})\delta\sum_{E_{j}\in\Gamma(v_{i})\cap C_{3}}x_{j}^{}\right)$ $\displaystyle=$ $\displaystyle\exp\left(\left(-k(1-\epsilon)+(1-\epsilon^{k})\right)\delta\sum_{E_{j}\in\Gamma(v_{i})\cap C_{2}}x_{j}^{}\right)\cdot\exp\left(-(1-\epsilon^{k})\delta\sum_{E_{j}\in\Gamma(v_{i})\cap\left(C_{2}\cup C_{3}\right)}x_{j}^{}\right).$ Since $1-\epsilon^{k}=(1-\epsilon)\sum_{i=0}^{k-1}\epsilon^{i}\leq k(1-\epsilon)$, we have $-k(1-\epsilon)+1-\epsilon^{k}\leq 0$. It follows that $\displaystyle\Pr(Y_{i}=0)$ $\displaystyle\leq$ $\displaystyle\exp\left(-k(1-\epsilon)+(1-\epsilon^{k})\right)\cdot\exp\left(-(1-\epsilon^{k})\delta\right)$ $\displaystyle=$ $\displaystyle\exp\left(-2a_{k,\epsilon}\right).$ Therefore $\displaystyle\mathbb{E}(Y)$ $\displaystyle=\sum_{i=1}^{n}\Pr(Y_{i}=1)=\sum_{i=1}^{n}(1-\Pr(Y_{i}=0))$ $\displaystyle\geq\sum_{i=1}^{n}(1-\exp\left(-2a_{k,\epsilon}\right))$ $\displaystyle\geq(1-\exp\left(-2a_{k,\epsilon}\right))n.$ (ii) Since $Y=\sum_{i=1}^{n}Y_{i}\leq n,$ we have $\mathbb{E}(Y^{2})\leq n^{2}.$ Thus, $\displaystyle{\rm Var}(Y)$ $\displaystyle=\mathbb{E}(Y^{2})-\mathbb{E}(Y)^{2}\leq n^{2}-(1-\exp\left(-2a_{k,\epsilon}\right))^{2}n^{2}$ $\displaystyle\leq n^{2}\left(1-(1-\exp\left(-2a_{k,\epsilon}\right))^{2}\right)$ $\displaystyle\leq 2n^{2}\exp\left(-2a_{k,\epsilon}\right).$ (iii) Let $E_{j}$ be an edge from $C_{2}$. By Lemma 2 we have $\frac{1}{\delta}\leq x^{}_{j}<\frac{2}{\delta+1}$. Recall that we include independently the edge $E_{j}$ in the cover $C$, with probability $\lambda_{0}x_{j}^{}$, $k$ times. Since $\frac{\delta-1}{2\delta}\leq\epsilon$, we have $1-\epsilon\leq\lambda_{0}x^{}_{j}<\frac{2}{\delta+1}(1-\epsilon)\delta\leq\frac{2}{\delta+1}(1-\frac{\delta-1}{2\delta})\delta=1$. Furthermore with $\epsilon\leq\left(\frac{\delta-1}{2\delta}\right)^{\frac{1}{k}}$ we have $\left(1-\epsilon^{k}\right)\delta\geq\left(1-\frac{\delta-1}{2\delta}\right)\delta=\frac{\delta+1}{2}=\lambda$. Then $\lambda\leq\left(1-\epsilon^{k}\right)\delta.$ (3) Clearly $\Pr\left(X_{j}=1\right)=1-\left(1-\lambda_{0}x^{}_{j}\right)^{k}$. Define the function $f$ by $f(x)=\frac{1-(1-x)^{k}}{x}$. $f$ is strictly decreasing on $(0,1]$. Therefore, $\ \frac{1-\left(1-\lambda_{0}x^{}_{j}\right)^{k}}{\lambda_{0}x^{}_{j}}\leq\frac{1-\left(1-(1-\epsilon)\right)^{k}}{1-\epsilon}=\frac{1-\epsilon^{k}}{1-\epsilon}$. It follows that $\Pr\left(X_{j}=1\right)\leq\frac{1-\epsilon^{k}}{1-\epsilon}\cdot\lambda_{0}x^{}_{j}$. Then $\Pr\left(X_{j}=1\right)\leq\left(1-\epsilon^{k}\right)\delta x^{}_{j}.$ (4) By using the LP relaxation and the definition of the sets $C_{1}$ and $C_{2}$, and since $\lambda x^{}_{j}\geq 1$ for all ${E_{j}\in C_{1}}$, we get $\displaystyle\mathbb{E}(X)$ $\displaystyle=$ $\displaystyle\|C_{1}\|+\sum_{E_{j}\in C_{2}}\Pr\left(X_{j}=1\right)+\sum_{E_{j}\in C_{3}}\Pr\left(X_{j}=1\right)$ $\displaystyle\overset{(\ref{probability5})}{\leq}$ $\displaystyle\sum_{E_{j}\in C_{1}}\lambda x^{}_{j}+\sum_{E_{j}\in C_{2}}(1-\epsilon^{k})\delta x^{}_{j}+\sum_{E_{j}\in C_{3}}(1-\epsilon^{k})\delta x^{}_{j}$ $\displaystyle\overset{(\ref{probability4})}{\leq}$ $\displaystyle(1-\epsilon^{k})\delta\sum_{E_{j}\in\mathcal{E}}x^{}_{j}$ $\displaystyle\leq$ $\displaystyle(1-\epsilon^{k})\delta\mathrm{Opt}^{}.$ (iv) Let us consider $\tilde{\mathcal{H}}$ the subhypergraph induced by $C_{1}$ in which degree equality gives $\sum_{i\in V}d(i)=\sum_{E_{j}\in C_{1}}\|E_{j}\|.$ As the minimum vertex degree in the subhypergraph $\tilde{\mathcal{H}}$ is $b-1$ with $b:=\min_{i\in[n]}b_{i}$, we have $(b-1)n\leq\sum_{i\in V}d(i)=\sum_{E\in C_{1}}\|E_{j}\|\leq\ell\|C_{1}\|.$ Therefore $\frac{(b-1)n}{\ell}\leq\|C_{1}\|.$ Since $\|C_{1}\|<\alpha\cdot\mathrm{Opt}^{}$ we obtain $\displaystyle\frac{(b-1)n}{\alpha\ell}<\mathrm{Opt}^{}.$ ###### Theorem 5 Let $\mathcal{H}$ be a hypergraph with fixed maximum vertex degree $\Delta$ and maximum edge size $\ell$. Let $\alpha=\frac{(b-1)\delta\epsilon^{k}}{6\ell}\times\exp\left(a_{k,\epsilon}\right)$ with $a_{k,\epsilon}=\frac{k(1-\epsilon)+(\delta-1)(1-\epsilon^{k})}{2}$ and $\epsilon\in\left[\frac{\delta-1}{2\delta},\left(\frac{\delta-1}{2\delta}\right)^{\frac{1}{k}}\right]$ as in Algorithm 1. The Algorithm 1 returns a set multicover $C$ such that $\|C\|<\max\left\\{\left(1-\frac{1}{2}\left(1-\epsilon\right)\epsilon^{k}\right)\delta,\left(1-\frac{(b-1)\epsilon^{k}}{18\ell}\times\exp\left(a_{k,\epsilon}\right)\right)\delta\right\\}\cdot\mathrm{Opt}^{}$ with probability greater than $0.65$. ###### Proof 4 Let $\mathcal{C}$ be the event that the inequality $\|C\|<\left(1-\frac{1}{2}\left(1-\epsilon\right)\epsilon^{k}\right)\delta\cdot\mathrm{Opt}^{}$ is satisfied. It suffices to prove that event $\mathcal{C}$ holds with the given probability in the case $\|C_{1}\|<\alpha\cdot\mathrm{Opt}^{}$ since the opposite case is discussed in Theorem 4. For this purpose, we estimate both the concentration of $X$ and $Y$ around their expectation. Choose $t=2n\exp\left(-a_{k,\epsilon}\right)$ and consider $\mathcal{A}$ the event $Y\leq n(1-\exp\left(-2a_{k,\epsilon}\right))-t$. This involves $\displaystyle n\exp\left(-2a_{k,\epsilon}\right)+t$ $\displaystyle=$ $\displaystyle n\exp\left(-2a_{k,\epsilon}\right)+2n\exp\left(-a_{k,\epsilon}\right)$ $\displaystyle\leq$ $\displaystyle 3n\exp\left(-a_{k,\epsilon}\right)$ $\displaystyle=$ $\displaystyle\frac{n(b-1)}{\ell}\cdot\frac{6\ell}{(b-1)\delta\epsilon^{k}\exp\left(a_{k,\epsilon}\right)}\cdot\frac{1}{2}\epsilon^{k}\delta$ $\displaystyle=$ $\displaystyle\frac{n(b-1)}{\alpha\ell}\cdot\frac{1}{2}\epsilon^{k}\delta$ $\displaystyle\overset{\textrm{Lem }~{}\ref{lemma:random}(iv)}{\leq}$ $\displaystyle\frac{1}{2}\epsilon^{k}\delta\cdot\mathrm{Opt}^{}.$ And by Lemma 3(ii) we have $\frac{t^{2}}{{\rm Var}(Y)}\geq\frac{4n^{2}\exp\left(-2a_{k,\epsilon}\right)}{2n^{2}\exp\left(-2a_{k,\epsilon}\right)}=2$. Therefore $\displaystyle\Pr\left(\mathcal{A}\right)$ $\displaystyle\leq$ $\displaystyle\Pr\left(Y\leq\mathbb{E}(Y)-t\right)$ $\displaystyle\overset{\textrm{Th }~{}\ref{Che-Can}}{\leq}$ $\displaystyle\frac{{\rm Var}(Y)}{{\rm Var}(Y)+t^{2}}$ $\displaystyle=$ $\displaystyle\frac{1}{1+\frac{t^{2}}{{\rm Var}(Y)}}$ $\displaystyle\leq$ $\displaystyle\frac{1}{3}.$ Consider now $\mathcal{B}$ the event $X\geq\left(1-\left(1-\frac{1}{2}\epsilon\right)\epsilon^{k}\right)\delta\mathrm{Opt}^{}$. Our basic assumption is to consider $\delta$, $k$ and $\epsilon$ constants, and we can certainly assume that $n\geq\frac{16\exp\left(a_{k,\epsilon}\right)}{\epsilon^{k+2}}$, since otherwise we obtain an optimal solution for the set multicover problem in polynomial time. Choosing $\beta=\frac{1}{2}\epsilon^{k+1}$ we have $\displaystyle(1+\beta)(1-\epsilon^{k})$ $\displaystyle=$ $\displaystyle 1-\epsilon^{k}+\frac{1}{2}\epsilon^{k+1}-\frac{1}{2}\epsilon^{2k+1}$ $\displaystyle=$ $\displaystyle 1-\epsilon^{k}\left(1-\frac{1}{2}\epsilon+\frac{1}{2}\epsilon^{k+1}\right)$ $\displaystyle\leq$ $\displaystyle 1-\left(1-\frac{1}{2}\epsilon\right)\epsilon^{k}.$ Note that $\epsilon\in\left[\frac{\delta-1}{2\delta},\left(\frac{\delta-1}{2\delta}\right)^{\frac{1}{k}}\right]$ therewith $1-\epsilon^{k}\geq 1-\frac{\delta-1}{2\delta}=\frac{\delta+1}{2\delta}>\frac{1}{2}$. We thus get $\displaystyle\Pr\left(\mathcal{B}\right)$ $\displaystyle\leq$ $\displaystyle\Pr\left(X\geq(1+\beta)\cdot(1-\epsilon^{k})\delta\mathrm{Opt}^{}\right)$ $\displaystyle\overset{\textrm{Th }~{}\ref{Doerr}}{\leq}$ $\displaystyle\exp\left(-\frac{\beta^{2}(1-\epsilon^{k})\delta\mathrm{Opt}^{}}{3}\right)$ $\displaystyle\overset{\textrm{Lem.}\ref{lemma:random}(iv)}{\leq}$ $\displaystyle\exp\left(-\frac{\epsilon^{2k+2}(1-\epsilon^{k})\delta n(b-1)}{12\ell}\cdot\frac{6\ell}{(b-1)\delta\epsilon^{k}\times\exp\left(a_{k,\epsilon}\right)}\right)$ $\displaystyle\leq$ $\displaystyle\exp\left(-\frac{\epsilon^{k+2}(1-\epsilon^{k})n}{2\exp\left(a_{k,\epsilon}\right)}\right)$ $\displaystyle\leq$ $\displaystyle\exp\left(-\frac{\epsilon^{k+2}n}{4\exp\left(a_{k,\epsilon}\right)}\right)$ $\displaystyle\leq$ $\displaystyle\exp\left(-4\right).$ Therefore it holds that $\displaystyle\Pr\left(\overline{\mathcal{A}}\cap\overline{\mathcal{B}}\right)$ $\displaystyle\geq$ $\displaystyle 1-\left(\frac{1}{3}+\exp\left(-4\right)\right),$ (5) where $\overline{\mathcal{A}}$ and $\overline{\mathcal{B}}$ denote the complement events of $\mathcal{A}$ and $\mathcal{B}$ respectively. We conclude that $\displaystyle\Pr\left(\mathcal{C}\right)$ $\displaystyle=$ $\displaystyle\Pr\left(\|C\|\leq\left(1-\frac{1}{2}\left(1-\epsilon\right)\epsilon^{k}\right)\delta\mathrm{Opt}^{}\right)$ $\displaystyle=$ $\displaystyle\Pr\left(\|C\|\leq\left(1-\left(1-\frac{1}{2}\epsilon\right)\epsilon^{k}+\frac{1}{2}\epsilon^{k}\right)\delta\mathrm{Opt}^{}\right)$ $\displaystyle\overset{(\ref{expection})}{\geq}$ $\displaystyle\Pr\left(X+n-Y\leq\left(1-\left(1-\frac{1}{2}\epsilon\right)\epsilon^{k}+\frac{1}{2}\epsilon^{k}\right)\delta\mathrm{Opt}^{}\right)$ $\displaystyle\geq$ $\displaystyle\Pr\left(X\leq\left(1-\left(1-\frac{1}{2}\epsilon\right)\epsilon^{k}\right)\delta\mathrm{Opt}^{}\text{\ and\ }n-Y\leq\frac{1}{2}\epsilon^{k}\delta\mathrm{Opt}^{}\right)$ $\displaystyle\geq$ $\displaystyle\Pr\left(X\leq\left(1-\left(1-\frac{1}{2}\epsilon\right)\epsilon^{k}\right)\delta\mathrm{Opt}^{}\text{\ and\ }Y\geq n-\frac{1}{2}\epsilon^{k}\delta\mathrm{Opt}^{}\right)$ $\displaystyle\geq$ $\displaystyle\Pr\left(X\leq\left(1-\left(1-\frac{1}{2}\epsilon\right)\epsilon^{k}\right)\delta\mathrm{Opt}^{}\text{\ and\ }Y\geq n-n\exp\left(-2a_{k,\epsilon}\right)-t\right)$ $\displaystyle\geq$ $\displaystyle\Pr\left(X\leq\left(1-\left(1-\frac{1}{2}\epsilon\right)\epsilon^{k}\right)\delta\mathrm{Opt}^{}\text{\ and\ }Y\geq n(1-\exp\left(-2a_{k,\epsilon}\right))-t\right)$ $\displaystyle\overset{(\ref{Intersection})}{\geq}$ $\displaystyle 1-\left(\frac{1}{3}+\exp\left(-4\right)\right)$ $\displaystyle\geq$ $\displaystyle 0.65.$ $\Box$ Remark 2. The proof above gives for $k=2$ and $\epsilon=\frac{1}{2}$ an approximation ratio of $\max\left\\{\frac{15}{16}\delta,\left(1-\frac{(b-1)\exp\left(\frac{3\delta+1}{8}\right)}{72\ell}\right)\delta\right\\}$. Note that $\frac{\delta-1}{2\delta}<\frac{1}{2}<\left(\frac{\delta-1}{2\delta}\right)^{\frac{1}{2}}$ therewith the condition of Theorem 5 on $\epsilon$ is satisfied. As mentioned above our performance guaranty improves over the ratio presented by Srivastav et al [7], and this without restriction on the parameter $\ell$. Namely, for $\delta\geq 13$ we have $\displaystyle 11(\delta-1)<\exp\left(\frac{3\delta+1}{8}\right)$ $\displaystyle\Rightarrow$ $\displaystyle\frac{11(\delta-1)}{72\ell}<\frac{\exp\left(\frac{3\delta+1}{8}\right)}{72\ell}$ $\displaystyle\overset{b-1\geq 1}{\Rightarrow}$ $\displaystyle\frac{11(\Delta-b)}{72\ell}<\frac{(b-1)\exp\left(\frac{3\delta+1}{8}\right)}{72\ell}$ $\displaystyle\Rightarrow$ $\displaystyle\left(1-\frac{(b-1)\exp\left(\frac{3\delta+1}{8}\right)}{72\ell}\right)\delta<\left(1-\frac{11(\Delta-b)}{72\ell}\right)\delta.$ ## 4 Lower Bound One of the features of the proof is the duality of hypergraphs. In dual hypergraphs, vertices and edges just swap the roles. So the set multicover problem in dual hypergraphs becomes as follows: find a minimum cardinality set $C\subseteq V$ such that for every $E\in\mathcal{E}$ it holds $\|E\cap C\|\geq b$. This problem is known as the $b$-vertex cover problem and we have that the set multicover problem in $\Delta$-regular hypergraphs is equivalent to the $b$-vertex cover problem in $\Delta$-uniform hypergraphs. ###### Theorem 6 Let $\epsilon>0$, $\Delta$ and $\bf b\in\mathbb{N}_{0}^{n}$ be given and $b=\min_{i}b_{i}$. Then, it is NP-hard to approximate the set multicover problem on $\Delta$-regular hypergraphs within a factor of $\Delta-b-\epsilon$. _Proof_. Assume, for a contradiction, that the theorem is false. Then there exists an algorithm $\mathcal{A}$ that returns a $(\Delta-b-\epsilon)$-approximation in polynomial time for the $b$-vertex cover problem on $\Delta$-uniform hypergraphs. We give a reduction of the minimum vertex cover problem on $\Delta$-uniform hypergraphs to the $b$-vertex cover problem on $\Delta+b-1$-uniform hypergraphs. Let $\tilde{\mathcal{H}}=(V,\mathcal{E})$ be a $\Delta$-uniform hypergraph and let $\alpha=\frac{2}{\epsilon}(\Delta-1-\epsilon)(b-1)$. Now we consider the following algorithm: * 1. Consider all subsets $T\subseteq V$ with $\|T\|\leq\alpha$. Check if any of these subsets is a vertex cover in $\tilde{\mathcal{H}}$. If it’s the case then return the smallest one of them, else go to step 2. * 2. Add $b-1$ vertices $v_{1},\ldots,v_{b-1}$ to $V$. Define for every hyper-edge $E$ a new edge $E^{\ast}:=E\cup\\{v_{1},\ldots,v_{b-1}\\}$ and the set $\mathcal{E}^{\ast}:=\\{E^{\ast}\|E\in\mathcal{E}\\}$. Finally set $\mathcal{H}=(V\cup\\{v_{1},\ldots,v_{b-1}\\},\mathcal{E}^{\ast})$. We execute $\mathcal{A}$ on $\mathcal{H}$. Return $T:=\mathcal{A}(\mathcal{H})\cap V$. #### Claim The algorithm given above returns a vertex cover in $\tilde{\mathcal{H}}$ in polynomial-time with an approximation ratio of $\Delta-1-\frac{\epsilon}{2}$. _Proof_. Correctness and approximation ratio. If $T$ is selected by the algorithm in step $1$ then $T$ is an optimal vertex cover in $\tilde{\mathcal{H}}$. If $T$ is selected by the algorithm in step $2$ then $\mathcal{A}(\mathcal{H})=T\cup K$ for some $K\subset\\{v_{1},\ldots,v_{b-1}\\}$. Note that $T$ and $K$ are disjoint sets. Consider an edge $E\in\mathcal{E}$. Because $\mathcal{A}(\mathcal{H})$ is a $b$-vertex cover in $\mathcal{H}$, we have $\|\mathcal{A}(\mathcal{H})\cap E^{\ast}\|=\|T\cap E^{\ast}\|+\|K\cap E^{\ast}\|\geq b$. Since $T\cap E^{\ast}=T\cap E$ and $\|K\cap E^{\ast}\|\leq b-1$, it follows that $\|T\cap E\|\geq 1$. Hence $T$ is a vertex cover in $\tilde{\mathcal{H}}$. Now, let $C$ and $C^{\prime}$ denote a minimum vertex cover in $\tilde{\mathcal{H}}$ and a minimum $b$-vertex cover in $\mathcal{H}$, respectively. Since $D^{{}^{\prime}}:=C\cup\\{v_{1},\ldots,v_{b-1}\\}$ is a feasible $b$-vertex cover in $\mathcal{H}$, it holds that $\|C^{\prime}\|\leq\|C\|+b-1$. On the other hand, it is clear that $\mathcal{H}$ is a $\left(\Delta+b-1\right)$-uniform hypergraph, and by the assumption we get $\displaystyle\|\mathcal{A}(\mathcal{H})\|$ $\displaystyle\leq\left((\Delta+b-1)-b-\epsilon\right)\|C^{\prime}\|$ $\displaystyle\leq(\Delta-1-\epsilon)\|C\|+(\Delta-1-\epsilon)(b-1)=(\Delta-1-\epsilon)\|C\|+\frac{\epsilon}{2}\alpha$ $\displaystyle\overset{\|C\|\geq\alpha}{\leq}(\Delta-1-\frac{\epsilon}{2})\|C\|.$ Since $\|T\|=\|\mathcal{A}(\mathcal{H})\cap V\|\leq\|\mathcal{A}(\mathcal{H})\|$, it follows that $\|T\|\leq(\Delta-1-\frac{\epsilon}{2})\|C\|$. Running time. In step 1 we test at most $n^{\alpha}$ sets of vertices to be a vertex cover in $\tilde{\mathcal{H}}$. Since $\alpha=\frac{2}{\epsilon}(\Delta-1-\epsilon)(b-1)$ is a constant, the running time in this step is polynomial. In step 2 we add a constant number of vertices to $V$ and execute the algorithm $\mathcal{A}$. Hence the algorithm runs in polynomial time in both steps. With Claim 1 there is a factor $\Delta-1-\frac{\epsilon}{2}$ approximation algorithm for the minimum vertex cover problem on $\Delta$-uniform hypergraphs, which contradicts the statement of Theorem 3. $\Box$ ## 5 The $\frac{\ln_{2}(n+1)}{2b}$-Integrality Gap The integrality gap for set multicover problem is defined as the supremum of the ratio $\frac{{\rm Opt}_{{\bf b}}(\mathcal{H})}{{\rm Opt}^{}_{{\bf b}}(\mathcal{H})}$ over all instances $\cal{H}$ of the problem. In this section we give a slight modification of the proof presented in [22] for the integrality gap. We present in the following a specific class of instances of the set multicover problem, where ${\bf b}:=(b,\ldots,b)\in\mathbb{N}^{n}$ for which the integrality gap is at least $\frac{\ln_{2}(n+1)}{2b}$. ###### Theorem 7 let ${\bf b}:=(b,\ldots,b)\in\mathbb{N}^{n}$. The integrality gap of the set multicover problem is at least $\frac{\ln_{2}(n+1)}{2b}$. Define $V=F_{2}^{k}\backslash\\{0\\}$ as the set of all $k$-dimensional non- zero vectors with component values of $\mathbb{Z}_{2}=\\{0,1\\}$ for a fixed integer $k$ and we define ${\cal E}$ as a collection of the sets $E_{v}=\\{u\in V:<v,u>\equiv 1[2]\\}$ for each $v\in V$, where $<.\,,.>$ is the usual dot product in $V$. We remark that each element $v\in V$ is contained in exactly half of the sets of ${\cal E}$ therewith the hypergraph ${\cal H}=(V,{\cal E})$ is regular and $n=\|V\|=2^{k}-1$. ###### Lemma 4 Let ${\cal H}=(V,{\cal E})$ the hypergraph defined and ${\bf b}\in\mathbb{N}_{\geq 1}^{n}$. It holds that the vector $x=(\frac{2b}{\|{\cal E}\|},\ldots,\frac{2b}{\|{\cal E}\|})$ is a feasible solution for LP($\Delta,\,{\bf b}$). _Proof_. It is clear that $x=(\frac{2b}{\|{\cal E}\|},\ldots,\frac{2b}{\|{\cal E}\|})$ is a feasible solution for lP$(\Delta,{\bf b})$, namely since $\mathcal{H}$ is regular with $\Delta=\frac{\|{\cal E}\|}{2}$ we have for every $i\in\\{1,\ldots,n\\}$ $\sum_{E\in\Gamma(v_{i})}\frac{2b}{\|{\cal E}\|}=\frac{2b}{\|{\cal E}\|}\cdot\Delta=\frac{2b}{\|{\cal E}\|}\cdot\frac{\|{\cal E}\|}{2}\geq b$ therewith $\mathrm{Opt}^{}\leq 2b$. $\Box$ ###### Lemma 5 The optimal integral solution to the previous LP formulation of the set multicover problem requires at least $k$ sets. _Proof_. Let $\\{E_{v_{1}},E_{v_{2}}\ldots E_{v_{t}}\\}$ a collection of sets such that ${\bigcup}_{i\in[t]}E_{v_{i}}=F_{2}^{k}\backslash\\{0\\}$. This implies that the intersection of their complements contains exactly the zero vector, i.e., ${\bigcap}_{i\in[t]}E_{v_{i}}^{C}=\\{0\\}$. It follows that $0$ is the only solution in $F_{2}^{k}$ of the system $<x,v_{i}>\equiv 0[2],\quad\forall i\in[t]$ Then it holds that $t\geq k$, since the dimension of $F_{2}^{k}$ is $k$ while the number of the equations in the system is $t$. From this we conclude $\mathrm{Opt}\geq\ln_{2}(n+1)$. $\Box$ Proof of Theorem 7. Theorem 7 follows from Lemma 4 and Lemma 5. $\Box$ ## 6 Future Work We believe now that the conjecture of Peleg et al. holds in the general setting. Hence proving the trueness of the conjecture remains a big challenge for our future works. ## References * [1] R. Bar-Yehuda. _Using Homogeneous Weights for Approximating the Partial Cover Problem._ Journal of Algorithms, 39(2):137–144, 2001. * [2] I. Dinur, V. Guruswami, S. Khot, O. Regev, _A new multilayered PCP and the hardness of hypergraph vertex cover_ , SIAM J. Comput. 34 (5) 1129–1146, 2005. * [3] B. Doerr, F. (Eds.) Neumann. _Theory of Evolutionary Computation: Recent Developments in Discrete Optimization_. 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Srinivasan. _Approximation Algorithms for Partial Covering Problems._ Journal of Algorithms, 53(1):55–84, 2004. * [11] N.G. Hall, D.S. Hochbaum. _A Fast Approximation Algorithm for the Multicovering Problem._ Discrete Applied Mathematics, 15:35–40, 1986. * [12] D.S. Hochbaum. _Approximation Algorithms for the Set Covering and Vertex Cover Problems._ SIAM J. Comput, 11(3):555–556, August 1982. * [13] D. S. Johnson. _Approximation Algorithms for Combinatorial Problems._ Journal of Computer and System Sciences, 9:256–278, 1974. * [14] R. KARP, _Reducibility among combinatorial problems._ In R.E. Miller and J.W. Thatcher, editors, Complexity of Computer Computations, pp. 85–103. Plenum Press, New York, NY, 1972. * [15] S. Khot and O. Regev. _Vertex Cover Might be Hard to Approximate to Within 2-epsilon._ Journal of Computer and System Sciences, 74(3):335–349, 2008. * [16] C. Koufogiannakis, N.E. Young. _Greedy $\Delta$-approximation algorithm for covering with arbitrary constraints and submodular cost_. Algorithmica, 66(1), 113–152, 2013. * [17] L. Lovász. _On the Ratio of Optimal Integral and Fractional Covers._ Discrete Mathematics, 13(4):383–390, 1975. * [18] R. Motwani, P. Raghavan. _Randomized Algorithms_. Cambridge University Press 1995. * [19] D. Peleg, G. Schechtman, A. Wool. _Randomized Approximation of Bounded Multicovering Problems._ Algorithmica, 18(1):44–66, 1997. * [20] D. Peleg, G. Schechtman, A. Wool. _Approximating bounded 0-1 integer linear programs._ In Proc. 2nd Israel Symp. on Theory of Computing Systems, pp. 69–77, Netanya, 1993. * [21] S. Rajagopalan, V. V. Vazirani. _Primal-dual RNC approximation algorithms for set cover and covering integer programs_. SIAM J. Comput., 28(2), 525–540, 1998. * [22] V. V. Vazirani. _Approximation Algorithms_ , pp. 108–112, Springer 2001.
# Convergence Analysis of Projection Method for Variational Inequalities Yekini Shehu111Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, People’s Republic of China; Institute of Science and Technology (IST), Am Campus 1, 3400, Klosterneuburg, Vienna, Austria; e-mail: <EMAIL_ADDRESS>Olaniyi. S. Iyiola222Department of Mathematics, Minnesota State University-Moorhead, Minnesota, USA; e-mail: <EMAIL_ADDRESS>Xiao-Huan Li333College of Science, Civil Aviation University of China, Tianjin 300300, China.; e-mail<EMAIL_ADDRESS>and Qiao-Li Dong444College of Science, Civil Aviation University of China, Tianjin 300300, China.; e-mail<EMAIL_ADDRESS> (January 22, 2021) ###### Abstract The main contributions of this paper are the proposition and the convergence analysis of a class of inertial projection-type algorithm for solving variational inequality problems in real Hilbert spaces where the underline operator is monotone and uniformly continuous. We carry out a unified analysis of the proposed method under very mild assumptions. In particular, weak convergence of the generated sequence is established and nonasymptotic $O(1/n)$ rate of convergence is established, where $n$ denotes the iteration counter. We also present some experimental results to illustrate the profits gained by introducing the inertial extrapolation steps. ## 1 Introduction We first state the formal definition of some classes of functions that play an essential role in this paper. Let $H$ be a real Hilbert space and $X\subseteq H$ be a nonempty subset. ###### Definition 1.1. A mapping $F:X\to H$ is called * (a) monotone on $X$ if $\langle F(x)-F(y),x-y\rangle\geq 0$ for all $x,y\in X$; * (b) Lipschitz continuous on $X$ if there exists a constant $L>0$ such that $\\|F(x)-F(y)\\|\leq L\\|x-y\\|,\ \forall x,y\in X.$ * (c) sequentially weakly continuous if for each sequence $\\{x_{n}\\}$ we have: $\\{x_{n}\\}$ converges weakly to $x$ implies $\\{F(x_{n})\\}$ converges weakly to $F(x)$. Let $C$ be a nonempty, closed and convex subset of $H$ and $F:C\rightarrow H$ be a continuous mapping. The variational inequality problem (for short, VI($F,C$)) is defined as: find $x\in C$ such that $\displaystyle\langle F(x),y-x\rangle\geq 0,\quad\forall y\in C.$ (1) Let SOL denote the solution set of VI($F,C$) (1). Variational inequality theory is an important tool in economics, engineering mechanics, mathematical programming, transportation, and so on (see, for example, [7, 8, 22, 29, 30, 31, 38]). A well-known projection-type method for solving VI($F,C$) (1) is the extragradient method introduced by Korpelevich in [32]. It is well known that the extragradient method requires two projections onto the set $C$ and two evaluations of $F$ per iteration. One important hallmark in the design of numerical methods related to the extragradient method is to minimize the number of evaluations of $P_{C}$ per iteration because if $C$ is a general closed and convex set, then a minimal distance problem has to be solved (twice) in order to obtain the next iterate. This has the capacity to seriously affect the efficiency of the extragradient method in a situation, where a projection onto $C$ is hard to evaluate and therefore computationally costly. An attempt in this direction was initiated by Censor et al.[18], who modified extragradient method by replacing the second projection onto the closed and convex subset $C$ with the one onto a subgradient half-space. Their method, which therefore uses only one projection onto $C$, is called the subgradient extragradient method: $x_{1}\in H$, $\displaystyle\left\\{\begin{array}[]{llll}&y_{n}=P_{C}(x_{n}-\lambda F(x_{n})),\\\ &T_{n}:=\\{w\in H:\langle x_{n}-\lambda F(x_{n})-y_{n},w-y_{n}\rangle\leq 0\\},\\\ &x_{n+1}=P_{T_{n}}(x_{n}-\lambda F(y_{n})).\end{array}\right.$ (5) Using (5), Censor et al. [18] proved weak convergence result for VI($F,C$) (1) with a monotone and $L$-Lipschitz-continuous mapping $F$ where $\lambda\in(0,\frac{1}{L})$. Several other related methods to extragradient method and (5) for solving VI($F,C$) (1) in real Hilbert spaces when $F$ is monotone and $L$-Lipschitz-continuous mapping have been studied in the literature (see, for example, [15, 16, 17, 21, 26, 35, 36, 37, 40, 46]). Motivated the result of Alvarez and Attouch in [2] and Censor et al. in [18], Thong and Hieu [45] introduced an algorithm which is a combination of (5) and inertial method for solving VI($F,C$) (1) in real Hilbert space: $x_{0},x_{1}\in H$, $\displaystyle\left\\{\begin{array}[]{llll}&w_{n}=x_{n}+\alpha_{n}(x_{n}-x_{n-1}),\\\ &y_{n}=P_{C}(w_{n}-\lambda F(w_{n})),\\\ &T_{n}:=\\{w\in H:\langle w_{n}-\lambda F(w_{n})-y_{n},w-y_{n}\rangle\leq 0\\},\\\ &x_{n+1}=P_{T_{n}}(w_{n}-\lambda F(y_{n})).\end{array}\right.$ (10) Thong and Hieu [45] proved that the sequence $\\{x_{n}\\}$ generated by (10) converges weakly to a solution of VI($F,C$) (1) with a monotone and $L$-Lipschitz-continuous mapping $F$ where $0<\lambda L\leq\frac{\frac{1}{2}-2\alpha-\frac{1}{2}\alpha^{2}-\delta}{\frac{1}{2}-\alpha+\frac{1}{2}\alpha^{2}}$ for some $0<\delta<\frac{1}{2}-2\alpha-\frac{1}{2}\alpha^{2}$ and $\\{\alpha_{n}\\}$ is a non-decreasing sequence with $0\leq\alpha_{n}\leq\alpha<\sqrt{5}-2$. One the main features of the above mentioned methods (5), (10) and other related methods is the computational issue, for example, step-sizes. The step- sizes in these above methods are bounded by the inverse of the Lipschitz constant which is quite inefficient, because in most cases a global Lipschitz constant (if it indeed exists) of $F$ cannot be accurately estimated, and is usually overestimated, thereby resulting in too small step-sizes. This, of course, is not practical. Therefore, algorithms (5) and (10) are not applicable in most cases of interest. The usual approach to overcome this difficulty consists in some prediction of a step-size with its further correction (see [29, 38]) or in a usage of an Armijo type line search procedure along a feasible direction (see [43]). In terms of computations, the latter approach is more effective, since very often the former approach requires too many projections onto the feasible set per iteration. This paper focuses on the analysis and development of computational projection-type algorithm with inertial extrapolation step for solving VI($F,C$) (1) when the underline operator $F$ is monotone and uniformly continuous when the feasible set $C$ is a nonempty closed affine subset. We obtain weak convergence of the sequence generated by our method. We provide theoretical analysis of our result with weaker assumption on the underline operator $F$ unlike [17, 18, 35, 36] and many other related results on monotone variational inequalities. We also establish the nonasymptotic $O(1/n)$ rate of convergence, which is not given before in other previous inertial type projection methods for VI($F,C$) (1) (see, e.g.,[21, 45]) and give carefully designed computational experiments to illustrate our results. Our computational results show that our proposed methods outperform the iterative methods (5) and (10). Furthermore, our result complements some recent results on inertial type algorithms (see, e.g., [2, 3, 4, 5, 6, 10, 12, 13, 19, 33, 34, 41, 42]). The paper is organized as follows: We first recall some basic definitions and results in Section 2. Some discussions about the proposed inertial projection- type method are given in Section 3. The weak convergence analysis of our algorithm is then investigated in Section 4. We give the rate of convergence of our proposed method in Section 5 and some numerical experiments can be found in Section 6. We conclude with some final remarks in Section 7. ## 2 Preliminaries First, we recall some properties of the projection, cf. [9] for more details. For any point $u\in H$, there exists a unique point $P_{C}u\in C$ such that $\\|u-P_{C}u\\|\leq\\|u-y\\|,\leavevmode\nobreak\ \forall y\in C.$ $P_{C}$ is called the metric projection of $H$ onto $C$. We know that $P_{C}$ is a nonexpansive mapping of $H$ onto $C$. It is also known that $P_{C}$ satisfies $\langle x-y,P_{C}x-P_{C}y\rangle\geq\\|P_{C}x-P_{C}y\\|^{2},\leavevmode\nobreak\ \leavevmode\nobreak\ \forall x,y\in H.$ (11) In particular, we get from (11) that $\langle x-y,x-P_{C}y\rangle\geq\\|x-P_{C}y\\|^{2},\leavevmode\nobreak\ \leavevmode\nobreak\ \forall x\in C,y\in H.$ (12) Furthermore, $P_{C}x$ is characterized by the properties $P_{C}x\in C\quad\text{and}\quad\langle x-P_{C}x,P_{C}x-y\rangle\geq 0,\leavevmode\nobreak\ \forall y\in C.$ (13) Further properties of the metric projection can be found, for example, in Section 3 of [23]. The following lemmas will be used in our convergence analysis. ###### Lemma 2.1. The following statements hold in $H$: * (a) $\\|x+y\\|^{2}=\\|x\\|^{2}+2\langle x,y\rangle+\\|y\\|^{2}$ for all $x,y\in H$; * (b) $2\langle x-y,x-z\rangle=\\|x-y\\|^{2}+\\|x-z\\|^{2}-\\|y-z\\|^{2}$ for all $x,y,z\in H$; * (c) $\\|\alpha x+(1-\alpha)y\\|^{2}=\alpha\\|x\\|^{2}+(1-\alpha)\\|y\\|^{2}-\alpha(1-\alpha)\\|x-y\\|^{2}$ for all $x,y\in H$ and $\alpha\in\mathbb{R}$. ###### Lemma 2.2. (see [1, Lem. 3]) Let $\\{\psi_{n}\\}$, $\\{\delta_{n}\\}$ and $\\{\alpha_{n}\\}$ be the sequences in $[0,+\infty)$ such that $\psi_{n+1}\leq\psi_{n}+\alpha_{n}(\psi_{n}-\psi_{n-1})+\delta_{n}$ for all $n\geq 1$, $\sum_{n=1}^{\infty}\delta_{n}<+\infty$ and there exists a real number $\alpha$ with $0\leq\alpha_{n}\leq\alpha<1$ for all $n\geq 1$. Then the following hold: $(i)\leavevmode\nobreak\ \leavevmode\nobreak\ \sum_{n\geq 1}[\psi_{n}-\psi_{n-1}]_{+}<+\infty$, where $[t]_{+}=\max\\{t,0\\}$; (ii) there exists $\psi^{}\in[0,+\infty)$ such that $\lim_{n\rightarrow+\infty}\psi_{n}=\psi^{}$. ###### Lemma 2.3. (see [9, Lem. 2.39]) Let $C$ be a nonempty set of $H$ and $\\{x_{n}\\}$ be a sequence in $H$ such that the following two conditions hold: (i) for any $x\in C$, $\lim_{n\rightarrow\infty}\\|x_{n}-x\\|$ exists; (ii) every sequential weak cluster point of $\\{x_{n}\\}$ is in $C$. Then $\\{x_{n}\\}$ converges weakly to a point in $C$. The following lemmas were given in $\mathbb{R}^{n}$ in [25]. The proof of the lemmas are the same if given in infinite dimensional real Hilbert spaces. Hence, we state the lemmas and omit the proof in real Hilbert spaces. ###### Lemma 2.4. Let $C$ be a nonempty closed and convex subset of $H$. Let $h$ be a real- valued function on $H$ and define $K:=\\{x:h(x)\leq 0\\}$. If $K$ is nonempty and $h$ is Lipschitz continuous on $C$ with modulus $\theta>0$, then ${\rm dist}(x,K)\geq\theta^{-1}\max\\{h(x),0\\},\leavevmode\nobreak\ \forall x\in C,$ where ${\rm dist}(x,K)$ denotes the distance function from $x$ to $K$. ###### Lemma 2.5. Let $C$ be a nonempty closed and convex subset of $H$, $y:=P_{C}(x)$ and $x^{}\in C$. Then $\\|y-x^{}\\|^{2}\leq\\|x-x^{}\\|^{2}-\\|x-y\\|^{2}.$ (14) The following lemma was stated in [28, Prop. 2.11], see also [27, Prop. 4]. ###### Lemma 2.6. Let $H_{1}$ and $H_{2}$ be two real Hilbert spaces. Suppose $F:H_{1}\rightarrow H_{2}$ is uniformly continuous on bounded subsets of $H_{1}$ and $M$ is a bounded subset of $H_{1}$. Then $F(M)$ is bounded. Finally, the following result states the equivalence between a primal and a weak form of variational inequality for continuous, monotone operators. ###### Lemma 2.7. ([44, Lem. 7.1.7]) Let $C$ be a nonempty, closed, and convex subset of $H$. Let $F:C\rightarrow H$ be a continuous, monotone mapping and $z\in C$. Then $z\in{\rm SOL}\Longleftrightarrow\langle F(x),x-z\rangle\geq 0\quad\text{for all }x\in C.$ ## 3 Inertial Projection-type Method Let us first state the assumptions that we will assume to hold for the rest of this paper. ###### Assumption 3.1. Suppose that the following hold: (a) The feasible set $C$ is a nonempty closed affine subset of the real Hilbert space $H$. * (b) $F:C\to H$ is monotone and uniformly continuous on bounded subsets of $H$. * (c) The solution set SOL of VI$(F,C)$ is nonempty. ###### Assumption 3.2. Suppose the real sequence $\\{\alpha_{n}\\}$ and constants $\beta,\delta,\sigma>0$ satisfy the following conditions: * (a) $\\{\alpha_{n}\\}\subset(0,1)$ with $0\leq\alpha_{n}\leq\alpha_{n+1}\leq\alpha<1$ for all $n$. * (b) $\delta>\frac{\alpha(1+\alpha)(\alpha+\delta\sigma)+\alpha\sigma\delta(\alpha+\delta\sigma)}{\sigma}$ and $\beta<\frac{\delta\sigma}{\alpha+\delta\sigma}-\alpha(1+\alpha)-\alpha\sigma\delta$. Let $r(x):=x-P_{C}(x-F(x))$ stand for the residual equation. Observe that if we take $y=x-F(x)$ in (12), then we have $\langle F(x),r(x)\rangle\geq\\|r(x)\\|^{2},\leavevmode\nobreak\ \forall x\in C.$ (15) We now introduce our proposed method below. Algorithm 1 Inertial Projection-type Method 1:Choose sequence $\\{\alpha_{n}\\}$ and $\sigma\in(0,1)$ such that the conditions from Assumption 3.2 hold, and take $\gamma\in(0,1)$. Let $x_{0}=x_{1}\in H$ be a given starting point. Set $n:=1$. 2:Set $w_{n}:=x_{n}+\alpha_{n}(x_{n}-x_{n-1}).$ Compute $z_{n}:=P_{C}(w_{n}-F(w_{n}))$. If $r(w_{n})=w_{n}-z_{n}=0$: STOP. 3:Compute $y_{n}=w_{n}-\gamma^{m_{n}}r(w_{n})$, where $m_{n}$ is the smallest nonnegative integer satisfying $\langle F(y_{n}),r(w_{n})\rangle\geq\frac{\sigma}{2}\\|r(w_{n})\\|^{2}.$ (16) Set $\eta_{n}:=\gamma^{m_{n}}$. 4:Compute $x_{n+1}=P_{C_{n}}(w_{n}),$ (17) where $C_{n}=\\{x:h_{n}(x)\leq 0\\}$ and $h_{n}(x):=\langle F(y_{n}),x-y_{n}\rangle.$ (18) 5:Set $n\leftarrow n+1$ and goto 2. It is clear that $r(w_{n})=0$ implies that we are at a solution of the variational inequality. In our convergence theory, we will implicitly assume that this does not occur after finitely many iterations, so that Algorithm 1 generates an infinite sequence satisfying, in particular, $r(w_{n})\neq 0$ for all $n\in\mathbb{N}$. We will see that this property implies that Algorithm 1 is well defined. ###### Remark 3.3. (a) Algorithm 1 requires, at each iteration, only one projection onto the feasible set $C$ and another projection onto the half-space $C_{n}$ (see [14] for formula for computing projection onto half-space), which is less expensive than the extragradient method especially for the case when computing the projection onto the feasible set $C$ is a dominating task during iteration. (b) Our Algorithm 1 is much more applicable than (5) and (10) in the sense that algorithm (5) and (10) are applicable only for monotone and $L$-Lipschitz-continuous mapping $F$. Thus, the $L$-Lipschitz constant of $F$ or an estimate of it is needed in order to implement the iterative method (5) but our Algorithm 1 is applicable for a much more general class of monotone and uniformly continuous mapping $F$. (c) We observe that the step-size rule in Step 3 involves a couple of evaluations of $F$, but these are often much less expensive than projections onto $C$ which was considered in [29, 38]. Furthermore, using the fact that $F$ is continuous and (15), we can see that Step 3 in Algorithm 1 is well- defined. $\Diamond$ ###### Lemma 3.4. Let the function $h_{n}$ be defined by (18). Then $h_{n}(w_{n})\geq\frac{\sigma\eta_{n}}{2}\\|w_{n}-z_{n}\\|^{2}.$ In particular, if $w_{n}\neq z_{n}$, then $h_{n}(w_{n})>0$. If $x^{}\in\text{SOL}$, then $h_{n}(x^{})\leq 0$. ###### Proof. Since $y_{n}=w_{n}-\eta_{n}(w_{n}-z_{n})$, using (16) we have $\displaystyle h_{n}(w_{n})$ $\displaystyle=\langle F(y_{n}),w_{n}-y_{n}\rangle$ $\displaystyle=\eta_{n}\langle F(y_{n}),w_{n}-z_{n}\rangle\geq\eta_{n}\frac{\sigma}{2}\\|w_{n}-z_{n}\\|^{2}\geq 0.$ If $w_{n}\neq z_{n}$, then $h_{n}(w_{n})>0$. Furthermore, suppose $x^{}\in\text{SOL}$. Then by Lemma 2.7 we have $\langle F(x),x-x^{}\rangle\geq 0\quad\text{for all }x\in C.$ In particular, $\langle F(y_{n}),y_{n}-x^{}\rangle\geq 0$ and hence $h_{n}(x^{})\leq 0.$ ∎ Observe that, in finding $\eta_{n}$, the operator $F$ is evaluated (possibly) many times, but no extra projections onto the set $C$ are needed. This is in contrast to a couple of related algorithms for the solution of monotone variational inequalities where the calculation of a suitable step-size requires (possibly) many projections onto $C$, see, e.g., [20, 29, 46]. ## 4 Convergence Analysis We present our main result in this section. To this end, we begin with a result that shows that the sequence $\\{x_{n}\\}$ generated by Algorithm 1 is bounded under the given assumptions. ###### Lemma 4.1. Let $\\{x_{n}\\}$ be generated by Algorithm 1. Then under Assumptions 3.1 and 3.2, we have that $\\{x_{n}\\}$ is bounded. ###### Proof. Let $x^{}\in\text{SOL}$. By Lemma 2.5 we get (since $x^{}\in C_{n}$) that $\displaystyle\\|x_{n+1}-x^{}\\|^{2}$ $\displaystyle=$ $\displaystyle\\|P_{C_{n}}(w_{n})-x^{}\\|^{2}\leq\\|w_{n}-x^{}\\|^{2}-\\|x_{n+1}-w_{n}\\|^{2}$ $\displaystyle=$ $\displaystyle\\|w_{n}-x^{}\\|^{2}-{\rm dist}^{2}(w_{n},C_{n}).$ Now, using Lemma 2.1 (c), we have $\displaystyle\\|w_{n}-x^{}\\|^{2}$ $\displaystyle=$ $\displaystyle\\|(1+\alpha_{n})(x_{n}-x^{})-\alpha_{n}(x_{n-1}-x^{})\\|^{2}$ (20) $\displaystyle=$ $\displaystyle(1+\alpha_{n})\\|x_{n}-x^{}\\|^{2}-\alpha_{n}\\|x_{n-1}-x^{}\\|$ $\displaystyle+\alpha_{n}(1+\alpha_{n})\\|x_{n}-x_{n-1}\\|^{2}.$ Substituting (20) into (4), we have $\displaystyle\\|x_{n+1}-x^{}\\|^{2}$ $\displaystyle\leq$ $\displaystyle(1+\alpha_{n})\\|x_{n}-x^{}\\|^{2}-\alpha_{n}\\|x_{n-1}-x^{}\\|^{2}+\alpha_{n}(1+\alpha_{n})\\|x_{n}-x_{n-1}\\|^{2}$ (21) $\displaystyle-\\|x_{n+1}-w_{n}\\|^{2}.$ We also have (using Lemma 2.1 (a)) $\displaystyle\\|x_{n+1}-w_{n}\\|^{2}$ $\displaystyle=$ $\displaystyle\\|(x_{x_{n}+1}-x_{n})-\alpha_{n}(x_{n}-x_{n-1})\\|^{2}$ (22) $\displaystyle=$ $\displaystyle\\|x_{x_{n}+1}-x_{n}\\|^{2}+\alpha^{2}\\|x_{n}-x_{n-1}\\|^{2}$ $\displaystyle-2\alpha_{n}\langle x_{x_{n}+1}-x_{n},x_{n}-x_{n-1}\rangle$ $\displaystyle\geq$ $\displaystyle\\|x_{x_{n}+1}-x_{n}\\|^{2}+\alpha^{2}\\|x_{n}-x_{n-1}\\|^{2}$ $\displaystyle+\alpha_{n}\Big{(}-\rho_{n}\\|x_{x_{n}+1}-x_{n}\\|^{2}-\frac{1}{\rho_{n}}\\|x_{n}-x_{n-1}\\|^{2}\Big{)},$ where $\rho_{n}:=\frac{1}{\alpha_{n}+\delta\sigma}$. Combining (21) and (22), we get $\displaystyle\\|x_{n+1}-x^{}\\|^{2}-(1+\alpha_{n})\\|x_{n}-x^{}\\|^{2}+\alpha_{n}\\|x_{n-1}-x^{}\\|^{2}$ (23) $\displaystyle\leq$ $\displaystyle(\alpha_{n}\rho_{n}-1)\\|x_{n+1}-x_{n}\\|^{2}+\lambda_{n}\\|x_{n}-x_{n-1}\\|^{2},$ where $\displaystyle\lambda_{n}:=\alpha_{n}(1+\alpha_{n})+\alpha_{n}\frac{1-\alpha_{n}\rho_{n}}{\rho_{n}}\geq 0$ (24) since $\alpha_{n}\rho_{n}<1$. Taking into account the choice of $\rho_{n}$, we have $\delta=\frac{1-\alpha_{n}\rho_{n}}{\sigma\rho_{n}}$ and from (24), it follows that $\displaystyle\lambda_{n}$ $\displaystyle=$ $\displaystyle\alpha_{n}(1+\alpha_{n})+\alpha_{n}\frac{1-\alpha_{n}\rho_{n}}{\rho_{n}}$ (25) $\displaystyle\leq$ $\displaystyle\alpha(1+\alpha)+\alpha\sigma\delta.$ Following the same arguments as in [1, 2, 11], we define $\varphi_{n}:=\\|x_{n}-x^{}\\|^{2},n\geq 1$ and $\varepsilon_{n}:=\varphi_{n}-\alpha_{n}\varphi_{n-1}+\lambda_{n}\\|x_{n}-x_{n-1}\\|^{2},n\geq 1.$ By the monotonicity of $\\{\alpha_{n}\\}$ and the fact that $\varphi_{n}\geq 0$, we have $\varepsilon_{n+1}-\varepsilon_{n}\leq\varphi_{n+1}-(1+\alpha_{n})\varphi_{n}+\alpha_{n}\varphi_{n-1}+\lambda_{n+1}\\|x_{n+1}-x_{n}\\|^{2}-\lambda_{n}\\|x_{n}-x_{n-1}\\|^{2}.$ Using (23), we have $\displaystyle\varepsilon_{n+1}-\varepsilon_{n}$ $\displaystyle\leq$ $\displaystyle(\alpha_{n}\rho_{n}-1)\\|x_{n+1}-x_{n}\\|^{2}+\lambda_{n}\\|x_{n}-x_{n-1}\\|^{2}$ (26) $\displaystyle+\lambda_{n+1}\\|x_{n+1}-x_{n}\\|^{2}-\lambda_{n}\\|x_{n}-x_{n-1}\\|^{2}$ $\displaystyle=$ $\displaystyle(\alpha_{n}\rho_{n}-1+\lambda_{n+1})\\|x_{n+1}-x_{n}\\|^{2}.$ We now claim that $\displaystyle\alpha_{n}\rho_{n}-1+\lambda_{n+1}\leq-\beta.$ (27) Indeed by the choice of $\rho_{n}$, we have $\displaystyle\alpha_{n}\rho_{n}-1+\lambda_{n+1}\leq-\beta$ $\displaystyle\Leftrightarrow$ $\displaystyle\alpha_{n}\rho_{n}-1+\lambda_{n+1}+\beta\leq 0$ $\displaystyle\Leftrightarrow$ $\displaystyle\lambda_{n+1}+\beta+\frac{\alpha_{n}}{\alpha_{n}+\delta\sigma}-1\leq 0$ $\displaystyle\Leftrightarrow$ $\displaystyle\lambda_{n+1}+\beta-\frac{\delta\sigma}{\alpha_{n}+\delta\sigma}\leq 0$ $\displaystyle\Leftrightarrow$ $\displaystyle(\alpha_{n}+\delta\sigma)(\lambda_{n+1}+\beta)\leq\delta\sigma$ Now, using (25), we have $(\alpha_{n}+\delta\sigma)(\lambda_{n+1}+\beta)\leq((\alpha+\delta\sigma)(\alpha(1+\alpha)\alpha\delta\sigma+\beta)\leq\delta\sigma,$ where the last inequality follows from Assumption 3.2 (b). Hence, the claim in (27) is true. Thus, it follows from (26) and (27) that $\displaystyle\varepsilon_{n+1}-\varepsilon_{n}\leq-\beta\\|x_{n+1}-x_{n}\\|^{2}.$ (28) The sequence $\\{\varepsilon_{n}\\}$ is non-increasing and the bounds of $\\{\alpha_{n}\\}$ delivers $\displaystyle-\alpha\varphi_{n-1}\leq\varphi_{n}-\alpha\varphi_{n-1}\leq\varepsilon_{n}\leq\varepsilon_{1},n\geq 1.$ (29) It then follows that $\displaystyle\varphi_{n}\leq\alpha^{n}\varphi_{0}+\varepsilon_{1}\sum_{k=0}^{n-1}\alpha^{k}\leq\alpha^{n}\varphi_{0}+\frac{\varepsilon_{1}}{1-\alpha},n\geq 1.$ (30) Combining (28) and (34), we get $\displaystyle\beta\sum_{k=1}^{n}\\|x_{k+1}-x_{k}\\|^{2}$ $\displaystyle\leq$ $\displaystyle\varepsilon_{1}-\varepsilon_{n+1}$ (31) $\displaystyle\leq$ $\displaystyle\varepsilon_{1}+\alpha\varphi_{n}$ $\displaystyle\leq$ $\displaystyle\alpha^{n+1}\varphi_{0}+\frac{\varepsilon_{1}}{1-\alpha}$ $\displaystyle\leq$ $\displaystyle\varphi_{0}+\frac{\varepsilon_{1}}{1-\alpha},$ which shows that $\displaystyle\sum_{k=1}^{\infty}\\|x_{k+1}-x_{k}\\|^{2}<\infty.$ (32) Thus, $\underset{n\rightarrow\infty}{\lim}\\|x_{n+1}-x_{n}\\|=0$. From $w_{n}=x_{n}+\alpha_{n}(x_{n}-x_{n-1})$, we have $\displaystyle\\|w_{n}-x_{n}\\|$ $\displaystyle\leq$ $\displaystyle\alpha_{n}\\|x_{n}-x_{n-1}\\|$ $\displaystyle\leq$ $\displaystyle\alpha\\|x_{n}-x_{n-1}\\|\rightarrow 0,n\rightarrow\infty.$ Similarly, $\\|x_{n+1}-w_{n}\\|\leq\\|x_{n+1}-x_{n}\\|+\\|x_{n}-w_{n}\\|\rightarrow 0,n\rightarrow\infty.$ Using Lemma 2.2, (23), (25) and (32), we have that $\underset{n\rightarrow\infty}{\lim}\\|x_{n}-x^{}\\|$ exists. Hence, $\\{x_{n}\\}$ is bounded. ∎ In the next two lemmas, we show that certain subsequences obtained in Algorithm 1 are null subsequences. These two lemmas are necessary in order to show that the weak limit of $\\{x_{n}\\}$ is an element of $SOL$ and for our weak convergence in Theorem 4.4 below. ###### Lemma 4.2. Let $\\{x_{n}\\}$ generated by Algorithm 1 above and Assumptions 3.1 and 3.2 hold. Then (a) $\displaystyle\lim_{n\rightarrow\infty}\eta_{n}\\|w_{n}-z_{n}\\|^{2}=0$; * (b) $\displaystyle\lim_{n\rightarrow\infty}\\|w_{n}-z_{n}\\|=0.$ ###### Proof. Let $x^{}\in\text{SOL}$. Since $F$ is uniformly continuous on bounded subsets of $X$, then $\\{F(x_{n})\\},\\{z_{n}\\},\\{w_{n}\\}$ and $\\{F(y_{n})\\}$ are bounded. In particular, there exists $M>0$ such that $\\|F(y_{n})\\|\leq M$ for all $n\in\mathbb{N}$. Combining Lemma 2.4 and Lemma 3.4, we get $\displaystyle\\|x_{n+1}-x^{}\\|^{2}$ $\displaystyle=$ $\displaystyle\\|P_{C_{n}}(w_{n})-x^{}\\|^{2}\leq\\|w_{n}-x^{}\\|^{2}-\\|x_{n+1}-w_{n}\\|^{2}$ (33) $\displaystyle=$ $\displaystyle\\|w_{n}-x^{}\\|^{2}-{\rm dist}^{2}(w_{n},C_{n})$ $\displaystyle\leq$ $\displaystyle\\|w_{n}-x^{}\\|^{2}-\Big{(}\frac{1}{M}h_{n}(w_{n})\Big{)}^{2}$ $\displaystyle\leq$ $\displaystyle\\|w_{n}-x^{}\\|^{2}-\Big{(}\frac{1}{2M}\sigma\eta_{n}\\|r(w_{n})\\|^{2}\Big{)}^{2}$ $\displaystyle=$ $\displaystyle\\|w_{n}-x^{}\\|^{2}-\Big{(}\frac{1}{2M}\sigma\eta_{n}\\|w_{n}-z_{n}\\|^{2}\Big{)}^{2}.$ Since $\\{x_{n}\\}$ is bounded, we obtain from (33) that $\displaystyle\Big{(}\frac{1}{2M}\sigma\eta_{n}\\|w_{n}-z_{n}\\|^{2}\Big{)}^{2}$ $\displaystyle\leq$ $\displaystyle\\|w_{n}-x^{}\\|^{2}-\\|x_{n+1}-x^{}\\|^{2}$ (34) $\displaystyle=$ $\displaystyle\Big{(}\\|w_{n}-x^{}\\|-\\|x_{n+1}-x^{}\\|\Big{)}\Big{(}\\|w_{n}-x^{}\\|+\\|x_{n+1}-x^{}\\|\Big{)}$ $\displaystyle\leq$ $\displaystyle\\|w_{n}-x^{}\\|-\\|x_{n+1}-x^{}\\|M_{1}$ $\displaystyle\leq$ $\displaystyle\\|w_{n}-x_{n+1}\\|M_{1},$ where $M_{1}:=\sup_{n\geq 1}\\{\\|w_{n}-x^{}\\|+\\|x_{n+1}-x^{}\\|\\}$. This establishes (a). To establish (b), We distinguish two cases depending on the behaviour of (the bounded) sequence of step-sizes $\\{\eta_{n}\\}$. Case 1: Suppose that $\liminf_{n\to\infty}\eta_{n}>0$. Then $0\leq\\|r(w_{n})\\|^{2}=\frac{\eta_{n}\\|r(w_{n})\\|^{2}}{\eta_{n}}$ and this implies that $\displaystyle\limsup_{n\to\infty}\\|r(w_{n})\\|^{2}$ $\displaystyle\leq\limsup_{n\to\infty}\bigg{(}\eta_{n}\\|r(w_{n})\\|^{2}\bigg{)}\bigg{(}\limsup_{n\to\infty}\frac{1}{\eta_{n}}\bigg{)}$ $\displaystyle=\bigg{(}\limsup_{n\to\infty}\eta_{n}\\|r(w_{n})\\|^{2}\bigg{)}\frac{1}{\liminf_{n\to\infty}\eta_{n}}$ $\displaystyle=0.$ Hence, $\limsup_{n\to\infty}\\|r(w_{n})\\|=0$. Therefore, $\lim_{n\rightarrow\infty}\\|w_{n}-z_{n}\\|=\lim_{n\rightarrow\infty}\\|r(w_{n})\\|=0.$ Case 2: Suppose that $\liminf_{n\to\infty}\eta_{n}=0$. Subsequencing if necessary, we may assume without loss of generality that $\lim_{n\to\infty}\eta_{n}=0$ and $\lim_{n\rightarrow\infty}\\|w_{n}-z_{n}\\|=a\geq 0$. Define $\bar{y}_{n}:=\frac{1}{\gamma}\eta_{n}z_{n}+\Big{(}1-\frac{1}{\gamma}\eta_{n}\Big{)}w_{n}$ or, equivalently, $\bar{y}_{n}-w_{n}=\frac{1}{\gamma}\eta_{n}(z_{n}-w_{n})$. Since $\\{z_{n}-w_{n}\\}$ is bounded and since $\lim_{n\to\infty}\eta_{n}=0$ holds, it follows that $\lim_{n\to\infty}\\|\bar{y}_{n}-w_{n}\\|=0.$ (35) From the step-size rule and the definition of $\bar{y}_{k}$, we have $\langle F(\bar{y}_{n}),w_{n}-z_{n}\rangle<\frac{\sigma}{2}\\|w_{n}-z_{n}\\|^{2},\ \forall n\in\mathbb{N},$ or equivalently $2\langle F(w_{n}),w_{n}-z_{n}\rangle+2\langle F(\bar{y}_{n})-F(w_{n}),w_{n}-z_{n}\rangle<\sigma\\|w_{n}-z_{n}\\|^{2},\ \forall n\in\mathbb{N}.$ Setting $t_{n}:=w_{n}-F(w_{n})$, we obtain form the last inequality that $2\langle w_{n}-t_{n},w_{n}-z_{n}\rangle+2\langle F(\bar{y}_{n})-F(w_{n}),w_{n}-z_{n}\rangle<\sigma\\|w_{n}-z_{n}\\|^{2},\ \forall n\in\mathbb{N}.$ Using Lemma 2.1 (b) we get $2\langle w_{n}-t_{n},w_{n}-z_{n}\rangle=\\|w_{n}-z_{n}\\|^{2}+\\|w_{n}-t_{n}\\|^{2}-\\|z_{n}-t_{n}\\|^{2}.$ Therefore, $\\|w_{n}-t_{n}\\|^{2}-\\|z_{n}-t_{n}\\|^{2}<(\sigma-1)\\|w_{n}-z_{n}\\|^{2}-2\langle F(\bar{y}_{n})-F(w_{n}),w_{n}-z_{n}\rangle\ \forall n\in\mathbb{N}.$ Since $F$ is uniformly continuous on bounded subsets of $H$ and (35), if $a>0$ then the right hand side of the last inequality converges to $(\sigma-1)a<0$ as $n\to\infty$. From the last inequality we have $\limsup_{n\to\infty}\left(\\|w_{n}-t_{n}\\|^{2}-\\|z_{n}-t_{n}\\|^{2}\right)\leq(\sigma-1)a<0.$ For $\epsilon=-(\sigma-1)a/2>0$, there exists $N\in\mathbb{N}$ such that $\\|w_{n}-t_{n}\\|^{2}-\\|z_{n}-t_{n}\\|^{2}\leq(\sigma-1)a+\epsilon=(\sigma-1)a/2<0\quad\forall n\in\mathbb{N},n\geq N,$ leading to $\\|w_{n}-t_{n}\\|<\\|z_{n}-t_{n}\\|\quad\forall n\in\mathbb{N},n\geq N,$ which is a contradiction to the definition of $z_{n}=P_{C}(w_{n}-F(w_{n}))$. Hence $a=0$, which completes the proof. ∎ The boundedness of the sequence $\\{x_{n}\\}$ implies that there is at least one weak limit point. We show that such weak limit point belongs to $SOL$ in the next result. ###### Lemma 4.3. Let Assumptions 3.1 and 3.2 hold. Furthermore let $\\{x_{n_{k}}\\}$ be a subsequence of $\\{x_{n}\\}$ converging weakly to a limit point $p$. Then $p\in\text{SOL}$. ###### Proof. By the definition of $z_{n_{k}}$ together with (13), we have $\langle w_{n_{k}}-F(w_{n_{k}})-z_{n_{k}},x-z_{n_{k}}\rangle\leq 0,\ \forall x\in C,$ which implies that $\langle w_{n_{k}}-z_{n_{k}},x-z_{n_{k}}\rangle\leq\langle F(w_{n_{k}}),x-z_{n_{k}}\rangle,\ \forall x\in C.$ Hence, $\langle w_{n_{k}}-z_{n_{k}},x-z_{n_{k}}\rangle+\langle F(w_{n_{k}}),z_{n_{k}}-w_{n_{k}}\rangle\leq\langle F(w_{n_{k}}),x-w_{n_{k}}\rangle,\ \forall x\in C.$ (36) Fix $x\in C$ and let $k\rightarrow\infty$ in (47). Since $\lim_{k\to\infty}\\|w_{n_{k}}-z_{n_{k}}\\|=0$, we have $0\leq\liminf_{k\to\infty}\langle F(w_{n_{k}}),x-w_{n_{k}}\rangle$ (37) for all $x\in C$. It follows from (47) and the monotonicity of $F$ that $\displaystyle\langle w_{n_{k}}-z_{n_{k}},x-z_{n_{k}}\rangle+\langle F(w_{n_{k}}),z_{n_{k}}-w_{n_{k}}\rangle$ $\displaystyle\leq$ $\displaystyle\langle F(w_{n_{k}}),x-w_{n_{k}}\rangle$ $\displaystyle\leq$ $\displaystyle\langle F(x),x-w_{n_{k}}\rangle\quad\forall x\in C.$ Letting $k\to+\infty$ in the last inequality, remembering that $\lim_{k\to\infty}\\|w_{n_{k}}-z_{n_{k}}\\|=0$ for all $k$, we have $\langle F(x),x-p\rangle\geq 0\quad\forall x\in C.$ In view of Lemma 2.7, this implies $p\in\text{SOL}$. ∎ All is now set to give the weak convergence result in the theorem below. ###### Theorem 4.4. Let Assumptions 3.1 and 3.2 hold. Then the sequence $\\{x_{n}\\}$ generated by Algorithm 1 weakly converges to a point in SOL. ###### Proof. We have shown that (i) $\lim_{n\to\infty}\\|x_{n}-x^{}\\|$ exists; (ii) $\omega_{w}(x_{n})\subset\text{SOL}$, where $\omega_{w}(x_{n}):=\\{x:\exists x_{n_{j}}\rightharpoonup x\\}$ denotes the weak $\omega$-limit set of $\\{x_{n}\\}$. Then, by Lemma 2.3, we have that $\\{x_{n}\\}$ converges weakly to a point in SOL. ∎ We give some discussions on further contributions of this paper in the remark below. ###### Remark 4.5. (a) Our iterative Algorithm 1 is more applicable than some recent results on projection type methods with inertial extrapolation step for solving VI($F,C$) (1) in real Hilbert spaces. For instance, the proposed method in [21] can only be applied for a case when $F$ is monotone and $L$-Lipschitz continuous. Moreover, the Lipschitz constant or an estimate of it has to be known when implementing the Algorithm 3.1 of [21]. In this result, Algorithm 1 is applicable when $F$ is uniformly continuous and monotone operator. (b) In finite-dimensional spaces, the assumption that $F$ is uniformly continuous on bounded subsets of $C$ automatically holds when $F$ is continuous. Moreover, in this case, only continuity of $F$ is required and our weak convergence in Theorem 4.4 coincides with global convergence of sequence of iterates $\\{x_{n}\\}$ in $\mathbb{R}^{n}$. (c) Lemmas 3.5, 4.1, 4.2 and Theorem 4.4 still hold for a more general case of $F$ pseudo-monotone (i.e., for all $x,y\in H$, $\langle F(x),y-x\rangle\geq 0\Longrightarrow\langle F(y),y-x\rangle\geq 0;$). We give a version of Lemma 4.3 for the case of $F$ pseudo-monotone in the Appendix. $\Diamond$ ## 5 Rate of Convergence In this section we give the rate of convergence of the iterative method 1 proposed in Section 3. We show that the proposed method has sublinear rate of convergence and establish the nonasymptotic $O(1/n)$ convergence rate of the proposed method. To the best of our knowledge, there is no convergence rate result known in the literature without stronger assumptions for inertial projection-type Algorithm 1 for VI$(F,C)$ (1) in infinite dimensional Hilbert spaces. ###### Theorem 5.1. Let Assumptions 3.1 and 3.2 hold. Let the sequence $\\{x_{n}\\}$ be generated by Algorithm 1 and $x_{0}=x_{1}$. Then for any $x^{}\in\text{SOL}$ and for any positive integer $n$, it holds that $\underset{1\leq i\leq n}{\min}\\|x_{i+1}-w_{i}\\|^{2}\leq\frac{\Big{[}1+\Big{(}\frac{\alpha}{1-\alpha}+\frac{\alpha(1-\alpha)}{\beta}\Big{)}\Big{(}1+\frac{1-\alpha_{0}}{1-\alpha}\Big{)}\Big{]}\\|x_{0}-x^{}\\|^{2}}{n}.$ ###### Proof. From (21), we have $\displaystyle\\|x_{n+1}-x^{}\\|^{2}-\\|x_{n}-x^{}\\|^{2}-\alpha_{n}(\\|x_{n}-x^{}\\|^{2}-\\|x_{n-1}-x^{}\\|^{2})$ (38) $\displaystyle\leq$ $\displaystyle\alpha_{n}(1+\alpha_{n})\\|x_{n}-x_{n-1}\\|^{2}-\\|x_{n+1}-w_{n}\\|^{2}.$ This implies that $\displaystyle\\|x_{n+1}-w_{n}\\|^{2}$ $\displaystyle\leq$ $\displaystyle\varphi_{n}-\varphi_{n+1}+\alpha_{n}(\varphi_{n}-\varphi_{n-1})+\delta_{n}$ (39) $\displaystyle\leq$ $\displaystyle\varphi_{n}-\varphi_{n+1}+\alpha[V_{n}]_{+}+\delta_{n},$ where $\delta_{n}:=\alpha_{n}(1+\alpha_{n})\\|x_{n}-x_{n-1}\\|^{2}$, $V_{n}:=\varphi_{n}-\varphi_{n-1}$, $[V_{n}]_{+}:=\max\\{V_{n},0\\}$ and $\varphi_{n}:=\\|x_{n}-x^{}\\|^{2}$. Observe from (31) that $\sum_{n=1}^{\infty}\\|x_{n+1}-x_{n}\\|^{2}\leq\frac{1}{\beta}\Big{[}\varphi_{0}+\frac{\varepsilon_{1}}{1-\alpha}\Big{]}.$ So, $\displaystyle\sum_{n=1}^{\infty}\delta_{n}$ $\displaystyle=$ $\displaystyle\sum_{n=1}^{\infty}\alpha_{n}(1+\alpha_{n})\\|x_{n}-x_{n-1}\\|^{2}$ (40) $\displaystyle\leq$ $\displaystyle\sum_{n=1}^{\infty}\alpha(1+\alpha)\\|x_{n}-x_{n-1}\\|^{2}$ $\displaystyle=$ $\displaystyle\alpha(1+\alpha)\sum_{n=1}^{\infty}\\|x_{n}-x_{n-1}\\|^{2}$ $\displaystyle\leq$ $\displaystyle\frac{\alpha(1+\alpha)}{\beta}\Big{[}\varphi_{0}+\frac{\varepsilon_{1}}{1-\alpha}\Big{]}:=C_{1}.$ The inequality (38) implies that $\displaystyle V_{n+1}$ $\displaystyle\leq$ $\displaystyle\alpha_{n}V_{n}+\delta_{n}$ $\displaystyle\leq$ $\displaystyle\alpha[V_{n}]_{+}+\delta_{n}.$ Therefore, $\displaystyle[V_{n+1}]_{+}$ $\displaystyle\leq$ $\displaystyle\alpha[V_{n}]_{+}+\delta_{n}$ (41) $\displaystyle\leq$ $\displaystyle\alpha^{n}[V_{1}]_{+}+\sum_{j=1}^{n}\alpha^{j-1}\delta_{n+1-j}.$ Note that by our assumption $x_{0}=x_{1}$. This implies that $V_{1}=[V_{1}]_{+}=0$ and $\delta_{1}=0$. From (41), we get $\displaystyle\sum_{n=2}^{\infty}[V_{n}]_{+}$ $\displaystyle\leq$ $\displaystyle\frac{1}{1-\alpha}\sum_{n=1}^{\infty}\delta_{n}$ (42) $\displaystyle=$ $\displaystyle\frac{1}{1-\alpha}\sum_{n=2}^{\infty}\delta_{n}.$ From (39), we get $\displaystyle\sum_{i=1}^{n}\\|x_{i+1}-w_{i}\\|^{2}$ $\displaystyle\leq$ $\displaystyle\varphi_{1}-\varphi_{n}+\alpha\sum_{i=1}^{n}[V_{i}]_{+}+\sum_{i=2}^{n}\delta_{i}$ (43) $\displaystyle\leq$ $\displaystyle\varphi_{1}+\alpha C_{2}+C_{1},$ where $C_{2}=\frac{C_{1}}{1-\alpha}\geq\frac{1}{1-\alpha}\sum_{i=2}^{\infty}\delta_{i}\geq\sum_{i=2}^{\infty}[V_{i}]_{+}$ by (42). Now, since $\varepsilon_{1}=\varphi_{1}-\alpha_{1}\varphi_{0}=(1-\alpha_{1})\varphi_{1}$, we have $\displaystyle\varphi_{1}+\alpha C_{2}+C_{1}$ $\displaystyle=$ $\displaystyle\varphi_{0}+\frac{\alpha C_{1}}{1-\alpha}$ (44) $\displaystyle+\frac{\alpha(1+\alpha)}{\beta}\Big{[}\varphi_{0}+\frac{\varepsilon_{1}}{1-\alpha}\Big{]}$ $\displaystyle=$ $\displaystyle\varphi_{0}+\frac{\alpha C_{1}}{1-\alpha}$ $\displaystyle+\frac{\alpha(1+\alpha)}{\beta}\Big{[}1+\frac{1-\alpha_{0}}{1-\alpha}\Big{]}\varphi_{0}$ $\displaystyle=$ $\displaystyle\varphi_{0}+\frac{\alpha}{1-\alpha}\Big{[}1+\frac{1-\alpha_{0}}{1-\alpha}\Big{]}$ $\displaystyle+\frac{\alpha(1+\alpha)}{\beta}\Big{[}1+\frac{1-\alpha_{0}}{1-\alpha}\Big{]}\varphi_{0}$ $\displaystyle=$ $\displaystyle\Big{[}1+\Big{(}\frac{\alpha}{1-\alpha}+\frac{\alpha(1-\alpha)}{\beta}\Big{)}\Big{(}1+\frac{1-\alpha_{0}}{1-\alpha}\Big{)}\Big{]}\varphi_{0}.$ From (43) and (44), we obtain $\displaystyle\underset{1\leq i\leq n}{\min}\\|x_{i+1}-w_{i}\\|^{2}\leq\frac{\Big{[}1+\Big{(}\frac{\alpha}{1-\alpha}+\frac{\alpha(1-\alpha)}{\beta}\Big{)}\Big{(}1+\frac{1-\alpha_{0}}{1-\alpha}\Big{)}\Big{]}\\|x_{0}-x^{}\\|^{2}}{n}.$ (45) ∎ ###### Remark 5.2. (a) Note that $x_{n+1}=w_{n}$ implies that $w_{n}\in C_{n}$, where $C_{n}$ is as defined in Algorithm 1 and hence $h_{n}(w_{n})\leq 0$. By Lemma 3.4, we get $\frac{\sigma\eta_{n}}{2}\\|w_{n}-z_{n}\\|^{2}\leq h_{n}(w_{n})$. Therefore, $0\leq\frac{\sigma\eta_{n}}{2}\\|w_{n}-z_{n}\\|^{2}\leq h_{n}(w_{n})\leq 0,$ which implies that $w_{n}=z_{n}$. Thus, the equality $x_{n+1}=w_{n}$ implies that $x_{n+1}$ is already a solution of VI$(F,C)$ (1). In this sense, the error estimate given in Theorem 5.1 can be viewed as a convergence rate result of the inertial projection-type method 1. In particular, (45) implies that, to obtain an $\epsilon$-optimal solution in the sense that $\\|x_{n+1}-w_{n}\\|^{2}<\epsilon$, the upper bound of iterations required by inertial projection-type method 1 is $\frac{\Big{[}1+\Big{(}\frac{\alpha}{1-\alpha}+\frac{\alpha(1-\alpha)}{\beta}\Big{)}\Big{(}1+\frac{1-\alpha_{0}}{1-\alpha}\Big{)}\Big{]}\\|x_{0}-x^{}\\|^{2}}{\epsilon}$. We note that with the ” $\min_{1\leq i\leq n}$”, a nonasymptotic $O(1/n)$ convergence rate implies that an $\epsilon$-accuracy solution, in the sense that $\\|x_{n+1}-w_{n}\\|^{2}<\epsilon$, is obtainable within no more than $O(1/\epsilon)$ iterations. Furthermore, if $\alpha_{n}=0$ for all $n$, then the ”$\min_{1\leq i\leq n}$” can be removed by setting $i=n$ in Theorem 5.1. $\Diamond$ ## 6 Numerical Experiments In this section, we discuss the numerical behaviour of Algorithm 1 using different test examples taken from the literature which are describe below and compare our method with (5), (10) and the original Algorithm (when $\alpha_{n}=0$) of Algorithm 1. ###### Example 6.1. This first example (also considered in [35, 36]) is a classical example for which the usual gradient method does not converge. It is related to the unconstrained case of VI($F,C$) (1) where the feasible set is $C:=\mathbb{R}^{m}$ (for some positive even integer $m$) and $F:=(a_{ij})_{1\leq i,j\leq m}$ is the square matrix $m\times m$ whose terms are given by $\displaystyle a_{ij}=\left\\{\begin{array}[]{llll}&-1,\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm if}\leavevmode\nobreak\ \leavevmode\nobreak\ j=m+1-i\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm and}\leavevmode\nobreak\ \leavevmode\nobreak\ j>i\\\ &1,\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm if}\leavevmode\nobreak\ \leavevmode\nobreak\ j=m+1-i\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm and}\leavevmode\nobreak\ \leavevmode\nobreak\ j<i\\\ &0\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm otherwise}\end{array}\right.$ The zero vector $z=(0,\ldots,0)$ is the solution of this test example. The initial point $x_{0}$ is the unit vector. We choose $\gamma=0.1$, $\sigma=0.8$ and $\alpha_{n}=0.6$, $m=500$. Figure 1: Comparison of Algorithm 1 with the original Algorithm. The numerical result is listed in Figure 1, which illustrates that Algorithm 1 highly improves the original Algorithm. ###### Example 6.2. This example is taken from [24] and has been considered by many authors for numerical experiments (see, for example, [26, 37, 43]). The operator $A$ is defined by $A(x):=Mx+q$, where $M=BB^{T}+S+D$, where $B,S,D\in\mathbb{R}^{m\times m}$ are randomly generated matrices such that $S$ is skew-symmetric (hence the operator does not arise from an optimization problem), $D$ is a positive definite diagonal matrix (hence the variational inequality has a unique solution) and $q=0$. The feasible set $C$ is described by linear inequality constraints $Bx\leq b$ for some random matrix $B\in\mathbb{R}^{k\times m}$ and a random vector $b\in\mathbb{R}^{k}$ with nonnegative entries. Hence the zero vector is feasible and therefore the unique solution of the corresponding variational inequality. These projections are computed by solving a quadratic optimization problem using the MATLAB solver quadprog. Hence, for this class of problems, the evaluation of $A$ is relatively inexpensive, whereas projections are costly. We present the corresponding numerical results (number of iterations and CPU times in seconds) using four different dimensions $m$ and two different numbers of inequality constraints $k$. We compare our proposed Algorithm 1, original Algorithm, subgradient extragradient method (5) and the inertial subgradient extragradient method (10) using Example 6.2 and the numerical results are listed in Tables 1 -4 and shown in Figures 2-5 below. We take the initial point $x_{0}$ to be the unit vector in these algorithms. We use “OPM” to denote the original Algorithm, “SPM” to denote the subgradient extragradient method (5) and “iSPM” to denote inertial subgradient extragradient method (10). We choose the stopping criterion as $\\|x^{k}\\|\leq\epsilon=0.001.$ The size $k=30,50,80$ and $m=20,50,80,100$. The matrices $B,S,D$ and the vector $b$ are generated randomly. We choose $\gamma=0.1$, $\sigma=0.8$ and $\alpha_{n}=0.1$ in Algorithm (1). In (5), we choose $\sigma=0.8$, $\rho=0.1$, $\mu=0.2$. In iSPM (10), $\alpha=0.2,L=\\|M\\|,\tau=0.5\frac{\frac{1}{2}-2\alpha-\frac{1}{2}\alpha^{2}}{\frac{1}{2}-\alpha+\frac{1}{2}\alpha^{2}}$. We denote by “Iter.” the number of iterations and “InIt.” the number of total iterations of finding suitable step size in Tables 1 -4 below. Table 1: Comparison of Algorithm 1, original Algorithm and methods (5) and (10) for $k=50,\alpha_{n}=0.1.$ \| \| Iter. \| \| \| \| \| InIt. \| \| \| \| CPU in second \| ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- $m$ \| Alg.1 \| OPM \| SPM \| iSPM \| \| Alg.1 \| OPM \| SPM \| \| Alg.1 \| OPM \| SPM \| iSPM \| 20 \| 1044 \| 1162 \| 2891 \| 4783 \| \| 2376 \| 5022 \| 4283 \| \| 0.0781 \| 0.2188 \| 0.6563 \| 0.5781 \| 50 \| 4829 \| 5912 \| 25544 \| 29639 \| \| 13809 \| 30885 \| 123982 \| \| 0.5625 \| 0.6094 \| 7.5938 \| 2.1875 \| 80 \| 19803 \| 22129 \| 19736 \| 61322 \| \| 74066 \| 157061 \| 93047 \| \| 6.6094 \| 7.0313 \| 15.9063 \| 13.4844 \| 100 \| 26821 \| 31149 \| 35520 \| 92579 \| \| 101925 \| 220297 \| 173670 \| \| 12.1406 \| 15.8438 \| 67.4219 \| 34.6250 \| Table 2: Comparison of Algorithm 1, the original Algorithm and methods (5) and (10) for $k=80,\alpha_{n}=0.1.$ \| \| Iter. \| \| \| \| InIt. \| \| \| \| CPU in second \| ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- $m$ \| Alg.1 \| OPM \| SPM \| iSPM \| \| Alg.1 \| OPM \| SPM \| \| Alg.1 \| OPM \| SPM \| iSPM 20 \| 1479 \| 1651 \| 5096 \| 5306 \| \| 3676 \| 7783 \| 19071 \| \| 0.1875 \| 0.2188 \| 6.7869 \| 0.7500 50 \| 4152 \| 5088 \| 8144 \| 28033 \| \| 11955 \| 26594 \| 36342 \| \| 1.0625 \| 1.2344 \| 8.2344 \| 6.2813 80 \| 22711 \| 25864 \| 22281 \| 64588 \| \| 85176 \| 182177 \| 105237 \| \| 8.2188 \| 9.3438 \| 20.6719 \| 16.1563 100 \| 26314 \| 30568 \| 37588 \| 88138 \| \| 99998 \| 216162 \| 185430 \| \| 13.0625 \| 16.1250 \| 118.4375 \| 36.0625 Table 3: Comparison of Algorithm 1, the original Algorithm and methods (5) and (10) for $k=30,\alpha_{n}=0.6.$ \| \| Iter. \| \| \| \| InIt. \| \| \| \| CPU in second \| ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- $m$ \| Alg.1 \| OPM \| SPM \| iSPM \| \| Alg.1 \| OPM \| SPM \| \| Alg.1 \| OPM \| SPM \| iSPM 10 \| 197 \| 469 \| 509 \| 1327 \| \| 341 \| 1370 \| 1460 \| \| 1.21884 \| 0.0313 \| 0.2969 \| 0.2500 30 \| 1548 \| 4745 \| 4347 \| 13697 \| \| 4243 \| 17534 \| 16880 \| \| 10.8750 \| 0.4063 \| 1.0938 \| 0.8750 50 \| 1581 \| 4899 \| 8269 \| 26393 \| \| 4762 \| 18859 \| 37237 \| \| 12.9688 \| 0.5000 \| 2.4688 \| 1.6094 70 \| 6192 \| 6256 \| 7826 \| 56491 \| \| 22381 \| 81445 \| 31406 \| \| 62.4844 \| 5.5781 \| 9.3281 \| 9.1813 Table 4: Comparison of Algorithm 1, the original Algorithm and methods (5) and (10) for $k=50,\alpha_{n}=0.6.$ \| \| Iter. \| \| \| \| InIt. \| \| \| \| CPU in second \| ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- $m$ \| Alg.1 \| OPM \| SPM \| iSPM \| \| Alg.1 \| OPM \| SPM \| \| Alg.1 \| OPM \| SPM \| iSPM 10 \| 147 \| 419 \| 579 \| 1117 \| \| 253 \| 1024 \| 1119 \| \| 3 \| 0.0313 \| 0.5781 \| 0.3594 30 \| 1715 \| 5110 \| 6373 \| 13934 \| \| 4705 \| 19018 \| 25006 \| \| 30.0469 \| 0.4844 \| 1.5469 \| 1.1563 50 \| 1308 \| 4798 \| 9227 \| 31585 \| \| 4062 \| 17873 \| 41084 \| \| 41.4375 \| 0.4844 \| 3.5156 \| 2.0469 70 \| 5673 \| 14944 \| 8205 \| 52124 \| \| 20548 \| 74974 \| 33716 \| \| 393 \| 4.5469 \| 7.6406 \| 10.4375 Figure 2: Comparison of Algorithm 1, original Algorithm and methods (5) and (10). $k=50,\alpha_{n}=0.1.$ Figure 3: Comparison of Algorithm 1, original Algorithm and methods (5) and (10). $k=80,\alpha_{n}=0.1.$ Figure 4: Comparison of Algorithm 1, original Algorithm and methods (5) and (10). $k=30,\alpha_{n}=0.6.$ Figure 5: Comparison of Algorithm 1, original Algorithm and methods (5) and (10). $k=50,\alpha_{n}=0.6.$ Tables 1 -4 and Figures 2-5 show that Algorithm 1 improves the original Algorithm with respect to “Iter.”, “InIt.” and CPU time. It is also observed from Tables 1 -4 and Figures 2-5 that our proposed Algorithm 1 outperform the subgradient extragradient method (5) and the inertial subgradient extragradient method (10) with respect to the CPU time and the number of iterations when the feasible set $C$ is nonempty closed affine subset of $H$. ## 7 Final Remarks This paper presents a weak convergence result with inertial projection-type method for monotone variational inequality problems in real Hilbert spaces under very mild assumptions. This class of method is of inertial nature because at each iteration the projection-type is applied to a point extrapolated at the current iterate in the direction of last movement. Our proposed algorithm framework is not only more simple and intuitive, but also more general than some already proposed inertial projection type methods for solving variational inequality. Based on some pioneering analysis and Algorithm 1, we established certain nonasymptotic $O(1/n)$ convergence rate results. Our preliminary implementation of the algorithms and experimental results have shown that inertial algorithms are generally faster than the corresponding original un-accelerated ones. In our experiments, the extrapolation step-length $\alpha_{n}$ was set to be constant. How to select $\alpha_{n}$ adaptively such that the overall performance is stable and more efficient deserves further investigation. Interesting topics for future research may include relaxing the conditions on $\\{\alpha_{n}\\}$, improving the convergence results, and proposing modified inertial-type algorithms so that the extrapolation step-size can be significantly enlarged. 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Let $F$ be pseudo-monotone, uniformly continuous and sequentially weakly continuous on $H$. Assume that Assumption 3.2 holds. Furthermore let $\\{x_{n_{k}}\\}$ be a subsequence of $\\{x_{n}\\}$ converging weakly to a limit point $p$. Then $p\in\text{SOL}$. ###### Proof. By the definition of $z_{n_{k}}$ together with (13), we have $\langle w_{n_{k}}-F(w_{n_{k}})-z_{n_{k}},x-z_{n_{k}}\rangle\leq 0,\ \forall x\in C,$ which implies that $\langle w_{n_{k}}-z_{n_{k}},x-z_{n_{k}}\rangle\leq\langle F(w_{n_{k}}),x-z_{n_{k}}\rangle,\ \forall x\in C.$ Hence, $\langle w_{n_{k}}-z_{n_{k}},x-z_{n_{k}}\rangle+\langle F(w_{n_{k}}),z_{n_{k}}-w_{n_{k}}\rangle\leq\langle F(w_{n_{k}}),x-w_{n_{k}}\rangle,\ \forall x\in C.$ (47) Fix $x\in C$ and let $k\rightarrow\infty$ in (47). Since $\lim_{k\to\infty}\\|w_{n_{k}}-z_{n_{k}}\\|=0$, we have $0\leq\liminf_{k\to\infty}\langle F(w_{n_{k}}),x-w_{n_{k}}\rangle$ (48) for all $x\in C$. Now we choose a sequence $\\{\epsilon_{k}\\}_{k}$ of positive numbers decreasing and tending to $0$. For each $\epsilon_{k}$, we denote by $N_{k}$ the smallest positive integer such that $\left\langle F(w_{n_{j}}),x-w_{n_{j}}\right\rangle+\epsilon_{k}\geq 0\quad\forall j\geq N_{k},$ (49) where the existence of $N_{k}$ follows from (48). Since $\left\\{\epsilon_{k}\right\\}$ is decreasing, it is easy to see that the sequence $\left\\{N_{k}\right\\}$ is increasing. Furthermore, for each $k$, $F(w_{N_{k}})\not=0$ and, setting $v_{N_{k}}=\frac{F(w_{N_{k}})}{\\|F(w_{N_{k}}\\|^{2}},$ we have $\left\langle F(w_{N_{k}}),v_{N_{k}}\right\rangle=1$ for each $k$. Now we can deduce from (49) that for each $k$ $\left\langle F(w_{N_{k}}),x+\epsilon_{k}v_{N_{k}}-w_{N_{k}}\right\rangle\geq 0,$ and, since $F$ is pseudo-monotone, that $\left\langle F(x+\epsilon_{k}v_{N_{k}}),x+\epsilon_{k}v_{N_{k}}-w_{N_{k}}\right\rangle\geq 0.$ (50) On the other hand, we have that $\left\\{x_{n_{k}}\right\\}$ converges weakly to $p$ when $k\to\infty$. Since $F$ is sequentially weakly continuous on $C$, $\left\\{F(w_{n_{k}})\right\\}$ converges weakly to $F(p)$. We can suppose that $F(p)\not=0$ (otherwise, $p$ is a solution). Since the norm mapping is sequentially weakly lower semicontinuous, we have $0<\\|F(p)\\|\leq\lim\inf_{k\to\infty}\\|F(w_{n_{k}})\\|.$ Since $\left\\{w_{N_{k}}\right\\}\subset\left\\{w_{n_{k}}\right\\}$ and $\epsilon_{k}\to 0$ as $k\to\infty$, we obtain $\displaystyle 0$ $\displaystyle\leq$ $\displaystyle\lim\sup_{k\to\infty}\\|\epsilon_{k}v_{N_{k}}\\|=\lim\sup_{k\to\infty}\Big{(}\frac{\epsilon_{k}}{\\|F(w_{n_{k}})\\|}\Big{)}$ $\displaystyle\leq$ $\displaystyle\frac{\lim\sup_{k\to\infty}\epsilon_{k}}{\lim\inf_{k\to\infty}\\|F(w_{n_{k}})\\|}\leq\frac{0}{\\|F(p)\\|}=0,$ which implies that $\lim_{k\to\infty}\\|\epsilon_{k}v_{N_{k}}\\|=0$. Hence, taking the limit as $k\to\infty$ in (50), we obtain $\left\langle F(x),x-p\right\rangle\geq 0.$ Now, using Lemma 2.2 of [39], we have that $p\in\text{SOL}$. ∎
# Orthogonal subspace based fast iterative thresholding algorithms for joint sparsity recovery Ningning Han, Shidong Li, and Jian Lu _Member, IEEE_ This work was supported by the National Natural Science Foundation of China under grants 61972265, 11871348 and 61373087, by the Natural Science Foundation of Guangdong Province of China under grant 2020B1515310008, by the Educational Commission of Guangdong Province of China under grant 2019KZDZX1007, and by the Guangdong Key Laboratory of Intelligent Information Processing, China, and the NSF of USA (DMS-1615288). Ningning Han<EMAIL_ADDRESS>and Jian Lu (corresponding author<EMAIL_ADDRESS>are with Shenzhen Key Laboratory of Advanced Machine Learning and Applications, College of Mathematics and Statistics, Shenzhen University, Shenzhen, 518060.Shidong Li <EMAIL_ADDRESS>is with Department of Mathematics, San Francisco State University, San Francisco, CA94132. ###### Abstract Sparse signal recoveries from multiple measurement vectors (MMV) with joint sparsity property have many applications in signal, image, and video processing. The problem becomes much more involved when snapshots of the signal matrix are temporally correlated. With signal’s temporal correlation in mind, we provide a framework of iterative MMV algorithms based on thresholding, functional feedback and null space tuning. Convergence analysis for exact recovery is established. Unlike most of iterative greedy algorithms that select indices in a measurement/solution space, we determine indices based on an orthogonal subspace spanned by the iterative sequence. In addition, a functional feedback that controls the amount of energy relocation from the “tails” is implemented and analyzed. It is seen that the principle of functional feedback is capable to lower the number of iteration and speed up the convergence of the algorithm. Numerical experiments demonstrate that the proposed algorithm has a clearly advantageous balance of efficiency, adaptivity and accuracy compared with other state-of-the-art algorithms. ###### Index Terms: Multiple measurement vectors, null space tuning, thresholding, feedback, orthogonal subspace. ## I Introduction In sparse reconstruction signal models with joint sparsity property, signals are sampled at $L$ time instances, resulting in the multiple measurement vector (MMV) model: $\begin{array}[]{l}Y=\Phi X+E,\end{array}$ (1) where $Y\in\mathbb{C}^{M\times L}$ is the observation matrix containing $L$ measurement/snapshot (column) vectors, $\Phi\in\mathbb{C}^{M\times N}$ is the measurement matrix governed by the specific physical system, and $X\in\mathbb{C}^{N\times L}$ is the underlying source signal matrix, to be recovered. $E\in\mathbb{C}^{M\times L}$ is an additive measurement noise matrix. In this system, $L$ measurements share the same row support and elements in each nonzero row of $X$ are temporally correlated. The solution problem to a noiseless MMV model can be formulated as $\begin{array}[]{l}\min\limits_{X}\\|X\\|_{0}~{}\text{s.t.}~{}Y=\Phi X,\end{array}$ (2) where $\\|X\\|_{0}=\|\text{supp}(X)\|$, $\text{supp}(X)=\\{1\leq i\leq N:X_{i\cdot}\neq 0\\}$, $X_{i\cdot}$ is the $i$-th row of $X$. In [1], the authors have shown that $X$ is the unique solution of (2) if $\begin{array}[]{l}\\|X\\|_{0}<\frac{\text{spark}(\Phi)+\text{rank}(Y)-1}{2},\end{array}$ (3) where spark$(\Phi)$ is the smallest number of linearly dependent columns of $\Phi$. A large majority of effective algorithms for solving (2) are based on two strategies: extending single measurement vector (SMV) algorithms or exploiting signal subspaces. Well-known algorithms of the first class include simultaneous orthogonal matching pursuit (SOMP) [2]-[5], mixed norm minimization techniques [6]-[14], simultaneous greedy algorithms [15, 16]. However, these algorithms, without exploiting subspace structures or temporal correlations, have not offered realistic improvements over performances than that of SMV cases. Recently, a multiple sparse Bayesian learning (MSBL) algorithm [17]-[21], as an extension of sparse Bayesian SMV algorithms, is seen to improve recovery performances by modeling temporal correlation of sparse vectors. Another strategy is to exploit subspace structures spanned by measurement vectors. Representative algorithms include, e.g., sequential compressive MUSIC (SeqCS-MUSIC) [22, 23], subspace-augmented MUSIC (SA- MUSIC+OSMP) [24], rank aware order recursive matching pursuit (RA-ORMP) [25, 26, 27], semi-supervised MUSIC (SS-MUSIC) [28] etc. In this report, we provide a computationally efficient “greedy” algorithm for joint sparsity signal recoveries from their multiple measurement vectors. The proposed algorithm combines procedures of hard thresholding (HT), functional feedback ($f$-FB) for “tail” energy shrinkage and enhanced feasibility, the null space tuning (NST), and a novel variable selection mechanism. The novel criterion of variable selection is based on estimations of significant coefficients in an orthogonal subspace of the iterative sequence. The cardinality of selected variables is determined by the feedback function $f$. Experimental results show that the proposed algorithm provides superior performances in terms of the efficiency and the critical sparsity (i.e., the maximum sparsity level at which the perfect recovery is guaranteed [29]). In fact, the rate of successful recovery of our algorithm has broken through the algebraic upper bound given in (3). --- Figure 1: Left: Frequency of exact recovery as a function of sparsity; right: running time as a function of sparsity. ## II Orthogonal subspace NST+HT+$f$-FB algorithm ### II-A Notations A submatrix of $\Phi$ with columns indexed by a set $I$ is denoted by $\Phi_{I}$ and a submatrix of $\Phi$ with rows indexed by a set $J$ is denoted by $\Phi_{(J)}$. We denote the $i$-th row and the $j$-th column of a matrix $\Phi$ by $\Phi_{i\cdot}$, and $\Phi_{\cdot j}$, respectively. $T\triangle T^{\prime}$ is the symmetric difference of $T$ and $T^{\prime}$, i.e., $T\triangle T^{\prime}=(T\setminus T^{\prime})\cup(T^{\prime}\setminus T)$. $\bm{H}_{T}(X)$ is a linear operator that sets all but elements belong to rows indexed by $T$ of $X$ to zero. Algorithm $1$ OSNST+HT+$f$-FB --- \| Input: $\Phi$, $Y$, $\epsilon$, $f(\cdot)$, $K$; \| Output: $W$; \| Initialize: $k=1$, $W^{0}=0$; \| While $\\|Y-\Phi W^{k-1}\\|_{2}>\epsilon$ and $k<K$ do \| $X^{k}=W^{k-1}+\Phi^{\ast}(\Phi\Phi^{\ast})^{-1}(Y-\Phi W^{k-1})$; \| $Q^{k}=$orth$(X^{k})$; \| $T_{k}=\\{$Indices of $f(k)$ largest $\\|Q^{k}_{i\cdot}\\|_{2}\\}$; \| $W_{T_{k}}^{k}=X_{T_{k}}^{k}+(\Phi_{T_{k}}^{\ast}\Phi_{T_{k}})^{-1}\Phi_{T_{k}}^{\ast}\Phi_{T^{c}_{k}}X_{T^{c}_{k}}^{k}$; \| $W_{T^{c}_{k}}^{k}=0$; \| $k=k+1$; \| end while; ### II-B Algorithm framework The iterative framework of approximation and null space tuning (NST) algorithms is as follows $\left\\{\begin{aligned} \begin{aligned} &W^{k}=\mathbb{D}(X^{k}),\\\ &X^{k+1}=X^{k}+\mathbb{P}(W^{k}-X^{k}).\\\ \end{aligned}\end{aligned}\right.$ Here $\mathbb{D}(X^{k})$ approximates the desired solution by various principles, and $\mathbb{P}:=I-\Phi^{\ast}(\Phi\Phi^{\ast})^{-1}\Phi$ is the orthogonal projection onto ker$(\Phi)$. Since the sequence $\\{X^{k}\\}$ is always feasible (i.e., $Y=\Phi X^{k}$) under the NST principle, one may split $Y$ as $Y=\Phi X=\Phi_{T_{k}}X_{(T_{k})}^{k}+\Phi_{T^{c}_{k}}X_{(T^{c}_{k})}^{k},$ where $T_{k}$ includes indices of $f(k)$ largest $\\|Q^{k}_{i\cdot}\\|_{2}$ ($i\in\\{1,\ldots,N\\}$), $f(\cdot)\geq 0$ is a non-decreasing function and columns of $Q^{k}$ are an orthonormal basis for the column space of $X^{k}$, i.e., $Q^{k}$=orth$(X^{k})$. The mechanism of feedback is to feed the contribution of $\Phi_{T^{c}_{k}}X_{(T^{c}_{k})}^{k}$ to $Y$ back to im($\Phi_{T_{k}}$), the image of $\Phi_{T_{k}}$. A straightforward way is to set $\Lambda^{k}=\arg\min\limits_{\Lambda}\\|\Phi_{T_{k}}\Lambda-\Phi_{T^{c}_{k}}X_{(T^{c}_{k})}^{k}\\|_{2},$ which has the best/least-square solution $\Lambda^{k}=(\Phi_{T_{k}}^{\ast}\Phi_{T_{k}})^{-1}\Phi_{T_{k}}^{\ast}\Phi_{T_{k}^{c}}X_{(T_{k}^{c})}^{k}.$ The orthogonal subspace iterative thresholding algorithm with functional feedback and null space tunning (OSNST+HT+$f$-FB) is then established in Algorithm 1. ### II-C Convergence analysis In this paper, we assume the number of snapshots is smaller than the dimension of measurement, i.e., $L<M$, and the measurement matrix $Y$ is full column rank, i.e., rank$(Y)=L$. We now turn to the convergence of OSNST+HT+$f$-FB. ###### Definition 1. [30]. For each integer $s=1,2,\cdots$, the restricted isometry constant (RIC) $\delta_{s}$ of a matrix $\Phi$ is defined as the smallest number $\delta_{s}$ such that $\begin{array}[]{l}(1-\delta_{s})\\|X\\|_{F}^{2}\leq\\|\Phi X\\|_{F}^{2}\leq(1+\delta_{s})\\|X\\|_{F}^{2}\end{array}$ holds for all $s$ row-sparse matrix $X$. Equivalently, it is given by $\begin{array}[]{l}\delta_{s}=\max\limits_{\|S\|\leq s}\\|I-\Phi_{S}^{\ast}\Phi_{S}\\|_{2}.\end{array}$ ###### Definition 2. [31]. For each integer $s=1,2,\cdots$ the preconditioned restricted isometry constant $\gamma_{s}$ of a matrix $A$ is defined as the smallest number $\gamma_{s}$ such that $\begin{array}[]{l}(1-\gamma_{s})\\|X\\|_{F}^{2}\leq\\|(\Phi\Phi^{\ast})^{-\frac{1}{2}}\Phi X\\|_{F}^{2}\end{array}$ holds for all $s$ row-sparse matrix $X$. In fact, the preconditioned restricted isometry constant $\gamma_{s}$ represents the restricted isometry property of the preconditioned matrix $(\Phi\Phi^{\ast})^{-\frac{1}{2}}\Phi$. Since $\begin{array}[]{l}\\|(\Phi\Phi^{\ast})^{-\frac{1}{2}}\Phi X\\|_{F}\leq\\|(\Phi\Phi^{\ast})^{-\frac{1}{2}}\Phi\\|_{2}\\|X\\|_{F}=\\|X\\|_{F},\end{array}$ $\gamma_{s}$ is actually the smallest number such that, for all $s$ row-sparse matrix $X$, $\begin{array}[]{l}(1-\gamma_{s})\\|X\\|_{F}^{2}\leq\\|(\Phi\Phi^{\ast})^{-\frac{1}{2}}\Phi X\\|_{F}^{2}\leq(1+\gamma_{s})\\|X\\|_{F}^{2}.\end{array}$ It indicates $\gamma_{s}(\Phi)=\delta_{s}((\Phi\Phi^{\ast})^{-\frac{1}{2}}\Phi)$. Equivalently, it is given by $\begin{array}[]{l}\gamma_{s}=\max\limits_{\|S\|\leq s}\\|I-\Phi_{S}^{\ast}(\Phi\Phi^{\ast})^{-1}\Phi_{S}\\|_{2}.\end{array}$ ###### Definition 3. Let the feasible solution space of (2) be $\mathcal{X}=\\{X\in\mathbb{C}^{N\times L}:Y=\Phi X\\}$. Define the modified matrix condition number of $\mathcal{X}$ by $\alpha=\max\limits_{X\in\mathcal{X}}\frac{\sigma_{\max}(X)}{\sigma_{\min}(X)}$, where $\sigma_{\min}(X)$ and $\sigma_{\max}(X)$ denote the smallest and the largest nonzero singular values of $X$, respectively. ###### Lemma 4. Let $U,V\in\mathbb{C}^{N\times L}$ with $\|supp(U)\cup supp(V)\|\leq t$, then $\|\langle U,(I-\Phi^{\ast}\Phi)V\rangle\|\leq\delta_{t}\\|U\\|_{F}\\|V\\|_{F}$. Suppose $\|R\cup supp(V)\|\leq t$, then $\\|[(I-\Phi^{\ast}\Phi)V]_{(R)}\\|_{F}\leq\delta_{t}\\|V\\|_{F}$. ###### Proof. Let $T=supp(U)\cup supp(V)$, we then have $\begin{array}[]{l}\|\langle U,(I-\Phi^{\ast}\Phi)V\rangle\|=\|\langle U,V\rangle-\langle\Phi U,\Phi V\rangle\|\\\ =\|\langle U_{(T)},V_{(T)}\rangle-\langle\Phi_{T}U_{(T)},\Phi_{T}V_{(T)}\rangle\|\\\ =\|\langle U_{(T)},(I-\Phi_{T}^{\ast}\Phi_{T})V_{(T)}\rangle\|\\\ \leq\\|U_{(T)}\\|_{F}\\|(I-\Phi_{T}^{\ast}\Phi_{T})V_{(T)}\\|_{F}\\\ \leq\\|U_{(T)}\\|_{F}\\|I-\Phi_{T}^{\ast}\Phi_{T}\\|_{2}\\|V_{(T)}\\|_{F}\\\ \leq\delta_{t}\\|U\\|_{F}\\|V\\|_{F}.\\\ \end{array}$ The first and the second inequalities are due to the Cauchy-Schwarz inequality, and the sub-multiplicativity of matrix norms, respectively. The last step is by Definition 1. It then follows that $\begin{array}[]{l}\\|[(I-\Phi^{\ast}\Phi)V]_{(R)}\\|_{F}^{2}=\langle(\bm{H}_{R}\left((I-\Phi^{\ast}\Phi)V\right),(I-\Phi^{\ast}\Phi)V\rangle\leq\delta_{t}\\|[(I-\Phi^{\ast}\Phi)V]_{(R)}\\|_{F}\\|V\\|_{F}.\end{array}$ Therefore, $\\|[(I-\Phi^{\ast}\Phi)V]_{(R)}\\|_{F}\leq\delta_{t}\\|V\\|_{F}$. ∎ ###### Remark 5. Let $\gamma_{t}$ be the P-RIP constant of $\Phi$ and $U,V\in\mathbb{C}^{N\times L}$ with $\|supp(U)\cup supp(V)\|\leq t$, then $\|\langle U,(I-\Phi^{\ast}(\Phi\Phi^{\ast})^{-1}\Phi)V\rangle\|\leq\gamma_{t}\\|U\\|_{F}\\|V\\|_{F}$. Suppose $\|R\cup supp(V)\|\leq t$, then $\\|[(I-\Phi^{\ast}(\Phi\Phi^{\ast})^{-1}\Phi)V]_{(R)}\\|_{F}\leq\gamma_{t}\\|V\\|_{F}$. ###### Lemma 6. For $E\in\mathbb{C}^{M\times L}$, $\\|[\Phi^{\ast}(\Phi\Phi^{\ast})^{-1}E]_{(T)}\\|_{F}\leq\sqrt{1+\theta_{t}}\\|E\\|_{F}$, where $\theta_{t}=\delta_{t}((\Phi\Phi^{\ast})^{-1}\Phi)$ and $\delta_{t}((\Phi\Phi^{\ast})^{-1}\Phi)$ is RIC of matrix $(\Phi\Phi^{\ast})^{-1}\Phi$. ###### Proof. $\begin{array}[]{l}\\|[\Phi^{\ast}(\Phi\Phi^{\ast})^{-1}E]_{(T)}\\|_{F}^{2}=\langle\Phi^{\ast}(\Phi\Phi^{\ast})^{-1}E,\bm{H}_{T}(\Phi^{\ast}(\Phi\Phi^{\ast})^{-1}E)\rangle\\\ =\langle E,(\Phi\Phi^{\ast})^{-1}\Phi\bm{H}_{T}(\Phi^{\ast}(\Phi\Phi^{\ast})^{-1}E)\rangle\\\ \leq\\|E\\|_{F}\sqrt{1+\theta_{t}}\\|[\Phi^{\ast}(\Phi\Phi^{\ast})^{-1}E]_{(T)}\\|_{F}.\end{array}$ Applying Definition 1 to the matrix $\Phi^{\ast}(\Phi\Phi^{\ast})^{-1}$ obtains the last step. Hence, for all $E\in\mathbb{C}^{M\times L}$, we have $\\|[\Phi^{\ast}(\Phi\Phi^{\ast})^{-1}E]_{(T)}\\|_{F}\leq\sqrt{1+\theta_{t}}\\|E\\|_{F}$. ∎ ###### Lemma 7. Let $Y=\Phi X+E$, where $X\in\mathbb{C}^{N\times L}$ is $s$ row-sparse with $S=$supp$(X)$ and $E\in\mathbb{C}^{M\times L}$ is the measurement error. If $\widetilde{W}\in\mathbb{C}^{N\times L}$ is $\widetilde{s}$ row-sparse, $\widetilde{X}=\widetilde{W}+\Phi^{\ast}(\Phi\Phi^{\ast})^{-1}(Y-\Phi\widetilde{W})$, $\widetilde{Q}=orth(\widetilde{X})$, and $T$ is an index set of $t\geq s$ largest $\\|\widetilde{Q}_{i\cdot}\\|_{2}$, then $\begin{array}[]{l}\\|X_{(T^{c})}\\|_{F}\leq\sqrt{2}\alpha(\gamma_{s+\widetilde{s}+t}\\|X-\widetilde{W}\\|_{F}+\sqrt{1+\theta_{t+s}}\\|E\\|_{F}),\end{array}$ where $\theta_{t+s}(\Phi)=\delta_{t+s}((\Phi\Phi^{\ast})^{-1}\Phi)$. ###### Proof. Since rank$(Y)=L$ and $Y=\Phi\widetilde{X}$, it is obvious that rank$(\widetilde{X})=L$. Consequently, the singular value decomposition of $\widetilde{X}$ can be denoted as $\widetilde{X}=\widetilde{U}_{\ell}\widetilde{\Sigma}_{(\ell)}\widetilde{V}^{\ast}$, where $\widetilde{U}_{\ell}$ is the first $L$ columns of $\widetilde{U}$ and $\widetilde{\Sigma}_{(\ell)}$ denotes the first $L$ rows of $\widetilde{\Sigma}$. Since $\widetilde{U}_{\ell}$ can be regarded as an orthonormal basis for the range of $\widetilde{X}$, without loss of generality, let $\widetilde{Q}=\widetilde{U}_{\ell}$, we have $\begin{array}[]{l}\\|[\widetilde{X}\widetilde{V}\widetilde{\Sigma}_{(\ell)}^{-1}]_{(T)}\\|_{F}\geq\\|[\widetilde{X}\widetilde{V}\widetilde{\Sigma}_{(\ell)}^{-1}]_{(S)}\\|_{F}.\\\ \end{array}$ It then follows that $\begin{array}[]{l}\widetilde{\sigma}_{\min}^{-1}\\|\widetilde{X}_{(T)}\\|_{F}\geq\widetilde{\sigma}_{\max}^{-1}\\|\widetilde{X}_{(S)}\\|_{F},\\\ \end{array}$ where $\widetilde{\sigma}_{\min}$ and $\widetilde{\sigma}_{\max}$ denote the smallest and the largest singular value of $\widetilde{\Sigma}_{(\ell)}$. Eliminating the common terms over $T\bigcap S$, we obtain $\begin{array}[]{l}\widetilde{\sigma}_{\min}^{-1}\\|[\widetilde{W}+\Phi^{\ast}(\Phi\Phi^{\ast})^{-1}(Y-\Phi\widetilde{W})]_{(T\setminus S)}\\|_{F}\geq\widetilde{\sigma}_{\max}^{-1}\\|[\widetilde{W}+\Phi^{\ast}(\Phi\Phi^{\ast})^{-1}(Y-\Phi\widetilde{W})]_{(S\setminus T)}\\|_{F}.\\\ \end{array}$ For the left hand, $\begin{array}[]{l}\widetilde{\sigma}_{\min}^{-1}\\|[\widetilde{W}+\Phi^{\ast}(\Phi\Phi^{\ast})^{-1}(Y-\Phi\widetilde{W})]_{(T\setminus S)}\\|_{F}\\\ =\widetilde{\sigma}_{\min}^{-1}\\|[\widetilde{W}-X+\Phi^{\ast}(\Phi\Phi^{\ast})^{-1}(\Phi X+E-\Phi\widetilde{W})]_{(T\setminus S)}\\|_{F}\\\ =\widetilde{\sigma}_{\min}^{-1}\\|[(I-\Phi^{\ast}(\Phi\Phi^{\ast})^{-1}\Phi)(\widetilde{W}-X)+\Phi^{\ast}(\Phi\Phi^{\ast})^{-1}E]_{(T\setminus S)}\\|_{F}.\\\ \end{array}$ The right hand satisfies $\begin{array}[]{l}\widetilde{\sigma}_{\max}^{-1}\\|[\widetilde{W}+\Phi^{\ast}(\Phi\Phi^{\ast})^{-1}(Y-\Phi\widetilde{W})]_{(S\setminus T)}\\|_{F}\\\ =\widetilde{\sigma}_{\max}^{-1}\\|[\widetilde{W}+\Phi^{\ast}(\Phi\Phi^{\ast})^{-1}(\Phi X+E-\Phi\widetilde{W})+X-X]_{(S\setminus T)}\\|_{F}\\\ \geq\widetilde{\sigma}_{\max}^{-1}\\|X_{(S\setminus T)}\\|_{F}-\widetilde{\sigma}_{\max}^{-1}\\|[(I-\Phi^{\ast}(\Phi\Phi^{\ast})^{-1}\Phi)(\widetilde{W}-X)+\Phi^{\ast}(\Phi\Phi^{\ast})^{-1}E]_{(S\setminus T)}\\|_{F}.\\\ \end{array}$ Therefore, we obtain $\begin{array}[]{l}\widetilde{\sigma}_{\max}^{-1}\\|X_{(S\setminus T)}\\|_{F}\\\ \leq\widetilde{\sigma}_{\max}^{-1}\\|[(I-\Phi^{\ast}(\Phi\Phi^{\ast})^{-1}\Phi)(\widetilde{W}-X)+\Phi^{\ast}(\Phi\Phi^{\ast})^{-1}E]_{(S\setminus T)}\\|_{F}\\\ +\widetilde{\sigma}_{\min}^{-1}\\|[(I-\Phi^{\ast}(\Phi\Phi^{\ast})^{-1}\Phi)(\widetilde{W}-X)+\Phi^{\ast}(\Phi\Phi^{\ast})^{-1}E]_{(T\setminus S)}\\|_{F}\\\ \leq\sqrt{2}\widetilde{\sigma}_{\min}^{-1}\\|[(I-\Phi^{\ast}(\Phi\Phi^{\ast})^{-1}\Phi)(\widetilde{W}-X)+\Phi^{\ast}(\Phi\Phi^{\ast})^{-1}E]_{(T\triangle S)}\\|_{F}\\\ \leq\sqrt{2}\widetilde{\sigma}_{\min}^{-1}\\|[(I-\Phi^{\ast}(\Phi\Phi^{\ast})^{-1}\Phi)(\widetilde{W}-X)]_{(T\triangle S)}\\|_{F}\sqrt{2}\widetilde{\sigma}_{\min}^{-1}\\|[\Phi^{\ast}(\Phi\Phi^{\ast})^{-1}E]_{(T\triangle S)}\\|_{F}\\\ \leq\sqrt{2}\widetilde{\sigma}_{\min}^{-1}(\gamma_{s+\widetilde{s}+t}\\|X-\widetilde{W}\\|_{F}+\sqrt{1+\theta_{t+s}}\\|E\\|_{F}).\end{array}$ The last step is due to Remark 5 and Lemma 6. In view of Definition 3, we derive $\begin{array}[]{l}\\|X_{(S\setminus T)}\\|_{F}\leq\sqrt{2}\alpha(\gamma_{s+\widetilde{s}+t}\\|X-\widetilde{W}\\|_{F}+\sqrt{1+\theta_{t+s}}\\|E\\|_{F}).\end{array}$ ∎ ###### Lemma 8. Let $Y=\Phi X+E$, where $X\in\mathbb{C}^{N\times L}$ is $s$ row-sparse signal matrix, and $E\in\mathbb{C}^{M\times L}$ is the measurement error. Let $S=$supp$(X)$ be the index set of the $s$ sparse rows of $X$. Denote by $\widetilde{Q}=$orth$(\widetilde{X})$ the orthogonal basis of the row-space of $X$, and $T$ the index set of $t\geq s$ largest values of $\\|\widetilde{Q}_{i\cdot}\\|_{2}$. If $\overline{W}$ is the feedback of $\widetilde{X}$ given by $\overline{W}_{(T)}=\widetilde{X}_{(T)}+(\Phi_{T}^{\ast}\Phi_{T})^{-1}\Phi_{T}^{\ast}\Phi_{T^{c}}\widetilde{X}_{(T^{c})}$ and $\overline{W}_{(T^{c})}=0$, then $\begin{array}[]{l}\\|(X-\overline{W})\\|_{F}\leq\frac{\\|X_{(T^{c})}\\|_{F}}{\sqrt{1-\delta_{s+t}^{2}}}+\frac{\sqrt{1+\delta_{t}}\\|E\\|_{F}}{1-\delta_{s+t}}.\end{array}$ ###### Proof. For any $Z\in\mathbb{C}^{N\times L}$ supported on $T$, $\begin{array}[]{l}\langle\Phi\overline{W}-Y,\Phi Z\rangle\\\ =\langle\Phi_{T}\widetilde{X}_{(T)}+\Phi_{T}(\Phi_{T}^{\ast}\Phi_{T})^{-1}\Phi_{T}^{\ast}\Phi_{T^{c}}\widetilde{X}_{(T^{c})}-Y,\Phi_{T}Z_{(T)}\rangle\\\ =\langle\Phi_{T}^{\ast}(\Phi_{T}\widetilde{X}_{(T)}+\Phi_{T^{c}}\widetilde{X}_{(T^{c})}-Y),Z_{(T)}\rangle\\\ =\langle\Phi_{T}^{\ast}(\Phi\widetilde{X}-Y),Z_{(T)}\rangle\\\ =0.\\\ \end{array}$ The last step is due to the feasibility of $\widetilde{X}$. The inner product can also be written as $\langle\Phi\overline{W}-Y,\Phi Z\rangle=\langle(\Phi\overline{W}-\Phi X-E),\Phi Z\rangle=0$. Therefore, $\langle(\overline{W}-X),\Phi^{\ast}\Phi Z\rangle=\langle E,\Phi Z\rangle,~{}\forall~{}Z\in\mathbb{C}^{N\times L}$ supported on $T$. Since $(\overline{W}-X)_{T}$ is supported on $T$, one has $\langle(\overline{W}-X),\Phi^{\ast}\Phi_{T}(\overline{W}-X)_{(T)}\rangle=\langle E,\Phi_{T}(\overline{W}-X)_{(T)}\rangle.$ Consequently, $\begin{array}[]{l}\\|(\overline{W}-X)_{(T)}\\|_{F}^{2}=\langle(\overline{W}-X),\bm{H}_{T}(\overline{W}-X)\rangle\\\ =\|\langle(X-\overline{W}),(I-\Phi^{\ast}\Phi)\bm{H}_{T}(X-\overline{W})\rangle+\|\langle E,\Phi\bm{H}_{T}(X-\overline{W})\rangle\|\\\ \leq\delta_{s+t}\\|X-\overline{W}\\|_{F}\\|(X-\overline{W})_{(T)}\\|_{F}+\sqrt{1+\delta_{t}}\\|E\\|_{F}\\|(X-\overline{W})_{(T)}\\|_{F}.\\\ \end{array}$ The last step is due to Lemma 4 and Definition 1. We can obtain $\begin{array}[]{l}\\|(X-\overline{W})_{(T)}\\|_{F}\leq\delta_{s+t}\\|X-\overline{W}\\|_{F}+\sqrt{1+\delta_{t}}\\|E\\|_{F}.\end{array}$ It then follows that $\begin{array}[]{l}\\|(X-\overline{W})\\|_{F}^{2}=\\|(X-\overline{W})_{(T)}\\|_{F}^{2}+\\|(X-\overline{W})_{(T^{c})}\\|_{F}^{2}\\\ \leq(\delta_{s+t}\\|X-\overline{W}\\|_{F}+\sqrt{1+\delta_{t}}\\|E\\|_{F})^{2}+\\|X_{(T^{c})}\\|_{F}^{2}.\\\ \end{array}$ This in turn implies $p(\\|X-\widetilde{W}\\|_{F})\leq 0$, where $p(\cdot)$ is a quadratic polynomial, defined by $\begin{array}[]{l}p(x)=(1-\delta_{s+t}^{2})x^{2}-2\delta_{s+t}\sqrt{1+\delta_{t}}\\|E\\|_{F}x-(1+\delta_{t})\\|E\\|_{F}^{2}-\\|X_{(T^{c})}\\|_{F}^{2}.\end{array}$ Since $(1-\delta_{s+t}^{2})\geq 0$, it means that $\\|(X-\overline{W})\\|_{F}$ is smaller than the largest root of $p(\cdot)$ $\begin{array}[]{l}\\|(X-\overline{W})\\|_{F}\leq\frac{\delta_{s+t}\sqrt{1+\delta_{t}}\\|E\\|_{F}+\sqrt{(1+\delta_{t})\\|E\\|_{F}^{2}+({1-\delta_{s+t}^{2})\\|X_{(T^{c})}\\|_{F}^{2}}}}{1-\delta_{s+t}^{2}}\\\ \leq\frac{\\|X_{(T^{c})}\\|_{F}}{\sqrt{1-\delta_{s+t}^{2}}}+\frac{\sqrt{1+\delta_{t}}\\|E\\|_{F}}{1-\delta_{s+t}}.\\\ \end{array}$ ∎ ###### Theorem 9. Let $Y=\Phi X+E$, where $X$ is the $s$ row-sparse signal matrix. Then the sequence $\\{W^{k}\\}$ produced by OSNST+HT+$f$-FB satisfies $\begin{array}[]{l}\\|(X-W^{k})\\|_{F}\leq\rho_{s+f(k)+f(k-1)}^{k}\\|X-W^{0}\\|_{F}+\frac{\kappa_{s+f(k)+f(k-1)}(1-\rho_{s+f(k)+f(k-1)}^{k})}{1-\rho_{s+f(k)+f(k-1)}}\\|E\\|_{F},\\\ \end{array}$ where $\rho_{\ell}=\sqrt{\frac{2\alpha^{2}\gamma_{\ell}^{2}}{1-\delta_{\ell}^{2}}}$ and $\kappa_{\ell}=(\frac{\sqrt{1+\delta_{\ell}}}{1-\delta_{\ell}}+\frac{\sqrt{2\alpha^{2}(1+\theta_{\ell})}}{\sqrt{1-\delta_{\ell}^{2}}})$. ###### Proof. Applying Lemma 7 to $\widetilde{W}=W^{k-1}$ and $T=T_{k}$ gives $\begin{array}[]{l}\\|X_{(T^{c}_{k})}\\|_{F}\leq\sqrt{2}\alpha(\gamma_{s+f(k-1)+f(k)}\\|X-W^{k-1}\\|_{F}+\sqrt{1+\theta_{s+f(k)}}\\|E\\|_{F}),\end{array}$ and setting $\overline{W}=W^{k}$ and $T=T_{k}$ in Lemma 8 obtains $\begin{array}[]{l}\\|(X-W^{k})\\|_{F}\leq\frac{\\|X_{(T^{c}_{k})}\\|_{F}}{\sqrt{(1-\delta_{s+f(k)}^{2})}}+\frac{\sqrt{1+\delta_{f(k)}}\\|E\\|_{F}}{1-\delta_{s+f(k)}}.\end{array}$ Combining these two inequalities, we have $\begin{array}[]{l}\\|(X-W^{k})\\|_{F}\leq\sqrt{\frac{2\alpha^{2}\gamma_{s+f(k)+f(k-1)}^{2}}{(1-\delta_{s+f(k)}^{2})}}\\|X-W^{k-1}\\|_{F}+(\frac{\sqrt{1+\delta_{f(k)}}}{1-\delta_{s+f(k)}}+\frac{\sqrt{2\alpha^{2}(1+\theta_{s+f(k)})}}{\sqrt{1-\delta_{s+f(k)}^{2}}})\\|E\\|_{F}.\end{array}$ Since $\delta_{\ell}$ and $\gamma_{\ell}$ are all non-decreasing [30], $\rho_{\ell}$ and $\kappa_{\ell}$ are also all non-decreasing as $\ell$ increases for all integer $\ell$. Note that $f(\ell)$ is also a nondecreasing function, it then follows that $\begin{array}[]{l}\\|(X-W^{k})\\|_{F}\leq\rho_{s+f(k)+f(k-1)}^{k}\\|X-W^{0}\\|_{F}+\frac{\kappa_{s+f(k)+f(k-1)}(1-\rho_{s+f(k)+f(k-1)}^{k})}{1-\rho_{s+f(k)+f(k-1)}}\\|E\\|_{F}.\end{array}$ ∎ Consequently, if the RIP and the P-RIP of the matrix $\Phi$ obeys $2\alpha^{2}\gamma_{s+f(k)+f(k-1)}^{2}+\delta_{s+f(k)+f(k-1)}^{2}<1$, the OSNST+HT+FB algorithm is guaranteed to converge. --- Figure 2: Left: Frequency of exact recovery as a function of sparsity; right: running time as a function of sparsity. ## III EXPERIMENTS In this experiment, the measurement matrix $\Phi$ is an $300\times 1000$ Gaussian random matrix and the number of snapshots is $10$. To model the temporal correlation of MMV problem, we employ an autoregressive process of order $1$, AR(1). As a result, the $j$-th snapshot $X_{\cdot j}$ is generated according to the model $\begin{array}[]{l}X_{\cdot j}=\beta X_{\cdot(j-1)}+(1-\beta)\epsilon_{j},\end{array}$ where $\beta$ is the AR model parameter controlling the temporal correlation and $\epsilon_{j}$ is the level of white Gaussian perturbation. The support of a sparse signal is also chosen randomly and the nonzero entries of Gaussian sparse signals are drawn independently from the Gaussian distribution with zero mean and unit variance. A successful recovery is recorded when $\\|X-\widehat{X}\\|_{F}/\\|X\\|_{F}\leq 10^{-4}$, where $X$ is the exact signal matrix and $\widehat{X}$ denotes the recovered signal. Each experiment is tested for $100$ (random) trials. A matlab implementation of the proposed algorithm is also available at https://www.dropbox.com/s/2avudk770m4c6rz/OSNST%2BHT%2Bf-FB.zip?dl=0. We first study the mechanisms of $f$-feedback by introducing six particular index selection functions: $f(x)=x$, $f(x)=3x$, $f(x)=6x$, $f(x)=9x$, $f(x)=12x$ and $f(x)=x^{2}$. As discussed, higher critical sparsity represents better empirical recovery performance. Figure 1 shows the frequency of exact recovery and the running time as functions of the sparsity levels $s$. As shown, linear functions with modest gradients present similar performance, which is better than the quadratic function $f(x)=x^{2}$. In addition, one can accelerate the convergence of the class of OSNST+HT+$f$-FB algorithms by adjusting the cardinality of indices per iteration. Also presented are comparisons among our OSNST+HT+$f$-FB and state-of-the-art techniques such as SOMP [2], $\ell_{2,1}$ norm [8], SHTP [15, 16], RA-ORMP [10], TMSBL [17], SA-MUSIC+OSMP [24], SeqCS-MUSIC [22, 23] in terms of frequency of exact recovery and running time. 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11institutetext: Radboud University Nijmegen, Email: 11email<EMAIL_ADDRESS>22institutetext: University of Groningen # Privacy Friendly E-Ticketing For Public Transport††thanks: Version: Fri Jan 22 10:58:06 2021 +0100 / arXiv / ov-pet.tex Jaap-Henk Hoepman 1122 ###### Abstract This paper studies how to implement a privacy friendly form of ticketing for public transport in practice. The protocols described are inspired by current (privacy invasive) public transport ticketing systems used around the world. The first protocol emulates paper based tickets. The second protocol implements a pay-as-you-go approach, with fares determined when users check-in and check-out. Both protocols assume the use of a smart phone as the main user device to store tickets or travel credit. We see this research as a step towards investigating how to design commonly used infrastructure in a privacy friendly manner in practice, paying particular attention to how to deal with failures. ## 1 Introduction At the turn of the century, several countries transitioned from paper based tickets for public transport to electronic forms of ticketing, either for public transport in a metropolitan area like the London Underground (the so- called Oyster card111https://oyster.tfl.gov.uk/ ) and Hong Kong public transport (the Octopus Card222https://www.octopus.com.hk/en ), or for all public transport in an entiry country (like the OV-chipkaart333https://www.ov- chipkaart.nl/ in the Netherlands). These e-ticketing systems are typically based on contactless smart cards. Some of these systems exhibit significant weaknesses in terms of security [11] (because the smart cards used need to be cheap and therefore contain weaker security features) and privacy [14] (because these smart cards often contain a unique fixed identifier [12, 19]).444Note that these security weaknesses are not necessarily easily exploitable in practice [24]. In this paper we focus on the privacy issues in e-ticketing for public transport. This problem has been studied in the past for e-ticketing in particular [15, 16], but also for related problems like electronic toll collection systems [10] or more general road pricing systems [17]. Compared to the work of Heydt-Benjamin et al. [15, 16] (which relies on anonymous e-cash and anonymous credentials as building blocks) we use (partially) blind signatures instead to create either unlinkable travel tickets or unlinkable travel credit. This makes the protocols less complex and more efficient as there is no need to spend several e-cash coins to pay the exact fare. Moreover, our approach is inspired by the work of Stubblebine et al. [22], which studied unlikable transactions in practice, with a particular focus on dealing with failures. We present two privacy friendly e-ticketing protocols for public transport. The first protocol emulates paper based tickets. The second protocol implements a pay-as-you-go approach, with fares determined when users check-in and check-out. Both protocols assume the use of a smart phone as the main user device to store tickets or travel credit, without relying on any tamper-proof component. We wish to stress that this means that we put no trust assumptions on the device the user uses to pay for public transport. In both cases we pay particular attention to possible failures and how to graciously deal with them. The remainder of the paper is structured as follows. We introduce the system model, requirements, threat model and other assumptions in section 2. We discuss protecting privacy in practice in section 3, especially the assumption to use smartphones as the primary user device. Section 4 discusses the primitives used in our protocols, that are then presented in section 5 and section 6. Our conclusions are presented in section 7. ## 2 Problem statement We assume a system that supports many modes of transportation. This means we distinguish several _public transport operators (PTOs)_ that offer public transport services. _Users (U)_ travelling by public transport make trips that may consist of several legs each using a different mode of transportation offered by a different PTO. _Inspectors_ on the trains and busses verify that everybody on board has a valid ticket. A central _public transport clearinghouse (PTC)_ provides the public transport ticketing infrastructure (or at least the APIs to connect to this infrastructure), and distributes financial compensation to the PTOs for services rendered. Separate _payment service providers (PSPs)_ handle payments from banks initiated by users. Users have a digital _token_ that enables them to travel by public transport. Instead of relying on a smart card (to store tickets or other information needed to verify whether someone is entitled to a certain mode of public transportation) we assume most people own a sufficiently modern and capable smartphone, and are willing to use it for public transport (we will discuss this further in section 2.2). Any entity (in particular any of the PTOs) can offer an app for this purpose. ### 2.1 Requirements Any public transport ticketing scheme for the model outlined above should satisfy the following requirements. * • Users should pay for trips, where the fare depends on when and where a user travels, the distance travelled, and whether the user has subscribed to a public transport pass that offers reduced fares (during certain times of the day, or on certain tracks). * • Public transport operators should receive compensation for their services, which (partly) depends on all the actual trips made bu users that traversed part of their infrastructure. In other words, the amount of compensation can depend on how many passengers travelled on which particular track on which day. * • The scheme should be privacy friendly: no party should be able to link any number of trips to each other (as belonging to one, unknown, person) or to any one particular person. In other words, the previous requirement only allows the PTO to learn _how many_ passengers travelled a certain track at a certain time of the day, not _who_ they were, or whether they were the _same people_ that set out on some other trip earlier. * • The scheme should be secure: it should prevent or detect fraud by users (e.g. creating fake tickets, paying less than the required fare) and prevent fraud by operators (e.g. claiming more trips than actually took place over their infrastructure). Travelling without a valid ‘ticket’ should be discouraged by regular inspection and appropriate fines. * • The scheme should be fast enough to process large volumes of travellers at peak hours. Checking travellers by conductors should be fast, e.g., take less than a second. Checking in or out to enter or exit public transport (like in the London Oystercard system or the Dutch OV-chipkaart system) should take only a few hundred milliseconds at most. We note that the timing constraints mentioned in the last requirement are important in practice, but performance measurements are unfortunately out of scope for this paper. ### 2.2 Threat model We assume users may try to actively defraud the system (travelling by public transport while paying less than required or nothing at all), if the probability of being caught is low. They will root their smartphones and install fraudulent apps if there is a clear benefit. This means that in terms of smartphones we cannot assume any trusted environment to store secrets (we rule out the possibility that a public transport app gets to use a secure enclave on the smartphone). In other words: the device and the app are untrusted from the PTO perspective. This means our system is weaker than one relying on smart cards that _can_ be used to store secrets and keep them confidential, preventing their users from accessing (and perhaps copying) them. We assume banks, PSPs, PTOs and the PTC will actively (and collectively) try to break privacy and recover trip details from their users, using any information they can get their hands on. They are untrusted from the user perspective. We do assume however that PTOs do _not_ try to break privacy by writing their apps in such a way that the information provided by the user through the app, but shielded from the central PTO servers by the protocol, is surreptitiously sent to the PTOs regardless. The PTOs could in theory do this. We can mitigate this by offering third party (open source) apps, requiring external audits and analysis, or through the vetting procedures enforced by the smartphone app stores. (This, by the way, is another reason why we cannot assume that the smartphone or app can store secrets.) We assume that PTOs will try to defraud the system and claim more compensation from the PTC than warranted. The PTC is trusted, in the sense that it does not favour one PTO over the other, and that at the of the day all money received must be spent (on compensating PTOs or on the cost of running the clearinghouse and its ticketing infrastructure). Audits can be used to ensure this. We assume the cryptographic primitives used cannot be broken, and that entities keep their secrets secret (unless they could benefit from not doing so). ### 2.3 Other assumptions We assume secure, i.e. authenticated and encrypted, connections between all entities. Clearly the user is not authenticated. We assume fares are course enough to ensure that the price associated with a trip does not reveal the actual trip itself. For example, trip prices could be set at fixed amounts for every ten kilometres travelled, with a fixed ceiling fare for all trips longer than a certain distance. (Care should be taken to ensure that for every possible fare the number of different trips with that fare is sufficiently high to guarantee a reasonable degree of anonymity.) We also assume that local device to device communication is using only ephemeral identifiers (if any) to prevent linking devices over longer periods of time. This means WiFi or Bluetooth are using properly randomised MAC addresses, or random anti-collision identifiers if NFC is used. This also implies that we assume apps do not have access to any other permanent, unique, device specific identifier.555The operating systems of these smartphones should, could and sometimes actually do prevent apps from having access to such a persistent identifier. Preventing the app itself to generate such an identifier itself and store it locally is of course not possible (although audits may reveal this). For normal (long range) internet connections between the smartphone and the servers of the other entities we cannot make such an assumption: it is rather trivial to track users based on their often fixed IP addresses. We discuss this in the next section. ## 3 Protecting privacy in practice Protecting privacy in practice is a major challenge, for several reasons. First of all, practical considerations may rule out certain solutions or may make it impossible to make simplifying assumptions. For example a complex tariff system may lead to a situation where particular fares correspond to one or perhaps only a few particular trips. This is not the case when the tariff system is very simple (e.g., two or three different zones in a metro network). This issue is exacerbated when people are forced to pay for individual trips separately (see the first protocol that emulates paper tickets in section 5) while the payment protocol is not anonymous. Except for cash payments (and certain privacy friendly crypto currencies perhaps) existing and widely accepted payment methods (credit card, debit card or e-banking apps) are account based and thus identifying. Even if this were not the case, the protocols detailed below rely extensively on Internet connectivity that by its very nature is identifying. Strategies to shield the user’s IP address from the other parties involved in the ticketing system (like using Tor [9] or mix networks [7])) should be used, but are probably impractical to use extensively and reliably at scale. Then again, letting users use a trusted VPN would solve most of the problems as this would hide all users behind the IP address of the VPN provider. We will have something more to say about this later on. The biggest paradox, from a privacy perspective, is of course the use of a smartphone as the basic user device for buying, storing and using tickets. On the one hand it is an entirely personal device, capable enough to orchestrate the interactions with all other parties using complex privacy friendly protocols, with the possibility of a nice user friendly interface to boot. Moreover, people expect their smartphone to support their day to day activities, like paying in shops, these days. This makes a smartphone the natural, in fact unavoidable, choice as the user device.666Although a fall back option should always be available those people that cannot afford to own a (recently modern) smartphone. But clearly the use of smartphones comes with severe privacy risks. By design, mobile phone operators know the approximate location of all their subscribers (and can zoom in using a process called triangulation). With GPS, standard on smartphones, location is also readily available to the phone itself as well as all apps that were granted permission to location services. With the increasing complexity of smartphones and the huge app ecosystem, users have very little reason to trust their smartphone or to expect it to protect their privacy. With these caveats in mind we follow a pragmatic approach in this paper, aiming for a strong enough technical protection of privacy under reasonable assumptions. No coalition of PSPs, PTOs and the PTC can link trips777Either as bought in the protocol that emulates paper based tickets, or as implied by check-in and corresponding check-out events to users, beyond what can be ascertained by observing the financial transactions of the users, knowledge of the tariff structure, and (partial) apriori knowledge of the travel patterns of a subset of these users. We do not solely rely on technical mechanisms however, but also depend on legal, societal and market incentives to keep the different stakeholders in check. All measures combined should ensure that the cost of obtaining privacy sensitive information in general outweighs the (business) benefit. The tacit assumption in this work is that it is much safer, from a privacy perspective, to collect personal data locally on the user device, instead of centrally on the servers of the service providers. Clearly a malicious public transport app can collect and upload all this personal data surreptitiously. The assumption is that this cannot or will not happen. ## 4 Primitives Our protocols for privacy friendly ticketing for public transport are based on three primitives, that we will describe in this section: partially blind signatures, attribute based credentials, and a mechanism to implement a form of privacy friendly payment with receipt. ### 4.1 Partially blind signatures Blind signatures were introduced by David Chaum almost four decades ago [6], as the fundamental building block to implement a form of untraceable digital cash. His proposal was to represent each digital coin as a unique serial number blindly signed by the issuing bank. The unique serial number embedded in the coin would prevent double spending, while the blind signature over the coin would guarantee both _untraceability_ (by not knowing which coin was signed) and _unforgeability_ (by signing the coins in the first place). In the protocols below we use a generalisation of this idea called partially blind signatures, introduced by Abe and Fujisaki [1] and further investigated and optimised by Abe and Okamoto [2, 20]. In a partially blind signature scheme the messages to be signed consists of a secret part (only known to the user) and a public part (known to both the user and the signer). Issuing a blind signature involves an interactive protocol between the user and the signer, where the user blinds the secret in order to hide it from the signer. In the protocols below we use these partially blind signatures to issue receipts and/or tickets where the receipt number or the trip details are kept secret. Because such receipts and tickets are only used once (in fact, we need to enforce that they are not used more than once), using simple signatures instead of full blown attribute based credentials (to be discussed further on) suffices. When describing our protocols we write $[\hbox{\pagecolor{lightgray}\rule[-1.5pt]{0.0pt}{8.5pt}$\mathit{secret}$}\,\|\,\mathit{public}]_{{k_{\mathit{}}}}$ when issuing a blind signature over secret part $\mathit{secret}$ and public part $\mathit{public}$ (using private key ${k_{\mathit{}}}$ of the signer to sign it), and write $[\mathit{secret}\,\|\,\mathit{public}]_{{k_{\mathit{}}}}$ when subsequently using it (revealing both the secret and the public part). #### Two faces of blindness Chaum explained blind signatures intuitively by showing how a blind signature could be implemented in a traditional, non digital, setting using carbon paper inside paper envelopes. To obtain a blind signature on a secret message, a user could send the message inside a sealed envelope to the signer, with the inside of the envelope covered with carbon paper. The carbon paper ensures that if the signer signs the envelope from the outside, the carbon paper transfers this signature to the secret message inside the envelope. When the signer returns the still sealed envelope (proving it didn’t see the message) all the user needs to do is to open the envelope to obtain the blindly signed message. This intuitive explanation clearly shows that the message stays hidden from the signer. But this by itself is not enough to prevent a bank from tracing a digital coin signed this way, even if it prevents the bank from learning its serial number. In fact, if the bank signs each envelope in a slightly different way, and remembers which way of signing it used to sign each envelope, it can link actual signatures on messages to the particular envelope on which he put the exact same signature. In other words, in order to guarantee untraceability (sometimes also called _unlinkability_), blind signatures need to guarantee two separate blindness properties: message hiding The message to be signed is hidden from the signer. signature unlinkability Given a final blind signature on a message, the signer cannot determine when it generated that particular signature. To see that these are indeed different properties, observe that a scheme where signing the cryptographic hash of message $m$ (without revealing $m$ itself to the signer) is message hiding but clearly not unlinkable. In the protocols below we rely on both these properties to hold. Most (partially) blind signature schemes in fact satisfy both of them. This is in particular the case888The (partial) blindness property is defined using a game where two messages $m_{0}$ and $m_{1}$ are randomly assigned to two users (based on a random bit $b$). Each user then requests a blind signature on its message from the signer. The signer is then given both signatures (and for each the corresponding message) and asked to guess the value of $b$. If it could distinguish which signature corresponds to which user, it could for sure determine the value of $b$. for the schemes of Abe and Okamoto [2, 20] (but not for the blind signature scheme underlying the Idemix attribute based credential scheme [18, 4]). #### Dealing with failures In the protocols below, partially blind signatures are used to represent receipts received after a successful payment, or as public transport tickets received in exchange for a valid receipt. In both cases a kind of ‘fair exchange’ [21] is required between a user and a signer, and there should be a way to recover from errors in case messages are dropped, connections fail, or system components crash, to ensure that either the exchange takes place completely, or that the exchange is cancelled and both parties return to the state before they started the exchange. Recall from section 2.2 that we assume users to be malicious while service providers (the signers in this case) are honest (but curious). This assumption makes it possible and relatively easy to implement a fair exchange in this particular case. Details will vary depending on the particular blind signature scheme used. For example, the partial blind signature scheme of Okamoto [20] consists of the following phases when creating the blind signature $[\hbox{\pagecolor{lightgray}\rule[-1.5pt]{0.0pt}{8.5pt}$s$}\,\|\,p]_{{k_{\mathit{}}}}$. 1. 1. The user blinds the secret part $s$ using some randomness $r_{u}$ as $b=\mathit{blind}(s,r_{u})$ and sends this to the signer. 2. 2. The user proves to the signer that she knows $s$ and $r_{u}$ used to construct $\mathit{blind}(s,r_{u})$ using a three messages zero-knowledge protocol. The signing protocol aborts if this proof fails. 3. 3. The signer generates some randomness $r_{s}$ and creates an intermediate signature $i=\mathit{intermediate}(b,p,r_{s},{k_{\mathit{}}})$ using its private key ${k_{\mathit{}}}$ over the blinded information $b$ received from the user, the public part $p$ of the to be signed message, and the randomness $r_{s}$ it just generated. The signer sends this intermediate signature to the user. 4. 4. The user transforms this intermediate signature $i$ to the final partially blind signature $[\hbox{\pagecolor{lightgray}\rule[-1.5pt]{0.0pt}{8.5pt}$s$}\,\|\,p]_{{k_{\mathit{}}}}$. The user acknowledges this to the signer Both the user and the signer keep a record of the values of all local variables used and messages exchanged during the signing protocol, and keep track of when they aborted the protocol. Current values of local variables must be safely stored before sending any message that depends on them. If both parties successfully complete the protocol, both can destroy the record for the protocol. Observe that the only dispute that can occur is when a user claims not to have received a blind signature in return for a payment or a receipt.999This uses the fact that the signer is honest. The idea is that if the signer claims not to have received the payment or the receipt, then any clearing and settlement of the payment or use of such a receipt will be detected later, and would lead to legal measures. Then the following cases have to be considered. * • If the signer aborts before sending $i=\mathit{intermediate}(b,p,r_{s},{k_{\mathit{}}})$ (the intermediate signature), then the protocol can be restarted from scratch. This results in a different blind signature, possibly for a different blind secret input $s$, but the same public input $p$. But since it is guaranteed that the sender never sent the intermediate signature, we are certain the user was never able to obtain a blind signature in the aborted run. * • If the signer aborted after generating the intermediate signature $i=\mathit{intermediate}(b,p,r_{s},{k_{\mathit{}}})$ (and this intermediate signature may or may not have been received by the user), then the protocol must be picked up from this point, with the user using the stored values for the variables used in step 1 and 2 (which should exist by assumption that local variables must be safely stored before sending messages that depend on them. This means that the previously generated intermediate signature $\mathit{intermediate}(b,p,r_{s},{k_{\mathit{}}})$ is sent to the user. This results in possibly a different blind signature, but for the same blind secret input $s$ and same public input $p$ that were used in the aborted run. We conclude that the above sketched dispute resolution protocol allows the user to obtain a valid blind signature of her choice (if the dispute resolution protocol itself does not abort of course), while guaranteeing that a (dishonest) user is never able to obtain two different blind signatures for two different values $s$ and $s^{\prime}$. ### 4.2 Attribute based credentials Partially blind signatures allow the user to hide (part of) the contents of a message to be signed, but must always reveal the full contents of the signed message to allow the signature to be verified. This means that such signatures _only_ break the link between the signing and the verification of the messages, meaning that the act of signing and the act of verifying is unlinkable. Unfortunately, any two acts of verification can still be linked (using the unique data embedded in each signature). For so-called multi-show unlinkability full blown attributed based credentials are required [18, 4]. We will not go into the details here, but only describe the functionality offered by such credentials, and the privacy properties they entertain. Such attributed based credentials are used in the protocols below to implement travel passes and seasonal tickets that offer reduced fares and that, by their very nature, are on the one hand tied to a particular person while on the other hand need to be presented continually to claim a reduced fare. An attribute based credential is a secure container for one or more attributes $a_{1},\ldots,a_{m}$. Credentials are bound to a particular person, and the attribute(s) it contains describe certain properties of that person. (In the current context, it describes the eligibility to certain fare reductions, for example because the person is more than 65 years old, or because the person is a student.) The values for the attributes are negotiated by the requesting person and the _issuer_ $I$ (under the assumption that the issuer knows or can verify that a particular property holds for the person to which the credential is being issued). The issuer also signs the credential, to prevent fraud. We write $C_{I}(a_{1},\ldots,a_{m})$ for the resulting credential that the person obtains. Typically the credential also contains a hidden private ${k_{\mathit{U}}}$ key known only to the user that is hidden from the issuer when the credential is being issued, somewhat similar to how partially blind signatures work. Tying this private key to the credential and requiring its use when showing the credential later (see below) aims to prevent users from sharing their credentials to commit fraud (e.g., when a student allows her younger, non student, brother to use her credential to obtain a reduced fare ticket). We note that such techniques to bind people to their credentials are not fool proof [4]. To prove a certain attribute, the user engages in a so called interactive _showing_ protocol with a verifier using one or more of such credentials. This showing protocol is typically _selective_ : the user can decide which attributes to reveal to (and which ones to hide from) the verifier. This means that the verifier never gets to see the full credential, which would be a bad idea anyway as every credential signature is unique and therefore would allow subsequent uses of the same credential by the same user to be linked. As we want multi show unlinkability, the user and the verifier instead engage in an (interactive) zero knowledge protocol where the user proves to the verifier that she owns a credential signed by a certain issuer, containing a selection of the revealed attributes $A_{r}\subseteq\\{a_{1},\ldots,a_{m}\\}$. This proof also requires the user to know the embedded private key ${k_{\mathit{U}}}$ (without revealing it of course). This reveals the issuer and the attribute values, and nothing more, to the verifier. We write ${k_{\mathit{U}}},C_{I}(a_{1},\ldots,a_{m})\leftrightarrow I,A_{r}\subseteq\\{a_{1},\ldots,a_{m}\\}$ (where the left hand side shows the input of the user, and the right hand side shows what the verifier learns (provided it knows the public key of the issuer needed to verify the proof). ### 4.3 Privacy friendly payment with receipt A basic mechanism used throughout our protocols is the possibility to pay a certain fare ${f}$ to a payment service provider (PSP) and to receive a receipt ${R}$ for this payment in return.101010The PSP could be your bank (provided it knows how to issue receipts as explained below), or a separate entity that lets bank process the payment and generates a receipt when the payment was successful. The receipt can subsequently be used at a (public transport) service provider to pay for transport. The idea is that such a payment mechanism can be implemented in many different (more or less privacy friendly) ways, with only the receipt being standardised for use in the protocols below. To maximise privacy protection in case the payment itself is less privacy friendly, the receipt ${R}=[\hbox{\pagecolor{lightgray}\rule[-1.5pt]{0.0pt}{8.5pt}${\textit{rs}}$}\,\|\,{f}]_{{k_{\mathit{\textrm{PSP}}}}}$ is a blind signature over the public fare ${f}$ paid as well as a blind receipt sequence number rs provided by the user, signed by the payment service provider PSP that processed the payment. To make explicit at which particular service the receipt can be used, the name of the service can be added, blindly, by the user as well. The user should ensure that each receipt has a different sequence number. This sequence number is used to prevent reuse of receipts: the sequence numbers in redeemed receipts are recorded as spent. (Which also shows why users have every reason to ensure that sequence numbers are indeed different.) Using a blind signature in this way guarantees that users cannot create fake receipts, while the receipt sequence number cannot be linked to the payment (and hence to the user making the payment). Users are expected to properly protect their receipts and keep them securely stored until use. In the protocols below, the paid fare is first collected by the PSP, then forwarded to a public transport clearinghouse (PTC) that later redistributes the paid fares to the PTOs based on submitted receipts the PTOs have collected. Each fare is recorded by the payment service provider (PSP) as a separate payment transaction for the specified amount with the clearinghouse as the recipient. If the payment transaction involves the bank account of the user (see below), care should be taken to _not_ include the bank account details of the user in the transaction towards the clearinghouse. This happens more or less automatically if the PSP is a separate entity independent of the bank (in which case the transaction will transfer the fare amount from the user bank account to that of the PSP). If the bank itself serves as PSP, an internal bank offset account should be used that aggregates individual payments to the PTC with only the daily or weekly totals being transferred to the actual PTC account. This prevents the clearinghouse from learning the bank account (and hence the identity) of all people travelling with public transport, including how often they travel and an indication of the distance they travel (given that the fare is often a good indication of this). One possible way to implement payment when using a smartphone based public transport ticketing app is to redirect the payment phase to a separate payment app on the user’s smartphone, and let PSP forward the resulting receipt back to the transport ticketing app. A more privacy friendly option is to allow travellers to pay with cash at designated kiosks at public transport stations. Or to support the payment of fares using some kind of online privacy friendly payment scheme (like Digicash [8], or Zcash [3]). ### 4.4 Notation When describing the knowledge acquired by parties involved in the (figures depicting the) protocols below, we use expressions like $(a,b,c)$ to denote that a party learns the values $a$, $b$, and $c$, and moreover learns that they are linked and thus belong together. Values in different tuples are not linked, but can however be correlated based on their actual values: if a party learns a specific fare ${f}$ was paid by user $U$ (i.e., it knows $(U,{f})$) and later sees a ticket with that particular fare for a trip $T$ (i.e., it also knows $(\langle r,d\rangle,{f})$, then it may conclude user $U$ travelled route $r$ on date $d$. We use $\hat{U}$ to denote the possibly static IP network address of the user visible to the other parties.111111This equals the VPN server address or the Tor exit node address in case any of these services are used by the user. \begin{overpic}[abs,unit=1pt]{./fig/prot-papertickets-x.eps} \put(233.81453,4.66342){learns $({\textit{rs}},{f})$} \put(60.2531,39.82378){$[\hbox{\pagecolor{lightgray}\rule[-1.5pt]{0.0pt}{8.5pt}$T,\textit{ts}$}\,\|\,{f}]_{{k_{\mathit{\textrm{PTO}}}}}$} \put(145.611,75.79517){$[T,\textit{ts}\,\|\,{f}]_{{k_{\mathit{\textrm{PTO}}}}}$} \put(111.2948,98.93964){\hbox to0.0pt{\hss$[T,\textit{ts}\,\|\,{f}]_{{k_{\mathit{\textrm{PTO}}}}}$}} \put(233.09183,49.10245){\hbox to0.0pt{\hss$[{\textit{rs}}\,\|\,{f}]_{{k_{\mathit{\textrm{PSP}}}}}$}} \put(226.93683,167.32111){sender anonymous} \put(227.17271,151.09951){not anonymous} \put(6.56553,13.4382){knows $(U,\hat{U},{f},T,{\textit{rs}},\textit{ts})$} \put(3.49004,167.45662){learns $({f},U,\hat{U})$} \put(6.00142,97.24129){${f}$} \put(55.83259,134.22948){${k_{\mathit{\textrm{PSP}}}}$} \put(40.33569,86.21409){$[\hbox{\pagecolor{lightgray}\rule[-1.5pt]{0.0pt}{8.5pt}${\textit{rs}}$}\,\|\,{f}]_{{k_{\mathit{\textrm{PSP}}}}}$} \put(123.97216,124.16488){learns $(T,\textit{ts},{f})$} \put(175.13129,18.64164){${k_{\mathit{\textrm{PTO}}}}$} \put(110.73671,62.74341){\hbox to0.0pt{\hss$[{\textit{rs}}\,\|\,{f}]_{{k_{\mathit{\textrm{PSP}}}}}$}} \put(95.89024,4.79793){learns $(\hat{U},{\textit{rs}},{f}),(T,\textit{ts},{f})$} \end{overpic} Figure 1: Protocol emulating paper tickets ## 5 Solution 1: Emulating paper tickets One way to achieve privacy in public transport ticketing is to emulate the traditional use of paper tickets in public transport. The basic idea is to first buy the ticket online, and subsequently use it for public transport later, in such a way that the financial transaction used to pay for the ticket cannot be linked to the actual trip being made. The protocol assumes that the public transport app on the user’s smartphone contains a database with all possible trips that can be made by public transportation, together with the corresponding fares to be paid. ### 5.1 Detailed protocol The protocol, graphically represented in figure 1, runs as follows. Phase 1: Obtaining a ticket * • The user selects the route $r$ she wants to travel, and the day $d$ on which she wishes to travel. This defines the trip $T=\langle r,d\rangle$. * • The user calculates the fare ${f}=\textit{fare}(T)$ for the trip. (Incorrectly calculated fares will be detected later.) * • The user starts a payment for this fare, and receives a receipt ${R}=[\hbox{\pagecolor{lightgray}\rule[-1.5pt]{0.0pt}{8.5pt}${\textit{rs}}$}\,\|\,{f}]_{{k_{\mathit{\textrm{PSP}}}}}$ in return. (See section 4.3 above for details.) The paid fare is credited to the PTC account. * • The user sends this receipt to the PTO. The PTO verifies the signature on the receipt, and submits it for clearing and settlement to the PTC. The PTC also checks the signature on the receipt, and checks whether a receipt with sequence number rs has been submitted before. If so, the receipt is rejected. Otherwise, the PTC accepts the receipt and records rs as submitted. * • The user engages in a partially blind signature issuing protocol with the PTO in order to obtain a ticket $T$ for the trip. The user blindly provides the trip $T$ as well as a blind and fresh ticket sequence number ts. The PTO provides the (unblinded) fare ${f}$ present in the receipt it received in the previous step. As a result the user receives the ticket $T=[\hbox{\pagecolor{lightgray}\rule[-1.5pt]{0.0pt}{8.5pt}$T,\textit{ts}$}\,\|\,{f}]_{{k_{\mathit{\textrm{PTO}}}}}$, signed by the PTO. Phase 2: Travelling by public transport Public transport operators need to verify that all users that travel with them have a valid ticket, with the correct fare. The traveller and the ticket inspector engage in the following protocol for that purpose. * • The user sends the ticket $T=[T,\textit{ts}\,\|\,{f}]_{{k_{\mathit{\textrm{PTO}}}}}$ (revealing all its contents) to the ticket inspector. One way to do so in a sender anonymous fashion is to let the public transport app display the ticket as a QR code on the smartphone display, and let the inspector scan this QR code. Many public transport operators use similar schemes to inspect ‘home print’ paper based tickets. * • The ticket inspector verifies the signature on the ticket, whether fare ${f}$ is correct for trip $T=\langle r,d\rangle$, whether the ticket sequence number ts is not invalidated, whether the date $d$ in trip $T$ is today, and whether the route $r$ in trip $T$ covers the leg (of the total trip) where the ticket inspector asks the user to provide a ticket. If the user cannot provide a valid ticket, a fine is issued. * • The ticket inspector verifies the ticket with the PTO. The PTO also checks the signature on the ticket, and checks whether a ticket with sequence number ts has been submitted before. If so, the ticket is rejected. Otherwise, the PTO accepts the ticket and records ts as submitted. (If trips consist of several legs, the same ticket should be accepted for different legs of the trip.) Note that the fact that ticket sequence numbers must be verified and invalidated in real time implies that the equipment of the inspector must be online. As tickets are only valid for a single day, PTOs may choose to forfeit on this strict form of checking (hence relaxing system requirements), relying on the fact that ticket can still only be used (perhaps multiple times) on a single day. Phase 3: Clearing and settlement PTOs are reimbursed based on the payment receipts received in phase 1, after submitting them to the clearinghouse PTC. For each receipt, the PTC verifies the signature, and verifies that the sequence number in the receipt rs is yet unclaimed. If so, the sequence number is recorded as claimed, and the PTC proceeds to pay the fare specified in the receipt to the PTO. Otherwise the claim is rejected. ### 5.2 Analysis To what extent does this solution fit the requirements set out above? Users obviously pay have to for their trips, and the fare depends on the distance travelled. Inspectors and sufficiently high fines are necessary to keep users honest and disincentivise travelling without a valid ticket. Public transport operators get paid based on the payment receipts they collect when issuing tickets. To get (statistical) information about actual trips made they need to have enough conductors to check the tickets of all their passengers when travelling (as this is the only time when the actual trip details are revealed). If multiple PTOs are involved in a particular trip, proper reimbursement can only be achieved if the app splits up the trip in different legs, one for each PTO the user needs to travel with. The level of privacy protection is reasonable, depending on the properties of the network being used. The protocol prevents trip details to be linked to users, in the following sense: for all the tickets a particular PTO sells for a particular fare ${f}$ it learns the set of IP addresses of users that bought a ticket for this particular fare on the one hand, and the set of trips made for this fare (through inspection) on the other hand, but it can never link a particular user address to a particular trip. In the setup described, the PSP and PTOs could however learn how many tickets you buy, and for which amount (ie for which distance), if they would try to identify you based on your (fixed) IP address. Note that this problem becomes much less significant if the payment receipts issued by the PSP can be used for many different types of purchases, i.e., if they are used as a type of generic digital currency. Even if PTOs and PSP collude, they would not be able to link users with actual trips made, but timing analysis linking payment times with ticket issuing times could be used by the PTO to be more certain about your identity. If you usually buy your tickets on the same day or the day before your trip, your PTO could learn when you travel. The PTO could learn whether you are using public transport a lot, or not. Many short trips on the same day may reveal you are in a city; certain patterns of distances may correspond to popular tourist routes (and hence reveal the city you are in). This limited level of privacy protection may already be a threat for people that engage in protests or civil disobedience, like the Hong Kong protesters or the Extinction Rebellion activists. All these problems can be avoided if users can use cash to buy tickets, at special digital kiosks. Communication between the user and inspector is sender anonymous. This means ticket inspection reveals no personal information. The system is secure: tickets are only issued by the PTO when given a payment receipt for a certain amount, signed by a bank. Only banks can create such a signed proof of payment. The amount paid for a ticket is checked by the conductor when inspecting a ticket. Only PTOs can create a valid ticket (signed in partially blind fashion). This signature is also checked by the conductor. Finally the conductor checks whether the ticket entitles a person to travel when and where the conductor inspected her ticket. Failure of one of these tests means the ticket is invalid. The sequence number of the ticket (embedded to guarantee one-time use) is checked in real-time with an online database of sequence numbers of already inspected tickets. If the sequence number is already in the database, the ticket is invalid. Otherwise, the sequence number is added to the database. ### 5.3 Dealing with failures and disputes Dealing with failures is always a challenge, but this is particularly the case in privacy friendly protocols where often the link between a user and her actions is deliberately broken. This means extra care needs to be taken to create some evidence that allows an entity to challenge a failure, while not eroding the privacy of the users. Below we describe some possible failures, and how they could be dealt with. See also [22] for additional measures that can be taken, and the general strategy to deal with failures during the issuing of blind signatures (like users not receiving a payment receipt after payment, or not receiving a ticket after submitting a payment receipt) outlined in section 4.1. The user wants to cancel a payment The user can return the payment receipt (which contains a unique sequence number) to the PSP to rewind the transaction. The PSP then forwards the payment receipt to the PTC signalling not to accept this payment receipt when a user requests a ticket to be issued. Also, the transfer of money from the bank to the PTC will be reversed. The user receives a valid but incorrect ticket This can happen if the user entered the wrong trip details, or if some internal error caused the wrong ticket to be issued. The user can ’return’ the ticket to the PTO, essentially running the showing protocol normally run when a conductor inspects the ticket. This invalidates the ticket. Using the same payment receipt she can start restart the issuing step, now with the correct trip details (assuming the fare is the same). The user wishes to cancel a ticket issued to her After ’returning’ the ticket the PTO as described in the previous case, she can then proceed to cancel the payment to the bank. The user receives an invalid ticket This is more tricky. Ideally the issuing protocol should guarantee that a valid ticket is issued. If this is impossible, at least the issuer should somehow be able to tell, from the logs, that the user indeed did not receive a valid ticket. Otherwise bogus claims for invalid tickets could be submitted. This all very much depends on the particular issuing protocol used. A valid ticket fails conductor inspection Ideally this should not happen. However, the user or conductor device may malfunction, and the communication between the two devices may be erroneous. If the ticket is valid, and the user app operates correctly, the user should at some point be able to convince the PTO she had a valid ticket when travelling. The app crashes or malfunctions This can be mitigated by ensuring that the app can be reinstalled without loosing any stored tickets, or turning them invalid. This requires operating system support, e.g., allowing data to be restored from data associated with a previous install of the application. The user looses or deletes a ticket There is no way to recover from this situation. (Loosing a ticket could happen when inadvertently deleting the whole app together with all its data.) ### 5.4 Variations and extensions #### Using actual paper tickets, or smart cards Instead of relying on users having smartphones, tickets could actually be printed on paper,121212This may sound pedantic, but in fact when trying to emulate something digitally based on how it was done physically, one always has to consider the option that the original, physical, approach simply works better. or be stored on contactless smart cards instead. In this case, a ticket kiosk needs to be used to allow users to select the ticket they need, allow them to pay (by cash or card), and to print the ticket or issue the ticket to the smart card. In the first case, the ticket (with its signature) is printed as QR code, which the inspectors can scan with their smartphone. In the second case, inspectors need to carry NFC enabled smartphones that allow them to scan the smart card and read the ticket (with its signature) from the smart card. This is certainly possible even with cheap smart cards (as it is not involved in any complex cryptographic operation: the inspector checks the signature locally on the device, and the blind signature is generated by the kiosk where the user buys the ticket). Paper tickets can also be obtained at home through a website (web app) that essentially emulates the functionality of the user smartphone app with respect to obtaining a ticket, but at the end of this phase prints the ticket as a QR code instead of storing it. #### Supporting seasonal tickets Reduced fares for public transport pass subscribers or holders of seasonal tickets can be catered for in a privacy friendly manners using attribute based credentials, in which case the attributes in the credential encode the fare reductions the holder is entitled to. The user can obtain such a credential using a protocol similar to that of buying a single ticket, except that in the last step the PTO issues a full blown credential instead.131313A simple blind signature as used for ordinary tickets will not do as the credential will have to be shown multiple times while retaining the desired privacy properties. The issuing protocol would run like this. * • The user selects which type of seasonal ticket she wishes to buy. This defines a set of attributes $a_{1},\ldots,a_{m}$ that define which type of reduction she is entitled to. * • The user calculates the total price ${f}$ for this seasonal ticket. * • The user starts a payment for this amount, and receives a receipt ${R}=[\hbox{\pagecolor{lightgray}\rule[-1.5pt]{0.0pt}{8.5pt}${\textit{rs}}$}\,\|\,{f}]_{{k_{\mathit{\textrm{PSP}}}}}$ in return. (Again see section 4.3 above for details.) The paid amount is credited to the PTC account. * • The user sends this receipt to the PTO. The user also sends the list of attributes $a_{1},\ldots,a_{m}$ to the PTO. The PTO verifies that the price ${f}$ present in the receipt corresponds to the amount due for this particular set of attributes. The PTO verifies the signature on the receipt, and submits it for clearing and settlement to the PTC. The PTC also checks the signature on the receipt, and checks whether a receipt with sequence number rs has been submitted before. If some of these tests fail, the receipt is rejected. Otherwise, the PTC accepts the receipt and records rs as submitted. * • The user and the PTO engage in a credential issuing protocol for this set of attributes. As a result the user receives the credential $C_{\textrm{PTO}}(a_{1},\ldots,a_{m})$, signed by the PTO. Such a credential can subsequently be used to travel by public transport with a reduced fare. Interestingly enough, the credential is actually irrelevant when buying a ticket (except that the user needs to apply the correct fare reduction based on the particular credential she owns), because the PTO blindly issues a ticket for a particular fare without learning the actual trip the ticket is for. Correctness of the fare paid is only verified at inspection time, when the inspector gets to see the full ticket containing both the trip and the corresponding fare. To prove that the user is entitled to a reduced fare, the inspection protocol needs to incorporate verification of the necessary attributes in the credential as well. As the original inspection protocol is sender anonymous, the credential verification protocol needs to be sender anonymous as well. This can be achieved by using a fully non- interactive credential showing protocol. Idemix [18], for example, uses a non- interactive proof of knowledge, but relies on a verifier generated nonce to guarantee freshness of the proof. Such a verifier generated nonce would be hard to incorporate in our setting, as it would require the equipment of the inspector to send something to the user device (which would either break sender anonymity or would require cumbersome approaches where the user needs to also scan a QR code on the inspector device). Luckily there is a way out of this dilemma: we can use the cryptographic hash of the randomly chosen ticket sequence number ts already present in the ticket as the nonce instead. The fact that in this case the nonce is generated by the prover is not a problem but actually a feature: the proof is now neatly tied to the ticket for which a reduced fare is claimed, and the original ticket inspection protocol already ensures that the same ticket sequence number cannot be used twice. This forces the user to pick a fresh sequence number.141414Note that the user is by no means forced to select the ticker sequence number _randomly_. Hashing it to derive the actual nonce to be used in the credential showing protocol however ensures that the protocol remains secure when the underlying credential showing protocol relies on actual randomness (and not merely freshness) of the nonce. A problem with the approach outlined above is that there is nothing inherently preventing users to pool and share a single credential (offering reduced fares) with a group of users that each ’prove’ possession of the credential to the inspector when necessary. Unless the private key associated with the credential is securely embedded in the user device (using e.g., a piece of trusted hardware to ensure that even the device owner cannot get access to it), this by itself does prevent such credential pooling attacks. This is a general problem of attribute based credentials, and indeed a problem of online digital identity management in general as securely binding actual persons to their online credentials is hard [4].151515Even embedding the private key in a secure enclave does not strictly speaking prevent the owner of the smartphone to share the phone itself with others (although it is surely not an enticing proposition to be without your private phone for several hours). ## 6 Solution 2: Pay as you go, with credit on device A fundamentally different, and increasingly popular approach for letting people pay for public transportation is to store credit on a contactless smart card serving as a public transport pass. People can (re)charge their passes at special kiosk (essentially transferring money from their bank account to their public transport pass) and subsequently pay when entering or leaving their chosen mode of transportation. This typically involves ‘checking in’ at a gate or turnstile when entering the station, or on the platform or in the bus itself, and ‘checking out’ when arriving at the destination or when changing connections. When people check-in, a check is performed to see whether there is enough credit left on the card. If so the location of the check-in is recorded on the card. When checking out, this check-in location is retrieved, and based on the check-out location the fare is computed and deducted from the credit on the card. To detect fare dodgers that travel without checking in, inspection on the trains or the bus is often still necessary, because it is hard to enforce an air-tight system that forces people to check-in or check- out at all times. The main challenge in implementing such a scheme is to ensure that the check-out operation is performed as fast and reliably as possible (given that at busy transportation hubs many people have to the check-out at the same time, and that a transaction involving a contactless public transport pass is prone to interference and failures). If current public transportation pass systems would actually work as just described, there would be no need to study privacy friendly forms of public transport ticketing: if all that the cards contain is user credit, there would not by any privacy issues with such a system. Unfortunately, this is not the case. All systems mentioned above involve cards with unique serial numbers that are recorded when checking in and when checking out, and stored in a central database. As these serial numbers are static, this allows users to be singled out and their public transportation travel patterns to be recorded over the years. What’s worse: these passes are almost always bound to a particular user (either because they are tied to a personal public transport account, or simply because they were recharged using the bank account of the user). The main reason for adding such tracing of passes is to be able to detect fraud and block passes that appear to be spending more credit than they should be spending based on the amounts used to charge them. Here we aim to emulate such a credit-based system in a privacy friendly manner, without needing to rely on tamper proof hardware or secure execution environments to prevent users from committing fraud by tampering with the credit on their tokens (i.e., their smartphones) in their possession. ### 6.1 Detailed protocol \begin{overpic}[abs,unit=1pt]{./fig/prot-PAYG-v2-x.eps} \put(187.93512,11.81414){${k_{\mathit{\textrm{PTO}}}}$} \put(284.3413,30.66255){${k_{\mathit{\textrm{PTC}}}}$} \put(117.1577,149.26766){\hbox to0.0pt{\hss$[\textit{cs}_{i}\,\|\,v_{i}]_{{k_{\mathit{\textrm{PTC}}}}}$}} \put(62.03376,69.41835){$[\hbox{\pagecolor{lightgray}\rule[-1.5pt]{0.0pt}{8.5pt}$\textit{cs}_{i+1}$}\,\|\,v_{i+1}]_{{k_{\mathit{\textrm{PTC}}}}}$} \put(62.48745,126.33398){$[\textit{cs}_{i},\ell,t]_{{k_{\mathit{\textrm{PTO}}}}}$} \put(75.93469,92.36206){\vbox{$[\textit{cs}_{i},\ell,t]_{{k_{\mathit{\textrm{PTO}}}}}$\\\ $\quad[{\textit{rs}}\,\|\,v]_{{k_{\mathit{\textrm{PSP}}}}}$}} \put(187.32685,126.33398){accept/reject} \put(235.47272,259.36298){sender anonymous} \put(235.7086,243.14238){not anonymous} \put(186.9926,69.41835){$[\hbox{\pagecolor{lightgray}\rule[-1.5pt]{0.0pt}{8.5pt}$\textit{cs}_{i+1}$}\,\|\,v_{i+1}]_{{k_{\mathit{\textrm{PTC}}}}}$} \put(250.88631,171.17952){\vbox{learns\\\ $(\ell,t)$ \\\ $(\textit{cs}_{i},v_{i},{f},{\textit{rs}},v,v_{i+1})$ }} \put(9.1803,261.63647){learns $(U,\hat{U},v)$} \put(14.53731,191.77547){$v$} \put(65.50573,228.91522){${k_{\mathit{\textrm{PSP}}}}$} \put(48.87158,180.39395){$[\hbox{\pagecolor{lightgray}\rule[-1.5pt]{0.0pt}{8.5pt}${\textit{rs}}$}\,\|\,v]_{{k_{\mathit{\textrm{PSP}}}}}$} \put(131.68397,210.16317){learns $(\textit{cs}_{i},\ell,t)$} \put(240.34894,149.09602){\hbox to0.0pt{\hss$[\textit{cs}_{i}\,\|\,v_{i}]_{{k_{\mathit{\textrm{PTC}}}}}$}} \put(240.34894,92.05391){\hbox to0.0pt{\hss${f},\textit{cs}_{i},[{\textit{rs}}\,\|\,v]_{{k_{\mathit{\textrm{PSP}}}}}$}} \put(181.21301,188.80437){$[\textit{cs}_{i},\ell,t]_{{k_{\mathit{\textrm{PTO}}}}}$} \put(98.19686,171.85806){$[\textit{cs}_{i},\ell,t]_{{k_{\mathit{\textrm{PTO}}}}}$} \put(3.49004,32.43116){\vtop{learns\\\ $(\textit{cs}_{i},v_{i},\ell,t,{f},\ldots$\\\ $\ldots,\text{checkout location},{\textit{rs}},v,v_{i+1})$ }} \end{overpic} Figure 2: Protocol “pay as you go” Each user maintains travel credit on their own device. As the device is not assumed to be trusted or tamper resistant, care must be taken to ensure that users cannot create counterfeit credit, or spend more than they have credit. Therefore, travel credit $C=[\hbox{\pagecolor{lightgray}\rule[-1.5pt]{0.0pt}{8.5pt}$\textit{cs}$}\,\|\,v]_{{k_{\mathit{\textrm{PTC}}}}}$ is represented by a blind signature of the Public Transport Clearinghouse PTC over the secret (blinded) sequence number cs and the known credit value $v$. Once the credit token is used, the sequence number cs becomes known. In the protocol below the PTC uses this to record the ’state’ of such a token as either checked-in or spent. We assume in this protocol that check-in and check-out use a sender anonymous form of communication, for example by using near field communication with a randomised anti-collision identifier. The protocol runs as follows. Phase 1: Obtaining credit * • The user starts a payment for the amount $v$ she wishes to obtain credit for, and receives a receipt ${R}=[\hbox{\pagecolor{lightgray}\rule[-1.5pt]{0.0pt}{8.5pt}${\textit{rs}}$}\,\|\,v]_{{k_{\mathit{\textrm{PSP}}}}}$ in return. (See section 4.3 above for details.) The paid amount is credited to the PTC account. * • The user can use this receipt to add the credit to her device when checking out (see phase 3 below).161616A separate protocol between the user device and the PTC to add credit is also possible, but is not discussed here. Phase 2: Check-in to start a trip * • The user sends her credit token $C=[\textit{cs}_{i}\,\|\,v_{i}]_{{k_{\mathit{\textrm{PTC}}}}}$ to the check-in device. * • The check-in device verifies the signature on the credit token, checks that the stored value $v_{i}$ is larger than some minimum credit required,171717This is necessary to prevent users to accrue (too much) negative credit by checking in with hardly any credit and going on an expensive trip. and submits it to the PTC. The PTC verifies the signature on the credit token, and checks whether $\textit{cs}_{i}$ is recorded as spent or checked-in. If so, the credit token is rejected. Otherwise it is accepted and the PTC records $\textit{cs}_{i}$ as checked-in, and records the associated value $v_{i}$ necessary when issuing a new credit token at check out. * • The check-in device sends the user a check-in token $\textrm{I}=[\textit{cs}_{i},\ell,t]_{{k_{\mathit{\textrm{PTO}}}}}$, containing the check-in location $\ell$, the check-in time $t$ and the credit token sequence number $\textit{cs}_{i}$, all signed by the PTO. The PTO logs the tokens for bookkeeping purposes. The user stores the check-in token. Phase 3: Check out to finish a trip * • The user sends her check-in token $\textrm{I}=[\textit{cs}_{i},\ell,t]_{{k_{\mathit{\textrm{PTO}}}}}$ to the check-out device. * • If the user wants to add additional credit to her device, she also submits a receipt ${R}=[\hbox{\pagecolor{lightgray}\rule[-1.5pt]{0.0pt}{8.5pt}${\textit{rs}}$}\,\|\,v]_{{k_{\mathit{\textrm{PSP}}}}}$ obtained earlier. * • The check-out device verifies the signatures on both tokens, and validates the time on the check-in token (i.e., checks whether the check-in time $t$ and check-in location $\ell$ make sense given the check-out time and the check-out location). * • Given the check-in time $t$, the check-in location $\ell$, the check-out time and the check-out location, the check-out device computes the fare $f$. * • The check-out device then submits the fare ${f}$, the credit sequence number $\textit{cs}_{i}$, and the (optional) receipt $[\hbox{\pagecolor{lightgray}\rule[-1.5pt]{0.0pt}{8.5pt}${\textit{rs}}$}\,\|\,v]_{{k_{\mathit{\textrm{PSP}}}}}$ to the PTC. (The PTO signs this transfer). The PTC also verifies the signature on the receipt, and checks whether $\textit{cs}_{i}$ is recorded as checked- in. If not, the check in is rejected. Otherwise it is accepted and the PTC records $\textit{cs}_{i}$ as spent. The PTC retrieves the associated credit $v_{i}$ (stored when the credit token was submitted at check-in) and computes the new credit $v_{i+1}=v_{i}+v-{f}$. (Negative credit is possible, but controlled through the credit check at check-in.) * • The user engages in a partially blind signature issuing protocol with the PTC, using the check-out device as a relay, in order to obtain an updated credit token $C_{i+1}$. The user provides a blind and fresh credit sequence number $\textit{cs}_{i+1}$. The PTC provides the (unblinded) credit value $v_{i+1}$ it just computed. As a result the user receives the new credit token $C_{i+1}=[\hbox{\pagecolor{lightgray}\rule[-1.5pt]{0.0pt}{8.5pt}$\textit{cs}_{i+1}$}\,\|\,v_{i+1}]_{{k_{\mathit{\textrm{PTC}}}}}$. * • The PTC proceeds to pay the fare to the PTO. (This can be done in bulk.) * • The user stores the new credit token. The user also logs the check-in in a local trip history (that can be consulted to resolve disputes). It may verify locally whether the deducted fare is correct. Phase 4: Inspection The inspector needs to verify that every person travelling has a valid check- in token. * • The user sends her check-in token $[\textit{cs}_{i},\ell,t]_{{k_{\mathit{\textrm{PTO}}}}}$ to the inspector. * • The inspector verifies the signatures on the token, validates the time on the check-in token (i.e., checks whether the check-in time $t$ and check-in location $\ell$ make sense given the inspection location, and submits the check-in token to the PTC. The PTC also verifies the signature on the check-in token, and checks whether $\textit{cs}_{i}$ is recorded as checked-in. If not, the credit token is rejected. Otherwise it is accepted and the PTC records $\textit{cs}_{i}$ as inspected. (If this particular token is encountered by a different inspector, on a leg of the trip that is inconsistent with earlier inspections of the same token, then fraud is assumed.) Note that the fact that credit must be verified in real time implies that the equipment of the inspector must be online, communicating with the PTC (and not the PTO). Phase 6: Clearing and settlement The PTO logs all check-in and credit tokens submitted during check out. The PTC pays the fare as soon as it receives the check-out token and computes the new credit token. (It may accumulate fares to pay the total amount every day or week.) The PTO verifies the payments it receives with the logs it keeps. ### 6.2 Practical considerations As discussed in the introduction of this section, the main challenge in practice of is to make check-in and check-out as fast (and reliable) as possible. Reliability can be improved in the above protocol by adding an acknowledgement message back from the user to the check-in or check-out device whenever the check-in or check-out token have been received in good order, and letting the check-in or check-out device generate an appropriate sound as confirmation. The user device itself could confirm proper check-in or check-out immediately after receiving the token (and sending the acknowledgement), or sound an alarm when the expected token is not received within a short timeout. But an additional message does increase the time needed to check-in or check-out, and adds another point of failure as well: what to do if the acknowledgement message itself is not delivered? Check-in speed is constrained both by the real time connection between the user device and the check-in device, and the real time connection between the check-in device and the PTC which needs to verify that the credit token is not double-spent. This check could be made asynchronous, and the check-in token be issues optimistically, at the expense of ramping up inspection within the public transportation system to detect people that checked in with such a double spent credit token. Alternatively, when there are not too many check out devices, optimistically issued check-in tokens can be revoked when necessary by blacklisting the embedded credit sequence number cs and sending this to all check-out devices. The check-in token itself involves computing a basic signature over the credit sequence number sent by the user device, after verifying the blind signature over the credit token. This should not prove to be an issue in practice. Check-out is more complex as it involves issuing a blind signature over the new credit, where the check-out device works as a relay between the user device and the PTC. Check-out speed can be significantly improved by decoupling the issuing of the check-out token from updating the credit on the user device, doing it ’lazily’ after check-out with a separate protocol that runs between the user device and the PTC. In this case a basic check-out token containing the fare can be issued by the check-out device, with an ordinary signature (instead of a blind one). To protect user privacy however, care needs to be taken to then hide the user address from the PTC to prevent it from linking the previous credit sequence number $\textit{cs}_{i}$ to this used address (as this allows the full trip to be linked to a particular user). ### 6.3 Dealing with failures and disputes Beyond the failures and disputes for protocol 1 discussed in section 5.3, the use of of check-in and check-out devices poses additional challenges. Also the fact that credit is stored on the user device makes the solution more fragile and risky for the user. Dispute resolution depends on clear information about what happened about the time a failure occurred. Unfortunately, due to their privacy friendly nature, the protocols retain very little useful information by themselves. Adding timestamps to local logs of each protocol step, by the PTC, the PTO, and the user device will help compare logs in case of disputes (and detect possible fraud). Creating append only logs (using hash chaining techniques) increases their integrity, especially if occasional public commitments to the current state of the log are recorded. A hash of the log on a user device can be submitted when checking in and checking out, and be included in the check in and check out token (that are signed by the PTO). This poses no linkability as the log will be updated with every check in and check out, provided such updates always contain some private information from the user device (e.g. the serial number used in the next credit token). To aid dispute resolution, the PTO could also issue a separate check-out $[\textit{cs}_{i},\ell,t,{f}]_{{k_{\mathit{\textrm{PTO}}}}}$ to the user when she checks out, containing the credit sequence number, check-out location, time, and fare, signed by the PTO. This allows the user to verify the correctness of the new credit token she receives when checking out (and allows her to check that the correct check-out information is used to compute the fare). Check in fails If it is a communication error in the first step, the user can try again. Otherwise, if the credit token fails to verify the user needs to start a dispute resolution (if she believes the credit token should be valid). If the credit token is accepted, but subsequent steps fail, dispute resolution should clear the recorded serial number for the credit token from the clearinghouse database to ensure it is valid the next time the user checks in. Check out fails If it is a communication error, the user can try again. Otherwise, if the check-in token (or payment receipt) fails to verify the user needs to start a dispute resolution (if she believes the check-in token should be valid). If the check in token is not accepted, dispute resolution needs to determine whether the user actually tried to check in earlier, or did not. If the user did not get an error when checking in, for sure the PTO log will contain the serial number of the current credit token. Fare dispute After checking out user discovers that the fare paid does not correspond to the fare due for the trip made. The user should submit a piece of the log with all entries involving the check in and corresponding check out for this trip (which should follow each other immediately in the user device log, and are thus linked through the internal hash chain). This is then matched with the corresponding logs of the PTO and clearinghouse. Any discrepancy can be compensated by adding it to the current credit on the device by issuing a new credit token. This can be done even after the user has made other, more recent, trips. ### 6.4 Analysis The security analysis is similar to that of the previous protocol, as presented in section 5.2. We therefore focus on the privacy aspects here. A significant improvement over the previous protocol is that neither the PTO nor the PTC obtains information about the user identity: communication with the inspector and the check-in or check-out devices is sender-anonymous. PTOs can link check-in and check-out location (and hence trips) to credit sequence numbers, but PTOs cannot link these to anything else (either on their own, or when colluding with others). Credit sequence numbers are in essence ephemeral identifiers. The situation changes slightly when a user decides to buy additional credit and to add it when checking out. In that case, the PTC learns also the value $v$ of the additional credit which can be linked to a particular user when thew PTC colludes with the PSP and the particular credit bought is more or less unique. This can be mitigated by allowing users only to buy predefined values of credit, thus ensuring a reasonable anonymity set of users all buying the same credit at roughly the same time. (We implicitly assume here that a user buys credit well in advance to prevent timing correlation attacks.) The situation also changes when the credit values stored in a credit token are unique. This would allow the PTO to link a check-in with credit $v_{i}$ with a subsequent check-in with credit value $v_{i+1}$. With a bit of ’luck’ a PTO might be able to link several trips made by the same user this way. This chain is severed as soon as a common credit value is reached,181818Under reasonable assumptions this would not be a concern in practice. Suppose the maximum credit is $€100$ and fares are multiples of $10$ cents, then there are $1000$ different possible credit values. If there are one million users, the anonymity set would on average contain $1000$ people (although the distribution is probably skewed with larger anonymity sets for smaller credit values). The system could also define some default credit value options (like $€25$, $€50$, and $€100$) and nudge users to always top-up their credit to these defaults. The anonymity sets for these particular values would then be much larger. or when a user decides to travel with a different, non colluding, PTO. ### 6.5 Pay as you go, paying later Given the potential benefits of ‘pay as you go’, it would especially be nice to allow users to pay for their trips afterwards, instead of forcing them to lock significant funds on the device itself. However, introducing a pay later option creates a risk for PTOs as users may fail to pay their debts, so mitigation strategies need to be considered. The basic idea is use to the same protocol, but allowing negative credit. The main risk is that users use their device up to the maximum negative credit, then de-install the app from their smartphone, and then reinstall a fresh one with a balance of zero. To counter such sybil-like attacks, reinstalling an app should be hard. One way to do so is to tie the install to your device identity or app store identity. In that case the app provider or even the app store itself could start asking questions when someone repeatedly installs the app. But this is not as straightforward as it seems, because ideally we want to allow arbitrary third parties to provide public transport apps (to increase trust). One idea is that any (third party) app must be ’blessed’, by the clearinghouse, with an ’admission credential’. In other words, a user can install any app he or she desires, but all protocols outlined above first verify whether the user has a valid admission credential. The user can obtain this credential, through the app, by registering the app with the clearinghouse.191919However, there should be a way to tie this credential to the specific device being used, to prevent cloning. This registration process requires the user to prove his or her identity (for example using a government wide digital identity scheme). Note that relying on such an approach is risky, as it undermines the main message that the public transport app is privacy friendly: if that is supposed to be the case, why does it require me to sign in with a government approved digital identity? The admission credential is special, because it can be blacklisted: the clearinghouse keeps information about all credentials it issued so that when a user wants to obtain a new admission credential (because he or she claims to have lost their phone, reinstalled the app or whatever), then the previous admission credential becomes blacklisted. Information about the blacklisted credential is sent to all PTOs so that when they check whether some user has a valid admission credential (in the first step of each protocol), this will fail for all blacklisted credentials. Note however that this will not deteriorate the privacy protection offered by the protocols, at least not for users without blacklisted credentials: for every credential that is _not_ blacklisted, the PTOs have no way to trace or link valid admission credentials that are not yet blacklisted. The exact privacy properties depend on the specific method to blacklist credentials: a naive scheme might allow the clearinghouse to share blacklisting information about _all_ users to the PTO to make them all traceable. The most privacy friendly scheme doesn’t even allow blacklisted users to be linked or identified [5, 23]. ## 7 Conclusions In this paper we explored options how to implement privacy friendly ticketing for public transport in practice. We show that this certainly possible, with certain constraints (or issues that deserve further study, see below). Starting point is the observation that from a privacy perspective it is better to collect personal data locally on the user device, instead of centrally on the servers of the service providers. Two different approaches (buying tickets beforehand, and pay as you go) have been studied. We show that these can be implemented with reasonably good privacy properties, under reasonably practical assumptions. In particular we show that an untrusted smartphone can be used as the ’token’ to carry tickets or travel credit. This allows third parties to provide the apps for that purpose, which should increase the (perceived) trust of the overall system. For the second protocol, there are rather strict requirements on the maximum checking in and checking out time (in the order of 200-300 milliseconds); actual implementations of the protocols proposed are necessary to verify whether these requirements can be met. One meta conclusion of this work is that we need an efficient, frictionless, way to provide sender anonymity on the Internet, similar to the use of randomised MAC addresses on local networks. A VPN is too weak (the VPN provider sees everything its users do), yet Tor is too strong (there is no need to protect against a NSA like adversary) given the impact on performance. If randomised client IP addresses could be used by default to set up a TCP connection between a client and a server, that would already provide a tremendous boost in privacy on the Internet as servers can no longer trace their users based on their IP address. There are some proposals for temporary IPv6 addresses that partially address this issue [13], but these only apply to larger subnets and do nothing to hide the often fixed IP addresses of private xDSL connections. The second meta conclusion of this work is that there is a need to make apps (or data in apps) _uncloneable_ , so that they can be used in similar contexts and with similar properties as smart cards. Moreover, there should be a secure way to establish that the person holding the phone and/or using the app is indeed the owner of the phone (and not someone that uses the phone with permission of the real owner). These properties are also mandatory to increase the security of attribute credentials, in particular to prevent the attributes in them being pooled or shared. This seems challenging if at the same time we want the apps to be open source. One idea is to use either the SIM card present in most smartphones, or to use the secure element present in most modern smartphones. ## References * [1] M. Abe and E. 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###### Abstract The paper presents the simulation studies of 10 $\mu$m pitch microstrips on a fully depleted monolithic active CMOS technology and describes their potential to provide a new and cost-effective solution for particle tracking and timing applications. The Fully Depleted Monolithic Active Microstrip Sensors (FD- MAMS) described in this work, which are developed within the framework of the ARCADIA project, are compliant with commercial CMOS fabrication processes. A TCAD simulation campaign was performed in the perspective of an upcoming engineering production run with the aim of designing FD-MAMS, studying their electrical characteristics and optimising the sensor layout for enhanced performance in terms of low capacitance, fast charge collection and low-power operation. A very fine pitch of 10 $\mu$m was chosen to provide very high spatial resolution. This small pitch still allows readout electronics to be monolithically integrated in the inter-strip regions, enabling the segmentation of long strips and the implementation of distributed readout architectures. The effects of surface radiation damage expected for total ionising doses of the order of 10 to 105 krad were also modelled in the simulations. The results of the simulations exhibit promising performance in terms of timing and low power consumption and motivate R&D efforts to further develop FD-MAMS; the results will be experimentally verified through measurements on the test structures that will be available at the beginning of 2021. ###### keywords: Particle detectors; silicon detectors; monolithic sensors; microstrip sensors; CMOS; TCAD simulations; fast timing. xx 1 5 Received: date; Accepted: date; Published: date Fully Depleted Monolithic Active Microstrip Sensors: TCAD simulation study of an innovative design concept Lorenzo De Cilladi 1,2,, Thomas Corradino 3,4, Gian-Franco Dalla Betta 3,4, Coralie Neubüser 4 and Lucio Pancheri 3,4 on behalf of the ARCADIA collaboration Firstname Lastname, Firstname Lastname and Firstname Lastname Correspondence<EMAIL_ADDRESS> ## 1 Introduction Charged particle tracking and timing are fundamental tools for both physics research and for numerous applications. Although a number of detection techniques are available, silicon detectors have become largely employed due to their versatility and to the parallel strong developments of the semiconductor industry. Various flavours of silicon sensors have been developed to meet the specific requirements of different experiments and applications, such as high spatial resolution, fast charge collection, low power consumption, high radiation tolerance and low cost per unit area. Silicon detectors are divided in two categories, namely hybrid and monolithic detectors. The former are made of two separate silicon elements, the sensor and the chip, which are interconnected through external bump or wire bonding. While the sensor hosts the sensing volume only, the chip integrates the front- end readout electronics. On the contrary, monolithic sensors, which are emerging as a valid alternative to hybrid detectors, embed the front-end electronics in the same silicon substrate which hosts the sensing volume, with benefits in terms of material budget, production yield and fabrication cost, as they are produced with commercial microelectronics processes wermes (2009, 2019); garcia (2018). Due to their characteristics, monolithic sensors have recently raised a wide interest in different research fields; studies, proposals and developments have been made for applications in high energy physics (HEP) mager (2016); pernegger (2017); wang (2017), X-ray imaging wunderer (2014); hatsui (2013), medical particle imaging mattiazzo (2018) and space experiments scotti (2019). The state of the art includes three main types of monolithic sensors. The first type, called Depleted Field Effect Transistors (DEPFETs), is capable of low noise operation thanks to low sensor input capacitance rummel (2009). DEPFET detectors have been developed and used for HEP applications marinas (2011), for X-ray imaging in space treis (2010) and for free electron laser experiments lutz (2010). The main limitation of DEPFETs is the need to reset their internal gate which can be quickly saturated by the leakage lutz (2016), thus making this technology not suitable for environments with high levels of non-ionising radiation. A second approach consists in the SOI (Silicon-On-Insulator) monolithic sensors. SOI sensors embed a buried-oxide layer separating a thin low- resistivity silicon layer, which hosts the integrated readout circuitry, from a thicker high-resistivity substrate, which serves as the sensitive detection region kucewicz (2005); lan (2015). This technology allows a low capacitance to be obtained lan (2015); however, SOI sensors suffer from back-gate effect and have a reduced radiation hardness, due to accumulation of positive holes charges in the buried oxide layer after irradiation hagino (2020). Strategies have been found to overcome these limitations and to recover from the Total Ionising Dose (TID) asano (2016), but, as a consequence, the fabrication process of SOI sensors have become highly specialised and not compliant with standard microelectronics production processes. This results in increased cost per unit area, which is a critical issue for large-area detector applications. A third flavour of monolithic sensors is represented by CMOS sensors turchetta (2003). CMOS sensors were already in use for light detection when they were first proposed for charged particle tracking at the beginning of the 2000s turchetta (2001). Over the last years, important advancements in CMOS sensors allowed them to be employed in many applications, eventually leading to very large scale productions for particle trackers at collider experiments. The STAR pixel detector, which took data at the Relativistic Heavy Ion Collider (RHIC) from 2014 to 2016, was the first large area monolithic pixel tracker ever built, for a total of 0.16 m2 contin (2018). These dimensions have been exceeded by the newly-constructed Inner Tracking System of the ALICE experiment at CERN, in which a total detector surface of about 10 m2 is covered by ALPIDE CMOS monolithic active111A monolithic sensors is called “active” if it integrates a signal amplifier inside each pixel or strip. pixel sensors (MAPS) mager (2016). These achievements demonstrate the level of maturity and reliability that CMOS sensors have recently reached. However, there is still room for further improvements, especially in terms of charge collection speed and radiation hardness, and possibility to push previous limits in terms of low power density, high spatial resolution and SNR. Pixel detectors are the first choice for small scale applications and for vertex trackers at collider experiments wermes (2009) as they have an intrinsic capability of providing a two-dimensional position information rossi (2006). On the other hand, microstrip sensors peisert (1992) are largely used as particle detectors for space applications and are a competitive option for particle trackers due to their high spatial resolution, simpler readout and much lower power density (i.e. power consumption per unit area) compared to pixel detectors. Large experiments at particle colliders have largely employed silicon hybrid strip sensors in the past, and are still developing and assembling new large-area trackers based on this technology, as in the case of the Phase-2 Upgrades of the CMS Outer Tracker chowdhury (2020) and of the ATLAS Strip Inner Tracker david (2018). Recent space experiments equipped with silicon hybrid microstrip trackers include FERMI-LAT atwood (2007), DAMPE azzarello (2016), PAMELA adriani (2003) and AMS-02 lubelsmeyer (2011). Strip- like sensors integrated in a monolithic technology have been proposed by combining the outputs of 55 $\mu$m $\times$ 55$\mu$m benhammadi (2019) or 40 $\mu$m $\times$ 600$\mu$m han (2020) pixels in each column or row of a pixel matrix. Spatial resolution of 1.25-1.3 $\mu$m was achieved using hybrid silicon microstrip sensors with 25 $\mu$m pitch straver (1994). However, it has recently been demonstrated with fully depleted double-SOI monolithic pixel sensors that the 1 $\mu$m limit can be exceeded by semiconductor detectors sekigawa (2017). The keys to a high spatial resolution with analogue readout are a fine microstrip pitch, a low sensor thickness to reduce Coulomb scattering and delta-ray emission, and an increased SNR, which can be achieved by reducing the leakage current and the sensor input capacitance to the readout electronics, but which is ultimately limited by the noise of the front-end electronics turchetta (1993); peisert (1992); straver (1994). This paper presents the first investigation, design and simulation studies of CMOS Fully Depleted Monolithic Active Microstrip Sensors with 10 $\mu$m pitch for charged particle detection. Properly optimised sensor layouts may allow sub-micron resolution, improved radiation hardness and fast timing performance thanks to full depletion wang (2017); snoeys (2017) in a power-saving and cost-effective commercial technology. Moreover, a further advantage of monolithic microstrips is the potential complexity reduction of the detector assembly compared to hybrid microstrip detectors. In fact, since many readout functions can be monolithically integrated on the same chip which hosts the sensing volume, 1-by-1 strip bonding to the external readout electronics would not be needed anymore. We have hence studied and designed the FD-MAMS within the framework of the INFN ARCADIA project in order to provide an innovative solution for satellite-based space trackers and for large area particle detectors at future collider experiments. The results of the Technology Computer-Aided Design (TCAD) simulation campaign222The TCAD simulations were produced using the Synopsys ® Sentaurus (Version O-2018.06-SP2) software. which allowed different MAMS design flavours to be compared in terms of sensor capacitance, reference voltage values, leakage current and charge collection time and efficiency are presented; the effects of the inclusion of a silicon dioxide (SiO2) layer on top of the sensor and of surface radiation damage on the sensor operating parameters are explored; the study of charge sharing between groups of adjacent strips when particles with different Linear Energy Transfer (LET) traverse the sensor is reported. A selection of the MAMS presented in this paper is going to be implemented in test structures which were submitted in November 2020 for an engineering production run. The paper is organised as follows: Section 2 presents the sensor concept for the ARCADIA fully depleted CMOS monolithic microstrip sensors and illustrates the simulation campaign that was performed for the sensor design optimisation; Section 3 describes and discusses the results of the simulations; Section 4 presents the conclusions, the future perspectives and the planned tests for the ARCADIA monolithic microstrip sensors. ## 2 The ARCADIA sensor concept The ARCADIA project and its precursor, SEED (Sensor with Embedded Electronics Development), designed an innovative sensor concept pancheri (2019, 2020) based on a modified 110 nm CMOS process developed in collaboration with LFoundry and compatible with their standard 110 nm CMOS process. Up to 6 metal layers can be stacked on top of the sensor, for a total metal and insulator thickness of about 4-5 $\mu$m. The ARCADIA collaboration is developing a scalable event-driven readout architecture to cover large detection surfaces (O(cm2)) while maintaining ultra-low power consumption. The target for pixel sensors is 10-20 mW/cm2 at high rates O(100 MHz), but for less dense particle environments (e.g. in space applications) a dedicated low-power operation mode implements a cyclic pulling of the data packets from each section of the pixel matrix and disables most of the serialisers and data transceivers, further reducing the total power consumption of the chip. In our project, an n-on-n sensor concept enabling full substrate depletion over tens or hundreds of microns and allowing full CMOS electronics to be implemented was employed. A simplified view of the sensor cross section is visible in Figure 1. The process allows to achieve sensor thicknesses from 50 to 400 $\mu$m. A high resistivity n-type substrate was used and constitutes the active volume. The sensing n-well node, located on top of the sensor, collects the electrons produced by ionisation due to particles traversing the active detection volume. N-doped and p-doped wells intended to host pMOSFETs and nMOSFETs respectively are shielded by a deep p-well, which allows the integration of full CMOS electronics and, hence, more complex digital functions, when necessary. In fact, the deep p-well prevents the n-wells hosting pMOSFETs from competing with the n-doped sensing node in the collection of the charge, thus avoiding loss of charge collection efficiency. Figure 1: ARCADIA monolithic sensor concept. The dotted arrows indicate the drift path of electrons (e) and holes (h) generated by a particle crossing the sensor. The voltages Vnwell and Vback applied to the sensor contacts are shown in green. A p+ boron-doped region sits at the backside of the n-substrate, thus forming a pn-junction; when a negative bias voltage Vback is applied to the backside p+ contact, sensor depletion starts from the pn-junction at the bottom of the sensor and eventually extends to the whole sensor, if the backside voltage is sufficiently large. Since the high voltage needed for sensor depletion is applied at the backside, it is possible to maintain the voltage Vnwell applied to the front n-well electrode below 1 V and to use low-voltage integrated electronics (1.2 V transistors) which is more radiation-resistant and has lower noise. Full sensor depletion allows fast charge collection by drift (beneficial to enhance the timing performance), higher charge collection efficiency, deeper collection depth and larger SNR; it also leads to improved radiation tolerance, as charge losses by trapping are reduced pernegger (2017). Since thicker sensors need higher backside bias voltage to reach full depletion, termination structures composed of multiple floating guard rings are used to avoid early breakdown at the edges of the pn-junction. An additional n-type epitaxial layer, with lower resistivity than the substrate, is integrated between the n-type substrate and the deep p-wells. Its aim is to better control the potential barrier below the deep p-well, in order to delay the onset of the punch through current described in details in Paragraph 2.2.2. The feasibility of this sensor concept and approach to Fully Depleted monolithic CMOS sensors was proven in the framework of the SEED project pancheri (2019, 2020). The upcoming ARCADIA engineering run will include different design flavours of FD-CMOS monolithic sensors, both pixelated and strip-like. Large-area ($1.3\times 1.3$ cm2) pixel demonstrators with embedded CMOS electronics and pixel test structures ($0.5\times 0.5$ and $1.5\times 1.5$ mm2) without integrated readout circuitry neubueser (2020) are foreseen, with pitches ranging from 10 to 50 $\mu$m. The test structures will include as well the innovative MAMS and will allow a detailed characterisation of these sensors. The 3D TCAD simulation campaign performed to design the first FD-MAMS will be presented and discussed in the following. ### 2.1 TCAD simulations 3D TCAD simulations were employed as a tool to optimise the sensor layout and performance. The use of 3D simulations is necessary to have a more realistic domain and results which are more accurate and less affected by boundary conditions. Furthermore, we were also interested in studying the charge collection dynamics after a particle crosses the sensor, and this is more straightforward with 3D simulations. A fine pitch of 10 $\mu$m was chosen for the microstrips in order to explore the characteristics and performance of a sensor layout which pushes the requirements on both spatial and timing resolution. Different sensor thicknesses foreseen for the production runs were simulated. Variations in the sensor layout and operating parameters were tested to study and optimise the sensor response. The simulated sensor flavours take into account the limitations imposed by the foundry’s sensors fabrication process, especially for the n-well and p-well sizes. The strip simulations investigated sensor flavours which pushed the design to the limits of the process requirements. Figure 2: Example TCAD 3D sensor domains for ARCADIA microstrips (top row) and corresponding cross sections (bottom row). (a) Standard simulation domain for sensors with the deep p-well. (b) Addition of n-wells above the deep p-wells. (c) Simulated ARCADIA microstrips without deep p-wells. All the TCAD simulations were performed at a temperature of 300 K. A standard simulation domain including three 50 $\mu$m long, 50 $\mu$m thick, 10 $\mu$m pitch microstrips is shown as an example in Figure 2 (a). The n-doped substrate is shown in green, the epitaxial layer in yellow, the microstrip sensing n-wells in red, the p-wells in blue and the less doped deep p-well in light blue. The default value for Vnwell is 0.8 V. The p-wells, instead, are kept at a voltage V${}_{pwell}=0$ V. One of the simulated sensor flavours has been specifically designed to allow for CMOS digital library cells to be integrated along the strips and is shown in Figure 2 (b). This sensor variant would allow the deployment of complex CMOS digital functions along the strip for distributed signal processing. We observed that the n-wells dedicated to the implementation of PMOS transistors and shielded by the deep p-well do not significantly influence the electrical characteristics of the detector in the TCAD simulation results. Therefore, we did not include them in the simulations. The deep p-well can be removed in the test structures that will be used to characterise the sensor (see Figure 2, c), and the necessary CMOS front-end electronics can be deployed at the end of the strips in the chip periphery. Sensors without the deep p-well were simulated as well. Different n-well, p-well and deep p-well sizes were considered to find the optimal layout in terms of sensor performance. Simulations were also employed to predict the effects that possible production uncertainties can have on the sensor operating parameters and electrical characteristics. For instance, the thickness and resistivity of the epitaxial layer may vary within a confidence range around their typical specified value (see Appendix A). 3D simulations for the different cases were run and compared. Some simulation parameters were fine-tuned using characterization results from a previous set of test structures, produced in the framework of the SEED project pancheri (2019). ### 2.2 Electrical and transient simulation In this Section, the simulations performed to extract the sensor electrical characteristics and to study the charge collection dynamics are briefly illustrated. Shared definitions and conventions on simulation setups and operating parameters were agreed for the whole ARCADIA simulation campaign and are also described in neubueser (2020). The strip length in the upcoming production run will be 1.2 cm. However, MAMS with lengths of 50 $\mu$m were simulated in order to run a large set of TCAD simulations in a reasonable computational time. The results were then scaled to the desired length. #### 2.2.1 Depletion voltage Sensor depletion starts at the backside, where the pn-junction between the n-type substrate and the p+ contact is located. If no negative bias voltage is applied to the backside contact, the sensor is not fully depleted and the collection n-wells are not isolated. This means that a resistive path exists between the n-type sensing nodes (see Figure 3, on the left). Therefore, if a voltage difference is applied between two adjacent n-wells, a current will flow between them. As the negative voltage applied to the backside contact increases, the space charge region enlarges through the high resistivity substrate, eventually merging with the depletion volume which surrounds the pn-junctions formed between the n-type substrate or epitaxial layer and the deep p-wells. At this point, the sensor is fully depleted, the resistive path between the sensing nodes is closed and the collection n-wells are isolated; this is shown in Figure 3, on the right. In this condition, no current (except for the leakage current) will flow among adjacent n-wells even when different voltages are applied to them. Figure 3: Depletion process in ARCADIA microstrips. On the left: cross section of a sensor before full depletion is reached. On the right: cross section of a fully depleted MAMS. The orange lines indicate the edge of the depletion region. This behaviour can be observed in the orange example IV curve in Figure 4 ($I_{nwell,unbalanced}$). The simulated domain shown in Figure 2 (left) was used. In this simulation, a voltage unbalance of 10 mV was applied between adjacent strips: the first n-well was biased at 0.79 V, the central one at 0.8 V and the third one at 0.81 V. The curve shows the current measured at the sensing node of the central strip as a function of $\|V_{back}\|$. A current of about 1 nA is measured at $V_{back}=0$ V. As the backside voltages increases and the space charge region enlarges, the current starts decreasing, and eventually reaches a plateau at a current of about $10^{-5}$ nA. This baseline corresponds to the leakage current (green IV curve, $I_{nwell,leakage}$). The backside voltage at which the single microstrips become isolated and the plateau is reached is the sensor depletion voltage $V_{dpl}$; this voltage is evaluated as the intersection point between the exponential decay fitting of the IV curve decreasing segment and the baseline. Figure 4: Example characteristic IV and CV curves extracted from TCAD simulations of ARCADIA monolithic sensors. The red vertical axis refer to the sensor capacitance ($C_{sens}$) CV curve. Figure 5 shows the simulated electrostatic potential and electric field maps at $V_{back}=V_{dpl}$ in a cross section of a 3-strip domain with all the n-wells at $V_{nwell}=0.8$ V. Electric field lines are plotted on top of both the electrostatic potential and the electric field maps. Figure 5: Electrostatic potential map (left) and electric field map (right) for a group of three ARCADIA microstrip sensors at $V_{back}=V_{dpl}$. The electric field lines are plotted on top of both maps. #### 2.2.2 Punch-through If $V_{back}$ exceeds a certain value, a hole current flowing between the shallow p-doped backside region and the (deep) p-well exponentially increases. This condition is known as punch-through and the hole current is the punch- through current chu (1972). We define the voltage corresponding to the onset of the punch-through as $V_{pt}$. The onset of the punch-through currents can be observed from the blue IV curve in Figure 4 ($I_{pwell}$), which shows the absolute value of the current measured at the top p-well contacts as a function of $\|V_{back}\|$. The dip in the curve, corresponding to the point of sign inversion of the current, was defined as $V_{pt}$. The simulation domain includes three 50 $\mu$m long, 50 $\mu$m thick, 10 $\mu$m pitch microstrips. In this case, the n-wells are all biased at $V_{nwell}=0.8$ V, which is the default value. Sensor operation in low punch-through regime can be tolerated, whereas a too large punch-through current ought to be avoided, as it determines a substantial increase in the power consumption of the whole detector. For this reason, we chose $V_{back}$ = $V_{pt}$ as a safe reference sensor operating voltage; this is the operating point for all the results shown in the following, if not stated differently. The sensor power density can be defined as $pd=\frac{V_{back}\cdot(I_{pwell}+I_{nwell})}{A}$, where $I_{nwell}$ and $I_{pwell}$ are the currents flowing at the sensing node and at the top p-well contacts respectively, and $A$ is the top surface area of the simulated microstrip domain. In order to quantify the maximum acceptable backside bias voltage that limits the absorbed power density, the value $V_{pd}$ at which $pd=0.1$ mW/cm2 was extracted from the simulated IV curves (see Figure 4). Figure 6 shows the hole current density at two different $\|V_{back}\|$ > $\|V_{pt}\|$ in the simulation domain used to extract the $I_{pwell}$ curve of Figure 4. On the left, a backside voltage exceeding $V_{pt}$ by 1 V was chosen, while on the right $V_{back}$ was set to $V_{pd}$. An increase in the hole current density of several orders of magnitude can be observed below the deep p-wells and in the substrate. Figure 6: Hole current density in a simulated sensor domain including three microstrips in punch-through condition at two different $V_{back}$. Care had to be taken to ensure that $\|V_{dpl}\|$ < $\|V_{pt}\|$ in the designed sensors. In this way, full depletion is reached before the onset of the punch- through. Moreover, the voltage operating range between $V_{dpl}$ and $V_{pt}$, defined as $\Delta V_{op}=\|V_{pt}-V_{dpl}\|$, should be large enough to ensure safe operation in full depletion before the onset of the punch-through even if deviations from the simulated design occur in the sensor fabrication process. #### 2.2.3 Leakage current The same sensor domain and n-well voltage configuration used for the extraction of $V_{pt}$ was also used to evaluate the sensor leakage current $I_{leak}$. The leakage current is defined as the current flowing at the collection nodes in full depletion and in absence of external stimuli, such as particles or radiation. The leakage current as a function of the backside bias voltage is shown in Figure 4 as a green curve ($I_{leak}$). In the example shown in Figure 4, a value of 10 fA was extracted for $I_{leak}$ at $V_{back}=V_{pt}$. #### 2.2.4 Sensor capacitance The sensor CV curve was simulated through AC simulations with a frequency of 10 kHz using the same sensor domain employed for $V_{pt}$ and $I_{leak}$ evaluation, with $V_{nwell}=0.8$ V. The major contribution to the sensor capacitance $C_{sens}$, which is the input capacitance seen by the DC-coupled front-end electronics, originates from the lateral capacitance between the collection n-well and the surrounding p-wells. It is thus important to minimize this contribution by a careful selection of the distance between the edge of the collection n-well and the p-wells; we call this distance "gap" (see Figure 1). An example CV curve is shown in red in Figure 4, with the capacitance per unit length considered. In the example of Figure 4, a value of about 0.33 fF/$\mu$m was obtained at $V_{back}=V_{pt}$. It has to be mentioned that in these sensors the depletion voltage does not necessarily correspond to the voltage of minimum capacitance. The reason for this is the presence of the epitaxial layer, which is located far from the backside pn-junction and has a lower resistivity than the substrate. Therefore, the depletion of the epitaxial layer begins after the depletion of the substrate and progresses more slowly with voltage. Full depletion of the whole sensor, including the epitaxial layer, and minimum capacitance are only reached at $\|V_{back}\|>\|V_{dpl}\|$. From this point, both capacitance and leakage current values will be intended at $V_{back}=V_{pt}$. A central focus of the layout optimisation was the minimisation of the sensor capacitance. In fact, low input capacitance to the DC-coupled CMOS readout electronics allows for low-noise readout, low analog power pernegger (2017) and, in particular, SNR maximisation. Large input capacitance worsens the noise levels and the speed of the front-end electronics wang (2017). #### 2.2.5 Surface radiation damage In the simulation campaign performed to study the properties of MAMS, a silicon dioxide (SiO2) layer was added on the top-side of the sensor. In addition to this, surface damage was modeled to evaluate the effects of Total Ionising Dose (TID) on the sensor electrical properties. The impact of surface radiation damage was modeled following the AIDA-2020-D7.4 report passeri (2019). The model introduces fixed positive oxide charges and band-gap acceptor/donor defect levels (trap states) at the Si-SiO2 interface. The concentrations of oxide charges and defect levels start from a fixed value before irradiation (i.e. with the only inclusion of the SiO2 surface layer, at $dose=0$) and increase with the dose provided to the sensors. The dependence of the oxide charge density $Q_{ox}$ [charges $\cdot$ cm-2], of the acceptor integrated interface trap state density $N_{int}^{acc}$ [cm-2] and of the donor integrated interface trap state density $N_{int}^{don}$ [cm-2] on the dose is shown in Figure 7. Pre-irradiation values, shown as dotted horizontal lines in Figure 7, are $Q_{ox}=6.5\cdot 10^{10}$ charges $\cdot$ cm-2, $N_{int}^{acc}=2.0\cdot 10^{9}$ cm-2 and $N_{int}^{don}=2.0\cdot 10^{9}$ cm-2. Figure 7: Dependence of the oxide charge density $Q_{ox}$, acceptor integrated interface trap state density $N_{int}^{acc}$ and donor integrated interface trap state density $N_{int}^{don}$ on the dose for the surface radiation damage model described in passeri (2019). Pre-irradiation values are shown as horizontal dotted lines. In the simulation campaign, the effects of the inclusion of the SiO2 layer and of the radiation damage on the leakage current, sensor capacitance, depletion voltage and punch-through voltage were investigated and will be discussed in Section 3. #### 2.2.6 Transient simulations TCAD transient simulations were run to study the sensor charge collection process in response to particles traversing the simulated microstrip domain. These simulations also let us identify the most relevant layout parameters to be optimised for improving the sensor performance in terms of fast and uniform charge collection irrespective of the particle incidence position. The transient simulations employ the Synopsys ® Sentaurus TCAD HeavyIon model, described in sentaurus (2018). The HeavyIon model gives an analytical description of the amount of charge generated within a 3D cylindrical distribution along the incident particle track. Two main parameters have to be passed to the HeavyIon model: the linear Energy Transfer (LET), defined as the average deposited charge per unit length, and the transverse size of the charge deposition volume generated around the particle trajectory. We chose the charge transverse distribution profile to be gaussian around the particle track. Figure 8: Best-case and worst-case scenarios considered in the TCAD transient simulations. The microstrips are represented as adjacent grey blocks and the particle traversing the domain is shown as an orange cylinder. The nomenclature used to identify the microstrips (from 1 to 5) is illustrated. Two extreme cases in terms of particle impact position were studied to evaluate the uniformity of charge collection time and charge collection efficiency. Particle trajectories perpendicular to the sensor surface were considered. In the best-case scenario, the particle impact point corresponds to the centre of a microstrip, which is the centre of a collection n-well. On the contrary, in the worst-case scenario, the particle traverses the sensor at the edge between two adjacent microstrips, i.e. in the middle of a p-well. In Figure 8 the two cases and the corresponding numbering of the strips are illustrated. This conventional strip nomenclature will be used in the following when referring to transient simulations. In order to save computational time, a reduced TCAD simulation domain that employs the symmetries was used. This reduced domain corresponds to a quarter of the full domain, with the particle incident in the corner of the domain instead of in the centre. An example for the best-case scenario is shown in Figure 9. The collected charge and current signals were then scaled to reproduce the full domain case, which includes nine or ten 100$\mu$m long microstrips in the best-case and worst-case scenario respectively (Figure 8). These numbers and size of strips guarantee that that the amount of deposited charge reaching the borders of the simulation domain is negligible. The correctness of this strategy was verified and confirmed by comparing the results of a simulation with a quarter domain and of a simulation with full domain. Figure 9: Example reduced TCAD domain used in transient simulations (best-case scenario). The microstrips are labelled following the nomenclature illustrated in Figure 8. A crossing particle is represented as an orange cylinder hitting the corner of the simulated reduced domain. An example of current signals $I_{nwell}(t)$ measured at the microstrip sensing nodes when a particle crosses the microstrip domain is shown in Figure 10 (left). We defined as charge collection efficiency for the i-th strip (CCEi) the integral of the current signal $I_{nwell,i}(t)$ extracted from the i-th strip and normalised at the total charge $Q_{tot}$ deposited in the sensor by the particle, according to the formula $CCE_{i}(t)=\frac{\int_{0}^{t}I_{nwell,i}(t\textquoteright)\,dt\textquoteright}{Q_{tot}}=\frac{\int_{0}^{t}I_{nwell,i}(t\textquoteright)\,dt\textquoteright}{LET\cdot d_{Si}}$ (1) where $d_{Si}$ is the sensor thickness. The total charge collection efficiency CCE for the whole simulated domain is defined as $CCE(t)=\sum_{i=1}^{N_{strips}}CCE_{i}(t)$ (2) where $N_{strips}$ is the total number of strips in the simulated domain. The total CCE at the end of the charge collection process (i.e. at $t=t_{end}=30$ ns, which was observed to be large enough for complete charge collection) has to be equal to 100% in the absence of recombination: $CCE(t=t_{max})=100\%$ (3) The CCEi as a function of time is shown in Figure 10, on the right, for strip number 1. The times needed for collecting the 95% and 99% of the total deposited charge were evaluated and referred to as $t_{95}$ and $t_{99}$, respectively. These values were compared for different design options and used to select the layouts of the fastest sensor flavours. The spatial mesh of the transient simulations was forced to be finer around the particle trajectory to more accurately simulate the charge deposition and the drift of electrons and holes from their generation points along the particle track towards the electrodes. Additionally, the time step of the transient simulations was fine tuned to guarantee the necessary accuracy while keeping the computational time requirement economical. We observed that these adjustments prevented the simulations from giving unphysical results. Figure 10: Simulated current signals (left) and corresponding charge collection efficiency CCEi (right) in the best-case and worst-case scenarios for strip number 1 in an example 50 $\mu$m thick microstrip domain. A particle track with an LET of $1.28\cdot 10^{-5}$ pC/$\mu$m was simulated. ### 2.3 Determination of the LET for heavy nuclei Since MAMS are an interesting candidate for tracking detectors in space applications, the charge collection was studied not only for minimum ionising particles (MIPs) but also for heavy nuclei of interest for in-orbit astroparticle experiments. The LET values of carbon and oxygen ions were studied in Geant4 (version 10.6 patch 01) simulations, and the typically used olive (2014) LET of 80 electron-hole (e-h) pairs per $\mu$m, or $1.28\cdot 10^{-5}$ pC per $\mu$m in silicon for MIPs could be reproduced. The Geant4 simulation setup included a 50 $\mu$m thick silicon layer immersed in air and with a transverse size of 1 $\times$ 1 cm2. The particle gun was positioned 15 cm in front of the centre of the silicon layer. The G4EmPenelopePhysics physics list was used to model the electromagnetic processes and the necessary precision on the energy deposited within the silicon was achieved with a maximum step size of 1 $\mu$m muonsilicon (2020). The LETs for carbon (C12+) and oxygen (O16+) ions at their minimum ionisation were computed from their most probable energy loss (i.e. the most probable value of the straggling or Landau functions tanabashi (2018); bichsel (2006)). Figure 11 shows the LET as a function of the particle energy obtained for C and O ions traversing 50 $\mu$m of silicon. The energies $E_{min}$ at which C and O ions are at the minimum of ionisation were found to be 35 GeV and 60 GeV, respectively. The corresponding LETs are $45.6\cdot 10^{-5}$ pC/$\mu$m and $83.0\cdot 10^{-5}$ pC/$\mu$m, which results in 36 and 65 times the MIP value. This is consistent with the expected scaling from the Bethe-Bloch formula. We were especially interested in studying the charge sharing among the microstrips surrounding the particle impact point and the charge collection time at different LETs. This will be reported and discussed in Section 3. Figure 11: Dependence of the LET on the energy of carbon ions (C12+, blue) and oxygen ions (O16+, orange) incident on 50 $\mu$m thick silicon. The LET values were evaluated through Geant4 simulations. The red vertical lines indicate the minimum ionisation energies for the two particle species. ## 3 Results and discussions In this section, the results of the TCAD simulation campaign will be presented. Their implications will be discussed and their connections to the design objectives will be highlighted. As mentioned in Section 1, the main targets of the FD-MAMS design were the following. 1. 1. To enhance the spatial resolution. A very fine pitch of 10 $\mu$m was chosen to reach this goal. Intrinsic spatial resolution in case of digital readout would be equal to $\frac{pitch}{\sqrt{12}}=\frac{10\,\mu m}{\sqrt{12}}\simeq 2.9$ $\mu$m, which can be further improved thanks to charge sharing and with an analog readout. 2. 2. To minimise the sensor capacitance $C_{sens}$ at $V_{back}=V_{pt}$. A low sensor capacitance is particularly important to keep low electronic noise and, consequently, to maximise the SNR. 3. 3. To obtain fast and uniform charge collection, irrespective of the particle incidence position. This will enhance the sensor timing capabilities and will reduce the dead-time between successive particle detections. For reasons of space available for MAMS in the first ARCADIA engineering run, only a few sensor flavours could be included. Hence, a simulation campaign was needed to identify the best performing sensor layouts. The deep p-well, when present, was kept the same size as the p-well. The expression "p-well and deep p-well" will be contracted and referred to as "(deep) p-well". In the legends of the figures, the abbreviation "dpw" will be used for deep p-well. ### 3.1 SiO2 layer and surface damage A first group of TCAD simulation studies was aimed at investigating the effects of the SiO2 layer and of surface TID damage on the FD-MAMS characteristics. The model that we employed was presented in Paragraph 2.2.5. As can be seen from Figure 12, for one of the selected 50 $\mu$m thick microstrip layouts, the inclusion of the SiO2 layer with a minimum concentration of traps and oxide charges ($dose=0$) determines a small increase of about 5% in the leakage current $I_{leak}$ from 20.8 fA to 22.0 fA. The sensor capacitance $C_{sens}$ is strongly affected by the inclusion of the SiO2 layer, as it increases by 31% from 0.26 fF/$\mu$m to 0.34 fF/$\mu$m. Both $I_{leak}$ and $C_{sens}$ are found to rise with increasing dose. The minimum dose that we considered is 50 krad, as the model is not validated for lower doses passeri (2019). Figure 13, instead, shows the effect of the SiO2 layer and of the TID on $V_{dpl}$ and on $V_{pt}$. The effect of the dose on these two values is smaller than in the case of $I_{leak}$ and $C_{sens}$. Furthermore, $V_{dpl}$ and $V_{pt}$ are influenced by the dose in opposite directions, which results in a slight increase in the operating range $\Delta$Vop with increasing dose. Figure 12: Leakage current $I_{leak}$ (green) and sensor capacitance $C_{sens}$ (red) as a function of the total ionising dose for a 50 $\mu$m thick microstrip sensor. The values obtained in simulations with and without the SiO2 layer in the absence of irradiation are shown as horizontal lines and referred to as "dose = 0" and "no SiO2 layer" respectively. Figure 13: Depletion voltage $V_{dpl}$ (blue) and punch-through voltage $V_{pt}$ (orange) as a function of the total ionising dose for a 50 $\mu$m thick microstrip sensor. The values obtained in simulations with and without the silicon dioxide layer in the absence of irradiation are shown as horizontal lines. #### 3.1.1 Effect on sensor capacitance The reason for the significant capacitance increase even after the simple inclusion of the SiO2 layer was found to be due to the introduction of positive oxide charges at the Si-SiO2 interface neubueser (2020). In fact, the model that we adopted foresees a significant positive oxide charge concentration $Q_{ox}$ = 6.5 $\cdot$ $10^{10}$ charges/cm-2 already at dose = 0. These positive oxide charges attract free electrons from the n-type silicon epitaxial layer towards the Si-SiO2 interface in the gap and determine an increase in the electron concentration around the heavily n-doped collection well, as illustrated in Figures 14 and 15. This electron accumulation behaves as an extension of the collection n-well. Figure 14: Schematic illustration of the electron accumulation in the gap between the collection n-well and the surrounding p-wells due to the positive oxide charges introduced at the Si-SiO2 interface. Figure 15: Electron density in an example microstrip simulation domain without (left) and with (right) the SiO2 layer on top of the sensors. ### 3.2 Capacitance minimisation The sizes of both the collection n-well and of the gap were found to contribute to $C_{sens}$. Therefore, both n-well and (deep) p-well sizes were adjusted to find the optimal layout for $C_{sens}$ minimisation. It was observed the inclusion of the SiO2 layer influences $C_{sens}$ in different ways for different gap sizes. Hence, $C_{sens}$ with and without the SiO2 layer was evaluated. Figure 16 shows the trend of $C_{sens}$ as a function of the gap size for 50 $\mu$m thick microstrips. The different sensor thicknesses considered (50, 100 and 300 $\mu$m) were found not to influence the sensor capacitance. Both the case with fixed minimum-size n-well and variable (deep) p-well (blue curves) and the case with fixed minimum-size (deep) p-well and variable n-well (orange curve) were studied. The dash-dotted lines refer to simulations without the surface SiO2 layer, whereas solid lines to the case with SiO2 layer included with minimal oxide charge and trap concentration. The reason for which smaller gaps with fixed n-wells could not be investigated is referred to as channel choking, a condition that inhibits sensor operation; this condition is explained in Section 3.3. The vertical grey band in Figure 16 and in the following ones corresponds to the forbidden region due to the constraints on n-well and (deep) p-well minimum sizes imposed by the fabrication process. The leftmost limit of the grey band is still permitted. Variations of n-well and of (deep) p-well size do not lead to the same $C_{sens}$ for the same gap size. A fixed n-well size with SiO2 layer included shows a trend that is not monotonic, but has a minimum at slightly less than 0.34 fF/$\mu$m. This effect is caused by the electron accumulation in the gap at the Si-SiO2 interface. However, the difference in $C_{sens}$ between the minimum-capacitance option and the sensor layout at the edge of the forbidden region is lower than 2%. There was, as expected, no benefit found from having large n-wells. The sensor capacitance increases with the n-well size, as can be seen from the blue curve in Figure 16. Therefore, we chose the best layout for minimum $C_{sens}$ to have the smallest possible n-well size and sufficiently small (deep) p-well. Figure 16: $C_{sens}$ as a function of the gap size for different sensor layout configuration. The vertical grey band is the forbidden region due to fabrication constraints; its leftmost limit is still permitted. Figure 17 compares the sensor capacitance for layouts with deep p-well (orange curve) and without deep p-well (green curve). All the sensor flavours feature the minimum n-well size permitted by the fabrication process. On the one hand, removing the deep p-well could help in further reducing the sensor capacitance. On the other hand, this choice would strongly affect the sensor bias voltage operating range, as discussed in the following section. Figure 17: Sensor capacitance $C_{sens}$ as a function of the gap size for different sensor layout configurations with and without the deep p-well. ### 3.3 Reference and operating voltages We found the influence of the n-well size on the operating voltages to be negligible compared to the effect of the (deep) p-well size. Therefore, for the sake of capacitance minimisation, we fixed the n-well size at the smallest possible value. With this assumption, Figure 18 presents the effect of the (deep) p-well size effect on $V_{dpl}$ and on $V_{pt}$ for 50 $\mu$m thick sensors. Both the cases with (orange curves) and without deep p-well (green curves) were considered and compared. The voltage values are reported for the case of dose = 0. In all the layouts considered in Figure 18, the onset of the punch through happens at voltages sufficiently larger than the depletion voltage. Outside of the forbidden region (grey band), the operating range $\Delta V_{op}$ is always between 4.2 V and 6.2 V, or between the 23% and the 41% of $V_{dpl}$. This is a sufficiently large operating range for safe sensor operation, even in the hypothesis of possible doping inhomogeneities among adjacent microstrips or slight deviations from the doping design values. Similar observations on $\Delta V_{op}$ have been made for 100 $\mu$m thick and 300 $\mu$m thick sensors. Figure 18: Sensor depletion voltage $V_{dpl}$ and punch-through voltage $V_{pt}$ as a function of the gap size for different sensor layout configurations. The orange region indicates the forbidden region due to the observed channel choking. As a general trend, it can be observed in Figure 18 that smaller (deep) p-wells result in larger $V_{dpl}$ and $V_{pt}$. This can be interpreted as follows. Large p-doped surfaces below the (deep) p-wells create wider pn- junctions with the n-doped epitaxial layer, thus facilitating the depletion of the underlying epitaxial layer at lower voltages. On the other hand, large (deep) p-wells also lower the potential barrier that prevents the direct flow of holes towards the substrate. This results in the earlier onset of the punch through hole current between the (deep) p-wells and the backside p+ region. Sensors without the deep p-well showed higher reference voltages. In fact, the presence of a deep p-well reduces the epitaxial layer thickness below the p-wells, thus requiring a lower voltage to achieve both full depletion and the onset of punch-through currents. Finally, for sensors with too large deep p-well, a phenomenon that we defined as channel choking was observed. This consists in the closure of the conductive channel below the collection n-well due to the lateral merging of the closely adjacent depletion regions formed at the junctions between the deep p-wells and the n-epitaxial layer. In this situation, in the simulations performed to extract $V_{dpl}$, no current flows among the n-wells at low values of $V_{back}$, even though the space charge region of the backside junction has not reached the surface yet. In this condition, the $I_{nwell,unbalanced}$ curve, that corresponds to the orange curve shown in Figure 4, appears flat and no $V_{dpl}$ can be extracted. This means that the n-wells are already isolated from one another at $V_{back}=0$ V and that the process of charge collection, which generates the current $I_{nwell}$ measured at the sensing node, is inhibited by the strong potential barrier present below the n-wells. No channel choking was observed for sensor layouts without the deep p-well. For completeness, Figure 19 (left) illustrates the dependence of $V_{pt}$ and $V_{dpl}$ on the sensor thickness for the sensor layout with minimum sizes for the n-well and for the (deep) p-well. The trend is linear over a wide range of thicknesses, both with and without the deep p-well. Also the operating voltage $\Delta V_{op}=V_{pt}-V_{dpl}$ linearly increases with the sensor thickness, as shown in Figure 19 (right). The sensor thickness investigated was extended down to 20 $\mu$m, well below the smallest thickness (i.e. 50 $\mu$m) of the sensors that will be produced in the first ARCADIA engineering run. The reason for this will become clear in Paragraph 3.5.1, as the study of very thin sensors was functional for enhancing the speed of the charge collection process and, consequently, for improving the sensor timing performance. Figure 19: Dependence of $V_{dpl}$ and $V_{pt}$ (left) and of the operating voltage range $\Delta V_{op}$ (right) on the sensor thickness. The voltage $V_{pd}$ at which the power density is 0.1 mW/cm2 was found to be about 4-5 V above $V_{pt}$ for 50 $\mu$m thick microstrips, 7-8 V for 100 $\mu$m thick microstrips and 18-20 V for 300 $\mu$m thick microstrips when the deep p-well was included. ### 3.4 Effects of Vnwell The $V_{nwell}$ voltage was varied with the aim of finding possible improvements in the sensor performances. The results are shown in Figure 20 (left), where the vertical red line indicates the default value of 0.8 V. A minimum $V_{nwell}$ of about 0.5 V is necessary to satisfy the condition $\|V_{pt}\|>\|V_{dpl}\|$. Moreover, an increase in $V_{nwell}$ has several interesting effects. First of all, it allows the sensor full depletion to be reached at lower (in absolute value) backside voltages. Secondly, it also shifts the onset of the punch through towards larger $\|V_{back}\|$, thus increasing the operating range $\Delta V_{op}$. Finally, as shown in Figure 20 (right), larger $V_{nwell}$ implies lower sensor capacitance. Figure 20: $V_{dpl}$ and $V_{pt}$ (left) and sensor capacitance (right) as a function of $V_{nwell}$. The vertical red line indicates the default value of $V_{nwell}=0.8$ V. ### 3.5 Charge collection studies As described in Paragraph 2.2.6, TCAD transient simulations were used to study the charge collection dynamics. In order to select the layouts with the optimal performance in terms of fast and uniform charge collection, the effect of the (deep) p-well size on the charge collection time at $V_{back}=V_{pt}$ was evaluated. The time $t_{95}$ needed to collect 95% of the total charge deposited in the simulated sensor domain is plotted in Figure 21 for 50 $\mu$m thick sensors and LET = $1.28\cdot 10^{-5}$ pC/$\mu$m (1 MIP) as a function of the gap size and with fixed minimum n-well size. Figure 21: $t_{95}$ as a function of the gap size for best-case and worst-case scenarios. Microstrips with large gaps, hence small (deep) p-wells, are to be preferred for fast charge collection. The reason for this is a higher $\|V_{pt}\|$, which enables sensor operation at a larger $\|V_{back}\|$. The consequent stronger electric field in the sensor results in higher charge velocity in the silicon substrate. For the same reason, microstrip sensors without deep p-well revealed a significantly faster charge collection in both the best-case and the worst-case scenario. Flavours with small (deep) p-wells also show very uniform charge collection for different particle incidence positions. The difference in $t_{95}$ for the best-case and worst-case scenarios is below 0.1 ns for the fastest permitted options. This result is also achieved thanks to the fine microstrip pitch of 10 $\mu$m. The channel choking, as described in Section 3.3, limits the deep p-well size as the potential barrier below the sensing node slows down the electron collection. This problem, as shown in Figure 21, can be avoided by removing the deep p-well. Figure 22 demonstrates that the proposed MAMS guarantee fast sensor response also under heavily ionising particles. The charge collection time is only weakly proportional to the charge deposited by the incident particle within an LET range of [1.28; 128] $\cdot$ 10-5 pC/$\mu$m. A 50 $\mu$m thick sensor was considered in Figure 22, and the LET values corresponding to 1 MIP, carbon (C) ion and oxygen (O) ion at their minimum of ionisation are highlighted as vertical green lines. Moreover, $t_{99}$ is added to show that the time needed for complete charge collection is only slightly larger than $t_{95}$, due to a small fraction of charge collected by the strips adjacent to the central one. However, $t_{95}$ and $t_{99}$ were never found to exceed 2 ns and 3 ns respectively in 50 $\mu$m thick sensors. Figure 22: $t_{95}$ (blue) and $t_{99}$ (red) as a function of the LET for best-case and worst-case scenarios. #### 3.5.1 Further enhancements for fast timing performance As we discussed in Section 3.5, the first strategy for improving the timing performance of the proposed microstrip sensors is to remove the deep p-well in order to obtain larger $\|V_{pt}\|$. However, we also investigated other ways to increase $\|V_{pt}\|$ and to speed up the charge collection. In particular, as shown in Section 3.4, a larger $V_{nwell}$ is capable of shifting the onset of the punch-through current towards larger $\|V_{back}\|$. Therefore, we explored the effects of $V_{nwell}$ on the charge collection time. In a strip readout system, timing information can be retrieved only from the strips collecting most of the charge (i.e. strip(s) number 1, following the nomenclature of Figure 8), as they provide a signal with sufficiently large SNR. Therefore, in order to study the sensor timing performance and after verifying through $t_{95}$ that the total deposited charge is quickly collected in the whole simulation domain, we considered the time $t_{95}^{central}$ needed to collect 95% of the charge in the central strip(s). Figure 23 shows the dependence of $t_{95}^{central}$ at $V_{back}=V_{pt}$ and with LET = $1.28\cdot 10^{-5}$ pC/$\mu$m on the voltage applied to the sensing node. A 50 $\mu$m thick sensor with a layout optimised for fast charge collection was considered. A significant improvement could be reached at larger $V_{nwell}$. For the option without deep p-well and at $V_{nwell}=3$ V, $t_{95}^{central}$ is 0.84 ns in the best-case and 0.94 ns in the worst-case scenario. If we assume an electron drift saturation velocity of $\sim 1\cdot 10^{7}$ cm/s in silicon at a temperature of 300 K canali (1975), the minimum drift time for electrons that have to cover a 50 $\mu$m distance is 0.5 ns. This explains the saturation observed in Figure 23 and demonstrates the fast charge collection and the promising timing capabilities of the proposed MAMS. Figure 23: $t_{95}^{central}$ as a function of the voltage $V_{nwell}$ applied to the sensing node for best-case and worst-case scenarios. The vertical red line indicates the default value of $V_{nwell}=0.8$ V. A way to further reduce the collection time is to explore thinner sensors. Figure 24 demonstrates that the charge collection time $t_{95}^{central}$ is proportional to the sensor thickness. For these simulations, $V_{nwell}$ was set to the 0.8 V and a 1 MIP LET was considered. Even at thicknesses as large as 300 $\mu$m, $t_{95}^{central}$ does not exceed 6 ns. In the best-case scenario without the deep p-well, reducing the sensor thickness from 50 $\mu$m to 40 $\mu$m, 30 $\mu$m and 20 $\mu$m results in a decrease in $t_{95}^{central}$ of 15%, 33% and 50%, respectively. Analogous proportionality was observed for $t_{95}$. Therefore, for future production runs, thinner sensors could be considered for the enhancement of the timing performance. Figure 24: $t_{95}^{central}$ as a function of the sensor thickness for best- case and worst-case scenarios. ### 3.6 Charge sharing A set of TCAD simulations was dedicated to study the charge sharing among adjacent microstrips when particles with different LETs traverse the sensor. Charge sharing is relevant for improving the spatial resolution, especially with analog readout, and is enhanced by fine microstrip pitches and large sensor thicknesses. On the contrary, it is reduced at higher $V_{back}$ for a fixed sensor thickness. In Figure 25, the case of a 300 $\mu$m thick sensor at $V_{back}=V_{pt}$ is presented for the best-case scenario. The total charge collected by each strip (identified using the nomenclature of Figure 8) is plotted versus the LET. The black horizontal line indicates a possible charge threshold corresponding to 10% of a MIP at the single strip level. A comparison with the sensors that will be produced in the first ARCADIA engineering run will allow deeper investigation on the charge sharing, a fine tuning of the simulations and studies aimed at evaluating the spatial resolution of 10 $\mu$m pitch MAMS. Figure 25: Charge sharing among adjacent microstrips. The total charge collected by strips 1 to 5 (following the nomenclature illustrated in Figure 8) is shown as a function of the LET. ## 4 Conclusions In this work, we presented detailed TCAD simulations of CMOS-based FD-MAMS, which may find use for tracking and timing in particle and nuclear physics, space and medical applications. The results of the TCAD simulation campaign, performed to design the 10 $\mu$m pitch FD-MAMS, demonstrate their very fast and uniform charge collection, which encourages their practicality for various applications, even under heavily ionizing particles. The effect of surface ionizing radiation damage was investigated, and the layout parameters were optimized to achieve a minimum capacitance, beneficial for electronic noise reduction. The possibility to operate the sensor in full depletion and at low- power density (i.e. before the onset of the punch through current) was verified in the simulations. A preference for small collection diodes and small (deep) p-wells emerged for obtaining lower capacitance and faster sensor response. Additionally, these simulations confirmed the possibility of monolithically integrating readout architectures in the inter-strip regions for strips of 10 $\mu$m pitch. The first FD-MAMS samples will be produced in the upcoming ARCADIA engineering production run at the beginning of 2021 and will allow the simulation results to be compared with experimental data from electrical characterisation, laser and beam irradiation tests. The promising results of the first simulation campaign on FD-MAMS will translate into further R&D activities to enhance the sensor performance in terms of low capacitance and high timing and spatial resolution. yes ## Appendix A Expected effects from epitaxial layer thicknesses Possible variations in the epitaxial layer thickness of [-15%; +30%] communicated by the foundry with respect to the reference value induced us to investigate their effect on the operating parameters. While the sensor capacitance was observed not to be influenced, both $V_{dpl}$ and $V_{pt}$ showed a linear dependence on the epitaxial layer thickness. This behaviour is presented in Figure 26. Figure 26: $V_{dpl}$ and $V_{pt}$ as a function of the epitaxial layer thickness, expressed as percentage variation with respect to the reference thickness. ###### Acknowledgements. The research activity presented in this article has been carried out in the framework of the ARCADIA experiment funded by the Istituto Nazionale di Fisica Nucleare (INFN), CSN5. The activity has also been supported by the project "Dipartimento di Eccellenza", Physics Department of the University of Torino (Dipartimento di Fisica - Università degli Studi di Torino), Italy, funded by MUR. Data curation, Lorenzo de Cilladi; Formal analysis, Lorenzo de Cilladi; Investigation, Lorenzo de Cilladi; Supervision, Coralie Neubüser and Lucio Pancheri; Writing – original draft, Lorenzo de Cilladi; Writing – review & editing, Thomas Corradino, Gian-Franco Dalla Betta, Coralie Neubüser and Lucio Pancheri. The authors declare no conflict of interest. References ## References garcia (2018) Garcia-Sciveres, M., and Wermes, N., A review of advances in pixel detectors for experiments with high rate and radiation. Rep. Prog. 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# Recurrent Sums and Partition Identities Roudy El Haddad Université La Sagesse, Faculté de génie, Polytech ###### Abstract Sums of the form $\sum_{N_{m}=q}^{n}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{(m);N_{m}}\cdots a_{(1);N_{1}}}}$ where the $a_{(k);N_{k}}$’s are same or distinct sequences appear quite often in mathematics. We will refer to them as recurrent sums. In this paper, we introduce a variety of formulas to help manipulate and work with this type of sums. We begin by developing variation formulas that allow the variation of a recurrent sum of order $m$ to be expressed in terms of lower order recurrent sums. We then proceed to derive theorems (which we will call inversion formulas) which show how to interchange the order of summation in a multitude of ways. Later, we introduce a set of new partition identities in order to then prove a reduction theorem which permits the expression of a recurrent sum in terms of a combination of non-recurrent sums. Finally, we apply this reduction theorem to a recurrent form of two famous types of sums: The $p$-series and the sum of powers. ###### Keywords. Recurrent sums, Partitions, Stirling numbers of the first kind, Bell polynomials, Multiple harmonic series, Riemann zeta function, Bernoulli numbers, Faulhaber formula. MSC 2020: primary 11P84 secondary 11B73, 11M32, 05A18 ## 1 Introduction and Notation The harmonic series was first studied and proven to diverge in the 14th century by Nicole Oresme [32]. Later, in the 17th century, new proofs for this divergence were provided by Pietro Mengoli [29], Johann Bernoulli [5], and Jacob Bernoulli [3, 4]. However, a more general form of this series does converge. Euler was the first to study such sums of the form $\zeta(s)=\sum_{n=1}^{\infty}{\frac{1}{n^{s}}}$ where $s$ is a real number. In the famous Basel problem, Euler proved that $\zeta(2)=\frac{\pi^{2}}{6}$ (see [12, 13, 15]. Fourteen additional proofs can be found in [11]). He later provided a general formula for this zeta function for positive even values of $s$. Euler’s definition was then extended to a complex variable $s$ by Riemann in his 1859 article “On the Number of Primes Less Than a Given Magnitude”. More recently, the multiple harmonic series, an even more general form of the zeta function, has been introduced and studied. Note that Euler was the first to study these multiple harmonic series for length 2 in [14]. A multiple harmonic series (MHS) or multiple zeta values (MZV) is defined as: $\zeta(s_{1},s_{2},\ldots,s_{k})=\sum_{1\leq N_{1}<N_{2}<\cdots<N_{k}}{\frac{1}{N_{1}^{s_{1}}N_{2}^{s_{2}}\cdots N_{k}^{s_{k}}}}.$ A very important variant of the MHS (see [27, 25, 30]) often referred to as multiple zeta star values MZSV or multiple harmonic star series MHSS (or simply multiple zeta values) is defined by: $\zeta^{\star}(s_{1},s_{2},\ldots,s_{k})=\sum_{1\leq N_{1}\leq N_{2}\leq\cdots\leq N_{k}}{\frac{1}{N_{1}^{s_{1}}N_{2}^{s_{2}}\cdots N_{k}^{s_{k}}}}.$ This variant of the multiple harmonic series is directly related to the Riemann zeta function $\zeta(s)$ [23, 18]. Additionally, it is involved in a variety of sums and series including the Arakawa–Kaneko zeta function [37] and Euler sums. Such sums have tremendous importance in number theory. They have been of interest to mathematicians for a long time and have been systematically studied since the 1990s with the work of Hoffman [23, 24] and Zagier [38]. However, their importance is not limited to Number Theory. In fact, such sums/series have appeared in physics even before the phrase “multiple zeta values” had been coined. As an example, the number $\zeta(\overline{6},\overline{2})$ appeared in the quantum field theory literature in 1986 [8]. They play a major role in the connection of knot theory with quantum field theory [9, 26]. MZVs and MZSVs became even more important after they became needed for higher order calculations in quantum electrodynamics (QED) and quantum chromodynamics (QCD) [7, 6]. These sums are a particular case of what we called recurrent sums as they are of the form $\sum_{1\leq N_{1}\leq\cdots\leq N_{m}\leq n}{a_{(m);N_{m}}\cdots a_{(1);N_{1}}}$ with $a_{(i);N_{i}}=\frac{1}{N_{i}^{s_{i}}}$ for all $i$. The particular case has been extensively studied while the general case received much less interest. Although there are hundreds if not thousands of formulae to help in the study of multiple harmonic star sums and multiple zeta star values, barely any formulae can be found for its general counterpart. In this article, we are interested in studying this more general form which is expressed as follows: $\sum_{1\leq N_{1}\leq\cdots\leq N_{m}\leq n}{a_{(m);N_{m}}\cdots a_{(1);N_{1}}}.$ We will also consider the particular case where all sequences are the same: $\sum_{1\leq N_{1}\leq\cdots\leq N_{m}\leq n}{a_{N_{m}}\cdots a_{N_{1}}}.$ This structure of sums appears in a variety of areas of mathematics. The objective is to develop formulae to improve and facilitate the way we work with recurrent sums. This includes deriving formulae to calculate the variation of such sums, formulae to interchange the order of summation as well as formulae to represent recurrent sums in terms of a combination of non- recurrent sums. Note that this type of sums is intimately related to partitions as they appear in the representation of recurrent sums as a combination of simple non-recurrent sums. Therefore, this article will also focus on partition identities that are needed to prove the previously stated theorems as well as the ones that can be derived from these same theorems. Among these partition identities that can be found through these theorems, a definition of binomial coefficients in terms of a sum over partitions will be presented. Similarly, we produce some identities involving special sums, over partitions, of Bernoulli numbers. Furthermore, we are also interested in applying the formulae develop for the general case to some particular cases. First, we will apply our results to the multiple sums of powers in order to generalize Faulhaber’s formula. Then, we will go back to the most famous particular case which is the MZSV and show how our results on the general case can improve in this case. A particularly beautiful identity that we will present is the following which relates the recurrent sum of $\frac{1}{N^{2}}$ to the zeta function for positive even values: $\sum_{N_{m}=1}^{\infty}{\cdots\sum_{N_{1}=1}^{N_{2}}{\frac{1}{N_{m}^{2}\cdots N_{1}^{2}}}}=\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!i^{y_{k,i}}}\left(\zeta(2i)\right)^{y_{k,i}}}}=\left(2-\frac{1}{2^{2(m-1)}}\right)\zeta(2m).$ Although this paper focuses on the generalized version of the multiple zeta star values, the multiple zeta values itself is a particular form of a type of sums presented in [20] and which is closely related to the recurrent sums by the relations also presented in that cited article. The main theorems of this paper have potential applications such as the following: Surprisingly, this form appears in the general formula for the $n$-th integral of $x^{m}(\ln x)^{m^{\prime}}$. In the unpublished paper [19], the relations presented in this paper will be used to derive and prove this general formula for the $n$-th integral of $x^{m}(\ln x)^{m^{\prime}}$. In paper [20], the partition identities here presented are combined with additional partition identities in order to produce identities for odd and even partitions. Let us now introduce some notation in order to facilitate the representation of such sums in this paper: For any $m,q,n\in\mathbb{N}$ where $n\geq q$ and for any set of sequences $a_{(1);N_{1}},\cdots,a_{(m);N_{m}}$ defined in the interval $[q,n]$, let $R_{m,q,n}(a_{(1);N_{1}},\cdots,a_{(m);N_{m}})$ represent the general recurrent sum of order $m$ for the sequences $a_{(1);N_{1}},\cdots,a_{(m);N_{m}}$ with lower an upper bounds respectively $q$ and $n$. For simplicity, however, we will denote it simply as $R_{m,q,n}$. $\begin{split}R_{m,q,n}&=\sum_{N_{m}=q}^{n}{a_{(m);N_{m}}\cdots\sum_{N_{2}=q}^{N_{3}}{a_{(2);N_{2}}\sum_{N_{1}=q}^{N_{2}}{a_{(1);N_{1}}}}}\\\ &=\sum_{N_{m}=q}^{n}{\cdots\sum_{N_{2}=q}^{N_{3}}{\sum_{N_{1}=q}^{N_{2}}{a_{(m);N_{m}}\cdots a_{(2);N_{2}}a_{(1);N_{1}}}}}\\\ &=\sum_{q\leq N_{1}\leq\cdots\leq N_{m}\leq n}{a_{(m);N_{m}}\cdots a_{(2);N_{2}}a_{(1);N_{1}}}.\\\ \end{split}$ (1) The most common case of a recurrent sum is that where all sequences are the same, $\begin{split}R_{m,q,n}(a_{N_{1}}\cdots a_{N_{m}})&=\sum_{N_{m}=q}^{n}{a_{N_{m}}\cdots\sum_{N_{2}=q}^{N_{3}}{a_{N_{2}}\sum_{N_{1}=q}^{N_{2}}{a_{N_{1}}}}}\\\ &=\sum_{N_{m}=q}^{n}{\cdots\sum_{N_{2}=q}^{N_{3}}{\sum_{N_{1}=q}^{N_{2}}{a_{N_{m}}\cdots a_{N_{2}}a_{N_{1}}}}}\\\ &=\sum_{q\leq N_{1}\leq\cdots\leq N_{m}\leq n}{a_{N_{m}}\cdots a_{N_{2}}a_{N_{1}}}.\\\ \end{split}$ (2) For simplicity, we will denote it as $\hat{R}_{m,q,n}$. This type of sums is described as recurrent because they can also be expressed using the following recurrent form: $\begin{cases}R_{m,q,n}=\sum_{N_{m}=q}^{n}{a_{(m);N_{m}}R_{m-1,q,N_{m}}}\\\ R_{0,i,j}\,\,\,=1\,\,\forall i,j\in\mathbb{N}.\end{cases}$ (3) ###### Remark. A recurrent sum of order $0$ is always equal to $1$. It is not equivalent to an empty sum $($which is equal to $0)$. In this paper, this type of sums will be studied. In Section 2, formulas for the calculation of variation of these sums in terms of lower order recurrent sums will be presented. Then, in Section 3, inversion formulas will be presented, which will allow the interchange of the order of summation in such sums. Finally, in Section 4, we will present a reduction formula that allows the representation of a recurrent sum as a combination of simple (non- recurrent) sums. These relations will be, then, used to calculate certain special sums such as the recurrent harmonic sum and the recurrent equivalent to the Faulhaber formula. ## 2 Variation Formulas In this section, we will develop formulas to express the variation of a recurrent sum of order $m$ ($R_{m,q,n+1}-R_{m,q,n}$) in terms of lower order recurrent sums. Equivalently, these formulas can be used to express $R_{m,q,n+1}$ in terms of $R_{m,q,n}$ and lower order recurrent sums. ### 2.1 Simple expression We start by proving the most basic form for the variation formula as illustrated by the following Lemma. This is needed in order to prove the general form of this formula. ###### Lemma 2.1. For any $m,q,n\in\mathbb{N}$, we have that $R_{m+1,q,n+1}=a_{(m+1);n+1}R_{m,q,n+1}+R_{m+1,q,n}.$ ###### Proof. $\begin{split}R_{m+1,q,n+1}&=\sum_{N_{m+1}=q}^{n+1}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{(m+1);N_{m+1}}\cdots a_{(1);N_{1}}}}\\\ &=a_{(m+1);n+1}\sum_{N_{m}=q}^{n+1}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{(m);N_{m}}\cdots a_{(1);N_{1}}}}+\sum_{N_{m+1}=q}^{n}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{(m+1);N_{m+1}}\cdots a_{(1);N_{1}}}}\\\ &=a_{(m+1);n+1}R_{m,q,n+1}+R_{m+1,q,n}.\end{split}$ ∎ Now we apply the basic case from Lemma 2.1 to show the general variation formula that allows $R_{m,q,n+1}$ to be expressed in terms of $R_{m,q,n}$ and of recurrent sums of order going from $0$ to $(m-1)$. ###### Theorem 2.1. For any $m,q,n\in\mathbb{N}$ where $n\geq q$ and for any set of sequences $a_{(1);N_{1}},\cdots,a_{(m);N_{m}}$ defined in the interval $[q,n+1]$, we have that $\sum_{N_{m}=q}^{n+1}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{(m);N_{m}}\cdots a_{(1);N_{1}}}}=\sum_{k=0}^{m}{\left(\prod_{j=0}^{m-k-1}{a_{(m-j);n+1}}\right)\left(\sum_{N_{k}=q}^{n}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{(k);N_{k}}\cdots a_{(1);N_{1}}}}\right)}.$ Using the notation from Eq. (1), this theorem can be written as $R_{m,q,n+1}=\sum_{k=0}^{m}{\left(\prod_{j=0}^{m-k-1}{a_{(m-j);n+1}}\right)R_{k,q,n}}.$ ###### Proof. 1\. Base Case: verify true for $m=1$. $\begin{split}\sum_{k=0}^{1}{\left(\prod_{j=0}^{-k}{a_{(1-j);n+1}}\right)R_{k,q,n}}&=\left(\prod_{j=0}^{0}{a_{(1-j);n+1}}\right)R_{0,q,n}+\left(\prod_{j=0}^{-1}{a_{(1-j);n+1}}\right)R_{1,q,n}\\\ &=(a_{(1);n+1})(1)+(1)\left(\sum_{N_{1}=q}^{n}{a_{(1);N_{1}}}\right)\\\ &=R_{1,q,n+1}.\end{split}$ 2\. Induction hypothesis: assume the statement is true until $m$. $R_{m,q,n+1}=\sum_{k=0}^{m}{\left(\prod_{j=0}^{m-k-1}{a_{(m-j);n+1}}\right)R_{k,q,n}}.$ 3\. Induction step: we will show that this statement is true for ($m+1$). We have to show the following statement to be true: $R_{m+1,q,n+1}=\sum_{k=0}^{m+1}{\left(\prod_{j=0}^{m-k}{a_{(m+1-j);n+1}}\right)R_{k,q,n}}.$ From Lemma 2.1, $R_{m+1,q,n+1}=a_{(m+1);n+1}R_{m,q,n+1}+R_{m+1,q,n}.$ By applying the induction hypothesis, $\begin{split}R_{m+1,q,n+1}&=a_{(m+1);n+1}\sum_{k=0}^{m}{\left(\prod_{j=0}^{m-k-1}{a_{(m-j);n+1}}\right)R_{k,q,n}}+R_{m+1,q,n}\\\ &=a_{(m+1);n+1}\sum_{k=0}^{m}{\left(\prod_{j=1}^{m-k}{a_{(m+1-j);n+1}}\right)R_{k,q,n}}+R_{m+1,q,n}\\\ &=\sum_{k=0}^{m}{\left(\prod_{j=0}^{m-k}{a_{(m+1-j);n+1}}\right)R_{k,q,n}}+R_{m+1,q,n}.\end{split}$ Noticing that $\sum_{k=m+1}^{m+1}{\left(\prod_{j=0}^{m-k}{a_{(m+1-j);n+1}}\right)R_{k,q,n}}=R_{m+1,q,n}$ hence, $R_{m+1,q,n+1}=\sum_{k=0}^{m+1}{\left(\prod_{j=0}^{m-k}{a_{(m+1-j);n+1}}\right)R_{k,q,n}}.$ Hence, the theorem is proven by induction. ∎ ###### Corollary 2.1. If all sequences are the same, Theorem 2.1 will be reduced to the following form, $\sum_{N_{m}=q}^{n+1}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{m}}\cdots a_{N_{1}}}}=\sum_{k=0}^{m}{\left(a_{n+1}\right)^{m-k}\left(\sum_{N_{k}=q}^{n}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{k}}\cdots a_{N_{1}}}}\right)}.$ Using the notation from Eq. (2), this theorem can be written as $\hat{R}_{m,q,n+1}=\sum_{k=0}^{m}{\left(a_{n+1}\right)^{m-k}\hat{R}_{k,q,n}}.$ ###### Example 2.1. Consider that $m=2$, we have the two following cases: * • If all sequences are distinct, $\sum_{N_{2}=q}^{n+1}{b_{N_{2}}\sum_{N_{1}=q}^{N_{2}}{a_{N_{1}}}}-\sum_{N_{2}=q}^{n}{b_{N_{2}}\sum_{N_{1}=q}^{N_{2}}{a_{N_{1}}}}=(b_{n+1})\sum_{N_{1}=q}^{n}{a_{N_{1}}}+(b_{n+1})(a_{n+1}).$ * • If all sequences are the same, $\sum_{N_{2}=q}^{n+1}{a_{N_{2}}\sum_{N_{1}=q}^{N_{2}}{a_{N_{1}}}}-\sum_{N_{2}=q}^{n}{a_{N_{2}}\sum_{N_{1}=q}^{N_{2}}{a_{N_{1}}}}=(a_{n+1})\sum_{N_{1}=q}^{n}{a_{N_{1}}}+(a_{n+1})^{2}.$ ###### Remark. Set $a_{(m);N}=\cdots=a_{(2);N}=1$, Theorem 2.1 becomes $\sum_{N_{m}=q}^{n+1}{\sum_{N_{m-1}=q}^{N_{m}}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{1}}}}}=\sum_{k=1}^{m}{\left(\sum_{N_{k}=q}^{n}{\sum_{N_{k-1}=q}^{N_{k}}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{1}}}}}\right)}+a_{n+1}.$ ### 2.2 Simple recurrent expression A recursive form of Theorem 2.1 can be obtained by expanding and factoring the theorem’s expression. ###### Theorem 2.2. For any $m,q,n\in\mathbb{N}$ where $n\geq q$ and for any set of sequences $a_{(1);N_{1}},\cdots,a_{(m);N_{m}}$ defined in the interval $[q,n+1]$, we have that $\sum_{N_{m}=q}^{n+1}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{(m);N_{m}}\cdots a_{(1);N_{1}}}}-\sum_{N_{m}=q}^{n}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{(m);N_{m}}\cdots a_{(1);N_{1}}}}=a_{(m);n+1}\left\\{a_{(m-1);n+1}\left[\cdots a_{(2);n+1}\left(a_{(1);n+1}(1)+\sum_{N_{1}=q}^{n}{a_{(1);N_{1}}}\right)+\sum_{N_{2}=q}^{n}{\sum_{N_{1}=q}^{N_{2}}{a_{(2);N_{2}}a_{(1);N_{1}}}}\right]+\sum_{N_{m-1}=q}^{n}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{(m-1);N_{m-1}}\cdots a_{(1);N_{1}}}}\right\\}.$ Using the notation from Eq. (1), this theorem can be written as $R_{m,q,n+1}=a_{(m);n+1}\left\\{a_{(m-1);n+1}\left[\cdots a_{(2);n+1}\left(a_{(1);n+1}\left(R_{0,q,n}\right)+R_{1,q,n}\right)+R_{2,q,n}\right]+R_{m-1,q,n}\right\\}+R_{m,q,n}$ where $R_{0,q,n}=1$. ###### Proof. 1\. Base Case: verify true for $m=1$. $a_{(1);n+1}(R_{0,q,n})+R_{1,q,n}=a_{(1);n+1}(1)+\sum_{N_{1}=q}^{n}{a_{(1);N_{1}}}=\sum_{N_{1}=q}^{n+1}{a_{(1);N_{1}}}=R_{1,q,n+1}.$ 2\. Induction Hypothesis: assume the statement is true until $m$. $R_{m,q,n+1}=a_{(m);n+1}\left\\{a_{(m-1);n+1}\left[\cdots a_{(2);n+1}\left(a_{(1);n+1}\left(R_{0,q,n}\right)+R_{1,q,n}\right)+R_{2,q,n}\right]+R_{m-1,q,n}\right\\}+R_{m,q,n}.$ 3\. Induction Step: we will show that this statement is true for $(m+1)$. We have to show the following statement to be true: $R_{m+1,q,n+1}=a_{(m+1);n+1}\left\\{a_{(m);n+1}\left[\cdots a_{(2);n+1}\left(a_{(1);n+1}\left(R_{0,q,n}\right)+R_{1,q,n}\right)+R_{2,q,n}\right]+R_{m,q,n}\right\\}+R_{m+1,q,n}.$ From Lemma 2.1, $R_{m+1,q,n+1}=a_{(m+1);n+1}R_{m,q,n+1}+R_{m+1,q,n}.$ By applying the induction hypothesis, $R_{m+1,q,n+1}=a_{(m+1);n+1}\left\\{a_{(m);n+1}\left[\cdots a_{(2);n+1}\left(a_{(1);n+1}\left(R_{0,q,n}\right)+R_{1,q,n}\right)+R_{2,q,n}\right]+R_{m,q,n}\right\\}+R_{m+1,q,n}.$ Hence, the theorem is proven by induction. ∎ ###### Corollary 2.2. If all sequences are the same, Theorem 2.2 will be reduced to the following form, $\sum_{N_{m}=q}^{n+1}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{m}}\cdots a_{N_{1}}}}-\sum_{N_{m}=q}^{n}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{m}}\cdots a_{N_{1}}}}=a_{n+1}\left\\{a_{n+1}\left[\cdots a_{n+1}\left(a_{n+1}(1)+\sum_{N_{1}=q}^{n}{a_{N_{1}}}\right)+\sum_{N_{2}=q}^{n}{\sum_{N_{1}=q}^{N_{2}}{a_{N_{2}}a_{N_{1}}}}\right]+\sum_{N_{m-1}=q}^{n}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{m-1}}\cdots a_{N_{1}}}}\right\\}.$ Using the notation from Eq. (2), this theorem can be written as $\hat{R}_{m,q,n+1}=a_{n+1}\left\\{a_{n+1}\left[\cdots a_{n+1}\left(a_{n+1}\left(\hat{R}_{0,q,n}\right)+\hat{R}_{1,q,n}\right)+\hat{R}_{2,q,n}\right]+\hat{R}_{m-1,q,n}\right\\}+\hat{R}_{m,q,n}$ where $\hat{R}_{0,q,n}=1$. ###### Example 2.2. Consider that $m=2$, we have the two following cases: * • If all sequences are distinct, $\sum_{N_{2}=q}^{n+1}{b_{N_{2}}\sum_{N_{1}=q}^{N_{2}}{a_{N_{1}}}}-\sum_{N_{2}=q}^{n}{b_{N_{2}}\sum_{N_{1}=q}^{N_{2}}{a_{N_{1}}}}=(b_{n+1})\left\\{a_{n+1}(1)+\sum_{N_{1}=q}^{n}{a_{N_{1}}}\right\\}.$ * • If all sequences are the same, $\sum_{N_{2}=q}^{n+1}{a_{N_{2}}\sum_{N_{1}=q}^{N_{2}}{a_{N_{1}}}}-\sum_{N_{2}=q}^{n}{a_{N_{2}}\sum_{N_{1}=q}^{N_{2}}{a_{N_{1}}}}=(a_{n+1})\left\\{a_{n+1}(1)+\sum_{N_{1}=q}^{n}{a_{N_{1}}}\right\\}.$ ### 2.3 General expression The variation of a recurrent sum can also be expressed in terms of only a certain range of lower order recurrent sums. In other words, $R_{m,q,n+1}$ can be expressed in terms of $R_{m,q,n}$ and of recurrent sums of order going only from $p$ to $(m-1)$. To do so, we develop the following theorem. ###### Theorem 2.3. For any $m,q,n\in\mathbb{N}$ where $n\geq q$, for any $p\in[0,m]$, and for any set of sequences $a_{(1);N_{1}},\cdots,a_{(m);N_{m}}$ defined in the interval $[q,n+1]$, we have that $\sum_{N_{m}=q}^{n+1}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{(m);N_{m}}\cdots a_{(1);N_{1}}}}=\sum_{k=p+1}^{m}{\left(\prod_{j=0}^{m-k-1}{a_{(m-j);n+1}}\right)\left(\sum_{N_{k}=q}^{n}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{(k);N_{k}}\cdots a_{(1);N_{1}}}}\right)}+\left(\prod_{j=0}^{m-p-1}{a_{(m-j);n+1}}\right)\left(\sum_{N_{p}=q}^{n+1}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{(p);N_{p}}\cdots a_{(1);N_{1}}}}\right).$ Using the notation from Eq. (1), this theorem can be written as $R_{m,q,n+1}=\sum_{k=p+1}^{m}{\left(\prod_{j=0}^{m-k-1}{a_{(m-j);n+1}}\right)R_{k,q,n}}+\left(\prod_{j=0}^{m-p-1}{a_{(m-j);n+1}}\right)R_{p,q,n+1}.$ ###### Proof. By applying Theorem 2.1, $\begin{split}R_{m,q,n+1}&=\sum_{k=0}^{m}{\left(\prod_{j=0}^{m-k-1}{a_{(m-j);n+1}}\right)R_{k,q,n}}\\\ &=\sum_{k=p+1}^{m}{\left(\prod_{j=0}^{m-k-1}{a_{(m-j);n+1}}\right)R_{k,q,n}}+\sum_{k=0}^{p}{\left(\prod_{j=0}^{m-k-1}{a_{(m-j);n+1}}\right)R_{k,q,n}}\\\ &=\sum_{k=p+1}^{m}{\left(\prod_{j=0}^{m-k-1}{a_{(m-j);n+1}}\right)R_{k,q,n}}+\left(\prod_{j=0}^{m-p-1}{a_{(m-j);n+1}}\right)\sum_{k=0}^{p}{\left(\prod_{j=m-p}^{m-k-1}{a_{(m-j);n+1}}\right)R_{k,q,n}}.\end{split}$ From Theorem 2.1, with $m$ substituted by $p$, we have $R_{p,q,n+1}=\sum_{k=0}^{p}{\left(\prod_{j=0}^{p-k-1}{a_{(p-j);n+1}}\right)R_{k,q,n}}=\sum_{k=0}^{p}{\left(\prod_{j=m-p}^{m-k-1}{a_{(m-j);n+1}}\right)R_{k,q,n}}.$ Hence, by substituting, we get $R_{m,q,n+1}=\sum_{k=p+1}^{m}{\left(\prod_{j=0}^{m-k-1}{a_{(m-j);n+1}}\right)R_{k,q,n}}+\left(\prod_{j=0}^{m-p-1}{a_{(m-j);n+1}}\right)R_{p,q,n+1}.$ ∎ ###### Corollary 2.3. If all sequences are the same, Theorem 2.3 will be reduced to the following form, $\sum_{N_{m}=q}^{n+1}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{m}}\cdots a_{N_{1}}}}=\sum_{k=p+1}^{m}{\left(a_{n+1}\right)^{m-k}\left(\sum_{N_{k}=q}^{n}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{k}}\cdots a_{N_{1}}}}\right)}+\left(a_{n+1}\right)^{m-p}\left(\sum_{N_{p}=q}^{n+1}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{p}}\cdots a_{N_{1}}}}\right).$ Using the notation from Eq. (2), this theorem can be written as $\hat{R}_{m,q,n+1}=\sum_{k=p+1}^{m}{\left(a_{n+1}\right)^{m-k}\hat{R}_{k,q,n}}+\left(a_{n+1}\right)^{m-p}\hat{R}_{p,q,n+1}.$ ###### Example 2.3. For $p=2$ and if the sequences are the same: $\sum_{N_{m}=q}^{n+1}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{m}}\cdots a_{N_{1}}}}=\sum_{k=3}^{m}{\left(a_{n+1}\right)^{m-k}\left(\sum_{N_{k}=q}^{n}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{k}}\cdots a_{N_{1}}}}\right)}+\left(a_{n+1}\right)^{m-2}\left(\sum_{N_{2}=q}^{n+1}{\sum_{N_{1}=q}^{N_{2}}{a_{N_{2}}a_{N_{1}}}}\right).$ ###### Example 2.4. For $p=m-2$ and if the sequences are the same: $\sum_{N_{m}=q}^{n+1}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{m}}\cdots a_{N_{1}}}}=\left(\sum_{N_{m}=q}^{n}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{m}}\cdots a_{N_{1}}}}\right)+\left(a_{n+1}\right)\left(\sum_{N_{m-1}=q}^{n}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{m-1}}\cdots a_{N_{1}}}}\right)+\left(a_{n+1}\right)^{2}\left(\sum_{N_{m-2}=q}^{n+1}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{m-2}}\cdots a_{N_{1}}}}\right).$ ###### Remark. Set $a_{(m);N}=\cdots=a_{(2);N}=1$, Theorem 2.3 becomes $\sum_{N_{m}=q}^{n+1}{\sum_{N_{m-1}=q}^{N_{m}}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{1}}}}}=\sum_{k=p+1}^{m}{\left(\sum_{N_{k}=q}^{n}{\sum_{N_{k-1}=q}^{N_{k}}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{1}}}}}\right)}+\sum_{N_{p}=q}^{n+1}{\sum_{N_{p-1}=q}^{N_{p}}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{1}}}}}.$ ### 2.4 General recurrent expression Similarly, the theorem introduced in the previous section can be reformulated in a recursive form by expanding and factoring the expression of Theorem 2.3 to obtain the following expression. ###### Theorem 2.4. For any $m,q,n\in\mathbb{N}$ where $n\geq q$, for any $p\in[0,m]$, and for any set of sequences $a_{(1);N_{1}},\cdots,a_{(m);N_{m}}$ defined in the interval $[q,n+1]$, we have that $R_{m,q,n+1}=a_{(m);n+1}\left\\{a_{(m-1);n+1}\left[\cdots a_{(p+2);n+1}\left(a_{(p+1);n+1}\left(R_{p,q,n+1}\right)+R_{p+1,q,n}\right)+R_{p+2,q,n}\right]+R_{m-1,q,n}\right\\}+R_{m,q,n}.$ ###### Proof. From Theorem 2.2, with $m$ substituted by $p$, we have $R_{p,q,n+1}=a_{(p);n+1}\left\\{a_{(p-1);n+1}\left[\cdots a_{(2);n+1}\left(a_{(1);n+1}\left(R_{0,q,n}\right)+R_{1,q,n}\right)+R_{2,q,n}\right]+R_{p-1,q,n}\right\\}+R_{p,q,n}$ where $R_{0,q,n}=1$. Substituting into the expression of Theorem 2.2, the inner part becomes $R_{p,q,n+1}$ and we get the desired formula. ∎ ###### Corollary 2.4. If all sequences are the same, Theorem 2.4 will be reduced to the following form, $\hat{R}_{m,q,n+1}=a_{n+1}\left\\{a_{n+1}\left[\cdots a_{n+1}\left(a_{n+1}\left(\hat{R}_{p,q,n+1}\right)+\hat{R}_{p+1,q,n}\right)+\hat{R}_{p+2,q,n}\right]+\hat{R}_{m-1,q,n}\right\\}+\hat{R}_{m,q,n}.$ ###### Example 2.5. For $p=1$ and if the sequences are the same: $\sum_{N_{m}=q}^{n+1}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{m}}\cdots a_{N_{1}}}}-\sum_{N_{m}=q}^{n}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{m}}\cdots a_{N_{1}}}}=a_{n+1}\left\\{a_{n+1}\left[\cdots a_{n+1}\left(\sum_{N_{1}=q}^{n+1}{a_{N_{1}}}\right)+\sum_{N_{2}=q}^{n}{\sum_{N_{1}=q}^{N_{2}}{a_{N_{2}}a_{N_{1}}}}\right]+\sum_{N_{m-1}=q}^{n}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{m-1}}\cdots a_{N_{1}}}}\right\\}.$ ###### Example 2.6. For $p=m-2$ and if the sequences are the same: $\sum_{N_{m}=q}^{n+1}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{m}}\cdots a_{N_{1}}}}-\sum_{N_{m}=q}^{n}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{m}}\cdots a_{N_{1}}}}=a_{n+1}\left\\{a_{n+1}\left[\sum_{N_{m-2}=q}^{n+1}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{m-2}}\cdots a_{N_{1}}}}\right]+\sum_{N_{m-1}=q}^{n}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{m-1}}\cdots a_{N_{1}}}}\right\\}.$ ## 3 Inversion Formulas In this section, we will develop formulas to interchange the order of summation in a recurrent sum. ### 3.1 Particular case (for 2 sequences) We start by proving the inversion formula with $2$ sequences which is required in order to prove the more general inversion formula with $m$ sequences. ###### Theorem 3.1. For $m,q,n\in\mathbb{N}$ where $n\geq q$ and for any 2 sequences $a_{N_{1}}$ and $b_{N_{2}}$ defined in the interval $[q,n]$, we have that $\sum_{N_{2}=q}^{n}{b_{N_{2}}\sum_{N_{1}=q}^{N_{2}}{a_{N_{1}}}}=\sum_{N_{1}=q}^{n}{a_{N_{1}}\sum_{N_{2}=N_{1}}^{n}{b_{N_{2}}}}.$ ###### Proof. By expanding the sum, we get $\sum_{N_{2}=q}^{n}{b_{N_{2}}\sum_{N_{1}=q}^{N_{2}}{a_{N_{1}}}}=b_{q}\left(\sum_{N_{1}=q}^{q}{a_{N_{1}}}\right)+b_{q+1}\left(\sum_{N_{1}=q}^{q+1}{a_{N_{1}}}\right)+\cdots+b_{n-1}\left(\sum_{N_{1}=q}^{n-1}{a_{N_{1}}}\right)+b_{n}\left(\sum_{N_{1}=q}^{n}{a_{N_{1}}}\right)=b_{q}\left(a_{q}\right)+b_{q+1}\left(a_{q}+a_{q+1}\right)+\cdots+b_{n-1}\left(a_{q}+\cdots+a_{n-1}\right)+b_{n}\left(a_{q}+\cdots+a_{n}\right).$ By regrouping the $b_{N}$ terms instead of the $a_{N}$ terms, the expression becomes $\sum_{N_{2}=q}^{n}{b_{N_{2}}\sum_{N_{1}=q}^{N_{2}}{a_{N_{1}}}}=a_{q}\left(b_{q}+\cdots+b_{n}\right)+a_{q+1}\left(b_{q+1}+\cdots+b_{n}\right)+\cdots+a_{n-1}\left(b_{n-1}+b_{n}\right)+a_{n}\left(b_{n}\right)=a_{q}\left(\sum_{N_{2}=q}^{n}{b_{N_{2}}}\right)+a_{q+1}\left(\sum_{N_{2}=q+1}^{n}{b_{N_{2}}}\right)+\cdots+a_{n-1}\left(\sum_{N_{2}=n-1}^{n}{b_{N_{2}}}\right)+a_{n}\left(\sum_{N_{2}=n}^{n}{b_{N_{2}}}\right)=\sum_{N_{1}=q}^{n}{a_{N_{1}}\sum_{N_{2}=N_{1}}^{n}{b_{N_{2}}}}.$ ∎ ###### Corollary 3.1. If all sequences are the same, Theorem 3.1 becomes $\sum_{N_{2}=q}^{n}{a_{N_{2}}\sum_{N_{1}=q}^{N_{2}}{a_{N_{1}}}}=\sum_{N_{1}=q}^{n}{a_{N_{1}}\sum_{N_{2}=N_{1}}^{n}{a_{N_{2}}}}.$ ### 3.2 General case (for m sequences) We now prove the more general inversion formula with $m$ sequences which allows us to invert the order of summation for a recurrent sum of order $m$. ###### Theorem 3.2. For any $m,q,n\in\mathbb{N}$ where $n\geq q$ and for any set of sequences $a_{(1);N_{1}},\cdots,a_{(m);N_{m}}$ defined in the interval $[q,n]$, we have that $\sum_{N_{m}=q}^{n}{a_{(m);N_{m}}\cdots\sum_{N_{2}=q}^{N_{3}}{a_{(2);N_{2}}\sum_{N_{1}=q}^{N_{2}}{a_{(1);N_{1}}}}}=\sum_{N_{1}=q}^{n}{a_{(1);N_{1}}\sum_{N_{2}=N_{1}}^{n}{a_{(2);N_{2}}\cdots\sum_{N_{m}=N_{m-1}}^{n}{a_{(m);N_{m}}}}}.$ ###### Proof. 1\. Base Case: verify true for $m=2$. This statement is true as proven in Theorem 3.1. 2\. Induction hypothesis: assume the statement is true until $m$. $\sum_{N_{m}=q}^{n}{a_{(m);N_{m}}\cdots\sum_{N_{2}=q}^{N_{3}}{a_{(2);N_{2}}\sum_{N_{1}=q}^{N_{2}}{a_{(1);N_{1}}}}}=\sum_{N_{1}=q}^{n}{a_{(1);N_{1}}\sum_{N_{2}=N_{1}}^{n}{a_{(2);N_{2}}\cdots\sum_{N_{m}=N_{m-1}}^{n}{a_{(m);N_{m}}}}}.$ 3\. Induction step: we will show that this statement is true for $(m+1)$. We have to show the following statement to be true: $\sum_{N_{m+1}=q}^{n}{a_{(m+1);N_{m+1}}\cdots\sum_{N_{2}=q}^{N_{3}}{a_{(2);N_{2}}\sum_{N_{1}=q}^{N_{2}}{a_{(1);N_{1}}}}}=\sum_{N_{1}=q}^{n}{a_{(1);N_{1}}\sum_{N_{2}=N_{1}}^{n}{a_{(2);N_{2}}\cdots\sum_{N_{m+1}=N_{m}}^{n}{a_{(m+1);N_{m+1}}}}}.$ $\sum_{N_{m+1}=q}^{n}{a_{(m+1);N_{m+1}}\cdots\sum_{N_{2}=q}^{N_{3}}{a_{(2);N_{2}}\sum_{N_{1}=q}^{N_{2}}{a_{(1);N_{1}}}}}=\sum_{N_{m+1}=q}^{n}{a_{(m+1);N_{m+1}}\left(\sum_{N_{m}=q}^{N_{m+1}}{a_{(m);N_{m}}\cdots\sum_{N_{2}=q}^{N_{3}}{a_{(2);N_{2}}\sum_{N_{1}=q}^{N_{2}}{a_{(1);N_{1}}}}}\right)}.$ Let $b_{N_{m}}$ be the following sequence (that dependents only on $N_{m}$), $b_{N_{m}}=a_{(m);N_{m}}\sum_{N_{m-1}=q}^{N_{m}}{a_{(m-1);N_{m-1}}\cdots\sum_{N_{2}=q}^{N_{3}}{a_{(2);N_{2}}\sum_{N_{1}=q}^{N_{2}}{a_{(1);N_{1}}}}}.$ By applying this substitution in the previous expression, we obtain a recurrent sum of order 2 that contains the 2 sequences $a_{(m+1);N_{m+1}}$ and $b_{N_{m}}$. Then, we apply the inversion formula for the case of 2 sequences (Theorem 3.1) to get the following, $\sum_{N_{m+1}=q}^{n}{a_{(m+1);N_{m+1}}\cdots\sum_{N_{2}=q}^{N_{3}}{a_{(2);N_{2}}\sum_{N_{1}=q}^{N_{2}}{a_{(1);N_{1}}}}}=\sum_{N_{m+1}=q}^{n}{a_{(m+1);N_{m+1}}\left(\sum_{N_{m}=q}^{N_{m+1}}{b_{N_{m}}}\right)}=\sum_{N_{m}=q}^{n}{b_{N_{m}}\left(\sum_{N_{m+1}=N_{m}}^{n}{a_{(m+1);N_{m+1}}}\right)}=\sum_{N_{m}=q}^{n}{a_{(m);N_{m}}\cdots\sum_{N_{2}=q}^{N_{3}}{a_{(2);N_{2}}\sum_{N_{1}=q}^{N_{2}}{a_{(1);N_{1}}\left(\sum_{N_{m+1}=N_{m}}^{n}{a_{(m+1);N_{m+1}}}\right)}}}.$ The sum of $a_{(m+1);N_{m+1}}$ has $N_{m}$ and $n$ as lower and upper bounds. Thus, knowing that $n$ is a constant, the sum of $a_{(m+1);N_{m+1}}$ depends only on $N_{m}$. This allows us to extract this sum from the inner sums to get $\sum_{N_{m+1}=q}^{n}{a_{(m+1);N_{m+1}}\cdots\sum_{N_{2}=q}^{N_{3}}{a_{(2);N_{2}}\sum_{N_{1}=q}^{N_{2}}{a_{(1);N_{1}}}}}=\sum_{N_{m}=q}^{n}{\left(a_{(m);N_{m}}\sum_{N_{m+1}=N_{m}}^{n}{a_{(m+1);N_{m+1}}}\right)\cdots\sum_{N_{2}=q}^{N_{3}}{a_{(2);N_{2}}\sum_{N_{1}=q}^{N_{2}}{a_{(1);N_{1}}}}}.$ Let $A_{N_{m}}$ be the following sequence (that only depends on $N_{m}$), $A_{N_{m}}=a_{(m);N_{m}}\sum_{N_{m+1}=N_{m}}^{n}{a_{(m+1);N_{m+1}}}.$ By substituting $A_{N_{m}}$ into the previous expression, we get a recurrent sum of order $m$ in terms of the following $m$ sequences: $A_{N_{m}},a_{(m-1);N_{m-1}},\cdots,a_{(1);N_{1}}$. Then the inversion formula for the case of $m$ sequences (which was assumed to be true in the induction hypothesis) is applied, $\sum_{N_{m+1}=q}^{n}{a_{(m+1);N_{m+1}}\cdots\sum_{N_{2}=q}^{N_{3}}{a_{(2);N_{2}}\sum_{N_{1}=q}^{N_{2}}{a_{(1);N_{1}}}}}=\sum_{N_{m}=q}^{n}{A_{N_{m}}\cdots\sum_{N_{2}=q}^{N_{3}}{a_{(2);N_{2}}\sum_{N_{1}=q}^{N_{2}}{a_{(1);N_{1}}}}}=\sum_{N_{1}=q}^{n}{a_{(1);N_{1}}\sum_{N_{2}=N_{1}}^{n}{a_{(2);N_{2}}\cdots\sum_{N_{m}=N_{m-1}}^{n}{A_{N_{m}}}}}=\sum_{N_{1}=q}^{n}{a_{(1);N_{1}}\sum_{N_{2}=N_{1}}^{n}{a_{(2);N_{2}}\cdots\sum_{N_{m}=N_{m-1}}^{n}{a_{(m);N_{m}}\sum_{N_{m+1}=N_{m}}^{n}{a_{(m+1);N_{m+1}}}}}}.$ We conclude that it must hold for all $m\geq 2$. ∎ ###### Corollary 3.2. If all sequences are the same, Theorem 3.2 becomes $\sum_{N_{m}=q}^{n}{a_{N_{m}}\cdots\sum_{N_{2}=q}^{N_{3}}{a_{N_{2}}\sum_{N_{1}=q}^{N_{2}}{a_{N_{1}}}}}=\sum_{N_{1}=q}^{n}{a_{N_{1}}\sum_{N_{2}=N_{1}}^{n}{a_{N_{2}}\cdots\sum_{N_{m}=N_{m-1}}^{n}{a_{N_{m}}}}}.$ Similarly, the innermost summation can be turned into the outermost summation as illustrated by Theorem 3.3. ###### Theorem 3.3. For any $m,q,n\in\mathbb{N}$ where $n\geq q$ and for any set of sequences $a_{(1);N_{1}},\cdots,a_{(m);N_{m}}$ defined in the interval $[q,n]$, we have that $\sum_{N_{m}=q}^{n}{a_{(m);N_{m}}\cdots\sum_{N_{2}=q}^{N_{3}}{a_{(2);N_{2}}\sum_{N_{1}=q}^{N_{2}}{a_{(1);N_{1}}}}}=\sum_{N_{1}=q}^{n}{a_{(1);N_{1}}\sum_{N_{m}=N_{1}}^{n}{a_{(m);N_{m}}\cdots\sum_{N_{3}=N_{1}}^{N_{4}}{a_{(3);N_{3}}\sum_{N_{2}=N_{1}}^{N_{3}}{a_{(2);N_{2}}}}}}.$ ###### Proof. From Theorem 3.2, $\sum_{N_{m}=q}^{n}{a_{(m);N_{m}}\cdots\sum_{N_{2}=q}^{N_{3}}{a_{(2);N_{2}}\sum_{N_{1}=q}^{N_{2}}{a_{(1);N_{1}}}}}=\sum_{N_{1}=q}^{n}{a_{(1);N_{1}}\sum_{N_{2}=N_{1}}^{n}{a_{(2);N_{2}}\cdots\sum_{N_{m}=N_{m-1}}^{n}{a_{(m);N_{m}}}}}.$ Applying Theorem 3.2 to the inner part of the right side sum would transform it as follows $\sum_{N_{2}=N_{1}}^{n}{a_{(2);N_{2}}\cdots\sum_{N_{m}=N_{m-1}}^{n}{a_{(m);N_{m}}}}=\sum_{N_{m}=N_{1}}^{n}{a_{(m);N_{m}}\cdots\sum_{N_{3}=N_{1}}^{N_{4}}{a_{(3);N_{3}}\sum_{N_{2}=N_{1}}^{N_{3}}{a_{(2);N_{2}}}}}.$ Hence, substituting back into Theorem 3.2 would give us the desired formula. ∎ ###### Corollary 3.3. If all sequences are the same, Theorem 3.3 becomes $\sum_{N_{m}=q}^{n}{a_{N_{m}}\cdots\sum_{N_{2}=q}^{N_{3}}{a_{N_{2}}\sum_{N_{1}=q}^{N_{2}}{a_{N_{1}}}}}=\sum_{N_{1}=q}^{n}{a_{N_{1}}\sum_{N_{m}=N_{1}}^{n}{a_{N_{m}}\cdots\sum_{N_{3}=N_{1}}^{N_{4}}{a_{N_{3}}\sum_{N_{2}=N_{1}}^{N_{3}}{a_{N_{2}}}}}}.$ ### 3.3 Inversion of p sequences from m sequences Finally, as we will show in this section, it is possible to partially invert the order of summation for a recurrent sum. In other words, as shown by the following theorem, it is possible to invert the order of summation of only the $p$ innermost summations from $m$ summations. ###### Theorem 3.4. For any $m,q,n\in\mathbb{N}$ where $n\geq q$, for any $p\in[0,m]$, and for any set of sequences $a_{(1);N_{1}},\cdots,a_{(m);N_{m}}$ defined in the interval $[q,n]$, we have that $\sum_{N_{m}=q}^{n}{a_{(m);N_{m}}\cdots\sum_{N_{p}=q}^{N_{p+1}}{a_{(p);N_{p}}\cdots\sum_{N_{1}=q}^{N_{2}}{a_{(1);N_{1}}}}}=\sum_{N_{m}=q}^{n}{a_{(m);N_{m}}\cdots\sum_{N_{p+1}=q}^{N_{p+2}}{a_{(p+1);N_{p+1}}\sum_{N_{1}=q}^{N_{p+1}}{a_{(1);N_{1}}\sum_{N_{2}=N_{1}}^{N_{p+1}}{a_{(2);N_{2}}\cdots\sum_{N_{p}=N_{p-1}}^{N_{p+1}}{a_{(p);N_{p}}}}}}}.$ ###### Proof. By replacing $m$ by $p$ and $n$ by $N_{p+1}$ in Theorem 3.2, we get the following relation, $\sum_{N_{p}=q}^{N_{p+1}}{a_{(p);N_{p}}\cdots\sum_{N_{2}=q}^{N_{3}}{a_{(2);N_{2}}\sum_{N_{1}=q}^{N_{2}}{a_{(1);N_{1}}}}}=\sum_{N_{1}=q}^{N_{p+1}}{a_{(1);N_{1}}\sum_{N_{2}=N_{1}}^{N_{p+1}}{a_{(2);N_{2}}\cdots\sum_{N_{p}=N_{p-1}}^{N_{p+1}}{a_{(p);N_{p}}}}}.$ Thus, $\sum_{N_{m}=q}^{n}{a_{(m);N_{m}}\cdots\sum_{N_{p}=q}^{N_{p+1}}{a_{(p);N_{p}}\cdots\sum_{N_{1}=q}^{N_{2}}{a_{(1);N_{1}}}}}=\sum_{N_{m}=q}^{n}{a_{(m);N_{m}}\cdots\sum_{N_{p+1}}^{N_{p+2}}{a_{(p+1);N_{p+1}}\left(\sum_{N_{p}=q}^{N_{p+1}}{a_{(p);N_{p}}\cdots\sum_{N_{1}=q}^{N_{2}}{a_{(1);N_{1}}}}\right)}}=\sum_{N_{m}=q}^{n}{a_{(m);N_{m}}\cdots\sum_{N_{p+1}=q}^{N_{p+2}}{a_{(p+1);N_{p+1}}\left(\sum_{N_{1}=q}^{N_{p+1}}{a_{(1);N_{1}}\sum_{N_{2}=N_{1}}^{N_{p+1}}{a_{(2);N_{2}}\cdots\sum_{N_{p}=N_{p-1}}^{N_{p+1}}{a_{(p);N_{p}}}}}\right)}}.$ ∎ ###### Corollary 3.4. If all sequences are the same, Theorem 3.4 becomes $\sum_{N_{m}=q}^{n}{a_{N_{m}}\cdots\sum_{N_{p}=q}^{N_{p+1}}{a_{N_{p}}\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{1}}}}}=\sum_{N_{m}=q}^{n}{a_{N_{m}}\cdots\sum_{N_{p+1}=q}^{N_{p+2}}{a_{N_{p+1}}\sum_{N_{1}=q}^{N_{p+1}}{a_{N_{1}}\sum_{N_{2}=N_{1}}^{N_{p+1}}{a_{N_{2}}\cdots\sum_{N_{p}=N_{p-1}}^{N_{p+1}}{a_{N_{p}}}}}}}.$ Similarly, the innermost summation can be pulled back to the $p$-th position as illustrated by Theorem 3.5. ###### Theorem 3.5. For any $m,q,n\in\mathbb{N}$ where $n\geq q$, for any $p\in[0,m]$, and for any set of sequences $a_{(1);N_{1}},\cdots,a_{(m);N_{m}}$ defined in the interval $[q,n]$, we have that $\sum_{N_{m}=q}^{n}{a_{(m);N_{m}}\cdots\sum_{N_{p}=q}^{N_{p+1}}{a_{(p);N_{p}}\cdots\sum_{N_{1}=q}^{N_{2}}{a_{(1);N_{1}}}}}=\sum_{N_{m}=q}^{n}{a_{(m);N_{m}}\cdots\sum_{N_{p+1}=q}^{N_{p+2}}{a_{(p+1);N_{p+1}}\sum_{N_{1}=q}^{N_{p+1}}{a_{(1);N_{1}}\sum_{N_{p}=N_{1}}^{N_{p+1}}{a_{(p);N_{p}}\sum_{N_{p-1}=N_{1}}^{N_{p}}{a_{(p-1);N_{p-1}}\cdots\sum_{N_{2}=N_{1}}^{N_{3}}{a_{(2);N_{2}}}}}}}}.$ ###### Proof. By applying Theorem 3.3 (with $m$ substituted by $p$ and $n$ substituted by $N_{p+1}$) to Theorem 3.4, we get the desired theorem. ∎ ###### Corollary 3.5. If all sequences are the same, Theorem 3.5 becomes $\sum_{N_{m}=q}^{n}{a_{N_{m}}\cdots\sum_{N_{p}=q}^{N_{p+1}}{a_{N_{p}}\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{1}}}}}=\sum_{N_{m}=q}^{n}{a_{N_{m}}\cdots\sum_{N_{p+1}=q}^{N_{p+2}}{a_{N_{p+1}}\sum_{N_{1}=q}^{N_{p+1}}{a_{N_{1}}\sum_{N_{p}=N_{1}}^{N_{p+1}}{a_{N_{p}}\sum_{N_{p-1}=N_{1}}^{N_{p}}{a_{N_{p-1}}\cdots\sum_{N_{2}=N_{1}}^{N_{3}}{a_{N_{2}}}}}}}}.$ ## 4 Reduction Formulas The objective of this section is to introduce formulas which can be used to reduce recurrent sums from their originally recurrent form $\left(\sum_{N_{m}=q}^{n}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{m}}\cdots a_{N_{1}}}}\right)$ to a form containing only simple non-recurrent sums $\left(\left(\sum_{N=q}^{n}{(a_{N})^{i}}\right)^{y}\right)$. ### 4.1 A brief introduction to partitions In this paper, partitions are involved in the reduction formula for a recurrent sum. For this reason, in this section, we will present a brief introduction to partitions. ###### Definition. A partition of a non-negative integer $m$ is a set of positive integers whose sum equals $m$. We can represent a partition of $m$ as a vector $(y_{k,1},\cdots,y_{k,m})$ that verifies $\displaystyle\begin{pmatrix}y_{k,1}\\\ \vdots\\\ y_{k,m}\\\ \end{pmatrix}\cdot\begin{pmatrix}1\\\ \vdots\\\ m\\\ \end{pmatrix}=y_{k,1}+2y_{k,2}+\cdots+my_{k,m}=m.$ (4) The set of partitions of a non-negative integer $m$ is the set of vectors $(y_{k,1},\cdots,y_{k,m})$ that verify the previous identity. We will denote this set by $P$. The cardinality of this set is equal to the number of partitions of $m$ (which is the partition function denoted by $p(m)$), $\text{Card}(P)=p(m).$ (5) Hence, the set of partitions of $m$ is the set of vectors $\\{(y_{1,1},\cdots,y_{1,m}),(y_{2,1},\cdots,y_{2,m}),\cdots\\}$ which consists of $p(m)$ vectors. The value of $p(m)$ is obtained from the generating function developed by Euler in the mid-eighteen century [16], $\sum_{m=0}^{\infty}{p(m)x^{m}}=\prod_{j=1}^{\infty}{\frac{1}{1-x^{j}}}.$ (6) Euler also showed that this relation implies the following recurrent definition for $p(m)$, $p(m)=\sum_{j=1}^{\infty}{(-1)^{j-1}\left(p\left(m-\frac{j(3j-1)}{2}\right)-p\left(m-\frac{j(3j+1)}{2}\right)\right)}.$ (7) In 1918, Hardy and Ramanujan provided an asymptotic expression for $p(m)$ in [22]. Later, in 1937, Rademacher was able to improve on Hardy and Ramanujan’s formula by proving the following expression for $p(m)$ in [34], $p(m)=\frac{1}{\pi\sqrt{2}}\sum_{k=1}^{\infty}{\sqrt{k}A_{k}(m)\frac{d}{dm}\left[\frac{\sinh\left(\frac{\pi}{k}\sqrt{\frac{2}{3}\left(m-\frac{1}{24}\right)}\right)}{\sqrt{m-\frac{1}{24}}}\right]}$ (8) where $A_{k}(m)$ is a Kloosterman type sum, $A_{k}(m)=\sum_{\begin{subarray}{c}0\leq h<k\\\ gcd(h,k)=1\end{subarray}}{e^{\pi i(s(h,k)-2mh/k)}}$ (9) and where the notation $s(m,k)$ represents a Dedekind sum. However, this formula has the disadvantage of being an infinite sum. This formula remained the only exact explicit formula for $p(m)$ until Ono and Bruinier presented a new formula for $p(m)$ as a finite sum [10]. Additionally, two of the most famous ways of representing a partition are using Ferrers diagrams or using Young diagrams. Similarly, there exists some variants of Ferrers diagrams that are used (see [33]). ###### Remark. For readers intrested in a more detailed explanation of partition, see [1]. ### 4.2 Reduction Theorem and Partition Identities We will start this section by proving several lemmas which are needed in order to prove the main theorem of this section (Theorem 4.1, which we will call the reduction theorem). However, some of these lemmas are important on their own as they provide relations governing partitions. We start by proving the following trivial lemma. ###### Lemma 4.1. No partition of a non-negative integer $m$ constructed from a sum of $r$ terms (positive integers) can contain an integer larger or equal to $m-r+2$. ###### Proof. The smallest sum of $r$ positive integers containing $i$ is $i+\underbrace{(1+\cdots+1)}_{(r-1)}=i+(r-1)$. If $i\geq m-r+2$ then $i+(r-1)\geq m-r+2+r-1=m+1>m$. Hence, such a sum, being strictly larger than $m$, cannot be a partition of $m$. ∎ Before we can proceed to prove the other needed lemmas, we need to define the following notation: Let $[x^{r}]\left(P(x)\right)$ represent the coefficient of $x^{r}$ in $P(x)$. Let $x^{\overline{m}}=x(x+1)\cdots(x+m-1)$ represent the Rising factorial. Let $(x)_{m}=x(x-1)\cdots(x-m+1)$ represent the Falling factorial. The original definition of Stirling numbers of the first kind $S(m,r)$ was as the coefficients in the expansion of $(x)_{m}$: $(x)_{m}=\sum_{k=0}^{m}{S(m,k)x^{k}}\,\,\,\,\,\,\,\,or\,\,\,\,\,\,\,\,S(m,r)=[x^{r}](x)_{m}.$ (10) In a similar way, the unsigned Stirling numbers of the first kind, denoted $\|S(m,r)\|$ or ${m\brack r}$, can be expressed in terms of the Rising factorial $x^{\overline{m}}$: $x^{\overline{m}}=\sum_{k=0}^{m}{{m\brack k}x^{k}}\,\,\,\,\,\,\,\,or\,\,\,\,\,\,\,\,{m\brack r}=[x^{r}]\left(x^{\overline{m}}\right).$ (11) From this definition, the famous finite sum of the unsigned Stirling numbers of the first kind can be directly deduced by substituting $x$ by 1 to get $\sum_{k=0}^{m}{{m\brack k}}=1(1+1)\cdots(1+m-1)=m!.$ (12) Note that $\|S(m,r)\|$ can also be defined as the number of permutations of $m$ elements with $r$ disjoint cycles. Similarly, the previous relation can be obtained by noticing that permutations are partitioned by number of cycles. ###### Remark. More details on Stirling numbers of the first kind can be found in [28]. For simplicity, we define $\sum{f(i)}$ to mean $\sum_{i=1}^{m}{f(i)}$. In particular, $\sum{i.y_{k,i}}=\sum_{i=1}^{m}{i.y_{k,i}}$ and $\sum{y_{k,i}}=\sum_{i=1}^{m}{y_{k,i}}$. Additionally, let a partition of $m$ of length $r$ refer to a partition $(y_{k,1},\cdots,y_{k,m})$ of $m$ such that $\sum{y_{k,i}}=r$. Now that we have defined the needed notation, we can continue proving the required lemmas. ###### Lemma 4.2. Let $m$ and $r$ be two non-negative integers with $r\leq m$, the following sum over partitions of $m$ of length $r$ can be expressed in terms of the unsigned Stirling numbers of the first kind as follows, $\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\\\ \sum{y_{k,i}}=r\end{subarray}}{\prod_{i=1}^{m}{\frac{1}{i^{y_{k,i}}(y_{k,i})!}}}=\frac{1}{m!}{m\brack r}.$ ###### Proof. A Bell polynomial is defined as follows $B_{m,r}(x_{1},x_{2},\cdots,x_{m-r+1})=\sum_{\begin{subarray}{c}y_{1}+2y_{2}+\cdots+(m-r+1)y_{m-r+1}=m\\\ y_{1}+y_{2}+\cdots+y_{m-r+1}=r\end{subarray}}{\frac{m!}{y_{1}!y_{2}!\cdots y_{m-r+1}!}\left(\frac{x_{1}}{1!}\right)^{y_{1}}\left(\frac{x_{2}}{2!}\right)^{y_{2}}\cdots\left(\frac{x_{m-r+1}}{(m-r+1)!}\right)^{y_{m-r+1}}}.$ These polynomials can also be rewritten more compactly as $B_{m,r}(x_{1},x_{2},\cdots,x_{m-r+1})=m!\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\\\ \sum{y_{k,i}}=r\end{subarray}}{\prod_{i=1}^{m-r+1}{\frac{1}{y_{k,i}!}\left(\frac{x_{i}}{i!}\right)^{y_{k,i}}}}.$ A property of the Bell polynomial, shown in [36], is that the value of the Bell polynomial on the sequence of factorials equals an unsigned Stirling number of the first kind, $B_{m,r}(0!,1!,\cdots,(m-r)!)=\|S(m,r)\|={m\brack r}.$ Likewise, by a numerical substitution into the definition of Bell polynomials, we have $B_{m,r}(0!,1!,\cdots,(m-r)!)=m!\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\\\ \sum{y_{k,i}}=r\end{subarray}}{\prod_{i=1}^{m-r+1}{\frac{1}{i^{y_{k,i}}(y_{k,i})!}}}.$ Hence, by equating, we get $\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\\\ \sum{y_{k,i}}=r\end{subarray}}{\prod_{i=1}^{m-r+1}{\frac{1}{i^{y_{k,i}}(y_{k,i})!}}}=\frac{1}{m!}{m\brack r}.$ From Lemma 4.1, we know that the biggest integer that can appear in a partition of an integer $m$ using $r$ terms is $m-r+1$ (which means that $y_{k,m-r}=\cdots=y_{k,m}=0$). Thus, we get $\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\\\ \sum{y_{k,i}}=r\end{subarray}}{\prod_{i=1}^{m}{\frac{1}{i^{y_{k,i}}(y_{k,i})!}}}=\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\\\ \sum{y_{k,i}}=r\end{subarray}}{\prod_{i=1}^{m-r+1}{\frac{1}{i^{y_{k,i}}(y_{k,i})!}}}=\frac{1}{m!}{m\brack r}.$ ∎ By Adding the arguments of the sum from Lemma 4.2 for all possible partition lengths, we obtain the following identity. ###### Lemma 4.3. Let $m$ be a non-negative integer, the following sum over all partitions of $m$ can be shown to equal $1$ independently of the value of $m$, $\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\end{subarray}}{\prod_{i=1}^{m}{\frac{1}{i^{y_{k,i}}(y_{k,i})!}}}=1.$ ###### Proof. From Lemma 4.2, we have $\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\\\ \sum{y_{k,i}}=r\end{subarray}}{\prod_{i=1}^{m}{\frac{1}{i^{y_{k,i}}(y_{k,i})!}}}=\frac{1}{m!}{m\brack r}.$ Hence, $\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\end{subarray}}{\prod_{i=1}^{m}{\frac{1}{i^{y_{k,i}}(y_{k,i})!}}}=\sum_{r=0}^{m}{\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\\\ \sum{y_{k,i}}=r\end{subarray}}{\prod_{i=1}^{m}{\frac{1}{i^{y_{k,i}}(y_{k,i})!}}}}=\sum_{r=0}^{m}{\frac{1}{m!}{m\brack r}}=\frac{1}{m!}\sum_{r=0}^{m}{{m\brack r}}.$ However, we have already shown that the finite sum of ${m\brack r}$ is equal to $m!$. Hence, $\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\end{subarray}}{\prod_{i=1}^{m}{\frac{1}{i^{y_{k,i}}(y_{k,i})!}}}=1.$ ∎ A more general form of Lemma 4.2 is illustrated in the following lemma. ###### Lemma 4.4. Let $(\varphi_{1},\cdots,\varphi_{m})$ be a partition of $\varphi\leq m$ such that $\sum{\varphi_{i}}=r_{\varphi}$. Let $(y_{k,1},\cdots,y_{k,m})=\\{(y_{1,1},\cdots,y_{1,m}),(y_{2,1},\cdots,y_{2,m}),\cdots\\}$ be the set of all partitions of $m$. $\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\\\ \sum{y_{k,i}}=r\end{subarray}}{\prod_{i=1}^{m}{\frac{\binom{y_{k,i}}{\varphi_{i}}}{i^{y_{k,i}}(y_{k,i})!}}}=\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\\\ \sum{y_{k,i}}=r\\\ y_{k,i}\geq\varphi_{i}\end{subarray}}{\prod_{i=1}^{m}{\frac{\binom{y_{k,i}}{\varphi_{i}}}{i^{y_{k,i}}(y_{k,i})!}}}=\frac{1}{(m-\varphi)!}{m-\varphi\brack r-r_{\varphi}}\prod_{i=1}^{m}{\frac{1}{i^{\varphi_{i}}(\varphi_{i})!}}.$ ###### Remark. Knowing that the largest element of a partition of $\varphi$ is $\varphi$, we can rewrite it as follows $\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\\\ \sum{y_{k,i}}=r\end{subarray}}{\prod_{i=1}^{m}{\frac{\binom{y_{k,i}}{\varphi_{i}}}{i^{y_{k,i}}(y_{k,i})!}}}=\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\\\ \sum{y_{k,i}}=r\\\ y_{k,i}\geq\varphi_{i}\end{subarray}}{\prod_{i=1}^{m}{\frac{\binom{y_{k,i}}{\varphi_{i}}}{i^{y_{k,i}}(y_{k,i})!}}}=\frac{1}{(m-\varphi)!}{m-\varphi\brack r-r_{\varphi}}\prod_{i=1}^{\varphi}{\frac{1}{i^{\varphi_{i}}(\varphi_{i})!}}.$ ###### Proof. Knowing that $\binom{n}{k}$ is zero if $n<k$, then $\binom{y_{k,i}}{\varphi_{i}}=0$ if $\exists i\in\mathbb{N},y_{k,i}<\varphi_{i}$. Hence, $\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\\\ \sum{y_{k,i}}=r\end{subarray}}{\prod_{i=1}^{m}{\frac{\binom{y_{k,i}}{\varphi_{i}}}{i^{y_{k,i}}(y_{k,i})!}}}=\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\\\ \sum{y_{k,i}}=r\\\ \exists i,y_{k,i}<\varphi_{i}\end{subarray}}{\prod_{i=1}^{m}{\frac{\binom{y_{k,i}}{\varphi_{i}}}{i^{y_{k,i}}(y_{k,i})!}}}+\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\\\ \sum{y_{k,i}}=r\\\ y_{k,i}\geq\varphi_{i}\end{subarray}}{\prod_{i=1}^{m}{\frac{\binom{y_{k,i}}{\varphi_{i}}}{i^{y_{k,i}}(y_{k,i})!}}}=\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\\\ \sum{y_{k,i}}=r\\\ y_{k,i}\geq\varphi_{i}\end{subarray}}{\prod_{i=1}^{m}{\frac{\binom{y_{k,i}}{\varphi_{i}}}{i^{y_{k,i}}(y_{k,i})!}}}.$ The first part of the proof is complete. $\begin{split}\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\\\ \sum{y_{k,i}}=r\end{subarray}}{\prod_{i=1}^{m}{\frac{\binom{y_{k,i}}{\varphi_{i}}}{i^{y_{k,i}}(y_{k,i})!}}}&=\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\\\ \sum{y_{k,i}}=r\end{subarray}}{\prod_{i=1}^{m}{\frac{1}{i^{y_{k,i}}(y_{k,i})!}.\frac{y_{k,i}!}{\varphi_{i}!(y_{k,i}-\varphi_{i})!}}}\\\ &=\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\\\ \sum{y_{k,i}}=r\end{subarray}}{\prod_{i=1}^{m}{\frac{1}{i^{y_{k,i}}}.\frac{1}{\varphi_{i}!(y_{k,i}-\varphi_{i})!}}}\\\ &=\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\\\ \sum{y_{k,i}}=r\end{subarray}}{\prod_{i=1}^{m}{\frac{1}{i^{\varphi_{i}}\varphi_{i}!}.\frac{1}{i^{y_{k,i}-\varphi_{i}}(y_{k,i}-\varphi_{i})!}}}\\\ &=\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\\\ \sum{y_{k,i}}=r\end{subarray}}{\prod_{i=1}^{m}{\frac{1}{i^{\varphi_{i}}\varphi_{i}!}}\prod_{i=1}^{m}{\frac{1}{i^{y_{k,i}-\varphi_{i}}(y_{k,i}-\varphi_{i})!}}}.\end{split}$ As $\varphi_{1},\cdots,\varphi_{m}$ are all constants then $\prod_{i=1}^{m}{\frac{1}{i^{\varphi_{i}}\varphi_{i}!}}$ is constant. This factor is constant and is common to all terms of the sum, therefore, we can factor it and take it outside the sum. $\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\\\ \sum{y_{k,i}}=r\end{subarray}}{\prod_{i=1}^{m}{\frac{\binom{y_{k,i}}{\varphi_{i}}}{i^{y_{k,i}}(y_{k,i})!}}}=\left(\prod_{i=1}^{m}{\frac{1}{i^{\varphi_{i}}\varphi_{i}!}}\right)\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\\\ \sum{y_{k,i}}=r\end{subarray}}{\prod_{i=1}^{m}{\frac{1}{i^{y_{k,i}-\varphi_{i}}(y_{k,i}-\varphi_{i})!}}}.$ Having that $(\varphi_{1},\cdots,\varphi_{m})$ is a partition of $\varphi\leq m$, hence, $\sum{i.\varphi_{i}}=\varphi\leq m$. Thus, the condition $\sum{i.y_{k,i}}=m$ can be replaced by $\sum{i.(y_{k,i}-\varphi_{i})}=\sum{i.y_{k,i}}-\sum{i.\varphi_{i}}=m-\varphi$. Similarly, $r_{\varphi}=\sum{\varphi_{i}}$, hence, the condition $\sum{y_{k,i}}=r$ can be replaced by $\sum{(y_{k,i}-\varphi_{i})}=\sum{y_{k,i}}-\sum{\varphi_{i}}=r-r_{\varphi}$. Hence, $\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\\\ \sum{y_{k,i}}=r\end{subarray}}{\prod_{i=1}^{m}{\frac{\binom{y_{k,i}}{\varphi_{i}}}{i^{y_{k,i}}(y_{k,i})!}}}=\left(\prod_{i=1}^{m}{\frac{1}{i^{\varphi_{i}}\varphi_{i}!}}\right)\sum_{\begin{subarray}{c}k\\\ \sum{i.(y_{k,i}-\varphi_{i})}=m-\varphi\\\ \sum{(y_{k,i}-\varphi)}=r-r_{\varphi}\end{subarray}}{\prod_{i=1}^{m}{\frac{1}{i^{y_{k,i}-\varphi_{i}}(y_{k,i}-\varphi_{i})!}}}.$ Let $Y_{k,i}=y_{k,i}-\varphi_{i}$, $\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\\\ \sum{y_{k,i}}=r\end{subarray}}{\prod_{i=1}^{m}{\frac{\binom{y_{k,i}}{\varphi_{i}}}{i^{y_{k,i}}(y_{k,i})!}}}=\left(\prod_{i=1}^{m}{\frac{1}{i^{\varphi_{i}}\varphi_{i}!}}\right)\sum_{\begin{subarray}{c}k\\\ \sum{i.Y_{k,i}}=m-\varphi\\\ \sum{Y_{k,i}}=r-r_{\varphi}\end{subarray}}{\prod_{i=1}^{m}{\frac{1}{i^{Y_{k,i}}Y_{k,i}!}}}.$ Knowing that the largest element of a partition of $(m-\varphi)$ is $(m-\varphi)$, hence, $\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\\\ \sum{y_{k,i}}=r\end{subarray}}{\prod_{i=1}^{m}{\frac{\binom{y_{k,i}}{\varphi_{i}}}{i^{y_{k,i}}(y_{k,i})!}}}=\left(\prod_{i=1}^{m}{\frac{1}{i^{\varphi_{i}}\varphi_{i}!}}\right)\sum_{\begin{subarray}{c}k\\\ \sum{i.Y_{k,i}}=m-\varphi\\\ \sum{Y_{k,i}}=r-r_{\varphi}\end{subarray}}{\prod_{i=1}^{m-\varphi}{\frac{1}{i^{Y_{k,i}}Y_{k,i}!}}}.$ Applying Lemma 4.2, with $y_{k,i}$ substituted by $Y_{k,i}$, $m$ substituted by $m-\varphi$, and $r$ substituted by $r-r_{\varphi}$, we get $\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\\\ \sum{y_{k,i}}=r\end{subarray}}{\prod_{i=1}^{m}{\frac{\binom{y_{k,i}}{\varphi_{i}}}{i^{y_{k,i}}(y_{k,i})!}}}=\left(\prod_{i=1}^{m}{\frac{1}{i^{\varphi_{i}}\varphi_{i}!}}\right)\frac{1}{(m-\varphi)!}{m-\varphi\brack r-r_{\varphi}}.$ The proof is complete. ∎ ###### Remark. If $\varphi>m$, then $\sum{i.Y_{k,i}}=m-\varphi<0$ which makes Lemma 4.2 invalid which then makes this lemma invalid. Similarly, a more general form of Lemma 4.3 is illustrated in the following lemma. ###### Lemma 4.5. Let $(y_{k,1},\cdots,y_{k,m})=\\{(y_{1,1},\cdots,y_{1,m}),(y_{2,1},\cdots,y_{2,m}),\cdots\\}$ be the set of all partitions of $m$. Let $(\varphi_{1},\cdots,\varphi_{m})$ be a partition of $r\leq m$. $\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\end{subarray}}{\prod_{i=1}^{m}{\frac{\binom{y_{k,i}}{\varphi_{i}}}{i^{y_{k,i}}(y_{k,i})!}}}=\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\\\ y_{k,i}\geq\varphi_{i}\end{subarray}}{\prod_{i=1}^{m}{\frac{\binom{y_{k,i}}{\varphi_{i}}}{i^{y_{k,i}}(y_{k,i})!}}}=\prod_{i=1}^{m}{\frac{1}{i^{\varphi_{i}}(\varphi_{i})!}}.$ ###### Remark. Knowing that the largest element of a partition of $r$ is $r$, we can rewrite it as follows $\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\end{subarray}}{\prod_{i=1}^{m}{\frac{\binom{y_{k,i}}{\varphi_{i}}}{i^{y_{k,i}}(y_{k,i})!}}}=\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\\\ y_{k,i}\geq\varphi_{i}\end{subarray}}{\prod_{i=1}^{m}{\frac{\binom{y_{k,i}}{\varphi_{i}}}{i^{y_{k,i}}(y_{k,i})!}}}=\prod_{i=1}^{r}{\frac{1}{i^{\varphi_{i}}(\varphi_{i})!}}.$ ###### Proof. Knowing that $\binom{n}{k}$ is zero if $n<k$, then $\binom{y_{k,i}}{\varphi_{i}}=0$ if $\exists i\in\mathbb{N},y_{k,i}<\varphi_{i}$. Hence, $\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\end{subarray}}{\prod_{i=1}^{m}{\frac{\binom{y_{k,i}}{\varphi_{i}}}{i^{y_{k,i}}(y_{k,i})!}}}=\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\\\ \exists i,y_{k,i}<\varphi_{i}\end{subarray}}{\prod_{i=1}^{m}{\frac{\binom{y_{k,i}}{\varphi_{i}}}{i^{y_{k,i}}(y_{k,i})!}}}+\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\\\ y_{k,i}\geq\varphi_{i}\end{subarray}}{\prod_{i=1}^{m}{\frac{\binom{y_{k,i}}{\varphi_{i}}}{i^{y_{k,i}}(y_{k,i})!}}}=\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\\\ y_{k,i}\geq\varphi_{i}\end{subarray}}{\prod_{i=1}^{m}{\frac{\binom{y_{k,i}}{\varphi_{i}}}{i^{y_{k,i}}(y_{k,i})!}}}.$ The first part of the proof is complete. $\begin{split}\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\end{subarray}}{\prod_{i=1}^{m}{\frac{\binom{y_{k,i}}{\varphi_{i}}}{i^{y_{k,i}}(y_{k,i})!}}}&=\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\end{subarray}}{\prod_{i=1}^{m}{\frac{1}{i^{y_{k,i}}(y_{k,i})!}.\frac{y_{k,i}!}{\varphi_{i}!(y_{k,i}-\varphi_{i})!}}}\\\ &=\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\end{subarray}}{\prod_{i=1}^{m}{\frac{1}{i^{y_{k,i}}}.\frac{1}{\varphi_{i}!(y_{k,i}-\varphi_{i})!}}}\\\ &=\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\end{subarray}}{\prod_{i=1}^{m}{\frac{1}{i^{\varphi_{i}}\varphi_{i}!}.\frac{1}{i^{y_{k,i}-\varphi_{i}}(y_{k,i}-\varphi_{i})!}}}\\\ &=\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\end{subarray}}{\prod_{i=1}^{m}{\frac{1}{i^{\varphi_{i}}\varphi_{i}!}}\prod_{i=1}^{m}{\frac{1}{i^{y_{k,i}-\varphi_{i}}(y_{k,i}-\varphi_{i})!}}}.\end{split}$ As $\varphi_{1},\cdots,\varphi_{m}$ are all constants then $\prod_{i=1}^{m}{\frac{1}{i^{\varphi_{i}}\varphi_{i}!}}$ is constant. This factor is constant and is common to all terms of the sum, therefore, we can factor it and take it outside the sum. $\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\end{subarray}}{\prod_{i=1}^{m}{\frac{\binom{y_{k,i}}{\varphi_{i}}}{i^{y_{k,i}}(y_{k,i})!}}}=\left(\prod_{i=1}^{m}{\frac{1}{i^{\varphi_{i}}\varphi_{i}!}}\right)\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\end{subarray}}{\prod_{i=1}^{m}{\frac{1}{i^{y_{k,i}-\varphi_{i}}(y_{k,i}-\varphi_{i})!}}}.$ Having that $(\varphi_{1},\cdots,\varphi_{m})$ is a partition of $r\leq m$, hence, $\sum{i.\varphi_{i}}=r\leq m$. Thus, the condition $\sum{i.y_{k,i}}=m$ can be replaced by $\sum{i.(y_{k,i}-\varphi_{i})}=\sum{i.y_{k,i}}-\sum{i.\varphi_{i}}=m-r(\geq 0)$. Hence, $\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\end{subarray}}{\prod_{i=1}^{m}{\frac{\binom{y_{k,i}}{\varphi_{i}}}{i^{y_{k,i}}(y_{k,i})!}}}=\left(\prod_{i=1}^{m}{\frac{1}{i^{\varphi_{i}}\varphi_{i}!}}\right)\sum_{\begin{subarray}{c}k\\\ \sum{i.(y_{k,i}-\varphi_{i})}=m-r\end{subarray}}{\prod_{i=1}^{m}{\frac{1}{i^{y_{k,i}-\varphi_{i}}(y_{k,i}-\varphi_{i})!}}}.$ Let $Y_{k,i}=y_{k,i}-\varphi_{i}$, $\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\end{subarray}}{\prod_{i=1}^{m}{\frac{\binom{y_{k,i}}{\varphi_{i}}}{i^{y_{k,i}}(y_{k,i})!}}}=\left(\prod_{i=1}^{m}{\frac{1}{i^{\varphi_{i}}\varphi_{i}!}}\right)\sum_{\begin{subarray}{c}k\\\ \sum{i.Y_{k,i}}=m-r\end{subarray}}{\prod_{i=1}^{m}{\frac{1}{i^{Y_{k,i}}Y_{k,i}!}}}.$ Knowing that the largest element of a partition of $(m-r)$ is $(m-r)$, hence, $\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\end{subarray}}{\prod_{i=1}^{m}{\frac{\binom{y_{k,i}}{\varphi_{i}}}{i^{y_{k,i}}(y_{k,i})!}}}=\left(\prod_{i=1}^{m}{\frac{1}{i^{\varphi_{i}}\varphi_{i}!}}\right)\sum_{\begin{subarray}{c}k\\\ \sum{i.Y_{k,i}}=m-r\end{subarray}}{\prod_{i=1}^{m-r}{\frac{1}{i^{Y_{k,i}}Y_{k,i}!}}}.$ Applying Lemma 4.3, with $y_{k,i}$ substituted by $Y_{k,i}$ and $m$ substituted by $m-r$, we get $\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\end{subarray}}{\prod_{i=1}^{m}{\frac{\binom{y_{k,i}}{\varphi_{i}}}{i^{y_{k,i}}(y_{k,i})!}}}=\left(\prod_{i=1}^{m}{\frac{1}{i^{\varphi_{i}}\varphi_{i}!}}\right).$ The proof is complete. ∎ ###### Remark. If $r>m$, then $\sum{i.Y_{k,i}}=m-r<0$ which makes Lemma 4.3 invalid which then makes this lemma invalid. ###### Proposition 4.1. Let $B_{m,r}(x_{1},\cdots,x_{m-r+1})$ be the partial Bell polynomial and $B_{m}(x_{1},\cdots,x_{m})$ be the complete Bell polynomial, $\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!}\left(\frac{1}{i}\sum_{N=q}^{n}{(a_{N})^{i}}\right)^{y_{k,i}}}}=\frac{1}{m!}\sum_{r=0}^{m}{B_{m,r}(x_{1},\cdots,x_{m-r+1})}=\frac{1}{m!}B_{m}(x_{1},\cdots,x_{m})$ where $x_{i}=(i-1)!(\sum_{N=q}^{n}{(a_{N})^{i}})$. ###### Proof. From Lemma 4.1, we can write $\sum_{\begin{subarray}{c}\sum{i.y_{k,i}}=m\\\ \sum{y_{k,i}}=r\end{subarray}}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!}\left(\frac{1}{i}\sum_{N=q}^{n}{(a_{N})^{i}}\right)^{y_{k,i}}}}=\sum_{\begin{subarray}{c}\sum{i.y_{k,i}}=m\\\ \sum{y_{k,i}}=r\end{subarray}}{\prod_{i=1}^{m-r+1}{\frac{1}{(y_{k,i})!}\left(\frac{1}{i}\sum_{N=q}^{n}{(a_{N})^{i}}\right)^{y_{k,i}}}}.$ We can notice that the right side term of the previous expression corresponds to a multiple of a special value of the partial Bell polynomial where $x_{i}=(i-1)!(\sum_{N=q}^{n}{(a_{N})^{i}}),\forall i\in[1,m]$. Hence, $\sum_{\begin{subarray}{c}\sum{i.y_{k,i}}=m\\\ \sum{y_{k,i}}=r\end{subarray}}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!}\left(\frac{1}{i}\sum_{N=q}^{n}{(a_{N})^{i}}\right)^{y_{k,i}}}}=\frac{1}{m!}B_{m,r}(x_{1},\cdots,x_{m-r+1}).$ Additionally, the sum over the partitions of $m$ is equivalent to the sum for $r$ going from $0$ to $m$ of the sums over the partitions of $m$ of length $r$. Thus, $\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!}\left(\frac{1}{i}\sum_{N=q}^{n}{(a_{N})^{i}}\right)^{y_{k,i}}}}=\sum_{r=0}^{m}{\sum_{\begin{subarray}{c}\sum{i.y_{k,i}}=m\\\ \sum{y_{k,i}}=r\end{subarray}}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!}\left(\frac{1}{i}\sum_{N=q}^{n}{(a_{N})^{i}}\right)^{y_{k,i}}}}}=\frac{1}{m!}\sum_{r=0}^{m}{B_{m,r}(x_{1},\cdots,x_{m-r+1})}.$ Applying the definition of a complete Bell polynomial, we get $\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!}\left(\frac{1}{i}\sum_{N=q}^{n}{(a_{N})^{i}}\right)^{y_{k,i}}}}=\frac{1}{m!}B_{m}(x_{1},\cdots,x_{m}).$ ∎ Now that all the required lemmas have been proven, we show the following theorem which allows the representation of a recurrent sum in terms of non- recurrent sums. ###### Theorem 4.1 (Reduction Theorem). Let $m$ be a non-negative integer, $k$ be the index of the $k$-th partition of $m$ $(1\leq k\leq p(m))$, $i$ be an integer between $1$ and $m$, and $y_{k,i}$ be the multiplicity of $i$ in the $k$-th partition of $m$. The reduction theorem for recurrent sums is stated as follow: $\sum_{N_{m}=q}^{n}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{m}}\cdots a_{N_{1}}}}=\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\end{subarray}}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!}\left(\frac{1}{i}\sum_{N=q}^{n}{(a_{N})^{i}}\right)^{y_{k,i}}}}.$ ###### Proof. 1\. Base Case: verify true for $n=q$, $\forall m\in\mathbb{N}$. $\begin{split}\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!}\left(\frac{1}{i}\sum_{N=q}^{q}{(a_{N})^{i}}\right)^{y_{k,i}}}}&=\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!i^{y_{k,i}}}\left(a_{q}\right)^{i.y_{k,i}}}}\\\ &=\sum_{\sum{i.y_{k,i}}=m}{\left(a_{q}\right)^{\sum{i.y_{k,i}}}\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!i^{y_{k,i}}}}}\\\ &=\left(a_{q}\right)^{m}\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!i^{y_{k,i}}}}}.\end{split}$ By applying Lemma 4.3, we get $\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!}\left(\frac{1}{i}\sum_{N=q}^{q}{(a_{N})^{i}}\right)^{y_{k,i}}}}=\left(a_{q}\right)^{m}.$ Likewise, $\sum_{N_{m}=q}^{q}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{m}}\cdots a_{N_{1}}}}=a_{q}\cdots a_{q}=\left(a_{q}\right)^{m}.$ 2\. Induction hypothesis: assume the statement is true until $n$, $\forall m\in\mathbb{N}$. $\sum_{N_{m}=q}^{n}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{m}}\cdots a_{N_{1}}}}=\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!}\left(\frac{1}{i}\sum_{N=q}^{n}{(a_{N})^{i}}\right)^{y_{k,i}}}}.$ 3\. Induction step: we will show that this statement is true for $(n+1)$, $\forall m\in\mathbb{N}$. We have to show the following statement to be true: $\sum_{N_{m}=q}^{n+1}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{m}}\cdots a_{N_{1}}}}=\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!}\left(\frac{1}{i}\sum_{N=q}^{n+1}{(a_{N})^{i}}\right)^{y_{k,i}}}}.$ $\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!}\left(\frac{1}{i}\sum_{N=q}^{n+1}{(a_{N})^{i}}\right)^{y_{k,i}}}}=\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!i^{y_{k,i}}}\left(\sum_{N=q}^{n}{(a_{N})^{i}}+(a_{n+1})^{i}\right)^{y_{k,i}}}}.$ The binomial theorem states that $(a+b)^{n}=\sum_{\varphi=0}^{n}{\binom{n}{\varphi}a^{n-\varphi}b^{\varphi}}.$ Hence, $\begin{split}\left(\sum_{N=q}^{n}{(a_{N})^{i}}+(a_{n+1})^{i}\right)^{y_{k,i}}&=\sum_{\varphi=0}^{y_{k,i}}{\binom{y_{k,i}}{\varphi}{\left(\sum_{N=q}^{n}{(a_{N})^{i}}\right)}^{\varphi}{\left((a_{n+1})^{i}\right)}^{y_{k,i}-\varphi}}.\end{split}$ Thus, $\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!}\left(\frac{1}{i}\sum_{N=q}^{n+1}{(a_{N})^{i}}\right)^{y_{k,i}}}}=\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!i^{y_{k,i}}}\sum_{\varphi=0}^{y_{k,i}}{\binom{y_{k,i}}{\varphi}{\left(\sum_{N=q}^{n}{(a_{N})^{i}}\right)}^{\varphi}{\left((a_{n+1})^{i}\right)}^{y_{k,i}-\varphi}}}}=\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\sum_{\varphi=0}^{y_{k,i}}{\frac{1}{(y_{k,i})!i^{y_{k,i}}}\binom{y_{k,i}}{\varphi}{\left(\sum_{N=q}^{n}{(a_{N})^{i}}\right)}^{\varphi}{\left(a_{n+1}\right)}^{i.y_{k,i}-i.\varphi}}}}.$ Let $A_{\varphi,i,k}={\frac{1}{(y_{k,i})!i^{y_{k,i}}}\binom{y_{k,i}}{\varphi}{\left(\sum_{N=q}^{n}{(a_{N})^{i}}\right)}^{\varphi}{\left(a_{n+1}\right)}^{i.y_{k,i}-i.\varphi}}$. By expanding then regrouping, it can be seen that $\prod_{i=1}^{m}{\sum_{\varphi=0}^{y_{k,i}}{A_{\varphi,i,k}}}=\sum_{\varphi_{m}=0}^{y_{k,m}}{\cdots\sum_{\varphi_{1}=0}^{y_{k,1}}{\prod_{i=1}^{m}{A_{\varphi_{i},i,k}}}}.$ This is because, for any given $k$, by expanding the product of sums (the left hand side term), we will get a sum of products of the form $A_{\varphi_{1},1}A_{\varphi_{2},2}\cdots A_{\varphi_{m},m}$ ($\prod_{i=1}^{m}{A_{\varphi_{i},i}}$) for all combinations of $\varphi_{1},\varphi_{2},\cdots,\varphi_{m}$ such that $0\leq\varphi_{1}\leq y_{k,1},\cdots,0\leq\varphi_{m}\leq y_{k,m}$, which is equivalent to the right hand side term. Hence, $\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!}\left(\frac{1}{i}\sum_{N=q}^{n+1}{(a_{N})^{i}}\right)^{y_{k,i}}}}=\sum_{\sum{i.y_{k,i}}=m}{\sum_{\varphi_{m}=0}^{y_{k,m}}{\cdots\sum_{\varphi_{1}=0}^{y_{k,1}}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!i^{y_{k,i}}}\binom{y_{k,i}}{\varphi_{i}}{\left(\sum_{N=q}^{n}{(a_{N})^{i}}\right)}^{\varphi_{i}}{\left(a_{n+1}\right)}^{i.y_{k,i}-i.\varphi_{i}}}}}}.$ A more compact way of writing the repeated sum over the $\varphi_{i}$’s is by expressing it with one sum that combines all the conditions. The set of conditions $0\leq\varphi_{1}\leq y_{k,1},\cdots,0\leq\varphi_{m}\leq y_{k,m}$ can be expressed as the condition $0\leq\varphi_{i}\leq y_{k,i}$ for $i\in[1,m]$. $\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!}\left(\frac{1}{i}\sum_{N=q}^{n+1}{(a_{N})^{i}}\right)^{y_{k,i}}}}=\sum_{\sum{i.y_{k,i}}=m}{\sum_{0\leq\varphi_{i}\leq y_{k,i}}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!i^{y_{k,i}}}\binom{y_{k,i}}{\varphi_{i}}{\left(\sum_{N=q}^{n}{(a_{N})^{i}}\right)}^{\varphi_{i}}{\left(a_{n+1}\right)}^{i.y_{k,i}-i.\varphi_{i}}}}}.$ Similarly, let $j$ represent $\sum{i.\varphi_{i}}$. Hence, we can add the trivial condition that is $j=\sum{i.\varphi_{i}}$ to the sum over $\varphi_{i}$. Additionally, $\sum{i.\varphi_{i}}=j$ is minimal when $\varphi_{1}=0,\cdots,\varphi_{m}=0$. Hence $j_{min}=0$. $\sum{i.\varphi_{i}}=j$ is maximal when $\varphi_{1}=y_{k,1},\cdots,\varphi_{m}=y_{k,m}$. Hence $j_{max}=\sum{i.y_{k,i}}=m$. Therefore, we have that $0\leq j\leq m$ or equivalently that $j$ can go from $0$ to $m$. Hence, knowing that adding a true statement to a condition does not change the condition, we can add this additional condition to get $\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!}\left(\frac{1}{i}\sum_{N=q}^{n+1}{(a_{N})^{i}}\right)^{y_{k,i}}}}=\sum_{\sum{i.y_{k,i}}=m}{\sum_{\begin{subarray}{c}j=0\\\ \sum{i.\varphi_{i}}=j\\\ 0\leq\varphi_{i}\leq y_{k,i}\end{subarray}}^{m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!i^{y_{k,i}}}\binom{y_{k,i}}{\varphi_{i}}{\left(\sum_{N=q}^{n}{(a_{N})^{i}}\right)}^{\varphi_{i}}{\left(a_{n+1}\right)}^{i.y_{k,i}-i.\varphi_{i}}}}}.$ Knowing that $\binom{y_{k,i}}{\varphi_{i}}=0$ if $\varphi_{i}>y_{k,i}$, hence, the terms produced for $\varphi_{i}>y_{k,i}$ would be zero. Thus, we can remove the condition $0\leq\varphi_{i}\leq y_{k,i}$ because terms that do not satisfy this condition will be zeros and, therefore, would not change the value of the sum. $\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!}\left(\frac{1}{i}\sum_{N=q}^{n+1}{(a_{N})^{i}}\right)^{y_{k,i}}}}=\sum_{\sum{i.y_{k,i}}=m}{\sum_{\begin{subarray}{c}j=0\\\ \sum{i.\varphi_{i}}=j\end{subarray}}^{m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!i^{y_{k,i}}}\binom{y_{k,i}}{\varphi_{i}}{\left(\sum_{N=q}^{n}{(a_{N})^{i}}\right)}^{\varphi_{i}}{\left(a_{n+1}\right)}^{i.y_{k,i}-i.\varphi_{i}}}}}.$ We expand the expression then, from all values of $k$ (from every partitions $(y_{k,1},\cdots,y_{k,m})$ of $m$), we regroup together the terms having a combination of exponents $(\varphi_{1},\cdots,\varphi_{m})$ that forms a partition of the same integer $j$ and we do so $\forall j\in[0,m]$. Hence, performing this manipulation allows us to interchange the sum over $k$ (over $\sum_{i}{i.y_{k,i}}=m$) with the sums over $j$. Thus, the expression becomes as follows, $\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!}\left(\frac{1}{i}\sum_{N=q}^{n+1}{(a_{N})^{i}}\right)^{y_{k,i}}}}=\sum_{\begin{subarray}{c}j=0\\\ \sum{i.\varphi_{i}}=j\end{subarray}}^{m}{\sum_{\begin{subarray}{c}\sum{i.y_{k,i}}=m\end{subarray}}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!i^{y_{k,i}}}\binom{y_{k,i}}{\varphi_{i}}{\left(\sum_{N=q}^{n}{(a_{N})^{i}}\right)}^{\varphi_{i}}{\left(a_{n+1}\right)}^{i.y_{k,i}-i.\varphi_{i}}}}}=\sum_{\begin{subarray}{c}j=0\\\ \sum{i.\varphi_{i}}=j\end{subarray}}^{m}{\sum_{\begin{subarray}{c}\sum{i.y_{k,i}}=m\end{subarray}}{{\left(a_{n+1}\right)}^{\sum{i.y_{k,i}}-\sum{i.\varphi_{i}}}\left[\prod_{i=1}^{m}{{\left(\sum_{N=q}^{n}{(a_{N})^{i}}\right)}^{\varphi_{i}}}\right]\left[\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!i^{y_{k,i}}}\binom{y_{k,i}}{\varphi_{i}}}\right]}}=\sum_{\begin{subarray}{c}j=0\\\ \sum{i.\varphi_{i}}=j\end{subarray}}^{m}{{\left(a_{n+1}\right)}^{m-j}\left[\prod_{i=1}^{m}{{\left(\sum_{N=q}^{n}{(a_{N})^{i}}\right)}^{\varphi_{i}}}\right]\left(\sum_{\begin{subarray}{c}\sum{i.y_{k,i}}=m\end{subarray}}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!i^{y_{k,i}}}\binom{y_{k,i}}{\varphi_{i}}}}\right)}.$ Applying Lemma 4.5, we get $\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!}\left(\frac{1}{i}\sum_{N=q}^{n+1}{(a_{N})^{i}}\right)^{y_{k,i}}}}=\sum_{\begin{subarray}{c}j=0\\\ \sum{i.\varphi_{i}}=j\end{subarray}}^{m}{{\left(a_{n+1}\right)}^{m-j}\left[\prod_{i=1}^{m}{{\left(\sum_{N=q}^{n}{(a_{N})^{i}}\right)}^{\varphi_{i}}}\right]\left(\prod_{i=1}^{m}{\frac{1}{i^{\varphi_{i}}(\varphi_{i})!}}\right)}=\sum_{\begin{subarray}{c}j=0\\\ \sum{i.\varphi_{i}}=j\end{subarray}}^{m}{{\left(a_{n+1}\right)}^{m-j}\left(\prod_{i=1}^{m}{\frac{1}{i^{\varphi_{i}}(\varphi_{i})!}}{\left(\sum_{N=q}^{n}{(a_{N})^{i}}\right)}^{\varphi_{i}}\right)}.$ Knowing that for any given value of $j$ there is multiple combinations of $\varphi_{1},\cdots,\varphi_{m}$ that satisfy $\sum{i.\varphi_{i}}=j$. Hence, every value of $j$ corresponds to a sum of the sum’s argument for all partitions of $j$ (for all combinations of $\varphi_{1},\cdots,\varphi_{m}$ satisfying $\sum{i.\varphi_{i}}=j$). Therefore, we can split the outer sum with two conditions into two sums each with one of the conditions as follows, $\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!}\left(\frac{1}{i}\sum_{N=q}^{n+1}{(a_{N})^{i}}\right)^{y_{k,i}}}}=\sum_{j=0}^{m}{{\left(a_{n+1}\right)}^{m-j}\sum_{\sum{i.\varphi_{i}}=j}{\left(\prod_{i=1}^{m}{\frac{1}{i^{\varphi_{i}}(\varphi_{i})!}}{\left(\sum_{N=q}^{n}{(a_{N})^{i}}\right)}^{\varphi_{i}}\right)}}.$ Knowing that the largest element of a partition of $j$ is $j$, $\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!}\left(\frac{1}{i}\sum_{N=q}^{n+1}{(a_{N})^{i}}\right)^{y_{k,i}}}}=\sum_{j=0}^{m}{{\left(a_{n+1}\right)}^{m-j}\left(\sum_{\sum{i.\varphi_{i}}=j}{\prod_{i=1}^{j}{\frac{1}{i^{\varphi_{i}}(\varphi_{i})!}}{\left(\sum_{N=q}^{n}{(a_{N})^{i}}\right)}^{\varphi_{i}}}\right)}.$ By using the induction hypothesis, the expression becomes $\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!}\left(\frac{1}{i}\sum_{N=q}^{n+1}{(a_{N})^{i}}\right)^{y_{k,i}}}}=\sum_{j=0}^{m}{\left(a_{n+1}\right)^{m-j}\left(\sum_{N_{j}=q}^{n}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{j}}\cdots a_{N_{1}}}}\right)}.$ Using Corollary 2.1, we get $\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!}\left(\frac{1}{i}\sum_{N=q}^{n+1}{(a_{N})^{i}}\right)^{y_{k,i}}}}=\sum_{N_{m}=q}^{n+1}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{N_{m}}\cdots a_{N_{1}}}}.$ The theorem is proven by induction. ∎ ###### Corollary 4.1. If the recurrent sum starts at 1, Theorem 4.1 becomes $\sum_{N_{m}=1}^{n}{\cdots\sum_{N_{1}=1}^{N_{2}}{a_{N_{m}}\cdots a_{N_{1}}}}=\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!}\left(\frac{1}{i}\sum_{N=1}^{n}{(a_{N})^{i}}\right)^{y_{k,i}}}}.$ An additional partition identity that can be deduced from Theorem 4.1 is as follows. ###### Corollary 4.2. For any $m,n\in\mathbb{N}$, we have that $\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!}\left(\frac{n}{i}\right)^{y_{k,i}}}}=\binom{n+m-1}{m}.$ ###### Proof. From paper [21], we have the following relation, $\sum_{N_{m}=1}^{n}{\cdots\sum_{N_{1}=1}^{N_{2}}{1}}=\binom{n+m-1}{m}.$ By applying Theorem 4.1, we get $\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!}\left(\frac{1}{i}\sum_{N=1}^{n}{1}\right)^{y_{k,i}}}}=\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!}\left(\frac{n}{i}\right)^{y_{k,i}}}}=\binom{n+m-1}{m}.$ ∎ ###### Example 4.1. For $n=1$, Corollary 4.2 gives $\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!i^{y_{k,i}}}}}=\binom{m}{m}=1.$ ###### Example 4.2. For $n=2$, Corollary 4.2 gives $\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{2^{y_{k,i}}}{(y_{k,i})!i^{y_{k,i}}}}}=\binom{m+1}{m}=m+1.$ ###### Example 4.3. For $n=3$, Corollary 4.2 gives $\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{3^{y_{k,i}}}{(y_{k,i})!i^{y_{k,i}}}}}=\binom{m+2}{m}=\frac{(m+1)(m+2)}{2}.$ ### 4.3 Particular cases In this section, we will apply the reduction formula for the cases of $m$ from $1$ to $4$. These cases were independently proven using two distinct methods (which are omitted here for simplicity). Similarly, these formulas were verified for a certain range of $n$ using a computer program which calculated the right expression as well as the left expression then checks that they are equal. * • For $m=1$ $\sum_{N_{1}=1}^{n}{a_{N_{1}}}=\sum_{N=1}^{n}{a_{N}}.$ * • For $m=2$ $\sum_{N_{2}=1}^{n}{\sum_{N_{1}=1}^{N_{2}}{a_{N_{2}}a_{N_{1}}}}=\frac{1}{2}\left(\sum_{N=1}^{n}{a_{N}}\right)^{2}+\frac{1}{2}\left(\sum_{N=1}^{n}{\left(a_{N}\right)^{2}}\right).$ * • For $m=3$ $\sum_{N_{3}=1}^{n}{\sum_{N_{2}=1}^{N_{3}}{\sum_{N_{1}=1}^{N_{2}}{a_{N_{3}}a_{N_{2}}a_{N_{1}}}}}=\frac{1}{6}\left(\sum_{N=1}^{n}{a_{N}}\right)^{3}+\frac{1}{2}\left(\sum_{N=1}^{n}{a_{N}}\right)\left(\sum_{N=1}^{n}{\left(a_{N}\right)^{2}}\right)+\frac{1}{3}\left(\sum_{N=1}^{n}{\left(a_{N}\right)^{3}}\right).$ * • For $m=4$ $\sum_{N_{4}=1}^{n}{\sum_{N_{3}=1}^{N_{4}}{\sum_{N_{2}=1}^{N_{3}}{\sum_{N_{1}=1}^{N_{2}}{a_{N_{4}}a_{N_{3}}a_{N_{2}}a_{N_{1}}}}}}=\frac{1}{24}\left(\sum_{N=1}^{n}{a_{N}}\right)^{4}+\frac{1}{4}\left(\sum_{N=1}^{n}{a_{N}}\right)^{2}\left(\sum_{N=1}^{n}{\left(a_{N}\right)^{2}}\right)+\frac{1}{3}\left(\sum_{N=1}^{n}{a_{N}}\right)\left(\sum_{N=1}^{n}{\left(a_{N}\right)^{3}}\right)+\frac{1}{8}\left(\sum_{N=1}^{n}{\left(a_{N}\right)^{2}}\right)^{2}+\frac{1}{4}\left(\sum_{N=1}^{n}{\left(a_{N}\right)^{4}}\right).$ ### 4.4 General Reduction Theorem We define the notation $\|A\|$ as the number of elements in the set $A$. Note that if $A$ is a set of sets then $\|A\|$ represents the number of sets in $A$ as they are considered the elements of $A$. Let $m$ be a non-negative integer and let $\\{(y_{1,1},\cdots,y_{1,m}),(y_{2,1},\cdots,y_{2,m}),\cdots\\}$ be the set of all partitions of $m$. Let us consider the set $M=\\{1,\cdots,m\\}$. The permutation group $S_{m}$ is the set of all permutations of the set $\\{1,\cdots,m\\}$. Let $\sigma\in S_{m}$ be a permutation of the set $\\{1,\cdots,m\\}$ and let $\sigma(i)$ represent the $i$-th element of this given permutation. The number of such permutations is given by $\|S_{m}\|=m!.$ (13) The cycle-type of a permutation $\sigma$ is the ordered set where the $i$-th element represents the number of cycles of size $i$ in the cycle decomposition of $\sigma$. The number of ways of arranging $i$ elements cyclically is $(i-1)!$. The number of possible combinations of $y_{k,i}$ cycles of size $i$ is $[(i-1)!]^{y_{k,i}}$. Hence, the number of permutations having cycle-type $(y_{k,1},\cdots,y_{k,m})$ is given by $\prod_{i=1}^{m}{[(i-1)!]^{y_{k,i}}}.$ (14) A partition $P$ of a set $M$ is a set of non-empty disjoint subsets of $M$ such that every element of $M$ is present in exactly one of the subsets. Let $P=\\{\underbrace{P_{1,1},\cdots,P_{1,y_{1}}}_{y_{1}\,\,sets},\cdots,\underbrace{P_{m,1},\cdots,P_{m,y_{m}}}_{y_{m}\,\,sets}\\}$ represent a partition of a set of $m$ elements (for our purpose let it be the set $\\{1,\cdots,m\\}$). $P_{i,y}$ represents the $y$-th subset of order (size) $i$. $y_{i}$ represents the number of subsets of size $i$ contained in this partition of the set. It is interesting to note that $(y_{1},\cdots,y_{m})$ will always form a partition of the non-negative integer $m$. However, the number of partitions of $m$ is different from the number of partitions of a set of $m$ elements because there are more than one partition of the set of $m$ elements that can be associated with a given partition of $m$. In fact, we can easily determine that the number of partitions of a set of $m$ elements associated with the partition $(y_{1},\cdots,y_{m})$ is given by $\|\Omega_{k}\|=\frac{m!}{1!^{y_{k,1}}\cdots m!^{y_{k,m}}(y_{k,1})!\cdots(y_{k,m})!}=\frac{m!}{\prod_{i=1}^{m}{i!^{y_{k,i}}y_{k,i}!}}.$ (15) where $\Omega_{k}$ is the set of all partitions of the set of $m$ elements associated the partition $(y_{k,i},\cdots,y_{k,m})$. This is because the number of ways to divide $m$ objects into $l_{1}$ groups of $1$ element, $l_{2}$ groups of $2$ elements, $\cdots$, and $l_{m}$ groups of $m$ elements is given by $\frac{m!}{1!^{l_{1}}\cdots m!^{l_{m}}l_{1}!\cdots l_{m}!}=\frac{m!}{\prod_{i=1}^{m}{i!^{l_{i}}l_{i}!}}.$ (16) We will denote by $\Omega$ the set of all partitions of the set of $m$ elements. Finally, a partition $P$ of a set $M$ is a refinement of a partition $\rho$ of the same set $M$ if every element in $P$ is a subset of an element in $\rho$. We denote this as $P\succeq\rho$. Using the notation introduced, we can formulate a generalization of Theorem 4.1 where all sequences are distinct. ###### Theorem 4.2. Let $m,n,q\in\mathbb{N}$ such that $n\geq q$. Let $a_{(1);N},\cdots,a_{(m);N}$ be $m$ sequences defined in the interval $[q,n]$. we have that $\begin{split}\sum_{\sigma\in S_{m}}{\left(\sum_{N_{m}=q}^{n}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{(\sigma(m));N_{m}}\cdots a_{(\sigma(1));N_{1}}}}\right)}=\sum_{\begin{subarray}{c}P\in\Omega\end{subarray}}{\prod_{i=1}^{m}{[(i-1)!]^{y_{k,i}}\left[\prod_{g=1}^{y_{k,i}}{\left(\sum_{N=q}^{n}{\prod_{h\in P_{i,g}}{a_{(h);N}}}\right)}\right]}}.\end{split}$ ###### Remark. The theorem can also be written as $\begin{split}&\sum_{\sigma\in S_{m}}{\left(\sum_{N_{m}=q}^{n}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{(\sigma(m));N_{m}}\cdots a_{(\sigma(1));N_{1}}}}\right)}\\\ &\,\,=\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\end{subarray}}{\sum_{\Omega_{k}}{\prod_{i=1}^{m}{[(i-1)!]^{y_{k,i}}\left[\prod_{g=1}^{y_{k,i}}{\left(\sum_{N=q}^{n}{\prod_{h\in P_{i,g}}{a_{(h);N}}}\right)}\right]}}}\\\ &\,\,=\|S_{m}\|\sum_{\begin{subarray}{c}k\\\ \sum{i.y_{k,i}}=m\end{subarray}}{\frac{1}{\|\Omega_{k}\|}\sum_{\Omega_{k}}{\prod_{i=1}^{m}{\frac{1}{y_{k,i}!i^{y_{k,i}}}\left[\prod_{g=1}^{y_{k,i}}{\left(\sum_{N=q}^{n}{\prod_{h\in P_{i,g}}{a_{(h);N}}}\right)}\right]}}}.\end{split}$ As every partition of a set of $m$ elements is associated with a given partition of $m$, hence, adding up all the partitions of the set for ever given partition of $m$ is equivalent to adding up all partitions of the set. The first form is obtained by regrouping together, from the set of all partitions of the set $\\{1,\cdots,m\\}$, those who are associated with a given partition of $m$. The second expression is obtained by noting that $\frac{\|S_{m}\|}{\|\Omega_{k}\|}\prod_{i=1}^{m}{\frac{1}{y_{k,i}!i^{y_{k,i}}}}=\prod_{i=1}^{m}{{[(i-1)!]^{y_{k,i}}}}$. These forms are shown as they can be more easily used to show that this theorem reduces to Theorem 4.1 if all sequences are the same. ###### Proof. Both sides of the equation produce all combinations of terms which are products of the $m$ sequences. Hence, the strategy of this proof is to show that every combination appear with the same multiplicity on both sides. We can assume the sequences to all be distinct without lost of generality. We can write $\sum_{\sigma\in S_{m}}{\left(\sum_{N_{m}=q}^{n}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{(\sigma(m));N_{m}}\cdots a_{(\sigma(1));N_{1}}}}\right)}=\sum_{\sigma\in S_{m}}{\left(\sum_{N_{m}=q}^{n}{\cdots\sum_{N_{1}=q}^{N_{2}}{a_{(m);N_{\sigma(m)}}\cdots a_{(1);N_{\sigma(1)}}}}\right)}.$ Hence, we can consider the symmetric group $S_{m}$ as acting on $N=(N_{1},\cdots,N_{m})$. $N=(N_{1},\cdots,N_{m})$ has an isotropy group $S_{m}(N)$ and an associated partition $\rho$ of the set of $m$ elements. The partition $\rho$ is the set of all equivalence classes of the relation given by $a\sim b$ if and only if $N_{a}=N_{b}$ and $S_{m}(N)=\\{\sigma\in S_{m}\,\,\|\,\,\sigma(i)\sim i\,\,\forall i\\}$. Thus, $a_{(m);N_{m}}\cdots a_{(1);N_{1}}$ (17) appears $\|S_{m}(N)\|$ times in the expansion of the left hand side of the theorem. Likewise, in the right hand side, (17) can only appears in the terms corresponding to partitions $P$ which are refinements of $\rho$. (17) appears $\sum_{P\succeq\rho}{\prod_{i=1}^{m}{[(i-1)!]^{y_{k,i}}}}$ (18) times in the right hand side of the theorem. Also let us notice that $[(i-1)!]^{y_{k,i}}$ corresponds to $(\|P_{i,1}\|-1)!\cdots(\|P_{i,y_{k,i}}\|-1)!$ because $\|P_{i,1}\|=\cdots=\|P_{i,y_{k,i}}\|=i$. Hence, $\prod_{i=1}^{m}{[(i-1)!]^{y_{k,i}}}$ corresponds to $\prod_{P_{h,g}\subset P}{(\|P_{h,g}\|-1)!}$ which is equal to the number of permutations having cycle- type specified by $P$. Knowing that any element of $S_{m}(N)$ has a unique cycle-type specified by a partition that refines $\rho$, hence, we conclude that $\sum_{P\succeq\rho}{\prod_{i=1}^{m}{[(i-1)!]^{y_{k,i}}}}=\|S_{m}(N)\|.$ (19) As both sides of the theorem produce the same terms and with the same multiplicity, we can say that these sides are equal to each other. ∎ ###### Example 4.4. For $m=2$, Theorem 4.2 gives the following, $\sum_{N_{2}=q}^{n}{\sum_{N_{1}=q}^{N_{2}}{a_{N_{2}}b_{N_{1}}}}+\sum_{N_{2}=q}^{n}{\sum_{N_{1}=q}^{N_{2}}{b_{N_{2}}a_{N_{1}}}}=\left(\sum_{N=q}^{n}{a_{N}}\right)\left(\sum_{N=q}^{n}{b_{N}}\right)+\left(\sum_{N=q}^{n}{a_{N}b_{N}}\right).$ ###### Example 4.5. For $m=3$, Theorem 4.2 gives the following, $\sum_{\sigma\in S_{3}}{\left(\sum_{N_{3}=q}^{n}{\sum_{N_{2}=q}^{N_{3}}{\sum_{N_{1}=q}^{N_{2}}{a_{(\sigma(3));N_{3}}a_{(\sigma(2));N_{2}}a_{(\sigma(1));N_{1}}}}}\right)}=\left(\sum_{N=q}^{n}{a_{(1);N}}\right)\left(\sum_{N=q}^{n}{a_{(2);N}}\right)\left(\sum_{N=q}^{n}{a_{(3);N}}\right)+\left(\sum_{N=q}^{n}{a_{(1);N}}\right)\left(\sum_{N=q}^{n}{a_{(2);N}a_{(3);N}}\right)+\left(\sum_{N=q}^{n}{a_{(2);N}}\right)\left(\sum_{N=q}^{n}{a_{(1);N}a_{(3);N}}\right)+\left(\sum_{N=q}^{n}{a_{(3);N}}\right)\left(\sum_{N=q}^{n}{a_{(1);N}a_{(2);N}}\right)+2\left(\sum_{N=q}^{n}{a_{(1);N}a_{(2);N}a_{(3);N}}\right).$ ### 4.5 Example applications In this section, we will apply the reduction formula presented in Theorem 4.1 to simplify certain special recurrent sums. The first special sum that we will simplify is a recurrent sum of $N^{p}$ which will produce a recurrent form of the Faulhaber formula. The second special sum is the recurrent harmonic series as well as the recurrent $p$-series for positive even values of $p$. #### 4.5.1 Recurrent Faulhaber Formula The Faulhaber formula is a formula developed by Faulhaber in a 1631 edition of Academia Algebrae [17] to calculate sums of powers $(N^{p})$. The Faulhaber formula is as follows $\sum_{N=1}^{n}{N^{p}}=\frac{1}{p+1}\sum_{j=0}^{p}{(-1)^{j}\binom{p+1}{j}B_{j}n^{p+1-j}}$ (20) where $B_{j}$ are the Bernoulli numbers of the first kind. ###### Remark. See [31] for details on the history of Bernoulli numbers. In this section, we will use the reduction formula for recurrent sums to develop a formula for a recurrent form of the Faulhaber formula. ###### Theorem 4.3. For any $m,n,p\in\mathbb{N}$, we have that $\begin{split}\sum_{N_{m}=1}^{n}{\cdots\sum_{N_{1}=1}^{N_{2}}{{N_{m}}^{p}\cdots{N_{1}}^{p}}}&=\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!i^{y_{k,i}}}\left(\sum_{N=1}^{n}{N^{ip}}\right)^{y_{k,i}}}}\\\ &=\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!i^{y_{k,i}}}\left(\frac{n^{ip+1}}{ip+1}\sum_{j=0}^{ip}{(-1)^{j}\binom{ip+1}{j}\frac{B_{j}}{n^{j}}}\right)^{y_{k,i}}}}\\\ \end{split}$ where $B_{j}$ are the Bernoulli numbers of the first kind. ###### Proof. This theorem is obtained by applying Theorem 4.1 and then applying Faulhaber’s formula. ∎ ###### Corollary 4.3. For $p=1$, Theorem 4.3 becomes $\begin{split}\sum_{N_{m}=1}^{n}{\cdots\sum_{N_{1}=1}^{N_{2}}{{N_{m}}\cdots{N_{1}}}}&=\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!i^{y_{k,i}}}\left(\sum_{N=1}^{n}{N^{i}}\right)^{y_{k,i}}}}\\\ &=\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!i^{y_{k,i}}}\left(\frac{n^{i+1}}{i+1}\sum_{j=0}^{i}{(-1)^{j}\binom{i+1}{j}\frac{B_{j}}{n^{j}}}\right)^{y_{k,i}}}}.\end{split}$ Where $B_{j}$ are the Bernoulli numbers of the first kind. Let us now consider a few particular cases: * • For $m=2$ $\begin{split}&\sum_{N_{2}=1}^{n}{\sum_{N_{1}=1}^{N_{2}}{{N_{2}}^{p}{N_{1}}^{p}}}\\\ &\,\,=\frac{1}{2}\left(\sum_{N=1}^{n}{{N}^{p}}\right)^{2}+\frac{1}{2}\left(\sum_{N=1}^{n}{{N}^{2p}}\right)\\\ &\,\,=\frac{1}{2}\left[\left(\frac{n^{p+1}}{p+1}\sum_{j=0}^{p}{(-1)^{j}\binom{p+1}{j}\frac{B_{j}}{n^{j}}}\right)^{2}+\left(\frac{n^{2p+1}}{2p+1}\sum_{j=0}^{2p}{(-1)^{j}\binom{2p+1}{j}\frac{B_{j}}{n^{j}}}\right)\right].\end{split}$ ###### Example 4.6. For $p=1$, by applying this theorem and exploiting Faulhaber’s formula, we can get the following formula $\sum_{N_{2}=1}^{n}{\sum_{N_{1}=1}^{N_{2}}{{N_{2}}{N_{1}}}}=\frac{n(n+1)(n+2)(3n+1)}{24}.$ ###### Example 4.7. For $p=2$, by applying this theorem and exploiting Faulhaber’s formula, we can get the following formula $\sum_{N_{2}=1}^{n}{\sum_{N_{1}=1}^{N_{2}}{{N_{2}}^{2}{N_{1}}^{2}}}=\frac{n(n+1)(n+2)(2n+1)(2n+3)(5n-1)}{360}.$ * • For $m=3$ $\sum_{N_{3}=1}^{n}{\sum_{N_{2}=1}^{N_{3}}{\sum_{N_{1}=1}^{N_{2}}{{N_{3}}^{p}{N_{2}}^{p}{N_{1}}^{p}}}}=\frac{1}{6}\left(\sum_{N=1}^{n}{a_{N}}\right)^{3}+\frac{1}{2}\left(\sum_{N=1}^{n}{a_{N}}\right)\left(\sum_{N=1}^{n}{\left(a_{N}\right)^{2}}\right)+\frac{1}{3}\left(\sum_{N=1}^{n}{\left(a_{N}\right)^{3}}\right).$ ###### Example 4.8. For $p=1$, by applying this theorem and exploiting Faulhaber’s formula, we can get the following formula $\sum_{N_{3}=1}^{n}{\sum_{N_{2}=1}^{N_{3}}{\sum_{N_{1}=1}^{N_{2}}{{N_{3}}{N_{2}}{N_{1}}}}}=\frac{n^{2}(n+1)^{2}(n+2)(n+3)}{48}=\left(\sum_{N=1}^{n}{N}\right)\left[\frac{n(n+1)(n+2)(n+3)}{4!}\right].$ #### 4.5.2 Recurrent p-series and harmonic series In this section, using the formula developed by Euler and the reduction theorem (Theorem 4.1), we will prove an expression which can be used to calculate a recurrent form of the zeta function for positive even values. Then we will conjecture a solution for a more general form of the Basel problem. We start by applying Theorem 4.1 and using the expression of the zeta function for positive even values to get an expression for the recurrent series of $\frac{1}{N^{2p}}$ (or recurrent harmonic series). ###### Theorem 4.4. For any $m,p\in\mathbb{N}$, we have that $\begin{split}\sum_{N_{m}=1}^{\infty}{\cdots\sum_{N_{1}=1}^{N_{2}}{\frac{1}{N_{m}^{2p}\cdots N_{1}^{2p}}}}&=\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!i^{y_{k,i}}}\left(\zeta(2ip)\right)^{y_{k,i}}}}\\\ &=(-1)^{pm}(2\pi)^{2pm}\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{(-1)^{y_{k,i}}}{(y_{k,i})!}\left(\frac{B_{2ip}}{(2i)(2ip)!}\right)^{y_{k,i}}}}.\end{split}$ ###### Proof. By applying Theorem 4.1, $\begin{split}\sum_{N_{m}=1}^{\infty}{\cdots\sum_{N_{1}=1}^{N_{2}}{\frac{1}{N_{m}^{2p}\cdots N_{1}^{2p}}}}&=\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!i^{y_{k,i}}}\left(\sum_{N=1}^{\infty}{\left(\frac{1}{N^{2p}}\right)^{i}}\right)^{y_{k,i}}}}\\\ &=\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!i^{y_{k,i}}}\left(\zeta(2ip)\right)^{y_{k,i}}}}.\end{split}$ Euler proved that, for $m\geq 1$ (see [2] for a proof), $\zeta(2m)=\frac{(-1)^{m+1}(2\pi)^{2m}}{2(2m)!}B_{2m}.$ Hence, $\begin{split}\sum_{N_{m}=1}^{\infty}{\cdots\sum_{N_{1}=1}^{N_{2}}{\frac{1}{N_{m}^{2p}\cdots N_{1}^{2p}}}}&=\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!i^{y_{k,i}}}\left((-1)^{ip+1}\frac{B_{2ip}(2\pi)^{2ip}}{2(2ip)!}\right)^{y_{k,i}}}}\\\ &=(-1)^{pm}(2\pi)^{2pm}\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{(-1)^{y_{k,i}}}{(y_{k,i})!}\left(\frac{B_{2ip}}{(2i)(2ip)!}\right)^{y_{k,i}}}}.\end{split}$ ∎ The following table summarizes some values of the zeta function for positive even arguments, $\zeta(2)=\frac{\pi^{2}}{6}\,\,\,\,\,\,\zeta(4)=\frac{\pi^{4}}{90}\,\,\,\,\,\,\zeta(6)=\frac{\pi^{6}}{945}\,\,\,\,\,\,\zeta(8)=\frac{\pi^{8}}{9450}\,\,\,\,\,\,\zeta(10)=\frac{\pi^{10}}{93555}\,\,\,\,\,\,$ $\zeta(12)=\frac{691\pi^{12}}{638512875}\,\,\,\,\,\,\zeta(14)=\frac{2\pi^{14}}{18243225}\,\,\,\,\,\,\zeta(16)=\frac{3617\pi^{16}}{325641566250}\,\,\,\,\,\,.$ By using the values in the above table as well as Theorem 4.4 and playing with different values, we can notice some identities. In particular, we can conjecture the following statement for the recurrent sum of $\frac{1}{N^{2}}$ (recurrent harmonic series with $2p=2$) for different values of $m$ (for different numbers of summations). This represents a generalization of the Basel Problem solved by Euler. However, this conjecture has already been proven, hence, we will directly use it to develop additional identities. ###### Theorem 4.5. For any $m\in\mathbb{N}$, we have that $\sum_{N_{m}=1}^{\infty}{\cdots\sum_{N_{1}=1}^{N_{2}}{\frac{1}{N_{m}^{2}\cdots N_{1}^{2}}}}=\frac{(-1)^{m+1}2\left(2^{2m-1}-1\right)B_{2m}\pi^{2m}}{(2m)!}=\left(2-\frac{1}{2^{2(m-1)}}\right)\zeta(2m)$ or identically (from Theorem 4.1), $\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{1}{(y_{k,i})!i^{y_{k,i}}}\left(\zeta(2i)\right)^{y_{k,i}}}}=\frac{(-1)^{m+1}2\left(2^{2m-1}-1\right)B_{2m}\pi^{2m}}{(2m)!}=\left(2-\frac{1}{2^{2(m-1)}}\right)\zeta(2m).$ ###### Proof. In [35], the following relation was proven but in another notation, $\sum_{1\leq N_{1}\leq\cdots\leq N_{m}}{\frac{1}{N_{m}^{2}\cdots N_{1}^{2}}}=\left(\frac{2^{2m-1}-1}{2^{2m-2}}\right)\zeta(2m).$ Hence, $\sum_{N_{m}=1}^{\infty}{\cdots\sum_{N_{1}=1}^{N_{2}}{\frac{1}{N_{m}^{2}\cdots N_{1}^{2}}}}=\left(\frac{2^{2m-1}-1}{2^{2m-2}}\right)\zeta(2m)=\left(2-\frac{1}{2^{2(m-1)}}\right)\zeta(2m).$ Euler proved that, for $m\geq 1$, $B_{2m}=\frac{(-1)^{m+1}2(2m)!}{(2\pi)^{2m}}\zeta(2m)\,\,\,\,\,\,\,\,or\,\,\,\,\,\,\,\,\zeta(2m)=\frac{(-1)^{m+1}(2\pi)^{2m}}{2(2m)!}B_{2m}.$ Hence, by substituting, we get $\sum_{N_{m}=1}^{\infty}{\cdots\sum_{N_{1}=1}^{N_{2}}{\frac{1}{N_{m}^{2}\cdots N_{1}^{2}}}}=\frac{(-1)^{m+1}2\left(2^{2m-1}-1\right)B_{2m}\pi^{2m}}{(2m)!}.$ The proof of the first equation is completed. Applying Theorem 4.1, we get the second equation. ∎ ###### Corollary 4.4. For any $m\in\mathbb{N}$, we have that $\sum_{\sum{i.y_{k,i}}=m}{\prod_{i=1}^{m}{\frac{(-1)^{y_{k,i}}}{(y_{k,i})!}\left(\frac{B_{2ip}}{(2i)(2ip)!}\right)^{y_{k,i}}}}=\left(\frac{1}{2^{2m-1}}-1\right)\frac{B_{2m}}{(2m)!}$ ###### Proof. By applying Theorem 4.4 with $p=1$ to Theorem 4.5, we obtain the theorem. ∎ We will use this to prove that this recurrent harmonic series (or recurrent $p$-series) with $2p=2$ will converge to 2 as the number of summations $m$ goes to infinity. ###### Theorem 4.6. For any $m\in\mathbb{N}$, we have that $\lim_{m\to\infty}{{\left(\sum_{N_{m}=1}^{\infty}{\cdots\sum_{N_{1}=1}^{N_{2}}{\frac{1}{N_{m}^{2}\cdots N_{1}^{2}}}}\right)}}=2.$ ###### Proof. It is known that $\lim_{m\to\infty}\zeta(2m)=1$. By applying Theorem 4.5, $\lim_{m\to\infty}{{\left(\sum_{N_{m}=1}^{\infty}{\cdots\sum_{N_{1}=1}^{N_{2}}{\frac{1}{N_{m}^{2}\cdots N_{1}^{2}}}}\right)}}=\lim_{m\to\infty}{\left(2-\frac{1}{2^{2(m-1)}}\right)}\times\lim_{m\to\infty}{\zeta(2m)}=2.$ ∎ ###### Example 4.9. For $m=4$, we have $\sum_{N_{4}=1}^{\infty}{\sum_{N_{3}=1}^{N_{4}}{\sum_{N_{2}=1}^{N_{3}}{\sum_{N_{1}=1}^{N_{2}}{\frac{1}{N_{4}^{2}N_{3}^{2}N_{2}^{2}N_{1}^{2}}}}}}=\frac{1}{24}\left(\sum_{N=1}^{\infty}{\frac{1}{N^{2}}}\right)^{4}+\frac{1}{4}\left(\sum_{N=1}^{\infty}{\frac{1}{N^{2}}}\right)^{2}\left(\sum_{N=1}^{\infty}{\frac{1}{N^{4}}}\right)+\frac{1}{3}\left(\sum_{N=1}^{\infty}{\frac{1}{N^{2}}}\right)\left(\sum_{N=1}^{\infty}{\frac{1}{N^{6}}}\right)+\frac{1}{8}\left(\sum_{N=1}^{\infty}{\frac{1}{N^{4}}}\right)^{2}+\frac{1}{4}\left(\sum_{N=1}^{\infty}{\frac{1}{N^{8}}}\right)=\frac{127\pi^{8}}{604800}{=\left(2-\frac{1}{2^{2(3)}}\right)\zeta(8)\approx 1.992466004.}$ Similarly, we will use this to show that the sum (over all non-negative values of $m$) of the recurrent harmonic series with $2p=2$ will diverge. ###### Theorem 4.7. We have that, $\sum_{m=0}^{\infty}{\left(\sum_{N_{m}=1}^{\infty}{\cdots\sum_{N_{1}=1}^{N_{2}}{\frac{1}{N_{m}^{2}\cdots N_{1}^{2}}}}\right)}\to\infty.$ ###### Proof. Applying Theorem 4.5, $\sum_{m=0}^{\infty}{\left(\sum_{N_{m}=1}^{\infty}{\cdots\sum_{N_{1}=1}^{N_{2}}{\frac{1}{N_{m}^{2}\cdots N_{1}^{2}}}}\right)}=\sum_{m=0}^{\infty}{\left(2-\frac{1}{2^{2(m-1)}}\right)\zeta(2m)}.$ For $m=0$, we have $\left(2-\frac{1}{2^{2(m-1)}}\right)\zeta(2m)=(2-4)(-1/2)=1$. Knowing that $\zeta(2m)\geq 1$ for $m\geq 1$ and noticing the following identity for $m\geq 1$, $1\leq\left(2-\frac{1}{2^{2(m-1)}}\right)\leq 2.$ Hence, for $m\geq 0$, $\left(2-\frac{1}{2^{2(m-1)}}\right)\zeta(2m)\geq 1.$ Thus, $\lim_{n\to\infty}{\sum_{m=0}^{n}{\left(\sum_{N_{m}=1}^{\infty}{\cdots\sum_{N_{1}=1}^{N_{2}}{\frac{1}{N_{m}^{2}\cdots N_{1}^{2}}}}\right)}}\geq\lim_{n\to\infty}{\sum_{m=0}^{n}{1}}=\infty.$ Hence, this sums is infinite. ∎ ## References * Andrews [1998] Andrews, G. E. (1998). The theory of partitions. 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15mm15mm15mm15mm # Thermally driven fission of protocells Romain Attal Cité des Sciences et de l’Industrie 30, avenue Corentin-Cariou 75019 Paris, France<EMAIL_ADDRESS> ###### Abstract. We propose a simple mechanism for the self-replication of protocells. Our main hypothesis is that the amphiphilic molecules composing the membrane bilayer are synthesised inside the protocell through globally exothermic chemical reactions. The slow increase of the inner temperature forces the hottest molecules to move from the inner leaflet to the outer leaflet of the bilayer. This asymmetric translocation process makes the outer leaflet grow faster than the inner leaflet. This differential growth increases the mean curvature and amplifies any local shrinking of the protocell until it splits in two. ###### Key words and phrases: protocell, bilayer, translocation, self-replication, thermodynamical instability. ###### Contents 1. 1 Protocells and metabolism 2. 2 Hypotheses of our model 3. 3 Flows, forces and energy dissipation 1. 3.1 Main irreversible processes 2. 3.2 Flows associated to each irreversible process 3. 3.3 Thermodynamical forces 4. 3.4 Conductance matrix 5. 3.5 Entropy production and stability 4. 4 Membrane geometry and growth equation 1. 4.1 Conservation of matter and exponential growth 2. 4.2 Cylindrical growth in steady state 5. 5 Thermal instability of cylindrical growth 1. 5.1 The Squeezed Sausage Theorem (SST) 2. 5.2 Fluctuations, translocation and heat transfer 3. 5.3 Thermal balance 6. 6 Translocation between leaflets 1. 6.1 An effective potential for translocation 2. 6.2 Computation of $L_{m\theta}$ 3. 6.3 Computation of $L_{mm}$ 4. 6.4 Estimation of $L_{\theta\theta}$ 5. 6.5 Destabilisation 7. 7 Conclusions and perspectives 8. A The mean curvature of the membrane 9. B Solutions of the growth equation 1. B.1 Case 1 : $\eta\neq 1$ and $\tau>\tau_{+}(\eta)$ or $\tau<\tau_{-}(\eta)$ 2. B.2 Case 2 : $\eta\neq 1$ and $\tau\in\\{\tau_{+}(\eta),\tau_{-}(\eta)\\}$ 3. B.3 Case 3 : $\tau_{-}(\eta)<\tau<\tau_{+}(\eta)$ 10. C Smooth perturbation of cylindrical growth 1. C.1 Isovolumic variation of the area 2. C.2 Isovolumic variation of the total mean curvature 11. D Asymptotic expansion of $F(a)$ ## 1\. Protocells and metabolism The objects modeled in the present article are protocells, the putative ancestors of modern living cells [23, 34]. In the absence of fossils [38], we ignore their detailed properties. However, we can sketch a minimalist functional diagram of protocells (FIG. 1). FoodWasteMetabolismHeatBiomass Protocells initiate the fundamental process of life : Food $\to$ Biomass + Heat + Waste. The protocell is a vesicle bounded by a bilayer made of amphiphilic molecules. Nutrient molecules (food) enter by mere diffusion, since they are consumed inside, where their concentration is lower than outside. Conversely, waste molecules have a larger concentration inside and therefore diffuse passively to the outside. The metabolism is a network of unknown chemical reactions taking place only inside the protocell. The net reaction is supposed to be exothermic, since living matter is hotter than abiotic matter (under the same external thermodynamical conditions). Let us compare this scheme to actual evolved cells. The growth of bacteria in a nutrient rich medium follows a species dependent periodic process [5, 12]. At regular time intervals, each cell splits to form two daughter cells. This requires the synchronization of numerous biochemical and mechanical processes inside the cell, involving cytoskeletal structures positioned at the locus of the future cut (septum). However, in the history of life, such complex structures are a high-tech luxury and must have appeared much later than the ability to split. Protocells must have used a simple splitting mechanism to ensure their reproduction, before the appearance of genes, RNA, enzymes and all the complex organelles present today even in the most rudimentary forms of autonomous life [34]. In this article, we present a simple model for the growth and self-replication of a protocell, following the laws of irreversible thermodynamics near equilibrium. Our guide is the rate of entropy production, which is minimal in a steady state [32, 20]. A key point of our approach is that the heat produced by the metabolism of the protocell is approximately proportional to its volume, whereas the heat flow that it can loose is proportional to the area of its membrane. In a rod-shaped cell (bacillus) growing linearly, these two quantities are approximately proportional so that each increment of the membrane area should be sufficient, ideally, to evacuate the heat produced by the corresponding increment of the cell volume. However, the irreversible physical and chemical processes produce heat more quickly than the growing tubular membrane can dissipate to the outside. This increases slowly the inner temperature and enhances the fluctuations of the shape of the membrane, of the various concentrations and of the local electric field. In a growing spherical protocell, the maximal heat flow that the membrane can expell to the outside without overheating the inside puts an upper limit to the radius of the protocell. Indeed, the formation of two small protocells from a big one releases work [33], so that large protocells are mechanically unstable. However, neither [33] nor [8] provides a path to follow to realise this deformation. In our model, we start from a cylindrical shape to simplify the computations. As the inner temperature increases, the growth of the outer leaflet of the membrane becomes more probable than the growth of the inner leaflet. If a random thermal fluctuation lowers slightly the radius of this cylinder, then its area increases more quickly than during the steady state cylindrical growth (FIG. 1). $\downarrow$$\downarrow$$\downarrow$hotcold Splitting a cylindrical protocell. This reduction of the radius induces a loss of convexity of the membrane. This favors the outflow of heat and the ratio area/volume increases slightly, compared to a convex cylindrical shape. The plan of the article goes as follows. In Section II, in order to formulate these ideas mathematically, we state all the physical hypotheses of our model of protocells. In Section III, we define the various flows of matter and energy and their associated thermodynamical forces. In the linear approximation, the rate of entropy production is the scalar product of these flows and forces and is minimal in a steady state [32]. In Section IV, we derive a differential equation for the evolution of the area and the integral of the mean curvature of the membrane, starting from the advancement of the chemical reaction for the synthesis of the membrane molecules. This linear equation admits a solution growing exponentially. In Section V, we use variational calculus [10] to prove that the local reduction of the radius of the cell increases its length and its area, if its volume is kept constant. This intuitive property implies that heat is more easily released when the protocell is squeezed. In Section VI, we propose a molecular mechanism for the increase of the mean curvature of the membrane associated to this squeezing. The position of each amphiphilic membrane molecule is reduced to a single degree of freedom : the distance from the polar head to the middle of the hydrophobic slice. We use a double well effective potential to describe the trapping of these molecules in the membrane. Due to the temperature difference between the inner and outer sides, the membrane molecules go from the inner leaflet to the outer leaflet more often than in the opposite direction. This asymmetry forces the membrane to curve and shrink around the middle of the protocell and initiates its splitting. Our main mathematical result (Proposition VI.I) states that a stability condition, $L_{m\theta}^{2}<L_{mm}L_{\theta\theta}$, can not be satisfied at high temperature, because the squared crossed conductance, $L_{m\theta}^{2}$, increases more quickly than the product of the diffusion coefficients, $L_{mm}$ for membrane molecules and $L_{\theta\theta}$ for heat. Hence, the cylindrical growth process is unstable when the temperature difference is sufficiently high. We conclude in Section VII with a proposition of an experimental test for our model. The appendices contain the detailed computations of our model. The mathematical notions involved are elementary (linear differential equations and geometry of surfaces). ## 2\. Hypotheses of our model Let us state more precisely the hypotheses underlying our model : 1. (1) Our protocells are made of a membrane of average thickness $2\varepsilon$, bounding a cytosol of finite volume $\mathcal{V}(t)$. 2. (2) The cytosol contains unknown specific molecules (reactants, catalysers, chromophores, …) which participate to a network of chemical reactions. We suppose that the concentrations are constant and uniform in the volume $\mathcal{V}$. 3. (3) The protocell starts with a cylindrical shape closed by two hemispherical caps of fixed radius, $R_{0}$. The total length, $\ell(t)+2R_{0}+2\varepsilon$, increases with time due to the synthesis of membrane molecules (FIG. 3). $\ell(t)+2R_{0}+2\varepsilon$$2\varepsilon$${2(R_{0}-\varepsilon)}$${2(R_{0}+\varepsilon)}$$\ell(t)$ Geometry of an idealised cylindrical protocell. This may seem a rather drastic hypothesis, but the computations could be made for a generic, approximately spherical shape using an expansion in spherical harmonics. This would add to the model an unnecessary mathematical complexity that would hide the main physical phenomena. The use of cylindrical, rotation invariant shapes allows us to reduce the problem to one dimension. Moreover, this is a best case scenario for the release of heat in steady state, since the ratio volume/area can be held constant in a steady growth. 4. (4) Due to the surface tension of the membrane, its mean curvature has an upper bound, $H_{\max}=\mfrac{1}{R_{0}}$. Indeed, due to the attractive forces between the polar heads of the membrane molecules, and due to their geometry, they can not form structures arbitrarily small [27]. 5. (5) Food (nutrients and water) enters the protocell by mere passive diffusion through the membrane. Waste and heat also diffuse passively but in the opposite direction. Protocells did not use specialized membrane molecules for an active transport through the membrane. 6. (6) The membrane molecules are synthesised inside the protocell in an unknown network of chemical reactions. It might use some encapsulated catalyzers or chromophores trapped in the volume and catching part of the ambient light [23], but we will make no hypothesis on the details of this network. 7. (7) These metabolic reactions generate heat to be evacuated and increase slowly the internal temperature, $T_{1}$, whereas the external temperature, $T_{0}$, remains fixed. 8. (8) The characteristic time of the variations of $T_{1}(t)$ is much larger than the characteristic times of chemical reactions and diffusion processes across the membrane. 9. (9) The cytosol is homogenous and contains no organelles, no cytoskeleton, no enzymes, no RNA/DNA. Just simple chemical reactants uniformly distributed. (Rashevsky’s model [33] allows for a slight radial variation of concentrations due to the diffusion of food and waste through the membrane). 10. (10) The membrane is a bilayer made of unspecified amphiphilic molecules. We presume that their hydrophobic tails are long enough (10-12 carbon atoms) to form a stable bilayer, but not too bulky in order to allow flip-flop (or translocation) processes between the two leaflets. We do not include sterol molecules because they are the product of a long biochemical selection process [27], and a high-tech luxury for protocells. 11. (11) The inner leaflet (L1) is at temperature $T_{1}$ whereas the outer leaflet (L0) is at temperature $T_{0}<T_{1}$. This temperature drop allows the bilayer to undergo coupled transport phenomena (food and waste diffusion, including water leaks, heat diffusion, flip-flop, etc.). 12. (12) The membrane may contain other molecules, in small concentrations, but we don’t need them to transport food, waste or any molecule through the membrane. The validity of these hypotheses will depend on the agreement of their predictions with the results of future experiments made with real protocells. ## 3\. Flows, forces and energy dissipation In any living system, some processes release energy whereas other processes consume energy. Globally, the system takes usable energy from the outside and rejects unusable energy, in the form of heat and waste, that can be used by other living systems. In order to describe such a system, we must define the various flows of matter and energy and the forces causing these flows. Any gradient of concentration, pressure, temperature, etc. will cause a current of particles, fluid, heat, etc. These processes are generally irreversible and dissipate energy to inaccessible degrees of freedom. This dissipation of a conserved quantity is measured by the entropy function, which increases as time passes. The study of irreversible thermodynamical processes near equilibrium [29, 30, 35, 20] is based on the rate of entropy production, represented by a bilinear function of flows (chemical reaction speed, thermal current, particle current, electric current, etc.) and forces (chemical affinity, temperature gradient, concentration gradient, electric tension, etc.). In a first approximation, flows and forces are related linearly, as in Ohm’s law : (1) $\displaystyle\text{electric current }=\text{ conductivity }\times\text{ electric field}$ and the power dissipated is a quadratic function of the tension : (2) power dissipated $\displaystyle=\text{ tension }\times\text{ current}$ $\displaystyle=\text{ conductance }\times\text{ tension}^{2}.$ Similarly, in viscous fluids : (3) power dissipated $\displaystyle=\text{ friction coefficient }\times\text{ velocity}^{2}.$ We suppose that the protocell metabolism is in a steady state not too far from equilibrium, so that the various flows, $J_{i}$, and the thermodynamic forces, $X_{k}$, are linearly related : (4) $\displaystyle J_{i}=\sum_{k}L_{ik}X_{k}$ and the entropy rate is a quadratic function of $X$ : (5) $\displaystyle\sigma:=XJ=\sum_{ik}X_{i}L_{ik}X_{k}.$ The coefficients $L_{ik}$ are called phenomenological because their computation depends on the chosen model of microscopic dynamics (kinetic theory) and their numerical value has to be compared to a measurement in the real world to (in)validate this model and the linearity hypothesis. An important property of the phenomenological coefficients is provided by Onsager’s relations [29, 30, 20, 35]. Under the hypotheses of microscopic reversibility and parity of the variables under time reversal (in particular, in the absence of magnetic coupling and vorticity), the matrix $L$ is symmetric : (6) $\displaystyle L_{ik}=L_{ki}.$ This important law has been checked experimentally for various systems near equilibrium and is satisfied quite accurately in many cases. ### 3.1. Main irreversible processes To each irreversible physical or chemical process are associated a flow of matter or energy and a thermodynamical force, just as an electric current and an electric tension correspond to each branch of an electric network. If we identify the main processes that take place during the growth of a protocell, we can compute the global rate of dissipation of energy, or entropy creation. According to Prigogine’s Theorem [32, 11], this rate reaches a minimum when the system is in a steady state. In order to compute this dissipation, we need to define the various compartiments containing energy. In the sequel of this article, the subscript $0$ (resp. $1$) will denote the variables outside (resp. inside) the protocell. The physical and chemical processes are grouped as follows : $f_{0}\to f_{1}\,$ : food molecules (nutrients $+$ water) diffuse into the protocell through the membrane. $f\to m+c+w\,$ : food is transformed into membrane, cytosol and waste, inside the protocell. This is a global process, a superposition of catabolism and anabolism. Taking into account the stoichiometric coefficients, we can write more precisely : (7) $\displaystyle\sum_{i}\nu_{f_{i}}f_{i}\ \longrightarrow\ \sum_{j}\nu_{m_{j}}m_{j}+\sum_{k}\nu_{c_{k}}c_{k}+\sum_{l}\nu_{w_{l}}w_{l}$ where $f_{i}$ denotes the food molecules of type $i$, $m_{j}$ the membrane molecules of type $j$, $c_{k}$ the cytosol molecules of type $k$ and $w_{l}$ the waste molecules of type $l$. If $N_{\alpha}$ is the number of molecules of type $\alpha$, the advancement of this reaction, $\xi$, is defined by : (8) $\displaystyle\mathrm{d}\xi:=\frac{\mathrm{d}N_{\alpha}}{\pm\nu_{\alpha}}$ where the stoichiometric coefficients, $\nu_{\alpha}$, are counted positively for the products and negatively for the reactants. Note that our definition of $\xi$ involves $N_{\alpha}$ instead of the volumic concentration, $C_{\alpha}=N_{\alpha}/\mathcal{V}$, because the volume is not fixed. $w_{1}\to w_{0}\,$ : waste molecules diffuse out of the protocell through the membrane. $m_{1}\leftrightarrows m_{0}\,$ : molecules of the membrane bilayer go from one side to the other. In modern cells, this process is catalysed by enzymes (flippase for $0\to 1$ and floppase for $1\to 0$), but in protocells such a complex machinery did not exist yet [34]. If we suppose that the first membranes were not as thick as today (most phospholipids in modern and healthy cell walls have hydrophobic chains made of $\sim$ 16-22 atoms of carbon [27]), the exchange of molecules between the two leaflets could have been possible in a reasonable time to allow spontaneous splitting. Medium length lipids (10-14 atoms of carbon) could be good candidates to make stable, flippable and not too porous protocells. We isolate the process of translocation ($\restrictwand\xleftrightharpoons{}\restrictwandup$) because the ratio ${N_{m0}}/{N_{m1}}$ of the numbers of membrane molecules on each side is related to the mean curvature of the bilayer, which is the geometric parameter monitoring the splitting process. $q_{1}\to q_{0}\,$ : electric charges can be transfered from one side of the membrane to the other, by an ionic bound on the polar head of the membrane molecules. This electric current builds up an electric tension, $U_{01}$, counteracted by possible ionic leaks through the membrane. If we suppose that the membrane molecules are monovalent fatty acids, each one can carry a monocation (H+, Na+, K+, …). This cotransport process could be the ancestor of the modern sodium-potassium pump. Anions also can participate to this transmembrane electric current, by leaking throuh water pores [14]. ### 3.2. Flows associated to each irreversible process The main processes of our model are described by the following flows in the protocell (see FIG. 3.2) : $J_{f}\,$ : the flow of food entering the protocell through its membrane (molecules per unit time per unit area). $J_{w}\,$ : the flow of waste exiting the protocell through its membrane (molecules per unit time per unit area). $J_{\theta}\,$ : the heat flow exiting the protocell by diffusion through its membrane (energy per unit time per unit area). $J_{mab}\,$ : the flow of membrane molecules from $a$ to $b$ (molecules per unit time per unit area). The possible values of $a$ and $b$ are : $c\,$ : the cytosol ; $1\,$ : the inner leaflet of the membrane (L1) ; $0\,$ : the outer leaflet of the membrane (L0) ; The net flow of membrane molecules is usually unidirectional, ${\mathbf{C}}\to{\mathbf{L}}_{1}\to{\mathbf{L}}_{0}$, hence $J_{mc1}>0$ and $J_{m}:=J_{m10}-J_{m01}>0$. $J_{r}\,$ : the speed of the synthesis reaction inside the cytosol (molecules per unit time per unit volume). $\xi$ being the advancement of the reaction $f\to m+c+w$, defined above, then $J_{r}$ is the time derivative of $\xi$ : (9) $\displaystyle J_{r}:=\frac{\mathrm{d}\xi}{\mathrm{d}t}.$ $J_{q}\,$ : some ions can be transported from one side to the other, bounded to the polar head of the membrane molecules. L0L1$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwand$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$\restrictwandup$$J_{m}$$J_{q}$ions$J_{f}$${\it{f}=}$food$f\,$$\xlongrightarrow{J_{r}}\ m+c+w$$J_{mc1}$${\it{w}=}$waste$J_{w}$$J_{\theta}$ heat diffusion$\textbf{E}=\text{environment}$$T_{0}$$\boxed{\textbf{C}=\text{cytosol}}$$T_{1}(t)$ Main flows of energy and matter in our model. We then have the following linear flow diagram for the synthesis and motion of membrane molecules : (10) $\displaystyle{\mathbf{E}}\xlongrightarrow{J_{f}}{\mathbf{C}}\xlongrightarrow{J_{r}}{\mathbf{C}}\xlongrightarrow{\color[rgb]{.1,.6,1}J_{mc1}}{\mathbf{L}}_{1}\xlongrightarrow{\color[rgb]{.1,.6,1}J_{m10}}{\mathbf{L}}_{0}.$ This picture is however slightly misleading. Indeed, the amphiphilic molecules being in a liquid phase, their positions fluctuate in each leaflet (transversal diffusion) and they undergo perpendicular motions (protrusion) and translocations from one leaflet to the other. The pictures obtained by molecular dynamics simulations [14, 15, 4, 1] give us a more precise representation of real world membranes. ### 3.3. Thermodynamical forces The thermodynamical forces associated to these processes are defined as follows : $X_{\theta}\,$ : the thermal force is the difference of the inverse temperatures inside and outside the protocell : (11) $\displaystyle X_{\theta}:=\frac{1}{T_{0}}-\frac{1}{T_{1}}>0.$ $X_{f_{i}}\,$ : the chemical force driving the food molecules, $f_{i}$, is the difference of the ratios $-\mu_{f_{i}}/T$ outside and inside the protocell : (12) $\displaystyle X_{f_{i}}:=\frac{\mu_{f_{i}0}}{T_{0}}-\frac{\mu_{f_{i}1}}{T_{1}}.$ The influx of food is guided by mere diffusion through the membrane (dedicated channel and intrinsic proteins did not exist yet in protocells). Since food is consumed inside the protocell, $[f_{i}]_{1}<[f_{i}]_{0}$. For a spherical protocell, the profile of the concentration of each molecule (as a function of the distance to the center) can be computed by solving the diffusion equation [33]. An important result of this computation is the existence of a discontinuity in the concentration of each molecule, $f_{i}$, proportional to the radius, $R$, of the protocell, to the rate of the reaction, $q_{i}$ (concentration/time), and inversely proportional to the permeability, $h_{i}$ (length/time), of the membrane for this molecule : $[f_{i}]_{1}-[f_{i}]_{0}\propto\mfrac{q_{i}R}{h_{i}}$. $X_{w_{j}}\,$ : the force driving the waste molecules to the outside of the protocell is the difference of chemical potentials divided by the temperature : (13) $\displaystyle X_{w_{j}}:=\frac{\mu_{w_{j}0}}{T_{0}}-\frac{\mu_{w_{j}1}}{T_{1}}.$ Note that $X_{w_{j}}$ and $X_{f_{i}}$ must have different signs for waste and food to go in opposite directions. $X_{r}\,$ : the chemical force driving the synthesis reactions (metabolism) is the chemical reaction affinity, $A_{r}$, of the global process $(f\to m+c+w)$, divided by the inner temperature of the protocell : (14) $\displaystyle X_{r}:=\frac{A_{r}}{T_{1}}.$ This affinity is a linear combination of the chemical potentials of the synthesis equation, weighted by the stoichiometric coefficients, counted positively for the reactants $(f)$ and negatively for the products $(m,c,w)$ : (15) $\displaystyle A_{r}=\sum_{i}\nu_{f_{i}}\mu_{f_{i}}-\sum_{j}\nu_{m_{j}}\mu_{m_{j}}-\sum_{k}\nu_{c_{k}}\mu_{c_{k}}-\sum_{l}\nu_{w_{l}}\mu_{w_{l}}.$ $X_{m^{\prime}}\,$ : The membrane molecules are synthesised in the cytosol at temperature $T_{1}$. Their hydrophobic tail enforces the spontaneous organisation of these molecules into a bilayer. We suppose that the temperature varies only across the membrane. The driving force of this isothermal process is the affinity of the reaction $m_{c}\to m_{1}$, divided by the inner temperature, $T_{1}$ : (16) $\displaystyle X_{m^{\prime}}=\frac{A_{mc1}}{T_{1}}=\frac{\mu_{mc}-\mu_{m1}}{T_{1}}.$ Here, $\mu_{mc}$ is the chemical potential of the free membrane molecules inside the cytosol and $\mu_{m1}$ is their chemical potential in the inner leaflet. The heat released to the inner leaflet during this process is : (17) $\displaystyle Q_{mc1}=\mu_{mc}-\mu_{m1}=T_{1}X_{m^{\prime}}.$ $X_{m}\,$ : the membrane molecules are transfered from the inner layer, at temperature $T_{1}$, to the outer leaflet, at temperature $T_{0}<T_{1}$, releasing the heat $Q_{m10}$ into the environmental thermostat, at temperature $T_{0}$. The thermodynamical force of this process is : (18) $\displaystyle X_{m}=\frac{\mu_{m1}}{T_{1}}-\frac{\mu_{m0}}{T_{0}}.$ $X_{q}\,$ : the thermodynamical force driving the ions of species $i$, of charge $z_{i}e$, across the membrane is the difference of electrochemical potentials [2] : (19) $\displaystyle X_{qi}$ $\displaystyle=\tilde{\mu}_{i1}-\tilde{\mu}_{i0}$ $\displaystyle=\big{(}\mu_{i1}+z_{i}e\psi_{1}\big{)}-\big{(}\mu_{i0}+z_{i}e\psi_{0}\big{)}$ $\displaystyle=\mu_{i1}-\mu_{i0}+z_{i}eU_{10}.$ where $\psi$ denotes the electrostatic potential and $U_{10}:=\psi_{1}-\psi_{0}$ is the electric tension across the membrane. Among these forces, only $X_{\theta}$ is a linear function of the small temperature difference, $\Delta T=T_{1}-T_{0}$. The others have, generically, a supplementary constant term, of order $0$ in $\Delta T$. ### 3.4. Conductance matrix The phenomenological coefficients, $L_{ik}$, which couple all the irreversible processes of our linear model, can be put in a $7\times 7$ matrix : (20) $\displaystyle L=\begin{pmatrix}L_{\theta\theta}&L_{\theta f}&L_{\theta w}&L_{\theta m}&L_{\theta m^{\prime}}&L_{\theta q}&L_{\theta r}\\\ L_{f\theta}&L_{ff}&L_{fw}&L_{fm}&L_{fm^{\prime}}&L_{fq}&L_{fr}\\\ L_{w\theta}&L_{wf}&L_{ww}&L_{wm}&L_{wm^{\prime}}&L_{wq}&L_{wr}\\\ L_{m\theta}&L_{mf}&L_{mw}&L_{mm}&L_{mm^{\prime}}&L_{mq}&L_{mr}\\\ L_{m^{\prime}\theta}&L_{m^{\prime}f}&L_{m^{\prime}w}&L_{m^{\prime}m}&L_{m^{\prime}m^{\prime}}&L_{m^{\prime}q}&L_{m^{\prime}r}\\\ L_{q\theta}&L_{qf}&L_{qw}&L_{qm}&L_{qm^{\prime}}&L_{qq}&L_{qr}\\\ L_{r\theta}&L_{rf}&L_{rw}&L_{rm}&L_{rm^{\prime}}&L_{rq}&L_{rr}\\\ \end{pmatrix}.$ In a first approximation, some coefficients can be set equal to zero : (21) $\displaystyle L\simeq\begin{pmatrix}L_{\theta\theta}&L_{\theta f}&L_{\theta w}&L_{\theta m}&0&L_{\theta q}&0&\\\ L_{f\theta}&L_{ff}&0&0&0&0&0\\\ L_{w\theta}&0&L_{ww}&0&0&0&0\\\ L_{m\theta}&0&0&L_{mm}&0&L_{mq}&0\\\ 0&0&0&0&L_{m^{\prime}m^{\prime}}&0&0\\\ L_{q\theta}&0&0&L_{qm}&0&L_{qq}&0\\\ 0&0&0&0&0&0&L_{rr}\\\ \end{pmatrix}.$ The diagonal coefficients of $L$ are positive but we let $L_{\bullet r}=0=L_{r\bullet}$ because the synthesis reactions take place in the cytosol and are decoupled from the transport processes across the membrane. Similarly, we let $L_{\bullet m^{\prime}}=0=L_{m^{\prime}\bullet}$, because the transfer of membrane molecules from the cytosol to the inner leaflet is decoupled from the other processes. Since the diffusion processes of different molecules (food, waste, ions or membrane constituents) across the membrane are supposed to be decoupled, we put $L_{fw}=0=L_{wf}$, $L_{fm}=0=L_{mf}$ and $L_{wm}=0=L_{mw}$. $L_{\theta\theta}$ is the thermal diffusion coefficient across the membrane. $L_{m^{\prime}m^{\prime}}$ is the diffusion coefficient for the transport of membrane molecules from the cytosol to the inner leaflet of the membrane. $L_{ff}$, $L_{ww}$, and $L_{mm}$, are the conductance coefficients of food, waste and membrane molecules through the membrane. We suppose that all these diagonal coefficients are constant and uniform across the cytosol or the membrane, because protocells could not rely on local specialised channel molecules (intrinsic proteins, in evolved cells) to supply their food and evacuate their waste. We also suppose that food and waste molecules are electrically neutral and that the electric current is entirely due to the transport of small ions with the help of the translocation process and water pores. The off-diagonal coefficients, $L_{f\theta}=L_{\theta f}$, $L_{w\theta}=L_{\theta w}$, $L_{m\theta}=L_{\theta m}$ and $L_{q\theta}=L_{\theta q}$, depend on the heat capacity of the molecules transported and on the rate constants of this transport. They couple the transport of matter and the heat flow. For our purpose, the most interesting off-diagonal coefficient is $L_{\theta m}$. It can be viewed as the ratio of heat flow, $J_{\theta}$, to the affinity $X_{m}$ when $T_{0}=T_{1}$ and in the absence of food and waste driving forces : (22) $\displaystyle L_{\theta m}=\left(\frac{J_{\theta}}{X_{m}}\right)_{(X_{\theta},X_{f},X_{w},X_{m^{\prime}})=0}.$ In this case, the thermal flow is due only to the asymmetry of the membrane, induced by its bending. This phenomenon is similar to the Dufour effect [20]. If one can prove experimentally that a bending of the membrane induces a heat flow through it, this means that $L_{\theta m}\neq 0$, hence, by Onsager’s reciprocity relations, $L_{m\theta}\neq 0$, i.e. a heat flow modifies the bending. Indeed, we also have the relation : (23) $\displaystyle L_{m\theta}=\left(\frac{J_{m}}{X_{\theta}}\right)_{(X_{m},X_{f},X_{w},X_{m^{\prime}})=0}.$ Hence, $L_{m\theta}$ measures the effect of a slight temperature difference (between both sides of the membrane) on the induced flow of molecules between the leaflets, which implies a modification of its mean curvature. This phenomenon is similar to the thermodiffusion or Soret effect [20]. It is reciprocal to the previous effect and might be easier to observe and measure. ### 3.5. Entropy production and stability Just as the power dissipated by Joule effect in an ohmic conductor is (24) Power dissipated $\displaystyle=\text{Current}\times\text{Voltage}$ $\displaystyle=\text{Conductance}\times\text{Voltage}^{2},$ the rate of dissipation of energy, or entropy creation, in a general chemical system out of equilibrium is a quadratic function of the thermodynamical forces acting in the system [32, 20] : (25) Rate of entropy produced $\displaystyle=\text{Flows}\times\text{Forces}$ $\displaystyle=\text{Forces}\times\text{Conductance matrix}\times\text{Forces}.$ This relation rests on a linearity hypothesis supposed to be valid only in the neighbourhood of an equilibrium state. The main difference between the ohmic conductor and the chemical system is that, in the latter, the conductance is not a single number but a matrix which, in the general case, couples all the currents. Taking into account the various thermodynamical forces defined previously, the rate of entropy production inside the protocell has to the following expression : (26) $\displaystyle\sigma(X)$ $\displaystyle=L_{ff}X_{f}^{2}+L_{ww}X_{w}^{2}+L_{mm}X_{m}^{2}+L_{m^{\prime}m^{\prime}}X_{m^{\prime}}^{2}+L_{rr}X_{r}^{2}+L_{\theta\theta}X_{\theta}^{2}+2X_{\theta}(L_{\theta f}X_{f}+L_{\theta w}X_{w}+L_{\theta m}X_{m})$ $\displaystyle=L_{ff}\left(\frac{\mu_{f0}}{T_{0}}-\frac{\mu_{f1}}{T_{1}}\right)^{2}+L_{ww}\left(\frac{\mu_{w0}}{T_{0}}-\frac{\mu_{w1}}{T_{1}}\right)^{2}+L_{mm}\left(\frac{\mu_{m0}}{T_{0}}-\frac{\mu_{m1}}{T_{1}}\right)^{2}$ $\displaystyle\qquad+L_{\theta\theta}\left(\frac{1}{T_{0}}-\frac{1}{T_{1}}\right)^{2}+L_{m^{\prime}m^{\prime}}\left(\frac{\mu_{mc}-\mu_{m1}}{T_{1}}\right)^{2}+L_{rr}\left(\frac{A_{r}}{T_{1}}\right)^{2}$ $\displaystyle\qquad+2\left(\frac{1}{T_{0}}-\frac{1}{T_{1}}\right)\left(L_{\theta f}\left(\frac{\mu_{f0}}{T_{0}}-\frac{\mu_{f1}}{T_{1}}\right)+L_{\theta w}\left(\frac{\mu_{w0}}{T_{0}}-\frac{\mu_{w1}}{T_{1}}\right)+L_{\theta m}\left(\frac{\mu_{m0}}{T_{0}}-\frac{\mu_{m1}}{T_{1}}\right)\right).$ The stability of this steady state is equivalent to the positivity of the matrix $L$, which is also the matrix of second order derivatives of $\sigma$ in the coordinate system $X=(X_{\theta},X_{f},X_{w},X_{m},X_{m^{\prime}},X_{r})$ : (27) $\displaystyle L_{ik}=\frac{1}{2}\,\frac{\partial^{2}\sigma}{\partial X_{i}\partial X_{k}}.$ If $P$ is a $n\times n$ matrix with real coefficients, the positivity of $P$, defined by : (28) $\displaystyle u^{\text{t}}Pu>0\qquad\forall u\in\mathbb{R}^{n}$ implies the following inequalities : (29) $\displaystyle P_{ii}>0\quad\forall\,i\qquad\text{and}\qquad P_{ii}P_{jj}>\left(\frac{P_{ij}+P_{ji}}{2}\right)^{2}\quad\forall\,i,j.$ These conditions are necessary but not sufficient to ensure the positivity of $P$. In the present case, $L$ being symmetric, we have, in particular : (30) $\displaystyle L_{ii}$ $\displaystyle>0$ $\displaystyle L_{\theta\theta}L_{ff}$ $\displaystyle>L_{\theta f}^{2}$ $\displaystyle L_{\theta\theta}L_{ww}$ $\displaystyle>L_{\theta w}^{2}$ $\displaystyle L_{\theta\theta}L_{qq}$ $\displaystyle>L_{\theta q}^{2}$ $\displaystyle L_{mm}L_{qq}$ $\displaystyle>L_{mq}^{2}$ $\displaystyle L_{\theta\theta}L_{mm}$ $\displaystyle>L_{\theta m}^{2}.$ If one of these inequalities is not satisfied, the growth process is destabilized. In Section VI, we will prove that the last one can be reversed as the inner temperature of the protocell increases. In order to prove this proposition, we must first write down evolution equations for the geometry of the cell. ## 4\. Membrane geometry and growth equation Just as the growth of a child depends on his diet, the evolution of the geometric parameters of a protocell depends on the flow of molecules to its membrane. This flow is determined by the food intake and by the rate of the synthesis of these structural molecules. In this section, we establish the differential equations governing the growth of the volume and area of a cylindrical protocell by relating them to the flows of matter. ### 4.1. Conservation of matter and exponential growth The advancement, $\xi$, of the overall synthesis reaction, $f\to m+c+w$, is the internal clock of the protocell. The corresponding flow of matter, $J_{r}=\mfrac{\mathrm{d}\xi}{\mathrm{d}t}$, is channeled to all the other processes in the protocell. In particular, it determines the flux of matter to the inner leaflet and the growth speed of the membrane. By writing the equations of conservation of matter, we can then determine the evolution of the size of the protocell. Let $a\in\\{c,1,0\\}$ denote the possible position of a membrane molecule : either in the cytosol $(c)$, or the inner leaflet $(1)$ or the outer leaflet $(0)$. Let $N_{ma}$ be the number of membrane molecules in each of them. The time derivatives of these functions are related to the flows defined previously : (31) $\displaystyle\frac{\mathrm{d}N_{mc}}{\mathrm{d}t}$ $\displaystyle=-J_{mc1}\mathcal{A}_{1}+J_{rm}\mathcal{V}$ $\displaystyle\frac{\mathrm{d}N_{m1}}{\mathrm{d}t}$ $\displaystyle=J_{mc1}\mathcal{A}_{1}-J_{m10}\mathcal{A}$ $\displaystyle\frac{\mathrm{d}N_{m0}}{\mathrm{d}t}$ $\displaystyle=J_{m10}\mathcal{A}.$ Similarly, the number of food (resp. cytosol and waste) molecules, $N_{f}$ (resp. $N_{c}$ and $N_{w}$), evolves according to the following relations : (32) $\displaystyle\frac{\mathrm{d}N_{f}}{\mathrm{d}t}$ $\displaystyle=J_{f}\mathcal{A}_{0}-J_{rf}\mathcal{V}$ $\displaystyle\frac{\mathrm{d}N_{c}}{\mathrm{d}t}$ $\displaystyle=J_{rc}\mathcal{V}$ $\displaystyle\frac{\mathrm{d}N_{w}}{\mathrm{d}t}$ $\displaystyle=-J_{w}\mathcal{A}_{1}+J_{rw}\mathcal{V}$ where the flows $J_{r\bullet}$ are defined by : (33) $\displaystyle J_{rm}$ $\displaystyle:=\nu_{m}\frac{\mathrm{d}\xi}{\mathrm{d}t}$ $\displaystyle J_{rf}$ $\displaystyle:=\nu_{f}\frac{\mathrm{d}\xi}{\mathrm{d}t}\,=\,\frac{\nu_{f}}{\nu_{m}}\,J_{rm}$ $\displaystyle J_{rc}$ $\displaystyle:=\nu_{c}\frac{\mathrm{d}\xi}{\mathrm{d}t}\,=\,\frac{\nu_{c}}{\nu_{m}}\,J_{rm}$ $\displaystyle J_{rw}$ $\displaystyle:=\nu_{w}\frac{\mathrm{d}\xi}{\mathrm{d}t}\,=\,\frac{\nu_{w}}{\nu_{m}}\,J_{rm}.$ In a steady state, the concentration of membrane molecules in the cytosol is constant : (34) $\displaystyle C_{mc}:=\frac{N_{mc}}{\mathcal{V}}=\text{cst.}$ Let $c_{m0}$ and $c_{m1}$ be the average number of membrane molecules per unit area in each leaflet : (35) $\displaystyle c_{m0}:=\frac{N_{m0}}{\mathcal{A}_{0}}\qquad\text{and}\qquad c_{m1}:=\frac{N_{m1}}{\mathcal{A}_{1}}.$ The conservation equations for $m$ imply the evolution equations of the geometry of the protocell : (36) $\displaystyle c_{m0}\frac{\mathrm{d}\mathcal{A}_{0}}{\mathrm{d}t}$ $\displaystyle=J_{m10}\,\frac{\mathcal{A}_{0}+\mathcal{A}_{1}}{2}$ $\displaystyle c_{m1}\frac{\mathrm{d}\mathcal{A}_{1}}{\mathrm{d}t}$ $\displaystyle=J_{mc1}\mathcal{A}_{1}-c_{m0}\frac{\mathrm{d}\mathcal{A}_{0}}{\mathrm{d}t}$ $\displaystyle c_{mc}\frac{\mathrm{d}\mathcal{V}}{\mathrm{d}t}$ $\displaystyle=J_{rm}\mathcal{V}-J_{mc1}\mathcal{A}_{1}.$ Let us introduce the following parameters : (37) $\displaystyle 2\varepsilon$ $\displaystyle:=\text{average thickness of the membrane}$ $\displaystyle\eta$ $\displaystyle:=\frac{c_{m1}}{c_{m0}}\quad\text{(layer density ratio $\simeq 1$)}$ $\displaystyle\tau$ $\displaystyle:=\frac{J_{m10}}{J_{mc1}}\quad\text{(transmission rate {through} the membrane)}$ $\displaystyle t_{1}$ $\displaystyle:=\frac{c_{m1}}{J_{mc1}}\quad\text{(inner leaflet characteristic time)}$ $\displaystyle\tau_{c}$ $\displaystyle:=\frac{J_{mc1}}{J_{rm}}\quad\text{(transmission rate {to} the membrane)}$ $\displaystyle t_{c}$ $\displaystyle:=\frac{c_{mc}}{J_{rm}}\quad\text{(cytosol characteristic time)}.$ The transmission ratio, $\tau$, can be written in terms of thermodynamical forces : (38) $\displaystyle\tau:=\frac{J_{m}}{J_{mc1}}=\frac{L_{mm}X_{m}+L_{m\theta}X_{\theta}+{\ldots}}{L_{m^{\prime}m^{\prime}}X_{m^{\prime}}}.$ Let $\mathcal{U}={\mathcal{V}}/{\varepsilon}$ and $\dot{X}=t_{1}\mfrac{\mathrm{d}X}{\mathrm{d}t}$. We obtain the following system of differential equations : (39) $\displaystyle\dot{\mathcal{A}}_{0}$ $\displaystyle=\frac{\eta\tau}{2}\,(\mathcal{A}_{0}+\mathcal{A}_{1})\,=\,\eta\tau\mathcal{A}$ $\displaystyle\dot{\mathcal{A}}_{1}$ $\displaystyle=-\frac{\tau}{2}\,\mathcal{A}_{0}+\left(1-\frac{\tau}{2}\right)\,\mathcal{A}_{1}\,=\,(1-\tau)\mathcal{A}-\mathcal{B}$ $\displaystyle\dot{\mathcal{U}}$ $\displaystyle=\frac{t_{1}}{t_{c}}\,\mathcal{U}-\frac{c_{m1}}{\varepsilon c_{mc}}\,\mathcal{A}_{1}.$ In matrix form : (40) $\displaystyle\dot{X}$ $\displaystyle=\begin{pmatrix}\dot{\mathcal{A}}_{0}\\\ \dot{\mathcal{A}}_{1}\\\ \dot{\mathcal{U}}\\\ \end{pmatrix}=\begin{pmatrix}\frac{\eta\tau}{2}&\frac{\eta\tau}{2}&0\\\ -\frac{\tau}{2}&\frac{2-\tau}{2}&0\\\ 0&-\frac{c_{m1}}{\varepsilon c_{mc}}&\frac{t_{1}}{t_{c}}\\\ \end{pmatrix}\begin{pmatrix}\mathcal{A}_{0}\\\ \mathcal{A}_{1}\\\ \mathcal{U}\\\ \end{pmatrix}=MX$ $\displaystyle M$ $\displaystyle:=\begin{pmatrix}\frac{\eta\tau}{2}&\frac{\eta\tau}{2}&0\\\ -\frac{\tau}{2}&\frac{2-\tau}{2}&0\\\ 0&-\frac{c_{m1}}{\varepsilon c_{mc}}&\frac{t_{1}}{t_{c}}\\\ \end{pmatrix}\quad\text{and}\quad X:=\begin{pmatrix}\mathcal{A}_{0}\\\ \mathcal{A}_{1}\\\ \mathcal{U}\\\ \end{pmatrix}.$ This growth equation is solved in Appendix B. The matrix $M$ has a block diagonal form, hence $\mathcal{A}_{0}$ and $\mathcal{A}_{1}$ evolve independently of $\mathcal{U}$, whereas the equation for $\mathcal{U}$ contains terms linear in $\mathcal{A}_{0}$ and $\mathcal{A}_{1}$. The upper left $2\times 2$ block is not diagonal, hence $\mathcal{A}_{0}$ and $\mathcal{A}_{1}$ are linear combinations of exponential functions of time (multiplied by an affine function of $t$ in the degenerate, non diagonalisable case). The rates of growth of these exponential functions are the eigenvalues of this $2\times 2$ block, plus an exponential of growth rate $\mfrac{t_{1}}{t_{c}}$ for $\mathcal{U}$. ### 4.2. Cylindrical growth in steady state When we meet an ordinary differential equation, describing the time evolution of a dynamical system, a first reflex is to search for constant solutions or at least steady state solutions, where the speed is constant. In the present case, we can look for a solution where the length increases steadily whereas the radius is constant. This corresponds to the observed growth of some bacterial species in difficult environments [28]. When the sludge content of wastewater is too high or when the composition is lopsided, a higher percentage of bacteria adopt a filamentous growth strategy which allows them to survive in harsher conditions, by catching food more easily. If the protocell grows like a cylinder of radius $R_{0}$, we have $\varepsilon\mathcal{A}=R_{0}\mathcal{B}$, hence $\mfrac{\mathrm{d}\mathcal{A}}{\mathcal{A}}=\mfrac{\mathrm{d}\mathcal{B}}{\mathcal{B}}$ and (41) $\displaystyle x$ $\displaystyle:=\frac{R_{0}}{\varepsilon}=\frac{\mathcal{A}}{\mathcal{B}}=\frac{\mathrm{d}\mathcal{A}}{\mathrm{d}\mathcal{B}}=\frac{\dot{\mathcal{A}}}{\dot{\mathcal{B}}}$ $\displaystyle=\frac{\big{(}(\eta-1)\tau+1\big{)}\mathcal{A}-\mathcal{B}}{\big{(}(\eta+1)\tau-1\big{)}\mathcal{A}+\mathcal{B}}$ $\displaystyle=\frac{\alpha_{+}\mathcal{A}-\mathcal{B}}{\alpha_{-}\mathcal{A}+\mathcal{B}}$ where $\alpha_{\pm}=(\eta\mp 1)\tau\pm 1$. Therefore, $x$ satisfies the fixed point equation : (42) $\displaystyle x=\frac{\alpha_{+}x-1}{\alpha_{-}x+1}\qquad\text{{\it i.e. }}\qquad\alpha_{-}x^{2}-(\alpha_{+}-1)x+1=0.$ The discriminant of this quadratic equation is (43) $\displaystyle(\alpha_{+}-1)^{2}-4\alpha_{-}$ $\displaystyle=(\eta-1)^{2}\tau^{2}-4(\eta+1)\tau+4$ $\displaystyle=4\Delta(\eta,\tau)$ (cf. Appendix B) and its roots, $x_{\pm}$, are related to the eigenvalues, $\lambda_{\pm}$, of the matrix $M$ (Eq. 40) : (44) $\displaystyle x_{\pm}$ $\displaystyle=\frac{1}{2\alpha_{-}}\left(\alpha_{+}-1\pm\sqrt{(\alpha_{+}-1)^{2}-4\alpha_{-}}\right)$ $\displaystyle=\frac{(\alpha_{+}-1)\pm 2\sqrt{\Delta(\eta,\tau)}}{2\alpha_{-}}$ $\displaystyle=\frac{2\lambda_{\pm}-1}{\alpha_{-}}.$ Consequently, the radius, $R_{0}$, of the cylinder whose length increases in a steady state is determined by the flows $(J_{mc1},J_{m10},J_{rm})$ and the concentrations $(C_{mc},c_{m1},c_{m0})$, via the coefficients $(\varepsilon,\eta,\tau)$ : (45) $\displaystyle R_{0}$ $\displaystyle=\varepsilon x_{\pm}=\varepsilon\,\frac{\lambda_{\pm}-1}{2\alpha_{-}}=\frac{\varepsilon\big{(}(\eta+1)\tau\pm 2\sqrt{\Delta}\big{)}}{2\big{(}(\eta+1)\tau-1\big{)}}$ $\displaystyle=\frac{\varepsilon}{2}\,\frac{(\eta+1)\tau\pm\sqrt{(\eta-1)^{2}\tau^{2}-(\eta+1)\tau+1}}{(\eta+1)\tau-1}$ ## 5\. Thermal instability of cylindrical growth As long as the protocell grows by increasing only its length, keeping a cylindrical shape of fixed radius, $R_{0}$, its volume and its membrane area grow proportionally, i.e. $\dot{\mathcal{A}}=\text{cst.}\times\dot{\mathcal{B}}$. If the heat generated by the metabolic reactions were exactly proportional to the volume increment, the increase of the area of the membrane would be sufficient to evacuate steadily the heat generated by the chemical reactions taking place inside the newly created volume. However, the heat generated by all these irreversible processes adds up to that coming from the exothermic metabolic reactions and the inner temperature must therefore increase. This overheating generates larger fluctuations of all the physical parameters which destabilize the initial steady state of cylindrical growth. We will see below that the geometrical parameters $(\mathcal{A},\mathcal{B},\mathcal{V})$ can follow a path leading to a more efficient release of heat, by reducing the radius $R_{0}$. ### 5.1. The Squeezed Sausage Theorem (SST) When we squeeze a sausage, its length increases as well as its area. Indeed, the stuffing being incompressible, the squeezing is an isovolumic deformation. The stuffing is pushed longitudinally, away from the squeezed zone, and increases the length of the sausage, thanks to the elasticity of the gut. The area of the slice of reduced radius increases consequently to bound the same volume. Let us prove this mathematically. A length $\delta x$ of cylinder of radius $R_{0}$ has volume $\delta\mathcal{V}$ and boundary area $\delta A$ given by : (46) $\displaystyle\delta\mathcal{V}$ $\displaystyle=\pi R_{0}^{2}\,\delta x$ $\displaystyle\delta\mathcal{A}$ $\displaystyle=2\pi R_{0}\,\delta x$ Let us suppose that this cylindrical growth is perturbed by a small, local radius variation, which can be positive (anevrism) or negative (stenosis). We study here a triangular perturbation and, in the appendix, a smooth $(\mathcal{C}^{2})$, rotation invariant perturbation of the cylinder. To keep it simple, we suppose that this perturbation is piecewise linear and symmetric, with an extremum $\delta R$ at $x=0$, and vanishes outside of the interval $\left[-\mfrac{\delta x^{\prime}}{2},\mfrac{\delta x^{\prime}}{2}\right]$. FIG. 5.1 represents the resulting isovolumic deformation according with the sign of $\delta R$. $2R_{0}$$\delta x$$\delta\mathcal{V}$$\delta\mathcal{A}$$\delta x^{\prime}>\delta x$ $2(R_{0}+\delta R)<2R_{0}$ $\delta\mathcal{V}^{\prime}=\delta\mathcal{V}$ $\delta\mathcal{A}^{\prime}>\delta A$ $2(R_{0}+\delta R)>2R_{0}$ $\delta\mathcal{V}^{\prime}=\delta V$ $\delta x^{\prime}<\delta x$$\delta\mathcal{A}^{\prime}<\delta A$ Isovolumic variation of the area of a cylinder under a small triangular deformation. In the second and third pictures of FIG. 5.1, the Gaussian curvature is concentrated on the circular sections at $x=0$ and at $x=\pm\mfrac{\delta x^{\prime}}{2}$ (dotted lines), where the mean curvature has a finite discontinuity. The volume and lateral membrane area of this slice of thickness $\delta x^{\prime}$ (contained between the dotted lines) are therefore : (47) $\displaystyle\delta\mathcal{V}^{\prime}$ $\displaystyle=\pi\left(R_{0}+\frac{\delta R}{2}\right)^{2}\,\delta x^{\prime}$ $\displaystyle\delta\mathcal{A}^{\prime}$ $\displaystyle=2\pi\left(R_{0}+\frac{\delta R}{2}\right)\,\delta x^{\prime}.$ The straight slice and the deformed slice have equal volumes $(\delta\mathcal{V}=\delta\mathcal{V}^{\prime})$ if their thicknesses satisfy : (48) $\displaystyle\frac{\delta x^{\prime}}{\delta x}=\left(1+\frac{\delta R}{2R_{0}}\right)^{-2}\simeq\left(1-\frac{\delta R}{R_{0}}\right).$ Hence the ratio of their areas is (49) $\displaystyle\frac{\delta\mathcal{A}^{\prime}}{\delta\mathcal{A}}\simeq\left(1+\frac{\delta R}{2R_{0}}\right)\left(1-\frac{\delta R}{R_{0}}\right)\simeq 1-\frac{\delta R}{2R_{0}}.$ The heat flows through these surfaces are, respectively : (50) $\displaystyle\delta q$ $\displaystyle=L_{\theta\theta}\left(\frac{1}{T_{0}}-\frac{1}{T_{1}}\right)\,\delta\mathcal{A}$ $\displaystyle\delta q^{\prime}$ $\displaystyle=L_{\theta\theta}\left(\frac{1}{T_{0}}-\frac{1}{T_{1}}\right)\,\delta\mathcal{A}^{\prime}$ hence their ratio is the same as for the areas : (51) $\displaystyle\frac{\delta q^{\prime}}{\delta q}=\frac{\delta\mathcal{A}^{\prime}}{\delta\mathcal{A}}=1-\frac{\delta R}{2R_{0}}.$ When $\delta R<0$, this ratio is larger than $1$. Consequently, the inner volume being held fixed, a small stenosis of a cylindrical protocell evacuates heat more efficiently than a small anevrism. This local reduction of the radius of the protocell increases its mean curvature. For this deformation to happen, the outer leaflet must grow more rapidly than the inner leaflet. Therefore, the equilibrium $m_{1}\leftrightarrows m_{0}$ must be shifted towards $m_{0}$ in order to have $\delta R<0$. This is possible if $T_{1}$ increases slightly and $m_{1}\to m_{0}$ is exothermic. We propose that the translocation of membrane molecules to the outer leaflet [14, 15, 4, 1] can be triggered by the increase of the inner temperature, $T_{1}(t)$. The area of the outer leaflet then increases more quickly than the area of the inner leaflet, which leads to the bending of the membrane until the total splitting of the protocell into two daughters. ### 5.2. Fluctuations, translocation and heat transfer In order to increase $L_{m\theta}$ and destabilise the cylindrical growth, the transfer coefficient, $\tau$, must also increase. In [14, 15], the authors present a detailed mechanism for the transfer of membrane molecules between the leaflets. Due to the fluctuations of ionic densities in the neighbourhood of the membranes, the local electric field fluctuates strongly enough to push molecules of water into the membrane, via the field-dipole interaction force (dielectrophoresis). When it is sufficiently strong, this force can create a transient water pore that is stable enough to let some membrane molecules dive into this water pore and join the other side. The increase of the inner temperature can also enhance these ionic density fluctuations and favor this translocation process from the hot side to the cold side, since the hottest, most agitated molecules have a higher probability to dive into the water pore than the colder molecules. This asymmetric flow of hot molecules to the cold side enhances the outgoing heat flow and cools down the protocell. During this process, the shape of the hydrophobic tails is not important, as long as they remain in the hydrophobic zone, surrounded by siblings. The only energetic cost is for the hydrophilic head surrounded by these aliphatic chains, and some clandestine water molecules forming the water pore (not represented below). The shape of the tail is irrelevant since the energy depends only on the position of the polar head (FIG. 5.2). $\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$hydrophilic zonehydrophilic zonehydrophobic zone Translocation of a membrane molecule from one leaflet to the other. ### 5.3. Thermal balance Let us make a thermal balance of the whole growth process. After heating its cold nutrient molecules from $T_{0}$ to $T_{1}$ and processing the isothermal inner chemical reactions $(J_{r})$, our protocell disposes of its hot waste (including some water flowing through the water pores) and loses heat by translocation of membrane molecules from the inside to the outside, and by diffusion $(J_{\theta})$ without mass transfer. Let $q_{i}$ be the heat exported out of the protocell by each molecule of type $i$. Cold entering molecules and hot outgoing molecules both have $q_{i}>0$. Let $\kappa_{i}$ be the heat capacity of the molecules of type $i$. Let $J_{h}$ be the outgoing heat flow (energy/(time $\times$ area)). The heat flow exported by the cold entering food and water molecules is : (52) $\displaystyle J_{f}q_{f}$ $\displaystyle=J_{f}\kappa_{f}(T_{1}-T_{0}).$ Similarly, the heat flow exported by the outgoing waste and water molecules is : (53) $\displaystyle J_{w}q_{w}$ $\displaystyle=J_{w}\kappa_{w}(T_{1}-T_{0}).$ And the heat flow exported by the net translocation of membrane molecules is : (54) $\displaystyle J_{m}q_{m}$ $\displaystyle=J_{m}\kappa_{m}(T_{1}-T_{0})$ if we suppose that they immediately thermalise from $T_{1}$ to $T_{0}$ once they reach the outer leaflet. The contact of the hydrophobic tails inside the membrane allows for a diffusive heat flow : (55) $\displaystyle J_{\theta}$ $\displaystyle=\sum_{i}L_{\theta k}X_{k}.$ The total heat flow is the sum of these terms : (56) $\displaystyle J_{h}$ $\displaystyle:=(J_{f}q_{f}+J_{w}q_{w}+J_{m}q_{m})+J_{\theta}$ $\displaystyle=\big{(}J_{f}\kappa_{f}+J_{w}\kappa_{w}\big{)}(T_{1}-T_{0})$ $\displaystyle\quad+(L_{mm}X_{m}+L_{m\theta}X_{\theta}+{\ldots})\kappa_{m}(T_{1}-T_{0})$ $\displaystyle\quad+(L_{\theta\theta}X_{\theta}+L_{\theta m}X_{m}+{\ldots}).$ Since (57) $\displaystyle T_{1}-T_{0}=T_{0}T_{1}X_{\theta}=\frac{T_{0}^{2}X_{\theta}}{1-T_{0}X_{\theta}}$ $L_{m\theta}$ appears as a factor of $X_{\theta}^{2}$ in the convective term, $J_{m}$, whereas $L_{\theta m}$ is a factor of $X_{m}$ in the diffusive term, $J_{\theta}$. Moreover, $X_{m}$ increases linearly with $X_{\theta}$ : (58) $\displaystyle X_{m}$ $\displaystyle=\frac{\mu_{m0}}{T_{0}}-\frac{\mu_{m1}}{T_{1}}$ $\displaystyle=\frac{\mu_{m}^{\circ}}{T_{0}}-\frac{\mu_{m}^{\circ}}{T_{1}}+k_{B}\ln\left(\frac{a_{m0}}{a_{m1}}\right)$ $\displaystyle=\mu_{m}^{\circ}X_{\theta}+k_{B}\ln\left(\frac{a_{m0}}{a_{m1}}\right)$ where $\mu^{\circ}$ denotes the standard chemical potential, at temperature $298$ K and pressure $1$ atm [2]. The $L_{m\theta}$-dependent term in $J_{h}$ becomes : (59) $\displaystyle J_{h}=L_{m\theta}\left(\frac{\kappa_{m}T_{0}^{2}X_{\theta}^{2}}{1-T_{0}X_{\theta}}+\mu_{m}^{\circ}X_{\theta}\right)+{\ldots}$ Consequently, as the cytosol heats up, $J_{h}$ increases more quickly by translocation ($\kappa_{m}$ term) than by diffusion ($\mu_{m}^{\circ}$ term). Translocation is a particular kind of heat convection and by analogy with the Rayleigh-Bénard instability [35], we conjecture the existence of a transition from a diffusive regime to a convective regime, where translocation overtakes diffusion and expells heat more efficiently. ## 6\. Translocation between leaflets The energetic barrier, of width $2\varepsilon^{\prime}$ and height $E_{\ast}$, is difficult to penetrate for the hydrophilic head since this guarantees the stability of the bilayer under ordinary thermal fluctuations. When the ratio of concentrations, $\eta=\mfrac{c_{m1}}{c_{m0}}$, becomes too large compared to unity, the mechanical constraint on the inner leaflet is released by pushing molecules to the outer leaflet. Conversely, when the outer leaflet is stretched and the inner leaflet compressed, $\eta$ is slightly greater than unity (FIG. 6). $\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup$$c_{m1}$compressed$c_{m0}<c_{m1}$stretched$\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand$ Mechanical constraints modify the ratio, $\eta$, of leaflet concentrations. To facilitate this process, some water molecules can leak through the hydrophobic zone and ease the passage of the hydrophilic head. This leakage of water lowers the activation energy, $E_{\ast}$, and realizes an aqueous catalysis of the translocation process [14, 15, 4, 1]. If we suppose that the density, $n_{p}$, of water pores in the membrane is constant for fixed temperatures, $T_{0}$ and $T_{1}$, then $J_{m10}$ depends only on this density and on the net number, $j_{mp}$, of membrane molecules translocated from $\mathbf{L}_{1}$ to $\mathbf{L}_{0}$ during the lifetime of the pores : (60) $\displaystyle n_{p}$ $\displaystyle:=\text{ number of water pores per unit area }$ $\displaystyle j_{mp}$ $\displaystyle:=\text{ net number of translocations}$ through each water pore $\displaystyle J_{m}$ $\displaystyle=n_{p}j_{mp}.$ This first approximation is based on the hypothesis that the pores have the same size, the same lifetime and the same number of net translocations during their short life. However, to be more realistic, we must take into account the fact that larger pores live longer and leak more (over the same duration) than smaller short lived pores. We integrate over the interval of possible lifetimes $(t_{p})$ the density of water pores of lifetime $t_{p}$ created per unit time $(n_{p}(t_{p}))$ multiplied by the net number $(\nu_{mp}(t_{p}))$ of molecules each pore of lifetime $t_{p}$ translocates from the inside to the outside during its existence : (61) $\displaystyle J_{m}=\int_{0}^{\infty}\mathrm{d}t_{p}\,n_{p}(t_{p})\nu_{mp}(t_{p}).$ The increase of $X_{\theta}$ enhances at the same time the rate of formation of pores, hence $n_{p}$, and the net number of translocated molecules, due to larger thermal fluctuations. Therefore, $J_{m10}$ increases more than linearly as a function of $X_{\theta}$. Consequently, the crossed conductivity coefficient, $L_{m\theta}$, increases with $X_{\theta}$. On the other side of the inequality, $L_{\theta\theta}$ and $L_{mm}$ depend more weakly on the temperature. Indeed, the heat diffusion coefficient, $L_{\theta\theta}$, involves the (temperature independant) number of interacting degrees of freedom between the hydrophobic tails inside the hydrophobic layer, and the molecular diffusion coefficient : (62) $\displaystyle L_{mm}=T_{0}\left(\frac{J_{m}}{\mu_{m1}-\mu_{m0}}\right)_{(X_{\theta},X_{f},X_{w},X_{m^{\prime}})=0}$ depends mainly on the ratio of concentrations between the two leaflets, i.e. on $\eta$. In order to know if the initial inequality, $L_{m\theta}^{2}<L_{\theta\theta}L_{mm}$, can be reversed, the temperature dependance of the convective coefficient, $L_{m\theta}$, must be computed and compared to that of the diffusion coefficients, $L_{\theta\theta}$ and $L_{mm}$. This necessitates a microscopic model of the interactions of membrane molecules and water and a precise description of the translocation process, to go beyond the linear response theory. In the sequel, we adopt a simple mean field approach where each molecule evolves in the same energetic landscape as the others. ### 6.1. An effective potential for translocation The exact shape and position of each membrane molecule is described by dozens of parameters specifying the position of each atom and the orientation of each interatomic bond. It would be cumbersome to take them all into account to describe mathematically the evolution of a single molecule inside the membrane. However, we can make a simplifying approximation by remarking that the main energetic cost is in the displacement of the hydrophilic head into the hydrophobic layer or the protrusion of this head outside of the membrane, which forces the tail to go into the hydrophilic zone. We can make a mean- field approximation by considering only the position, $z$, of the hydrophilic head as a dynamical variable, and defining an adequate effective potential energy, $U(z)$, that traps the head inside the membrane. In the sequel of this article, we will use a double well effective potential to compute the net flow, $J_{m}$, across a plane bilayer subject to a difference of temperatures. By differentiation, we obtain the coefficients $L_{m\theta}$ and $L_{mm}$ and, in particular, their dependence on temperature. This model suggests that the inequality $L_{m\theta}^{2}<L_{mm}L_{\theta\theta}$ can be reversed if the inner temperature increases sufficiently. Our hypotheses are the following ones : 1. (1) The membrane molecules have length $\varepsilon=\varepsilon^{\prime}+\varepsilon^{\prime\prime}$, where $\varepsilon^{\prime\prime}$ is the size of the hydrophilic head and $\varepsilon^{\prime}$ is the length of the hydrophobic tail. 2. (2) The translocation process is described by only one parameter : the position of the center of mass of the hydrophilic head, varying between $-\varepsilon$ and $+\varepsilon$. 3. (3) On each side of the membrane, the distribution of velocities of the heads follows a Maxwell-Boltzmann law [35]. The probability of finding a molecule with velocity $v$ perpendicularly to the membrane is : (63) $\displaystyle p_{i}(v)=\sqrt{\frac{m}{2\pi k_{B}T_{i}}}\exp\left(-\frac{mv^{2}}{2k_{B}T_{i}}\right).$ 4. (4) The translocation requires an energy $E^{\ast}$ and the head of the molecule evolves in an effective double well potential (FIG. 4). hydrophilic zonepolar heads\+ water\+ ionshydrophobic zonealiphatic chainshydrophilic zonepolar heads\+ water\+ ions$z$$-\varepsilon$$-\varepsilon^{\prime}$$T_{1}$$\varepsilon$$\varepsilon^{\prime}$$T_{0}$$E^{\ast}$$\color[rgb]{1,0,0}U(z)$ Potential energy of the hydrophilic head 5. (5) The hydrophilic heads trapped in the well $[-\varepsilon,-\varepsilon^{\prime}]$ have temperature $T_{1}$, whereas those trapped in the well $[\varepsilon^{\prime},\varepsilon]$ have temperature $T_{0}$. The thermalisation processes for the motion along the $z$ axis occur only once the head is trapped in the arrival well. This drastic hypothesis simplifies the computations and should be refined in a more realistic model. In reality, the motions of the hydrophobic tails between $z=-\varepsilon^{\prime}$ and $z=+\varepsilon^{\prime}$ can thermalise the molecule during the travel across the membrane and this affects the translocation time. Only half of the molecules of kinetic energy $E>E_{\ast}$ can escape from a well to the other side. The time it takes them to go through the barrier is given by : (64) $\displaystyle t_{f}$ $\displaystyle=\int_{-\varepsilon^{\prime}}^{+\varepsilon^{\prime}}\,\mathrm{d}z\sqrt{\frac{m}{2(E-E_{\ast})}}$ $\displaystyle=2\varepsilon^{\prime}\sqrt{\frac{m}{2(E-E_{\ast})}}$ $\displaystyle=2\varepsilon^{\prime}\sqrt{\frac{m}{mv^{2}-2E_{\ast}}}.$ The flow of molecules of velocity belonging to the interval $[v,v+\mathrm{d}v]$, with $v>v_{\ast}:=\sqrt{\mfrac{2E_{\ast}}{m}}$, going from side $1$ to side $0$, is proportional to the surface density of molecules, $c_{m1}$, to the Maxwell-Boltzmann weight, $p_{1}(v)\mathrm{d}v$, of this velocity interval, and to the reciprocal of the translocation time : (65) $\displaystyle J_{m10}$ $\displaystyle=\int_{v_{\ast}}^{+\infty}\mathrm{d}v\,\frac{c_{m1}p_{1}(v)}{t_{f}}$ $\displaystyle=\int_{E_{\ast}}^{+\infty}\frac{\mathrm{d}E}{\sqrt{2mE}}\,\frac{1}{2\varepsilon^{\prime}}\,\sqrt{\frac{2(E-E_{\ast})}{m}}\,\frac{c_{m1}e^{-E/k_{B}T_{1}}}{\sqrt{\frac{2\pi k_{B}T_{1}}{m}}}$ $\displaystyle=\frac{1}{2\varepsilon^{\prime}\sqrt{\pi}}\int_{E_{\ast}}^{+\infty}\frac{\mathrm{d}E}{\sqrt{2mE}}\sqrt{\frac{E-E_{\ast}}{k_{B}T_{1}}}\,c_{m1}e^{-E/k_{B}T_{1}}.$ The net flow of molecules from leaflet $1$ to leaflet $0$ is : (66) $\displaystyle J_{m}$ $\displaystyle:=J_{m10}-J_{m01}$ $\displaystyle=\frac{1}{2\varepsilon^{\prime}\sqrt{\pi}}\int_{E_{\ast}}^{+\infty}\frac{\mathrm{d}E}{\sqrt{2mE}}\sqrt{\frac{E-E_{\ast}}{k_{B}T_{1}}}\,c_{m1}e^{-E/k_{B}T_{1}}$ $\displaystyle\quad-\frac{1}{2\varepsilon^{\prime}\sqrt{\pi}}\int_{E_{\ast}}^{+\infty}\frac{\mathrm{d}E}{\sqrt{2mE}}\sqrt{\frac{E-E_{\ast}}{k_{B}T_{0}}}\,c_{m0}e^{-E/k_{B}T_{0}}.$ ### 6.2. Computation of $L_{m\theta}$ The temperature $T_{0}$ being fixed, we have : (67) $\displaystyle L_{m\theta}$ $\displaystyle=\frac{\partial J_{m}}{\partial X_{\theta}}=-\frac{\partial J_{m}}{\partial T_{1}^{-1}}$ $\displaystyle=-\frac{c_{m1}}{2\varepsilon^{\prime}\sqrt{\pi}}\int_{E_{\ast}}^{+\infty}\mathrm{d}E\,\sqrt{\frac{E-E_{\ast}}{2mE}}\,\frac{\partial}{\partial T_{1}^{-1}}\left(\frac{e^{-E/k_{B}T_{1}}}{\sqrt{k_{B}T_{1}}}\right)$ $\displaystyle=\frac{c_{m1}}{2\varepsilon^{\prime}k_{B}\sqrt{2\pi mk_{B}T_{1}}}\int_{E_{\ast}}^{+\infty}\mathrm{d}E\,\sqrt{\frac{E-E_{\ast}}{E}}\,\left(E-\frac{k_{B}T_{1}}{2}\right)\,{e^{-E/k_{B}T_{1}}}.$ We set $u_{\ast 1}:=\mfrac{E_{\ast}}{k_{B}T_{1}}$ and change the variable of integration from $E$ to $s:=\mfrac{E}{E_{\ast}}$ : (68) $\displaystyle L_{m\theta}$ $\displaystyle=\frac{c_{m1}E_{\ast}\sqrt{k_{B}T_{1}}}{4\varepsilon^{\prime}k_{B}\sqrt{2\pi m}}\int_{1}^{+\infty}\mathrm{d}s\,\sqrt{1-\frac{1}{s}}\,(2su_{\ast 1}-1)\,{e^{-su_{\ast 1}}}$ $\displaystyle=\alpha_{1}F(u_{\ast 1})$ $\displaystyle\alpha_{1}$ $\displaystyle:=\frac{c_{m1}E_{\ast}\sqrt{k_{B}T_{1}}}{4\varepsilon^{\prime}k_{B}\sqrt{2\pi m}}$ where the function $F$ is defined by : (69) $\displaystyle F(a):=\int_{1}^{+\infty}\mathrm{d}s\,\sqrt{1-\frac{1}{s}}\,(2as-1)e^{-as}.$ We can now compute the relative variations of $L_{m\theta}$ with respect to relative variations of temperature. Since $L_{m\theta}$ depends on $T_{1}$ through $E_{\ast}$ and $F(u_{\ast 1})$, we have : (70) $\displaystyle\frac{\partial\ln L_{m\theta}}{\partial\ln T_{1}}$ $\displaystyle=\frac{\partial\ln\alpha_{1}}{\partial\ln T_{1}}+\frac{\partial\ln F}{\partial\ln T_{1}}$ $\displaystyle=\frac{1}{2}+\frac{\partial\ln E_{\ast}}{\partial\ln T_{1}}+\frac{\partial\ln u_{\ast 1}}{\partial\ln T_{1}}\,\frac{\partial\ln F}{\partial\ln u_{\ast 1}}$ $\displaystyle=\frac{1}{2}+\frac{\partial\ln E_{\ast}}{\partial\ln T_{1}}+\left(\frac{\partial\ln E_{\ast}}{\partial\ln T_{1}}-1\right)\,\frac{\partial\ln F}{\partial\ln u_{\ast 1}}$ $\displaystyle=\frac{1}{2}+\frac{\partial\ln E_{\ast}}{\partial\ln T_{1}}\left(1+\frac{\partial\ln F}{\partial\ln u_{\ast 1}}\right)-\frac{\partial\ln F}{\partial\ln u_{\ast 1}}.$ $\mfrac{\partial\ln E_{\ast}}{\partial\ln T_{1}}$ can not be computed in the present model, because it depends on the microscopic details of the formation of water pores. However, we know that $E_{\ast}$ diminishes as $T_{1}$ increases, since the water pores become more frequent (and, probably, larger and more durable) when the ionic density fluctuations increase [14, 15]. Consequently, we have : (71) $\displaystyle\frac{\partial\ln E_{\ast}}{\partial\ln T_{1}}<0.$ In Appendix C, we prove that $1+\mfrac{\partial\ln F}{\partial\ln u_{\ast 1}}$ is slightly negative at high temperature. Since $\mfrac{\partial\ln E_{\ast}}{\partial\ln T_{1}}$ is also negative, we obtain the following estimate : (72) $\displaystyle\frac{\partial\ln L_{m\theta}}{\partial\ln T_{1}}\gtrsim\frac{3}{2}\qquad\text{at high temperature.}$ ### 6.3. Computation of $L_{mm}$ $L_{mm}$ is obtained by differentiating $J_{m}$ with respect to $X_{m}=\mfrac{\mu_{m1}-\mu_{m0}}{T_{0}}$ while keeping the other thermodynamical forces equal to zero : (73) $\displaystyle L_{mm}$ $\displaystyle=\left(\frac{\partial J_{m}}{\partial X_{m}}\right)_{(X_{\theta},X_{f},X_{w},X_{m^{\prime}})=0}$ $\displaystyle=T_{0}\left(\frac{\partial J_{m}}{\partial(\mu_{m1}-\mu_{m0})}\right)_{(X_{\theta},X_{f},X_{w},X_{m^{\prime}})=0}$ $\displaystyle=\frac{1}{k_{B}}\left(\frac{\partial J_{m}}{\partial\ln(a_{1}/a_{0})}\right)_{(X_{\theta},X_{f},X_{w},X_{m^{\prime}})=0}.$ In our model, based on the double well effective potential, the activities of the membrane molecules in each leaflet are equal to their respective concentrations. A more accurate model, taking into account the attractive interactions inside each leaflet, is necessary to improve this first approximation. Replacing $\mfrac{a_{1}}{a_{0}}$ by $\mfrac{c_{m1}}{c_{m0}}=\eta$, we obtain : (74) $\displaystyle L_{mm}=\frac{1}{k_{B}}\left(\frac{\partial J_{m}}{\partial\ln\eta}\right)_{T_{1}=T_{0}}.$ $J_{m}$ is a linear combination of the leaflet concentrations : (75) $\displaystyle J_{m}$ $\displaystyle=\zeta(T_{1})c_{m1}-\zeta(T_{0})c_{m0}$ $\displaystyle\zeta(T)$ $\displaystyle:=\frac{E_{\ast}e^{-u_{\ast}}}{2\varepsilon^{\prime}\sqrt{2\pi mk_{B}T}}\int_{0}^{+\infty}\mathrm{d}x\,e^{-u_{\ast}x}\sqrt{\frac{x}{x+1}}$ $\displaystyle u_{\ast}$ $\displaystyle:=\frac{E_{\ast}}{k_{B}T}.$ If the temperatures of both leaflets are equal, then $J_{m}$ is simply proportional to the difference of their concentrations : (76) $\displaystyle\big{(}J_{m}\big{)}_{T_{0}=T_{1}=T}=\zeta(T)(c_{m1}-c_{m0})$ and its derivative with respect to $\ln\eta$, while $c_{m0}$ is held fixed, is : (77) $\displaystyle\left(\frac{\partial J_{m}}{\partial\ln\eta}\right)_{c_{m0}=\text{cst.}}=\zeta(T_{1})c_{m1}=k_{B}L_{mm}.$ Since (78) $\displaystyle\int_{0}^{+\infty}\mathrm{d}x\,e^{-ax}\sqrt{\frac{x}{x+1}}=\frac{1}{a}-\frac{\ln(a)}{2}+\mathcal{O}(1)\qquad(a\to 0^{+})$ the high temperature expansion of $L_{mm}$ gives : (79) $\displaystyle\left(\frac{\partial\ln L_{mm}}{\partial\ln T}\right)_{T_{1}=T_{0}=T}=\left(\frac{\partial\ln\zeta}{\partial\ln T}\right)_{T_{1}=T_{0}=T}=\frac{1}{2}+o(1).$ ### 6.4. Estimation of $L_{\theta\theta}$ The heat diffusion coefficient, $L_{\theta\theta}$, depends only on the number of degrees of freedom that interact in the membrane bilayer. As long as the structure of the membrane is unchanged, the same hydrophobic tails interact similarly at any temperature. Therefore, we conjecture that $L_{\theta\theta}$ is independant of the temperature in the liquid disordered phase [27]. Therefore : (80) $\displaystyle\frac{\partial\ln L_{\theta\theta}}{\partial\ln T}\simeq 0.$ ### 6.5. Destabilisation Putting together the scaling laws for $L_{m\theta}$, $L_{mm}$ and $L_{\theta\theta}$, we obtain : (81) $\displaystyle\frac{\partial}{\partial\ln T_{1}}\left(\frac{L_{m\theta}^{2}}{L_{mm}L_{\theta\theta}}\right)=3-\frac{1}{2}-0=\frac{5}{2}$ The main mathematical proposition of the present article is the following. ###### Proposition 6.1. Since $\mfrac{L_{m\theta}^{2}}{L_{mm}L_{\theta\theta}}$ grows as $T_{1}^{5/2}$, the stability condition, $L_{m\theta}^{2}<L_{mm}L_{\theta\theta}$, can not be satisfied at high temperature. The exact value of $T_{1}$ for which this transition occurs can not be computed in our simple model, but the only characteristic temperature being $\mfrac{E_{\ast}}{k_{B}}$, the critical temperature must be of this order of magnitude. This destabilisation of the steady growth regime is comparable with the onset of heat convection in a fluid subject to a strong temperature gradient. In fine, the self-replication of protocells could be interpreted as a convective phenomenon inside their membrane, triggered by their metabolic activity. ## 7\. Conclusions and perspectives We have proposed a toy model of protocell growth, fission and reproduction. The scenario thus described can be viewed as the ancestor of mitosis. The main force driving this irreversible process is the temperature difference between the inside and the outside of the protocell, due to the inner chemical activity. We propose that the increase of the inner temperature, due to a rudimentary inner metabolism, enhances the transfer of membrane molecules from the inner leaflet to the outer leaflet, as described in silico by models of molecular dynamics [14, 15]. Due to this transfer of molecules, coupled to a heat transfer, the difference of their areas and the total mean curvature of the median surface increase. The cylindrical growth becomes unstable and any slight local reduction of the radius of the initial cylinder increases until the protocell is cut into two daughter protocells, each one containing reactants and catalysers to continue the growth and fission process. The cut occurs near the hottest zone, around the middle. This model is based on the idea [23] that the early forms of life were simple vesicles containing a particular network of chemical reactions, precursor of modern cellular metabolism : Protolife = Cellularity + Inner Metabolism. With a large supply of reactants in the so-called prebiotic soup [31, 16, 23], and with an optimal salinity and pH, these ingredients are sufficient to induce an exponential growth of prebiomass and make possible the exploration of a large number of chemical reactions in these miniature chemical factories. The possibility to sythesize complex molecules (sterols, RNA, DNA, proteins, etc.) comes later, once these factories self-replicate and thrive. In order to test our model experimentally, we have to manipulate vesicles that can be heated from within in a controled way. Let us imagine, in a solution maintained at temperature $T_{0}$, vesicles containing molecules of type $A$ able to absorb visible radiation, with which the surrounding molecules do not interact. Let us suppose that $A$ re-emits radiation in the near infrared. The heat thus generated inside the vesicle creates a controled temperature difference, $T_{1}-T_{0}>0$, between both sides of the membrane. If $L_{m\theta}$ is large enough, we should observe a bending of the membrane of the vesicles due to the transfer of the hottest molecules from the inner leaflet to the outer leaflet. Another experimental test of our model can be made by observing eukaryotic cells, where the mitochondria are the main source of heat. It seems possible to measure their temperature variations using fluorescent molecules [3]. Although the very notion of temperature at this scale and far from a thermodynamical equilibrium is not clear, the measurement of the temperature variations inside the cell during its life cycle could be correlated with the onset of mitosis and with the shape of mitochondrial network [21]. Our model is obviously oversimplified since the polar heads of membrane molecules are treated as an ideal gas in a box. In particular, we haven’t taken into account the interaction between these molecules and the surrounding solution. This calls for the development of a better model to treat the effect of these interactions on the temperature dependence of the conductance coefficients. The scaling law of the ratio $\mfrac{L_{m\theta}^{2}}{L_{mm}L_{\theta\theta}}$ at temperatures higher than $\mfrac{E_{\ast}}{k_{B}}$ is the key argument that explains the splitting of the protocell. Future investigations and experiments will decide of the plausibility of this proposition. Acknowledgments : We thank Jorgelindo Da Veiga Moreira (Université de Montréal), Marc Henry (Université de Strasbourg), Olivier Lafitte (Institut Galilée, Université Paris XIII), Kirone Mallick (Institut de Physique Théorique, CEA, Saclay), Laurent Schwartz (AP-HP) and Jean-Yves Trosset (SupBiotech, Villefuif) for their advice and helpful discussions. ## Appendix A The mean curvature of the membrane Let $\Sigma_{t}$ be a family of surfaces, indexed by a time parameter $t\in[t_{0},+\infty[$. We suppose that each $\Sigma_{t}$ is a smooth, orientable and closed (compact, without boundary) hence diffeomorphic to the standard $2$-sphere. At each point $P\in\Sigma_{t}$, the Taylor expansion of the distance from $Q\in\Sigma_{t}$ to the tangent plane, $T_{P}\Sigma_{t}$, defines a quadratic form whose eigenvalues (homogenous to the inverse of a length) do not depend on the coordinate system in the neighbourhood of $P$. We denote them $R_{-}$ and $R_{+}$. The mean curvature of $\Sigma_{t}$ at $P$ is the arithmetic mean of the principal curvatures : (82) $\displaystyle H:=\frac{1}{2}\left(\frac{1}{R_{+}}+\frac{1}{R_{-}}\right)$ and the gaussian curvature is their product : (83) $\displaystyle K:=\frac{1}{R_{+}R_{-}}.$ In the case of a cylinder, $R_{+}=+\infty$ and $R_{-}=R_{0}=$ its radius, hence $H_{\text{cyl.}}(P)=\mfrac{1}{2R_{0}}$ and $K(P)=0$ at every point $P\in\Sigma_{t}$ (except on the end hemispheres). Let $\Sigma_{t0}$ and $\Sigma_{t1}$ be the surfaces obtained by shifting $\Sigma_{t}$ in the normal direction, over an infinitesimal distance $\varepsilon$ on both sides of $\Sigma_{t}$. Let $\mathcal{A}_{0}(t)$ (resp. $\mathcal{A}_{1}(t)$) be the average area of the outer (resp. inner) layer of the membrane, measured at the hydrophilic heads, and $\mathcal{A}=\mfrac 12(\mathcal{A}_{0}+\mathcal{A}_{1})$ the average area of the median surface, where the hydrophobic tails join. The difference of their areas, $\mathcal{A}_{1}-\mathcal{A}_{0}$, is given by the first term of Weyl’s Tube Formula [13] : (84) $\displaystyle\mathcal{A}_{0}-\mathcal{A}_{1}=4\varepsilon\int_{\Sigma_{t}}H\,\mathrm{d}A+\mathcal{O}(\varepsilon^{2}).$ Let $\mathcal{B}$ be the infinitesimal variation of area along the outer normal : (85) $\displaystyle\mathcal{B}:=2\varepsilon\int_{\Sigma_{t}}H\,\mathrm{d}A.$ $\mathcal{A}_{0}$ and $\mathcal{A}_{1}$ can also be written as functions of $\mathcal{A}$ and $\mathcal{B}$ : (86) $\displaystyle\mathcal{A}_{0}=\mathcal{A}+\mathcal{B}\qquad\text{and}\qquad\mathcal{A}_{1}=\mathcal{A}-\mathcal{B}.$ Our dynamical variables are the area of the median surface, $\mathcal{A}(t)=\int_{\Sigma_{t}}\mathrm{d}A$, the volume of the cytosol, $\mathcal{V}(t)$, and the variation of area, $\mathcal{B}(t)=2\varepsilon\int_{\Sigma_{t}}H\,\mathrm{d}A$. In the next section, we will establish their evolution equations as a consequence of the balance equations for the number of membrane molecules. Remark : In the case of a cylinder of radius $R_{0}$, we have $H=\mfrac{1}{2R_{0}}$ and (87) $\displaystyle\mathcal{A}_{0}-\mathcal{A}_{1}=2\mathcal{B}=4\varepsilon H\mathcal{A}=\frac{2\varepsilon\mathcal{A}}{R_{0}}.$ Since $2\varepsilon\,\mfrac{\mathcal{A}_{0}+\mathcal{A}_{1}}{2}=2\varepsilon\mathcal{A}$ is also the volume, $v$, of this normal thickening of $\Sigma_{t}$, we have : (88) $\displaystyle v=\frac{2\mathcal{B}}{H}=4\mathcal{B}R_{0}.$ ## Appendix B Solutions of the growth equation In this appendix, we solve the growth equation, using basic linear algebra and standard results about linear differential equations [19]. The matrix form of the growth equation is (89) $\displaystyle\dot{X}=\begin{pmatrix}\dot{\mathcal{A}}_{0}\\\ \dot{\mathcal{A}}_{1}\\\ \dot{\mathcal{U}}\\\ \end{pmatrix}=\begin{pmatrix}\frac{\eta\tau}{2}&\frac{\eta\tau}{2}&0\\\ -\frac{\tau}{2}&\frac{2-\tau}{2}&0\\\ 0&-\frac{c_{m1}}{\varepsilon c_{mc}}&\frac{t_{1}}{t_{c}}\\\ \end{pmatrix}\begin{pmatrix}\mathcal{A}_{0}\\\ \mathcal{A}_{1}\\\ \mathcal{U}\\\ \end{pmatrix}=MX.$ As long as no flow vanishes, the determinant of $M$ is non-zero : (90) $\displaystyle\det(M)=\frac{\eta\tau t_{1}}{2t_{c}}=\frac{c_{m1}^{2}J_{m10}J_{rm}}{2c_{m0}c_{mc}J_{mc1}^{2}}$ and the protocell grows exponentially : (91) $\displaystyle X(t)=e^{\frac{t}{t_{1}}M}X(0).$ In general, the two leaflets of the membrane grow at different speeds. Indeed, the characteristic polynomial of $M$ is : (92) $\displaystyle\det(M-\lambda\,\mathrm{Id})$ $\displaystyle=\begin{vmatrix}\frac{\eta\tau}{2}-\lambda&\frac{\eta\tau}{2}&0\\\ -\frac{\tau}{2}&\frac{2-\tau}{2}-\lambda&0\\\ 0&-\frac{c_{m1}}{\varepsilon c_{mc}}&\frac{t_{1}}{t_{c}}-\lambda\\\ \end{vmatrix}$ $\displaystyle=\left(\lambda^{2}-\lambda\left(1+\frac{(\eta-1)\tau}{2}\right)+\frac{\eta\tau}{2}\right)\left(\frac{t_{1}}{t_{c}}-\lambda\right).$ Its roots are $\mfrac{t_{1}}{t_{c}}$ and the two roots, $\lambda_{\pm}(\eta,\tau)$, of the polynomial $\lambda^{2}-\lambda\big{(}1+(\eta-1)\mfrac{\tau}{2})\big{)}+\mfrac{\eta\tau}{2}$ : (93) $\displaystyle\lambda_{\pm}(\eta,\tau)$ $\displaystyle:=\frac{1}{2}\left(1+\frac{(\eta-1)\tau}{2}\pm\sqrt{\Delta(\eta,\tau)}\right)$ $\displaystyle\Delta(\eta,\tau)$ $\displaystyle:=\frac{1}{4}(\eta-1)^{2}\tau^{2}-(\eta+1)\tau+1.$ $\bullet$ If $\eta=1$, i.e. if both leaflets have the same density, then $\Delta$ is an affine function of $\tau$ : (94) $\displaystyle\Delta(1,\tau)=1-2\tau\qquad\text{and}\qquad\lambda_{\pm}(1,\tau)=\frac{1\pm\sqrt{1-2\tau}}{2}.$ If, moreover, $\tau=\mfrac 12$, i.e. the inner leaflet transmits half of the incoming membrane molecules to the outer leaflet, then (95) $\displaystyle\Delta\left(1,\frac{1}{2}\right)=0\qquad\text{and}\qquad\lambda_{\pm}\left(1,\frac{1}{2}\right)=\frac{1}{2}$ and both leaflets grow at the same speed. $\bullet$ If $\eta\neq 1$, then $\Delta(\eta,\tau)$ is a quadratic function of $\tau$, bounded from below, of discriminant (96) $\displaystyle\delta=(\eta+1)^{2}-(\eta-1)^{2}=4\eta>0$ and has distinct roots : (97) $\displaystyle\tau_{\pm}(\eta)=2\,\frac{\eta+1\pm\sqrt{4\eta}}{(\eta-1)^{2}}=2\left(\frac{\sqrt{\eta}\pm 1}{\eta-1}\right)^{2}=\frac{2}{(\sqrt{\eta}\mp 1)^{2}}.$ Physically, $\eta\simeq 1$ and $\tau\simeq\mfrac 12$. If $\eta=1+h$, with $0<h\ll 1$ then $\tau_{+}\simeq\mfrac{8}{h^{2}}\gg 1>\tau_{-}$ and (98) $\displaystyle\tau_{-}\simeq\frac{2}{\left(2+\frac{h}{2}\right)^{2}}\simeq\frac{1}{2}-\frac{h}{4}.$ Consequently, $\tau$ stays $<\tau_{-}$ (FIG. B). $\tau_{-}$$\mfrac 12$physicalregionnon-physical region$\Delta>0$$\Delta<0$$\Delta>0$$\tau_{+}\gg\tau_{-}$ Exponential growth necessitates to keep $\tau<\tau_{-}$. Mathematically, we have three possibilities : $\displaystyle(1)$ $\displaystyle\tau<\tau_{-}(\eta)\quad\text{or}\quad\tau>\tau_{+}(\eta)\ \to\ \Delta(\eta,\tau)>0$ $\displaystyle\quad\lambda_{+}(\eta,\tau)\neq\lambda_{-}(\eta,\tau)\quad\text{(real numbers) ;}$ $\displaystyle(2)$ $\displaystyle\tau=\tau_{-}(\eta)\quad\text{or}\quad\tau=\tau_{+}(\eta)\ \to\ \Delta(\eta,\tau_{\pm})=0$ $\displaystyle\quad\lambda_{+}\big{(}\eta,\tau_{\pm}(\eta)\big{)}=\lambda_{-}\big{(}\eta,\tau_{\pm}(\eta)\big{)}=\frac{\sqrt{\eta}}{\sqrt{\eta}\pm 1};$ $\displaystyle(3)$ $\displaystyle\tau_{-}(\eta)<\tau<\tau_{+}(\eta)\ \to\ \Delta(\eta,\tau)<0$ $\displaystyle\quad\lambda_{+}(\eta,\tau)=\overline{\lambda}_{-}(\eta,\tau)\quad\text{(complex numbers).}$ ### B.1. Case 1 : $\eta\neq 1$ and $\tau>\tau_{+}(\eta)$ or $\tau<\tau_{-}(\eta)$ In these intervals, $N$ is diagonalisable and a basis of eigenvectors of $N$ is given by : (99) $\displaystyle\mathcal{B}_{\pm}$ $\displaystyle=\begin{pmatrix}\frac{1}{2}\\\ \frac{\lambda_{\pm}(\eta,\tau)}{\eta\tau}-\frac{1}{2}\end{pmatrix}$ $\displaystyle=\frac{\mathcal{A}_{0}-\mathcal{A}_{1}}{2}+\frac{\lambda_{\pm}(\eta,\tau)}{\eta\tau}\,\mathcal{A}_{1}$ $\displaystyle=\mathcal{B}+\frac{\lambda_{\pm}(\eta,\tau)}{\eta\tau}\,\mathcal{A}_{1}$ i.e. $\mathcal{B}_{+}$ and $\mathcal{B}_{-}$ grow exponentially, with a rate of growth $\lambda_{\pm}/t_{1}$, respectively : (100) $\displaystyle\mathcal{B}_{\pm}(t)=\mathcal{B}_{\pm}(0)\,\exp\left(\frac{\lambda_{\pm}(\eta,\tau)}{t_{1}}\,t\right).$ The area of the inner leaflet is : (101) $\displaystyle\mathcal{A}_{1}(t)$ $\displaystyle=\frac{\mathcal{B}_{+}(t)-\mathcal{B}_{-}(t)}{\frac{\lambda_{+}}{\eta\tau}-\frac{\lambda_{-}}{\eta\tau}}$ $\displaystyle=\frac{\eta\tau}{\sqrt{\Delta}}\left(\mathcal{B}_{+}(0)e^{t\lambda_{+}/t_{1}}-\mathcal{B}_{-}(0)e^{t\lambda_{-}/t_{1}}\right).$ The area of the outer leaflet is : (102) $\displaystyle\mathcal{A}_{0}(t)$ $\displaystyle=\frac{(2\lambda_{+}-\eta\tau)\mathcal{B}_{-}(t)-(2\lambda_{-}-\eta\tau)\mathcal{B}_{+}(t)}{\lambda_{+}-\lambda_{-}}$ $\displaystyle=\frac{2\lambda_{+}-\eta\tau}{\sqrt{\Delta}}\mathcal{B}_{-}(0)e^{t\lambda_{-}/t_{1}}-\frac{2\lambda_{-}-\eta\tau}{\sqrt{\Delta}}\mathcal{B}_{+}(0)e^{t\lambda_{+}/t_{1}}.$ And $\mathcal{U}(t)$ is obtained from $\mathcal{A}_{1}(t)$ : (103) $\displaystyle e^{tt_{1}/t_{c}}\,\frac{\mathrm{d}}{\mathrm{d}t}\left(\mathcal{U}(t)e^{-tt_{1}/t_{c}}\right)$ $\displaystyle=-\frac{c_{m1}}{\varepsilon c_{mc}}\,\mathcal{A}_{1}(t)$ (104) $\displaystyle\mathcal{U}(t)\,e^{-tt_{1}/t_{c}}$ $\displaystyle=-\frac{\eta\tau c_{m1}\mathcal{B}_{+}(0)}{\varepsilon c_{mc}\sqrt{\Delta}\left(\frac{\lambda_{+}}{t_{1}}-\frac{t_{1}}{t_{c}}\right)}\,e^{t\,\left({\frac{\lambda_{+}}{t_{1}}-\frac{t_{1}}{t_{c}}}\right)}$ $\displaystyle+\frac{\eta\tau c_{m1}\mathcal{B}_{-}(0)}{\varepsilon c_{mc}\sqrt{\Delta}\left(\frac{\lambda_{-}}{t_{1}}-\frac{t_{1}}{t_{c}}\right)}\,e^{t\,\left({\frac{\lambda_{-}}{t_{1}}-\frac{t_{1}}{t_{c}}}\right)}+\text{cst.}$ $\displaystyle\mathcal{U}(t)$ $\displaystyle=\frac{\eta\tau c_{m1}}{\varepsilon c_{mc}\sqrt{\Delta}}\left(\frac{e^{t\lambda_{-}/t_{1}}}{\frac{\lambda_{-}}{t_{1}}-\frac{t_{1}}{t_{c}}}-\frac{e^{t\lambda_{+}/t_{1}}}{\frac{\lambda_{+}}{t_{1}}-\frac{t_{1}}{t_{c}}}\right)$ $\displaystyle\ +\text{cst.}\,e^{tt_{1}/t_{c}}$ where the integration constant is determined by $\mathcal{U}(0)$. ### B.2. Case 2 : $\eta\neq 1$ and $\tau\in\\{\tau_{+}(\eta),\tau_{-}(\eta)\\}$ In this singular case, the upper-left $2\times 2$ submatrix is not diagonalisable but conjugate to a lower triangular matrix of Jordan form : (105) $\displaystyle N:=\frac{1}{2}\begin{pmatrix}\eta\tau&\eta\tau\\\ -\tau&2-\tau\\\ \end{pmatrix}=T\begin{pmatrix}\Lambda_{\pm}(\eta)&0\\\ 1&\Lambda_{\pm}(\eta)\\\ \end{pmatrix}T^{-1}$ where $\Lambda_{\pm}(\eta)$ is the single eigenvalue of $N$ when $\tau$ is fixed equal to $\tau_{+}(\eta)$ or $\tau_{-}(\eta)$ : (106) $\displaystyle\Lambda_{\pm}(\eta)$ $\displaystyle:=\lambda\big{(}\eta,\tau_{\pm}(\eta)\big{)}=\frac{2+(\eta-1)\tau_{\pm}(\eta)}{4}$ $\displaystyle=\frac{2+\frac{2(\eta-1)}{(\sqrt{\eta}\mp 1)^{2}}}{4}=\frac{\eta\pm\sqrt{\eta}}{\eta-1}=\frac{\sqrt{\eta}}{\sqrt{\eta}\pm 1}.$ An easy computation gives us : (107) $\displaystyle\frac{2\Lambda_{+}}{\eta\tau}-1=\frac{1-3\sqrt{\eta}}{\eta+\sqrt{\eta}}\quad\text{and}\quad\frac{2\Lambda_{-}}{\eta\tau}-1=\frac{1+3\sqrt{\eta}}{\eta-\sqrt{\eta}}.$ $N$ has a unique proper line, generated by the vector (108) $\displaystyle\mathcal{B}_{\ast}^{\pm}$ $\displaystyle:=\begin{pmatrix}\frac{1}{2}\\\ \frac{\Lambda_{\pm}}{\eta\tau_{\pm}}-\frac{1}{2}\end{pmatrix}=\frac{1}{2}\begin{pmatrix}1\\\ \frac{1\mp 3\sqrt{\eta}}{\eta\pm\sqrt{\eta}}\end{pmatrix}$ $\displaystyle=\frac{\mathcal{A}_{0}}{2}+\left(\frac{1\mp 3\sqrt{\eta}}{\eta\pm\sqrt{\eta}}\right)\frac{\mathcal{A}_{1}}{2}$ where the lower $\ast$ means that $\lambda_{+}=\lambda_{-}$, whereas the upper $\pm$ depends on the choice between $\tau=\tau_{+}(\eta)$ and $\tau=\tau_{-}(\eta)$. Since (109) $\displaystyle\mathcal{B}_{\ast}^{\pm}(t)=\mathcal{B}_{\ast}^{\pm}(0)\,e^{t\Lambda_{\pm}/t_{1}}$ we obtain : (110) $\displaystyle\mathcal{A}_{0}(t)+\left(\frac{3\eta\mp 1}{\eta\pm\sqrt{\eta}}\right)\,\mathcal{A}_{1}(t)=2\,\mathcal{B}_{\ast}^{\pm}(0)\,e^{t\Lambda_{\pm}/t_{1}}.$ Since $\mathcal{B}_{\ast}^{\pm}=T{0\choose 1}$, the vector $\mathcal{B}_{\ast}^{\pm}$ is the right column of $T$. The left column of $T$ is the vector $\mathcal{B}_{\bullet}^{\pm}={x\choose y}$ which satisfies the equation $(N-\Lambda_{\pm})\mathcal{B}_{\bullet}^{\pm}=\mathcal{B}_{\ast}^{\pm}$, or in extenso : (111) $\displaystyle\left(\frac{\eta\tau}{2}-\Lambda_{\pm}\right)x+\left(\frac{\eta\tau}{2}\right)y$ $\displaystyle=\frac{1}{2}$ $\displaystyle-\frac{\tau}{2}\,x+\frac{2-\tau-2\Lambda_{\pm}}{2}\,y$ $\displaystyle=\frac{\Lambda_{\pm}}{\eta\tau}-\frac{1}{2}.$ Taking $x=0$ and $y=\mfrac{1}{\eta\tau}$ gives a solution : (112) $\displaystyle\big{(}N-\Lambda_{\pm}\big{)}\mathcal{B}_{\bullet}^{\pm}$ $\displaystyle=\begin{pmatrix}\frac{\eta\tau}{2}-\Lambda_{\pm}&\frac{\eta\tau}{2}\\\ -\frac{\tau}{2}&\frac{2-\tau}{2}-\Lambda_{\pm}\end{pmatrix}\begin{pmatrix}0\\\ \frac{1}{\eta\tau}\end{pmatrix}$ $\displaystyle=\begin{pmatrix}\frac{1}{2}\\\ \frac{2-\tau-2\Lambda_{\pm}}{\eta\tau}\end{pmatrix}=\begin{pmatrix}\frac{1}{2}\\\ \frac{2\Lambda_{\pm}-\eta\tau}{\eta\tau}\end{pmatrix}=\mathcal{B}_{\ast}^{\pm}.$ The matrix $T$ and its inverse, $T^{-1}$, are therefore : (113) $\displaystyle T=\begin{pmatrix}0&1\\\ \frac{1}{\eta\tau}&\frac{2\Lambda-\eta\tau}{2\eta\tau}\end{pmatrix}\qquad\text{and}\qquad T^{-1}=\begin{pmatrix}\frac{\eta\tau-2\Lambda}{2}&\eta\tau\\\ 1&0\end{pmatrix}.$ Since $\mathcal{B}_{\bullet}^{\pm}=\mfrac{\mathcal{A}_{1}}{\eta\tau_{\pm}(\eta)}$, we have $\mathcal{B}_{\bullet}^{\pm}(t)=\mathcal{B}_{\ast}(0)\,\mfrac{t}{t_{1}}\,e^{t\Lambda_{\pm}/t_{1}}$ and : (114) $\displaystyle\mathcal{A}_{1}(t)=\eta\tau_{\pm}(\eta)\,\mathcal{B}_{\ast}^{\pm}(0)\,\frac{t}{t_{1}}\,e^{t\Lambda_{\pm}/t_{1}}$ ### B.3. Case 3 : $\tau_{-}(\eta)<\tau<\tau_{+}(\eta)$ In this interval, $\Delta<0$ and $M$ has two distinct complex conjugated eigenvalues, $\lambda$ and $\overline{\lambda}$, functions of $\eta$ and $\tau$. Let $\alpha,\beta\in\mathbb{R}$ be the real and imaginary parts of $\lambda$ : (115) $\displaystyle\alpha$ $\displaystyle:=\frac{2+(\eta-1)\tau}{4}\,>0$ $\displaystyle\beta$ $\displaystyle:=\frac{\sqrt{-\Delta}}{2}\,>0$ $\displaystyle\lambda(\eta,\tau)$ $\displaystyle=\alpha+\mathbf{i}\,\beta\qquad(\mathbf{i}\,^{2}=-1).$ Let $V$ (resp. $\overline{V}$) be a complex eigenvector of $N$, of eigenvalue $\lambda$ (resp. $\overline{\lambda}$), for instance : (116) $\displaystyle V:=\mathcal{B}+\frac{\lambda}{\eta\tau}\,\mathcal{A}_{1}\qquad\text{and}\qquad\overline{V}:=\mathcal{B}+\frac{\overline{\lambda}}{\eta\tau}\,\mathcal{A}_{1}$ then the real and imaginary parts of $V$, defined by $V^{\prime}:=\mfrac 12(V+\overline{V})$ and $V^{\prime\prime}:=\mfrac{1}{2\mathbf{i}\,}(V-\overline{V})$, form a basis of $\mathbb{R}^{2}$ on which $N$ acts as an orthogonal matrix [19] : (117) $\displaystyle\frac{1}{2}\begin{pmatrix}\eta\tau&\eta\tau\\\ -\tau&2-\tau\\\ \end{pmatrix}$ $\displaystyle=U\begin{pmatrix}\alpha&\beta\\\ -\beta&\alpha\\\ \end{pmatrix}U^{-1}$ $\displaystyle V^{\prime}=U\begin{pmatrix}1\\\ 0\end{pmatrix}=\begin{pmatrix}\frac{1}{2}\\\ \frac{\alpha}{\eta\tau}-\frac{1}{2}\end{pmatrix}$ $\displaystyle\quad V^{\prime\prime}=U\begin{pmatrix}0\\\ 1\end{pmatrix}=\begin{pmatrix}0\\\ \frac{\beta}{\eta\tau}\end{pmatrix}$ i.e. the matrix $U$ has $V^{\prime}$ and $V^{\prime\prime}$ as columns : (118) $\displaystyle U=\begin{pmatrix}\frac{1}{2}&0\\\ \frac{\alpha}{\eta\tau}-\frac{1}{2}&\frac{\beta}{\eta\tau}\end{pmatrix}=\begin{pmatrix}\frac{1}{2}&0\\\ \frac{2-(\eta+1)\tau}{4\eta\tau}&\frac{\sqrt{-\Delta}}{2\eta\tau}\end{pmatrix}.$ Let $s=\mfrac{t}{t_{1}}$. Since our evolution operator, the exponential of $sN$, is : (119) $\displaystyle e^{sN}=e^{\alpha s}U\begin{pmatrix}\cos(\beta s)&\sin(\beta s)\\\ -\sin(\beta s)&\cos(\beta s)\end{pmatrix}U^{-1}$ we have : (120) $\displaystyle V^{\prime}(s)$ $\displaystyle=e^{sN}V^{\prime}(0)=e^{\alpha s}U\begin{pmatrix}\cos(\beta s)&\sin(\beta s)\\\ -\sin(\beta s)&\cos(\beta s)\end{pmatrix}\begin{pmatrix}\frac{1}{2}\\\ \frac{\alpha}{\eta\tau}-\frac{1}{2}\end{pmatrix}$ $\displaystyle=\frac{e^{\alpha s}}{2\eta\tau}\begin{pmatrix}\frac{1}{2}&0\\\ \frac{\alpha}{\eta\tau}-\frac{1}{2}&\frac{\beta}{\eta\tau}\end{pmatrix}\begin{pmatrix}\eta\tau\cos(\beta s)+(2\alpha-\eta\tau)\sin(\beta s)\\\ -\eta\tau\sin(\beta s)+(2\alpha-\eta\tau)\cos(\beta s)\end{pmatrix}$ $\displaystyle=\frac{e^{\alpha s}}{2\eta\tau}\begin{pmatrix}\frac{\eta\tau}{2}\cos(\beta s)+\frac{2\alpha-\eta\tau}{2}\sin(\beta s)\\\ \frac{(2\alpha-\eta\tau)(2\beta+\eta\tau)}{2\eta\tau}\cos(\beta s)+\left(\frac{(2\alpha-\eta\tau)^{2}}{2\eta\tau}-\beta\right)\sin(\beta s)\end{pmatrix}.$ Similarly, we have the expression of $V^{\prime\prime}(s)$ : (121) $\displaystyle V^{\prime\prime}(s)$ $\displaystyle=e^{sN}V^{\prime\prime}(0)$ $\displaystyle=\frac{\beta e^{\alpha s}}{\eta\tau}\begin{pmatrix}\frac{1}{2}\sin(\beta s)\\\ \left(\frac{2\alpha-\eta\tau}{2\eta\tau}\right)\sin(\beta s)+\frac{\beta}{\eta\tau}\cos(\beta s)\end{pmatrix}.$ Finally, $\mathcal{A}_{1}$ and $\mathcal{B}$ are obtained from $V^{\prime}$ and $V^{\prime\prime}$ by the linear relations : (122) $\displaystyle\mathcal{B}(s)$ $\displaystyle=\frac{\beta V^{\prime}(s)-\alpha V^{\prime\prime}(s)}{\beta-\alpha}$ $\displaystyle\mathcal{A}_{1}(s)$ $\displaystyle=\frac{\eta\tau}{\alpha-\beta}\big{(}V^{\prime}(s)-V^{\prime\prime}(s)\big{)}.$ ## Appendix C Smooth perturbation of cylindrical growth In this appendix, we compute the variation of the area and of the total mean curvature of a surface of revolution under a small variation of its generating curve. We will work in an orthonormal system of coordinates $(x,y,z)$. Let us suppose now that $\Sigma$ is a revolution surface whose generating curve, rotated around the axis $\\{y=0=z\\}$, is given by : (123) $\displaystyle\sqrt{y^{2}+z^{2}}=R(x)=R_{0}+\delta R(x)$ with $\|\delta R(x)\|\ll R_{0}$. The function $\delta R$ represents an infinitesimal normal perturbation around the cylindrical shape. The variable $x$ satisfies $0\leq x\leq\ell$ and the deformed cylinder is glued smoothly with two hemispherical caps of radius $R_{0}$. In other words, we suppose that (124) $\displaystyle\delta R(0)=\delta R(\ell)$ $\displaystyle=0$ $\displaystyle\delta R^{\prime}(0)=\delta R^{\prime}(\ell)$ $\displaystyle=0.$ Let us compute the variations of area, $\delta\mathcal{A}$, of length, $\delta\ell$, and of total mean curvature, $\delta\mathcal{H}$, for a fixed volume. ### C.1. Isovolumic variation of the area $\mathcal{A}$ is a functional of the length, $\ell$, the radius, $R$, and its derivative, $R^{\prime}$ : (125) $\displaystyle\mathcal{A}(\ell,R,R^{\prime})=\int_{0}^{\ell}\mathrm{d}x\,2\pi R\sqrt{1+R^{\prime 2}}.$ Its variation under infinitesimal changes of $\ell$ and $R$ is : (126) $\displaystyle\delta\mathcal{A}$ $\displaystyle=2\pi R_{0}\,\delta\ell$ $\displaystyle+2\pi\int_{0}^{\ell}\mathrm{d}x\,\delta R\left(\sqrt{1+R^{\prime 2}}-\frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{RR^{\prime}}{\sqrt{1+R^{\prime 2}}}\right)\right).$ Since (127) $\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{RR^{\prime}}{\sqrt{1+R^{\prime 2}}}\right)$ $\displaystyle=\frac{RR^{\prime\prime}+R^{\prime 2}}{\sqrt{1+R^{\prime 2}}}-R^{\prime}R^{\prime\prime}\frac{RR^{\prime}}{\big{(}1+R^{\prime 2}\big{)}^{3/2}}$ $\displaystyle=(1+R^{\prime 2})^{-3/2}\left(\big{(}RR^{\prime\prime}+R^{\prime 2}\big{)}\big{(}1+R^{\prime 2}\big{)}-RR^{\prime 2}R^{\prime\prime}\right)$ $\displaystyle=(1+R^{\prime 2})^{-3/2}\big{(}RR^{\prime\prime}+R^{\prime 2}+R^{\prime 4}\big{)}$ we obtain (128) $\displaystyle\delta\mathcal{A}$ $\displaystyle=2\pi R_{0}\,\delta\ell$ $\displaystyle+2\pi\int_{0}^{\ell}\mathrm{d}x\,\delta R\big{(}1+R^{\prime 2}\big{)}^{-3/2}\big{(}1-RR^{\prime\prime}-R^{\prime 4}\big{)}.$ Similarly, the volume, $\mathcal{V}$, is a functional of $\ell$ and $R$ : (129) $\displaystyle\mathcal{V}(\ell,R)=\frac{4\pi R_{0}^{3}}{3}+\int_{0}^{\ell}\mathrm{d}x\,\pi R^{2}$ and its variation under infinitesimal changes of $\ell$ and $R$ is : (130) $\displaystyle\delta\mathcal{V}=\pi R_{0}^{2}\,\delta\ell+2\pi\int_{0}^{\ell}\mathrm{d}x\,R\,\delta R.$ If $\mathcal{V}$ is held constant, then $\delta\mathcal{V}=0$ and : (131) $\displaystyle\big{(}\delta\ell\big{)}_{\mathcal{V}=\text{cst.}}=-\frac{2}{R_{0}^{2}}\int_{0}^{\ell}\mathrm{d}x\,R\,\delta R.$ Inserting this expression of $\delta\ell$ into that of $\delta\mathcal{A}$, we obtain the isovolumic variation of area : (132) $\displaystyle(\delta\mathcal{A})_{\mathcal{V}=\text{cst.}}$ $\displaystyle=2\pi\int_{0}^{\ell}\mathrm{d}x\,\delta R\left(\big{(}1+R^{\prime 2}\big{)}^{-3/2}\big{(}1-RR^{\prime\prime}-R^{\prime 4}\big{)}-\frac{2R}{R_{0}}\right).$ ###### Theorem C.1. The isovolumic variational derivatives of the length and of the area of a (nearly cylindrical) closed revolution surface are negative : (133) $\displaystyle\left(\frac{\delta\ell}{\delta R}\right)_{\mathcal{V}=\text{cst.}}<0\qquad\text{and}\qquad\left(\frac{\delta\mathcal{A}}{\delta R}\right)_{\mathcal{V}=\text{cst.}}<0.$ In other words, since the stuffing is incompressible whereas the gut is elastic, the length and the area of a squeezed sausage increase. We call this simple statement the Squeezed Sausage Theorem (SST). ### C.2. Isovolumic variation of the total mean curvature The circles $\\{x=\text{cst.}\\}$ and the meridians, obtained by rotating the generating curve of equation $z^{2}=R^{2}(x)$, form an orthogonal system of geodesics [7], and the mean curvature of $\Sigma$ is given by : (134) $\displaystyle H=\frac{1}{2}\left(\frac{1}{R\sqrt{1+R^{\prime 2}}}+\frac{R^{\prime\prime}}{\big{(}1+R^{\prime 2}\big{)}^{3/2}}\right).$ The lateral area of a slice of width $\mathrm{d}x$, perpendicular to the axis of the surface, is : (135) $\displaystyle\mathrm{d}A=2\pi R\sqrt{1+R^{\prime 2}}\,\mathrm{d}x$ and the total mean curvature is : (136) $\displaystyle\mathcal{H}$ $\displaystyle:=\int_{\Sigma}H\,\mathrm{d}A$ $\displaystyle=\int_{\text{caps}}H\,\mathrm{d}A+\int_{0}^{\ell}2\pi R\sqrt{1+R^{\prime 2}}\,\mathrm{d}x$ $\displaystyle=4\pi R_{0}^{2}\cdot\frac{1}{R_{0}}+2\pi\int_{0}^{\ell}\frac{1}{2}\left(1+\frac{R\,R^{\prime\prime}}{1+R^{\prime 2}}\right)\mathrm{d}x$ $\displaystyle=4\pi R_{0}+\pi\ell+\pi\int_{0}^{\ell}\frac{R\,R^{\prime\prime}}{1+R^{\prime 2}}\,\mathrm{d}x.$ Since $\mathcal{H}$ is a functional of $\ell$, $R$, $R^{\prime}$ and $R^{\prime\prime}$, its variation under a change $\delta R$ of the radius of gyration and a change of length $\delta\ell$, is obtained after a double integration by parts [10] : (137) $\displaystyle\delta\mathcal{H}$ $\displaystyle=\pi\delta\ell+\pi\int_{0}^{\ell}\mathrm{d}x\ \frac{R^{\prime\prime}}{1+R^{\prime 2}}+\frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{2RR^{\prime}R^{\prime\prime}}{\big{(}1+R^{\prime 2}\big{)}^{2}}\right)$ $\displaystyle\hskip 85.35826pt+\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}\left(\frac{R}{1+R^{\prime 2}}\right)\,\delta R.$ Instead of computing each term of the integrand, let us make the approximation $R^{\prime 2}\ll 1$, valid when the initial cylinder is only slightly deformed. The expression of $\delta\mathcal{H}$ then simplifies to (138) $\displaystyle\delta\mathcal{H}$ $\displaystyle\simeq\pi\delta\ell+\pi\delta\int_{0}^{\ell}\mathrm{d}x\,RR^{\prime\prime}$ $\displaystyle\simeq\pi\delta\ell+2\pi\int_{0}^{\ell}\mathrm{d}x\,R^{\prime\prime}\,\delta R.$ Using the expression of $\delta\ell=-\mfrac{2}{R_{0}^{2}}\int_{0}^{\ell}\mathrm{d}x\,R\,\delta R$ when $\mathcal{V}$ is held constant, we obtain : (139) $\displaystyle\big{(}\delta\mathcal{H}\big{)}_{\mathcal{V}=\text{cst.}}\simeq 2\pi\int_{0}^{\ell}\mathrm{d}x\left(R^{\prime\prime}-\frac{R}{R_{0}^{2}}\right)\,\delta R.$ As long as $R_{0}^{2}\|R^{\prime\prime}\|\ll R$, the isovolumic variational derivative of $\mathcal{H}$ with respect to $R$ is negative : (140) $\displaystyle\left(\frac{\delta\mathcal{H}}{\delta R}\right)_{\mathcal{V}=\text{cst.}}<0\qquad\text{if}\quad R^{\prime 2}\ll 1\quad\text{and}\quad R_{0}^{2}\|R^{\prime\prime}\|\ll R.$ When $\delta R$ approaches $-R_{0}$ and the protocell is ready to split, the two radii of curvature are small compared to $R_{0}$ but have opposite sign, hence the Gaussian curvature around the septum is large and negative. After the cut, when the two caps are formed, the mean curvature and the Gaussian curvature are positive again. ## Appendix D Asymptotic expansion of $F(a)$ The change of variable $t=\sqrt{a(s-1)}$ in the integral defining $F$ gives us : (141) $\displaystyle F(a)$ $\displaystyle=\frac{e^{-a}}{a}\int_{0}^{+\infty}\mathrm{d}t\,f(a,t)$ $\displaystyle f(a,t)$ $\displaystyle:=2t^{2}\,e^{-t^{2}}\left(2\sqrt{t^{2}+a}-\frac{1}{\sqrt{t^{2}+a}}\right).$ Let (142) $\displaystyle G(a):=\int_{0}^{+\infty}\mathrm{d}t\,f(a,t)=ae^{a}F(a).$ The function $f(0,\cdot)$ is integrable over the half line $[0,+\infty[$ and (143) $\displaystyle G(0)$ $\displaystyle=\int_{0}^{+\infty}\mathrm{d}t\,f(0,t)$ $\displaystyle=\int_{0}^{+\infty}\mathrm{d}t\,2t\,e^{-t^{2}}(2t^{2}-1)$ $\displaystyle=\int_{0}^{+\infty}\mathrm{d}u\,e^{-u}(2u-1)=1.$ Let us compute the asymptotic expansion of $G(a)$ when $a\to 0^{+}$ : (144) $\displaystyle{\,}G(a)-G(0)$ $\displaystyle=\int_{0}^{+\infty}\mathrm{d}t\,\big{(}f(a,t)-f(0,t)\big{)}$ $\displaystyle=2\int_{0}^{+\infty}\mathrm{d}t\,t^{2}e^{-t^{2}}\left(2\big{(}\sqrt{t^{2}+a}-t\big{)}-\left(\frac{1}{\sqrt{t^{2}+a}}-\frac{1}{t}\right)\right)$ $\displaystyle=\int_{0}^{+\infty}2t\,\mathrm{d}t\,e^{-t^{2}}\big{(}\sqrt{t^{2}+a}-t\big{)}\left(2t+\frac{1}{\sqrt{t^{2}+a}}\right)$ $\displaystyle=\int_{0}^{+\infty}\mathrm{d}u\,e^{-u}\big{(}\sqrt{u+a}-\sqrt{u}\big{)}\left(2\sqrt{u}+\frac{1}{\sqrt{u+a}}\right)$ $\displaystyle=2\int_{0}^{+\infty}\mathrm{d}u\,e^{-u}\sqrt{u(u+a)}+\int_{0}^{+\infty}\mathrm{d}u\,e^{-u}(1-2u)-\int_{0}^{+\infty}\mathrm{d}u\,e^{-u}\sqrt{\frac{u}{u+a}}$ $\displaystyle=2\int_{0}^{+\infty}\mathrm{d}u\,e^{-u}\sqrt{u(u+a)}-1-\int_{0}^{+\infty}\mathrm{d}u\,e^{-u}\sqrt{\frac{u}{u+a}}.$ Hence : (145) $\displaystyle G(a)=\varphi(a)-\varphi^{\prime}(a)$ where (146) $\displaystyle\varphi(a)$ $\displaystyle:=2\int_{0}^{+\infty}\mathrm{d}u\,e^{-u}\sqrt{u(u+a)}$ $\displaystyle=2a^{2}\int_{0}^{+\infty}\mathrm{d}x\,e^{-ax}\sqrt{x(x+1)}.$ $\varphi(a)$ being the Laplace transform of the function $x\mapsto 2a^{2}\sqrt{x(x+1)}$, its expansion as $0^{+}$ is given by integrating the expansion of $\sqrt{x(x+1)}$ at $+\infty$ term by term : (147) $\displaystyle\sqrt{x(x+1)}$ $\displaystyle=x+\frac{1}{2}-\frac{1}{8x}+\mathcal{O}(x^{-2})$ $\displaystyle\varphi(a)$ $\displaystyle=2a^{2}\left(\frac{1}{a^{2}}+\frac{1}{2a}-\frac{1}{8}\int_{1}^{+\infty}\mathrm{d}x\,\frac{e^{-ax}}{x}+\mathcal{O}(1)\right)$ $\displaystyle=2+a-\frac{a^{2}}{4}\ln(a)+\mathcal{O}(a^{2}).$ Similarly, for $\varphi^{\prime}(a)$ we have : (148) $\displaystyle\sqrt{\frac{x}{x+1}}$ $\displaystyle=1-\frac{1}{2x}+\frac{3}{8x^{2}}+\mathcal{O}(x^{-3})\qquad(x\to+\infty)$ $\displaystyle\varphi^{\prime}(a)$ $\displaystyle=a\int_{0}^{1}\mathrm{d}x\,e^{-ax}\sqrt{\frac{x}{x+1}}$ $\displaystyle+a\int_{1}^{+\infty}\mathrm{d}x\,e^{-ax}\sqrt{\frac{x}{x+1}}$ $\displaystyle=a\int_{0}^{1}+a\left(\frac{e^{-a}}{a}-\frac{1}{2}\int_{1}^{+\infty}\mathrm{d}x\,\frac{e^{-ax}}{x}+\mathcal{O}(1)\right)$ $\displaystyle=1-\frac{a\ln(a)}{2}+\mathcal{O}(a).$ Consequently : (149) $\displaystyle G(a)$ $\displaystyle=1+\frac{a\ln(a)}{2}+\mathcal{O}(a)$ and (150) $\displaystyle F(a)$ $\displaystyle=\frac{e^{-a}}{a}+\frac{e^{-a}\ln(a)}{2}+\mathcal{O}(a)$ $\displaystyle=\frac{1}{a}+\frac{1}{2}\ln(a)+o(1).$ The asymptotic expansion of $\frac{\partial\ln F}{\partial\ln a}$ is therefore : (151) $\displaystyle\frac{\partial\ln F}{\partial\ln a}=-1+\frac{a\ln(a)}{2}+\mathcal{O}(a).$ In particular, since $a\ln(a)<0$ for $0<a<1$, we have (152) $\displaystyle\frac{\partial\ln F}{\partial\ln a}<-1\qquad(a\to 0^{+}).$ ## References * [1] J. 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# Quantum lock-in detection of a vector light shift Kosuke Shibata<EMAIL_ADDRESS>Naota Sekiguchi Takuya Hirano Department of Physics, Gakushuin University, Tokyo, Japan ###### Abstract We demonstrate detection of a vector light shift (VLS) using the quantum lock- in method. The method offers precise and accurate VLS measurement without being affected by real magnetic field fluctuations. We detect a VLS on a Bose–Einstein condensate (BEC) of 87Rb atoms caused by an optical trap beam with a resolution less than 1 Hz. We also demonstrate elimination of a VLS by controlling the beam polarization to realize a long coherence time of a transversally polarized $F$ = 2 BEC. Quantum lock-in VLS detection should find wide application, including the study of spinor BECs, electric-dipole moment searches, and precise magnetometry. ## I Introduction The a.c. Stark shift or light shift plays significant roles in atomic physics. One example is the optical trap Grimm et al. (2000), which has been extensively used in cold atom experiments and has been the subject of intriguing and important research, including low-dimensional Görlitz et al. (2001) and uniform gases Gaunt et al. (2013), and atoms in an optical lattice with applications to quantum simulation Bloch et al. (2008) and atomic clocks Derevianko and Katori (2011); Katori (2011). It has also enabled the study of multi-component gases and, in particular, spinor Bose–Einstein condensates (BECs) Stamper-Kurn and Ueda (2013). The light shift has vector and tensor components and hence is state-dependent in general Deutsch and Jessen (1998); Geremia et al. (2006); Deutsch and Jessen (2010). The state dependence has been exploited for realizing state- selective transport Mandel et al. (2003a, b) and confinement Heinz et al. (2020). However, a state-dependent shift is often undesirable for situations in which well-controlled spin evolution is required. Escaping from a vector light shift (VLS), which is equivalent to a fictitious magnetic field, has been an important issue in precise measurements, such as the search for an atomic electric-dipole moment Romalis and Fortson (1999) and exotic spin- dependent interactions Jackson Kimball et al. (2017a). Reducing the VLS is also important in atomic magnetometers, in which the VLS introduces systematic errors. The quantum noise associated with the light shift due to the probe field ultimately limits the sensitivity Fleischhauer et al. (2000). The VLS restricts the potential use of optically trapped atoms for magnetically sensitive experiments. While its effect can be diminished by applying a bias magnetic field in a direction orthogonal to the wavevector, the VLS can still be a significant noise source in precise measurements Romalis and Fortson (1999). It is necessary to reduce the VLS when an ultralow magnetic field is required. In addition, the relative direction cannot be chosen satisfactorily in some situations, such as in 3D optical lattice experiments. In order to eliminate the VLS caused by optical trapping beams, the light polarization should be precisely controlled, because the VLS is proportional to the intensity of a circularly polarized component Grimm et al. (2000); Deutsch and Jessen (1998); Geremia et al. (2006); Deutsch and Jessen (2010). However, it is a formidable task to precisely extinguish the circular component at the atom position located in a vacuum cell. Polarization measurements and control outside the cell do not assure the degree of linear polarization due to the stress-induced birefringence of the vacuum windows Jellison (1999). Therefore, a sensitive and robust polarization measurement method using atoms themselves as a probe is important. Most effective polarization measurements are accomplished by using atoms themselves as a probe. Polarization measurements with an atomic gas have been performed with various methods including Larmor precession measurement Zhu et al. (2013), precise microwave spectroscopy Steffen et al. (2013), and frequency modulation nonlinear magneto-optical rotation Jackson Kimball et al. (2017b). Differential Ramsey interferometry has been developed for spinor condensates Wood et al. (2016). Polarization measurements by fluorescence detection have been recently demonstrated for ions Yuan et al. (2019). In this paper, we demonstrate VLS detection by applying the quantum lock-in method Kotler et al. (2011); de Lange et al. (2011). The measurement is immune to environmental magnetic field noise, and thus achieves excellent precision and accuracy. We detect a VLS induced by an optical trap beam on a BEC of 87Rb atoms with a resolution less than 1 Hz. This detection method is feasible to implement and should have wide applications in various research areas involved with optical fields. The paper is organized as follows. In Sec. II, our experimental method and setup are presented. The experimental results are described in Sec. III. We discuss the applications and potential performance of the quantum lock-in VLS detection in Sec. IV. We conclude the paper in Sec. V. ## II Experimental method and setup We produce a BEC in a vacuum glass cell. A BEC of $3\times 10^{5}$ atoms in the hyperfine spin $F=2$ state is trapped in a crossed optical trap. The trap consists of an axial beam at the wavelength of 850 nm and a radial beam at 1064 nm. The axial and radial beam waists are $\approx$ 30 $\mu$m and 70 $\mu$m, respectively. A magnetic bias field $B$ of 15 $\mu$T is applied along the axial beam to define the quantization axis, as shown in Fig. 1(a). The atoms are initially in the $\|F,m_{F}\rangle=\|2,2\rangle$ state, where $m_{F}$ denotes the magnetic sublevel. The ellipticity of the axial beam at the atomic position is controlled with a quarter waveplate (QWP) in the VLS measurement described below. The QWP is located between a polarization beam splitter for polarization cleaning and the cell. The angle of the QWP is adjusted with a precise manual rotation stage. The minimum scale of the rotation stage is 0.28 mrad. Figure 1: (color online) (a) Experimental configuration. A BEC is trapped in the axial trap beam along the $z$ axis and the radial trap beam along the $x$ axis (not shown). (b) Typical TOF image of a BEC measured after rf pulses for the detection. The spin components ($m_{F}=-2,-1,0,1,2$) are spatially resolved by the Stern–Gerlach method. (c) Time sequence for the quantum lock- in detection of a VLS. The beam power, $P(t)$, is modulated with a frequency $\omega_{m}$. The phase of the spin vector evolves with an angular frequency of $\omega(t)$. The accumulated phase, $\Phi=\int_{0}^{T}\omega(t)dt$, is finally measured. The time sequence for the quantum lock-in detection of a VLS is shown in Fig. 1(c). The lock-in technique enables enhanced sensitivity at the modulation frequency while reducing the effect of unwanted noise. We measure a VLS induced by the axial optical trap beam with multiple rf pulses. The trap beam power, $P(t)$, is modulated with a frequency $\omega_{m}$ during the pulse application as $P(t)=P_{0}\left(1+p\sin(\omega_{m}t)\right)\equiv P_{0}+P_{1}\sin(\omega_{m}t),$ (1) where $P_{0}$ is the mean power and $p$ is the modulation index. $P_{1}$ can be negative by changing the modulation phase by $\pi$. $\omega_{m}$ is set to be sufficiently higher than twice the trapping frequency to avoid parametric heating of the atoms. The modulation generates an a.c. fictitious magnetic field to be measured, given by $B_{\mathrm{fic}}=-\frac{1}{4}\alpha^{(\mathrm{1})}\mathcal{C}I_{1}\sin(\omega_{m}t)\equiv B_{1}\sin(\omega_{m}t),$ (2) where $\alpha^{(\mathrm{1})}$ is the a.c. vector polarizability, $\mathcal{C}$ is the degree of the circularity and $I_{1}$ is the beam intensity corresponding to $P_{1}$. The pulse set consists of an initial $\pi/2$ pulse at $t=0$, an odd number ($N$) of $\pi$-pulses, and a readout $\pi/2$ pulse. The pulses are equally spaced by $\Delta T$. The spacing satisfies $\omega_{m}$$=\pi/\Delta T$ so that the evolved phase due to the fictitious field is constructively accumulated. The relative phase, $\Delta\varphi$, between the initial and read-out pulses is set to $\pi/2$ for maximum sensitivity to small changes in the accumulated phase, $\Phi$. $\Phi$ is explicitly given by $\Phi=\frac{2}{\pi}\frac{g_{F}\mu_{B}B_{1}}{\hbar}T\equiv\frac{2}{\pi}\omega_{1}T,$ (3) where $g_{F}$ is the Landé g-factor, $\mu_{B}$ is the Bohr magneton, $\hbar$ is the reduced Planck constant and $T=(N+1)\Delta T$ is the phase accumulation time. $\omega_{1}/(2\pi)$ represents the VLS corresponding to $B_{1}$ in units of frequency. The read-out pulse converts $\Phi$ into the magnetization, $m$, as $m\equiv\frac{\sum_{i}iN_{i}}{N_{\mathrm{tot}}}=\mathcal{V}F\sin\Phi,$ (4) where $N_{i}$ is the atom number in the $\|F,m_{F}=i\rangle$ state $(i=-2,-1,0,1,2)$ after the read-out pulse, $N_{\mathrm{tot}}=\sum_{i}N_{i}$ is the total atom number, and $\mathcal{V}$ is the visibility. $\mathcal{V}$ is ideally $1$, but in practice it is less than $1$ due to magnetic field noise Kotler et al. (2011). Imperfections in the initial state preparation and spin manipulation also decrease $\mathcal{V}$. The magnetization is measured by standard absorption imaging after a time-of-flight with Stern–Gerlach spin separation (see Fig. 1(b)). ## III Results Figure 2: (color online) Detection of the VLS. The error bars represent the sample standard deviation. The red solid line is the fitting curve by $\mathcal{V}F\sin(ap)$. The right axis represents $B_{1}$. It should be noted that the right axis scale is not linear since $B_{1}$ is proportional to $\arcsin(\frac{m}{\mathcal{V}F})$. We first confirm the validity of the detection scheme. We perform a lock-in detection with $\omega_{m}=2\pi\times 2$ kHz ($\Delta T=0.25$ ms) and $N=27$, and hence $T=7$ ms. $P_{0}$ is fixed to 11 mW. The change in $m$ is observed as $p$ is varied. The result is plotted in Fig. 2. Here, the angle of the QWP axis, $\theta$, is approximately 4∘ apart from the optimal angle, $\theta^{}$, minimizing the VLS. The experimental determination of $\theta^{}$ is described below. $m$ is well fitted by a sinusoidal function $\mathcal{V}F\sin(ap)$, indicating the VLS was successfully detected. The visibility in this detection setting is found to be $\mathcal{V}=0.746(42)$ from an independent measurement with no modulation ($p=0$) where $\Delta\varphi$ is scanned. The detection is used to minimize the VLS. We control the VLS by changing $\theta$ with $p$ fixed to 0.32. The $\theta$-dependence of $m$ is shown in Fig. 3(a). Because $\mathcal{C}\approx\sin 2(\theta-\theta^{})\equiv\sin 2\Delta\theta$ when the birefringence in the optical path is small Wood et al. (2016) and $\|\Delta\theta\|\ll 1$, we fit $m$ by $\mathcal{V}F\sin(\beta_{1}(\theta-\theta^{}))$. The fit gives $\beta_{1}=6.2(2)$, which is in reasonable agreement with the calculation. $\theta^{}$ is found to be -6.6(1.1) mrad. The VLS resolution is evaluated as $\delta\omega=\beta_{1}\delta\theta^{}/T=2\pi\times 0.16$ Hz, where $\delta\theta^{}$ is the uncertainty in the $\theta^{}$ estimation. Figure 3: (color online) Polarization dependence of the signal. (a) Measurement result with $T$ = 7 ms. (b) Measurement result with $T$ = 28.2 ms. The blue circles and red squares represent $m_{+}$ and $m_{-}$, respectively. (c) $\Delta m$ as a function of $\theta$. $\Delta B_{1}$ is the difference between the fictitious magnetic fields for positive and negative $P_{1}$. The solid lines in (a) and (c) are the fitting curves. We perform a fine estimation of $\theta^{}$ by extending $T$ to 27.2 ms and applying a larger modulation. In this experiment, $\omega_{m}$ and $N$ are $2\pi\times 625$ Hz and $33$, respectively. We measure $m$ for $P_{1}=\pm 13$ mW, referred to as $m_{\pm}$, respectively. In finding $\theta^{}$, we use $\Delta m=m_{+}-m_{-}$ to cancel the offset due to the background field and the systematic error in the spin measurement. The results are shown in Figs. 3(b) and (c). $\Delta m$ is fitted by $4\mathcal{V}F\sin(\beta_{2}(\theta-\theta^{}))$, giving $\beta_{2}=117(16)$ and $\theta^{}=0.06$ mrad with $\delta\theta^{}$ = 0.40 mrad. The angle resolution is improved 2.8 times. We observe a larger variance in $m$ in the experiments for the fine $\theta^{}$ estimation. The standard deviation of $\Delta m$ is on average 0.64, while that for the reference data without modulation is $0.09$. Therefore, a further improvement by a factor of at least 7 is possible, because $\Delta m$ should ideally be independent of $T$ and the modulation strength. We ascribe the increased variance to the actual variation of the vector shift over the experimental runs, caused by beam polarization fluctuation. The result of the sensitive detection implies that the beam circularity varies with the standard deviation of approximately $3\times 10^{-3}$. On the other hand, from an independent experiment, we expect that the retardance of the QWP should vary by several mrad due to the temperature change in our experimental room (within $\approx$ 0.6 K with a period of around 20 minutes). The BEC is subject to a fictitious magnetic field gradient without the VLS cancellation, because it is located at the shoulder of the optical trap beam due to gravity sag. While the observed fictitious magnetic field is small, the gradient in the fictitious field can be on the order of 100 $\mu$T/m. The gradient displaces the trap potential for each spin state other than the $m_{F}$ = $0$ state, thereby driving the spin dependent motion. We observe an actual motion in a transversally-spin-polarized BEC in the hyperfine spin $F=2$ state, prepared after the initial $\pi/2$ pulse. We plot the vertical displacement of the spin components in the TOF image, which reflects the momentum, in Figs. 4(a)–(d). The direction of the motion inverts depending on the sign of $\Delta\theta$ and the motion becomes small at $\Delta\theta\approx 0$. These observations indicate that the motion is induced by the fictitious magnetic field. Figure 4: (color online) Effects of the fictitious magnetic field on a transversally polarized BEC. (a)–(d) Vertical displacement of the center of mass in the TOF image. The panels show the data for $\Delta\theta$ = (-1, +0.02, +1, +4) degrees, respectively. The solid and dashed lines are guides for the eyes. (e)–(h) Population evolution corresponding to (a)–(d). The fictitious magnetic field gradient also causes nonlinear spin evolution and thus a population change, as does the real magnetic field gradient Eto et al. (2014). The initial polarized atomic spin state breaks due to the spin mixing seeded by the nonlinear spin evolution. We show $p_{0}=N_{0}/N_{\mathrm{tot}}$, $p_{1}=(N_{-1}+N_{+1})/(2N_{\mathrm{tot}})$, and $p_{2}=(N_{-2}+N_{+2})/(2N_{\mathrm{tot}})$ in Figs. 4(e)–(h). The population changes are observed at an earlier time ($t<100$ ms) except for the case $\Delta\theta\approx 0$. These changes can be attributed to the fictitious magnetic field gradient. The faster population change for $\Delta\theta=4^{\circ}$ is consistent with a qualitative estimation of the characteristic time for the change of $t_{}\propto b^{-2/3}$, where $b$ is the magnetic field gradient Eto et al. (2014). A slow population change, which occurs regardless of $\Delta\theta$, is caused by a residual axial magnetic field gradient, $\partial B_{z}/\partial z$. The existence of the axial gradient in these data is confirmed by the fact that the spin components separate in the axial direction at later times. Figure 5: (color online) Atom number losses. The solid line is an exponential fit to the data with the optimized field gradient. The inset shows the population evolution for the optimized case. The dotted lines are the prediction curves of the mean-field driven evolution without including the inelastic losses Kronjäger et al. (2008). We next observed the change in the atom loss rate. Figure 5 shows the evolution of the atom numbers, corresponding to the data in Figs. 4(b) and (d). The decay is faster when $\Delta=4^{\circ}$ than for $\Delta\approx 0^{\circ}$. In the latter case, the decay rate starts to increase from around $t$ = 150 ms, where the population changes occur (see Fig. 4(f)). No increase in the loss rate is observed at the later time when we optimize $\theta$ and reduce the axial real magnetic field gradient. The $1/e$ time for the optimized condition is found to be 742 (31) ms. The change of the loss rate can be understood from the property of the inelastic collisions in the $F=2$ state Tojo et al. (2009). Note that the inelastic collisional loss in the polarized state is inhibited due to the restriction of the angular momentum conservation. The break of the polarized state due to the field gradient results in rapid atom losses. However, the loss still occurs for the optimized condition. Although the remaining loss may be due to the residual field inhomogeneity, it is associated with the spin mixing induced by the quadratic Zeeman energy Kronjäger et al. (2006, 2008). The model presented in Kronjäger et al. (2006, 2008), however, needs to be modified to explain the observed population conservation, shown in the inset of Fig.5. According to Kronjäger et al. (2008), the population evolution in the limit of small quadratic Zeeman energy, $q$, is approximately given by $\displaystyle p_{0}=$ $\displaystyle\frac{3}{8}\left[1+\frac{q}{2g_{1}n}(1-\cos(4g_{1}nt/\hbar))\right],$ (5) $\displaystyle p_{1}=$ $\displaystyle\frac{1}{4},$ (6) $\displaystyle p_{2}=$ $\displaystyle\frac{1}{16}\left[1-\frac{3q}{2g_{1}n}(1-\cos(4g_{1}nt/\hbar))\right],$ (7) where $g_{1}=\frac{4\pi\hbar^{2}}{m}\frac{a_{4}-a_{2}}{7}$ is the interaction strength with $a_{\mathcal{F}}$ being the $s$-wave scattering length for the collisional channel of the total angular momentum $\mathcal{F}$ and $n$ is the mean atomic density. Following these equations, $p_{0}$ and $p_{2}$ would undergo oscillations, which is not in agreement with the observed experimental result. We therefore attribute the population conservation to polarization purification by inelastic collisional losses Eto et al. (2019). It should be noted that the observed population conservation contrasts with the case of the $F=1$ state, in which the magnitude of the polarization modulates Jasperse et al. (2017). ## IV Discussion The quantum lock-in VLS detection is of practical use in cold atom experiments. It can be used for evaluating the degree of circular polarization of an optical trap beam at the atomic position, as we have shown. As the vacuum window birefringence introduces a maximum ellipticity of $10^{-2}$ or $10^{-1}$ Steffen et al. (2013), a beam with no special care taken with respect to the in vacuo polarization may generate a fictitious field of several nT or a VLS of tens of Hz, even with a shallow trap for ultracold atom experiments. Quantum lock-in detection is sensitive enough to ensure better linear polarization at the atomic position and therefore will greatly improve the magnetic conditions in cold atom experiments. The sensitivity is sufficient to suppress the VLS below the requirements for magnetically sensitive experiments, including studies of spinor BECs. Although a homogeneous linear Zeeman shift does not affect the spinor physics due to spin conservation Stamper-Kurn and Ketterle (2001), a magnetic field gradient below several $\mu$T/m is typically required to prevent magnetic polarization and observe the intrinsic magnetic ground state Stenger et al. (1998) or dynamics. The sub-Hz VLS resolution of quantum lock-in detection meets this challenging demand. Reducing the VLS is also important for precise measurements. In addition to a direct energy shift, an inhomogeneous fictitious field is also detrimental to measurement accuracy Cates et al. (1988). VLS reduction leads to a long coherence time, which is a mandatory requirement for highly sensitive measurements. We have constructed a precise BEC magnetometer using a transversally polarized $F=2$ BEC with a long coherence time, realized using VLS elimination as we have shown. The detail of the $F=2$ BEC magnetometer will be presented elsewhere Sekiguchi et al. (2020). We finally discuss the sensitivity limitations. The sensitivity of the quantum lock-in detection is essentially the same as that of a Ramsey interferometer with an equal phase accumulation time. As the atom shot noise is dominant over the photon shot noise in typical absorption imaging, the standard quantum limit in the VLS measurement is given by Giovannetti et al. (2004, 2006) $\delta\omega=\frac{1}{T\sqrt{N_{\mathrm{tot}}}}.$ (8) Here we replace the factor $\frac{2}{\pi}$ in Eq. (3) due to the sinusoidal modulation with the maximal value of 1, which is realized with a rectangular waveform modulation. Substituting $N_{\mathrm{tot}}=3\times 10^{5}$ and $T$ = $30$ ms into Eq. (8), we obtain $\delta\omega=2\pi\times 10$ mHz. 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# Integral morphisms and log blow-ups Fumiharu Kato ###### Abstract. This paper is a revision of the author’s old preprint “Exactness, integrality, and log modifications”. We will prove that any quasi-compact morphism of fs log schemes can be modified locally on the base to an integral morphism by base change by fs log blow-ups. ###### 2010 Mathematics Subject Classification: Primary: 14A99, Secondary: 14E05 ## 1\. Introduction The aim of this paper is to prove the following theorem: ###### Theorem 1.1. Let $f\colon X\rightarrow Y$ be a quasi-compact morphism of fs $(=$ fine and saturated$)$ log schemes. Then for any $y\in Y$ there exists an étale neighborhood $V\rightarrow Y$ of $\overline{y}$ $(=$ a separable closure of $y)$ and an fs log blow-up $V^{\prime}:=\operatorname{Bl}_{\mathcal{K}}(V)\rightarrow V$ along a coherent log ideal $\mathcal{K}\subset\mathcal{M}_{V}$, by which the fs base change $f_{V^{\prime}}\colon X_{V^{\prime}}\rightarrow V^{\prime}$ is integral. Here, by fs log blow-up (resp. fs base change) we mean a log blow-up (resp. base change) in the category of fs log schemes; cf. Remark 3.6. This theorem has been announced and proved in somewhat incomplete and inaccurate form in the author’s old preprint [5], a first draft of which has actually been written in 1997, and afterwards put in the arXiv in 1999. Since then, mainly because the author has been away from log geometry, the paper has been kept unpublished; the author apologizes for all inconvenience caused thereby. While there have been much progress and many new results in log geometry last two decades, the paper has sometimes been referred to. Moreover, it seems, to the best of the author’s knowledge, that the theorem itself has not yet been written anywhere, even in the foundational book [11] by Ogus, and became folklore among experts. In fact, the theorem is nowadays a consequence of combination of known results. For example, Luc Illusie, Kazuya Kato, and Chikara Nakayama proved in [4] (see also [11, III.2.6.7]) a weak version of the theorem, where “integral” is replaced by “$\mathbb{Q}$-integral”. Then by a further fs log blow-up of the base to make the log structure free (i.e., to make each stalk of $\overline{\mathcal{M}}=\mathcal{M}/\mathcal{O}^{\times}$ a free monoid), the resulting map becomes integral (cf. [11, I.4.7.5]). Since, due to Nizioł [10, 4.11], the composition of fs log blow-ups is again an fs log blow-up, this actually suffices to prove the theorem. In the mean time, in August 2020, Michael Temkin asked the author some questions on the preprint, and suggested the final form of the theorem presented as above. Based upon the fact that the theorem has to be referred to in a recent work [2] of him and his coauthors, and that the theorem has not yet been presented in published form, he encouraged the author to revise the old preprint for publishing. This is the situation from whence the present paper comes out, where we keep the original proof based on the technique of toric flattening, the original idea of which is attributed to Takeshi Kajiwara. Let us mention some consequences of Theorem 1.1. Tsuji [13, II.3.4] proved that any integral and quasi-compact morphism between fs log schemes can be made saturated by fs base change by “multiplication-by-$n$” map. Combined with our result, this yields the following: ###### Corollary 1.2. Let $f\colon X\rightarrow Y$ be a quasi-compact morphism between fs log schemes. Then for any $y\in Y$ there exists a diagram $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 14.38945pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{V^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 44.98112pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 0.0pt\raise-29.49998pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 44.98112pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{V^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 60.57668pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 91.16835pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 52.7789pt\raise-29.49998pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 91.16835pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\operatorname{Bl}_{K}(V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 109.59393pt\raise-30.38885pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 158.6112pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 158.6112pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 165.63898pt\raise-30.38885pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 203.25844pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 203.25844pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{Y}$}}}}}}}{\hbox{\kern-14.38945pt\raise-40.88885pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\operatorname{S}(Q^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 21.28053pt\raise-46.07634pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.8264pt\hbox{$\scriptstyle{\mu}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 38.38945pt\raise-40.88885pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 38.38945pt\raise-40.88885pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\operatorname{S}(Q^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 67.16837pt\raise-40.88885pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 91.24336pt\raise-40.88885pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 91.24336pt\raise-40.88885pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\operatorname{Bl}_{K}(Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 132.62137pt\raise-45.3958pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\pi}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 152.01952pt\raise-40.88885pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 152.01952pt\raise-40.88885pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\operatorname{S}(Q)}$}}}}}}}\ignorespaces}}}}\ignorespaces,$ with all vertical arrows strict and all squares cartesian in the category of fs log schemes, such that * • $V\rightarrow Y$ is an étale neighborhood of $\overline{y}$; * • $\pi$ is an fs log blow-up of $\operatorname{S}(Q)=\operatorname{Spec}\mathbb{Z}[Q]$ $($with the log structure by $Q\rightarrow\mathbb{Z}[Q]$$)$ along an ideal $K\subset Q$; * • $\operatorname{S}(Q^{\prime})\hookrightarrow\operatorname{Bl}_{K}(Q)$ is an arbitrary affine patch of $\operatorname{Bl}_{K}(Q)$; * • $\mu$ is the morphism induced from the multiplication-by-$n$ homomorphism $Q^{\prime}\rightarrow Q^{\prime}$ for some positive integrer $n$, and that the fs base change $f_{V^{\prime\prime}}\colon X\times_{Y}V^{\prime\prime}\rightarrow V^{\prime\prime}$ is saturated. Our second application is to log flatness. ###### Corollary 1.3. Let $f\colon X\rightarrow Y$ be a log flat and finitely presented morphism between fs log schemes. Then for any $y\in Y$ there exists an étale neoghborhood $V\rightarrow Y$ of $\overline{y}$ and an fs log blow-up $V^{\prime}\rightarrow V$ such that the underlying morphism of the fs base change $f_{V^{\prime}}\colon X\times_{Y}V^{\prime}\rightarrow V^{\prime}$ is flat. Indeed, we may assume that $f$ is integral, log flat, and of finite presentation. We may further assume that $f$ has a global chart by $h\colon Q\rightarrow P$, which is “neat” at a point $x\in X$ (as in Lemma 2.4 below) such that $Q\cong\overline{\mathcal{M}}_{Y,\overline{y}}$ ($y=f(x)$) and $\overline{P}\cong\overline{\mathcal{M}}_{X,\overline{x}}$. Then, since $\overline{h}\colon\overline{Q}=Q\rightarrow\overline{P}$ is integral, so is $h$ (cf. Lemma 2.2 (5)). Since $Q$ is sharp (i.e., $Q^{\times}=\\{1\\}$) and $h$ is local (i.e., $h^{-1}(P^{\times})=Q^{\times}$), the ring homomorphism $\mathbb{Z}[P]\rightarrow\mathbb{Z}[Q]$ is flat (cf. Lemma 2.2 (6)). Since the log flatness implies that $X$ is flat over $Y\otimes_{\mathbb{Z}[Q]}\mathbb{Z}[P]$ ([12, 4.15]), we deduce that $f$ itself is flat. As for the interaction between log flatness and usual flatness, much has been studied recently by some authors. Among them, we refer to a preprint by Gillam [3]. It seems that, combining our result with many of the results therein, we can deduce several useful consequences. Finally, let us remark that, if $Y$ in Theorem 1.1 is log regular, then $V$ can be equal to $Y$ itself, i.e., one can find an fs log blow-up of $Y$ itself that makes the morphism $f$ integral by fs base change. This version of the theorem has been proven independently in [2, 3.6.11], which we will include, with a few comments, at the end of this paper (Proposition 4.2). ###### Remark 1.4. The original paper [5] included an “exactness version” of the theorem, where “integral” is replaced by “exact”, which we do not include in the present paper, since it follows immediately from the result in [4] mentioned above. The composition of this paper is as follows. In the next section, we will collect some basics of integral morphisms and neat charts. In Section 3, we will overview log blow-ups. We will then prove the main theorem in Section 4. The original version of the paper owes much to Richard Pink, Takeshi Kajiwara, and Max Planck Institute für Mathematik in Bonn, Germany. In addition, the preparation of the present version owes much to Michael Temkin for his encouragement, and to Chikara Nakayama for valuable discussions. ### 1.5. Notation and conventions All rings and monoids are assumed to be commutative. For a monoid $M$, we denote by $M^{\times}$ and $M^{\mathrm{gp}}$ the subgroup of invertible elements and the associated group, respectively, and write $\overline{M}=M/M^{\times}$. All sheaves on schemes are considered with respect to the étale topology. For a point $x$ of a scheme, we denote by $\overline{x}$ a separable closure of $x$. For a log scheme $X$, we denote by $\alpha_{X}\colon\mathcal{M}_{X}\rightarrow\mathcal{O}_{X}$ the log structure of $X$, and write $\overline{\mathcal{M}}_{X}=\mathcal{M}_{X}/\mathcal{O}^{\times}_{X}$. We denote by $\underline{X}$ the underlying scheme of $X$, which is also considered as a log scheme with the trivial log structure. For a monoid $P$, we denote by $\operatorname{S}(P)$ the log scheme whose underlying scheme is the affine scheme $\operatorname{Spec}\mathbb{Z}[P]$ with the log structure induced from $P\rightarrow\mathbb{Z}[P]$. ($\operatorname{S}(P)$ is denoted by $\mathsf{A}_{P}$ in [11].) ## 2\. Integral homomorphisms Let us first recall the definition of integral homomorphisms. ###### Definition 2.1. A homomorphism $h\colon Q\rightarrow P$ of integral monoids is said to be integral if, for any integral monoid $Q^{\prime}$ and any homomorphism $Q\rightarrow Q^{\prime}$, the push-out $P\oplus_{Q}Q^{\prime}$ in the category of monoids is an integral monoid. It can be shown ([6, (4.1)][11, I.4.6.2]) that $h\colon Q\rightarrow P$ is integral if and only if it has the following property: if $h(a_{1})b_{1}=h(a_{2})b_{2}$ for $a_{1},a_{2}\in Q$ and $b_{1},b_{2}\in P$, there exists $a_{3},a_{4}\in Q$ and $b\in P$ such that $b_{1}=h(a_{3})b$, $b_{2}=h(a_{4})b$ and $a_{1}a_{3}=a_{2}a_{4}$. ###### Lemma 2.2. $(1)$ The composition of integral homomorphisms is integral. For a diagram $Q\stackrel{{\scriptstyle h}}{{\rightarrow}}P\stackrel{{\scriptstyle k}}{{\rightarrow}}R$ of integral monoids, if $k\circ h$ is integral and $k$ is exact, then $h$ is integral; if $k\circ h$ is integral and $h$ is surjective, then $k$ is integral. $(2)$ The pushout of an integral homomorphism in the category of monoids is integral. $(3)$ If $P$ is an integral monoid, and $N\subset P$ is a submonoid, then the canonical map $P\rightarrow P/N$ is integral. $(4)$ An integral and local homomorphism of integral monoids is exact. $(5)$ A homomorphism $h\colon Q\rightarrow P$ of integral monoids is integral if and only if $\overline{h}\colon\overline{Q}\rightarrow\overline{P}$ is integral. $(6)$ A homomorphism $h\colon Q\rightarrow P$ of integral monoids is integral if the homomorphism of monoid algebras $\mathbb{Z}[Q]\rightarrow\mathbb{Z}[P]$ is flat. The converse is true, if $h$ is local and $Q$ is sharp. Recall that a homomorphism $h\colon Q\rightarrow P$ of integral monoids said to be exact if $(h^{\mathrm{gp}})^{-1}(P)=Q$, where $h^{\mathrm{gp}}\colon Q^{\mathrm{gp}}\rightarrow P^{\mathrm{gp}}$ is the associated group homomorphism. Recall also that, for a monoid $P$ and a submonoid $N\subset P$, the quotient monoid $P/N$ is given by $P/\sim$ (endowed with the natural monoid structure), where $a\sim b$ if and only if $ac=bd$ for some $c,d\in N$. ###### Proof. For (1), (4), and (6), see [11, I.4.6.3 & I.4.6.7]. (2) is immediate from the definition of integral homomorphisms. (5) follows from (1), (3), and the fact that a homomorphism of integral monoids of the form $Q\rightarrow\overline{Q}$ is always exact. Hence it suffices to show (3). To show that $\pi\colon P\rightarrow P/N$ ($a\mapsto\overline{a}$) is integral, take $a_{1},a_{2},b_{1},b_{2}\in P$ such that $\overline{a_{1}}\overline{b_{1}}=\overline{a_{2}}\overline{b_{2}}$; the last equality means $a_{1}b_{1}c_{1}=a_{2}b_{2}c_{2}$ for $c_{1},c_{2}\in N$, and if we set $a_{3}=b_{1}c_{1}$, $a_{4}=b_{2}c_{2}$ and $b=\overline{1}$, then we have $a_{1}a_{3}=a_{2}a_{4}$, $\overline{b_{1}}=\overline{a_{3}}b$ and $\overline{b_{2}}=\overline{a_{4}}b$, which shows that $\pi$ is integral. ∎ ###### Definition 2.3. A morphism $f\colon X\rightarrow Y$ of integral log schemes is said to be integral at $x\in X$ if the monoid homomorphism $\overline{\mathcal{M}}_{Y,\overline{y}}\rightarrow\overline{\mathcal{M}}_{X,\overline{x}}$, where $y=f(x)$, is integral, or equivalently, $\mathcal{M}_{Y,\overline{y}}\rightarrow\mathcal{M}_{X,\overline{x}}$ is integral (cf. Lemma 2.2 (6)). We say $f$ is integral if it is integral at all points of $X$. By Lemma 2.2 (2), integral morphisms are stable under base change in the category of fine log schemes. It is, however, not true that integral morphisms are stable under base change in the category of fs log schemes, cf. [11, I.4.6.5]. So it is often convenient to refer to a base change (or a fiber product) in the category of fs log schemes as fs base change for emphasis. Let us finally mention some technical facts on charts. ###### Lemma 2.4. Let $f\colon X\rightarrow Y$ be a morphism of fs log schemes. Then, for any $x\in X$ and $y=f(x)$, there exists commutative diagram $\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{\operatorname{S}(P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{S}(Q)}$ comprised of an étale neighborhood $V\rightarrow Y$ of $\overline{y}$, an fppf neighborhood $U\rightarrow X$ of $\overline{x}$, and a chart $Q\rightarrow P$ of $g\colon U\rightarrow V$ such that the following conditions are satisfied: * (a) $P$ and $Q$ are fs monoids; * (b) $Q\cong\overline{\mathcal{M}}_{Y,\overline{y}}$, and $\overline{P}\cong\overline{\mathcal{M}}_{X,\overline{x}}$; * (c) $Q^{\mathrm{gp}}\rightarrow P^{\mathrm{gp}}$ is injective and $P^{\mathrm{gp}}/Q^{\mathrm{gp}}\cong\mathcal{M}^{\mathrm{gp}}_{X/Y,\overline{x}}$. $($Note that, in this situation, $Q$ is sharp, and $Q\rightarrow P$ is local.$)$ ###### Proof. Since $Y$ is fs, one can take on an étale neighborhood of $\overline{y}$ a chart subordinate to the fs monoid $Q=\overline{\mathcal{M}}_{Y,\overline{y}}$. Then one can construct a local chart of $f$ as above according to the recipe as in [11, III.1.2.7], where one can take $P$ to be an fs monoid by the construction as in [11, II.2.4.4] ∎ ###### Lemma 2.5. Let $f\colon X\rightarrow Y$ be a morphism of fine log schemes. $(1)$ Suppose $f$ has a $($global$)$ chart by a homomorphism $h\colon Q\rightarrow P$ of fine monoids, such that the conditions (b) and (c) in Lemma 2.4 is satisfied. Then $f$ is integral if and only if the homomorphism $h$ is integral. $(2)$ If $f$ is integral at $x$, then it is integral at all points in an open neighborhood of $x$. ###### Proof. See [11, III.2.5.2]. ∎ ## 3\. Log blow-ups In this section, we briefly recall the notion of log blow-ups and their basic properties. Recall first that an ideal of a monoid $M$ is a subset $K\subset M$ such that $x\in K$ and $a\in M$ imply $ax\in K$. Trivial ideals are $\emptyset$ and $M$ itself. It follows from Dickson’s lemma that any ideal of a finitely generated monoid is finitely generated. If $\pi\colon M\rightarrow\overline{M}$ is the canonical map, the map $K\mapsto\pi(K)$ gives a bijection from the set of ideals of $M$ to the set of ideals of $\overline{M}$. Let $X$ be a log scheme. A log ideal of $X$ is a sheaf of ideals $\mathcal{K}$ of $\mathcal{M}_{X}$. We denote by $\overline{\mathcal{K}}$ the corresponding ideal of $\overline{\mathcal{M}}_{X}$. For a morphism $f\colon X\rightarrow Y$ of fine log schemes and a log ideal $\mathcal{K}$ of $Y$, one has the extension of the log ideal $\mathcal{K}\mathcal{M}_{X}=(f^{-1}\mathcal{K})\mathcal{M}_{X}$, which is a log ideal of $X$. ###### Example 3.1. Let $P$ be a monoid and $K\subset P$ an ideal. One has the log ideal $\widetilde{K}$ associated to $K$ on $X=\operatorname{S}(P)$, constructed as follows. For any open subset $U\subset X$, we have a monoid homomorphism $P\rightarrow\mathcal{M}_{X}(U)$, and hence the extension ideal $K\mathcal{M}_{X}(U)$ of $\mathcal{M}_{X}(U)$. Then $\widetilde{K}$ is the sheafification of the subpresheaf of ideals of $\mathcal{M}_{X}$ given by $U\mapsto K\mathcal{M}_{X}(U)$. Note that, for any $x\in X$, we have $\widetilde{K}_{\overline{x}}=K\mathcal{M}_{X,\overline{x}}$. ###### Definition 3.2. A log ideal $\mathcal{K}$ of $X$ is called coherent at $x\in X$ if there exists a local chart $U\rightarrow\operatorname{S}(P)$, where $U$ is an étale neighborhood around $\overline{x}$, and an ideal $K\subset P$ such that $\mathcal{K}\|_{U}=\widetilde{K}\mathcal{M}_{U}$ (let us say, in this situation, that $K$ is a chart of $\mathcal{K}$ over $U$). A log ideal is called coherent if it is coherent at all points. ###### Remark 3.3 (cf. [11, II.2.6.2]). If a log ideal $\mathcal{K}$ of $X$ is coherent at $x\in X$, then, for any local chart $U\rightarrow\operatorname{S}(P)$ around $\overline{x}$, the pullback ideal $K\subset P$ of $\overline{\mathcal{K}}_{\overline{x}}$ by $P\rightarrow\mathcal{M}_{X,\overline{x}}\rightarrow\overline{\mathcal{M}}_{X,\overline{x}}$ generates $\mathcal{K}$ around $\overline{x}$; i.e., $\widetilde{K}\mathcal{M}_{U^{\prime}}=\mathcal{K}\|_{U^{\prime}}$ over an étale neighborhood $U^{\prime}$ of $\overline{x}$ contained in $U$. A coherent log ideal $\mathcal{K}$ of a log scheme $X$ is said to be invertible if, for any $x\in X$, $\overline{\mathcal{K}}_{\overline{x}}$ is a principal (i.e. generated by a single element) ideal of $\overline{\mathcal{M}}_{X,\overline{x}}$, or equivalently, there exist étale locally a chart $U\rightarrow\operatorname{S}(P)$ and an ideal $K\subset P$ as in Definition 3.2 with $K$ being principal. ###### Definition 3.4. A morphism $f\colon X^{\prime}\rightarrow X$ of fine (resp. fs) log schemes is called a log blow-up along a coherent log ideal $\mathcal{K}$ if it has the following universal mapping property: * (a) $\mathcal{K}\mathcal{M}_{X^{\prime}}$ is an invertible log ideal of $\mathcal{M}_{X^{\prime}}$; * (b) If $g\colon T\rightarrow X$ is a morphism of fine (resp. fs) log schemes such that $\mathcal{K}\mathcal{M}_{T}$ is invertible, then there exists a uniquely morphism $T\rightarrow X^{\prime}$ such that the diagram $\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{X}$ commutes. The log blow-up along a coherent log ideal is unique up to isomorphisms. Since every extension of invertible log ideal is again invertible, we have: ###### Lemma 3.5. The family of log blow-ups is stable under base change. More precisely, if $X_{\mathcal{K}}\rightarrow X$ is a log blow-up of a fine log scheme $X$ along a coherent log ideal $\mathcal{K}$, and if $Y\rightarrow X$ is a morphism of fine log schemes, then $X_{\mathcal{K}}\times_{X}Y\rightarrow Y$ is a log blow-up of $Y$ along $\mathcal{K}\mathcal{M}_{Y}$. If $X$ is an fs log scheme, and $f\colon X^{\prime}\rightarrow X$ is a log blow-up in the category of fine log schemes, then the saturation $f^{\mathrm{sat}}\colon X^{\prime\mathrm{sat}}\rightarrow X$ gives a log blow- up of $X$ along the same coherent log ideal in the category of fs log schemes. Hence, to show the existence of log blow-ups, it suffices to deal with the case of fine log schemes. ###### Remark 3.6. As indicated above, the log blow-ups in the category of fs log schemes are rather similar to normalized blow-ups, i.e., blow-up followed by normalization. To make clear the distinction, we will call log blow-ups taken in the category of fs log schemes fs log blow-ups. The following construction of log blow-ups is due to Kazuya Kato [7, (1.3.3)] (cf. [11, III.2.6]): We first construct the log blow-up $\operatorname{Bl}_{K}(P)\longrightarrow\operatorname{S}(P)$ of $\operatorname{S}(P)$, where $P$ is a fine monoid, along the coherent log ideal $\widetilde{K}$ constructed from an ideal $K\subset P$. Let $I(K)$ be the ideal of $\mathbb{Z}[P]$ generated by $K$, and consider the natural morphism $\operatorname{Proj}\bigoplus_{n}I(K)^{n}\rightarrow\operatorname{Spec}\mathbb{Z}[P]$. $\operatorname{Proj}\bigoplus_{n}I(K)^{n}$ has the affine open covering $\operatorname{Proj}\bigoplus_{n}I(K)^{n}=\bigcup_{t\in K}\operatorname{Spec}\mathbb{Z}[P\langle t^{-1}K\rangle].$ Here, $P\langle E\rangle$ for a subset $E$ of $P^{\mathrm{gp}}$ denotes the smallest fine submonoid in $P^{\mathrm{gp}}$ that contains $P$ and $E$. The canonical log structures given by $P\langle t^{-1}K\rangle\rightarrow\mathbb{Z}[P\langle t^{-1}K\rangle]$ glue to a fine log structure on $\operatorname{Proj}\bigoplus_{n}I(K)^{n}$. Then it follows that $\operatorname{Bl}_{K}(P):=\operatorname{Proj}\bigoplus_{n}I(K)^{n}\rightarrow\operatorname{S}(P)$ gives a log blow-up of $\operatorname{S}(P)$ along $\widetilde{K}$. To give a more explicit local description, take generators $t_{0},\ldots,t_{r}$ of $K$. Then $\operatorname{Bl}_{K}(P)$ is the union of the affine log schemes $\operatorname{Spec}\mathbb{Z}\big{[}P\big{\langle}\textrm{{\footnotesize$\frac{t_{0}}{t_{i}},\ldots,\frac{t_{r}}{t_{i}}$}}\big{\rangle}\big{]},$ with the log structure induced from $P\langle t_{0}/t_{i},\ldots,t_{r}/t_{i}\rangle\rightarrow\mathbb{Z}[P\langle t_{0}/t_{i},\ldots,t_{r}/t_{i}\rangle]$, i.e., the affine log schemes $\operatorname{S}(P\langle t_{0}/t_{i},\ldots,t_{r}/t_{i}\rangle)$, for $i=0,\ldots,r$. Let $X$ be a fine log scheme, and $\mathcal{K}$ a coherent log ideal of $X$. Suppose there exist a chart $\lambda\colon X\rightarrow\operatorname{S}(P)$ modeled on a fine monoid $P$ and an ideal $K\subset P$ such that $\mathcal{K}=\widetilde{K}\mathcal{M}_{X}$. Then, by Lemma 3.5, $\operatorname{Bl}_{\mathcal{K}}(X)=X\times_{\operatorname{S}(P)}\operatorname{Bl}_{K}(P)\longrightarrow X$ gives a log blow-up of $X$ along $\mathcal{K}$. In general, we take an étale covering $\\{U_{\alpha}\\}_{\alpha\in L}$ of $X$ such that each $U_{\alpha}$ allow a chart $U_{\alpha}\rightarrow\operatorname{S}(P_{\alpha})$ with an ideal $K_{\alpha}\subset P_{\alpha}$ satisfying $\mathcal{K}\|_{U_{\alpha}}=\widetilde{K}_{\alpha}\mathcal{M}_{U_{\alpha}}$. Then, by the universality of log blow-ups, the local log blow-ups $\operatorname{Bl}_{\mathcal{K}_{\alpha}}(U_{\alpha})\rightarrow U_{\alpha}$ constructed as above glue to a log blow-up of $X$ along $\mathcal{K}$. ###### Example 3.7. Let $P$ be a sharp fs monoid, and set $X=\operatorname{S}(P)$. Set $M=P^{\mathrm{gp}}$ and $N=\operatorname{Hom}_{\mathbb{Z}}(M,\mathbb{Z})$. The scheme $X$ is an affine toric variety corresponding to the corn $\sigma$ in $N_{\mathbb{R}}$ such that $\sigma^{\vee}\cap M=P$; i.e., $\sigma$ is the dual corn of the corn in $M_{\mathbb{R}}$ generated by $P$. Let $\phi\colon\sigma\rightarrow\mathbb{R}_{\geq 0}$ be a continuous convex piecewise linear function satisfying the following conditions (cf. [9, p.27]): * (a) $\phi(\lambda x)=\lambda\phi(x)$ for $x\in\sigma$ and $\lambda\in\mathbb{R}_{\geq 0}$; * (b) $\phi(N\cap\sigma)\subset\mathbb{Z}$. The function $\phi$ induces an ideal $K_{\phi}$ of $P$ given by $K_{\phi}=\\{m\in M\mid\langle x,m\rangle\geq\phi(x)\ \textrm{for all}\ x\in\sigma\\}.$ Then the fs log blow-up of $X$ along the log ideal $\widetilde{K}_{\phi}$ is the normalization of the blow-up of the toric variety $X=X_{\sigma}$ obtained from the coarsest subdividing fan $\Sigma_{\phi}$ of the cone $\sigma$ such that $\phi$ is linear on each cone in $\Sigma_{\phi}$; cf. [9, p.31, Theorem 10]. ## 4\. Proof of the theorem ###### Lemma 4.1. Let $P,Q$ be sharp fs monoids, and $h\colon Q\hookrightarrow P$ an injective homomorphism. Consider the induced morphism $f\colon X=\operatorname{S}(P)\rightarrow Y=\operatorname{S}(Q)$ of fs log schemes. Then there exists an ideal $K\subset Q$ such that the following conditions are satisfied: if $\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\operatorname{Bl}_{K}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y}$ is the fs base change of $f$ by the fs log blow-up $\operatorname{Bl}_{K}(Y)\rightarrow Y$, then the underlying scheme-theoretic morphism of $f^{\prime}$ is equidimensional. Note that $X^{\prime}\rightarrow X$ is isomorphic to the fs log blow-up along $\widetilde{KP}=\widetilde{K}\mathcal{M}_{X}$, i.e., $X^{\prime}\cong\operatorname{Bl}_{KP}(X)$. ###### Proof. In this proof, we follow the original argument in [5, 3.16] based on the idea of T. Kajiwara, which we note is similar to the argument in [1, Lemma 4.3]. We use the following notation: * • $M_{Q}=Q^{\mathrm{gp}}$, $M_{P}=P^{\mathrm{gp}}$; * • $N_{Q}=\operatorname{Hom}_{\mathbb{Z}}(M_{Q},\mathbb{Z})$, $N_{P}=\operatorname{Hom}_{\mathbb{Z}}(M_{P},\mathbb{Z})$; * • $\sigma_{Q}$ $($resp. $\sigma_{P})$ $=$ the cone in $N_{Q}$ $($resp. $N_{P})$ such that $Q=\sigma^{\vee}_{Q}\cap M_{Q}$ $($resp. $P=\sigma^{\vee}_{P}\cap M_{P})$. * • $\Sigma_{Q}$ (resp. $\Sigma_{P}$) $=$ the fan made up from the faces of the cone $\sigma_{Q}$ (resp. $\sigma_{P}$). Note that we have a map $\Phi\colon\Sigma_{P}\rightarrow\Sigma_{Q}$ of fans that induces the morphism of affine toric schemes $\operatorname{Spec}\mathbb{Z}[P]\rightarrow\operatorname{Spec}\mathbb{Z}[Q]$. Consider the subset $\Sigma_{P,1}\subset\Sigma_{P}$ (resp. $\Sigma_{Q,1}\subset\Sigma_{Q}$) of rays, i.e., cones of dimension $1$. Each $\rho\in\Sigma_{P,1}$ is mapped by $\Phi$ onto a ray in $\Sigma_{Q}$ or to a point (i.e. the origin of $N_{Q}$). If $\rho$ is mapped onto a ray, then take the primitive base $n_{1}\in N_{Q}$ of $\Phi(\rho)$, and extend it to a $\mathbb{Z}$-base $n_{1},\ldots,n_{r}$ of $N_{Q}$. The $r+1$-rays spanned by $n_{1},\ldots,n_{r},-(n_{1}+\cdots+n_{r})$ defines the projective $r$-space $\mathbb{P}^{r}_{\mathbb{Z}}$, and hence the ideal $\mathcal{O}(-1)$ gives rise to a support function, denoted by $\phi_{\rho}$, i.e., a continuous convex piecewise linear function $N_{Q,\mathbb{R}}\rightarrow\mathbb{R}_{\geq 0}$ satisfying the conditions (a) and (b) in Example 3.7; we denote the restriction of $\phi_{\rho}$ onto $\sigma_{Q}$ by the same symbol. If $\Phi(\rho)$ is a point, then set $\phi_{\rho}=0$. Set $\phi=\sum_{\rho\in\Sigma_{P,1}}\phi_{\rho},$ and let $\Sigma^{\prime}_{Q}$ be the coarsest fan that subdivides $\Sigma_{Q}$ such that $\phi$ is linear on each cone in $\Sigma^{\prime}_{Q}$. (The author learned this way of constructing $\phi$ from T. Kajiwara.) Now, let $K\subset Q$ be an ideal constructed from $\phi$ as in Example 3.7. Consider the fs log blow-up $\operatorname{Bl}_{K}(Y)\rightarrow Y$, and the fs base change $f^{\prime}\colon X^{\prime}:=X\times_{Y}\operatorname{Bl}_{K}(Y)\rightarrow\operatorname{Bl}_{K}(Y)$ of $f$. Then, $X^{\prime}\rightarrow X$ is the log blow-up of $X$ along $\widetilde{K}\mathcal{M}_{X}$, which is the normalized toric blow-up induced from the piecewise linear function on $N_{P,\mathbb{R}}$ given by the pull- back of $\phi$. If we denote the corresponding fan of $X\times_{Y}\operatorname{Bl}_{K}(Y)$ by $\Sigma^{\prime}_{P}$, then the induced map $\Phi^{\prime}\colon\Sigma^{\prime}_{P}\rightarrow\Sigma^{\prime}_{Q}$ maps each ray onto either a ray or a point (the origin), and hence mapping each cone onto a cone. Therefore, the morphism $f^{\prime}\colon X^{\prime}\rightarrow\operatorname{Bl}_{K}(Y)$ is equidimensional by [1, Lemma 4.1]. ∎ ###### Proof of Theorem 1.1. Let $Q=\overline{\mathcal{M}}_{Y,\overline{y}}$, and take an étale local chart $Y\leftarrow V\rightarrow\operatorname{S}(Q)$ around $\overline{y}$. For any $x\in X_{V}=X\times_{Y}V$, take an fppf local chart $X_{V}\leftarrow U_{x}\rightarrow\operatorname{S}(P_{x})$ around $\overline{x}$, where $P_{x}$ is an fs monoid, which extends to a local chart of $f$ as in Lemma 2.4. Since $f$ is quasi-compact, one can take finitely many $x_{1},\ldots,x_{n}\in X_{V}$ such that $X_{V}$ is covered by the union of $U_{i}:=U_{x_{i}}$ for $i=1,\ldots,n$. We set $P_{i}=P_{x_{i}}$ for $i=1,\ldots,n$. For $i=1,\ldots,n$, there exists by Lemma 4.1 an ideal $K_{i}\subset Q$ such that the fs base change $\operatorname{Bl}_{K_{i}P_{i}}(P_{i})=\operatorname{S}(P_{i})\times_{\operatorname{S}(Q)}\operatorname{Bl}_{K_{i}}(Q)\rightarrow\operatorname{Bl}_{K_{i}}(Q)$ by the corresponding log blow-up is equidimensional. Set $K=K_{1}\cdots K_{n}$. Then $\operatorname{Bl}_{KP_{i}}(P_{i})=\operatorname{S}(P_{i})\times_{\operatorname{S}(Q)}\operatorname{Bl}_{K}(Q)\rightarrow\operatorname{Bl}_{K}(Q)$ is equidimensional for any $i=1,\ldots,n$. One can further perform a toric blow-up of the toric scheme $\operatorname{Bl}_{K}(Q)$ so that the resulting toric scheme is smooth over $\mathbb{Z}$ (cf. [9, p.32, Theorem 11]). Since the composition of fs log blow-ups is again an fs log blow-up ([10, 4.11]), we may assume that there exists an ideal $K\subset Q$ such that * (a) the induced morphism $None$ $\operatorname{Bl}_{KP_{i}}(P_{i})=\operatorname{S}(P_{i})\times_{\operatorname{S}(Q)}\operatorname{Bl}_{K}(Q)\longrightarrow\operatorname{Bl}_{K}(Q)$ is equidimensional for each $i=1,\ldots,n$; * (b) $\operatorname{Bl}_{K}(Q)$ is smooth over $\mathbb{Z}$. We claim that $(\ast)_{i}$ is integral for each $i=1,\ldots,n$. Since toric schemes are Cohen-Macaulay, the properties (a) and (b) imply that the underlying scheme-theoretic morphism of $(\ast)_{i}$ is flat. Thus, for any cones $\sigma$ from the fan of $\operatorname{Bl}_{K}(Q)$ and $\tau$ from the fan of $\operatorname{Bl}_{KP_{i}}(P_{i})$ such that $\tau$ is mapped to $\sigma$, the affine portion of $(\ast)_{i}$ $\operatorname{S}(P^{\prime}_{i})\longrightarrow\operatorname{S}(Q^{\prime})$ where $Q^{\prime}=\sigma^{\prime}\cap M_{Q}$ and $P^{\prime}_{i}=\tau^{\prime}\cap M_{P}$, is flat, and hence is integral by Lemma 2.2 (6) and Lemma 2.5 (2). This means $(\ast)_{i}$ is integral. Now, by Lemma 2.5 (2), we deduce that $U_{i}\rightarrow V$ is integral for any $i=1,\ldots,n$, and hence $X_{V}\rightarrow V$ is integral. ∎ Let us finally remark that, the argument of the above proofs shows that, if we start from a toroidal morphism (in the sense as in [1, §1]) $f\colon X\rightarrow Y$, then, since $f$ is described globally by a morphism of polyhedral complexes $f_{\Delta}\colon\Delta_{X}\rightarrow\Delta_{Y}$ of K. Kato’s fans (cf. [8]), one can actually do the above argument globally on $Y$; cf. [1, 4.4]. Since the only question here lies as to whether one can take a global log blow-up of $Y$, one can slightly generalize the situation to $Y$ being log regular but without assuming $f$ to be toroidal. This situation has been treated in [2], which we include here for the reader’s convenience: ###### Proposition 4.2 ([2, 3.6.11]). Let $f\colon X\rightarrow Y$ be a quasi-compact morphism of fs log schemes, where $Y$ is log regular. Then there exists an fs log blow-up $Y^{\prime}:=\operatorname{Bl}_{\mathcal{K}}(Y)\rightarrow Y$ along a coherent log ideal $\mathcal{K}\subset\mathcal{M}_{Y}$ such that the fs base change $f_{Y^{\prime}}\colon X_{Y^{\prime}}\rightarrow Y^{\prime}$ is integral. ## References * [1] Abramovich, D.; Karu, K.: Weak semistable reduction in characteristic $0$, Invent. Math. 139 (2000), no. 2, 241–273. * [2] Abramovich, D.; Temkin, M.; Wlodarczyk, J.: Relative Desingularization and principalization of ideals, preprint, arXiv:2003.03659. * [3] Gillam, W.D.: Logarithmic flatness, preprint, arXiv:1601.02422. * [4] Illusie, L.; Kato, K.; Nakayama, C.: Quasi-unipotent Logarithmic Riemann-Hilbert Correspondences, J. Math. Sci. Univ. Tokyo 12 (2005), 1–66. * [5] Kato, F.: Exactness, integrality, and log modifications, preprint, arXiv:math/9907124. * [6] Kato, K.: Logarithmic structures of Fontaine–Illusie, in Algebraic Analysis, Geometry and Number Theory (J.-I. Igusa, ed.). Johns Hopkins Univ., 1988, 191–224. * [7] Kato, K.: Logarithmic degeneration and Dieudonne theory, preprint. * [8] Kato, K.: Toric singularities, Amer. J. Math. 116 (1994), no. 5, 1073–1099. * [9] Kempf, G., Knudsen, F., Mumford, D., Saint-Donat, B.: Toroidal Embeddings I, Lecture Note in Math. 339, Springer-Verlag, Berlin, Heidelberg, New York (1973). * [10] Nizioł, W.: Toric singularities: log-blow-ups and global resolutions, Journal of Algebraic Geometry 15 (2006), 1–29. * [11] Ogus, A.: Lectures on Logarithmic Algebraic Geometry, Cambridge University Press, Nov. 2018. * [12] Olsson, M.C.: Logarithmic geometry and algebraic stacks, Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 5, 747–791. * [13] Tsuji, T.: Saturated morphisms of logarithmic schemes, Tunis. J. Math. 1 (2019), no. 2, 185–220. Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro, Tokyo 152-8551, Japan (e-mail<EMAIL_ADDRESS>
Decay width modeling of Higgs boson within THDM model T.V. Obikhod and I.A. Petrenko Institute for Nuclear Research National Academy of Sciences of Ukraine 03068 Kiev, Ukraine e-mail<EMAIL_ADDRESS> Abstract As part of the search for new physics beyond the Standard Model, we chose the determination of the Higgs boson decay width as one of the least experimentally determined values. The decay widths into the four fermions of the lightest and heaviest CP-even Higgs bosons of the THDM model were calculated, taking into account QCD and electroweak corrections in the NLO approximation. To achieve this goal, the program Monte Carlo Prophecy 4f with special scenarios of parameters, 7B1 and 5B1 were used. It was found that the decay width of the heavier CP-even Higgs boson, H differs from HSM by 1227.93 times and changes to a negative value when deviating from the standard scenarios. Scale factors k${}^{2}_{Z}$ and k${}^{2}_{W}$ showed the predominance of the associated with Z boson production cross section of CP- even Higgs boson over the associated with W production cross section. 1\. Introduction In light of the latest experimental data on the searches for new physics beyond the Standard model (SM), Higgs boson remains the only candidate for a window into new physics, [1]. This task is related to the experimental study and theoretical predictions of the properties of the Higgs boson: the production cross sections, the partial decay width, coupling measurements, $k_{i}$. The crucial role for the investigation of the Higgs boson properties is played by the Higgs branching ratios and decay widths, [2]. The Higgs particle is a massive scalar boson with zero spin, no electric charge, and no colour charge is very unstable, decaying immediately into other particles. As all the channels of decay of the Higgs boson as well as possible new particles with certain masses have not yet been studied, there are uncertainties in the properties of the coupling constants and, accordingly, in the decay width of this particle. This fact is demonstrated by the deviation of the predicted SM Higgs decay width of about $4.07\cdot 10^{-3}$ GeV from the experimental data, which are presented in Table 1, [3, 4]. _Table 1._ Run 1 observed (expected) direct 95% CL constraints on the width of the 125 GeV resonance from fits to the $\gamma\gamma$ and ZZ mass spectra. The CMS measurement from the 4l mass line-shape was performed using Run 2 data. Experiment \| $M_{\gamma\gamma}$ \| $M_{4l}$ \| ---\|---\|---\|--- ATLAS \| $\prec 5.0(6.2)$ GeV \| $\prec 2.6(6.2)$ GeV \| CMS \| $\prec 2.4(3.1)$ GeV \| $\prec 1.1(1.6)$ GeV \| The purpose of our paper is to calculate decay widths of lightest, h, and CP- even, H, Higgs bosons of Two Higgs doublet model (THDM), [5] as well as the value of the deviation from SM of the sum of the partial Higgs decay widths compared to the SM, $\kappa_{H}^{2}$, through computer modeling with the help of Monte Carlo program Prophecy 4f 3.0 [6]. 2\. The calculations of decay width and scale factors The Standard Model predicts a very small width of about $4$ MeV for a $126$ GeV Higgs boson. But the error of the energy measurement at the LHC is hundreds of times greater, of the order of $1$ GeV, and it will not be possible to significantly reduce it. As a result, measuring the width of the Higgs boson directly is unrealistic. However, it is possible to accumulate data on the production and decay of the Higgs boson at significantly higher energies - not in the vicinity of $126$ GeV, but, say, above $300$ GeV, [7]. This process will look like the birth and decay of a virtual Higgs boson in this mass range. It is, of course, strongly weakened in comparison with the main process at the resonance peak, but it can be quite measurable. As THDM model predicts the existence of five Higgs bosons, we will carry out our calculations for two bosons: lightest Higgs boson, h and CP-even Higgs boson, H as the analog of virtual Higgs boson described above. Thus, the idea of theorists - to accumulate data on the production and decay of the Higgs boson at significantly higher energies can be realized. The efficiency of this method can be estimated by comparing the calculations of decay widths for the lightest and heaviest bosons. The precise experimental investigation of the Higgs boson and theoretical searches for deviations from the predictions of $SM$ requires precise Monte Carlo computer modeling. Prophecy 4f computes the inclusive partial decay widths and differential distributions of the decay products, where unweighted events for leptonic final states are provided. The advantage of the Prophecy4f program is that it allows the calculations for the Higgs decays into four fermions including full electroweak and QCD next-to-leading order (NLO) corrections with interference contributions between different $WW$/$ZZ$ channels, and inclusion of all off-shell effects of intermediate $W/Z$ bosons. We’ll consider the processes, LO Feynman diagram of which is in the form of Fig. 1 _Fig.1._ _Generic diagram for decay of $H\rightarrow 4f$ where $V=W,Z$, from [6]._ The total state width of Higgs boson is equal to the sum of the partial channel widths [6]: $\Gamma_{H\rightarrow 4f}=\Gamma^{{total}}=\Gamma^{{leptonic}}+\Gamma^{{semi- leptonic}}+\Gamma^{{hadronic}},$ The total width can be presented via $ZZ$, $WW$ decays and their interference: $\Gamma_{H\rightarrow 4f}=\Gamma_{H\rightarrow W^{}W^{}\rightarrow 4f}+\\\ +\Gamma_{H\rightarrow Z^{}Z^{}\rightarrow 4f}+\Gamma_{WW/ZZ-int}\ ,$ where the components are defined in terms of specific final states: $\Gamma_{H\rightarrow W^{\star}W^{\star}\rightarrow 4f}=9\cdot\Gamma_{H\rightarrow\nu_{e}\overline{e}\mu^{-}\nu_{\mu}}+12\cdot\Gamma_{H\rightarrow\nu_{e}\overline{e}d\overline{u}}+4\cdot\Gamma_{H\rightarrow u\overline{d}s\overline{c}}\ ,$ $\Gamma_{H\rightarrow Z^{}Z^{}\rightarrow 4f}=3\cdot\Gamma_{H\rightarrow\nu_{e}\overline{\nu_{e}}\nu_{\mu}\overline{\nu_{\mu}}}+3\cdot\Gamma_{H\rightarrow e\overline{e}\mu\mu^{+}}\\\ \hskip 73.97733pt+9\cdot\Gamma_{H\rightarrow\nu_{e}\overline{\nu_{e}}\mu\mu^{+}}+3\cdot\Gamma_{H\rightarrow\nu_{e}\overline{\nu_{e}}\nu_{e}\overline{\nu_{e}}}\\\ \hskip 73.97733pt+3\cdot\Gamma_{H\rightarrow e\overline{e}e\overline{e}}+6\cdot\Gamma_{H\rightarrow\nu_{e}\overline{\nu_{e}}u\overline{u}}\\\ \hskip 73.97733pt+9\cdot\Gamma_{H\rightarrow\nu_{e}\overline{\nu_{e}}d\overline{d}}+6\cdot\Gamma_{H\rightarrow u\overline{u}e\overline{e}}\\\ \hskip 73.97733pt+9\cdot\Gamma_{H\rightarrow d\overline{d}e\overline{e}}+1\cdot\Gamma_{H\rightarrow u\overline{u}c\overline{c}}\\\ \hskip 73.97733pt+3\cdot\Gamma_{H\rightarrow d\overline{d}s\overline{s}}+6\cdot\Gamma_{H\rightarrow u\overline{u}s\overline{s}}\\\ \hskip 73.97733pt+2\cdot\Gamma_{H\rightarrow u\overline{u}u\overline{u}}+3\cdot\Gamma_{H\rightarrow d\overline{d}d\overline{d}}\ ,$ $\Gamma_{WW/ZZ- int}=3\cdot\Gamma_{H\rightarrow\nu_{e}\overline{e}e\overline{\nu_{e}}}-3\cdot\Gamma_{H\rightarrow\nu_{e}\overline{\nu_{e}}\mu\mu^{+}}-3\cdot\Gamma_{H\rightarrow\nu_{e}\overline{e}\mu\overline{\nu_{\mu}}}+\\\ \hskip 88.2037pt2\cdot\Gamma_{H\rightarrow u\overline{d}d\overline{u}}-2\cdot\Gamma_{H\rightarrow u\overline{u}s\overline{s}}-2\cdot\Gamma_{H\rightarrow u\overline{d}s\overline{c}}\ .$ Using scenarios obtained from the experimental measurements, [8], we presented the calculated NLO results on the four-fermion decays of light $CP$-even Higgs boson, $h$, Table 2 _Table 2._ Decay widths of lightest Higgs boson, $h$. \| Full decay width of lightest Higgs boson, $h$, (MeV) \| $\Gamma\rightarrow WW$ \| $\Gamma\rightarrow ZZ$ \| $\Gamma^{int}$ ---\|---\|---\|---\|--- 5-B1 \| 0.92852 \| 0.8326 \| 0.1007 \| -0.00478 7-B1 \| 0.93026 \| 0.8311 \| 0.104 \| -0.00484 We also perform calculations of CP-even Higgs boson with the different parameters presented below, in Table 3 and decay widths, Table 4 _Table 3._ THDM input parameters. \| $M_{H}$, GeV \| $M_{H+}$, GeV \| $M_{A}$, GeV \| $\lambda_{5}$ \| $\tan\beta$ \| $c_{\alpha\beta}$ ---\|---\|---\|---\|---\|---\|--- I \| 360 \| 690 \| 420 \| -1.9 \| 4.5 \| 0.15 II \| 600 \| 690 \| 690 \| -1.9 \| 4.5 \| 0.15 _Table 4._ Decay width of $CP$-even Higgs boson, $H$. Full decay width of lightest Higgs boson, $h$, (MeV) \| $\Gamma\rightarrow WW$ \| $\Gamma\rightarrow ZZ$ \| $\Gamma^{int}$ ---\|---\|---\|--- -54.487 \| -71.47 \| 17.203 \| -0.22 1176.36 \| 789.98 \| 385.74 \| 0.64 The calculations of SM Higgs boson decay width give us the following result, Table 5 _Table 5._ Decay width of SM Higgs boson, $H_{SM}$ Full decay width of lightest Higgs boson, $h$, (MeV) \| $\Gamma\rightarrow WW$ \| $\Gamma\rightarrow ZZ$ \| $\Gamma^{int}$ ---\|---\|---\|--- 0.958 \| 0.858 \| 0.10724 \| -0.00724 In the absence of beyond SM (BSM) Higgs decay modes, total scale factor $\kappa_{H}^{2}$ is the value of the deviation of the sum of the partial Higgs decay widths compared to the SM total width $\Gamma^{SM}_{H}$, [9, 10]: $\kappa_{H}^{2}\left(\kappa_{i},m_{H}\right)=\sum\limits_{{j=WW^{\star},ZZ^{\star},b\overline{b},\tau^{-}\tau^{+},\gamma\gamma,Z\gamma,gg,t\overline{t},c\overline{c},s\overline{s},\mu^{-}\mu^{+}}}\frac{\Gamma_{j}(\kappa_{i},m_{H})}{\Gamma^{SM}_{H}(m_{H})}\ .$ Since the identification of four leptons is the most detectable decay mode in comparison with other decay channels, the optimal direction of the search for new physics will be finding and comparison of the factor $\kappa_{H}^{2}$ for the decays of two Higgs bosons — the lightest and the heaviest one into $WW$ or $ZZ$ bosons. So, the scale factor $\kappa_{H}^{2}$ in this case is the following: $\kappa_{H}^{2}\left(\kappa_{i},m_{H}\right)=\sum\limits_{{j=WW^{\star},\ ZZ^{\star}}}\frac{\Gamma_{j}(\kappa_{i},m_{H})}{\Gamma^{SM}_{H}(m_{H})}\ .$ It is also interesting to calculate scale factors $\kappa_{W}^{2}$ and $\kappa_{Z}^{2}$ $\frac{\Gamma_{WW^{}}}{\Gamma^{SM}_{WW^{}}}=\kappa_{W}^{2},$ $\frac{\Gamma_{ZZ^{}}}{\Gamma^{SM}_{ZZ^{}}}=\kappa_{Z}^{2},$ which allow probing for BSM contributions in the loops for each channel separately. Moreover, these factors make it possible to calculate the deviations from the SM of the associated production cross sections in accordance with the formulas: $\frac{\sigma_{WH}}{\sigma^{SM}_{WH}}=\kappa_{W}^{2},$ $\frac{\sigma_{ZH}}{\sigma^{SM}_{ZH}}=\kappa_{Z}^{2}.$ The results of our calculations are performed in the Table 6: _Table 6._ Scaling factors of two Higgs bosons Higgs boson \| Scenario \| $\kappa_{W}^{2}$ \| $\kappa_{Z}^{2}$ \| $\kappa_{H}^{2}$ \| ---\|---\|---\|---\|---\|--- \| $\kappa_{H}^{2}$ \| \| \| w/o int \| \| w int \| \| \| \| h \| 5-B1 \| 0.97 \| 0.939 \| 0.967 \| 0.969 h \| 7-B1 \| 0.968 \| 0.97 \| 0.969 \| 0.971 H \| II \| 921 \| 3597 \| 1218 \| 1228 From the comparison of the data from Table 6 we see the slight change in $\kappa_{H}^{2}$ factor for 5-B1 and 7-B1 scenarios and huge increase compared to SM one for scenario II. Moreover, we can see the increasing of $\kappa_{H}^{2}$ factor for all scenarios with inclusion of interference. The BSM contributions in the loops for $WW$ channel are larger in 5-B1 scenario but for 7-B1 scenario the larger contribution in the loops are for $ZZ$ channel. Therefore, the chose of renormalization schema is also essential to the final result. The sharp jump in the $\kappa_{H}^{2}$ factor for heavier $CP$-even Higgs boson indicates about significant deviation from the $SM$ for scenario 7-B1. The difference in factor $\kappa_{Z}^{2}$ compared to $\kappa_{W}^{2}$ by almost four times indicates the predominance of the associated with $Z$ boson production cross section of $CP$-even Higgs boson over the associated with $W$ production cross section. 3\. Conclusions The searches for BSM physics are connected with studying of Higgs boson properties. The way of the realization of this purpose is connected with the decay widths measurements and theoretical predictions of Higgs boson properties. The most perspective and convenient Higgs boson decay channel into four fermions is one of the interesting way of its investigation. For the precise measurements of the decay width is proposed THDM model in the paper. We have considered lightest and CP-even heavier Higgs bosons, h and H correspondingly and modeled their decay widths into four fermions with the help of Monte Carlo program Prophecy 4f 3.0. The results of our calculations led us to the following conclusions connected with the searches of deviations from SM: * • decay widths of lightest Higgs boson, $h$ and $H_{SM}$ almost do not differ from each other; * • the scale factor $\kappa_{H}^{2}$ of $CP$-even Higgs boson, $H$ equal to 1228; * • the calculations of decay widths strongly depend on the parameter space and can take negative values as the masses of the $CP$-even and $CP$-odd Higgs bosons decrease by almost two times from the parameters of the 7B1 scenario; * • the interference account leads to an insignificant increase in decay widths; * • the difference in factor $\kappa_{Z}^{2}$ compared to $\kappa_{W}^{2}$ by almost four times indicates the predominance of the associated with $Z$ boson production cross section of $CP$-even Higgs boson, $H$ over the associated with $W$ production cross section. * • BSM contributions in the loops for $WW$ and $ZZ$ channels are vary depending on renormalization schema. ## References * [1] CERNweb. The Higgs boson as a probe for new physics // URL: https://ep-news.web.cern.ch/higgs-boson-probe-new-physics. * [2] M. Spira. Higgs boson production and decay at hadron colliders // Progress in Particle and Nuclear Physics 2017, V.95, p. 98-159. * [3] ATLAS Collaboration. Measurement of the Higgs boson mass from the $H{\rightarrow}{\gamma}{\gamma}$ and $H{\rightarrow}Z{Z}^{}{\rightarrow}4l$ channels in $pp$ collisions at center-of-mass energies of 7 and 8 TeV with the ATLAS detector // Phys. Rev. D 2014, V.90, p. 052004. [4] CMS Collaboration. Measurements of properties of the Higgs boson decaying into four leptons in pp collisions at $\sqrt{s}=13$ TeV // CMS-PAS-HIG-16-041, 2017. * [5] G.C. Branco et al. Theory and phenomenology of two-Higgs-doublet models // Phys. Rep. 2012, V.156, p. 1-102. * [6] A. Denner, S. Dittmaier, A. Muck. Prophecy4f 3.0: A Monte Carlo program for Higgs-boson decays into four-fermion final states in and beyond the Standard Model // Comput. Phys. Commun. 2020, V. 254, p. 107336. * [7] elementy$\\_ru$ // URL:https://elementy.ru/novosti$\\_nauki/432235.$ * [8] A. Denner, S. Dittmaier, J.-N. Lang. Renormalization of mixing angles // JHEP 2018, V. 11, p. 104, arXiv:1808.03466 [hep-ph]. * [9] The LHC Higgs Cross Section Working Group. Handbook of LHC Higgs Cross Sections: 3. Higgs Properties // CERN-2013-004, arXiv:1307.1347 [hep-ph] * [10] D. de Florian et al. Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector // CERN Yellow Reports: Monographs Volume 2/2017 (CERN–2017–002-M), arXiv:1610.07922 [hep-ph].
# Quantum theory cannot violate a causal inequality Tom Purves<EMAIL_ADDRESS>H.H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol, BS8 1TL, U.K. Anthony J. Short <EMAIL_ADDRESS>H.H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol, BS8 1TL, U.K. ###### Abstract Within quantum theory, we can create superpositions of different causal orders of events, and observe interference between them. This raises the question of whether quantum theory can produce results that would be impossible to replicate with any classical causal model, thereby violating a causal inequality. This would be a temporal analogue of Bell inequality violation, which proves that no local hidden variable model can replicate quantum results. However, unlike the case of non-locality, we show that quantum experiments _can_ be simulated by a classical causal model, and therefore cannot violate a causal inequality. _Introduction_.– A fascinating aspect of quantum theory that has been investigated recently is the possibility for the causal order of events to be placed into superposition Chiribella2013 ; Chiribella2012 ; Branciard2016 ; Zych2019 ; Barrett2020 , leading to ‘causal indefiniteness’ about the order with which events have taken place. This phenomenon has been tested experimentally Procopio2015 ; White2018 ; Goswami2020 , and can be exploited to gain advantages within quantum theory. For example, setups based on the quantum switch Chiribella2013 can help to determine whether unknown unitaries commute or anticommute Chiribella2012 . An interesting question is whether quantum theory can generate results which could not be simulated by any classical causal model. Such results would violate a Causal Inequality Oreshkov2012 ; Brukner2014 ; Branciard2015 ; Abbott2016 . These are the temporal analogues of Bell Inequalities Bell1964 , and the violation of such an inequality in nature would call into question the elementary properties that scientists regularly invoke when talking about cause and effect relationships. In this paper we focus on the relationship between the type of causal indefiniteness present in quantum theory and the type needed to violate causal inequalities. We show that despite allowing causally indefinite processes, the correlations generated by quantum theory can be simulated by a classical causal model. This means that quantum theory cannot violate causal inequalities, and hence cannot yield an advantage over classical causal processes for tasks defined in a theory-independent way (such as ‘guess your neighbour’s input’ Almeida2010 ; Branciard2015 ). Previous works in this direction have shown that particular switch-type scenarios cannot violate causal inequalities Arajo2015 , and that causal order cannot be placed in a pure superposition Costa2020 ; Yokojima2021 . It has also been shown that causal inequality violations are possible when we condition on measurement outcomes of one party Milz2020 . However, our results imply that such violations are not possible for general quantum setups without conditioning. Indefinite causal structure is often studied via process matrices Oreshkov2012 , which assume that local laboratories obey standard quantum theory, but allow any connections between them consistent with this. This may include processes which are not achievable in standard quantum theory, or in nature more generally. Here we focus on what is possible in standard quantum theory, using quantum control of different parties’ operations to generate superpositions of causal order, in a similar way to Colnaghi2012 ; Araujo2014 . As process matrices can yield causal inequality violation, a corollary of our result is that all process matrices cannot be implemented in standard quantum theory. _Results_.– Before considering quantum processes, we first define causal processes, which are those which could be realised classically by a set of parties in separate laboratories passing systems between them pearl ; Barrett2020 . First consider two parties, Alice and Bob, with measurement settings $x$ and $y$ and measurement results $a$ and $b$ respectively. During the experiment, depicted in figure 1, each party sees a system enter their laboratory exactly once, performs a measurement on it with their corresponding measurement setting (which may also modify the system), and records their result. They then pass the system out of their laboratory. Apart from the systems entering and leaving their laboratories, the two parties cannot communicate with each other, but the system leaving one laboratory may be later sent into the other. Alice and Bob’s joint measurement results can be described by a conditional probability distribution $p(ab\|xy)$. However, not all such probability distributions can be achieved by a causal process. Figure 1: An example of a causal process in which Alice goes before Bob. Note that the system which is passed from Alice’s to Bob’s laboratory could encode information about $a$ and $x$. The most general causal process in this case would be to first choose randomly whether Alice or Bob would go first (with probabilities $p(\textrm{Alice first})$ or $p(\textrm{Bob first})$). If Alice goes first, then her measurement result can depend on her measurement setting but not on Bob’s, who hasn’t acted yet, so is given by $p(a\|x)$. She can then encode her measurement setting and result in the system and pass it out of her laboratory. This system then enters Bob’s laboratory, where his measurement result can depend on all of the other variables, given by $p(b\|a,x,y)$. Considering the other causal order in which Bob goes first in the same way, we obtain Branciard2015 $\displaystyle p^{\mathrm{causal}}(ab\|xy)=$ $\displaystyle p(\textrm{Alice first})p(a\|x)p(b\|a,x,y)$ $\displaystyle+p(\textrm{Bob first})p(b\|y)p(a\|b,x,y)$ (1) For the multiparty generalisation Oreshkov2016multi ; Abbott2016 , observe that the above causal probability contains two types of terms. The first, such as $p(\textrm{Alice first})$, determines the order in which the parties act, and the second, such as $p(a\|x)$ or $p(b\|a,x,y)$, determines the outcome probabilities of their measurements, constrained by their causal order. We now extend these ‘who is next?’ and ‘what did they see?’ type probabilities to an arbitrary number of parties. We use $l_{k}$ to denote the $k{\textrm{th}}$ party that receives the system (or equivalently, the $k{\textrm{th}}$ laboratory the system enters), and denote the probability for this to occur by $p_{k}(l_{k}\|H_{k-1})$. The conditional on $H_{k-1}$ represents the history (including all previous parties that have measured, and their inputs and outputs) for it should be permitted for parties in the causal past of $l_{k}$ to affect who is the next party to act. As a simple example of this, consider a tripartite experiment, with Alice, Bob and Charlie participating. If Charlie comes first, the system could be passed to Alice or Bob next, based on the outcome of his measurement. Here, $p_{2}(\text{Alice next}\|\text{Charlie got outcome = }1)$ may not be equal to $p_{2}(\text{Alice next}\|\text{Charlie got outcome = }0)$. Scenarios of this form this are what $p_{k}(l_{k}\|H_{k-1})$ accounts for. The probability for $l_{k}$ to obtain given results may also depend on this history (but, importantly, not on the causal future), and of course on the measurement setting, denoted $x_{l_{k}}$. We write this probability as $p_{k}(a_{l_{k}}\|H_{k-1},x_{l_{k}})$. A causal model is then the summation over all available parties at all stages of the measurement procedure, under the assumption that each party only acts once in the entire procedure. ###### Definition 1 A causal probabilistic model can be written as $\displaystyle p^{\mathrm{causal}}(\vec{a}\|\vec{x})=\sum_{l_{1}\notin\mathcal{L}_{0}}...\sum_{l_{N}\notin\mathcal{L}_{N}-1}$ $\displaystyle p_{1}(l_{1}\|H_{0})p_{1}(a_{l_{1}}\|H_{0},x_{l_{1}})...$ $\displaystyle\quad...p_{N}(l_{N}\|H_{N-1}$ $\displaystyle)p_{N}(a_{l_{N}}\|H_{N-1},x_{l_{N}})$ (2) where the $p_{k}(l_{k}\|H_{k-1})$ terms represent probabilities for party $l_{k}$ to act at stage $k$ of the causal order, and $p_{k}(a_{l_{k}}\|H_{k-1},x_{l_{k}})$ terms represent probabilities for party $l_{k}$, who has acted at stage $k$ of the causal order to obtain measurement result $a_{l_{k}}$. Both of the above probabilities are conditional on a history, $H_{k-1}$, which contains all of the information about previous inputs, outputs and party order. In particular, the history $H_{k}=(h_{1},...,h_{k})$ is the ordered list of triples $h_{i}=(l_{i},a_{l_{i}},x_{l_{i}})$. The summations are performed over all possible next parties, excluding parties who have already acted, which are stored in the unordered sets $\mathcal{L}_{k}=\\{l_{1},...,l_{k}\\}$. To emphasise the symmetry between the terms we include $H_{0}$ and $\mathcal{L}_{0}$, which are defined as empty sets, as no parties have acted at that point. This definition leads to a convex polytope of causal probability distributions $p^{\mathrm{causal}}(\vec{a}\|\vec{x})$. Note that although the notation differs, this generates the same set of probabilities as was previously defined in Abbott2016 ; Oreshkov2016multi . Linear constraints on these probabilities which are satisfied by all $p^{\mathrm{causal}}(\vec{a}\|\vec{x})$ but which could be violated by some arbitrary probability distribution $p(\vec{a}\|\vec{x})$ are known as ‘causal inequalities’, and are a temporal analogue of the Bell inequalities which have been widely studied in the context of quantum non-locality. By definition, any $p^{causal}(a,b\|x,y)$ cannot violate a causal inequality. A violation of a causal inequality, by observation in experiment or by calculation in theory, proves that those experimental results or predictions do not have a causal explanation of the type defined above. _Quantum Processes_.– A general representation of quantum theory is provided by the quantum circuit model. However, if we construct a circuit with the parties’ actions at fixed locations, then there is no causal indefiniteness and a causal inequality cannot be violated 111We could space out the circuit such that there is at most one lab at each time-step, and then pass the full quantum state between the labs as a classical hidden variable which would allow us to recover the same correlations in a causal way. Even to capture all classical causal processes, we need to be able to alter when different parties act. This can be achieved in the circuit model by representing the parties’ actions by controlled quantum gates. Such gates could be constructed within standard quantum theory (e.g. by sending a system into the lab when the control is in the appropriate state and not otherwise), and are effectively what has been used in experiments probing quantum causality White2018 . Quantum circuits involving controlled lab gates appear sufficient to represent any processes achievable within standard quantum theory. For simplicity, we consider a setup involving a single quantum control which can trigger any of the labs. However this is equivalent to considering any quantum circuit which can be constructed from any number of individual controlled lab operations and other unitary gates (see the appendix for more details). The key idea is to consider $N$ parties, each of whom will interact with a quantum system exactly once, but in an order that is controlled coherently via the quantum control. We allow arbitrary unitary transformations of the system and control between each party’s action, so that the ordering of later parties can be modified by earlier actions. Figure 2: Illustration of the quantum protocol. The system interacts with the different parties via a sequence of controlled entangling unitaries. Quantum control of causal order is achieved by a series of unitaries $U_{n}$ on the control and system wires. To allow the maximum possible interference, and avoid ‘collapses’ which would prevent interference between different causal orders, we model each party’s measurement as a unitary interaction between the system and a local measurement register. This corresponds to the case in which there is no record in the measuring device of the time at which the measurement was performed. At the end of the experiment, all parties read off their measurement results from their local measurement registers (which can be modelled by a standard projective measurement). Each party also has a ‘flag’ which keeps track of how many times they have interacted with the system. At the end of the protocol we require that each party has interacted with the system exactly once. Formally, the Hilbert space can be decomposed into the following components * • An arbitrary quantum system $\mathcal{H}_{s}$, which is passed between parties. * • A quantum control $\mathcal{H}_{c}$ which has dimension $N+1$. The basis states $\ket{1}\ldots\ket{N}$ denote which party will measure next, while $\ket{0}$ is treated as a ‘do nothing’ command. By considering superpositions of these basis states, we can superpose different causal orders. * • A result register $\mathcal{H}_{r_{i}}$ for each of the $N$ parties. The different results are represented by orthonormal basis states $\ket{a_{i}}$ with $a_{i}\in\mathcal{A}_{i}$, leading to the result register having dimension $\|\mathcal{A}_{i}\|$. We choose one of these states as a starting state for the results register and denote it by $\ket{0}_{r_{i}}$. * • A ‘flag’ $\mathcal{H}_{f_{i}}$ for each of the $N$ parties, indicating how many times they have interacted with the system. For simplicity, we take each of these to be infinite dimensional, with basis states labelled by the integers. When the party interacts with the system the value of the flag is unitarily raised by the operator $\Gamma=\sum_{n}\ket{n+1}_{f_{i}}\bra{n}_{f_{i}}$. Each flag starts in the $\ket{0}_{f_{i}}$ state, and at the end of the protocol, we require them all to be in the $\ket{1}_{f_{i}}$ state. Note that we do not include separate local quantum ancillas for the parties, as these can always be incorporated in $\mathcal{H}_{s}$. We denote the combined result and flag spaces by $\mathcal{H}_{r}=\bigotimes{\mathcal{H}_{r_{i}}}$ and $\mathcal{H}_{f}=\bigotimes{\mathcal{H}_{f_{i}}}$ respectively. We consider quantum protocols as follows. Firstly, the initial state $\displaystyle\ket{0}=\ket{0}_{s}\ket{0}_{c}\ket{0}_{r}^{N}\ket{0}_{f}^{N}\in\mathcal{H}_{s}\otimes\mathcal{H}_{c}\otimes\mathcal{H}_{r}\otimes\mathcal{H}_{f}.$ (3) is prepared, and each party $l$ either chooses or is distributed their individual classical measurement setting $x_{l}$. The protocol then consists of $T$ time-steps, each of which is composed of two operations. Firstly, an arbitrary unitary transformation $U_{t}$ is applied to the system and control, which can depend on the time $t$. Secondly, a fixed controlled lab-activation unitary $V$ is applied, which activates whichever party is specified by the control. This is given by $V=\ket{0}\bra{0}_{c}\otimes I+\sum_{l=1}^{N}\ket{l}\bra{l}_{c}\otimes V_{s,r_{l}}(x_{l})\otimes\Gamma_{f_{l}}\otimes I$ (4) where the identities are over all remaining subsystems. $V_{s,r_{l}}(x_{l})$ is a unitary which implements the measurement of party $l$ on the system specified by the measurement setting $x_{l}$, and stores the result in the register $r_{l}$. For example, two different values of $x_{l}$ could correspond to party $l$ measuring the system in either the computational or the Fourier basis. Note that by incorporating ancillas within the system, any local quantum measurement (i.e a POVM) is realisable within this paradigm. Ancillas can also be used to generate arbitrary mixed states if required (via purification). The unitary operator $\Gamma_{f_{l}}$ raises the flag system of the party making their measurement. At the end of the protocol, we require that the flags are in the state $\ket{1}_{f}^{N}$ (i.e. that each party has measured the system once). This places constraints on the possible protocols which can be constructed. Note that each party does not have access to an operation which resets the flag, aside from the initialisation operation at the start of the protocol. They therefore always ‘remember’ if they have made a measurement or not. Also, we do not allow circuits involving the controlled inverse of a party’s action (which would lower their flag and erase their memory), as this would enlarge the set of causal possibilities even classically. The total unitary for the protocol is given by $\mathcal{U}=VU_{T}VU_{T-1}...VU_{1}.$ (5) At the end of the protocol, each party performs a projective measurement on their results register to obtain their final result 222Note that from a many- worlds perspective Everett such an additional step would not be necessary. However, we include it here to maintain connection with standard quantum theory and give an explicit formula for the outcome probabilities. The output probability distribution of the quantum protocol is therefore given by $\displaystyle p^{\mathrm{quantum}}(\vec{a}\|\vec{x})=\|(\ket{\vec{a}}\bra{\vec{a}}_{r}\otimes I)\mathcal{U}\ket{0}\|^{2}.$ (6) The full protocol is illustrated as a quantum circuit in figure 2. The main result of this paper is that any probability distribution which can be generated within quantum theory, as described above, can also be obtained via a classical causal process. ###### Theorem 1 Any quantum probability distribution $p^{\mathrm{quantum}}(\vec{a}\|\vec{x})$ can be exactly replicated by a classically causal process $p^{\mathrm{causal}}(\vec{a}\|\vec{x})$. Hence quantum theory cannot violate a causal inequality. In particular, we now show how to construct an explicit classical causal process which replicates the results of any quantum protocol, together with a sketch of the proof of Theorem 1. The full proof of the theorem can be found in the appendix. We first define notation for describing states at each stage of the quantum protocol, and then show how to use these to construct the probabilities in the corresponding classical model. ###### Definition 2 The (un-normalised) state with a History $H_{k-1}$, at a time $t$, with the control set to trigger the action of party $l_{k}$ is given by $\displaystyle\ket{\psi_{(l_{k},t,H_{k-1})}}=(\ket{l_{k}}\bra{l_{k}}_{c}\\!\otimes\\!\pi^{H_{k-1}}_{rf}\\!\otimes\\!I_{s})U_{t}VU_{t-1}...VU_{1}\ket{0}.$ (7) The projector onto the result and flag spaces is given by $\pi^{H_{k-1}}_{rf}=\bigotimes_{i=1}^{N}\left(\pi^{H_{k-1}}_{r_{i}f_{i}}\right),$ where $\displaystyle\pi^{H_{k-1}}_{r_{i}f_{i}}=\begin{cases}\ket{a_{i}}\bra{a_{i}}_{r_{i}}\otimes\ket{1}\bra{1}_{f_{i}}&\text{ if }(i,a_{i},x_{i})\in H_{k-1},\\\ I_{r_{i}}\otimes\ket{0}\bra{0}_{f_{i}}&\text{ otherwise }.\end{cases}$ (8) This notation describes states which are about to be measured by the parties (i.e., a $V$ type operator is about to act on them). We also set up some notation for states which have just been measured, in a similar fashion. ###### Definition 3 The (un-normalised) state with a History $H_{k}$, at a time $t$, in which party $l_{k}$ has just acted is given by $\displaystyle\ket{\phi_{(l_{k},t,H_{k})}}=(\ket{a_{l_{k}}}\bra{a_{l_{k}}}_{r_{k}}\otimes I)V\ket{\psi_{(l_{k},t,H_{k-1})}}.$ (9) With these definitions, we can associate the states in this quantum process with the probabilities in our classical causal model. ###### Definition 4 The probability for party $l_{k}$ to act next, given a history $H_{k-1}$ is given by: $\displaystyle p_{k}(l_{k}\|H_{k-1})=\frac{\sum_{t_{k}=1}^{T}\|\ket{\psi_{(l_{k},t_{k},H_{k-1})}}\|^{2}}{\sum_{l_{k}^{\prime}\notin\mathcal{L}_{k-1}}\sum_{t_{k}^{\prime}=1}^{T}\|\ket{\psi_{(l_{k}^{\prime},t_{k}^{\prime},H_{k-1})}}\|^{2}}.$ (10) We have summed over time Note2 , because it is possible within the quantum paradigm to conduct the $k^{th}$ measurement at different times according to a background clock (which we note the labs have no access to). Note that states at different times combine incoherently, but different sequences leading to the same set of historical measurement results combine coherently inside $\ket{\psi_{(l_{k},t_{k},H_{k-1})}}$. The form of equation (10) makes it a valid probability distribution, as it is non-negative, and obeys the correct normalisation that $\sum_{l_{k}\notin\mathcal{L}_{k-1}}p_{k}(l_{k}\|H_{k-1})=1$. Also note that it depends on only those input variables $x_{i}$ which appear in the history $H_{k-1}$. Next, we specify similar probabilities for seeing measurement results based on a given history. ###### Definition 5 The probability for party $l_{k}$ to obtain the measurement result $a_{l_{k}}$, given a history $H_{k-1}$, and an input variable $x_{l_{k}}$ is given by: $\displaystyle p_{k}(a_{l_{k}}\|H_{k-1},x_{l_{k}})=\frac{\sum_{t_{k}=1}^{T}\|\ket{\phi_{(l_{k},t_{k},H_{k})}}\|^{2}}{\sum_{a^{\prime}_{l_{k}}\in\mathcal{A}_{l_{k}}}\sum_{t_{k}^{\prime}=1}^{T}\|\ket{\phi_{(l_{k},t_{k}^{\prime},H_{k}^{\prime})}}\|^{2}},$ (11) where $H_{k}=(H_{k-1},(l_{k},a_{l_{k}},x_{l_{k}}))$ and $H_{k}^{\prime}=(H_{k-1},(l_{k},a^{\prime}_{l_{k}},x_{l_{k}}))$ This is again a valid probability distribution, since $\sum_{a_{l_{k}}\in\mathcal{A}_{l_{k}}}p_{k}(a_{l_{k}}\|H_{k-1},x_{l_{k}})=1$. In the numerator, we have simply taken sum of the modulus squared of all of the states which have the correct historical results, the control in the correct state, and the results register containing the result we want to calculate the probability for. To prove Theorem 1, We begin by inserting $p_{k}(l_{k}\|H_{k-1})$ (from (10)) and $p_{k}(a_{l_{k}}\|H_{k-1},x_{l_{k}})$ (from (11)) into the definition of a causal model (1). We are then able to straightforwardly cancel the numerator of the ‘who is next?’ type probabilities with the denominator of the ‘what did they see?’ probabilities for the probabilities evaluated at the same stage of the causal order. Next, we show that a sum over the last party to measure in the numerator at one stage of the causal order, cancels with the denominator at the next stage of the causal order. We then make the observation that for the first stage of the causal order, the denominator of $p_{1}(l_{1}\|H_{0})$ is equal to one (which corresponds to the fact that someone must measure first in the quantum circuit). Finally, we note that the numerator of the final term, summed over all parties, represents exactly the probabilities $p^{\mathrm{quantum}}(\vec{a}\|\vec{x})$ arising from the quantum protocol. This allows us to simulate the results of the quantum protocol via the classically causal model given in (1). Given that it can be replicated by a causal model, it follows that quantum theory cannot violate a causal inequality. In the appendix, we give an example of how these results can be applied in practice, based on the quantum switch Chiribella2013 . This involves the causal order of two parties becoming entangled with the control. A third party then performs a measurement which leads to interference between the two causal orders. It has already been shown that this simple setup cannot be used to violate a causal inequality Arajo2015 ; Oreshkov2012 . However, it is instructive to see how it fits into our framework. Despite the quantum setup including interference, our results give an explicit classical causal process which generates the same behaviour (i.e. the same $p(a,b,c\|x,y,z)$). _Conclusions_.– By using a quantum control to determine when different parties measure, and treating these measurements as coherent unitary processes, quantum theory allows us to generate superpositions of causal orders and to observe interference between them. At the level of the theory, such processes do not arise from a single causal order, or even a mixture of orders. However, we have shown that the probabilities $p(\vec{a}\|\vec{x})$ generated by any quantum protocol _can_ be simulated by a classical causal process. This means that quantum theory cannot violate a causal inequality, and thus one could not convince a sceptic that nature deviates from classical notions of causality. This is in sharp distinction to non-locality, where not only does the theory appear non-local (e.g. via entangled states) but we can also prove that some quantum probabilities cannot be replicated by any local hidden variable model. By violating a Bell inequality we can therefore prove non-locality experimentally. Although our framework is very general, one key requirement is that each party interacts with the system once (which leads to a requirement on the final flag state). This is the normal setup for causal inequalities, and allows us to assign a single input and output to each party, and to represent the experimental results via $p(\vec{a}\|\vec{x})$. However, it would be interesting to lift this assumption in future research. For example, could we obtain a violation of causality if parties are allowed to measure twice, or a variable number of times, or to forget they have measured? We also have a technical assumption that the protocol takes finite time (i.e. that it terminates after a finite number of steps). This seems physically reasonable, but it might be interesting to investigate lifting this assumption, as well as to consider extending the results to continuous time. Finally, it would be interesting to consider a network structure in the causal scenario, in the non-local case this is known to generate non-linear Bell inequalities, and sets of non convex probability polytopes Brunner2020 . Investigation of causal indefiniteness and causal inequalities in these type of scenarios might prove of general interest. Finally, our framework assumes standard quantum theory. If the theory changes significantly to incorporate quantum gravity we might expect new possibilities for causal inequality violation, although not necessarily Zych2019 (note that even classical general relativity allows for the existence of closed time-like curves, which appear to violate the simple classical causal models we have considered here Lobo2010 ; Barrett2020cyclic ; Arajo2017 ). We hope that the framework and tools developed here will prove helpful in discussing these interesting issues, and in highlighting differences from the standard case. T.Purves acknowledges support from the EPSRC. _Note added -_ Independently obtained related results using the process matrix formalism Wechs2021 appeared on the ArXiv on the same day as this paper. ## References * (1) Giulio Chiribella, Giacomo Mauro D’Ariano, Paolo Perinotti, and Benoit Valiron. Quantum computations without definite causal structure. Physical Review A, 88(2), 2013. * (2) Giulio Chiribella. 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Abbott and Cyril Branciard. Quantum circuits with classical versus quantum control of causal order arXiv:2101.08796. ## Appendix A Equivalence of combined and individual controlled lab gates In this section we show that the framework for quantum processes in the main text is equivalent to considering any quantum circuit built up of standard unitary gates and controlled gates for individual laboratories, in terms of the probability distributions they can generate. The key ingredient is to show how to construct individual controlled lab gates from $V$, and conversely how to construct the operation $V$ from individual controlled lab gates. To map any circuit involving individual controlled lab gates into our framework, we first space out the gates in the circuit, so that there is only one gate per time-step (this will increase the depth, but not affect the results). If an individual lab gate acts on only part of the system, we extend it such that it acts on the entire system, taking the action to be trivial (i.e. tensored with the identity) on any part of the system which was not initially included. We can then replace each individual controlled lab gate by a circuit fragment involving one use of $V$, using the approach described below. Finally we merge all unitary gates between instances of $V$ into the unitaries $U_{t}$. This will lead to a circuit in our framework yielding exactly the same results as the original circuit. To go in the other direction, we simply replace each instance of $V$ with its construction in terms of individual controlled lab gates. Figures 1 and 2 show how to construct an individual controlled lab gate using $V$. Figure 3 shows how to construct $V$ from individual controlled lab gates. @C=1.5em @R=1.2em &c_l 1 s 2V_s,r_l(x_l) r_l V_s,r_l(x_l) f_l V_s,r_l(x_l) Figure 3: A controlled lab gate for an individual laboratory @C=1.5em @R=1.2em &—0⟩_c W_l 2 W_l^† c_l -1 -1 s 2V r V f V Figure 4: An equivalent circuit to the individual controlled lab gate above, built from a single instance of $V$, where $W_{l}\ket{0}=\ket{l}$. Note that the individual wires may represent composite subsystems rather than individual qubits. @C=1.2em @R=1.2em &c 1 2 3⋯ 3 3⋯ 3 3⋯ 2 1 —0⟩ 3 —0⟩ 2 —0⟩ 1 s 2V_s,r_1(x_1) 4V_s,r_2(x_2) 7V_s,r_n(x_n) r_1 V_s,r_1(x_1) f_1 V_s,r_1(x_1) r_2 V_s,r_n(x_n) f_2 V_s,r_n(x_n) ⋯ ⋯ r_n V_s,r_n(x_n) f_n V_s,r_n(x_n) Figure 5: V, built from individual controled lab gates. The first and last $CNOT$ gate are controlled from the state $\ket{1}_{c}$, the second and second from last are controlled from $\ket{2}_{c}$, and so on until the $n$’th and $n+1$’th $CNOT$, which are controlled from state $\ket{n}_{c}$. ## Appendix B Proof of the Main Result In this appendix section, we give the full proof of the main result, that the probabilities generated by a quantum protocol can be replicated by a classical causal model, and therefore cannot violate a causal inequality. We begin with recalling a few definitions from the main text, together with some convenient derived quantities. Note that we assume throughout that laboratory labels $l$ are non-zero. ###### Definition 6 A causal probabilistic model can be written as $\displaystyle p(\vec{a}\|\vec{x})=\sum_{l_{1}\notin\mathcal{L}_{0}}\sum_{l_{2}\notin\mathcal{L}_{1}}...\sum_{l_{N}\notin\mathcal{L}_{N}-1}p_{1}(l_{1}\|H_{0})p_{1}(a_{l_{1}}\|H_{0},x_{l_{1}})p_{2}(l_{2}\|H_{1})p_{2}(a_{l_{1}}\|H_{1},x_{l_{1}})...p_{N}(l_{N}\|H_{N-1})p_{N}(a_{l_{N}}\|H_{N-1},x_{l_{N}})$ (12) where $p_{k}(l_{k}\|H_{k-1})$ terms represent probabilities for party $l_{k}$ to act at stage $k$ of the causal order, and $p_{k}(a_{l_{k}}\|H_{k-1},x_{l_{k}})$ terms represent probabilities for party $l_{k}$, who has acted at stage $k$ of the causal order to obtain measurement result $a_{l_{k}}$. Both of the above probabilities are conditional on a history, $H_{k-1}$, which contains all of the information about previous inputs, outputs and party order. In particular, the history $H_{k}=(h_{1},...,h_{k})$ is the ordered collection of triples $h_{k}=(l_{k},a_{l_{k}},x_{l_{k}})$. The summations are performed over all possible next parties, excluding parties who have already acted, which are stored in the unordered sets $\mathcal{L}_{k}=\\{l_{1},...,l_{k}\\}$. To emphasise the symmetry between the terms we include $H_{0}$ and $\mathcal{L}_{0}$, which are defined as empty sets, as no parties have acted at that point. ###### Definition 7 The state of the system with a History $H_{k-1}$, at a time given by $t$, with the control set to trigger the action of party $l_{k}$ is given by $\displaystyle\ket{\psi_{(l_{k},t,H_{k-1})}}=(\ket{l_{k}}\bra{l_{k}}_{c}\otimes\pi^{H_{k-1}}_{rf}\otimes I)U_{t}VU_{t-1}...VU_{1}\ket{0}.$ (13) The projector onto the result and flag spaces is given by $\pi^{H_{k-1}}_{rf}=\bigotimes_{i=1}^{N}\left(\pi^{H_{k-1}}_{r_{i}f_{i}}\right),$ where $\displaystyle\pi^{H_{k-1}}_{r_{i}f_{i}}=\begin{cases}\ket{a_{i}}\bra{a_{i}}_{r_{i}}\otimes\ket{1}\bra{1}_{f_{i}}&\text{ if }(i,a_{i},x_{i})\in H_{k-1},\\\ I_{r_{i}}\otimes\ket{0}\bra{0}_{f_{i}}&\text{ otherwise }.\end{cases}$ (14) We also define the same state evolved to the end of protocol to be $\displaystyle{\ket{\bar{\psi}_{(l_{k},t,H_{k-1})}}}=VU_{T}VU_{T-1}...U_{t+1}V\ket{\psi_{((l_{k},t,H_{k-1}))}}.$ (15) It will also be convenient for the proof to define $\ket{\psi_{(0,t,H_{k-1})}}$ and ${\ket{\bar{\psi}_{(0,t,H_{k-1})}}}$, which are the same as the above states, but with $l_{k}=0$ (i.e. the control in the ‘do nothing’ setting). ###### Definition 8 The state of the system with a History $H_{k}$, at a time given by $t$, in which party $l_{k}$ has just acted is given by $\displaystyle\ket{\phi_{(l_{k},t,H_{k})}}=(\ket{a_{l_{k}}}\bra{a_{l_{k}}}_{r_{k}}\otimes I)V\ket{\psi_{(l_{k},t,H_{k-1})}}.$ (16) We also define the same state evolved to the end of protocol to be $\displaystyle{\ket{\bar{\phi}_{(l_{k},t,H_{k})}}}=VU_{T}VU_{T-1}...U_{t+1}\ket{\phi_{((l_{k},t,H_{k}))}}.$ (17) It will also be convenient for the proof to define $\ket{\phi_{(0,t,H_{k})}}=V\ket{\psi_{(0,t,H_{k})}}$ and ${\ket{\bar{\phi}_{(0,t,H_{k})}}}=VU_{T}VU_{T-1}...U_{t+1}\ket{\phi_{((0,t,H_{k}))}}.$ ###### Definition 9 The probability for party $l_{k}$ to act next, given a history $H_{k-1}$ is given by: $\displaystyle p_{k}(l_{k}\|H_{k-1})=\frac{\sum_{t_{k}=1}^{T}\|\ket{\psi_{(l_{k},t_{k},H_{k-1})}}\|^{2}}{\sum_{l_{k}^{\prime}\notin\mathcal{L}_{k-1}}\sum_{t_{k}^{\prime}=1}^{T}\|\ket{\psi_{(l_{k}^{\prime},t_{k}^{\prime},H_{k-1})}}\|^{2}}$ (18) ###### Definition 10 The probability for party $l_{k}$ to obtain the measurement result $a_{l_{k}}$, given a history $H_{k-1}$, and an input variable $x_{l_{k}}$ is given by: $\displaystyle p_{k}(a_{l_{k}}\|H_{k-1},x_{l_{k}})=\frac{\sum_{t_{k}=1}^{T}\|\ket{\phi_{(l_{k},t_{k},H_{k})}}\|^{2}}{\sum_{a^{\prime}_{l_{k}}\in\mathcal{A}_{l_{k}}}\sum_{t_{k}^{\prime}=1}^{T}\|\ket{\phi_{(l_{k},t_{k}^{\prime},H_{k}^{\prime})}}\|^{2}}$ (19) where $H^{\prime}_{k}=(H_{k-1},(l_{k},a^{\prime}_{l_{k}},x_{l_{k}}))$ (i.e. $H_{k}$ with $a_{l_{k}}$ replaced by $a^{\prime}_{l_{k}}$). ###### Definition 11 The quantum protocol consists of preparing an initial state $\ket{0}$, then acting with an alternating sequence of unitaries $U_{t}$ that act on the system and the control, and unitaries $V$ that act on the system, results and flag spaces as specified by the control. The total unitary for the protocol is given by $\displaystyle\mathcal{U}=VU_{T}VU_{T-1}V...VU_{N}V...VU_{1}$ (20) where we note that for an $N$ party protocol, $T\geq N$. Finally, the results registers are measured in the computational basis, giving the outcome probability distribution $\displaystyle p^{\mathrm{quantum}}(\vec{a}\|\vec{x})=\|(\ket{\vec{a}}\bra{\vec{a}}_{r}\otimes I)\mathcal{U}\ket{0}\|^{2}.$ (21) With these definitions in place, we first prove some useful orthogonality lemmas concerning the barred states. ###### Lemma 1 We have that $\displaystyle\Braket{\bar{\psi}_{l^{\prime},t^{\prime},H}}{\bar{\psi}_{l,t,H}}=0$ (22) unless $l=l^{\prime}$ and $t^{\prime}=t$. Proof: consider first that $t=t^{\prime}$ and $l\neq l^{\prime}$. Then we have that $\braket{\bar{\psi}_{l^{\prime},t,H}}{\bar{\psi}_{l,t,H}}=\braket{{\psi}_{l^{\prime},t,H}}{{\psi}_{l,t,H}}=0$, since $\ket{{\psi}_{l^{\prime},t,H}}$ and $\ket{{\psi}_{l,t,H}}$ are orthogonal on the control $\mathcal{H}_{c}$. Next, consider that $t<t^{\prime}$. Then $\braket{\bar{\psi}_{l^{\prime},t^{\prime},H}}{\bar{\psi}_{l,t,H}}=\bra{{\psi}_{l^{\prime},t^{\prime},H}}U_{t^{\prime}}V...U_{t+1}V\ket{{\psi}_{l,t,H}}=0$ since $V\ket{{\psi}_{l,t,H}}$ contains a raised $l$ flag that is not raised in $\bra{{\psi}_{l^{\prime},t^{\prime},H}}$, and there is no operator connecting the two which can lower this flag. The case with $t>t^{\prime}$ follows from the $t<t^{\prime}$ case by noting that $\Braket{\bar{\psi}_{l^{\prime},t^{\prime},H}}{\bar{\psi}_{l,t,H}}=\Braket{\bar{\psi}_{l,t,H}}{\bar{\psi}_{l^{\prime},t^{\prime},H}}^{}$. ###### Lemma 2 We have that $\displaystyle\braket{\bar{\phi}_{l^{\prime},t^{\prime},H}}{\bar{\phi}_{l,t,H}}=0$ (23) unless $l=l^{\prime}$ and $t=t^{\prime}$. Proof: consider first that $t=t^{\prime}$ and $l\neq l^{\prime}$. Then we have that $\braket{\bar{\phi}_{l^{\prime},t,H}}{\bar{\phi}_{l,t,H}}=\braket{{\phi}_{l^{\prime},t,H}}{{\phi}_{l,t,H}}=0$, since $\ket{{\phi}_{l^{\prime},t,H}}$ and $\ket{{\phi}_{l,t,H}}$ are orthogonal on the control $\mathcal{H}_{c}$. Next, consider that $t<t^{\prime}$. Then $\braket{\bar{\phi}_{l^{\prime},t^{\prime},H}}{\bar{\phi}_{l,t,H}}=\bra{{\phi}_{l^{\prime},t^{\prime},H}}VU_{t^{\prime}}V...U_{t+1}\ket{{\phi}_{l,t,H}}=0$, since the leftmost $V$ either raises a flag not in the history $H$, or the control at this point is set to zero, either of which will give the desired orthogonality. The case with $t>t^{\prime}$ follows from the $t<t^{\prime}$ case by noting that $\Braket{\bar{\phi}_{l^{\prime},t^{\prime},H}}{\bar{\phi}_{l,t,H}}=\Braket{\bar{\phi}_{l,t,H}}{\bar{\phi}_{l^{\prime},t^{\prime},H}}^{}$. We now move onto proving the main result. This will consist of four stages, the first concerns a cancellation within terms of the same causal order stage, which allows us to rewrite the causal model in a nice way. The second and third results concern the initial and final terms in the inductive proof. The former corresponds to the fact that ‘somebody has to measure first’ in the quantum protocol, and the latter that the final term in the causal model has sufficient expressive power to capture the quantum measurement probabilities in their entirety. Finally, the fourth result concerns cancellations between terms at subsequent stages of the causal order. This leads to our main result which ties all of this together for a full proof that $p(\vec{a}\|\vec{x})=\|\bra{0}\mathcal{U}\ket{0}\|^{2}$ is causal. ###### Result 1 There is an equality between the numerator of the ‘who is next’ type probabilities $p_{k}(l_{k}\|H_{k-1})$, and the denominator of the ‘results’ type probabilities $p_{k}(a_{l_{k}}\|H_{k-1},x_{l_{k}})$, allowing us to write the product of these probabilities in a nice way as $\displaystyle p_{k}(l_{k}\|H_{k-1})p_{k}(a_{l_{k}}\|x_{l_{k}},H_{k-1})=\frac{\sum_{t_{k}=1}^{T}\|\ket{\phi_{(l_{k},t_{k},H_{k})}}\|^{2}}{\sum_{l^{\prime}_{k}\notin\mathcal{L}_{k-1}}\sum_{t^{\prime}_{k}=1}^{T}\|\ket{\psi_{(l^{\prime}_{k},t_{k}^{\prime},H_{k-1})}}\|^{2}}.$ (24) . Proof: Starting with the denominator of the ‘results’ probability $\displaystyle\sum_{a^{\prime}_{l_{k}}\in\mathcal{A}_{l_{k}}}\sum_{t_{k}=1}^{T}\left\|\ket{\phi_{(l_{k},t_{k},H^{\prime}_{k})}}\right\|^{2}$ $\displaystyle=\sum_{a^{\prime}_{l_{k}}\in\mathcal{A}_{l_{k}}}\sum_{t_{k}=1}^{T}\left\|(\ket{a^{\prime}_{l_{k}}}\bra{a^{\prime}_{l_{k}}}_{r_{k}}\otimes\mathcal{I})V\ket{\psi_{(l_{k},t_{k},H_{k-1})}}\right\|^{2}$ $\displaystyle=\sum_{t_{k}=1}^{T}\left\|\sum_{a^{\prime}_{l_{k}}\in\mathcal{A}_{l_{k}}}(\ket{a^{\prime}_{l_{k}}}\bra{a^{\prime}_{l_{k}}}_{r_{k}}\otimes\mathcal{I})V\ket{\psi_{(l_{k},t_{k},H_{k-1})}}\right\|^{2}.$ $\displaystyle=\sum_{t_{k}=1}^{T}\|V\ket{\psi_{(l_{k},t_{k},H_{k-1})}}\|^{2}$ (25) $\displaystyle=\sum_{t_{k}=1}^{T}\|\ket{\psi_{(l_{k},t_{k},H_{k-1})}}\|^{2},$ (26) we obtain the numerator of the ‘who is next’ probabilities. In the second line we have used orthogonality on the results register, in the third line we have used the fact that after a measurement by party $l_{k}$, some result in $\mathcal{A}_{l_{k}}$ must have been obtained, and in the final line we have used unitarity. Using this to cancel the numerator of (18) with the denominator of (19) we obtain the desired result. ###### Result 2 The denominator of the first term $p_{1}(l_{1}\|H_{0})$ satisfies $\displaystyle\sum_{l_{1}\notin\mathcal{L}_{0}}\sum_{t_{1}=1}^{T}\|\ket{\psi_{(l_{1},t_{1},H_{0})}}\|^{2}=1.$ (27) Proof: by first using unitarity and then Lemma 1 we have $\displaystyle\sum_{l_{1}\notin\mathcal{L}_{0}}\sum_{t_{1}=1}^{T}\|\ket{\psi_{(l_{1},t_{1},H_{0})}}\|^{2}=\sum_{l_{1}\notin\mathcal{L}_{0}}\sum_{t_{1}=1}^{T}\|\ket{\bar{\psi}_{(l_{1},t_{1},H_{0}})}\|^{2}$ $\displaystyle=\|\sum_{l_{1}\notin\mathcal{L}_{0}}\sum_{t_{1}=1}^{T}\ket{\bar{\psi}_{(l_{1},t_{1},H_{0}})}\|^{2}.$ (28) To simplify this further, consider evolving the state $\ket{\psi_{0,t_{1}-1,H_{0}}}$ forward for a full time-step using $U_{t_{1}}V$. As the control is in state $0$, no measurement occurs during $V$, and the unitary $U_{1}$ creates a superposition in which the control takes any possible state. Symbolically, $\displaystyle U_{t_{1}}V\ket{\psi_{0,t_{1}-1,H_{0}}}=\sum_{l_{1}\notin\mathcal{L}_{0}}\ket{\psi_{(l_{1},t_{1},H_{0}})}+\ket{\psi_{(0,t_{1},H_{0}})}.$ (29) By applying $VU_{T}V\ldots U_{t_{1}+1}V$ to this equation, we can obtain a similar form for the barred states, $\displaystyle\ket{\bar{\psi}_{0,t_{1}-1,H_{0}}}=\sum_{l_{1}\notin\mathcal{L}_{0}}\ket{\bar{\psi}_{(l_{1},t_{1},H_{0}})}+\ket{\bar{\psi}_{(0,t_{1},H_{0}})}.$ (30) We can rearrange this equation and substitute for the sum over $l_{1}$ on the right-hand side of (B) to obtain $\displaystyle\sum_{l_{1}\notin\mathcal{L}_{0}}\sum_{t_{1}=1}^{T}\|\ket{\psi_{(l_{1},t_{1},H_{0})}}\|^{2}=\|\sum_{t_{1}=1}^{T}\left(\ket{\bar{\psi}_{(0,t_{1}-1,H_{0}})}-\ket{\bar{\psi}_{(0,t_{1},H_{0})}}\right)\|^{2}$ (31) By expanding the summation on the right-hand side we find that only the first and last terms remain, giving $\displaystyle\sum_{l_{1}\notin\mathcal{L}_{0}}\sum_{t_{1}=1}^{T}\|\ket{\psi_{(l_{1},t_{1},H_{0})}}\|^{2}=\|\ket{\bar{\psi}_{(0,0,H_{0}})}-\ket{\bar{\psi}_{(0,T,H_{0}})}\|^{2}$ (32) Note that it is impossible by the requirements of our protocol that no-one has measured by time $t=T$. Such a scenario would violate the assumption that there are exactly $N$ flags raised at the end of the protocol. Therefore, $\ket{\bar{\psi}_{(0,T,H_{0}})}=0$, and we find $\displaystyle\sum_{l_{1}\notin\mathcal{L}_{0}}\sum_{t_{1}=1}^{T}\|\ket{\psi_{(l_{1},t_{1},H_{0})}}\|^{2}=\|\ket{\bar{\psi}_{(0,0,H_{0}})}\|^{2}=\|\mathcal{U}\ket{0}\|^{2}=1$ (33) as desired. ###### Result 3 The outcome statistics in the numerator of the final term in the causal probabilistic model represent the quantum probabilities arising from the protocol. In other words, $\displaystyle\sum_{l_{N}\in\mathcal{L}_{N}}\sum_{t_{N}=1}^{T}\|\ket{\phi_{(l_{N},t_{N},H_{N})}}\|^{2}=\|\left(\ket{\vec{a}}\bra{\vec{a}}\otimes I\right)\mathcal{U}\ket{0}\|^{2}$ (34) Proof: Firstly, note that by unitarity and Lemma 2 we have that $\displaystyle\sum_{l_{N}\in\mathcal{L}_{N}}\sum_{t_{N}=1}^{T}\|\ket{\phi_{(l_{N},t_{N},H_{N})}}\|^{2}$ $\displaystyle=\sum_{l_{N}\in\mathcal{L}_{N}}\sum_{t_{N}=1}^{T}\|\ket{\bar{\phi}_{(l_{N},t_{N},H_{N})}}\|^{2}$ $\displaystyle=\|\sum_{l_{N}\in\mathcal{L}_{N}}\sum_{t_{N}=1}^{T}\ket{\bar{\phi}_{(l_{N},t_{N},H_{N})}}\|^{2}.$ (35) The history $H_{N}$ represents a case in which all parties have already measured. At time $t<T$, what are the possible ways that this history can be filled? Either, nobody has measured in the previous time-step, or the last party to be filled into the history (subject to the requirement every party must enter the history exactly once) has just measured. In any case, evolving the linear combination of these states forward a time-step must produce a state at $t+1$ which contains an empty control (i.e., no-one else is left to measure, so don’t trigger them!). This gives us the key relation $\displaystyle VU_{t+1}\left(\sum_{l_{N}\in\mathcal{L}_{N}}\ket{\phi_{(l_{N},t,H_{N}})}+\ket{\phi_{(0,t,H_{N})}}\right)=\ket{\phi_{(0,t+1,H_{N}})}$ (36) which holds for $t<T$. Applying $VU_{T}V\ldots U_{t+2}$ we obtain $\displaystyle\left(\sum_{l_{N}\in\mathcal{L}_{N}}\ket{\bar{\phi}_{(l_{N},t,H_{N}})}+\ket{\bar{\phi}_{(0,t,H_{N})}}\right)=\ket{\bar{\phi}_{(0,t+1,H_{N}})}$ (37) which we can then rearrange to get $\displaystyle\sum_{l_{N}\in\mathcal{L}_{N}}\ket{\bar{\phi}_{(l_{N},t,H_{N})}}=\ket{\bar{\phi}_{(0,t+1,H_{N})}}-\ket{\bar{\phi}_{(0,t,H_{N})}}.$ (38) By separating out the $t_{N}=T$ term in equation (B) and then substituting this in the remaining terms , we find that $\displaystyle\sum_{l_{N}\in\mathcal{L}_{N}}\sum_{t_{N}=1}^{T}\|\ket{\phi_{(l_{N},t_{N},H_{N})}}\|^{2}$ $\displaystyle=\|\sum_{l_{N}\in\mathcal{L_{N}}}\ket{\bar{\phi}_{(l_{N},T,H_{N})}}+\sum_{t_{N}=1}^{T-1}\sum_{l_{N}\in\mathcal{L_{N}}}\ket{\bar{\phi}_{(l_{N},t_{N},H_{N})}}\|^{2}$ $\displaystyle=\|\sum_{l_{N}\in\mathcal{L_{N}}}\ket{\bar{\phi}_{(l_{N},T,H_{N})}}+\sum_{t_{N}=1}^{T-1}\left(\ket{\bar{\phi}_{(0,t_{N}+1,H_{N})}}-\ket{\bar{\phi}_{(0,t_{N},H_{N})}}\right)\|^{2}$ $\displaystyle=\|\sum_{l_{N}\in\mathcal{L_{N}}}\ket{\bar{\phi}_{(l_{N},T,H_{N})}}+\ket{\bar{\phi}_{(0,T,H_{N})}}-\ket{\bar{\phi}_{(0,1,H_{N})}}\|^{2}.$ (39) Now we note that $\ket{\bar{\phi}_{(0,1,H_{N})}}=0$ since it would be impossible for all parties to have measured in one time-step, and for the control to be in the zero state. Then we note that $\sum_{l_{N}\in\mathcal{L_{N}}}\ket{\bar{\phi}_{(l_{N},T,H_{N})}}+\ket{\bar{\phi}_{(0,T,H_{N})}}=(\pi^{H_{N}}_{rf}\otimes I)\mathcal{U}\ket{0}$, which is to say that these are simply the possible states at the end of the protocol, containing the measurement results we want to calculate the probabilities for in the history. Therefore $\displaystyle\sum_{l_{N}\in\mathcal{L}_{N}}\sum_{t_{N}=1}^{T}\|\ket{\phi_{(l_{N},t_{N},H_{N})}}\|^{2}$ $\displaystyle=\|(\pi^{H_{N}}_{rf}\otimes I)\mathcal{U}\ket{0}\|^{2}$ $\displaystyle=\|(\ket{\vec{a}}\bra{\vec{a}}\otimes I)\mathcal{U}\ket{0}\|^{2}$ (40) as desired. ###### Result 4 This is a technical result which establishes an equality between the states after measurement at causal order stage $k$ and the states before measurement at the next stage of the causal order. Namely, for $1\leq k<N$ that: $\displaystyle\sum_{l_{k}\in\mathcal{L}_{k}}\sum_{t=1}^{T}\|\ket{\phi_{(l_{k},t,H_{k})}}\|^{2}=\sum_{t^{\prime}=1}^{T}\sum_{l^{\prime}_{k+1}\notin\mathcal{L}_{k}}\|\ket{{\psi}_{(l^{\prime}_{k+1},t^{\prime},H_{k})}}\|^{2}.$ (41) Proof: Firstly, by unitarity and Lemma 2 we have that $\displaystyle\sum_{l_{k}\in\mathcal{L}_{k}}\sum_{t=1}^{T}\|\ket{\phi_{(l_{k},t,H_{k})}}\|^{2}$ $\displaystyle=\sum_{l_{k}\in\mathcal{L}_{k}}\sum_{t=1}^{T}\|\ket{\bar{\phi}_{(l_{k},t,H_{k})}}\|^{2}$ $\displaystyle=\|\sum_{l_{k}\in\mathcal{L}_{k}}\sum_{t=1}^{T}\ket{\bar{\phi}_{(l_{k},t,H_{k})}}\|^{2}.$ (42) Consider time-evolving a state just after the $t^{\textrm{th}}$ measurement step, in which history $H_{k}$ has been obtained (either by the last party just having measured, or by all parties in $H_{k}$ having measured previously), by $U_{t+1}$. This links states of the form $\ket{\phi_{(l_{k},t,H_{k})}}$ and $\ket{\psi_{(l^{\prime}_{k},t+1,H_{k})}}$ via $\displaystyle U_{t+1}\left(\sum_{l_{k}\in\mathcal{L}_{k}}\ket{\phi_{(l_{k},t,H_{k})}}+\ket{\phi_{(0,t,H_{k})}}\right)=\sum_{l^{\prime}_{k+1}\notin\mathcal{L}_{k}}\ket{\psi_{(l^{\prime}_{k+1},t+1,H_{k})}}+\ket{\psi_{(0,t+1,H_{k})}}$ (43) for $t<T$. Applying $VU_{T}V\ldots U_{t+2}V$ we obtain a very similar result for barred states; $\displaystyle\sum_{l_{k}\in\mathcal{L}_{k}}\ket{\bar{\phi}_{(l_{k},t,H_{k})}}+\ket{\bar{\phi}_{(0,t,H_{k})}}=\sum_{l^{\prime}_{k+1}\notin\mathcal{L}_{k}}\ket{\bar{\psi}_{(l^{\prime}_{k+1},t+1,H_{k})}}+\ket{\bar{\psi}_{(0,t+1,H_{k})}}.$ (44) We also note that $\ket{\phi_{(0,t,H_{k})}}=V\ket{\psi_{(0,t,H_{k})}}$ and hence that $\ket{\bar{\phi}_{(0,t,H_{k})}}=\ket{\bar{\psi}_{(0,t,H_{k})}}$. Making this substitution and rearranging a little we get $\displaystyle\sum_{l_{k}\in\mathcal{L}_{k}}\ket{\bar{\phi}_{(l_{k},t,H_{k})}}=\sum_{l^{\prime}_{k+1}\notin\mathcal{L}_{k}}\ket{\bar{\psi}_{(l^{\prime}_{k+1},t+1,H_{k})}}+\ket{\bar{\psi}_{(0,t+1,H_{k})}}-\ket{\bar{\psi}_{(0,t,H_{k})}}.$ (45) By substituting (45) into (B) for $t<T$, we arrive at $\displaystyle\sum_{l_{k}\in\mathcal{L}_{k}}\sum_{t=1}^{T}\left\|\ket{\phi_{(l_{k},t,H_{k})}}\right\|^{2}$ $\displaystyle=\left\|\sum_{l_{k}\in\mathcal{L}_{k}}\ket{\bar{\phi}_{(l_{k},T,H_{k})}}+\sum_{t=1}^{T-1}\left(\sum_{l^{\prime}_{k+1}\notin\mathcal{L}_{k}}\ket{\bar{\psi}_{(l^{\prime}_{k+1},t+1,H_{k})}}+\ket{\bar{\psi}_{(0,t+1,H_{k})}}-\ket{\bar{\psi}_{(0,t,H_{k})}}\right)\right\|^{2}$ $\displaystyle=\left\|\sum_{l_{k}\in\mathcal{L}_{k}}\ket{\bar{\phi}_{(l_{k},T,H_{k})}}+\sum_{t=1}^{T-1}\sum_{l^{\prime}_{k+1}\notin\mathcal{L}_{k}}\ket{\bar{\psi}_{(l^{\prime}_{k+1},t+1,H_{k})}}+\ket{\bar{\psi}_{(0,T,H_{k})}}-\ket{\bar{\psi}_{(0,1,H_{k})}}\right\|^{2}$ (46) where for the sums over time in the last two terms only the states with maximal and minimal times remain. Given that $k<N$ and all parties must have measured by the end of the protocol, it must be the case that $\ket{\bar{\phi}_{(l_{k},T,H_{k})}}=0$ and $\ket{\bar{\psi}_{(0,T,H_{k})}}=0$. Also as $k\geq 1$ it must be the case that $\ket{\bar{\psi}_{(0,1,H_{k})}}=0$ and $\ket{\bar{\psi}_{(l^{\prime}_{k+1},1,H_{k})}}=0$, as these states are just before the first measurement and hence must have no history. Using these results in equation (46) and setting $t^{\prime}=t+1$, we obtain $\displaystyle\sum_{l_{k}\in\mathcal{L}_{k}}\sum_{t=1}^{T}\|\ket{\phi_{(l_{k},t,H_{k})}}\|^{2}=\left\|\sum_{t^{\prime}=1}^{T}\sum_{l^{\prime}_{k+1}\notin\mathcal{L}_{k}}\ket{\bar{\psi}_{(l^{\prime}_{k+1},t^{\prime},H_{k})}}\right\|^{2}.$ (47) Finally, using Lemma 1 and unitarity we arrive at $\displaystyle\sum_{l_{k}\in\mathcal{L}_{k}}\sum_{t=1}^{T}\|\ket{\phi_{(l_{k},t,H_{k})}}\|^{2}$ $\displaystyle=\sum_{t^{\prime}=1}^{T}\sum_{l^{\prime}_{k+1}\notin\mathcal{L}_{k}}\left\|\ket{\bar{\psi}_{(l^{\prime}_{k+1},t^{\prime},H_{k})}}\right\|^{2}$ $\displaystyle=\sum_{t^{\prime}=1}^{T}\sum_{l^{\prime}_{k+1}\notin\mathcal{L}_{k}}\left\|\ket{{\psi}_{(l^{\prime}_{k+1},t^{\prime},H_{k})}}\right\|^{2}$ (48) as required. ###### Result 5 We will now show that the results of the quantum protocol can be replicated by a causal process. In particular $\displaystyle p(\vec{a}\|\vec{x})=\|\left(\ket{\vec{a}}\bra{\vec{a}}\otimes I\right)\mathcal{U}\ket{0}\|^{2}$ $\displaystyle=$ $\displaystyle\sum_{l_{1}\notin\mathcal{L}_{0}}\sum_{l_{2}\notin\mathcal{L}_{1}}$ $\displaystyle...\sum_{l_{N}\notin\mathcal{L}_{N}-1}p_{1}(l_{1}\|H_{0})p_{1}(a_{l_{1}}\|H_{0},x_{l_{1}})p_{2}(l_{2}\|H_{1})p_{2}(a_{l_{1}}\|H_{1},x_{l_{1}})...p_{N}(l_{N}\|H_{N-1})p_{N}(a_{l_{N}}\|H_{N-1},x_{l_{N}})$ (49) and as such, the outcome statistics $p(\vec{a}\|\vec{x})$ cannot violate a causal inequality. Proof: Firstly, substituting definitions 9 and 10 into the causal model (6), and then using Result 1, we can re-write the probability distribution for the entire causal model as $\displaystyle p(\vec{a}\|\vec{x})=\sum_{l_{1}\notin\mathcal{L}_{0}}\sum_{l_{2}\notin\mathcal{L}_{1}}...\sum_{l_{N}\notin\mathcal{L}_{N}-1}\frac{\sum_{t_{1}=1}^{T}\|\ket{\phi_{(l_{1},t_{1},H_{1})}}\|^{2}}{\sum_{l_{1}^{\prime}\notin\mathcal{L}_{0}}\sum_{t_{1}^{\prime}=1}^{T}\|\ket{\psi_{(l_{1}^{\prime},t_{1}^{\prime},H_{0})}}\|^{2}}$ $\displaystyle\frac{\sum_{t_{2}=1}^{T}\|\ket{\phi_{(l_{2},t_{2},H_{2})}}\|^{2}}{\sum_{l_{2}^{\prime}\notin\mathcal{L}_{1}}\sum_{t_{2}^{\prime}=1}^{T}\|\ket{\psi_{(l_{2}^{\prime},t_{2}^{\prime},H_{1})}}\|^{2}}...$ $\displaystyle...$ $\displaystyle\frac{\sum_{t_{N}=1}^{T}\|\ket{\phi_{(l_{N},t_{N},H_{N})}}\|^{2}}{\sum_{l_{N}^{\prime}\notin\mathcal{L}_{N-1}}\sum_{t_{N}^{\prime}=1}^{T}\|\ket{\psi_{(l_{N}^{\prime},t_{N}^{\prime},H_{N-1})}}\|^{2}}$ (50) Let us begin by performing a simple reshuffling of (B)’s numerators and denominators, by writing the denominator of the term associated to causal order stage $k$ as the denominator of the term associated to $k-1$. $\displaystyle p(\vec{a}\|\vec{x})=\frac{1}{\sum_{l_{1}^{\prime}\notin\mathcal{L}_{0}}\sum_{t_{1}^{\prime}=1}^{T}\|\ket{\psi_{(l_{1}^{\prime},t_{1}^{\prime},H_{0})}}\|^{2}}\sum_{l_{1}\notin\mathcal{L}_{0}}\sum_{l_{2}\notin\mathcal{L}_{1}}...\sum_{l_{N}\notin\mathcal{L}_{N}-1}$ $\displaystyle\frac{\sum_{t_{1}=1}^{T}\|\ket{\phi_{(l_{1},t_{1},H_{1})}}\|^{2}}{\sum_{l_{2}^{\prime}\notin\mathcal{L}_{1}}\sum_{t_{2}^{\prime}=1}^{T}\|\ket{\psi_{(l_{2}^{\prime},t_{2}^{\prime},H_{1})}}\|^{2}}\frac{\sum_{t_{2}=1}^{T}\|\ket{\phi_{(l_{2},t_{2},H_{2})}}\|^{2}}{\sum_{l_{3}^{\prime}\notin\mathcal{L}_{2}}\sum_{t_{3}^{\prime}=1}^{T}\|\ket{\psi_{(l_{3}^{\prime},t_{3}^{\prime},H_{2})}}\|^{2}}...$ $\displaystyle...\frac{\sum_{t_{k-1}=1}^{T}\|\ket{\phi_{(l_{k-1},t_{k-1},H_{k-1})}}\|^{2}}{\sum_{l_{k}^{\prime}\notin\mathcal{L}_{k-1}}\sum_{t_{k}^{\prime}=1}^{T}\|\ket{\psi_{(l_{k}^{\prime},t_{k}^{\prime},H_{k-1})}}\|^{2}}...\sum_{t_{N}=1}^{T}\|\ket{\phi_{(l_{N},t_{N},H_{N})}}\|^{2}$ (51) by using Result 2 we have $\displaystyle p(\vec{a}\|\vec{x})$ $\displaystyle=\sum_{l_{1}\notin\mathcal{L}_{0}}\sum_{l_{2}\notin\mathcal{L}_{1}}...\sum_{l_{N}\notin\mathcal{L}_{N}-1}\frac{\sum_{t_{1}=1}^{T}\|\ket{\phi_{(l_{1},t_{1},H_{1})}}\|^{2}}{\sum_{l_{2}^{\prime}\notin\mathcal{L}_{1}}\sum_{t_{2}^{\prime}=1}^{T}\|\ket{\psi_{(l_{2}^{\prime},t_{2}^{\prime},H_{1})}}\|^{2}}\frac{\sum_{t_{2}=1}^{T}\|\ket{\phi_{(l_{2},t_{2},H_{2})}}\|^{2}}{\sum_{l_{3}^{\prime}\notin\mathcal{L}_{2}}\sum_{t_{3}^{\prime}=1}^{T}\|\ket{\psi_{(l_{3}^{\prime},t_{3}^{\prime},H_{2})}}\|^{2}}...\sum_{t_{N}=1}^{T}\|\ket{\phi_{(l_{N},t_{N},H_{N})}}\|^{2}$ (52) now note that we can rewrite the leftmost sum as $\displaystyle\sum_{l_{1}\notin\mathcal{L}_{0}}=\sum_{\mathcal{L}_{1}}\sum_{l_{1}\in\mathcal{L}_{1}},$ (53) where the sum over $\mathcal{L}_{1}$ is over all singleton sets $\\{l_{1}\\}$ (and hence has $N$ terms), and the subsequent sum over ${l_{1}\in\mathcal{L}_{1}}$ contains just a single term. We can use this to rewrite the probability distribution as $\displaystyle p(\vec{a}\|\vec{x})$ $\displaystyle=\sum_{\mathcal{L}_{1}}\frac{\sum_{l_{1}\in\mathcal{L}_{1}}\sum_{t_{1}=1}^{T}\|\ket{\phi_{(l_{1},t_{1},H_{1})}}\|^{2}}{\sum_{l_{2}^{\prime}\notin\mathcal{L}_{1}}\sum_{t_{2}^{\prime}=1}^{T}\|\ket{\psi_{(l_{2}^{\prime},t_{2}^{\prime},H_{1})}}\|^{2}}\sum_{l_{2}\notin\mathcal{L}_{1}}...\sum_{l_{N}\notin\mathcal{L}_{N}-1}\frac{\sum_{t_{2}=1}^{T}\|\ket{\phi_{(l_{2},t_{2},H_{2})}}\|^{2}}{\sum_{l_{3}^{\prime}\notin\mathcal{L}_{2}}\sum_{t_{3}^{\prime}=1}^{T}\|\ket{\psi_{(l_{3}^{\prime},t_{3}^{\prime},H_{2})}}\|^{2}}...\sum_{t_{N}=1}^{T}\|\ket{\phi_{(l_{N},t_{N},H_{N})}}\|^{2}$ (54) where we have used the fact that the first numerator and denominator do not depend on $\\{l_{2},\ldots l_{N}\\}$. By application of Result 4 this is just $\displaystyle p(\vec{a}\|\vec{x})$ $\displaystyle=\sum_{\mathcal{L}_{1}}\sum_{l_{2}\notin\mathcal{L}_{1}}...\sum_{l_{N}\notin\mathcal{L}_{N}-1}\frac{\sum_{t_{2}=1}^{T}\|\ket{\phi_{(l_{2},t_{2},H_{2})}}\|^{2}}{\sum_{l_{3}^{\prime}\notin\mathcal{L}_{2}}\sum_{t_{3}^{\prime}=1}^{T}\|\ket{\psi_{(l_{3}^{\prime},t_{3}^{\prime},H_{2})}}\|^{2}}...\sum_{t_{N}=1}^{T}\|\ket{\phi_{(l_{N},t_{N},H_{N})}}\|^{2}$ (55) we can iterate the same process again using $\displaystyle\sum_{\mathcal{L}_{1}}\sum_{l_{2}\notin\mathcal{L}_{1}}=\sum_{\mathcal{L}_{2}}\sum_{l_{2}\in\mathcal{L}_{2}}.$ (56) The left-hand side corresponds to first picking $l_{1}$ (with $N$ possibilities) and then picking a different $l_{2}$ ($N-1$ possibilities), whereas the right-hand side corresponds to first picking a pair of distinct labs $\mathcal{L}_{2}$ (with $N(N-1)/2$ possibilities) and then picking which of them was last ($2$ possibilities). This gives $\displaystyle p(\vec{a}\|\vec{x})$ $\displaystyle=\sum_{\mathcal{L}_{2}}\sum_{l_{2}\in\mathcal{L}_{2}}...\sum_{l_{N}\notin\mathcal{L}_{N}-1}\frac{\sum_{t_{2}=1}^{T}\|\ket{\phi_{(l_{2},t_{2},H_{2})}}\|^{2}}{\sum_{l_{3}^{\prime}\notin\mathcal{L}_{2}}\sum_{t_{3}^{\prime}=1}^{T}\|\ket{\psi_{(l_{3}^{\prime},t_{3}^{\prime},H_{2})}}\|^{2}}...\sum_{t_{N}=1}^{T}\|\ket{\phi_{(l_{N},t_{N},H_{N})}}\|^{2}$ (57) which is just $\displaystyle p(\vec{a}\|\vec{x})$ $\displaystyle=\sum_{\mathcal{L}_{2}}\frac{\sum_{l_{2}\in\mathcal{L}_{2}}\sum_{t_{2}=1}^{T}\|\ket{\phi_{(l_{2},t_{2},H_{2})}}\|^{2}}{\sum_{l_{3}^{\prime}\notin\mathcal{L}_{2}}\sum_{t_{3}^{\prime}=1}^{T}\|\ket{\psi_{(l_{3}^{\prime},t_{3}^{\prime},H_{2})}}\|^{2}}\sum_{l_{3}\notin\mathcal{L}_{2}}...\sum_{l_{N}\notin\mathcal{L}_{N}-1}\frac{\sum_{t_{3}=1}^{T}\|\ket{\phi_{l_{3},t_{3},H_{3}}}\|^{2}}{\sum_{l_{4}^{\prime}\notin\mathcal{L}_{3}}\sum_{t_{4}^{\prime}=1}^{T}\|\ket{\psi_{(l_{4}^{\prime},t_{4}^{\prime},H_{3})}}\|^{2}}...\sum_{t_{N}=1}^{T}\|\ket{\phi_{(l_{N},t_{N},H_{N})}}\|^{2}$ (58) application of Result 4 leads to another cancellation, so that we may write now $\displaystyle p(\vec{a}\|\vec{x})$ $\displaystyle=\sum_{\mathcal{L}_{2}}\sum_{l_{3}\notin\mathcal{L}_{2}}...\sum_{l_{N}\notin\mathcal{L}_{N}-1}\frac{\sum_{t_{3}=1}^{T}\|\ket{\phi_{l_{3},t_{3},H_{3}}}\|^{2}}{\sum_{l_{4}^{\prime}\notin\mathcal{L}_{3}}\sum_{t_{4}^{\prime}=1}^{T}\|\ket{\psi_{(l_{4}^{\prime},t_{4}^{\prime},H_{3})}}\|^{2}}...\sum_{t_{N}=1}^{T}\|\ket{\phi_{(l_{N},t_{N},H_{N})}}\|^{2}$ (59) We can then iterate this process by applying the general result that $\displaystyle\sum_{\mathcal{L}_{k}}\sum_{l_{k+1}\notin\mathcal{L}_{k}}=\sum_{\mathcal{L}_{k+1}}\;\sum_{l_{k+1}\in\mathcal{L}_{k+1}},$ (60) and cancelling one of the numerators and denominators using Result 4 until we are left with the final term, $\displaystyle p(\vec{a}\|\vec{x})$ $\displaystyle=\sum_{\mathcal{L}_{N}}\sum_{l_{n}\in\mathcal{L}_{N}}\sum_{t_{N}=1}^{T}\|\ket{\phi_{(l_{N},t_{N},H_{N})}}\|^{2}$ (61) the summation $\sum_{\mathcal{L}_{N}}=1$, as the only term corresponds to $\mathcal{L}_{N}=\\{1,2,\ldots N\\}$. Application of Result 3 then shows that this causal model indeed reproduces the quantum probabilities.i.e. that $\displaystyle p(\vec{a}\|\vec{x})=\|\left(\ket{\vec{a}}\bra{\vec{a}}\otimes I\right)\mathcal{U}\ket{0}\|^{2}$ (62) as desired. ## Appendix C Example We now give an example of how our results apply in practice, based on the quantum switch Chiribella2013 . This involves using a quantum control to determine the order in which two operations are applied to another quantum system. The switch can be modelled in number of different ways (e.g. as a process matrix that exhibits causal non-separability Branciard2016 ) but here the basic idea is to prepare a superposition state of the form $\frac{1}{\sqrt{2}}\left(\ket{1}_{c}\otimes U_{A}U_{B}\ket{0}_{sfr}+\ket{2}_{c}\otimes U_{B}U_{A}\ket{0}_{sfr}\right)$ (63) where $U_{A}$ and $U_{B}$ are unitary transformations by Alice and Bob (representing their measurements). If a third party, Charlie, measures the control in a basis consisting of superpositions of $\ket{1}$ and $\ket{2}$ this will introduce interference between the two causal orders in which either Alice or Bob goes first. It has already been shown that this simple setup cannot be used to violate a causal inequality Arajo2015 ; Oreshkov2012 . However, it is instructive to see how it fits into our framework. @C=1.5em @R=1.2em &—0⟩_c U_1 1 U_2 1 2U_3 1 —0⟩_s_1 3V 3V U_3 3V —0⟩_s_2 V V U_3 V —0⟩_r V V V —0⟩_f V V V Figure 6: Realisation of the quantum switch in our framework through a quantum circuit. Because parties cannot directly measure the control in our framework, we transfer the state of the control into the system before Charlie’s measurement, and split the system into two qubits to facilitate this. Overall, the circuit we consider is shown in figure 6, where $\displaystyle U_{1}\ket{0}_{c}$ $\displaystyle=\frac{1}{\sqrt{2}}\left(\ket{1}_{c}+\ket{2}_{c}\right),$ $\displaystyle U_{2}\ket{1}_{c}$ $\displaystyle=\ket{2}_{c}$ $\displaystyle U_{2}\ket{2}_{c}$ $\displaystyle=\ket{1}_{c}$ $\displaystyle U_{3}\ket{1}_{c}\ket{\psi}_{s_{1}}\ket{0}_{s_{2}}$ $\displaystyle=\ket{3}_{c}\ket{0}_{s_{1}}\ket{\psi}_{s_{2}},$ $\displaystyle U_{3}\ket{2}_{c}\ket{\psi}_{s_{1}}\ket{0}_{s_{2}}$ $\displaystyle=\ket{3}_{c}\ket{1}_{s_{1}}\ket{\psi}_{s_{2}},$ (64) and $V$ is as given in equation 4 of the main body text. By considering the outcome statistics generated by this switch setup, we find that they differ from those which would be obtained from an equal mixture of the causal orders $A\rightarrow B\rightarrow C$ and $B\rightarrow A\rightarrow C$, due to the presence of interference. We proceed with an explicit calculation for the setup in figure 6. We assume that all parties have two possible measurements, hence their input variables $x,y,z$ are bits. When their input bit is zero, they measure the first part of the system in the computational basis and output the result in $a,b,c$. When their input bit is one, they instead measure the first part of the system in the Fourier basis (composed of the states $\ket{\pm}=\frac{1}{\sqrt{2}}\left(\ket{0}\pm\ket{1}\right)$, and output zero if they obtain the state $\ket{+}$ and one if they obtain the state $\ket{-}$. All of these measurements are implemented via unitary operations between the system and result register. e.g. $\displaystyle V_{s_{1},r_{1}}(x_{1}=1)$ $\displaystyle=\ket{+}\bra{+}_{s_{1}}\otimes I_{r_{1}}+\ket{-}\bra{-}_{s_{1}}\otimes\left(\ket{0}\bra{1}_{r_{1}}+\ket{1}\bra{0}_{r_{1}}\right)$ (65) Consider the case where the input variables are $x=0,y=1,z=1$. After some calculation, we find the state at the end of the protocol to be $\displaystyle\ket{3}_{c}\bigg{(}$ $\displaystyle\ket{+}_{s_{1}}(\frac{1}{2\sqrt{2}}\ket{+}_{s_{2}}+\frac{1}{4}\ket{0}_{s_{2}})\ket{000}_{r}+\ket{-}_{s_{1}}(\frac{1}{4}\ket{0}_{s_{2}}-\frac{1}{2\sqrt{2}}\ket{+}_{s_{2}})\ket{001}_{r}$ $\displaystyle+$ $\displaystyle\ket{+}_{s_{1}}(\frac{1}{2\sqrt{2}}\ket{-}_{s_{2}}+\frac{1}{4}\ket{0}_{s_{2}})\ket{010}_{r}+\ket{-}_{s_{1}}(\frac{1}{4}\ket{0}_{s_{2}}-\frac{1}{2\sqrt{2}}\ket{-}_{s_{2}})\ket{011}_{r}$ $\displaystyle+$ $\displaystyle\frac{1}{4}\ket{+}_{s_{1}}\ket{1}_{s_{2}}\ket{100}_{r}+\frac{1}{4}\ket{-}_{s_{1}}\ket{1}_{s_{2}}\ket{101}_{r}-\frac{1}{4}\ket{+}_{s_{1}}\ket{1}_{s_{2}}\ket{110}_{r}-\frac{1}{4}\ket{-}_{s_{1}}\ket{1}_{s_{2}}\ket{111}_{r}\bigg{)}\ket{111}_{f},$ (66) where we adopt the convention that $\ket{000}_{r}=\ket{a=0,b=0,c=0}_{r}$, et cetera. The probabilities to observe different outcomes in this measurement setting can then be obtained from this state. For example, $p(000\|011)=\|\frac{1}{2\sqrt{2}}\ket{+}_{s_{1}}\ket{+}_{s_{2}}+\frac{1}{4}\ket{+}_{s_{1}}\ket{0}_{s_{2}}\|^{2}=5/16$. Such probabilities notably involve interference between different causal orders. In particular, they differ from those which would be obtained in the naive classical case, in which we first flip a coin to determine which of the two causal orders we will place ourselves in, and then perform the measurements in this causal order. We now calculate this ‘naive causal’ probability $p^{\text{nc}}(000\|011)$. One half of the time, when we are in the causal order $A\rightarrow B\rightarrow C$, Alice measures $0$ with certainty and Bob, and Charlie have each a $50:50$ chance to measure either $0$ or $1$. The other half of the time, we are in the causal order $B\rightarrow A\rightarrow C$ all parties have a $50:50$ chance to measure either $0$ or $1$ (since Bob’s measurement in the Fourier basis, which occurs first, now makes Alice completely uncertain of her outcome). Putting this all together we find $p^{\text{nc}}(000\|011)=\frac{1}{2}\times 1\times\frac{1}{2}\times\frac{1}{2}+\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}=3/16\neq p(000\|011)$. Nevertheless, our results show that we can find some classical causal model which generates the same outcome distribution $p(abc\|xyz)$ as the quantum case. Let’s do this explicitly, to see how our proof translates in practice. We find by direct substitution into the definitions 4, and 5 in the main body that: $\displaystyle p_{1}(l_{1}=\text{Alice}\|H_{0})=1/2\quad$ $\displaystyle\qquad\qquad p_{1}(l_{1}=\text{Bob}\|H_{0})=1/2$ $\displaystyle p_{2}(l_{2}=\text{Bob}\|H_{1}=(1,0,0))=1\quad$ $\displaystyle\qquad\qquad p_{2}(l_{2}=\text{Alice}\|H_{1}=(2,0,1))=1$ $\displaystyle p_{3}(l_{3}=\text{Charlie}\|H_{2}=((1,0,0),(2,0,1))=1\quad$ $\displaystyle\qquad\qquad p_{3}(l_{3}=\text{Charlie}\|H_{2}=((2,0,1),(1,0,0))=1$ $\displaystyle p_{1}(a=0\|H_{0},x=0)=1\quad$ $\displaystyle\qquad\qquad p_{1}(b=0\|H_{0},y=1)=1/2$ $\displaystyle p_{2}(a=0\|H_{1}=(1,0,0),x=0)=1/2\quad$ $\displaystyle\qquad\qquad p_{2}(b=0\|H_{1}=(2,0,1),y=1)=1/2$ which are all the same as the naive classical case. However, the results-type probability for Charlie differs from the naive case. In particular we find $\displaystyle p_{3}(c=0\|H_{2}=((1,0,0),(2,0,1),z=1)=\frac{5}{6}\quad$ $\displaystyle\qquad\qquad p_{3}(c=0\|H_{2}=((2,0,1),(1,0,0),z=1)=\frac{5}{6}.$ (68) Despite the ordering of the history for the classical protocol being different in these two cases, the quantum calculation given by definition 5 is the same for both (as it only depends on the flags raised and results obtained before Charlie measures). This leads to interference between the two causal orders of $A$ and $B$ in $\ket{\phi_{(l_{3}=\text{Charlie},3,H_{2})}}$. This alternative classical procedure which emulates the quantum experiment therefore gives $p^{\text{ac}}(000\|011)=\frac{1}{2}\times\frac{1}{2}\times\frac{5}{6}+\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}\times\frac{5}{6}=5/16$ as desired. Although we have focused on only one probability here, the same method can be used to generate a full classical strategy which replicates the quantum experiment for all input and output choices.
# Uniquely orderable interval graphs Marta Fiori-Carones Instytut Matematyki, Uniwersytet Warszawski, Banacha 2, 02-097 Warszawa — Poland<EMAIL_ADDRESS>and Alberto Marcone Dipartimento di scienze matematiche, informatiche e fisiche, Università di Udine, Via delle Scienze 208, 33100 Udine — Italy<EMAIL_ADDRESS> ###### Abstract. Interval graphs and interval orders are deeply linked. In fact, edges of an interval graphs represent the incomparability relation of an interval order, and in general, of different interval orders. The question about the conditions under which a given interval graph is associated to a unique interval order (up to duality) arises naturally. Fishburn provided a characterisation for uniquely orderable finite connected interval graphs. We show, by an entirely new proof, that the same characterisation holds also for infinite connected interval graphs. Using tools from reverse mathematics, we explain why the characterisation cannot be lifted from the finite to the infinite by compactness, as it often happens. ###### Key words and phrases: Interval graphs, infinite graphs, unique orderability, reverse mathematics ###### 2020 Mathematics Subject Classification: Primary 05C63; Secondary 05C75, 03B30, 05C62 Both authors were partially supported by the Italian PRIN 2017 Grant “Mathematical Logic: models, sets, computability”. ## 1\. Introduction An _interval graph_ is a graph whose vertices can be mapped (by an _interval representation_) to nonempty intervals of a linear order in such a way that two vertices are adjacent if and only if the intervals associated to them intersect (it is thus convenient to assume that the adjacency relation is reflexive). Consequently, if two vertices are incomparable in the graph, the corresponding intervals are placed one before the other in the linear order. The definition of interval graphs leads to an analogous concept for partial orders. In fact, a partial order $<_{P}$ is an _interval order_ if its points can be mapped to nonempty intervals of a linear order in such a way that $x<_{P}y$ if and only if the interval associated to $x$ completely precedes the interval associated to $y$. Thus interval graphs are the incomparability graphs of interval orders, i.e. two vertices are adjacent in the graph if and only if they are incomparable in the partial order. Norbert Wiener was probably the first to pay attention to interval orders, disguised under the less familiar name ‘relations of complete sequence’, in [Wie14]. Interval graphs and interval orders were rediscovered and given the current name in [Fis70]. There is now an extensive literature on the topic: [Tro97] provides a survey for many result in this area, focusing primarily on finite structures. Interval graphs and interval orders are extensively employed in diverse fields like psychology, archaeology and physics, just to mention a few. Wiener himself noticed that interval orders are useful for the analysis of temporal events and in the representation of measures subject to a margin of error. Interval orders actually occur in many digital calendars, where hours and days form a linear order and a rectangle covers the time assigned to an appointment: if two rectangles intersect, we better choose which event we will miss. Intervals are also suitable for representations of measurements of physical properties which are subject to error, since they can take into account the accuracy of the measuring device much better than a representation with points. In psychology and economics the overlap between two intervals often indicates that the corresponding stimuli or preferences are indistinguishable. In the first paragraph we described how to build an interval order from an interval representation of an interval graph. In general, an interval graph leads to many different interval orders on its vertices: an extreme example is a totally disconnected graph which is associated to any total order on its vertices. This paper deals with the situation were the interval graph is _uniquely orderable_ , i.e. there is essentially only one interval order associated to the given interval graph. (The “essentially” in the previous sentence is due to the obvious observation that if an interval order is associated to a graph, then the same is true for the reverse partial order.) Here the extreme example is a complete graph, which is associated to a unique partial order, the antichain of its vertices. The question of which interval graphs are uniquely orderable is easily settled for non connected graphs. It is in fact immediate that a non connected interval graph is uniquely orderable if and only if it has at most two components each of which is complete. We can thus restrict our attention to connected interval graphs. In this context, Fishburn [Fis85, §3.6], building on results proved in [Han82], provides two characterizations of unique orderability for finite graphs. Indeed, some steps of the proof heavily rely on the finiteness of the graph. This is in contrast with the rest of Fishburn’s monograph, where results are systematically proved for arbitrary interval graphs and orders; we thus believe that Fishburn did not know whether his result held for infinite interval graphs as well. The main result of this paper solves this issue by extending Fishburn’s characterizations to arbitrary interval graphs by an entirely different proof (for undefined notions see §2 below): ###### Theorem 1. Let $(V,E)$ be a (possibly infinite) connected interval graph. Let $W=\\{(a,b)\in V\times V\mid\neg a\,E\,b\\}$ and $(a,b)\,Q\,(c,d)\iff a\,E\,c\land b\,E\,d$. The following are equivalent: 1. (1) $(V,E)$ is uniquely orderable; 2. (2) $(V,E)$ does not contain a buried subgraph; 3. (3) the graph $(W,Q)$ has two components. Fishburn’s statement is slightly different from ours, since it is formulated for connected interval graphs without universal vertices. Since universal vertices (i.e. those adjacent to all vertices of the graph) are incomparable to all other vertices in any partial order associated to an interval graph, removing all universal vertices does not change the unique orderability of the graph. We prefer our formulation of the result since it highlights the connectedness of the graph, which is the central property characterising the class of interval graphs for which 1 holds. A typical method to lift a result from finite structures to arbitrary ones is compactness. Hence, once 1 is proved for finite interval graphs, the first attempt to generalise it to the infinite is to argue by compactness. This is not obvious and, using tools from mathematical logic, we are able to show that it is in fact impossible. To this end we work in the framework of reverse mathematics, a research program whose goal is to establish the minimal axioms needed to prove a theorem. In this framework compactness is embodied by the formal system $\mathsf{WKL}_{0}$. We first indicate, with results which parallel those obtained in [Mar07] about interval orders, that all the basic aspects of the theory of interval orders can be developed in $\mathsf{WKL}_{0}$. On the other hand we prove the following: ###### Theorem 2. Over the base system $\mathsf{RCA}_{0}$, the following are equivalent: 1. (1) $\mathsf{ACA}_{0}$, 2. (2) a countable connected interval graph $(V,E)$ is uniquely orderable if and only if does not contain a buried subgraph. Since $\mathsf{ACA}_{0}$ is properly stronger than $\mathsf{WKL}_{0}$ this shows that compactness does not suffice to prove 1. Section 2 establishes notation and terminology, while Section 3 is devoted to the proof of 1. Section 4 gives an overview of the reverse mathematics of interval graphs: the first author’s PhD thesis [FC19] includes full proofs. The last section is devoted to the proof of 2. ## 2\. Preliminaries In this section we establish the terminology used in the paper and underline some properties of interval graphs that turn out to be useful in the next section. All the graphs $(V,E)$ in this paper are such that $E\subseteq V\times V$ is a symmetric relation (we do not ask $E$ to be irreflexive, as in some cases it is convenient to have reflexivity). As usual, we write $v\,E\,u$ to mean $(v,u)\in E$ and, if $V^{\prime}\subseteq V$, we write $(V^{\prime},E)$ in place of $(V^{\prime},E\cap(V^{\prime}\times V^{\prime}))$. We denote by $(V,\overline{E})$ the _complementary graph_ of $(V,E)$: for $u,v\in V$ we have $u\,\overline{E}\,v$ if and only if $u\,E\,v$ does not hold. Paths and cycles are defined as usual, and their length is the number of their edges. A _simple cycle_ $v_{0}\,E\,\dots\,E\,v_{n}$ is a cycle such that the vertices in $v_{0},\dots,v_{n-1}$ are distinct. A _chord_ of a cycle $v_{0}\,E\,v_{1}\,E\,\dots\,E\,v_{n}$ is an edge $(v_{i},v_{j})$ with $2\leq j-i\leq n-2$. The chord is _triangular_ if either $j-i=2$ or $j-i=n-2$. ###### Definition 2.1. If $(V,\prec)$ is a strict partial order, the _comparability graph of $(V,\prec)$_ is the graph $(V,E)$ such that for $v,u\in V$ it holds that $v\,E\,u$ if and only if either $v\prec u$ or $u\prec v$. The _incomparability graph of $(V,\prec)$_ is the complementary graph of the comparability graph, so that two vertices are adjacent if and only if they coincide or are $\prec$-incomparable. While the comparability graph of a strict partial order is irreflexive, its incomparability graph is reflexive. Notice that a graph $(V,E)$ can be the incomparability graph of more than one partial order: we say that each such partial order is _associated to $(V,E)$_. In particular, $\prec$ and the dual of $\prec$ (i.e. $\prec^{\prime}$ such that $u\prec^{\prime}v$ iff $v\prec u$) are associated to the same incomparability graph. ###### Definition 2.2. A graph $(V,E)$ is _uniquely orderable_ if it is the incomparability graph of a partial order $\prec$ and the only other partial order associated to $(V,E)$ is the dual order of $\prec$; in other words, there exists a unique (up to duality) partial order $\prec$ such that for each $v,u\in V$ it holds that $\neg u\,E\,v$ if and only if $u\prec v$ or $v\prec u$. The following definition formalises the intuitive idea of interval graph given in the previous pages. ###### Definition 2.3. A graph $(V,E)$ is an _interval graph_ if it is reflexive and there exist a linear order $(L,<_{L})$ and a map $F\colon V\to\wp(L)$ such that for all $v,u\in V$, $F(v)$ is an interval in $(L,<_{L})$ (i.e. if $\ell<_{L}\ell^{\prime}<_{L}\ell^{\prime\prime}$ and $\ell,\ell^{\prime\prime}\in F(v)$, then also $\ell^{\prime}\in F(v)$) and $v\,E\,u\Leftrightarrow F(v)\cap F(u)\neq\emptyset.$ It is well-known that we may in fact assume that there exist functions $f_{L},f_{R}\colon V\to L$ such that $F(v)=\\{\ell\in L\mid f_{L}(v)\leq_{L}\ell\leq_{L}f_{R}(v)\\}$ for all $v\in V$ (this is the definition given in [Fis85]). We say that $(L,<_{L},f_{L},f_{R})$ (but often only $(f_{L},f_{R})$ or just $F$) is a _representation_ of $(V,E)$. To decide whether two vertices $u$ and $v$ are adjacent in an interval graph with representation $(f_{L},f_{R})$ we can assume without loss of generality that $f_{L}(v)\leq_{L}f_{L}(u)$ and then simply check whether $f_{L}(u)\leq_{L}f_{R}(v)$. In the context of a representation $(f_{L},f_{R})$ of an interval graph, we write $F(v)<_{L}F(u)$ in place of $f_{R}(v)<_{L}f_{L}(u)$. Then $\neg v\,E\,u$ means that either $F(v)<_{L}F(u)$ or $F(u)<_{L}F(v)$. Figure 1 provides an example of interval graph, while the graph in Figure 2 does not have an interval representation (in the figures self loops are not shown for clarity). abcd Figure 1. An example of interval graph with its representation acbxyz? Figure 2. A graph which is not an interval graph, with a partial representation A classical characterization of interval graphs is the following ([LB62], see [Fis85, Theorem 3.6]). ###### Definition 2.4. A graph $(V,E)$ is _triangulated_ if every simple cycle of length four or more has a chord. An _asteroidal triple_ in $(V,E)$ is an independent set of three vertices (i.e. a set of pairwise non adjacent vertices) of $V$ such that any two of them are connected by a path that avoids the vertices adjacent to the third. ###### Theorem 2.5. A reflexive graph $(V,E)$ is an interval graph if and only it is triangulated and has no asteroidal triples. ###### Proposition 2.6. Let $v_{0}\,E\,\dots\,E\,v_{n}$ be a path in the interval graph $(V,E)$ with representation $F$, and suppose $w\in V$ is such that $F(w)\nless_{L}F(v_{0})$ and $F(v_{n})\nless_{L}F(w)$. Then $v_{i}\,E\,w$ for some $i\leq n$, and hence $v_{0}\,E\,\dots\,E\,v_{i}\,E\,w$ and $w\,E\,v_{i}\,E\,\dots\,E\,v_{n}$ are paths. ###### Proof. Let $i\leq n$ be maximum such that $F(w)\nless_{L}F(v_{i})$. If $i=n$, then $F(w)\nless_{L}F(v_{n})$ and $F(v_{n})\nless_{L}F(w)$ imply $v_{n}\,E\,w$. If $i<n$, then $F(w)<_{L}F(v_{i+1})$ and $F(v_{i})\nless_{L}F(v_{i+1})$ (because $v_{i}\,E\,v_{i+1}$) imply $F(v_{i})\nless_{L}F(w)$. This, together with $F(w)\nless_{L}F(v_{i})$, yields $v_{i}\,E\,w$. ∎ ###### Definition 2.7. Let $(V,E)$ be a graph. A path $v_{0}\,E\,\dots\,E\,v_{n}$ is a _minimal path_ if $\neg v_{i}\,E\,v_{j}$ for every $i,j$ such that $i+1<j\leq n$. Notice that if $v_{0}\,E\,\dots\,E\,v_{n}$ is a path of minimal length among the paths connecting $v_{0}$ and $v_{n}$, then it is a minimal path, but the reverse implication does not hold. ###### Property 2.8. Let $(V,E)$ be a graph. Then each path can be refined to a minimal path. ###### Proof. The statement follows immediately from the following observation: if $v_{0}\,E\,\dots\,E\,v_{n}$ is a path and $v_{i}\,E\,v_{j}$ with $i+1<j\leq n$, then $v_{0}\,E\,\dots\,E\,v_{i}\,E\,v_{j}\,E\,\dots\,E\,v_{n}$ is still a path. ∎ ###### Property 2.9. Let $(V,E)$ be an interval graph with representation $(L,<_{L},f_{L},f_{R})$ and suppose that $v_{0}\,E\,\dots\,E\,v_{n}$ is a minimal path with $F(v_{0})<_{L}F(v_{n})$. 1. (i) Then $f_{R}(v_{i})<_{L}f_{R}(v_{i+1})$ for each $i<n-1$ and $f_{L}(v_{j})<_{L}f_{L}(v_{j+1})$ for each $j>0$; 2. (ii) if $F(v)<_{L}F(v_{0})$, then $\neg v_{i}\,E\,v$ for every $i\neq 1$; symmetrically, if $F(v_{n})<_{L}F(v)$, then $\neg v_{i}\,E\,v$ for every $i\neq n-1$. ###### Proof. To check the first conjunct of (i), suppose $i<n-1$ is least such that $f_{R}(v_{i+1})\leq_{L}f_{R}(v_{i})$. Since $i<n-1$ it holds that $\neg v_{j}\,E\,v_{n}$ for each $j\leq i$ by definition of minimal path. An easy induction, starting with our assumption $F(v_{0})<_{L}F(v_{n})$, shows that $F(v_{k})<_{L}F(v_{n})$ for each $k\leq i$. Thus, in particular it holds that $f_{R}(v_{i})<_{L}f_{R}(v_{n})$. Let $m\leq n$ be least such that $f_{R}(v_{i})<_{L}f_{R}(v_{m})$ and notice that $m>i+1$ by choice of $i$. By choice of $m$ it holds that $f_{R}(v_{m-1})\leq_{L}f_{R}(v_{i})<_{L}f_{R}(v_{m})$, and so that $f_{L}(v_{m})\leq_{L}f_{R}(v_{m-1})$ because $v_{m-1}\,E\,v_{m}$. To summarise we get that $f_{L}(v_{m})\leq_{L}f_{R}(v_{i})<_{L}f_{R}(v_{m})$, namely that $v_{i}\,E\,v_{m}$ contrary to the definition of minimal path. The second conjunct of (i) follows from the first considering the interval representation given by the linear order $(L,>_{L})$ and by the maps $f_{L}$ and $f_{R}$. For (ii), let $v_{0}\,E\,\dots\,E\,v_{n}$ be a minimal path and $F(v)<_{L}F(v_{0})<_{L}F(v_{n})$. Assume $v\,E\,v_{i}$, for some $i>1$ (notice that $\neg v\,E\,v_{0}$ by assumption). Since $f_{R}(v)<_{L}f_{L}(v_{0})$ by assumption, $f_{R}(v_{0})<_{L}f_{R}(v_{i})$ by (i), and $f_{L}(v_{i})<_{L}f_{R}(v)$ by $v\,E\,v_{i}$, it holds that $v_{0}\,E\,v_{i}$, contrary to the definition of minimal path. ∎ ## 3\. Uniquely orderable connected interval graphs In this section we prove 1. Suppose $(V,E)$ is a connected incomparability graph. Saying that $(V,E)$ is not uniquely orderable amounts to check that there are two partial orders $\prec$ and $\prec^{\prime}$ associated to $(V,E)$ and three vertices $a,b,c\in V$ such that $a\prec b\prec c$ and $b\prec^{\prime}a\prec^{\prime}c$. The vertices $a$ and $b$ can be reordered regardless, so to speak, the order of $c$. The connected graph pictured (by one of its interval representations) in Figure 3 is an example of a non uniquely orderable connected interval graph (in fact the intervals for $a$ and $b$ can be swapped without changing their relationship with the intervals $c$ and $k$). $\scriptstyle{a}$$\scriptstyle{b}$$\scriptstyle{c}$$\scriptstyle{k}$ Figure 3. An interval representation of a non uniquely orderable connected interval graph The first characterization of uniquely orderable interval graphs exploits the above observation to identify subgraphs which are forbidden in uniquely orderable interval graphs. ###### Definition 3.1. Let $(V,E)$ be a graph. For $B\subseteq V$ let $K(B)=\\{v\in V\mid\forall b\in B\,(v\,E\,b)\\}$ and $R(B)=V\setminus(B\cup K(B))$. We say that $B$ is a _buried subgraph_ of $(V,E)$ if the following hold: 1. (i) there exist $a,b\in B$ such that $\neg a\,E\,b$, 2. (ii) $K(B)\cap B=\emptyset$ and $R(B)\neq\emptyset$, 3. (iii) if $b\in B$ and $r\in R(B)$, then $\neg b\,E\,r$. The last point in the previous definition implies that any path between a vertex in $B$ and a vertex outside $B$ must go through a vertex in $K(B)$. The main consequence of (iii), which we use many times without mention, is that if $v\in V$ is such that there exist $a,b\in B$ such that $v\,E\,a$ and $\neg v\,E\,b$, then $v\in B$ (because $\neg v\,E\,b$ implies $v\notin K(B)$, while $v\,E\,a$ and (iii) imply $v\notin R(B)$). Our definition of buried subgraph is slightly different from the one in [Fis85], but it is equivalent for the class of graphs studied by Fishburn, i.e. connected interval graphs without universal vertices. Since we allow universal vertices, in condition (ii) we substituted $K(B)\neq\emptyset$ with $R(B)\neq\emptyset$ (the former condition implies the latter if there are no universal vertices, the reverse implication holds if the graph is connected by (iii)). Moreover we restated condition (iii) in simpler, yet equivalent, terms. The other main character of 1 is the graph $(W,Q)$. ###### Definition 3.2. If $(V,E)$ is a graph we let $W=\\{(a,b)\in V\times V\mid\neg a\,E\,b\\}$ and (writing $ab$ in place of $(a,b)$ for concision) $ab\,Q\,cd$ if and only if $a\,E\,c$ and $b\,E\,d$. If $ab$ and $cd$ are elements of $W$ which are connected by a path in $(W,Q)$ we write $ab\,\bar{Q}\,cd$. ###### Proposition 3.3. Let $(V,E)$ be an interval graph and $\prec$ a partial order associated to $(V,E)$. If $ab,cd\in W$ are such that $ab\,\bar{Q}\,cd$ and $a\prec b$, then $c\prec d$. In particular we have $\neg ab\,\bar{Q}\,ba$. ###### Proof. Suppose first that $ab\,Q\,cd$, so that $a\,E\,c$ and $b\,E\,d$. Notice that $b\,E\,c$ and $a\,E\,d$ cannot both hold. In fact, if $a,b,c,d$ are not all distinct, then this would contradict $ab\in W$ or $cd\in W$. Otherwise, $a\,E\,c\,E\,b\,E\,d\,E\,a$ would be a simple cycle of length four without chords, against Theorem 2.5. If $\neg b\,E\,c$, then $c\prec b$ because $a\prec b$ and $a\,E\,c$. From this we obtain $c\prec d$, since $d\,E\,b$. If instead $\neg a\,E\,d$ we obtain first $a\prec d$ and then again $c\prec d$. To derive $c\prec d$ from $ab\,\bar{Q}\,cd$ it suffices to apply the transitivity of $\prec$ to a $Q$-path connecting $ab$ with $cd$. ∎ The last part of the previous proposition implies that if $W\neq\emptyset$ (which is equivalent to $(V,E)$ being not complete), then $(W,Q)$ has at least two components. Moreover, if $(W,Q)$ has more than two (and so at least four) components, then for every partial order $\prec$ associated to $(V,E)$ there exist $ab,cd\in W$ such that $a\prec b$ and $c\prec d$, yet $ab\,\bar{Q}\,cd$ fails. We split, as originally done by Fishburn, the proof of 1 in three steps corresponding to (1) implies (2) (Lemma 3.4), (3) implies (1) (Lemma 3.5), and (2) implies (3) (Theorem 3.12). The proof of the first implication in [Fis85] is not completely accurate, and we apply Fishburn’s idea after a preliminary step which is necessary even when the graph is finite. The second implication is straightforward and applies to interval graphs of any cardinality. The proof of the last implication is completely new and requires more work. The connectedness of the graph is not needed in the first two implications. Moreover, the hypotheses of Lemma 3.4 could be further relaxed, as the proof applies to arbitrary incomparability graphs. ###### Lemma 3.4. Every uniquely orderable interval graph does not contain a buried subgraph. ###### Proof. Let $(V,E)$ be an interval graph with a buried subgraph $B$. Fix a partial order $\prec_{0}$ associated to $(V,E)$ and some $b_{0}\in B$. We define a new binary relation $\prec$ on $V$ as follows: when either $u,v\in B$ or $u,v\notin B$ set $u\prec v$ if and only if $u\prec_{0}v$; when $b\in B$ and $v\notin B$ set $b\prec v$ if and only if $b_{0}\prec_{0}v$, and $v\prec b$ if and only if $v\prec_{0}b_{0}$. Thus the whole $B$ is $\prec$-above the elements not in $B$ which are $\prec_{0}$-below $b_{0}$ and $\prec$-below the elements not in $B$ which are $\prec_{0}$-above $b_{0}$. Using the fact that the vertices not in $B$ are either $\prec_{0}$-incomparable to every vertex of $B$ or $\prec_{0}$-comparable to every vertex of $B$, it is straightforward to check that $\prec$ is transitive, and hence a partial order. For the same reason $\prec$ is associated to $(V,E)$. The key feature of $\prec$ (not necessarily shared by $\prec_{0}$) is that $B$ is $\prec$-convex, i.e. if $b\prec v\prec b^{\prime}$ with $b,b^{\prime}\in B$, then $v\in B$ as well. Indeed, if $v\notin B$, then $b\prec v$ implies $b_{0}\prec_{0}v$ and $v\prec b^{\prime}$ implies $v\prec_{0}b_{0}$. Following now [Fis85], let $\prec^{\prime}$ be such that the restrictions of $\prec$ and $\prec^{\prime}$ to $B$ are dual, while $\prec^{\prime}$ and $\prec$ coincide on $V\setminus B$ and between elements of $B$ and $V\setminus B$. Formally, $u\prec^{\prime}v$ if and only if either $u,v\in B$ and $v\prec u$, or if at least one of $u$ and $v$ does not belong to $B$ and $u\prec v$. The transitivity of $\prec^{\prime}$ is a consequence of the $\prec$-convexity of $B$ (an observation lacking in the proof given in [Fis85]) and hence $\prec^{\prime}$ is a partial order associated to $(V,E)$. If $x,y\in B$ are such that $x\prec y$ and $v\in R(B)$ (these elements exist by Definition 3.1) we have either $x\prec y\prec v$ or $v\prec x\prec y$. In the first case $y\prec^{\prime}x\prec^{\prime}v$, in the second case $v\prec^{\prime}y\prec^{\prime}x$, witnessing that $\prec^{\prime}$ is neither $\prec$ nor the dual order of $\prec$. ∎ ###### Lemma 3.5. Let $(V,E)$ be an interval graph. If ($W,Q$) has two components, then $(V,E)$ is uniquely orderable. ###### Proof. This follows easily from Proposition 3.3. ∎ For the proof of Theorem 3.12 we describe a construction that, starting from a pair of non-adjacent vertices, attempts to build the minimal buried subgraph containing those two vertices. We then show that if this attempt always fails, then for any $ab,cd\in W$ either $ab\,\bar{Q}\,cd$ or $ab\,\bar{Q}\,dc$. ###### Construction 3.6. Let $(V,E)$ be a connected interval graph and $v,u\in V$ be such that $\neg v\,E\,u$. We define recursively $B_{n}(v,u)\subseteq V$: $\displaystyle B_{0}(v,u)$ $\displaystyle=\\{v,u\\}$ $\displaystyle B_{n+1}(v,u)$ $\displaystyle=\\{w\in V\mid\exists z,z^{\prime}\in B_{n}(v,u)\,(z\,E\,w\land\neg z^{\prime}\,E\,w)\\}$ We then set $B(v,u)=\bigcup B_{n}(v,u)$. If $w\in B(v,u)$ let $e_{w}$ be the least $n$ such that $w\in B_{n}(v,u)$ (formally we should write $e_{w}^{v,u}$ but we omit the superscript as $v$ and $u$ will always be understood). A straightforward induction shows that $B_{n}(v,u)\subseteq B_{n+1}(v,u)$, for each $n\in\mathbb{N}$ (for the base step recall that interval graphs are reflexive, so that $v$ and $u$ themselves witness that $v,u\in B_{1}(v,u)$). We now show that $B(v,u)$ is close to being a buried subgraph. ###### Property 3.7. In the situation of 3.6, $B(v,u)$ is a buried subgraph if and only if $R(B(v,u))\neq\emptyset$. ###### Proof. Notice that Condition (i) of Definition 3.1 is witnessed by $v$ and $u$. Condition (3) is obvious, because if $r\notin K(B(u,v))$ but $b\,E\,r$ for some $b\in B(u,v)$, then $r\in B(u,v)$. Moreover, if $k\in K(B(v,u))$, then $k\in K(B_{n}(u,v))$ for every $n$ and hence $k\notin B(u,v)$; hence $B(v,u)\cap K(B(v,u))=\emptyset$. Therefore, to verify that $B(v,u)$ is a buried subgraph it suffices that $R(B(v,u))\neq\emptyset$. ∎ In the next propositions, we will always consider a connected interval graph $(V,E)$ with representation $(L,<_{L},f_{L},f_{R})$, fix $v,u\in V$ with $\neg v\,E\,u$ and $F(v)<_{L}F(u)$ and define $B(v,u)$ as in 3.6. For brevity, we call this set of hypotheses $(\maltese)$ and indicate it next to the proposition number. ###### Proposition 3.8 (✠). Let $x,y\in B(v,u)$. If $F(u)<_{L}F(x)$ and $f_{L}(x)\leq_{L}f_{L}(y)$, then $e_{x}\leq e_{y}$. Analogously, if $F(x)<_{L}F(v)$ and $f_{R}(y)\leq_{L}f_{R}(x)$, then $e_{x}\leq e_{y}$ as well. ###### Proof. We prove the first half of the statement by induction on $e_{y}$. The base case is trivially satisfied since there is no $y\in B_{0}(v,u)$ satisfying the hypotheses. Assume $e_{y}>0$ and let $z\in B_{e_{y}-1}(v,u)$ be such that $z\,E\,y$. If $f_{L}(x)\leq_{L}f_{L}(z)$, then $e_{x}\leq e_{z}<e_{y}$ by induction hypothesis. Otherwise, $f_{L}(z)<_{L}f_{L}(x)\leq f_{L}(y)\leq_{L}f_{R}(z)$ given that $z\,E\,y$. This means that $z\,E\,x$, which implies that $x\in B_{e_{z}+1}(v,u)$ since $u\in B_{e_{z}}(v,u)$ is such that $\neg u\,E\,x$. Hence $e_{x}\leq e_{z}+1\leq e_{y}$ The second half of the statement follows considering the representation $(L,>_{L},f_{L},f_{R})$. ∎ ###### Proposition 3.9 (✠). Let $w\in B(v,u)$. If $F(w)\nless_{L}F(u)$, then there exists a path $u\,E\,b_{1}\,E\,\dots b_{k}\,E\,w$ such that $e_{b_{i}}<e_{w}$ for all $i\leq k$. Analogously, if $F(v)\nless_{L}F(w)$, then there exists a path $v\,E\,b_{1}\,E\,\dots b_{k}\,E\,w$ such that $e_{b_{i}}<e_{w}$ for all $i\leq k$. ###### Proof. By definition of $B_{e_{w}}(v,u)$ there exists a path $b_{0}\,E\,b_{1}\,E\,\dots\,E\,b_{k}\,E\,w$ where $b_{i}\in B(v,u)$ and $0=e_{b_{0}}<e_{b_{1}}<\dots<e_{b_{k}}<e_{w}$. Hence $b_{0}\in\\{u,v\\}$ and, since $F(w)\nless_{L}F(u)$ and $F(u)\nless_{L}F(v)$, by Proposition 2.6 we can assume that $b_{0}=u$. The second half of the statement follows from the first one as usual. ∎ ###### Proposition 3.10 (✠). Let $x,z\in B(v,u)$ and $m=\max\\{e_{x},e_{z}\\}$. Assume $F(z)<_{L}F(x)$ and $F(v)\nless_{L}F(x)$ (this implies $m>0$). Then there exists a minimal path $z\,E\,v_{1}\,E\,\dots\,E\,v_{n}\,E\,x$ and $s\in B(v,u)$ with $e_{s}<m$ such that $e_{v_{i}}<m$ and $F(v_{i})<_{L}F(s)$ for each $i\leq n$. Analogously, if $F(x)<_{L}F(z)$ and $F(x)\nless_{L}F(u)$ there exists a minimal path $x\,E\,v_{1}\,E\,\dots\,E\,v_{n}\,E\,z$ and $s\in B(v,u)$ with $e_{s}<m$ such that $e_{v_{i}}<m$ and $F(s)<_{L}F(v_{i})$ for each $i\leq n$. ###### Proof. Notice that once we find the minimal path $z\,E\,v_{1}\,E\,\dots\,E\,v_{n}\,E\,x$ and $s\in B(v,u)$ with $e_{s}<m$ such that $e_{v_{i}}<m$ for all $i\leq n$ it suffices to prove that $F(v_{n})<_{L}F(s)$, since then $F(v_{i})<_{L}F(s)$ for $i<n$ follows from 2.9.i. We can apply Proposition 3.9 to both $x$ and $z$ obtaining paths connecting $v$ to $x$ and $v$ to $z$ and with $e_{b}<m$ for all vertices $b$, distinct from $x$ and $z$, occurring in the paths. Joining these paths and then using 2.8 we obtain a minimal path $z\,E\,v_{1}\,E\,\dots\,E\,v_{n}\,E\,x$ with $e_{v_{i}}<m$. Notice that $n>0$ as $\neg z\,E\,x$. Let $j<m$ be such that $e_{v_{n}}=j$: we may assume $j$ is least for which such a minimal path exists. If $j=0$, then we claim that we can assume $v_{n}=v$ and hence we can choose $s=u$. In fact, if $v_{n}=u$, then $F(x)<_{L}F(v)$ is impossible and we have $v\,E\,x$. The hypotheses imply $F(v)\nless_{L}F(z)$ and, since $F(u)\nless_{L}F(v)$, by Proposition 2.6 we can find $i<n$ such that $v\,E\,v_{i}$ and consider the path $z\,E\,v_{1}\,E\,\dots\,E\,v_{i}\,E\,v\,E\,x$. We now assume $j>0$: there exists $s\in B(v,u)$ such that $e_{s}<j$ and $\neg s\,E\,v_{n}$. We claim that $F(v_{n})<_{L}F(s)$, completing the proof. Suppose on the contrary that $F(s)<_{L}F(v_{n})$ ($F(v_{n})\cap F(s)\neq\emptyset$ cannot hold because $\neg s\,E\,v_{n}$). In this case we have $F(s)<_{L}F(x)$ because $f_{L}(v_{n})<_{L}f_{L}(x)$ by 2.9.i. Hence $F(v)\nless_{L}F(s)$ and we can use Proposition 3.9 and 2.8 to obtain a minimal path $s\,E\,u_{1}\,E\,\dots\,E\,u_{\ell}$, with $u_{\ell}=v$ and $e_{u_{i}}<e_{s}$. Since $F(x)\nless_{L}F(s)$ and $F(v)\nless_{L}F(x)$ by Proposition 2.6 there exists $k\leq\ell$ such that $u_{k}\,E\,x$. We distinguish two cases: $F(z)\nless_{L}F(s)$ and $F(z)<_{L}F(s)$. In the first case we apply Proposition 2.6 to the path $s\,E\,u_{1}\,E\,\dots\,E\,u_{k}\,E\,x$: there exists $h\leq k$ such that $z\,E\,u_{h}$. Since $z\,E\,u_{h}\,E\,\dots\,E\,u_{k}\,E\,x$ can be refined to a minimal path and $e_{u_{i}}<j$, the minimality of $j$ is contradicted. In the second case we apply Proposition 2.6 to the path $z\,E\,v_{1}\,E\,\dots\,E\,v_{n}\,E\,x$: there exists $h<n$ (recall that $\neg v_{n}\,E\,s$) such that $v_{h}\,E\,s$. Then $z\,E\,v_{1}\,E\,\dots\,E\,v_{h}\,E\,s\,E\,u_{1}\,E\,\dots\,E\,u_{k}\,E\,x$ can be refined to a path, which can then be refined to a minimal path $z\,E\,w_{1}\,E\,\dots\,E\,w_{r}\,E\,x$. Notice that $w_{r}=u_{p}$, for some $p\leq k$ because $\neg v_{i}\,E\,x$ for every $i\leq h<n$, by minimality of the path $z\,E\,v_{1}\,E\,\dots\,E\,v_{n}\,E\,x$, and $F(s)<_{L}F(x)$. Since $e_{w_{r}}<j$ we contradict again the minimality of $j$. The second half of the statement follows from the first one as usual. ∎ ###### Lemma 3.11 (✠). If $x,y\in B(v,u)$, $f_{R}(x)\leq_{L}f_{R}(v)$ and $f_{L}(u)\leq_{L}f_{L}(y)$, then $vu\,\bar{Q}\,xy$. ###### Proof. The proof is by induction on $e_{x}+e_{y}$. If $e_{x}+e_{y}=0$, then $x=v$ and $y=u$, so that the conclusion is immediate (recall that $Q$ is reflexive). Now assume that $e_{x}+e_{y}>0$ and suppose $e_{x}\leq e_{y}$ (if $e_{y}<e_{x}$ we can employ the usual trick of reversing the representation) and hence $e_{y}>0$. If $u\,E\,y$, then $xu\,Q\,xy$ and, since the induction hypothesis implies $vu\,\bar{Q}\,xu$ (because $e_{u}=0$), we obtain $vu\,\bar{Q}\,xy$. Thus we assume $\neg u\,E\,y$ and hence $F(u)<_{L}F(y)$. Let $z\in B_{e_{y}-1}(v,u)$ be such that $y\,E\,z$. If $f_{L}(u)\leq_{L}f_{L}(z)$, then we can apply the induction hypothesis to $xz$ obtaining $vu\,\bar{Q}\,xz$. Since $xz\,Q\,xy$, we are done. We thus assume $f_{L}(z)<_{L}f_{L}(u)$ which, together with $F(u)<_{L}F(y)$ and $z\,E\,y$, implies $f_{R}(u)<_{L}f_{R}(z)$ and hence $z\,E\,u$. Notice moreover that $z\neq u$ and hence (since $z\neq v$ is obvious) $e_{y}>1$. If $\neg x\,E\,z$, then $xu\,Q\,xz\,Q\,xy$ and, since by induction hypothesis $vu\,\bar{Q}\,xu$, we have $vu\,\bar{Q}\,xy$. If instead $x\,E\,z$ we must have $f_{L}(z)\leq_{L}f_{R}(x)\leq_{L}f_{R}(v)$. Let $t\in B_{e_{y}-2}(v,u)$ be such that $\neg t\,E\,z$. If $F(z)<_{L}F(t)$, then $F(u)<_{L}F(t)$ and $f_{L}(y)<_{L}f_{L}(t)$, so that Proposition 3.8 implies $y\in B_{e_{y}-2}(v,u)$, which is impossible. Hence $F(t)<_{L}F(z)$. This implies $f_{R}(t)<_{L}f_{R}(x)$. It follows that $x\in B_{e_{y}-1}(v,u)$, either by Proposition 3.8, if $F(x)<_{L}F(v)$, or because $x\in B_{1}(v,u)$ if $x\,E\,v$, given that $\neg x\,E\,u$. Since $vu\,\bar{Q}\,tu$ holds by induction hypothesis and we have also $tu\,Q\,tz\,Q\,ty$ it suffices to show that $ty\,\bar{Q}\,xy$. If $t\,E\,x$ the conclusion is immediate, otherwise $F(t)<_{L}F(x)$. Since $F(v)\nless_{L}F(x)$ we can apply Proposition 3.10 finding a minimal path $t\,E\,u_{1}\,E\,\dots\,E\,u_{n}\,E\,x$ and $s\in B_{e_{y}-2}(v,u)$ such that $u_{i}\in B_{e_{y}-2}(v,u)$ and $F(u_{i})<_{L}F(s)$ for all $i\leq n$. We claim that $\neg u_{i}\,E\,y$, for each $i\leq n$, so that $ty\,Q\,u_{1}y\,Q\,\dots\,Q\,u_{n}y\,Q\,xy$ witnesses $ty\,\bar{Q}\,xy$. Indeed, if $u_{i}\,E\,y$, for some $i\leq n$, we would have $f_{L}(y)<_{L}f_{R}(u_{i})<_{L}f_{L}(s)$ and we could apply Proposition 3.8 to obtain $y\in B_{e_{y}-2}(v,u)$, which is impossible. ∎ ###### Theorem 3.12. Let $(V,E)$ be a connected interval graph. If $(V,E)$ does not contain a buried subgraph, then $(W,Q)$ has two components. ###### Proof. Fix a representation $(L,<_{L},f_{L},f_{R})$ of $(V,E)$ and assume that $(V,E)$ does not contain a buried subgraph. We show that if $ab,cd\in W$ are such that $F(a)<_{L}F(b)$ and $F(c)<_{L}F(d)$, then $ab\,\bar{Q}\,cd$. We can assume without loss of generality that $f_{R}(c)\leq_{L}f_{R}(a)$. We consider three cases: 1. Case 1: $f_{L}(b)<_{L}f_{L}(d)$: $B(a,b)$ (which satisfies the hypotheses of $(\maltese)$) is not a buried subgraph and hence by 3.7 we must have $B(a,b)=V$. In particular $c,d\in B(a,b)$ and we are in the hypotheses of Lemma 3.11: we conclude that $ab\,\bar{Q}\,cd$. 2. Case 2: $f_{R}(a)<_{L}f_{L}(d)\leq_{L}f_{L}(b)$: $B(a,d)$ (which satisfies the hypotheses of $(\maltese)$) is not a buried subgraph and hence by 3.7 $b,c\in B(a,d)$. Lemma 3.11 implies both $ad\,\bar{Q}\,ab$ and $ad\,\bar{Q}\,cd$. It follows that $ab\,\bar{Q}\,cd$. 3. Case 3: $f_{L}(d)\leq_{L}f_{R}(a)$: neither $B(a,b)$ nor $B(c,d)$ (which both satisfy the hypotheses of $(\maltese)$) is a buried subgraph. By 3.7 we have $c\in B(a,b)$, which implies $ab\,\bar{Q}\,cb$, and $b\in B(c,d)$, which together with $f_{L}(d)<_{L}f_{L}(b)$ yields $cd\,\bar{Q}\,cb$ (we use Lemma 3.11 in both cases). Thus $ab\,\bar{Q}\,cd$ also in this case.∎ ## 4\. Reverse mathematics and interval graphs Reverse mathematics is a research program, which dates back to the Seventies, whose goal is to find the exact axiomatic strength of theorems from different areas of mathematics. It deals with statements about countable, or countably representable, structures, using the framework of the formal system of second order arithmetic $\mathsf{Z}_{2}$. We do not introduce reverse mathematics here, but refer the reader to monographs such as [Sim09] and [Hir15]. The subsystems of second order arithmetic are obtained by limiting the comprehension and induction axioms of $\mathsf{Z}_{2}$ to specific classes of formulas. We mention only the subsystems we are going to use in this paper: $\mathsf{RCA}_{0}$ is the weak base theory corresponding to computable mathematics, $\mathsf{WKL}_{0}$ extends $\mathsf{RCA}_{0}$ by adding Weak König’s Lemma (each infinite binary tree has an infinite path), and $\mathsf{ACA}_{0}$ is even stronger allowing for definitions of sets by arithmetical comprehension. It is well-known that $\mathsf{WKL}_{0}$ is equivalent to many compactness principles and thus we can claim that a theorem not provable in $\mathsf{WKL}_{0}$ does not admit a proof by compactness. In particular this applies to 1, as 2 shows that it is not provable in $\mathsf{WKL}_{0}$. The second author studied the equivalence of different characterizations of interval orders from the reverse mathematics perspective in [Mar07]. A similar study can be carried out for interval graphs, and we summarize here the main results: full details and proofs are included in the first author’s PhD thesis [FC19], which includes also results about the subclass of indifference graphs (corresponding to proper interval orders studied in [Mar07]). As customary in reverse mathematics, the system in parenthesis indicates where the definition is given or the statement proved. Notice also that in this and in the next section we deal with countable graphs and orders, the only ones second order arithmetic and its subsystems can speak of. In the literature it is possible to find slightly different definitions of interval graphs and orders, which depend on the notion of interval employed. For example intervals may be required to be closed or not. We thus have five conceptually distinct definitions of interval graphs: ###### Definition 4.1 ($\mathsf{RCA}_{0}$). Let $(V,E)$ be a graph. * • $(V,E)$ is an _interval graph_ if there exist a linear order $(L,<_{L})$ and a relation $F\subseteq V\times L$ such that, abbreviating $\\{x\in L\mid(p,x)\in F\\}$ by $F(p)$, for all $p,q\in V$ the following hold: * (i1) $F(p)\neq\emptyset$ and $\forall x,y\in F(p)\,\forall z\in L\,(x<_{L}z<_{L}y\rightarrow z\in F(p))$, * (i2) $p\,E\,q\Leftrightarrow F(p)\cap F(q)\neq\emptyset$. * • $(V,E)$ is a _1-1 interval graph_ if it also satisfies * (i3) $F(p)\neq F(q)$ whenever $p\neq q$. * • $(V,E)$ is a _closed interval graph_ if there exist a linear order $(L,<_{L})$ and two functions $f_{L},f_{R}\colon V\to L$ such that for all $p,q\in V$ * (c1) $f_{L}(p)<_{L}f_{R}(p)$, * (c2) $p\,E\,q\Leftrightarrow f_{L}(p)\leq_{L}f_{R}(q)\leq_{L}f_{R}(p)\lor f_{L}(q)\leq_{L}f_{R}(p)\leq_{L}f_{R}(q)$ * • A closed interval graph $(V,E)$ is a _1-1 closed interval graph_ if we also have * (c3) $f_{R}(p)\neq f_{R}(q)\lor f_{L}(p)\neq f_{L}(q)$ whenever $p\neq q$. * • $(V,E)$ is a _distinguishing interval graph_ if (c1) and (c2) hold together with * (c4) $f_{i}(p)\neq f_{j}(q)$ whenever $p\neq q\lor i\neq j$. ### 4.1. Definitions and characterizations of interval graph In Definition 2.3 we mentioned that every interval graph is a closed interval graph: in fact all the notions introduced in Definition 4.1 are equivalent in a sufficiently strong theory. Our first results concern the systems where the implications between the notions introduced in Definition 4.1 can be proved. The same investigation for interval orders was carried out in [Mar07] and in this respect interval graphs and interval orders behave similarly. Indeed the proofs of the results we are going to state either mimic the corresponding proofs for interval orders or are easily derived from those results. Definition 4.1 enumerates increasingly strong conditions, so that the implications from a later to an earlier notion are easily proved in $\mathsf{RCA}_{0}$. Regarding the other implications we obtain that, as is the case for interval orders, there are three distinct notions of interval graphs in $\mathsf{RCA}_{0}$, namely that of interval, 1-1 interval and closed interval graph. ###### Theorem 4.2 ($\mathsf{RCA}_{0}$). Every closed interval graph is a distinguishing interval graph. ###### Theorem 4.3 ($\mathsf{RCA}_{0}$). The following are equivalent: 1. (1) $\mathsf{WKL}_{0}$; 2. (2) every interval graph is a 1-1 interval graph; 3. (3) every 1-1 interval graph is a closed interval graph; 4. (4) every interval graph is a closed interval graph. ### 4.2. Structural characterizations of interval graphs Since interval graphs are incomparability graphs (and Definition 2.1 can be given in $\mathsf{RCA}_{0}$) we first look at the most important structural characterization of comparability graphs. The first result is due to Jeff Hirst ([Hir87, Theorem 3.20]). ###### Lemma 4.4 ($\mathsf{RCA}_{0}$). The following are equivalent: 1. (1) $\mathsf{WKL}_{0}$; 2. (2) every irreflexive graph such that every cycle of odd length has a triangular chord is a comparability graph. We then consider two structural characterizations of interval graphs (notice that Definition 2.4 can be given in $\mathsf{RCA}_{0}$). The necessity of both conditions is provable in $\mathsf{RCA}_{0}$, but the sufficiency of one of them requires $\mathsf{WKL}_{0}$. ###### Theorem 4.5 ($\mathsf{RCA}_{0}$). Every interval graph is an incomparability graph such that every simple cycle of length four has a chord. Moreover, every interval graph is triangulated and has no asteroidal triples. Every incomparability graph such that every simple cycle of length four has a chord is an interval graph. ###### Theorem 4.6 ($\mathsf{RCA}_{0}$). The following are equivalent: 1. (1) $\mathsf{WKL}_{0}$; 2. (2) if a reflexive graph is triangulated and has no asteroidal triples, then it is an interval graph. Figure 4 summarizes the results about the different definitions and characterizations of interval graphs. The arrows correspond to provability in $\mathsf{RCA}_{0}$, while every implication from a notion below another is equivalent to $\mathsf{WKL}_{0}$. distinguishing interval1-1 closed intervalclosed interval1-1 intervalfour cycle + incomparabilityinterval graphtriangulated + no asteroidal triples Figure 4. Implications in $\mathsf{RCA}_{0}$ Schmerl [Sch05] claimed that the statement “A graph is an interval graph if and only if each finite subgraph is representable by intervals” is equivalent to $\mathsf{WKL}_{0}$. Theorem 4.6 confirms his claim and shows that compactness is necessary to prove the statement. On the other hand, the corresponding statement for interval orders, i.e. an order is an interval order if and only if each suborders is an interval order, is provable in $\mathsf{RCA}_{0}$ because the structural characterization of interval orders (as the partial orders not containing $2\oplus 2$) is provable in $\mathsf{RCA}_{0}$ [Mar07, Theorem 2.1]. The different strengths of the structural characterizations of interval graphs and orders can be traced to the fact that an interval order carries full information about the relative position of the intervals in its representations, while an interval graph does not. Lekkerkerker and Boland [LB62] provide another characterization of interval graphs listing all the forbidden subgraphs. It is routine to check in $\mathsf{RCA}_{0}$ that those graphs are a complete list of graphs whose cycles of length greater than four do not have chords or which contain an asteroidal triple. ### 4.3. Interval graphs and interval orders Different definitions for interval orders, mirroring those of Definition 4.1, were given and studied in [Mar07]. We give here only the most basic one, as the others can be easily guessed from this. ###### Definition 4.7 ($\mathsf{RCA}_{0}$). A partial order $(V,\preceq)$ is an _interval order_ if there exist a linear order $(L,<_{L})$ and a relation $F\subseteq V\times L$ such that, abbreviating $\\{x\in L\mid(p,x)\in F\\}$ by $F(p)$, for all $p,q\in V$ the following hold: * (i1) $F(p)\neq\emptyset$ and $\forall x,y\in F(p)\,\forall z\in L\,(x<_{L}z<_{L}y\rightarrow z\in F(p))$, * (i2) $p\preceq q\Leftrightarrow\forall x\in F(p)\,\forall y\in F(q)\,(x<_{L}y)$. We explore the strength of the statements that allow moving from interval graphs to interval orders and back. By the previous results (and the corresponding ones in [Mar07]) it suffices to consider three different notions on each side, and we concentrate on the relationship between corresponding notions. In one direction everything goes through in $\mathsf{RCA}_{0}$. ###### Theorem 4.8 ($\mathsf{RCA}_{0}$). Let $(V,E)$ be a graph and let $\mathcal{P}$ be any of “interval”, “1-1 interval”, “closed interval”. $(V,E)$ is a $\mathcal{P}$ graph if and only if there exists a $\mathcal{P}$ order $(V,\prec)$ such that $p\,E\,q\Leftrightarrow p\nprec q\land q\nprec p$ for all $p,q\in V$. The other direction is more interesting, as only in one case $\mathsf{RCA}_{0}$ suffices. The proofs of the reversals to $\mathsf{WKL}_{0}$ are modifications of the proof of [Mar07, Theorem 6.4]. ###### Theorem 4.9 ($\mathsf{RCA}_{0}$). Let $(V,\preceq)$ be a partial order. $(V,\preceq)$ is an interval order if and only if $(V,E)$, where $p\,E\,q\Leftrightarrow p\nprec q\land q\nprec p$ for all $p,q\in V$, is an interval graph. ###### Theorem 4.10 ($\mathsf{RCA}_{0}$). The following are equivalent: 1. (1) $\mathsf{WKL}_{0}$ 2. (2) Let $(V,\preceq)$ be a partial order. $(V,\preceq)$ is a 1-1 interval order if and only if $(V,E)$, where $p\,E\,q\Leftrightarrow p\nprec q\land q\nprec p$ for all $p,q\in V$, is a 1-1 interval graph. 3. (3) Let $(V,\preceq)$ be a partial order. $(V,\preceq)$ is a closed interval order if and only if $(V,E)$, where $p\,E\,q\Leftrightarrow p\nprec q\land q\nprec p$ for all $p,q\in V$, is a closed interval graph. ## 5\. Why compactness does not suffice It is immediate (using Theorem 4.5) that Lemmas 3.4 and 3.5 are provable in $\mathsf{RCA}_{0}$. On the other hand, we now show that Theorem 3.12 is much stronger, and indeed equivalent to $\mathsf{ACA}_{0}$. As mentioned in the introduction of the paper, this result explains why the attempts to prove it by compactness cannot succeed. ###### Lemma 5.1 ($\mathsf{ACA}_{0}$). Let $(V,E)$ be a connected reflexive graph which is triangulated and with no asteroidal triples. Suppose furthermore that $a,b,c,d\in V$ are such that $\neg ab\,\bar{Q}\,cd$ and $\neg ab\,\bar{Q}\,dc$. Then there exists a buried subgraph $B\subseteq V$ such that either $a,b\in B$ or $c,d\in B$, and no subgraph $A\subseteq B$, which contains either $a,b$ or $c,d$ respectively, is a buried subgraph. ###### Proof. By Theorems 4.3 and 4.6 $\mathsf{WKL}_{0}$, and a fortiori $\mathsf{ACA}_{0}$, suffices to prove that any connected graph which is triangulated and with no asteroidal triples has a closed interval representation. We then need to check that the proof of Theorem 3.12, which indeed provides a buried subgraph with the desired properties, goes through in $\mathsf{ACA}_{0}$. The first step is checking that, given $v,u\in V$ with $\neg v\,E\,u$, we can carry out 3.6 and define $B(v,u)$ and the various $B_{n}(v,u)$’s in $\mathsf{ACA}_{0}$. In fact the definition of each $B_{n}(v,u)$ in 3.6 uses an instance of arithmetical comprehension and thus the whole construction, as presented there, appears to require the system known as $\mathsf{ACA}_{0}^{+}$, which is properly stronger than $\mathsf{ACA}_{0}$. This problem can however be overcome in the following way. Given $v,u\in V$ as before, we can characterize $B(v,u)$ as the set of $w\in V$ such that there exists a finite tree $T\subseteq 2^{<\mathbb{N}}$ and a label function $\ell\colon T\to V$ with the following properties: * • $\ell(\emptyset)=w$ (here $\emptyset$ is the root of $T$); * • if $\sigma\in T$ is not a leaf of $T$, then $\sigma{}^{\smallfrown}0,\sigma{}^{\smallfrown}1\in T$, $\ell(\sigma)\,E\,\ell(\sigma{}^{\smallfrown}0)$ and $\neg\ell(\sigma)\,E\,\ell(\sigma{}^{\smallfrown}1)$; * • if $\sigma\in T$ is a leaf of $T$, then $\ell(\sigma)\in\\{v,u\\}$. In fact, the tree and its label function describe the ‘steps’ allowing $w$ to enter $B(v,u)$. Moreover $B_{n}(v,u)$ is the set of $w\in V$ such that there exists $T\subseteq 2^{<n}$ and $\ell$ witnessing $w\in B(v,u)$. These characterizations of $B(v,u)$ and $B_{n}(v,u)$ use $\Sigma^{0}_{1}$-formulas, and show that $\mathsf{ACA}_{0}$ suffices to prove the existence of the sets. Once $B(v,u)$ and each $B_{n}(v,u)$ are defined, it is straightforward to check that all subsequent steps in the proof of Theorem 3.12 can be carried out in $\mathsf{RCA}_{0}$. ∎ To prove that Theorem 3.12 implies $\mathsf{ACA}_{0}$ we use the following notions. Given an injective function $f\colon\mathbb{N}\to\mathbb{N}$ we say that $i$ is true for $f$ when $f(k)>f(i)$ for all $k>i$. It is easy to see that there exist infinitely many $i$ which are true for $f$. If $i$ is not true for $f$, i.e. if $f(k)<f(i)$ for some $k>i$, we say that $i$ is false for $f$. Moreover, we say that $i$ is true for $f$ at stage $s$ if $f(k)>f(i)$ whenever $i<k<s$, and that $i$ is false for $f$ at stage $s$ if $f(k)<f(i)$ for some $k$ with $i<k<s$. If the injective function $f$ is fixed, we omit “for $f$” from this terminology. The following Proposition is well-known (see e.g. the discussion after Definition 4.1 in [FHM+16]). ###### Proposition 5.2 ($\mathsf{RCA}_{0}$). The following are equivalent: 1. (1) $\mathsf{ACA}_{0}$; 2. (2) if $f\colon\mathbb{N}\to\mathbb{N}$ is an injective function there exists an infinite set $T$ such that every $i\in T$ is true for $f$. ###### Theorem 5.3 ($\mathsf{RCA}_{0}$). The following are equivalent: 1. (1) $\mathsf{ACA}_{0}$; 2. (2) let $(V,E)$ be a connected graph, triangulated and with no asteroidal triples; if $a,b,c,d\in V$ are such that $\neg ab\,\bar{Q}\,cd$ and $\neg ab\,\bar{Q}\,dc$, then there exists a buried subgraph $B\subseteq V$ such that either $a,b\in B$ or $c,d\in B$, and no subgraph $A\subseteq B$, which contains either $a,b$ or $c,d$ respectively, is a buried subgraph; 3. (3) let $(V,E)$ be a connected closed interval graph; if $(W,Q)$ has more than two components, then there exists a buried subgraph $B\subseteq V$; 4. (4) let $(V,E)$ be a connected closed interval graph; if $(V,E)$ is not uniquely orderable, then there exists a buried subgraph $B\subseteq V$. ###### Proof. $(1\Rightarrow 2)$ is Lemma 5.1. The implication $(2\Rightarrow 3)$ is trivial, while $(3\Rightarrow 4)$ follows directly from Lemma 3.5, which goes through in $\mathsf{RCA}_{0}$. To prove $(4\Rightarrow 1)$ we fix an injective function $f\colon\mathbb{N}\to\mathbb{N}$ and we define (within $\mathsf{RCA}_{0}$) a connected closed interval graph $(V,E)$ such that $(W,Q)$ has more than two components. We then prove, arguing in $\mathsf{RCA}_{0}$, that the unique buried subgraph $B\subseteq V$ codes the (necessarily infinite) set of numbers which are true for $f$. We let $V=\\{a,b,k,r\\}\cup\\{x_{i},y_{i}\mid i\in\mathbb{N}\\}$. Beside making sure that $(V,E)$ is reflexive, the definition of the edge relation is by stages: at stage $s$ we define $E$ on $V_{s}=\\{a,b,k,r\\}\cup\\{x_{i},y_{i}\mid i<s\\}\subseteq V$. At stage $0$ let $k$ be adjacent to $a$, $b$ and $r$ (and add no other edges). At stage $s+1$ we define the vertices adjacent to $x_{s}$ and $y_{s}$ by the following clauses: 1. (a) $a\,E\,x_{s}\,E\,b\,E\,y_{s}$ and $x_{s}\,E\,k\,E\,y_{s}$, 2. (b) $x_{s}\,E\,x_{i}$ and $y_{s}\,E\,y_{i}$ for each $i<s$, 3. (c) $x_{s}\,E\,y_{i}$ for each $i\leq s$, 4. (d) for $i\leq s$, $y_{s}\,E\,x_{i}$ if and only if $i$ is true for $f$ at stage $s+1$. It is immediate that $(V,E)$ is connected. To check that it is a closed interval graph we define a closed interval representation $f_{L},f_{R}:V\to L$ where $(L,<_{L})$ is a dense linear order. The definition of $f_{L}$ and $f_{R}$ reflects the construction of the graph by stages. At stage $0$ assign to the members of $V_{0}$ elements of $L$ satisfying $f_{L}(r)<_{L}f_{L}(k)<_{L}f_{R}(r)<_{L}f_{L}(a)<_{L}f_{R}(a)<_{L}f_{L}(b)<_{L}f_{R}(b)<_{L}f_{R}(k).$ This ensures that we are representing the restriction of the graph to $V_{0}$. At stage $s+1$, first let $f_{L}(x_{s})=f_{L}(a)$ and $f_{R}(y_{s})=f_{R}(b)$ (since this is done at every stage, we are respecting conditions (a) and (b)). We thus still need to define $f_{R}(x_{s})$ and $f_{L}(y_{s})$; first of all we make sure that $f_{L}(b)<_{L}f_{L}(y_{i})<_{L}f_{L}(y_{s})<_{L}f_{R}(x_{s})<_{L}f_{R}(b)$ for every $i<s$, so that (c) is also respected. To respect condition (d) as well we satisfy the following requirements: * • if $i<s$ is true at stage $s+1$, then $f_{R}(x_{s})<_{L}f_{R}(x_{i})$ (which implies $f_{L}(y_{s})<_{L}f_{R}(x_{i})$); * • if $j<s$ is false at stage $s+1$, then $f_{R}(x_{j})<_{L}f_{L}(y_{s})$. The existence of $f_{L}(y_{s})<_{L}f_{R}(x_{s})$ with these properties follows from the density of $L$ and from the fact that if $i<s$ is true at stage $s+1$ and $j<s$ is false at stage $s+1$, then $f_{R}(x_{j})<_{L}f_{R}(x_{i})$. To see this notice that: * • if $i<j$, then $i$ was also true at stage $j+1$ and we set $f_{R}(x_{j})<_{L}f_{R}(x_{i})$ then; * • if $j<i$, then $j$ was already false at stage $i+1$ (if $j$ was true at stage $i+1$, then $f(j)<f(i)$, and $i$ would be false at stage $s+1$ because $j$ is false at that stage), and hence we set $f_{R}(x_{j})<_{L}f_{L}(y_{i})<_{L}f_{R}(x_{i})$ at that stage. Figure 5 depicts a sample interval representation following this construction. $k$$a$$b$$r$$y_{0}$$x_{0}$$y_{1}$$x_{1}$$y_{2}$$x_{2}$$y_{3}$$x_{3}$ Figure 5. Interval representation of $V_{4}$ in case $1$ (and so $2$) becomes false at stage $3$. To check that $(V,E)$ is not uniquely orderable let $\prec_{1}$ be the partial order induced by the interval representation we just described: $v\prec_{1}u$ if and only if $f_{R}(v)<_{L}f_{L}(u)$. Define $\prec_{2}$ so that $\prec_{1}$ and $\prec_{2}$ coincide on $V\setminus\\{s\\}$ and $u\prec_{2}s$ for all $u\in V\setminus\\{r,k\\}$. It is immediate that both $\prec_{1}$ and $\prec_{2}$ are associated to $(V,E)$, and that $\prec_{2}$ is not dual of $\prec_{1}$. By (4) there exists a buried subgraph $B\subseteq V$. First of all notice that $k\in K(B)$ and hence $k\notin B$. Now observe that $r\in B$ implies, using Conditions (i) and (iii) of Definition 3.1, that either some $x_{n}$ or some $y_{n}$ belongs to $B$. From there, using Condition (iii) again, it is easy to see that $B=V\setminus\\{k\\}$ and hence $R(B)=\emptyset$, contradicting Condition (ii). Thus $r\notin B$. Then, in order to satisfy Condition (i), we must have either $a,b\in B$ or $a,y_{n}\in B$ or $x_{m},y_{n}\in B$, for some $n$ and some $m$ which is false at stage $n+1$. In any case we have $a\in B$: in the first two cases this is obvious, and in the latter case this follows from Condition (iii) because $a\,E\,x_{m}$ and $\neg a\,E\,y_{n}$ for every $n$ and $m$. But then, using $b\,E\,y_{n}$ and $\neg b\,E\,a$ we obtain $b\in B$ even in the second and third case. Thus we can conclude that $a,b\in B$. For each $n$ we have $y_{n}\,E\,b$ and $\neg y_{n}\,E\,a$ and therefore $y_{n}\in B$. Since $b\,E\,x_{n}\,E\,a$ for each $n\in\mathbb{N}$, then either $x_{n}\in K(B)$ or $x_{n}\in B$ depending whether $x_{n}$ is adjacent to every $y_{m}$ or not, namely whether $n$ is true or false for $f$. Therefore we showed $B=\\{a,b\\}\cup\\{y_{n}\mid n\in\mathbb{N}\\}\cup\\{x_{n}\mid n\text{ is false}\\},$ so that $K(B)=\\{k\\}\cup\\{x_{n}\mid n\text{ is true}\\}$ and $R(B)=\\{r\\}$. Then $T=\\{n\mid x_{n}\notin B\\}$ is the (necessarily infinite) set of all $n$ which are true for $f$. ∎ ## References * [FC19] Marta Fiori-Carones. Filling cages. Reverse mathematics and combinatorial principles. PhD thesis, Università di Udine, Italy, 2019. * [FHM+16] Emanuele Frittaion, Matthew Hendtlass, Alberto Marcone, Paul Shafer, and Jeroen Van der Meeren. Reverse mathematics, well-quasi-orders, and Noetherian spaces. Archive for Mathematical Logic, 55(3-4):431–459, 2016. * [Fis70] Peter C Fishburn. Intransitive indifference with unequal indifference intervals. Journal of Mathematical Psychology, 7(1):144–149, 1970. * [Fis85] Peter C. Fishburn. Interval Orders and Interval Graphs. Wiley, 1985. * [Han82] Philip Hanlon. Counting interval graphs. Transactions of the American Mathematical Society, 272(2):383–426, 1982. * [Hir87] Jeffry L. Hirst. Combinatorics in Subsystems of Second Order Arithmetic. PhD thesis, The Pennsylvania State University, 1987. * [Hir15] Denis R. Hirschfeldt. Slicing the Truth. World Scientific, 2015. * [LB62] Cornelis J. Lekkerkerker and Johan C. Boland. Representation of a finite graph by a set of intervals on the real line. Fundamenta Mathematicae, 51:45–64, 1962. * [Mar07] Alberto Marcone. Interval orders and reverse mathematics. Notre Dame Journal of Formal Logic, 48:425–448, 2007. * [Sch05] James H. Schmerl. Reverse mathematics and graph coloring: eliminating diagonalization. In S. Simpson, editor, Reverse Mathematics 2001, volume 21 of Lecture Notes in Logic, pages 331–348. Association of Symbolic Logic, La Jolla, CA, 2005. * [Sim09] Stephen G. Simpson. Subsystems of Second Order Arithmetic. Cambridge University Press, second edition, 2009. * [Tro97] William T. Trotter. New perspectives on interval orders and interval graphs. In Surveys in combinatorics, 1997 (London), volume 241 of London Math. Soc. Lecture Note Ser., pages 237–286. Cambridge Univ. Press, Cambridge, 1997. * [Wie14] Norbert Wiener. A contribution to the theory of relative position. Proc. Camb. Philos. Soc., 17:441–449, 1914.
# Extended theoretical transition data in C i – iv W. Li,1,3 A. M. Amarsi,2 A. Papoulia1,3 J. Ekman1 and P. Jönsson1 1Department of Materials Science and Applied Mathematics, Malmö University, SE-205 06, Malmö, Sweden 2Theoretical Astrophysics, Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20 Uppsala, Sweden 3Division of Mathematical Physics, Lund University, Post Office Box 118, SE-221 00 Lund, Sweden E-mail<EMAIL_ADDRESS> (Accepted XXX. Received YYY; in original form ZZZ) ###### Abstract Accurate atomic data are essential for opacity calculations and for abundance analyses of the Sun and other stars. The aim of this work is to provide accurate and extensive results of energy levels and transition data for C i – iv. The Multiconfiguration Dirac–Hartree–Fock and relativistic configuration interaction methods were used in the present work. To improve the quality of the wave functions and reduce the relative differences between length and velocity forms for transition data involving high Rydberg states, alternative computational strategies were employed by imposing restrictions on the electron substitutions when constructing the orbital basis for each atom and ion. Transition data, e.g., weighted oscillator strengths and transition probabilities, are given for radiative electric dipole (E1) transitions involving levels up to $\mathrm{1s^{2}2s^{2}2p6s}$ for C i, up to $\mathrm{1s^{2}2s^{2}7f}$ for C ii, up to $\mathrm{1s^{2}2s7f}$ for C iii, and up to $\mathrm{1s^{2}8g}$ for C iv. Using the difference between the transition rates in length and velocity gauges as an internal validation, the average uncertainties of all presented E1 transitions are estimated to be 8.05%, 7.20%, 1.77%, and 0.28%, respectively, for C i – iv. Extensive comparisons with available experimental and theoretical results are performed and good agreement is observed for most of the transitions. In addition, the C i data were employed in a reanalysis of the solar carbon abundance. The new transition data give a line-by-line dispersion similar to the one obtained when using transition data that are typically used in stellar spectroscopic applications today. ###### keywords: Atomic data — Atomic processes — Line: formation — Radiative transfer — Sun: abundances — Methods: numerical ††pubyear: 2020††pagerange: Extended theoretical transition data in C i – iv–A ## 1 Introduction Accurate atomic data are of fundamental importance to many different fields of astronomy and astrophysics. This is particularly true for carbon. As the fourth-most abundant metal in the cosmos (Asplund et al., 2009), carbon is a major source of opacity in the atmospheres and interiors of stars. Complete and reliable sets of atomic data for carbon are essential for stellar opacity calculations, because of their significant impact on stellar structure and evolution (e.g. VandenBerg et al., 2012; Chen et al., 2020). Accurate atomic data for carbon are also important in the context of spectroscopic abundance analyses and Galactic Archaeology. Carbon abundances measured in late-type stars help us to understand the nucleosynthesis of massive stars and AGB stars, and thus the Galactic chemical evolution (e.g. Franchini et al., 2020; Jofré et al., 2020; Stonkutė et al., 2020). In early- type stars, carbon abundances help constrain the present-day Cosmic Abundance Standard (e.g. Nieva & Przybilla, 2008, 2012; Alexeeva et al., 2019). In the Sun, the carbon abundance is precisely measured in order to put different cosmic objects onto a common scale (e.g. Caffau et al., 2010; Amarsi et al., 2019). In all of these cases, oscillator strengths for C i (cool stars) and for C i – iv (hot stars) underpin the spectroscopic analyses; this is especially the case for studies that relax the assumption of local thermodynamic equilibrium (LTE; e.g. Przybilla et al., 2001; Nieva & Przybilla, 2006), in which case much larger sets of reliable atomic data are needed. On the experimental side, a number of studies of transition data have been presented in the literature. Neutral C i transition probabilities for the $\mathrm{2p4p\rightarrow 2p3s}$ transition array have been studied by Miller et al. (1974) using a spectroscopic shock tube and by Jones & Wiese (1984) using a wall-stabilized arc. The measurements of relative oscillator strengths for $\mathrm{2p3p\rightarrow 2p3s}$, $\mathrm{2p3d\rightarrow 2p3p}$ and $\mathrm{2p4s\rightarrow 2p3p}$ have been performed by Musielok et al. (1997); Bacawski et al. (2001); Golly et al. (2003) using a wall-stabilized arc. Older measurements of oscillator strengths are also available using the same technique (Maecker, 1953; Richter, 1958; Foster, 1962; Boldt, 1963; Goldbach & Nollez, 1987; Goldbach et al., 1989). By analysing the high-resolution spectra obtained with the Goddard High Resolution Spectrograph on the Hubble Space Telescope, Federman & Zsargo (2001) derived oscillator strengths for C i lines below 1200 Å. For C ii, a number of measurements have also been performed. Träbert et al. (1999) measured the radiative decay rates for the intercombination (IC) transitions $\mathrm{2s2p^{2}~{}^{4}P\rightarrow 2s^{2}2p~{}^{2}P^{o}}$ at a heavy-ion storage ring, and the total measured radiative decay rates to the ground term were 125.8 $\pm$ 0.9 s-1 for $\mathrm{{}^{4}P_{1/2}}$, 9.61 $\pm$ 0.05 s-1 for $\mathrm{{}^{4}P_{3/2}}$, and 45.35 $\pm$ 0.15 s-1 for $\mathrm{{}^{4}P_{5/2}}$. The aforementioned results are, however, not in agreement with the values measured by Fang et al. (1993) using a radio- frequency ion trap, i.e., 146.4(+8.3, -9.2) s-1 for $\mathrm{{}^{4}P_{1/2}}$, 11.6(+0.8, -1.7) s-1 for $\mathrm{{}^{4}P_{3/2}}$, and 51.2(+2.6, -3.5) s-1 for $\mathrm{{}^{4}P_{5/2}}$. Goly & Weniger (1982) measured the transition probabilities from a helium-carbon arc for some multiplets of $\mathrm{\\{2p^{3},2s^{2}3p\\}\rightarrow 2s2p^{2}}$ and $\mathrm{2s^{2}4s\rightarrow 2s^{2}3p}$ with estimated relative uncertainty of 50%. Using an electric shock tube, Roberts & Eckerle (1967) provided the relative oscillator strengths of some C ii multiplets with relative uncertainties of 7%. Reistad et al. (1986) gave lifetimes for 11 C ii levels using the beam-foil excitation technique and extensive cascade analyses. For C iii, the IC decay rate of the $\mathrm{2s2p~{}^{3}P^{o}_{1}\rightarrow 2s^{2}~{}^{1}S_{0}}$ transition was measured to be 121.0 $\pm$ 7 s-1 by Kwong et al. (1993) using a radio-frequency ion trap and 102.94 $\pm$ 0.14 s-1 by Doerfert et al. (1997) using a heavy-ion storage ring. The discrepancy between the values obtained from the two different methods is quite large, i.e., of the order of 15%. The result given by the latter measurement is closer to earlier $ab~{}initio$ calculations ranging between 100 and 104 s-1 (Fleming et al., 1994; Fischer, 1994; Ynnerman & Fischer, 1995). Several measurements have also been performed for the lifetimes of the low-lying levels of C iii (Reistad & Martinson, 1986; Mickey, 1970; Nandi et al., 1996; Buchet-Poulizac & Buchet, 1973a). For the system of Li-like C iv, the transition probabilities of the $\mathrm{1s^{2}2p~{}^{2}P^{o}_{1/2,3/2}\rightarrow 1s^{2}2s~{}^{2}S_{1/2}}$ transitions were measured by Berkner et al. (1965) using the foil-excitation technique and by Knystautas et al. (1971) using the beam-foil technique, respectively. There are also a number of measurements of lifetimes in C iv using the beam-foil technique (Donnelly et al., 1978; Buchet-Poulizac & Buchet, 1973b; Jacques et al., 1980). On the theoretical side, Froese Fischer et al. have performed detailed studies of C i – iv, focusing on the low-lying levels. They carried out Multiconfiguration Hartree-Fock (MCHF) calculations and used the Breit-Pauli (MCHF-BP) approximation for computing energy levels and transition properties, e.g., transition probabilities, oscillator strengths, and lifetimes, in C i (Tachiev & Fischer, 2001; Fischer, 2006; Fischer & Tachiev, 2004), C ii (Tachiev & Fischer, 2000), C iii (Tachiev & Fischer, 1999; Fischer, 2000), and C iv (Godefroid et al., 2001; Fischer et al., 1998). Hibbert et al. have presented extensive calculations for optical transitions. They used the CIV3 code (Hibbert, 1975) to calculate oscillator strengths and transition probabilities in C i (Hibbert et al., 1993), C ii (Corrégé & Hibbert, 2004), and C iii (Kingston & Hibbert, 2000). In the calculations of Hibbert et al. (1993); Corrégé & Hibbert (2004), empirical adjustments were introduced to the diagonal matrix elements in order to accurately reproduce energy splittings. Their C i oscillator strengths are frequently used in the abundance analyses of cool stars (Sect. 5). A number of other authors have also presented theoretical transition data for carbon. Zatsarinny & Fischer (2002) calculated the oscillator strengths for transitions to high-lying excited states of C i using a spline frozen-cores method. Nussbaumer & Storey (1984) provided the radiative transition probabilities using the $LS$-coupling approximation and intermediate coupling approximation, respectively, for the six energetically lowest configurations of C i. Nussbaumer & Storey (1981) calculated the transition probabilities for C ii, from terms up to $\mathrm{2s^{2}4f~{}^{2}F^{o}}$, using the $LS$-coupling and close coupling (CC) approximation, respectively. In view of the great astrophysical interest for large sets of homogeneous atomic data, extensive spectrum calculations of transition data in the carbon atom and carbon ions were carried out under the umbrella of the Opacity Project using the CC approximation of the R-matrix theory, and the results are available in the Opacity Project online database (TOPbase; Cunto & Mendoza (1992); Cunto et al. (1993)). The latest compilation of C i transition probabilities was made available by Haris & Kramida (2017), and those of C ii- iv can be found in earlier compilations by Wiese & Fuhr (2007b); Wiese & Fuhr (2007a) and Fuhr (2006). In this context, the General-purpose Relativistic Atomic Structure Package (Grasp) has, more recently, been used by Aggarwal & Keenan (2015) to predict the radiative decay rates and lifetimes of 166 levels belonging to the n $\leq$ 5 configurations in C iii. Using an updated and extended version of this code (Grasp2K), Jönsson et al. (2010) determined transition data involving 26 levels in C ii. Although for the past decades a considerable amount of research has been conducted for carbon, there is still a need for extended sets of reliable theoretical transition data. To address this, we have carried out new calculations based on the fully relativistic Multiconfiguration Dirac–Hartree–Fock (MCDHF) and relativistic configuration interaction (RCI) methods, as implemented in the newest version of the Grasp code, Grasp2018 (Jönsson et al., 2013; Fischer et al., 2019). We performed energy spectrum calculations for 100, 69, 114, and 53 states, in C i – iv, respectively. Electric dipole (E1) transition data (wavelengths, transition probabilities, line strengths, and oscillator strengths) were computed along with the corresponding lifetimes of these states. This paper is structured into six sections, including the introduction. Our theoretical methods are described in Sect. 2, and computational details are given in Sect. 3. In Sect. 4, we present our results and the validation of the data. As a complementary method of validation, in Sect. 5, we use the derived data in a reanalysis of the solar carbon abundance. Finally, we present our conclusions in Sect. 6. ## 2 Theory In the Multiconfiguration Dirac-Hartree-Fock (MCDHF) method (Grant, 2007; Fischer et al., 2016), wave functions for atomic states $\gamma^{(j)}\,PJM$, $j=1,2,\ldots,N$ with angular momentum quantum numbers $JM$ and parity $P$ are expanded over ${N_{\mathrm{CSFs}}}$ configuration state functions $\Psi(\gamma^{(j)}\,PJM)=\sum_{i}^{{N_{\mathrm{CSFs}}}}c^{(j)}_{i}\,\Phi(\gamma_{i}\,PJM).$ (1) The configuration state functions (CSFs) are $jj$-coupled many-electron functions, recursively built from products of one-electron Dirac orbitals. As for the notation, $\gamma_{i}$ specifies the occupied subshells of the CSF with their complete angular coupling tree information. The radial large and small components of the one-electron orbitals and the expansion coefficients {$c^{(j)}_{i}$} of the CSFs are obtained, for a number of targeted states, by solving the Dirac-Hartree-Fock radial equations and the configuration interaction eigenvalue problem resulting from applying the variational principle on the statistically weighted energy functional of the targeted states with terms added for preserving the orthonormality of the one-electron orbitals. The energy functional is based on the Dirac-Coulomb (DC) Hamiltonian and accounts for relativistic kinematic effects. Once the radial components of the one-electron orbitals are determined, higher-order interactions, such as the transverse photon interaction and quantum electrodynamic (QED) effects (vacuum polarization and self-energy), are added to the Dirac-Coulomb Hamiltonian. Keeping the radial components fixed, the expansion coefficients {$c^{(j)}_{i}$} of the CSFs for the targeted states are obtained by solving the configuration interaction eigenvalue problem. The evaluation of radiative E1 transition data (transition probabilities, oscillator strengths) between two states: $\gamma^{\prime}P^{\prime}J^{\prime}M^{\prime}$ and $\gamma PJM$ is non- trivial. The transition data can be expressed in terms of reduced matrix elements of the transition operator ${\bf T}^{(1)}$: $\displaystyle\langle\,\Psi(\gamma PJ)\,\\|{\bf T}^{(1)}\\|\,\Psi(\gamma^{\prime}P^{\prime}J^{\prime})\,\rangle$ $\displaystyle=$ $\displaystyle\sum_{j,k}c_{j}c^{\prime}_{k}\;\langle\,\Phi(\gamma_{j}PJ)\,\\|{\bf T}^{(1)}\\|\,\Phi(\gamma^{\prime}_{k}P^{\prime}J^{\prime})\,\rangle,$ (2) where $c_{j}$ and $c^{\prime}_{k}$ are, respectively, the expansion coefficients of the CSFs for the lower and upper states, and the summation occurs over all the CSFs for the lower and upper states. The reduced matrix elements are expressed via spin-angular coefficients $d^{(1)}_{ab}$ and operator strengths as: $\displaystyle\langle\,\Phi(\gamma_{j}PJ)\,\\|{\bf T}^{(1)}\\|\,\Phi(\gamma^{\prime}_{k}P^{\prime}J^{\prime})\,\rangle$ $\displaystyle=$ $\displaystyle\sum_{a,b}d^{(1)}_{ab}\;\langle\,n_{a}l_{a}j_{a}\,\\|{\bf T}^{(1)}\\|\,n_{b}l_{b}j_{b}\,\rangle.$ (3) Allowing for the fact that we are now using Brink-and-Satchler type reduced matrix elements, we have $\displaystyle\langle\,n_{a}l_{a}j_{a}\,\\|{\bf T}^{(1)}\\|\,n_{b}l_{b}j_{b}\,\rangle$ $\displaystyle=$ $\displaystyle\left(\frac{(2j_{b}+1)\omega}{\pi c}\right)^{1/2}(-1)^{j_{a}-1/2}\begin{pmatrix}j_{a}~{}~{}~{}~{}1~{}~{}~{}~{}j_{b}\\\ \frac{1}{2}~{}~{}~{}~{}0~{}-\frac{1}{2}\end{pmatrix}\overline{M_{ab}},$ (4) where $\overline{M_{ab}}$ is the radiative transition integral defined by Grant (1974). The factor in front of $\overline{M_{ab}}$ is the Wigner 3-j symbol that gives the angular part of the matrix element. The $\overline{M_{ab}}$ integral can be written $\overline{M_{ab}}=\overline{M^{e}_{ab}}+G\overline{M^{l}_{ab}}$, where $G$ is the gauge parameter. When $G=0$ we get the Coulomb gauge, whereas for $G=\sqrt{2}$ we get the Babushkin gauge. The Babushkin gauge corresponds to the length gauge in the non-relativistic limit and puts weight on the outer part of the wave functions (Grant, 1974; Hibbert, 1974). The Coulomb gauge corresponds to the velocity gauge and puts more weight on the inner part of the wave functions (Papoulia et al., 2019). For E1 transitions, the Babushkin and Coulomb gauges give the same value of the transition moment for exact solutions of the Dirac-equation (Grant, 1974). For approximate solutions, the transition moments differ, and the quantity $dT$, defined as (Froese Fischer, 2009; Ekman et al., 2014) $dT=\frac{\|A_{l}-A_{v}\|}{\max(A_{l},A_{v})},$ (5) where $A_{l}$ and $A_{v}$ are transition rates in length and velocity form, can be used as an estimation of the uncertainty of the computed rate. ## 3 Computational schemes Calculations were performed in the extended optimal level (EOL) scheme (Dyall et al., 1989) for the weighted average of the even and odd parity states. The CSF expansions were determined using the multireference-single-double (MR-SD) method, allowing single and double (SD) substitutions from a set of important configurations, referred to as the MR, to orbitals in an active set (AS) (Olsen et al., 1988; Sturesson et al., 2007; Fischer et al., 2016). The orbitals in the AS are divided into spectroscopic orbitals, which build the configurations in the MR, and correlation orbitals, which are introduced to correct the initially obtained wave functions. During the different steps of the calculations for C i – iv, the CSF expansions were systematically enlarged by adding layers of correlation orbitals. MCDHF calculations aim to generate an orbital set. The orbital set is then used in RCI calculations based on CSF expansions that can be enlarged to capture additional electron correlation effects. For the same CSF expansion, different orbital sets give different results for both energy levels and transition data. Conventionally, MCDHF calculations are performed for CSF expansions obtained by allowing substitutions not only from the valence subshells, but also from the subshells deeper in the core, accounting for valence-valence (VV), core-valence (CV), and core-core (CC) electron correlation effects. Using orbital sets from such calculations, Pehlivan Rhodin et al. (2017) predicted large $dT$ values for transitions between low- lying states and high Rydberg states, indicating substantial uncertainties in the corresponding transition data. For transitions involving high Rydberg states, it was shown that the velocity gauge gave the more accurate results, which is contradictory to the general belief that the length gauge is the preferred one (Hibbert, 1974). Analyzing the situation more carefully, Papoulia et al. (2019) found that correlation orbitals resulting from MCDHF calculations based on CSF expansions obtained by allowing substitutions from deeper subshells are very contracted in comparison with the outer Rydberg orbitals. As a consequence, the outer parts of the wave functions for the Rydberg states are not accurately described. Thus, the length form that probes the outer part of the wave functions does not produce trustworthy results, while the velocity form that probes the inner part of the wave functions yields more reliable transition rates. In the same work, the authors showed how transition rates that are only weakly sensitive to the choice of gauge can be obtained, by paying close attention to the CSF generation strategies for the MCDHF calculations. In the present work, following the suggestion by Papoulia et al. (2019), the MCDHF calculations were based on CSF expansions for which we impose restrictions on the substitutions from the inner subshells and obtain, as a consequence, correlation orbitals that overlap more with the spectroscopic orbitals of the higher Rydberg states, adding to a better representation of the outer parts of the corresponding wave functions. The MR and orbital sets for each atom and ion are presented in Table 1. The computational scheme, including CSF generation strategies, for each atom and ion is discussed in detail below. The MCDHF calculations were followed by RCI calculations, including the Breit interaction and leading QED effects. Table 1: Summary of the computational schemes for C i – iv. The first column displays the configurations of the targeted states. MR and AS, respectively, denote the multireference sets and the active sets of orbitals used in the MCDHF and RCI calculations, and ${N_{\mathrm{CSFs}}}$ are the numbers of generated CSFs in the final RCI calculations, for the even (e) and the odd (o) parity states. Targeted configurations \| MR \| AS \| $N_{\mathrm{{CSFs}}}$ \| \| ---\|---\|---\|---\|---\|--- \| C i, $\mathrm{N_{levels}}=100$ \| \| \| $\mathrm{2s2p^{3}}$ \| $\mathrm{2s2p^{3}}$ \| {11s,10p,10d,9f, \| e: 14 941 842 \| \| $\mathrm{2s^{2}2p}\\{n_{1}\mathrm{s},n_{2}\mathrm{p},n_{3}\mathrm{d,4f}\\}$ \| $\mathrm{2s^{2}2p}\\{n_{1}\mathrm{s},n_{2}\mathrm{p},n_{3}\mathrm{d,4f}\\}$ \| 7g,6h} \| o: 15 572 953 \| \| ($3\leq n_{1}\leq 6$, $2\leq n_{2}\leq 5$, $3\leq n_{3}\leq 5$) \| ($3\leq n_{1}\leq 6$, $2\leq n_{2}\leq 6$, $3\leq n_{3}\leq 5$) \| \| \| \| \| $\mathrm{2p^{3}}\\{n_{1}\mathrm{s},n_{2}\mathrm{p},n_{3}\mathrm{d}\\}$ \| \| \| \| \| ($3\leq n_{1}\leq 6$, $3\leq n_{2}\leq 5$, $3\leq n_{3}\leq 6$) \| \| \| \| \| $\mathrm{2s2p^{2}\\{3s,3p,4p,6p,6d,7s\\}}$ \| \| \| \| \| $\mathrm{2s2p}\\{n_{1}\mathrm{s},n_{2}\mathrm{p},n_{3}\mathrm{d,4f\\}6d}$ \| \| \| \| \| ($3\leq n_{1}\leq 6$, $3\leq n_{2}\leq 5$, $3\leq n_{3}\leq 5$) \| \| \| \| \| C ii, $\mathrm{N_{levels}}=69$ \| \| \| $\mathrm{2s^{2}}nl$($n\leq 6,l\leq 4$) \| $\mathrm{2s2p^{2}}$, $\mathrm{2s^{2}}\\{n_{1}\mathrm{s},n_{2}\mathrm{p},n_{3}\mathrm{d},n_{4}\mathrm{f},n_{5}\mathrm{g}\\}$ \| {$\mathrm{{14s,14p,12d,12f}},$ \| e: 6 415 798 \| \| $\mathrm{2s^{2}7l}$($l\leq 3$) \| $(3\leq n_{1}\leq 9,2\leq n_{2}\leq 9,3\leq n_{3}\leq 7$, \| $\mathrm{10g,8h}$} \| o: 4 988 973 \| \| $\mathrm{2s2p^{2}}$, $\mathrm{2p^{3}}$, \| $4\leq n_{4}\leq 7,5\leq n_{5}\leq 6)$ \| \| \| \| $\mathrm{2s2p3s}$, $\mathrm{2s2p3p}$ \| $\mathrm{2p^{3}}$, $\mathrm{2p^{2}}\\{n_{1}\mathrm{s},n_{2}\mathrm{p},n_{3}\mathrm{d},n_{4}\mathrm{f},n_{5}\mathrm{g}\\}$ \| \| \| \| \| $(3\leq n_{1}\leq 9,4\leq n_{2}\leq 9,3\leq n_{3}\leq 7$, \| \| \| \| \| $4\leq n_{4}\leq 7,5\leq n_{5}\leq 6)$ \| \| \| \| \| $\mathrm{2s2p3s}$, $\mathrm{2s2p3p}$ \| \| \| \| \| C iii, $\mathrm{N_{levels}}=114$ \| \| \| $\mathrm{2s}nl$($n\leq 7,l\leq 4$) \| $\mathrm{2s}nl$ ($n\leq 7,l\leq 4$) \| {$\mathrm{12s,12p,12d,12f,}$ \| e: 1 578 620 \| \| $\mathrm{2p^{2}}$, $\mathrm{2p\\{3s,3p,3d\\}}$ \| $\mathrm{2p^{2}}$, $\mathrm{2p\\{3s,3p,3d\\}}$ \| $\mathrm{11g,8h}$} \| o: 1 274 147 \| \| \| C iv, $\mathrm{N_{levels}}=53$ \| \| \| $\mathrm{1s^{2}}nl$ ($n\leq 8,l\leq 4$) \| $\mathrm{1s^{2}}nl$ ($n\leq 8,l\leq 4$) \| {$\mathrm{14s,14p,14d,12f,12g,}$ \| e: 1 077 872 \| \| $\mathrm{1s^{2}6h}$ \| $\mathrm{1s^{2}6h}$ \| $\mathrm{8h,7i}$} \| o: 1 287 706 \| \| ### 3.1 C i As seen in Table 1, in the computations of neutral carbon, configurations with ${n=7~{}(l=\mathrm{s});6~{}(l=\mathrm{p,d})}$, which are not of direct relevance, were included in the MR set to obtain orbitals that are spatially extended, improving the quality of the outer parts of the wave functions of the higher Rydberg states. The MCDHF calculations were performed using CSF expansions that were produced by SD substitutions from the valence orbitals of the configurations in the MR to the active set of orbitals, with the restriction of allowing maximum one substitution from orbitals with $n=2$. The $\mathrm{1s^{2}}$ core was kept closed and, at this point, the expansions of the atomic states accounted for VV electron correlation. As a final step, an RCI calculation was performed for the largest SD valence expansion augmented by a CV expansion. The CV expansion was obtained by allowing SD substitutions from the valence orbitals and the $\mathrm{1s^{2}}$ core of the configurations in the MR, with the restriction that there should be at most one substitution from $\mathrm{1s^{2}}$. The numbers of CSFs in the final even and odd state expansions are, respectively, 14 941 842 and 15 572 953, distributed over the different $J$ symmetries. ### 3.2 C ii Similarly to the computations in C i, in the computations of the singly- ionized carbon, the configurations $\mathrm{2s^{2}\\{8s,8p,9s,9p\\}}$, which are not our prime targets, were included in the MR set (see also Table 1). In this manner, we generated orbitals that are localized farther from the atomic core. The MCDHF calculations were performed using CSF expansions obtained by allowing SD substitutions from the valence orbitals of the MR configurations. During this stage, the $\mathrm{1s^{2}}$ core remained frozen and the CSF expansions accounted for VV correlation. The final wave functions of the targeted states were determined in an RCI calculation, which included CSF expansions that were formed by allowing SD substitution from all subshells of the MR configurations, with the restriction that there should be at most one substitution from the $\mathrm{1s^{2}}$ core. The numbers of CSFs in the final even and odd state expansions are, respectively, 6 415 798 and 4 988 973, distributed over the different $J$ symmetries. ### 3.3 C iii In the computations of beryllium-like carbon, the MR simply consisted of the targeted configurations (see also Table 1). The CSF expansions used in the MCDHF calculations were obtained by allowing SD substitutions from the valence orbitals, accounting for VV correlation effects. The final wave functions of the targeted states were determined in subsequent RCI calculations, which included CSFs that were formed by allowing single, double, and triple (SDT) substitutions from all orbitals of the MR configurations, with the limitation of leaving no more than one hole in the $\mathrm{1s^{2}}$ atomic core. The final even and odd state expansions, respectively, contained 1 578 620 and 1 274 147 CSFs, distributed over the different $J$ symmetries. ### 3.4 C iv Likewise the computations in C iii, the MR in the computations of lithium-like carbon was solely represented by the targeted configurations (see also Table 1). In the MCDHF calculations, the CSF expansions were acquired by implementing SD electron substitutions from the configurations in the MR, with the restriction of allowing maximum one hole in the $\mathrm{1s^{2}}$ core. In this case, the shape of the correlation orbitals was established by CSFs accounting for valence (V) and CV correlation effects. In the subsequent RCI calculations, the CSF expansions were enlarged by enabling all SDT substitutions from the orbitals in the MR to the active set of orbitals. The final expansions of the atomic states gave rise to 1 077 872 CSFs with even parity and 1 287 706 CSFs with odd parity, respectively, shared among the different $J$ symmetry blocks. ## 4 Results Figure 1: Left panel: Comparison of computed energy levels in the present work with data from the NIST database, for C i – iv. The dashed lines indicate the $-$0.5% and 0.5% relative discrepancies. Right panel: The relative differences between the lifetimes in length and velocity forms, for C i – iv. The dashed and solid lines indicate the 5% and 10% relative differences, respectively. No., as label in the x-axis, corresponds to the No. in Table 4. The energy spectra and wave function composition in $LS$-coupling for the 100, 69, 114, and 53 lowest states, respectively, for C i – iv are given in Table 4. In the tables, the states are given with unique labels (Gaigalas et al., 2017), and the labelling is determined by the CSFs with the largest coefficient in the expansion of Eq. (1). We first summarise the results here, before discussing the individual ions in detail in Sects. 4.1 – 4.4, below. The accuracy of the wave functions from the present calculations was evaluated by comparing the calculated energy levels with experimental data provided via the National Institute of Standards and Technology (NIST) Atomic Spectra Database (Kramida et al., 2019). In the left panel of Fig. 1, energy levels computed in this work are compared with the NIST data. A closer inspection of the figure reveals that the relative discrepancies between the experimental and the computed in this work energies are, in most cases, about $-$0.35%, $-$0.08%, 0.03%, and 0.003%, respectively, for C i – iv. Only for levels of the $\mathrm{2s2p^{3}}$ configuration in C i, the disagreements are larger than 1.0%. The average difference of the computed energy levels relative to the energies from the NIST database is 0.41%, 0.081%, 0.041%, and 0.0044%, respectively, for C i – iv. In Table 4, lifetimes in length and velocity gauges are also presented. The right panel of Fig. 1 presents the relative differences between the lifetimes in length and velocity forms for C i – iv. Except for a few long-lived states that can decay to the ground state only through IC transitions, the relative differences are well below 5%. Table 2: Distribution of the uncertainties $dT$ (in %) of the computed transition rates in C i – iv depending on the magnitude of the rates. The transition rates are arranged in five groups based on the magnitude of the $A$ values (in s-1). The number of transitions, No., the mean $dT$, $\langle dT\rangle$, (in %), and the standard deviations, $\sigma$, are given for each group of transitions, in C i – iv, respectively. The last three rows show the proportions of the transitions with $dT$ less than 20%, 10%, and 5% in all the transitions with $A\geq$ $10^{2}$ s-1 for C i and C ii and $A\geq$ $10^{0}$ s-1 for C iii and C iv, respectively. \| C i \| \| C ii \| \| C iii \| \| C iv ---\|---\|---\|---\|---\|---\|---\|--- Group \| No. \| $\langle dT\rangle(\%)$ \| $\sigma$ \| \| No. \| $\langle dT\rangle(\%)$ \| $\sigma$ \| \| No. \| $\langle dT\rangle(\%)$ \| $\sigma$ \| \| No. \| $\langle dT\rangle(\%)$ \| $\sigma$ $<10^{0}$ \| 62 \| 52.6 \| 0.34 \| \| 80 \| 29.6 \| 0.32 \| \| 137 \| 10.8 \| 0.18 \| \| 20 \| 5.92 \| 0.061 $10^{0}-10^{2}$ \| 156 \| 34.0 \| 0.25 \| \| 134 \| 17.1 \| 0.24 \| \| 239 \| 5.57 \| 0.096 \| \| 10 \| 2.38 \| 0.017 $10^{2}-10^{4}$ \| 451 \| 13.2 \| 0.15 \| \| 128 \| 14.4 \| 0.19 \| \| 354 \| 2.48 \| 0.050 \| \| 6 \| 0.667 \| 0.0047 $10^{4}-10^{6}$ \| 600 \| 7.20 \| 0.11 \| \| 167 \| 11.8 \| 0.15 \| \| 360 \| 1.44 \| 0.034 \| \| 43 \| 0.267 \| 0.0035 $>10^{6}$ \| 284 \| 1.68 \| 0.020 \| \| 297 \| 1.53 \| 0.023 \| \| 715 \| 0.297 \| 0.010 \| \| 307 \| 0.205 \| 0.0041 $dT$ < 20% \| 87.4% \| \| 89.5% \| \| 98.4% \| \| 100% $dT$ < 10% \| 77.3% \| \| 80.7% \| \| 95.7% \| \| 100% $dT$ < 5% \| 62.0% \| \| 68.7% \| \| 91.7% \| \| 99.4% The accuracy of calculated transition rates can be estimated either by comparisons with other theoretical works and experimental results, when available, or by the quantity $dT$, which is defined in Eq. (5) as the agreement between the values in length and velocity gauges (Froese Fischer, 2009; Ekman et al., 2014). The latter is particularly useful when no experimental measurements are available. Transition data, e.g., wavenumbers; wavelengths; line strengths; weighted oscillator strengths; transition probabilities of E1 transitions; and the accuracy indicators $dT$, are given in Tables 5 – 8, respectively, for C i – iv. Note that the wavenumbers and wavelengths are adjusted to match the level energy values in the NIST database, which are critically evaluated by Haris & Kramida (2017) for C i and Moore & Gallagher (1993) for C ii-iv. When no NIST values are available, the wavenumbers and wavelengths are from the present MCDHF/RCI calculations and marked with * in the tables. To better display the uncertainties $dT$ of the computed transitions rates and their distribution in relation to the magnitude of the transition rate values $A$, the transitions are organized in five groups based on the magnitude of the $A$ values. A statistical analysis of the uncertainties $dT$ of the transitions is performed for the 1553, 806, 1805, and 386 E1 transitions, respectively, for C i – iv. In Table 2, the mean value of the uncertainties $\langle dT\rangle$ and standard deviations $\sigma$ are given for each group of transitions. As seen in Table 2, most of the estimated uncertainties $dT$ are well below 10%. Most of the strong transitions with $A$ > $10^{6}$ s-1 are associated with small uncertainties $dT$, less than 2%, especially for C iii and C iv, for which $\langle dT\rangle$ is 0.297% ($\sigma$ = 0.01) and 0.205% ($\sigma$ = 0.0041), respectively. It is worth noting that, by employing the alternative optimization scheme of the radial orbitals in the present calculations, the uncertainties $dT$ for transitions involving high Rydberg states are significantly reduced. Contrary to the strong transitions, the weaker transitions are associated with relatively large $dT$ values. This is even more pronounced for the first two groups of transitions in C i and C ii, where $A$ is less than $10^{2}$ s-1. These weak E1 transitions are either IC or two-electron one-photon transitions. The rates of the former transitions, in relativistic calculations, are small due to the strong cancellation contributions to the transition moment (Ynnerman & Fischer, 1995), whereas the rates of the latter transitions are identically zero in the simplest approximation of the wave function and only induced by correlation effects (Bogdanovich et al., 2007; Li et al., 2010). These types of transitions are extremely challenging, and therefore interesting from a theoretical point of view, and improved methodology is needed to further decrease the uncertainties of the respective transition data. Fortunately, the weak transitions tend to be of lesser astrophysical importance, either for opacity calculations, or for spectroscopic abundance analyses. Thus, only the transitions with $A\geq$ $10^{2}$ s-1 for C i and C ii, and $A\geq$ $10^{0}$ s-1 for C iii and C iv, are discussed in the paper; although the complete transition data tables, for all computed E1 transitions in C i – iv, are available online. The scatterplots of $dT$ versus $A$ are given in Fig. 2. The mean $dT$ for all presented E1 transitions shown in Fig. 2 is 8.05% ($\sigma$ = 0.12), 7.20% ($\sigma$ = 0.13), 1.77% ($\sigma$ = 0.05), and 0.28% ($\sigma$ = 0.0059), respectively, for C i – iv. A statistical analysis of the proportions of the transitions with $dT$ less than 20%, 10%, and 5% in all the presented E1 transitions is also performed and shown in the last three rows of Table 2. Finally, the present work can be compared with other theoretical calculations. In Fig. 3, $\log gf$ values from the present work are compared with results from MCHF-BP (Fischer, 2006; Tachiev & Fischer, 2000, 1999; Fischer et al., 1998), CIV3 (Hibbert et al., 1993; Corrégé & Hibbert, 2004), and TOPbase data (Cunto & Mendoza, 1992), when available. As shown in the figure, the differences between the $\log gf$ values computed in the present work and respective results from other sources are rather small for most of the transitions. Comparing the MCDHF/RCI results with those from CIV3 calculations by Hibbert et al. (1993), which are frequently used in the abundance analyses, 292(228) out of 378 transitions are in agreement within 20% (10%) for C i, and 78(66) out of 87 transitions are within the same range for C ii. The results from the MCDHF/RCI and MCHF-BP calculations are found to be in very good agreement for C iii–iv, with the relative differences being less than 5% for all the computed transitions. More details about the comparisons with other theoretical calculations, as well as with experimental results, are given in Sects 4.1 – 4.4. Figure 2: Scatterplot of d$T$ values vs. transition rates $A$ of E1 transitions, for C i – iv. The solid lines indicate the 10% relative agreement between the length and velocity gauges. Figure 3: Differences between the calculated $\log gf$ values in this work and results from other theoretical calculations: MCHF-BP (red asterisk), CIV3 (blue plus sign), and TOPbase (black point), for C i – iv. ### 4.1 C i The computed excitation energies, given in Table 4, are compared with results from NIST (Kramida et al., 2019). With the exception of the levels belonging to the $\mathrm{2s2p^{3}}$ configuration, for which the average relative difference between theory and experiment is 1.22%, the mean relative difference for the rest of the states is 0.35%. The complete transition data, for all computed E1 transitions in C i, can be found in Table 5. Based on the statistical analysis of the uncertainties $dT$ shown in table 2, out of the 1335 transitions with $A\geq$ $10^{2}$ s-1, the proportions of the transitions with $dT$ less than 20%, 10%, and 5% are, respectively, 87.4%, 77.3%, and 62.0%. In C i, experimental transition data are available for the $\mathrm{2p3p\rightarrow 2p3s}$, $\mathrm{2p3d\rightarrow 2p3p}$, and $\mathrm{2p4s\rightarrow 2p3p}$ transition arrays using a stabilized arc source (Musielok et al., 1997; Golly et al., 2003; Bacawski et al., 2001). In Table LABEL:tab:com_exp, the experimental relative line strengths, together with their uncertainties, are compared with the present MCDHF/RCI theoretical values and with values from the non-relativistic CIV3 calculations by Hibbert et al. (1993) that included semi-empirical diagonal energy shifts by $LS$ configuration in the interaction matrix in the determination of the wavefunctions. The estimated uncertainties $dT$ of the MCDHF/RCI line strengths are given as percentages in parentheses. In most cases, the theoretical values fall into, or only slightly outside, the range of the estimated uncertainties of the experimental values. Comparing the MCDHF/RCI results with the results from the CIV3 calculations by Hibbert et al. (1993), we see that 41 out of the 50 transitions in common are in good agreement, with the relative differences being less than 10% (see Table LABEL:tab:com_exp). For the $\mathrm{2p4s~{}^{3}P^{o}\rightarrow 2p3p~{}^{3}P}$ transitions and the $\mathrm{2p4s~{}^{3}P_{2}^{o}\rightarrow 2p3p~{}^{3}D_{1}}$ transition, the $S$ values deduced from the present MCDHF/RCI calculations differ substantially from the experimental values, i.e., by more than 20%, while the values from the CIV3 calculations appear to be in better agreement with the corresponding experimental values. Based on the agreement between the length and velocity forms, the estimated uncertainties $dT$ of the present MCDHF/RCI calculations for the above- mentioned transitions are of the order of 8.5% and 1.4%, respectively. For the $\mathrm{2p3d~{}^{3}P_{2}^{o}\rightarrow 2p3p~{}^{3}P_{1}}$, $\mathrm{2p4s~{}^{3}P_{2}^{o}\rightarrow 2p3p~{}^{3}D_{2}}$, and $\mathrm{2p3d~{}^{3}D_{2}^{o}\rightarrow 2p3p~{}^{3}D_{3}}$ transitions, both theoretical results are outside the range of the estimated uncertainties of the experimental values. For the $\mathrm{2p3d~{}^{3}D^{o}\rightarrow 2p3p~{}^{3}P}$ transitions, the evaluated relative line strengths by Golly et al. (2003) slightly differ from the observations by Bacawski et al. (2001). The latter seem to be in better overall agreement with the transition rates predicted by the present calculations. In Table LABEL:tab:CI_com, the computed line strengths and transition rates are compared with values from the spline frozen-cores (FCS) method by Zatsarinny & Fischer (2002) and the MCHF-BP calculations by Fischer (2006). Zatsarinny & Fischer (2002) presented oscillator strengths for transitions from the $\mathrm{2p^{2}~{}^{3}P}$ term to high-lying excited states, while Fischer (2006) considered only transitions from $\mathrm{2p^{2}~{}^{3}P}$, $\mathrm{{}^{1}D}$, and $\mathrm{{}^{1}S}$ to odd levels up to $\mathrm{2p3d~{}^{3}P^{o}}$. As seen in the table, the present MCDHF/RCI results seem to be in better agreement with the values from spline FCS calculations. 76 out of 98 transitions from Zatsarinny & Fischer (2002) agree with present values within 10%, while only 38 out of 78 transitions from Fischer (2006) are within the same range. The relatively large differences with Fischer (2006) may be due to the fact that limited electron correlations were included in their calculations. In the MCHF-BP calculations, two types of correlation, i.e., VV, CV, have been accounted for; however, the CC correlation has not been considered. Additionally, CSF expansions obtained from SD substitutions are not as large as the CSF expansions used in the present calculations. For the majority of the strong transitions with $A$ > $10^{6}$ s-1, there is a very good agreement between the MCDHF/RCI results and the spline FCS values, with the relative difference being less than 5%. On the other hand, for the $\mathrm{2p3d~{}^{3}F\rightarrow 2p^{2}~{}^{3}P}$ and $\mathrm{2p4s~{}^{1}P_{1}\rightarrow 2p^{2}~{}^{3}P}$ transitions, the observed discrepancies between these three methods, i.e., MCDHF/RCI, spline FCS, and MCHF-BP, are quite large. These transitions are all $LS$-forbidden transitions, the former is with $\Delta L$ = 2 and the latter is spin- forbidden transition; these types of transitions are challenging for computations and are always with large uncertainties. For example, for the $\mathrm{2p3d~{}^{3}F_{3}\rightarrow 2p^{2}~{}^{3}P_{2}}$ transition, the $A$ values from MCDHF/RCI, spline FCS, and MCHF-BP calculations are, respectively, 7.92E+06, 6.24E+06, and 1.14E+07 s-1, with the relative difference between each two of them being greater than 20%. Experimental data are, therefore, needed for validating these theoretical results. On the contrary, based on the agreement between the length and velocity forms displayed in the parentheses, the estimated uncertainties of the MCDHF/RCI calculations for the above- mentioned transitions are all less than 0.5%. ### 4.2 C ii The relative differences between theory and experiment for all the energy levels of $\mathrm{2s2p^{2}}$ are 0.16%, while the mean relative difference for the rest of the states is 0.071% (see Table 4). The complete transition data, for all computed E1 transitions in C ii, can be found in Table 6. Out of the presented 592 E1 transitions with $A\geq$ $10^{2}$ s-1, the proportions of the transitions with $dT$ less than 20%, 10%, and 5% are, respectively, 89.5%, 80.7%, and 68.7%. In Table LABEL:tab:com_exp, the lifetimes from the present MCDHF/RCI calculations are compared with available results from the MCHF-BP calculations by Tachiev & Fischer (2000) and observations by Reistad et al. (1986) and Träbert et al. (1999). Träbert et al. (1999) measured lifetimes for the three fine-structure components of the $\mathrm{2s2p^{2}~{}^{4}P}$ term in an ion storage ring. For the measured lifetimes by Reistad et al. (1986) of the doublets terms using the beam-foil technique, a single value for the two fine- structure levels is provided. It can be seen that, in all cases, the MCDHF/RCI computed lifetimes agree with the experimental values by Reistad et al. (1986) within the experimental errors. For the $\mathrm{2s2p^{2}~{}^{4}P_{1/2,3/2,5/2}}$ states, as discussed in Sect. 1, the discrepancies between the measured transition rates by Fang et al. (1993) and by Träbert et al. (1999) are quite large. It is found that the MCDHF/RCI values are in better agreement with the results given by the latter measurements, with a relative difference less than 3%. For these long-lived states, the measured lifetimes are better represented by the MCDHF/RCI results than by the MCHF-BP values. The computed line strengths and transition rates are compared with values from the MCHF-BP calculations by Tachiev & Fischer (2000) and the CIV3 calculations by Corrégé & Hibbert (2004) in Table LABEL:tab:CII_com. We note that the agreement between the present MCDHF/RCI and the MCHF-BP transition rates exhibits a broad variation. In the earlier MCHF-BP and our MCDHF/RCI calculations, the same correlation effects, i.e., VV and CV, have been accounted for. However, the CSF expansions obtained from SD substitutions in the MCHF-BP calculations are not as large as the CSF expansions used in the present calculations, and as a consequence, the $LS$-composition of the configurations might not be predicted as accurately in the former calculations. The MCDHF/RCI results seem to be in better overall agreement with the values from the CIV3 calculations, except for transitions from $\mathrm{2p^{3}~{}^{2}P^{o}}$ to $\mathrm{2s2p^{2}~{}\\{^{4}P,{{}^{2}}S\\}}$ and to $\mathrm{2s^{2}3d~{}^{2}D}$. For these transitions, involving $\mathrm{2p^{3}~{}^{2}P^{o}}$ as the upper level, the transition rates $A$ are of the order of $10^{2}$ – $10^{4}$ s-1. The $dT$ values are relatively large in the present calculations. This is due to the strong cancellation effects caused by, e.g., the strong mixing between the $\mathrm{2p^{3}~{}^{2}P^{o}}$ and $\mathrm{2s2p3s~{}^{2}P^{o}}$ levels for $\mathrm{2p^{3}~{}^{2}P^{o}}\rightarrow\mathrm{2s2p^{2}~{}{{}^{2}}S}$, and the mixing between the $\mathrm{2p^{3}~{}^{2}P^{o}}$ and $\mathrm{2s^{2}4p~{}^{2}P^{o}}$ levels for $\mathrm{2p^{3}~{}^{2}P^{o}}\rightarrow\mathrm{2s^{2}3d~{}{{}^{2}}D}$. Large discrepancies are also observed between the MCDHF/RCI and MCHF-BP results, as well as between the MCHF-BP and CIV3 results for these transitions. Experimental data are, therefore, crucial for validating the aforementioned theoretical results. On the contrary, for the majority of the strong transitions with $A$ > $10^{6}$ s-1, there is a very good agreement between the MCDHF/RCI results and those from the two previous calculations, with the relative differences being less than 5%. ### 4.3 C iii The average relative discrepancy between the computed excitation energies, shown in Table 4, and the NIST recommended values is 0.041%. The complete transition data, for all computed E1 transitions in C iii, can be found in Table 7. Out of the 1668 transitions with $A\geq$ $10^{0}$ s-1, 91.7% (98.4%) of them have $dT$ values less than $5\%$ (20%). Further, the mean $dT$ for all transitions with $A\geq$ $10^{0}$ s-1 is 1.8% with $\sigma$ = 0.05. The lifetimes of the $\mathrm{2s2p~{}^{1}P^{o}_{1}}$, $\mathrm{2p^{2}~{}\\{^{1}S_{0},^{1}D_{2}\\}}$, and $\mathrm{2s3s~{}^{1}S_{0}}$ states were measured by Reistad et al. (1986) using the beam-foil technique, and the oscillator strengths for the $\mathrm{2s2p~{}^{1}P^{o}_{1}\rightarrow 2s^{2}~{}^{1}S_{0}}$ and the $\mathrm{2p^{2}~{}\\{^{1}S_{0},^{1}D_{2}\\}\rightarrow 2s2p~{}^{1}P^{o}_{1}}$ transitions were also provided. Table LABEL:tab:com_exp gives the comparisons between the observed and computed oscillator strengths and lifetimes in C iii. Looking at the table, we see an excellent agreement between the present calculations and those from the MCHF-BP calculations (Tachiev & Fischer, 1999) with the relative difference being less than 0.7%. In all cases, the computed oscillator strengths and lifetimes agree with experiment within the experimental errors. The exceptions are the oscillator strength of the $\mathrm{2p^{2}~{}^{1}S_{0}\rightarrow 2s2p~{}^{1}P^{o}_{1}}$ transition and the lifetime of the $\mathrm{2p^{2}~{}^{1}S_{0}}$ state, for which the computed values slightly differ from the observations. In Table LABEL:tab:CIII_com, the computed line strengths and transition rates are compared with values from the MCHF-BP calculations by Tachiev & Fischer (1999) and the Grasp calculations by Aggarwal & Keenan (2015). For the majority of the transitions, there is an excellent agreement between the MCDHF/RCI and MCHF-BP values with the relative differences being less than 1%. Only 4 out of 60 transitions display discrepancies that are greater than 20%. These large discrepancies are observed for the IC transitions, e.g., $\mathrm{2s3d~{}^{3}D_{2}\rightarrow 2s2p~{}^{1}P^{o}_{1}}$ and $\mathrm{2s3d~{}^{3}D_{2}\rightarrow 2s2p~{}^{1}P^{o}_{1}}$, for which the $dT$ is relatively large. The discrepancies between the MCDHF/RCI and Grasp values are overall large; this is due to the fact that limited electron correlations were included in their calculations. Based on the excellent agreement between the MCDHF/RCI and MCHF-BP results as well as with experiment, we believe that the present transition rates together with the MCHF-BP transition data are more reliable than the ones provided by Aggarwal & Keenan (2015). ### 4.4 C iv The mean relative discrepancy between the computed excitation energies, given in Table 4, and the NIST values is 0.0044%. Out of the presented 366 transitions with $A\geq$ $10^{0}$ s-1 shown in Table 8, only two of them have $dT$ values greater than $5\%$; 94.0% of them with $dT$ being less than 1%. The mean $dT$ for all transitions with $A\geq$ $10^{0}$ s-1 is $0.28\%$ with $\sigma$ = 0.0059. For C iv, there are a number of measurements of transition properties. The transition rates of the $\mathrm{2p~{}^{2}P^{o}_{1/2,3/2}\rightarrow 2s~{}^{2}S_{1/2}}$ transitions were measured by Knystautas et al. (1971) using the beam-foil technique. By using the same technique, the lifetimes for a number of excited states were measured in four different experimental works (Donnelly et al., 1978; Buchet-Poulizac & Buchet, 1973b; Jacques et al., 1980; Peach et al., 1988). In Table LABEL:tab:com_exp, we compare the theoretical results, from present calculations and MCHF-BP calculations, with the NIST recommended values and observed values. The transition rates of the $\mathrm{2p~{}^{2}P^{o}_{1/2,3/2}\rightarrow 2s~{}^{2}S_{1/2}}$ transitions from the present work agree perfectly with the values from the MCHF-BP calculations by Fischer et al. (1998), while they are slightly smaller than the NIST data and the values by Knystautas et al. (1971). A comparison of the lifetimes of the $\mathrm{\\{3s,4s,2p,3p,4p,3d,4d,5d\\}}$ states is made with other theoretical results, i.e., from the MCHF-BP calculations and the Model Potential method. The agreements between these different theoretical results are better than 1% for all these states. Furthermore, the agreement between the computed values and those from observations is also very good except for the $\mathrm{3s~{}^{2}S_{1/2}}$ level, for which the MCDHF/RCI calculations give a slightly smaller lifetime of 0.2350 ns than the observed value of 0.25 $\pm$ 0.01 ns. In Table LABEL:tab:CIV_com, the computed line strengths and transition rates are compared with available values from the MCHF-BP calculations by Fischer et al. (1998). There is an excellent agreement between the two methods with the relative differences being less than 1% for all transitions. ## 5 Reanalysis of the solar carbon abundance Table 3: The $14$ permitted C i lines used as abundance diagnostics in Amarsi et al. (2019). Shown are the upper and lower configurations, oscillator strengths obtained from the present calculations, and oscillator strengths from NIST; the latter being based on the calculations from CIV3 (Hibbert et al., 1993). The estimated uncertainties $dT$ of the oscillator strengths are given as percentages in parentheses. The final two columns show the abundances derived in Amarsi et al. (2019), and the post-corrected values derived here based on the formula $\Delta\log\mathrm{\upvarepsilon_{C}}^{\text{line}}=-\Delta\log gf^{\text{line}}$. \| \| \| $\log gf$ \| \| \| ---\|---\|---\|---\|---\|---\|--- Upper \| Lower \| $\lambda_{\text{air}}$(nm) \| NIST \| \| MCDHF/RCI($dT$) \| \| $\log\upvarepsilon_{\mathrm{C}}^{\text{A19}}$ \| $\log\upvarepsilon_{\mathrm{C}}^{\text{L20}}$ $\mathrm{2p4p~{}^{1}D_{2}}$ \| $\mathrm{2p3s~{}^{1}P_{1}^{o}}$ \| 505.217 \| -1.30 \| \| -1.36(0.8%) \| \| 8.41 \| 8.47 $\mathrm{2p4p~{}^{1}P_{1}}$ \| $\mathrm{2p3s~{}^{1}P_{1}^{o}}$ \| 538.034 \| -1.62 \| \| -1.71(1.4%) \| \| 8.43 \| 8.52 $\mathrm{2p4d~{}^{1}P_{1}^{o}}$ \| $\mathrm{2p3p~{}^{1}P_{1}}$ \| 658.761 \| -1.00 \| \| -1.05(0.2%) \| \| 8.33 \| 8.38 $\mathrm{2p4d~{}^{3}F_{2}^{o}}$ \| $\mathrm{2p3p~{}^{3}D_{1}}$ \| 711.148 \| -1.08 \| \| -1.24(0.9%) \| \| 8.31 \| 8.47 $\mathrm{2p4d~{}^{3}F_{4}^{o}}$ \| $\mathrm{2p3p~{}^{3}D_{3}}$ \| 711.318 \| -0.77 \| \| -0.94(1.5%) \| \| 8.41 \| 8.58 $\mathrm{2p3p~{}^{3}D_{1}}$ \| $\mathrm{2p3s~{}^{3}P_{2}^{o}}$ \| 1075.40 \| -1.61 \| \| -1.62(1.3%) \| \| 8.49 \| 8.50 $\mathrm{2p3d~{}^{3}F_{2}^{o}}$ \| $\mathrm{2p3p~{}^{3}D_{2}}$ \| 1177.75 \| -0.52 \| \| -0.46(0.9%) \| \| 8.46 \| 8.40 $\mathrm{2p3d~{}^{3}P_{1}^{o}}$ \| $\mathrm{2p3p~{}^{3}P_{0}}$ \| 1254.95 \| -0.57 \| \| -0.65(3.3%) \| \| 8.51 \| 8.60 $\mathrm{2p3d~{}^{3}P_{0}^{o}}$ \| $\mathrm{2p3p~{}^{3}P_{1}}$ \| 1256.21 \| -0.52 \| \| -0.61(3.3%) \| \| 8.51 \| 8.60 $\mathrm{2p3d~{}^{3}P_{1}^{o}}$ \| $\mathrm{2p3p~{}^{3}P_{1}}$ \| 1256.90 \| -0.60 \| \| -0.70(3.2%) \| \| 8.46 \| 8.56 $\mathrm{2p3d~{}^{3}P_{2}^{o}}$ \| $\mathrm{2p3p~{}^{3}P_{1}}$ \| 1258.16 \| -0.54 \| \| -0.61(3.4%) \| \| 8.46 \| 8.53 $\mathrm{2p3d~{}^{1}P_{1}^{o}}$ \| $\mathrm{2p3p~{}^{1}S_{0}}$ \| 2102.31 \| -0.40 \| \| -0.39(0.5%) \| \| 8.47 \| 8.46 $\mathrm{2p4p~{}^{1}D_{2}}$ \| $\mathrm{2p3d~{}^{1}F_{3}^{o}}$ \| 3085.46 \| +0.10 \| \| +0.07(0.2%) \| \| 8.41 \| 8.44 $\mathrm{2p4d~{}^{1}D_{2}^{o}}$ \| $\mathrm{2p4p~{}^{1}P_{1}}$ \| 3406.58 \| +0.44 \| \| +0.45(3.1%) \| \| 8.47 \| 8.46 Figure 4: Inferred solar carbon abundances. Black points (A19) are the 3D non- LTE results of Amarsi et al. (2019) for $14$ permitted C i lines. Blue points (L20) are these same results but post-corrected using the new $\log gf$ data. Error bars reflect $\pm 5\%$ uncertainties in the measured equivalent widths as stipulated by those authors. The four lines between $1254\,\mathrm{nm}$ and $1259\,\mathrm{nm}$ discussed in the text have been highlighted in red. The unweighted means $\mu$ (including all $14$ lines) and the standard deviations of the samples $\sigma$ are stated in each panel. One can also attempt to verify the present atomic data empirically, in an astrophysical context. To demonstrate this, a solar carbon abundance analysis was carried out, based on permitted C i lines. Larger errors in the atomic data usually impart a larger dispersion in the line-by-line abundance results, as well as trends in the results with respect to the line parameters. The solar carbon abundance analysis recently presented in Amarsi et al. (2019) was taken as the starting point. Their analysis is based on equivalent widths measured in the solar disk-centre intensity, for $14$ permitted C i lines in the optical and near-infrared, as well as a single forbidden [C i] line at $872.7\,\mathrm{nm}$. Their analysis draws on a three-dimensional (3D) hydrodynamic model solar atmosphere and 3D non-local thermodynamic equilibrium (non-LTE) radiative transfer, that reflects the current state-of-the-art in stellar elemental abundance determinations (e.g. Asplund et al., 2009). For the $14$ permitted C i lines, the authors adopted transition probabilities from NIST, that are based on those of Hibbert et al. (1993) but normalized to a different scale (Haris & Kramida, 2017), corresponding to differences of the order $\pm 0.01\,\mathrm{dex}{}$. Here, we post-correct the solar carbon abundances inferred in Amarsi et al. (2019) from the $14$ permitted C i lines, using the new atomic data derived in the present study (see Table 3). To first-order, for a given spectral line, the change in the inferred abundances are related to the difference in the adopted transition probabilities simply as $\Delta\log\upvarepsilon_{\mathrm{C}}^{\text{line}}=-\Delta\log gf^{\text{line}}$. We briefly note that second-order effects on the inferred abundances, propagated forward from changes to the non-LTE statistical equilibrium when adopting the full set of new $\log gf$ data in the non-LTE model atom, were also tested; these were found to be negligible. The results of this post-correction are illustrated in Fig. 4. We find that the dispersion in the line-by-line abundance results are similar when using the new and the old sets of $\log gf$ data. We also find that the trends in the results with respect to the line parameters are of similar gradients. This is consistent with the finding in Sect. 4.1, that the precision of this new, much larger atomic data set is comparable to that of Hibbert et al. (1993). This new analysis implies a solar carbon abundance of $8.50\,\mathrm{dex}{}$, which is $0.06\,\mathrm{dex}{}$ larger than that inferred in Amarsi et al. (2019) from C i lines, and $0.07\,\mathrm{dex}{}$ larger than the current standard value from Asplund et al. (2009) that is based on C i lines as well as on molecular diagnostics. This increase in the mean abundance is due to $12$ of the $14$ permitted C i lines having lower oscillator strengths in the present calculations, compared to the NIST data set. Six of the lines give results that are larger than the mean ($\log\upvarepsilon\geq 8.51$); included in this set are all four of the lines between $1254\,\mathrm{nm}$ and $1259\,\mathrm{nm}$, which give rise to values of between $8.53$ and $8.60\,\mathrm{dex}$. These four lines have the same upper level configuration, $\mathrm{2p3d\,^{3}P^{o}}$, and a closer inspection of the $LS$-composition reveals that these states are strongly mixed (of the order of $26$%) with $\mathrm{2s2p^{3}\,{}^{3}P^{o}}$ states, which are less accurately described in the present calculations. As a consequence, as shown in Table 3, these transitions appear to be associated with slightly larger uncertainties $dT$ than most of the other lines. Omitting these four lines, or adopting NIST oscillator strengths for them, would reduce the mean abundance from $8.50$ to $8.47\,\mathrm{dex}$. Given that the scatter and trends in the results do not support one set of data over the other, we refrain from advocating a higher solar carbon abundance at this point. Nevertheless, this quite drastic change in the resulting solar carbon abundance highlights the importance of having accurate atomic data for abundance analyses. This is especially relevant in the context of the solar modelling problem, wherein standard models of the solar interior, adopting the solar chemical composition of Asplund et al. (2009), fail to reproduce key empirical constraints, including the depth of the convection zone and interior sound speed that are precisely inferred from helioseismic observations (Basu & Antia, 2008; Zhang et al., 2019). Extra opacity in the solar interior near the boundary of the convection zone would resolve the problem (Bailey et al., 2015). Carbon contributes about $5\%$ of the opacity in this region (Blancard et al., 2012), so a higher carbon abundance would help alleviate the problem, albeit only very slightly. ## 6 Conclusions In the present work, energy levels and transition data of E1 transitions are computed for C i – iv using the MCDHF and RCI methods. Special attention is paid to the computation of transition data involving high Rydberg states by employing an alternative orbital optimization approach. The accuracy of the predicted excitation energies is evaluated by comparing with experimental data provided by the NIST database. The average relative differences of the computed energy levels compared with the NIST data are 0.41%, 0.081%, 0.041%, and 0.0044%, respectively, for C i – iv. The accuracy of the transition data is evaluated based on the relative differences of the computed transition rates in the length and velocity gauges, which is given by the quantity $dT$, and by extensive comparisons with previous theoretical and experimental results. For most of the strong transitions in C i – iv, the $dT$ values are less than 5%. The mean $dT$ for all presented E1 transitions are 8.05% ($\sigma$ = 0.12), 7.20% ($\sigma$ = 0.13), 1.77% ($\sigma$ = 0.050), and 0.28% ($\sigma$ = 0.0059), respectively, for C i – iv. Particularly, for strong transitions with $A>$ $10^{6}$ s-1, the mean $dT$ is 1.68% ($\sigma$ = 0.020), 1.53% ($\sigma$ = 0.023), 0.297% ($\sigma$ = 0.010), and 0.205% ($\sigma$ = 0.0041), respectively, for C i – iv. By employing alternative optimization schemes of the radial orbitals, the uncertainties $dT$ of the computed transition data for transitions involving high Rydberg states are significantly reduced. The agreement between computed transition properties, e.g., line strengths, transition rates, and lifetimes, and experimental values is overall good. The exception is the weak transitions, e.g., the IC transitions, for which the strong cancellation effects are important; however, these effects cannot be properly considered in the present calculations. The present calculations are extended to high Rydberg states that are not covered by previous accurate calculations and this is of special importance in various astrophysical applications. The accurate and extensive sets of atomic data for C i – iv are publicly available for use by the astronomy community. These data should be useful for opacity calculations and for models of stellar structures and interiors. They should also be useful to non-LTE spectroscopic analyses of both early-type and late-type stars. ## Acknowledgements This work is supported by the Swedish research council under contracts 2015-04842, 2016-04185, 2016-03765, and 2020-03940, and by the Knut and Alice Wallenberg Foundation under the project grant KAW 2013.0052. Some of the computations were enabled by resources provided by the Swedish National Infrastructure for Computing (SNIC) at the Multidisciplinary Center for Advanced Computational Science (UPPMAX) and at the High Performance Computing Center North (HPC2N) partially funded by the Swedish Research Council through grant agreement no. 2018-05973. 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(2019) Zhang Q.-S., Li Y., Christensen-Dalsgaard J., 2019, ApJ, 881, 103 ## Appendix A Additional tables Table 4: Wave function composition (up to three $LS$ components with a contribution $>$ 0.02 of the total wave function) in $LS$-coupling, energy levels (in cm-1), and lifetimes (in s; given in length ($\tau_{l}$) and velocity ($\tau_{v}$) gauges) for C i – iv. Energy levels are given relative to the ground state and compared with NIST data (Kramida et al., 2019). The full table is available online. Species \| No. \| State \| $LS$-composition \| $E_{RCI}$ \| $E_{NIST}$ \| $\tau_{l}$ \| $\tau_{v}$ ---\|---\|---\|---\|---\|---\|---\|--- C I \| 1 \| $\mathrm{2s^{2}2p^{2}(^{3}_{2}P)~{}^{3}P_{0}}$ \| 0.88 + 0.03 $\mathrm{2s^{2}2p~{}^{2}P\,7p~{}^{3}P}$ \| 0 \| 0 \| \| C I \| 2 \| $\mathrm{2s^{2}2p^{2}(^{3}_{2}P)~{}^{3}P_{1}}$ \| 0.88 + 0.03 $\mathrm{2s^{2}2p~{}^{2}P\,7p~{}^{3}P}$ \| 16 \| 16 \| \| C I \| 3 \| $\mathrm{2s^{2}2p^{2}(^{3}_{2}P)~{}^{3}P_{2}}$ \| 0.88 + 0.03 $\mathrm{2s^{2}2p~{}^{2}P\,7p~{}^{3}P}$ \| 43 \| 43 \| \| C I \| 4 \| $\mathrm{2s^{2}2p^{2}(^{1}_{2}D)~{}^{1}D_{2}}$ \| 0.85 + 0.05 $\mathrm{2s^{2}2p~{}^{2}P\,7p~{}^{1}D}$ \+ 0.03 $\mathrm{2s^{2}2p~{}^{2}P\,3p~{}^{1}D}$ \| 10 275 \| 10 193 \| \| C I \| 5 \| $\mathrm{2s^{2}2p^{2}(^{1}_{0}S)~{}^{1}S_{0}}$ \| 0.78 + 0.06 $\mathrm{2s^{2}2p~{}^{2}P\,7p~{}^{1}S}$ \+ 0.06 $\mathrm{2p^{4}(^{1}_{0}S)~{}^{1}S}$ \| 21 775 \| 21 648 \| \| C I \| 6 \| $\mathrm{2s~{}^{2}S\,2p^{3}(^{4}_{3}S)~{}^{5}S_{2}^{\circ}}$ \| 0.93 + 0.04 $\mathrm{2s~{}^{2}S\,2p^{2}(^{3}_{2}P)~{}^{4}P\,7p~{}^{5}S^{\circ}}$ \| 33 859 \| 33 735 \| 3.00E-02 \| 1.26E-02 C I \| 7 \| $\mathrm{2s^{2}2p~{}^{2}P\,3s~{}^{3}P_{0}^{\circ}}$ \| 0.91 + 0.04 $\mathrm{2p^{3}(^{2}_{1}P)~{}^{2}P\,3s~{}^{3}P^{\circ}}$ \| 60 114 \| 60 333 \| 3.00E-09 \| 3.04E-09 C I \| 8 \| $\mathrm{2s^{2}2p~{}^{2}P\,3s~{}^{3}P_{1}^{\circ}}$ \| 0.91 + 0.04 $\mathrm{2p^{3}(^{2}_{1}P)~{}^{2}P\,3s~{}^{3}P^{\circ}}$ \| 60 133 \| 60 353 \| 3.00E-09 \| 3.04E-09 C I \| 9 \| $\mathrm{2s^{2}2p~{}^{2}P\,3s~{}^{3}P_{2}^{\circ}}$ \| 0.91 + 0.04 $\mathrm{2p^{3}(^{2}_{1}P)~{}^{2}P\,3s~{}^{3}P^{\circ}}$ \| 60 174 \| 60 393 \| 3.00E-09 \| 3.04E-09 C I \| 10 \| $\mathrm{2s^{2}2p~{}^{2}P\,3s~{}^{1}P_{1}^{\circ}}$ \| 0.92 + 0.04 $\mathrm{2p^{3}(^{2}_{1}P)~{}^{2}P\,3s~{}^{1}P^{\circ}}$ \| 61 750 \| 61 982 \| 2.78E-09 \| 2.83E-09 – \| – \| – \| – \| – \| – \| – \| – Table 5: Electric dipole transition data for C i from present calculations. Upper and lower states, wavenumber, $\Delta E$, wavelength, $\lambda$, line strength, $S$, weighted oscillator strength, $gf$, transition probability, $A$, together with the relative difference between two gauges of $A$ values, $dT$, provided by the present MCDHF/RCI calculations are shown in the table. Wavelength and wavenumber values are from the NIST database (Kramida et al., 2019) when available. Wavelengths and wavenumbers marked with * are from the present calculations. Only the first ten rows are shown; the full table is available online. Upper \| Lower \| $\Delta E$(cm-1) \| $\lambda$ (Å) \| $S$ (a.u. of a${}_{0}^{2}$e2) \| $gf$ \| $A$ (s-1) \| $dT$ ---\|---\|---\|---\|---\|---\|---\|--- $\mathrm{2s^{2}2p5d~{}^{3}D_{2}^{o}}$ \| $\mathrm{2s^{2}2p^{2}~{}^{3}P_{1}}$ \| 86373 \| 1157.769 \| 1.025E-01 \| 2.679E-02 \| 2.647E+07 \| 0.004 $\mathrm{2s^{2}2p5d~{}^{3}D_{1}^{o}}$ \| $\mathrm{2s^{2}2p^{2}~{}^{3}P_{0}}$ \| 86362 \| 1157.909 \| 8.568E-02 \| 2.240E-02 \| 3.689E+07 \| 0.003 $\mathrm{2s^{2}2p5d~{}^{3}D_{3}^{o}}$ \| $\mathrm{2s^{2}2p^{2}~{}^{3}P_{2}}$ \| 86354 \| 1158.018 \| 3.197E-01 \| 8.358E-02 \| 5.897E+07 \| 0.003 $\mathrm{2s^{2}2p6s~{}^{3}P_{2}^{o}}$ \| $\mathrm{2s^{2}2p^{2}~{}^{3}P_{1}}$ \| 86352 \| 1158.038 \| 1.273E-01 \| 3.328E-02 \| 3.287E+07 \| 0.002 $\mathrm{2s^{2}2p5d~{}^{3}D_{1}^{o}}$ \| $\mathrm{2s^{2}2p^{2}~{}^{3}P_{1}}$ \| 86346 \| 1158.130 \| 4.687E-02 \| 1.225E-02 \| 2.017E+07 \| 0.002 $\mathrm{2s^{2}2p5d~{}^{3}D_{2}^{o}}$ \| $\mathrm{2s^{2}2p^{2}~{}^{3}P_{2}}$ \| 86346 \| 1158.131 \| 1.049E-01 \| 2.742E-02 \| 2.708E+07 \| 0.000 $\mathrm{2s^{2}2p6s~{}^{3}P_{1}^{o}}$ \| $\mathrm{2s^{2}2p^{2}~{}^{3}P_{0}}$ \| 86331 \| 1158.324 \| 2.442E-02 \| 6.380E-03 \| 1.050E+07 \| 0.001 $\mathrm{2s^{2}2p6s~{}^{3}P_{2}^{o}}$ \| $\mathrm{2s^{2}2p^{2}~{}^{3}P_{2}}$ \| 86325 \| 1158.400 \| 2.619E-02 \| 6.844E-03 \| 6.756E+06 \| 0.001 $\mathrm{2s^{2}2p5d~{}^{3}D_{1}^{o}}$ \| $\mathrm{2s^{2}2p^{2}~{}^{3}P_{2}}$ \| 86319 \| 1158.492 \| 1.367E-03 \| 3.571E-04 \| 5.875E+05 \| 0.002 $\mathrm{2s^{2}2p6s~{}^{3}P_{1}^{o}}$ \| $\mathrm{2s^{2}2p^{2}~{}^{3}P_{1}}$ \| 86315 \| 1158.544 \| 6.630E-03 \| 1.732E-03 \| 2.849E+06 \| 0.004 – \| – \| – \| – \| – \| – \| – \| – Table 6: Electric dipole transition data for C ii from present calculations. Upper and lower states, wavenumber, $\Delta E$, wavelength, $\lambda$, line strength, $S$, weighted oscillator strength, $gf$, transition probability, $A$, together with the relative difference between two gauges of $A$ values, $dT$, provided by the present MCDHF/RCI calculations are shown in the table. Wavelength and wavenumber values are from the NIST database (Kramida et al., 2019) when available. Only the first ten rows are shown; the full table is available online. Upper \| Lower \| $\Delta E$(cm-1) \| $\lambda$ (Å) \| $S$ (a.u. of a${}_{0}^{2}$e2) \| $gf$ \| $A$ (s-1) \| $dT$ ---\|---\|---\|---\|---\|---\|---\|--- $\mathrm{2s2p3p~{}^{2}D_{3/2}}$ \| $\mathrm{2s^{2}2p~{}^{2}P_{1/2}^{o}}$ \| 188581 \| 530.275 \| 8.159E-02 \| 4.673E-02 \| 2.771E+08 \| 0.015 $\mathrm{2s2p3p~{}^{2}D_{5/2}}$ \| $\mathrm{2s^{2}2p~{}^{2}P_{3/2}^{o}}$ \| 188551 \| 530.359 \| 1.515E-01 \| 8.678E-02 \| 3.430E+08 \| 0.015 $\mathrm{2s2p3p~{}^{2}D_{3/2}}$ \| $\mathrm{2s^{2}2p~{}^{2}P_{3/2}^{o}}$ \| 188517 \| 530.454 \| 1.661E-02 \| 9.511E-03 \| 5.636E+07 \| 0.015 $\mathrm{2s^{2}7d~{}^{2}D_{3/2}}$ \| $\mathrm{2s^{2}2p~{}^{2}P_{1/2}^{o}}$ \| 187353 \| 533.752 \| 1.094E-01 \| 6.223E-02 \| 3.637E+08 \| 0.007 $\mathrm{2s^{2}7d~{}^{2}D_{5/2}}$ \| $\mathrm{2s^{2}2p~{}^{2}P_{3/2}^{o}}$ \| 187289 \| 533.933 \| 1.943E-01 \| 1.104E-01 \| 4.300E+08 \| 0.007 $\mathrm{2s^{2}7d~{}^{2}D_{3/2}}$ \| $\mathrm{2s^{2}2p~{}^{2}P_{3/2}^{o}}$ \| 187289 \| 533.933 \| 2.205E-02 \| 1.254E-02 \| 7.321E+07 \| 0.007 $\mathrm{2s2p3p~{}^{4}P_{3/2}}$ \| $\mathrm{2s^{2}2p~{}^{2}P_{1/2}^{o}}$ \| 186443 \| 536.355 \| 1.779E-07 \| 1.007E-07 \| 5.830E+02 \| 0.017 $\mathrm{2s2p3p~{}^{4}P_{1/2}}$ \| $\mathrm{2s^{2}2p~{}^{2}P_{1/2}^{o}}$ \| 186427 \| 536.402 \| 6.007E-07 \| 3.400E-07 \| 3.936E+03 \| 0.039 $\mathrm{2s2p3p~{}^{4}P_{5/2}}$ \| $\mathrm{2s^{2}2p~{}^{2}P_{3/2}^{o}}$ \| 186402 \| 536.473 \| 1.227E-05 \| 6.942E-06 \| 2.678E+04 \| 0.020 $\mathrm{2s2p3p~{}^{4}P_{3/2}}$ \| $\mathrm{2s^{2}2p~{}^{2}P_{3/2}^{o}}$ \| 186380 \| 536.537 \| 1.327E-06 \| 7.506E-07 \| 4.343E+03 \| 0.027 – \| – \| – \| – \| – \| – \| – \| – Table 7: Electric dipole transition data for C iii from present calculations. Upper and lower states, wavenumber, $\Delta E$, wavelength, $\lambda$, line strength, $S$, weighted oscillator strength, $gf$, transition probability, $A$, together with the relative difference between two gauges of $A$ values, $dT$, provided by the present MCDHF/RCI calculations are shown in the table. Wavelength and wavenumber values are from the NIST database (Kramida et al., 2019) when available. Wavelengths and wavenumbers marked with * are from the present calculations. Only the first ten rows are shown; the full table is available online. Upper \| Lower \| $\Delta{E}$(cm-1) \| $\lambda$ (Å) \| $S$ (a.u. of a${}_{0}^{2}$e2) \| $gf$ \| $A$ (s-1) \| $dT$ ---\|---\|---\|---\|---\|---\|---\|--- $\mathrm{2s7p~{}^{3}P_{1}^{o}}$ \| $\mathrm{2s^{2}~{}^{1}S_{0}}$ \| 365034* \| 273.947* \| 7.670E-07 \| 8.505E-07 \| 2.520E+04 \| 0.005 $\mathrm{2s7p~{}^{1}P_{1}^{o}}$ \| $\mathrm{2s^{2}~{}^{1}S_{0}}$ \| 364896 \| 274.051 \| 1.043E-02 \| 1.156E-02 \| 3.423E+08 \| 0.010 $\mathrm{2s6p~{}^{1}P_{1}^{o}}$ \| $\mathrm{2s^{2}~{}^{1}S_{0}}$ \| 357109 \| 280.026 \| 1.593E-02 \| 1.728E-02 \| 4.901E+08 \| 0.001 $\mathrm{2s6p~{}^{3}P_{1}^{o}}$ \| $\mathrm{2s^{2}~{}^{1}S_{0}}$ \| 357050 \| 280.073 \| 5.265E-06 \| 5.711E-06 \| 1.619E+05 \| 0.004 $\mathrm{2p3d~{}^{1}P_{1}^{o}}$ \| $\mathrm{2s^{2}~{}^{1}S_{0}}$ \| 346712 \| 288.423 \| 9.317E-04 \| 9.818E-04 \| 2.627E+07 \| 0.003 $\mathrm{2s5p~{}^{3}P_{1}^{o}}$ \| $\mathrm{2s^{2}~{}^{1}S_{0}}$ \| 344236 \| 290.498 \| 3.900E-07 \| 4.079E-07 \| 1.075E+04 \| 0.018 $\mathrm{2s5p~{}^{1}P_{1}^{o}}$ \| $\mathrm{2s^{2}~{}^{1}S_{0}}$ \| 343258 \| 291.326 \| 4.551E-02 \| 4.747E-02 \| 1.244E+09 \| 0.000 $\mathrm{2p3d~{}^{3}P_{1}^{o}}$ \| $\mathrm{2s^{2}~{}^{1}S_{0}}$ \| 340127 \| 294.007 \| 7.645E-07 \| 7.903E-07 \| 2.035E+04 \| 0.005 $\mathrm{2p3d~{}^{3}D_{1}^{o}}$ \| $\mathrm{2s^{2}~{}^{1}S_{0}}$ \| 337655 \| 296.159 \| 7.267E-07 \| 7.458E-07 \| 1.893E+04 \| 0.005 $\mathrm{2s4p~{}^{1}P_{1}^{o}}$ \| $\mathrm{2s^{2}~{}^{1}S_{0}}$ \| 322404 \| 310.170 \| 3.480E-02 \| 3.409E-02 \| 7.884E+08 \| 0.000 – \| – \| – \| – \| – \| – \| – \| – Table 8: Electric dipole transition data for C iv from present calculations. Upper and lower states, wavenumber, $\Delta E$, wavelength, $\lambda$, line strength, $S$, weighted oscillator strength, $gf$, transition probability, $A$, together with the relative difference between two gauges of $A$ values, $dT$, provided by the present MCDHF/RCI calculations are shown in the table. Wavelength and wavenumber values are from the NIST database (Kramida et al., 2019). Only the first ten rows are shown; the full table is available online. Upper \| Lower \| $\Delta E$(cm-1) \| $\lambda$ (Å) \| $S$ (a.u. of a${}_{0}^{2}$e2) \| $gf$ \| $A$ (s-1) \| $dT$ ---\|---\|---\|---\|---\|---\|---\|--- $\mathrm{8p~{}^{2}P_{3/2}^{o}}$ \| $\mathrm{2s~{}^{2}S_{1/2}}$ \| 492479 \| 203.054 \| 5.029E-03 \| 7.523E-03 \| 3.043E+08 \| 0.004 $\mathrm{8p~{}^{2}P_{1/2}^{o}}$ \| $\mathrm{2s~{}^{2}S_{1/2}}$ \| 492477 \| 203.055 \| 2.517E-03 \| 3.766E-03 \| 3.046E+08 \| 0.004 $\mathrm{7p~{}^{2}P_{3/2}^{o}}$ \| $\mathrm{2s~{}^{2}S_{1/2}}$ \| 483950 \| 206.633 \| 7.885E-03 \| 1.159E-02 \| 4.527E+08 \| 0.001 $\mathrm{7p~{}^{2}P_{1/2}^{o}}$ \| $\mathrm{2s~{}^{2}S_{1/2}}$ \| 483948 \| 206.634 \| 3.946E-03 \| 5.801E-03 \| 4.532E+08 \| 0.001 $\mathrm{6p~{}^{2}P_{3/2}^{o}}$ \| $\mathrm{2s~{}^{2}S_{1/2}}$ \| 470778 \| 212.414 \| 1.349E-02 \| 1.930E-02 \| 7.132E+08 \| 0.000 $\mathrm{6p~{}^{2}P_{1/2}^{o}}$ \| $\mathrm{2s~{}^{2}S_{1/2}}$ \| 470775 \| 212.416 \| 6.753E-03 \| 9.657E-03 \| 7.139E+08 \| 0.000 $\mathrm{5p~{}^{2}P_{3/2}^{o}}$ \| $\mathrm{2s~{}^{2}S_{1/2}}$ \| 448862 \| 222.785 \| 2.644E-02 \| 3.604E-02 \| 1.211E+09 \| 0.000 $\mathrm{5p~{}^{2}P_{1/2}^{o}}$ \| $\mathrm{2s~{}^{2}S_{1/2}}$ \| 448855 \| 222.789 \| 1.323E-02 \| 1.804E-02 \| 1.212E+09 \| 0.000 $\mathrm{8d~{}^{2}D_{3/2}}$ \| $\mathrm{2p~{}^{2}P_{1/2}^{o}}$ \| 428244 \| 233.511 \| 1.264E-02 \| 1.644E-02 \| 5.027E+08 \| 0.002 $\mathrm{8d~{}^{2}D_{5/2}}$ \| $\mathrm{2p~{}^{2}P_{3/2}^{o}}$ \| 428136 \| 233.570 \| 2.275E-02 \| 2.958E-02 \| 6.028E+08 \| 0.002 – \| – \| – \| – \| – \| – \| – \| – Table 9: Comparison of relative line strengths ($S$), weighted oscillator strengths ($gf$), and lifetimes ($\tau$), or transition probabilities ($A$), with other theoretical work and experimental results for C i – iv. The present values from the MCDHF/RCI calculations are given in the Babushkin(length) gauge. The values in the parentheses are the relative differences between the length and velocity gauges. The references for the experiments are shown in the last column. Note that the sums of the line strengths $S$ have been normalized to 100 for each multiplet in C i. \| \| C i \| \| \| \| \| ---\|---\|---\|---\|---\|---\|---\|--- Transition array \| Mult. \| $J_{u}-J_{l}$ \| $S$ (a.u. of a${}_{0}^{2}$e2) \| \| \| \| MCDHF/RCI \| CIV3(a) \| Expt. \| Expt. \| $\mathrm{2s^{2}2p3p-2s^{2}2p3s}$ \| $\mathrm{{}^{3}D-~{}^{3}P^{o}}$ \| 3 - 2 \| 46.67(1.3%) \| 46.72 \| 46.3 $\pm$ 1.8 (b) \| \| \| \| 2 - 1 \| 25.29(1.2%) \| 25.43 \| 25.5 $\pm$ 1.2 (b) \| \| \| \| 1 - 0 \| 11.25(1.2%) \| 11.29 \| 11.8 $\pm$ 0.5 (b) \| \| \| \| 2 - 2 \| 8.047(1.3%) \| 7.898 \| 7.67 $\pm$ 0.38 (b) \| \| \| \| 1 - 1 \| 8.213(1.3%) \| 8.153 \| 8.42 $\pm$ 0.46 (b) \| \| $\mathrm{2s^{2}2p3p-2s^{2}2p3s}$ \| $\mathrm{{}^{3}P-~{}^{3}P^{o}}$ \| 2 - 2 \| 42.15(0.2%) \| 41.92 \| 40.6 $\pm$ 0.9 (b) \| 41.3(d) \| \| \| 1 - 1 \| 7.812(0.3%) \| 7.873 \| 7.98 $\pm$ 0.23 (b) \| 8.1 (d) \| \| \| 1 - 2 \| 15.07(0.2%) \| 14.79 \| 15.1 $\pm$ 0.4 (b) \| 15.1(d) \| \| \| 0 - 1 \| 11.11(0.2%) \| 11.11 \| 11.3 $\pm$ 0.3 (b) \| 11.6(d) \| \| \| 2 - 1 \| 13.41(0.2%) \| 13.67 \| 14.0 $\pm$ 0.35 (b) \| 13.0(d) \| \| \| 1 - 0 \| 10.44(0.2%) \| 10.64 \| 10.9 $\pm$ 0.3 (b) \| 10.9(d) \| $\mathrm{2s^{2}2p3p-2s^{2}2p3s}$ \| $\mathrm{{}^{3}S-~{}^{3}P^{o}}$ \| 1 - 2 \| 51.96(0.3%) \| 51.43 \| 52.4 $\pm$ 1.1 (b) \| \| \| \| 1 - 1 \| 35.49(0.2%) \| 35.80 \| 34.8 $\pm$ 0.9 (b) \| \| \| \| 1 - 0 \| 12.54(0.2%) \| 12.77 \| 12.8 $\pm$ 0.38 (b) \| \| $\mathrm{2s^{2}2p3d-2s^{2}2p3p}$ \| $\mathrm{{}^{3}P^{o}-~{}^{3}S}$ \| 2 - 1 \| 57.27(3.0%) \| 56.85 \| 59.0$\pm$3.2 (c) \| \| \| \| 1 - 1 \| 32.30(2.9%) \| 32.56 \| 32.1$\pm$4.1 (c) \| \| \| \| 0 - 1 \| 10.43(2.9%) \| 10.59 \| 8.9$\pm$2.1 (c) \| \| $\mathrm{2s^{2}2p3d-2s^{2}2p3p}$ \| $\mathrm{{}^{3}P^{o}-~{}^{3}P}$ \| 2 - 2 \| 42.67(3.3%) \| 42.63 \| 43.5$\pm$0.4 (c) \| \| \| \| 1 - 1 \| 9.465(3.2%) \| 9.669 \| 10.2$\pm$0.6 (c) \| \| \| \| 1 - 2 \| 13.91(3.3%) \| 13.84 \| 14.9$\pm$0.8 (c) \| \| \| \| 0 - 1 \| 11.57(3.3%) \| 11.69 \| 11.2$\pm$0.5 (c) \| \| \| \| 2 - 1 \| 11.72(3.4%) \| 11.47 \| 10.2$\pm$0.5 (c) \| \| \| \| 1 - 0 \| 10.66(3.3%) \| 10.70 \| 10.0$\pm$0.4 (c) \| \| $\mathrm{2s^{2}2p4s-2s^{2}2p3p}$ \| $\mathrm{{}^{3}P^{o}-~{}^{3}P}$ \| 2 - 2 \| 62.05(5.7%) \| 50.76 \| 51.2$\pm$5.0 (c) \| \| \| \| 1 - 1 \| 9.312(7.1%) \| 8.778 \| 8.4$\pm$1.6 (c) \| \| \| \| 1 - 2 \| 13.79(7.0%) \| 14.08 \| 14.3$\pm$2.0 (c) \| \| \| \| 0 - 1 \| 7.335(8.7%) \| 9.670 \| 10.2$\pm$1.5 (c) \| \| \| \| 2 - 1 \| 3.113(11.9%) \| 8.440 \| 10.1$\pm$2.0 (c) \| \| \| \| 1 - 0 \| 4.397(10.3%) \| 8.271 \| 5.8$\pm$1.0 (c) \| \| $\mathrm{2s^{2}2p3d-2s^{2}2p3p}$ \| $\mathrm{{}^{3}D^{o}-~{}^{3}P}$ \| 3 - 2 \| 47.89(1.2%) \| 46.50 \| 45.5$\pm$2.0 (c) \| 45.1(d) \| \| \| 2 - 1 \| 26.70(1.2%) \| 26.59 \| 27.5$\pm$1.2 (c) \| 24.2(d) \| \| \| 1 - 0 \| 9.708(1.2%) \| 11.11 \| 11.0$\pm$0.6 (c) \| 13.6(d) \| \| \| 2 - 2 \| 8.170(0.9%) \| 7.559 \| 7.5$\pm$0.4 (c) \| 8.0 (d) \| \| \| 1 - 1 \| 7.100(1.1%) \| 7.792 \| 8.1$\pm$0.5 (c) \| 9.1 (d) \| $\mathrm{2s^{2}2p4s-2s^{2}2p3p}$ \| $\mathrm{{}^{3}P^{o}-~{}^{3}D}$ \| 2 - 3 \| 44.42(3.2%) \| 44.77 \| 44.5$\pm$2.0 (c) \| \| \| \| 1 - 2 \| 23.65(3.1%) \| 24.14 \| 24.8$\pm$1.2 (c) \| \| \| \| 0 - 1 \| 11.10(3.0%) \| 11.24 \| 11.5$\pm$0.6 (c) \| \| \| \| 2 - 2 \| 9.145(2.6%) \| 9.424 \| 8.4$\pm$0.6 (c) \| \| \| \| 1 - 1 \| 9.301(2.7%) \| 9.071 \| 9.5$\pm$0.5 (c) \| \| \| \| 2 - 1 \| 2.373(1.4%) \| 1.356 \| 1.3$\pm$0.2 (c) \| \| $\mathrm{2s^{2}2p3d-2s^{2}2p3p}$ \| $\mathrm{{}^{3}D^{o}-~{}^{3}D}$ \| 3 - 3 \| 50.31(1.5%) \| 49.19 \| 50.0$\pm$5.0 (c) \| \| \| \| 2 - 2 \| 25.28(1.6%) \| 25.72 \| 25.4$\pm$2.2 (c) \| \| \| \| 1 - 1 \| 11.50(1.7%) \| 12.68 \| 12.3$\pm$0.7 (c) \| \| \| \| 2 - 3 \| 8.250(0.9%) \| 7.291 \| 6.6$\pm$0.6 (c) \| \| \| \| 1 - 2 \| 4.043(1.8%) \| 4.621 \| 4.7$\pm$0.5 (c) \| \| $\mathrm{2s^{2}2p3d-2s^{2}2p3p}$ \| $\mathrm{{}^{3}F^{o}-~{}^{3}D}$ \| 4 - 3 \| 43.41(0.4%) \| 43.53 \| 44.9$\pm$2.0 (c) \| \| \| \| 3 - 2 \| 30.86(0.4%) \| 30.92 \| 30.0$\pm$1.5 (c) \| \| \| \| 2 - 1 \| 20.77(0.3%) \| 21.09 \| 20.4$\pm$1.0 (c) \| \| \| \| 3 - 3 \| 1.961(1.2%) \| 1.743 \| 1.9$\pm$0.2 (c) \| \| \| \| 2 - 2 \| 2.999(0.9%) \| 2.722 \| 2.7$\pm$0.2 (c) \| \| \| \| C ii \| \| \| \| \| Configuration \| Term \| $J$ \| $\tau$ (ns) \| Ref. \| \| \| \| MCDHF/RCI \| MCHF-BP(e) \| Expt. \| \| $\mathrm{2s2p^{2}}$ \| $\mathrm{~{}^{2}S}$ \| 1/2 \| 0.4497 (0.7%) \| 0.4523 \| 0.44$\pm$ 0.02 \| (f) \| $\mathrm{2s^{2}3s}$ \| $\mathrm{~{}^{2}S}$ \| 1/2 \| 2.292 (0.6%) \| 2.266 \| 2.4 $\pm$ 0.3 \| (f) \| $\mathrm{2s^{2}4s}$ \| $\mathrm{~{}^{2}S}$ \| 1/2 \| 2.017 (0.1%) \| \| 1.9 $\pm$ 0.1 \| (f) \| $\mathrm{2s^{2}5s}$ \| $\mathrm{~{}^{2}S}$ \| 1/2 \| 3.774 (0.1%) \| \| 3.7 $\pm$ 0.2 \| (f) \| $\mathrm{2s2p^{2}}$ \| $\mathrm{~{}^{2}P^{o}}$ \| 1/2 \| 0.2446(0.3%) \| 0.2445 \| 0.25$\pm$ 0.01 \| (f) \| \| \| 3/2 \| 0.2445(0.3%) \| 0.2449 \| 0.25$\pm$ 0.01 \| (f) \| $\mathrm{2s^{2}3p}$ \| $\mathrm{~{}^{2}P^{o}}$ \| 1/2 \| 9.265(0.7%) \| 8.973 \| 8.9 $\pm$ 0.4 \| (f) \| \| \| 3/2 \| 9.255(0.7%) \| 8.963 \| 8.9 $\pm$ 0.4 \| (f) \| $\mathrm{2s^{2}4p}$ \| $\mathrm{~{}^{2}P^{o}}$ \| 1/2 \| 3.838(1.3%) \| \| 3.8 $\pm$ 0.2 \| (f) \| \| \| 3/2 \| 3.854(1.3%) \| \| 3.8 $\pm$ 0.2 \| (f) \| $\mathrm{2p^{3}}$ \| $\mathrm{~{}^{2}P^{o}}$ \| 1/2 \| 0.4998(0.8%) \| 0.4966 \| 0.48$\pm$ 0.02 \| (f) \| \| \| 3/2 \| 0.4981(0.8%) \| 0.4944 \| 0.48$\pm$ 0.02 \| (f) \| $\mathrm{2s^{2}5p}$ \| $\mathrm{~{}^{2}P^{o}}$ \| 1/2 \| 5.044(0.2%) \| \| 5.2 $\pm$ 0.3 \| (f) \| \| \| 3/2 \| 5.099(0.2%) \| \| 5.2 $\pm$ 0.3 \| (f) \| $\mathrm{2s^{2}3d}$ \| $\mathrm{~{}^{2}D}$ \| 3/2 \| 0.3490(0.2%) \| 0.3493 \| 0.34$\pm$ 0.01 \| (f) \| \| \| 5/2 \| 0.3491(0.2%) \| 0.3494 \| 0.34$\pm$ 0.01 \| (f) \| $\mathrm{2s^{2}4d}$ \| $\mathrm{~{}^{2}D}$ \| 3/2 \| 0.7299(0.2%) \| \| 0.75$\pm$ 0.03 \| (f) \| \| \| 5/2 \| 0.7304(0.2%) \| \| 0.75$\pm$ 0.03 \| (f) \| Configuration \| Term \| $J$ \| $\tau$ (ms) \| Ref. \| \| \| \| MCDHF/RCI \| MCHF-BP(e) \| Expt. \| \| $\mathrm{2s2p^{2}}$ \| $\mathrm{~{}^{4}P}$ \| 1/2 \| 8.151 (47.6%) \| 7.654 \| 7.95$\pm$0.07 \| (g) \| \| \| 3/2 \| 106.1 (68.5%) \| 96.93 \| 104.1$\pm$0.5 \| (g) \| \| \| 5/2 \| 22.66 (48.0%) \| 22.34 \| 22.05$\pm$0.07 \| (g) \| \| \| C iii \| \| \| \| \| Transition array \| Mult. \| $J_{u}-J_{l}$ \| $gf$ \| Ref. \| \| \| \| MCDHF/RCI \| MCHF-BP(h) \| Expt. \| \| $\mathrm{2s2p}$ \- $\mathrm{2s^{2}}$ \| $\mathrm{{}^{1}P^{o}-~{}^{1}S}$ \| 1-0 \| 0.7592(0.1%) \| 0.7583 \| 0.75 $\pm$ 0.03 \| (f) \| $\mathrm{2p^{2}}$ \- $\mathrm{2s2p}$ \| $\mathrm{{}^{1}S-~{}^{1}P^{o}}$ \| 0-1 \| 0.1623($<$0.05%) \| 0.1622 \| 0.152 $\pm$ 0.009 \| (f) \| $\mathrm{2p^{2}}$ \- $\mathrm{2s2p}$ \| $\mathrm{{}^{1}D-~{}^{1}P^{o}}$ \| 2-1 \| 0.1815(0.5%) \| 0.1819 \| 0.183 $\pm$ 0.005 \| (f) \| Configuration \| Term \| $J$ \| $\tau$ (ns) \| Ref. \| \| \| \| MCDHF/RCI \| MCHF-BP(h) \| Expt. \| \| $\mathrm{2s2p}$ \| $\mathrm{~{}^{1}P^{o}}$ \| 1 \| 0.5638(0.1%) ns \| 0.5651 \| 0.57$\pm$ 0.02 \| (f) \| $\mathrm{2p^{2}}$ \| $\mathrm{~{}^{1}S}$ \| 0 \| 0.4766($<$0.05%) ns \| 0.4764 \| 0.51$\pm$ 0.01 \| (f) \| $\mathrm{2p^{2}}$ \| $\mathrm{~{}^{1}D}$ \| 2 \| 7.240(0.5%) ns \| 7.191 \| 7.2$\pm$ 0.2 \| (f) \| $\mathrm{2s3s}$ \| $\mathrm{~{}^{1}S}$ \| 0 \| 1.164($<$0.05%) ns \| 1.171 \| 1.17$\pm$ 0.05 \| (f) \| \| \| C iv \| \| \| \| \| Transition array \| Mult. \| $J_{u}-J_{l}$ \| $A(10^{8}\mathrm{s^{-1}})$ \| Ref. \| \| \| MCDHF/RCI \| MCHF-BP(j) \| NIST \| Expt. \| $\mathrm{2p-2s}$ \| $\mathrm{{}^{2}P^{o}-~{}^{2}S}$ \| 1/2 - 1/2 \| 2.632($<$0.05%) \| 2.6320 \| 2.65 \| 2.72$\pm$0.07 \| (k) $\mathrm{2p-2s}$ \| $\mathrm{{}^{2}P^{o}-~{}^{2}S}$ \| 3/2 - 1/2 \| 2.646($<$0.05%) \| 2.6459 \| 2.64 \| 2.71$\pm$0.07 \| (k) Configuration \| Term \| $J$ \| $\tau$ (ns) \| Ref. \| \| \| MCDHF/RCI \| MCHF-BP(j) \| Model Potential(o) \| Expt. \| $\mathrm{3s}$ \| $\mathrm{~{}^{2}S}$ \| 1/2 \| 0.2350($<$0.05%) \| 0.2350 \| 0.236 \| 0.25 $\pm$ 0.1 \| (l) $\mathrm{4s}$ \| $\mathrm{~{}^{2}S}$ \| 1/2 \| 0.3755($<$0.05%) \| 0.3747 \| 0.377 \| 0.34 $\pm$ 0.035 \| (m) $\mathrm{2p}$ \| $\mathrm{~{}^{2}P^{o}}$ \| 1/2 \| 3.799 ($<$0.05%) \| 3.799 \| 3.79 \| 3.7 $\pm$ 0.1 \| (k) \| \| 3/2 \| 3.779 ($<$0.05%) \| 3.779 \| 3.79 \| \| $\mathrm{3p}$ \| $\mathrm{~{}^{2}P^{o}}$ \| 1/2 \| 0.2146($<$0.05%) \| 0.2142 \| 0.216 \| 0.226 $\pm$ 0.03 \| (n) \| \| 3/2 \| 0.2149($<$0.05%) \| 0.2145 \| 0.216 \| \| $\mathrm{4p}$ \| $\mathrm{~{}^{2}P^{o}}$ \| 1/2 \| 0.3435($<$0.05%) \| \| 0.344 \| 0.32 $\pm$ 0.03 \| (m) \| \| 3/2 \| 0.3440($<$0.05%) \| \| 0.344 \| \| $\mathrm{3d}$ \| $\mathrm{~{}^{2}D}$ \| 3/2 \| 0.05717($<$0.05%) \| 0.05716 \| 0.0572 \| 0.0575$\pm$ 0.006 \| (n) \| \| 5/2 \| 0.05719($<$0.05%) \| 0.05719 \| 0.0572 \| \| $\mathrm{4d}$ \| $\mathrm{~{}^{2}D}$ \| 3/2 \| 0.1312 ($<$0.05%) \| \| 0.130 \| 0.14 $\pm$ 0.015 \| (m) \| \| 5/2 \| 0.1313 ($<$0.05%) \| \| 0.130 \| \| $\mathrm{5d}$ \| $\mathrm{~{}^{2}D}$ \| 3/2 \| 0.2511 ($<$0.05%) \| \| 0.251 \| 0.23 $\pm$ 0.023 \| (m) \| \| 5/2 \| 0.2512 ($<$0.05%) \| \| 0.251 \| \| (a)Hibbert et al. (1993); (b)Musielok et al. (1997); (c)Bacawski et al. (2001); (d)Golly et al. (2003); (e)Tachiev & Fischer (2000); (f)Reistad et al. (1986); (g)Träbert et al. (1999); (h)Tachiev & Fischer (1999); (j)Fischer et al. (1998); (k)Knystautas et al. (1971); (l)Donnelly et al. (1978); (m)Buchet- Poulizac & Buchet (1973b); (n)Jacques et al. (1980); (o)Peach et al. (1988). Table 10: Comparison of line strengths ($S$) and transition rates ($A$) with other theoretical results for C i. The present values from the MCDHF/RCI calculations are given in the Babushkin(length) gauge. The wavenumber $\Delta E$ and wavelength $\lambda$ values are taken from the NIST database. The estimated uncertainties $dT$ of the transition rates are given as percentages in parentheses. Transition array \| Mult. \| $J_{u}-J_{l}$ \| $\Delta E$ \| $\lambda$ \| MCDHF/RCI \| \| Spline FCS(a) \| \| MCHF-BP(b) ---\|---\|---\|---\|---\|---\|---\|---\|---\|--- \| \| \| ($\mathrm{cm^{-1}}$) \| (Å) \| $S$ (a.u. of a${}_{0}^{2}$e2) \| $A$ (s-1) \| \| $S$ (a.u. of a${}_{0}^{2}$e2) \| $A$ (s-1) \| \| $S$ (a.u. of a${}_{0}^{2}$e2) \| $A$ (s-1) $\mathrm{2s^{2}2p3d-2s^{2}2p^{2}}$ \| $\mathrm{{}^{3}F^{o}-~{}^{3}P}$ \| 3 - 2 \| 78172 \| 1279.228 \| 5.79E-02 \| 7.92E+06(0.1%) \| \| 4.52E-02 \| 6.24E+06 \| \| 8.20E-02 \| 1.14E+07 \| \| 2 - 2 \| 78155 \| 1279.498 \| 1.27E-02 \| 2.43E+06(0.3%) \| \| 9.62E-03 \| 1.86E+06 \| \| 1.16E-02 \| 2.25E+06 \| \| 2 - 1 \| 78182 \| 1279.056 \| 1.08E-02 \| 2.07E+06(0.1%) \| \| 8.95E-03 \| 1.73E+06 \| \| 2.08E-02 \| 4.04E+06 $\mathrm{2s^{2}2p4d-2s^{2}2p^{2}}$ \| $\mathrm{{}^{3}F^{o}-~{}^{3}P}$ \| 3 - 2 \| 83717 \| 1194.488 \| 4.48E-02 \| 7.52E+06(0.1%) \| \| 3.99E-02 \| 6.77E+06 \| \| \| \| \| 2 - 2 \| 83709 \| 1194.614 \| 1.59E-03 \| 3.74E+05(0.5%) \| \| 1.44E-03 \| 3.42E+05 \| \| \| \| \| 2 - 1 \| 83736 \| 1194.229 \| 1.97E-02 \| 4.64E+06(0.2%) \| \| 1.73E-02 \| 4.10E+06 \| \| \| $\mathrm{2s^{2}2p5d-2s^{2}2p^{2}}$ \| $\mathrm{{}^{3}F^{o}-~{}^{3}P}$ \| 3 - 2 \| 86283 \| 1158.966 \| 4.17E-02 \| 7.67E+06(0.3%) \| \| 3.88E-02 \| 7.21E+06 \| \| \| \| \| 2 - 2 \| 86274 \| 1159.094 \| 1.38E-03 \| 3.55E+05(0.3%) \| \| 1.36E-03 \| 3.54E+05 \| \| \| \| \| 2 - 1 \| 86301 \| 1158.731 \| 2.20E-02 \| 5.66E+06(0.3%) \| \| 1.96E-02 \| 5.11E+06 \| \| \| $\mathrm{2s^{2}2p3s-2s^{2}2p^{2}}$ \| $\mathrm{{}^{3}P^{o}-~{}^{3}P}$ \| 2 - 2 \| 60349 \| 1657.008 \| 2.84E+00 \| 2.50E+08(1.2%) \| \| 2.69E+00 \| 2.41E+08 \| \| 2.93E+00 \| 2.61E+08 \| \| 1 - 2 \| 60309 \| 1658.121 \| 9.46E-01 \| 1.39E+08(1.2%) \| \| 8.95E-01 \| 1.33E+08 \| \| 9.75E-01 \| 1.45E+08 \| \| 2 - 1 \| 60376 \| 1656.267 \| 9.47E-01 \| 8.35E+07(1.2%) \| \| 8.97E-01 \| 8.05E+07 \| \| 9.78E-01 \| 8.73E+07 \| \| 1 - 1 \| 60336 \| 1657.379 \| 5.67E-01 \| 8.32E+07(1.2%) \| \| 5.37E-01 \| 8.01E+07 \| \| 5.84E-01 \| 8.67E+07 \| \| 0 - 1 \| 60317 \| 1657.907 \| 7.57E-01 \| 3.33E+08(1.2%) \| \| 7.17E-01 \| 3.20E+08 \| \| 7.80E-01 \| 3.47E+08 \| \| 1 - 0 \| 60352 \| 1656.928 \| 7.57E-01 \| 1.11E+08(1.2%) \| \| 7.17E-01 \| 1.07E+08 \| \| 7.81E-01 \| 1.16E+08 $\mathrm{2s^{2}2p3s-2s^{2}2p^{2}}$ \| $\mathrm{{}^{1}P^{o}-~{}^{3}P}$ \| 1 - 2 \| 61938 \| 1614.507 \| 1.80E-04 \| 2.86E+04(1.6%) \| \| 2.72E-04 \| 4.40E+04 \| \| 1.85E-04 \| 2.97E+04 \| \| 1 - 1 \| 61965 \| 1613.803 \| 1.65E-04 \| 2.61E+04(1.0%) \| \| 1.62E-04 \| 2.62E+04 \| \| 1.74E-04 \| 2.81E+04 \| \| 1 - 0 \| 61981 \| 1613.376 \| 2.21E-04 \| 3.51E+04(1.1%) \| \| 2.34E-04 \| 3.79E+04 \| \| 2.26E-04 \| 3.65E+04 $\mathrm{2s2p^{3}-2s^{2}2p^{2}}$ \| $\mathrm{{}^{3}D^{o}-~{}^{3}P}$ \| 3 - 2 \| 64043 \| 1561.437 \| 1.53E+00 \| 1.22E+08(2.3%) \| \| 1.42E+00 \| 1.14E+08 \| \| 1.54E+00 \| 1.19E+08 \| \| 2 - 2 \| 64046 \| 1561.366 \| 2.72E-01 \| 3.03E+07(2.1%) \| \| 2.52E-01 \| 2.84E+07 \| \| 2.75E-01 \| 2.96E+07 \| \| 1 - 2 \| 64047 \| 1561.339 \| 1.81E-02 \| 3.35E+06(2.0%) \| \| 1.67E-02 \| 3.14E+06 \| \| 1.83E-02 \| 3.28E+06 \| \| 2 - 1 \| 64073 \| 1560.708 \| 8.22E-01 \| 9.14E+07(2.2%) \| \| 7.60E-01 \| 8.59E+07 \| \| 8.28E-01 \| 8.91E+07 \| \| 1 - 1 \| 64074 \| 1560.681 \| 2.73E-01 \| 5.06E+07(2.1%) \| \| 2.53E-01 \| 4.76E+07 \| \| 2.76E-01 \| 4.94E+07 \| \| 1 - 0 \| 64090 \| 1560.282 \| 3.65E-01 \| 6.78E+07(2.1%) \| \| 3.38E-01 \| 6.37E+07 \| \| 3.68E-01 \| 6.61E+07 $\mathrm{2s2p^{3}-2s^{2}2p^{2}}$ \| $\mathrm{{}^{3}P^{o}-~{}^{3}P}$ \| 2 - 2 \| 75212 \| 1329.562 \| 7.98E-01 \| 1.42E+08(3.1%) \| \| 7.44E-01 \| 1.35E+08 \| \| 9.54E-01 \| 1.66E+08 \| \| 1 - 2 \| 75210 \| 1329.600 \| 2.69E-01 \| 7.95E+07(3.1%) \| \| 2.51E-01 \| 7.57E+07 \| \| 3.20E-01 \| 9.28E+07 \| \| 2 - 1 \| 75239 \| 1329.085 \| 2.58E-01 \| 4.58E+07(3.1%) \| \| 2.40E-01 \| 4.34E+07 \| \| 3.12E-01 \| 5.44E+07 \| \| 1 - 1 \| 75237 \| 1329.123 \| 1.64E-01 \| 4.85E+07(3.1%) \| \| 1.53E-01 \| 4.63E+07 \| \| 1.93E-01 \| 5.60E+07 \| \| 0 - 1 \| 75238 \| 1329.100 \| 2.17E-01 \| 1.93E+08(3.1%) \| \| 2.04E-01 \| 1.85E+08 \| \| 2.57E-01 \| 2.24E+08 \| \| 1 - 0 \| 75254 \| 1328.833 \| 2.13E-01 \| 6.30E+07(3.1%) \| \| 1.99E-01 \| 6.00E+07 \| \| 2.54E-01 \| 7.37E+07 $\mathrm{2s^{2}2p3d-2s^{2}2p^{2}}$ \| $\mathrm{{}^{1}D^{o}-~{}^{3}P}$ \| 2 - 2 \| 77636 \| 1288.055 \| 4.63E-04 \| 8.69E+04(1.4%) \| \| 4.27E-04 \| 8.10E+04 \| \| 3.51E-04 \| 6.68E+04 \| \| 2 - 1 \| 77663 \| 1287.608 \| 8.40E-04 \| 1.58E+05(1.2%) \| \| 7.68E-04 \| 1.46E+05 \| \| 7.84E-04 \| 1.49E+05 $\mathrm{2s^{2}2p4s-2s^{2}2p^{2}}$ \| $\mathrm{{}^{3}P^{o}-~{}^{3}P}$ \| 2 - 2 \| 78104 \| 1280.333 \| 3.23E-01 \| 6.17E+07(0.7%) \| \| 3.21E-01 \| 6.21E+07 \| \| 3.30E-01 \| 6.40E+07 \| \| 1 - 2 \| 78073 \| 1280.847 \| 1.13E-01 \| 3.59E+07(0.5%) \| \| 1.10E-01 \| 3.55E+07 \| \| 1.13E-01 \| 3.63E+07 \| \| 2 - 1 \| 78131 \| 1279.890 \| 1.96E-01 \| 3.76E+07(0.5%) \| \| 1.81E-01 \| 3.50E+07 \| \| 1.77E-01 \| 3.44E+07 \| \| 1 - 1 \| 78100 \| 1280.404 \| 5.55E-02 \| 1.77E+07(0.6%) \| \| 5.57E-02 \| 1.79E+07 \| \| 5.83E-02 \| 1.88E+07 \| \| 0 - 1 \| 78088 \| 1280.597 \| 9.23E-02 \| 8.82E+07(0.5%) \| \| 8.91E-02 \| 8.61E+07 \| \| 9.14E-02 \| 8.84E+07 \| \| 1 - 0 \| 78116 \| 1280.135 \| 1.16E-01 \| 3.71E+07(0.5%) \| \| 1.11E-01 \| 3.57E+07 \| \| 1.10E-01 \| 3.56E+07 $\mathrm{2s^{2}2p3d-2s^{2}2p^{2}}$ \| $\mathrm{{}^{3}D^{o}-~{}^{3}P}$ \| 3 - 2 \| 78274 \| 1277.550 \| 1.68E+00 \| 2.31E+08(0.2%) \| \| 1.61E+00 \| 2.23E+08 \| \| 1.64E+00 \| 2.28E+08 \| \| 2 - 2 \| 78264 \| 1277.723 \| 3.44E-01 \| 6.61E+07(0.1%) \| \| 3.22E-01 \| 6.26E+07 \| \| 3.24E-01 \| 6.32E+07 \| \| 1 - 2 \| 78250 \| 1277.954 \| 1.47E-02 \| 4.70E+06($<$0.05%) \| \| 1.72E-02 \| 5.58E+06 \| \| 1.84E-02 \| 5.95E+06 \| \| 2 - 1 \| 78291 \| 1277.282 \| 8.75E-01 \| 1.68E+08(0.2%) \| \| 8.42E-01 \| 1.64E+08 \| \| 8.73E-01 \| 1.70E+08 \| \| 1 - 1 \| 78277 \| 1277.513 \| 2.43E-01 \| 7.80E+07(0.1%) \| \| 2.66E-01 \| 8.61E+07 \| \| 2.85E-01 \| 9.26E+07 \| \| 1 - 0 \| 78293 \| 1277.245 \| 3.42E-01 \| 1.10E+08(0.2%) \| \| 3.59E-01 \| 1.17E+08 \| \| 3.88E-01 \| 1.26E+08 $\mathrm{2s^{2}2p4s-2s^{2}2p^{2}}$ \| $\mathrm{{}^{1}P^{o}-~{}^{3}P}$ \| 1 - 2 \| 78296 \| 1277.190 \| 1.14E-02 \| 3.64E+06(0.4%) \| \| 6.78E-03 \| 2.20E+06 \| \| 5.59E-03 \| 1.82E+06 \| \| 1 - 1 \| 78323 \| 1276.750 \| 7.84E-02 \| 2.52E+07(0.2%) \| \| 3.90E-02 \| 1.27E+07 \| \| 2.89E-02 \| 9.41E+06 \| \| 1 - 0 \| 78340 \| 1276.482 \| 5.97E-02 \| 1.92E+07(0.1%) \| \| 2.48E-02 \| 8.05E+06 \| \| 1.55E-02 \| 5.05E+06 $\mathrm{2s^{2}2p3d-2s^{2}2p^{2}}$ \| $\mathrm{{}^{1}F^{o}-~{}^{3}P}$ \| 3 - 2 \| 78486 \| 1274.109 \| 1.00E-02 \| 1.39E+06(0.1%) \| \| 1.18E-02 \| 1.65E+06 \| \| 8.60E-03 \| 1.21E+06 $\mathrm{2s^{2}2p3d-2s^{2}2p^{2}}$ \| $\mathrm{{}^{1}P^{o}-~{}^{3}P}$ \| 1 - 2 \| 78687 \| 1270.844 \| 1.77E-06 \| 5.75E+02(0.4%) \| \| 3.03E-07 \| 9.97E+01 \| \| 1.28E-05 \| 4.23E+03 \| \| 1 - 1 \| 78714 \| 1270.408 \| 6.02E-04 \| 1.96E+05(0.3%) \| \| 5.66E-04 \| 1.87E+05 \| \| 6.50E-04 \| 2.15E+05 \| \| 1 - 0 \| 78731 \| 1270.143 \| 1.65E-03 \| 5.39E+05(0.1%) \| \| 1.61E-03 \| 5.32E+05 \| \| 1.38E-03 \| 4.55E+05 $\mathrm{2s^{2}2p3d-2s^{2}2p^{2}}$ \| $\mathrm{{}^{3}P^{o}-~{}^{3}P}$ \| 2 - 2 \| 79267 \| 1261.552 \| 8.17E-01 \| 1.66E+08(2.1%) \| \| 8.13E-01 \| 1.67E+08 \| \| 6.65E-01 \| 1.35E+08 \| \| 1 - 2 \| 79275 \| 1261.425 \| 2.74E-01 \| 9.26E+07(2.1%) \| \| 2.72E-01 \| 9.31E+07 \| \| 2.23E-01 \| 7.55E+07 \| \| 2 - 1 \| 79294 \| 1261.122 \| 2.54E-01 \| 5.15E+07(2.1%) \| \| 2.52E-01 \| 5.19E+07 \| \| 1.98E-01 \| 4.02E+07 \| \| 1 - 1 \| 79302 \| 1260.996 \| 1.69E-01 \| 5.72E+07(2.1%) \| \| 1.68E-01 \| 5.75E+07 \| \| 1.39E-01 \| 4.70E+07 \| \| 0 - 1 \| 79306 \| 1260.926 \| 2.20E-01 \| 2.24E+08(2.1%) \| \| 2.18E-01 \| 2.24E+08 \| \| 1.79E-01 \| 1.82E+08 \| \| 1 - 0 \| 79318 \| 1260.735 \| 2.12E-01 \| 7.19E+07(2.1%) \| \| 2.11E-01 \| 7.23E+07 \| \| 1.69E-01 \| 5.73E+07 $\mathrm{2s^{2}2p3d-2s^{2}2p^{2}}$ \| $\mathrm{{}^{3}F^{o}-~{}^{1}D}$ \| 2 - 2 \| 68006 \| 1470.449 \| 4.71E-04 \| 5.91E+04(1.4%) \| \| \| \| \| 4.22E-04 \| 5.36E+04 \| \| 3 - 2 \| 68022 \| 1470.094 \| 1.52E-02 \| 1.36E+06(0.2%) \| \| \| \| \| 1.55E-02 \| 1.41E+06 $\mathrm{2s^{2}2p3s-2s^{2}2p^{2}}$ \| $\mathrm{{}^{3}P^{o}-~{}^{1}D}$ \| 1 - 2 \| 50160 \| 1993.620 \| 1.04E-03 \| 8.67E+04(1.4%) \| \| \| \| \| 9.65E-04 \| 8.18E+04 \| \| 2 - 2 \| 50200 \| 1992.012 \| 1.97E-05 \| 9.90E+02(3.2%) \| \| \| \| \| 1.50E-05 \| 7.66E+02 $\mathrm{2s^{2}2p3s-2s^{2}2p^{2}}$ \| $\mathrm{{}^{1}P^{o}-~{}^{1}D}$ \| 1 - 2 \| 51789 \| 1930.905 \| 3.59E+00 \| 3.30E+08(1.5%) \| \| \| \| \| 3.62E+00 \| 3.37E+08 $\mathrm{2s2p^{3}-2s^{2}2p^{2}}$ \| $\mathrm{{}^{3}D^{o}-~{}^{1}D}$ \| 3 - 2 \| 53894 \| 1855.483 \| 9.91E-06 \| 4.71E+02(24.2%) \| \| \| \| \| 6.60E-06 \| 3.01E+02 $\mathrm{2s2p^{3}-2s^{2}2p^{2}}$ \| $\mathrm{{}^{3}P^{o}-~{}^{1}D}$ \| 1 - 2 \| 65061 \| 1537.011 \| 8.58E-06 \| 1.65E+03(1.7%) \| \| \| \| \| 8.59E-08 \| 1.60E+01 \| \| 2 - 2 \| 65063 \| 1536.960 \| 2.51E-05 \| 2.89E+03(6.9%) \| \| \| \| \| 7.15E-06 \| 8.02E+02 $\mathrm{2s^{2}2p3d-2s^{2}2p^{2}}$ \| $\mathrm{{}^{1}D^{o}-~{}^{1}D}$ \| 2 - 2 \| 67487 \| 1481.763 \| 3.03E-01 \| 3.72E+07(1.2%) \| \| \| \| \| 2.77E-01 \| 3.45E+07 $\mathrm{2s^{2}2p4s-2s^{2}2p^{2}}$ \| $\mathrm{{}^{3}P^{o}-~{}^{1}D}$ \| 1 - 2 \| 67924 \| 1472.231 \| 4.55E-03 \| 9.48E+05(1.6%) \| \| \| \| \| 3.68E-03 \| 7.77E+05 \| \| 2 - 2 \| 67955 \| 1471.552 \| 2.02E-04 \| 2.53E+04(1.0%) \| \| \| \| \| 1.77E-04 \| 2.25E+04 $\mathrm{2s^{2}2p3d-2s^{2}2p^{2}}$ \| $\mathrm{{}^{3}D^{o}-~{}^{1}D}$ \| 3 - 2 \| 68125 \| 1467.877 \| 7.46E-03 \| 6.72E+05(0.2%) \| \| \| \| \| 5.35E-03 \| 4.88E+05 \| \| 2 - 2 \| 68114 \| 1468.106 \| 5.36E-05 \| 6.76E+03(2.9%) \| \| \| \| \| 7.36E-05 \| 9.40E+03 \| \| 1 - 2 \| 68100 \| 1468.410 \| 4.59E-02 \| 9.64E+06(2.2%) \| \| \| \| \| 1.13E-02 \| 2.40E+06 $\mathrm{2s^{2}2p4s-2s^{2}2p^{2}}$ \| $\mathrm{{}^{1}P^{o}-~{}^{1}D}$ \| 1 - 2 \| 68147 \| 1467.402 \| 2.36E-01 \| 4.97E+07(2.0%) \| \| \| \| \| 2.57E-01 \| 5.48E+07 $\mathrm{2s^{2}2p3d-2s^{2}2p^{2}}$ \| $\mathrm{{}^{1}F^{o}-~{}^{1}D}$ \| 3 - 2 \| 68336 \| 1463.336 \| 1.99E+00 \| 1.81E+08(0.3%) \| \| \| \| \| 1.94E+00 \| 1.78E+08 $\mathrm{2s^{2}2p3d-2s^{2}2p^{2}}$ \| $\mathrm{{}^{1}P^{o}-~{}^{1}D}$ \| 1 - 2 \| 68538 \| 1459.031 \| 2.18E-01 \| 4.66E+07(0.1%) \| \| \| \| \| 2.51E-01 \| 5.44E+07 $\mathrm{2s^{2}2p3d-2s^{2}2p^{2}}$ \| $\mathrm{{}^{3}P^{o}-~{}^{1}D}$ \| 2 - 2 \| 69118 \| 1446.797 \| 1.83E-05 \| 2.46E+03(5.2%) \| \| \| \| \| 3.09E-05 \| 4.13E+03 \| \| 1 - 2 \| 69126 \| 1446.630 \| 4.50E-06 \| 1.01E+03(5.4%) \| \| \| \| \| 2.65E-06 \| 5.92E+02 $\mathrm{2s^{2}2p3s-2s^{2}2p^{2}}$ \| $\mathrm{{}^{3}P^{o}-~{}^{1}S}$ \| 1 - 0 \| 38704 \| 2583.670 \| 1.61E-04 \| 6.13E+03(5.2%) \| \| \| \| \| 1.48E-04 \| 5.72E+03 $\mathrm{2s^{2}2p3s-2s^{2}2p^{2}}$ \| $\mathrm{{}^{1}P^{o}-~{}^{1}S}$ \| 1 - 0 \| 40333 \| 2479.310 \| 6.69E-01 \| 2.89E+07(3.7%) \| \| \| \| \| 6.31E-01 \| 2.76E+07 $\mathrm{2s2p^{3}-2s^{2}2p^{2}}$ \| $\mathrm{{}^{3}P^{o}-~{}^{1}S}$ \| 1 - 0 \| 53605 \| 1865.464 \| 6.83E-06 \| 7.36E+02(23.2%) \| \| \| \| \| 2.71E-06 \| 2.83E+02 $\mathrm{2s^{2}2p4s-2s^{2}2p^{2}}$ \| $\mathrm{{}^{3}P^{o}-~{}^{1}S}$ \| 1 - 0 \| 56468 \| 1770.891 \| 9.52E-05 \| 1.13E+04(6.6%) \| \| \| \| \| 8.80E-05 \| 1.06E+04 $\mathrm{2s^{2}2p3d-2s^{2}2p^{2}}$ \| $\mathrm{{}^{3}D^{o}-~{}^{1}S}$ \| 1 - 0 \| 56645 \| 1765.366 \| 1.32E-02 \| 1.59E+06(2.9%) \| \| \| \| \| 6.33E-03 \| 7.72E+05 $\mathrm{2s^{2}2p4s-2s^{2}2p^{2}}$ \| $\mathrm{{}^{1}P^{o}-~{}^{1}S}$ \| 1 - 0 \| 56692 \| 1763.909 \| 1.72E-02 \| 2.07E+06(5.0%) \| \| \| \| \| 1.99E-02 \| 2.43E+06 $\mathrm{2s^{2}2p3d-2s^{2}2p^{2}}$ \| $\mathrm{{}^{1}P^{o}-~{}^{1}S}$ \| 1 - 0 \| 57083 \| 1751.827 \| 7.32E-01 \| 9.00E+07($<$0.05%) \| \| \| \| \| 6.67E-01 \| 8.33E+07 $\mathrm{2s^{2}2p3d-2s^{2}2p^{2}}$ \| $\mathrm{{}^{3}P^{o}-~{}^{1}S}$ \| 1 - 0 \| 57670 \| 1733.980 \| 4.99E-05 \| 6.48E+03(8.1%) \| \| \| \| \| 1.18E-04 \| 1.53E+04 $\mathrm{2s^{2}2p4d-2s^{2}2p^{2}}$ \| $\mathrm{{}^{1}D^{o}-~{}^{3}P}$ \| 2 - 2 \| 83454 \| 1198.262 \| 3.73E-04 \| 8.70E+04(0.8%) \| \| 2.87E-04 \| 6.75E+04 \| \| \| \| \| 2 - 1 \| 83481 \| 1197.875 \| 1.03E-03 \| 2.39E+05(0.8%) \| \| 8.64E-04 \| 2.04E+05 \| \| \| $\mathrm{2s^{2}2p5s-2s^{2}2p^{2}}$ \| $\mathrm{{}^{3}P^{o}-~{}^{3}P}$ \| 2 - 2 \| 83747 \| 1194.063 \| 1.18E-01 \| 2.78E+07(0.9%) \| \| 1.25E-01 \| 2.98E+07 \| \| \| \| \| 1 - 2 \| 83704 \| 1194.686 \| 4.34E-02 \| 1.70E+07(0.8%) \| \| 4.47E-02 \| 1.77E+07 \| \| \| \| \| 2 - 1 \| 83774 \| 1193.678 \| 1.17E-01 \| 2.75E+07(0.6%) \| \| 1.07E-01 \| 2.54E+07 \| \| \| \| \| 1 - 1 \| 83731 \| 1194.301 \| 1.95E-02 \| 7.67E+06(0.9%) \| \| 2.10E-02 \| 8.34E+06 \| \| \| \| \| 0 - 1 \| 83723 \| 1194.405 \| 3.64E-02 \| 4.29E+07(0.7%) \| \| 3.70E-02 \| 4.40E+07 \| \| \| \| \| 1 - 0 \| 83747 \| 1194.066 \| 4.98E-02 \| 1.95E+07(0.7%) \| \| 4.90E-02 \| 1.94E+07 \| \| \| $\mathrm{2s^{2}2p4d-2s^{2}2p^{2}}$ \| $\mathrm{{}^{3}D^{o}-~{}^{3}P}$ \| 3 - 2 \| 83805 \| 1193.240 \| 6.80E-01 \| 1.15E+08(0.1%) \| \| 6.52E-01 \| 1.11E+08 \| \| \| \| \| 2 - 2 \| 83794 \| 1193.393 \| 1.57E-01 \| 3.70E+07(0.1%) \| \| 1.45E-01 \| 3.46E+07 \| \| \| \| \| 1 - 2 \| 83776 \| 1193.649 \| 5.60E-03 \| 2.20E+06(0.1%) \| \| 5.67E-03 \| 2.25E+06 \| \| \| \| \| 2 - 1 \| 83821 \| 1193.009 \| 3.37E-01 \| 7.95E+07($<$0.05%) \| \| 3.31E-01 \| 7.90E+07 \| \| \| \| \| 1 - 1 \| 83803 \| 1193.264 \| 1.10E-01 \| 4.31E+07($<$0.05%) \| \| 1.08E-01 \| 4.29E+07 \| \| \| \| \| 1 - 0 \| 83820 \| 1193.030 \| 1.65E-01 \| 6.50E+07(0.1%) \| \| 1.61E-01 \| 6.39E+07 \| \| \| $\mathrm{2s^{2}2p5s-2s^{2}2p^{2}}$ \| $\mathrm{{}^{1}P^{o}-~{}^{3}P}$ \| 1 - 2 \| 83833 \| 1192.835 \| 6.48E-03 \| 2.55E+06(0.5%) \| \| 5.50E-03 \| 2.19E+06 \| \| \| \| \| 1 - 1 \| 83860 \| 1192.451 \| 2.38E-02 \| 9.38E+06(0.3%) \| \| 1.91E-02 \| 7.61E+06 \| \| \| \| \| 1 - 0 \| 83877 \| 1192.218 \| 7.36E-03 \| 2.90E+06(0.3%) \| \| 5.39E-03 \| 2.15E+06 \| \| \| $\mathrm{2s^{2}2p4d-2s^{2}2p^{2}}$ \| $\mathrm{{}^{1}F^{o}-~{}^{3}P}$ \| 3 - 2 \| 83903 \| 1191.841 \| 1.54E-02 \| 2.61E+06(0.1%) \| \| 1.70E-02 \| 2.90E+06 \| \| \| $\mathrm{2s^{2}2p4d-2s^{2}2p^{2}}$ \| $\mathrm{{}^{1}P^{o}-~{}^{3}P}$ \| 1 - 2 \| 83988 \| 1190.636 \| 3.47E-05 \| 1.37E+04(0.3%) \| \| 3.15E-05 \| 1.26E+04 \| \| \| \| \| 1 - 1 \| 84015 \| 1190.253 \| 1.25E-03 \| 4.93E+05(0.1%) \| \| 1.20E-03 \| 4.79E+05 \| \| \| \| \| 1 - 0 \| 84032 \| 1190.021 \| 2.39E-03 \| 9.48E+05($<$0.05%) \| \| 2.26E-03 \| 9.05E+05 \| \| \| $\mathrm{2s^{2}2p4d-2s^{2}2p^{2}}$ \| $\mathrm{{}^{3}P^{o}-~{}^{3}P}$ \| 2 - 2 \| 84059 \| 1189.631 \| 2.38E-01 \| 5.68E+07(1.0%) \| \| 2.19E-01 \| 5.28E+07 \| \| \| \| \| 1 - 2 \| 84072 \| 1189.447 \| 8.04E-02 \| 3.20E+07(1.0%) \| \| 7.37E-02 \| 2.96E+07 \| \| \| \| \| 2 - 1 \| 84086 \| 1189.249 \| 5.07E-02 \| 1.21E+07(1.1%) \| \| 4.53E-02 \| 1.09E+07 \| \| \| \| \| 1 - 1 \| 84099 \| 1189.065 \| 5.43E-02 \| 2.16E+07(1.0%) \| \| 5.01E-02 \| 2.02E+07 \| \| \| \| \| 0 - 1 \| 84104 \| 1188.993 \| 6.52E-02 \| 7.79E+07(1.0%) \| \| 5.96E-02 \| 7.19E+07 \| \| \| \| \| 1 - 0 \| 84116 \| 1188.833 \| 5.35E-02 \| 2.13E+07(1.1%) \| \| 4.86E-02 \| 1.96E+07 \| \| \| $\mathrm{2s^{2}2p5d-2s^{2}2p^{2}}$ \| $\mathrm{{}^{1}D^{o}-~{}^{3}P}$ \| 2 - 2 \| 86141 \| 1160.876 \| 6.52E-04 \| 1.67E+05($<$0.05%) \| \| 4.58E-04 \| 1.19E+05 \| \| \| \| \| 2 - 1 \| 86168 \| 1160.513 \| 1.50E-03 \| 3.86E+05(0.8%) \| \| 1.26E-03 \| 3.27E+05 \| \| \| $\mathrm{2s^{2}2p6s-2s^{2}2p^{2}}$ \| $\mathrm{{}^{3}P^{o}-~{}^{3}P}$ \| 2 - 2 \| 86325 \| 1158.400 \| 2.62E-02 \| 6.76E+06(0.1%) \| \| 3.15E-02 \| 8.19E+06 \| \| \| \| \| 1 - 2 \| 86288 \| 1158.907 \| 1.78E-02 \| 7.62E+06(0.2%) \| \| 1.93E-02 \| 8.36E+06 \| \| \| \| \| 2 - 1 \| 86352 \| 1158.038 \| 1.27E-01 \| 3.29E+07(0.2%) \| \| 1.19E-01 \| 3.10E+07 \| \| \| \| \| 1 - 1 \| 86315 \| 1158.544 \| 6.63E-03 \| 2.85E+06(0.4%) \| \| 7.73E-03 \| 3.36E+06 \| \| \| \| \| 0 - 1 \| 86305 \| 1158.674 \| 1.57E-02 \| 2.02E+07(0.3%) \| \| 1.68E-02 \| 2.19E+07 \| \| \| \| \| 1 - 0 \| 86331 \| 1158.324 \| 2.44E-02 \| 1.05E+07(0.1%) \| \| 2.50E-02 \| 1.09E+07 \| \| \| $\mathrm{2s^{2}2p5d-2s^{2}2p^{2}}$ \| $\mathrm{{}^{3}D^{o}-~{}^{3}P}$ \| 3 - 2 \| 86354 \| 1158.018 \| 3.20E-01 \| 5.90E+07(0.3%) \| \| 2.99E-01 \| 5.56E+07 \| \| \| \| \| 2 - 2 \| 86346 \| 1158.131 \| 1.05E-01 \| 2.71E+07($<$0.05%) \| \| 9.90E-02 \| 2.58E+07 \| \| \| \| \| 1 - 2 \| 86319 \| 1158.492 \| 1.37E-03 \| 5.88E+05(0.2%) \| \| 1.45E-03 \| 6.29E+05 \| \| \| \| \| 2 - 1 \| 86373 \| 1157.769 \| 1.03E-01 \| 2.65E+07(0.4%) \| \| 1.00E-01 \| 2.62E+07 \| \| \| \| \| 1 - 1 \| 86346 \| 1158.130 \| 4.69E-02 \| 2.02E+07(0.2%) \| \| 4.55E-02 \| 1.98E+07 \| \| \| \| \| 1 - 0 \| 86362 \| 1157.909 \| 8.57E-02 \| 3.69E+07(0.3%) \| \| 8.08E-02 \| 3.51E+07 \| \| \| Table 10: Continued. (a)Zatsarinny & Fischer (2002); (b)Fischer (2006). Table 11: Comparison of line strengths ($S$) and transition rates ($A$) with other theoretical results for C ii. The present values from the MCDHF/RCI calculations are given in the Babushkin(length) gauge. The wavenumber $\Delta E$ and wavelength $\lambda$ values are taken from the NIST database. The estimated uncertainties $dT$ of the transition rates are given as percentages in parentheses. Transition array \| Mult. \| $J_{u}-J_{l}$ \| $\Delta E$ \| $\lambda$ \| MCDHF/RCI \| \| MCHF-BP(a) \| \| CIV3(b) ---\|---\|---\|---\|---\|---\|---\|---\|---\|--- \| \| \| ($\mathrm{cm^{-1}}$) \| (Å) \| $S$ (a.u. of a${}_{0}^{2}$e2) \| $A$ (s-1) \| \| $S$ (a.u. of a${}_{0}^{2}$e2) \| $A$ (s-1) \| \| $A$ (s-1) $\mathrm{2s2p^{2}-2s^{2}2p}$ \| $\mathrm{{}^{2}D-~{}^{2}P^{o}}$ \| 5/2 - 3/2 \| 74866 \| 1335.708 \| 2.03E+00 \| 2.89E+08(0.2%) \| \| 2.03E+00 \| 2.90E+08 \| \| 2.89E+08 \| \| 3/2 - 3/2 \| 74869 \| 1335.663 \| 2.24E-01 \| 4.79E+07($<$0.05%) \| \| 2.24E-01 \| 4.80E+07 \| \| 4.79E+07 \| \| 3/2 - 1/2 \| 74932 \| 1334.532 \| 1.13E+00 \| 2.42E+08(0.1%) \| \| 1.13E+00 \| 2.43E+08 \| \| 2.42E+08 $\mathrm{2s2p^{2}-2s^{2}2p}$ \| $\mathrm{{}^{2}S-~{}^{2}P^{o}}$ \| 1/2 - 3/2 \| 96430 \| 1037.018 \| 1.62E+00 \| 1.48E+09(0.7%) \| \| 1.61E+00 \| 1.47E+09 \| \| 1.47E+09 \| \| 1/2 - 1/2 \| 96493 \| 1036.337 \| 8.18E-01 \| 7.48E+08(0.6%) \| \| 8.11E-01 \| 7.43E+08 \| \| 7.46E+08 $\mathrm{2s2p^{2}-2s^{2}2p}$ \| $\mathrm{{}^{2}P-~{}^{2}P^{o}}$ \| 1/2 - 3/2 \| 110560 \| 904.480 \| 9.96E-01 \| 1.37E+09(0.3%) \| \| 9.96E-01 \| 1.37E+09 \| \| 1.36E+09 \| \| 3/2 - 3/2 \| 110602 \| 904.142 \| 4.95E+00 \| 3.41E+09(0.4%) \| \| 4.94E+00 \| 3.41E+09 \| \| 3.38E+09 \| \| 1/2 - 1/2 \| 110624 \| 903.962 \| 1.97E+00 \| 2.72E+09(0.4%) \| \| 1.97E+00 \| 2.72E+09 \| \| 2.69E+09 \| \| 3/2 - 1/2 \| 110665 \| 903.623 \| 9.88E-01 \| 6.82E+08(0.4%) \| \| 9.87E-01 \| 6.82E+08 \| \| 6.76E+08 $\mathrm{2s^{2}3s-2s^{2}2p}$ \| $\mathrm{{}^{2}S-~{}^{2}P^{o}}$ \| 1/2 - 3/2 \| 116474 \| 858.559 \| 1.81E-01 \| 2.89E+08(0.6%) \| \| 1.83E-01 \| 2.93E+08 \| \| 2.83E+08 \| \| 1/2 - 1/2 \| 116537 \| 858.092 \| 9.19E-02 \| 1.47E+08(0.6%) \| \| 9.28E-02 \| 1.49E+08 \| \| 1.44E+08 $\mathrm{2s^{2}3d-2s^{2}2p}$ \| $\mathrm{{}^{2}D-~{}^{2}P^{o}}$ \| 3/2 - 3/2 \| 145485 \| 687.352 \| 3.03E-01 \| 4.71E+08(0.2%) \| \| 3.02E-01 \| 4.70E+08 \| \| 4.60E+08 \| \| 5/2 - 3/2 \| 145487 \| 687.345 \| 2.72E+00 \| 2.82E+09(0.2%) \| \| 2.71E+00 \| 2.82E+09 \| \| 2.76E+09 \| \| 3/2 - 1/2 \| 145549 \| 687.053 \| 1.51E+00 \| 2.35E+09(0.2%) \| \| 1.51E+00 \| 2.35E+09 \| \| 2.30E+09 $\mathrm{2s^{2}3p-2s2p^{2}}$ \| $\mathrm{{}^{2}P^{o}-~{}^{4}P}$ \| 3/2 - 5/2 \| 88681 \| 1127.626 \| 6.13E-07 \| 2.16E+02(2.0%) \| \| 5.04E-07 \| 1.79E+02 \| \| 1.80E+02 $\mathrm{2p^{3}-2s2p^{2}}$ \| $\mathrm{{}^{4}S^{o}-~{}^{4}P}$ \| 3/2 - 5/2 \| 98973 \| 1010.371 \| 3.45E+00 \| 1.70E+09(0.4%) \| \| 3.44E+00 \| 1.70E+09 \| \| 1.71E+09 \| \| 3/2 - 3/2 \| 99001 \| 1010.083 \| 2.30E+00 \| 1.13E+09(0.4%) \| \| 2.30E+00 \| 1.13E+09 \| \| 1.14E+09 \| \| 3/2 - 1/2 \| 99023 \| 1009.858 \| 1.15E+00 \| 5.67E+08(0.4%) \| \| 1.15E+00 \| 5.67E+08 \| \| 5.69E+08 $\mathrm{2p^{3}-2s2p^{2}}$ \| $\mathrm{{}^{2}D^{o}-~{}^{4}P}$ \| 5/2 - 5/2 \| 107407 \| 931.030 \| 4.41E-06 \| 1.86E+03(21.1%) \| \| 3.52E-06 \| 1.48E+03 \| \| 1.50E+03 \| \| 3/2 - 3/2 \| 107441 \| 930.740 \| 1.04E-06 \| 6.59E+02(22.2%) \| \| 8.44E-07 \| 5.33E+02 \| \| 5.39E+02 $\mathrm{2s^{2}3s-2s2p^{2}}$ \| $\mathrm{{}^{4}P^{o}-~{}^{4}P}$ \| 3/2 - 5/2 \| 123937 \| 806.861 \| 6.36E-01 \| 6.13E+08(0.4%) \| \| 3.93E-01 \| 3.79E+08 \| \| \| \| 1/2 - 3/2 \| 123941 \| 806.830 \| 5.88E-01 \| 1.13E+09(0.4%) \| \| 3.64E-01 \| 7.02E+08 \| \| \| \| 1/2 - 1/2 \| 123963 \| 806.687 \| 1.18E-01 \| 2.27E+08(0.4%) \| \| 7.27E-02 \| 1.40E+08 \| \| \| \| 3/2 - 3/2 \| 123965 \| 806.677 \| 1.88E-01 \| 1.81E+08(0.4%) \| \| 1.16E-01 \| 1.12E+08 \| \| \| \| 5/2 - 5/2 \| 123982 \| 806.568 \| 1.48E+00 \| 9.53E+08(0.4%) \| \| 9.16E-01 \| 5.90E+08 \| \| \| \| 3/2 - 1/2 \| 123987 \| 806.533 \| 5.88E-01 \| 5.67E+08(0.4%) \| \| 3.63E-01 \| 3.51E+08 \| \| \| \| 5/2 - 3/2 \| 124010 \| 806.384 \| 6.35E-01 \| 4.09E+08(0.4%) \| \| 3.92E-01 \| 2.53E+08 \| \| $\mathrm{2p^{3}-2s2p^{2}}$ \| $\mathrm{{}^{2}P^{o}-~{}^{4}P}$ \| 3/2 - 5/2 \| 125924 \| 794.125 \| 2.69E-05 \| 2.72E+04(3.2%) \| \| 7.77E-06 \| 7.83E+03 \| \| 1.72E+02 \| \| 1/2 - 3/2 \| 125953 \| 793.947 \| 4.39E-06 \| 8.88E+03(0.3%) \| \| 3.80E-06 \| 7.65E+03 \| \| 1.88E+02 \| \| 3/2 - 3/2 \| 125953 \| 793.947 \| 1.60E-06 \| 1.62E+03(8.6%) \| \| 8.35E-06 \| 8.42E+03 \| \| 1.57E+03 \| \| 1/2 - 1/2 \| 125975 \| 793.808 \| 1.51E-06 \| 3.06E+03(10.2%) \| \| 1.73E-06 \| 3.50E+03 \| \| 2.04E+01 \| \| 3/2 - 1/2 \| 125975 \| 793.808 \| 1.44E-05 \| 1.46E+04(2.7%) \| \| 2.48E-05 \| 2.50E+04 \| \| 6.67E+02 $\mathrm{2s^{2}3p-2s2p^{2}}$ \| $\mathrm{{}^{2}P^{o}-~{}^{2}D}$ \| 1/2 - 3/2 \| 56791 \| 1760.819 \| 2.23E-01 \| 4.09E+07(1.6%) \| \| 2.20E-01 \| 4.08E+07 \| \| 4.37E+07 \| \| 3/2 - 3/2 \| 56802 \| 1760.473 \| 4.45E-02 \| 4.09E+06(1.6%) \| \| 4.40E-02 \| 4.07E+06 \| \| 4.37E+06 \| \| 3/2 - 5/2 \| 56805 \| 1760.395 \| 4.01E-01 \| 3.68E+07(1.6%) \| \| 3.96E-01 \| 3.67E+07 \| \| 3.94E+07 $\mathrm{2p^{3}-2s2p^{2}}$ \| $\mathrm{{}^{2}D^{o}-~{}^{2}D}$ \| 5/2 - 3/2 \| 75528 \| 1323.995 \| 2.27E-01 \| 3.33E+07(0.2%) \| \| 2.25E-01 \| 3.28E+07 \| \| 3.50E+07 \| \| 5/2 - 5/2 \| 75531 \| 1323.951 \| 3.16E+00 \| 4.63E+08(0.1%) \| \| 3.13E+00 \| 4.56E+08 \| \| 4.88E+08 \| \| 3/2 - 3/2 \| 75534 \| 1323.906 \| 2.03E+00 \| 4.45E+08(0.1%) \| \| 2.01E+00 \| 4.39E+08 \| \| 4.69E+08 \| \| 3/2 - 5/2 \| 75536 \| 1323.862 \| 2.30E-01 \| 5.05E+07($<$0.05%) \| \| 2.27E-01 \| 4.96E+07 \| \| 5.31E+07 $\mathrm{2s^{2}3s-2s2p^{2}}$ \| $\mathrm{{}^{4}P^{o}-~{}^{2}D}$ \| 1/2 - 3/2 \| 92034 \| 1086.549 \| 1.55E-05 \| 1.21E+04(1.0%) \| \| 2.45E-05 \| 1.93E+04 \| \| \| \| 3/2 - 3/2 \| 92058 \| 1086.270 \| 1.50E-05 \| 5.88E+03(1.4%) \| \| 2.13E-05 \| 8.39E+03 \| \| \| \| 3/2 - 5/2 \| 92060 \| 1086.241 \| 9.46E-05 \| 3.71E+04(0.9%) \| \| 1.50E-04 \| 5.93E+04 \| \| \| \| 5/2 - 5/2 \| 92105 \| 1085.710 \| 3.20E-06 \| 8.38E+02(3.6%) \| \| 2.45E-06 \| 6.45E+02 \| \| $\mathrm{2p^{3}-2s2p^{2}}$ \| $\mathrm{{}^{2}P^{o}-~{}^{2}D}$ \| 1/2 - 3/2 \| 94045 \| 1063.313 \| 1.97E+00 \| 1.65E+09(0.9%) \| \| 1.73E+00 \| 1.45E+09 \| \| 1.63E+09 \| \| 3/2 - 3/2 \| 94045 \| 1063.313 \| 3.97E-01 \| 1.66E+08(0.9%) \| \| 3.44E-01 \| 1.44E+08 \| \| 1.64E+08 \| \| 3/2 - 5/2 \| 94048 \| 1063.284 \| 3.55E+00 \| 1.49E+09(0.8%) \| \| 3.11E+00 \| 1.30E+09 \| \| 1.46E+09 $\mathrm{2s^{2}3p-2s2p^{2}}$ \| $\mathrm{{}^{2}P^{o}-~{}^{2}S}$ \| 1/2 - 1/2 \| 35230 \| 2838.439 \| 7.05E-01 \| 3.06E+07(0.1%) \| \| 7.44E-01 \| 3.27E+07 \| \| 3.32E+07 \| \| 3/2 - 1/2 \| 35241 \| 2837.541 \| 1.41E+00 \| 3.06E+07(0.1%) \| \| 1.49E+00 \| 3.28E+07 \| \| 3.32E+07 $\mathrm{2p^{3}-2s2p^{2}}$ \| $\mathrm{{}^{2}D^{o}-~{}^{2}S}$ \| 3/2 - 1/2 \| 53972 \| 1852.780 \| 1.78E-05 \| 1.43E+03(1.8%) \| \| 1.55E-05 \| 1.23E+03 \| \| 1.46E+03 $\mathrm{2p^{3}-2s2p^{2}}$ \| $\mathrm{{}^{2}P^{o}-~{}^{2}S}$ \| 1/2 - 1/2 \| 72484 \| 1379.603 \| 1.55E-05 \| 5.92E+03(59.0%) \| \| 1.02E-01 \| 3.87E+07 \| \| 2.50E+04 \| \| 3/2 - 1/2 \| 72484 \| 1379.603 \| 1.78E-04 \| 3.39E+04(31.6%) \| \| 2.13E-01 \| 4.05E+07 \| \| 6.94E+04 $\mathrm{2s^{2}3p-2s2p^{2}}$ \| $\mathrm{{}^{2}P^{o}-~{}^{2}P}$ \| 1/2 - 3/2 \| 21058 \| 4748.606 \| 2.62E-03 \| 2.37E+04(5.9%) \| \| 2.46E-03 \| 2.31E+04 \| \| 2.52E+04 \| \| 3/2 - 3/2 \| 21069 \| 4746.093 \| 1.35E-02 \| 6.12E+04(5.6%) \| \| 1.26E-02 \| 5.92E+04 \| \| 6.53E+04 \| \| 1/2 - 1/2 \| 21100 \| 4739.292 \| 6.33E-03 \| 5.76E+04(5.2%) \| \| 5.75E-03 \| 5.42E+04 \| \| 6.12E+04 \| \| 3/2 - 1/2 \| 21111 \| 4736.789 \| 1.87E-03 \| 8.54E+03(6.5%) \| \| 1.90E-03 \| 8.96E+03 \| \| 9.23E+03 $\mathrm{2p^{3}-2s2p^{2}}$ \| $\mathrm{{}^{2}D^{o}-~{}^{2}P}$ \| 5/2 - 3/2 \| 39796 \| 2512.814 \| 2.68E+00 \| 5.73E+07(2.7%) \| \| 2.64E+00 \| 5.62E+07 \| \| 6.20E+07 \| \| 3/2 - 3/2 \| 39801 \| 2512.491 \| 2.95E-01 \| 9.45E+06(2.6%) \| \| 2.90E-01 \| 9.28E+06 \| \| 1.02E+07 \| \| 3/2 - 1/2 \| 39842 \| 2509.881 \| 1.49E+00 \| 4.81E+07(2.6%) \| \| 1.47E+00 \| 4.72E+07 \| \| 5.21E+07 $\mathrm{2s^{2}3s-2s2p^{2}}$ \| $\mathrm{{}^{4}P^{o}-~{}^{2}P}$ \| 1/2 - 3/2 \| 56301 \| 1776.149 \| 4.38E-06 \| 7.80E+02(0.4%) \| \| 9.53E-06 \| 1.71E+03 \| \| \| \| 3/2 - 3/2 \| 56325 \| 1775.405 \| 5.90E-05 \| 5.25E+03(0.4%) \| \| 1.53E-04 \| 1.37E+04 \| \| \| \| 1/2 - 1/2 \| 56342 \| 1774.845 \| 4.89E-06 \| 8.72E+02(0.5%) \| \| 1.55E-05 \| 2.78E+03 \| \| \| \| 3/2 - 1/2 \| 56366 \| 1774.102 \| 1.29E-05 \| 1.15E+03(0.3%) \| \| 3.19E-05 \| 2.88E+03 \| \| $\mathrm{2p^{3}-2s2p^{2}}$ \| $\mathrm{{}^{2}P^{o}-~{}^{2}P}$ \| 1/2 - 3/2 \| 58312 \| 1714.890 \| 5.79E-01 \| 1.14E+08(0.3%) \| \| 8.67E-01 \| 1.71E+08 \| \| 1.12E+08 \| \| 3/2 - 3/2 \| 58312 \| 1714.890 \| 2.91E+00 \| 2.88E+08(0.3%) \| \| 4.36E+00 \| 4.31E+08 \| \| 2.83E+08 \| \| 1/2 - 1/2 \| 58354 \| 1713.674 \| 1.16E+00 \| 2.30E+08(0.3%) \| \| 1.74E+00 \| 3.44E+08 \| \| 2.26E+08 \| \| 3/2 - 1/2 \| 58354 \| 1713.674 \| 5.79E-01 \| 5.74E+07(0.3%) \| \| 8.70E-01 \| 8.62E+07 \| \| 5.63E+07 $\mathrm{2s^{2}3p-2s^{2}3s}$ \| $\mathrm{{}^{2}P^{o}-~{}^{2}S}$ \| 1/2 - 1/2 \| 15186 \| 6584.700 \| 1.03E+01 \| 3.64E+07(0.2%) \| \| 1.03E+01 \| 3.78E+07 \| \| 3.70E+07 \| \| 3/2 - 1/2 \| 15197 \| 6579.869 \| 2.06E+01 \| 3.65E+07(0.3%) \| \| 2.06E+01 \| 3.79E+07 \| \| 3.71E+07 $\mathrm{2p^{3}-2s^{2}3s}$ \| $\mathrm{{}^{2}P^{o}-~{}^{2}S}$ \| 1/2 - 1/2 \| 52440 \| 1906.916 \| 2.19E-02 \| 3.19E+06(0.1%) \| \| 6.60E-02 \| 9.58E+06 \| \| 3.35E+06 \| \| 3/2 - 1/2 \| 52440 \| 1906.916 \| 4.31E-02 \| 3.15E+06(0.1%) \| \| 1.30E-01 \| 9.45E+06 \| \| 3.32E+06 $\mathrm{2s^{2}3d-2s^{2}3p}$ \| $\mathrm{{}^{2}D-~{}^{2}P^{o}}$ \| 3/2 - 3/2 \| 13813 \| 7239.164 \| 5.20E+00 \| 6.94E+06(0.6%) \| \| 5.21E+00 \| 6.73E+06 \| \| 7.09E+06 \| \| 5/2 - 3/2 \| 13815 \| 7238.415 \| 4.68E+01 \| 4.17E+07(0.6%) \| \| 4.69E+01 \| 4.04E+07 \| \| 4.25E+07 \| \| 3/2 - 1/2 \| 13824 \| 7233.325 \| 2.60E+01 \| 3.48E+07(0.6%) \| \| 2.60E+01 \| 3.37E+07 \| \| 3.55E+07 $\mathrm{2p^{3}-2s^{2}3d}$ \| $\mathrm{{}^{2}P^{o}-~{}^{2}D}$ \| 1/2 - 3/2 \| 23429 \| 4268.202 \| 9.99E-05 \| 1.30E+03(27.0%) \| \| 3.11E-01 \| 3.99E+06 \| \| 4.08E+04 \| \| 3/2 - 5/2 \| 23427 \| 4268.462 \| 1.15E-04 \| 7.49E+02(32.7%) \| \| 5.57E-01 \| 3.58E+06 \| \| 3.33E+04 Table 11: Continued. (a)Tachiev & Fischer (2000); (b)Corrégé & Hibbert (2004). Table 12: Comparison of line strengths ($S$) and transition rates ($A$) with other theoretical results for C iii. The present values from the MCDHF/RCI calculations are given in the Babushkin(length) gauge. The wavenumber $\Delta E$ and wavelength $\lambda$ values are taken from the NIST database. The estimated uncertainties $dT$ of the transition rates are given as percentages in parentheses. Transition array \| Mult. \| $J_{u}-J_{l}$ \| $\Delta E$ \| $\lambda$ \| MCDHF/RCI \| \| MCHF-BP(a) \| \| Grasp(b) ---\|---\|---\|---\|---\|---\|---\|---\|---\|--- \| \| \| ($\mathrm{cm^{-1}}$) \| (Å) \| $S$ (a.u. of a${}_{0}^{2}$e2) \| $A$ (s-1) \| \| $S$ (a.u. of a${}_{0}^{2}$e2) \| $A$ (s-1) \| \| $S$ (a.u. of a${}_{0}^{2}$e2) \| $A$ (s-1) $\mathrm{2s2p-2s^{2}}$ \| $\mathrm{{}^{1}P^{o}-~{}^{1}S}$ \| 1 - 0 \| 102352 \| 977.020 \| 2.44E+00 \| 1.77E+09(0.1%) \| \| 2.44E+00 \| 1.77E+09 \| \| 2.38E+00 \| 2.15E+09 $\mathrm{2s3p-2s^{2}}$ \| $\mathrm{{}^{1}P^{o}-~{}^{1}S}$ \| 1 - 0 \| 258931 \| 386.203 \| 3.06E-01 \| 3.59E+09($<$0.05%) \| \| 3.06E-01 \| 3.59E+09 \| \| 2.70E-01 \| 3.16E+09 $\mathrm{2s3p-2s^{2}}$ \| $\mathrm{{}^{3}P^{o}-~{}^{1}S}$ \| 1 - 0 \| 259711 \| 385.043 \| 4.29E-05 \| 5.08E+05(0.5%) \| \| 4.37E-05 \| 5.18E+05 \| \| 1.17E-02 \| 1.36E+08 $\mathrm{2p^{2}-2s2p}$ \| $\mathrm{{}^{3}P-~{}^{3}P^{o}}$ \| 1 - 2 \| 85007 \| 1176.370 \| 1.32E+00 \| 5.48E+08($<$0.05%) \| \| 1.32E+00 \| 5.50E+08 \| \| 1.33E+00 \| 5.95E+08 \| \| 0 - 1 \| 85034 \| 1175.987 \| 1.05E+00 \| 1.32E+09(0.1%) \| \| 1.05E+00 \| 1.32E+09 \| \| 1.07E+00 \| 1.43E+09 \| \| 2 - 2 \| 85054 \| 1175.711 \| 3.95E+00 \| 9.89E+08(0.1%) \| \| 3.95E+00 \| 9.91E+08 \| \| 4.00E+00 \| 1.07E+09 \| \| 1 - 1 \| 85063 \| 1175.590 \| 7.90E-01 \| 3.30E+08(0.1%) \| \| 7.89E-01 \| 3.31E+08 \| \| 8.00E-01 \| 3.58E+08 \| \| 1 - 0 \| 85087 \| 1175.263 \| 1.05E+00 \| 4.40E+08(0.1%) \| \| 1.05E+00 \| 4.41E+08 \| \| 1.07E+00 \| 4.78E+08 \| \| 2 - 1 \| 85111 \| 1174.933 \| 1.32E+00 \| 3.30E+08(0.1%) \| \| 1.32E+00 \| 3.31E+08 \| \| 1.33E+00 \| 3.59E+08 $\mathrm{2p^{2}-2s2p}$ \| $\mathrm{{}^{1}D-~{}^{3}P^{o}}$ \| 2 - 2 \| 93429 \| 1070.331 \| 9.42E-05 \| 3.13E+04(5.3%) \| \| 9.51E-05 \| 3.17E+04 \| \| 3.74E-05 \| 1.51E+04 \| \| 2 - 1 \| 93485 \| 1069.686 \| 1.36E-05 \| 4.52E+03(7.8%) \| \| 1.49E-05 \| 4.97E+03 \| \| 3.37E-06 \| 1.37E+03 $\mathrm{2p^{2}-2s2p}$ \| $\mathrm{{}^{1}S-~{}^{3}P^{o}}$ \| 0 - 1 \| 130129 \| 768.467 \| 4.58E-07 \| 2.06E+03(17.8%) \| \| 4.71E-07 \| 2.12E+03 \| \| 2.01E-07 \| 1.14E+03 $\mathrm{2s3s-2s2p}$ \| $\mathrm{{}^{3}S-~{}^{3}P^{o}}$ \| 1 - 2 \| 185765 \| 538.312 \| 4.73E-01 \| 2.05E+09(0.1%) \| \| 4.73E-01 \| 2.05E+09 \| \| 5.03E-01 \| 2.11E+09 \| \| 1 - 1 \| 185822 \| 538.149 \| 2.83E-01 \| 1.23E+09(0.1%) \| \| 2.84E-01 \| 1.23E+09 \| \| 3.01E-01 \| 1.27E+09 \| \| 1 - 0 \| 185845 \| 538.080 \| 9.43E-02 \| 4.09E+08(0.1%) \| \| 9.45E-02 \| 4.10E+08 \| \| 1.00E-01 \| 4.22E+08 $\mathrm{2s3s-2s2p}$ \| $\mathrm{{}^{1}S-~{}^{3}P^{o}}$ \| 0 - 1 \| 194779 \| 513.401 \| 1.45E-08 \| 2.17E+02(25.2%) \| \| 1.76E-08 \| 2.64E+02 \| \| 2.46E-08 \| 3.60E+02 $\mathrm{2s3d-2s2p}$ \| $\mathrm{{}^{3}D-~{}^{3}P^{o}}$ \| 1 - 2 \| 217563 \| 459.635 \| 4.24E-02 \| 2.95E+08($<$0.05%) \| \| 4.24E-02 \| 2.95E+08 \| \| 4.24E-02 \| 2.88E+08 \| \| 2 - 2 \| 217564 \| 459.633 \| 6.36E-01 \| 2.65E+09($<$0.05%) \| \| 6.36E-01 \| 2.66E+09 \| \| 6.36E-01 \| 2.60E+09 \| \| 3 - 2 \| 217567 \| 459.627 \| 3.56E+00 \| 1.06E+10($<$0.05%) \| \| 3.56E+00 \| 1.06E+10 \| \| 3.56E+00 \| 1.04E+10 \| \| 1 - 1 \| 217620 \| 459.516 \| 6.36E-01 \| 4.43E+09($<$0.05%) \| \| 6.36E-01 \| 4.43E+09 \| \| 6.36E-01 \| 4.33E+09 \| \| 2 - 1 \| 217621 \| 459.514 \| 1.91E+00 \| 7.96E+09($<$0.05%) \| \| 1.91E+00 \| 7.97E+09 \| \| 1.91E+00 \| 7.79E+09 \| \| 1 - 0 \| 217643 \| 459.466 \| 8.47E-01 \| 5.90E+09($<$0.05%) \| \| 8.47E-01 \| 5.91E+09 \| \| 8.48E-01 \| 5.77E+09 $\mathrm{2s3d-2s2p}$ \| $\mathrm{{}^{1}D-~{}^{3}P^{o}}$ \| 2 - 1 \| 224092 \| 446.245 \| 4.65E-07 \| 2.12E+03(0.3%) \| \| 2.80E-07 \| 1.28E+03 \| \| 2.48E-07 \| 1.14E+03 $\mathrm{2p^{2}-2s2p}$ \| $\mathrm{{}^{3}P-~{}^{1}P^{o}}$ \| 0 - 1 \| 35073 \| 2851.142 \| 2.69E-06 \| 2.36E+02(37.5%) \| \| 2.99E-06 \| 2.65E+02 \| \| 1.85E-06 \| 1.02E+02 \| \| 2 - 1 \| 35149 \| 2844.953 \| 7.79E-05 \| 1.37E+03(8.2%) \| \| 8.02E-05 \| 1.43E+03 \| \| 2.85E-05 \| 3.15E+02 $\mathrm{2p^{2}-2s2p}$ \| $\mathrm{{}^{1}D-~{}^{1}P^{o}}$ \| 2 - 1 \| 43524 \| 2297.578 \| 4.11E+00 \| 1.38E+08(0.5%) \| \| 4.11E+00 \| 1.39E+08 \| \| 4.18E+00 \| 1.34E+08 $\mathrm{2p^{2}-2s2p}$ \| $\mathrm{{}^{1}S-~{}^{1}P^{o}}$ \| 0 - 1 \| 80167 \| 1247.383 \| 1.99E+00 \| 2.10E+09($<$0.05%) \| \| 1.99E+00 \| 2.10E+09 \| \| 2.24E+00 \| 2.69E+09 $\mathrm{2s3s-2s2p}$ \| $\mathrm{{}^{3}S-~{}^{1}P^{o}}$ \| 1 - 1 \| 135860 \| 736.047 \| 4.69E-07 \| 7.92E+02(5.0%) \| \| 5.42E-07 \| 9.17E+02 \| \| 3.76E-07 \| 5.18E+02 $\mathrm{2s3s-2s2p}$ \| $\mathrm{{}^{1}S-~{}^{1}P^{o}}$ \| 0 - 1 \| 144818 \| 690.521 \| 1.40E-01 \| 8.59E+08(0.1%) \| \| 1.39E-01 \| 8.54E+08 \| \| 1.77E-01 \| 9.10E+08 $\mathrm{2s3d-2s2p}$ \| $\mathrm{{}^{3}D-~{}^{1}P^{o}}$ \| 1 - 1 \| 167658 \| 596.449 \| 5.91E-07 \| 1.88E+03(6.5%) \| \| 7.50E-07 \| 2.39E+03 \| \| 4.14E-07 \| 1.12E+03 \| \| 2 - 1 \| 167659 \| 596.446 \| 9.30E-07 \| 1.77E+03(6.2%) \| \| 4.89E-07 \| 9.34E+02 \| \| 3.79E-07 \| 6.15E+02 $\mathrm{2s3d-2s2p}$ \| $\mathrm{{}^{1}D-~{}^{1}P^{o}}$ \| 2 - 1 \| 174130 \| 574.281 \| 2.93E+00 \| 6.25E+09($<$0.05%) \| \| 2.92E+00 \| 6.25E+09 \| \| 3.37E+00 \| 6.40E+09 $\mathrm{2s3p-2p^{2}}$ \| $\mathrm{{}^{1}P^{o}-~{}^{3}P}$ \| 1 - 2 \| 121429 \| 823.525 \| 1.05E-05 \| 1.27E+04(0.3%) \| \| 1.07E-05 \| 1.29E+04 \| \| 7.12E-06 \| 7.96E+03 \| \| 1 - 1 \| 121476 \| 823.202 \| 1.76E-07 \| 2.12E+02(1.0%) \| \| 8.87E-08 \| 1.07E+02 \| \| 1.41E-05 \| 1.58E+04 $\mathrm{2s3p-2p^{2}}$ \| $\mathrm{{}^{3}P^{o}-~{}^{3}P}$ \| 1 - 2 \| 122209 \| 818.269 \| 7.20E-04 \| 8.84E+05(0.2%) \| \| 7.05E-04 \| 8.65E+05 \| \| 6.36E-04 \| 7.10E+05 \| \| 2 - 2 \| 122222 \| 818.181 \| 2.17E-03 \| 1.60E+06(0.2%) \| \| 2.15E-03 \| 1.58E+06 \| \| 1.91E-03 \| 1.28E+06 \| \| 0 - 1 \| 122251 \| 817.988 \| 5.76E-04 \| 2.13E+06(0.2%) \| \| 5.65E-04 \| 2.08E+06 \| \| 5.08E-04 \| 1.70E+06 \| \| 1 - 1 \| 122256 \| 817.950 \| 4.29E-04 \| 5.27E+05(0.2%) \| \| 4.26E-04 \| 5.23E+05 \| \| 3.63E-04 \| 4.06E+05 \| \| 2 - 1 \| 122269 \| 817.863 \| 7.30E-04 \| 5.39E+05(0.3%) \| \| 7.20E-04 \| 5.31E+05 \| \| 6.43E-04 \| 4.31E+05 \| \| 1 - 0 \| 122285 \| 817.758 \| 5.78E-04 \| 7.12E+05(0.2%) \| \| 5.69E-04 \| 7.00E+05 \| \| 4.86E-04 \| 5.44E+05 $\mathrm{2s3p-2p^{2}}$ \| $\mathrm{{}^{1}P^{o}-~{}^{1}D}$ \| 1 - 2 \| 113055 \| 884.524 \| 3.77E-01 \| 3.66E+08($<$0.05%) \| \| 3.69E-01 \| 3.59E+08 \| \| 7.16E-01 \| 5.67E+08 $\mathrm{2s3p-2p^{2}}$ \| $\mathrm{{}^{3}P^{o}-~{}^{1}D}$ \| 1 - 2 \| 113835 \| 878.464 \| 6.05E-05 \| 6.00E+04(0.1%) \| \| 6.32E-05 \| 6.27E+04 \| \| 3.07E-02 \| 2.43E+07 $\mathrm{2s3p-2p^{2}}$ \| $\mathrm{{}^{1}P^{o}-~{}^{1}S}$ \| 1 - 0 \| 76411 \| 1308.705 \| 7.24E-02 \| 2.16E+07(0.1%) \| \| 7.20E-02 \| 2.15E+07 \| \| 1.54E-01 \| 2.79E+07 $\mathrm{2s3p-2p^{2}}$ \| $\mathrm{{}^{3}P^{o}-~{}^{1}S}$ \| 1 - 0 \| 77191 \| 1295.482 \| 1.04E-05 \| 3.19E+03(0.2%) \| \| 1.05E-05 \| 3.23E+03 \| \| 6.65E-03 \| 1.20E+06 $\mathrm{2s3p-2s3s}$ \| $\mathrm{{}^{1}P^{o}-~{}^{3}S}$ \| 1 - 1 \| 20718 \| 4826.653 \| 1.65E-03 \| 9.96E+03(1.7%) \| \| 1.69E-03 \| 1.02E+04 \| \| 4.51E-01 \| 3.08E+06 $\mathrm{2s3p-2s3s}$ \| $\mathrm{{}^{3}P^{o}-~{}^{3}S}$ \| 0 - 1 \| 21492 \| 4652.775 \| 3.57E+00 \| 7.20E+07($<$0.05%) \| \| 3.57E+00 \| 7.19E+07 \| \| 3.65E+00 \| 7.44E+07 \| \| 1 - 1 \| 21498 \| 4651.548 \| 1.07E+01 \| 7.20E+07($<$0.05%) \| \| 1.07E+01 \| 7.20E+07 \| \| 1.05E+01 \| 7.14E+07 \| \| 2 - 1 \| 21511 \| 4648.720 \| 1.78E+01 \| 7.22E+07($<$0.05%) \| \| 1.78E+01 \| 7.21E+07 \| \| 1.83E+01 \| 7.46E+07 $\mathrm{2s3p-2s3s}$ \| $\mathrm{{}^{1}P^{o}-~{}^{1}S}$ \| 1 - 0 \| 11761 \| 8502.657 \| 9.25E+00 \| 1.02E+07($<$0.05%) \| \| 9.25E+00 \| 1.02E+07 \| \| 8.59E+00 \| 1.02E+07 $\mathrm{2s3p-2s3s}$ \| $\mathrm{{}^{3}P^{o}-~{}^{1}S}$ \| 1 - 0 \| 12540 \| 7973.871 \| 1.42E-03 \| 1.89E+03(2.6%) \| \| 1.45E-03 \| 1.92E+03 \| \| 3.69E-01 \| 4.33E+05 $\mathrm{2s3d-2s3p}$ \| $\mathrm{{}^{3}D-~{}^{1}P^{o}}$ \| 1 - 1 \| 11079 \| 9025.645 \| 6.66E-04 \| 6.10E+02(1.6%) \| \| 6.87E-04 \| 6.35E+02 \| \| 1.86E-01 \| 1.44E+05 \| \| 2 - 1 \| 11080 \| 9024.749 \| 2.12E-03 \| 1.17E+03(1.4%) \| \| 2.23E-03 \| 1.24E+03 \| \| 5.55E-01 \| 2.58E+05 $\mathrm{2s3d-2s3p}$ \| $\mathrm{{}^{1}D-~{}^{1}P^{o}}$ \| 2 - 1 \| 17551 \| 5697.496 \| 1.94E+01 \| 4.26E+07($<$0.05%) \| \| 1.94E+01 \| 4.28E+07 \| \| 1.88E+01 \| 5.15E+07 $\mathrm{2s3d-2s3p}$ \| $\mathrm{{}^{3}D-~{}^{3}P^{o}}$ \| 1 - 2 \| 10286 \| 9721.451 \| 2.94E-01 \| 2.16E+05($<$0.05%) \| \| 2.94E-01 \| 2.18E+05 \| \| 3.00E-01 \| 2.35E+05 \| \| 2 - 2 \| 10287 \| 9720.412 \| 4.40E+00 \| 1.95E+06($<$0.05%) \| \| 4.41E+00 \| 1.97E+06 \| \| 4.51E+00 \| 2.11E+06 \| \| 3 - 2 \| 10290 \| 9717.757 \| 2.47E+01 \| 7.79E+06($<$0.05%) \| \| 2.47E+01 \| 7.87E+06 \| \| 2.52E+01 \| 8.46E+06 \| \| 1 - 1 \| 10299 \| 9709.105 \| 4.40E+00 \| 3.25E+06($<$0.05%) \| \| 4.41E+00 \| 3.29E+06 \| \| 4.32E+00 \| 3.39E+06 \| \| 2 - 1 \| 10300 \| 9708.069 \| 1.32E+01 \| 5.86E+06($<$0.05%) \| \| 1.32E+01 \| 5.92E+06 \| \| 1.30E+01 \| 6.10E+06 \| \| 1 - 0 \| 10305 \| 9703.764 \| 5.87E+00 \| 4.35E+06($<$0.05%) \| \| 5.88E+00 \| 4.39E+06 \| \| 6.01E+00 \| 4.72E+06 $\mathrm{2s3d-2s3p}$ \| $\mathrm{{}^{1}D-~{}^{3}P^{o}}$ \| 2 - 1 \| 16771 \| 5962.446 \| 3.11E-03 \| 5.97E+03(0.4%) \| \| 3.23E-03 \| 6.24E+03 \| \| 8.04E-01 \| 2.22E+06 Table 12: Continued. (a)Tachiev & Fischer (1999); (b)Aggarwal & Keenan (2015). Table 13: Comparison of line strengths ($S$) and transition rates ($A$) with other theoretical results for C iv. The present values from the MCDHF/RCI calculations are given in the Babushkin(length) gauge. The wavenumber $\Delta E$ and wavelength $\lambda$ values are taken from the NIST database. The estimated uncertainties $dT$ of the transition rates are given as percentages in parentheses. Transition array \| Mult. \| $J_{u}-J_{l}$ \| $\Delta E$ \| $\lambda$ \| MCDHF/RCI \| \| MCHF-BP(a) ---\|---\|---\|---\|---\|---\|---\|--- \| \| \| ($\mathrm{cm^{-1}}$) \| (Å) \| $S$ (a.u. of a${}_{0}^{2}$e2) \| $A$ (s-1) \| \| $S$ (a.u. of a${}_{0}^{2}$e2) \| $A$ (s-1) $\mathrm{2p-2s}$ \| $\mathrm{{}^{2}P^{o}-~{}^{2}S}$ \| 1/2 - 1/2 \| 64484 \| 1550.772 \| 9.68E-01 \| 2.63E+08($<$0.05%) \| \| 9.68E-01 \| 2.63E+08 \| \| 3/2 - 1/2 \| 64591 \| 1548.187 \| 1.94E+00 \| 2.65E+08($<$0.05%) \| \| 1.94E+00 \| 2.65E+08 $\mathrm{3p-2s}$ \| $\mathrm{{}^{2}P^{o}-~{}^{2}S}$ \| 1/2 - 1/2 \| 320050 \| 312.451 \| 1.39E-01 \| 4.63E+09($<$0.05%) \| \| 1.40E-01 \| 4.64E+09 \| \| 3/2 - 1/2 \| 320081 \| 312.420 \| 2.78E-01 \| 4.62E+09($<$0.05%) \| \| 2.79E-01 \| 4.63E+09 $\mathrm{3s-2p}$ \| $\mathrm{{}^{2}S-~{}^{2}P^{o}}$ \| 1/2 - 3/2 \| 238257 \| 419.714 \| 2.07E-01 \| 2.84E+09($<$0.05%) \| \| 2.07E-01 \| 2.84E+09 \| \| 1/2 - 1/2 \| 238365 \| 419.525 \| 1.03E-01 \| 1.42E+09($<$0.05%) \| \| 1.03E-01 \| 1.42E+09 $\mathrm{3d-2p}$ \| $\mathrm{{}^{2}D-~{}^{2}P^{o}}$ \| 3/2 - 3/2 \| 260288 \| 384.190 \| 3.26E-01 \| 2.91E+09($<$0.05%) \| \| 3.26E-01 \| 2.91E+09 \| \| 5/2 - 3/2 \| 260298 \| 384.174 \| 2.94E+00 \| 1.75E+10($<$0.05%) \| \| 2.94E+00 \| 1.75E+10 \| \| 3/2 - 1/2 \| 260395 \| 384.031 \| 1.63E+00 \| 1.46E+10($<$0.05%) \| \| 1.63E+00 \| 1.46E+10 $\mathrm{4s-2p}$ \| $\mathrm{{}^{2}S-~{}^{2}P^{o}}$ \| 1/2 - 3/2 \| 336756 \| 296.951 \| 2.74E-02 \| 1.06E+09($<$0.05%) \| \| 2.75E-02 \| 1.06E+09 \| \| 1/2 - 1/2 \| 336864 \| 296.856 \| 1.37E-02 \| 5.30E+08($<$0.05%) \| \| 1.38E-02 \| 5.32E+08 $\mathrm{3p-3s}$ \| $\mathrm{{}^{2}P^{o}-~{}^{2}S}$ \| 1/2 - 1/2 \| 17201 \| 5813.582 \| 6.10E+00 \| 3.15E+07($<$0.05%) \| \| 6.11E+00 \| 3.15E+07 \| \| 3/2 - 1/2 \| 17232 \| 5802.921 \| 1.22E+01 \| 3.17E+07($<$0.05%) \| \| 1.22E+01 \| 3.16E+07 $\mathrm{3d-3p}$ \| $\mathrm{{}^{2}D-~{}^{2}P^{o}}$ \| 3/2 - 3/2 \| 4798 \| 20841.583 \| 1.71E+00 \| 9.54E+04($<$0.05%) \| \| 1.71E+00 \| 9.51E+04 \| \| 5/2 - 3/2 \| 4808 \| 20796.074 \| 1.54E+01 \| 5.76E+05($<$0.05%) \| \| 1.54E+01 \| 5.74E+05 \| \| 3/2 - 1/2 \| 4829 \| 20705.220 \| 8.54E+00 \| 4.87E+05($<$0.05%) \| \| 8.54E+00 \| 4.85E+05 $\mathrm{4s-3p}$ \| $\mathrm{{}^{2}S-~{}^{2}P^{o}}$ \| 1/2 - 3/2 \| 81266 \| 1230.521 \| 1.32E+00 \| 7.16E+08($<$0.05%) \| \| 1.31E+00 \| 7.14E+08 \| \| 1/2 - 1/2 \| 81298 \| 1230.043 \| 6.57E-01 \| 3.58E+08($<$0.05%) \| \| 6.57E-01 \| 3.57E+08 Table 13: Continued. * • (a)Fischer et al. (1998).
# Measurements and analysis of response function of cold atoms in optical molasses Subhajit Bhar1<EMAIL_ADDRESS>Maheswar Swar1 Urbashi Satpathi2 <EMAIL_ADDRESS>Supurna Sinha1 Rafael D. Sorkin1,3 Saptarishi Chaudhuri1 Sanjukta Roy1<EMAIL_ADDRESS>1 Raman Research Institute, C. V. Raman Avenue, Sadashivanagar, Bangalore-560080, India. 2International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore 560089, India 3Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON N2L 2Y5, Canada ###### Abstract We report our experimental measurements and theoretical analysis of the position response function of a cloud of cold atoms residing in the viscous medium of an optical molasses and confined by a magneto-optical trap (MOT). We measure the position response function by applying a transient homogeneous magnetic field as a perturbing force. We observe a transition from a damped oscillatory motion to an over-damped relaxation, stemming from a competition between the viscous drag provided by the optical molasses and the restoring force of the MOT. Our observations are in both qualitative and quantitative agreement with the predictions of a theoretical model based on the Langevin equation. As a consistency check, and as a prototype for future experiments, we also study the free diffusive spreading of the atomic cloud in our optical molasses with the confining magnetic field of the MOT turned off. We find that the measured value of the diffusion coefficient agrees with the value predicted by our Langevin model, using the damping coefficient. The damping coefficient was deduced from our measurements of the position response function at the same temperature. ## I Introduction The response of a physical system to an applied force can reveal intrinsic characteristics of the system such as electric polarisability, impedance of an electronic circuit, magnetic susceptibility and optical conductivity Kubo (1966); Mazenko (2006); Balescu (1975); Kumar _et al._ (2020); Pan _et al._ (2020). In a similar context, but without the applied force, the study of diffusive behaviour can provide crucial information regarding transport properties Barkai _et al._ (2014); Beilin _et al._ (2010); Sagi _et al._ (2012). In recent years, the diffusion of a Brownian particle in the presence of quantum zero-point fluctuations was analysed in Sinha and Sorkin (1992); Satpathi _et al._ (2017) starting from the fluctuation-dissipation theorem (FDT) Kubo (1966); Balescu (1975). The key input to the analysis presented in Satpathi _et al._ (2017) is the position response function that describes how the particle reacts to an externally applied force. The specific response function employed in that paper was suggested by the model of a viscous medium. In the present work, we study a concrete experimental realization of such a model, by utilising a three-dimensional configuration of laser beams known as ‘optical molasses’ which enables cooling as well as viscous confinement of the atomic cloud. We find agreement (in a classical regime) with the type of response function that was assumed in Satpathi _et al._ (2017). Aside from the intrinsic interest of a direct measurement of the response function, our experiment lays the groundwork for future experiments that would access the deep quantum regime, where some of the most interesting effects discussed in Satpathi _et al._ (2017) would show up. In this paper, we demonstrate a method to measure the position response function of a cold atomic cloud in a MOT by temporarily subjecting it to a homogeneous magnetic field (transient oscillation method Kim _et al._ (2005)). We observe a transition from a damped oscillatory motion to an over- damped motion of the atomic cloud. This transition stems from a competition between the reactive spring-like force coming from the magneto-optical trap and the viscous drag due to the optical molasses. By turning off the MOT magnetic field, we are also able to study the spatial diffusion of the cold atoms in the viscous medium of an optical molasses, and we verify the Stokes-Einstein-Smoluchowski relation, as described in more detail in Appendix B. The motion of a Brownian particle can be analysed in terms of either the Fluctuation-Dissipation theorem or the Langevin equation. The FDT (which holds both classically and quantum mechanically) relates the spontaneous position and velocity fluctuations of a system in thermal equilibrium to its linear response to an external perturbing force. This allows the spontaneous fluctuations to be determined from the time-dependent response-function and vice versa. The Langevin equationKubo (1966); Mazenko (2006); Ford _et al._ (1988); Balescu (1975); Hohmann _et al._ (2017); Volpe and Volpe (2013); Deng _et al._ (2007); Kessler and Barkai (2012); Graham (2000); Majumdar and Orland (2015); Vulpiani and Baldovin (2020) (in its classical, generalised, and quantum forms) offers a complementary approach which relates the response function directly to the fluctuating forces that drive the position- fluctuations. In this paper, we have adopted the Langevin equation as our starting point, since it enables easy identification of all the forces coming into play. In applying it to the theoretical analysis of the dynamics governing the motion of the cold atoms, we have treated the MOT as an interesting example of an out of equilibrium system, and we have studied it from the point of view of statistical physics, rather than from the viewpoint of cold atom experimenters for whom it serves as a valuable and well-documented source of cold atoms. This type of analysis can be extended to a variety of physical situations where one is interested in the motion of particles in a viscous medium. The paper is organised as follows: In Sec. II, we briefly describe our experimental setup and methods for preparation and detection of the cold atoms. Sec. III is devoted to the position response function of the cold atomic cloud. In Sec. III.1, we set up the theoretical perspective. In Sec. III.2, we describe our method for measuring the response function of the cold atoms, and in Sec. III.3, we compare the analytical results with the experimental observations. In Sec. IV, we present some concluding remarks and future perspective. There are two appendices. The first supplements our treatment of the response function in the body of the paper. The second presents our results on the spatial diffusion of the cold atoms. ## II Preparation and detection of cold atoms Our experiment uses a cold atomic cloud of 87Rb atoms trapped in a MOT inside an ultra-high vacuum (UHV) region ($\sim 10^{-11}$ mbar) in a glass cell. A schematic diagram of the experimental set-up is shown in Fig. 1. The MOT is vapour-loaded from a Rb getter source. An external cavity diode laser (ECDL) serves as the cooling laser, the laser beam being 12 MHz red-detuned from the $5S_{1/2},F=2\rightarrow 5P_{3/2},F^{\prime}=3$ ($D_{2}$) transition of 87Rb. Another ECDL, the repump laser, is tuned to the transition, $5S_{1/2},F=1\rightarrow 5P_{3/2},F^{\prime}=2$, and used to optically pump the atoms back into the cooling cycle. This is a standard procedure in laser cooling experiments. The detuning and intensity of the cooling and repump beams are controlled by acousto-optic modulators (AOM). The restoring force required to confine the cold atoms is provided by a pair of current carrying coils in a near ideal anti-Helmholtz configuration. The fiber coupled laser beams are expanded to have a Gaussian waist diameter of 10 mm and combined in a non-polarizing cube beam splitter. Thereafter, the combined cooling and repump beams are split into three pairs of beams using a combination of half wave plates and polarizing cube beam splitters. Each of the cooling beams is sent through the UHV glass cell and retro-reflected via a quarter waveplate and a mirror. The incoming cooling beams are kept slightly converging so as to account for the losses in the optical elements and to ensure that any radiation-pressure imbalance between the incoming and the retro-reflected beam is eliminated. The cold atoms are detected by a time-of-flight absorption imaging technique, using a short ($\sim$ 100 $\mu$sec) pulse of weak, resonant linearly polarised laser light tuned to the $5S_{1/2},F=2\rightarrow 5P_{3/2},F^{\prime}=3$ transition. The shadow cast by the atoms is imaged onto an ICCD camera with a magnification factor of 0.4. In a typical run of the experiment, we trap and cool about 5 $\times$ 107 atoms at a temperature of around 150 $\mu$K. Figure 1: Schematic diagram of the experimental setup where a cold atomic cloud is produced in a MOT in a glass cell. A magnified view near the cold atomic cloud is shown in the inset. The trajectory of the cold atomic cloud is shown as a series of atomic clouds at successive positions in the XY plane. The cooling beams are retro-reflected using a mirror and a quarter-wave plate. The cylindrical magnetic coils produce the quadrupole magnetic field used for the MOT, and the square coils produce the homogeneous magnetic field used in measuring the response function. The detection of the atomic cloud is done by means of absorption imaging using an ICCD camera. ## III Response Function of the cold atoms ### III.1 The Langevin Equation Our starting point is the Langevin Equation.111In this paper we have used the Langevin equation as the starting point, unlike Ref Sinha and Sorkin (1992); Satpathi _et al._ (2017) where the FDT was used. Note, however, that the crucial input to the Langevin equation is the noise-noise correlation function given in Eq. (2), and this rests entirely on the FDT. In its fully quantum mechanical form, it reads $\small M\ddot{x}+\int_{-\infty}^{t}dt^{\prime}\alpha(t-t^{\prime})\dot{x}(t^{\prime})+kx=\zeta(t)+f(t)$ (1) where $M$ is the mass of the particle, $\alpha(t)$ is the dissipation kernel, and $\zeta(t)$ is the noise related to the dissipation-kernel via the Fluctuation Dissipation Theorem (FDT) Ford _et al._ (1988) as follows: $\begin{split}\small\langle\zeta(t)\rangle=&0\\\ \langle\left\\{\zeta(t),\zeta(t^{\prime})\right\\}\rangle\,=\,&\frac{2}{\pi}\int_{0}^{\infty}d\omega\,\hbar\,\omega\,\text{coth}\left(\frac{\hbar\omega}{2k_{B}T}\right)\\\ &\text{Re}[\tilde{\alpha}(\omega)]\text{cos}(\omega(t-t^{\prime}))\end{split}$ (2) The position-operator, $x(t)$, of the particle at any time $t$ can be obtained by solving these equations. The experiments reported here are at high enough temperatures that the noise can be treated classically. We can therefore take the $\hbar\to 0$ limit of the previous equation, to obtain the noise correlators in their classical form: $\begin{split}\small\langle\zeta(t)\rangle=&0\\\ \langle\zeta(t)\zeta(t^{\prime})\rangle\,=\,&\frac{2k_{B}T}{\pi}\int_{0}^{\infty}d\omega\,\text{Re}[\tilde{\alpha}(\omega)]\text{cos}(\omega(t-t^{\prime}))\end{split}$ (3) (This form of Langevin equation is sometimes termed “generalized” to indicate that the dissipation-kernel is not restricted to being a delta-function.) The last term on the left-hand side of Eq. (1) corresponds to a harmonic force characterized by a spring constant $k$. In our present problem $kx$ corresponds to the restoring force of the MOT. The term $f(t)$ on the right- hand side is a perturbing force, which in this experiment is an additional magneto-optical force induced by the transient homogeneous magnetic field used to measure the position-response function. Taking the expectation value of Eq. (1), and substituting $\langle\zeta(t)\rangle=0$, we obtain a deterministic equation for $\langle x(t)\rangle$, whose Fourier transform is $\small-M\omega^{2}\tilde{x}(\omega)-i\omega\tilde{\alpha}(\omega)\tilde{x}(\omega)+k\tilde{x}(\omega)=\tilde{f}(\omega)$ (4) (Here we have used $x$ to denote the mean position of the particle.) Eq. (4) can be re-expressed as $\small\tilde{x}(\omega)=\tilde{R}(\omega)\tilde{f}(\omega)~{},$ (5) where $\small\tilde{R}(\omega)=\frac{1}{[-M\omega^{2}-i\omega\alpha+k]}$ (6) is the position response function of the particle (in this case the cold atomic cloud) in the frequency domain. Here we have set $\tilde{\alpha}(\omega)=\alpha$, corresponding to the choice of an Ohmic bath to which the system is coupled. This choice is motivated by the optical molasses in the cold-atom experimental setup. In fact, an Ohmic bath is equivalent to a force proportional to the velocity with a fixed coefficient of proportionality or damping coefficient. The present experiment serves as a test of this theoretical model of the molasses. The position response function in the time domain is given by: $\small{R}(t)=\frac{1}{2\pi}\int{\tilde{R}(\omega)e^{-i\omega t}d\omega}$ (7) The position response function obtained thereby from Eq. (6) for the Ohmic bath is $\small R(t)\,=\frac{2}{\alpha_{c}}e^{-\frac{\alpha t}{2M}}\,\text{sinh}\left(\frac{\alpha_{c}t}{2M}\right)$ (8) where $\alpha_{c}=\sqrt{\alpha^{2}-4kM}$. There are three qualitatively distinct cases. For $\alpha^{2}>4kM$, $\alpha_{c}$ is real and one gets an overdamped motion of the cold atomic cloud. For $\alpha^{2}=4kM$, the motion of the cold atomic cloud is critically damped, while for $\alpha^{2}<4kM$, $\alpha_{c}$ is imaginary and the motion of the cold atomic cloud is a damped oscillation. For $k=0$, Eq. (8) reduces to the position response function used in Satpathi _et al._ (2017). (Note that in the MOT, $k$ is always nonzero due to the presence of the non-zero magnetic field gradient.) In Sec. III.3, we will compare the analytically calculated motion of the cold atomic cloud that follows from $R(t)$ via equation (10) below with the experimentally observed oscillatory and damped motions of the cloud. By taking a time derivative of $R(t)$, one can also get the velocity response function, $\small\dot{R}(t)=\frac{1}{\alpha_{c}M}e^{-\frac{\alpha t}{2M}}\left(\alpha_{c}\text{cosh}\left(\frac{\alpha_{c}t}{2M}\right)-\alpha\,\text{sinh}\left(\frac{\alpha_{c}t}{2M}\right)\right)$ (9) It’s an interesting fact that the position response function $R(t)$ can be inferred directly from the mean velocity induced by a homogeneous force whose time-dependence is that of a step-function. By definition, the mean displacement $\langle x(t)\rangle$ is related to $R(t)$ by the equation, $\small\langle x(t)\rangle=\int_{-\infty}^{t}{R(t-t^{\prime})f(t^{\prime})dt^{\prime}}$ (10) (“linear response theory” Sinha and Sorkin (1992); Satpathi _et al._ (2017)). On differentiation, this gives the expectation value $\langle v(t)\rangle$ of the velocity: $\small\langle v(t)\rangle=\int_{-\infty}^{t}{\dot{R}(t-t^{\prime})f(t^{\prime})dt^{\prime}}$ (11) Here $f(t)$ is the external perturbing force, which in our experiment takes the form of a “top-hat function”, $f(t)=f_{0}\,\theta(t+w)\,\theta(-t)$: $f(t)=\begin{cases}f_{0},&\text{for}\;-w<t<0\\\ 0,&\text{for}\;t\leq-w,\;t\geq 0\end{cases}$ (12) (In our experiment $f(t)$ is induced by a bias field. The temporal profile of this field, together with an analysis of how $f(t)$ depends on it, is given in detail in Appendix A.1.) Substituting into Eq. (11), we get: $\displaystyle\langle v(t)\rangle$ $\displaystyle=$ $\displaystyle f_{0}\int_{-w}^{0}{\dot{R}(t-t^{\prime})dt^{\prime}}$ (13) $\displaystyle=$ $\displaystyle-f_{0}\left(R(t)-R(t+w)\right)$ (14) $\small R(t)=-\frac{1}{f_{0}}\langle v(t)\rangle+R(t+w)$ (15) For $w\rightarrow\infty$, $R(t+w)\rightarrow R(\infty)=0$. (This assumes that the MOT is turned on. When it is turned off and only the molasses is present, $R(\infty)$ will be nonzero.) Therefore we get $R(t)=-\frac{1}{f_{0}}\langle v(t)\rangle$ (16) This simple relationship means that one can measure the position response function directly, simply by measuring the expectation value of the velocity. ### III.2 Measurement of the position response function The theoretical expression (8) for the position response function of the cold atoms contains two unknown parameters, the damping-coefficient $\alpha$ and the spring-constant $k$. In order to test the theoretical model that leads to (8), and at the same time determine the values of the parameters $\alpha$ and $k$, one needs to observe how the cloud of cold atoms moves in response to an external force. In our experiment we apply a homogeneous magnetic field (bias field), and then follow the motion of the cloud of cold atoms after the field is switched off. We first prepare the laser-cooled ${}^{87}\text{Rb}$ atoms in a MOT as described in Sec. II. After that, we apply a homogeneous bias field, $B_{b}$. This shifts the trap center to the zero of the new magnetic field. The cold atoms experience a force towards the new center, and equilibrate there within a short interval of time. After 5 sec, we turn the bias field off, and the cold atoms return to the initial trap center, following a trajectory from which the position response function can be inferred. To trace the trajectory, we record the position of the cold atoms at regular intervals of time after turning off the bias field. Fig. 2 is a schematic diagram of the sequence of events in the experiment. We capture and cool the atoms in the MOT from a Rb getter source with a loading time of 15 sec. The cooling beams, having a Gaussian cross-section with a waist size of $10$ mm, are red-detuned by $2.2$ $\Gamma$ from the $5S_{1/2},F=2\rightarrow 5P_{3/2},F^{\prime}=3$ transition, where $\Gamma=38.11(6)\times{10}^{6}s^{-1}$ ($2\pi\times 6.065(9)$ MHz) is the decay rate (natural line-width) of the 87Rb $D_{2}$ transition. Different values of the MOT magnetic field gradient were used in different sets of measurements. For the oscillatory motion shown in Fig. 3, the gradient was $18$ Gauss/cm; for the over-damped motion shown in Fig. 4, it was $3.5$ Gauss/cm. After the preparation stage, we apply the bias field for $5$ seconds (its amplitude being $3$ Gauss for Fig. 3 and $0.6$ Gauss for Fig. 4). Thereafter, we turn the bias field off and wait for a variable time t, after which we switch off the quadrupole magnetic field and the cooling and repumper laser beams simultaneously, and take an absorption image after allowing the cloud to move ballistically for a time $\tau_{tof}=1.2$ ms. The mean position of the cold atomic cloud is inferred by fitting a Gaussian to the column-density profile of the cloud. Figure 2: Timing sequence for measuring the response function of the cold ${}^{87}\text{Rb}$ atoms. We prepare the cold atomic cloud by loading the MOT for 15 sec. Thereafter, we apply a homogeneous bias magnetic field for 5 sec ($w$). After the bias field is switched off, the cooling beam detuning and intensity are changed by a variable amount in order to explore a range of values of $\alpha$. The ensuing motion of the cloud is monitored via time-of- flight absorption imaging. In our experiment, time-of-flight($\tau_{tof}$) is 1.2 ms, the detection gate time ($t_{g}$) is 1 ms, and the pulse width of the imaging beam is 100 $\mu$s. The time separation between successive absorption images ($t_{w}$) is 1 sec. ### III.3 Experimental Results and comparisons with the theory #### III.3.1 Motion of the cold atoms In our experimental runs, we allow the cloud to move ballistically for a time $\tau_{tof}=1.2$ ms after switching off the MOT light beams and the quadrupole field (the bias field having been switched off earlier, of course). This delay lets us acquire the absorption image of the cloud in a magnetic field-free environment. However it introduces a small correction to the mean position of the cloud given by $\langle x_{observed}\rangle\,=\langle x(t)\rangle\,+\tau_{tof}\,\langle v(t)\rangle$ (17) The graphs in Fig. 3 and Fig. 4 show the time variation of $\langle x_{observed}\rangle$ after the bias field is turned off. Each data point shown is the average of three experimental runs, and the error bar is the standard deviation of the mean position, measured as described in Sec. III.2. Figure 3: Position of the cold atoms as a function of time after the homogeneous bias field is switched off, illustrating the underdamped regime. Cooling beam detuning: $-2.2\,\Gamma$, total intensity: $I=16.91\,I_{sat}$; MOT Magnetic field gradient: 18 G/cm; bias magnetic field: 3 Gauss with its direction at an angle to the image plane. The solid line is the best fit between the experimental data and the theoretical prediction from Eq. (26) with $\alpha\,=\,(1.04\pm 0.04)\times 10^{-22}$ kg/sec. Inset: A test fit of the data to Eq. (18) yielded an initial estimate of $\alpha=(1.06\pm 0.24)\times 10^{-22}$ kg/sec. Figure 4: Position of the cold atoms as a function of time after the homogeneous bias field is switched off, illustrating the overdamped regime. Detuning: $-2.2\,\Gamma$, total intensity $I$: 9.73 $I_{sat}$; MOT Magnetic field gradient: 3.5 G/cm; bias magnetic field: 0.6 Gauss with its direction along one of the Cartesian axes in the image plane. The solid line exhibits the best fit between the data and Eq. (26) with $\alpha\,=\,(1.57\pm 0.46)\times 10^{-22}$ kg/sec. Inset: A test fit to Eq. (18) yielded an initial estimate of $\alpha=(1.58\pm 0.24)\times 10^{-22}$ kg/sec. Figure 5: Damping coefficient ($\alpha$) as a function of the light shift. MOT magnetic field gradient: 3.5 G/cm. The data was taken in the overdamped regime exemplified by Fig. 4. --- Figure 6: Position response function deduced directly from velocity for (a) oscillatory motion with $\alpha\,=\,(1.04\pm 0.04)\times 10^{-22}$ kg/sec and (b) damped motion with $\alpha\,=\,(1.57\pm 0.46)\times 10^{-22}$ kg/sec. In both the graphs, solid lines represent the theoretical prediction of $R(t)$ given in Eq. (8) and the shaded region shows the $68\%$ confidence band. The experimental data points correspond to the scaled velocities -$\frac{1}{f_{0}}\langle v(t)\rangle$ of the cold atoms. In Fig. 3, we observe an underdamped oscillatory motion of the cold atomic cloud where the MOT magnetic field gradient is $18$ Gauss/cm and the magnitude of the bias field is $3$ Gauss along the $x$-direction as shown in Fig. 1. In Fig. 4, we observe an over-damped motion of the cold atomic cloud where the MOT magnetic field gradient is $3.5\,$ Gauss/cm and the magnitude of the bias field is $0.6$ Gauss along the $x$-direction. The theoretical curves shown in Fig. 3 and Fig. 4, were obtained by fitting the experimental data to the prediction Eq. (26) (with due regard to Eq. (17)). In these fits, there is only a single fitting parameter: the damping coefficient $\alpha$. In the insets to Fig. 3 and Fig. 4, we have fitted the experimental data to the solution of a damped-harmonic oscillator, $\langle x(t)\rangle\,=A\,e^{\dfrac{(-\alpha+\alpha_{c})t}{2M}}\,+\,B\,e^{\dfrac{-(\alpha+\alpha_{c})t}{2M}}$ (18) without assuming anything further about the form of the position response function. Here $\alpha_{c}$ is defined as earlier, and $M=1.44316060(11)\times 10^{-25}$ kg is the mass of the ${}^{87}Rb$ atom. The fitting parameters in this case were $A$, $B$, and $\alpha$. From these fits, we obtained our initial estimates of $\alpha$. An approach based on a similar 3-parameter fit to the motion of an atom in a MOT was presented in Kim _et al._ (2005). However, while being a correct approximation, it does not capture the details of the external perturbing force in their entirety. In contrast, our approach based on the response function can be used for any form of the perturbing force. Hence it offers a theoretical model which is versatile and widely applicable for this class of experiments. As discussed in Sec. III.1, the cold atomic cloud shows an underdamped oscillatory motion or an over-damped motion in response to the applied bias field depending on whether $\alpha^{2}<4kM$ or $\alpha^{2}>4kM$ respectively, i.e. whether the restoring force due to the magneto-optical trapping overwhelms the viscous force due to the optical molasses or vice versa. As always, $\alpha$ here denotes the damping coefficient and $k$ the spring constant corresponding to the MOT. Both $\alpha$ and $k$ can be calculated from 1D Doppler cooling theory Lett _et al._ (1989); Chang _et al._ (2014) as, $\displaystyle\small\alpha\,$ $\displaystyle=\,4\hbar{\kappa}^{2}\,s_{0}\,\dfrac{2\absolutevalue{\delta}/\Gamma}{\bigg{(}1+2\,s_{0}+\frac{4\delta^{2}}{\Gamma^{2}}\bigg{)}^{2}}$ (19) $\displaystyle k$ $\displaystyle=g\dfrac{\mu_{B}\lambda}{h}\,\alpha\,\dfrac{\partial B_{m}}{\partial x}$ (20) where $\lambda$ is the wavelength and $\kappa=2\pi/\lambda$ is the wavenumber of the cooling beams, $\delta$ is the detuning of the cooling beams from the atomic transition, $\mu_{B}$ is the Bohr magneton, $\dfrac{\partial B_{m}}{\partial x}$ is the MOT magnetic field gradient and $s_{0}$ is the the saturation parameter of the cooling beams defined as $I/I_{sat}$ where $I$ is the total intensity of the cooling beams and $I_{sat}$ is the saturation intensity ($I_{sat}=1.6\,mW/cm^{2}$ for 87Rb $5S_{1/2},F=2\rightarrow 5P_{3/2},F^{\prime}=3$ transition and $\sigma^{\pm}$ polarised light). Hence, the damping coefficient $\alpha$ depends on the detuning and intensity of the cooling beams of the MOT, while the spring constant $k$ has an additional dependence on the magnetic field gradient. Using the simplest possible assumption that the fluorescence from the trapped atoms accurately gives the damping co-efficient in our fitting algorithm described above, we obtain a normalised mean square residual of 8.2% and 5% for the data presented in Fig. 3 and Fig. 4 respectively. However, in the presence of a gradient magnetic field in the MOT and for a Gaussian atom number spatial distribution in the atomic cloud and a Gaussian spatial intensity profile of the cooling laser beams, this simple assumption is likely to be inaccurate. Therefore, we kept $\alpha$ to be a free fitting parameter and obtained a normalised mean square residual to be 2.1% and 2.6% for the data presented in Fig. 3 and Fig. 4 respectively. This indicates that while the fluorescence measurements can give a reasonable estimate of the damping coefficient of cold atoms in the MOT, more accurate values of the damping coefficient can be found using experimental measurements which is modelled well using our theoretical description presented in this paper. #### III.3.2 Estimation of the damping coefficient ($\alpha$) in the MOT In Fig. 5, the damping coefficients ($\alpha$) obtained from fitting the experimental data with the analytical expression given in Eq. (26) and Eq. (30) are plotted against the light shifts, where the light shift ($\Delta$) is given by: $\small\Delta\,=\hbar\,\absolutevalue{\delta}\,\,\dfrac{I/I_{sat}}{1\,+\,4\delta^{2}/\Gamma^{2}}$ (21) where $\delta$ is the detuning of the cooling beam from the atomic transition and $\Gamma$ is the natural linewidth of the atomic transition having transition wavelength $\simeq$ 780 nm. #### III.3.3 Position response function from velocity In Fig. 6a and Fig. 6b, we show comparisons between the theoretically obtained position response functions given in Eq. (8) (solid lines) and the experimentally obtained scaled velocities -$\frac{1}{f_{0}}\langle v(t)\rangle$ (circle with error bars) for the motion of the atomic clouds given in Fig. 3 (oscillatory motion) and Fig. 4 (damped motion). Note that the scaled velocity data agrees very well with the curves for the response functions, confirming Eq. (16), which is indeed a very good approximation to the exact response function (in other words, the top hat function approximates the exact bias field and in turn the perturbing force well). As we vary the molasses parameters and the MOT’s magnetic field-gradient in the experiment, we observe both oscillatory and monotonic motions of the cloud’s centroid $\langle x(t)\rangle$, indicating a transition from an underdamped to an over-damped regime. We did not attempt to explore all the parameters (intensity, detuning, magnetic field gradient) in sufficient detail to pin down the exact transition point between the two regimes. Nevertheless, in the reasonably large parameter space that we have explored, the two regimes appear clearly, as does more generally the systematic variation of the response function with the experimental parameters of the MOT. ## IV Conclusion and Outlook In this work we have measured the position response function of the cold atoms in a MOT by subjecting them to a transient homogeneous magnetic field. We have tested theoretical predictions regarding the nature of the response function, and we have done extensive theoretical analysis and numerical modelling of our experimental observations. One of the significant outcomes of the study has been the verification of the functional form of the position response function which was used as input to a recent theoretical studySatpathi _et al._ (2017) of diffusion, not only in the classical domain dominated by thermal fluctuations, but also in the still- to-be-explored quantum domain where zero point fluctuations are the main driver of the diffusion foo . Our study has led to an interesting experimental observation of a transition from an oscillatory to an over-damped behaviour of the response function as a result of a competition between elastic and dissipative effects. We find a good agreement between our experimental measurements and the theoretical model of a particle moving in a viscous medium and confined by a harmonic-oscillator potential. These measurements can be readily extended to lighter atomic species compared to Rb such as Na and K so as to access a larger range of parameter space to observe a smooth transition of the response function from an under-damped to an over-damped behaviour. We also studied the spatial diffusion of the cold atoms in the optical molasses (Appendix B), observing a behaviour which is consistent with a theoretical model based on the Langevin equation. In particular, the measured value of the diffusion coefficient agreed with the value predicted by the Langevin model, using the damping coefficient deduced from our measurements of the position response function at the same temperature. One novelty of our theoretical analysis is the observation that the position response function can be obtained directly from the velocity (Eq. (16)) if the temporal variation of the perturbing force is a step function. This is confirmed by our experiment. Our theoretical analysis also points out that in the MOT where the magnetic field is linearly proportional to the distance from the centre, the magneto- optical force can be written as the gradient of the square of the local magnitude of the magnetic field as shown in Eq. (23) of the first Appendix. This relationship simplified the theoretical modelling of the perturbing force in our experiment as seen in Eq. (24). Our study provides a general framework to analyse the motion of a particle in optical molasses combined with a restoring force, such as in a MOT, ion-traps in the presence of cooling laser beams Fan _et al._ (2019), or ultra-cold atoms in optical lattices in the presence of additional optical molasses Sherson _et al._ (2010); Bakr _et al._ (2009). These and other similar experimental systems are of current interest in the context of quantum technology devices Amico _et al._ (2017). This study paves the way for exploring spatial diffusion of ultra-cold atoms in the quantum regime where zero point fluctuations dominate over thermal ones Sinha and Sorkin (1992); Satpathi _et al._ (2017); Das _et al._ (2020). The central questions addressed in this paper are rooted in non-equilibrium statistical mechanics, and the fact that we address them using the tools of cold atom physics makes this study inherently interdisciplinary in nature. In future we intend to experimentally measure and analyse the zero point fluctuation driven diffusion in the quantum domain that has been predicted in Sinha and Sorkin (1992); Satpathi _et al._ (2017). In that context we will expand our perspective beyond the classical Langevin Equation to a fully quantum mechanical formulation (quantum Langevin equation). ###### Acknowledgements. This work was partially supported by the Ministry of Electronics and Information Technology (MeitY), Government of India, through the Center for Excellence in Quantum Technology, under Grant4(7)/2020-ITEA. S.R acknowledges funding from the Department of Science and Technology, India, via the WOS-A project grant no. SR/WOS-A/PM-59/2019. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Colleges and Universities. We acknowledge Hema Ramachandran, Meena M. S., Priyanka G. L. and RRI mechanical workshop for the instruments and assistance with the experiments. ## Appendix A Theoretical Modelling: Response function of the cold atoms ### A.1 Perturbing force on cold atoms subjected to a transient homogeneous magnetic field The temporal profile of the bias field used in our experiments is shown in Fig. 7. We fit this profile with the following equation: $\small B_{b}(t)\begin{cases}=0&\text{if $t\leq-w$}\\\\[15.0pt] =B_{0}\Bigg{(}1-e^{-\frac{t+\text{$w$}}{\tau_{1}}}\Bigg{)}&\text{if $-w\leq t\leq 0$}\\\\[15.0pt] =B_{0}\Bigg{(}1-e^{-\frac{\text{$w$}}{\tau_{1}}}\Bigg{)}\,e^{-\frac{t}{\tau_{2}}}&\text{if $t\geq 0$}\\\ \simeq B_{0}\,e^{-\frac{t}{\tau_{2}}}&(\text{for $w>>\tau_{1}$})\end{cases}$ (22) where $B_{0}$ is the magnitude and $w$ is the pulse width of the bias field, and where $\tau_{1}\,\text{ and }\,\tau_{2}$ are the rise time and fall time of the bias field. In our experiment $\tau_{1}\,\text{ and }\,\tau_{2}$ are $912\,\,\mu sec\text{ and }\,\,29.6\,\mu sec$ respectively. The approximation done in the last line of Eq. (22) is due to the fact that the time duration of the bias field ($w$ = 5 sec) is much larger than the $912\,\,\mu sec$ rise time of the bias field. The exact values of $\tau_{1}$ and $\tau_{2}$ depend on the design details of the fast switching circuit for the magnetic field coils in Helmholtz configuration producing the bias field Dedman _et al._ (2001). It is important to have a fast ‘switching off’ of the magnetic field so as to ensure that the measurements taken after switching off the magnetic field are not significantly affected by the time-constant $\tau_{2}$. In any case, we incorporate the effect of $\tau_{1}$ and $\tau_{2}$ on the motion of the atoms in our theoretical model. In a MOT, the $x$-component of the force on the cold atoms, which in Foot (2007) is expressed in terms of $x\,\partial{B_{m}}/\partial{x}$, can be recast as follows to show that the squared $B$-field acts like a potential energy for the atoms: $\begin{split}\small F_{MOT}\,&=-\alpha v-\,g\,\dfrac{\mu_{B}\lambda}{h}\,\alpha\,x\,\dfrac{\partial B_{m}}{\partial x}\\\\[5.0pt] &=-\alpha v-\,g\,\dfrac{\mu_{B}\lambda}{h}\,\alpha\,\dfrac{1}{2C_{m}}\,\dfrac{\partial B_{m}^{2}}{\partial x}\end{split}$ (23) Here, $g=g_{F^{\prime}}m_{F^{\prime}}-g_{F}m_{F}$ for transition between the hyperfine levels $\|F,m_{F}\rangle$ and $\|F^{\prime},m_{F}^{\prime}\rangle$, $\mu_{B}$ is the Bohr magneton, $\lambda$ is the wavelength of the cooling beams, $h$ is the Planck’s constant, and $\alpha$ is the damping coefficient. In the second line we have used that $B_{m}=C_{m}x$ with $C_{m}$ a constant, which implies that $x\dfrac{\partial B_{m}}{\partial x}\,=\,\dfrac{B_{m}}{C_{m}}\dfrac{\partial B_{m}}{\partial x}=\dfrac{1}{2C_{m}}\dfrac{\partial B_{m}^{2}}{\partial x}$ . In the presence of an additional bias field ($B_{b}$) along the negative $x$-direction, the force on an atom is given by (23) with the bias field added to $B_{m}$: $\begin{split}\small F_{net}\,&=-\alpha v-\,g\,\dfrac{\mu_{B}\lambda}{h}\,\alpha\,\dfrac{1}{2C_{m}}\,\dfrac{\partial(B_{m}-B_{b})^{2}}{\partial x}\\\\[5.0pt] &=-\alpha v-\,g\,\dfrac{\mu_{B}\lambda}{h}\,\alpha\,\dfrac{1}{2C_{m}}\,\bigg{(}\dfrac{\partial B_{m}^{2}}{\partial x}-\,2C_{m}B_{b}\bigg{)}\\\\[5.0pt] &=F_{MOT}\,+\,f(t)\end{split}$ (24) where we used that $\dfrac{\partial B_{b}}{\partial x}\,=\,0$. Therefore $f(t)=g\,\dfrac{\mu_{B}\lambda}{h}\,\alpha\,B_{b}$ (25) Figure 7: Temporal profile of the bias field. The black solid line is the experimental data recorded using a pick-up coil. Insets (a) and (b) show the growth and the decay, respectively, of the bias field as a function of time. After fitting the data using Eq. (22) we obtain $\tau_{1}=(912\pm 0.37)\,\mu sec\,\text{ and }\,\tau_{2}=(29.6\pm 0.056)\,\mu sec$. ### A.2 Mean displacement of the cold atoms Using the expression for the position response function in Eq. (8) and the perturbing force in Eq. (25), we get $\small\langle x(t)\rangle=Ae^{\dfrac{\left(-\alpha+\alpha_{c}\right)t}{2M}}+Be^{\dfrac{-\left(\alpha+\alpha_{c}\right)t}{2M}}+Ce^{\dfrac{-t}{\tau_{2}}}$ (26) where, $\displaystyle A$ $\displaystyle=$ $\displaystyle-\frac{2Mf_{0}}{\alpha_{c}}\left[\frac{\left(e^{\frac{\left(-\alpha+\alpha_{c}\right)w}{2M}}-1\right)}{\alpha-\alpha_{c}}-\frac{\tau_{1}\left(e^{\frac{\left(-\alpha+\alpha_{c}\right)w}{2M}}-e^{\frac{-w}{\tau_{1}}}\right)}{\alpha\tau_{1}-2M-\alpha_{c}\tau_{1}}\right.$ (27) $\displaystyle\left.+\dfrac{\tau_{2}\left(1-e^{\frac{-w}{\tau_{1}}}\right)}{\alpha\tau_{2}-2M-\alpha_{c}\tau_{2}}\right]$ $\displaystyle B$ $\displaystyle=$ $\displaystyle\frac{2Mf_{0}}{\alpha_{c}}\left[\frac{\left(e^{\frac{-\left(\alpha+\alpha_{c}\right)w}{2M}}-1\right)}{\alpha+\alpha_{c}}-\frac{\tau_{1}\left(e^{\frac{-\left(\alpha+\alpha_{c}\right)w}{2M}}-e^{\frac{-w}{\tau_{1}}}\right)}{\alpha\tau_{1}-2M+\alpha_{c}\tau_{1}}\right.$ (28) $\displaystyle\left.+\frac{\tau_{2}\left(1-e^{\frac{-w}{\tau_{1}}}\right)}{\alpha\tau_{2}-2M+\alpha_{c}\tau_{2}}\right]$ $\displaystyle C$ $\displaystyle=$ $\displaystyle\frac{4M\tau_{2}^{2}f_{0}\left(1-e^{\frac{-w}{\tau_{1}}}\right)}{\left(4M^{2}+\tau_{2}^{2}(\alpha^{2}-\alpha_{c}^{2})-4M\alpha\tau_{2}\right)}$ (29) Here, $f_{0}=g\,\dfrac{\mu_{B}\lambda}{h}\,\alpha\,B_{0}$ from Eq. (25). The mean velocity can also be obtained by taking a time derivative of $\langle x(t)\rangle$ in Eq. (26), $\displaystyle\langle v(t)\rangle$ $\displaystyle=$ $\displaystyle\frac{(\alpha_{c}-\alpha)A}{2M}e^{\dfrac{\left(-\alpha+\alpha_{c}\right)t}{2M}}-\frac{(\alpha+\alpha_{c})B}{2M}$ (30) $\displaystyle e^{\dfrac{-\left(\alpha+\alpha_{c}\right)t}{2M}}-\frac{C}{\tau_{2}}e^{\dfrac{-t}{\tau_{2}}}$ Note that for negligible $\tau_{1},\tau_{2}$ and for infinite width, i.e. $w\rightarrow\infty$, using Eq. (27), Eq. (28) and Eq. (29), $\frac{(\alpha_{c}-\alpha)A}{2M}\rightarrow-\frac{f_{0}}{\alpha_{c}},\frac{(\alpha+\alpha_{c})B}{2M}\rightarrow\,-\frac{f_{0}}{\alpha_{c}},\frac{C}{\tau_{2}}\rightarrow 0$ then, using Eq. (30) $\langle v(t)\rangle\rightarrow-f_{0}R(t)$ ($R(t)$ is given by Eq. (8)) and thus Eq. (16) is satisfied. ## Appendix B Spatial diffusion of cold atoms We study the diffusive behaviour of the cold atoms in the viscous medium provided by our optical molasses, exploring different temperatures as the atomic cloud is cooled to lower temperatures via sub-Doppler cooling. When the restoring force produced by the MOT magnetic field is absent, we are in the $k=0$ regime of the Langevin equation which defines our theoretical model. We continue to assume that the dissipation kernel $\alpha(t)$ is simply a delta-function in time, or equivalently that the force exerted on an atom by the optical molasses is $F_{OM}\,=\,-\alpha v$ (31) where v is the velocity of the atom. For consistency, one would hope that essentially the same value of $\alpha$ would explain both the position response function studied above and the diffusive spreading studied here. As is well known, the mean-square distance traveled by the diffusing atom can be determined from the Langevin equation together with the noise-correlator, i.e. from equations (1) and (3). The predicted time-dependence of the spreading depends on how the observation time $\tau$ compares with the “relaxation time” $M/\alpha$. When $\tau\gg M/\alpha$ one finds the familiar Brownian motion, with diffusion coefficient $D$ given by the Stokes-Einstein- Smoluchowski relation: $D=\frac{k_{B}T}{\alpha}$ (32) where $k_{B}$ is the Boltzmann constant and $T$ is the temperature of the cloud. However, for $\tau\ll M/\alpha$ one finds that the mean-square distance travelled grows like $t^{3}$ rather than $t$. In our experiment, the observation time of 20 ms is about 20 times bigger than $M/\alpha\sim 1$ ms. Although this is not enormously greater than unity, it seems sufficiently big for us to ignore the short-time crossover to $t^{3}$ spreading. We have therefore fitted the data under the assumption that we are in the regime of Brownian motion (Wiener process). Figure 8: The plot shows the atomic cloud size expanding in an optical molasses at a temperature of around $120\mu K$ . The solid line is a fit to the experimental data using Eq. (33) . To observe the diffusive spreading of the atoms in the cold atomic cloud, we first loaded the MOT from the background Rb vapour. Thereafter, the MOT magnetic field was switched off, and the cloud was allowed to diffuse in the presence of the cooling laser beams forming the optical molasses, but still in the absence of the MOT magnetic field. In the Brownian motion approximation, an atomic cloud of initial size of $\evaluated{(\Delta r)^{2}}_{t=0}$ expands to a size of $\evaluated{(\Delta r)^{2}}_{t=\tau}$ in time $\tau$ according to the relation: $\small\evaluated{(\Delta r)^{2}}_{t=\tau}\,=\evaluated{(\Delta r)^{2}}_{t=0}\,+\,4\,D\,\tau,$ (33) where, $\Delta r$ is the rms width of the cold atomic cloud. We obtained ${(\Delta r)}^{2}$ directly from the column density profile of the absorption image at time $t$. In other similar experiments Hodapp _et al._ (1995), the density profile was fitted to a Gaussian distribution, whereas the $(\Delta r)^{2}$ shown in Fig. 8 was obtained directly from the absorption images without assuming Gaussianity. This additional generality could become important in the quantum regime of logarithmic spreading, for which the analysis of Satpathi _et al._ (2017) furnishes $(\Delta r)^{2}$ but not the full probability distribution of $\Delta r$. (We know of no proof that the latter will be Gaussian when the diffusion is not classical.) Eq. (32) relates the damping coefficient $\alpha$ to the diffusion coefficient $D$ and thereby allows us to check for consistency between our direct measurement of D (Fig. 8) and the value of the $\alpha$ deduced from our earlier measurements of the position response function. For a temperature of around $120\mu K$ of the cold atomic cloud, the diffusion coefficient obtained from the measurement of the diffusive spreading of the atomic cloud was $(1.01\pm 0.15)\times 10^{-5}$ m2/s yielding a value of $(1.58\pm 0.25)\times 10^{-22}$ kg/s for $\alpha$. For the same temperature, the value of $\alpha$ obtained from the measurement of the position response function was $(1.57\pm 0.46)\times 10^{-22}$ kg/s. The agreement could not be better. ## References * Kubo (1966) R. Kubo, Reports on Progress in Physics 29, 306 (1966). * Mazenko (2006) G. F. 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# Softness, anomalous dynamics, and fractal-like energy landscape in model cell tissues Yan-Wei Li Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore School of Physics, Beijing Institute of Technology, Beijing 100081, China Leon Loh Yeong Wei Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore Matteo Paoluzzi Departament de Física de la Matèria Condensada, Universitat de Barcelona, C. Martí Franquès 1, 08028 Barcelona, Spain Massimo Pica Ciamarra<EMAIL_ADDRESS>Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore CNR–SPIN, Dipartimento di Scienze Fisiche, Università di Napoli Federico II, I-80126, Napoli, Italy ###### Abstract Epithelial cell tissues have a slow relaxation dynamics resembling that of supercooled liquids. Yet, they also have distinguishing features. These include an extended short-time sub-diffusive transient, as observed in some experiments and recent studies of model systems, and a sub-Arrhenius dependence of the relaxation time on temperature, as reported in numerical studies. Here we demonstrate that the anomalous glassy dynamics of epithelial tissues originates from the emergence of a fractal-like energy landscape, particles becoming virtually free to diffuse in specific phase space directions up to a small distance. Furthermore, we clarify that the stiffness of the cells tunes this anomalous behaviour, tissues of stiff cells having conventional glassy relaxation dynamics. ## I Introduction Cells in tissues rearrange in many biological processes, including embryonic development, wound healing, and tumour metastases Manli _et al._ (2012); Poujade _et al._ (2007); Basan _et al._ (2013); Trepat and Sahai (2018). The resulting tissue dynamics is slow and heterogeneous in both space and time, as cells in tissues may spend long transients in the cages formed by their neighbours, before relaxing through cooperative processes Schötz _et al._ (2013); Angelini _et al._ (2011); Bi _et al._ (2016). These observations evidence strong similarities between the dynamics of tissues Angelini _et al._ (2010); Schötz _et al._ (2013); Angelini _et al._ (2011); Bi _et al._ (2016); Sussman _et al._ (2018); Jordan _et al._ (2011); Tanaka and Ogishima (2015); Kalluri and Weinberg (2009) and that of supercooled liquids Debenedetti and Stillinger (2001); Binder and Kob (2011). However, the investigation of the relaxation dynamics of model cell tissue Sussman _et al._ (2018); Sadhukhan and Nandi (2020) where temperature-like stochastic forces drive particles, revealed distinct features not shared by ordinary supercooled liquids. In particular, the relaxation time was observed to grow as $\tau_{\alpha}\propto\exp(-E/T^{x})$ with $x<1$, a sub-Arrhenius behavior markedly distinct from the strong, $x=1$, or a super-Arrhenius, $x>1$, behavior of supercooled liquids Angell (1995). Besides, while in supercooled liquids particles do not diffuse during the transient caging-regime, the mean square displacement scaling as $t^{\alpha}$ with $\alpha\to 0$ at low temperature, cells in tissue may transiently exhibit a sub-diffusive behaviour, during which $1>\alpha>0$ is approximately constant. Experimental findings are compatible with this sub-diffusive behavior Rogers (2007); Schötz _et al._ (2013); Nixon-Abell _et al._ (2016); Armiger _et al._ (2018); Fodor _et al._ (2018), which occurs on a short time scale where the dynamics is likely to be dominated by thermal effects, rather than by self-propulsion of cells. These results are indicative of unusual features of the energy landscape of tissues, which have not yet been rationalized. It has not even been ascertained if the anomalous glassy dynamics ubiquitously occur in epithelial cell tissues, or rather if it depends on the mechanical properties of the cell, recently correlated to their geometrical features Park _et al._ (2015). In this paper, we show that the distinctive sub-diffusive and sub-Arrhenius glassy relaxation only occur in a tissue of highly deformable cells, while conversely a conventional glassy relaxation dynamics occurs. Furthermore, we rationalize that these distinct features signal the existence, in the energy landscape of highly-deformable epithelial tissues, of selected phase space directions along which the system moves almost freely, for short distances. Displacements along these phase space directions trigger cell rearrangement processes, or T1 transitions Weaire and Rivier (1984); Staple _et al._ (2010); Bi _et al._ (2014), that have a negligible energy cost. The physical mechanism leading to sub-diffusion establishes an unexpected connection between the dynamics of cell tissues and that of particles diffusing in random media, and indicates that the energy landscape of tissues is locally fractal- like as that of the Lorentz model close to the percolation threshold van Beijeren (1982); Höfling _et al._ (2006); Zeitz _et al._ (2017). ## II Voronoi model ### II.1 Numerical model We investigate the dynamics of a model of epithelial tissues Farhadifar _et al._ (2007); Staple _et al._ (2010); Bi _et al._ (2015); Manning _et al._ (2010); Fletcher _et al._ (2014); Bi _et al._ (2016), where the configurational degrees of freedom are the centers of mass of the cells, $\\{\mathbf{r}_{i}\\}$, and the shape of cell $i$ is that of the Voronoi cell centered in $\mathbf{r}_{i}$. Biological considerations Farhadifar _et al._ (2007); Staple _et al._ (2010); Bi _et al._ (2015); Manning _et al._ (2010); Fletcher _et al._ (2014); Bi _et al._ (2016); Moshe _et al._ (2018); Giavazzi _et al._ (2018) indicate that the mechanical energy of a cell depends on its area $A_{i}$ and perimeter $P_{i}$, $E_{i}=K_{A}(A_{i}-A_{0}^{i})^{2}+K_{P}(P_{i}-P_{0}^{i})^{2}$, where $A_{0}^{i}$ and $P_{0}^{i}$ are preferred values, while $K_{A}$ and $K_{P}$ are area and perimeter elastic constants. Hence, the dimensionless energy functional is $e=\sum_{i=1}^{N}[(a_{i}-a_{0}^{i})^{2}+r^{-1}(p_{i}-p_{0}^{i})^{2}],$ (1) where the sum runs over all $N=1024$ cells of the system, $a_{i}=A_{i}/l^{2}$ and $p_{i}=P_{i}/l$ with $l$ the unit of length which we have chosen so that $\langle a_{i}\rangle=1$. The preferred area $a_{0}^{i}$ is uniformly distributed in the range $0.8$–$1.2$, to avoid crystallization, while the preferred perimeter is fixed to $p_{0}^{i}=p_{0}\sqrt{a_{0}^{i}}$, with $p_{0}$ the target shape index. The non-dimensional energy then depends on the inverse perimeter modulus, $r=K_{A}l^{2}/K_{P}$, we fix to 1, and on $p_{0}$. This parameter determines the cell deformability, higher values of $p_{0}$ corresponding to more deformable cells Bi _et al._ (2015); Li and Ciamarra (2018). Simulations are performed using periodic boundary conditions. ### II.2 Connection with experiments The use of this model to simulate epithelial tissues poses two challenges. First, one would need to determine the values of the model parameters. While the physical and biological interpretation of the model’s parameters is clear Farhadifar _et al._ (2007); Staple _et al._ (2010), these have never been experimentally determined. Estimating these parameters, and in particular the elastic constants, would require probing the interaction of cells in a tissue, and take into account that the model focuses on a two-dimensional representation. Secondly, one needs to drive the cells via active forces. This is a issue as the features of the active forces inducing the dynamics of cell tissues are still unclear, and indeed different models have been proposed in the literature Barton _et al._ (2017); Bi _et al._ (2016). Specifically, the issue is whereas the biological process leading cell motion also induces aligning interactions between the cells Barton _et al._ (2017). To tackle these issues, we perform simulations in the NVT ensemble, where $T$ should be interpreted as an effective temperature. Specifically, we integrate the equations of motion via the Verlet algorithm, and fix the temperature using a Langevin thermostat Allen (1987). Furthermore, we relate the model parameters to experimental values considering their effect on the dynamics. While the diffusion coefficient of cells in epithelial tissues vary greatly with the control parameters, a typical order of magnitude estimate is $D\simeq 10^{-2}\mu m^{2}/{\rm min}$ (e.g, Armiger _et al._ (2018); Dieterich _et al._ (2008)). We investigate effective temperature values leading to a diffusion coefficients, which we estimate from the mean square displacement of our numerical model, of order $D_{\rm sim}=10^{-3}l^{2}/\tau_{0}$, with $l$ and $\tau$ are length and time units. This is also an order of magnitude estimate, as the diffusion depends on the temperature. Equating the numerical and the experimental diffusion coefficient, and fixing our length unit to the typical cell size, of order $10\mu m$, we estimate our time unit to be approximately $0.06$s. In the following, we investigate the dynamics for $t\leq 10^{5}\tau_{0}\simeq 4h$. This is a short time scale with respect to that of biological processes such as cell reproduction and apoptosis, which affect cell size Puliafito _et al._ (2012); Straetmans and Khain (2019), and with respect to the time scale of cell volume fluctuations Zehnder _et al._ (2015), which we therefore neglect. ## III Conventional and anomalous glassy dynamics: $p_{0}$ dependence Figure 1: Time dependence of the mean square displacement (a), of its log- slope (c) and the self-intermediate scattering function (e), at $p_{0}=3.0$. (b), (d) and (f) show the same quantities at $p_{0}=3.81$. Open symbols in (a)-(d) are results obtained via overdamped simulations, for the indicated low-temperatures values. In all panels, black dots mark the relaxation time. At zero temperature, on increasing the target shape index, this model exhibits a sharp crossover for $p_{0}\simeq 3.81$, which is reminiscent of a rigidity transition Bi _et al._ (2015); Li and Ciamarra (2018); Sussman and Merkel (2018). Here, we compare the relaxation dynamics at $p_{0}=3.0$ and at $p_{0}=3.81$, respectively in the solid phase and close to the crossover. We investigate the MSD $\left\langle\Delta r^{2}(t)\right\rangle=\left\langle\frac{1}{N}\sum_{i=1}^{N}\Delta\mathbf{r}_{i}(t)^{2}\right\rangle$, with $\Delta\mathbf{r}_{i}(t)$ displacement of particle $i$ at time $t$, its log-slope $\Delta(t)=\mathrm{d}\left(\ln\left\langle\Delta r^{2}(t)\right\rangle\right)/\mathrm{d}(\ln(t))$, and the self-intermediate scattering function (ISF) $F_{s}(q,t)=\left\langle\frac{1}{N}\sum_{j=1}^{N}e^{i\mathbf{q}\cdot\Delta\mathbf{r}_{j}(t)}\right\rangle$ with $q=\|\mathbf{q}\|$ the wavenumber of the first peak of the static structure factor. We define the relaxation time $\tau_{\alpha}$ as $F_{s}(q,\tau_{\alpha})=e^{-1}$. Stiff cells ($p_{0}=3.0$) exhibit a typical glassy behaviour Debenedetti and Stillinger (2001); Binder and Kob (2011); As the temperature decreases the MSD develops an increasingly long plateau during which $\Delta(t)$ attains a small value, and the ISF develops a two-step decay (Figs. 1(a), 1(c) and 1(e)). Conversely, soft cells ($p_{0}=3.81$) relax in a qualitatively different way. Although the dynamics slows down dramatically at low temperatures, the MSD does not exhibit a true plateau, if not at extremely low temperature, and the ISF does not decay in two steps (Figs. 1(b) and 1(f)). More importantly, an extended sub-diffusive behaviour follows the short-time ballistic regime one. Indeed, the log-slope $\Delta(t)$ of the mean square displacement develops an extended plateau, we show in Fig. 1(d). These results clarify that an anomalous glassy dynamics occurs for soft cells, as previously observed Sussman _et al._ (2018), but not for stiff ones. The stiffness of the cells, therefore, does not simply alters the energy scale for particle rearrangement, but rather qualitatively influences the relaxation dynamics. Figure 2: (a) Angell plot representation of the temperature dependence of the relaxation time, for different values of the target shape index $p_{0}$. $T_{g}$ is the glass transition temperature at which the relaxation time is $10^{4}$. (b) The Angell plot with the relaxation time defined as that where the mean square displacement attains a threshold value, as specified in the legend. The relaxation times are scaled so that they equal $\tau_{\alpha}$ at $T_{g}$. The high-$p_{0}$ regime where the anomalous diffusive behaviour occurs, is also that where the relaxation time exhibits a sub-Arrhenius temperature dependence Sussman _et al._ (2018), as we show in Fig. 2(a). Our results obtained in an extended $p_{0}$ range, however, demonstrate that a traditional super-Arrhenius behaviour occurs at low $p_{0}$. A similar crossover is found defining the relaxation time from the decay of correlation function of the area and of the perimeter of the cells, as discussed in Appendix A. A single parameter, $p_{0}$, thus controls the fragility and allows to transit from a sub- to a super-Arrhenius behaviour. We are not aware of other models with a similar crossover. We now clarify that the anomalous sub-diffusive regime and the unusual sub- Arrhenius behaviour are strongly tied. To this end, we define the relaxation time as that at which the mean square displacement reaches a threshold, $\langle\Delta r^{2}(\tau_{\Delta r^{2}_{\rm th}})\rangle=\Delta r^{2}_{\rm th}$. Figures 1(a) and 1(b) show that the mean square displacement at the relaxation time (circles) is $\Delta r^{2}_{\tau_{\alpha}}\simeq 0.1$, regardless of the temperature and of $p_{0}$. Hence, $\tau_{\Delta r^{2}_{\rm th}=0.1}\simeq\tau_{\alpha}$. The relaxation time $\tau_{\Delta r^{2}_{\rm th}}$ decreases with the threshold $\Delta r^{2}_{\rm th}$, becoming increasingly more influenced by the sub-diffusive regime rather than by the subsequent caging regime. When this occurs, the sub-Arrhenius behaviour becomes more apparent, as we illustrate in Fig. 2(b). We then conclude that the sub-diffusive behaviour induces the sub-Arrhenius one, while the caging regime contrasts it. ## IV Anomalous glassy dynamics: physical origin ### IV.1 Particle trajectories The above results demonstrate that cell tissues, for large values of the target shape index $p_{0}$, have a distinctive relaxation dynamics, which is quite different from that of conventional glassy systems. Why is this so? To begin addressing this question, we have repeated the above investigations via overdamped simulations, for selected low-temperature values, and show the results as open circles in Figs. 1(a)-(d). These simulations reproduce the anomalous sub-diffusive regime, demonstrating that this has not an inertial origin. Besides, we have also investigated the relaxation dynamics using cage- relative quantities Shiba _et al._ (2016); Illing _et al._ (2017); Vivek _et al._ (2017); Li _et al._ (2019a), an approach which allows filtering out the effect of long-wavelength fluctuations. These fluctuations might indeed be relevant in two spatial dimensions Mermin and Wagner (1966); Li _et al._ (2019a); Illing _et al._ (2017); Vivek _et al._ (2017); Shiba _et al._ (2016), influencing both the mean square displacement and the relaxation time. We illustrate in Appendix B that there are no considerable differences between the standard and the cage-relative relaxation dynamics; the anomalous sub- diffusive behaviour, therefore, is not the vestige of the vibrational dynamics of the system. Figure 3: Time dependence of the averaged eccentricity $\langle E(t)\rangle$ and of $\langle\rm{cos}\theta\rangle$ (see text) at $p_{0}=3.0$, (a) and (c), and at $p_{0}=3.81$, (b) and (d). The insets in (a) and (b) illustrate particle trajectories at the relaxation time. The horizontal dashed lines in (a) and (b) indicate the Brownian limit, $\langle E(t)\rangle=4/7$. To unveil the microscopic origin of the sub-diffusive behaviour, we then focus on the particle trajectories in the supercooled regime. Example trajectories, evaluated at the relaxation time, are in Figs. 3(a) for $p_{0}=3.0$, and 3(b) for $p_{0}=3.81$. We find that at small $p_{0}$, the trajectories have a round shape reflecting a caging regime, while conversely at large $p_{0}$ they are unusually stretched. To quantify this observation, we describe a trajectory as a sequence of $n_{v}=50$ points equally spaced in time. The eigenvectors of the gyration tensor of this set of points fix the spatial directions along which the fluctuations of the trajectory, as estimated by the squared eigenvalues $\lambda_{1}^{2}\geq\lambda_{2}^{2}$, are maximal and minimal. This allows associating to each trajectory an eccentricity, $E(t)=\left(\lambda_{1}^{2}(t)-\lambda_{2}^{2}(t)\right)^{2}/\left(\lambda_{1}^{2}(t)+\lambda_{2}^{2}(t)\right)^{2}$ Rudnick and Gaspari (1987); Ernst _et al._ (2012). Radially symmetric trajectories have $E=0$, straight lines $E=1$, while Brownian trajectories have $E=4/7$, in two dimensions Rudnick and Gaspari (1987). Consistently, the sample averaged eccentricity attains large values in the ballistic or super- diffusive regimes, reaches the Brownian limit at long times, and it is suppressed in the caging regime, as illustrated in Figs. 3(a) and 3(b). At $p_{0}=3.81$, an intermediate regime occurs between the ballistic and the caging one, where the eccentricity has a plateau. The time dependence of the eccentricity, henceforth, closely resembles that of the log-slope, as apparent comparing Figs. 3(a) and 3(b) with Figs. 1(c) and 1(d). More importantly, the trajectories reveal that the sub-diffusive behaviour results from an anisotropic motion of cells. These anisotropic motion does not correlate with the possible anisotropic shape of the cell, as we show in Appendix C. The stretched trajectories lead to a sub-diffusive dynamics, rather than to a super-diffusive dynamics as one might naively expected, due to the presence of anti-correlations in the motion of the cells. To highlight these correlations, we investigate the angle $\theta$ between consecutive displacements ${\bf r}(t_{0}+t)-{\bf r}(t_{0})$,${\bf r}(t_{0}+2t)-{\bf r}(t_{0}+t)$, over a time $t$. Both at low- and at high-$p_{0}$ values, the time evolution of $\langle\cos\theta\rangle$ resembles that of $\Delta(t)$ and of $E(t)$, as shown in Figs. 3(c) and 3(d). In particular, at large $p_{0}$, we observe an intermediate regime in between the ballistic and the caging ones. In this intermediate regime, $\langle\cos\theta\rangle\approx-0.1$ for a transient. Since this small value of $\langle\cos\theta\rangle$ occurs when the trajectories are elongated, we understand that sub-diffusion emerges as cells are transiently only slightly constrained. Figure 4: (a) Probability distribution of the cell edge length $l$ of energy minima configurations, and (b) dependence of the average energy barrier of T1 transitions on $l$, for $p_{0}=3.0$ (black squares) and $p_{0}=3.81$ (red circles). (a, inset): schematic of a T1 transition in which the edge connecting particles $\rm{a_{1}}$ and $\rm{a_{2}}$ disappears, and a novel edge connecting previously separated cells appears. (c) and (d) illustrate the time dependence of mean square displacement scaled by the temperature, respectively for $p_{0}=3.0$ and $p_{0}=3.81$. Colors indicate different temperature values, as in Fig. 1. ### IV.2 T1 transitions The structural relaxation dynamics is strongly correlated with the topology of the free-energy landscape in glassy systems Berthier and Biroli (2011). Indeed, we now show the existence of phase space directions along which the system is essentially free to diffuse, for short distances, considering the energetic cost of relaxation events involving cell rearrangements, or T1 transitions Weaire and Rivier (1984); Staple _et al._ (2010); Bi _et al._ (2014), one of which is schematically illustrated in the inset of Fig. 4(a). In a T1 transition a cell-edge of length $l$ disappears, as the system overcomes an energy barrier $\Delta e(l)$ we expect to increase with $l$, as observed in the Vertex model Bi _et al._ (2014). We have investigated the edge-length distribution $P(l)$ and the dependence of the average energy barrier on $l$, which are illustrated Figs. 4(a) and 4(b). We detail the procedure used to evaluate these quantities is Appendix D. At small $p_{0}$, $P(l)$ is Gaussian shaped, as observed in the Vertex model Bi _et al._ (2014), and the average energy barrier increases with $l$. At large $p_{0}$, $P(l)$ is broad and has almost a bi-modal shape, which is actually observed at even larger $p_{0}$ values not considered here Li and Ciamarra (2018). In particular, on increasing $p_{0}$ small $l$-values become more probable. The energy cost of T1 transitions involving small edges, e.g. $l\lesssim 0.2$, is sensibly smaller than the energy cost of the other edges, as apparent in Fig. 4(b). Hence, the system is essentially free to diffuse along the specific phase space directions that trigger the T1 transitions involving these small edges. To corroborate this picture we further consider that, since the free diffusion coefficient is proportional to $T$, the mean square displacement should scale as $T$ not only in the ballistic regime but also in the sub-diffusive one. We indeed observe in Figs. 4(c) and 4(d) that, at low $p_{0}$, plots of $\left\langle\Delta r^{2}(t)\right\rangle/T$ only collapse in the ballistic regime, while conversely at high-$p_{0}$ they also collapse in the sub-diffusive one. The emerging scenario reminds the diffusion of a particle in a random media as described by the Lorentz gas models van Beijeren (1982); Höfling _et al._ (2006); Bauer _et al._ (2010); Zeitz _et al._ (2017); Petersen and Franosch (2019), where a particle is free until it hits randomly placed obstacles, and sub-diffusion occurs below the correlation length of the fractal cluster of free space. Figure 5: Time dependence of probability of irreversible consecutive T1 transitions (black) and of the log-slope of mean square displacement (blue) for (a) $p_{0}=3.0$ and $T=0.06$ and for (b) $p_{0}=3.81$ and $T=0.002$. To further support the deep connection between anomalous dynamics and T1 transitions, we consider the probability $P_{\rm irr}$ that two consecutive T1 transitions of the same particle are not one the reverse of the other; this occurs if the two transitions lead to a change in the Voronoi neighbours of the particle. We illustrate in Fig. 11 the dependence of $P_{\rm irr}$ on the time interval $t$ separating the two transitions. To avoid cluttering of data, we consider in (a) $p_{0}=3.0$ and $T=0.06$, and in (b) $p_{0}=3.81$ and $T=0.002$, two state points having close relaxation time, and report results for other parameter values in Appendix E. In the figure, we also superimpose the log-slope $\Delta(t)$ of the mean square displacements. For small $p_{0}$, $P_{\rm irr}$ quickly attains a high, almost constant plateau value, characterizing the caging regime. $P_{\rm irr}$ then approaches $1$ as the system relaxes. For large $p_{0}$, $P_{\rm irr}$ grows essentially as a power-law during the sub-diffusive transient. An inflexion, reminiscent of a plateau in the caging-regime follows the power-law growth and the final approach to $1$. Hence, the sub-diffusive regime is characterized by a scarcity of irreversible transition. ## V Discussion Our study demonstrates that the relaxation dynamics of a model cell tissue qualitatively depends on the stiffness of the cells; while stiff cells exhibit a conventional glass-like relaxation dynamics, soft ones have an extended sub- diffusive transient and a sub-Arrhenius dependence on the relaxation time on the temperature. Consistently, dynamical heterogeneities grow on cooling for stiff cells, while they are almost temperature independence for soft cells, as we demonstrate in Appendix F. The qualitative changes in the relaxation dynamics originate from the emergence of phase space directions along which the system is essentially free to move, in soft cells, and establish an analogy between the energy landscape of cell tissues and the Lorentz model, on short length scales. We do not expect the sub-diffusive behavior we have discussed to be a universal feature of the dynamics of cell tissues. Its occurrence, indeed, might be hidden by the super-diffusive contribution to the mean square displacement of the active forces. To observe our finding one might suppress cell-motility, making the cell tissue dynamics thermal. In order for thermal forces alone to be able to induce the relaxation of the system, it migth be also convenient to consider soft tissues, as those close to the epithelial- mesenchymal transition Jordan _et al._ (2011); Tanaka and Ogishima (2015); Kalluri and Weinberg (2009). We remark, however that a sub-diffusive transient, $r^{2}\propto t^{\beta}$, with $\beta$ constant over an extended period of time, has been observed in some experiments Rogers (2007); Schötz _et al._ (2013); Nixon-Abell _et al._ (2016); Armiger _et al._ (2018); Fodor _et al._ (2018). Our results offer a possible explanation of these experimental findings because at short time the thermal contribution to the mean square displacement ($\propto t$), which is the one we have modeled, dominates over the active contribution ($\propto t^{2}$). ###### Acknowledgements. We acknowledge support from the Singapore Ministry of Education through the Academic Research Fund MOE2017-T2-1-066 (S), and are grateful to the National Supercomputing Centre (NSCC) of Singapore for providing computational resources. MP is supported by the H2020 program under the MSCA grant agreement No. 801370 and by the Secretary of Universities and Research of the Government of Catalonia through Beatriu de Pinós program Grant No. BP 00088 (2018). ## Appendix A Shape correlation functions Figure 6: Time dependence of perimeter (a) and of area (c) correlation functions at $p_{0}=3.0$. (b) and (d) are the time evolution of the same quantities at $p_{0}=3.81$. To prove that the anomalous dynamics are associated to changes in the shapes of the cells, as those one might expect T1 transitions to induce, we investigate the perimeter and area correlation functions. The perimeter correlation function is defined as $C_{P}(t)=\frac{\sum_{i=1}^{N}\left[p_{i}(t)-\langle p_{i}\rangle\right]\left[p_{i}(0)-\langle p_{i}\rangle\right]}{\sum_{i=1}^{N}\left[p_{i}(0)-\langle p_{i}\rangle\right]^{2}},$ (2) where $p_{i}(t)$ is the non-dimensional perimeter of cell $i$ at time $t$ and $\langle p_{i}\rangle$ is the time average value, which is cell dependent due to the polydispersity of our system. The area correlation function $C_{A}(t)$ is similarly defined. We illustrate the time dependence of $C_{P}(t)$ and of $C_{A}(t)$ at $p_{0}=3.0$ and at $p_{0}=3.81$ in Fig. 6. $C_{P}(t)$ and $C_{A}(t)$ demonstrate similar behavior. In particular, at $p_{0}=3.0$, both correlation functions exhibit a two-step decay, which is conversely not apparent at $p_{0}=3.81$. This $p_{0}$ dependence is consistent with that of the ISF (Fig. 1) and CR-ISF (Figs. 7(e) and 7(f)). We further extract from the shape correlation function the perimeter and the area relaxation time, $\tau_{P}$ and $\tau_{A}$, which satisfy $C_{P}(\tau_{P})=C_{A}(\tau_{A})=1/e$. Both relaxation time have a super-Arrhenius temperature dependence at small $p_{0}$, and a sub-Arrhenius one at large $p_{0}$, as we show in Fig. 8(b). The investigation of the relaxation dynamics via the shape-correlation function establishes a coupling between the geometrical properties of the cells and their displacement. Considering that the shape of the cell changes as a consequence of T1 transitions, this result indirectly links anomalous dynamics and T1 transitions. ## Appendix B Cage-relative dynamics Figure 7: Time dependence of the cage-relative mean square displacement (a), of its log-slope (c) and of the cage-relative self-intermediate scattering function (e) at $p_{0}=3.0$. (b), (d) and (f) show the time dependence of the same quantities at $p_{0}=3.81$. The full circles in all panels mark the cage- relative relaxation time $\tau_{\alpha}^{\rm CR}$, which is the time at which the cage-relative self-intermediate scattering function reaches $1/e$. Figure 8: Angell plots, as obtained using different definitions of the relaxation time. In (a), $\tau_{\alpha}^{\rm CR}$ is the cage-relative relaxation time. In (b), $\tau_{\rm P}$ and $\tau_{\rm A}$ are the perimeter and the area relaxation time. We have illustrated in Fig. 1 the mean square displacements (MSD), its log- slope, and the self-intermediate scattering function (ISF). We have additionally investigated the time dependence of these quantities using cage- relative (CR) measures. The CR measures differ from the standard ones in that the CR displacement $\Delta\mathbf{r}_{i}^{\rm CR}(t)=\Delta\mathbf{r}_{i}(t)-1/N_{i}\sum_{j=1}^{N_{i}}\Delta\mathbf{r}_{j}(t)$, where the sum is over the $N_{i}$ neighbors particle $i$ has at time $0$, replaces the displacement $\Delta\mathbf{r}_{i}(t)=\mathbf{r}_{i}(t)-\mathbf{r}_{i}(0)$. Particles moving coherently with their immediate neighbours have a large displacement, but a small CR displacement. Hence, CR measures filter out the effect of coherent displacements, and particularly the effect of long-wavelength fluctuations, which could affect the relaxation dynamics of two-dimensional systems Shiba _et al._ (2016); Vivek _et al._ (2017); Illing _et al._ (2017); Li _et al._ (2019a). Figure 7 shows that, for small $p_{0}$, the CR one reveals a typical glassy behaviour, including an extended plateau in CR-MSD and a two-step decay in CR- ISF at low temperatures, as the standard measure. Similarly, the anomalous sub-diffusive behaviour found at large $p_{0}$ persists when the relaxation dynamics is investigated using CR measures. Indeed, a region of anomalous diffusion is clearly observed in CR-MSD (Fig. 7(b)) and in its log-slope (Fig. 7(d)). This anomalous behaviour, and that observed in the standard quantities in Fig. 1 at the same $p_{0}$ value, occur on the same time scale. We further define the CR relaxation time $\tau_{\alpha}^{\rm CR}$ as the time at which CR-ISF reaches $1/e$, and illustrate the resulting Angell plot in Fig. 8(a). On increasing $p_{0}$, we observe a crossover from a super- to a sub-Arrhenius behaviour, as found in Fig. 2(a) using the standard measure. Overall, the investigation of the relaxation dynamics using CR quantities excludes the possibility that the observed anomalous behaviour occurring at large $p_{0}$ could originate from the emergence of collective particle displacements, like those induced by long-wavelength fluctuations. ## Appendix C Absence of correlation between shape and displacement of a cell Cells in tissue, being deformable objects, may acquire elongated shapes. A cell’s displacement could, therefore, correlate with its shape, e.g. in the anomalous diffusive regime at high $p_{0}$. To investigate this possibility, we first assume the eigenvector associated with the largest eigenvalue of the covariance matrix of the vertices of cell $i$ to identify its principal axis, $\mathbf{v}_{i}$. Next, we consider how the normalized cell displacement $\Delta\mathbf{r}_{i}(t)/\|\Delta\mathbf{r}_{i}(t)\|$ at time $t$ correlates with the principal axis at time $0$, studying $\cos\alpha_{i}(t)=\Delta\mathbf{r}_{i}(t)/\|\Delta\mathbf{r}_{i}(t)\|\cdot\mathbf{v_{i}}(0)$. Since the cell’s principal axis is defined up to an angle $\pi$, $\langle\cos\alpha(t)\rangle=0$. We, therefore, focus on $\langle\cos^{2}\alpha(t)\rangle$, which equal $1/2$ in the absence of correlations. In Fig. 9, we show that $\langle\cos^{2}\alpha(t)\rangle$ does equal $1/2$, regardless of the $p_{0}$ value and of the time. Analogous results are obtained at different temperatures. Accordingly, the shape of a cell at a given time does not correlate with its subsequent displacement. This result is consistent with our finding, according to which in the anomalous region cells move along direction inducing T1 transition associated with their short edges. Figure 9: Time dependence of $\langle\cos^{2}\alpha(t)\rangle$ for $p_{0}=3.0$ and $T=0.06$ (black squares) and for $p_{0}=3.81$ and $T=0.002$ (red circles). ## Appendix D T1 energy barrier Figure 10: (a) Illustration of tuning the edge length $l_{{\rm a}}$ between two selected neighboring cells (green) by separating them gradually so as to induce a T1 transition. (b) and (c) the edge length $l_{{\rm a}}$ dependence of the minimised total energy $e(l_{{\rm a}})$ at $p_{0}=3.0$ and $p_{0}=3.81$, respectively. Different colors are for different selected neighboring cell couples. The edge length $l_{{\rm a}}$ and the corresponding energy $e(l_{{\rm a}})$ for the snapshots shown in (a) are indicated in (b). We investigate the energy barrier for T1 transition to occur focusing on systems with $N=100$ cells quenched to their inherent state via the conjugate- gradient algorithm. In these systems, we randomly select two neighbouring cells and indicate with $l_{{\rm a}}$ the length of the Voronoi edge separating them. Then, we gradually increase the separation of the two cells, moving them by small steps along the direction connecting their centres. We fix the step size to $0.1$, $0.01$, and $0.001$ when the distance $dr$ between the cell centers is $dr>0.2$, $0.2>dr>0.1$, and $0.1>dr$, respectively. After each step, we minimize the energy of the tissue using the conjugate-gradient method, keeping fixed the positions of the selected cells. As the distance between the centers of selected cells increases, the length $l_{\rm a}$ of the Voronoi edge separating them decreases, and the energy of the system increases, as visualized in Fig. 10(a). As the length scale increases, the energy of the tissue grows, as illustrated for a few selected cell couples in Fig. 10(b) for $p_{0}=3.0$, and in Fig. 10(c) for $p_{0}=3.81$. The energy suddenly drops as the T1 transition separating the selected particles occurs, as $l_{{\rm a}}$ approaches $0$. The overall change in energy defines the energy barrier $\Delta e(l)=e(l_{{\rm a}}\rightarrow 0)-e(l)$. Figure 4(b) illustrates $\langle\Delta e(l)\rangle$ as a function of the initial edge length $l$. The data are obtained randomly by triggering 200 random T1 transitions, from 24 independent configurations. We note here that in a few instances we have observed drops in the dependence of the energy versus $l_{{\rm a}}$ due to T1 transitions which do not involve the displaced particles. Regardless, we operatively define $\Delta e(l)$ as the difference between the energy of the system as the separating particles undergo a T1 transition and the initial one. ## Appendix E T1 correlations In Fig. 11, we illustrate the time dependence of the probability to find irreversible consecutive T1 transitions, $P_{\rm irr}(t)$, at different temperatures for $p_{0}=3.0$ (panel (a)) and for $p_{0}=3.81$ (panel (b)). Figure 5 shows that data for $p_{0}=3.0$ and $T=0.06$ and for $p_{0}=3.81$ and $T=0.002$ are qualitatively different, and that $P_{\rm irr}(t)$ correlates with the MSD. Here, we notice that the temperature dependence of $P_{\rm irr}(t)$ is qualitatively the same, for different $p_{0}$ values. At higher temperature, it becomes increasingly more probable for two consecutive transitions separated by a small time interval $t$ not to be one the reverse of the other. Furthermore, as the temperature increases the plateau that $P_{\rm irr}(t)$ attains at long-time during the caging regime, reduces in extension and increases in value approaching $1$. Figure 11: Time dependence of the probability that two consecutive T1 transition of a same particle are not one the reverse of the other, for several selected values of temperatures at (a) $p_{0}=3.0$ and at (b) $p_{0}=3.81$. ## Appendix F Dynamical length scales Figure 12: Spatial-temporal correlation functions at time $t_{\rm max}$ for (a) $p_{0}=3.0$ and for (b) $p_{0}=3.81$. $t_{\rm max}$ is the time at which the corresponding four-point susceptibility reaches the maximum. The solid lines are exponential fits. (c) illustrates the dependence of the dynamical correlation length on the relaxation time for different $p_{0}$ values. The existence of a standard and of an anomalous glassy dynamics, respectively at small at a high $p_{0}$ values, suggests that the spatial temporal correlation between the particle displacement may likewise be strongly $p_{0}$ dependent. To investigate this issue, we focus on the decay of the spatial-temporal correlation function Pastore _et al._ (2011); Li _et al._ (2019b): $g_{4}(r_{ij},t)=\langle\omega_{i}(t)\omega_{j}(t)\rangle-\langle\omega_{i}(t)\rangle\langle\omega_{j}(t)\rangle.$ (3) Here $r_{ij}=\|\mathbf{r}_{i}(0)-\mathbf{r}_{j}(0)\|$ and $\omega_{i}(t)=1(0)$ if $\|\mathbf{r}_{i}(t)-\mathbf{r}_{i}(0)\|\leq$ ($>$) $l_{}$. We fix $l_{}=0.64$, the value at which the peak height of the corresponding four- point susceptibility $\chi_{4}(t)=1/N\sum_{i,j}g_{4}(r_{ij},t)$ is maximal, and fix the time $t=t_{\rm max}$ at which the corresponding $\chi_{4}(t)$ attains the maximum. 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# A Smoking Gun for Planetesimal Formation: Charge Driven Growth into a New Size Range Jens Teiser University of Duisburg-Essen Faculty of Physics Lotharstr. 1-21 D-47057 Duisburg, Germany Maximilian Kruss University of Duisburg-Essen Faculty of Physics Lotharstr. 1-21 D-47057 Duisburg, Germany Felix Jungmann University of Duisburg-Essen Faculty of Physics Lotharstr. 1-21 D-47057 Duisburg, Germany Gerhard Wurm University of Duisburg-Essen Faculty of Physics Lotharstr. 1-21 D-47057 Duisburg, Germany ###### Abstract Collisions electrically charge grains which promotes growth by coagulation. We present aggregation experiments with three large ensembles of basalt beads ($150\,\mu\mathrm{m}-180\,\mu\mathrm{m})$, two of which are charged, while one remains almost neutral as control system. In microgravity experiments, free collisions within these samples are induced with moderate collision velocities ($0-0.2\,\mathrm{m\,s}^{-1}$). In the control system, coagulation stops at (sub-)mm size while the charged grains continue to grow. A maximum agglomerate size of 5 cm is reached, limited only by bead depletion in the free volume. For the first time, charge-driven growth well into the centimeter range is directly proven by experiments. In protoplanetary disks, this agglomerate size is well beyond the critical size needed for hydrodynamic particle concentration as, e.g., by the streaming instabilities. ††journal: ApJL ## 1 Introduction The first stage of planet formation is dominated by hit-and-stick collisions between small dust and ice grains at small collision velocities (Wurm & Blum, 1998; Blum & Wurm, 2008; Johansen et al., 2014; Gundlach & Blum, 2015). Although this first step is fast and efficient, there are several obstacles, which stop this evolution. With increasing agglomerate size, the relative velocities between the colliding aggregates increase (Weidenschilling & Cuzzi, 1993). This leads to restructuring and compaction (Weidling et al., 2009; Meisner et al., 2012). If now two of these compact aggregates collide slowly ($<1\,\mathrm{m\,s}^{-1}$), they rather bounce off than stick to each other. This has been introduced as the bouncing barrier (Güttler et al., 2010; Zsom et al., 2010). Several experiments showed that self-consistent growth indeed comes to a halt at a particle size in the millimeter range (Kruss et al., 2016, 2017; Demirci et al., 2017). Slight shifts in aggregate size are possible depending on temperatures or magnetic fields (Kruss & Wurm, 2018, 2020; Demirci et al., 2019) but the bouncing barrier is a robust finding. Collisions and growth in a protoplanetary disk are governed by the interaction between gas and solids. Hydrodynamic processes therefore have a strong effect on particle evolution. Beyond inducing collisions, they can especially change the local particle concentration (Johansen et al., 2007; Johansen & Youdin, 2007; Chiang & Youdin, 2010; Squire & Hopkins, 2018). If a critical solid-to- gas ratio is reached, the mutual gravity between the solids might lead to the direct formation of a planetesimal (Youdin & Goodman, 2005; Simon et al., 2016; Klahr & Schreiber, 2020). This way, barriers in collisional growth could be prevented. However, while these drag instabilities can be very efficient, they require a minimum size of solids to be present (Dra̧żkowska & Dullemond, 2014; Carrera et al., 2015). Typically, they work best for so-called pebbles with the Stokes-number (ratio between the orbital period and gas-grain-coupling time) $St\sim 1$. Depending on the disk model and the location in the protoplanetary disk, this Stokes number typically translates into particle sizes of the order of a decimeter though somewhat smaller sizes might still work (Yang et al., 2017) under certain conditions. Obviously, to explain planet formation, a severe size gap must be bridged between the millimeter size resulting from the bouncing barrier and the decimeter required for the hydrodynamic processes to work. This bridge might be a charge dominated growth phase. Collisions and friction between particles lead to charge separation upon contact (Lacks & Mohan Sankaran, 2011). For a long time, this was attributed either to different materials in contact (different surface energies) or due to different sizes (Lee et al., 2015). Experiments showed that charge separation also occurs for particles of the same size and material (Jungmann et al., 2018). While the detailed physical processes are poorly understood, the resulting charge distributions in granular samples are well characterized. For a granular sample with particles of the same size and material, a broad charge distribution is the result. By first glimpse, it is similar to a Gaussian distribution, but with heavier tails (Haeberle et al., 2018). The peak position (mean charge) and the full width at half maximum (FWHM) are typically used to characterize the charge distribution of a granular sample. For multiple collisions of particles of the same size and same material the resulting charge distribution peaks at zero charge (Wurm et al., 2019). Within the scope of this paper, the term ”strongly charged” then refers to the FWHM of a corresponding charge distribution. Agitating a granular system for a duration of about 10 min establishes a charge distribution within the system which does not change significantly when the agitation is continued. It only depends on sample and atmospheric parameters as was shown in experiments with monodisperse basalt beads by Wurm et al. (2019). Larger beads are charged more strongly (larger FWHM) than smaller beads. Additionally, the charging of a granular sample strongly depends on the surrounding gas pressure. For a constant granular sample the width of the reached charge distribution follows a curve similar to Paschen’s law of electrical breakthrough in gases. The width (FWHM) of the charge distributions reaches a minimum at a characteristic pressure ($100\,\mathrm{Pa}\,-\,\mathrm{few}\,100\,\mathrm{Pa}$), depending on the sample. At larger pressures, the reached width increases gently, while it increases sharply for pressures smaller than the characteristic value. ## 2 Charge driven aggregation and stability It is obvious that two grains of opposite charge attract each other as a first step in aggregation. This effectively changes the collisional cross section. It is not obvious a priori though why an initial charge on individual grains should be beneficial for the aggregation process later on, once agglomerates have formed. In fact, once two grains of the same absolute charge, but opposite sign, collide and stick to each other this dimer is overall neutral. The long-range Coulomb interaction of net charges is no longer present and the collisional cross section will no longer be strongly enhanced. However, insulating grains do not discharge upon contact. This even holds for metal spheres, if their surface is not extremely cleaned from any contamination (Genc et al., 2019; Kaponig et al., 2020). A dimer or a more complex aggregate still hold charges on their surface. It has to be noted at this point that collisions lead to charge separation (not neutralization) in the first place and grains charged this way have a complex charge pattern with patches of negative and positive charges on their surface (Grosjean et al., 2020; Steinpilz et al., 2020b). This is also valid for grains which are net neutral. The typical configuration are therefore multipoles even on a single grain. Certainly, there are ways to discharge and neutralize grains, which depends on the environment (water content, gas pressure, temperature, radiation, material conductivity). In the experiments here, discharge takes hours, under protoplanetary disk conditions it might be years (Steinpilz et al., 2020a; Jungmann et al., 2018, 2021) (and running experiments by Steinpilz et al., personal communication). So, while these multipole configurations in aggregates might not attract other grains from far away, charges remain highly important during contact. As Coulomb forces decrease with distance of two charges $r_{c}$ as $1/r_{c}^{2}$, two oppositely charged spots on a surface close to the contact point can dominate the sticking force, independent of the net charge budget of the grains. Therefore, collisions lead to sticking at much higher collision velocities for charged grains (Jungmann et al., 2018). It is important to note that in an ensemble of grains net charge is only a proxy that is easily accessible to confirm that grains have a surface charge pattern. Nevertheless, the multipoles determine sticking forces. Aggregates are also more stable, i.e. higher collision velocities are required to destroy them compared to uncharged aggregates (Steinpilz et al., 2020a). Charge patches glue aggregates together. A simple analog for this situation is a salt crystal, which is overall neutral but the alternating charges still provide strong attraction. This way, we expect collisionally charged grains to grow far beyond the bouncing barrier. Therefore, there is a high potential in charge driven growth. It is still unclear though, how large agglomerates can grow this way. In Steinpilz et al. (2020a), cm-size charged agglomerates were observed but their direct formation from individual grains was not traced and took place prior to the free-floating phase in an agitated granular bed. The effects of charging were shown by means of numerical simulations matching the experiments in that case. The key question thus remains: Is charge driven coagulation able to provide the necessary aggregate sizes for drag instabilities to take over? ## 3 Experiment To investigate how large agglomerates can form by collisions of small charged particles, a microgravity experiment is currently being developed for sub- orbital platforms. Here, we report on the first shorter time microgravity experiments during this development, which were conducted at the Bremen Drop Tower (ZARM). Using the catapult mode, microgravity with residual acceleration of $<10^{-6}\,\mathrm{g}$ and a duration of $9.2\,\mathrm{s}$ could be used. ### 3.1 Experimental setup Figure 1: Schematic view of test cells 2 and 3. The upper cell (3) is a vacuum chamber ($p\approx 20\,\mathrm{Pa}$), while cell 2 is at normal pressure. The central part of the experimental setup consists of three test cells, two of which are shown in Fig. 1. All cells have the same geometry with a free volume of $50\,\mathrm{mm}\times 50\,\mathrm{mm}\times 42\,\mathrm{mm}$ and an additional particle reservoir of 14 mm depth and 25 mm length. While cells 2 and 3 are placed in the same unit, cell 1 is placed in a single unit with half the height. Test cell 3 (the upper one in Fig. 1) is designed as a vacuum chamber with a pressure of $20\,\mathrm{Pa}$, while the other two cells are at normal pressure. The side walls of the test cells are copper electrodes which are part of a capacitor at which a DC-voltage of 4 kV can be applied. The experiments are observed with a Raspberry Pi camera (30 frames/s, resolution: $1640\,\times 1230\,\mathrm{Px}$) using bright field illumination in a backlight configuration. The optical resolution is of the order of the particle size. Each sample consists of basalt beads with a size distribution between $150\,\mathrm{\mu m}$ and $180\,\mathrm{\mu m}$ diameter (Whitehouse Scientific). A total amount of 6 g is used in each cell. To reduce collisions between basalt beads and different materials, the top and the bottom of each test cell (including the particle reservoir) are coated with the same basalt beads. ### 3.2 Experiment protocol The test cells can be agitated with a harmonic motion using a voice coil mounted underneath. Prior to the catapult launch this agitation is used to charge the basalt beads for test cells 2 and 3 by collisions and friction due to constant agitation on ground. The duration is 20 min with a frequency of 14 Hz and an amplitude of 4.6 mm (peak to peak), similar to the experiment protocol in Steinpilz et al. (2020a). During this period the sample mostly remains within the sample reservoir and the beads are exposed to numerous collisions and friction among each other. Even in case the particles leave the reservoir, they are only exposed to particle-particle collisions, as the bottom of the test cells is coated with basalt beads and tilted by an angle of $4^{\circ}$, so basically no particles hit the side walls. The sample was left at rest for about 10 min between this agitation period and the catapult launch, which does not change the charge distribution significantly. In contrast to cells 2 and 3, test cell 1 was not agitated on ground, so a sample with minimum charge is used as control experiment. In microgravity, the agitation is used to distribute the sample at the beginning and to keep up a sufficient collision rate to see growth on the short timescales available at the drop tower. The experiment protocol has been changed for the different experiments, so that different aspects of the planned long-duration experiments could be tested on a short timescale. A high voltage at the capacitor plates can be used to estimate the charges in aggregates, but immediately stops further growth processes as the volume is cleared from particles. Agitation can be used to induce a high collision rate, but no charge measurement or particle tracking is possible during this agitation. The different parameters used in the presented experiments are described in section 4. ### 3.3 Collision velocities Due to the limitations of the optical system and the large particle concentration within the test cells, it is not possible to track single particles. However, the collision velocities can be estimated using the parameters of the agitation cycle. As the agitation follows a harmonic, frequency $f$ and maximum amplitude $A_{0}$ directly translate in a maximum velocity of the test cells via $v_{\rm{max}}=2\pi f\cdot A_{0}$. For the parameters used in Fig. 2 ($f=14\,\mathrm{Hz}$, $A_{0}=1.2\,\mathrm{mm}$) this results in a maximum velocity of $v_{\rm{max}}=0.11\,\mathrm{m\,s}^{-1}$. In case of perfectly elastic collisions between the particles and the experiment walls (top and bottom), a resting particle could get a maximum velocity of $v_{\rm{col}}=2v_{\rm{max}}$ or even larger in case of an initial velocity towards the wall. Indeed, non-charged basalt beads collide rather elastically with a coefficient of restitution $\epsilon=v_{\rm{after}}/v_{\rm{before}}$ of the order of $\epsilon=0.9$ (Bogdan et al., 2019). It has to be noted that smaller basalt beads are used and collision velocities are lower in the drop tower experiments compared to the work by Bogdan et al. (2019). Additionally, this situation is slightly more complicated. As the surfaces of the bottom (including the particle compartment) and the top are coated with the same basalt beads as the sample particles, all collisions are collisions between beads at a random impact parameter. Also the charges of the beads themselves influence the collision behavior (Jungmann et al., 2018), as the coefficient of restitution gets smaller for larger charges. Altogether we estimate the maximum collision velocities to be roughly the same as the maximum velocities of the test cells. Additionally, collisions in the free volume of the test cell will damp the original velocity distribution. We therefore assume a broad velocity distribution from almost zero to the maximum velocity of the test cell. ## 4 Results The experiments presented were planned to qualify an experiment hardware for suborbital flights and to test parts of the experiment protocol. Here, we present three different experiment protocols, which were used to show different aspects of the upcoming long-duration experiments. Figure 2: Evolution of the particle ensemble in cell 3 during one experimental run at different times, starting directly after the begin of microgravity and ending when the sample has reached a final state while the test cell is at rest. The width of the chamber of 50 mm can be used as a scale. ### 4.1 First growth The first experiment protocol was used to check if a rather homogeneous particle distribution can be generated by agitation in microgravity. Here, the sample was agitated with a frequency of 14 Hz, an amplitude of 1.2 mm, and a duration of 1 s, starting at the beginning of the microgravity phase. After a short break of 1 s, the agitation was then repeated for a duration of $1\,\mathrm{s}$. Afterwards, the test cells were not moved until the end of the microgravity. Fig. 2 shows the temporal evolution of the particles in the vacuum chamber (cell 3) for this experiment. It starts directly after the last agitation cycle from a well distributed sample, which blocks the illumination almost completely. While the test cell is kept at rest a clustering process can be observed. After around 5 s, the final state is reached as the grown clusters do not collide anymore. Of the three cells, the vacuum chamber shows the most homogeneous sample distribution in the initial state and the largest agglomerate sizes in the final state (see also Fig. 4). According to Wurm et al. (2019) basalt beads charge differently depending on the surrounding pressure. At pressures of about $100\,\mathrm{Pa\,-\,\mathrm{few}\,100\,\mathrm{Pa}}$ the charge distribution is narrowest. For lower pressure the width of the charge distribution rises steeply (Wurm et al., 2019). Additionally, particle motion is not damped significantly by gas drag, as the gas-grain coupling time exceeds the experiment duration. Therefore, this directly leads to larger particle velocities and a higher collision rate. However, the maximum size reached in this experiment run is still restricted to a few millimeters. This can be attributed to the declining collision rate, as the particle velocities are damped by collisions and/or gas drag (depending on the test cell) and therefore also the collision probability goes down. ### 4.2 Charges The charge distribution of single basalt beads cannot be obtained from the data available, as the spatial and temporal resolution of the camera system are not suitable for this. However, the agitation method and therefore the process of particle charging is almost identical to previous studies, either with glass beads (Jungmann et al., 2018, 2021; Steinpilz et al., 2020a) or with basalt beads (Wurm et al., 2019). The width (FWHM) of the corresponding charge distribution scales with the particle size (Wurm et al., 2019), with many studies treating charges on insulators as surface charges only (Lee et al., 2018; Grosjean et al., 2020; Steinpilz et al., 2020b). With a similar charge density on the surface as in Wurm et al. (2019), a charge distribution centered at zero charge with a FWHM of about $3\cdot 10^{-13}\,\mathrm{C}$ can be expected. To roughly estimate the charges on the agglomerates we performed an experiment in which the beads were shaken for 5 s during microgravity (f = 14 Hz, A = 1.2 mm). As shown in Fig. 4 (middle) cm-sized agglomerates form in cell 2. After agitation, a voltage of $\pm 2\,$kV is applied in that cell which accelerates all charged particles towards the electrodes. These single agglomerates are tracked manually and their acceleration is translated to the amount of charge they carry. For this the mass of the agglomerates and its error is estimated via their cross-sections in the images. Fig. 3 shows that the agglomerates are formed from several hundred up to thousands of single particles. Their typical charge is up to $10^{7}$ electron charges. We note that this is only an estimate of the order of magnitude as a detailed analysis is beyond the scope of this paper and not possible with the available data from the short-time experiments. However, this indicates that there are abundant charges on the clusters which might play a role in the agglomeration process. Figure 3: Absolute estimated charges of resulting clusters formed during microgravity. The large error bars result from the high uncertainty of the masses of the agglomerates. ### 4.3 Growing tall Figure 4: Final particle distributions with continuous agitation of 1 Hz under microgravity. Top: sample with minimum charge (no shaking in advance). Middle: 20 min shaking in advance, normal pressure. Bottom: 20 min shaking in advance, vacuum (20 Pa). The evolution presented in Fig. 2 shows that growth by collisions of charged basalt beads is possible in principle. On the other hand it becomes clear that the collision rate is crucial for the outcome and has to be kept on a high level. This was considered in the experiment shown in Fig. 4. Here, the experiment protocol was changed to maintain a certain collision rate, while observation of the grains is still possible. With the onset of microgravity, the test cells were agitated with $f=14\,\mathrm{Hz}$ and $A_{0}=1.2\,\mathrm{mm}$ amplitude for a duration of 3 s. Afterwards, the agitation frequency was reduced to $f=1\,\mathrm{Hz}$ for a duration of 4 s, resulting in an amplitude of 3 mm and a maximum velocity of $v_{\rm{max}}=0.02\,\mathrm{m\,s}^{-1}$. Afterwards, the test cells were at rest for the last 2 s of microgravity. Fig. 4 shows the final particle distributions during this experiment run. It also reveals the systematic differences between the three test cells. The least charged sample (top) only shows minor growth in comparison to the charged sample in the test cell with atmospheric pressure (middle) where larger entities evolve. The vacuum cell (bottom) shows a striking result. Almost all particles are incorporated into one large agglomerate, which therefore has a width of 5 cm (from wall to wall), a thickness of about 2 cm and a total mass of about $6\,\mathrm{g}$. The free volume between the larger agglomerates is also of great interest to interpret the result. In the least charged sample, the particles remain widely distributed in the entire volume. Also in the test cell with atmospheric pressure there is still a significant amount of single beads (or only very small agglomerates) in the free volume between the larger aggregates. This is totally different in cell 3, where the maximum charges can be expected. Single particles are either incorporated into the major agglomerate or stick to the walls of the cell. The free volume is almost completely cleared from small particles. Due to this particle depletion the growth process comes to a halt. Therefore, it can be assumed that the final distribution does not show the maximum sizes achievable by such collisions. ### 4.4 Stability When applying an electric field some large clusters are accelerated towards the electrodes and therefore reach high impact velocities. This can be used to estimate the stability of these clusters. Similar to chapter 4.2 these clusters were tracked manually and their impact velocities determined. An example of a cluster colliding and bouncing off the wall is shown in Fig. 5. Figure 5: Sequence of a 2.5 mm cluster (average diameter) colliding with the electrode. Its impact velocity is about 2 cm/s. The red arrow shows the direction of motion. Collisions between clusters and the (side) walls occur at impact velocities from $10^{-2}\,\mathrm{m\,s}^{-1}$ to $2.1\cdot 10^{-2}\,\mathrm{m\,s}^{-1}$, while the typical cluster sizes (average diameters) range from $2.5\,\mathrm{mm}$ to $5.3\,\mathrm{mm}$. No fragmentation was observed, only sticking or bouncing were observed. Relative velocities in protoplanetary disks depend on the sizes of the collision partners and are lowest for particles of similar size Weidenschilling & Cuzzi (1993). For equally sized particles, the collision velocities are $\leq 10^{-2}\,\mathrm{m\,s}^{-1}$, even for particles of a few cm in size (Weidenschilling & Cuzzi, 1993). As a collision with a solid wall is much more severe than mutual collisions between equally sized clusters, mutual collisions in protoplanetary disks will not destroy the grown agglomerates. The relative velocities are larger for particles of different size, so single particles hitting an agglomerate will be faster. However, the velocities are still in the range of $10^{-1}\,\mathrm{m\,s}^{-1}$, which is exactly in the velocity range of the single basalt beads during the agitation cycles, resulting in growth. ## 5 Conclusion and outlook Charge driven coagulation has already been presented by Steinpilz et al. (2020a) as a possible mechanism to overcome the bouncing barrier in planet formation. Although they could show that the size distribution of agglomerates in an ensemble changes when electrical charges are present, the maximum sizes found were still restricted. Of course, this first step might already be sufficient to help hydrodynamic processes to facilitate further growth (Yang et al., 2017; Schaffer et al., 2018). However, this process is still fragile with respect to the corresponding hydrodynamic models. Here, we present a smoking gun for coagulation to large agglomerate sizes. Although the detailed growth processes are still hidden due to the experimental limitations, there is the proof of concept that small charged grains indeed grow well into the centimeter range and possibly further starting from scratch, i.e. starting with a cloud of individual grains. We observed that growth is only possible if the grains are charged. The size of the largest agglomerates in the experiments presented here is limited by the particle supply (no further collisions) and a non-sufficient experiment duration. Charge driven growth might even continue into the decimeter range. It has to be noted though, that agglomerates in this size range are in danger of destruction by wind erosion, as they already move fast with respect to the surrounding gas. Future experiments will enable us to trace the charge driven coagulation in more detail. 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# Parity alternating permutations starting with an odd integer Frether Getachew Kebede111Corresponding author. Department of Mathematics, College of Natural and Computational Sciences, Addis Ababa University, P.O.Box 1176, Addis Ababa, Ethiopia; e-mail<EMAIL_ADDRESS>Fanja Rakotondrajao Département de Mathématiques et Informatique, BP 907 Université d’Antananarivo, 101 Antananarivo, Madagascar; e-mail<EMAIL_ADDRESS> ###### Abstract A Parity Alternating Permutation of the set $[n]=\\{1,2,\ldots,n\\}$ is a permutation with even and odd entries alternatively. We deal with parity alternating permutations having an odd entry in the first position, PAPs. We study the numbers that count the PAPs with even as well as odd parity. We also study a subclass of PAPs being derangements as well, Parity Alternating Derangements (PADs). Moreover, by considering the parity of these PADs we look into their statistical property of excedance. ###### keywords: parity , parity alternating permutation , parity alternating derangement , excedance ###### MSC: [2020] 05A05 , 05A15 , 05A19 ## 1 Introduction and preliminaries A permutation $\pi$ is a bijection from the set $[n]=\\{1,2,\ldots,n\\}$ to itself and we will write it in standard representation as $\pi=\pi(1)\,\pi(2)\,\cdots\,\pi(n)$, or as the product of disjoint cycles. The parity of a permutation $\pi$ is defined as the parity of the number of transpositions (cycles of length two) in any representation of $\pi$ as a product of transpositions. One way of determining the parity of $\pi$ is by obtaining the sign of $(-1)^{n-c}$, where $c$ is the number of cycles in the cycle representation of $\pi$. That is, if the sign of $\pi$ is -1, then $\pi$ is called an odd permutation, and an even permutation otherwise. For example, the permutation $4\,2\,1\,7\,8\,6\,3\,5=(1\,\,4\,\,7\,\,3)(2)(5\,\,8)(6)$, of length 8, is even since it has sign 1. All basic definitions and properties not explained here can be found in for example [8] and [4]. According to [9], a Parity Alternating Permutation over the set $[n]$ is a permutation, in standard form, with even and odd entries alternatively (in this general sense). The set $\mathcal{P}_{n}$ of all parity alternating permutations is a subgroup of the symmetric group $S_{n}$, the group of all permutations over $[n]$. The order of the set set $\mathcal{P}_{n}$ has been studied lately in relations to other number sequences such as Eulerian numbers (see [10, 9]). However, in this paper we will deal only with the parity alternating permutations which in addition have an odd entry in the first position; and we call them PAPs. It can be shown that the set $P_{n}$ containing all PAPs over $[n]$ is a subgroup of the symmetric group $S_{n}$ and also of the group $\mathcal{P}_{n}$. We consider this kind of permutations because, for odd $n$ there are no parity alternating permutations over $[n]$ beginning with an even integer. Avi Peretz determined the number sequence that count the number of PAPs (see A010551). Unfortunately, we could not find any details of his work. In A010551, we can also find the exponential generating function of these numbers due to Paul D. Hanna. Since there is no published proof of this formula we prove it here, as Theorem 2.2. Moreover, the numbers that count the PAPs with even parity and with odd parity (which were not studied before) are determined. By $p_{n}$ we denote the cardinality of the set $P_{n}$ of all PAPs over $[n]$. Let $\phi_{n}$ denote a map from $P_{n}$ to $S_{\lceil\frac{n}{2}\rceil}\times S_{\lfloor\frac{n}{2}\rfloor}$ that relates a PAP $\sigma$ to a pair of permutations $(\sigma_{1},\sigma_{2})$ in the set $S_{\lceil\frac{n}{2}\rceil}\times S_{\lfloor\frac{n}{2}\rfloor}$, in such a way that $\sigma_{1}(i)=\frac{\sigma(2i-1)+1}{2}$ and $\sigma_{2}(i)=\frac{\sigma(2i)}{2}$. It is easy to see that this map is a bijection. For example, the PAPs $5\,2\,1\,4\,3\,6\,7$ and $7\,4\,5\,6\,3\,2\,1$ over [7] are mapped to the pairs $(3\,1\,2\,4,\,1\,2\,3)$ and $(4\,3\,2\,1,\,2\,3\,1)$, respectively. If we consider a PAP $\sigma$ in cycle representations, then each cycle consists of integers of the same parity. Thus, we immediately get cycle representation of $\sigma_{1}$ and $\sigma_{2}$. For instance, the cycle form of the two PAPs above are $(1\,5\,3)(7)(2)(4)(6)$ and $(1\,7)(3\,5)(2\,4\,6)$ which correspond to the pairs $\left((1\,3\,2)(4),\,(1)(2)(3)\right)$ and $((1\,4)(2\,3),\,(1\,2\,3))$, respectively. (Unless stated otherwise we will always use (disjoint) cycle representation of permutations.) Another way of looking at the mapping $\phi_{n}$ is that $\sigma_{1}$ and $\sigma_{2}$ correspond to the parts that contain the odd and even integers in $\sigma$, respectively. Therefore, studying PAPs is similar to studying the two permutations that correspond to the even and the odd integers in the PAP separately and then combining the properties. In Table 1, we give a short summary of properties that permutations and PAPs satisfy (for detailed discussions, see Section 2). \| Permutations \| PAPs ---\|---\|--- Seq \| $1,1,2,6,24,120,\ldots.$ (A000142) \| $1,1,1,2,4,12,\ldots.$ (A010551) EGF \| $\frac{1}{1-x}$ \| $\frac{2\sqrt{4-x^{2}}+2\cos^{-1}\left(1-x^{2}/2\right)}{(2-x)\sqrt{4-x^{2}}}$ Even (seq) \| $1,1,1,3,12,60,\ldots.$ (A001710) \| $1,1,1,1,2,6,18,72,\ldots$ Odd (seq) \| $1,1,1,3,12,60,\ldots.$ (A001710) \| $0,0,0,1,2,6,18,72,\ldots$ Even (EGF) \| $\frac{2-x^{2}}{2-2x}$ \| $\frac{\sqrt{4-x^{2}}+\cos^{-1}\left(1-\frac{x^{2}}{2}\right)}{(2-x)\sqrt{4-x^{2}}}+\frac{x^{2}}{4}+\frac{x}{2}+\frac{1}{2}$ Odd (EGF) \| $\frac{x^{2}}{2-2x}$ \| $\frac{\sqrt{4-x^{2}}+\cos^{-1}\left(1-\frac{x^{2}}{2}\right)}{(2-x)\sqrt{4-x^{2}}}-\frac{x^{2}}{4}-\frac{x}{2}-\frac{1}{2}$ Table 1: A comparison table of permutations and PAPs (EGF mean exponential generating function). One interesting subset of $S_{n}$ is the set $D_{n}$ of derangements. For $d_{n}=\|D_{n}\|$, we have a well known relation $\displaystyle d_{n}=(n-1)[d_{n-1}+d_{n-2}],\,\,d_{0}=1\text{ and }d_{1}=0$ (1) for $n\geq 2$. A proof of this relation may be found in any textbook on combinatorics, but we will have later use of the following bijection due to Mantaci and Rakotondrajao ([6]). They define $\psi_{n}$ to be the bijection between $D_{n}$ and $[n-1]\times(D_{n-1}\cup D_{n-2})$ as follows: let $D_{n}^{(1)}$ denote the set of derangements over $[n]$ having the integer $n$ in a cycle of length greater than 2, and $D_{n}^{(2)}$ be the set of derangements over $[n]$ having $n$ in a transposition. These two sets are disjoint and their union is $D_{n}$. Then for $\delta\in D_{n}$ define $\psi_{n}(\delta)=(i,\delta^{\prime})$, where $i=\delta^{-1}(n)$ and $\delta^{\prime}$ is the derangement obtained from 1. $\bullet$ $\delta\in D_{n}^{(1)}$ by removing $n$ or 2. $\bullet$ $\delta\in D_{n}^{(2)}$ by removing the transposition $(i\,\,n)$ and then decreasing all integers greater than $i$ by 1. For instance, the pairs $(2,(1\,5\,2)(3\,4))$ and $(2,(1\,2)(3\,4))$ correspond to the derangements $(1\,5\,2\,6)(3\,4)$ and $(1\,3)(4\,5)(2\,6)$, respectively, for $n=6$. We denote the restricted bijections $\psi_{n}\|_{D_{n}^{(1)}}$ and $\psi_{n}\|_{D_{n}^{(2)}}$ by $\psi_{n}^{(1)}$ and $\psi_{n}^{(2)}$, respectively. Another important, and more difficult to prove, recurrence relation that the numbers $d_{n}$ satisfy is $\displaystyle d_{n}=n\,d_{n-1}+(-1)^{n},\,\,d_{0}=0$ (2) for $n\geq 1$. We will later make a use of the bijection $\tau_{n}:([n]\times D_{n-1})\backslash F_{n}\longrightarrow D_{n}\backslash E_{n}$ given by the second author ([7]) proving the recurrence. Where $E_{n}$ is the set containing the derangement $\Delta_{n}=(1\,2)(3\,4)\cdots(n-1\,\,\,n)$ for even $n$, and is empty for odd $n$. $F_{n}$ is the set containing the pair $(n,\,\Delta_{n-1})$ when $n$ is odd, and is empty when $n$ is even. Thus, the inverse $\zeta_{n}$ of $\tau_{n}$ relates an element of $[n-1]\times D_{n-1}$ with every derangement over $[n]$ that has the integer $n$ in a cycle of length greater than 2, and an element of $\\{n\\}\times D_{n-1}\backslash F_{n}$ with every derangement over $[n]$ in which $n$ lies in a transposition. Classifying derangements by their parity, we denote the number of even and odd derangements over $[n]$ by $d^{e}_{n}$ and $d^{o}_{n}$, respectively. Clearly $d_{n}=d^{e}_{n}+d^{o}_{n}$. Moreover, the numbers $d^{e}_{n}$ and $d^{o}_{n}$ satisfy the relations $\displaystyle d^{e}_{n}=(n-1)[d^{o}_{n-1}+d^{o}_{n-2}]\quad\text{ and }\quad d^{o}_{n}=(n-1)[d^{e}_{n-1}+d^{e}_{n-2}],$ (3) for $n\geq 2$ with initial conditions $d^{e}_{0}=1$, $d^{e}_{1}=0$, $d^{o}_{0}=0$, and $d^{o}_{1}=0$ ([6], Proposition 4.1). We will put a major interest on Parity Alternating Derangements (PADs) which are the derangements which also are parity alternating permutations starting with odd integers. Let $\mathfrak{d}_{n}$ denote cardinality of the set of PADs $\mathfrak{D}_{n}=D_{n}\cap P_{n}$. The restricted bijection $\Phi_{n}=\phi_{n}\|_{\mathfrak{D}_{n}}:\mathfrak{D}_{n}\longrightarrow D_{\lceil\frac{n}{2}\rceil}\times D_{\lfloor\frac{n}{2}\rfloor}$ will let us consider the odd parts and the even parts of any given PAD regarded as ordinary derangements with smaller length than the length of the PAD. The mapping $\Phi_{n}$ plays the central role in our investigations. In Table 2, we display the connection of ordinary derangements and PADs (for detailed discussions, see Section 3). Finding explicit expressions for some of the generating functions are still open questions. On the other hand the EGF for the PADs for example is the solution to an eighth order differential equation with polynomial coefficients, and also is expressible in terms of Hadamard products of some known generating functions. \| Derangements \| PADs ---\|---\|--- Seq \| $1,0,1,2,9,44,\ldots$ (A000166) \| $1,0,0,0,1,2,4,18,81,396,\ldots$ EGF \| $\frac{e^{-x}}{1-x}$ \| open RR \| $d_{n}=(n-1)[d_{n-1}+d_{n-2}]$ \| relation (4) RR \| $d_{n}=nd_{n-1}+(-1)^{n}$ \| relation (5) Even (seq) \| $1,0,0,2,3,24,130,\ldots$ (A003221) \| $1,0,0,0,1,0,4,6,45,192,976\ldots$ Odd (seq) \| $0,0,1,0,6,20,135,\ldots$ (A000387) \| $0,0,0,0,0,2,0,12,36,204,960,\ldots$ Even (EGF) \| $\frac{(2-x^{2})e^{-x}}{2(1-x)}$ \| open Odd (EGF) \| $\frac{x^{2}e^{-x}}{2(1-x)}$ \| open Even (RR) \| $d^{e}_{n}=(n-1)[d^{o}_{n-1}+d^{o}_{n-2}]$ \| relation (6) Odd (RR) \| $d^{o}_{n}=(n-1)[d^{e}_{n-1}+d^{e}_{n-2}]$ \| relation (7) Even - Odd \| $(-1)^{n-1}(n-1)$ \| $(-1)^{n-2}\Big{\lceil}\frac{n-2}{2}\Big{\rceil}\Big{\lfloor}\frac{n-2}{2}\Big{\rfloor}$ Table 2: A comparison table of derangements and PAPs, RR represents recurrence relation. In section 4, we study excedance distribution over PADs by means of the corresponding distributions for the two derangements obtained by $\Phi_{n}$. ## 2 Parity Alternating Permutations (PAPs) As we stated in the introduction, we use splitting method by the mapping $\phi_{n}$ in the study of PAPs. One application of this is that the number of PAPs of length $n$ is $\displaystyle p_{n}=\|S_{\lceil\frac{n}{2}\rceil}\|\|S_{\lfloor\frac{n}{2}\rfloor}\|=\lceil n/2\rceil!\lfloor n/2\rfloor!.$ $n$ \| 0 \| 1 \| 2 \| 3 \| 4 \| 5 \| 6 \| 7 \| 8 \| 9 \| 10 ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- $p_{n}$ \| 1 \| 1 \| 1 \| 2 \| 4 \| 12 \| 36 \| 144 \| 576 \| 2880 \| 14400 Table 3: First few terms of the sequence $\\{p_{n}\\}_{0}^{\infty}$. ###### Proposition 2.1. The numbers $p_{n}$ satisfy the recurrence relation $\displaystyle p_{n}=\lceil n/2\rceil p_{n-1},$ for $n\geq 1$ and $p_{0}=1$. ###### Proof. First let us define a mapping $\omega_{n}:S_{n}\longrightarrow[n]\times S_{n-1}$ by $\displaystyle\omega_{n}(\pi)=(i,\pi^{\prime}),$ where $\pi^{\prime}$ is obtained from $\pi\in S_{n}$ by removing the integer $n$, and $i=\pi^{-1}(n)$. One can easily see that $\omega_{n}$ is a bijection. Now let us take a PAP $\sigma$ over $[n]$. Then $\phi_{n}$ maps $\sigma$ to a pair $(\sigma_{1},\sigma_{2})$. Define then a mapping $\Omega:P_{n}\longrightarrow\Bigl{[}\big{\lceil}\frac{n}{2}\big{\rceil}\Bigr{]}\times P_{n-1}$ as follows: for $n=2m$ $\displaystyle\Omega(\sigma)=\left(i,\phi_{2m}^{-1}(\sigma_{1},\,\sigma_{2}^{\prime})\right),$ where $(i,\sigma_{2}^{\prime})=\omega_{m}(\sigma_{2})$, and for $n=2m+1$ $\displaystyle\Omega(\sigma)=\left(i,\phi_{2m+1}^{-1}(\sigma_{1}^{\prime},\,\sigma_{2})\right),$ where $(i,\sigma_{1}^{\prime})=\omega_{m+1}(\sigma_{1})$. The mapping $\Omega$ is a bijection since $\omega_{n}$ is a bijection for every $n\geq 1$. In any case, there are $\big{\lceil}\frac{n}{2}\big{\rceil}$ possibilities for $i$. ∎ As a consequence, we get the following theorem. ###### Theorem 2.2. The exponential generating function $P(x)=\sum_{n\geq 0}p_{n}\frac{x^{n}}{n!}$ of the sequence $\\{p_{n}\\}_{n=0}^{\infty}$ has the closed formula $P(x)=\displaystyle\frac{2}{2-x}+\displaystyle\frac{\cos^{-1}(1-\displaystyle\frac{x^{2}}{2})}{(2-x)\sqrt{1-\displaystyle\frac{x^{2}}{4}}}.$ ###### Proof. Based on the recurrence relation in Proposition 2.1, we obtain the following relations $\displaystyle P_{0}(x)=\frac{x}{2}P_{1}(x)+1\quad\text{ and }\quad P_{1}(x)=\frac{x}{2}P_{0}(x)+\frac{1}{2}\int_{0}^{x}P_{0}(t)\,dt,$ where $P_{0}(x)=\sum_{n\geq 0}p_{2n}\frac{x^{2n}}{(2n)!}$ and $P_{2}(x)=\sum_{n\geq 0}p_{2n+1}\frac{x^{2n+1}}{(2n+1)!}$. Clearly, $P(x)=P_{0}(x)+P_{1}(x)$. Additionally, $P_{0}(x)$ satisfies the differential equation $\displaystyle\left(1-\frac{x^{2}}{4}\right)P^{\prime}_{0}(x)=\frac{x^{2}+2}{2x}P_{0}(x)-\frac{1}{x}.$ Thus, we obtain the formulas $\displaystyle P_{0}(x)=\frac{4}{4-x^{2}}+\frac{4x\sin^{-1}\left(\frac{x}{2}\right)}{(4-x^{2})^{3/2}}\quad\text{ and }\quad P_{1}(x)=\frac{8}{4x-x^{3}}+\frac{8x\sin^{-1}\left(\frac{x}{2}\right)}{x(4-x^{2})^{3/2}}-\frac{2}{x}.$ Therefore, $P(x)=\displaystyle\frac{2}{2-x}+\displaystyle\frac{\cos^{-1}(1-\displaystyle\frac{x^{2}}{2})}{(2-x)\sqrt{1-\displaystyle\frac{x^{2}}{4}}}.\qed$ For classification of PAPs in terms of their parity, we use $P^{e}_{n}$ and $P^{o}_{n}$ to denote the set of even PAPs and odd PAPs, respectively, and $p_{n}^{e}$ and $p_{n}^{o}$ as their cardinality, respectively. Thus, $\displaystyle p_{n}=p^{e}_{n}+p^{o}_{n}.$ $n$ \| 0 \| 1 \| 2 \| 3 \| 4 \| 5 \| 6 \| 7 \| 8 \| 9 \| 10 ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- $p_{n}^{e}$ \| 1 \| 1 \| 1 \| 1 \| 2 \| 6 \| 18 \| 72 \| 288 \| 1440 \| 7200 $p_{n}^{o}$ \| 0 \| 0 \| 0 \| 1 \| 2 \| 6 \| 18 \| 72 \| 288 \| 1440 \| 7200 Table 4: First few terms of the sequences $\\{p_{n}^{e}\\}_{0}^{\infty}$ and $\\{p_{n}^{o}\\}_{0}^{\infty}$. Our goal is now to study the relationships between these two sequences. ###### Theorem 2.3. The numbers $p_{n}^{e}$ and $p_{n}^{o}$ satisfy the recurrence relations $\displaystyle p_{n}^{e}=\lfloor(n-1)/2\rfloor p_{n-1}^{o}+p_{n-1}^{e}$ $\displaystyle p_{n}^{o}=\lfloor(n-1)/2\rfloor p_{n-1}^{e}+p_{n-1}^{o},$ for $n\geq 1$, with initial conditions $p^{e}_{0}=1$ and $p^{o}_{0}=0$. ###### Proof. Let $S_{n}^{e}$ and $S_{n}^{o}$ be the set of even and odd permutations, respectively. Define two mappings $\omega_{n}^{e}:S_{n}^{e}\longrightarrow[n-1]\times S_{n-1}^{o}\cup S_{n-1}^{e}$ and $\omega_{n}^{o}:S_{n}^{o}\longrightarrow[n-1]\times S_{n-1}^{e}\cup S_{n-1}^{o}$ by $\displaystyle\omega_{n}^{e}(\pi)=\begin{cases}(i,\pi^{\prime}),\text{ if }i\neq n\\\ \pi^{\prime\prime},\text{ otherwise}\end{cases}\quad\text{ and }\quad\omega_{n}^{o}(\pi)=\begin{cases}(j,\pi^{\prime}),\text{ if }j\neq n\\\ \pi^{\prime\prime},\text{ otherwise}\end{cases},$ respectively, where $i=\pi^{-1}(n)$, $\pi^{\prime}$ is obtained from $\pi$ by removing the integer $n$, and $\pi^{\prime\prime}$ is obtained from $\pi$ by removing the cycle $(n)$, for $\pi\in S_{n}^{e}$. Similarly for $\omega_{n}^{o}$. It is easy to see that both mappings $\omega_{n}^{e}$ and $\omega_{n}^{o}$ are bijections. The mapping $\omega_{n}^{e}$ changes the parity of $\pi$ when it results in $\pi^{\prime}$ and preserves when it results in $\pi^{\prime\prime}$. This is because the signs of $\pi$, $\pi^{\prime}$ and $\pi^{\prime\prime}$ are $(-1)^{n-c}$, $(-1)^{n-1-c}$, and $(-1)^{n-1-c+1}$, respectively, where $c$ is the number of cycles in $\pi$. For the mapping $\omega_{n}^{o}$ we apply similar argument. Now consider a PAP $\sigma$ in $P_{n}$. Then $\phi_{n}$ maps $\sigma$ in to a pair $(\sigma_{1},\sigma_{2})$. Following the notation in the proof of Proposition 2.1, let us define two mappings $\Omega^{e}:P_{n}^{e}\longrightarrow\Bigl{[}\big{\lfloor}\frac{n-1}{2}\big{\rfloor}\Bigr{]}\times P_{n-1}^{o}\cup P_{n-1}^{e}$ and $\Omega^{o}:P_{n}^{o}\longrightarrow\Bigl{[}\big{\lfloor}\frac{n-1}{2}\big{\rfloor}\Bigr{]}\times P_{n-1}^{e}\cup P_{n-1}^{o}$ as follows: 1. 1. when $n$ is even $\displaystyle\Omega^{e}(\sigma)=\begin{cases}\left(i,\,\phi_{n}^{-1}(\sigma_{1},\sigma_{2}^{\prime})\right),\text{ if }i\neq\frac{n}{2}\\\ \phi_{n}^{-1}(\sigma_{1},\sigma_{2}^{\prime\prime}),\text{ otherwise}\end{cases}\text{and }$ $\displaystyle\Omega^{o}(\sigma)=\begin{cases}\left(i,\,\phi_{n}^{-1}(\sigma_{1},\sigma_{2}^{\prime})\right),\text{ if }i\neq\frac{n}{2}\\\ \phi_{n}^{-1}(\sigma_{1},\sigma_{2}^{\prime\prime}),\text{ otherwise}\end{cases},$ where $i=\sigma_{2}^{-1}(\frac{n}{2})$, and both $\sigma^{\prime}_{2}$, $\sigma^{\prime\prime}_{2}$ are obtained from $\sigma_{2}$ by the mapping $\omega_{\frac{n}{2}}^{e}$ when $\sigma\in P_{n}^{e}$ and by the mapping $\omega_{\frac{n}{2}}^{o}$ when $\sigma\in P_{n}^{o}$, 2. 2. when $n$ is odd $\displaystyle\Omega^{e}(\sigma)=\begin{cases}\left(j,\,\phi_{n}^{-1}(\sigma_{1}^{\prime},\sigma_{2})\right),\text{ if }j\neq\frac{n+1}{2}\\\ \phi_{n}^{-1}(\sigma_{1}^{\prime\prime},\sigma_{2}),\text{ otherwise}\end{cases}\text{and }$ $\displaystyle\Omega^{o}(\sigma)=\begin{cases}\left(j,\,\phi_{n}^{-1}(\sigma_{1}^{\prime},\sigma_{2})\right),\text{ if }j\neq\frac{n+1}{2}\\\ \phi_{n}^{-1}(\sigma_{1}^{\prime\prime},\sigma_{2}),\text{ otherwise}\end{cases},$ where $j=\sigma_{1}^{-1}(\frac{n+1}{2})$, and both $\sigma^{\prime}_{1}$, $\sigma^{\prime\prime}_{1}$ are obtained from $\sigma_{1}$ by the mapping $\omega_{\frac{n+1}{2}}^{e}$ when $\sigma\in P_{n}^{e}$ and by the mapping $\omega_{\frac{n+1}{2}}^{o}$ when $\sigma\in P_{n}^{o}$. Since $\omega_{n}$ is bijection for $n\geq 2$, both $\Omega^{e}$ and $\Omega^{o}$ are bijections too. Note that in both mappings there are $\big{\lfloor}\frac{n-1}{2}\big{\rfloor}$ possibilities for $i$ ($i\neq\frac{n}{2}$) and similarly for $j$ ($j\neq\frac{n+1}{2}$). ∎ ###### Proposition 2.4. For any positive integer $n\geq 3$, we have $\displaystyle p_{n}^{e}=p_{n}^{o}.$ ###### Proof. Multiplying a PAP by a transposition $(1,n)$ if $n$ is odd, or by $(1,n-1)$ if $n$ is even, we obtain a PAP having opposite parity. This multiplication means swapping the first and the last odd integer of a PAP in standard representation. It creates a bijection between $P^{e}_{n}$ and $P^{o}_{n}$. ∎ By applying Proposition 2.4 and considering $p(x)$, we get: ###### Corollary 2.5. The exponential generating functions of the sequences $\\{p_{n}^{e}\\}_{n\geq 0}$ and $\\{p_{n}^{o}\\}_{n\geq 0}$ have the closed forms $\displaystyle P^{e}(x)=\displaystyle\frac{1}{2}\left(P(x)+\frac{x^{2}}{2}+x+1\right)\quad\text{ and }\quad P^{o}(x)=\displaystyle\frac{1}{2}\left(P(x)-\frac{x^{2}}{2}-x-1\right).$ $\square$ ## 3 Parity Alternating Derangements (PADs) As a result of the bijection $\Phi_{n}$ in the introduction above, we can determine the number $\mathfrak{d}_{n}$ of PADs over $[n]$ as follows: $\displaystyle\mathfrak{d}_{n}=d_{\lceil n/2\rceil}d_{\lfloor n/2\rfloor}=\sum_{j=0}^{\lceil n/2\rceil}\sum_{i=0}^{\lfloor n/2\rfloor}\lceil n/2\rceil!\lfloor n/2\rfloor!\frac{(-1)^{i+j}}{j!\,i!}.$ $n$ \| 0 \| 1 \| 2 \| 3 \| 4 \| 5 \| 6 \| 7 \| 8 \| 9 ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- $d_{n}$ \| 1 \| 0 \| 1 \| 2 \| 9 \| 44 \| 265 \| 1854 \| 14833 \| 133496 $\mathfrak{d}_{n}$ \| 1 \| 0 \| 0 \| 0 \| 1 \| 2 \| 4 \| 18 \| 81 \| 396 Table 5: First few values of $d_{n}$ and $\mathfrak{d}_{n}$. In the next theorem we give a formula for the number of PADs, connected to the relation (1). ###### Theorem 3.6. The number of PADs over $[n]$ satisfy the recurrence relation $\displaystyle\mathfrak{d}_{n}=s\big{(}\mathfrak{d}_{n-1}+(n-2-s)\left(\mathfrak{d}_{n-3}+\mathfrak{d}_{n-4}\right)\big{)},$ (4) where $s=\big{\lfloor}\frac{n-1}{2}\big{\rfloor}=\frac{2n-3-(-1)^{n}}{4}$, for $n\geq 4$, with initial conditions $\mathfrak{d}_{0}=1$, $\mathfrak{d}_{1}=0$, $\mathfrak{d}_{2}=0$ and $\mathfrak{d}_{3}=0$. ###### Proof. The proof is splitted into two cases, for PADs over a set of odd and even sizes. Let $(\delta_{1},\delta_{2})$ be the corresponding pair of a PAD $\delta\in\mathfrak{D}_{n}$ under the mapping $\Phi_{n}$. Define a mapping $\Psi:\mathfrak{D}_{n}\longrightarrow[s]\times\left(\mathfrak{D}_{n-1}\cup[n-2-s]\times(\mathfrak{D}_{n-3}\cup\mathfrak{D}_{n-4})\right)$ as follows: Case I: for odd $n$, depending on the following two elective properties of $\delta$, the mapping $\Psi$ will be defined as: 1. 1. in the event of the largest entry $\frac{n+1}{2}$ of $\delta_{1}$ being in a cycle of length greater than 2, we let $\displaystyle\Psi(\delta)=\left(i,\,\Phi_{n}^{-1}(\delta^{\prime}_{1},\,\delta_{2})\right),$ where $(i,\delta^{\prime}_{1})=\psi_{\frac{n+1}{2}}^{(1)}(\delta_{1})$; 2. 2. in the event when $\frac{n+1}{2}$ lies in a transposition in $\delta_{1}$, we distinguish two cases: * (a) if the largest entry $\frac{n-1}{2}$ of $\delta_{2}$ is contained in a cycle of length greater than 2, then $\displaystyle\Psi(\delta)=\left(i,\,j,\,\Phi_{n}^{-1}(\delta^{\prime}_{1},\,\delta^{\prime}_{2})\right),$ where $(i,\,\delta^{\prime}_{1})=\psi_{\frac{n+1}{2}}^{(2)}(\delta_{1})$ and $(j,\,\delta^{\prime}_{2})=\psi_{\frac{n-1}{2}}^{(1)}(\delta_{2})$; * (b) if $\frac{n-1}{2}$ is contained in a cycle of length 2 in $\delta_{2}$, then $\displaystyle\Psi(\delta)=\left(i,\,j,\,\Phi_{n}^{-1}(\delta^{\prime}_{1},\,\delta^{\prime}_{2})\right),$ where $(i,\,\delta^{\prime}_{1})=\psi_{\frac{n+1}{2}}^{(2)}(\delta_{1})$ and $(j,\,\delta^{\prime}_{2})=\psi_{\frac{n-1}{2}}^{(2)}(\delta_{2})$. Case II: for even $n$, 1. 1. in the event of the largest entry $\frac{n}{2}$ of $\delta_{2}$ lies in a cycle of length greater than 2, we let $\displaystyle\Psi(\delta)=\left(i,\,\Phi_{n}^{-1}(\delta_{1},\,\delta^{\prime}_{2})\right),$ where $(i,\,\delta^{\prime}_{2})=\psi_{\frac{n}{2}}^{(1)}(\delta_{2})$; 2. 2. in the event when $\frac{n}{2}$ being in a transposition in $\delta_{2}$, we distinguish two cases: * (a) if the largest entry $\frac{n}{2}$ in $\delta_{1}$ contained in a cycle of length greater than 2, then $\displaystyle\Psi(\delta)=\left(i,\,j,\,\Phi_{n}^{-1}(\delta^{\prime}_{1},\,\delta^{\prime}_{2})\right),$ where $(i,\,\delta^{\prime}_{1})=\psi_{\frac{n}{2}}^{(1)}(\delta_{1})$ and $(j,\,\delta^{\prime}_{2})=\psi_{\frac{n}{2}}^{(2)}(\delta_{2})$; * (b) if $\frac{n}{2}$ contained in a cycle of length 2 in $\delta_{1}$, then $\displaystyle\Psi(\delta)=\left(i,\,j,\,\Phi_{2n}^{-1}(\delta^{\prime}_{1},\,\delta^{\prime}_{2})\right),$ where $(i,\,\delta^{\prime}_{1})=\psi_{\frac{n}{2}}^{(2)}(\delta_{1})$ and $(j,\,\delta^{\prime}_{2})=\psi_{\frac{n}{2}}^{(2)}(\delta_{2})$. Since $\psi_{n}$ is a bijection for any $n\geq 2$, one can easily conclude that $\Psi$ is a bijection too. Note that in both cases there are $\big{\lfloor}\frac{n-1}{2}\big{\rfloor}=s$ possibilities for $i$ and $\big{\lfloor}\frac{n-2}{2}\big{\rfloor}=n-2-s$ possibilities for $j$. Thus, the formula in the Theorem follows. ∎ The next theorem is connected to the relation (2). ###### Theorem 3.7. The number $\mathfrak{d}_{n}$ of PADs also satisfies the relation $\displaystyle\mathfrak{d}_{n}=s\mathfrak{d}_{n-1}+(-1)^{s}d_{n-s},$ (5) where $s=\big{\lceil}\frac{n}{2}\big{\rceil}=\frac{2n+1-(-1)^{n}}{4}$, for $n\geq 1$ with $d_{1}=0$, $d_{0}=1$ and $\mathfrak{d}_{0}=1$. ###### Proof. Distinguishing by means of the parity of $n$, we can write the relation (5) as: $\displaystyle\mathfrak{d}_{n}=d_{s}\big{(}s\,d_{s-1}+(-1)^{s}\big{)}\,\,\text{ when }n\text{ is even, and }$ $\displaystyle\mathfrak{d}_{n}=\big{(}s\,d_{s-1}+(-1)^{s}\big{)}\,d_{s-1}\,\,\text{ when }n\text{ is odd}.$ Now, take a PAD $\delta$ in $\mathfrak{D}_{n}$ and introduce two mappings as: * 1. $Z_{0}:\big{(}D_{s}\backslash E_{s}\big{)}\times D_{s}\longrightarrow[s]\times\big{(}D_{s-1}\backslash F_{s}\big{)}\times D_{s}$ by $\displaystyle\delta\xrightleftharpoons[\Phi^{-1}_{n}]{\Phi_{n}}(\delta_{1},\delta_{2})\xrightleftharpoons[id_{s}\times\tau_{s}]{id_{s}\times\zeta_{s}}\big{(}\delta_{1},(i,\delta^{\prime}_{2})\big{)}\xrightleftharpoons[h]{h}(i,(\delta_{1},\delta^{\prime}_{2}))\xrightleftharpoons[id\times\Phi_{n}]{id\times\Phi_{n}^{-1}}\big{(}i,\Phi_{n}^{-1}(\delta_{1},\delta^{\prime}_{2})\big{)},$ and * 2. $Z_{1}:\big{(}D_{s}\backslash E_{s}\big{)}\times D_{s-1}\longrightarrow[s]\times\big{(}D_{s-1}\backslash F_{s}\big{)}\times D_{s-1}$ by $\displaystyle\delta\xrightleftharpoons[\Phi^{-1}_{n}]{\Phi_{n}}(\delta_{1},\delta_{2})\xrightleftharpoons[\tau_{s}\times id_{s-1}]{\zeta_{s}\times id_{s-1}}\big{(}(i,\delta^{\prime}_{1}),\delta_{2}\big{)}\xrightleftharpoons[h_{2}]{h_{1}}\big{(}i,(\delta^{\prime}_{1},\delta_{2})\big{)}\xrightleftharpoons[id\times\Phi_{n}^{-1}]{id\times\Phi_{n}}\big{(}i,\Phi_{n}^{-1}(\delta^{\prime}_{1},\delta_{2})\big{)}.$ Note that $h$, $h_{1}$, and $h_{2}$ are the obvious recombination maps. Since all the functions we used to define the two mappings $Z_{0}$ and $Z_{1}$ are injective, both $Z_{0}$ and $Z_{1}$ are bijection mappings. ∎ In order to classify PADs with respect to their parity, we let $\mathfrak{D}_{n}^{e}$ and $\mathfrak{D}_{n}^{o}$ denote the set of even and odd PADs over $[n]$, respectively. Moreover, $\mathfrak{d}^{e}_{n}=\|\mathfrak{D}_{n}^{e}\|$ and $\mathfrak{d}^{o}_{n}=\|\mathfrak{D}_{n}^{o}\|$. Obviously, $\mathfrak{d}_{n}=\mathfrak{d}^{e}_{n}+\mathfrak{d}^{o}_{n}$. $n$ \| 0 \| 1 \| 2 \| 3 \| 4 \| 5 \| 6 \| 7 \| 8 \| 9 \| 10 ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- $\mathfrak{d}_{n}^{e}$ \| 1 \| 0 \| 0 \| 0 \| 1 \| 0 \| 4 \| 6 \| 45 \| 192 \| 976 $\mathfrak{d}_{n}^{o}$ \| 0 \| 0 \| 0 \| 0 \| 0 \| 2 \| 0 \| 12 \| 36 \| 204 \| 960 Table 6: First few values of the number of even and odd PADs. ###### Proposition 3.8. The numbers of even and odd PADs satisfy the relations $\displaystyle\mathfrak{d}^{e}_{n}=d^{e}_{\lfloor\frac{n}{2}\rfloor}d^{e}_{\lceil\frac{n}{2}\rceil}+d^{o}_{\lfloor\frac{n}{2}\rfloor}d^{o}_{\lceil\frac{n}{2}\rceil},$ $\displaystyle\mathfrak{d}^{o}_{n}=d^{e}_{\lfloor\frac{n}{2}\rfloor}d^{o}_{\lceil\frac{n}{2}\rceil}+d^{e}_{\lceil\frac{n}{2}\rceil}d^{o}_{\lfloor\frac{n}{2}\rfloor},$ for $n\geq 0$, with initial conditions $d^{e}_{0}=1$, $d^{e}_{1}=0$, $d^{o}_{0}=0$, and $d^{o}_{1}=0$. ###### Proof. Let $\delta$ be a PAD over $[n]$. Then, there exist $\delta_{1}\in D_{\lceil\frac{n}{2}\rceil}$ and $\delta_{2}\in D_{\lfloor\frac{n}{2}\rfloor}$ such that $\Phi_{n}(\delta)=(\delta_{1},\delta_{2})$. If $\delta\in\mathfrak{D}_{n}^{e}$, then $\delta_{1}$ and $\delta_{2}$ must have the same parity. Thus, $\mathfrak{d}^{e}_{n}=d^{e}_{\lfloor\frac{n}{2}\rfloor}d^{e}_{\lceil\frac{n}{2}\rceil}+d^{o}_{\lfloor\frac{n}{2}\rfloor}d^{o}_{\lceil\frac{n}{2}\rceil}$. If $\delta\in\mathfrak{D}_{n}^{e}$, then $\delta_{1}$ and $\delta_{2}$ must have opposite parities. Hence, $\mathfrak{d}^{o}_{n}=d^{e}_{\lfloor\frac{n}{2}\rfloor}d^{o}_{\lceil\frac{n}{2}\rceil}+d^{e}_{\lceil\frac{n}{2}\rceil}d^{o}_{\lfloor\frac{n}{2}\rfloor}$. ∎ ###### Corollary 3.9. The number of PADs with even parity and with odd parity satisfy the recurrence relations $\displaystyle\mathfrak{d}_{n}^{e}=s\left(\mathfrak{d}_{n-1}^{o}+(n-2-s)(\mathfrak{d}_{n-3}^{e}+\mathfrak{d}_{n-4}^{e})\right)$ (6) $\displaystyle\mathfrak{d}_{n}^{o}=s\left(\mathfrak{d}_{n-1}^{e}+(n-2-s)(\mathfrak{d}_{n-3}^{o}+\mathfrak{d}_{n-4}^{o})\right),$ (7) where $s=\big{\lfloor}\frac{n-1}{2}\big{\rfloor}=\frac{2n-3-(-1)^{n}}{4}$, for $n\geq 4$ with initial conditions $\mathfrak{d}_{0}^{e}=1$, $\mathfrak{d}_{0}^{o}=0$, and $\mathfrak{d}_{i}^{e}=\mathfrak{d}_{i}^{o}=0$ for $i=1,2,3$. ###### Proof. It is enough to clarify the effect of the bijection $\psi_{n}$ on the parity of a derangement over $[n]$, the rest is just applying the bijection $\Psi$ from the proof of Theorem 3.6. Letting $\delta$ be in $D_{n}$, $\delta^{\prime}$ has sign either $(-1)^{n-1-c}$ if $\delta\in D_{n}^{(1)}$, or $(-1)^{n-2-(c-1)}=(-1)^{n-1-c}$ if $\delta\in D_{n}^{(2)}$. Here $\delta^{\prime}$ is the derangement obtained from $\delta$ by applying $\psi_{n}$, and $c$ is the number of cycles in the cycle representation of $\delta$. This means, the bijection $\psi_{n}$ changes the parity of a derangement. ∎ ###### Definition 3.10. Let $\delta$ be a derangement over $[n]$ in standard cycle representation and let $C_{1}=(1\,\,a_{2}\,\,\cdots\,\,a_{m})$ be the first cycle. Following [3], an extraction point of $\delta$ is an entry $e\geq 2$ if $e$ is the smallest number in the set $\\{2,\ldots,n\\}\backslash\\{a_{2}\\}$ for which $C_{1}$ does not end with the numbers of $\\{2,\ldots,e\\}\backslash\\{a_{2}\\}$ written in decreasing order. The $(n-1)$ derangements, $\delta_{n,i}=(1\,\,i\,\,n\,\,n-1\,\,\cdots\,\,i+2\,\,i+1\,\,i-1\,\,i-2\,\,\cdots\,\,3\,\,2)$ for $i\in[2,n]$, that do not have extraction points are called the exceptional derangements and the set of exceptional derangements is denoted by $X_{n}$. Note that the extraction point (if it exists) must belong to the first or the second cycle. Following this approach we may introduce: ###### Definition 3.11. We call the PAD $\displaystyle\Phi_{n}^{-1}(\delta_{\lceil\frac{n}{2}\rceil,i},\,\delta_{\lfloor\frac{n}{2}\rfloor,j}),\text{ for }i\in\big{[}2,\lceil n/2\rceil\big{]}\text{ and }j\in\big{[}2,\lfloor n/2\rfloor\big{]}$ an exceptional PAD and we let $\mathcal{X}_{n}$ denote the set containing them. ###### Example 3.12. If $n=8$, then $\displaystyle\mathcal{X}_{8}$ $\displaystyle=\\{\Phi_{8}^{-1}(\delta_{4,2},\,\delta_{4,2}),\,\,\Phi_{8}^{-1}(\delta_{4,2},\,\delta_{4,3}),\,\,\Phi_{8}^{-1}(\delta_{4,2},\,\delta_{4,4}),\,\,\Phi_{8}^{-1}(\delta_{4,3},\,\delta_{4,2}),\,\,\Phi_{8}^{-1}(\delta_{4,3},\,\delta_{4,3}),$ $\displaystyle\quad\quad\Phi_{8}^{-1}(\delta_{4,3},\,\delta_{4,4}),\,\,\Phi_{8}^{-1}(\delta_{4,4},\,\delta_{4,2}),\,\,\Phi_{8}^{-1}(\delta_{4,4},\,\delta_{4,3}),\,\,\Phi_{8}^{-1}(\delta_{4,4},\,\delta_{4,4})\\}$ $\displaystyle=\\{(1\,3\,7\,5)(2\,4\,8\,6),\,\,(1\,3\,7\,5)(2\,6\,8\,4),\,\,(1\,3\,7\,5)(2\,8\,6\,4),\,\,(1\,5\,7\,3)(2\,4\,8\,6),\,\,(1\,5\,7\,3)$ $\displaystyle\quad\quad(2\,6\,8\,4),\,\,(1\,5\,7\,3)(2\,8\,6\,4),\,\,(1\,7\,5\,3)(2\,4\,8\,6),\,\,(1\,7\,5\,3)(2\,6\,8\,4),\,\,(1\,7\,5\,3)(2\,8\,6\,4)\\}.$ ###### Remark 3.13. Since the exceptional derangements over $[n]$ have sign $(-1)^{n-1}$, all the exceptional PADs in $\mathcal{X}_{n}$ have the same parity, with sign $(-1)^{n-2}=(-1)^{n}$. ###### Remark 3.14. As it was proved in [3], the number of the exceptional derangements in $X_{n}$ is the difference of the number of even and odd derangements, i.e., $d_{n}^{e}-d_{n}^{o}=(-1)^{n-1}(n-1)$. Chapman ([5]) also provide a bijective proof for the same formula. Below we give the idea of the proof due to Benjamin, Bennett, and Newberger ([3]). Let $f_{n}$ be the involution on $D_{n}\backslash X_{n}$ defined by $\displaystyle f_{n}(\pi)=f_{n}\big{(}(1\,\,a_{2}\,\,X\,\,e\,\,Y\,\,Z)\,\pi^{\prime}\big{)}=(1\,\,a_{2}\,\,X\,\,Z)(e\,\,Y)\,\pi^{\prime}$ for $\pi$ in $D_{n}\backslash X_{n}$ with the extraction point $e$ in the first cycle; and vice versa for the other $\pi$ in $D_{n}\backslash X_{n}$ with the extraction point $e$ in the second cycle. $a_{2}$ is the second element in the first cycle of $\pi$; $X$, $Y$, and $Z$ are ordered subsets of $[n]$, $Y\neq\emptyset$ and $Z$ consist the elements of $\\{2,3,\ldots,e-1\\}\backslash\\{a_{2}\\}$ written in decreasing order, and $\pi^{\prime}$ is the rest of the derangement in $\pi$. Since the number of the cycles in $\pi$ and $f_{n}(\pi)$ differ by one, they must have opposite parity. Labeling $\mathfrak{d}_{n}^{e}{-}\mathfrak{d}_{n}^{o}$ as $\mathfrak{f}_{n}$, we have the following result. ###### Proposition 3.15. The difference $\mathfrak{f}_{n}$ counts the number of exceptional PADs over $[n]$ and its closed formula is given by $\displaystyle\mathfrak{f}_{n}=(-1)^{n-2}\Big{\lceil}\frac{n-2}{2}\Big{\rceil}\Big{\lfloor}\frac{n-2}{2}\Big{\rfloor}.$ (8) ###### Proof. Let $\delta$ be in $\mathfrak{D}_{n}\backslash\mathcal{X}_{n}$. Then $\Phi_{n}$ map $\delta$ with the pair $(\delta_{1},\delta_{2})$. Define a mapping $F$ from $\mathfrak{D}_{n}\backslash\mathcal{X}_{n}$ to itself as $\displaystyle F(\delta)=\begin{cases}\Phi_{n}^{-1}\left(f_{\lceil\frac{n}{2}\rceil}(\delta_{1}),\,\delta_{2}\right)\text{ if }n\text{ is odd}\\\ \Phi_{n}^{-1}\left(\delta_{1},\,f_{\lfloor\frac{n}{2}\rfloor}(\delta_{2})\right)\text{ otherwise}\end{cases}$ Since $f_{n}$ is a bijection and changes parity, $F$ is a bijection and also $\delta$ and $F(\delta)$ have opposite parity. The leftovers, which are the PADs in $\mathcal{X}_{n}$ with sign $(-1)^{n-2}$, are counted by $\lceil\frac{n-2}{2}\rceil\lfloor\frac{n-2}{2}\rfloor$. ∎ $n$ \| 0 \| 1 \| 2 \| 3 \| 4 \| 5 \| 6 \| 7 \| 8 \| 9 \| 10 \| 11 \| 12 ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- $\mathfrak{f}_{n}$ \| 1 \| 0 \| 0 \| 0 \| 1 \| -2 \| 4 \| -6 \| 9 \| -12 \| 16 \| -20 \| 25 Table 7: First few values the difference $\mathfrak{f}_{n}=\mathfrak{d}_{n}^{e}-\mathfrak{d}_{n}^{o}$. This sequence looks like an alternating version of A002620, to which Paul Barry constructed an EGF. From [2], we learned that he used Mathematica to generate the formulas by taking the Inverse Laplace Transform of the ordinary generating function described in [1]. However, we propose the following more direct, constructive proof. ###### Theorem 3.16. The exponential generating function of the difference $\mathfrak{f}_{n}$ has the closed form $\displaystyle\frac{e^{x}}{8}+\frac{e^{-x}}{8}(2x^{2}+6x+7).$ ###### Proof. From the closed formula of $\mathfrak{f}_{n}$ in Proposition 3.15, we have $\displaystyle(n-1)^{2}=\mathfrak{f}_{2n}$ $\displaystyle=\mathfrak{d}_{2n}^{e}-\mathfrak{d}_{2n}^{o}$ $\displaystyle n(n-1)=\mathfrak{f}_{2n+1}$ $\displaystyle=-(\mathfrak{d}_{2n+1}^{e}-\mathfrak{d}_{2n+1}^{o}).$ Hence, $\displaystyle\sum_{n\geq 0}\mathfrak{f}_{2n}\frac{x^{n}}{(2n)!}=\sum_{n\geq 0}(n-1)^{2}\frac{x^{2n}}{(2n)!}=\frac{x^{2}-3x+4}{8}e^{x}+\frac{x^{2}+3x+4}{8}e^{-x},\quad\text{ and }$ $\displaystyle\sum_{n\geq 0}\mathfrak{f}_{2n+1}\frac{x^{2n+1}}{(2n+1)!}=-\sum_{n\geq 0}n(n-1)\frac{x^{n}}{n!}=-\frac{x^{2}-3x+3}{8}e^{x}+\frac{x^{2}+3x+3}{8}e^{-x}$ Thus, $\displaystyle\sum_{n\geq 0}\mathfrak{f}_{2n}\frac{x^{2n}}{(2n)!}+\sum_{n\geq 0}\mathfrak{f}_{2n+1}\frac{x^{2n+1}}{(2n+1)!}$ $\displaystyle=\frac{e^{x}}{8}+\frac{e^{-x}}{8}(2x^{2}+6x+7)$ is the desired formula. ∎ ## 4 Excedance distribution over PADs In this section, we focus on excedance distribution in PADs. ###### Definition 4.17. We say that a permutation $\sigma$ has an excedance on $i\in[n]$ if $\sigma(i)>i$. In this case, $i$ is said to be an excedant. We give the notation below in the study of this property: $\displaystyle\mathfrak{d}_{n,k}$ $\displaystyle=\|\\{\delta\in\mathfrak{D}_{n}:\,\,\delta\text{ has }k\text{ excedances}\\}\|,$ $\displaystyle\mathfrak{d}_{n,k}^{o}$ $\displaystyle=\|\\{\delta\in\mathfrak{D}^{o}(n):\,\,\delta\text{ has }k\text{ excedances}\\}\|,$ $\displaystyle\mathfrak{d}_{n,k}^{e}$ $\displaystyle=\|\\{\delta\in\mathfrak{D}^{e}(n):\,\,\delta\text{ has }k\text{ excedances}\\}\|.$ Mantaci and Rakotondrajao ([6]) studied the excedance distribution in derangements, i.e., the numbers $\displaystyle d_{n,k}$ $\displaystyle=\|\\{\delta\in D_{n}:\,\,\delta\text{ has }k\text{ excedances}\\}\|,$ $\displaystyle d^{e}_{n,k}$ $\displaystyle=\|\\{\delta\in D_{n}:\,\,\delta\text{ is an even derangemnt and }\text{ has }k\text{ excedances}\\}\|,$ $\displaystyle d^{o}_{n,k}$ $\displaystyle=\|\\{\delta\in D_{n}:\,\,\delta\text{ is an odd derangemnt and }\text{ has }k\text{ excedances}\\}\|.$ ###### Remark 4.18. Since the number of excedances in a derangement over $[n]$ is in the range $[1,n-1]$, the number of excedances of a PAD over $[n]$ is at least 2 and at most $n-2$. ###### Proposition 4.19. The numbers $\mathfrak{d}_{n,k}$, $\mathfrak{d}_{n,k}^{e}$, and $\mathfrak{d}_{n,k}^{o}$ are symmetric, that is $\displaystyle\mathfrak{d}_{n,k}=\mathfrak{d}_{n,n-k},\,\,\mathfrak{d}^{e}_{n,k}=\mathfrak{d}^{e}_{n,n-k},\text{ and }\,\,\mathfrak{d}^{o}_{n,k}=\mathfrak{d}^{o}_{n,n-k}.$ ###### Proof. The bijection from $\mathfrak{D}_{n}$ to it self, defined as $\delta\mapsto\delta^{-1}$ for $\delta\in\mathfrak{D}_{n}$, associates a PAD having $k$ excedances with a PAD having $n-k$ exeedances and also preserves parity. ∎ \| \| \| $\mathfrak{d}_{n,k}$ \| \| \| \| ---\|---\|---\|---\|---\|---\|---\|--- $n\setminus k$ \| 2 \| 3 \| 4 \| 5 \| 6 \| 7 \| 4 \| 1 \| \| \| \| \| \| 5 \| 1 \| 1 \| \| \| \| \| 6 \| 1 \| 2 \| 1 \| \| \| \| 7 \| 1 \| 8 \| 8 \| 1 \| \| \| 8 \| 1 \| 14 \| 51 \| 14 \| 1 \| \| 9 \| 1 \| 28 \| 169 \| 169 \| 28 \| 1 \| 10 \| 1 \| 42 \| 483 \| 884 \| 483 \| 42 \| 1 Table 8: First few terms of the number of PADs in terms of number of excedances \| \| \| $\mathfrak{d}^{e}_{n,k}$ \| \| \| \| ---\|---\|---\|---\|---\|---\|---\|--- $n\setminus k$ \| 2 \| 3 \| 4 \| 5 \| 6 \| 7 \| 4 \| 1 \| \| \| \| \| \| 5 \| 0 \| 0 \| \| \| \| \| 6 \| 1 \| 2 \| 1 \| \| \| \| 7 \| 0 \| 3 \| 3 \| 0 \| \| \| 8 \| 1 \| 8 \| 27 \| 8 \| 1 \| \| 9 \| 0 \| 13 \| 83 \| 83 \| 13 \| 0 \| 10 \| 1 \| 22 \| 243 \| 444 \| 243 \| 22 \| 1 \| \| \| $\mathfrak{d}^{o}_{n,k}$ \| \| \| \| ---\|---\|---\|---\|---\|---\|---\|--- $n\setminus k$ \| 2 \| 3 \| 4 \| 5 \| 6 \| 7 \| 4 \| 0 \| \| \| \| \| \| 5 \| 1 \| 1 \| \| \| \| \| 6 \| 0 \| 0 \| 0 \| \| \| \| 7 \| 1 \| 5 \| 5 \| 1 \| \| \| 8 \| 0 \| 6 \| 24 \| 6 \| 0 \| \| 9 \| 1 \| 15 \| 86 \| 86 \| 15 \| 1 \| 10 \| 0 \| 20 \| 240 \| 440 \| 240 \| 20 \| 0 Table 9: First few values of $\mathfrak{d}_{n,k}$ in terms of their parity ###### Proposition 4.20. The excedance distribution of a PAD is given by $\displaystyle\mathfrak{d}_{n,k}=\begin{cases}\sum_{i=1}^{k-1}d_{\lceil\frac{n}{2}\rceil,i}\,d_{\lfloor\frac{n}{2}\rfloor,k-i},\,\,\text{ if }\,2\leq k\leq\lfloor\frac{n}{2}\rfloor\\\ \sum_{i=1}^{n-k-1}d_{\lceil\frac{n}{2}\rceil,i}\,d_{\lfloor\frac{n}{2}\rfloor,n-k+i},\,\,\text{ if }\,\lfloor\frac{n}{2}\rfloor<k\leq n{-}2\end{cases}.$ ###### Proof. To find the number of excedances of a PAD $\delta$ over $[n]$, we sum up the number of exeedances in $\delta_{1}$ and in $\delta_{2}$, where $(\delta_{1},\,\delta_{2})$ is the image of $\delta$ defined in $\Phi_{n}$. Since there are $d_{m,i}$ derangements in $D_{m}$ having $i$ excedances, for $i\in[1,\,\,m-1]$, the products $d_{m,i}\,d_{l,k-i}$, for $m=\lceil\frac{n}{2}\rceil$ and $l=\lfloor\frac{n}{2}\rfloor$, determine the number of PADs over $[n]$ having $k$ excedances. Summing up the products over the range $i=1,2,\ldots,k{-}1$ will give the first formula. The second formula follows from Proposition 4.19. ∎ ###### Corollary 4.21. The excedance distribution of PADs in terms of their parity is given by: $\displaystyle\mathfrak{d}_{n,k}^{e}=\begin{cases}\sum_{i=1}^{k-1}(d^{e}_{\lceil\frac{n}{2}\rceil,i}\,d^{e}_{\lfloor\frac{n}{2}\rfloor,k-i}+d^{o}_{\lceil\frac{n}{2}\rceil,i}\,d^{o}_{\lfloor\frac{n}{2}\rfloor,k-i}),\,\,\text{ if }\,2\leq k\leq\lfloor n/2\rfloor\\\ \sum_{i=1}^{n-k-1}(d^{e}_{\lceil\frac{n}{2}\rceil,i}\,d^{e}_{\lfloor\frac{n}{2}\rfloor,n-k+i}+d^{o}_{\lceil\frac{n}{2}\rceil,i}\,d^{o}_{\lfloor\frac{n}{2}\rfloor,n-k+i}),\,\,\text{ if }\,\lfloor n/2\rfloor<k\leq n{-}2\end{cases},$ $\displaystyle\mathfrak{d}_{n,k}^{o}=\begin{cases}\sum_{i=1}^{k-1}(d^{e}_{\lceil\frac{n}{2}\rceil,i}\,d^{o}_{\lfloor\frac{n}{2}\rfloor,k-i}+d^{o}_{\lceil\frac{n}{2}\rceil,i}\,d^{e}_{\lfloor\frac{n}{2}\rfloor,k-i}),\,\,\text{ if }\,2\leq k\leq\lfloor n/2\rfloor\\\ \sum_{i=1}^{n-k-1}(d^{e}_{\lceil\frac{n}{2}\rceil,i}\,d^{o}_{\lfloor\frac{n}{2}\rfloor,n-k+i}+d^{o}_{\lceil\frac{n}{2}\rceil,i}\,d^{e}_{\lfloor\frac{n}{2}\rfloor,n-k+i}),\,\,\text{ if }\,\lfloor n/2\rfloor<k\leq n{-}2\end{cases}.$ $\square$ An immediate consequence of this Corollary is ###### Proposition 4.22. We have $\displaystyle\mathfrak{f}_{n,k}=\mathfrak{d}_{n,k}^{e}-\mathfrak{d}_{n,k}^{o}=(-1)^{n}\,\max\\{k-1,\,n{-}(k+1)\\}$ for $n\geq 4$ and $2\leq k\leq n{-}2$. ###### Proof. Mantaci and Rakotondrajao (see [6]) have proved the identity $d^{o}_{n,k}-d^{e}_{n,k}=(-1)^{n}$ using recursive argument. Applying this with the Corollary 4.21, we get the desired formula. ∎ \| \| \| $\mathfrak{f}_{n,k}$ \| \| \| \| ---\|---\|---\|---\|---\|---\|---\|--- $n\setminus k$ \| 2 \| 3 \| 4 \| 5 \| 6 \| 7 \| 4 \| 1 \| \| \| \| \| \| 5 \| -1 \| -1 \| \| \| \| \| 6 \| 1 \| 2 \| 1 \| \| \| \| 7 \| -1 \| -2 \| -2 \| -1 \| \| \| 8 \| 1 \| 2 \| 3 \| 2 \| 1 \| \| 9 \| -1 \| -2 \| -3 \| -3 \| -2 \| -1 \| 10 \| 1 \| 2 \| 3 \| 4 \| 3 \| 2 \| 1 Table 10: The first few values of the difference $\mathfrak{d}^{e}_{n,k}-\mathfrak{d}^{o}_{n,k}$. ###### Theorem 4.23. The exponential generating function for the sequence $\\{\mathfrak{f}_{n,k}\\}$ has the closed form $\displaystyle\frac{1}{(1-u)^{2}}\left(u^{2}e^{-x}+e^{-ux}-2u\cosh{\sqrt{u}x}+\frac{u+u^{2}}{\sqrt{u}}\sinh{\sqrt{u}x}-(1-u)^{2}\right).$ ###### Proof. Let $\mathfrak{f}_{n}(u)=\sum_{k=2}^{n-2}\mathfrak{f}_{n,k}u^{k}$ and $\mathfrak{f}(x,u)=\sum_{n\geq 4}\mathfrak{f}_{n}(u)\frac{x^{n}}{n!}$. From Proposition 4.22, we have $\displaystyle\mathfrak{f}_{2m,k}$ $\displaystyle=\begin{cases}k-1,\quad\text{if }2\leq k\leq m\\\ 2m-k-1,\quad\text{if }m<k\leq 2m{-}2\end{cases},$ $\displaystyle\mathfrak{f}_{2m+1,k}$ $\displaystyle=\begin{cases}-(k-1),\quad\text{if }2\leq k\leq m\\\ -(2m-k),\quad\text{if }m<k\leq 2m{-}1\end{cases},$ for $m\geq 2$. So, $\displaystyle\mathfrak{f}_{2m}(u)$ $\displaystyle=\sum_{k=2}^{m}(k-1)u^{k}+\sum_{k=m+1}^{2m-2}(2m-k-1)u^{k}=\frac{u^{2}-2u^{m+1}+u^{2m}}{(1-u)^{2}},$ $\displaystyle\mathfrak{f}_{2m+1}(u)$ $\displaystyle=\sum_{k=2}^{m}-(k-1)u^{k}+\sum_{k=m+1}^{2m-1}-(2m-k)u^{k}=\frac{u^{m+2}+u^{m+1}-u^{2}-u^{2m+1}}{(1-u)^{2}},$ $\displaystyle\mathfrak{f}(x,u)$ $\displaystyle=\sum_{m\geq 2}\mathfrak{f}_{2m}(u)\frac{x^{2m}}{(2m)!}+\sum_{m\geq 2}\mathfrak{f}_{2m+1}(u)\frac{x^{2m+1}}{(2m+1)!}$ $\displaystyle=\frac{1}{(1-u)^{2}}\left(u^{2}e^{-x}+e^{-ux}-2u\cosh{\sqrt{u}x}+\frac{u+u^{2}}{\sqrt{u}}\sinh{\sqrt{u}x}-(1-u)^{2}\right).$ ∎ ## Methodological remarks In this paper, most of our results are obtained in a way of splitting the permutations into two subwords. However, this method is not always applicable. One example is the number of PADs avoiding the pattern $p=1\,2$. The only derangement that avoid $p$ is $(1\,\,n\,)(2\,\,\,n{-}1\,)\cdots(\frac{n}{2}\,\,\,\frac{n+2}{2})$, for even $n$, that is, the derangement over $[n]$ with entries in decreasing order when written in linear representation. However, it does not exist if $n$ is odd, since $\frac{n+1}{2}$ is a fixed point. The PAD $\delta$ created from a pair $(\delta_{1},\delta_{2})$, by the mapping $\Phi_{n}^{-1}$, of two even length derangements that both avoids the pattern $p$ is $\delta=(1\,\,n{-}1\,)(3\,\,\,n{-}3\,)\cdots\big{(}\frac{n-2}{2}\,\,\,\frac{n+2}{2}\big{)}(2\,\,n\,)(4\,\,\,n{-}2\,)\cdots\big{(}\frac{n}{2}\,\,\,\frac{n+4}{2}\big{)}$, which is $n{-}1\,\,n\,\,n{-}3\,\,n{-}2\cdots 3\,\,4\,\,1\,\,2$ in linear form, has length $n\equiv 0\pmod{4}$. However, each pair $i\,\,i{+}1$, where $i$ is an entry in odd position, is a subword with the occurrence of the pattern $p$ in $\delta$. This indicates that $\delta_{1}$ and $\delta_{2}$ avoid $p$ but $\delta$ doesn’t. Things get even more complicated with patterns of length greater than 2. Final Remarks: As for now, we have not been successful in finding the recurrence relations and generating functions for the sequences $\\{\mathfrak{d}_{n,k}\\}_{n=0}^{\infty}$, $\\{\mathfrak{d}_{n,k}^{e}\\}_{n=0}^{\infty}$, and $\\{\mathfrak{d}_{n,k}^{o}\\}_{n=0}^{\infty}$. ## Acknowledgements The first author acknowledges the financial support extended by the cooperation agreement between International Science Program at Uppsala University and Addis Ababa University. Special thanks go to Prof. Jörgen Backelin, Prof. Paul Vaderlind and Dr. Per Alexandersson of Stockholm University - Dept. of Mathematics, for all their valuable inputs and suggestions. Many thanks to our colleagues from CoRS - Combinatorial Research Studio, for lively discussions and comments. ## References * [1] Paul Barry. On a central transform of integer sequences. arXiv preprint arXiv:2004.04577, 2020. * [2] Paul Barry. Private communication to the first author, $14^{th}$, December 2020. * [3] Arthur T Benjamin, Curtis T Bennett, and Florence Newberger. Recounting the odds of an even derangement. Mathematics Magazine, 78(5):387–390, 2005. * [4] Miklós Bóna. Combinatorics of permutations. CRC Press, 2012. * [5] Robin Chapman. An involution on derangements. Discrete Mathematics, 231(1):121–122, 2001. * [6] Roberto Mantaci and Fanja Rakotondrajao. Exceedingly deranging! Advances in Applied Mathematics, 30(1-2):177–188, 2003\. * [7] Fanja Rakotondrajao. k-fixed-points-permutations. Integer: Electronic Jornal of Combinatorial Number Theory, 7(A36):A36, 2007. * [8] Richard P Stanley. Enumerative combinatorics volume 1 second edition. Cambridge studies in advanced mathematics, 2011. * [9] Shinji Tanimoto. Combinatorics of the group of parity alternating permutations. Advances in Applied Mathematics, 44(3):225–230, 2010. * [10] Shinji Tanimoto. Parity alternating permutations and signed eulerian numbers. Annals of Combinatorics, 14(3):355–366, 2010.
# Will Artificial Intelligence supersede Earth System and Climate Models? Christopher Irrgang Helmholtz Centre Potsdam, German Research Centre for Geosciences GFZ, Potsdam, Germany Niklas Boers Department of Mathematics and Computer Science, Free University of Berlin, Germany Potsdam Institute for Climate Impact Research, Potsdam, Germany Department of Mathematics and Global Systems Institute, University of Exeter, UK Maike Sonnewald Program in Atmospheric and Oceanic Sciences, Princeton University, Princeton, NJ 08540, USA NOAA/OAR Geophysical Fluid Dynamics Laboratory, Ocean and Cryosphere Division, Princeton, NJ 08540, USA University of Washington, School of Oceanography, Seattle, WA, USA Elizabeth A. Barnes Colorado State University, Fort Collins, USA Christopher Kadow German Climate Computing Center DKRZ, Hamburg, Germany Joanna Staneva Helmholtz-Zentrum Geesthacht, Center for Material and Coastal Research HZG, Geesthacht, Germany Jan Saynisch-Wagner Helmholtz Centre Potsdam, German Research Centre for Geosciences GFZ, Potsdam, Germany ###### Abstract We outline a perspective of an entirely new research branch in Earth and climate sciences, where deep neural networks and Earth system models are dismantled as individual methodological approaches and reassembled as learning, self-validating, and interpretable Earth system model-network hybrids. Following this path, we coin the term ”Neural Earth System Modelling” (NESYM) and highlight the necessity of a transdisciplinary discussion platform, bringing together Earth and climate scientists, big data analysts, and AI experts. We examine the concurrent potential and pitfalls of Neural Earth System Modelling and discuss the open question whether artificial intelligence will not only infuse Earth system modelling, but ultimately render them obsolete. For decades, scientists have utilized mathematical equations to describe geophysical and climate processes and to construct deterministic computer simulations that allow for the analysis of such processes. Until recently, process-based models had been considered irreplaceable tools that helped to understand the complex interactions in the coupled Earth system and that provided the only tools to predict the Earth system’s response to anthropogenic climate change. The provocative thought that Earth system models (ESMs) might lose their fundamental importance in the advent of novel artificial intelligence (AI) tools has sparked both a gold-rush feeling and contempt in the scientific communities. On the one hand, deep neural networks have been developed that complement and have started to outperform the skill of process-based models in various applications, ranging from numerical weather prediction to climate research. On the other hand, most neural networks are trained for isolated applications and lack true process knowledge. Regardless, the daily increasing data streams from Earth system observation (ESO), increasing computational resources, and the availability and accessibility of powerful AI tools, particularly in machine learning (ML), have led to numerous innovative frontier applications in Earth and climate sciences. Based on the current state, recent achievements, and recognised limitations of both process-based modelling and AI in Earth and climate research, we draw a perspective on an imminent and profound methodological transformation, hereafter named Neural Earth System Modelling (NESYM). To solve the emerging challenges, we highlight the necessity of new transdisciplinary collaborations between the involved communities. ## Overview on Earth System Modelling and Earth System Observations Earth system models (ESMs)1 combine process-based models of the different sub- systems of the Earth system into an integrated numerical model that yields for a given state of the coupled system at time $t$ the tendencies associated with that state, i.e., a prediction of the system state for time $t+1$. The individual model components, or modules, describe sub-systems including the atmosphere, the oceans, the carbon and other biogeochemical cycles, radiation processes, as well as land surface and vegetation processes and marine ecosystems. These modules are then combined by a dynamic coupler to obtain a consistent state of the full system for each time step. For some parts of the Earth system, the primitive physical equations of motion are known explicitly, such as the Navier-Stokes equations that describe the fluid dynamics of the atmosphere and oceans (Fig. 1). In practise, it is impossible to numerically resolve all relevant scales of the dynamics and approximations have to be made. For example, the fluid dynamical equations for the atmosphere and oceans are integrated on discrete spatial grids, and all processes that operate below the grid resolution have to be parameterised to assure a closed description of the system. Since the multi-scale nature of the dynamics of geophysical fluids implies that the subgrid-scale processes interact with the larger scales that are resolved by the model, (stochastic) parameterization of subgrid-scale processes is a highly non-trivial, yet unavoidable, part of climate modelling 2, 3, 4. Figure 1: Symbolic representation of Earth system components and exemplary deterministic or stochastic coupling mechanisms on long and short spatio- temporal scales. For other parts of the Earth system, primitive equations of motion, such as the Navier-Stokes equations for atmospheric motion, do not exist. Essentially, this is due to the complexity of the Earth system, where many phenomena that emerge at a macroscopic level are not easily deducible from microscopic-scales that may or may not be well-understood. A typical example is given by ecosystems and the physiological processes governing the vegetation that covers vast parts of the land surface, as well as their interactions with the atmosphere, the carbon and other geochemical cycles. Also for these cases, approximations in terms of parameterizations of potentially crucial processes have to be made. Regardless of the specific process, such parameterizations induce free parameters in ESMs, for which suitable values have to be found empirically. The size of state-of-the-art ESMs mostly prohibits systematic calibration methods such as, e.g., the ones based on Bayesian inference, and the models are therefore often tuned manually. The quality of the calibration as well as the overall accuracy of the model can only be assessed with respect to relatively sparse observations of the last 170 years, at most, and there is no way to assess the models’ skill in predicting future climate conditions5. The inclusion of free parameters possibly causes biases or structural model errors and the example of the discretized spatial grid suggests that the higher the spatial resolution of an ESM, the smaller the potential errors. Likewise, it is expected that the models’ representation of the Earth system will become more accurate the more processes are resolved explicitly. The inclusion of a vastly increasing number of processes, together with continuously rising spatial resolution, have indeed led to the development of comprehensive ESMs that have become irreplaceable tools to analyse and predict the state of the Earth system. From the first assessment report of the Intergovernmental Panel on Climate Change (IPCC) in 1990 to the fifth phase of the Climate Model Intercomparison Project (CMIP5)6 and the associated fifth IPCC assessment report in 2014, the spatial resolution has increased from around 500km to up to 70km. In accordance, the CMIP results show that the models have, over the course of two decades, greatly improved in their accuracy to reproduce crucial characteristics of the Earth system, such as the evolution of the global mean temperatures (GMT) since the beginning of instrumental data in the second half of the 19th century, or the average present-day spatial distribution of temperature or precipitation 7, 8. Despite the tremendous success of ESMs, persistent problems and uncertainties remain: (1) A crucial quantity for the evaluation of ESMs is the equilibrium climate sensitivity (ECS), defined as the amount of equilibrium GMT increase that results from an instantaneous doubling of atmospheric carbon dioxide 9. There remains a large ECS range in current ESM projections and reducing these uncertainties, and hence the uncertainties of future climate projections, is one of the key challenges in the development of ESMs. Nevertheless, from CMIP5 to CMIP6, the likely range of ECS has widened from $2.$1–$4.7^{\circ}$C to $1.8$–$5.6^{\circ}$C10, 11. A highly promising line of research in this regard focuses on the identification of emergent constraints, which in principle allow to narrow down the projected range for a model variable of interest, given that the variable has a concise relationship with another model variable that can be validated against past observations 12, 13. The development of suitable data-driven techniques for this purpose is still in its infancy. (2) Both theoretical considerations and paleoclimate data suggest that several sub-systems of the Earth system can abruptly change their state in response to gradual changes in forcing14, 15. There is concern that current ESMs will not be capable of predicting future abrupt climate changes, because the instrumental era of less than two centuries has not experienced comparable transitions, and model validation against paleoclimate data evidencing such events remains impossible due to the length of the relevant time scales16. In an extensive search, many relatively abrupt transitions have been identified in future projections of CMIP5 models17, but due to the nature of these rare, high-risk events, the accuracy of ESM in predicting them remains untested. (3) Current ESMs are not yet suitable for assessing the efficacy or the environmental impact of carbon dioxide removal techniques, which are considered key mitigation options in pathways realizing the Paris Agreement 18. Further, ESMs still insufficiently represent key environmental processes such as the carbon cycle, water and nutrient availability, or interactions between land use and climate. This can impact the usefulness of land-based mitigation options that rely on actions such as biomass energy with carbon capture and storage or nature-based climate solutions 19, 20. (4) The distributions of time series encoding Earth system dynamics typically exhibit heavy tails. Extreme events such as heat waves and droughts, but also extreme precipitation events and associated floods, have always caused tremendous socio-economic damages. With ongoing anthropogenic climate change, such events are projected to become even more severe, and the attribution of extremes poses another outstanding challenge of Earth system science21. While current ESMs are very skilful in predicting average values of climatic quantities, there remains room for improvement in representing extremes. In addition to the possible solutions to these fundamental challenges, improvements of the overall accuracy of ESMs can be expected from more extensive and more systematic integration of the process-based numerical models with observational data. Earth system observations (ESOs) are central to ESMs, serving a multitude of purposes. ESOs are used to evaluate and compare process-based model performance, to generate model parameters and initial model states, or as boundary forcing of ESMs 22, 23. ESOs are also used to directly influence the model output by either tuning or nudging parameters that describe unmodeled processes, or by the more sophisticated methods of data assimilation that alter the model’s state variables to bring the model output in better agreement with the observations24. To incorporate uncertainty into model predictions, variational interfaces have been used25. Existing techniques for assimilating data into ESMs fall into two main categories, each with their own limitations. Gradient-based optimization, as in four-dimensional variational (4DVar) schemes, is the current state of the art for efficiency and accuracy, but currently requires time consuming design and implementation of adjoint calculation routines tailored to each model. Ensemble-based Kalman filter (EnKF) schemes are gradient-free but produce unphysical outputs and rely on strong statistical assumptions that are often unsatisfied, leading to biases and overconfident predictions26. The main problems of contemporary ESM data assimilation are 1) nonlinear dynamics and non-Gaussian error budgets in combination with the high dimensionality of many ESM components 27, 28, 29, and 2) constraining the governing processes over the different spatio-temporal scales found in coupled systems 30, 31. ML approaches can be used to combine the accuracy of 4DVar with the flexibility of EnKF, essentially allowing optimization-based assimilation in cases where gradients are currently unavailable. Furthermore, these traditional approaches of model and observation fusion have slowly been expanded or replaced by ML methods in recent years 32, 33, 34. ESOs cover a wide range of spatio-temporal scales and types, ranging from a couple of centimeters to tens of thousands of kilometers, and from seconds and decades to millennia. The types of observations range from in-situ measurements of irregular times and spaces (ship cruises, buoy arrays, etc.), over single time series (ice and sediment cores, tide gauges, etc.) to satellite-based global 2D or 3D data fields (altimetry, gravimetry, radio occultation, etc.). The amount of available observations is rapidly increasing and has reached a threshold where automated analysis becomes crucial. Yet, the available observational data pool still contains large gaps in time and space that prevent building a holistic observation-driven picture of the coupled Earth system, which result from insufficient spatio-temporal data resolution, too short observation time periods, and largely unobserved compartments of Earth systems like, for instance, abyssal oceans. The combination of these complex characteristics render Earth system observations both challenging and particularly interesting for AI applications. ## From Machine Learning-based Data Exploration Towards Learning Physics ML and other AI techniques have achieved stunning results in computer vision 35, speech and language models 36, medical science 37, 38, economical and societal analytics 39, and other disciplines 40, 41. Due to this wide-spread integration into both fundamental research and end-user products, and despite shortcomings and inherent limitations 42, 43, 44, 45, ML is already praised as a key disruptive technology of the 21st century 46. In contrast, the usage of ML in Earth and climate sciences is still in its infancy. A key observation is that ML concepts from computer vision and automated image analysis can be isomorphically transferred to ESO imagery. Pioneering studies demonstrated the feasibility of ML for remote sensing data analysis, classification tasks, and parameter inversion already in the 1990s 47, 48, 49, 50, and climate-model emulation in the early 2000s 51. The figurative Cambrian explosion of AI techniques in Earth and climate sciences, however, only began over the last five years and will rapidly continue throughout the coming decades. Under the overarching topic of ESO data exploration, ML has been applied for a huge variety of statistical and visual use cases. Classical prominent examples are pattern recognition in geo-spatial observations, climate data clustering, automated remote-sensing data analysis, and time series prediction 52, 32. In this context, ML has been applied across various spatial and temporal scales, ranging from short-term regional weather prediction to Earth-spanning climate phenomena. Significant progress has been made in developing purely data-driven weather prediction networks, which start to compete with process-based model forecasts 53, 54, 55. ML contributed to the pressing need to improve the predictability of natural hazards, for instance, by uncovering global extreme- rainfall teleconnections 56, or by improving long-term forecasts of the El Niño Southern Oscillation (ENSO)57, 58. ML-based image filling techniques were utilized to reconstruct missing climate information, allowing to correct previous global temperature records 59. Furthermore, ML was applied to analyze climate data sets, e.g., to extract specific forced signals from natural climate variability 60, 61 or to predict clustered weather patterns 62. In these applications, the ML tools function as highly specialized agents that help to uncover and categorize patterns in an automated way. A key methodological advantage of ML in comparison to covariance-based spatial analysis lies in the possibility to map nonlinear processes 63, 64. At the same time, such trained neural networks lack actual physical process knowledge, as they solely function through identifying and generalizing statistical relations by minimizing pre-defined loss measures for a specific task 65. Consequently, research on ML in Earth and climate science differs fundamentally from the previously described efforts of advancing ESMs in terms of methodological development and applicability. Concepts of utilizing ML not only for physics-blind data analyses, but also as surrogates and methodological extensions for ESMs have only recently started to shape 66. Scientists started pursuing the aim that ML methods learn aspects of Earth and climate physics, or at least plausibly relate cause and effect. The combination of ML with process-based modelling is the essential distinction from the previous ESO data exploration. Lifting ML from purely diagnosis-driven usage towards the prediction of geophysical processes will also be crucial for aiding climate change research and the development of mitigation strategies 67. Following this reasoning, ML methods can be trained with process-based model data to inherit a specific geophysical causation or even emulate and accelerate entire forward simulations. For instance, ML has been used in combination with ESMs and ESOs to invert space-borne oceanic magnetic field observations to determine the global ocean heat content 33. Similarly, a neural network has been trained with a continental hydrology model to recover high-resolution terrestrial water storage from satellite gravimetry 34. ML plays an important role for upscaling unevenly distributed carbon flux measurements to improve global carbon monitoring systems68. As such, the eddy covariance technique was combined with ML to measure the net ecosystem exchange of $\text{CO}_{2}$ between ecosystems and the atmosphere, offering a unique opportunity to study ecosystem responses to climate change 69. ML has shown remarkable success in representing subgrid-scale processes and other parameterizations of ESMs, given that sufficient training data were available. As such, neural networks were applied to approximate turbulent processes in ocean models 70 and atmospheric subgrid processes in climate models 71. Several studies highlight the potential for ML-based parameterization schemes 72, 73, 74, 75, 76, helping step-by-step to gradually remove numerically and human-induced simplifications and other biases of ESMs 77. While some well-trained ML tools and simple hybrids have shown higher predictive power than traditional state-of-the-art process-based models, only the surface of new possibilities, but also of new scientific challenges, has been scratched. So far, ML, ESMs, and ESO have largely been independent tools. Yet, we have reached the understanding that applications of physics-aware ML and model-network hybrids pose huge benefits by filling up niches where purely process-based models persistently lack reliability. ## The Fusion of Process-based Models and Artificial Intelligence Figure 2: Successive stages of the fusion process of Earth system models and artificial intelligence. The idea of hybrids of process-based and ML models is not new 78. So far, hybrids have almost exclusively been thought of as numerical models that are enhanced by ML to either improve the models’ performance in the sense of a useful metric, or to accelerate the forward simulation time in exchange for a decrease in simulation accuracy. Along with the general advance regarding the individual capabilities and limitations of ESMs and ML methods, respectively, also the understanding of how ML can enhance process-based modelling has evolved. This progress allows ML to take over more and more components of ESMs, gradually blending the so far strict distinction between process-based modelling and data-driven ML approaches. Even more so, entirely new methodological concepts are dawning that justify acknowledging Neural Earth System Modelling as a distinct research branch (Fig. 2). The long-term goal will be to consistently integrate the recently discovered advantages of ML into the already decade-long source of process knowledge in Earth system science. However, this evolution does not come without methodological caveats, which need to be investigated carefully. For the sake of comparability, we distinguish between weakly coupled NESYM hybrids, i.e., an ESM or AI technique benefits from information from the respective other, and strongly coupled NESYM hybrids, i.e., fully coupled model-network combinations that dynamically exchange information between each other. The emergent development of weak hybrids is predominantly driven by the aim to resolve the previously described ESM limitations, particularly unresolved and especially sub-grid scale processes. Neural networks can emulate such processes after careful training with simulation data from a high-resolution model that resolves the processes of interest, or with relevant ESO data. The next methodological milestone will be the integration of such trained neural networks into ESMs for operational usage. First tests have indicated that the choice of the AI technique, e.g., neural networks versus random forests, seems to be crucial for the implementation of learning parameterization schemes, as they can significantly deteriorate the ESM’s numerical stability 79. Thus, it is not only important to identify how neural networks can be trained to resolve ESM limitations, but also how such ML-based schemes can be stabilized in the model physics context and how their effect on the process-based simulation can be evaluated and interpreted 80. The limitations of ML-based parameterization approaches can vary widely for different problems or utilized models and, consequently, should be considered for each learning task individually 81. Nevertheless, several ideas have been proposed to stabilize ML parameterizations, e.g., by enforcing physical consistency through customized loss functions in neural networks and specific network architectures 82, 75, or by optimizing the considered high-resolution model training data 76. In addition, an ESM blueprint has been proposed, in which learning parameterizations can be targeted through searching an optimal fit of statistical measures between ESMs, observations, and high-resolution simulations 83. In this context, further efforts have been made to enhance an ESM not with ML directly, but in combination with a data assimilation system 24. For instance, emulating a Kalman filter scheme with ML has been investigated 84, 85, an ML-based estimation of atmospheric forcing uncertainties used as error covariance information in data assimilation has been proposed 86, as well as further types of Kalman-network hybrids 87, 88. In the second class of weak hybrids, the model and AI tasks are transposed, such that the information flow is directed from the model towards the AI tool. Here, neural networks are trained directly with model state variables, their trajectories, or with more abstract information like seasonal signals, interannual cycles, or coupling mechanisms. The goal of the ML application might not only be model emulation, but also inverting non-linear geophysical processes 33, learning geophysical causation 89, or predicting extreme events 90, 91. In addition to these inference and generalization tasks, a key question in this sub-discipline is whether a neural network can learn to outperform the utilized process-based trainer model in terms of physical consistency or predictive power. ESOs play a vital role in this context, as they can serve as additional training constraints for a neural network training, allowing it to build independent self-evaluation measures 34. The given examples generally work well for validation and prediction scenarios within the given training distribution. Out-of-distribution samples, in contrast, pose a huge challenge for supervised learning, which renders the “learning from the past” principle particularly ill-posed for prediction tasks in NESYM. As a consequence, purely data-driven AI methods will not be able to perform accurate climate projections on their own, because of the both naturally and anthropogenically induced non-stationarity of the climate and Earth system. Overcoming these limitations requires a deeper holistic integration in terms of strongly coupled hybrids and the consideration of further, less constrained training techniques like unsupervised training 92 and generative AI methods 93, 73, 94. For example, problems of pure AI methods with non-stationary training data can be attenuated by combining them with physical equations describing the changing energy-balance of the Earth system due to anthropogenic greenhouse-gas emissions 95. In addition, first steps towards physics-informed AI have been made by ML-based and data-driven discovery of physical equations 96 and by the implementation of neural partial differential equations 97, 98 into the context of climate modelling 99. Continuous maturing of the methodological fusion process will allow building hybrids of neural networks, ESMs and ESOs that dynamically exchange information. ESMs will soon utilize output from supervised and unsupervised neural networks to optimize their physical consistency and, in turn, feed back improved information content to the ML component. ESOs form another core element and function as constraining ground truth of the AI-infused process prediction. Similar to the adversarial game of generative networks 100, or coupling mechanisms in an ESM 101, also strongly coupled NESYM hybrids will require innovative interfaces that control the exchange of information that are, so far, not available. In addition, we formulate key characteristics and goals of this next stage: (1) Hybrids can better reproduce and predict out-of-distribution samples and extreme events, (2) hybrids perform constrained and consistent simulations that obey physical conservation laws despite potential shortcomings of the hybrids’ individual components, (3) hybrids include integrated adaptive measures for self-validation and self- correction, and (4) NESYM allows replicability and interpretability. We believe that cross-discipline collaborations between Earth system and AI scientists will become more important than ever to achieve these milestones. Frontier applications of Neural Earth system models are manifold. Yet, ultimately, NESYM hybrids need to drastically improve the current forecast limits of geophysical processes and contribute towards understanding the Earth’s susceptible state in a changing climate. Consequently, not only the fusion of ESM and AI will be in the research focus, but also AI interpretability and resolving the common notion of a black box (Fig. 3). ## Peering into the Black Box Figure 3: Qualitative comparison of isolated (AI - Artificial Intelligence, ESM - Earth System Model, ESO - Earth System Observation) and hybrid methodological approaches. The respective approaches are represented as trajectories in a meta space of hybrid coupling degree, interpretability, and prediction skill. The goal of Neural Earth System Modelling (NESYM) is to integrate the interpretability and to exceed the prediction skill of the respective isolated approaches. In this meta space, the necessary research increment to achieve this goal can be described through an increase in the degree of hybrid coupling (Fusion of AI and ESM) and an increase in interpretability (XAI - explainable AI, IAI - interpretable AI). ML has emerged as a set of methods based on the combination of statistics, applied mathematics and computer science, but it comes with a unique set of hurdles. Peering into the black box and understanding the decision making process of the ML method, termed explainable AI (XAI), is critical to the use of these tools. Especially in the physical sciences, adaptation of ML suffers from a lack of interpretability, particularly supervised ML. In contrast and in addition to XAI stands the call for interpretable AI (IAI), i.e., building specifically interpretable ML models from the beginning on, instead of explaining ML predictions through post-process diagnostics 102. Ensuring that what is ‘learned’ by the machine is physically tractable or causal, and not due to trivial coincidences 103, is important before ML tools are used, e.g., in an ESM setting targeted at decision making. Thus, explainability provides the user with trust in the ML output, improving its transparency. This is critical for ML use in the policy-relevant area of climate science as society is making it increasingly clear that understanding the source of AI predictive skill is of crucial importance104, 105. Ensuring the ML method is getting the right answers for the right reasons is essential given the transient nature of the climate system. As the climate continues to respond to anthropogenic climate change, NESYM will be required to make predictions of continually evolving underlying distributions and XAI/IAI will be critical to ensuring that the skill of the ML method can be explained, and inspire trust in its extrapolation to future climate regimes. There are many ML tools at our disposal, and XAI can assist researchers in choosing the optimal ML architecture, inputs, outputs, etc. By analyzing the decision making process, climate scientists will be able to better incorporate their own physical knowledge into the ML method, ultimately leading to improved results. Perhaps least appreciated in geoscientific applications thus far is the use of XAI to discover new science 106. When the ML method is capable of making a prediction, XAI allows us to ask “what did it learn?”. In this way, ML becomes more than just a prediction and allows scientists to ask “why?” as they normally would, but now with the power of ML. Explaining the source of an ML applications skill can be done retrospectively102. The power of XAI for climate and weather applications has very recently been demonstrated 107, 106, 108. For example, neural networks coupled with the XAI attribution method known as layerwise relevance propagation (LRP)109, 110 have revealed modes of variability within the climate system, sources of predictability across a range of timescales, and indicator patterns of climate change 106, 61. There is also evidence that XAI methods can be used to evaluate climate models against observations, identifying the most important climate model biases for the specific prediction task 111. However, these methods are in their infancy and there is vast room for advancements in their application, making it explicitly appropriate to employ them within the physical sciences. Unsupervised ML can be intuitively IAI through the design of experiments. For example, applying clustering on closed model budgets of momentum ensures all relevant physics are represented, and can be interpreted in terms of the statistically dominant balances. In this manner, different regimes can be discovered 112, 92. Adversarial learning has been an effective tool for generating super-resolution fields of atmospheric variables in climate models 94. Furthermore, unsupervised ML approaches have been proposed for discovering and quantifying causal interdependencies and dynamical links inside a system, such as the Earth’s climate 89, 113. The development of ESMs is increasingly turning to process-oriented diagnostics (PODs)114, where a certain process is targeted and used as a benchmark for model improvement. A revolution of analysis tools has been called for, and ML is poised to be part of this change 115, 116, 66. For instance, the POD approach has been applied to evaluate the ability of ESM projections to simulate atmospheric interactions and to constrain climate projection uncertainties 117. Given the importance of both explainability and interpretability for improving ML generalization and scientific discovery, we need climate scientists working together with AI scientists to develop methods that are tailored to the field’s needs. This is not just an interesting exercise - it is essential for the proper use of AI for NESYM development and use (Fig. 3). Earth and climate scientists can aid the development of consistent benchmarks that allow evaluating both stand-alone ML and NESYM hybrids in terms of geophysical consistency 118. However, help of the AI community is needed to resolve other recently highlighted ML pitfalls, for instance, translating the concepts of adversarial examples and deep learning artifacts 119 into the ESM context or finding new measures to identify and avoid shortcut learning 45 in NESYM hybrids. In summary, only combined efforts and continuous development of both ESM and AI will evolve Neural Earth System Modelling. Our perspective should not only be seen as the outline of a promising scientific pathway to achieve a better understanding of the Earth’s present and future state, but also as an answer to the recent support call from the AI community 120. Based on the recent advances in applying AI to Earth system and climate sciences, it seems to be a logical succession that AI will take over more and more tasks of traditional statistical and numerical ESM methods. Yet, in its current stage, it also seems unthinkable that AI alone can solve the climate prediction problem. In the forthcoming years, AI will necessarily need to rely on the geophysical determinism of process-based modelling and on careful human evaluation. However, once we find solutions to the foreseeable limitations described above and can build interpretable and geophysically consistent AI tools, this next evolutionary step will seem much more likely. ## Acknowledgements This study was funded by the Helmholtz Association and by the Initiative and Networking Fund of the Helmholtz Association through the project Advanced Earth System Modelling Capacity (ESM). NB acknowledges funding by the Volskwagen foundation and the European Union’s Horizon 2020 research and innovation program under grant agreement No 820970 (TiPES). ## Authors’ contributions CI conceived the paper and organized the collaboration. All authors contributed to writing and revising all chapters of this manuscript. 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msam10 # Bridging the gap between collisional and collisionless shock waves Antoine Bret1,2 and Asaf Pe’er3 Email address for correspondence: <EMAIL_ADDRESS>1ETSI Industriales, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain 2Instituto de Investigaciones Energéticas y Aplicaciones Industriales, Campus Universitario de Ciudad Real, 13071 Ciudad Real, Spain 3Department of Physics, Bar-Ilan University, Ramat-Gan, 52900, Israel (?; revised ?; accepted ?. - To be entered by editorial office) ###### Abstract While the front of a fluid shock is a few mean-free-paths thick, the front of a collisionless shock can be orders of magnitude thinner. By bridging between a collisional and a collisionless formalism, we assess the transition between these two regimes. We consider non-relativistic, un-magnetized, planar shocks in electron/ion plasmas. In addition, our treatment of the collisionless regime is restricted to high Mach number electrostatic shocks. We find that the transition can be parameterized by the upstream plasma parameter $\Lambda$ which measures the coupling of the upstream medium. For $\Lambda\lesssim 1.12$, the upstream is collisional, i.e. strongly coupled, and the strong shock front is about $\mathcal{M}_{1}\lambda_{\mathrm{mfp},1}$ thick, where $\lambda_{\mathrm{mfp},1}$ and $\mathcal{M}_{1}$ are the upstream mean-free- path and Mach number respectively. A transition occurs for $\Lambda\sim 1.12$ beyond which the front is $\sim\mathcal{M}_{1}\lambda_{\mathrm{mfp},1}\ln\Lambda/\Lambda$ thick for $\Lambda\gtrsim 1.12$. Considering $\Lambda$ can reach billions in astrophysical settings, this allows to understand how the front of a collisionless shock can be orders of magnitude smaller than the mean-free- path, and how physics transitions continuously between these 2 extremes. ## 1 Introduction Shock waves are very common in systems that involve fluid flows. Such systems occur on very different scales - from the microphysical scale to astronomical scales. As such, the properties of the shocks can vary considerably, depending on the environment. From the microphysical point of view, it is useful to discriminate between collisional and collisionless shocks. Collisional shock waves, first discovered in the $19^{th}$ century (Salas, 2007), can occur in any fluid as the result of a steepening of a large amplitude sound wave, or collision of two media (Zel’dovich & Raizer, 2002). The front of a collisional shock is necessarily at least a few mean-free-paths thick, as the dissipation from the downstream to the upstream occurs via binary collisions. Collisionless shock waves were discovered later and can only form in plasma (Petschek, 1958; Buneman, 1964; Sagdeev, 1966). The dissipation is provided by collective plasma phenomena instead of binary collisions. As a result, the front of such shocks can be orders of magnitude thinner than the mean-free- path. For example, the front of the bow shock of the earth magnetosphere in the solar wind is some 100 km thick (Bale et al., 2003; Schwartz et al., 2011). Yet, the proton mean-free-path at this location is about the Sun-Earth distance, nearly 7 orders of magnitude longer. Hence, if the earth bow shock were collisional, its front would be about 1 a.u. thick (see also Balogh & Treumann (2013) §2.1.3 and references therein). Is it possible to bridge between these two regimes? How does a shock switches from a regime where its front is a few mean-free-paths thick, to another regime where its front is million times smaller? Exploring the intermediate case, bridging between collisional and collisionless shocks, is the aim of this article. On the collisionless side, shock-accelerated particles which can enhance the density jump, or external magnetization which can reduce it, will be ignored (Berezhko & Ellison, 1999; Bret & Narayan, 2018, 2019; Bret, 2020). The method implemented is explained in Section 2. The big picture is as follows: we first present an evaluation of the shock front thickness in the collisional regime, then in the collisionless regime. The first task is achieved in Section 3 using the Mott-Smith _ansatz_ (Mott-Smith, 1951), which writes the distribution function at any place along the shock profile, as a linear combination of the upstream and downstream Maxwellians. We then follow Tidman (1967) in Section 4 for the collisionless case before we bridge between the 2 expressions of the front thickness in Section 5, to propose an expression of the front thickness valid from the collisional to the collisionless regime. Figure 1: Setup and notations. ## 2 Method As previously stated, a fluid shock is mediated by collisions while a collisionless shock is mediated by collective effects. For a plasma where only electrostatic fields are active (such is the case for an electrostatic shock, the kinetic equation accounting for both kinds of effects would formally read (Kulsrud (2005), p. 9), $\frac{\partial F}{\partial t}+\mathbf{v}\cdot\frac{\partial F}{\partial\mathbf{r}}+\frac{q\mathbf{E}}{m}\cdot\frac{\partial F}{\partial\mathbf{v}}=\left(\frac{\partial F}{\partial t}\right)_{c}+\left(\frac{\partial F}{\partial t}\right)_{w},$ (1) where $q$ and $m$ are the charge and the mass of the species considered. The first term of the right-hand-side, namely $(\partial F/\partial t)_{c}$, stands for the rate of change of the distribution $F$ due to collisions. It is typically given by the Fokker-Planck operator. The second term, $(\partial F/\partial t)_{w}$, accounts for the effects of the waves and is given, for example, by the quasi-linear operator. In principle, accounting at once for these two collision terms with appropriate collision operators, should allow to describe a shock wave from the collisional to the fully collisionless regimes. Resolving the shock front requires a formalism capable of resolving the entire shock profile. This is a notoriously difficult problem which has been greatly aided by the introduction of the so-called Mott-Smith _ansatz_ (Mott-Smith, 1951). Initially introduced for a neutral fluid, this _ansatz_ consisted in approximating the molecular distribution function $F$ along the shock profile by a linear combination of the upstream and downstream drifting Maxwellians, $\displaystyle F(\mathbf{v})=$ $\displaystyle n_{1}(x)\left(\frac{m}{2\pi k_{B}T_{1}}\right)^{3/2}\exp\left(-\frac{m}{2k_{B}T_{1}}(\mathbf{v}-\mathbf{U}_{1})^{2}\right)$ (2) $\displaystyle+~{}n_{2}(x)\left(\frac{m}{2\pi k_{B}T_{2}}\right)^{3/2}\exp\left(-\frac{m}{2k_{B}T_{2}}(\mathbf{v}-\mathbf{U}_{2})^{2}\right),$ where $T_{1,2}$ and $\mathbf{U}_{1,2}$ are the upstream (subscript 1) and downstream (subscript 2) temperatures and velocities respectively, determined by the Rankine-Hugoniot (RH) jump conditions (see figure 1)111All temperatures are not always considered constant in the main articles cited here (Mott- Smith, 1951; Tidman, 1958, 1967). Yet, they are considered so when it comes to computing the shock profile.. The boundary conditions for the functions $n_{1,2}(x)$ are, $\displaystyle n_{1}(+\infty)$ $\displaystyle=N_{1,0}~{}~{}~{}~{}$ $\displaystyle n_{1}(-\infty)=0,$ $\displaystyle n_{2}(+\infty)$ $\displaystyle=0~{}~{}~{}~{}$ $\displaystyle n_{2}(-\infty)=N_{2,0},$ (3) where again $N_{1,0}$ and $N_{2,0}$ fulfill the RH jump conditions. Taking then the appropriate moments of the dispersion equation gives a differential equation which allows to determine the respective weights of the 2 Maxwellians in terms of $x$, hence the shock profile together with its front thickness (Mott-Smith, 1951). The method implemented here consists in dealing with the collisional and the collisionless regimes separately. * • We study the _collisional regime_ in Section 3. There we apply the Mott-Smith _ansatz_ using the BGK collision term (Bhatnagar et al., 1954) as a collision operator for $(\partial F/\partial t)_{c}$ in Eq. (1), with $(\partial F/\partial t)_{w}=0$. Notably, Bhatnagar et al. (1954) presented 4 different collision operators through their Eqs. (3, 4, 5-6, 15-19). Those given by Eqs. (3, 4, 5-6), like $\nu(f-f_{0})$222Here $\nu$ is a collision frequency, $f$ the distribution function and $f_{0}$ the equilibrium distribution function., have been widely used although they do not conserve all 3 quantities: particle number and/or momentum and/or energy. In Bhatnagar et al. (1954), only the operator of Eqs. (15-19) does conserve all 3, hence this is the one used here. Tidman (1958) used the Fokker-Planck operator to deal with the problem, considering Eq. (1) with $(\partial F/\partial t)_{w}=0$ and, $\left(\frac{\partial F}{\partial t}\right)_{c}=\frac{4\pi e^{4}}{m_{i}^{2}}\ln\Lambda\left(\frac{\partial F}{\partial t}\right)_{c,FP},$ (4) where $(\partial F/\partial t)_{c,FP}$ is the Fokker-Planck collision operator, $m_{i}$ the ion mass, and $\Lambda$ the number of particles in the Debye sphere, that is, the co-called “plasma parameter” which measures the coupling of the plasma. As we shall see in Section 6, the present treatment provides a more adequate bridging to the collisionless regime than Tidman (1958)’s Fokker-Planck result. * • For the _collisionless regime_ we follow in Section 4 the collisionless result of Tidman (1967) who also used the Mott-Smith _ansatz_. In recent years, the correctness of this approximation, namely that the distribution function is well approximated by superimposed drifting Maxwellians, was validated numerically using Particle-In-Cell simulations (Spitkovsky, 2008). Tidman (1967) considered Eq. (1) with $(\partial F/\partial t)_{c}=0$, describing $(\partial F/\partial t)_{w}=0$ by the quasi-linear operator. Having assessed the width of the shock front in the collisional and the collisionless regimes, we then bridge between the 2 expressions of the shock thickness in Section 5. ## 3 Collisional regime: applying Bhatnagar et al. (1954) collision term to the Mott-Smith _ansatz_ We switch to the reference frame of the shock and assume steady state in this frame. As in Tidman (1958), we consider the distributions are functions of $(v_{x},v_{y},v_{z})$ and assume quantities only vary with the $x$ coordinate. We therefore set $\partial_{y,z,t}=0$ and $E_{y,z}=0$ so that equation (1), with $(\partial F/\partial t)_{w}=0$, reads for the ion distribution $F$, $v_{x}\frac{\partial F}{\partial x}+\frac{eE_{x}}{m_{i}}\frac{\partial F}{\partial v_{x}}=\left(\frac{\partial F}{\partial t}\right)_{c,BGK},$ (5) where $m_{i}$ is the ion mass. The BGK collision term now reads (Bhatnagar et al., 1954; Gross & Krook, 1956), $\left(\frac{\partial F}{\partial t}\right)_{c,BGK}=\frac{1}{\sigma}\left(N^{2}\Phi-NF\right),$ (6) which vanishes for a Maxwellian distribution. According to Bhatnagar et al. (1954), $``\sigma^{-1}$ $\times$ a density” is a collision frequency $\nu$. In the present setting we define, $\frac{N_{1,0}}{\sigma}=\nu\sim\frac{v_{\mathrm{thi},1}}{\lambda_{\mathrm{mfp},1}}~{}~{}\Rightarrow~{}~{}\sigma=\frac{N_{1,0}\lambda_{\mathrm{mfp},1}}{v_{\mathrm{thi},1}},$ (7) where $v_{\mathrm{thi},1}$ and $\lambda_{\mathrm{mfp},1}$ are the upstream thermal velocity and mean-free-path respectively. Then $N$ and $\Phi$ are given by Eqs. (15-19) of Bhatnagar et al. (1954), $\displaystyle N$ $\displaystyle=$ $\displaystyle\int Fd^{3}v=n_{1}(x)+n_{2}(x),$ (8) $\displaystyle\Phi$ $\displaystyle=$ $\displaystyle\left(\frac{m_{i}}{2\pi k_{B}T}\right)^{3/2}\exp\left(-\frac{m_{i}}{2k_{B}T}(\mathbf{v}-\mathbf{q})^{2}\right),$ (9) $\displaystyle\mathbf{q}$ $\displaystyle=$ $\displaystyle\frac{1}{N}\int\mathbf{v}Fd^{3}v,$ (10) $\displaystyle\frac{3k_{B}T}{m_{i}}$ $\displaystyle=$ $\displaystyle\frac{1}{N}\int(\mathbf{v}-\mathbf{q})^{2}Fd^{3}v,$ (11) where $F$ has been considered of the form (2). Multiplying equation (5) by $v_{y}^{2}$ and integrating over $d^{3}v$ (see detailed calculation reported in Appendix A) gives an exact, simple result, $U_{1}\frac{k_{B}T_{1}}{m_{i}}~{}\frac{\partial n_{1}}{\partial x}+U_{2}\frac{k_{B}T_{2}}{m_{i}}~{}\frac{\partial n_{2}}{\partial x}=\frac{1}{3\sigma}(U_{1}-U_{2})^{2}~{}n_{1}n_{2}.$ (12) This differential equation is structurally identical to the ones found in Mott-Smith (1951); Tidman (1958). We show in Appendix B how it yields density profiles like the ones pictured in Figure 1, of the form, $\displaystyle n_{1}(x)$ $\displaystyle=$ $\displaystyle N_{1,0}\frac{1}{1+e^{-x/\ell}},$ $\displaystyle n_{2}(x)$ $\displaystyle=$ $\displaystyle N_{2,0}\frac{e^{-x/\ell}}{1+e^{-x/\ell}},$ (13) implicitly defining the shock width $\ell$. From Eq. (39) we find the thickness of the shock according to the present formalism, $\ell=3\sigma\frac{k_{B}(T_{1}-T_{2})U_{2}}{N_{1,0}m_{i}(U_{1}-U_{2})^{2}}=3\frac{N_{1,0}\lambda_{\mathrm{mfp},1}}{v_{\mathrm{thi},1}}\frac{k_{B}(T_{1}-T_{2})U_{2}}{N_{1,0}m_{i}(U_{1}-U_{2})^{2}}.$ (14) It is now convenient to use the RH jump conditions to express $\ell$ in terms of the upstream quantities, like the upstream Mach number and mean-free-path. The calculations reported in Appendix C give, $\displaystyle\ell$ $\displaystyle=$ $\displaystyle\lambda_{\mathrm{mfp},1}\frac{U_{1}}{v_{\mathrm{thi},1}}\frac{(\mathcal{M}_{1}^{2}+3)(5\mathcal{M}_{1}^{2}(\mathcal{M}_{1}^{2}+2)-3)}{5\mathcal{M}_{1}^{2}(\mathcal{M}_{1}^{2}-1)^{2}},$ (15) $\displaystyle=$ $\displaystyle\lambda_{\mathrm{mfp},1}\frac{(\mathcal{M}_{1}^{2}+3)(5\mathcal{M}_{1}^{2}(\mathcal{M}_{1}^{2}+2)-3)}{5\mathcal{M}_{1}(\mathcal{M}_{1}^{2}-1)^{2}},$ where $\mathcal{M}_{1}$ is the upstream Mach number333Here we set $v_{\mathrm{thi},1}\sim c_{s1}$ in order to write $U_{1}/v_{\mathrm{thi},1}\sim\mathcal{M}_{1}$, where $c_{s1}$ is the upstream sound speed. An exact calculation only changes the end result by a factor of order unity. Moreover, the same factor also modifies the collisionless shock width (18). Therefore, the critical plasma parameter $\Lambda_{c}$ defined by Eq. (20) for the collisional/collisionless transition, remains unchanged when considering $v_{\mathrm{thi},1}\sim c_{s1}$.. We eventually obtain the following limits for the shock width $\ell$, $\ell=\lambda_{\mathrm{mfp},1}\times\left\\{\begin{array}[]{r}\frac{12}{5}(\mathcal{M}_{1}-1)^{-2}~{}~{}\mathrm{for}~{}~{}\mathcal{M}_{1}\sim 1,\\\ \mathcal{M}_{1}~{}~{}\mathrm{for}~{}~{}\mathcal{M}_{1}\rightarrow\infty.\end{array}\right.$ (16) Figure 2: Shock front thickness in units of the mean-free-path, $\ell/\lambda_{\mathrm{mfp},1}$, in terms of the upstream Mach number $\mathcal{M}_{1}$. The function $\ell/\lambda_{\mathrm{mfp},1}$ is plotted in Figure 2 in terms of the Mach number. It reaches a minimum for $\mathcal{M}_{1}=3.53$ with $\ell/\lambda_{\mathrm{mfp},1}=6$. Such a “U” shape has also been found in Tidman (1958). We shall further comment on Tidman (1958) in Section 6. ## 4 Collisionless regime Our expression (16) of the shock width cannot be used to bridge all the way to collisionless shocks since it has been derived from the kinetic equation (1) without the $(\partial F/\partial t)_{w}$ collision term. Yet, collisionless shocks are sustained by the mechanism described by this very term. Tidman (1967) treated the problem of a collisionless shock by setting $(\partial F/\partial t)_{c}=0$ in Eq. (1) and considering the quasi-linear operator for $(\partial F/\partial t)_{w}$. The Mott-Smith _ansatz_ was also implemented in this study. Tidman (1967) could not derive an equation of the form (12) allowing to extract an analytical shock profile. Further analysis in Biskamp & Pfirsch (1969) and Tidman & Krall (1969) concluded that the quasi- linear formalism is not non-linear enough to fully render a shock. Yet, Tidman (1967) could derive the following estimate of the width of the front, $\displaystyle\ell$ $\displaystyle=$ $\displaystyle A\frac{U_{1}}{\omega_{pi,1}}=A\frac{U_{1}}{v_{\mathrm{thi},1}}\frac{v_{\mathrm{thi},1}}{\omega_{pi,1}}$ (17) $\displaystyle=$ $\displaystyle A\mathcal{M}_{1}\lambda_{\mathrm{Di},1},$ where $\lambda_{\mathrm{Di},1}$ is the upstream ionic Debye length and $A$ is a parameter expected to be of order $\mathcal{O}(10)$. We can eventually cast this result under the form, $\ell=A\mathcal{M}_{1}\frac{\ln\Lambda}{\Lambda}\lambda_{\mathrm{mfp,1}},$ (18) where $\Lambda$ is the plasma parameter already introduced in Eq. (4) and we have used (Fitzpatrick (2014), p. 10), $\lambda_{\mathrm{mfp,1}}=\frac{\Lambda}{\ln\Lambda}\lambda_{\mathrm{Di},1}.$ (19) Notably, Tidman (1967) only addressed high Mach numbers turbulent shocks triggered by electrostatic instabilities. The forthcoming bridging between the 2 regimes is therefore only valid for such shocks. Weibel shocks sustained by electromagnetic instabilities are therefore excluded (Stockem et al., 2014; Ruyer et al., 2017). Figure 3: Plot of $\ell(\mathcal{M}_{1}\lambda_{\mathrm{mfp},1})^{-1}$ in terms of $\Lambda$. For a collisional plasma with small $\Lambda$, the front thickness $\ell$ is given by Eq. (16). The collisionless thickness is given by Eq. (18). The width of the front for any plasma parameter $\Lambda$ is given by the red curve. Only valid for strong shock (see end of Section 4). Figure 4: Value of $\Lambda_{c}$ for which Eqs. (16 & 18) intersect, in terms of $A$ defined through Eq. (17). ## 5 Bridging between the 2 regimes Figure 3 shows the collisional and collisionless expressions of $\ell(\mathcal{M}_{1}\lambda_{\mathrm{mfp},1})^{-1}$ from Eqs. (16, 18). For upstream Mach number $\mathcal{M}_{1}>$ a few (4-5), these 2 expressions intersect for a critical plasma parameter $\Lambda_{c}$ defined by, $\lambda_{\mathrm{mfp},1}\mathcal{M}_{1}=A\mathcal{M}_{1}\frac{\ln\Lambda_{c}}{\Lambda_{c}}\lambda_{\mathrm{mfp,1}}~{}~{}\Rightarrow~{}~{}1=A\frac{\ln\Lambda_{c}}{\Lambda_{c}},$ (20) fulfilled for $\Lambda_{c}\sim 1.12$ and then for $\Lambda_{c}\sim 35$ (for $A=10$). For $\Lambda<1.12$, the upstream is strongly coupled, that is, collisional, and the width of the front will be given by the collisional result (16). For $\Lambda>1.12$, the upstream is weakly coupled, that is, collisionless, and the relevant front width is therefore the collisionless result (18). Hence, the larger value of $\Lambda_{c}\sim 35$ where the 2 expressions intersect again is not physically meaningful. For such values of $\Lambda$, the upstream is collisionless so that the collisionless result applies. The transition between the 2 regimes occurs therefore for a critical plasma parameter $\Lambda_{c}=1.12$, coinciding with the transition of the upstream from the strongly coupled/collisional regime, to the weakly coupled/collisionless regime. Although this value of $\Lambda_{c}$ has been computed for $A=10$, Figure 4 shows it is poorly sensitive to $A$ as long as $A=\mathcal{O}(10)$. Note that this value of $\Lambda_{c}=1.12$ is only indicative. For example, Lee & More (1984) developed an electron conductivity model for _dense_ plasmas requiring $\ln\Lambda\geq 2$, i.e, $\Lambda\geq e^{2}=7.39$. Therefore, while $\Lambda=35$ probably pertains to weakly collisional plasmas, the value $\Lambda_{c}=1.12$ only gives a general idea of where the transition occurs. The width of the front for any plasma parameter $\Lambda$ is eventually given by the red curve in Figure 3. Simply put, the nature of the shock is the same as the nature of the upstream. Both are collisional or collisionless together. The non-monotonic behavior in the collisionless regime is just the consequence of the non-monotonic variation of the mean-free-path in terms of the plasma parameter. The function $g(x)=A\ln x/x$ reaches a max for $x=e$ with $g(e)=3.67$, still for $A=10$. ## 6 Comparison with Tidman (1958) A calculation parallel to the one performed in Section 3 for the collisional regime was achieved in Tidman (1958). However, as we show here, the bridging it provides to the collisionless regime is inadequate. For the ion distribution function $F$, Tidman (1958) used the Fokker-Planck operator for $(\partial F/\partial t)_{c}$ in Eq. (1), set $(\partial F/\partial t)_{w}=0$, and found for strong shocks444See Eq. (6.6) of Tidman (1958) where $V$ is the sound speed and $K$ is the Mach number., $\ell_{T}=\alpha\frac{c_{s}^{4}}{N_{1,0}\Gamma}\mathcal{M}_{1}^{4},$ (21) where $\alpha=29.1/512\pi$ and $\Gamma=\frac{4\pi e^{4}}{m_{i}^{2}}\ln\Lambda$. We can recast this result under the form, $\ell_{T}=4\pi\alpha~{}\lambda_{\mathrm{mfp,1}}\mathcal{M}_{1}^{4}\sim 0.23~{}\lambda_{\mathrm{mfp,1}}\mathcal{M}_{1}^{4},$ (22) where we have used Eq. (19). As a consequence, bridging the collisional result of Tidman (1958) with the collisionless result of Tidman (1967), that is, bridging Eq. (22) with Eq. (18), implicitly defines a critical plasma parameter $\Lambda_{c}$ through, $\frac{4\pi\alpha}{A}\mathcal{M}_{1}^{3}=\frac{\ln\Lambda_{c}}{\Lambda_{c}},$ (23) yielding a Mach number-dependent value of $\Lambda_{c}$ and having no solution if the left-hand-side is larger than the maximum of the right hand-side, that is, for $\mathcal{M}_{1}>2.51$ (considering $A=10$). As opposed to that, the $\propto\mathcal{M}_{1}$ scaling of the collisional $\ell$ given by BGK-derived Eq. (16) is essential to give a value of $\Lambda_{c}$ independent of the upstream Mach number $\mathcal{M}_{1}$, with a switch from the collisional to the collisionless regime when the upstream becomes collisionless. We therefore find that BGK provides a better bridging to the collisionless regime than Fokker-Planck. Hazeltine (1998) already noted the capacity of the BGK operator to behave adequately in the collisionless limit. Computing the moments of the kinetic equation with the BGK operator, he could derive a _non- local_ expression of the heat flux in the collisionless regime, as expected when the mean-free-path becomes large (Hammett & Perkins, 1990; Hazeltine, 1998). Indeed, the BGK operator was specifically designed to provide an operator capable of giving an adequate description of low-density plasmas (Bhatnagar et al., 1954). The physical reason for the better behavior of the BGK operator when the mean free path becomes large could be that regardless of the mean free path, BGK assumes the equilibrium distribution function is a Maxwellian, since the collision term (6) vanishes for $F=N\phi$, where $\phi$ is a Maxwellian (see Eq. 9). In contrast, the Fokker-Planck operator does not assume any a priori form of the equilibrium distribution function. It can even be used to prove that such a function is a Maxwellian. Yet, the collision rate is implicitly assumed large compared to the dynamic terms $v/L$ in the Fokker-Planck equation (Kulsrud (2005), p. 213) since the derivation of the Fokker-Planck operator involves a Taylor expansion in time, implicitly assuming collisions are frequent enough (Kulsrud (2005), Eq. 29-30, p. 204 or Chandrasekhar (1943), §II.4). Therefore, when collisions become scarce, the BGK formalism keeps forcing, by design, a Maxwellian equilibrium, while Fokker-Planck progressively loses validity. ## 7 Conclusion We propose a bridging between collisional and collisionless shocks. The collisional “leg” is worked out using the Moot-Smith _ansatz_ (Mott-Smith, 1951) with the “full” BGK collision term which behaves correctly in the large mean-free-path limit (Bhatnagar et al., 1954; Gross & Krook, 1956; Hazeltine, 1998). The collisionless part is from Tidman (1967), valid for strong turbulent electrostatic shocks. The result makes perfect physical sense. As long as the upstream is strongly coupled, that is, collisional with $\Lambda\lesssim 1.12$, the strong shock is collisional with a front thickness $\sim\mathcal{M}_{1}\lambda_{\mathrm{mfp},1}$ given by Eq. (16). From $\Lambda\gtrsim 1.12$, the shock switches to the collisionless regime, with a front thickness $\ell\sim\mathcal{M}_{1}\lambda_{\mathrm{mfp},1}\ln\Lambda/\Lambda$, given by Eq. (18). We show that the BGK treatment of the collisional regime provides a better bridge to the collisionless regime than the Fokker-Planck model. Nevertheless, a confusing feature remains: in the collisional limit, one would expect the BGK and the Fokker-Planck treatments to merge. Yet, they don’t, as evidenced by their different $\mathcal{M}_{1}$ scaling for the strong shock width ($\propto\mathcal{M}_{1}$ for BGK vs. $\propto\mathcal{M}_{1}^{4}$ for Fokker- Planck). The reason for this could be that the collision frequency used in BGK (Eq. 7) does not depend on the particle velocity. However, this is still unclear to us. A smoother transition between the 2 regimes could be assessed from Eq. (1) considering both $(\partial F/\partial t)_{c}$ and $(\partial F/\partial t)_{w}$ at once, whereas we here switched them on and off according to the regime considered. The Mott-Smith _ansatz_ could still be applied, while using BGK for $(\partial F/\partial t)_{c}$ and the operator proposed by Dupree (1966) (as suggested in Tidman (1967)) or Baalrud et al. (2008), for $(\partial F/\partial t)_{w}$. Although the present theory is formally restricted to high Mach number, un- magnetized, electrostatic shocks, it may help understand how the value of $\Lambda\sim 10^{10}$ observed in the solar wind (see for example Fitzpatrick (2014), p. 8) yields an earth bow shock thickness orders of magnitude shorter than the mean-free-path. ## 8 Acknowledgments A.B. acknowledges support by grants ENE2016-75703-R from the Spanish Ministerio de Economía y Competitividad and SBPLY/17/180501/000264 from the Junta de Comunidades de Castilla-La Mancha. A. P. acknowledges support from the European Research Council via ERC consolidating grant #773062 (acronym O.M.J.). Thanks are due to Anatoly Spitkovsky, Bill Dorland, Ian Hutchinson, Ellen Zweibel and Richard Halzeltine for valuable inputs. ## Appendix A Proof of Eq. (12) Equation (12) is the $v_{y}^{2}$ moment of Eq. (5). The left-hand-side is calculated in Tidman (1958). Note that the term proportional to $E_{x}$ vanishes in this moment. We only detail here the calculation proper to the present work, that is, that of the right-hand-side. For this we need $\Phi$, hence $\mathbf{q}$ and $T$ defined by Eqs. (8-11). According to Eq. (10), $\mathbf{q}$ is given by, $\mathbf{q}=\frac{1}{N}\int\mathbf{v}Fd^{3}v=\frac{1}{n_{1}+n_{2}}\int\mathbf{v}Fd^{3}v.$ (24) Since $F$ is the sum of 2 drifting Maxwellians given by Eq. (2), we find for $\mathbf{q}$, $\mathbf{q}=\frac{n_{1}\mathbf{U}_{1}+n_{2}\mathbf{U}_{2}}{n_{1}+n_{2}}\equiv q~{}\mathbf{e}_{x},$ (25) where $\mathbf{e}_{x}$ is the unit vector of the $x$ axis. For $T$ we then get from (11)555The factor 2 in the second term of Eq. (A) comes from $\int v_{z}^{2}F=\int v_{y}^{2}F$., $\displaystyle\frac{3k_{B}T}{m_{i}}$ $\displaystyle=$ $\displaystyle\frac{1}{n_{1}+n_{2}}\int(\mathbf{v}-\mathbf{q})^{2}Fd^{3}v$ $\displaystyle=$ $\displaystyle\frac{1}{n_{1}+n_{2}}\int[(v_{x}-q)^{2}+v_{y}^{2}+v_{z}^{2}]Fd^{3}v$ $\displaystyle=$ $\displaystyle\frac{1}{n_{1}+n_{2}}\int(v_{x}-q)^{2}Fd^{3}v+\frac{2}{n_{1}+n_{2}}\int v_{y}^{2}Fd^{3}v$ $\displaystyle=$ $\displaystyle\frac{1}{n_{1}+n_{2}}\int(v_{x}-q)^{2}Fd^{3}v+\frac{2(k_{B}T_{1}n_{1}+k_{B}T_{2}n_{2})}{m_{i}(n_{1}+n_{2})}$ $\displaystyle\Rightarrow k_{B}T$ $\displaystyle=$ $\displaystyle\frac{n_{1}k_{B}T_{1}+n_{2}k_{B}T_{2}}{n_{1}+n_{2}}+\frac{n_{1}n_{2}}{3(n_{1}+n_{2})^{2}}m_{i}(U_{1}-U_{2})^{2}.$ (27) Let us now write explicitly the $v_{y}^{2}$ moment of the right-hand-side ($rhs$) of Eq. (5), $rhs=\frac{1}{\sigma}\int v_{y}^{2}(-NF+N^{2}\Phi)d^{3}v=\underbrace{\frac{N^{2}}{\sigma}\int v_{y}^{2}\Phi d^{3}v}_{\mathbf{1}}-\underbrace{\frac{N}{\sigma}\int v_{y}^{2}Fd^{3}v}_{\mathbf{2}}.$ (28) From (8) we see $N$ does not depend on $\mathbf{v}$. It can therefore be taken out of the integrals. Computing 2 we find, $\displaystyle\frac{N}{\sigma}\int v_{y}^{2}Fd^{3}v$ $\displaystyle=$ $\displaystyle\frac{n_{1}+n_{2}}{\sigma}\left(n_{1}\frac{k_{B}T_{1}}{m_{i}}+n_{2}\frac{k_{B}T_{2}}{m_{i}}\right),$ (29) $\displaystyle=$ $\displaystyle\frac{k_{B}}{m_{i}\sigma}(n_{1}+n_{2})(n_{1}T_{1}+n_{2}T_{2}).$ Then we compute 1. $\displaystyle\frac{N^{2}}{\sigma}\int v_{y}^{2}\Phi d^{3}v$ $\displaystyle=$ $\displaystyle\frac{(n_{1}+n_{2})^{2}}{\sigma}\int v_{y}^{2}\left(\frac{m_{i}}{2\pi k_{B}T}\right)^{3/2}\exp\left(-\frac{m_{i}}{2k_{B}T}(\mathbf{v}-\mathbf{q})^{2}\right)d^{3}v,$ $\displaystyle=$ $\displaystyle\frac{(n_{1}+n_{2})^{2}}{\sigma}\left(\frac{m_{i}}{2\pi k_{B}T}\right)^{3/2}\underbrace{\int v_{y}^{2}\exp\left(-\frac{m_{i}}{2k_{B}T}(\mathbf{v}-\mathbf{q})^{2}\right)d^{3}v}_{\mathbf{3}}.$ For 3 we get, $\mathbf{3}=(2\pi)^{3/2}\left(\frac{k_{B}T}{m_{i}}\right)^{5/2},$ (30) so that 1 gives, $\displaystyle\mathbf{1}$ $\displaystyle=$ $\displaystyle\frac{(n_{1}+n_{2})^{2}}{\sigma}\left(\frac{m_{i}}{2\pi k_{B}T}\right)^{3/2}(2\pi)^{3/2}\left(\frac{k_{B}T}{m_{i}}\right)^{5/2}$ (31) $\displaystyle=$ $\displaystyle\frac{(n_{1}+n_{2})^{2}}{\sigma}\frac{k_{B}T}{m_{i}}.$ Finally, Eq. (28) simplifies nicely and reads, $\displaystyle rhs$ $\displaystyle=$ $\displaystyle\frac{(n_{1}+n_{2})^{2}}{\sigma}\frac{k_{B}T}{m_{i}}-\frac{k_{B}}{m_{i}\sigma}(n_{1}+n_{2})(n_{1}T_{1}+n_{2}T_{2}),$ (32) $\displaystyle=$ $\displaystyle\frac{1}{3\sigma}n_{1}n_{2}(U_{1}-U_{2})^{2},$ in agreement with the right-hand-side of Eq. (12). ## Appendix B Derivation of the density profiles (3) from Eq. (12) Let us define $\alpha,\beta,\gamma$ from Eq. (12) by, $\underbrace{U_{1}\frac{k_{B}T_{1}}{m_{i}}}_{\alpha}\frac{\partial n_{1}}{\partial x}+\underbrace{U_{2}\frac{k_{B}T_{2}}{m_{i}}}_{\beta}\frac{\partial n_{2}}{\partial x}=\underbrace{\frac{1}{3\sigma}(U_{1}-U_{2})^{2}}_{\gamma}n_{1}n_{2}.$ (33) Consider now the matter conservation equation obtained equating the $v_{x}$ moments of (2) between any $x$ and $x=+\infty$, $n_{1}(x)U_{1}+n_{2}(x)U_{2}=N_{1,0}U_{1}.$ (34) Differentiate with respect to $x$ gives, $\frac{\partial n_{1}(x)}{\partial x}U_{1}+\frac{\partial n_{2}(x)}{\partial x}U_{2}=0~{}~{}\Rightarrow~{}~{}\frac{\partial n_{2}(x)}{\partial x}=-\frac{\partial n_{1}(x)}{\partial x}\frac{U_{1}}{U_{2}},$ (35) and use the result to eliminate $\partial n_{2}/\partial x$ in (33), $\frac{\partial n_{1}}{\partial x}\left(\alpha-\beta\frac{U_{1}}{U_{2}}\right)=\gamma n_{1}n_{2}~{}~{}\Rightarrow~{}~{}\frac{\partial n_{1}}{\partial x}\frac{1}{n_{1}n_{2}}=\frac{\gamma}{\alpha-\beta\frac{U_{1}}{U_{2}}}.$ (36) Making now use again of the conservation equation (34) to write, $n_{2}=(N_{1,0}-n_{1})\frac{U_{1}}{U_{2}},$ (37) one gets, $\frac{U_{2}}{U_{1}}\frac{\partial n_{1}}{\partial x}\frac{1}{n_{1}(N_{1,0}-n_{1})}=\frac{U_{2}}{U_{1}}\frac{\partial n_{1}}{\partial x}\frac{1}{N_{1,0}}\left(\frac{1}{n_{1}}+\frac{1}{N_{1,0}-n_{1}}\right)=\frac{\gamma}{\alpha-\beta\frac{U_{1}}{U_{2}}}.$ (38) We eventually obtain, $\frac{\partial n_{1}}{\partial x}\left(\frac{1}{n_{1}}+\frac{1}{N_{1,0}-n_{1}}\right)=N_{1,0}\frac{\gamma}{\alpha-\beta\frac{U_{1}}{U_{2}}}\frac{U_{1}}{U_{2}}\equiv-\ell^{-1},$ (39) where $\ell$ is the shock thickness since the solution accounting for the boundary conditions (2) is, $n_{1}(x)=N_{1,0}\frac{1}{1+e^{-x/\ell}}.$ (40) From (34) one then obtains for $n_{2}(x)$, $n_{2}(x)=N_{2,0}\frac{e^{-x/\ell}}{1+e^{-x/\ell}}.$ (41) ## Appendix C Derivation of Eq. (15) from Eq. (14) We first cast Eq. (14) under the form, $\ell=3\sigma\frac{k_{B}T_{1}(1-T_{2}/T_{1})(U_{2}/U_{1})U_{1}}{N_{1,0}m_{i}U_{1}^{2}(1-U_{2}/U_{1})^{2}}.$ (42) We then use the RH jump conditions (see for example Fitzpatrick (2014) p. 216, or Thorne & Blandford (2017) p. 905), $\left(\frac{U_{2}}{U_{1}}\right)^{-1}=\frac{N_{2,0}}{N_{1,0}}=\frac{\gamma+1}{\gamma-1+2\mathcal{M}_{1}^{-2}},$ (43) and, $\frac{T_{2}}{T_{1}}=\frac{P_{2}}{P_{1}}\frac{N_{1,0}}{N_{2,0}}$ (44) with, $\frac{P_{2}}{P_{1}}=\frac{2\gamma\mathcal{M}_{1}^{2}-\gamma+1}{\gamma+1}.$ (45) Substituting these ratios and setting, $\mathcal{M}_{1}^{2}=\frac{U_{1}^{2}}{\gamma P_{1}/N_{1,0}}$ (46) we get to Eq. (15) with $\gamma=5/3$. ## References * Baalrud et al. (2008) Baalrud, S. D., Callen, J. D. & Hegna, C. C. 2008 A kinetic equation for unstable plasmas in a finite space-time domain. Physics of Plasmas 15 (9), 092111. * Bale et al. (2003) Bale, S. D., Mozer, F. S. & Horbury, T. 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# Traffic Flow Estimation using LTE Radio Frequency Counters and Machine Learning Forough Yaghoubi Ericsson ABStockholmSweden<EMAIL_ADDRESS>, Armin Catovic Schibsted Media GroupStockholmSweden <EMAIL_ADDRESS>, Arthur Gusmao Ericsson ABStockholmSweden <EMAIL_ADDRESS>, Jan Pieczkowski Ericsson ABStockholmSweden <EMAIL_ADDRESS>and Peter Boros Ericsson ABStockholmSweden <EMAIL_ADDRESS> ###### Abstract. As the demand for vehicles continues to outpace construction of new roads, it becomes imperative we implement strategies that improve utilization of existing transport infrastructure. Traffic sensors form a crucial part of many such strategies, giving us valuable insights into road utilization. However, due to cost and lead time associated with installation and maintenance of traffic sensors, municipalities and traffic authorities look toward cheaper and more scalable alternatives. Due to their ubiquitous nature and wide global deployment, cellular networks offer one such alternative. In this paper we present a novel method for traffic flow estimation using standardized LTE/4G radio frequency performance measurement counters. The problem is cast as a supervised regression task using both classical and deep learning methods. We further apply transfer learning to compensate that many locations lack traffic sensor data that could be used for training. We show that our approach benefits from applying transfer learning to generalize the solution not only in time but also in space (i.e., various parts of the city). The results are very promising and, unlike competing solutions, our approach utilizes aggregate LTE radio frequency counter data that is inherently privacy- preserving, readily available, and scales globally without any additional network impact. Intelligent Transportation Systems, Traffic Flow, LTE, Radio Frequency, Machine Learning, Transfer Learning ††copyright: none††doi: ††isbn: ††conference: ; 2021††journalyear: ;††price: ## 1\. Introduction The increasing number of vehicles in the public roadway network, relative to the limited construction of new roads, has caused recurring congestion in the U.S. and throughout the industrialized world (Lawrence A. Klein, 2006). In the U.S. alone, the total cost of lost productivity caused by traffic congestion was estimated at $87 billion in 2018 (Fleming, [n.d.]). While one solution is to build new and expand existing roads, this is costly and takes time. A complementary approach is to implement strategies that improve the utilization of existing transport infrastructure. These strategies are found in Intelligent Transportation Systems (ITS) roadway and transit programs that have among their goals reducing travel time, easing delay and congestion, improving safety, and reducing pollutant emissions (Lawrence A. Klein, 2006). Traffic flow sensor technology forms a key component of ITS. Traffic sensors can be categorized as in-roadway (e.g. inductive loop sensors and magnetometers), or over-roadway (e.g. traffic cameras, radar, infrared and laser sensors). More recently there has been a surge of ad-hoc over-roadway sensor technology, including road-side cellular network masts, Bluetooth and Wi-Fi sensors, as well as telemetry collected from connected vehicles, smartphones and GPS devices. Cellular network masts are particularly appealing, combining ubiquity of cellular network technology (e.g. LTE or more specifically E-UTRA), with strict high availability requirements. While there have been previous approaches in utilizing cellular networks for understanding traffic flow, they’ve either been intrusive due to using user data, or not practical from a network operations perspective. In this paper we present a novel method for traffic flow estimation that leverages standard LTE/E-UTRA performance management (PM) counters, as defined by the 3rd Generation Partnership Project (3GPP) (3GPP, 2019). Namely, we utilize two radio frequency (RF) measurements - path loss distribution, and timing advance distribution counters, aggregated over 15 minute intervals. These counters are inherently privacy-preserving, and are continuously collected by nearly all LTE networks around the world, independent of network vendor. Thus our solution is non-invasive, and highly practical as it can be scaled across vast geographic regions with no live network impact. Our contributions in this paper are threefold: 1. (1) We present a novel method for estimating traffic flow using classical and deep learning regression models trained on E-UTRA RF counters (features) and vehicle counts from actual traffic sensors (targets). 2. (2) We evaluate the performance of our models by applying the learned model to different time samples, referred to as temporal generalization in this paper. 3. (3) We evaluate the performance of our models by applying the learned model to different road segments lacking ground truth data, referred to as spatial generalization in this paper; it is shown that due to difference in traffic distribution, the performance of our models is sub-optimal, hence we improve the accuracy using two transfer learning approaches. The rest of the paper is organized as follows: in section 2 we summarize the related works in the area of traffic flow estimation; our overall solution including feature selection/transformation and learning algorithms is explained in section 3; section 4 introduces two different transfer learning approaches; in section 5 we evaluate the performance of our models in terms of temporal and spatial generalization; ethical aspects are considered in section 6; finally the key takeaways are summarized in section 7. ## 2\. Related Works Traffic flow estimation has and continues to be a popular research topic. In (Zewei et al., 2015; George et al., 2013; Ma et al., 2013; Haferkamp et al., 2017; Nam et al., 2020) traffic flow estimation approaches are presented using data gathered from different sources such as cameras, acoustic sensors, magnetometers and spatially separated magnetic sensors. These solutions are not efficient due to coverage limitations and effort required in terms of installation and maintenance. To cope with these problems (Hansapalangkul et al., 2007; Pattara-Atikom and Peachavanish, 2007; Hongsakham et al., 2008; Caceres et al., 2012; Xing et al., 2019; Ji and Hong, 2019; Wang et al., 2020) propose the use of mobile subscriber data in traffic flow estimation. In (Hansapalangkul et al., 2007; Pattara-Atikom and Peachavanish, 2007; Hongsakham et al., 2008) the cell dwelling time and global positioning system (GPS) coordinates of a mobile subscriber are used to estimate the traffic congestion. These methods however have the disadvantage of high power consumption on a mobile device due to constant use of GPS, and are inherently intrusive. Another type of cellular data is considered in (Caceres et al., 2012) where authors propose a traffic flow estimation algorithm based on the number of subscribers in cars making a voice call. With today’s heavy usage of streaming and social media services however, voice calls are hardly representative of the traffic density, which limits the accuracy of such an approach. The authors in (Xing et al., 2019) use the travel trajectory of different mobile subscribers to detect in-vehicle users and henceforth compute the number of vehicles on a specific road. Tracking individual mobile users however is highly contentious and in most countries any user-identifiable or user-sensitive information limits the real-time usage of such data. In more recent work (Ji and Hong, 2019), the authors propose a method to predict the traffic speed and direction using wireless communication access logs including S1 application protocol (S1AP) collected from multiple radio base stations (RBS) located within a predetermined distance from the road. Due to high- intensity nature of S1AP signalling, tracing on the S1 interface in every single RBS leads to increase in network load, which is undesirable as it could lead to network overload with potentially catastrophic consequences. Furthermore, S1AP exposes potentially sensitive subscriber information allowing for the user to be fingerprinted or tracked. Another approach as presented in (Wang et al., 2020), describes a ”data fusion” approach, i.e. combining taxi GPS data with vehicle counts from license plate recognition (LPR) devices. The scalability of such approach is constrained due to limited availability of LPR and taxi GPS data. Therefore in this paper, we propose a new solution to traffic flow estimation problem based on aggregate LTE/E-UTRA radio frequency counter data that is inherently privacy preserving, readily available, and does not impose any extra load on the network. ## 3\. Method In this paper we describe two different approaches to using E-UTRA RF counters for traffic flow estimation. One approach involves using uplink path loss distribution as a feature vector in our model. Here we reason that different number of vehicles, i.e. obstacles on the road, can be represented by different path loss distributions. By optimizing for the number of vehicles the model should be able to discriminate between vehicles and all other users in the vicinity. In the second approach we use radio propagation delay, or more specifically timing advance (TA), as our feature vector. LTE radio base stations (eNBs) estimate the propagation delay on every random access (RA) initiated by a user. These propagation delays are aggregated from all RAs and represented as a distribution over discretized distances, where every bin represents a certain distance range from the eNB. By selecting only the bins corresponding to the known distances between the eNB and the relevant road segments, we can directly capture the road users, i.e. vehicles. Figure 1. Traffic flow estimation system presented in this paper. Path loss and timing advance features are described in detail in section 3.1. In both cases we use supervised regression techniques to train and evaluate our models. Fig. 1 shows the high level view of our system. In this paper we work in two domains: the source domain consists of training and validation data - the feature and target variables; the second domain, referred to as target domain, is where we perform inference using only the feature variables - however we use ground truth data for evaluation purposes. Our solution depends on the following assumptions: * • The eNB, or more specifically the sector antenna, is located within line-of- sight (LOS) of the relevant road segment, * • The distance between the road segment and the sector antenna is known, * • The relevant road segment consists of predominantly vehicular traffic, * • Traffic sensors used to supply ground truth data completely capture the traffic flow along the relevant road segment. In the following subsections we describe our feature and target variables, and learning algorithms. ### 3.1. Feature and Target Variables Figure 2. An example of a laser-based traffic sensor used in this paper; photograph by Holger Ellgaard, distributed under a CC BY-SA 3.0 license. #### 3.1.1. Traffic Sensor Data Target variables, i.e. ground truth data, consist of total vehicle counts aggregated over 15-minute intervals. The data is collected from a number of laser-based traffic sensors around inner Stockholm. Fig. 2 shows an example of such a sensor. Each sensor scans one lane of the road. For each road segment we sum the values from each lane to obtain total vehicle counts per 15-minute interval. We remove any samples where one of the sensor’s values are missing (e.g. due to a malfunction). #### 3.1.2. Path Loss Features Path loss (PL) is the attenuation of electromagnetic wave caused by free-space losses, absorption (e.g. by atmospheric particles), and scattering off various obstacles and surfaces. Radio propagation models attempt to account for this attenuation, and are a pivotal component in cellular network planning. Hata- Okamura (Hata, 1980) models are one such family of radio propagation models used to approximate cellular network coverage in different environments. In LTE, eNBs estimate PL values for all the scheduled users on every transmit time interval (TTI), which is typically 1ms. These estimates, represented as decibel (dB) values are then placed into discretized PL bins; in our case we have 21 bins, where each bin covers a range of 5dB, starting from $<$50dB and going up to $>$140dB. These estimates are done per-frequency band; in our case we have three bands, 800MHz, 1800MHz (two separate antennas working in this band) and 2600MHz, so we concatenate PL bins for all bands, resulting in total of 4 x 21 = 48 PL features. We don’t apply any filtering or transformation to PL features, and we treat all PL bins equally. Even though PL estimates are done every 1ms, the actual data available to us is aggregated in 15-minute intervals. The motivation behind our use of PL features is that different traffic conditions will result in different radio wave scattering characteristics, leading to different path loss distributions. A condition where there are no vehicles on the road will be represented by path loss distribution $PL_{a}$, which would be representative of radio wave losses due to predominantly indoor users and pedestrians. On the other hand a condition where the traffic flow is greater than zero would result in path loss distribution $PL_{b}$ where $PL_{b}\neq PL_{a}$, since radio wave scattering off vehicle surfaces would yield a different path loss ”signature”. Our trained algorithms should be able to discriminate between such conditions. Figure 3. Spatial granularity of an eNB. #### 3.1.3. Timing Advance Features Timing advance (TA) is estimated for every user connection request, or more specifically on every random access. TA estimation is dependent on successful completion of an RRC Connection Request procedure, and the 11-bit TA command. The name is a slight misnomer, since TA features are actually represented as discretized distance bins/ranges, representing the distance between the user and the sector antenna. In our case we have 35 bins, starting from $<$80m up to 100km; typically only the first few bins are incremented, as users are normally within 500m of the antenna (otherwise they will be handed over to another sector, or another eNB). Fig. 3 shows the spatial granularity of an eNB and how the TA features may be represented. Unlike PL features, we do actually apply a distance selection filter to TA features. Our aim is to consider only road users (vehicles), which means selecting TA bins/features that represent the known distances between the relevant road segment and the sector antenna. Lets assume that TA value ranges are indicated by $M$ bins where bin $ta_{i}$ corresponds to a distance interval shown by $[d_{i};d_{i+1})$ given a known distance $d$ between the road segment and the sector antenna, we choose the TA bin index $ta_{i}$ where $d_{i}\leq d\leq d_{i+1}$. Just like traffic sensor data and PL features, TA features are also aggregated in 15-minute intervals. #### 3.1.4. Cyclic Time Features As traffic exhibits strong seasonality, it is beneficial to give our models temporal information. To encode this information, a common method is to transform the date-time representation into cyclic time features using a $\sin$ and $\cos$ transformation as follows: $\displaystyle x_{\sin}=$ $\displaystyle\sin(\frac{2\pi x}{\max(x)})$ $\displaystyle x_{\cos}=$ $\displaystyle\cos(\frac{2\pi x}{\max(x)})$ where $x$ can be hour, day and month. By using the above equation, we convert time-of-day, day-of-week and week-of-month to the corresponding cyclic time features. As the time granularity for our data is in minutes, we set the $\max(x)$ to $2460$, $72460$, and $472460$ respectively. #### 3.1.5. Road-dependent Features As traffic flow depends on the road characteristics, we also apply different road-dependent features. These features are easily extracted from e.g. OpenStreetMap services. In this paper we use the following road-dependent features: number of lanes, maximum speed limit, and road category, i.e. highway, large city road and small city road. ### 3.2. Learning Algorithms In this paper we compare two supervised learning approaches for traffic flow estimation. In the first approach we evaluate a number of different classical regression algorithms. In the second approach we take into account the history of time samples using gradient based Long Short-term Memory (LSTM). #### 3.2.1. Classical Regression Models Classical regression assumes independence between time samples. Since we’re working with fairly coarse 15-min aggregate intervals, it is reasonable to assume this independence. Regression then amounts to estimating a function $f(\bm{x};\bm{\theta})$, which transforms a feature vector $\bm{x}$ to a target variable $y$. Function parameters $\bm{\theta}=(\theta_{0},\theta_{1},...,\theta_{k})$ are found by minimizing expected loss, typically a mean squared error (MSE) of the form $MSE(\bm{\theta})=\frac{1}{N}\sum_{n=1}^{N}(y_{n}-f(\bm{x_{n}};\bm{\theta}))^{2}$, where $N$ corresponds to total number of 15-min aggregate samples. We evaluate a number of different regression algorithms including Support Vector Regressor (SVR), Kernel Ridge (KR), Decision Tree (DT), and Random Forest (RF). Each algorithm also requires setting its internal parameters, or hyperparameters. Since the total number of hyperparameters is small, we use grid search method to exhaustively search through the hyperparameter space and pick the combination of parameters that yield the best performance. We apply a time- dependent train/test split, e.g. by selecting the first 6 weeks for training, and the following 2 weeks for testing; compared to a random assignment of train/test data, our approach is more in line with how the algorithm would be used in practice, and is more representative of the generalization capability in the real-world setting. #### 3.2.2. LSTM LSTM is a specific kind of recurrent neural network (RNN) that has the ability to capture long-term time dependencies and bridge time intervals in excess of 1000 steps even in case of noisy, in-compressible input sequences (Du et al., 2017). Similar to other types of RNNs, LSTM has a chain structure with modified repeating modules. In each module, instead of having a single neural network layer, there are four layers that interact with each other. More detailed information about LSTM architecture can be found in (Smagulova and James, 2019). The architecture of our LSTM based traffic flow estimator consists of two LSTM layers, followed by a dropout regularization layer, and then finally the two fully-connected (FC) layers. The two LSTM, as well as the two FC layers, use the rectified linear unit (ReLU) activation function, while the output layer activates with the linear function. ## 4\. Transfer Learning Approaches The learning approaches mentioned above optimize the model for temporal generalization where we use all available locations in our training set, but withhold a contiguous period of time (e.g. two weeks) for test purposes. However, we would like our models to generalize well across all possible locations, even never-before-seen locations, which may potentially have completely different traffic patterns/distributions. We refer to this problem as spatial generalization. To cope with this problem we use transfer learning (TL) approaches. TL focuses on transferring the knowledge between different domains and can be a promising solution to overcome the spatial generalization problem. Recently, there has been lot of work focusing on transfer learning and proposing efficient solutions (Zhuang et al., 2019; Pan and Yang, 2009; Tan et al., 2018). These studies categorize TL into three subcategories based on different situations involving source and target domain data and the tasks, including inductive, transductive, and unsupervised transfer learning. Our work can be fitted into transductive transfer learning where the source label data are available while no label data for target domain is provided. Here the assumption is that the task between target and source domain is the same, but the domain marginal or conditional distributions are different. Among the proposed transductive TL algorithms, we evaluate two approaches - one based on instant weighting and the second one based on deep domain adaptation. We explain each of the algorithms in detail in the following sections. ### 4.1. Instant Weighting The data-based TL approaches, such as instant weighting, focus on transferring the knowledge by adjustment of the source data. Assuming that the source and target domain only differ in marginal distribution, a simple idea for transformation is to assign weights to source domain data equal to the ratio of source and target domain marginal distribution. Therefore the general loss function of the learning algorithm is given by: (1) $\min_{\theta}\frac{1}{N_{s}}\sum_{1}^{N_{s}}\alpha_{i}J(\theta(x_{i}^{s}),y_{i}^{s})+\lambda\gamma(\theta)$ where $J$ represents the loss of source data and $\alpha_{i}$ is the weighting parameter and is equal to: (2) $\alpha_{i}=\frac{P^{T}(x)}{P^{S}(x)}.$ In the literature, there exist many ways to compute $\alpha_{i}$; in (Huang et al., 2006) the authors used Kernel Mean Matching (KMM) to estimate the ratio by matching the means of target and source domain data in the reproducing- kernel Hilbert space where the problem of finding weights can be written as follows: (3) $\displaystyle\min_{\alpha}\frac{1}{2}\alpha^{T}K\alpha-\kappa\alpha$ $\displaystyle s.t\sum_{i}^{N_{s}}\alpha_{i}-N_{s}\leq N_{s}\epsilon$ $\displaystyle\alpha\in[0,B]$ where $N_{s}$ shows the number of sample in source domain data and $K$ is kernel matrix and is defined as: (4) $K=\begin{bmatrix}k_{ss}&k_{st}\\\ k_{ts}&k_{tt}\end{bmatrix},$ while $k_{ss}=k(x_{s},x_{s})$ and $\kappa_{i}=\frac{N_{s}}{N_{T}}\sum_{i}^{N_{T}}k(x_{i},x_{Tj})$. ### 4.2. Domain Adaptation Deep learning algorithms have received lot of attention from researchers having successfully outperformed many traditional machine learning methods in tasks such as computer vision and natural language processing (NLP). Therefore in the TL area many researchers also utilize deep learning techniques. In this paper, we use discrepancy-based domain adaptation, where a deep neural network is used to learn the domain-independent feature representations. In deep neural networks, the early layers tends to learn more generic transferable features, while domain-dependent features are extracted in the terminal layers. Therefore, to decrease the gap between the distribution in the last layers, we add multiple adaptation layers with discrepancy loss as regularizer. The deep learning model used for feature extraction is the LSTM model explained in previous section. The pretrained LSTM model will be used to extract the features for both source and target domains. After that the primary goal is to reduce the difference between target and source domain distribution. The term maximum mean discrepancy (MMD) is widely used in TL literature as a metric to compute the distance between two distribution (Wang et al., 2017; Gretton et al., 2012). Fig. 4 shows the architecture of our domain adaptation network based on LSTM. Figure 4. The domain adaptation network based on LSTM. Let $f$ denote the function for feature representation of our pretrained model, then the distance between the feature distribution of source and target domain is given by: (5) $d(p,q)=\sup_{f\in F}{E_{p}\\{f(x)\\}-E_{q}\\{f(y)\\}}$ where $\sup$ defines the supremum, $E$ denotes the expectation and $x$ and $y$ are independently and identically distributed (i.i.d) samples from $p$ and $q$, respectively. The above equation can be easily computed using the kernel trick where it can be expressed by expectation of kernel functions. Therefore, the square of equation (5) can be reformulated as follows: (6) $d^{2}_{k}(p,q)=E_{x_{p}^{s}x_{p}^{s}k(x_{p}^{s},x_{p}^{s})}+E_{x_{q}^{t}x_{q}^{t}k(x_{q}^{t},x_{q}^{t})}-2E_{x_{p}^{s}x_{q}^{t}k(x_{p}^{s},x_{q}^{t})},$ where $x_{p}^{s}$ and $x_{q}^{t}$ are the samples from source and target domain respectively, and $k$ is the kernel defined as $\exp(\frac{-\left\lVert x_{i}-x_{j}\right\rVert^{2}}{\gamma})$. To adapt the pretrained model for the target data samples, the objective function of our TL algorithm is given by (Long et al., 2015): (7) $\min_{\theta}\frac{1}{N_{s}}\sum_{1}^{N_{s}}J(\theta(x_{i}^{s}),y_{i}^{s})+\lambda\sum_{l=l_{1}}^{l_{2}}d_{k}^{2}(D^{s}_{l},D^{t}_{l}),$ where $J$ is the loss for source domain in LSTM network, $l_{1}$ and $l_{2}$ indicate the layer indices between which the regularization is effective, and $D^{s}_{l}$ and $D^{t}_{l}$ are $l$ layer representation of the source and target samples, respectively. The parameter $\lambda$ is a trade off term so that the objective function can benefit both from TL and deep learning. ## 5\. Results We use approximately 8 weeks worth of data, where every data sample corresponds to a 15-min interval, so we have $\scriptstyle\sim$ 96 * 7 * 8 = 5376 data samples. The data are collected from six different locations around inner Stockholm; each location corresponds to a road segment with a traffic sensor and a nearby LTE eNB. We evaluate models using PL and TA features independently and across a range of regression algorithms. When evaluating temporal generalization we use all locations during training and split the data into 80/20 train/test sets, which corresponds to approximately 6 weeks of contiguous training data, and 2 weeks of test data. When evaluating spatial generalization we use all time samples for training but we randomly assign road segments into source and target domains. For evaluation purposes we use coefficient of determination $R^{2}$ defined as follows: (8) $R^{2}=1-\frac{SS_{res}}{SS_{tot}}$ where $\displaystyle SS_{tot}=\sum_{i}\left(y_{i}-\bar{y}\right)^{2}$ (9) $\displaystyle SS_{res}=\sum_{i}\left(y_{i}-\hat{y}_{i}\right)^{2}$ $SS_{tot}$ represents total sum of squares, and $SS_{res}$ represents residual sum of squares, while $\bar{y}$ and $\hat{y_{i}}$ are the observed data mean and the predicted traffic flow respectively. A model that always predicts observed data mean will have $R^{2}=0$; models with observations worse than the observed data mean will have negative values; the most optimal value is $R^{2}=1$, so we want our models to be as close to 1 as possible. The set of classical regression algorithms used for training are Support Vector Regression (SVR), Kernel Ridge (KR), Decision Trees (DT) and Random Forest (RF). We also train a deep learning model with two LSTM layers followed by a dropout layer and two fully-connected layers activated with the ReLU function. The hyperparameters providing the best $R^{2}$ score on the test set for our models are found using grid search and presented in Table 1. The corresponding results for both temporal and spatial generalization performance are shown in Table 2. Table 1. Hyperparameters used in this paper. Models \| Parameters ---\|--- \| Kernel = rbf SVR \| $C$ = 10 \| $\gamma$ = 0.001 \| Kernel = rbf KR \| $\alpha$ = 1 \| $\gamma$ = 0.01 DT \| Maximum depth = 10 RF \| Maximum depth = 30 \| Learning rate = 0.0009 LSTM \| Hidden size = 100 \| Epochs = 300 \| Dropout rate = 0.2 \| Window = 5 Table 2. $R^{2}$ score for temporal and spatial generalization performance using TA and PL features. Higher scores are better. Models \| Temporal Generalization \| Spatial Generalization ---\|---\|--- TA \| PL \| TA \| PL SVR \| 0.754 \| 0.786 \| 0.12 \| -0.62 KR \| 0.862 \| 0.888 \| -0.79 \| -0.63 DT \| 0.938 \| 0.946 \| -3.22 \| -0.37 RF \| 0.946 \| 0.959 \| -0.96 \| 0.017 LSTM \| 0.845 \| 0.901 \| 0.087 \| -1.67 The results in Table 2 indicate that all regression algorithms perform reasonably well in terms of temporal generalization, using either TA or PL features. The Random Forest (RF) model outperforms all the others, including the LSTM model, with an average $R^{2}$ score of 0.95. These results validate our initial assumption that due to a fairly coarse 15-min aggregate interval, it is safe to assume independence between time steps, hence deep learning based LSTM does not add any additional value. A more visual representation of the RF algorithm performance is shown in Fig. 5 where we compare traffic flow estimates from our model against the actual values across three different locations. The algorithm does not always capture the peaks - our hypothesis is that more training samples with varied traffic flow distributions are needed for the model to generalize even better. (a) (b) (c) Figure 5. Traffic flow estimates in terms of number of vehicles per 15-min interval using our Random Forest (RF) model compared to the ground truth, for 5(a) Road 1, 5(b) Road 2 and 5(c) Road 3. Despite good temporal generalization performance, the average $R^{2}$ score for spatial generalization is very low for all regression models. This poor performance is due to inherent difference between the source and target domain distributions. In order to improve spatial generalization we use two types of transfer learning (TL) algorithms, namely instant weighting and deep domain adaptation. In the first approach we implement the instant weighting for classical regression. For each test location, we compute the weights solving the quadratic optimization problem, and then retrain the model using these weights. Since the RF model yields the highest $R^{2}$ score on temporal generalization we apply instant weighting to RF only. Table 3 presents the $R^{2}$ scores of RF model for both TA and PL features with and without applying the instant weighting. The results indicate that instant weighting can only improve the performance when TA features are used. Since the TA features represent the road users more explicitly we expect there to be some minimum similarity between all domain distributions. On the other hand PL represents all users, including indoor users, and therefore PL features are highly sensitive to physical layout of the environment, i.e. number of buildings, thickness of walls, heights of buildings etc. Table 3. $R^{2}$ score for spatial generalization using the Random Forest (RF) model with and without transfer learning (TL). Higher scores are better. Test Road \| PL Features \| TA Features ---\|---\|--- No TL \| TL \| No TL \| TL 1 \| 0.02 \| -0.47 \| -0.96 \| 0.71 2 \| 0.02 \| -0.75 \| -0.96 \| 0.72 3 \| 0.02 \| -0.86 \| -0.96 \| 0.42 Mean \| 0.02 \| -0.69 \| -0.96 \| 0.62 Table 4. $R^{2}$ score for spatial generalization using the LSTM model with and without transfer learning (TL). Higher scores are better. Test Road \| PL Features \| TA Features ---\|---\|--- No TL \| TL \| No TL \| TL 1 \| -0.20 \| 0.24 \| 0.24 \| 0.66 2 \| -1.73 \| -1.02 \| -0.62 \| 0.61 3 \| -3.09 \| -0.52 \| -0.13 \| 0.61 Mean \| -1.67 \| -0.43 \| -0.17 \| 0.63 In the second approach, we implement the deep domain adaptation algorithm as shown in Fig. 4. We freeze the two LSTM layers and the two fully-connected layers using the pre-trained weights, while we train the final two fully- connected layers using the MMD regularizer. As there is no target domain label data available only the source output is considered in the loss function. Table 4 shows the performance of spatial generalization using the LSTM and deep domain adaptation. The LSTM model performs reasonably well using TA features, with average $R^{2}$ score very similar to what we saw using RF and instant weighting. ## 6\. Ethical Considerations One of the main motivations for the work presented in this paper concerns user privacy and integrity. Traffic cameras and automated license plate recognition devices allow for unprecedented levels of identification and tracking. This is all the more true for user data obtained from cellular networks and mobile devices. Our approach as presented in this paper uses data that is inherently privacy-preserving - we use readily available radio frequency counters that are aggregated on cell level and per definition do not contain any information about individual users, nor could this information be reconstructed. It is therefore impossible to identify or track any individual user based on this data. With that in mind we can state that the work presented in this paper does not raise any ethical issues. ## 7\. Conclusion Traffic flow estimation has traditionally involved forecasting methods based on observations from dedicated traffic sensors. Firstly these methods don’t scale well since we require large number of sensors. Secondly we would need a separate forecasting model for every road, since roads don’t exhibit homogeneous behaviour. Finally our traffic estimation performance would be susceptible to drastic changes in driver behaviour or road conditions, such as traffic accidents and road works. To overcome these limitations alternative approaches have been proposed, including using various forms of cellular network data to estimate traffic flow. However existing approaches are either user invasive, or can potentially result in adverse operational impacts to cellular networks. In this paper we propose a traffic flow estimation method using inherently anonymous and widely available LTE/E-UTRA radio frequency counters, namely path loss and timing advance counters, effectively turning LTE eNBs into traffic sensors. We cast traffic flow estimation as a supervised regression problem, where path loss and timing advance counters are used as primary features, and vehicle counts from actual traffic sensors as target or ground truth variables. We demonstrated excellent performance using both Random Forest and LSTM regression models. Since we have limited amount of ground truth data, i.e. we only had access to six different locations, we also evaluated the performance of two different transfer learning approaches, namely instant weighting, and deep domain adaptation. With transfer learning we demonstrated reasonable performance using either Random Forests or LSTMs, but using only timing advance features. Our hypothesis is that with more data and more locations the performance will improve further still. While our models are not perfect estimators, they are still extremely useful - they capture the shape of the traffic very well, and for most purposes provide a good-enough estimate of the traffic flow. The output of these models can be used for anomaly detection, for example for detecting traffic congestion or accidents. All this can be achieved without having to install any additional sensors - we simply re-use LTE radio base stations that are permanently fixed in their locations with near 100% uptime. ## Acknowledgments We would like to thank the following people for their support throughout the project, and for facilitating the network and traffic sensor data without which none of this would be possible: Elin Allison, Madeleine Körling and Jyrki Lehtinen from Telia Company AB; Anders Broberg and Tobias Johansson from City of Stockholm; Annika Engström from KTH Royal Institute of Technology and Digital Demo Stockholm; Chris Deakin and Chris Holmes from WM5G Limited; Mo Elhabiby and Mike Grogan from Vodafone UK. 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# Streaming Models for Joint Speech Recognition and Translation Orion Weller1, Matthias Sperber2, Christian Gollan2, Joris Kluivers2 1Brigham Young University 2Apple oweller<EMAIL_ADDRESS> ###### Abstract Using end-to-end models for speech translation (ST) has increasingly been the focus of the ST community. These models condense the previously cascaded systems by directly converting sound waves into translated text. However, cascaded models have the advantage of including automatic speech recognition output, useful for a variety of practical ST systems that often display transcripts to the user alongside the translations. To bridge this gap, recent work has shown initial progress into the feasibility for end-to-end models to produce both of these outputs. However, all previous work has only looked at this problem from the consecutive perspective, leaving uncertainty on whether these approaches are effective in the more challenging streaming setting. We develop an end-to-end streaming ST model based on a re-translation approach and compare against standard cascading approaches. We also introduce a novel inference method for the joint case, interleaving both transcript and translation in generation and removing the need to use separate decoders. Our evaluation across a range of metrics capturing accuracy, latency, and consistency shows that our end-to-end models are statistically similar to cascading models, while having half the number of parameters. We also find that both systems provide strong translation quality at low latency, keeping 99% of consecutive quality at a lag of just under a second. ††Work done during an internship with Apple ## 1 Introduction Speech translation (ST) is the process of translating acoustic sound waves into text in a different language than was originally spoken in. This paper focuses on ST in a particular setting, as described by two characteristics: (1) We desire models that translate in a streaming fashion, where users desire the translation before the speaker has finished. This setting poses additional difficulties compared to consecutive translation, forcing systems to translate without knowing what the speaker will say in the future. (2) Furthermore, the speaker may want to verify that their speech is being processed correctly, intuitively seeing a streaming transcript while they speak Fügen (2008); Hsiao et al. (2006). For this reason, we consider models that produce both transcripts and translation jointly.111This corresponds to the mandatory transcript case in the proposed categorization by Sperber and Paulik (2020). Previous approaches to streaming ST have typically utilized a cascaded system that pipelines the output of an automatic speech recognition (ASR) system through a machine translation (MT) model for the final result. These systems have been the preeminent strategy, taking the top place in recent streaming ST competitions Pham et al. (2019); Jan et al. (2019); Elbayad et al. (2020); Ansari et al. (2020). Despite the strong performance of these cascaded systems, there are also some problems: error propagation from ASR output to MT input Ruiz and Federico (2014); ASR/MT training data mismatch and loss of access to prosodic/paralinguistic speech information at the translation stage Sperber and Paulik (2020); and potentially sub-optimal latencies in the streaming context. End-to-end (E2E) models for ST have been proposed to remedy these problems, leveraging the simplicity of a single model to sidestep these issues. E2E models are also appealing from computational and engineering standpoints, reducing model complexity and decreasing parameter count. Although initial research has explored E2E models for joint speech recognition and translation, no previous works have examined them in the streaming case, a crucial step in using them for many real-world applications. To understand this area more fully, we develop an E2E model to compare with its cascading counterpart in this simultaneous joint task. We build off the models proposed by Sperber et al. (2020) in the consecutive case, extending them for use in the streaming setting. We also use the re-translation technique introduced by Niehues et al. (2018) to maintain simplicity while streaming. To reduce model size, we introduce a new method for E2E inference, producing both transcript and translation in an interleaved fashion with one decoder. As this task requires a multi-faceted evaluation along several axes, we provide a suite of evaluations to highlight the differences of these major design decisions. This suite includes assessing translation quality, transcription quality, lag of the streaming process, output flicker, and consistency between the transcription and translation. We find that our E2E model performs similarly to the cascaded model, indicating that E2E networks are a feasible and promising direction for streaming ST. ## 2 Proposed Method ### Network Architecture In the ST survey provided by Sperber et al. (2020), they introduce several E2E models that could be used for the joint setting. As our work focuses on providing a simple but effective approach to streaming ST, we focus on the CONCAT model, which generates both the transcript and translation in a concatenated fashion. We compare this E2E model against the standard cascading approach, following the architecture and hyperparameter choices used in Sperber et al. (2020). All audio input models use the same multi-layer bidirectional LSTM architecture, stacking and downsampling the audio by a factor of three before processing. We note that although bidirectional encoders are unusual with standard ASR architectures, re-translation makes them possible. The cascaded model’s textual encoder follows the architecture described in Vaswani et al. (2017) but replaces self-attention blocks with LSTMs. Decoder networks are similar, but use unidirectional LSTMs. More implementation details can be found in Appendix A. In order to reduce model size and inference time for E2E networks, we introduce a novel method for interleaving both transcript and translation in generation, removing the need to use separate decoders. This method extends the CONCAT model proposed by Sperber et al. (2020) to jointly decode according to the ratio given by the parameter $\gamma$ (Figure 1). When $\gamma=0.0$, we generate the transcript tokens until completion, followed by the translation tokens (vice versa for $\gamma=1.0$). At $\gamma=0.0$, our model is equivalent to the previously proposed model. Defining $\mathrm{count_{i}}$ as the count of $i$ tokens previously generated, transcription tokens as st and translation tokens as tt, we generate the next token as a transcription token if: $\displaystyle(1.0-\gamma)(1+\mathrm{count_{tt}})>\gamma(1+\mathrm{count_{st}})$ This approach enables us to produce tokens in an interleaving fashion, given the hyperparameter $\gamma$. Figure 1: Example token representations (En→De) for three different interleaving parameters (Section 2). Language tokens indicate whether the data corresponds to the source transcript or the target translation and are used with a learned embedding that is summed with the word embeddings, as described in Sperber et al. (2020). Figure 2: Left: average lag in seconds vs BLEU score. Right: average lag in seconds vs WER score. All points are the mean of each configuration’s score across the eight target languages. Configurations are the cross product of the values for $K$ and $F$, see Section 2: Inference. Note that points near 1.0 AL have appx. 99% of the unconstrained BLEU score. Results for the E2E model use $\gamma=0.5$. . Metric \| Params \| Model \| De \| Es \| Fr \| It \| Nl \| Pt \| Ro \| Ru \| Average ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- BLEU $\uparrow$ \| 217M \| Cascade \| 18.8 \| 22.7 \| 27.0 \| 18.9 \| 22.5 \| 21.9 \| 17.9 \| 13.0 \| 20.3 \| 107M \| E2E $\gamma$=0.0 \| 18.1 \| 23.1 \| 27.0 \| 18.7 \| 22.3 \| 22.2 \| 17.6 \| 12.2 \| 20.2 \| 107M \| E2E $\gamma$=0.3 \| 17.7 \| 22.6 \| 26.3 \| 18.0 \| 21.5 \| 21.5 \| 17.0 \| 12.1 \| 19.6 \| 107M \| E2E $\gamma$=0.5 \| 18.2 \| 22.8 \| 27.0 \| 18.6 \| 21.9 \| 21.9 \| 17.1 \| 12.0 \| 19.9 \| 107M \| E2E $\gamma$=1.0 \| 18.2 \| 22.8 \| 27.1 \| 18.9 \| 22.2 \| 22.3 \| 17.6 \| 12.7 \| 20.2 WER $\downarrow$ \| 217M \| Cascade \| 25.9 \| 24.0 \| 23.1 \| 25.6 \| 28.5 \| 26.4 \| 24.4 \| 23.1 \| 25.1 \| 107M \| E2E $\gamma$=0.0 \| 24.2 \| 23.5 \| 23.3 \| 23.0 \| 23.4 \| 25.3 \| 24.1 \| 23.6 \| 23.8 \| 107M \| E2E $\gamma$=0.3 \| 24.1 \| 23.6 \| 22.9 \| 23.8 \| 23.4 \| 25.7 \| 24.1 \| 24.1 \| 24.0 \| 107M \| E2E $\gamma$=0.5 \| 24.5 \| 23.9 \| 22.9 \| 23.8 \| 23.4 \| 25.7 \| 24.3 \| 23.6 \| 24.0 \| 107M \| E2E $\gamma$=1.0 \| 23.6 \| 22.9 \| 22.3 \| 23.0 \| 22.4 \| 24.7 \| 23.4 \| 22.7 \| 23.1 Table 1: BLEU and WER scores for models trained on different target languages. Bold scores indicate results that are statistically similar to the best score using a bootstrap permutation test with $\alpha=0.05$. ### Re-translation We use the re-translation method Niehues et al. (2018); Arivazhagan et al. (2020a, b) as it provides a simple way to handle the streaming case. This method works by simply re-translating the utterance as new data arrives, updating its former prediction. As we are generating both transcript and translation, this avoids the challenging issue of combining the requirements for both components: streaming speech models need to manage the audio signal variability across time while streaming translation models need to overcome issues with reordering and lack of future context. Alternative strategies to the re-translation approach include the chunk-based strategy explored by Liu et al. (2020), which commits to all previous output chunks and Ren et al. (2020) who utilize an additional segmenter model trained via CTC Graves et al. (2006) to create segments that are translated via wait-k Ma et al. (2019). Although these approaches show effective results, they add additional complexity without addressing issues particular to streaming transcription. ### Inference In order to generate quality-latency curves, we use several techniques to reduce latency and flicker at the cost of quality. The first is the mask-k method proposed by Arivazhagan et al. (2020b), masking the last $K$ output tokens. The second method is a form of constrained decoding: we define a hyperparameter $F$ that sets the number of free tokens allowed to change in the next re-translation. Thus, we constrain future output to match the first $\textit{len}(\textit{tokens})-F$ tokens of the current output. All models use values $\\{0,1,2,3,4,5,7,10,100\\}$ for $K$ and $\\{0,1,2,3,4,5,7,10,15,20,25,100\\}$ for $F$. For interleaving models, we set $K$ and $F$ on both transcript and translation tokens. Model \| En→De Incr. \| En→De Full \| En→Es Incr. \| En→Es Full \| Mean Incr. \| Mean Full ---\|---\|---\|---\|---\|---\|--- Cascade \| 13.8 \| 13.2 \| 12.2 \| 11.6 \| 14.1 \| 13.4 Concat $\gamma$=0.0 \| 17.6 \| 16.7 \| 14.9 \| 13.8 \| 17.0 \| 16.0 Concat $\gamma$=0.3 \| 17.2 \| 16.6 \| 14.3 \| 13.7 \| 16.6 \| 15.8 Concat $\gamma$=0.5 \| 17.8 \| 16.5 \| 14.8 \| 13.3 \| 17.3 \| 15.7 Concat $\gamma$=1.0 \| 17.3 \| 16.8 \| 14.9 \| 13.7 \| 16.9 \| 15.8 Table 2: Consistency scores for En→De, En→Es, and average results over all languages; lower is better (see Sperber et al. (2020)). _Incr._ stands for the incremental consistency score, or the average consistency throughout re- translation. Bold scores indicate results that are statistically similar to the best score using a bootstrap permutation test with $\alpha=0.05$. ## 3 Experimental Settings ### Data We use the MuST-C corpus di Gangi et al. (2019) since it is the largest publicly available ST corpus, consisting of TED talks with their English transcripts and translations into eight other language pairs. The dataset consists of at least 385 hours of audio for each target language. We utilize the log Mel filterbank speech features provided with the corpus as input for the ASR and E2E models. To prepare the textual data, we remove non- speech artifacts (e.g. “(laughter)” and speaker identification) and perform subword tokenization using SentencePiece Kudo and Richardson (2018) on the unigram setting. Following previous work for E2E ST models, we use a relatively small vocabulary and share transcription and translation vocabularies. We use MuST-C dev for validation and report results on tst- COMMON, utilizing the segments provided (Appendix D). ### Prefix Sampling We implement techniques developed by Niehues et al. (2018); Arivazhagan et al. (2020b) for improving streaming ST, sampling a random proportion of each training instance as additional data to teach our models to work with partial input. See Appendix C for implementation details. ### Metrics We evaluate these models on a comprehensive suite of metrics: sacrebleu (_BLEU_ , Post (2019)) for translation quality, word error rate (_WER_ , Fiscus (1997)) for transcription quality, average lag (_AL_ , Ma et al. (2019)) for the lag between model input and output, and normalized erasure (_NE_ , Arivazhagan et al. (2020a)) for output flicker. Measuring consistency is a nascent area of research; we use the robust and simple lexical consistency metric defined by Sperber et al. (2020), which uses word-level translation probabilities. To show how consistent these results are while streaming, we compute an incremental consistency score, averaging the consistency of each re-translation. ## 4 Results Results for the quality-latency curves created by the use of constrained decoding and mask-k (Section 3) are shown in Figure 2. Unconstrained settings are used for all results in table form. For convenience, bold scores indicate the highest performing models in each metric according to a bootstrap permutation test. ### Translation Quality We see in Table 1 that the cascaded model slightly outperforms some E2E models, while achieving statistically similar performance to the $\gamma=1.0$ model. We note however, that the cascaded model has nearly twice as many parameters as the E2E models (217M vs 107M). When we examine these models under a variety of different inference conditions (using constrained decoding and mask-k as in Arivazhagan et al. (2020a)), we further see this trend illustrated through the quality vs latency trade-off (left of Figure 2), with both models retaining 99% of their BLEU at less than 1.0 AL. ### Transcription Quality Conversely, Table 1 and the right of Figure 2 show that the $\gamma=1.0$ E2E model performs similarly or slightly better than the cascaded model across all inference parameters and all target languages. With an AL of 1.5, the E2E model loses only 3% of its performance. ### Consistency The E2E models perform worse than the cascaded on consistency, with the best models being approximately 18% less consistent (Table 2). The cascaded model also maintains better scores through each re-translation (_Incr._).222Initial experiments indicate that the triangle E2E architecture Sperber et al. (2020) model may perform better on consistency in our streaming setting, but due to time constraints we were not able to explore this further. Future work exploring alternative architectures or decoding techniques Le et al. (2020) may provide fruitful avenues of research. ### Flicker We note that the flicker scores for cascade and E2E models are similar, with both having normalized erasure scores of less than 1 and the majority of inference settings having less than the “few-revision” threshold of 0.2 (proposed by Arivazhagan et al. (2020a)). More NE details are found in Appendix B. ### Interleaving Rate Table 1 also shows us the overall results for different interleaving rates. We see that interleaving at a rate of 1.0 has the best quality scores (0.7 less WER than the next best rate, the base $\gamma=0.0$ model) but the worst consistency (Table 2). Conversely, $\gamma=0.3$ has the worst quality scores but the best consistency. ## 5 Conclusion We focus on the task of streaming speech translation, producing both a target translation and a source transcript from an audio source. 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Consistent Transcription and Translation of Speech. _Transactions of the Association for Computational Linguistics (TACL)_. * Vaswani et al. (2017) Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Łukasz Kaiser, and Illia Polosukhin. 2017. Attention Is All You Need. In _Neural Information Processing Systems Conference (NIPS)_ , pages 5998–6008, Long Beach, USA. ## Appendix A Model Details In this section we will describe implementation details of the model architectures (shown in Figure 3) and training processes. ### Model Architectures Unless otherwise noted, the same hyperparameters are used for all models. Weights for the speech encoder are initialized based on a pre-trained attentional ASR task that is identical to the ASR part of the direct multitask model. Other weights are initialized according to Glorot and Bengio (2010). The speech encoder is a 5-layer bidirectional LSTM with 700 dimensions per direction. Attentional decoders consist of 2 Transformer blocks Vaswani et al. (2017) but use 1024-dimensional unidirectional LSTMs instead of self- attention, except for the CONCAT model, which uses 3 layers. For the cascade’s MT model, encoder/decoder both contain 6 layers with 1024-dimensional LSTMs. Subword embeddings are size 1024. We regularize using LSTM dropout with $p=0.3$, decoder input word-type dropout Gal and Ghahramani (2016), and attention dropout, both $p=0.1$. We apply label smoothing with strength $\epsilon=0.1$. Figure 3: Architectures of the cascade and concatenated model ### Training We optimize using Adam Kingma and Ba (2014) with $\alpha=0.0005$, $\beta_{1}=0.9$, $\beta_{2}=0.98$, 4000 warm-up steps, and learning rate decay by using the inverse square root of the iteration of each instance. We set the batch size dynamically based on the sentence length, such that the average batch size is 128 utterances. The training is stopped when the validation score has not improved over 10 epochs, where the validation score is corpus- level translation BLEU score (for the E2E and MT models) and corpus-level WER for the cascade’s ASR model. Figure 4: Left: average lag in seconds vs NE score. Right: NE vs WER score. All values are the mean of the results from the eight target languages. For decoding and generating n-best lists, we use beam size 5 and polynomial length normalization with exponent 1.5. Our implementation is based on PyTorch Paszke et al. (2019) and XNMT Neubig et al. (2018), and all models are trained in single-GPU environments, employing Tesla V100 GPUs with 32 GB memory. Most E2E and ASR models converged after approximately 30 epochs or 5 days of training. MT models converged after approximately 50 epochs or 2 days of training. ## Appendix B Normalized Erasure (Output Flicker) We see similar curves for both the cascaded model and the E2E model when comparing normalized erasure in Figure 4. We see that most settings have an NE score of less than 0.2, while virtually all settings are less than 1. We note that a proportion of 0.2 for NE means that, on average, 1/5 of the tokens change once before they settle to their final state. ## Appendix C Prefix Training We used prefix training to increase stability and reduce flickering in the streaming setting. We conducted this by utilizing each training instance twice in each epoch: one as normal and the other with only the prefix. The length of the prefixes were randomly sampled from [0, 1]. We found that this additional data augmentation was particularly helpful; without it, the models would hallucinate the rest of a partial sentence. We further found that starting the prefix sampling data augmentation too late in training was also negative. After testing initial models on the dev set, we found that starting this additional augmentation 15 epochs after training was best. ## Appendix D Utterance Segmentation We follow the audio segments provided in the MuST-C corpus, created through a use of human alignment and XNMT Neubig et al. (2018). We note that there exist a variety of methods for creating segments for such models, however, we leave additional exploration of E2E alignment methods as future work.
# Virtual laser scanning with HELIOS++: A novel take on ray tracing-based simulation of topographic 3D laser scanning Lukas Winiwarter<EMAIL_ADDRESS>Alberto Manuel Esmorís Pena Hannah Weiser Katharina Anders Jorge Martínez Sanchez Mark Searle Bernhard Höfle<EMAIL_ADDRESS>3DGeo Research Group, Institute of Geography, Heidelberg University, Germany Centro Singular de Investigación en Tecnoloxías Intelixentes, CiTIUS, USC, Spain Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University, Germany ###### Abstract Topographic laser scanning is a remote sensing method to create detailed 3D point cloud representations of the Earth’s surface. Since data acquisition is expensive, simulations can complement real data given certain premises are available: i) a model of 3D scene and scanner, ii) a model of the beam-scene interaction, simplified to a computationally feasible while physically realistic level, and iii) an application for which simulated data is fit for use. A number of laser scanning simulators for different purposes exist, which we enrich by presenting HELIOS++. HELIOS++ is an open-source simulation framework for terrestrial static, mobile, UAV-based and airborne laser scanning implemented in C++. The HELIOS++ concept provides a flexible solution for the trade-off between physical accuracy (realism) and computational complexity (runtime, memory footprint), as well as ease of use and of configuration. Unique features of HELIOS++ include the availability of Python bindings (`pyhelios`) for controlling simulations, and a range of model types for 3D scene representation. Such model types include meshes, digital terrain models, point clouds and partially transmissive voxels, which are especially useful in laser scanning simulations of vegetation. In a scene, object models of different types can be combined, so that representations spanning multiple spatial scales in different resolutions and levels of detail are possible. We aim for a modular design, where the core components of platform, scene, and scanner can be individually interchanged, and easily configured by working with XML files and Python bindings. Virtually scanned point clouds may be used for a broad range of applications. Our literature review of publications employing virtual laser scanning revealed the four categories of use cases prevailing at present: data acquisition planning, method evaluation, method training and sensing experimentation. To enable direct interaction with 3D point cloud processing and GIS software, we support standard data formats for input models (Wavefront Objects, GeoTIFFs, ASCII xyz point clouds) and output point clouds (LAS/LAZ and ASCII). HELIOS++ further allows the simulation of beam divergence using a subsampling strategy, and is able to create full- waveform outputs as a basis for detailed analysis. As generation and analysis of waveforms can strongly impact runtimes, the user may set the level of detail for the subsampling, or optionally disable full-waveform output altogether. A detailed assessment of computational considerations and a comparison of HELIOS++ to its predecessor, HELIOS, reveal reduced runtimes by up to 83 %. At the same time, memory requirements are reduced by up to 94 %, allowing for much larger (i.e. more complex) 3D scenes to be loaded into memory and hence to be virtually acquired by laser scanning simulation. ###### keywords: software , LiDAR simulation , point cloud , data generation , voxel , vegetation modelling , diffuse media ††journal: Remote Sensing of Environment ## 1 Introduction Simulation of physical processes is often carried out when experiments are not feasible or simply impossible, or to find parameters that produce a certain outcome if inversion is non-trivial. In virtual laser scanning (VLS), simulations of LiDAR (Light Detection and Ranging) create 3D point clouds from models of scenes, platforms, and scanners (Figure 1), that aim to recreate real-world scenarios of laser scanning acquisitions. Such simulated point clouds may, for certain use cases, replace real data, and may even allow for analyses where real data capture is not feasible, e.g. due to technical, economical or logistic constraints, or when simulating hardware which is not yet existing. However, there are use cases where VLS is not appropriate, for example, when analysing effects only partially modelled in the simulation such as penetration of the laser into opaque objects. In a similar argument, (passive) photogrammetry is inadequate for the reconstruction of a non- textured flat area, but still a useful method for many other tasks. Therefore, VLS can be seen as a tool to acquire 3D geospatial data under certain _premises_. These include: 1. 1. an adequate model of the 3D scene and the scanner, as well as the platform behaviour, 2. 2. a simplification of the real-world beam-scene interactions to a computationally feasible and physically realistic level, and finally, 3. 3. an application for which VLS data is fit for use. VLS can easily and cheaply produce large amounts of data with very well defined properties and known ground truth. Parameters (e.g. tree attributes such as crown base height) can be extracted from the scene model (e.g. a mesh object that is being scanned) automatically and without errors, and these parameters can in turn be used for training or validation of algorithms that attempt to extract them from point cloud data. Due to the low cost compared to real data acquisitions, VLS can be combined with Monte-Carlo simulations to solve non-continuous optimisation problems on scan settings and acquisition strategies. In a research workflow, VLS experiments may be employed to identify promising candidate settings before carrying out a selected number of real experiments that are used to answer the respective research questions. Figure 1: Schematic concept of HELIOS++, showcasing platforms (boxed labels) and object models composing a scene (non-boxed labels). A variety of model types to represent 3D scenes are supported: terrain models, voxel models (custom .vox format or XYZ point clouds) and mesh models. For platforms, four options are currently supported: airplane, multicopter, ground vehicle and static tripod. A schematic diverging laser beam and its corresponding waveform (magenta) is shown being emitted from the airplane and interacting with a mesh model tree and the rasterised ground surface. In this paper, we present a novel take on ray tracing-based VLS covering the given premises (1-3). This is implemented as the open source software package HELIOS++ (Heidelberg LiDAR Operations Simulator ++)111HELIOS++ is available on GitHub (https://github.com/3dgeo-heidelberg/helios) and is licensed under both GNU GPL and GNU LGPL. HELIOS++ is also indexed with Zenodo (Winiwarter et al., 2021).. HELIOS++ is the successor of HELIOS (Bechtold and Höfle, 2016), with a completely new code base, implemented in C++ (whereas the former version was implemented in Java). HELIOS++ improves over HELIOS in terms of memory footprint and runtime, and also in functionality, correctness of implemented algorithms, and usability, making it versatile and highly performant to end users. We first motivate the need for a novel VLS framework by a survey of previous methodologies to laser scanning simulation and their applications, and point out the unique features of HELIOS++ (Section 2). In Section 3, we present the architecture and design considerations of HELIOS++. We then show different types of applications and conduct a systematic literature survey of uses of HELIOS in Section 4. Technical considerations concerning the handling of big 3D scenes and ray tracing are dealt with in Section 5 and conclusions are drawn in Section 6. ## 2 Existing implementations and state of the art virtual laser scanning Simulating a process within a system always involves a simplified substitute of reality. The complexity of this substitute depends on the process understanding, on computational considerations and on the specific problem that is to be solved. Approaches with different levels of simulation complexity exist regarding i) the type of input scene model (e.g. 2.5D digital elevation models (DEMs) vs. 3D meshes) and ii) how the interaction of beam and object is modeled (e.g. single ray/echo vs. full-waveform). An overview of publications of these approaches is listed in Table 1. Publication \| Platforms \| \| Beam --- div. FWF \| Scene \| Comments North (1996), North et al. (2010) \| satellite \| ✓ \| ✓ \| 3D \| FLIGHT model Lewis (1999) \| ALS \| ✓ \| ✓ \| 3D \| \| used by --- Calders et al. (2013) & Disney et al. (2010) Tulldahl and Steinvall (1999) \| ALS \| ✓ \| ✓ \| 3D \| for bathymetry Ranson and Sun (2000) \| \| ALS --- (nadir) ✓ \| ✓ \| 3D \| Holmgren et al. (2003) \| ALS \| \| \| 3D \| Goodwin et al. (2007) \| ALS \| \| \| 3D \| LITE model Lohani and Mishra (2007) \| ALS \| \| \| 2.5D \| Morsdorf et al. (2007) \| ALS \| ✓ \| ✓ \| 3D \| using POVray Kim et al. (2009) \| ALS \| \| \| 3D \| Kukko and Hyyppä (2009) \| ALS, MLS \| ✓ \| ✓ \| 2.5D \| Hodge (2010) \| TLS \| ✓ \| ✓ \| 2.5D \| Kim et al. (2012) \| ALS \| ✓ \| ✓ \| 3D \| Wang et al. (2013) \| TLS \| \| \| 3D \| Gastellu-Etchegorry et al. (2015) \| \| satellite, --- ALS, TLS ✓ \| ✓ \| 3D \| DART model Bechtold and Höfle (2016) \| \| ALS, TLS, --- MLS, ULS ✓ \| ✓ \| 3D \| HELIOS Table 1: Overview of virtual laser scanning simulators and associated publications. For each simulator, a check mark (✓) is added if they support simulation of finite (non-zero) beam divergence (”Beam div.”) and full waveforms (”FWF”). Scene representation may be in full 3D or 2.5D, i.e. raster-based. A simple airborne laser scanning (ALS) simulator is presented by Lohani and Mishra (2007) for use in research and education. Their tool comes with a user- friendly graphical user interface (GUI) and allows selecting different scanner and trajectory configurations. This simulator models the laser ray as an infinitesimal beam with zero divergence to simplify the ray tracing procedure. The scene is represented by a 2.5D elevation raster, which allows only a simplified representation of the Earth’s surface. In a different application, the interactions between laser beams and forest canopies are investigated by Goodwin et al. (2007), Holmgren et al. (2003) and Lovell et al. (2005). They present approaches of combining 3D forest modelling and ALS simulation using ray tracing. As in Lohani and Mishra (2007), their simulators all model the laser beam as an infinite straight line, which intersects the scene in one distinct point. Full-waveform laser scanners can record the full waveform of the backscattered signal, providing information about the objects in a scene that are illuminated by the conic beam. This waveform from a finite footprint was specifically simulated by Ranson and Sun (2000) in the forestry context and by Tulldahl and Steinvall (1999) for airborne LiDAR bathymetry. Morsdorf et al. (2007) use the ray tracing software POV-Ray222http://www.povray.org/ to model the waveform of laser scans of a 3D tree model from a combination of intensity and depth images. Kukko and Hyyppä (2009) aim for a more complete and universal LiDAR simulator that considers platform and beam orientation, pulse transmission depending on power distribution and laser beam divergence, beam interaction with the scene, and full-waveform recording. Their approach models the physics involved in LiDAR measurements in high detail, and is demonstrated for a use case in forestry. Similar to the work by Lohani and Mishra (2007), they use 2.5D elevation maps to represent the scene. This makes the simulator useful for airborne simulations over terrain or building models, where the scene is scanned from above and scene elements are assumed to be solid and opaque. However, it is less suited for penetrable 3D objects such as vegetation or overhanging geometries, especially for the simulation of ground-based acquisitions which are less in a bird’s-eye view perspective. Kim et al. (2012) present a similarly detailed simulator, which includes radiometric simulation and recording of the waveform and includes explicit 3D object representations. It is unclear if it also supports static platforms such as TLS. TLS simulations are the sole focus of some studies for specific applications, such as leaf area index inversion (Wang et al., 2013) or TLS measurement error quantification (Hodge, 2010). While Wang et al. (2013) use a more simple model assuming no beam divergence, the simulation described by Hodge (2010) includes both the modelling of beam divergence and recording of the waveform. Their simulation again uses 2.5D elevation models to represent the scene, which is appropriate for their particular objective of error quantification in high- resolution, short-range TLS of natural surfaces, specifically fluvial sediment deposits, of small scenes (area of $1\text{\times}1\text{\,}\mathrm{m}$). Established Monte-Carlo ray tracing simulator are used and being extended for airborne and satellite laser scanning simulations. The librat model (Calders et al., 2013; Disney et al., 2009, 2010), a modular development of ARARAT (Lewis and Muller, 1993) is such a Monte-Carlo simulator. Similarly, North et al. (2010) extend the 3D radiative transfer model FLIGHT (North, 1996) to model satellite LiDAR waveforms. Monte-Carlo methods represent a simple, robust and versatile set of techniques to solve multi-dimensional problems by repeatedly sampling from a probability density function describing the system that is investigated (Disney et al., 2000). These stochastic methods are useful for simulating multi-scattering processes, e.g. for modeling canopy reflectance. The main drawback of Monte-Carlo ray tracing methods are high computation times to simulate sufficient photons for the scattering model to converge to an accurate solution (Disney et al., 2000; Gastellu-Etchegorry et al., 2016). The LiDAR extension of the Discrete Anisotropic Radiative Transfer (DART) model attempts to alleviate these restrictions by quickly selecting scattering directions of simulated photons using the so-called Box method and modeling their propagation and interaction using a Ray Carlo method, which combines classical Monte-Carlo and ray tracing methods (Gastellu-Etchegorry et al., 2016). A comprehensive review of simulators for the generation of point cloud data, including LiDAR simulators, is presented in Schlager et al. (2020) with focus on their applicability to generate data in the context of driver assistance systems and autonomous driving vehicles. In this context, they analyse algorithms with respect to their fidelity, operating principles, considered effects and possible improvements. In contrast to most of the previously mentioned approaches, HELIOS++ provides a framework for full 3D laser scanning simulation with multiple platforms (terrestrial (TLS), mobile (MLS), UAV-borne (ULS) and airborne (ALS)), and a flexible system to represent scenes, which allows combination of input data from multiple sources and data formats (Figure 1). The simulation of beam divergence and full waveform recording are supported. While HELIOS++ may not be as realistic in terms of physical accuracy regarding the energy budget of a single laser shot as, e.g., DART, it provides a sensible trade-off between computational efforts and resulting point cloud quality. Users can simulate VLS over a large range of scales, and even combine different scales in one scene. For example, a highly detailed tree model with individual leaves might be placed in a forest scene represented by (transmissive) voxels (Weiser et al., 2021), while using a rasterised digital terrain model as ground surface. This allows to model the influence of the surrounding of a particular object of interest on the derived VLS point clouds. Furthermore, HELIOS++ aims for high usability by providing a comprehensible set of parameters, while not overwhelming the users with options. These parameters represent the state of the art and are supported by peer-reviewed literature. Since HELIOS++ can be used from the command line and from within Python, workflows integrating HELIOS++ can be easily scripted and automated, as well as linked to external software (e.g. GIS, 3D point cloud processing software, Jupyter Notebooks, and others). HELIOS++ comes with an extensive documentation of all algorithms that are used in the simulations, including examples and references to the relevant literature describing the implemented methods. Furthermore, the open source implementation allows any user to i) inspect and ii) alter/adapt the source code of the program. For ease of use, we provide pre-compiled versions that are ready to use for major operating systems (Microsoft Windows 10 and Debian Linux 10.7 Buster), and the option to use Python as a scripting language to create, manipulate and simulate VLS data acquisition with HELIOS++. ## 3 Implementation of HELIOS++ This section introduces the concepts and interfaces of HELIOS++, for which the important components are the overall architecture and modules (Section 3.1), platforms (Section 3.2), scanners and laser beam deflectors (Section 3.3), the waveform simulation (Section 3.4), input formats (Section 3.5) and output formats (Section 3.6), and the aspect of randomness and repeatability of results (Section 3.7). ### 3.1 Architecture and modules The central element in HELIOS++ simulations is a _survey_. A survey contains links to the _scene_ , which defines the objects that are scanned, the _platform_ , on which the virtual scanner is mounted and moved through the scene, and to the _scanner_ itself. Furthermore, a survey contains a number of _legs_ , which represent waypoints for the platform. The scene consists of a number of _parts_. Each part represents one input source, for example, a 3D mesh file, or a voxel file. Multiple parts may be combined in a scene, and no limitation to the combination of different data source types is imposed. HELIOS++ uses internationally accepted standard file formats for input and output. These elements are defined through Extensible Markup Language (XML) files, that are referenced using (relative) file paths. XML is a text-based format, which can easily be manipulated using a text editor or an XML editor. Figure 2 presents the different files along with a subset of the parameters that can be set in the respective files. Figure 2: File structure of HELIOS++ survey, scene, platform, and scanner. A survey consists of one or more legs, and a single scanner, platform, and scene, respectively. A scene is built up from one or more parts, which can be of different data type. A survey may then be run through either i) the command line, where an executable is provided or ii) the Python bindings `pyhelios`. The Python bindings allow access to the simulation parameters as parsed from the XML files, including changing the parameters programmatically for each simulation run. For example, different scanners can be exchanged automatically and simulated in sequence in a single Python script without changing any other input and settings of the simulation. `pyhelios` furthermore allows access to the simulation result, i.e. the point cloud and the platform trajectory, and converts them into a NumPy array either at the end of the simulation run or through a callback function that can be executed every $n$-th cast laser ray. In this way, a live preview of the point cloud acquisition can be implemented in Python to give a visual impression of the ongoing simulation. A Python script distributed with HELIOS++ acts as such a visualiser, and is called with the same commands as the standalone executable. A user may thus quickly switch between using the pure C++ implementation without visualisation or the Python bindings with visualisation, as presented in Listing 1. Listing 1: Comparison between running HELIOS++ as an executable and through the Python wrapper providing an interactive viewer ⬇ 1run\helios.exe data\surveys\arbaro_demo.xml \--lasOutput \--writeWaveform 2python pyhelios\helios.py data\surveys\arbaro_demo.xml \--lasOutput \--writeWaveform With `pyhelios`, it is also possible to combine HELIOS++ with tools like _Jupyter Notebooks_ , allowing for explanations along-side code and figures. We include sample notebooks in the documentation of HELIOS++. One of these samples is shown in Figure 3. Figure 3: Screenshot of a Jupyter Notebook showcasing the Python bindings of HELIOS++ by plotting the trajectory of a simulation over flat terrain. ### 3.2 Supported laser scanning platforms Both static and dynamic platforms are supported by HELIOS++, which can resemble an airplane (`LinearPath` platform), a multicopter (`Multicopter` platform), a ground-based vehicle (`GroundVehicle` platform) or a static tripod (`Static` platform). In the case of the `LinearPath` platform, the vehicle is moved with a constant speed from one waypoint (leg) to the next one. The orientation of the platform is always towards the next waypoint. The `Multicopter` platform additionally simulates acceleration and deceleration of the platform. In the turn mode _smooth_ , the platform banks to make more smooth turns at the waypoints instead of stopping and turning on the spot. This mode simulates the _banked angle turns_ available in the flight protocols of major drone companies (e.g. DJI). Custom yaw angles for the beginning and the end of each leg can be provided. The `GroundVehicle` platform is bound to elements of the scene defined as ground in the respective material file (cf. Section 3.5). Furthermore, it considers maximum turn radii and implements three-point-turns to resemble a car or tractor on which a scanner is mounted (i.e. MLS). The different platforms and scene types (Section 3.5) are shown in Figure 1. As expected, choice of platform influences the resulting pointcloud, which is shown in Figure 4. Figure 4: Point clouds resulting from virtual laser scanning of the scene shown in Figure 1, using (a) an airplane, (b) a multicopter, (c) a ground vehicle and (d) a static tripod as platform. Since HELIOS++ records which objects are generating which return, the points can be perfectly assigned to the objects, here illustrated by distinct colouring. ### 3.3 Supported laser scanners and scan deflectors The core component of the simulation is the laser scanner, which includes a model for the scan deflector. The choice of the deflector influences the resulting scan pattern (Fig. 5). Figure 5: Different scan patterns depending on the deflector used in the simulation: a) rotating mirror, b) fibre-optic line scanner, c) oscillating mirror and d) slanted rotating mirror (Palmer scanner). The patterns shown here result from simulations with HELIOS++ using the respective deflectors, albeit with unrealistic settings of pulse repetition rate and scanning frequency, in order to show the patterns more clearly. HELIOS++ supports polygonal mirror deflection, resulting in parallel scan lines with even point density (Fig. 5a), fibre-optics, where the beam is fed through fibreoptic cables to point in different directions, resulting in a similar scan pattern to (a), but without the need of mechanically moving parts (Fig. 5b), a swinging mirror, which results in a zig-zag-pattern with increased point densities at the extrema (Fig. 5c) and rotating slanted mirrors, resulting in a conical point pattern at a constant scan angle off- nadir, also referred to as Palmer scanner (Fig. 5d). ### 3.4 Waveform simulation During real LiDAR acquisitions, a laser beam sent out from the laser scanner has a finite footprint, i.e. a non-zero area that intersects with the scene. For every infinitesimal point in this area, energy is transmitted back to the detector where it is recorded as the integral of intensities over the area. By considering this received intensity over time, it is possible to extract multiple echoes from one laser pulse, corresponding to multiple targets that were hit by parts of the intersection area, respectively. In HELIOS++, the non-zero beam divergence is simulated by subrays, that are sampled in a regular pattern around the central ray (Fig. 6). Every subray has its own base intensity, which is calculated according to Equation 1, representing a 2D Gaussian power distribution (Carlsson et al., 2001). $I=I_{0}\exp\left(-2r^{2}/w^{2}\right)$ (1) where $I_{0}$ [$\mathrm{W}$] is the peak power, $w$ [$\mathrm{m}$] the local beam divergence and $r$ [$\mathrm{m}$] the radial distance from the power maximum, i.e. the ray centre. $w$ is calculated using Equation 2 from the beam waist radius $w_{0}$ [$\mathrm{m}$], and the helper values $\omega=\frac{\lambda R}{\pi w_{0}^{2}}$ and $\omega_{0}=\frac{\lambda R_{0}}{\pi w_{0}^{2}}$ with $\lambda$ [$\mathrm{nm}$] as the wavelength, $R$ [$\mathrm{m}$] the range of the target and $R_{0}$ [$\mathrm{m}$] the focusing length of the laser. $w=w_{0}\sqrt{\omega_{0}^{2}+\omega^{2}}$ (2) Every subray is individually cast into the scene and intersected with objects. If it hits an object, a return is generated and recorded by the detector. The respective subray does not continue in the scene through transmission or reflection. In the case of the transmissive voxel model (cf. Section 3.5), a subray may either fully traverse a voxel or produce a return, but never both. Therefore, the property of transmissivity of a scene part is coupled to the use of multiple subrays. The power returned from the object is further dependent on the material specified in the scene definition (Section 3.5). The number of subrays generated can be set by the user by providing the `beamSampleQuality` parameter in the XML file of the scanner or the survey. The `beamSampleQuality` corresponds to the number of concentric circles where subrays are sampled. For each circle, the number of subrays is defined as $\lfloor 2\pi i\rfloor$ where $i=1,\dots,$`beamSampleQuality` is the circle index. This ensures that the angular distance between adjacent subrays is approximately constant, i.e. each subray represents a solid angle of equal size. A central subray is always added. Figure 6 shows the subray distribution for three different beam sample qualities. Figure 6: Subray configurations for beam sample qualities of (a) 2 (7 subrays), (b) 5 (93 subrays), and (c) 9 (279 subrays). The colour of the subrays corresponds to the normalised intensity at this location within the beam cone: low (purple) to high (yellow). The black circle represents the single beam divergence (at the $1/e^{2}$ points, here: $0.3\text{\,}\mathrm{mrad}$), the gray circle twice the beam divergence. The pulse shape in time is approximated using bins of a regular, user-defined size via the parameter `binWidth_ns`. Each bin’s power is calculated according to Equation 3, where $t$ [$\mathrm{ns}$] is the time and $\tau$ [$\mathrm{ns}$] is the pulse length of the scanner divided by 1.75 (Carlsson et al., 2001). $I$ [$\mathrm{W}$] is calculated for each subray according to Equation 1. $P(t)=I\left(\frac{t}{\tau}\right)^{2}\exp\left(-\frac{t}{\tau}\right)$ (3) For every pulse, the subrays are collected as a representation of the full returned waveform, and the recorded power for each bin is taken as the sum of the subray’s waveforms, shifted according to the different ranges of the subrays. Range difference is converted to time difference by using the speed of light as a constant ($c=\;$$299,792,458\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$). A local maximum filter is then used on the summed waveform to detect peaks, which are regarded as echoes and exported as points. Optionally, a Gaussian may be fitted to the resulting waveform, to get a measure of echo width (i.e. standard deviation of the Gaussian). The position of the point along the range (time) axis, however, is taken from the local maximum, not from the fitted Gaussian, as the outgoing waveform (Eq. 3) is not Gaussian. ### 3.5 Input scene models HELIOS++ supports data formats that are (de-facto) standards in the field, and can be created and manipulated with free software such as QGIS333https://qgis.org/en/site/, CloudCompare444https://www.danielgm.net/cc/, Blender555https://www.blender.org/ or AmapVOX666Vincent et al. (2017), http://amap-dev.cirad.fr/projects/amapvox. #### 3.5.1 Wavefront objects In wavefront object files (file extension .obj), meshes are represented as lists of triangles (triples of point IDs) and points (triples of coordinates). This allows opaque 3D models of arbitrary complexity. In addition, material properties can be assigned to the objects in so-called material files (file extension .mtl; Wavefront Technologies, 1992). #### 3.5.2 GeoTIFF models Raster models created with common GIS tools, such as a digital elevation or a digital surface model, can be included. The raster is converted to a triangular mesh on import, where the pixel centres are interpreted as points. Invalid pixels (i.e. no-data values) are ignored in the triangulation, resulting in holes in the mesh. This data type allows an simple representation of the Earth’s surface. As prescribed by the file format, only 2.5D-data is supported. #### 3.5.3 Point clouds In ASCII xyz files, point clouds can be used as a raw input to HELIOS++. On import, the point clouds are voxelised using a voxel size defined by the user. In addition, a normal vector used for intensity calculation can be assigned to the voxels (by nearest neighbour or mean reducing) or calculated from the point cloud on the spot (cf. Section 5). Alternatively, all laser rays can be set to have an incidence angle of 0°, or the rays are intersected with the actual faces of the cube representing the voxel. Material properties can be defined in the respective scene part definition of the XML file. #### 3.5.4 Transmissive voxels Especially for modelling vegetation, we include support for voxel data containing plant area density information (e.g. created using the AmapVOX software (Vincent et al., 2017), file extension .vox). Multiple modes are supported in this case: * 1. opaque voxels, where the voxels are represented by solid cubes with a fixed side length equal to voxel resolution, * 2. adaptive scaling of opaque voxel cubes (with optional random but reproducible shift to avoid regular patterns in the result), where the scaling (side length of a voxel $a$) is dependent on the plant area density $PAD$, the user-defined scaling factor $\alpha$, the base voxel size $a_{0}$ and the maximum $PAD$, related by Equation 4, and * 3. transmissive voxels (explained in more detail in the following). Point clouds resulting from a tree simulation using these different modes are shown in Figure 7. The opaque voxel models also support the definition of material properties through the scene part’s definition. A comprehensive study on the effects of different levels of detail in modelling forests using the first two modes on the extraction of forestry and point cloud parameters is performed by Weiser et al. (2021). $a=a_{0}\left(\frac{PAD}{PAD_{max}}\right)^{\alpha}$ (4) Here, $a$ is the side length of the resulting cube, $a_{0}$ the base voxel size, $PAD$ the plant area density of each individual voxel, $PAD_{max}$ a maximum value for the plant area density and $\alpha$ a user-defined scaling factor. Figure 7: VLS point cloud of a tree based on different voxel modes, using a UAV as a platform. Point clouds are simulated using input voxel models with different fixed sizes (a-c), PAD-dependent scaled voxels (d and e) and transmissive voxels (f). The transmissive voxels use an extinction approach as presented by North (1996). The extinction coefficient $\sigma$ is calculated following Equation 5: $\sigma=\frac{\mu_{L}}{2\pi}\int_{0}^{2\pi}g_{L}\left\|\Omega^{\prime}\cdot\Omega_{L}\right\|d\Omega_{L}$ (5) Here, $\mu_{L}$ is the plant area density (PAD) as defined in the .vox file, $g_{L}$ is the probability taken from the leaf angle distribution at direction L, and $\left\|\Omega^{\prime}\cdot\Omega_{L}\right\|$ is the cosine of the angle between the incident ray $\Omega^{\prime}$ and the direction $\Omega_{L}$. This leaf angle distribution is supplied via a look-up-table as shown in Table 2, and examples for planophile, erectophile, plagiophile, extremophile, spherical and uniform distributions are provided. Subsequently, a random number $R$ is drawn from a uniform distribution $\in[0,1)$. The expectation of the intersection of a subray with a voxel given the extinction coefficient $\sigma$ is then given as in Equation 6, where $s$ is the distance of an echo after entering the voxel (i.e. traversed distance). $s=\frac{-\ln(R)}{\sigma}$ (6) If the actual length of the subray traversing the voxel is larger than $s$, no echo is recorded for this ray in the voxel, and it is assumed to have transmitted through the voxel, potentially intersecting the next voxel or another object. Otherwise, an echo is created at distance $s$ from where the subray first entered the respective voxel, and the subray is not continued through the scene (North, 1996). Irrespective of the input type, each scene part can be individually scaled, rotated and translated within the scene, allowing the re-use of models to create multiple instances of similar objects. These transformations can also be manipulated using the Python bindings. For rotations, both extrinsic (”global”, fixed coordinate axes) and intrinsic (”local”, rotating coordinate axes) modes are supported. HELIOS++ allows arbitrary combinations of all possible object models in a scene. For example, a GeoTIFF can be used as digital terrain model on which a vehicle is moving, combined with transmissive voxel models of trees, mesh models for tree stems and a point cloud of buildings, which is voxelised by HELIOS++ (Figure 1). Such combination allows highly versatile use of the scene definition, while considering shadowing and overlapping effects between the different types of object models, and is one of the key advantages over other VLS software. ### 3.6 Output format and options The simulated point clouds can be written either as LAS file (Version 1.0, point format 1), according to the format definition by the American Society of Photogrammetry and Remote Sensing (ASPRS, 2011), as LAZ file, which is a lossless compression of LAS provided by the LASzip library777https://laszip.org/, Isenburg (2013), or as ASCII file, where the point coordinates and attributes are written in columns. The LAS/LAZ formats allow for smaller file sizes and faster read/write access (cf. Section 5.4), while the ASCII format can be parsed by any program working with text files. In addition to creating compressed LAS files, HELIOS++ can also compress output ASCII files, if smaller output sizes are required. Uncompressing LAZ or compressed ASCII files is possible by invoking HELIOS++ using the `--unzip` flag. In the simulation of full waveforms (see Section 3.4), the echo width is optionally estimated for every recorded return. Since this estimation uses a non-linear least squares method and is hence expensive to calculate, the estimation has to be switched on explicitly by the user. In typical use cases, this increases the runtime of the simulation by a factor of 1.3. If the full waveform is to be exported, a separate ASCII file will be written. This file contains information about every beam that generated at least one hit consisting in the beam origin and the beam direction, as well as the sampled returned waveform. The bin size as well as the maximum length of this sampled waveform can be defined by the user (Section 3.4). To connect the point cloud output with the full waveform output, the point cloud has an attribute `fullwaveIndex`, which corresponds to the `fullwaveIndex` in the ASCII waveform output. Multiple points, if coming from the same beam, may have the same `fullwaveIndex`. The output created by HELIOS++ can be easily converted to standard waveform formats such as PulseWaves and LAS 1.4 WDP. Furthermore, HELIOS++ comes with an option to export the trajectory of the platform. The time interval between successive platform positions can be defined by the user in the survey XML file. In addition to the time and the position of the platform, the attitude angles of roll, pitch and yaw are exported. ### 3.7 Randomness and repeatability The trajectory of the platform, the scanner definitions and the ray-scene interactions with the exception of transmissive voxels are deterministic. To allow for more realistic point clouds, random noise sources may be introduced at various points of the simulation. A single distance measurement is attributed with a random ranging error, drawn from a normal distribution with parameters defined for each scanner. Additionally, random platform noise can be added to simulate trajectory estimation or scan position localisation errors. By default, the system time is used to generate randomness, which results in different outcomes for every simulation run. However, the ability to produce the identical results in repeated simulation runs is a major advantage of VLS over real data acquisitions and may be aspired for certain use cases. To allow for repeatable survey results while using randomness, there are additional options to define a custom seed for pseudo-randomness generation. Still, when using multithreading, each thread accesses the randomness generator in a non-deterministic order, resulting in different outputs for every simulation run. This can be avoided by running HELIOS++ in single-threaded mode and with a fixed seed. This will generate the exact same result in repeated runs, even for transmissive voxels, as the random number drawn from the uniform distribution is also created based on the seed. ## 4 Applications of laser scanning simulation To investigate the usability of the HELIOS++ concept of providing a sensible trade-off between physical reality and computational complexity, along with the support of generic input files to create the 3D scene, we present applications of laser scanning simulation. This literature survey is based on studies that use HELIOS++’s predecessor, HELIOS (Bechtold and Höfle, 2016) or pre-release versions of HELIOS++. Since the functionality of HELIOS, apart from the visualisation module, is fully integrated in HELIOS++, the publications presented in the following provide an adequate review of simulation applications. The analysis is conducted on all publications that cite the original publication of Bechtold and Höfle (2016), and that actively use HELIOS in their research according to the published article. It consists of journal papers, conference contributions and posters. The following questions are posed to analyse the contents of each publication. * 1. What is the scientific target of the simulation? * 2. How many simulations are carried out on which platform (ALS, TLS, ULS, MLS)? * 3. Which type of model is employed to compose the scene for the simulation? * 4. Which parameters are extracted from the point cloud, if applicable? * 5. What is the resulting VLS point cloud compared to? The synthesis of the literature review allows a grouping of the publications and of the purposes of laser scanning simulation into four main categories: (1) optimising or analysing different scan settings and acquisition modes, (2) comparing parameters extracted from simulated data to error-free ground truth, (3) generating training data for supervised machine learning, and (4) testing and development of novel or future algorithms and sensors. ### 4.1 Data acquisition planning and scan setting effects Data acquisition planning at its core is an optimisation problem. The goal is to acquire data that is fit for the purpose of the analysis by minimal effort. This can be a minimum number of scan positions, flight lines, etc., that is sufficient for the specific requirement to the data, such as coverage of the scene regarding occlusion or a certain point density. While flight planning tools and visibility analysis-based methods can be used to create potential acquisition plans, they usually lack methods to verify the fitness for use of obtained data. With virtual laser scanning, assuming that at least a coarse model of the area of interest is available, a 3D point cloud can be generated and tested for its fitness directly in the planned application. There is no need to define proxy metrics like target point density, required accuracy and overlap as required by simple survey planning tools. This way, users can ensure that the acquired data will meet all requirements such as coverage, adequate representation of geometry, and resolution before actually going to the field to collect the data, by running their analyses on the simulated point clouds and interpreting the results. Similarly, the effects of different scan settings on parameters extracted from the simulated point clouds can be studied with HELIOS++, as single variables can easily be manipulated in a way that is isolated from all other influences, including environmental influences, which are very difficult to control in repeated real-world acquisitions. For example, the effects of flying height and maximum scan angle on the resulting ground point density and resolution (i.e. illuminated area per beam) may be analysed. Similarly, TLS scans of different resolution can be simulated and the effect of resolution on the results of data analysis (e.g. the quality of extraction of tree stems) can be quantified. While the quality of the simulation in terms of being physically realistic highly depends on the input models, even a coarse model can be useful to estimate occlusion and resulting point densities. Such analyses, carried out prior to real data acquisitions, can save valuable time in the field. Using a Monte-Carlo approach, multiple simulations using different parameters are carried out to create the optimal (in terms of number of positions, time, etc.) acquisition plan. Existing publications in this category include Backes et al. (2020), who use a digital surface model created from photogrammetry to simulate acquisition of an alpine valley by TLS and ULS. They estimate the minimal detectable change, to optimise scan positions and trajectories for change analysis. Similar analyses, but with focus on resulting point density and completeness of data acquisition, are carried out based on a DEM provided by public agencies by Lin and Wang (2019). They are able to show that a downsampled scene model can provide accurate measures for simulated point densities when also reducing the pulse frequency. This validates the simulation’s representation of spatial scales. A validation of scan position planning based on viewshed analysis is carried out with respect to achieved accuracy, point density and completeness by Previtali et al. (2019). Their use cases are acquisitions of complex archaeological sites, where complete coverage is often difficult to achieve due to occlusions. In two examples, they optimise the positions of 97 and 16 scan positions, respectively, to scan a basilica and part of an ancient food storage in Italy. The effect of scan settings on point cloud-based parameters is analysed for forestry settings by Hämmerle et al. (2017), who extract understory tree heights from TLS and ULS data. They use 3D models created with the Arbaro tree generator888http://arbaro.sourceforge.net/ (Weber and Penn, 1995), on which they densely sample points to create a reference point cloud as ground truth. They find a favourable trade-off between acquisition effort and accuracy of results for a number of three TLS scan positions around a tree object of interest (Hämmerle et al., 2017). Similarly, Li et al. (2020) create 3D tree models using ONYXTREE999http://www.onyxtree.com/ and evaluate the influence of scan parameters on extracted values of diameter at breast height (DBH), tree height, stem curve and crown volume. They succeed in reducing the root-mean- squared error on these values by iteratively adapting scan parameters. Weiser et al. (2021) analyse the different opaque voxel models at different scales and their effect on common tree metrics, showing that a scaled voxel model requires much less complexity (i.e., allows for a larger voxel size) than binary voxels at a fixed scale. Schäfer et al. (2019) use the novel transmissive voxel-support of HELIOS++ for similar analyses with ALS and ULS data. ### 4.2 Algorithm and method evaluation: validation and calibration In numerous point cloud analysis methods, the objective is to extract certain parameters describing the objects of interest. For example, in forestry applications, ALS and TLS point clouds are commonly used to derive the diameter at breast height of trees, crown radii, tree heights, and tree species (Giannetti et al., 2018). To calibrate the extraction algorithms as well as to validate the results, in-situ measurements are required, which are laborious and costly to acquire, and not free of error. An alternative to this in-situ data acquisition can be provided by HELIOS++, if the parameters to be extracted from the point cloud can be derived from the objects within the scene, or the scene is created dynamically according to the parameters. For example, a method may be designed to measure DBH from a point cloud. In VLS, the true values of DBH can be derived from the stem models, or the virtual tree models themselves may be generated according to given values of DBH. In addition to being error-free, the domain of the parameters (e.g. the range of DBH values) used in the simulation can be defined by the user. A simulated example can therefore be picked to have exactly the properties needed in a specific application (say, use case-specific DBH values between $20\text{\,}\mathrm{cm}25\text{\,}\mathrm{cm}$), whereas real objects with the required properties as ground truth may be hard to find. In addition to parameter extraction, perfect ground truth can be used to evaluate the performance of classification algorithms. Currently, classification methods are difficult to compare, as authors use different evaluation approaches, mainly due to the lack of semantic reference data. Since in VLS the interaction between the laser beam and the scene can be attributed to the exact mesh face that is hit during ray casting, and therefore to a specific object, the ground truth data is free of error even in highly complex and multi-echo scenarios. This has compelling advantages: First, reference data can be created automatically, drastically cutting costs and providing the possibility of generating massive amounts. And second, it removes labelling errors, as manual labelling will always include some degree of human error. Considerable research using HELIOS to validate or calibrate extracted parameters has been undertaken in multiple forestry applications. Liu et al. (2019a) and Liu et al. (2019b) estimated leaf angles on trees, where ground truth is practically impossible to obtain, because in real settings, leaves are continuously moving due to wind. VLS allows to validate their approach of leaf angle estimation and subsequently apply it to real data. Wang et al. (2020) validate their calculation of photon recollision probability over spatial locations using VLS data. Zhu et al. (2020) use two different methods to assess Leaf Area Index (LAI) and compare these methods with ground truth obtained from the object models, allowing them a comparison with the true LAI of the input objects. For tree segmentation, ground truth is also difficult to obtain, as tree canopies often intersect each other. By using HELIOS, Wang (2020) and Xiao et al. (2019) obtain perfect ground truth for training and validation of their segmentation methods. Wang (2020) achieve 2.9 % and 19.8 % RMSE for tree height and crown diameter estimates. In a non-forestry application, road curves are reconstructed and the reconstruction is compared with the model parameters (Zhang et al., 2019), achieving a relative accuracy of 0.6 % in circle radii estimation using VLS data. Requirements for Building Information Modelling (BIM) are evaluated by (Rebolj et al., 2017), who generate around 100 point clouds to ascertain the influence of parameter values. They define accuracy criteria for the successful identification of building elements in the scans. Bechtold et al. (2016) test a segmentation tool for rock outcrops by using HELIOS as a simulator. They show the value of simulated test data for method development by easily generating point clouds with different occlusions and scan settings, resulting in a multitude of point densities and point patterns. ### 4.3 Method training With the advent of neural networks as supervised machine learning method in geospatial domains, the need for training data has grown almost indefinitely. While raster-based approaches can make use of existing pre-trained networks by domain transfer (Pires de Lima and Marfurt, 2019), no such networks exist for point-based deep learning such as PointNet/PointNet++ (Qi et al., 2017). Simulated data, though only replicating parts of reality, can be used by neural networks to learn basic descriptors which describe point cloud neighbourhoods, e.g. planes, corners or edges. From these descriptors, higher- level features are derived, which are subsequently used in classification or regression (Winiwarter et al., 2019). Once a network has learned to represent data in form of these features, it can be adjusted to real data by adding relatively small amounts of training data, in approaches shown to work for the image domain (Danielczuk et al., 2019). Further research is needed on this approach especially for point cloud data, but HELIOS++ allows easy and fast generation of labelled training data, which does not suffer from ground truth errors. As an example application of this purpose, Martínez Sánchez et al. (2019) use HELIOS-simulated data to train and evaluate a semantic classification of an urban scene created from OpenStreetMap101010https://www.openstreetmap.org/ models. The use of VLS allowed a quantification of their classifier’s total error, which amounts to 0.5 % on simulated point clouds. ### 4.4 Sensing experimentation The fourth category summarises publications concerned with the development of novel sensors and methods, such as Park et al. (2020), who present a new Time- of-Flight sensor and compare its results to a HELIOS simulation. In general, the parameters of the virtual sensors can be tuned to resemble a non-existent sensor, the performance of which can then be simulated without the need of actually building a prototype. Especially when looking for potential improvements of current sensors, this enables the identification of weak links or bottlenecks for certain use cases. More in-depth experimentation is also conceivable, where e.g. a novel deflector model shall be simulated. Due to the open-source license, a developer may take the HELIOS++ framework and would only have to implement a new deflection method, whereas the scene and all other components can be used as is to run a simulation, testing the usability of the novel deflector model. Especially when considering the short lifecycle of current hardware, simulation may be the only way to ensure fitness for use of the sensor. Equivalent simulators are widely used in remote sensing, as especially tools working with data acquired by satellites can be developed using the simulated data, and are then ready to use as soon as the first real data is delivered. ## 5 Computational considerations Since modern laser scanners can measure millions of points per second, it is crucial to have an efficient implementation for the ray-scene intersection. In this section, we first present theoretical considerations on the ray tracing implemented in HELIOS++, the modelling of vegetation and options of generating large and complex scenes. In Section 5.4, a comparison between HELIOS++ and its predecessor HELIOS is carried out. ### 5.1 Ray tracing implementation using a kD-Tree A scene $S$ in HELIOS++ context can be mathematically described as a set of primitives, so $S=\\{P_{1},\ldots,P_{n}\\}$. For each primitive $P$ its boundaries are defined considering an axis-aligned bounding box, its centroid, and the set of vertices $V=\\{v_{1},\ldots,v_{m}\\}$ composing the primitive. Each primitive supports rotation, scaling and translation together with a material specification defining its reflectance and specularity. Certain primitives, such as transmissive voxels, also support a look-up table which can be used for vegetation modelling, as explained in Section 5.2. The scene building process consists in generating the set of primitives composing the scene. Multiple input sources are supported (cf. Section 3.5), so it is possible to build a scene considering Wavefront Object files, point clouds as ASCII files specifying $(X,Y,Z)$ coordinates for each point, GeoTIFF files, or a custom voxel file format based on AMAPVox (Vincent et al., 2017). When building a scene, different sources can be considered, as each one is associated to its own scene part. It is also possible to apply aforementioned affine transformations to an entire scene part. This can be used, for instance, to load the same object multiple times in one scene and placing it in different locations through translations. Saving and loading already built scenes is relying on boost serialisation technology (Ramey, 2004). Ray intersections are computed through a recursive search performed over a kD- Tree containing all primitives (Bentley, 1975). Let $O$ be the ray origin and $\hat{v}$ the normalised ray direction vector. When recursively searching through the kD-Tree starting at $O$, $\hat{v}$ is used to consider which node must be visited until a leaf node is reached. For each recursive search operation $s$, coordinates are analysed, so $s\equiv 0\mod 3$ means the $X$ coordinate splits the space, $s\equiv 1\mod 3$ means the Y coordinate splits the space, and $s\equiv 2\mod 3$ means the Z coordinate splits the space. Once inside a leaf node, ray intersections with respect to each primitive are computed. The minimum distance intersection $t_{0}$ is the time the ray needed to enter the primitive while $t_{1}$ is the time the ray needed to leave it. It is possible to have only $t_{0}$ determined, as is the case for triangles, since they are only intersected once per ray. For the special case of primitives which support multiple ray intersections, such as transmissive voxels, the process is repeated considering consecutive origins. Suppose we have a transmissive voxel intersected by a ray $\\{O_{1},\hat{v}\\}$: If this voxel lets the ray pass through it, then the next ray intersection will be found considering the ray $\\{O_{2},\hat{v}\\}$, with $O_{2}=O_{1}+(t_{1}+\varepsilon)\hat{v}$, where $\varepsilon$ is a small decimal number to assure getting out of the previously intersected primitive. ### 5.2 Vegetation modelling: Transmissive voxels Beam horizontal component \| Beam vertical component \| Hit probability $g_{L}$ ---\|---\|--- 1.000000 \| 0.000000 \| 0.424413 0.999683 \| 0.025180 \| 0.424682 0.998732 \| 0.050345 \| 0.425489 $\vdots$ \| $\vdots$ \| $\vdots$ 0.009444 \| 0.999955 \| 0.848789 0.000000 \| 1.000000 \| 0.848822 Table 2: Values from a look-up table (LUT) for the hit probability $g_{L}$ at a given beam direction $L$, represented by horizontal and vertical component. The numbers here correspond to an erectophile distribution and are obtained using a numerical integration method on the formulae from North (1996). The probability $g_{L}$ is normalised over $2\pi$. One of the new functionalities of HELIOS++ is vegetation modelling through transmissive voxel primitives. For this purpose, a look-up table for leaf angle distribution is used to compute ray intensity with respect to a $\sigma$ (cross-section) value obtained from it. The values in this look-up table represent a hit probability $g_{L}$ given the horizontal and vertical component of the beam vector (Tab. 2). The intensity $I$ is calculated according to Equation 7 where $P$ is the emitted power from Equation 3 (Carlsson et al., 2001), $d$ is the distance between ray origin and intersection point, $\alpha^{2}$ is the square of the scanner receiver diameter, $\beta^{2}$ is the square of the scanner beam divergence and $\lambda$ is the product between atmospheric factor and scanner efficiency. The $\sigma$ value can be seen as function of ray direction vector $\sigma(\hat{v})$, so it will have a different value depending on ray incidence. $I\propto\lambda\sigma\frac{P\alpha^{2}}{4{\pi}d^{4}\beta^{2}}$ (7) Voxels can operate in transmissive mode. This implies the voxel will let the ray continue and not return a signal after intersection if $\sigma=0$. If $\sigma>0$, then the return of the ray is randomly determined by sampling a value $u$ from a uniform distribution $\in[0,1)$ and computing $s=\frac{-\log(u)}{\sigma}$. If the value of $s$ is greater than the distance between both intersection points at the voxel, the ray will continue. Otherwise, the ray will stop at the voxel. For the non-transmissive mode, only voxels with transmittance $1.0$ will allow rays to continue. Thus, by using detailed voxels operating in transmissive mode together with an appropriate look-up table specification, HELIOS++ is capable of simulating laser scanning of vegetation from a precomputed leaf angle distribution following North (1996). To achieve high-quality output, it is recommended to apply individual leaf angle distributions for each vegetation type within a scene. ### 5.3 Strategy to handle large and complex scenes When loading point clouds from ASCII xyz files, big files might not be entirely containable in memory. For this purpose, a two-stage algorithm is used to digest point clouds of arbitrary size. The first stage simply finds the minimum and maximum values for each coordinate and counts the total number of points. The second stage builds all necessary voxels to represent the point cloud inside HELIOS++. For this purpose, a voxel grid is allocated. Then, voxels which have points inside them are built as HELIOS++ primitives. For each voxel its spatial coordinates $(X,Y,Z)$, normal vector components $(N_{x},N_{y},N_{z})$ and colour components $(R,G,B)$ are considered, if available, as may be the case for photogrammetric point clouds. The colour for the voxel is computed as the average of each colour component for all points inside the voxel. We approximate the sRGB color space by averaging the squares of the red, green, and blue values, and taking the square root of this average as the resulting value for the voxel. If normals are provided, the voxel normal can be determined either as the normal of the point which is closest to the voxel centre or the average of each normal component for all points. In case no normal vectors are provided, HELIOS++ can estimate them using singular value decomposition (SVD, Golub and Kahan (1965)). All points inside a voxel are considered to build a matrix of coordinates, for which singular values and singular vectors are obtained. Then, the singular vector of the smallest singular value is the orthonormal vector defining the plane which best fits the point set in terms of the smallest sum of squared orthogonal residuals. It can hence be understood as the voxel normal vector. The normal estimation method can be applied to the entire input point cloud in a single stage for small point clouds. If the size of input point cloud is too big with respect to RAM, the normal estimation is performed in batch mode, dividing the workload into smaller parts. Each batch extracts points inside its voxels while ignoring points outside its scope. ### 5.4 Performance comparison As the direct successor of HELIOS (Bechtold and Höfle, 2016), it is interesting to compare the performance of HELIOS++ with the original implementation in Java. However, HELIOS++ was not just a port of the code, but also comes with multiple computational improvements over its predecessor. One of these improvements concerns the ability for the user to better leverage accuracy at the cost of runtime, or vice versa, by setting parameters accordingly. For some tasks, a rough, thereby faster simulation may suffice, whereas for other tasks very detailed simulation is required, but smaller sample sizes can be used or long processing times are acceptable. Another improvement concerns the binning mechanism for the full-waveform simulation, which is used for maxima detection even if the waveform is not written to an output file. In HELIOS++, we use the parameters `binSize_ns` and `maxFullwaveRange_ns` for this binning, where the bin size is used for both the outgoing pulse and the returned waveform. To limit the impact on performance of very low-incidence rays with a high sampling quality, the user can provide a maximum length of the recorded waveform, beyond which any further echoes are discarded. To compare the performances of HELIOS++ (Version 1.0.0) and HELIOS (Version 2018-09-24), we carry out a number of simulations using different parameter settings, and record the runtime as well as peak memory usage. We present three different scenes, at three different complexity levels, and make use of the option in HELIOS++ to write different file types and to skip the echo width determination, if not required. The first example scenario represents an ALS survey over terrain, which is created by loading a digital terrain model in GeoTIFF format as described in Section 3.5. Since the GeoTIFF loader in the Java version was faulty and this version is no longer maintained, we created a mesh in Wavefront Object format as an input for the Java version. We simulate two flight lines at an altitude of $1,500\text{\,}\mathrm{m}\,\mathrm{a}\mathrm{s}\mathrm{l}$ over terrain with an extent of $26\text{\times}17.8\text{\,}\mathrm{km}$. As scanner we use a _Leica ALS50-II_ , and a Cessna SR-22 as platform. The scene is shown in Figure 8. Figure 8: Visualisation of example survey for airborne laser scanning (ALS), using a digital terrain model as object and a standard ALS instrument as scanner. The 3”-SRTM model by the USGS is used as input raster. The second scenario represents a mobile laser scan of an urban area, where a car is driving through downtown buildings modelled as prisms, cylinders and pyramids (Figure 9). The trajectory of the car is $208\text{\,}\mathrm{m}$ long and its average speed is $20\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$. The scanner is an oblique- mounted _RIEGL VUX-1UAV_. Figure 9: Visualisation of example survey for mobile laser scanning (MLS), with a car driving between objects of simple geometry. The yellow points show the acquired point cloud of scanned parts. In the third and final scenario, we present a TLS survey of a vegetation scene with a large potential for multi-echoes, as it represents two trees generated with the Arbaro Tree Simulator (Weber and Penn, 1995), scanned using a _RIEGL VZ-400_ from two static positions. This simulation has a large number of geometric primitives. A visual is depicted in Figure 10. Figure 10: Visualisation of example survey for terrestrial laser scanning (TLS), scanning two highly complex tree models created using the Arbaro Tree Simulator. The yellow points show the acquired point cloud of scanned parts. Each simulation scenario was carried out three times on an Intel-i9-7900 @ $3.3\text{\,}\mathrm{GHz}$ with $64\text{\,}\mathrm{GB}$ of RAM, and I/O on an SSD connected via SATA. The average runtime and memory consumption values are given in Table 3. \| \| HELIOS (Java) --- Version 2018-09-24 \| HELIOS++ --- Version 1.0.0 Echo width \| ✓ \| ✓ \| \| Waveform output \| ✓ \| ✓ \| ✓ \| File format \| XYZ \| XYZ \| XYZ \| LAS \| Scene 1 --- (ALS, GeoTIFF) \| $4,712.1\pm 613.8\text{\,}\mathrm{s}$ --- $6,570\text{\,}\mathrm{MB}$ \| $2,773.0\pm 53.8\text{\,}\mathrm{s}$ --- $2,246\text{\,}\mathrm{MB}$ \| $2,403.8\pm 18.0\text{\,}\mathrm{s}$ --- $2,246\text{\,}\mathrm{MB}$ \| $1,912.1\pm 20.22\text{\,}\mathrm{s}$ --- $2,246\text{\,}\mathrm{MB}$ \| Scene 2 --- (MLS, geometric primitives) \| $68.0\pm 2.3\text{\,}\mathrm{s}$ --- $560\text{\,}\mathrm{MB}$ \| $23.9\pm 0.2\text{\,}\mathrm{s}$ --- $24\text{\,}\mathrm{MB}$ \| $23.0\pm 0.3\text{\,}\mathrm{s}$ --- $24\text{\,}\mathrm{MB}$ \| $16.1\pm 0.6\text{\,}\mathrm{s}$ --- $24\text{\,}\mathrm{MB}$ \| Scene 3 --- (TLS, tree models) \| $362.6\pm 6.7\text{\,}\mathrm{s}$ --- $5,360\text{\,}\mathrm{MB}$ \| $97.2\pm 1.3\text{\,}\mathrm{s}$ --- $314\text{\,}\mathrm{MB}$ \| $74.4\pm 1.0\text{\,}\mathrm{s}$ --- $314\text{\,}\mathrm{MB}$ \| $62.4\pm 0.3\text{\,}\mathrm{s}$ --- $314\text{\,}\mathrm{MB}$ Table 3: Performance comparison of HELIOS and HELIOS++ with different options. We use default parameters for waveform modelling (``beamSampleQuality=3— and ``binSize_ns=0.25—; ``numBins=100— and ``numFullwaveBins=200— for HELIOS++ and HELIOS, respectively). Runtimes are average of three runs ($\pm$ standard deviation), memory footprint is the highest value (maximum) during the full run. In the case with largest improvement over HELIOS, runtimes of HELIOS++ are lower by 83 % (TLS) while using only 6 % of the previously required memory. Especially for large scenes, the issue of memory footprint has been a limiting factor for the usability of HELIOS. In the case of the ALS scene, the memory consumption can be reduced by 66 %, from more than $6\text{\,}\mathrm{GB}$ to just above $2\text{\,}\mathrm{GB}$, with a runtime reduction of 51 %. For the MLS scene using the simple geometric shapes, the reduced processing overhead leads to a reduction of 96 % in memory usage and 76 % in runtime. From these results we deduce a significant improvement both in runtime and memory footprint when comparing any configuration of HELIOS++ to the previous Java version. ## 6 Conclusions With HELIOS++, we present an open-source laser scanning simulation framework that enables highly performant virtual laser scanning (VLS). In its C++ implementation and with the possibility to use the `pyhelios` package in Python to manipulate simulation parameters, we opt for an efficient and easy- to-use software. While physical accuracy and realism may be superseded by complementing simulation software, HELIOS++ provides a flexible solution to balance computational requirements (runtime, memory footprint) and quality of results (physical realism), while being easy to use and configure for users. Different studies using the HELIOS concept demonstrate the usefulness of virtually acquired laser scanning data for analyses in different categories of purpose. The main categories are (a) planning of flight patterns (ULS, ALS) or scan positions (TLS); (b) generation of ground-truth data for validation of algorithms that extract parameters (i.a. for forestry) from point clouds, (c) generation of training data for supervised machine learning, and (d) testing and development of novel sensors and algorithms. The novel object model of HELIOS++, the transmissive voxel, allows a stochastic simulation of penetrable objects, such as vegetation canopy, without the need for a highly detailed 3D mesh model. Generally, HELIOS++ performs up to of $5.8\,\times$ faster than its predecessor HELIOS on common scenes, and allows the user to omit intensive calculations if not required (e.g. the calculation of the echo width for returned pulses). To summarise, HELIOS++ is a versatile, easy-to-use, well documented scientific software for virtual laser scanning simulations that provides a tool to generate VLS data, thereby complementing data obtained by real-world laser scanning. The framework invites users to experiment and develop new ideas, while being able to rely on established algorithms from the literature. #### Acknowledgements The authors wish to acknowledge the contribution of Patrick Herbers (Ruhr- Universität Bochum), for the improvement the triangle-ray intersection in C++, which helped to significantly reduce the runtime of HELIOS++. Figures 1, 4, 5, and 8 show an airplane model CC-BY Emmanuel Beranger. Figures 1 and 4 show a house model by free3d.com user `gerhald3d`, and a drone model by cgtrader.com user `CGaxr`. ## References * ASPRS (2011) ASPRS, 2011. LAS specification. https://www.asprs.org/wp-content/uploads/2010/12/LAS_1_4_r13.pdf. * Backes et al. 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# Thermodynamics of light emission Antoine Rignon-Bret<EMAIL_ADDRESS>École Normale Supérieure, 45 rue d’Ulm, F-75230 Paris, France ###### Abstract Some interactions between classical or quantum fields and matter are known to be irreversible processes. Here we associate an entropy to the electromagnetic field from well-known notions of statistical quantum mechanics, in particular the notion of diagonal entropy. We base our work on the study of spontaneous emission and light diffusion. We obtain a quantity which allows to quantify irreversibility for a quantum and classical description of the electromagnetic field, that we can study and interpret from a thermodynamical point of view. ## I Introduction Quantification of irreversibility has been a main issue for a long time in physics, and becomes even more important nowadays with the development of new branches as quantum information or quantum thermodynamics. Among all the physical phenomena involving irreversible processes, we will focus here on some well known process appearing in field theory. The equations governing classical or quantum field theories are known to be reversible, however some processes described by these theories exhibit an irreversible behavior. For instance, we can think about radiation damping in classical electrodynamics [1], emission of gravitational waves in relativistic gravitation [2] or spontaneous emission in quantum electrodynamics [3]. The way of this irreversible behavior arises from reversible field equations is very well understood and belongs to the established knowledge of physics. However, in this paper, we will try to quantify explicitly the degree of irreversibility of such matter-radiation interactions, using the framework of thermodynamics and statistical mechanics. We will focus here on the electromagnetic field and we will show that we can associate an entropy variation to such processes, which vanishes when there is no asymmetry between emission of light and absorption, but which is strictly positive in the opposite case. The situation is analog to the fall of a ball in a gravity field. The ball falls because of gravity, but if there were no dissipation, it could bounce back and reach the initial height. However, it does not, because by hitting the ground, some part of the mechanical energy of the ball go to the atoms in the ground. They get excited and so the final state of the ball is the state of maximal entropy or minimal energy. If we had a precise enough thermometer, we could observe a temperature variation because of this excitation, and this is the signature of an increasing entropy variation. We aim to think some matter-radiations interactions, as the spontaneous emission process, in the same way, which means that we have to find the corresponding vibrating degrees of freedom which makes the ground state stable unlike the excited states, and get an entropy from them. By doing this, we will associate some entropy to the electromagnetic field. The idea of associating an entropy to the electromagnetic field is old, as old as thermodynamics and electromagnetism are. Actually it is by confronting these two theories together that Einstein understood that radiation was made of indivisible quanta of energy [4], which lead to quantum mechanics, with the success we know. However, if Einstein worked with a thermostatted box where radiation was trapped and reached equilibrium with matter, the aim in the following work is different. We focus on quantum or classical electromagnetic processes, possibly involving a single atom and a single photon, so there is no associated thermal equilibrium and we are not in the thermodynamic limit. However, as entropy is an extensive quantity, unlike temperature, the notion still makes sense as long as we can count states. Finding the ”good” microscopic definition of entropy is still a challenge. Aside from the well known Von Neumann entropy $S=-k_{B}Tr\big{(}\hat{\rho}ln\hat{\rho}\big{)}$, more general frameworks have been developed [5]. For this work, we will be particularly interested in the diagonal entropy [6], which has been introduced to quantify the degree of irreversibility of some transformations in an isolated system. Indeed, the classical Von Neumann entropy associated to quantum systems always vanishes for unitary evolutions. In the following, after reminding the main features of spontaneous emission in section II, we will give in section III an appropriate definition of the entropy of the electromagnetic field, that we will relate to diagonal entropy. In particular, we will give a thermodynamical interpretation to the formula and we will check that it verifies the basic requirements for an entropy. In section IV, we will talk a bit about classical electrodynamics and we will show that the electromagnetic field entropy formula obtained in the previous sections is still valid and makes sense. ## II The framework of spontaneous emission To fully understand our purpose, it may be useful to return to the general framework of quantum electrodynamics and spontaneous emission. We will only remind some basic results and make some comments, further details or proofs are given in Scully’s book [3]. The general hamiltonian is given by : $\hat{H}=\hat{H}_{atom}+\hat{H}_{em}+\hat{H}_{int}$ (1) $\hat{H}=\frac{\hbar\omega_{0}}{2}(Id+\hat{\sigma}_{z})+\sum_{(\textbf{k},s)}\hbar\omega_{\textbf{k}}\hat{a}_{\textbf{k},s}^{\dagger}\hat{a}_{\textbf{k},s}+\sum_{(\textbf{k},s)}\hbar\lambda_{\textbf{k},s}(\hat{\sigma}_{+}\hat{a}_{\textbf{k},s}+\hat{\sigma}_{-}\hat{a}_{\textbf{k},s}^{\dagger})$ (2) where : $\lambda_{\textbf{k},s}=\textbf{d}\cdot\textbf{u}_{\textbf{k},s}\sqrt{\frac{\hbar\omega_{\textbf{k}}}{2\epsilon_{0}V}}$ (3) is a result obtained from quantum field theory. Here, d is the dipole moment operator matrix element between the atom ground state and excited state, $\textbf{u}_{\textbf{k},s}$ is a unit vector in Fourier space in the direction k and polarization $s$, $\hat{a}_{\textbf{k},s}$($\hat{a}^{\dagger}_{\textbf{k},s}$) being the electromagnetic field state $(\textbf{k},s)$ annihilation (creation) operator, $\hat{\sigma}_{+}$($\hat{\sigma}_{-}$) the atom quantum energy creation (annihilation) operator, and finally $V=L^{3}$ the volume of the cavity which encloses the atom. As we study the desexcitation of an atom, there is at most one photon in the cavity, and the state of the system atom + field is the following : $\ket{\Psi(t)}=c_{0}(t)\ket{e}\ket{0}+\sum_{(\textbf{k},s)}c_{(\textbf{k},s)}(t)\ket{g}\ket{\textbf{k},s}$ (4) where $\ket{e}$($\ket{g}$) in the atom excited (ground) state. By using Schrodinger equation with hamiltonian (2) and projecting it onto proper states we get the following equation system : $\displaystyle\dot{c}_{0}(t)=-i\omega_{0}c_{0}(t)-i\sum_{(\textbf{k},s)}\lambda_{\textbf{k},s}c_{\textbf{k},s}(t)$ (5) $\displaystyle\dot{c}_{\textbf{k},s}(t)=-i\omega_{\textbf{k}}c_{\textbf{k},s}(t)-i\lambda_{\textbf{k},s}c_{0}(t)$ (6) Here is the point. Until now, we considered that the atom was enclosed in a box of finite volume $V=L^{3}$. The size of the box is relevant because it constrains the electromagnetic modes that can propagate in the box by the quantification of the wave vector k. To lead the calculation further, we will assume that $L$ is very big such that the sums on k can be replaced by integrals. Then, after using the Weiskoppf-Wigner approximation, we can show that : ${c}_{0}(t)=e^{-\frac{\Gamma}{2}t-i\omega_{0}t}$ (7) With : $\Gamma=\frac{d^{2}\omega_{0}^{3}}{3\pi\hbar\epsilon_{0}c^{3}}$ (8) From (7), the probability to find the atom in its excited state after time $t$ is simply : $P_{e}(t)=e^{-\Gamma t}$ (9) The process is here clearly irreversible, the atom will stay forever in its ground state after de-excitation. The reason why we get an irreversible behavior starting from the Schrodinger equation which is clearly reversible is that we considered that the box was very big. If the box is relatively small, after emission, the photon can reflect on the cavity mirrors and come back to the atom in order to be reabsorbed, so with finite $L$, we do not get a decaying exponential but a periodic function, the photon being emitted, reflected, reabsorbed, then re-emitted, etc. If $L$ goes to infinity, the photon is never reflected by the walls of the cavity and the process is irreversible. Mathematically, a discrete sum of periodic function is still a periodic function, while a continuous sum (integral) of periodic function is not a periodic function anymore, so by doing the approximation of replacing sums by integrals we exhibited the irreversibility in the calculations. Now, we are interested in the electromagnetic wave packet components, so in the values of the $c_{k,s}$. We can inject (7) into (6) to find (when $t>>\Gamma$) : $c_{\textbf{k},s}=\frac{\lambda_{\textbf{k},s}}{(\omega_{k}-\omega_{0})+i\frac{\Gamma}{2}}e^{-i\omega_{\textbf{k}}t}$ (10) By taking the norm squared and by summing on the k of constant modulus, we get the $\omega$ probability distribution : $P(\omega)=\sum_{(\textbf{k},s)}\lvert c_{\textbf{k},s}\rvert^{2}\delta(\omega_{k}-\omega)=\frac{1}{\pi}\frac{\frac{\Gamma}{2}}{(\omega-\omega_{0})^{2}+\frac{\Gamma^{2}}{4}}$ (11) It is a lorentzian distribution, with standard deviation $\Delta\omega=\frac{\Gamma}{2}$ which is also its frequency width. Its time length is $\frac{1}{\Gamma}$, exactly as is the atom excited state lifetime. ## III Electromagnetic field entropy ### III.1 Statistical point of view As Eq.(4) shows, the system atom + photon is always in a pure state. The matrix density of the system reads : $\hat{\rho}=\ket{\Psi(t)}\bra{\Psi(t)}=\hat{U}_{\hat{H}}(t,0)\ket{\Psi(0)}\bra{\Psi(0)}\hat{U}^{\dagger}_{\hat{H}}(t,0)$ (12) where $\hat{U}_{\hat{H}}(t,0)$ is the unitary evolution operator between times $0$ and $t$. The Von Neumann entropy $S=-k_{B}Tr\big{(}\hat{\rho}ln\hat{\rho}\big{)}$ always vanishes at all time because $\hat{\rho}$ is always in a pure state and it seems that we can’t describe the irreversibility of the spontaneous emission process by a corresponding amount of entropy. It is true as long as we stay with the pure Von Neumann entropy. However, we can introduce a little trick. The system atom + electromagnetic field is not completely isolated, because there is a reflecting cavity which surrounds it. This cavity itself is placed in a dynamic universe and interact with it. Indeed, because of long-range interaction as gravity that we can never really remove, such tiny effects will play a role if we wait a long time enough. For instance, as the photon wave vector $k$ is related to its impulsion, and when the wave packet reflects on the cavity walls, it transfers some momentum to the box, that we can schematically write as : $\displaystyle\ket{\Psi(t)}=\bigg{(}c_{0}(t)\ket{e}\ket{0}+\sum_{(\textbf{k},s)}c_{(\textbf{k},s)}(t)\ket{g}\ket{\textbf{k},s}\bigg{)}\otimes\ket{box}\longrightarrow$ (13) $\displaystyle c_{0}(t)\ket{e}\ket{0}\otimes\ket{box}+\sum_{(\textbf{k},s)}c_{(\textbf{k},s)}(t)\ket{g}\ket{\textbf{-k},s}\otimes\ket{2\textbf{k}}_{box}$ (14) By momentum conservation. Afterwards, the box can entangle with, for instance, dust present in the environment which starts a Von Neumann infinite regress [7]. If we are interested only in the system atom + electromagnetic field, we trace out the environment comprised of the box, dust, and any exterior system in the universe which interacts with our box. We get from (4): $\displaystyle\hat{\rho}_{sys}=Tr_{env}\hat{\rho}_{sys+env}$ (15) $\displaystyle=\lvert c_{0}(t)\rvert^{2}\ket{e}\ket{0}\bra{0}\bra{e}+\sum_{(\textbf{k},s)}\lvert c_{\textbf{k},s}\rvert^{2}\ket{g}\ket{\textbf{k},s}\bra{\textbf{k},s}\bra{g}$ (16) Which is a diagonal density matrix representing a mixed state, and in consequence has non-vanishing Von Neumann entropy. We used here the framework of quantum decoherence [7, 8], which ensures that after a long enough time, all the off diagonal entries of the density matrix vanish, because of entanglement with an always existing environment. Likewise, diagonal entropy has been introduced by A.Polkovnikov [6] in order to study formally the irreversibility processes in quantum systems which are meant to be closed. The definition of this diagonal entropy is : $S_{d}=-\sum_{n}\rho_{nn}ln\rho_{nn}$ (17) where $\rho_{nn}$ is the $nn$-entry of the density matrix. Polkovnikov showed that it had all the good properties that we can expect from a suitable definition of entropy. This diagonal entropy has already been investigated by many authors [9, 10, 11, 12, 13]. We can directly show that the diagonal entropy of the pure state (4) is the von Neumann entropy of the density matrix (16) obtained from a decoherent process. It reads : $S=-\lvert c_{0}\rvert^{2}ln\lvert c_{0}\rvert^{2}-\sum_{(\textbf{k},s)}\lvert c_{\textbf{k},s}\rvert^{2}ln\lvert c_{\textbf{k},s}\rvert^{2}$ (18) As from (7) $\lvert c_{0}(t)\rvert$ goes from $1$ to $0$ when $t$ goes from $0$ to infinity, we can define the spontaneous emission entropy variation as $t\longrightarrow\infty$ as: $\Delta S=-\sum_{(\textbf{k},s)}\lvert c_{\textbf{k},s}\rvert^{2}ln\lvert c_{\textbf{k},s}\rvert^{2}$ (19) which is a strictly positive quantity. ### III.2 Explicit calculation of entropy We can calculate (18) and (19) directly from the values of (10). However, we will make an approximation here to get a simpler result. Indeed, first, from (3) we know that $\lambda_{k,s}$ is proportional to $sin\theta$, where $\theta$ is the angle between the dipole moment vector d and the wave vector k. This gives us that, typically, k has non-zero probability distribution in the following solid angle (for a given norm of the wave vector $k_{0}$) : $\Omega_{k_{0}}=\int_{0}^{\pi}\int_{0}^{2\pi}k_{0}^{2}sin^{3}\theta\mathrm{d}\theta\mathrm{d}\phi=\frac{8\pi\omega_{0}^{2}}{3c^{2}}$ (20) The first approximation that we will make is setting that the distribution of the wave vector k is uniform in this angular distribution $\Omega_{k_{0}}$ in Fourier space, and vanishes elsewhere Secondly, from (11), the frequency distribution is lorentzian. As we saw, the characteristic size of the frequency width is $\frac{\Gamma}{2}<<\omega_{0}$ which is the lorentzian mean for standard values of the two quantities. So we will replace the lorentzian distribution by a uniform distribution centered on $\omega_{0}$ and of width $\delta\omega=\frac{\Gamma}{2}$. Thus, we can replace $\sum_{(\textbf{k},s)}\longrightarrow\int\frac{\mathrm{d^{3}}k}{(\frac{2\pi}{L})^{3}}$ in (19) and get from it : $\Delta S\simeq\int_{-\infty}^{+\infty}\frac{1}{\pi}\frac{\frac{\Gamma}{2}}{(\omega-\omega_{0})^{2}+\frac{\Gamma^{2}}{4}}ln\bigg{(}\frac{8\pi\omega_{0}^{2}}{3c^{3}}\bigg{(}\frac{L}{2\pi}\bigg{)}^{3}\pi\frac{(\omega-\omega_{0})^{2}+\frac{\Gamma^{2}}{4}}{\frac{\Gamma}{2}}\bigg{)}{\mathrm{d}\omega}$ (21) $\Delta S\simeq ln\frac{V\omega_{0}^{2}\delta\omega}{3\pi c^{3}}$ (22) The volume $K_{0}=\frac{\omega_{0}^{2}\delta\omega}{3\pi c^{3}}$ is the typical volume in Fourier space of the available Fourier modes of the photon emitted. The more they are, the more ”irreversible” is the desexcitation. We see that it corresponds to a volume $V_{0}$ in the real space such as : $V_{0}=\frac{3\pi c^{3}}{\omega_{0}^{2}\delta\omega}$ (23) And from now we will write : $\Delta S=ln\frac{V}{V_{0}}$ (24) The interaction between the electromagnetic field and the atom broadens the frequency range in Fourier space available for the photon. Irreversibility comes from the fact that there is not only one mode $\omega_{0}$ of the electromagnetic field that can be excited but many of them. However, the formula (24) gives us another interpretation. To make it clearer, let consider first the 1D case. In the same way as we derived (22), we can show that for a one-dimensional cavity we can associate to the electromagnetic field the entropy : $\Delta S^{(1D)}=ln\frac{L\delta\omega}{2\pi c}$ (25) Where $\delta\omega$ is the standard deviation of the one dimensional lorentzian frequency distribution. The wave packet typical length $\delta x$ can be obtained from the Heisenberg relation $\delta x\delta k\simeq\frac{1}{2}$, so we can write (25) as : $\Delta S^{(1D)}\simeq ln\frac{L}{\delta x}$ (26) Up to a irrelevant $4\pi$ factor. Actually, it is true as long as we can set $c_{0}=0$ (remember that it is rigorously true only for $L\longrightarrow\infty$). In order to understand why, let consider the the typical time of de-excitation $\tau_{em}$, which is the inverse of the spontaneous emission rate. The typical time taken by the photon to explore the whole box is $\frac{L}{c}$. Furthermore, if we define $\tau_{ps}$ as the time taken by the energy quantum to explore the whole phase space, we should consider the time $\frac{L}{c}$ when the energy quantum is propagating freely in the box and the time $\tau_{em}$, when the energy quantum is inside the atom (when the atom is in the excited state). So : $\tau_{ps}=\frac{L}{c}+\tau_{em}$ (27) as long as $L>>\delta x$, we can neglect $\tau_{em}$ before $\frac{L}{c}$ and the entropy is indeed given by (26). But in general we expect that the entropy is given by : $\Delta S^{(1D)}=ln\frac{\tau_{ps}}{\tau_{em}}$ (28) because the entropy counts the number of states accessible to the energy quantum, and the time that the atom spends in the excited state $\tau_{em}$ is the same time as the wave packet spends in the mesh of size $\delta x\simeq c\tau_{em}$. The formula (28) enhances the fact that irreversibility is just a matter of time scale. Now, if we decide to decrease the size of the cavity up to it becomes on the same order as the typical length of the wave packet, the coefficient $c_{0}$ cannot be neglected anymore. In this case we get from (27) : $\tau_{ps}\simeq\frac{L}{c}+\tau_{em}\simeq 2\tau_{em}$ (29) Therefore, when $L\simeq\delta x$ : $\Delta S^{(1D)}=ln\frac{2\tau_{em}}{\tau_{em}}=ln2$ (30) In that case, the atom will be half of the time in the excited state and half of the time in the ground state, and the process will almost seem to be reversible. We can make an analogy with a one particule Joule Gay-Lussac expansion, initially contained in a box of length $\frac{L}{2}$ and which can propagate in the whole box of size $L$ when the constraint is released. In the three-dimensional case, the volume $V_{0}$ given by (23) is not the typical volume of the wave packet, which is approximatly, for $r=ct>>\frac{c}{\delta\omega}>>\frac{c}{\omega_{0}}$ : $V_{\textbf{r}}\simeq\frac{8\pi}{3}r^{2}\frac{1}{2\delta k}$ (31) However, our theory tells us that if we wait a long time compared to the decoherence time, everything happens as if the wavepacket occupies a volume $V_{0}$ during a time $\tau_{em}=\frac{1}{\Gamma}=\frac{1}{2\delta\omega}$. In other words, our phase space, which is the total volume $V$ is discretized in a mesh of volumes $V_{0}$, and the wave packet explores it. Actually, in this framework, the wave packet looks more like a particule than a wave packet, and the formula (24) is exactly the ideal gas entropy for one particule. If we add more excited atoms in the box (with the same energy gap), we should get : $S=Nln\frac{V}{V_{0}}$ (32) as the photons do not interact. The philosophy here is we can see the spontaneous emission here as a purely thermodynamic process. Let suppose that at the beginning we ”turned-off” the interaction between the atom and the electromagnetic field. The atom is forced to stay in the excited state and there is no entropy. At $t=0$, we turn on the interaction and the energy quantum can now move freely in the whole box. Here the matter-radiation interaction plays the role of the partition, and when this constraint is released, the system can reach a new equilibrium with entropy given by (32). Seen in this way, spontaneous emission is analog to a classical Joule Gay- Lussac expansion and the state where the energy quantum is inside the atom (the state when the atom is in its excites state) is just a particular state among others. ## IV Classical electrodynamics Let consider an oscillating electron hooked to a spring, with pulsation $\omega_{0}$. If we ignore the electrodynamic laws, the electron movement is just : $\textbf{r}(t)=\textbf{r}_{0}e^{i\omega_{0}t}$ (33) However, from the Maxwell equations, we can recover the Larmor formula giving the radiated power at time $t$ and at distance $r$ from the oscillator [1] : $P_{ray}(r,t)=\frac{1}{6\pi\epsilon_{0}c^{3}}\ddot{d}^{2}(t-\frac{r}{c})$ (34) This involves that the electron mechanical energy decays. We get : $\displaystyle\frac{d\langle E\rangle_{T}}{dt}_{ray}=\frac{-e^{2}}{6\pi\epsilon_{0}c^{3}}\frac{1}{T}\int_{t}^{t+T}a(t^{\prime})\frac{dv(t^{\prime})}{dt^{\prime}}\mathrm{dt^{\prime}}$ (35) $\displaystyle=\frac{+e^{2}}{6\pi\epsilon_{0}c^{3}}\frac{1}{T}\bigg{(}\int_{t}^{t+T}v(t^{\prime})\frac{da(t^{\prime})}{dt^{\prime}}-[v(t^{\prime})a(t^{\prime})]_{t}^{t+T}\bigg{)}$ (36) where $T=\frac{2\pi}{\omega_{0}}$. But $[v(t^{\prime})a(t^{\prime})]_{t}^{t+T}\simeq 0$ because the electron trajectory is almost periodic. Therefore the electron is submitted to a radiative force : $F_{ray}(r,t)=\frac{e^{2}}{6\pi\epsilon_{0}c^{3}}\frac{da(t)}{dt}$ (37) which in near harmonic regime with pseudo pulsation $\omega_{0}$ becomes : $F_{ray}(r,t)=-m\frac{v(t)}{\tau}$ (38) where : $\frac{1}{\tau}=\frac{e^{2}\omega_{0}^{2}}{6\pi m\epsilon_{0}c^{3}}$ (39) Therefore, at time $t>0$, the electron trajectory is : $\textbf{r}(t>0)=\textbf{r}_{0}e^{-\frac{t}{2\tau}+i\omega_{0}t}$ (40) Thus, by comparing Eq (33) and (40), we understand that the interaction matter-radius allows new ways of vibrating, because the frequency range broadens. Of course, it is very similar to the spontaneous emission process we studied in the previous sections. But it is also similar to the ball which hits the ground and transforms its mechanical energy into heat. As their total entropy increases because they have new ways of vibrating, the entropy of the electromagnetic field rises because the electron can excite new Fourier modes of the electromagnetic field. Therefore the electromagnetic field entropy increasing should be equal to : $\Delta S=-\sum_{(\textbf{k},s)}p(\textbf{k},s)lnp(\textbf{k},s)$ (41) where $p(\textbf{k},s)$ is the amount of energy of the emitted signal going into the Fourier mode $(\textbf{k},s)$ (divided by the total energy). Of course, the Fourier transform of (40) is easy to calculate, and its amplitude squared gives the energy contained in the Fourier modes. The calculations lead again to a lorentzian distribution with mean value $\omega_{0}$ and standard deviation $\frac{1}{2\tau}$. Thus, if we enclose our oscillator in a cubic box of volume $V=L^{3}$, the density of states is $(\frac{L}{2\pi})^{3}$. Following the same steps and the same approximations as in the previous sections, we get at the end : $\Delta S=ln\frac{V}{V_{0}}$ (42) With : $V_{0}=\frac{6\pi\tau c^{3}}{\omega_{0}^{2}}$ (43) Of course, it is totally similar to what we found previously. It’s not surprising, the entropy formula (22) we found previously does not involve any purely quantum quantity, and it should also apply to the classical case. The oscillator energy splits into many components, because there are many available modes of the electromagnetic field, and this splitting is responsible of the irreversibility. Thanks to (42), we can interpret the radiation force (37) as an entropic force. As for the spontaneous emission, the entropy associated to this splitting can be interpreted as the perfect gaz entropy, which enhances a description of the Fourier modes in terms of particules. Therefore, a corpuscular description of the Fourier modes seems to be relevant for matter-radiation interaction. ## V Conclusion We found a quantity for the classical and quantum electromagnetic field which measures the degree of irreversibility of the matter-radiation process. We claimed that we could associate an entropy production to all irreversible processes, and in particular to matter-radiation processes. If we focused on the electromagnetic field here, we can think about irreversibility processes in other field theories, as gravity. Indeed, the gravitational wave emission is analog to the electromagnetic wave emission of the dipole, except that the gravitational wave dipole moment vanishes, and is replaced by the quadrupole moment. However, gravity is a more complex topic than electromagnetism. First, gravitation is not linear. Second, the volume $V$ of the ”box” containing the gravitationnal system is itself solution of the field equations. ## VI Acknowledgements I want to thank my friends Martin Caelen, Helmy Chekir, Léonard Ferdinand and Pierre Vallet for their feedback on this work and for having encouraged me to write it down. I want espacially to thank Mark T. Mitchison for his careful reading and his invaluable advice. ## References * Feynman _et al._ [1965] R. P. Feynman, R. B. Leighton, and M. Sands, American Journal of Physics 33, 750 (1965). * Einstein and Rosen [1937] A. Einstein and N. Rosen, Journal of the Franklin Institute 223, 43 (1937). * Scully and Zubairy [1999] M. O. Scully and M. S. Zubairy, “Quantum optics,” (1999). * Einstein [1905] A. Einstein, Annalen der Physik , 1 (1905). * Šafránek _et al._ [2020] D. Šafránek, A. Aguirre, J. Schindler, and J. Deutsch, arXiv preprint arXiv:2008.04409 (2020). * Polkovnikov [2011] A. Polkovnikov, Annals of Physics 326, 486 (2011). * Laloë [2001] F. Laloë, American Journal of Physics 69, 655 (2001). * Zurek [2003] W. H. Zurek, Reviews of modern physics 75, 715 (2003). * Santos _et al._ [2011] L. F. Santos, A. Polkovnikov, and M. Rigol, Physical review letters 107, 040601 (2011). * Giraud and García-Mata [2016] O. Giraud and I. García-Mata, Physical Review E 94, 012122 (2016). * Piroli _et al._ [2017] L. Piroli, E. Vernier, P. Calabrese, and M. Rigol, Physical Review B 95, 054308 (2017). * Wang _et al._ [2020] Z. Wang, Z.-H. Sun, Y. Zeng, H. Lang, Q. Hong, J. Cui, and H. Fan, Physics Letters A , 126333 (2020). * Sun _et al._ [2020] Z.-H. Sun, J. Cui, and H. Fan, Physical Review Research 2, 013163 (2020).
# SUTRA: An Approach to Modelling Pandemics with Undetected Patients, and Applications to COVID-19 Manindra Agrawal, Madhuri Kanitkar, Deepu Phillip, Tanima Hajra, Arti Singh, Avaneesh Singh, Prabal Pratap Singh and Mathukumalli Vidyasagar MA, DP, TH, ArS, AvS, PPS are at Indian Institute of Technology Kanpur, Kanpur, UP 208016; MK is the Vice Chancellor of Maharashtra University of Health Sciences, Nashik, MH 422004; MV is with the Department of Artificial Intelligence, Indian Institute of Technology Hyderabad, Kandi, TS 502284; MA is the corresponding author. Email: <EMAIL_ADDRESS> ###### Abstract The Covid-19 pandemic has two key properties: (i) asymptomatic cases (both detected and undetected) that can result in new infections, and (ii) time- varying characteristics due to new variants, Non-Pharmaceutical Interventions etc. We develop a model called SUTRA (Susceptible, Undetected though infected, Tested positive, and Removed Analysis) that takes into account both of these two key properties. While applying the model to a region, two parameters of the model can be learnt from the number of daily new cases found in the region. Using the learnt values of the parameters the model can predict the number of daily new cases so long as the learnt parameters do not change substantially. Whenever any of the two parameters changes due to the key property (ii) above, the SUTRA model can detect that the values of one or both of the parameters have changed. Further, the model has the capability to relearn the changed parameter values, and then use these to carry out the prediction of the trajectory of the pandemic for the region of concern. The SUTRA approach can be applied at various levels of granularity, from an entire country to a district, more specifically, to any large enough region for which the data of daily new cases are available. We have applied the SUTRA model to thirty-two countries, covering more than half of the world’s population. Our conclusions are: (i) The model is able to capture the past trajectories very well. Moreover, the parameter values, which we can estimate robustly, help quantify the impact of changes in the pandemic characteristics. (ii) Unless the pandemic characteristics change significantly, the model has good predictive capability. (iii) Natural immunity provides significantly better protection against infection than the currently available vaccines. These properties of the model make it useful for policy makers to plan logistics and interventions. ## 1 Introduction The COVID-19 pandemic caused by the SARS-CoV-2 virus has by now led to more than 600 million reported cases and more than six million deaths worldwide, as of October 1, 2022 [1]. By way of comparison, the infuenza epidemic of 1957 led to 20,000 deaths in the UK and 80,000 deaths in the USA, while the 1968 influenza pandemic led to 30,000 deaths in the UK and 100,000 deaths in the USA [2]. In contrast, the COVID-19 pandemic has already led to more than one million deaths in the USA and more than 170,000 deaths in the UK [2]. Therefore the COVID-19 pandemic is the most deadly since the Spanish Flu pandemic which started in 1918. In the USA, 675,000 people, or 0.64% of the population, died in that pandemic [3], compared to 0.33% of the population in the current pandemic. In order to cope with a health crisis of this magnitude, governments everywhere require accurate projections of the progress of the pandemic, both in space and over time, and at various levels of granularity. In addition, decision-makers also require assessments of the relative effectiveness of different non-pharmaceutical interventions (NPIs) such as lockdowns. Over the past century or so, various mathematical models have been developed to predict the trajectory of a pandemic. These can be classified in three broad categories: statistical models, state-space models, and empirical models [4]. The most popular among these are compartment models, a type of state- space models. These models divide population into disjoint compartments representing different stages of infection, and are based on the premise that the disease spreads when an infected person comes into contact with a susceptible person. In the initial SIR model [5], the population was divided into three compartments: S (Susceptible), I (Infected), and R (Removed / Recovered). Subsequently an intermediate compartment of E (Exposed) was introduced between S and I [6]. In SEIR model, interactions between S and E do not lead to fresh infections. For pandemics like COVID-19, that have significant number of asymptomatic patients, instead of E, compartment A (Asymptomatic) with interactions between S and A also leading to new infections, is more suited [7]. For COVID-19 pandemic, a number of models have been introduced to capture its trajectory. Two significant features of this pandemic are the presence of a large number of undetected cases and time-varying parameter values due to the emergence of various mutants, lockdowns etc. For any model to capture the trajectory well, it must take into account these two features. Some of the proposed models have large number of compartments in order to make the model biologically more realistic (for example [8]). These models have a large number of parameters, and estimating their values reliably is not possible from reported data due to the well-known phenomenon of the bias-variance tradeoff in statistics. This is even more true when the parameter values change with time. Besides the ones mentioned above, there are other models work with four compartments (and consequently a small number of parameters). In these models, parameter estimation is easier (for example [9, 10, 11]). However, even these models have to make certain assumptions about how the parameter values change. For example, [9] assumes the parameter values are derived from other considerations and do not change with time, while both [10] and [11] assume that parameters change following a specific equation. ## 2 Our Contributions Against this background, in this paper we propose a four compartment model called SUTRA (Susceptible, Undetected, Tested positive, and Removed, Approach).111In Sanskrit, the word Sutra also means an aphorism. Sutras are a genre of ancient and medieval Hindu texts, and depict a code strung together by a genre. The components are same as in [10], however, there are differences in the dynamics (with better epidemiological justification, see 3). This has following consequences. ### 2.1 Fundamental Equation The model admits a time invariant relationship (called fundamental equation) between detected new cases (${\mathcal{N}}_{T}$), detected active cases (${\mathcal{T}}$), and total detected cases (${\mathcal{C}}_{T}$): ${\mathcal{T}}(t)=\frac{1}{\tilde{\beta}}{\mathcal{N}}_{T}(t+1)+\frac{1}{\tilde{\rho}P_{0}}{\mathcal{C}}_{T}(t){\mathcal{T}}(t)$ where $\tilde{\beta}$ and $\tilde{\rho}$ are parameters of the model and $P_{0}$ the population of region under study (see section 3). ### 2.2 Estimation of $\tilde{\beta}$ and $\tilde{\rho}$ We can efficiently estimate values of the two parameters $\tilde{\beta}$ and $\tilde{\rho}$ for the entire duration of the pandemic using the fundamental equation and simple linear regression (see section 5). These parameters have the following interpretation: * • Parameter $\tilde{\beta}\approx\beta$, a standard parameter denoting contact rate (also called transmission rate). * • We may interpret $\tilde{\rho}\approx\epsilon$ where $\epsilon$ is detection ratio, the ratio of detected new cases to total number of new cases (as done in an earlier version of our model [12]), however, it is not satisfactory since in every region, value of $\tilde{\rho}$ is observed to increase by a very large factor ($1000$ or more) in the initial couple of months of pandemic before becoming much less volatile (see section 9). A better interpretation is $\tilde{\rho}\approx\epsilon\rho$ where $\rho P_{0}$ is effective population under the pandemic influence (see section 3). The large initial increase then makes sense since the effective population under pandemic at the beginning is a very small fraction of $P_{0}$ and increases very rapidly. ### 2.3 Phases of Pandemic The value of parameter $\tilde{\beta}$ reduces when restrictive measures like lockdowns are imposed, and increases when these measures are lifted or a more infectious mutant arrives. Change in the value of parameter $\tilde{\rho}$ happens for many reasons. For example, when pandemic spreads to newer regions, or is completely eliminated from a region, or a part of susceptible population gets vaccine-induced immunity, or a part of population with immunity (acquired through vaccination or prior infection) loses it (see section 6). When the value of $\tilde{\beta}$ or $\tilde{\rho}$ changes significantly, the trajectory of the pandemic changes. The model captures it as a phase change and recomputes the new values. As explained in section 6, the model can detect when the values of $\tilde{\beta}$ or $\tilde{\rho}$ are changing, and when do they stabilize. ### 2.4 Future Projection With the knowledge of $\tilde{\rho}$ and $\tilde{\beta}$, model can efficiently compute trajectory of the pandemic for the entire duration. The computed trajectory is a good estimate for future also as long as parameters do not change significantly (see section 7). ### 2.5 Estimation of $\rho$ and $\epsilon$ To understand the impact of pandemic better, it is desirable to estimate values of $\rho$ and $\epsilon$ separately instead of their product. We show (Theorem 1) that given values of $\tilde{\beta}$ and $\tilde{\rho}$ along with number of total and active infections on starting date of the simulation, there are only finitely many possible values for $\rho$ and $\epsilon$. Further, there is a unique canonical value for both at each time instant. Since infections count at the start of simulation is not known, one requires a more realistic condition to be able to compute $\rho$ and $\epsilon$ values. Towards this, we show that the condition can be replaced by knowledge of value of $\epsilon$ or $\rho$ at any one time instant (section 8). Former requirement is met with a good serosurvey of the region at any point in time. Latter requirement is achievable if we can identify the time when pandemic has spread all over the region making $\rho$ close to $1$. One also requires that there is little vaccine-induced immunity at the time, since vaccine immunity reduces $\rho$ (see Lemma 2). For COVID-19, as the Omicron mutant arrived nearly eighteen months after pandemic started and is supposed to have bypassed vaccine-immunity nearly completely (as we also show in section 10), we can assume $\rho\approx 1$ sometime after Omicron reached a region that did not implement strict control measures at the time. For such regions (that cover almost the entire world barring exceptions like China), we can estimate values of $\rho$ and $\epsilon$. ### 2.6 Analysis of the Past The computed parameter values provide a quantification of impact of various events during the course of the pandemic. This includes impact of lockdowns and other restriction measures and arrival of new mutants. Section 9 does this analysis for four countries. ### 2.7 Analysis of Immunity Loss The Omicron mutant caused widespread loss of immunity. Applying the model on thirty-two countries covering all continents and more than half the world’s population, we deduce that loss of vaccine-immunity was significantly more than natural immunity conferred by prior exposure to any variant (see section 10). This, coupled with the fact that vaccines continue to protect against severe infection, strongly suggests that the best strategy to manage the pandemic is to allow it to spread after vaccinating the population. ## 3 Model Formulation Perhaps the earliest paper to propose a pandemic model incorporating asymptomatic patients is [7]. In this paper, the population is divided into four compartments: $S$, $A$ (for Asymptomatic), $I$ (for Infected) and $R$. Interactions between members of $S$ and $A$, as well as between members of $S$ and $I$, can lead to fresh infections. In that paper, it is assumed that almost all persons in $A$ escape detection, while almost all persons in $I$ are detected by the health authorities. While the SAIR model of [7] is a good starting point for modeling diseases with asymptomatic patients, and has been used in a few models for COVID-19 ([9] for example), it is not a good fit for COVID-19 for the following reasons: (i) due to contact tracing, some fraction of $A$ does get detected and is often of similar order as detected symptomatic ones, (ii) many symptomatic cases are not detected. Therefore, the size of $I$ cannot be estimated well. In the present paper we propose a different grouping, namely: $S$ = Susceptible Population, $U$ = Undetected cases in the population, $T$ = Tested Positive, either asymptomatic or symptomatic, and $R$ = Removed, either through recovery or death. This leads to the SUTRA model, where the last A in SUTRA stands for “approach.” Same division is used for models in [10, 11]. As is standard, we use symbols $S$, $U$, $T$, $R$ to also represent (time varying) fractional size of the four compartments. The category $R$ of removed can be further subdivided into $R_{U}$ denoting those who are removed from $U$, and $R_{T}$ denoting those who are removed from $T$. As in the conventional SAIR model [7], interactions between members of $S$ on the and members of $U$ or $T$, can lead to the person in S getting infected with a certain likelihood. $S$$U$$T$$R$$\beta SU$$\epsilon\beta SU$$\gamma T$$\gamma U$ Figure 1: Flowchart of the SUTRA model A compartmental diagram of the SUTRA model is shown in Figure 1. Typically, to handle undetected cases, models assume that the size of $T$ is $\epsilon$ fraction of size of $U+T$ (see for example, [10, 11]). This is essentially equivalent to the assumption that detected new cases are $\epsilon$ fraction of new infections, as assumed in our model. Epidemiologically, all the new cases (but for rare exceptions) will remain undetected for a few days (until the symptoms appear). Therefore, one needs to justify the choice of $\epsilon\beta SU$ for detected new cases. We argue as follows: * • Recently infected persons have higher chances of getting detected for two reasons. For symptomatic cases, the symptoms appears within a few days. For asymptomatic cases, they are detected through contact tracing which mostly starts with a symptomatic case and the asymptomatic cases detected would all be infected after the initiating symptomatic case. * • Number of new cases do not change dramatically over a few days and so number of detected cases over past few days can be taken to be proportional to $\beta SU$, number of most recent cases. A few additional reasonable assumptions have been made to simplify parameter estimation. Specifically, * • It is assumed that the removal rate for both compartments $T$ and $U$ is the same. This can be justified because, due to contact tracing, a significant fraction of patients in $T$ are asymptomatic, and those people recover at the same rate as the asymptomatic people in $U$. Even for the small fraction in $T$ who develop complications and pass away, the time duration is very close to that of those who recover. * • There is no interaction shown between the $T$ and $S$ compartments. In most countries, those who test positive (whether symptomatic or not) are either kept in institutional quarantine, or told to self-quarantine. In reality, there might still be a small amount of contact between $T$ and $S$. However, neglecting this does not significantly change the dynamics of the model, and greatly simplifies the parameter estimation. With these considerations, the governing equations for the SUTRA model are: $\dot{S}=-\beta SU,$ (1) $\dot{U}=\beta SU-\epsilon\beta SU-\gamma U,\dot{T}=\epsilon\beta SU-\gamma T,$ (2) $\dot{R}_{U}=\gamma U,\dot{R}_{T}=\gamma T.$ (3) Since these quantities denote the fraction of the population within each compartment, we have $S+U+T+R_{U}+R_{T}=1.$ There are three parameters in above equations, namely $\beta$, $\gamma$, and $\epsilon$. The interpretation of these parameters is as follows: * • $\beta$ = The expected number of susceptible persons infected by an infected person in one day; it is called the contact rate or transmission rate. * • $\gamma$ = Removal rate, the rate at which infected people are removed including both recoveries and deaths. * • $\epsilon$ = Rate at which infected patients in $U$ move over to $T$. As shown later, it also equals the ratio $T/(U+T)$ most of the time, and is thus called the detection rate. Later, we introduce two more parameters $\rho$ and $c$, and derive expressions for $\tilde{\beta}$ and $\tilde{\rho}$ in terms of $\beta$, $\epsilon$, $\rho$, and $c$. ### 3.1 Analyzing Model Equations Defining $M=U+T$, $R=R_{U}+R_{T}$, we get from equations (2) and (3) that $\dot{M}+\dot{R}=\beta SU=\frac{1}{\epsilon}(\dot{T}+\dot{R}_{T}),$ (4) resulting in $M+R=\frac{1}{\epsilon}(T+R_{T})+c$ (5) for an appropriate constant of integration $c$. Adding equations (2) gives $\dot{M}=\beta SU-\gamma M=\frac{1}{\epsilon}(\dot{T}+\gamma T)-\gamma M,$ or $\frac{d(Me^{\gamma t})}{dt}=\frac{1}{\epsilon}\frac{d(Te^{\gamma t})}{dt},$ (6) resulting in $M=\frac{1}{\epsilon}T+de^{-\gamma t}$ (7) for some constant $d$. Since $e^{-\gamma t}$ is a decaying exponential, it follows that, except for an initial transient period, the relationship $M=\frac{1}{\epsilon}I$ holds. This in turn implies that $U=M-T=\frac{1-\epsilon}{\epsilon}T$. How long is the transient period? Observe that the constant $d$ equals $M(0)-\frac{1}{\epsilon}T(0)$ which is close to zero since fraction of infected cases at the start of pandemic is very small. Therefore the transient period will not last more than a few days. As we will see later, such transient periods will recur at various stages of pandemic and all of them remain small. Define $N_{T}=\dot{T}+\dot{R}_{T}=\epsilon\beta SU$, the fraction of population detected to be positive at time $t$, and $C_{T}=T+R_{T}$, the fraction of population detected to be infected up to time $t$. The above simplifications allow us to rewrite equation (2) as: $\begin{split}N_{T}&=\epsilon\beta SU=\beta(1-\epsilon)ST\\\ &=\beta(1-\epsilon)(1-(M+R))T\\\ &=\beta(1-\epsilon)(1-\frac{1}{\epsilon}(T+R_{T})-c)T\\\ &=\beta(1-\epsilon)(1-c)T-\frac{\beta(1-\epsilon)}{\epsilon}C_{T}T\end{split}$ (8) Rearrange (8) as $T=\frac{1}{\tilde{\beta}}N_{T}+\frac{1}{\epsilon(1-c)}C_{T}T,$ (9) where $\tilde{\beta}=\beta(1-\epsilon)(1-c).$ ### 3.2 Discretization of the Model Relationships The progression of a pandemic is typically reported via two daily statistics: The number of people who test positive, and the number of people who are removed (including both recoveries and deaths). The second statistics has a problem though: there is no agreement on when to classify an infected person as removed. Some do it when RTPCR test is negative, some do it when symptoms are gone for a certain period, and some others do it after a fixed period of time. For the purpose of modeling, this classification needs to be done at the time when an infected person is no longer capable of infecting others. This is hard to decide, and so is almost never done. Further, some countries do not report second statistics at all (UK for example). In such a situation, we cannot rely on reported data, and instead compute $R_{T}$ by fixing $\gamma$ to an appropriate value as discussed in section 4. Let ${\mathcal{T}}(t)$ denote the number of active detected cases on day $t$, ${\mathcal{R}}_{T}(t)$ denote the number of detected cases that are removed on or before day $t$, and ${\mathcal{N}}_{T}(t)$ denote the number of cases detected on day $t$. Note that all three are integers, and $t$ is also a discrete counter. In contrast, in the SUTRA model, $T$, $R_{T}$ and $N_{T}$ are fractions in $[0,1]$, while $t$ is a continuum. Therefore, ${\mathcal{T}}(t)=P\int_{t-1}^{t}T(s)ds,{\mathcal{R}}_{T}=P\int_{t-1}^{t}R_{T}(s)ds,{\mathcal{N}}_{T}=P\int_{t-1}^{t}N_{T}(s)ds$ where $P$ is the effective population that is potentially affected by the pandemic. Now we introduce the parameter measuring the spread of the pandemic. Define number $\rho$, called the reach, which equals $P/P_{0}$, where $P$ is the effective population and $P_{0}$ is the total population of the group under study, e.g., the entire country, or an individual state, or a district (this parameter is also introduced and studied in [11]). The reach parameter $\rho$ is usually nondecreasing, starts at $0$, and increases towards $1$ over time (situations where it decreases are discussed later). While the underlying population $P_{0}$ is known, the reach $\rho$ is not known and must be inferred from the data. Substituting $P=\rho P_{0}$, and integrating equation (9) over a day gives a relationship that involves only measurable and computable quantities ${\mathcal{T}}$, ${\mathcal{C}}_{T}={\mathcal{T}}+{\mathcal{R}}_{T}$, and ${\mathcal{N}}_{T}$, and the parameters of the model, namely ${\mathcal{T}}(t)=\frac{1}{\tilde{\beta}}{\mathcal{N}}_{T}(t+1)+\frac{1}{\tilde{\rho}P_{0}}{\mathcal{C}}_{T}(t){\mathcal{T}}(t),$ (10) where $\tilde{\rho}=\epsilon\rho(1-c).$ Note that ${\mathcal{N}}_{T}$ is shifted forward by one day since new infections reported on day $t+1$ are determined by active infections and susceptible population on day $t$. Eq. (10) is the fundamental equation governing the pandemic. It establishes a linear relationship between ${\mathcal{N}}_{T}$, ${\mathcal{T}}$, and ${\mathcal{C}}_{T}{\mathcal{T}}$, that can be computed using the fundamental equation and the first three equations below (after fixing $\gamma$). In addition to the fundamental equation, we will need discrete forms of other equations of the model to compute all quantities. We group them in two—first the quantities that can be computed from ${\mathcal{N}}_{T}$: $\begin{split}{\mathcal{T}}(t)&={\mathcal{N}}_{T}(t)+(1-\gamma){\mathcal{T}}(t-1)\\\ {\mathcal{R}}_{T}(t)&={\mathcal{R}}_{T}(t-1)+\gamma{\mathcal{T}}(t-1)\\\ {\mathcal{C}}_{T}(t)&={\mathcal{T}}(t)+{\mathcal{R}}_{T}(t)={\mathcal{N}}_{T}(t)+{\mathcal{C}}_{T}(t-1)\end{split}$ (11) The second group is of equations that involve numbers that cannot be computed from reported data: $\begin{split}{\mathcal{N}}(t)&=\beta(1-\epsilon)S(t-1){\mathcal{M}}(t-1)\\\ {\mathcal{M}}(t)&={\mathcal{N}}(t)+(1-\gamma){\mathcal{M}}(t-1)\\\ {\mathcal{R}}(t)&={\mathcal{R}}(t-1)+\gamma{\mathcal{M}}(t-1)\\\ {\mathcal{C}}(t)&={\mathcal{M}}(t)+{\mathcal{R}}(t)={\mathcal{N}}(t)+{\mathcal{C}}(t-1)\\\ {\mathcal{U}}(t)&={\mathcal{M}}(t)-{\mathcal{T}}(t)\\\ {\mathcal{R}}_{U}(t)&={\mathcal{R}}(t)-{\mathcal{R}}_{T}(t)\\\ S(t)&=1-\frac{{\mathcal{C}}(t)}{\rho P_{0}}\end{split}$ (12) It is easy to see that all the quantities can be computed using above equations in addition to the fundamental equation once the parameter values used in the equations are available. ## 4 Fixing $\gamma$ As discussed in the previous section, reported removal data does not provide a good estimate for $\gamma$. In [13], median duration of infection for asymptomatic cases was estimated in the range $[6.5,9.5]$ and mean duration for symptomatic cases in the range $[10.9,15.8]$ days with a caveat that the duration reduces when children are included. In [14], infection duration for symptomatic cases was observed to be less than $10$ days. Since our groups $U$ and $T$ consist of a mix of asymptomatic and symptomatic cases, and it is likely that an infected person stops infecting others before becoming RTPCR negative, we take the mean duration of infection for both groups to be $10$ days, implying $\gamma=0.1$. All our simulations are done using the above value of $\gamma$ and show a good fit with the actual trajectories. ## 5 Estimation of $\tilde{\beta}$ and $\tilde{\rho}$ One of the distinctive features of our approach is a methodology for estimating the values of all the parameters in the pandemic model from reported raw data on the number of daily new cases. The model has five parameters $\gamma$, $\beta$, $\epsilon$, $\rho$, and $c$. At a first glance, these appear all independent, however, we show in section 8 that last four are essentially determined by $\tilde{\beta}$ and $\tilde{\rho}$. In this section, we show how to estimate $\tilde{\beta}$ and $\tilde{\rho}$ from reported data using the fundamental equation. Let $\widehat{{\mathcal{N}}}_{T}(t)$ be the reported new infections on day $t$. Note that $\widehat{{\mathcal{N}}}_{T}(t)$ may not be the same as detected new infections on day $t$ since there may be delays in reporting detected cases. Moreover, weekends often see fewer tests being done, causing unexpected variations in $\widehat{{\mathcal{N}}}_{T}$. To remove latter, we average $\widehat{{\mathcal{N}}}_{T}(t)$ over a week, and let $\widetilde{{\mathcal{N}}}_{T}(t)=\frac{1}{7}\sum_{j=0}^{6}\widehat{{\mathcal{N}}}_{T}(t-j).$ Let $\widetilde{{\mathcal{C}}}_{T}(t)=\sum_{s=0}^{t}\widetilde{{\mathcal{N}}}_{T}(s)$, the total number of reported cases until day $t$, and $\widetilde{{\mathcal{T}}}(t)$ be the number of reported active cases on day $t$ computed inductively using equation $\widetilde{{\mathcal{T}}}(t)=\widetilde{{\mathcal{N}}}_{T}(t)+(1-\gamma)\widetilde{{\mathcal{T}}}(t-1)$. Fix a time interval $[t_{0},t_{1}]$. Define ($t_{1}-t_{0}$)-dimensional vectors ${\mathbf{u}}$, ${\mathbf{v}}$, ${\mathbf{w}}$ as follows: ${\mathbf{u}}(t-t_{0})=\widetilde{{\mathcal{T}}}(t),t_{0}\leq t<t_{1},$ ${\mathbf{v}}(t-t_{0})=\widetilde{{\mathcal{N}}}_{T}(t+1),t_{0}\leq t<t_{1},$ ${\mathbf{w}}(t-t_{0})=\frac{1}{P_{0}}\widetilde{{\mathcal{C}}}_{T}(t)\widetilde{{\mathcal{T}}}(t),t_{0}\leq t<t_{1}.$ Then the following linear regression problem is solved: $\min_{\tilde{\beta},\tilde{\rho}}\|\|{\mathbf{u}}-\frac{1}{\tilde{\beta}}{\mathbf{v}}-\frac{1}{\tilde{\rho}}{\mathbf{w}}\|\|^{2}.$ The quality of the fit parameter, usually denoted by $R^{2}$, is computed as follows: $R^{2}=1-\frac{\|\|{\mathbf{u}}-\frac{1}{\tilde{\beta}}{\mathbf{v}}-\frac{1}{\tilde{\rho}}{\mathbf{w}}\|\|^{2}}{\|\|{\mathbf{u}}\|\|^{2}},$ with the optimal parameter choices. The closer $R^{2}$ is to one, the better is the quality of the fit. At times, when there are relatively few data points ($t_{1}-t_{0}$ is small), or the data has significant errors, above linear regression method fails to work (e.g., estimated parameter value becomes negative). In such situations we use a different method for estimation that is more tolerant to errors as described below. Let $\displaystyle R^{2}_{\beta}$ $\displaystyle=$ $\displaystyle 1-\frac{\|{\mathbf{u}}-\frac{1}{\tilde{\beta}}{\mathbf{v}}-\frac{1}{\tilde{\rho}}{\mathbf{w}}\|^{2}}{\|{\mathbf{u}}-\frac{1}{\tilde{\rho}}{\mathbf{w}}\|^{2}}$ $\displaystyle R^{2}_{\rho}$ $\displaystyle=$ $\displaystyle 1-\frac{\|{\mathbf{u}}-\frac{1}{\tilde{\beta}}{\mathbf{v}}-\frac{1}{\tilde{\rho}}{\mathbf{w}}\|^{2}}{\|{\mathbf{u}}-\frac{1}{\tilde{\beta}}{\mathbf{v}}\|^{2}}$ Find values of $\tilde{\beta}>0$ and $\tilde{\rho}>0$ that maximize the product $R^{2}=R^{2}_{\beta}\cdot R^{2}_{\rho}$. This choice ensures that both $\tilde{\beta}$ and $\tilde{\rho}$ play almost equally significant roles in minimizing the error. Further, the desired maximum of $R^{2}_{\beta}R^{2}_{\rho}$ is guaranteed to exist: ###### Lemma 1. When ${\mathbf{u}}$ is independent of ${\mathbf{v}}$ as well as ${\mathbf{w}}$, there is a maxima of $R^{2}$ with $R^{2}_{\beta},R^{2}_{\rho},\tilde{\beta},\tilde{\rho}>0$. The only situation when the above method will not yield the desired maxima of $R^{2}$ is when ${\mathbf{u}}$ is dependent on either ${\mathbf{v}}$ or ${\mathbf{w}}$. Former implies that ${\mathcal{T}}$ is proportional to ${\mathcal{N}}_{T}$ over the time period, or equivalently, $S$ does not change over the period. This implies ${\mathcal{N}}=0={\mathcal{N}}_{T}={\mathcal{T}}$ for the period. Similarly, latter implies that ${\mathcal{T}}$ is proportional to ${\mathcal{C}}_{T}{\mathcal{T}}$ for the duration, or equivalently, ${\mathcal{C}}_{T}$ does not change over the period. This also implies that ${\mathcal{N}}_{T}=0={\mathcal{N}}$. Either case occurs when the pandemic has effectively ended and there are no new cases for an extended period. The uncertainty in the parameter estimation is computed using the standard mean-square error formula for linear regression. We use it to compute $95$% confidence interval ranges for $\tilde{\beta}$ and $\tilde{\rho}$ values. ## 6 Phases of the Pandemic The parameters $\rho$, $\beta$ and $\epsilon$ are not constant, and vary over time. This causes changes in $\tilde{\beta}$ and $\tilde{\rho}$ as well. The contact rate $\beta$ changes for following reasons: * • Emergence of new and more infectious variants of the virus, which would spread faster than its predecessor. It takes time for the new variant to overtake whatever existed previously, which is why this factor would cause $\beta$ to increase over a period. * • Non-compliance with COVID guidelines. The $\beta$ parameter measures the likelihood of infection when an infected person (from either $U$ or $T$) meets a susceptible person from $S$. Thus $\beta$ increases if people do not wear masks, or fail to maintain social distancing, and the like. * • The parameter can also decrease suddenly, with almost a step change, due to non-pharmaceutical interventions such as lockdowns. The reach $\rho$ changes for following reasons: * • Spread of the pandemic to parts of the region that were previously untouched by it causes $\rho$ to increase. The parts may even be physically co-located with parts already touched by the pandemic comprising of those people who had completely isolated themselves. * • Elimination of the pandemic from parts of the region that were under its influence causes $\rho$ to decrease by the fraction of still susceptible population of the parts. * • Vaccination of susceptible people causes $\rho$ to decrease, as these people moving out of susceptible compartment can be viewed as effective population under the pandemic reducing. Similarly, loss of immunity among immune population causes $\rho$ to increase as this can be viewed as effective population under the pandemic increasing. This is formalized by the following lemma. ###### Lemma 2. Suppose $\rho_{\text{g}ain}$ is the fraction of susceptible population that became immune via vaccination, and $\rho_{\text{l}oss}$ is the fraction of immune population that lost immunity over a specified period of time. Then the new trajectory of the pandemic is obtained by multiplying both $\beta$ and $\rho$ (equivalently both $\tilde{\beta}$ and $\tilde{\rho}$) by $1+\frac{\rho_{\text{l}oss}-\rho_{\text{g}ain}}{\rho}$. Finally, the detection rate $\epsilon$ may increase due to more comprehensive testing, and may decrease due to reduction in testing. The changes in parameter values occur either as a slow drift over an extended period of time, or as sudden rise and fall. We divide the entire timeline of the pandemic into phases, such that within each phase, the parameters are (nearly) constant. A phase change occurs when one or more parameter values change significantly. It could be due to a quick change for reasons listed above, or accumulated slow change over an extended period. By convention, we include the duration of change in a parameter as part of new phase and call it drift period of the phase. The remaining duration of a phase is called stable period of the phase. When the value of $\epsilon$ changes, then the relationship $T=\epsilon M$ breaks down. The following lemma shows that $T$ converges to $\epsilon M$ as soon as $\epsilon$ stabilizes to its new value. ###### Lemma 3. Suppose a new phase begins at time $t_{0}$ with a drift period of $d$ days. Further, suppose that the value of parameter $\epsilon$ changes from $\epsilon_{0}$ to $\epsilon_{1}$ during the drift period. Then, ${\mathcal{M}}(t_{0}+d)=\frac{1}{\epsilon_{1}}{\mathcal{T}}(t_{0}+d)$. The above analysis leads to the following methodology of phase identification and parameter estimation for phases: 1. 1. Suppose first phase starts at $t=0$. Consider a small initial drift period $d$ (we start with $d=10$) and a small time interval $[0,t_{1}]$, and compute the values of $\tilde{\beta}$ and $\tilde{\rho}$ for this interval. 2. 2. Increase the value of $t_{1}$ and adjust the value of $d$ until value of $R^{2}$ stabilizes. Freeze the computed values of $\tilde{\beta}$ and $\tilde{\rho}$ for the phase. 3. 3. Increase the value of $t_{1}$ further until the fundamental equation has significant errors. This indicates that a new phase has started. 4. 4. Repeat the same with every subsequent phase. We demonstrate the above methodology for one phase (phase $\\#9$) in India: when the delta-variant started spreading rapidly in the country during April 2021. In Appendix A, we have plotted points $(\widetilde{{\mathcal{T}}}-\frac{1}{\tilde{\beta}}\widetilde{{\mathcal{N}}}_{T},\frac{1}{P_{0}}\widetilde{{\mathcal{C}}}_{T}\widetilde{{\mathcal{T}}})$ for different values of $t_{1}$. During the drift period of a phase, when the parameter values are changing, the points continuously drift away from a line passing through the origin (Figures 14, 14, 16) indicating that equation 10 is not satisfied. When the phase stabilizes, the points corresponding to the period line up nicely (Figures 16, 18, 18, 20, 20) indicating that the equation 10 is now satisfied. The plots also show that values of $\tilde{\beta}$ and $\tilde{\rho}$ are changing quickly during the drift period, and do not change much during stable period. This leads to easy identification of phases and stable period within. ### 6.1 Parameter values during drift period We have so far seen how to estimate values of $\tilde{\beta}$ and $\tilde{\rho}$ during stable period of every phase. However, in order to simulate the course of the pandemic, it is necessary to have the values of the parameters during the drift period as well. Suppose $d$ is the number of days in drift period, and $b_{0}$ and $b_{1}$ are the computed values of a parameter in the previous and the current phases. Then its value will move from $b_{0}$ to $b_{1}$ during the drift period. A natural way of fixing its value during the period is to use either arithmetic or geometric progression. That is, on $i$th day in the drift period the value is set to $b_{0}+\frac{i}{d}\cdot(b_{1}-b_{0})$ or $b_{0}\cdot(\frac{b_{1}}{b_{0}})^{i/d}$ respectively. Among these, geometric progression captures the way parameters change better: • When a new, more infectious, mutant spreads in a population, its infections grow exponentially initially. This corresponds to a multiplicative increase in $\beta$. * • Similarly, a new virus spreads in a region exponentially at the beginning. This corresponds to a multiplicative increase in $\rho$. * • A lockdown typically restricts movement sharply causing a multiplicative decrease in $\beta$. * • A change in testing strategy typically gets implement fast in a region, causing a multiplicative change in $\epsilon$. For these reasons, we assume that changes in parameters $\beta(1-\epsilon)$ (this is the effective contact rate due to quarantining of detected cases), $\rho$, and $\epsilon$ are multiplicative. Further, changes in parameter $c$ are additive as it is constant of integration ensuring continuity between two phases. Therefore, we may assume that changes in $1-c$ are multiplicative. This leads to the conclusion that changes in $\tilde{\beta}$ and $\tilde{\rho}$ are also multiplicative. Having defined how the parameters change during drift periods, we assume that the equations (10), (11), and (12) hold on all days. When in drift period, even the parameter values in the equations change daily as defined above. Subsequent sections show that our model with these assumptions is able to capture the trajectory of the pandemic very well. ## 7 Future Projections Once the quantities $\tilde{\beta},\tilde{\rho}$ are estimated as above for current phase, equations (10) and (11) can be used to compute values of ${\mathcal{N}}_{T}$, ${\mathcal{C}}_{T}$, and ${\mathcal{T}}$ for the entire phase duration. If the model captures the dynamics well, the predictions for daily new cases ${\mathcal{N}}_{T}$ should match closely with averaged reported numbers $\widetilde{{\mathcal{N}}}_{T}$ after the phase enters stable period as long as parameters do not change significantly. Indeed, this is confirmed by our simulations of trajectories in multiple countries. For example, for the phase $\\#9$ of India discussed in the previous section, predicted trajectory changed rapidly when the phase was in drift period, and stabilized when it transitioned to stable period (see Figure 2). Figure 2: Predicted Trajectories for India during April-June, 2021 This property allows one to accurately predict the future course of the pandemic once the present phase stabilizes. We used this to make several successful predictions in the past. Some notable ones were predicting the timing and height of the peak of second wave of India ten days in advance [15], predicting timing of the peak of third wave in India as well as many states of the country [16], predicting timing and height of the peak of Delta- wave in UK ten days in advance [17], and predicting timing and height of the peak of Delta-wave in US more than a month in advance [18]. The predictions for India and its states were useful to the policy-makers in planning the required capacity for providing health care, and scheduling nonpharmaceutical interventions such as school reopenings. ## 8 Estimation of $\rho$ and $\epsilon$ After fixing values of $\gamma$, the model has four parameters left: $\beta$, $\rho$, $\epsilon$ and $c$. We have seen how to estimate values of composite parameters $\tilde{\beta}$ and $\tilde{\rho}$ at all times, which allows us to compute the trajectory of daily new detected cases for a region. We can obtain more information about the pandemic if values of $\rho$ and $\epsilon$ can be estimated separately. For example, $\epsilon$ will enable us to estimate trajectory of total daily new cases, including undetected ones. We provide more applications in the next two sections. Without any additional information, besides the daily new detected infections time series, it is not possible to estimate value of $\epsilon$: ###### Lemma 4. Given detected new cases trajectory, ${\mathcal{N}}_{T}(t)$, $0\leq t\leq t_{F}$, there exist infinitely many total new cases trajectories and corresponding values of $\epsilon$ consistent with ${\mathcal{N}}_{T}$. In this section, we show that with just one additional data point—${\mathcal{C}}(0)$ and ${\mathcal{M}}(0)$, the total number of cases up to time $t=0$ and total active cases at $t=0$—-the number of possible trajectories for ${\mathcal{N}}(t)$, consistent with the given data, becomes finite: ###### Theorem 1. Given detected new cases trajectory, ${\mathcal{N}}_{T}(t)$, $0\leq t\leq t_{F}$ and ${\mathcal{C}}(0)$, there exist only finitely many trajectories for ${\mathcal{N}}(t)$ consistent with ${\mathcal{N}}_{T}$. Further, a good estimate for all the trajectories can be obtained efficiently. As is shown in the proof of above theorem (see Appendix C), the trajectory for ${\mathcal{N}}(t)$ is unique for the first phase, but there may be multiple ones for subsequent phases, identified by a unique value for the pair $(\epsilon,c)$ for each. Using the observation that the value of $\epsilon$ from one phase to next will not change significantly, we can identify a unique canonical trajectory for total new cases: for each phase, given the possible values of $\epsilon$ that give rise to consistent trajectories for ${\mathcal{N}}(t)$, choose the canonical value of $\epsilon$ to be the one closest to the canonical value of $\epsilon$ of previous phase (for first phase, there is anyway a unique value of $\epsilon$). The corresponding trajectory for ${\mathcal{N}}(t)$ is called canonical trajectory. As the theorem also states, the canonical value of $\epsilon$ and corresponding value of $c$ can be efficiently estimated. This, in turn, provides values of $\beta=\frac{\tilde{\beta}}{(1-\epsilon)(1-c)}$ and $\rho=\frac{\tilde{\rho}}{\epsilon(1-c)}$ for the phase. In this way, we get the values of all parameters at all times. ### 8.1 Calibrating the Model Above shows how to estimate parameter values for all phases, provided we know the values of ${\mathcal{C}}(0)$. This is equivalent to finding out the values of parameters $\epsilon$ for the first phase, say $\epsilon_{1}$, since ${\mathcal{C}}(0)=\frac{1}{\epsilon_{1}}{\mathcal{C}}_{T}(0)$ (since $c_{1}=0$ as shown in the proof of theorem 1). While this is good in theory, we do not know $\epsilon_{1}$ in practice. Moreover, time $t=0$ when the data becomes available for the first time is unlikely to be the time when the pandemic begins, and hence we may not have $c_{1}=0$. However, $c_{1}=0$ will still be a good estimate since $R(0)$ is likely to be very small. We estimate value of $\epsilon_{1}$ from other information available about the pandemic. This is called calibrating the model. We can calibrate the model in two ways: * • A sero-survey at time $t_{0}$ provides a good estimate of ${\mathcal{C}}(t_{0}-\delta)$, where $\delta$ equals the time taken for antibodies to develop. Once we accurately estimate $\epsilon_{1}$, the model can compute ${\mathcal{C}}(t)$ at all times $t$. We choose a suitable value of $\epsilon_{1}$ ensuring that model computation matches with the sero-survey result at time $t-\delta$. * • When the pandemic has been active long enough in a region without major, long- term restrictions, we may assume that it has reached all sections of society, making $\rho$ close to $1$. Again, we can choose $\epsilon_{1}$ that ensures that the reach of the pandemic is close to $1$ at suitable time. While using the above two methods for calibrating the model, following points need to be kept in mind: Using serosurveys. Many serosurveys suffer from significant sampling biases. For example, if a survey is done using residual sera from a period of high infection numbers, it is likely to significantly overestimate the seroprevalence because a large fraction of uninfected persons would not venture to give blood sample in such a period. In order to minimize sampling biases, therefore, one should use serosurveys done during a period of low infection numbers. Even then, some uninfected people may not participate making the estimates higher than actual. To further reduce bias, one should ideally be able to use multiple serosurveys as well as use the fact that reach is close to $1$ by a given time. Using reach. As observed earlier (Lemma 2), parameter $\rho$ is impacted by several factors, including gain and loss of immunity. Therefore, $\rho$ may not be close to $1$ even when the pandemic has spread over entire population. To capture this, we define $\rho_{\text{a}ctual}$ to denote the actual reach of pandemic, so $\rho=\rho_{\text{a}ctual}\cdot(1+\frac{\rho_{\text{l}oss}-\rho_{\text{g}ain}}{\rho_{\text{a}ctual}})=\rho_{\text{a}ctual}+\rho_{\text{l}oss}-\rho_{\text{g}ain}$. To calibrate the model using $\rho_{\text{a}ctual}\approx 1$ at certain time, we need an estimate of $\rho_{\text{l}oss}-\rho_{\text{g}ain}$ too, which introduces more errors in calibration. There are regions where neither an accurate sero-survey is available, and it is evident that reach is nowhere close to $1$. For such regions, calibration cannot be done with any confidence, and so estimation of all parameter values is not possible. ## 9 Analysis of the Past The parameter table of a country enables us to quantify the impacts of various events like the arrival of a new mutant, or a lockdown. Moreover, through the reach parameter, we can also explain the somewhat mysterious phenomenon of multiple peaks occurring in rapid succession that was observed in many countries. Below, we discuss in detail the progression of the pandemic in four countries. The time series data for India was sourced from [19], and for rest of the countries from [1]. ### 9.1 India Model computed trajectory for detected cases shows an excellent match with the reported trajectory: Figure 3: Predicted and Actual Trajectories for India For estimating all parameters, the calibration was done using sero survey done in December 2020 [20], a period of low infection. Estimates of seropositivity computed by the model were matched with two other serosurveys [21, 22], and very good agreement was found. Further, our model showed that the reach was close to maximum by December 2021, a very likely scenario. Interestingly, the detection rate $\epsilon$ stayed almost unchanged at $1/32$ throughout the course of the pandemic. Comparing the timeline of the pandemic [23] with parameter table below, we observe the following. Table 1: Parameter Table for India Ph No \| Start \| Drift \| $\beta$ \| $1/\epsilon$ \| $\rho$ ---\|---\|---\|---\|---\|--- 1 \| 03-03-2020 \| 4 \| $0.29\pm 0.04$ \| $32$ \| $0\pm 0$ 2 \| 19-03-2020 \| 1 \| $0.33\pm 0.02$ \| $32\pm 0$ \| $0\pm 0$ 3 \| 12-04-2020 \| 4 \| $0.16\pm 0$ \| $32\pm 0$ \| $0.033\pm 0.003$ 4 \| 17-06-2020 \| 30 \| $0.16\pm 0.01$ \| $32\pm 0$ \| $0.204\pm 0.02$ 5 \| 20-08-2020 \| 17 \| $0.16\pm 0$ \| $32\pm 0$ \| $0.364\pm 0.012$ 6 \| 29-10-2020 \| 10 \| $0.18\pm 0$ \| $32\pm 0$ \| $0.423\pm 0.008$ 7 \| 18-12-2020 \| 20 \| $0.19\pm 0.01$ \| $32\pm 0$ \| $0.448\pm 0.005$ 8 \| 11-02-2021 \| 35 \| $0.38\pm 0.01$ \| $32\pm 0$ \| $0.462\pm 0.008$ 9 \| 30-03-2021 \| 25 \| $0.28\pm 0.01$ \| $32\pm 0$ \| $0.828\pm 0.009$ 10 \| 25-05-2021 \| 0 \| $0.27\pm 0$ \| $32\pm 0$ \| $0.859\pm 0.026$ 11 \| 20-06-2021 \| 38 \| $0.51\pm 0.01$ \| $32\pm 0$ \| $0.92\pm 0.001$ 12 \| 20-08-2021 \| 2 \| $0.58\pm 0.02$ \| $32\pm 0$ \| $0.92\pm 0.004$ 13 \| 01-11-2021 \| 35 \| $0.61\pm 0.01$ \| $32\pm 0$ \| $0.949\pm 0.001$ 14 \| 26-12-2021 \| 9 \| $1.56\pm 0.21$ \| $32\pm 0$ \| $1.03\pm 0.031$ 15 \| 10-01-2022 \| 7 \| $1.18\pm 0.02$ \| $32.1\pm 0.1$ \| $1.035\pm 0.022$ 16 \| 06-02-2022 \| 1 \| $1.56\pm 0.01$ \| $32.1\pm 0$ \| $1.02\pm 0.011$ 17 \| 24-03-2022 \| 20 \| $3.45\pm 0.26$ \| $32.1\pm 0$ \| $1.044\pm 0.003$ 18 \| 02-06-2022 \| 5 \| $2.89\pm 0.06$ \| $32.4\pm 2.1$ \| $1.078\pm 0.113$ First wave (March to October 2020): The strict lockdown imposed at the end of March 2020 brought down the contact rate $\beta$ by a factor of two. The reach was very small until May ($\approx 0.03$) but increased to $0.36$ between the end of June and the end of August. This was caused by reverse migration of workers and a partial lifting of lockdown that happened during this period. Second wave (February to July 2021): The arrival of the Delta variant caused the value of $\beta$ to rise to $0.38$ in February 2021. As the variant began to spread in different parts of the country, most states imposed restrictions, which reduced the nationwide $\beta$ to $0.28$ by April. In the same month, $\rho$ increased sharply to $0.83$. Note that while increase in $\rho$ was clearly due to Delta variant, the increase happened more than a month after increase in $\beta$. We have observed this delayed increase phenomenon in $\rho$ repeatedly. The removal of all restrictions by August caused $\beta$ to increase to $0.58$. This suggests that the Delta variant was more infectious by a factor of $\approx 2$ compared to original variant. Third wave (December 2021 to March 2022): The arrival of the Omicron variant caused $\beta$ to increase sharply to $1.56$ and $\rho$ to increase to $1.03$ (from $0.95$) by the end of December. In January, mild restrictions were imposed across the country, causing $\beta$ to drop to $1.18$. These were lifted in February, and $\beta$ went back up to $1.56$ in February. A ripple (April 2022 to September 2022): The value of $\beta$ increased to around $3$ by June. This, coupled with an increase in $\rho$ from $1.02$ to $1.08$ (indicating around $6\%$ population losing natural immunity) caused a ripple that peaked in July. At present, around $98\%$ of population is estimated to have natural immunity. ### 9.2 UK Model computed trajectory for detected cases shows a good match with the reported trajectory: Figure 4: Predicted and Actual Trajectories for UK For estimating all parameters, the calibration was done using two of the three serosurveys reported in [24]. During the period of serosurveys, the infection numbers were going up and down, which made calibration a little tricky. We used the numbers from the last two surveys as well as the observation that reach has remained stationary from November 2021 (suggesting that $\rho_{\text{a}ctual}$ has been around $1$ since then) for calibration. The detection rate started at $1/9.3$ and over time increased to almost one in three cases. Comparing the timeline of the pandemic [25] with parameter table below, we observe the following. Table 2: Parameter Table for UK Ph No \| Start \| Drift \| $\beta$ \| $1/\epsilon$ \| $\rho$ ---\|---\|---\|---\|---\|--- 1 \| 14-03-2020 \| 10 \| $0.26\pm 0.01$ \| $9.3$ \| $0.02\pm 0.001$ 2 \| 17-04-2020 \| 0 \| $0.15\pm 0$ \| $9.3\pm 0$ \| $0.063\pm 0.003$ 3 \| 07-07-2020 \| 10 \| $0.24\pm 0.01$ \| $9.3\pm 0$ \| $0.08\pm 0.002$ 4 \| 01-09-2020 \| 7 \| $0.27\pm 0.01$ \| $9.3\pm 0$ \| $0.122\pm 0.004$ 5 \| 30-09-2020 \| 0 \| $0.21\pm 0.01$ \| $9.3\pm 0$ \| $0.296\pm 0.032$ 6 \| 09-11-2020 \| 5 \| $0.26\pm 0.01$ \| $9.3\pm 0$ \| $0.3\pm 0.021$ 7 \| 03-12-2020 \| 25 \| $0.32\pm 0.01$ \| $8.7\pm 1.2$ \| $0.594\pm 0.087$ 8 \| 29-01-2021 \| 40 \| $0.68\pm 0.04$ \| $8.7\pm 0.3$ \| $0.616\pm 0.053$ 9 \| 15-05-2021 \| 25 \| $0.61\pm 0.03$ \| $7.7\pm 1.9$ \| $0.763\pm 0.116$ 10 \| 03-08-2021 \| 28 \| $0.37\pm 0.01$ \| $6.5\pm 0.1$ \| $0.921\pm 0.043$ 11 \| 18-09-2021 \| 27 \| $0.5\pm 0.02$ \| $5.9\pm 0.2$ \| $0.956\pm 0.032$ 12 \| 06-11-2021 \| 27 \| $0.54\pm 0.01$ \| $5.4\pm 0.1$ \| $1.046\pm 0.034$ 13 \| 13-12-2021 \| 20 \| $0.74\pm 0$ \| $4.2\pm 0$ \| $1.014\pm 0.001$ 14 \| 18-01-2022 \| 20 \| $0.87\pm 0.19$ \| $3.5\pm 0.2$ \| $1.013\pm 0.035$ 15 \| 28-02-2022 \| 15 \| $0.93\pm 0.03$ \| $3.1\pm 0.1$ \| $1.048\pm 0.007$ 16 \| 14-04-2022 \| 10 \| $0.73\pm 0.09$ \| $2.9\pm 1.6$ \| $1.019\pm 0.343$ 17 \| 01-06-2022 \| 3 \| $1.71\pm 0.09$ \| $2.8\pm 0.4$ \| $1.02\pm 0.106$ First wave (March to July 2020): The strict lockdown imposed in March 2020 brought down the contact rate $\beta$ from $0.26$ to $0.15$ in mid-April. However, almost simultaneously, $\rho$ increased three-fold causing another peak. By July, $\beta$ was back up to $0.24$ after removal of restrictions. Second wave (September 2020 to January 2021): This wave was primarily caused by increase in value of $\rho$ from $0.08$ to $0.6$. This increase in $\rho$ was a natural consequence of very small effective population until August (less than one percent) and easing of lockdown from July (as noted above, increase in $\rho$ happens with a lag). As the numbers started increasing, fresh restrictions were put in place bringing $\beta$ down by $20\%$ by September-end. This caused the cases to peak by October-end (by that time $\rho$ increased to $0.3$). As the lockdown was eased before Christmas, both $\beta$ and $\rho$ started increasing causing and second bigger peak in January 2021. The rise in case number caused another lockdown, but this time $\beta$ did not decrease. Note that a new variant, called Alpha, started spreading in UK rapidly in December 2020. It was believed to be significantly more infectious than earlier one. This appears to be the reason why the value of $\beta$ did not decrease in January, and went up slightly instead. Third wave (February 2021 to October 2021): Lockdown was eased during February-March which resulted in a significant rise in $\beta$ to $0.68$ by second half of March. This jump, however, caused only a slight change in trajectory because reach stayed around $0.6$ and more than $85\%$ of population within reach had natural immunity by then. Numbers started rising from mid-June due to increase in $\rho$ (again a delayed increase). There were three peaks in quick succession: The first caused by increase in $\rho$ to $0.76$ in July, the second caused by further increase in $\rho$ to $0.92$ in August (when $\beta$ came down to $0.37$ during this period, likely caused by precautions taken by people due to high numbers), and the third caused by increase in $\beta$ to $0.5$ in addition to a slight increase in $\rho$. This increase in $\beta$ was likely due to the Delta variant now active in the country. Fourth wave (November 2021 to August 2022): In November the Omicron variant arrived causing $\beta$ to increase further. The wave had four peaks (although the second one got a bit messed up due to reporting of very large numbers on 31st January of backlog cases). These peaks were all caused by increase in $\beta$ – to $0.74$ in December, to $0.87$ in January-end, to $0.93$ in March, and finally to $1.71$ in June. The stepwise increase is connected to levels of restrictions imposed. At present, around $92\%$ of population is estimated to have natural immunity. ### 9.3 US Model computed trajectory for detected cases shows an excellent match with the reported trajectory: Figure 5: Predicted and Actual Trajectories for USA For estimating all parameters, the calibration was done using the serosurvey [26]. The samples were taken from life insurance applications. The calibration was further supported by the fact that $\rho$ has not changed since December, suggesting that $\rho_{\text{a}ctual}$ was close to $1$ at the time. The detection rate has slowly decreased from $1/3.5$ to $1/4$ during the course of the pandemic. Comparing the timeline of the pandemic [27] with parameter table below, we observe the following. Table 3: Parameter Table for US Ph No \| Start \| Drift \| $\beta$ \| $1/\epsilon$ \| $\rho$ ---\|---\|---\|---\|---\|--- 1 \| 15-03-2020 \| 3 \| $0.31\pm 0.02$ \| $3.5$ \| $0.007\pm 0.001$ 2 \| 13-04-2020 \| 40 \| $0.18\pm 0.01$ \| $3.5\pm 0$ \| $0.038\pm 0.003$ 3 \| 11-06-2020 \| 12 \| $0.18\pm 0.01$ \| $3.5\pm 0$ \| $0.113\pm 0.006$ 4 \| 03-09-2020 \| 65 \| $0.24\pm 0.01$ \| $3.5\pm 0$ \| $0.255\pm 0.019$ 5 \| 01-12-2020 \| 10 \| $0.24\pm 0$ \| $3.5\pm 0$ \| $0.325\pm 0.007$ 6 \| 30-12-2020 \| 5 \| $0.26\pm 0.01$ \| $3.5\pm 0$ \| $0.391\pm 0.022$ 7 \| 19-02-2021 \| 7 \| $0.23\pm 0.01$ \| $3.5\pm 0$ \| $0.462\pm 0.009$ 8 \| 08-03-2021 \| 16 \| $0.45\pm 0.02$ \| $3.6\pm 0.6$ \| $0.434\pm 0.12$ 9 \| 06-06-2021 \| 10 \| $0.38\pm 0.01$ \| $3.6\pm 0$ \| $0.459\pm 0.002$ 10 \| 26-06-2021 \| 21 \| $0.65\pm 0.01$ \| $3.7\pm 0.1$ \| $0.512\pm 0.044$ 11 \| 11-08-2021 \| 3 \| $0.46\pm 0.01$ \| $3.8\pm 0.1$ \| $0.583\pm 0.036$ 12 \| 11-09-2021 \| 0 \| $0.31\pm 0.01$ \| $3.8\pm 0.1$ \| $0.697\pm 0.09$ 13 \| 17-10-2021 \| 28 \| $0.42\pm 0.01$ \| $3.9\pm 0$ \| $0.756\pm 0.007$ 14 \| 28-11-2021 \| 4 \| $0.6\pm 0.01$ \| $3.9\pm 0$ \| $0.734\pm 0.003$ 15 \| 22-12-2021 \| 6 \| $0.53\pm 0.01$ \| $4.2\pm 0.2$ \| $1.088\pm 0.038$ 16 \| 24-02-2022 \| 39 \| $1.87\pm 0.02$ \| $4.3\pm 0$ \| $1.083\pm 0.026$ 17 \| 06-04-2022 \| 35 \| $1.02\pm 0.02$ \| $4.2\pm 0.2$ \| $1.195\pm 0.032$ 18 \| 24-06-2022 \| 5 \| $1.03\pm 0.68$ \| $4.1\pm 0$ \| $1.201\pm 0.056$ 19 \| 08-07-2022 \| 5 \| $0.71\pm 0.07$ \| $4\pm 0$ \| $1.279\pm 0.018$ 19 \| 08-07-2022 \| 5 \| $0.58\pm 0.06$ \| $4\pm 0$ \| $1.317\pm 0.024$ First wave (March to August 2020): Restrictions imposed in April 2020 brought down the contact rate $\beta$ from $0.31$ to $0.18$ by mid-May. However, almost simultaneously, $\rho$ increased to $0.04$ causing a flat trajectory. In June, most restrictions were lifted. This increased $\rho$ further to $0.11$ causing a peak in July-end. The value of $\beta$, however, did not increase. This could be due to precautions taken by a large number of people. Second wave (September 2020 to February 2021): By October, $\beta$ went up to $0.24$ and stayed around this value until the end of the wave. The value of $\rho$ increased in three steps: to $0.26$ during September-October period, to $0.33$ in December, and to $0.39$ in January. This causes three successive peaks in November, December, and January. Third wave (March 2021 to November 2021): There were two peaks separated by more than four months in this period. The Delta variant appeared to have arrived in March causing $\beta$ to increase to $0.45$. However, it caused only a small peak since $\rho$ stayed around $0.5$ until July, and more than $75\%$ of population under reach had natural immunity. The reach started increasing in August to eventually become $0.7$ by mid-September causing another peak (yet another case of delayed increase in $\rho$). Fourth wave (December 2021 to March 2022): The Omicron variant started spreading in December causing $\beta$ to increase to $0.6$, but the numbers did not increase much by December-end, since $\rho$ did not change by much. Then the reach increased substantially to $1.08$ in a short time leading to a very sharp and high peak. By February, the wave subsided, and even though $\beta$ jumped to $1.87$ in March, it did not cause cases to increase as more than $90\%$ of population was immune by then. Fifth wave (April 2022 to September 2022): The primary cause of this wave appears to be a loss of natural immunity. By May, $\rho$ increased to $1.2$ and is close to $1.3$ at present. This implies that more than $20\%$ of population has lost natural immunity in past six months. Immunity loss at such a scale has not been observed in the other three countries discussed here. Reasons for this are not clear. At present, around $85\%$ of population is estimated to have natural immunity. ### 9.4 South Africa Model computed trajectory for detected cases shows an excellent match with the reported trajectory: Figure 6: Predicted and Actual Trajectories for South Africa For estimating all parameters, the calibration was done using the serosurvey [28]. The calibration was further supported by the fact that $\rho$ has not changed since November suggesting that $\rho_{\text{a}ctual}$ has been close to $1$ since then. The detection rate has remained almost unchanged at $1/17$ during the course of the pandemic. Comparing the timeline of the pandemic [29] with parameter table below, we observe the following. Table 4: Parameter Table for South Africa Ph No \| Start \| Drift \| $\beta$ \| $1/\epsilon$ \| $\rho$ ---\|---\|---\|---\|---\|--- 1 \| 16-04-2020 \| 15 \| $0.18\pm 0.01$ \| $17$ \| $0.036\pm 0.009$ 2 \| 03-06-2020 \| 25 \| $0.19\pm 0.01$ \| $17.1\pm 0$ \| $0.241\pm 0.005$ 3 \| 21-08-2020 \| 10 \| $0.16\pm 0.01$ \| $17.1\pm 0$ \| $0.288\pm 0.007$ 4 \| 11-09-2020 \| 15 \| $0.25\pm 0.01$ \| $17.1\pm 0$ \| $0.329\pm 0.007$ 5 \| 08-11-2020 \| 5 \| $0.27\pm 0.01$ \| $17.1\pm 0$ \| $0.4\pm 0.007$ 6 \| 03-12-2020 \| 5 \| $0.31\pm 0$ \| $17.1\pm 0$ \| $0.5\pm 0.037$ 7 \| 28-12-2020 \| 12 \| $0.46\pm 0.01$ \| $17.9\pm 0.1$ \| $0.485\pm 0.009$ 8 \| 11-02-2021 \| 40 \| $0.65\pm 0.02$ \| $18\pm 0$ \| $0.536\pm 0.002$ 9 \| 11-04-2021 \| 36 \| $0.4\pm 0.01$ \| $18.1\pm 0$ \| $0.752\pm 0.004$ 10 \| 06-06-2021 \| 10 \| $0.43\pm 0.01$ \| $19.6\pm 0.4$ \| $0.938\pm 0.037$ 11 \| 24-07-2021 \| 30 \| $0.99\pm 0.02$ \| $18.2\pm 2.1$ \| $0.923\pm 0.108$ 12 \| 01-11-2021 \| 22 \| $1.58\pm 0.05$ \| $17.5\pm 1.1$ \| $1.033\pm 0.026$ 13 \| 31-12-2021 \| 7 \| $1.27\pm 0.01$ \| $16.4\pm 0.1$ \| $1.01\pm 0.01$ 14 \| 20-01-2022 \| 25 \| $1.56\pm 0.01$ \| $16.2\pm 0$ \| $1.044\pm 0.001$ 15 \| 11-03-2022 \| 30 \| $4.35\pm 0.04$ \| $16.1\pm 0$ \| $1.027\pm 0$ 16 \| 16-04-2022 \| 15 \| $2.99\pm 0.02$ \| $15.8\pm 0.2$ \| $1.06\pm 0.004$ 17 \| 12-06-2022 \| 38 \| $2.56\pm 0.15$ \| $15.6\pm 0$ \| $1.076\pm 0.002$ First wave (April to August 2020): Restrictions imposed in March and April 2020 brought down the contact rate $\beta$ to around $0.2$. A significant increase in $\rho$ to $0.24$ by June- end caused the first wave that peaked in July-end. Second wave (September 2020 to February 2021): Restrictions were lowered in September causing increase in $\beta$ value to $0.25$. Reach also continued to increase slowly to $0.5$. This caused only a slow rise since more than $40\%$ of population under the reach was already immune. The Beta variant arrived in December causing an immediate jump in $\beta$ value to $0.46$. This caused the second peak in January. Restrictions were reimposed in December to control the rise in the numbers due to Beta variant. The fact that $\beta$ still increased substantially shows that infectiousness of this variant was quite high. Third wave (March 2021 to October 2021): With removal of restrictions measures by March, value of $\beta$ further increased to $0.65$. However, since increase in $\rho$ is typically delayed and more than $90\%$ of population within reach was already immune, the rise in $\beta$ did not cause increase in numbers. The numbers started rising when $\rho$ started increasing in April to become $0.75$ by mid-May. Restrictions were partly brought back causing $\beta$ to come down to around $0.4$. The value of $\rho$ further went up to $0.94$ causing a peak in July. As the numbers started coming down from the peak in August, an unusual phenomenon occurred. Cases started increasing once again, there was a short peak in second half of August, and then the numbers came down once again but with a slightly less steep slope than before. Our model shows that this happened due to a sharp increase in value of $\beta$ to nearly $1$ from $0.43$. Note that restrictions were being increased during June-July and were relaxed only from September, so the increase in $\beta$ was not due to relaxations. Was this caused by Omicron variant that was detected later in South Africa? The sharp increase in $\beta$ which then stayed high certainly suggests so. Fourth wave (November 2021 to March 2022): In November, with more and more relaxations, Omicron caused $\beta$ to further increase to $1.58$ and $\rho$ to $1.03$ resulting in a high peak. No restrictions were imposed this time, and so the numbers rose and fell sharply. Fifth wave (March 2022 to June 2022): A further increase in $\beta$ to around $3$ by April resulted in another peak in mid-May. This peak was, however, a small one since reach was stationary around $1.05$ and more than $95\%$ of population had natural immunity. At present, around $98\%$ of population is estimated to have natural immunity. ## 10 Analysis of the Immunity Loss After the South African authorities announced the emergence of a new variant of concern (VOC), later named Omicron, the epidemiology community started analysing the ability of the Omicron variant to bypass immunity provided by vaccination, or prior exposure, or both. Our objective in this section is to provide a quantitative analysis using the SUTRA model. But before that, we give a brief summary of the vast literature based on laboratory (as opposed to population-level) studies. Everywhere in the world where it was discovered, the Omicron VOC soon replaced all other variants and was responsible for a massive increase in cases. This was due to high transmissibility conferred by the mutation, ensuring a tight binding to the ACE 2 receptor facilitating immune escape [30]. The immune escape phenomenon was reported by many groups studying the neutralization activity of sera from both infected and vaccinated individuals; see [31, 32, 33, 34, 35]. The immunity conferred by complete vaccination decreased from 80% for the Delta variant to about 30% for the Omicron variant. People infected with the Delta were better off than those infected with the initial Beta variant. There was a complete loss of neutralizing antibodies in over 50% of the vaccinated individuals and the decrease in titres varied from 43-122 fold between vaccines [36]. A booster Pfizer dose could generate an anti-Omicron neutralizing response, but titres were 6-23 fold lower than those for Delta variant. Sera from vaccinated individual of the Pfizer or Astra Zeneca vaccine barely inhibited the Omicron variant five months after complete vaccination [37]. In addition, Omicron was completely or partially resistant to neutralization by all monoclonal antibodies tested [30]. Overall, most studies confirmed that sera from convalescent as well as fully vaccinated individuals irrespective of the vaccine (BNT162b2, mRNA-1273, Ad26.COV2.5 or ChAdOx1-nCoV19, Sputnik V or BBIBP-CorV) contained very low to undetectable levels of nAbs against Omicron. A booster with a third dose of mRNA vaccine appeared to restore neutralizing activity but the duration over which this effect may last has not been confirmed. Double vaccination followed by Delta breakthrough infection, or prior infection followed by mRNA vaccine double vaccination, appear to generate increased protective levels of neutralizing antibodies [38]. Viral escape from neutralising antibodies can facilitate breakthrough infections in vaccinated and convalescent individuals; however, pre-existing cellular and innate immunity could protect from severe disease [38, 39]. Mutations in Omicron can knock out or substantially reduce neutralization by most of the large panel of potent monoclonal antibodies and antibodies under commercial development. Studies also showed that neutralizing antibody titers against BA.2 were similar to those against the BA.1 variant. A third dose of the vaccine was needed for induction of consistent neutralizing antibody titers against either the BA.1 or BA.2.3,4 variants, suggesting a substantial degree of cross-reactive natural immunity [40]. The studies above indicate that vaccine immunity was lost substantially against Omicron, and natural immunity provided better protection. All the studies were done in laboratories or in a small section of population, and our analysis in this section complements them as it is based on population-wise data. ### 10.1 Vaccine Immunity before Omicron We first analyze the gain in immunity due to vaccination before the arrival of Omicron. For this propose, we downloaded an extensive list of serosurveys, carried out in various countries and maintained by the site [41], eliminated surveys that were not done at national level, or had small sample sizes, or had high risk of bias. Nineteen countries remained after this pruning. These sero-surveys were used together with the SUTRA model to capture the pandemic trajectories and estimate parameter values in these countries. We identify two values for each country: 1. 1. Value $\frac{1}{\rho}-1$ before arrival of Omicron. All the nineteen countries had restrictions removed well before Omicron and therefore, it is reasonable to expect that $\rho_{\text{a}ctual}$ was close to $1$ by the arrival of Omicron in the country. As shown in Lemma 2, $\rho=\rho_{\text{a}ctual}+\rho_{\text{l}oss}-\rho_{\text{g}ain}\approx 1+\rho_{\text{l}oss}-\rho_{\text{g}ain}$ at the time. In other words, $\rho_{\text{g}ain}-\rho_{\text{l}oss}\approx 1-\rho$. Since calibration of the model provides only an approximate value of $\rho$, we use fractional gain in immunity $\frac{\rho_{\text{g}ain}-\rho_{\text{l}oss}}{\rho}\approx\frac{1}{\rho}-1$ which is likely to be more robust. 2. 2. Fraction of uninfected population in the country that has received at least one dose of vaccination at the onset of Omicron wave. To estimate this number, we assume that the two types of immunity, vaccine and natural, are independent random variables, implying that the fraction with hybrid immunity is the product of vaccine immunity and natural immunity fractions. With this assumption, and using vaccination data from [42], we can estimate the required value. Figure 7 plots the above two numbers. It shows a very strong correlation between the two numbers implying that vaccination provided excellent immunity before Omicron. Figure 7: $\frac{1}{\rho}-1$ vs. Percentage of Vaccinated & Uninfected Population ### 10.2 Loss of Vaccine Immunity after Omicron We can measure immunity loss due to Omicron by comparing the value of $\rho$ after Omicron arrives in a country with the value before its arrival. This change will be almost entirely due to immunity loss since $\rho_{\text{a}ctual}\approx 1$ before Omicron as discussed above. We first compare it with vaccine-only immunity present in the population to get an estimate of how much of it was lost. Figure 8 plots these two numbers. Again, a very strong correlation is observed between the numbers. This, and the fact that the slope of best-fit line is close to $1$, suggests that almost all of immunity loss was due to loss of vaccination immunity. Figure 8: Immunity Loss after Omicron vs. Vaccinated & Uninfected Population The above conclusion is further strengthened by the next plot where we compare immunity loss due to Omicron with the natural immunity present in the population before the arrival of mutation. Figure 9 plots these two numbers. It shows a very strong negative correlation implying that natural immunity provided excellent protection against Omicron. Figure 9: Immunity Loss after Omicron vs. Naturally Immune Population ### 10.3 Incorporating More Countries To make our conclusions more broad-based, we include seventeen more countries based on following criteria: 1. 1. All continents are represented well (five from Africa, two from North America, four from South America, thirteen from Asia, eleven from Europe, and one from Australia) 2. 2. Populous countries are simulated (except China for which it is not possible to calibrate the model). More than half the world’s population lives in these countries. 3. 3. It is likely that $\rho_{\text{a}ctual}$ was close to maximum in these countries at the time of Omicron’s arrival, allowing us to calibrate the model. Adding these countries to the plots, we find little change in the correlations (see Figures 10, 11, 12), further strengthening the conclusions. Figure 10: $\frac{1}{\rho}-1$ vs. Percentage of Vaccinated & Uninfected Population Figure 11: Immunity Loss after Omicron vs. Vaccinated & Uninfected Population Figure 12: Immunity Loss after Omicron vs. Naturally Immune Population Taken together, these plots show that Omicron bypassed vaccination immunity almost completely, but natural immunity provided excellent protection. A clear conclusion is that countries that followed zero-COVID strategy – strictly control the spread and vaccinate entire population – suffered maximum during the Omicron wave. 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New England Journal of Medicine 386, 1579–1580. * [41] Arora RK, Joseph A, Wyk JV et al.. 2022 Serotracker. https://serotracker.com/en. * [42] Our World in Data. 2022 Vaccination Statistics. https://ourworldindata.org/covid-vaccinations. ## Appendix A India Phase 9 Plots All the plots in this section have $\frac{1}{P_{0}}\widetilde{{\mathcal{C}}}_{T}\widetilde{{\mathcal{T}}}$ on $x$-axis and $\widetilde{{\mathcal{T}}}-\frac{1}{\tilde{\beta}}\widetilde{{\mathcal{N}}}_{T}$ on $y$-axis. Figure 13: Phase Plot on April 21, 2021 Figure 14: Phase Plot on April 23, 2021 Figure 15: Phase Plot on April 25, 2021 Figure 16: Phase Plot on April 27, 2021 Figure 17: Phase Plot on May 07, 2021 Figure 18: Phase Plot on May 17, 2021 Figure 19: Phase Plot on May 27, 2021 Figure 20: Phase Plot on June 07, 2021 ## Appendix B Pandemic Status in Countries at the Time of Omicron Arrival In this section, we provide the data used for analysis in section 10. The table below lists, for $32$ countries, the percentage of population that had vaccine immunity (given at least one shot of vaccine fifteen days before) when Omicron arrived in the country. It also lists percentage of population with natural immunity (estimated by the model), with hybrid immunity (assuming two types of immunity are independent), with vaccine-only immunity (difference vaccine immunity and hybrid immunity), and within reach of the pandemic at the time. Table 5: Status of Pandemic at the Time of Omicron Arrival \| Serosurvey \| Vaccination \| Natural \| Hybrid \| Only \| Increase ---\|---\|---\|---\|---\|---\|--- Country \| Available \| Immunity % \| Immunity % \| Immunity % \| Vaccination \| in Reach % \| \| \| \| \| Immunity % \| Australia \| N \| 78.8 \| 2.7 \| 2.1 \| 76.7 \| 96.9 Bangladesh \| N \| 60.0 \| 79.0 \| 47.4 \| 12.6 \| 5.7 Brazil \| N \| 80.4 \| 63.7 \| 51.2 \| 29.2 \| 12.8 Canada \| Y \| 84.8 \| 46.9 \| 39.8 \| 45.0 \| 31.2 Chile \| N \| 90.1 \| 58.3 \| 52.5 \| 37.6 \| 31.8 Croatia \| Y \| 52.7 \| 39.6 \| 20.9 \| 31.8 \| 31.8 Ecuador \| N \| 79.9 \| 62.7 \| 50.1 \| 29.8 \| 21.8 Ethiopia \| N \| 7.7 \| 79.2 \| 6.1 \| 1.6 \| 17.2 France \| Y \| 75.6 \| 34.0 \| 25.7 \| 49.9 \| 62.2 Greece \| Y \| 71.2 \| 29.6 \| 21.1 \| 50.1 \| 54.7 India \| Y \| 60.7 \| 81.1 \| 49.2 \| 11.5 \| 11.3 Indonesia \| Y \| 55.5 \| 80.3 \| 44.6 \| 10.9 \| 1.7 Iran \| N \| 73.5 \| 60.2 \| 44.2 \| 29.3 \| 24.2 Israel \| Y \| 69.3 \| 35.6 \| 24.7 \| 44.6 \| 63.4 Italy \| Y \| 77.7 \| 45.4 \| 35.3 \| 42.4 \| 43.3 Japan \| N \| 80.5 \| 4.5 \| 3.6 \| 76.9 \| 84.3 Jordan \| Y \| 43.5 \| 74.3 \| 32.3 \| 11.2 \| 15.2 Kenya \| Y \| 8.7 \| 82.9 \| 7.2 \| 1.5 \| 7.8 Lithuania \| Y \| 60.0 \| 81.6 \| 49.0 \| 11.0 \| 24.8 Mexico \| Y \| 63.5 \| 61.0 \| 38.7 \| 24.8 \| 15.7 Nigeria \| N \| 2.8 \| 87.3 \| 2.4 \| 0.4 \| 0.3 Norway \| Y \| 78.2 \| 4.6 \| 3.6 \| 74.6 \| 93.3 Oman \| Y \| 63.1 \| 73.3 \| 46.3 \| 16.8 \| 11.9 Pakistan \| N \| 53.0 \| 76.7 \| 40.7 \| 12.3 \| 4.6 Philippines \| N \| 57.2 \| 79.4 \| 45.4 \| 11.8 \| 14.1 Portugal \| Y \| 89.3 \| 18.5 \| 16.5 \| 72.8 \| 62.5 Singapore \| N \| 86.9 \| 16.1 \| 14.0 \| 72.9 \| 73.3 South Africa \| Y \| 26.0 \| 73.8 \| 19.2 \| 6.8 \| 10.7 Spain \| Y \| 81.2 \| 46.6 \| 37.8 \| 43.4 \| 42.4 UK \| Y \| 78.3 \| 83.3 \| 65.2 \| 13.1 \| 3.4 US \| Y \| 73.4 \| 55.9 \| 41.0 \| 32.4 \| 32.5 Vietnam \| N \| 81.3 \| 18.8 \| 15.3 \| 66.0 \| 56.7 ## Appendix C Proofs In this Appendix, we provide proofs of all lemmas and theorems stated in the paper. * Lemma 1 When ${\mathbf{u}}$ is independent of ${\mathbf{v}}$ as well as ${\mathbf{w}}$, there is a maxima of $R^{2}$ with $R^{2}_{\beta},R^{2}_{\rho},\tilde{\beta},\tilde{\rho}>0$. ###### Proof. Let $x=\frac{1}{\tilde{\beta}}$ and $y=\frac{1}{\tilde{\rho}}$. Then we have: $\begin{split}R^{2}&=\frac{xy(2{\mathbf{v}}^{T}{\mathbf{u}}-y{\mathbf{v}}^{T}{\mathbf{w}}-x{\mathbf{v}}^{T}{\mathbf{v}})(2{\mathbf{w}}^{T}{\mathbf{u}}-x{\mathbf{w}}^{T}{\mathbf{v}}-y{\mathbf{w}}^{T}{\mathbf{w}})}{\|{\mathbf{u}}-y{\mathbf{w}}\|^{2}\|{\mathbf{u}}-x{\mathbf{v}}\|^{2}}\end{split}$ (13) with $R^{2}_{\beta}>0$ iff $2{\mathbf{w}}^{T}{\mathbf{u}}-x{\mathbf{w}}^{T}{\mathbf{v}}-y{\mathbf{w}}^{T}{\mathbf{w}}>0$ and $R^{2}_{\epsilon}>0$ iff $2{\mathbf{v}}^{T}{\mathbf{u}}-y{\mathbf{v}}^{T}{\mathbf{w}}-x{\mathbf{v}}^{T}{\mathbf{v}}>0$. The denominator of equation (13) is always positive since ${\mathbf{u}}$ is independent of ${\mathbf{v}}$ as well as ${\mathbf{w}}$. The numerator is a product of four linear terms in the unknowns $x$ and $y$. Therefore the value of $R^{2}$ is positive inside the polygon defined by: $\displaystyle x$ $\displaystyle\geq$ $\displaystyle 0$ $\displaystyle y$ $\displaystyle\geq$ $\displaystyle 0$ $\displaystyle 2{\mathbf{v}}^{T}{\mathbf{u}}-y{\mathbf{v}}^{T}{\mathbf{w}}-x{\mathbf{v}}^{T}{\mathbf{v}}$ $\displaystyle\geq$ $\displaystyle 0$ $\displaystyle 2{\mathbf{w}}^{T}{\mathbf{u}}-x{\mathbf{w}}^{T}{\mathbf{v}}-y{\mathbf{w}}^{T}{\mathbf{w}}$ $\displaystyle\geq$ $\displaystyle 0$ and is zero on the boundaries. This guarantees that there exists at least one maxima inside the polygon. ∎ * Lemma 2 Suppose $\rho_{\text{g}ain}$ is the fraction of susceptible population that became immune via vaccination, and $\rho_{\text{l}oss}$ is the fraction of immune population that lost immunity over a specified period of time. Then the new trajectory of the pandemic is obtained by multiplying both $\beta$ and $\rho$ (equivalently both $\tilde{\beta}$ and $\tilde{\rho}$) by $1+\frac{\rho_{\text{l}oss}-\rho_{\text{g}ain}}{\rho}$. ###### Proof. During the course of the pandemic, first few phases did not have any vaccination or immunity loss. Consider the first phase with either immunity loss or gain through vaccination or both. Suppose fraction of population that gains immunity through vaccination in this phase is $\rho_{\text{g}ain}$ and the fraction of population that loses immunity is $\rho_{\text{l}oss}$. Consider a time instant $t$ in the stable period of the phase when the changes in immunity have already taken place. Then the fraction of removed population would be ${\mathcal{R}}(t)+\rho_{\text{g}ain}P_{0}-\rho_{\text{l}oss}P_{0}$ where ${\mathcal{R}}(t)$ is the fraction of removed population if there was no change in immunity levels. Therefore, the fundamental equation (10) changes to: $\displaystyle{\mathcal{N}}_{T}(t+1)$ $\displaystyle=$ $\displaystyle\beta(1-\epsilon)S{\mathcal{T}}$ $\displaystyle=$ $\displaystyle\beta(1-\epsilon)(1-\frac{1}{\rho P_{0}}({\mathcal{M}}+{\mathcal{R}}+\rho_{\text{g}ain}P_{0}-\rho_{\text{l}oss}P_{0})){\mathcal{T}}$ $\displaystyle=$ $\displaystyle\beta(1-\epsilon)(1-\frac{\rho_{\text{g}ain}-\rho_{\text{l}oss}}{\rho}-\frac{1}{\rho P_{0}}({\mathcal{M}}+{\mathcal{R}})){\mathcal{T}}$ $\displaystyle=$ $\displaystyle\beta(1-\epsilon)f(1-\frac{1}{\rho fP_{0}}({\mathcal{M}}+{\mathcal{R}})){\mathcal{T}}$ $\displaystyle=$ $\displaystyle\beta(1-\epsilon)f(1-c-\frac{1}{\epsilon\rho fP_{0}}({\mathcal{T}}+{\mathcal{R}}_{T})){\mathcal{T}}$ $\displaystyle=$ $\displaystyle\tilde{\beta}f(1-\frac{1}{\tilde{\rho}fP_{0}}({\mathcal{T}}+{\mathcal{R}}_{T})){\mathcal{T}}$ where $f=1-\frac{\rho_{\text{g}ain}-\rho_{\text{l}oss}}{\rho}$. Therefore, fundamental equation now holds with values of $\beta$ and $\rho$ multiplied by $f$. The other equations of the model (11 and 12) can easily be seen to hold with the same change in values of $\beta$ and $\rho$. ∎ * Lemma 3 Suppose a new phase begins at time $t_{0}$ with a drift period of $d$ days. Further, suppose that the value of parameter $\epsilon$ changes from $\epsilon_{0}$ to $\epsilon_{1}$ during the new phase. Then, ${\mathcal{M}}(t_{0}+d)=\frac{1}{\epsilon_{1}}{\mathcal{T}}(t_{0}+d)$. ###### Proof. The model parameters $\beta(1-\epsilon)$, $\epsilon$, $\rho$ and $1-c$ change multiplicatively until they stabilize to new values. Suppose that in one day, $\beta(1-\epsilon)$ changes by a factor of $f_{b}$, $\epsilon$ by a factor of $f_{e}$, $\rho$ by a factor of $f_{r}$ and $1-c$ by a factor of $f_{c}$. Then, composite parameter $\tilde{\beta}=\beta(1-\epsilon)(1-c)$ will change by a factor of $f_{b}f_{c}$ and $\tilde{\rho}=\epsilon\rho(1-c)$ will change by a factor of $f_{e}f_{r}f_{c}$. We know that after the drift period, $\tilde{\beta}$ and $\tilde{\rho}$ stabilize. Suppose that $\epsilon$ continues to change even after the drift period of $d$ days and stabilizes after $D$ days in the phase. We now consider following cases: * • $\tilde{\beta}$ changes during the drift period and $\beta(1-\epsilon)$ changes on day $D-1$. In that case, change in $\tilde{\beta}$ on day $D-1$ equals $f_{b}f_{c}$ which is not equal to $1$ since $\tilde{\beta}$ changes during drift period. This is not possible. * • $\tilde{\beta}$ changes during the drift period and $\beta(1-\epsilon)$ does not change on day $D-1$. Then change in $\tilde{\beta}$ on day $D-1$ equals $f_{c}$, and since $\tilde{\beta}$ does not change on day $D-1$, $f_{c}=1$. Computation leading up to derivation of equation (15), as in the proof of Theorem 1, can be carried out for day $D$ (since all parameters stabilize by then), and substituting $y=\frac{1}{f_{c}}=1$ in the equation (15) we get $f_{e}=x=1$ (note that $\tilde{\rho}_{D}=\tilde{\rho}_{D-1}$). This contradictions the assumption that $\epsilon$ changes during the phase. * • $\tilde{\rho}$ changes during drift period and $\rho$ changes on day $D-1$. Then change in $\tilde{\rho}$ on day $D-1$ equals $f_{r}f_{e}f_{c}$ which is not equal to $1$ since $\tilde{\rho}$ changes during drift period. This is not possible. * • $\tilde{\rho}$ changes during drift period and $\rho$ does not change on day $D-1$. Then change in $\tilde{\rho}$ on day $D-1$ equals $f_{e}f_{c}$ which must be equal to $1$ since $\tilde{\rho}$ does not change after $d$ days. Therefore, $f_{e}=1/f_{c}$. Going back to equation (15) and substituting $x=f_{e}=1/f_{c}=y$, we again get $f_{e}=1=f_{c}$, contradicting the assumption that $\epsilon$ changes during the phase. Together, the cases above cover all possibilities and hence we conclude that $D=d$ and therefore, ${\mathcal{M}}(t_{0}+d)=\frac{1}{\epsilon_{1}}{\mathcal{T}}(t_{0}+d)$. ∎ * Lemma 4 Given detected new cases trajectory, ${\mathcal{N}}_{T}(t)$, $0\leq t\leq t_{F}$, there exist infinitely many total new cases trajectories and corresponding values of $\epsilon$ consistent with ${\mathcal{N}}_{T}$. ###### Proof. Given ${\mathcal{N}}_{T}(t)$, $0\leq t\leq t_{F}$, we can compute phases of the trajectory and values of $\tilde{\beta}$ and $\tilde{\rho}$ for all phases as shown in section 5, as well as ${\mathcal{C}}_{T}(t)$, ${\mathcal{T}}(t)$ and ${\mathcal{R}}_{T}(t)$ for the entire duration. Choose any value $\epsilon_{0}$ in the range $[0.9,1.0]$. Fix $\epsilon=\epsilon_{0}$ and $c=0$. This allows us to compute the values of $\beta$ and $\rho$ for all phases (the value of $\rho$ will be at most $\frac{1}{0.9}$ times the value of $\tilde{\rho}$ at any time). Let ${\mathcal{N}}(t)=\frac{1}{\epsilon}{\mathcal{N}}_{T}(t)$, and ${\mathcal{M}}(t)=\frac{1}{\epsilon}{\mathcal{T}}(t)$ for $0\leq t\leq t_{F}$. Setting ${\mathcal{R}}(0)=0$, and using the equation (12) for $R$, we get that ${\mathcal{R}}(t)=\gamma\sum_{s=0}^{t-1}{\mathcal{M}}(s)=\frac{\gamma}{\epsilon}\sum_{s=0}^{t-1}{\mathcal{T}}(s)=\frac{1}{\epsilon}{\mathcal{R}}_{T}(t)$. Then, for all $t$, $0\leq t\leq t_{F}$: $\displaystyle{\mathcal{N}}(t)$ $\displaystyle=$ $\displaystyle\frac{1}{\epsilon}{\mathcal{N}}_{T}(t)$ $\displaystyle=$ $\displaystyle\frac{1}{\epsilon}\tilde{\beta}(1-\frac{1}{\tilde{\rho}P_{0}}({\mathcal{T}}(t-1)+{\mathcal{R}}_{T}(t-1))){\mathcal{T}}(t-1)$ $\displaystyle=$ $\displaystyle\tilde{\beta}(1-\frac{1}{\rho P_{0}}({\mathcal{M}}(t-1)+{\mathcal{R}}(t-1))){\mathcal{M}}(t-1)$ $\displaystyle=$ $\displaystyle\beta(1-\epsilon)S(t-1){\mathcal{M}}(t-1).$ It is straightforward to see that the remaining model equations (12) are also satisfied. As there are infinitely many values in the range $[0.9,1.0]$, the proof is complete. ∎ * Theorem 1 Given detected new cases trajectory, ${\mathcal{N}}_{T}(t)$, $0\leq t\leq t_{F}$ and ${\mathcal{C}}(0)$, there exist only finitely many trajectories for ${\mathcal{N}}(t)$ consistent with ${\mathcal{N}}_{T}$. Further, a good estimate for all the trajectories can be obtained efficiently. ###### Proof. Proof is by induction on the number of phases. In the base case we have only one phase. For this phase, there is no drift period since there are no previous values of parameters. Therefore, parameter values stay the same throughout the phase duration. Let $\beta_{1}$, $\rho_{1}$, $\epsilon_{1}$ and $c_{1}$ be the parameter values governing the actual trajectory for this phase. Therefore, ${\mathcal{M}}(t)=\frac{1}{\epsilon_{1}}{\mathcal{T}}(t)$ and ${\mathcal{R}}(t)=\frac{1}{\epsilon_{1}}{\mathcal{R}}_{T}(t)+c_{1}\rho_{1}P_{0}$ for the entire phase. Note that ${\mathcal{R}}(0)=0$ since at time $t=0$, when the pandemic starts, there are no recoveries. Hence, $c_{1}=0$. Further, $\epsilon_{1}={\mathcal{T}}(0)/{\mathcal{M}}(0)={\mathcal{T}}(0)/{\mathcal{C}}(0)$. From this, we can compute $\displaystyle\rho_{1}$ $\displaystyle=$ $\displaystyle\frac{1}{\epsilon_{1}}\tilde{\rho}_{1}$ $\displaystyle\beta_{1}$ $\displaystyle=$ $\displaystyle\tilde{\beta}_{1}/(1-\epsilon_{1})$ giving values of all parameters for first phase, using which the trajectory can be computed for the first phase uniquely. Suppose there are finitely trajectories up to phase $i-1$. Fix any one trajectory with values of four parameters in phase $i-1$ being $\beta_{0}$, $\rho_{0}$, $\epsilon_{0}$ and $c_{0}$. Let $t_{0}$ be the time when phase $i$ starts. We have: $\begin{bmatrix}{\mathcal{M}}(t_{0})\\\ {\mathcal{C}}(t_{0})\end{bmatrix}=\frac{1}{\epsilon_{0}}\begin{bmatrix}{\mathcal{T}}(t_{0})\\\ {\mathcal{C}}_{T}(t_{0})\end{bmatrix}+c_{0}\rho_{0}\begin{bmatrix}0\\\ P_{0}\end{bmatrix}.$ Suppose phase $i$ has a drift period of $d$ days. In the model, the parameter values change multiplicatively during the drift period. Let $\epsilon_{j}=\epsilon_{0}x^{j}$, and $1-c_{j}=(1-c_{0})/y^{j}$ for $1\leq j\leq d$, where $x$ and $y$ are unknown multipliers by which the two parameters change every day. The final value of the parameters will be $\epsilon_{d}=\epsilon_{0}x^{d}$ and $1-c_{d}=(1-c_{0})/y^{d}$. Let $\tilde{\beta}_{j}=\tilde{\beta}_{0}(\frac{\tilde{\beta}_{d}}{\tilde{\beta}_{0}})^{j/d}$ and $\tilde{\rho}_{j}=\tilde{\rho}_{0}(\frac{\tilde{\rho}_{d}}{\tilde{\rho}_{0}})^{j/d}$, for $1\leq j\leq d$. These numbers can be computed since $\tilde{\beta}_{0}$, $\tilde{\beta}_{d}$, $\tilde{\rho}_{0}$, and $\tilde{\rho}_{d}$ are known. Let $\beta_{j}=\frac{\tilde{\beta}_{j}}{(1-\epsilon_{j})(1-c_{j})}$ and $\rho_{j}=\frac{\tilde{\rho}_{j}}{\epsilon_{j}(1-c_{j})}$ for $1\leq j\leq d$. Then we can write: $\begin{bmatrix}{\mathcal{M}}(t_{0}+j)\\\ {\mathcal{R}}(t_{0}+j)\end{bmatrix}=\begin{bmatrix}g_{j}&0\\\ \gamma&1\end{bmatrix}\cdot\begin{bmatrix}{\mathcal{M}}(t_{0}+j-1)\\\ {\mathcal{R}}(t_{0}+j-1)\end{bmatrix}$ where $\displaystyle g_{j}$ $\displaystyle=$ $\displaystyle\beta_{j-1}(1-\epsilon_{j-1})(1-\frac{{\mathcal{M}}(t_{0}+j-1)+{\mathcal{R}}(t_{0}+j-1)}{\rho_{j-1}P_{0}})-\gamma+1$ $\displaystyle=$ $\displaystyle\tilde{\beta}_{j-1}(\frac{1}{1-c_{j-1}}-\frac{\epsilon_{j-1}}{\tilde{\rho}_{j-1}}\frac{{\mathcal{M}}(t_{0}+j-1)+{\mathcal{R}}(t_{0}+j-1)}{P_{0}})-\gamma+1$ $\displaystyle=$ $\displaystyle\tilde{\beta}_{j-1}(\frac{y^{j-1}}{1-c_{0}}-\frac{\epsilon_{0}x^{j-1}}{\tilde{\rho}_{j-1}}\frac{{\mathcal{M}}(t_{0}+j-1)+{\mathcal{R}}(t_{0}+j-1)}{P_{0}})-\gamma+1$ Therefore, both ${\mathcal{M}}(t_{0}+j)$ and ${\mathcal{R}}(t_{0}+j)$ are polynomials in $x$ and $y$. It is straightforward to show that the degrees of ${\mathcal{M}}(t_{0}+j)$ and ${\mathcal{R}}(t_{0}+j)$ equal $2^{j}-j-1$ and $2^{j-1}-j-2$ respectively. At the end of drift period, we have: $\displaystyle\begin{bmatrix}{\mathcal{M}}(t_{0}+d)\\\ {\mathcal{R}}(t_{0}+d)\end{bmatrix}$ $\displaystyle=$ $\displaystyle\frac{1}{\epsilon_{d}}\begin{bmatrix}{\mathcal{T}}(t_{0}+d)\\\ {\mathcal{R}}_{T}(t_{0}+d)\end{bmatrix}+c_{d}\rho_{d}\begin{bmatrix}0\\\ P_{0}\end{bmatrix}$ (14) $\displaystyle=$ $\displaystyle\frac{1}{\epsilon_{d}}\begin{bmatrix}{\mathcal{T}}(t_{0}+d)\\\ {\mathcal{R}}_{T}(t_{0}+d)\end{bmatrix}+c_{d}\frac{\tilde{\rho}_{d}}{\epsilon_{d}(1-c_{d})}\begin{bmatrix}0\\\ P_{0}\end{bmatrix}$ $\displaystyle=$ $\displaystyle\frac{1}{\epsilon_{0}x^{d}}\begin{bmatrix}{\mathcal{T}}(t_{0}+d)\\\ {\mathcal{R}}_{T}(t_{0}+d)\end{bmatrix}+\frac{\tilde{\rho}_{d}}{\epsilon_{0}x^{d}}(\frac{y^{d}}{1-c_{0}}-1)\begin{bmatrix}0\\\ P_{0}\end{bmatrix}\mbox{~{}~{}~{}~{}~{}~{}}$ For what values of unknown multipliers $x$ and $y$ are the relationships in equation (14) satisfied? To see this, we analyze the quantities ${\mathcal{N}}(t_{0}+d)$, ${\mathcal{M}}(t_{0}+d)$, and ${\mathcal{C}}(t_{0}+d)$. From (12) we have: $\displaystyle{\mathcal{C}}(t_{0}+d)$ $\displaystyle=$ $\displaystyle{\mathcal{C}}(t_{0}+d-1)+{\mathcal{N}}(t_{0}+d)$ $\displaystyle{\mathcal{M}}(t_{0}+d)$ $\displaystyle=$ $\displaystyle{\mathcal{N}}(t_{0}+d)+(1-\gamma){\mathcal{M}}(t_{0}+d-1)$ $\displaystyle{\mathcal{N}}(t_{0}+d)$ $\displaystyle=$ $\displaystyle\beta_{d-1}(1-\epsilon_{d-1})S(t_{0}+d-1){\mathcal{M}}(t_{0}+d-1)$ $\displaystyle=$ $\displaystyle\beta_{d-1}(1-\epsilon_{d-1})\left(1-\frac{1}{\rho_{d-1}P_{0}}{\mathcal{C}}(t_{0}+d-1)\right){\mathcal{M}}(t_{0}+d-1)$ Therefore, $\displaystyle{\mathcal{C}}(t_{0}+d-1)$ $\displaystyle=$ $\displaystyle\frac{1}{\epsilon_{d}}{\mathcal{C}}_{T}(t_{0}+d)+c_{d}\rho_{d}P_{0}-\frac{1}{\epsilon_{d}}{\mathcal{N}}_{T}(t_{0}+d)$ $\displaystyle=$ $\displaystyle\frac{1}{\epsilon_{d}}{\mathcal{C}}_{T}(t_{0}+d-1)+c_{d}\rho_{d}P_{0}$ $\displaystyle{\mathcal{M}}(t_{0}+d-1)$ $\displaystyle=$ $\displaystyle\frac{1}{\epsilon(1-\gamma)}({\mathcal{T}}(t_{0}+d)-{\mathcal{N}}(t_{0}+d))$ $\displaystyle=$ $\displaystyle\frac{1}{\epsilon}{\mathcal{T}}(t_{0}+d-1)$ $\displaystyle{\mathcal{N}}(t_{0}+d)$ $\displaystyle=$ $\displaystyle\beta_{d-1}(1-\epsilon_{d-1})\left(1-\frac{1}{\epsilon_{d}\rho_{d-1}P_{0}}{\mathcal{C}}_{T}(t_{0}+d-1)-\frac{c_{d}\rho_{d}}{\rho_{d-1}}\right)\frac{1}{\epsilon_{d}}{\mathcal{T}}(t_{0}+d-1)$ $\displaystyle=$ $\displaystyle\frac{\tilde{\beta}_{d-1}}{\epsilon_{d}(1-c_{d-1})}\left(1-\frac{1}{\epsilon_{d}\rho_{d-1}P_{0}}{\mathcal{C}}_{T}(t_{0}+d-1)-\frac{c_{d}\rho_{d}}{\rho_{d-1}}\right){\mathcal{T}}(t_{0}+d-1).$ Since ${\mathcal{N}}(t_{0}+d)=\frac{1}{\epsilon_{d}}{\mathcal{N}}_{T}(t_{0}+d)=\frac{\tilde{\beta}_{d-1}}{\epsilon_{d}}\left(1-\frac{1}{\tilde{\rho}_{d-1}P_{0}}{\mathcal{C}}_{T}(t_{0}+d-1)\right){\mathcal{T}}(t_{0}+d-1)$ where the second equality is from fundamental equation (10), we have: $\displaystyle 1-\frac{1}{\tilde{\rho}_{d-1}P_{0}}{\mathcal{C}}_{T}(t_{0}+d-1)$ $\displaystyle=$ $\displaystyle\frac{1}{1-c_{d-1}}\left(1-\frac{1}{\epsilon_{d}\rho_{d-1}P_{0}}{\mathcal{C}}_{T}(t_{0}+d-1)-\frac{c_{d}\rho_{d}}{\rho_{d-1}}\right)$ $\displaystyle=$ $\displaystyle\frac{1}{1-c_{d-1}}-\frac{c_{d}\rho_{d}}{(1-c_{d-1})\rho_{d-1}}-\frac{1}{\epsilon_{d}\rho_{d-1}(1-c_{d-1})P_{0}}{\mathcal{C}}_{T}(t_{0}+d-1)$ $\displaystyle=$ $\displaystyle\frac{y^{d-1}}{1-c_{0}}-\frac{\epsilon_{d-1}c_{d}\rho_{d}}{\tilde{\rho}_{d-1}}-\frac{1}{x\tilde{\rho}_{d-1}P_{0}}{\mathcal{C}}_{T}(t_{0}+d-1)$ $\displaystyle=$ $\displaystyle\frac{y^{d-1}}{1-c_{0}}-\frac{c_{d}\tilde{\rho}_{d}}{x(1-c_{d})\tilde{\rho}_{d-1}}-\frac{1}{x\tilde{\rho}_{d-1}P_{0}}{\mathcal{C}}_{T}(t_{0}+d-1)$ $\displaystyle=$ $\displaystyle\frac{y^{d-1}}{1-c_{0}}+\frac{\tilde{\rho}_{d}}{x\tilde{\rho}_{d-1}}-\frac{y^{d}\tilde{\rho}_{d}}{x(1-c_{0})\tilde{\rho}_{d-1}}-\frac{1}{x\tilde{\rho}_{d-1}P_{0}}{\mathcal{C}}_{T}(t_{0}+d-1).$ Therefore, $\displaystyle x$ $\displaystyle=$ $\displaystyle\frac{\left(\frac{\tilde{\rho}_{d}}{\tilde{\rho}_{d-1}}-\frac{y^{d}\tilde{\rho}_{d}}{(1-c_{0})\tilde{\rho}_{d-1}}-\frac{1}{\tilde{\rho}_{d-1}P_{0}}{\mathcal{C}}_{T}(t_{0}+d-1)\right)}{\left(1-\frac{y^{d-1}}{1-c_{0}}-\frac{1}{\tilde{\rho}_{d-1}P_{0}}{\mathcal{C}}_{T}(t_{0}+d-1)\right)}.$ (15) Equation (15) expresses $x$ as a degree $d$ rational function of $y$. Substituting this in the polynomial ${\mathcal{M}}(t_{0}+d)$, we obtain a degree $d(2^{d}-d-1)$ rational function in $y$. Equating it to $\frac{1}{\epsilon_{d}}{\mathcal{T}}(t_{0}+d)=\frac{1}{\epsilon_{0}x^{d}}{\mathcal{T}}(t_{0}+d)$ results in a polynomial in $y$ of degree bounded by $d(2^{d}-d)$, say $P(y)$, whose roots are the possible values of $y$. Therefore, there are at most $d(2^{d}-d)$ many values of $(x,y)$ that satisfy all the required equations. While the potential number of solutions is very large for even moderate values of $d$ (say $d>20$), most of the solutions are likely to be infeasible since a feasible solution needs to satisfy the conditions that $\epsilon_{d}$ must be in the range $[0,1]$ and $c_{d}$ in the range $[-1,1]$. Further, there is a simple and efficient way to list out good estimates for all the solutions: since $-1<c_{d}<1$, one can step through possible values of $c_{d}$ in the range using small discrete steps, compute the value of $y$ for the chosen value of $c_{d}$, compute value of $x$ using equation (15), and then check if $P(y)$ is close to zero. This will list out good estimates of all feasible solutions. ∎
# Topology and complexity of the hydrogen bond network in classical models of water Fausto Martelli IBM Research Europe, Hartree Centre, Daresbury, WA4 4AD, United Kingdom<EMAIL_ADDRESS>Department of Physics and CNR Institute of Complex Systems, Sapienza University of Rome, P.le Aldo Moro 5, 00185 Roma, Italy<EMAIL_ADDRESS> ###### Abstract Over the years, plenty of classical interaction potentials for water have been developed and tested against structural, dynamical and thermodynamic properties. On the other hands, it has been recently observed (F. Martelli et. al, ACS Nano, 14, 8616–8623, 2020) that the topology of the hydrogen bond network (HBN) is a very sensitive measure that should be considered when developing new interaction potentials. Here we report a thorough comparison of 11 popular non polarizable classical water models against their HBN, which is at the root of water properties. We probe the topology of the HBN using the ring statistics and we evaluate the quality of the network inspecting the percentage of broken and intact HBs. For each water model, we assess the tendency to develop hexagonal rings (that promote crystallization at low temperatures) and pentagonal rings (known to frustrate against crystallization at low temperatures). We then introduce the _network complexity index_ , a general descriptor to quantify how much the topology of a given network deviates from that of the ground state, namely of hexagonal or cubic ice. Remarkably, we find that the network complexity index allows us to relate, for the first time, the dynamical properties of different water models with their underlying topology of the HBN. Our study provides a benchmark against which the performances of new models should be tested against, and introduces a general way to quantify the complexity of a network which can be transferred to other materials and that links the topology of the HBN with dynamical properties. Finally, our study introduces a new perspective that can help in rationalizing the transformations among the different phases of water and of other materials. Water models, Classical potentials, Network topology, Network complexity, Hydrogen bonds ††preprint: AIP/123-QED ## I Introduction On our planet, water is the only substance that can be found co-existing in the solid, liquid and vapour phases outside research laboratories. Its molecular simplicity hides a remarkably wide list of anomalous behaviors that stretch over the the most complex phase diagram of any pure substance Salzmann (2019), and whose origin lies in a critical point located at low temperatures and low pressures Palmer _et al._ (2014); Sellberg _et al._ (2014); Debenedetti, Sciortino, and Zerze (2020); Kringle _et al._ (2020); Kim _et al._ (2020). Nonetheless, water plays a primary role in many industrial, biological and geological processes. Therefore, there is a great interest in developing classical interaction potentials able to embrace water’s complex nature. This intent is aggravated by the wide span of thermodynamic conditions at which water exists (from low temperatures and low pressures of the interstellar medium to high pressures and high temperatures in the core of planets), and by the broad range of timescales required for several processes to occur (from heterogeneous nucleation to protein folding to geological processes). After a long-lasting debate Poole _et al._ (1992); Liu _et al._ (2010); Limmer and Chandler (2011); Wikfeldt, Nilsson, and Pettersson (2011); Palmer, Car, and Debenedetti (2013); Limmer and Chandler (2013); Palmer _et al._ (2014); Limmer and Chandler (2014); Chandler (2016); Palmer _et al._ (2016a, 2018, b), it is now accepted that liquid water is a mixture Palmer _et al._ (2014); Sellberg _et al._ (2014); Palmer _et al._ (2018); Debenedetti, Sciortino, and Zerze (2020); Kim _et al._ (2020); Kringle _et al._ (2020) of molecules whose local neighborhood constantly change between an ordered tetrahedral state with local lower density, and a more distorted tetrahedral state with local higher density Martelli _et al._ (2020); Shi and Tanaka (2020a, b); Shi, Russo, and Tanaka (2018a); Russo and Tanaka (2014a); Akahane and Tanaka (2018); Santra _et al._ (2015); Huang _et al._ (2009); Nilsson and Pettersson (2015); Wikfeldt, Nilsson, and Pettersson (2011); De Marzio _et al._ (2017); Martelli (2019). These two local structures continuously interchange with each other, giving rise to a complex network of bonds that actively determines the properties of water at the macro scale Martelli (2019). The percentage of such local environments depends on the thermodynamic conditions Martelli (2019); Wikfeldt, Nilsson, and Pettersson (2011), and eventually liquid water becomes a 1:1 mixture at low temperatures and pressures Palmer _et al._ (2014); Debenedetti, Sciortino, and Zerze (2020). Classical interaction potentials have been developed over the years to reproduce experimental structural, dynamical and/or thermodynamic properties of water (over a relatively reduced set of thermodynamic points), and several studies have compared them against a plethora of observables Palmer _et al._ (2016b); Pekka and Lennart (2001); Mao and Zhang (2012); Lee and Kim (2019); Jorgensen _et al._ (1983); Harrach and Drossel (2014); González _et al._ (2010); Zielkiewicz (2005); Steinczinger, Jóvári, and Pusztai (2017); Dix, Lue, and Carbone (2018). On the other hand, even though Bernal and Fowler first recognized almost 100 years ago that water molecules build a complex network of bonds that is at the heart of water’s anomalous behavior Bernal and Fowler (1933), the pivotal role of the hydrogen bond network (HBN) has so far been showcased only in a handful of important cases involving transformations between complex phases of water Tse _et al._ (1999); Marton̆ák, Donadio, and Parrinello (2004, 2005); Palmer _et al._ (2014); Shephard _et al._ (2017); Martelli _et al._ (2018); Martelli (2019), the mutual interactions between water and biological membranes Martelli, Crain, and Franzese (2020), and between water and graphene sheets for technological purposes Chiricotto _et al._ . In particular, the inspection of the topology of the HBN in biological environments is opening new avenues in enhancing the efficacy of new drugs/vaccines Martelli, Calero, and Franzese (2021). The HBN has never been used as a metric to test water models. Two reasons for this lack of comparison are: (i) it is very hard to have a direct experimental description of the HBN in liquid phases, and (ii) it is hard to probe the HBN from a computational perspective. Here we fill the gap of the point (ii) above by comparing the HBN of 11 popular and widely adopted classical water models at ambient conditions. We study the TIP3P Jorgensen _et al._ (1983) the SPC Berendsen _et al._ (1981), the SPC/E Berendsen, Grigera, and Straatsma (1987) and the flexible SPC Toukan and Rahman (1985); Amira, Spángberg, and Hermansson (2004) as 3-points models; the TIP4P Jorgensen _et al._ (1983), the TIP4P-Ice Abascal _et al._ (2005), the TIP4P/2005 Abascal and Vega (2005), the flexible TIP4P/2005 González and Abascala (2011) and the TIP4P-Ew Horn _et al._ (2004) as 4-points models; the TIP5P Mahoney and Jorgensen (2000) and the TIP5P-Ew Rick (2004) as 5-points models. We probe the topology of the HBN using the ring statistics, a theoretical tool that has seen increasingly high relevance in determining the properties of bulk water Martelli _et al._ (2016, 2018); Martelli (2019); Formanek and Martelli (2020); Palmer _et al._ (2014); Santra _et al._ (2015); Leoni _et al._ (2019); Camisasca _et al._ (2019); Marton̆ák, Donadio, and Parrinello (2004, 2005); Russo and Tanaka (2014b); Palmer _et al._ (2014); Fitzner _et al._ (2019); Shi and Tanaka (2018), of aqueous solutions Bakó _et al._ (2017); Pothoczki, Pusztai, and Bakó (2018, 2019); Li _et al._ (2020a) and of water under confinement Martelli, Crain, and Franzese (2020); Chiricotto _et al._ . We then measure the quality of the HBN for each water model in terms of broken and intact HBs, a measure intimately linked to the fluidity and tetrahedrality of water DiStasio Jr. _et al._ (2014); Martelli, Crain, and Franzese (2020). We also introduce the concept of _network complexity index_ $\xi$ that allows to quantify how much a given HBN deviates from the HBN of water in the crystalline phase at low temperatures and ambient pressure, i.e., hexagonal or cubic ice. As a showcase, we apply this index $\xi$ to one of the rings definition and counting scheme here adopted. Remarkably, the index $\xi$ allows us to link dynamical properties of water with the topology of the underlying HBN and the corresponding structural properties. Finally, we show that the HBN topology is more sensitive to the size of the simulation box with respect to, e.g., structural properties measured _via_ the two-bodies pair correlation function. Therefore, the inspections of the network topology should always be considered when simulating network-forming materials. This issue becomes particularly relevant when dealing with _ab initio_ molecular dynamics simulations which are restricted to small simulation cells. The article is organized as follows. In Section II we report the details of the numerical simulations, the ring definitions and counting schemes, and we introduce the network complexity index. In Section III we report our main findings for all 11 water models here inspected. Conclusions and final remarks are reported in Section IV. ## II Computational details In this section we describe the numerical setup, the protocols implemented to count rings and we introduce the definition of the network complexity index. ### II.1 Numerical simulations Our study is based on classical molecular dynamics simulations of systems composed of $N=1100$ water molecules described by different 11 interaction potentials in the isobaric ($N$$p$$T$) ensemble. We have employed Nosé-Hoover thermostat Nosé (1984); Hoover (1985) with 0.2 ps relaxation time to maintain constant temperature at $T=300$ K, and Parrinello-Rahman barostat Parrinello and Rahman (1981) with 2 ps relaxation time to maintain constant pressure at 1 bar. We have performed simulations with the GROMACS 18.0.1 package Abraham _et al._ (2015). All simulations have been equilibrated for 1 ns. The production runs achieved 3 ns. For each water model we have averaged over 10 independent trajectories. Our analysis investigates the properties of liquid water in the presence of thermal noise. ### II.2 Ring statistics and network complexity index In order to compute the ring statistics it is necessary to follow two steps. First of all, it is necessary to define the link between atoms/molecules. Possible definitions can be based on the formation of bonds, interaction energies, geometric distances, etc.. The second step is the definition of ring and the corresponding counting scheme. This task is of particular relevance in directional networks, like water or silica, were the donor/acceptor nature of the bonds breaks the symmetry in the linker search path. Several definitions of rings and counting schemes have been reported in the literature King (1967); Rahman and Stillinger (1973); Guttman (1990); Franzblau (1991); Wooten (2002); Yuan and Cormack (2002); Roux and Jund (2010). Such different definitions have yielded to different interpretations of numerical results even for the simplest crystalline structures, not to mention more complicated networks such as amorphous silicate structures Jin _et al._ (1994); Hobbs _et al._ (1998); Guttman (1990); Marians and Hobbs (1988, 1990); Yuan and Cormack (2002) and water Fitzner _et al._ (2019); Camisasca _et al._ (2019); Martelli (2019); Santra _et al._ (2015). In the case of water, these inconsistencies have been recently reconciled by Formanek and Martelli showing that they were caused by different counting schemes Formanek and Martelli (2020). Here, we report a thorough analysis of the HBN for the 11 classical non- polarizable models of water. This benchmark study adopts the three ring definitions and counting schemes reported in Ref. Formanek and Martelli (2020). The definition of HB follows the geometric construction described in Ref. Luzar and Chandler (1996). In this regard, any quantitative measure of HBs in liquid water is somewhat ambiguous, since the notion of an HB itself is not uniquely defined. However, qualitative agreement between the definition here adopted and several other proposed definitions have been deemed satisfactory over a wide range of thermodynamic conditions, including the one here investigated Prada-Gracia, Shevchuk, and Rao (2013); Shi, Russo, and Tanaka (2018b). We construct rings by starting from a tagged water molecule (marked as 1 in fig. 1) and recursively traversing the HBN until the starting point is reached again or the path exceeds the maximal ring size considered (12 water molecules in our case). In the first scheme sketched in fig. 1 a), molecule 1 _donates_ a HB emphasized by the red arrow, and we restrict our counting only to the shortest rings King (1967), i.e., to rings that can not be further decomposed into smaller ones (the 6-folded ring in this case). We will refer to this scheme to as d1. This scheme emphasizes the directional nature of the HBs resulting in an enhanced hexagonal character of the HBN Formanek and Martelli (2020). It is therefore well suited for investigating phenomena such as nucleation, where hexagonal rings are the elemental building blocks of the final, crystal network. In the second scheme sketched in fig. 1 b), we consider only rings formed when molecule 1 _accepts_ a HB and we restrict our counting only to the shortest ring King (1967) (the 5-folded ring in this case). We will refer to this scheme to as d2. This scheme emphasizes the formation of pentagonal rings in the network Formanek and Martelli (2020), which are known to play an important role at supercooled conditions frustrating against crystallization Russo and Tanaka (2014a); Shi, Russo, and Tanaka (2018a); Martelli (2019), as well as in promoting the crystallization of clathrate structures Li _et al._ (2020b). These two counting schemes d1 and d2 provide drastically different distributions Formanek and Martelli (2020) because of the different energies involved in accepting and donating a HB. As we will show in the forthcoming discussion, such difference translates into distinct percentages of coordination defects of the kind $\textit{A}_{2}\textit{D}_{1}$ and $\textit{A}_{1}\textit{D}_{2}$, where $\textit{A}_{x}\textit{D}_{y}$ indicates that a water molecules accepts $x$ and donates $y$ HBs. In the third scheme sketched in fig. 1 c) we loosen the previous restrictions. Molecule 1 can now either accept or donate a HB, hence counting both the 6-folded and the 5-folded ring. In the following, we will refer to this counting scheme to as d3. As shown in Ref. Formanek and Martelli (2020), these three different counting schemes carry different, but complementary physical information and, therefore, allow us to make a proper comparison on the topology of the HBN generated by different classical models of water. Figure 1: Schematic representation of the three counting schemes. Red filled circles represent oxygen atoms, while red empty circles represent hydrogen atoms. The network of HBs is represented by the arrows Water molecule labeled as 1 is the starting molecule from which rings are counted. a): Molecule donates a HB (red arrow) and the shortest ring is the hexagonal one. b): Molecule 1 can accept a HB (one of the two red arrows) and the shortest ring is the pentagonal one c): Molecule 1 can either donate one HB or accept a HB, generating both the short hexagonal and pentagonal ring. When studying the topology of a network via the ring statistics, the probability distribution P(n) of having a n-folded ring is a normalized quantity that does not reflect the overall number of each n-folded ring. On the other hand, the actual number of rings is important to understand the degree of complexity in a network (for a given number of molecules). The stable crystalline phase of water at ambient pressure is the hexagonal(cubic) ice, Ih(c), characterized by an hexagonal network of bonds. In a sample of liquid water at higher temperatures, the thermal noise allows water molecules to explore a larger configurational space and, hence, the underlying HBN hosts also shorter and longer rings. Intuitively, such network is more ”complex” with respect to the HBN of the ground state, and its fluctuations are related to the dynamical properties (translational and rotational diffusion) of water molecules. Knowing the symmetry of the HBN at the ground state, we can measure the deviation from it. We introduce the _network complexity index_ $\xi$ defined as the ratio between the number of 6-folded rings and the total number of rings: $\xi=\frac{n_{6}}{\sum_{i=3}^{12}n_{i}}$ (1) where $n_{i}$ is the number of the $i-$th ring. The sum on the denominator of eq. 1 runs from n=3, the shortest ring length possible, to n=12, the longest ring length here considered. The network complexity index $\xi$ encodes the number of rings in a given network. For the ground state Ih(c), the network complexity index is trivially $\xi=1$. We can define the case of maximal disorder in the network when each ring length occur with the same frequency in the network. This case corresponds to a flat distribution in P(n) and, since in our case we consider 10 possible ring lengths (from n=3 to n=12), $\xi=0.1$. Here, we compute $\xi$ for the counting scheme d3, scheme c) in fig. 1 which, compared to the other schemes, accounts for the presence in the network of longer rings. The definition of network complexity index can be extended to other networks with different ground states (i.e., non-hexagonal networks), in which case the nominator in eq. 1 will be $n_{j}$, $j$ being the length of the characteristic ring length at the ground state. However, when comparing the index $\xi$ in different simulations, care must be taken in order to ensure that (i) the same definition of ring and counting schemes are used, (ii) the same maximum search path (or maximal ring size) is implemented. It is worthy to remark, at this point, that our analysis occurs in the presence of thermal noise. As recently shown by Montes de Oca et al., the percentage of broken and intact bonds drastically changes upon removal of the thermal noise de Oca _et al._ (2020), i.e., in correspondence with the inherent potential energy surfaces (IPES). While we do expect quantitative differences in correspondence with the IPES with respect to the results that we will show in Section III, we are confident that the overall trend should be preserved. ## III Results In this section, we report and discuss our results in terms of topology and complexity of the HBNs, as well as their quality in terms of broken and intact HBs for the water 11 models. We then compare the network complexity indices and we relate them to the dynamical properties of each water model. Finally, we show and discuss the effects of the simulation box size on the topology of the HBN. ### III.1 3 points models we start our analysis by comparing, in fig. 2, the oxygen-oxygen two-bodies pair correlation functions g2(r) for 3-points models with the g2(r) obtained from various scattering experiments Skinner _et al._ (2013); Soper and Benmore (2008) (open circles and squares) and _ab initio_ molecular dynamics (AIMD) simulations DiStasio Jr. _et al._ (2014) (open triangles) at the PBE0 level of theory accounting for non-local van der Waals/dispersion interactions DiStasio Jr. _et al._ (2014). It is worth to mention, at this point, that the PBE0+vdw level of theory gives an accurate g2(r) compared to the experimental one, but does not capture the correct density difference between liquid water at ambient conditions and Ih. We can observe that the TIP3P model (black line) has the first peak at $\sim 2.8$ nm with intensity comparable with that of scattering experiments and _ab initio_ simulations. On the other hand, after the first minimum the distribution is almost flat with no peaks at larger distances. This is indicative of a lack of structurization at larger distances. The SPC model (red line) is able to perform better than the TIP3 model, with a g2(r) showing hints of a second and a third peak, though shifted compared to the experimental and _ab initio_ g2(r). On the other hand, the intensity of the first peak overcomes the experimental and AIMD first peak. The SPC/E model (green line) results in a better g2(r) in terms of intensity and position of the second and third peak with respect to the experimental and the _ab initio_ g2(r), reflecting the better performances in density and diffusion constant than the SPC model Berendsen, Grigera, and Straatsma (1987). On the other hand, the intensity of the first peak is further enhanced. The flexible SPC water model (blue line) is a re-parametrization of the SPC water model in which the O–H stretching is made anharmonic, and thus the dynamical behavior is well described and bulk density and permettivity are correct Praprotnik, Janežič, and Mavri (2004). The g2(r) of the flexible SPC model is characterized by a high intensity first peak and a deeper first minimum with respect to the rigid SPC model, and correctly captures the profile of the g2(r) at larger distances. Overall, the sequence TIP3, SPC, SPC/E and flexible SPC is characterized by an increment in the height of the first peak that also shifts slightly towards lower distances, a corresponding deepening of the first minimum and a gradual appearance of a second and third peak that tend to overlap with the experimental and with AIMD ones for the SPC/E and the flexible SPC models. Figure 2: The oxygen-oxygen two-bodies pair correlation, g2(r), of liquid water for the TIP3P (black), SPC (red), SPC/E (green) and the flexible SPC model (blue). The g2(r) obtained from various scattering experiments Skinner _et al._ (2013); Soper and Benmore (2008) and _ab initio_ molecular dynamics simulations DiStasio Jr. _et al._ (2014) are reported for comparison with open symbols. In fig. 3 we report the probability distribution P(n) of having a n-folded ring, with n$\in[3,12]$ for the classical 3 points models. The distribution for the TIP3P model is reported as open circles, SPC as open squares, SPC/E as open diamonds and flexible SPC as open triangles. It is worth to remark, at this point, that the P(n) is a normalized distribution and, therefore, it does not reflect the actual number of each ring. Panel a) reports the P(n) according to the definition d1 sketched in fig. 1 a). According to this counting scheme that emphasizes the directionality of the HBs, all networks have a dominating hexagonal character. Such character is milder in the TIP3P model (black open circles), which shows a very broad P(n) and whose network accommodates also 10- and 11-folded rings. The hexagonal character of the network grows moving to the SPC model (red open squares), with a corresponding reduction of longer (n$>$8) rings. The reduction of longer rings develops along with an enhancement of 5- and 7-folded rings, which are comparable to 6-folded rings in terms of energy Camisasca _et al._ (2019). In particular, the HBN of the SPC model described with this counting scheme is almost completely deprived of rings with n$>$10\. The hexagonal character of the network further grows in the SPC/E (green open diamonds) and in the flexible SPC (open blue triangles). These models show almost identical P(n)s. In particular, besides the enhanced hexagonal character, the HBNs of these models have an enhanced pentagonal and heptagonal character. Consequently, the contribution from longer rings, namely rings with n=8 and n=9, is less marked. Therefore, loosening the holonomic constraints and allowing the (re-parametrized) SPC model to vibrate has a major effect not just on the structural properties (as shown from the g2(r), fig. 2) but also on the topology of the HBN. Panel b) reports the P(n) according to the definition d2 sketched in fig. 1 b). As shown in Ref. Formanek and Martelli (2020), this counting scheme emphasizes the pentagonal character of the HBN, as the starting water molecule must accept one HB instead of donating. Interestingly, the TIP3P model shows, according to this counting scheme, an almost equal pentagonal and hexagonal character with a tail populating configurations up to n=10. The HBN of the SPC model, on the other hand, shows an improved pentagonal character with a slight increase in the hexagonal character and a reduction of longer rings which mostly disappear at n$>$9\. The pentagonal character of the HBN over the hexagonal one become particularly dominant in the SPC/E and in the flexible SPC models, for which rings characterized by n$>$8 are mostly absent. Panel c) reports the P(n) according to the definition d3 sketched in fig. 1 c). Since this counting scheme does not discern among the donor/acceptor character of the HBs and does not implement the shortest path criterion King (1967), the resulting topology is more complex with respect to the previous counting schemes. Therefore, for this scheme we also compute the network topology index $\xi$ (eq. 1) and report them in table 1. According to this counting scheme, the HBN for all the inspected models show a similar character in terms of hexagonal and heptagonal rings. The TIP3P model shows a quite broad distribution with a considerable contribution of longer (n$>$7) rings. The presence of rings with n=12 reflects the more complex topology of the network with respect to the previous counting schemes. Interestingly, such distribution is comparable with that of the TIP3P model optimized MacKerell Jr. _et al._ (1998) for biological simulations Martelli, Crain, and Franzese (2020). Because of the very broad character of P(n), the TIP3P model has a low value of network complexity index $\xi=0.1496$, not too far from the value $\xi=0.1$ that characterizes (as described in Section II.2) a flat distribution P(n) with equal populations. Moving to the SPC model, we observe a slight enhancement of n=5, and a more pronounced enhancement of n=6 and n=7 which, as for the TIP3P model, equally characterize the HBN. Contrarily, the network is deprived of longer rings, namely of n=10, n=11 and n=12 rings. The slight increase in n=6 causes a small increment in the network complexity index to $\xi=0.1665$. Moving to the SPC/E model, we observe a mild increment in n=5, with a relevant increment in n=6 and n=7, as well as a marked depletion of longer rings. The complexity index for the SPC/E model further increases to $\xi=0.1863$. A similar distribution characterizes the HBN of the flexible SPC model, which almost overlaps with the P(n) of the SPC/E model. The distribution of the flexible SPC model suggests that, as mentioned above, the introduction of flexibility plays a very important role in shaping the network. The complexity index for the flexible SPC model rises to $\xi=0.1922$. Figure 3: Probability distributions of the hydrogen-bonded n-folded rings, P(n), for liquid water at ambient conditions described by the TIP3P (black open circles), the SPC (red open squares), the SPC/E (green open diamonds) and the flexible SPC (blue open triangles) models. Panel a) reports the P(n) according to the definition sketched in fig. 1 a); panel b) reports the p(n) according to the definition sketched in fig. 1 b); panel c) reports the P(n) according to the definition sketched in fig. 1 c). \| TIP3P \| SPC \| SPC/E \| SPC-Flex ---\|---\|---\|---\|--- $\xi$ \| 0.1496 \| 0.1665 \| 0.1863 \| 0.1922 Table 1: Values of the network complexity index $\xi$ computed for the rings counting scheme d3 for the 3 points models of water. In fig. 4, we report the percentage of broken and intact HBs for the 3 point models of water and, for comparison, the _ab initio_ liquid water at the PBE0 level of theory with vdW long range interactions at a temperature of 330K to account for nuclear quantum effects DiStasio Jr. _et al._ (2014). We adopt the following syntax: $\textit{A}_{x}\textit{D}_{y}$ indicates the number of acceptors ($\textit{A}_{x}$) and donors ($\textit{D}_{y}$) HBs. The network in _ab initio_ liquid water (black open circles) is dominated by a intact HBs ($\textit{A}_{2}\textit{D}_{2}$), which account for $\sim 48\%$. The second highest configuration is the $\textit{A}_{1}\textit{D}_{2}$, with $\sim 20\%$, followed by $\textit{A}_{2}\textit{D}_{1}$ ($12\%$), $\textit{A}_{2}\textit{D}_{2}$ ($\sim 10\%$) and $\textit{A}_{3}\textit{D}_{2}$ ($\sim 5\%$). The percentage of this last configuration, i.e., the configuration $\textit{A}_{3}\textit{D}_{2}$, is shared with all 3 points models. The percentage of broken and intact HBs in the TIP3P model (red open squares) quantitatively differs from the distribution of _ab initio_ liquid water, but qualitatively shows a similar behaviour. In particular, the distribution for the TIP3P model is dominated by a markedly reduced amount of intact HBs with a percentage $\textit{A}_{2}\textit{D}_{2}\sim 33\%$, followed by $\textit{A}_{1}\textit{D}_{2}$ with $\sim 23\%$, $\textit{A}_{2}\textit{D}_{1}$ with $\sim 12\%$ and an almost equal percentage of $\textit{A}_{1}\textit{D}_{1}$. The low percentage of intact HBs explains the very broad distribution of rings (fig. 3) as well as the absence of structurization beyond the first hydration shell in the g2(r) (fig. 2). Moving from the TIP3 model to the SPC model (open green diamonds) we observe a slight decrease in the configuration $\textit{A}_{1}\textit{D}_{1}$ to $\sim 10\%$ and an increment in the percentage of intact HBs to $\sim 37\%$. Therefore, the SPC model is characterized by a slightly enhanced percentage of fully coordinated configurations which cause an enhancement in 5-, 6-, and 7-folded rings, as reported in fig. 3, and are responsible for the appearance of the second peak in the g2(r) (fig. 2). The modified SPC models, namely the SPC/E (open blue triangles) and the flexible SPC model (open orange left triangles), are both characterized by a distribution that mostly overlap with the _ab initio_ liquid water potential. The high percentage of intact HBs reaches values in the order of $\sim 45-48\%$ for both SPC/E and the flexible SPC models, and this enhanced tetra-coordinated character explains the enhanced number of 5-, 6-, and 7-folded rings and the corresponding decrease in longer rings, as reported in fig. 3, as well as an enhancement in the intensity of the first peak in the g2(r) and a corresponding deepening of the following minimum. Figure 4: Percentage-wise decomposition of the intact HBs per water molecule into acceptor-(A) and donor-(D) for _ab initio_ liquid water at T=330 K as black open circles, and for the 3 points models. The TIP3P model is reported as red open squares, the SPC as green open diamonds, the SPC/E as blue open triangles and the flexible SPC as orange open left triangles. The $x$-axis labels $\textit{A}_{x}\textit{D}_{y}$ indicate the number of acceptor ($\textit{A}_{x}$) and donor ($\textit{D}_{y}$) HBs. For clarity we omit combinations with minor contributions, e.g., $\textit{A}_{3}\textit{D}_{1}$, $\textit{A}_{0}\textit{D}_{y}$, $\textit{A}_{x}\textit{D}_{0}$, etc. ### III.2 4 points models In fig. 5 we report the g2(r) for the 4 points models, namely the TIP4 (black line), the TIP4P-Ew (red line), the TIP4P/2005 (green line), the flexible TIP4P/2005 (blue line) and the TIP4P-ice (orange line) models. As for the 3 points models, the open symbols represent the g2(r) from experimental data Skinner _et al._ (2013); Soper and Benmore (2008) and from _ab initio_ molecular dynamics simulations DiStasio Jr. _et al._ (2014), and serve as a comparison. The TIP4P model is characterized by the intensity of the first peak as high as $\sim 3$, the closest to the experimental/_ab initio_ ones. The depth of the following minimum is slightly more pronounced than the experimental/AIMD. At larger distances, on the other hand, the g2(r) almost overlap with the experimental/AIMD ones. The TIP4P-Ew (red line) has been optimized for simulating water in biological environments Horn _et al._ (2004). With respect to the TIP4P model, the TIP4P-Ew shows a higher intensity in the first peak of the g2(r), reaching $\sim 3.2$. Other features of the g2(r) are qualitatively overlapping with the experimental/AIMD distribution functions. A similar trend characterizes the g2(r) for the TIP4P/2005 (green line) and for the flexible TIP4P/2005 (blue line) which are mostly indistinguishable from the distribution function of the TIP4P-Ew function. The TIP4P-Ice, on the other hand, shows a g2(r) more structured with respect to the previous ones. The intensity of the first peak reaches values as close as $\sim 3.5$, and the following minimum shows a deeper depth. Finally, the intensity of the second peak is slightly more pronounced with respect to the experimental/AIMD ones. Such ”over”-structurization is not surprising considering that this water model has been developed to study crystalline and non crystalline solid forms of ice Abascal _et al._ (2005). Overall, moving from the TIP4P model to the TIP4P-Ew, TIP4P/2005, flexible TIP4P/2005 and the TIP4P-Ice we observe a systematic enhancement of the first peak which also slightly shifts towards larger values. All models well capture the second and third peaks. Figure 5: The oxygen-oxygen two-bodies pair correlation, g2(r), of liquid water for the TIP4P (black), TIP4P-Ew (red), TIP4P/2005 (green), the flexible TIP4P/2005 (blue), and the TIP4P-Ice (orange) models. The g2(r) obtained from various scattering experiments Skinner _et al._ (2013); Soper and Benmore (2008) and _ab initio_ molecular dynamics simulations DiStasio Jr. _et al._ (2014) are reported for comparison with open symbols. In fig. 6 we report the probability distribution P(n) of having a n-folded ring, with n$\in[3,12]$ for the TIP4P (black open circles), the TIP4P-Ew (red open squares), the TIP4P/2005 (green open diamonds), the flexible TIP4P/2005 (blue open triangles) and the TIP4P-Ice (orange open left triangles) models. In panel a) we report the P(n) according to the definition d1 sketched in fig. 1 a). All classical models of water show a P(n) maximized at n=6. In particular, the TIP4P model shows, among all the 4 points models here inspected, the broader distribution, with an almost equal contribution of 5-folded and 7-folded rings. Interestingly, the distribution for the TIP4P model is less broad compared with the distribution for the TIP3P model reported in fig. 3 a), with almost no rings longer than n=9. The HBN of the TIP4P-Ew model shows a slight increase in the pentagonal and in the hexagonal character and a corresponding lower character in rings longer than n=7. Very similar distributions occur for the TIP4P/2005 and for the flexible TIP4P/2005 models. The hexagonal character of the HBN is further emphasized in the TIP4P-Ice model for which we can observe also a slight increment in the pentagonal character and a depletion of rings longer than n=7. This enhanced hexagonal character of the HBN is a consequence of the parametrization of this model which has been optimized to reproduce crystalline and non-crystalline solid forms of ice. In panel b) of fig. 6 we report the P(n) computed using the rings and counting scheme definition d2 sketched in fig. 1 b). According to such counting scheme, all distributions are maximized over n=5. In particular, the P(n) for the TIP4P model shows the broader distribution, with some contribution from n=8, while the contributions to the HBN from longer rings are mostly negligible. The TIP4P-Ew model shows an enhanced pentagonal character and a slightly increment of the hexagonal character as well, with a reduction of the contribution coming from rings longer than n=6. Such tendency becomes more pronounced moving to the TIP4P/2005 and to the flexible TIP4P/2005 models, while differences among these three models are minimal. On the other hand, the pentagonal character of the HBN is particularly enhanced in the TIP4P-Ice model whose HBN, according to this counting scheme, dose not account for rings longer than n=7. In panel c) we report the P(n) computed using the counting scheme d3 sketched in fig. 1 c). For this scheme, we also compute the complexity indices $\xi$, reported in table 2. We can observe that the distribution P(n) for the TIP4P model is fairly broad and mostly dominated by an almost equal amount of 6- and 7-folded rings followed by 5-folded rings, while the amount of longer rings decreases with increasing the rings lengths. The complexity index for the TIP4P model is $\xi=0.1721$, which is larger compared to the TIP3 and the SPC models, but lower compared to the other 3 points models here inspected. The network of TIP4P-Ew model is characterized by a consistent increment in the hexagonal character, followed by the heptagonal character and a smaller increment in the pentagonal character. Longer rings weight less with respect to the TIP4P model. The complexity index for the TIP4P-Ew is $\xi=0.1887$, larger compared to the TIP4P model. The distribution P(n) for the TIP4P/2005 and the flexible TIP4P/2005 models mostly overlap with the P(n) for the TIP4P-Ew, and have a similar complexity index, namely $\xi=0.1915$ for the TIP4P/2005 model and $\xi=0.1984$ for the flexible TIP4P/2005 model. Interestingly, the P(n) for the TIP4P-Ew, for the TIP4P/2005 and for the flexible TIP4P/2005 models show a breaking in the equal hexagonal and heptagonal character of the network, with a slight predominance of the hexagonal character which justifies the higher values of the index $\xi$. The network of the TIP4P-Ice model is characterized by a further enhancement of the hexagonal character, followed by the heptagonal and hexagonal characters, with a depletion of longer rings. Such enhancement in the hexagonal character is reflected by the complexity index $\xi=0.2155$, the closest to the value of ice among all models inspected in this work. The similar distributions of rings for the TIP4P-Ew, TIP4P/2005 and flexible TIP4P/2005 in all counting schemes suggest that such models perform equally in terms of HBN, and this reflects the very similar g2(r) (fig. 5). On the other hand, the TIP4P-Ice, which shows a more structured g2(r) (fig. 5), tends to favour a network with a shorter connectivity. Figure 6: Probability distributions of the hydrogen-bonded n-folded rings, P(n), for liquid water at ambient conditions described by the TIP4P (black open circles), the TIP4P-EW (red open squares), the TIP4P/2005 (green open diamonds), the flexible TIP4P/2005 (blue open triangles), and the TIP4P-Ice (open orange triangles) models. \| TIP4P \| TIP4P-Ew \| TIP4P/2005 \| TIP4P/2005-Flex \| TIP4P-Ice ---\|---\|---\|---\|---\|--- $\xi$ \| 0.1721 \| 0.1887 \| 0.1915 \| 0.1984 \| 0.2155 Table 2: Values of the network complexity index $\xi$ computed for the rings counting scheme d3 for the 4 points models of water. In fig. 7 we report the percentage of broken and intact HBs for the 4 points classical models and for _ab initio_ liquid water. We can observe that the percentage of broken and intact HBs for all 4 points models qualitatively resembles that of _ab initio_ liquid water (open black circles). With respect to AIMD water, the TIP4P model (open red squares) underestimates the percentage of $\textit{A}_{2}\textit{D}_{2}$ configurations to $\sim 43\%$ and a slightly higher percentage of $\textit{A}_{1}\textit{D}_{2}$ configurations ($\sim 22\%$). All other configurations mostly overlap with the configurations of AIMD water. The percentage of broken and intact HBs for the TIP4P-Ew (open green diamonds), for the TIP4P/2005 (open blue triangles) and the flexible TIP4P/2005 (open orange left triangles) mostly overlap with each other, reflecting the almost overlapping distributions of rings and the similar values of the complexity index $\xi$ (fig. 6). In particular, we observe almost no differences between the distribution for the TIP4P/2005 and the flexible TIP4P/2005 models. With respect to the TIP4P model, they recover the percentage of $\textit{A}_{2}\textit{D}_{2}$ for the AIMD liquid water, while showing a small reduction of $\textit{A}_{1}\textit{D}_{1}$ defects ($\sim 7\%$). Such increment in the percentage of $\textit{A}_{2}\textit{D}_{2}$ to $\sim 48\%$ reflects the stronger 5-, 6- and 7-folded character of the HBN described in fig. 6. The TIP4P-Ice model, on the other hand, overestimate the percentage of intact HBs with respect to AIMD water, reaching $\sim 55\%$ of the total configurations. This enhanced 4-folded coordination explains the distribution of rings (fig.6) describing an HBN particularly enriched in 6- and 7-folded rings, as well as the higher intensity in the first peak of the g2(r) (fig 5). Overall, we can state that, overall, the 4 points models are characterized by similar HBNs, with the exception of the TIP4P model which shows a broad distribution of rings caused by a lower percentage of intact HBs, and the TIP4P-Ice model, whose high percentage of intact HBs causes generates a network characterized by a strong hexagonal and heptagonal character with a marked reduction of longer rings. Figure 7: Percentage-wise decomposition of the intact HBs per water molecule into acceptor-(A) and donor-(D) for _ab initio_ liquid water at T=330 K as black open circles, and for the 4 points models. The TIP4P model is reported as red open squares, the TIP4P-Ew model as green open diamonds, the TIP4P/2005 model as blue open triangles, the flexible TIP4P/2005 as orange open left triangles, and the TIP4P-Ice model as brown open lower triangles. The distribution for the TIP4P/2005 model almost perfectly overlaps with the distribution for the flexible TIP4P/2005 model. ### III.3 5 points models In fig. 8 we compare the g2(r) of two 5 points models, namely the TIP5P (black line) and the TIP5P-E (red line) models with the g2(r) obtained from various scattering experiments Skinner _et al._ (2013); Soper and Benmore (2008) and _ab initio_ molecular dynamics simulations DiStasio Jr. _et al._ (2014) reported as open symbols. We can observe that the g2(r) for both 5 points models mostly overlap. With respect to the 3 points and to the 4 points models, the g2(r) of both 5 points models are more closer to the experimental and to the AIMD g2(r). The intensity of the first peak for the TIP5P and the TIP5P-E models is below 3, namely $\sim$2.9. The depth of the first minimum is slightly more pronounced with respect to the experimental and the AIMD g2(r), while experimental and AIMD peaks at longer distances are well captured. Figure 8: The oxygen-oxygen two-bodies pair correlation, g2(r), of liquid water for the TIP5P (black) and the TIP5P-EW (red) models. The g2(r) obtained from various scattering experiments Skinner _et al._ (2013); Soper and Benmore (2008) and _ab initio_ molecular dynamics simulations DiStasio Jr. _et al._ (2014) are reported for comparison with open symbols. In fig. 9 we report the probability distribution P(n) of having a n-folded ring, with n$\in[3,12]$ for the TIP5P model (open black circles) and for the TIP5P-E model (open red squares). In panel a) we report the P(n) computed using the counting scheme d1 sketched in fig. 1 a) and which emphasizes the directionality of the HBN. Both the TIP5P and the TIP5P-E models provide similar distributions which are slightly maximized at n=6, and contributions of rings up to n=11. In panel b) we report the P(n) computed using, as a counting scheme, the definition d2 sketched in fig. 1 b). As for the previous case, also in this case the two distributions are mostly indistinguishable and with an almost equal pentagonal and hexagonal character. Overall, the pentagonal character of both HBNs is less pronounced with respect to the four points models, and qualitatively resemble the P(n) for the three points model TIP3P (fig. 3 b)). In panel c) we report the P(n) computed using the definition d3 and sketched in fig. 1 c). For this counting scheme, we also compute the complexity indices reported in table 3. As for the previous counting schemes, the differences between the two P(n)’s are minimal, indicating that the two model provide similar HBNs. Such similarity can be quantified observing that both models have almost the same complexity index $\xi$, i.e., $\xi=0.1774$ for the TIP5P model and $\xi=0.1754$ for the TIP4P-Ew model. The values of $\xi$ for both networks are comparable with that of the TIP4P model but, with respect to the TIP4P model, the hexagonal character of the HBN is more pronounced for both the TIP5P and the TIP5P-E models. Besides the TIP4P-Ice model, among all other interaction potentials here inspected, the TIP5P and the TIP5P-E models are the only one for which the counting scheme d3 is (slightly) maximized towards n=6. Figure 9: Probability distributions of the hydrogen-bonded n-folded rings, P(n), for liquid water at ambient conditions described by the TIP5P (black open circles) and the TIP5P-EW (red open squares). \| TIP5P \| TIP5P-E ---\|---\|--- $\xi$ \| 0.1774 \| 0.1754 Table 3: Values of the network complexity index $\xi$ computed for the rings counting scheme d3 for the 5 points models of water. In fig. 10 we report the percentage of broken and intact HBs for the 5 points classical models and for _ab initio_ liquid water. With respect to the AIMD liquid water (open black circles), both the TIP5P (open red squares) and the TIP5P-E (open green diamonds) models are characterized by a markedly lower percentage of intact HBs ($\sim 36\%$), while low-coordinated defects occur in higher percentages. The $\textit{A}_{1}\textit{D}_{2}$ configuration account for the $\sim 22\%$ of the total configurations, followed by $\textit{A}_{2}\textit{D}_{1}$ configurations with a percentage of $\sim 16\%$ and $\textit{A}_{1}\textit{D}_{1}$ configurations with $\sim 12\%$ of the total configurations. The low amount of intact HBs (compared with the AIMD water), reflects the broad distribution of rings and the corresponding low value of the network complexity indices $\xi$ (fig. 9). Figure 10: Percentage-wise decomposition of the intact HBs per water molecule into acceptor-(A) and donor-(D) for _ab initio_ liquid water at T=330 K as black open circles, and for the 5 points models. The TIP5P model is reported as red open squares and the TIP5P-EW model as green open diamonds. ### III.4 Relation between network complexity and dynamical properties We here show how the network complexity is linked to dynamical properties, namely the translational and rotational diffusion. This link comes from the observation that bonding can be viewed as a competition between the energy gained from the formation of a bond, and the entropy loss due to the reduction in configurational volume that occurs when two particles are constrained to stay close relative to each other. The establishment of an extended network of bonds occurs when the energy gain (that controls the lifetime of bonds) balances the entropy loss. For each water model we have computed the diffusion coefficient and the rotational relaxation time ($\tau_{rot}$), and we have reported them against the network complexity index $\xi$. We have computed the diffusion coefficient from the mean squared displacement, and the rotational relaxation time $\tau_{rot}$ from the integral of the rotational autocorrelation function as reported in Refs Calero, Stanley, and Franzese (2016); Martelli, Calero, and Franzese (2021) $C_{rot}(t)=\left<\textit{OH}(t)\cdot\textit{OH}(0)\right>$, i.e., $\tau_{rot}=\int_{0}^{+\infty}C_{rot}(t)dt$. Fig. 11 show the values of these three observables in a three dimensional plot, with projections on the corresponding two dimensional spaces. The values for 3-points models are reported in red, the values for 4-points models are reported in green, and the values of 5-points models are reported in blue. We can observe a clear correlation between the complexity of the HBN and the dynamical properties of water molecules. The models with the highest diffusion coefficients and the fastest rotational relaxation times are the TIP3P and the SPC models, which are also characterized by the lowest values of the index $\xi$. Contrarily, the TIP4P-Ice model is the model with the lowest diffusion coefficient and the slowest rotational relaxation time, and the highest index $\xi$. Overall, we can observe that for all models of water the higher the value of $\xi$ (reported in ascending order in the tables 1, 2, 3), the slower the diffusion coefficient and the rotational relaxation time. Therefore, we can assert that there is a clear correlation between the complexity of the HBN and dynamical properties, a relation never observed before. Such correlation suggests that faster diffusion and rotations allow water molecules to increase the possible connections between each other, hence increasing the configurational space that the network can explore resulting in a more complex topology able to host a larger amount of longer rings. Figure 11: Three dimensional plot reporting the projection on the corresponding two dimensions of the values acquired by the 11 classical models of water. We report the values for the network complexity index $\xi$, the diffusion coefficient and the rotational relaxation time $\tau_{rot}$. Data for 3-points models are reported in red, while data for the 4-points models are reported in green and data for the 5-points models are reported in blue. ### III.5 System size dependence We now turn our attention to the study of finite size effects, commonly inspected when computing physical quantities to check whether a system suffers from periodicity artifacts. In fig. 12 a) we report the oxygen-oxygen g2(r) for a simulation box containing 500 water molecules (black continuous line), 1000 water molecules (red dashed line) and 1500 water molecules (green dotted- dashed line) interacting with the TIP4P classical interaction potential. We can observe that the three g2(r) perfectly overlap. In fig. 12 b) we report the probability distribution P(n) of having a n-folded ring for the three cases inspected above and computed according to the ring definition and counting scheme d3. We can observe that the distribution computed for the smaller simulation box with N=500 molecules is remarkably different from the distributions with N=1000 and N=1500 molecules. In particular, we observe a strong enhancement of n=12 rings which causes a reduced contribution of shorter rings to the P(n). This result indicates that a search path of n=12 water molecules is too long for a small simulation box with only N=500 water molecules, and the increment in n=12 is caused by periodicity in the simulation box. It is worth to mention that the definitions d1 and d2 do not show such behavior (data not reported), as the network investigated with these definitions does not host rings as long as n=12 (see fig. 6 upper and middle panels). Therefore, although the g2(r) for N=500 is the same as the g2(r) for larger simulation boxes, care must be taken when inspecting the network topology and in how such inspection is performed. Figure 12: Panel a): system size dependence on the oxygen-oxygen two bodies pair correlation function for the TIP4P model using a box containing 500 water molecules (black continuous line), 1000 water molecules (red dashed line) and 1500 water molecules (green dotted-dashed line). Panel b): ring distribution for a system described by the TIP4P model in a box containing 500 water molecules (black pluses), 1000 water molecules (red crosses) and 1500 water molecules (green stars). ## IV Conclusions In this article we have tested 11 popular non polarizable classical interaction potentials for water against their hydrogen bond networks (HBNs). We have probed the topology of the HBN using three schemes that emphasize different physical features. We have evaluated the quality of the HBNs in terms of broken and intact HBs, and we have linked our results to structural properties measured _via_ the two bodies pair correlation function g2(r). We have then introduced the network complexity index $\xi$ that measures how much the topology of a HBN deviates from that of the ground state, and we have tested it to one of the three rings counting schemes. We have shown that the index $\xi$ is directly related to dynamical properties, hence establishing a clear cause-effect relationship between molecular motions and network connectivity. Finally, we have inspected how periodicity artifacts can influence the topology of the HBN. We have performed all studies at ambient conditions, i.e., T=300 K and p=1 bar. Although different water models have (very) different melting points, their network topology –and hence their network complexity– remain roughly unchanged away from the limit of supercooling Formanek and Martelli (2020). On the other hand, when water is under confinement, water molecules in the proximity of the surfaces undergo a drastic change in the dynamics Samatas _et al._ (2018); Martelli, Crain, and Franzese (2020); Chiricotto _et al._ ; Gallo, Rovere, and Chen (2010); Camisasca, Marzio, and Gallo (2020); Iorio, Camisasca, and Gallo (2019); Iorio _et al._ (2020); Tenuzzo, Camisasca, and Gallo (2020); Calero and Franzese (2020) and in network topology Martelli, Crain, and Franzese (2020); Chiricotto _et al._ . Therefore, the choice of a given interaction potential becomes of particular relevance. In the class of 3 points models, we have tested the TIP3P, the SPC, the SPC/E and the flexible SPC models. We have found that the TIP3P model is characterized by the less structured network, with broad distributions or rings and a network rich in coordination defects which allows the network to arrange in long rings. In particular, the counting scheme d3 gives a low value of complexity index $\xi$ which reflects the large distance from the HBN of crystalline ice for which $\xi=1.0$. The broad rings distribution and the low percentage of intact HBs explain the absence of a second hydration peak in the g2(r). The percentage of intact HBs increases in the SPC model which is, therefore, characterized by an HBN with fewer longer rings and by a g2(r) with signatures of a second hydration peak. The SPC/E and the flexible SPC models are characterized by a further increase in the percentage of intact HBs, comparable with that of _ab initio_ liquid water. The resulting HBNs accommodate an even lower percentage of longer rings and, therefore, the network complexity index is higher compared to the previous models, indicating a closer (but still very far) HBN to the HBN of Ih(c), in agreement with the more structured g2(r). It is of particular interest to observe how the introduction of flexibility in the SPC model drastically affects the topology of the HBN. In the class of 4 points models, we have tested the TIP4P, the TIP4P-Ew, the TIP4P/2005, the flexible TIP4P/2005 and the TIP4P-Ice models. The TIP4P model is the only model whose network accommodates a lower percentage of intact HBs with respect to _ab initio_ liquid water. The topology of the corresponding HBN is hence the most complex, i.e., with a low complexity index. The TIP4P-Ew, the TIP4P/2005 and the flexible TIP4P/2005 models are characterized by a similar percentage of intact HBs, comparable with that of _ab initio_ liquid water. The corresponding HBNs have comparable topologies and values of complexity indices. The TIP4P-Ice model, finally, shows a higher percentage of intact HBs with respect to _ab initio_ liquid water. The topology of the corresponding HBN is the less complex among all models here studied, with small contributions of longer rings and the highest value of $\xi$ index. Such results explain the over structured g2(r) with respect to both _ab initio_ water and experimental results. In the class of 5 points models, we have tested the TIP5P and the TIP5P-E models. Both models have similar HBNs, characterized by a lower percentage of broken HBs with respect to _ab initio_ liquid water. Both networks are fairly complex, accommodating longer rings causing low values of the network complexity index. Overall, the balance between intact HBs and network topology allows the 5 points models to be the better models in reproducing the _ab initio_ and experimental water g2(r). Overall, we have shown that water models endowed with the fastest dynamics are able to establish more complex networks, while models with the slowest dynamics establishes networks more closely related to that of cubic or hexagonal ice. In particular, among the 11 models here inspected the TIP3P and the SPC models are the ones with the fastest dynamics and with the network deviating the most from that of ice. On the other hand, the TIP4P-Ice model is the one with the slowest dynamics and, hence, with a network of HB the closer to that of ice. Finally, we have shown that the topology of the HBN might be affected by finite size effects when other observables such as, e.g., the two body pair correlation function, do not show such sensitivity. In conclusion, the topology of the HBN and its quality in terms of broken and intact HBs are more sensitive quantities than other physical observables Martelli, Crain, and Franzese (2020); Chiricotto _et al._ . Therefore, the properties of the HBN should be inspected along with all other properties such as, e.g., structural, dynamical and thermodynamic properties when developing new interaction potentials. Our study provides a benchmark evaluated following three different ring definitions and counting schemes. New interaction potentials should be tested against such results. Nonetheless, the network complexity index provides a direct quantitative measure of how much a HBN is complex and far from the HBN at the ground state, and a direct link to the dynamical properties. Such quantity can be transferred to other materials. The effect of polarization on the topology of the HBN and on its quality should be investigated. ###### Acknowledgements. We acknowledge support from the STFC Hartree Centre’s Innovation Return on Research programme, funded by the Department for Business, Energy and Industrial Strategy. ## References * Salzmann (2019) C. G. Salzmann, J. Chem. 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top=1in, bottom=1in, left=1in, right=1in universitá degli studi di pavia dipartimento di fisica corso di laurea magistrale in scienze fisiche Bit Commitment in Operational Probabilistic Theories Tesi per la Laurea Magistrale di Lorenzo Giannelli Chiar.mo Prof. Giacomo Mauro D'Ariano Dott. Alessandro Tosini Anno Accademico 2019-2020 A Piero The aim of this thesis is to investigate the bit commitment protocol in the framework of operational probabilistic theories. In particular a careful study is carried on the feasibility of bit commitment in the non-local boxes theory and in order to do this new aspects of the theory are presented. Lo scopo di questa tesi è di investigare il protocollo di bit commitment all'interno delle teorie probabilistiche operazionali. In particolare si è analizzato attentamente la fattibilità del protocollo all'interno della teoria dei non-local boxes e i nuovi aspetti della teoria emersi in questa analisi sono presentati. CHAPTER: INTRODUCTION The study of quantum foundations is a discipline of science that seeks to understand the most characterizing aspects of quantum theory, to reformulate it and even propose new generalizations. An active area of research in quantum foundations is therefore to find alternative formulations of quantum theory which rely on physically compelling principles in attempt to find a re-derivation of the quantum formalism in terms of operational axioms. One of the most interesting effort in this direction is made investigating the relations between the current operational axioms and the main results of quantum information theory. As a recent example it has been proved in Ref. [1] that the no information without disturbance (NIWD) theorem, i.e. the impossibility in quantum theory to extract information without disturbing the state of the system or its correlations with other systems, is independent of both local discriminability and purification, two of the defining axioms of quantum theory. Especially the latter, as we will see thoroughly in the first Chapter, is considered as a characteristic and distinctive quantum trait but now NIWD can be exhibited in absence of it and also of most of the principles of quantum theory. The NIWD property spawns other no-go theorems, which represent some of the most famous and classic results in quantum information theory. Among these results one can certainly list the no-cloning theorem, the no-programming and the impossibility of perfectly secure bit commitment. In our thesis our efforts are focused on the latter. A bit commitment (BC) protocol is meant to allow one party, Alice, to send a bit to a second party, Bob, in such a way that Bob cannot read the bit until Alice allows for its disclosure, while Alice cannot change the value of the bit after she encoded it. The bit commitment protocol is a very important primitive in cryptography, and perfectly secure protocols are known to be impossible in classical information theory. This is the case in quantum theory as well. The proof involves a very important characterization theorem for general theories with purification [2]. Numerous bit commitment protocols have been proposed in literature and this cryptography primitive has been deeply studied, especially the possibility of unconditionally secure bit commitment, both in quantum and classical information theory, due to its importance in practical applications. We would like to investigate the relations between the theorem of no-bit commitment and the operational axioms that characterize quantum theory. A general strategy to apprehend the nature of these links is to test the validity of the theorem in a theory that lacks one or more principles; naturally the first try has to be made excluding the purification, the most quantum feature. The answer of how to start in the analysis comes directly from the literature on bit commitment protocol itself. In fact, after it was proved to be impossible in quantum theory the protocol has been tested in more general scenarios, in particular in more non-local scenarios. To understand what it means we need to take a step back to the two parties of the protocol, Alice e Bob. Assume that they are not able to communicate but have access to physical states that they can use to generate joint correlations. In this experiment the outcome of the measurements on the state of their local systems are given by random variables. Obviously, causality constrains the correlations to be non signalling, and on the other side quantum theory prevents the strength of the non-local correlation to violate Bell's inequalities [3], where the maximal value is known as Cirel'son's bound [4]. A well-known variant of a Bell inequality is the Clauser, Horne, Shimony & Holt (CHSH) inequality [5], which can be expressed as [6] \begin{equation} \sum_{x,y\in\{0,1\}}\mathsf{Pr}(a_x\oplus b_y=x\cdot y)\le2\sqrt{2}\,. \end{equation} Where $x$ and $y$ denote the choice of Alice's and Bob's measurement, respectively, $a_x\in\{0,1\}$, $b_y\in\{0,1\}$ the respective binary outcomes, and $\oplus$ addition modulo 2. However, if we only care about the causality constrain, Cirel'son's bound can be violated up to the maximal value of 4. Popescu and Rohrlich, who first noticed [7], raised the question of why nature is not more non-local and why does quantum mechanics not allow a stronger violation of the CHSH inequality. Following this lead, bit commitment has been studied in Popescu Rohrlich (PR) non-local boxes theory. It has been claimed, as in Ref. [8], that bit commitment was admissible in PR-box theory but a counter-proof reached from Short, Gisin, and Popescu [9]. It was formulated on the behalf that, since non-locality is the main reason to prevent bit commitment in quantum theory, it would not be possible that in a theory more non-local than the quantum one BC would work. However, thereafter a new protocol was proposed by Buhrman et al. [10] and the argument of the counter-proof of Short, Gisin and Popescu is not able to deny it. The first aim of our work is to bring some clarity, pointing out the limitations of the framework of validity about the results that have been claimed until now. In our path new aspects of PR-box theory emerged and have been studied. Even if the theory is still far to be considered complete, important characterizations have been made. We will show that all the BC protocols proposed so far respect the limitations of PR-box theory under which we are able to prove a no-bit commitment theorem. Furthermore we are able to create, with very similar arguments, a counter-proof of the protocol proposed in Ref. [10]. Finally, a surprising result is observed. Relaxing the constraint of PR-box theory and including some recent developments, a scheme of bit commitment that is perfectly secure seems possible. Even if there are actually no "operational" evidences within the theory to deny it, by the same fundamental reason expressed by Short, Gisin, and Popescu, we think that future advancements of the theory would reconsider the protocol as cheatable. However, as a matter of fact, now the answer to the existence of a theory with entanglement that admits bit commitment, seems to be “Yes!”. A synopsis of the thesis is the following: In the first Chapter the framework of operational probabilistic theories (OPTs) is presented by first introducing the operational language that expresses the possible connections between events, and then by dressing the elements of the language with a probabilistic structure. After that, the principles for the OPT of quantum mechanics are stated (given the framework, the rule of connectivity among events are given). In the second Chapter, we will discuss the bit commitment protocol. We will start with an historical perspective and then we will rigorously define the protocol in the language of OPT. We will mainly deal only with perfectly secure bit commitment since our analysis is carried in the OP framework and there are not yet the technical tools to analyse unconditionally secure BC in OPTs. In the third section of the Chapter, we will study the proof of the impossibility of perfectly secure bit commitment in quantum theory done in Ref. [2]. This is a very elegant and solid demonstration and our purpose is to adapt the proof in order to comprehend other theories than the quantum one. As we will see, non-locality and entanglement are the key reasons that plays in favor of the impossibility of BC and so it seems reasonable to try to extend the impossibility proof also to other non-local theories. In the third Chapter, we will analyse the probabilistic theory corresponding to the popular PR-box model. A comprehensive study has never been made and an organic theory is not available. We propose to bring clarity to the actual model considering only bipartite correlated boxes, highlighting its limitations, and some progress by analysing new aspects such as perfect discriminability and states purification. We will also add some considerations and prevision on $N$-partite correlated boxes. Finally, in the fourth Chapter we propose a proof of impossibility of perfectly secure bit commitment in PR-boxes. Even if the PR-box model does not constitute a proper theory it has been often used in numerous application in literature, such as in protocol of perfectly or unconditionally secure BC. However this led to neglect important elements of the theory and in return numerous results published can be proved false when contextualized in the PR-box OPT (also if it is still incomplete). As the last remark, we conjecture that including tripartite correlated boxes in the theory, secure bit commitment would be possible. The reason is the following. We will see that what prevents BC in quantum theory is that every state can be purified and its purification is unique up to local reversible transformations. Exactly these local operations allow the cheating in the protocol. When we deal with PR-box theory limited to bipartite correlated boxes, only one internal state can be purified, but its purification is unique and this is enough to allow cheating. On the contrary, when we admit tripartite correlated boxes, the uniqueness of purification is lost and exactly this uncommon feature could open the door to the possibility of secure bit commitment. CHAPTER: OPERATIONAL PROBABILISTIC THEORY The purpose of this Chapter is to introduce the framework of operational probabilistic theories (OPTs) and to express quantum theory as an OPT. In this Chapter we will follow Ref. [11]. The framework of operational probabilistic theories consists of two distinct conceptual ingredients: an operational structure, describing circuits that produce outcomes, and a probabilistic structure, which assigns probabilities to the outcomes in a consistent way. The operational structure summarizes all the possible circuits that can be constructed in a given physical theory, in this setting a rigorous formulation of the elements of the circuits: systems, transformations, and their composition is given, which constitutes the grammar for the probabilistic description of an experiment. However, it is only the probabilistic structure that promotes the operational language from a merely descriptive tool to a framework for predictions, the predictive power being the crucial requirement for any scientific theory and for its testability - the essence of science itself. Different OPTs will have different rules for assigning the joint probabilities of events. Working in this framework allows us to deal with a wide range of probabilistic theories, including not only quantum and classical theories, but also the theory of Popescu-Rohrlich (PR), or non-local, boxes. In the first part of the Chapter the framework is provided by first introducing the operational language that expresses the possible connections between events, and then by dressing the elements of the language with a probabilistic structure. Then, in the second part of the Chapter we formulate the principles for the OPT of quantum theory. In fact, once the framework is defined, the rule of connectivity among events are given. § THE FRAMEWORK A theory for making predictions about joint events depending on their reciprocal connections is what we call an operational probabilistic theory. We see that OPT is a non-trivial extension of probability theory. To the joint events we associate not only their joint probability, but also a circuit that connects them. When the events in the circuit have a well-defined order, the circuit is mathematically described by a directed acyclic graph (a graph with directed edges and without loops). The basic element of an OPT - the notion of event - gets dressed with wires that allow us to connect it with other events. Such wires are the systems of the theory. In agreement with the directed nature of the graph, there are input and output systems. The events are the transformations, whereas the transformations with no input system are the states (corresponding to preparations of systems), and those with no output system are the effects (corresponding to observations of systems). Since the purpose of a single event is to describe a process connecting an input with an output, the full circuit associated to a probability is a closed one, namely a circuit with no input and no output. The circuit framework is mathematically formalized in the language of category theory. In this language, an OPT is a category, whose systems and events are objects and arrows, respectively. Every arrow has an input and an output object, and arrows can be sequentially composed. The associativity, existence of a trivial system, and commutativity of the parallel composition of systems of quantum theory technically correspond to having a strict symmetric monoidal category. §.§ Primitive notions and notation The primitive notions of any operational theory are those of test, event, and system. A test $\{\tA_i\}_{i\in \rI}$ is the collection of events $\tA_i$, where $i$ labels the element of the outcome space I. In addition to comprising a collection of events, the notion of test carries also the event connectivity of the theory that is achieved by the systems. These can represent the input and the output of the test. The resulting representation of a test is the following diagram: \begin{equation} \begin{aligned} \Qcircuit @C=1em @R=.7em { \poloFantasmaCn{\rA}\qw& \gate{\{\tE_x\}_{x\in \rX}}& \poloFantasmaCn{\rB}\qw& \qw \end{aligned} \end{equation} The wire on the left labeled as A represents the input system, whereas the wire on the right labeled as B represents the output system. The same diagrammatic representation is also used for any of the events, namely for $x\in \rX$ \begin{equation} \begin{aligned} \Qcircuit @C=1em @R=.7em { \poloFantasmaCn{\rA}\qw& \gate{\tE_x}& \poloFantasmaCn{\rB}\qw& \qw \end{aligned} \end{equation} In the following, the systems will be denoted by capital Roman letters A,B,...,Z, whereas the events by capital calligraphic letters $\mathcal{A,B,}...,\mathcal{Z}$. Different tests can be combined in a circuit, which is a directed acyclic graph where the links are the systems (oriented from left to right, namely from input to output) and the nodes are the boxes of the tests. The same graph can be built up for a single test instance, namely with the network nodes being events instead of tests, corresponding to a joint outcome for all tests. The circuit graph is obtained precisely by using the following rules. Sequential Composition of Test When the output system of test $\{\mathcal{C}_x\}_{x\in \rX}$ and the input system of test $\{\tD_y\}_{y\in \rY}$ coincide, the two tests can be composed in sequence as follows: \begin{equation} \begin{aligned} \Qcircuit @C=1em @R=.7em { \poloFantasmaCn{\rA}\qw& \gate{\{\tC_x\}_{x\in \rX}}& \poloFantasmaCn{\rB}\qw& \gate{\{\tD_y\}_{y\in \rY}}& \poloFantasmaCn{\rC}\qw& \qw \, \eqqcolon \, \Qcircuit @C=1em @R=.7em { \poloFantasmaCn{\rA}\qw& \gate{\{\tE_{(x,y)}\}_{(x,y)\in \rX\times \rY}}& \poloFantasmaCn{\rC}\qw& \qw \end{aligned} \end{equation} resulting in the test $\{\tE_{(x,y)}\}_{(x,y)\in \rX\times \rY}$ called sequential composition of $\{\tC_x\}_{x\in \rX}$ and $\{\tD_y\}_{y\in \rY}$. In formulas we will also write $\tE_{(x,y)}:=\tD_y\tC_x$. Identity Test For every system A, one can perform the identity test (shortly identity) that "leaves the system alone". Formally, this is the deterministic test $\mathcal{I}_\textnormal{A}$ with the property \begin{equation} \begin{aligned} \Qcircuit @C=1em @R=.7em @! R { \poloFantasmaCn{\rA}\qw& \gate{\mathcal{I}_\rA}& \poloFantasmaCn{\rA}\qw& \gate{\mathcal{C}}& \poloFantasmaCn{\rB}\qw& \qw \end{aligned} \, =\, \begin{aligned} \Qcircuit @C=1em @R=.7em @! R { \poloFantasmaCn{\rA}\qw& \gate{\mathcal{C}}& \poloFantasmaCn{\rB}\qw& \qw \end{aligned}\,, \end{equation} \begin{equation} \begin{aligned} \Qcircuit @C=1em @R=.7em @! R { \poloFantasmaCn{\rB}\qw& \gate{\mathcal{D}}& \poloFantasmaCn{\rA}\qw& \gate{\mathcal{I}_\rA}& \poloFantasmaCn{\rA}\qw& \qw \end{aligned} \, =\, \begin{aligned} \Qcircuit @C=1em @R=.7em @! R { \poloFantasmaCn{\rB}\qw& \gate{\mathcal{D}}& \poloFantasmaCn{\rA}\qw& \qw \end{aligned}\,, \end{equation} where the above identities must hold for any event $ \Qcircuit @C=.5em @R=.5em { & \poloFantasmaCn{\rA} \qw & \gate{\mathcal{C}} & \poloFantasmaCn{\rB} \qw &\qw} $ and $ \Qcircuit @C=.5em @R=.5em { & \poloFantasmaCn{\rB} \qw & \gate{\mathcal{D}} & \poloFantasmaCn{\rA} \qw &\qw} $,respectively. The sub-index A will be dropped from $\mathcal{I}_\text{A}$ where there is no ambiguity. Operationally Equivalent Systems We say that two systems A and $\text{A}^\prime$ are operationally equivalent - denoted as $\text{A}^\prime\simeq \text{A}$ or just $\text{A}^\prime=\text{A}$ - if there exist two deterministic events $ \Qcircuit @C=.5em @R=.5em { & \poloFantasmaCn{\rA} \qw & \gate{\mathcal{U}} & \poloFantasmaCn{\rA^\prime} \qw &\qw} $ and $ \Qcircuit @C=.5em @R=.5em { & \poloFantasmaCn{\rA^\prime} \qw & \gate{\mathcal{V}} & \poloFantasmaCn{\rA} \qw &\qw} $ such that \begin{equation} \begin{aligned} \Qcircuit @C=1em @R=.7em @! R {& \poloFantasmaCn{\rA} \qw & \gate{\mathcal{U}} & \poloFantasmaCn{\rA^\prime} \qw & \gate{\mathcal{V}} & \poloFantasmaCn{\rA} \qw & \qw} \end{aligned} \ =\ \begin{aligned} \Qcircuit @C=1em @R=.7em @! R {& \poloFantasmaCn{\rA} \qw & \gate{\mathcal{I}} & \poloFantasmaCn{\rA} \qw & \qw} \end{aligned}\,, \end{equation} \begin{equation} \begin{aligned} \Qcircuit @C=1em @R=.7em @! R {& \poloFantasmaCn{\rA^\prime} \qw & \gate{\mathcal{V}} & \poloFantasmaCn{\rA} \qw & \gate{\mathcal{U}} & \poloFantasmaCn{\rA^\prime} \qw & \qw} \end{aligned} \ =\ \begin{aligned} \Qcircuit @C=1em @R=.7em @! R {& \poloFantasmaCn{\rA^\prime} \qw & \gate{\mathcal{I}} & \poloFantasmaCn{\rA^\prime} \qw & \qw} \end{aligned}\,. \end{equation} Accordingly, if $\{\tC_x\}_{x\in \rX}$ is any test for system A, performing an equivalent test on system $\text{A}^\prime$ means performing the test $\{\tC^\prime_x\}_{x\in \rX}$ defined as \begin{equation} \begin{aligned} \Qcircuit @C=1em @R=.7em @! R {& \poloFantasmaCn{\rA^\prime} \qw & \gate{\mathcal{C}^\prime_x} & \poloFantasmaCn{\rA^\prime} \qw & \qw} \end{aligned} \ =\ \begin{aligned} \Qcircuit @C=1em @R=.7em @! R {& \poloFantasmaCn{\rA^\prime} \qw & \gate{\mathcal{V}} & \poloFantasmaCn{\rA} \qw & \gate{\mathcal{C}_x} & \poloFantasmaCn{\rA} \qw & \gate{\mathcal{U}} & \poloFantasmaCn{\rA^\prime} \qw & \qw} \end{aligned}\,. \end{equation} Composite System Given two systems A and B, one can join them into the single composite system AB. As a rule, the system AB is operationally equivalent to the system BA, and we will identify them in the following. This means that the system composition is commutative, \begin{equation} \text{AB}=\text{BA}. \end{equation} We will call a system trivial system, reserving for it the letter I, if it corresponds to the identity in the system composition, namely \begin{equation} \text{AI}=\text{IA}=\text{A}. \end{equation} The trivial system corresponds to having no system, namely I carries no information. Finally we require the composition of systems to be associative, namely \begin{equation} \text{A(BC)}=\text{(AB)C}. \end{equation} In other words, if we iterate composition on many systems we always end up with a composite system that only depends on the components, and not on the particular composition sequence according to which they have been composed. Systems then make an Abelian monoid. A test with input system AB and output system CD represents an interaction process (see the parallel composition of tests in the following). Parallel Composition of Tests Any two tests $ \Qcircuit @C=.5em @R=.5em { & \poloFantasmaCn{\rA} \qw & \gate{\{\mathcal{C}_x\}_{x\in \rX}} & \poloFantasmaCn{\rB} \qw &\qw} $ $ \Qcircuit @C=.5em @R=.5em { & \poloFantasmaCn{\rC} \qw & \gate{\{\mathcal{D}_y\}_{y\in \rY}} & \poloFantasmaCn{\rD} \qw &\qw} $ can be composed in parallel as follows: \begin{equation} \begin{aligned} \Qcircuit @C=1em @R=.7em @! R {& \poloFantasmaCn{\rA} \qw & \gate{\{\mathcal{C}_x\}_{x\in \rX}} & \poloFantasmaCn{\rB} \qw &\qw\\ & \poloFantasmaCn{\rC} \qw & \gate{\{\mathcal{D}_y\}_{y\in \rY}} & \poloFantasmaCn{\rD} \qw &\qw} \end{aligned} \ =:\ \begin{aligned} \Qcircuit @C=1em @R=.7em @! R { & \poloFantasmaCn{\rA\rC} \qw & \gate{\{\tF_{(x,y)}\}_{(x,y)\in \rX\times \rY}} & \poloFantasmaCn{\rB\rD} \qw &\qw } \end{aligned}\,. \end{equation} The test $ \Qcircuit @C=.8em @R=.5em { & \poloFantasmaCn{\rA\rC} \qw & \gate{\{\tF_{(x,y)}\}_{(x,y)\in \rX\times \rY}} & \poloFantasmaCn{\rB\rD} \qw &\qw } $ is the parallel composition of tests $ \Qcircuit @C=.5em @R=.5em { & \poloFantasmaCn{\rA} \qw & \gate{\{\mathcal{C}_x\}_{x\in \rX}} & \poloFantasmaCn{\rB} \qw &\qw} $ and $ \Qcircuit @C=.5em @R=.5em { & \poloFantasmaCn{\rC} \qw & \gate{\{\mathcal{D}_y\}_{y\in \rY}} & \poloFantasmaCn{\rD} \qw &\qw} $, where $\{\tF_{(x,y)}\}_{(x,y)\in \rX\times \rY}\equiv\{\tC_x\otimes\tD_y\}_{(x,y)\in \rX\times \rY}$. Parallel and sequential composition of tests commute, namely one has \begin{equation} \label{c:parallel composition} \begin{aligned} \Qcircuit @C=1em @R=.7em @! R { & \poloFantasmaCn{\rA} \qw & \gate{\tC_z} & \poloFantasmaCn{\rB} \qw & \gate{\tA_x} & \poloFantasmaCn{\rC} \qw & \qw \\ & \poloFantasmaCn{\rD} \qw & \gate{\tB_y} & \poloFantasmaCn{\rE} \qw & \gate{\tD_w} & \poloFantasmaCn{\rF} \qw & \qw \gategroup{1}{3}{2}{3}{.7em}{--} \gategroup{1}{5}{2}{5}{.7em}{--}} \end{aligned} \ =\ \begin{aligned} \Qcircuit @C=1em @R=1.2em @! R { & \poloFantasmaCn{\rA} \qw & \gate{\tC_z} & \poloFantasmaCn{\rB} \qw & \gate{\tA_x} & \poloFantasmaCn{\rC} \qw & \qw \\ & \poloFantasmaCn{\rD} \qw & \gate{\tB_y} & \poloFantasmaCn{\rE} \qw & \gate{\tD_w} & \poloFantasmaCn{\rF} \qw & \qw \gategroup{1}{3}{1}{5}{.7em}{--} \gategroup{2}{3}{2}{5}{.7em}{--}} \end{aligned}\,. \end{equation} When one of the two operation is the identity, we wull omit the identity box and drawn only a straight line: \begin{equation} \begin{aligned} \Qcircuit @C=1em @R=.7em @! R { & \poloFantasmaCn{\rA} \qw & \gate{\tC_x} & \poloFantasmaCn{\rB} \qw & \qw \\ & \qw & \poloFantasmaCn{\rC} \qw & \qw & \qw \end{aligned} \end{equation} Therefore, as a consequence of commutation between sequetial and parallel composition, we have the following identity: \begin{equation} \begin{aligned} \Qcircuit @C=1em @R=.7em @! R { & \poloFantasmaCn{\rA} \qw & \gate{\tC_x} & \qw & \poloFantasmaCn{\rB} \qw & \qw \\ & \poloFantasmaCn{\rC} \qw & \qw & \gate{\tD_y} & \poloFantasmaCn{\rD} \qw & \qw \end{aligned} \ =\ \begin{aligned} \Qcircuit @C=1em @R=1.2em @! R { & \poloFantasmaCn{\rA} \qw & \qw & \gate{\tC_x} & \poloFantasmaCn{\rB} \qw & \qw \\ & \poloFantasmaCn{\rC} \qw & \gate{\tD_y} & \qw & \poloFantasmaCn{\rD} \qw & \qw \end{aligned}\,. \end{equation} Preparation Tests and Observation Tests Tests with a trivial input system are called preparation tests, and tests with a trivial output system are called observation tests. They will be represented as follows: \begin{equation} \begin{aligned} \Qcircuit @C=1em @R=.7em @! R {& \prepareC {\{\rho_x\}_{x\in \rX}} & \poloFantasmaCn{\rB} \qw & \qw \end{aligned} \ :=\ \begin{aligned} \Qcircuit @C=1em @R=.7em @! R { & \poloFantasmaCn{\rI} \qw & \gate{\{\rho_x\}_{x\in \rX}} & \poloFantasmaCn{\rB} \qw & \qw \end{aligned}\,, \end{equation} \begin{equation} \begin{aligned} \Qcircuit @C=1em @R=.7em @! R { & & \poloFantasmaCn{\rA} \qw & \measureD{\{a_y\}_{y\in \rY}} \end{aligned} \ :=\ \begin{aligned} \Qcircuit @C=1em @R=.7em @! R { & \poloFantasmaCn{\rA} \qw & \gate{\{a_y\}_{y\in \rY}} & \poloFantasmaCn{\rI} \qw & \qw \end{aligned}\,. \end{equation} The corresponding events will be called preparation events and observation events. In formulas we will also write $\|\rho_i)_\text{A}$ to denote a preparation event and $(a_j\|_\text{A}$ to denote an observation event. Closed Circuits Using the above rules we can build up closed circuits, i.e. circuits with no input and no output system. An example is given by the following circuit: \begin{equation} \label{c:closed1} \begin{aligned} \Qcircuit @C=1em @R=.7em @! R { \multiprepareC{3}{\{\Psi_{i}\}}& \qw\poloFantasmaCn{\rA}& \multigate{1}{\{\tA_{j}\}}& \qw\poloFantasmaCn{\rB}& \gate{\{\tC_{l}\}}& \qw\poloFantasmaCn{\rC}& \multigate{1}{\{\tE_{n}\}}& \qw\poloFantasmaCn{\rD}& \multimeasureD{2}{\{\tG_{q}\}} \\ \pureghost{\{\Psi_{i}\}}& \qw\poloFantasmaCn{\rE}& \ghost{\{\tA_{j}\}}&\qw\poloFantasmaCn{\rF}& \multigate{1}{\{\tD_{m}\}}& \qw\poloFantasmaCn{\rG}& \ghost{\{\tE_{n}\}} \\ \pureghost{\{\Psi_{i}\}}& \qw\poloFantasmaCn{\rH}& \multigate{1}{\{\tB_{k}\}}& \qw\poloFantasmaCn{\rL}& \ghost{\{\tD_{m}\}}&\qw\poloFantasmaCn{\rM}& \multigate{1}{\{\tF_{p}\}}& \qw\poloFantasmaCn{\rN}&\pureghost{\{\tG_{q}\}}\qw \\ \pureghost{\{\Psi_{i}\}}& \qw\poloFantasmaCn{\rO}& \ghost{\{\tB_{k}\}}& \qw& \qw\poloFantasmaCn{\rP}& \qw& \ghost{\{\tF_{p}\}} \\ \end{aligned} \end{equation} where we omitted the probability spaces of each test. Independent Systems For any (generally open) circuit constructed according to the above rules we call a set of systems independent if for each couple of systems in the set the two are not connected by a unidirected path (i.e. following the arrow from the input to the output). For example, in Eq. (<ref>) the sets {A,E}, {H,O}, {A,E,H,O}, {A,L}, {A,E,L,P} are independent, whereas e.g. the sets {A,M}, {A,B}, {A,E,N} are not. A maximal set of independent systems is called a slice. We are now in position to move towards the general purpose of an operational probabilistic theory: predicting and accounting for the joint probability of events corresponding to a particular circuit of connections. Given a closed circuit, as in Eq. (<ref>), we are left with just a joint probability distribution. Therefore, to a closed circuit of event as the following: \begin{equation} \label{c:closed2} \begin{aligned} \Qcircuit @C=1em @R=.7em @! R { \multiprepareC{3}{\Psi_{i}}& \qw\poloFantasmaCn{\rA}& \multigate{1}{\tA_{j}}& \qw\poloFantasmaCn{\rB}& \gate{\tC_{l}}& \qw\poloFantasmaCn{\rC}& \multigate{1}{\tE_{n}}& \qw\poloFantasmaCn{\rD}& \multimeasureD{2}{\tG_{q}} \\ \pureghost{\Psi_{i}}& \qw\poloFantasmaCn{\rE}& \ghost{\tA_{j}}&\qw\poloFantasmaCn{\rF}& \multigate{1}{\tD_{m}}& \qw\poloFantasmaCn{\rG}& \ghost{\tE_{n}} \\ \pureghost{\Psi_{i}}& \qw\poloFantasmaCn{\rH}& \multigate{1}{\tB_{k}}& \qw\poloFantasmaCn{\rL}& \ghost{\tD_{m}}&\qw\poloFantasmaCn{\rM}& \multigate{1}{\tF_{p}}& \qw\poloFantasmaCn{\rN}&\pureghost{\tG_{q}}\qw \\ \pureghost{\Psi_{i}}& \qw\poloFantasmaCn{\rO}& \ghost{\tB_{k}}& \qw& \qw\poloFantasmaCn{\rP}& \qw& \ghost{\tF_{p}} \\ \end{aligned} \end{equation} we will associate a joint probability $p(i,j,k,l,m,n,p,q)$ which we will consider as parametrically dependent on the circuit, namely, for a different choice of events and/or different connections we will have a different joint probability. Since we are interested only in the joint probabilities and their corresponding circuits, we will build up probabilistic equivalence classes, and define: Two events from system A to system B are equivalent if they occur with the same joint probability with the other events within any circuit. We will call transformation from A to B - denoted as $\tA\in\mathsf{Transf}(\rA\rightarrow\rB)$ - the equivalence class of events from A to B that are equivalent in the above sense. Likewise we will call instrument an equivalence class of tests, state an equivalence class of preparation events, and effect an equivalence class of observation events. We will denote the set of states of system A as $\mathsf{St}(\rA)$, and the set of its effects as $\mathsf{Eff}(\rA)$. Clearly, the input systems belonging to two different elements of an equivalence class will be operationally equivalent, and likewise for output systems. We now can define an operational probabilistic theory as follows: An operational probabilistic theory (OPT) is a collection of systems and transformations, along with rules for composition of systems and parallel and sequential composition of transformations. The OPT assigns a joint probability to each closed circuit. Therefore, in an OPT every test from the trivial system I to itself is a probability distribution $\{p_i\}_{i\in \rX}$ for the set of joint outcomes X, with $p(i):=p_i\in\left[0,1\right]$ and $\sum_{i\in\rX}p(i)=1$. Compound events from the trivial system to itself are independent, namely their joint probability is given by the product of the respective probabilities for both the parallel and the sequential composition, namely \begin{equation} \begin{aligned} \Qcircuit @C=1em @R=.7em @! R {& \prepareC{\rho_{i_1}} & \poloFantasmaCn{\rA} \qw & \measureD{a_{i_2}}\\ & \prepareC{\sigma_{j_1}} & \poloFantasmaCn{\rB} \qw & \measureD{b_{j_2}} \end{aligned} \ =\ \begin{aligned} \Qcircuit @C=1em @R=.7em @! R { \prepareC{\rho_{i_1}} & \poloFantasmaCn{\rA} \qw & \measureD{a_{i_2}} & \prepareC{\sigma_{j_1}} & \poloFantasmaCn{\rB} \qw & \measureD{b_{j_2}} \end{aligned} \,. \end{equation} A special case of OPT is the deterministic OPT, where all probabilities are 0 or 1. §.§ States and effects Using the parallel and sequential composition of transformation it follows that any closed circuit can be regarded as the composition of a preparation event and an observation event, for example the circuit in Eq. (<ref>) can be cut along a slice as follows: \begin{equation} \begin{aligned} \Qcircuit @C=1em @R=.7em @! R { \multiprepareC{3}{\Psi_{i}}& \qw\poloFantasmaCn{\rA}& \multigate{1}{\tA_{j}}& \qw\poloFantasmaCn{\rB}& \qw \\ \pureghost{\Psi_{i}}& \qw\poloFantasmaCn{\rE}& \ghost{\tA_{j}}&\qw\poloFantasmaCn{\rF}& \multigate{1}{\tD_{m}}& \qw\poloFantasmaCn{\rG}& \\ \pureghost{\Psi_{i}}& \qw\poloFantasmaCn{\rH}& \multigate{1}{\tB_{k}}& \qw\poloFantasmaCn{\rL}& \ghost{\tD_{m}}&\qw\poloFantasmaCn{\rM}& \multigate{1}{\tF_{p}}& \qw \\ \pureghost{\Psi_{i}}& \qw\poloFantasmaCn{\rO}& \ghost{\tB_{k}}& \qw& \qw\poloFantasmaCn{\rP}& \qw& \ghost{\tF_{p}} \\ \end{aligned} \ + \ \begin{aligned} \Qcircuit @C=1em @R=.7em @! R { & \qw\poloFantasmaCn{\rB}& \gate{\tC_{l}}& \qw\poloFantasmaCn{\rC}& \multigate{1}{\tE_{n}}& \qw\poloFantasmaCn{\rD}& \multimeasureD{2}{\tG_{q}} \\ & & & \qw\poloFantasmaCn{\rG}& \ghost{\tE_{n}} \\ & & & & & \qw\poloFantasmaCn{\rN}& \pureghost{\tG_{q}}\qw \\ \\ \end{aligned} \end{equation} and thus is equivalent to the following state-effect circuit: \begin{equation} \begin{aligned} \Qcircuit @C=1em @R=.7em @! R {& \prepareC{(\Psi_i,\tA_j,\tB_k,\tD_m,\tF_p)} & \poloFantasmaCn{\rB\rG\rN} \qw & \measureD{(\tC_l,\tE_n,\tG_q)} \end{aligned} \end{equation} Therefore, a state $\rho\in\st{A}$ is a functional over effects $\eff{A}$, the functional being denoted with the pairing $(a\|\rho)$ with $a\in\eff{A}$ and analogously an effect $a\in\eff{A}$ is a functional over states $\st{A}$. By taking linear combinations of functionals we see that $\st[R]{A}:=\mathsf{Span}_\mathbb{R}\left[\st{A}\right]$ and $\eff[R]{A}:=\mathsf{Span}_\mathbb{R}\left[\eff{A}\right]$ are dual spaces, and states are positive linear functionals over effects, and effects are positive linear functional over states ($\st[R]{A}$ and $\eff[R]{A}$ are assume finite dimensional and we denote as $D_\rA:=\text{dim}\st[R]{A}\equiv\text{dim}\eff[R]{A}$ also called size of system A). In the following we also denote by $\mathsf{St}_1(\rA)$ and $\mathsf{Eff}_1(\rA)$ the sets of deterministic states and effects, respectively. According to the above definition, two states are different if and only if there exists an effect which occurrs on them with different joint probabilities. We also have that two effects are different if and only if there exists a state on which they have different probabilities. In particular, given two states $\rho_0\ne\rho_1\in\st{A}$ we will say that an effect $a\in\eff{A}$ separates the state $\rho_0$ and $\rho_1$ when $(\rho_1\|a)\ne(\rho_0\|a)$, namely when the effect occurs with different joint probabilities over the two states (the analogous relation holds for separable states respect to effects). Therefore we conclude that: States are separating for effects and effects are separating for states. It is possible to demonstrate that in any convex OPT if two states (effects) $\rho_0$,$\rho_1\in\st{A}$ ($a_0,a_1\in\eff{A}$) are distinct, then one can discriminate them with error probability strictly smaller than $\frac{1}{2}$. §.§ Transformations From what we said before, the following circuit is a state of system BFHO: \begin{equation} \begin{aligned} \Qcircuit @C=1em @R=.7em @! R { \multiprepareC{3}{\Psi}& \qw\poloFantasmaCn{\rA}& \multigate{1}{\tA}& \qw\poloFantasmaCn{\rB}& \qw \\ \pureghost{\Psi}& \qw\poloFantasmaCn{\rE}& \ghost{\tA}&\qw\poloFantasmaCn{\rF}& \qw \\ \pureghost{\Psi}& \qw\poloFantasmaCn{\rH}& \qw \\ \pureghost{\Psi}& \qw\poloFantasmaCn{\rO}& \qw& \\ \end{aligned} \end{equation} This means that any transformation connected to some output systems of a state maps the state into another state of generally different systems. Thus, while states and effects are linear functionals over each other, we can always regard a transformation as a map between states. In particular, a transformation $\tT\in\transf{A}{B}$ is always associated to a map $\hat{\tT}$ from $\st{A}$ to $\st{B}$, uniquely defined as \begin{equation} \hat{\tT}:\|\rho)\in\st{A}\mapsto\hat{\tT}\|\rho)=\|\tT\rho)\in\st{B}. \end{equation} Similarly the transformation can be associated to a map from $\eff{A}$ to $\eff{B}$. The map $\hat{\tT}$ can be linearly extended to a map from $\st[R]{A}$ to $\st[R]{B}$. Notice that the linear extension of $\tT$ (which we will denote by the same symbol) is well defined. In fact, a linear combination of states of A is null - in formula $\sum_{i}c_i\|rho_i)=0$ - if and only if $\sum_{i}c_i(a\|rho_i)=0$ for every $a\in\eff{A}$, and since for every $b\in\eff{B}$ we have $(b\|\tT\in\eff{A}$, then $(b\|\tT(\sum_{i}c_i\|\rho_i))=\sum_{i}c_i(b\|\tT\|\rho_i)=(b\|\sum_{i}c_i\tT\|\rho_i)=0$, and finally We want to stress that if two transformations $\tT,\tT^\prime\in\transf{A}{B}$ correspond to the same map $\hat{\tT}$ from $\st{A}$ to $\st{B}$, this does not mean that the two transformations are the same, since as an equivalence class, they must occur with the same joint probability in all possible circuits. In terms of state mappings, the same definition of the transformation as equivalence class corresponds to say that $\tT,\tT^\prime\in\transf{A}{B}$ as maps from states of AR to states of BR are the same for all possible systems R of the theory, namely $\tT=\tT^\prime\in\transf{A}{B}$ if and only if \begin{equation} \label{eq:equal maps} \forall\rR,\;\forall\Psi\in\st{AR}\quad \begin{aligned} \Qcircuit @C=1em @R=.7em @! R { \multiprepareC{1}{\Psi}& \qw\poloFantasmaCn{\rA}& \gate{\tT}& \qw\poloFantasmaCn{\rB}& \qw \\ \pureghost{\Psi}& \qw\poloFantasmaCn{\rR}& \qw&\qw&\qw \end{aligned} \, = \, \begin{aligned} \Qcircuit @C=1em @R=.7em @! R { \multiprepareC{1}{\Psi}& \qw\poloFantasmaCn{\rA}& \gate{\tT^\prime}& \qw\poloFantasmaCn{\rB}& \qw \\ \pureghost{\Psi}& \qw\poloFantasmaCn{\rR}& \qw&\qw&\qw \end{aligned}\,. \end{equation} Indeed, there exist cases of OPT where there are transformations $\tT=\tT^\prime\in\transf{A}{B}$ corresponding to the same map when applied to $\st{A}$ and not when applied to $\st{AR}$ for some system R, a relevant example is fermionic theory [12, 13]. Since we can take linear combinations of linear transformations, $\transf{A}{B}$ can be embedded in the vector space $\transf{A}{B}$. The deterministic transformations, whose set will be denoted as $\mathsf{Transf}_1(\rA\rightarrow\rB)$, will be also called channels. Finally, a transformation $\tU\in\transf{A}{B}$ is reversible if there exists another transformation $\tU^{-1}\in\transf{B}{A}$ such that $\tU^{-1}\tU = \tI_\rA$ and $\tU\tU^{-1}=\tI_\rB$. The set of reversible transformations from A to B will be denoted by $\textsf{RevTransf}(\rA\rightarrow\rB)$. §.§ Coarse-graining and refinement When dealing with probabilistic events, a natural notion is that of coarse-graining, corresponding to merging events into a single event. According to probability theory, the probability of a coarse-grained event $\rS\subseteq\rX$ subset of the outcome space X is the sum of probabilities of the elements of S, namely $p(\rS)=\sum_{i\in\rS}p(i)$. We then correspondingly have that the coarse-grained event $\tT_\rS$ of a test $\{\tT_i\}_{i\in\rX}$ will be given by \begin{equation} \label{eq:coarse-grained event} \tT_\rS=\sum_{i\in\rS}\tT_i. \end{equation} We stress that the equal sign in Eq. (<ref>) is to be meant in the sense of Eq. (<ref>). In addition to the notion of coarse-grained event we have also that of coarse-grained test, corresponding to the collection of a coarse-grained events $\{\tT_{\rX_l}\}_{l\in\rZ}$ from a partition $\rX=\cup_{l\in\rZ}\rX_l$ of the outcome space $\rX$, with $\rX_i\cap\rX_j=\emptyset$ for $i\neq j$. The converse procedure of coarse-graining is what we call refinement. If $\tT_\rS$ the coarse-graining in Eq. (<ref>), we call any sum $\sum_{i\in\rS^\prime}\tT_i$ with $\rS^\prime\subseteq\rS$ a refinement of $\tT_\rS$. The same notion can be analogously considered for a test. Intuitively, a test that refines another is a test that extracts more detailed information, namely it is a test with better "resolving power". The notion of refinement is translated to transformations (hence also to states, and effects), as equivalence classes of events. Refinement and coarse-graining define a partial ordering in the set of transformations $\transf{A}{B}$, writing $\tD\prec\tC$ if $\tD$ is a refinement of $\tC$. A transformation $\tC$ is atomic if it has only trivial refinement, namely $\tC_i$ refines $\tC$ implies that $\tC_i=p\tC$ for some probability $p\ge0$. A test that consists of atomic transformations is a test whose "resolving power" cannot be further improved. It is often useful to refer to the set of all possible refinements of a given event $\tC$. This set is called refinement set of the event $\tC\in\transf{A}{B}$, and is denoted by $\textsf{RefSet}(\tC)$. In formula, $\textsf{RefSet}(\tC):=\{\tD\in\transf{A}{B}\|\tD\prec\tC\}.$ In the special case of states, we will use the word pure as a synonym of atomic. A pure state describes an event providing maximal knowledge about the system's preparation, namely a knowledge that cannot be further refined (we will denote with $\mathsf{PurSt}(\rA)$ the set of pure states of system $\rA$). As usual, a state that is not pure will be called mixed. An important notion is that of internal state. A state is called internal when any other state can refine it: precisely, $\omega\in\st{A}$ is internal if for every $\rho\in\st{A}$ there is a non-zero probability $p>0$ such that $p\rho$ is a refinement of $\omega$, i.e. $p\rho\in\mathsf{RefSet}(\omega)$. The adjective "internal" has a precise geometric connotation, since the state cannot belong to the border of $\st{A}$. An internal state describes a situation in which there is no definite knowledge about the system preparation, namely a priori we cannot in principle exclude any possible preparation. § QUANTUM THEORY AS AN OPT In this section we provide an overview of the six principles used for constructing quantum theory as an OPT. All features of quantum theory - ranging from the superposition principle, entanglement, no cloning, teleportation, Bell's inequalities violation, quantum cryptography - can be understood and proved using only the principles, without using Hilbert spaces. However, our aim is only to introduce the principle and analyse them from an operative point of view. All the six principles are operational, in that they stipulate whether or not certain tasks can be accomplished: they set the rules of the game for all the experiments and all the protocols that can be carried out in the theory. They also provide a great insight into the worldview at which quantum theory hints. We review the list of the principles: * Atomicity of composition * Perfect discriminability * Ideal compression * Causality * Local discriminability * Purification All six principles, with the exception of purification, express standard features that are shared by both classical and quantum theory. The principle of purification picks up uniquely quantum theory among the theories allowed by the first five, partly explaining the magic of quantum information. §.§ Atomicity of composition In the general framework we encountered the notions of coarse-grained and atomic operation. A coarse-grained operation is obtained by joining together outcomes of a test, corresponding to neglect some information. The inverse process of coarse-graining is that of refining. An atomic operation is one where no information has been neglected, namely an operation that cannot be refined. When the operation is atomic, the experimenter has maximal knowledge of what’s happening in the lab. A test consisting of atomic operations represents the highest level of control achievable according to our theory. The principle of atomicity of composition states that it possible to maintain such a level of control throughout a sequence of experiments, stating precisely what follows: [Atomicity of composition] The sequence of two atomic operations is an atomic operation. One of the immediate consequences granted by atomicity of composition is the following: Given two pure states $\alpha\in\st{A}$ and $\beta\in\st{B}$, the parallel composition of $\alpha$ and $\beta$ is a pure state of $\st{AB}$. §.§ Perfect discriminability Two deterministic states $\rho_0$ and $\rho_1$ are perfectly discriminable if there exists a measurement $\{m_y\}_{y\in\{0,1\}}$ such that $$(m_y\|\rho_x)=\delta_{xy}\quad\forall x,y\in\{0,1\}.$$ The existence of perfectly discriminable states is important, because these states can be used to communicate classical information without errors. In a communication protocol, the sender can encode the value of a bit $x$ into the state $\rho_x$ and then transmit the system to the receiver, who can decode the value of the bit using the measurement $\{m_y\}_{y\in\{0,1\}}$. The perfect discriminability axiom ensures that our ability to discriminate states is as sharp as it could possibly be: except for trivial cases, every state can be perfectly discriminated from some other state. The "trivial cases" are those states that cannot be discriminated from anything else because they contain every other state in their convex decomposition. We can call them internal, or completely mixed. [Perfect discriminability] Every deterministic state that is not completely mixed is perfectly discriminable from some other state. As anticipated, the perfect discriminability axiom guarantees that every non-trivial system has at least two perfectly discriminable states: In a theory satisfying perfect discriminability, every physical system has at least two perfectly discriminable states, unless the system is trivial (i.e. it has only one deterministic state). Pick a pure state $\alpha\in\st{A}$. If $\alpha$ is not internal, then perfect discriminability guarantees that $\alpha$ is perfectly discriminable from some other state $\alpha^\prime$, hence A has two perfectly discriminable states. If $\alpha$ is internal every pure state belongs to its refinement set. Moreover, since it is also pure, i.e. extremal, one has that every other deterministic state $\rho_1\in\mathsf{St}_1(\rA)$ must be equal to $\alpha$, i.e. A has only one deterministic state. An easy consequence of this result is that the theory can describe noiseless classical communication. §.§ Ideal compression Ideal compression garantes that information can be transferred faithfully from one system to another. Namely, suppose that Alice has a preparation device, which prepares system A in some state $\alpha$. Alice does not know the state $\alpha$, but she knows that on average the device prepares the deterministic state $\rho\in\mathsf{St}_1(\rA)$. Now, suppose Alice wants to transfer the state of her system to Bob's laboratory, but unfortunately she cannot send system A directly. Instead, she has to encode the state $\alpha$ into the state of another system B, by applying a suitable deterministic operation $\tE$ (the encoding), which transforms the state $\alpha$ into the state We say that the encoding is lossless for the state $\rho$ iff there exists another deterministic operation $\tD$ (the decoding) such that $$\tD\tE\alpha=\alpha\quad\forall\alpha\in F_\rho$$ where $F_\rho$ is the refinement set of $\rho$, which is made of the set of all states $\alpha$ that are compatible with $\rho$ (on the convex set of states this would be the face to which $\rho$ belongs). This third axiom establishes the possibility of a particular type of lossless encoding, called ideal compression. The ultimate limit to the lossless compression of a given state $\rho$ is reached when every state of the encoding system B is a codeword for some state in $F_\rho$, namely when every state $\beta\in\st{B}$ is of the form $\tE\alpha$ for some $\alpha\in F_\rho$. When this is the case, we say that the compression is efficient, and we call the triple $(\rB,\tE,\tD)$ an ideal compression protocol. [Ideal Compression] Every state can be compressed in a lossless and efficient way. §.§ Causality The causality axiom identifies the input–output ordering of a circuit with the direction along which information flows, identifying such ordering with a proper-time arrow, corresponding to the request that future choices cannot influence the present. The probability of the outcome of a preparation test is independent of the choice of observation tests connected at its output. To better understand the statement it is useful to consider the joint test consisting of a preparation test $\tX=\{\rho_i\}_{i\in\rX}\subset\st{A}$ followed by the observation test $\tY=\{a_j\}_{j\in\rY}\subset\eff{A}$ performed on system A: \begin{equation} \begin{aligned} \Qcircuit @C=1em @R=.7em @! R { \prepareC{\tX}&\qw& \qw\poloFantasmaCn{\rA}&\qw& \measureD{\tY} \end{aligned} \end{equation} The joint probability of preparation $\rho_i$ and observation $a_j$ is given by \begin{equation} \begin{aligned} \Qcircuit @C=1em @R=.7em @! R { \prepareC{\rho_i}&\qw& \qw\poloFantasmaCn{\rA}&\qw& \measureD{a_j} \end{aligned} \end{equation} The marginal probability of the preparation alone does not depend on the outcome $j$. Yet, it generally depends on which observation test $\tY$ is performed, namely $$\sum_{a_j\in\rY}(a_j\|\rho_i)\eqqcolon p(i\|\tX,\tY)\,.$$ The marginal probability of preparation $\rho_i$ is then generally conditioned on the choice of the observation test $\tY$. What the causality axiom states is that $p(i\|\tX,\tY)$ is actually independent of $\tY$, namely for any two different observation tests $\tY=\{a_j\}_{j\in\rY}$ and $\tZ=\{b_k\}_{k\in\rZ}$ one has In a causal OPT the choice of a test on a system can be conditioned on the outcomes of a preceding test, since causality guarantees that the probability distribution of the preceding test is independent of the choice of the following test. This leads us to introduce the notion of conditioned test. If $\{\tA_i\}_{i\in\rX}$ is a test from $\rA$ to $\rB$, and $\{\tB_j^{(i)}\}_{j\in\rY_i}$ is a test from $\rB$ to $\rC$ for every $i\in\rX$, then the conditioned test is a test from $\rA$ to $\rC$, with outcomes $(i,j)\in\rZ:=\cup_i\{\{i\}\times\rY_i\}$, and events $\{\tB_j^{(i)}\circ\tA_i\}_{(i,j)\in\rZ}$. Diagrammatically, the events $\tB_j^{(i)}\circ\tA_i$ are represented as follows: \begin{equation} \begin{aligned} \Qcircuit @C=1em @R=.7em @! R { & \qw\poloFantasmaCn{\rA}& \gate{\tA_i}& \qw\poloFantasmaCn{\rB}& \gate{\tB_j^{(i)}}& \poloFantasmaCn{\rC}\qw& \qw \end{aligned} \end{equation} Among conditioned test, a special role is played by the observe-and-prepare test,where the "connecting" system is the null system I. They are thus made of a preparation test conditioned by an observation test, as follows: \begin{equation} \begin{aligned} \Qcircuit @C=.5em @R=.7em @! R { & \qw\poloFantasmaCn{\rA}& \measureD{l_i}& \poloFantasmaCn{\rC}\qw& \qw \end{aligned} \end{equation} which can be also represented as $\{\|\omega^{(i)})(l_i\|\}_{i\in\rX}$. Another remarkable way to characterize causal theories is to require the unicity of the deterministic effect, the equivalence of this formulation with Axiom <ref> is given by the following lemma. An OPT is causal if and only if for every system $\rA$ there is a unique deterministic effect. We will prove the two directions separately, namely: (1) if the probability of preparation of states is independent of the observation test, then the deterministic effect is unique; (2) vice versa. (1) The probability of the preparation $\rho$ is given by the marginal of the joint probability with the observation, namely $p(\rho)=\sum_{i\in\rX}(a_i\|\rho)$. Upon denoting the deterministic effects of two different tests as $a=\sum_{i\in\rX}a_i$ and $b=\sum_{j\in\rY}b_j$, the statement that the preparation probability is independent of the observation tests translates to $(a\|\rho) = (b\|\rho)$ for every preparation $\rho\in\st{A}$, which implies that $a=b$, since the set of states is separating for events. (2) Uniqueness of the deterministic effect implies that the preparation probability of each state is independent of the test, since the effect $a=\sum_{i\in\rX}a_i$ for any test $\{a_i\}_{i\in\rX}$ is deterministic, and $(a\|\rho)$ for any deterministic effect $a\in\eff{a}$ is the probability of preparation $\rho$. We will denote the unique deterministic effect for system A as $e_\rA$, and the subindex will be dropped when no confusion can arise. In the following we will use the notation $\le$ to denote the partial ordering between effects, defined as follows: $$a,b\in\eff{A},\,a\le b\quad\Leftrightarrow\quad(a\|\rho)\le(b\|\rho),\,\forall\rho\in\st{A}\,.$$ It is immediate to show that the causality condition of Lemma <ref> spawns the following lemmas. Causality is equivalent to the following statements regarding tests: * Completeness of observation tests: For any system $\rA$ and for every observation test $\{a_i\}_{i\in\rX}$ one has * Completeness of tests: For any systems $\rA$, $\rB$ and for every test $\{\tC_i\}_{i\in\rX}$ from $\rA$ to $\rB$ one has * Domination of transformations: For any systems $\rA$, $\rB$ a transformation $\tC\in\transf{A}{B}$ satisfies the condition with the equality if and only if $\tC$ is a channel, i.e. a deterministic transformation corresponding to a single-outcome test. * Domination of effects: For any system $\rA$ all effects are dominated by a unique effect $e_\rA$ which is deterministic $$\forall a\in\eff{A},\quad0\le a\le e_\rA\,.$$ An immediate consequence of uniqueness of the deterministic effect is the identification of all transformations of the form for any observation test $\{a_i\}_{i\in\rX}$ of system A. In particular, we have the factorization of the deterministic effect of composite systems $$e_{\rA\rB}=e_\rA\otimes e_\rB.$$ The uniqueness of the deterministic effect naturally leads to the relevant notion of marginal state or also called local state. The marginal state of $\|\sigma)_{\rA\rB}$ on system $\rA$ is the state represented by the diagram \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{\sigma_{\rA\rB}}& \poloFantasmaCn{\rA}\qw& \qw \\ \pureghost{\sigma_{\rA\rB}}& \poloFantasmaCn{\rB}\qw& \measureD{e_\rB} \\ \end{aligned}\, \eqqcolon \, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \prepareC{\rho_{\rA}}& \poloFantasmaCn{\rA}\qw& \qw \end{aligned} \end{equation} Finally, the last implication of causality we would outline is the impossibility of signaling without interaction, i.e. by just performing local tests. In a causal OPT it is impossible to send signals by performing only local tests. Suppose the general situation in which two "distant" parties Alice and Bob share a bipartite state $\|\Psi)_{\rA\rB}$ of systems A and B. Alice performs her local test $\{\tA_i\}_{i\in\rX}$ on system A and similarly Bob performs his local test $\{\tB_j\}_{j\in\rY}$ on system B. The joint probability of their outcomes is The marginal probabilities $p_i^\rA$ at Alice and $p_j^\rB$ at Bob are given by $$p_i^\rA\coloneqq\sum_{j}p_{ij},\quad p_j^\rB\coloneqq\sum_{i}p_{ij}.$$ Alice's marginal does not depend on the choice of test $\{\tB_j\}$ of Bob, since \begin{equation} \begin{aligned} \end{aligned} \end{equation} where we used Eq. (<ref>) and the normalization condition $\sum_{j}(e\|_{\rB}\tB_j=(e\|_{\rB}$. The same argument holds for Bob’s marginal. §.§ Local discriminability Now we introduce the principle of local discriminability, which stipulates the possibility of discriminating states of composite systems via local measurements on the component systems. [Local discriminability] It is possible to discriminate any pair of states of composite systems using only local measurements. Mathematically the axiom asserts that for every two joint states $\rho,\sigma\in\st{AB}$, with $\rho\ne\sigma$, there exist effects $a\in\eff{A}$ and $b\in\eff{B}$ such that the joint probabilities for the two states are different, namely, in circuits \begin{equation} \label{eq:loc-discr} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{\rho}& \poloFantasmaCn{\rA}\qw& \qw \\ \pureghost{\rho}& \poloFantasmaCn{\rB}\qw& \qw \\ \end{aligned} \,\ne\, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{\sigma}& \poloFantasmaCn{\rA}\qw& \qw \\ \pureghost{\sigma}& \poloFantasmaCn{\rB}\qw& \qw \\ \end{aligned} \, \Longrightarrow \, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{\rho}& \poloFantasmaCn{\rA}\qw& \measureD{a} \\ \pureghost{\rho}& \poloFantasmaCn{\rB}\qw& \measureD{b} \\ \end{aligned} \,\ne\, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{\sigma}& \poloFantasmaCn{\rA}\qw& \measureD{a} \\ \pureghost{\sigma}& \poloFantasmaCn{\rB}\qw& \measureD{b} \\ \end{aligned}\,. \end{equation} We can now prove one of the main theorem following from the principle of local discriminability. A theory satisfies local discriminability if and only if, for every composite system $\rA\rB$, one has \begin{equation} \label{eq:product-rule-for-composite-systems} D_{\rA\rB}=D_\rA D_\rB\,. \end{equation} By Eq. (<ref>), a theory satisfies local discriminability if and only if local effects $a\otimes b\in\eff{AB}$, with $a\in\eff{A}$ and $b\in\eff{B}$, are separating for joint states $\st{AB}$. Equivalently, the set $T\coloneqq\{a\otimes b\|a\in\eff{A},b\in\eff{B}\}$ is a spanning set for $\mathsf{Eff}_\mathbb{R}(\rA\rB)$. Since the dimension of $\mathsf{Span}_\mathbb{R}(T)$ is $D_\rA D_\rB$ and the spaces of states and effects have the same dimension, we have $D_{\rA\rB}=D_\rA D_\rB$. Conversely, if Eq. (<ref>) holds, then the product effects are a spanning set for the vector space $\mathsf{Eff}_\mathbb{R}(\rA\rB)$, hence they are separating, and local discriminability holds. Along with the axiom of local discriminability we introduce the notion of entangled and separable states, where entangled states are defined, by negation, as those states that are not separable. Given $n$ systems $\rA_1,\rA_2,\dots,\rA_n$, the separable states of the composite system $\rA_1\rA_2\dots\rA_n$ are those of the form \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{3}{\Sigma}& \poloFantasmaCn{\rA_1}\qw& \qw \\ \pureghost{\Sigma}& \poloFantasmaCn{\rA_2}\qw& \qw \\ \pureghost{\Sigma}& \vdots&& \\ \pureghost{\Sigma}& \poloFantasmaCn{\rA_n}\qw& \qw \\ \end{aligned} \,=\, \sum_{i\in\rX}p_i \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \prepareC{\alpha_{1_i}}& \poloFantasmaCn{\rA}\qw& \qw \\ \prepareC{\alpha_{2_i}}& \poloFantasmaCn{\rA}\qw& \qw \\ \vdots \\ \prepareC{\alpha_{n_i}}& \poloFantasmaCn{\rA}\qw& \qw \end{aligned} \end{equation} where for $j=1,2,\dots,n$, $\alpha_{j_i}\in\st{A_j}$ $\forall i\in\rX$. §.§ Purification Purification is the really distinctive and fundamental trait of quantum theory, in the sense that purification allows to distinguish it between all the other possible theories (all the ones we can think of). The statement of the axiom is the following. For every system $\rA$ and for every state $\rho\in\st{A}$, there exists a system $\rB$ and a pure state $\Psi\in\mathsf{PurSt}(\rA\rB)$ such that \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \prepareC{\rho}& \poloFantasmaCn{\rA}\qw& \qw \end{aligned} \,=\, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{\Psi}& \poloFantasmaCn{\rA}\qw& \qw \\ \pureghost{\Psi}& \poloFantasmaCn{\rB}\qw& \measureD{e} \\ \end{aligned}\,. \end{equation} If two pure states $\Psi$ and $\Psi^\prime$ satisfy \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{\Psi^\prime}& \poloFantasmaCn{\rA}\qw& \qw\\ \pureghost{\Psi^\prime}& \poloFantasmaCn{\rB}\qw& \measureD{e}\\ \end{aligned} \,=\, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{\Psi}& \poloFantasmaCn{\rA}\qw& \qw \\ \pureghost{\Psi}& \poloFantasmaCn{\rB}\qw& \measureD{e} \\ \end{aligned}\,, \end{equation} then there exists a reversible transformation $\tU$, acting only on system $\rB$, such that \begin{equation} \label{eq:uniq purification} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{\Psi^\prime}& \poloFantasmaCn{\rA}\qw& \qw\\ \pureghost{\Psi^\prime}& \poloFantasmaCn{\rB}\qw& \qw\\ \end{aligned} \,=\, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{\Psi}& \qw& \poloFantasmaCn{\rA}\qw& \qw& \qw \\ \pureghost{\Psi}& \poloFantasmaCn{\rB}\qw& \gate{\tU}& \poloFantasmaCn{\rB}\qw& \qw\\ \end{aligned}\,\,. \end{equation} Here we say that $\Psi$ is a purification of $\rho$ and that $\rB$ is the purifying system. The property stated in Eq. <ref> is called uniqueness of purification and refers to the case in which the two purifications have the same purifying system. It can be easily generalized (the purifying systems are different): If two pure states $\Psi\in\mathsf{PurSt}(\rA\rB)$ and $\Psi^\prime\in\mathsf{PurSt}(\rA\rB^\prime)$ are purifications of the same mixed state, then \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{\Psi^\prime}& \poloFantasmaCn{\rA}\qw& \qw\\ \pureghost{\Psi^\prime}& \poloFantasmaCn{\rB^\prime}\qw& \qw\\ \end{aligned} \,=\, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{\Psi}& \qw& \poloFantasmaCn{\rA}\qw& \qw& \qw \\ \pureghost{\Psi}& \poloFantasmaCn{\rB}\qw& \gate{\tC}& \poloFantasmaCn{\rB^\prime}\qw& \qw\\ \end{aligned} \end{equation} for some deterministic transformation $\tC$ transforming system $\rB$ into system $\rB^\prime$. Pick two pure states $\beta\in\mathsf{PurSt}(\rB)$ and $\beta^\prime\in\mathsf{PurSt}(\rB^\prime)$. Since $\Psi\otimes\beta^\prime$ and $\Psi^\prime\otimes\beta$ are purifications of the same state on $\rA$, the uniqueness of purification implies \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{\Psi^\prime}& \poloFantasmaCn{\rA}\qw& \qw\\ \pureghost{\Psi^\prime}& \poloFantasmaCn{\rB^\prime}\qw& \qw\\ \prepareC{\beta}& \poloFantasmaCn{\rB}\qw& \qw\\ \end{aligned} \,=\, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{\Psi}& \qw& \poloFantasmaCn{\rA}\qw& \qw& \qw\\ \pureghost{\Psi}& \poloFantasmaCn{\rB}\qw& \multigate{1}{\tU}& \poloFantasmaCn{\rB^\prime}\qw& \qw\\ \prepareC{\beta^\prime}& \poloFantasmaCn{\rB^\prime}\qw& \ghost{\tU}& \poloFantasmaCn{\rB}\qw& \qw\\ \end{aligned} \end{equation} for some reversible transformation $\tU$ (we have also assumed local discriminability). Discarding system $\rB$ on both sides we then obtain $\Psi^\prime = (\tI_\rA\otimes\tC)\Psi$, where $\rC$ is the deterministic transformation defined by \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { & \poloFantasmaCn{\rB}\qw& \gate{\tC}& \poloFantasmaCn{\rB^\prime}\qw& \qw \end{aligned} \,\coloneqq \, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { & \poloFantasmaCn{\rB}\qw& \multigate{1}{\tU}& \poloFantasmaCn{\rB^\prime}\qw& \qw\\ \prepareC{\beta}& \poloFantasmaCn{\rB^\prime}\qw& \ghost{\tU}& \poloFantasmaCn{\rB}\qw& \measureD{e}\\ \end{aligned}\,. \end{equation} CHAPTER: BIT COMMITMENT Bit commitment is a cryptographic primitive involving two mistrustful parties, conventionally called Alice and Bob. Alice is supposed to submit an encoded bit of information to Bob in such a way that Bob has (in principle) no chance to identify the bit before Alice later decodes it for him, whereas Alice has (in principle) no way of changing the value of the bit once she has submitted it. In other words, Bob is interested in binding Alice to some commitment, whereas Alice would like to conceal her commitment from Bob. In the first two sections of this Chapter we will describe the protocol: we will start from an historical perspective, from the first article published by Blum in 1983 [14] to the most recent developments of the last decade that focus on the impossibility of bit commitment in quantum theory [15, 16]. Then, we will rigorously define the protocol in the language of OPTs. As every other cryptographic primitive, bit commitment does not need to be perfectly secure, i.e. probability of cheating equals to 0 for Alice and 1/2 for Bob (who can always randomly guess), to be efficient. In fact, even if a greater probability of error is admitted, iterating the protocol the error probability can be generally asymptotically reduced. A protocol that admits this possibility is called unconditionally secure, with a vast literature on the subject. However in our thesis we will only deal with perfectly secure bit commitment. Since we are considering the protocol within the operational framework, there are not yet the technical tools to analyse unconditionally secure bit commitment in OPTs and, anyway, the perfectly secure protocol should be the starting point for a rigorous analysis. The exceptional thing is that in our analysis we will be anyway able to build a cheating scheme also for some unconditionally secure bit commitment protocol, as we will see in Section <ref>. Finally, in the third part of the Chapter, we will study the proof of the impossibility of perfectly secure bit commitment in OP quantum theory done in Ref. [2]. We will review if all of the six axioms of quantum theory introduced in Section <ref> are really necessary conditions. Some of them will be neglected and other will be replaced with weaker hypothesis. With the new sufficient conditions that we will find, the proof of impossibility of perfectly secure bit commitment can be extended to other theories than the quantum one. Some applications hypothesis are the fermionic theory and the real quantum theory and in particular the PR-box theory. To the latter will be focused the next two Chapter, in fact, in literature numerous bit commitment protocols (and more generally quantum key distributions protocols) have been studied in the context of non-local correlated box, i.e. PR-boxes. So our analysis will provide a solid (operational probabilistic) point of view from which study all these protocols that have been prosed in the past years. § FROM COIN TOSSING TO NO-GO THEOREM The bit commitment protocol was conceived for the first time by Blum in 1983 as a building block for secure coin tossing. To cite the abstract of the original work [14]: Alice and Bob want to flip a coin by telephone. (They have just divorced, live in different cities, want to decide who gets the car.) Bob would not like to tell Alice HEADS and hear Alice (at the other end of the line) say “Here goes... I'm flipping the coin .... You lost!”. A standard example to illustrate bit commitment is for Alice to write the bit down on a piece of paper, which is then locked in a safe and sent to Bob, whereas Alice keeps the key. At a later time, she will unveil it by handing over the key to Bob. However, Bob has a well-equipped toolbox at home and may have been able to open the safe in the meantime. So while this scheme may offer reasonably good practical security, it is in principle insecure. Yet all bit commitment schemes that have wide currency today rely on such technological constraints: not on strongboxes and keys, but on unproven assumptions that certain computations are hard to perform. The first example of quantum bit commitment was first proposed by Bennet and Brassard in their famous paper of 1984 [17] as a primitive for implementing coin tossing. In their scheme, Alice commits to a bit value by preparing a sequence of photons in either of two mutually unbiased bases, in a way that the resulting quantum states are indistinguishable to Bob. The authors show that their protocol is secure against so-called passive cheating, in which Alice initially commits to the bit value $k$ and then tries to unveil $1-k$ later. However, they also prove that Alice can cheat with a more sophisticated strategy, in which she initially prepares pairs of maximally entangled states instead, keeps one particle of each pair in her laboratory and sends the second particle to Bob. For the first time, entanglement is recognized as a crucial factor in preventing perfectly secure bit commitment. Subsequent proposals for bit commitment schemes tried to evade this type of attack by forcing the players to carry out measurements and communicate classically as they go through the protocol. At a 1993 conference Brassard et al. presented a bit commitment protocol [18] that was claimed and generally accepted to be unconditionally secure. In 1996 Lo and Chau [19], and Mayers [20] realized that all previously proposed bit commitment protocols were vulnerable to a generalized version of the (EPR) attack that renders the BB84 proposal insecure, a result that they slightly extended to cover quantum bit commitment protocols in general. In OP terms, the determinant factor in the impossibility of bit commitment shifts from entanglement to the purification principle, this will be fully proved in Ref [2]. Their basic argument is the following. At the end of the commitment phase, Bob will hold one out of two quantum states $\psi_k$ as proof of Alice's commitment to the bit value $k\in\{0, 1\}$. Alice holds its purification $\Psi_k$, which she will later pass on to Bob to unveil. For the protocol to be concealing, the two states $k$ should be (almost) indistinguishable, $\psi_0\approx\psi_1$. But Uhlmann's theorem then implies the existence of a unitary transformation $\tU$ that (nearly) rotates the purification of $\psi_0$ into the purification of $\psi_1$. Since $\tU$ is localized on the purifying system only, which is entirely under Alice's control, Lo-Chau-Mayers argue that Alice can switch at will between the two states, and is not in any way bound to her commitment. As a consequence, any concealing bit commitment protocol is argued to be necessarily non-binding. Starting from 2000 the Lo-Chau-Mayers no-go theorem has been continually challenged, arguing that the impossibility proof does not exhaust all conceivable quantum bit commitment protocols. Several protocols have been proposed and claimed to circumvent the no-go theorem. These protocols seek to strengthen Bob's position with the help of "secret parameters" or "anonymous states", so that Alice lacks some information needed to cheat successfully: while Uhlmann's theorem would still imply the existence of a unitary cheating transformation as described above, this transformation might be unknown to Alice. However, the above attempts to build up a secure quantum bit commitment protocol have motivated the thorough analysis of Ref. [15], which provided a strengthened and explicit impossibility proof exhausting all conceivable protocols in which classical and quantum information is exchanged between two parties, including the possibility of protocol aborts and resets. This proof encompasses protocols even with unbounded number of communication rounds (it is only required that the expected number of rounds is finite), and with quantum systems on infinite-dimensional Hilbert spaces. However, the considerable length of this proof made it hard to follow, lacking a synthetic intuition of the impossibility proof. Finally in 2009 Chiribella et al. in Ref. [16] provided a new short impossibility proof of quantum bit commitment. In Ref. [2] a similar demonstrative structure is used to prove the impossibility of perfectly secure bit commitment in an operational framework, not only in quantum theory, but in a wide range of theories with purification. This proof will be the one that we will study in the third Section of this Chapter. § A FORMAL DEFINITION A rigorous definition is in order for a twofold reason. If it is true that it is necessary to define a framework where to operate, it is at the same time important to clearly remark the definition of bit commitment to which the statement "bit commitment is impossible" refers to. Despite we already referred to the word protocol, we did not linger to define what we mean with it, therefore we will start by the notion of protocol and we will state the bit commitment in the OPT language and its key properties only thereafter. §.§ The protocol A protocol regulates the exchange of messages between participants, defining what are the honest strategies that they can adopt, so that at every stage it is clear what type of message is expected from the participants, although, of course, their content is not fixed. The expected message types can be either classical or quantum or a combination thereof. In any bit commitment protocol, we can distinguish two main phases: the first is the commitment phase, in which Alice and Bob exchange classical and quantum messages in order to commit the bit. The second is the opening phase where Alice will send to Bob some classical or quantum information in order to to reveal the bit value. Commitment phase this phase can end either with a successful commitment, or with an abort, in which the two parties irrevocably give up the purpose of committing the bit (of course, in a well designed protocol, if both parties are honest the probability of abort should be vanishingly small). If no abort took place, the bit value is considered to be committed to Bob but, supposedly, concealed from him. Since bit commitment is a two-party protocol and trusted third parties are not allowed, the starting state necessarily has to be originated by one of the two parties. Moreover, since we can always include in the protocol null steps (in which no information, classical or quantum, is exchanged), without loss of generality, we can restrict our attention to protocols that are started by Alice. Opening phase in the case of abort during the commitment, this is just a null step, whereas, in the case of successful commitment, at the opening Alice will send to Bob some classical or quantum information in order to to reveal the bit value. Taking both Alice's message and his own (classical and quantum) records, Bob will then perform a suitable verification measurement. His measurement will result in either a successful readout of the committed bit, or in a failure, e.g. due to the detection of an attempted cheat. Again, in a well-designed protocol the probability of failure should be vanishingly small. §.§ OP bit commitment Alice wants to commit a classical bit $b\in\{0,1\}$ to Bob. * as we have seen before, the first phase is the commitment phase, in which Alice and Bob can perform any sequence of operations. Depending on whether Alice intended to commit $b=0$ or $b=1$, the state $\Psi_0,\,\Psi_1\in\st{AB}$ is selected, respectively. We will assume that Alice and Bob's systems at the end of the commitment phase are A and B respectively, so that the two possible pure states that can be transmitted to Bob are $(e\|_\rA\Psi_0,\,(e\|_\rA\Psi_1\in\st{B}$; * between the commitment and the opening phase Alice and Bob can perform only local operations on their systems A and B, respectively (together with every other ancillary system they control); * during the opening phase, Alice transmits her system A to Bob who can perform any measurement on the joint system AB to know the bit $b$ and to check if it is compatible with the commitment of Alice. In general, to do this Bob can perform a two outcome measurement (Positive Operator Valued Measure, POVM) $\{a_0,a_1\}$ on the joint system AB. From now on, when we will refer to bit commitment we mean a protocol that is included in the previous scheme. Already at the beginning of this Chapter we intuitively mention the two way of cheating that can occur. Being a two party protocol, we can have Alice's cheating, i.e. she changes the bit after the commitment (the protocol is not binding), and Bob's cheating, i.e. Bob discovers the bit committed before the opening (the protocol is not concealing). In this way we have now identified two key properties of any bit commitment protocol. However there is a third key property that is often omitted in the literature: the correctness of the protocol that guarantees the correct verification of the bit committed in the opening phase. To recapitulate with a properer language, we will say that a bit commitment protocol is * binding: if, for honest Bob, Alice should not be able to change the bit she committed. More precisely, assume that a possibly dishonest Alice committed $b$ but she wants to reveal $b^\prime\ne b$, then it must be \begin{equation} \label{eq:binding} \textsf{Pr}\left[\text{ Bob accepts }\|\text{ Alice reveals }b^\prime\,\right]<1\,; \end{equation} * concealing: if, for honest Alice, Bob should not be able to know the bit that Alice committed until she reveals it; * correct: for honest Alice and Bob, if Alice commits $b$ and later reveals $b$ to Bob, then Bob accepts with probability grater then $1/2$: \begin{equation} \label{eq:correctness} \textsf{Pr}\left[\text{ Bob accepts }\|\text{ Alice reveals }b\,\right]>\frac{1}{2}\,. \end{equation} Furthermore, we will say that a bit commitment protocol is perfect if * it is perfectly binding, namely for honest Bob, Alice cannot switch between $\Psi_0$ and $\Psi_1$ in such a way that Bob cannot detect the switch with certainty (i.e. Eq. (<ref>) becomes $\textsf{Pr}\left[\text{ Bob accepts }\|\text{ Alice reveals }b^\prime\,\right]=0$). Namely it does not exists a reversible channel $\tU$ such that \begin{equation} \label{eq:perfectly-binding} \end{equation} * it is perfectly concealing, namely for honest Alice, Bob would not be able to perform some measurement of his system B and gain at least partial information about which bit Alice committed \begin{equation} \label{eq:perfectly-concealing} \end{equation} * it is correct with probability one, namely Bob accepts with probability one. So Eq. (<ref>) becomes \begin{equation} \textsf{Pr}\left[\text{ Bob accepts }\|\text{ Alice reveals }b\,\right]=1\,. \end{equation} In the perfect implementation of the protocol the probability of cheating of Bob must be $1/2$, since in the worst case he can always make a random guess of the committed bit. The cheating probability of Alice is instead equal to zero. As we have anticipated in Section <ref>, perfectly secure bit commitment protocol is impossible in quantum and classic theory. However, if the perfect concealing and perfect biding conditions are relaxed as follows \begin{equation} \begin{aligned} \end{aligned} \end{equation} interesting level of security relevant for concrete applications emerges. Within this scenario the main effort is in quantifying the cheating probabilities and their trade-off in operational terms. The goal is to achieve a protocol that is asymptotically binding and concealing (both the Alice and Bob cheating probability can be made arbitrarily small) against an adversary that has no restrictions on the computational resources. In this case one has unconditionally secure bit commitment. § LIGHTENING NO BIT COMMITMENT In this Section we will analyse the necessary assumptions to prove the impossibility of perfectly secure bit commitment. In the literature, a no-go theorem in quantum theory has been proven in Ref. [15, 16] but we found that not all of the axioms of quantum theory are necessary. So, using weaker assumption, we will generalize the impossibility of bit commitment to a larger set of operational probabilistic theories. Clearly the causality Axiom <ref> is essential, otherwise the very protocol could not be defined properly (there would not be a given order in the succeeding of the phases). In our analysis we also assume Axiom <ref>, atomicity of composition, that is a sufficient condition to grant Corollary <ref>. Finally, instead of the purification Axiom <ref>, we take the following weaker assumption. Before stating it, we introduce the notion of dynamically faithful state. A state $\sigma\in\st{AC}$ is dynamically faithful for system $\rA$ if for any couple of transformations $\tA,\,\tA^\prime\in\transf{A}{B}$ one has \begin{equation} \tA\|\sigma)_{\rA\rC}=\tA^\prime\|\sigma)_{\rA\rC}\Longrightarrow\tA=\tA^\prime\, . \end{equation} We are now in position to state the "new" axiom. For every system $\rA$ there exist a system $\tilde{\rA}$ and a pure state $\Psi^{(\rA)}\in\mathsf{St}(\rA\tilde{\rA})$ that is dynamically faithful for system $\rA$. Furthermore, for every system $\rB$ and for every bipartite state $R\in\mathsf{St}(\rB\tilde{\rA})$ such that \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{R}& \poloFantasmaCn{\rB}\qw& \measureD{e}\\ \pureghost{R}& \poloFantasmaCn{\tilde{\rA}}\qw& \qw\\ \end{aligned} \, = \, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{\Psi^{(\rA)}}& \poloFantasmaCn{\rA}\qw& \measureD{e}\\ \pureghost{\Psi^{(\rA)}}& \poloFantasmaCn{\tilde{\rA}}\qw& \qw\\ \end{aligned}\,, \end{equation} there exists a purification of $R$. If $\Phi$, $\Phi^\prime\in\mathsf{St}(\rC\rB\tilde{\rA})$ are two purifications of $R$ then they are connected by a reversible transformation $\tU\in\mathsf{Transf}(\rC)$. In quantum theory, the existence of mixed faithful states is a direct consequence of local discriminability. In a theory with both local discriminability and purification there exist also dynamically faithful states that are pure. In Axiom <ref> we do not require local discriminability nor purification but the existence of dynamically faithful pure states. In addition we also require that a purification exists only for those states that have the same marginal of the dynamically faithful pure ones and that this purification is unique up to a reversible transformation on the purifying system. An immediate consequence of Axioms <ref> and <ref> is the following lemma on the uniqueness of purification. Let $\Psi\in\st{AB}$ and $\Psi^\prime\in\st{AC}$ be two purification of $\rho\in\st{A}$. Then there exists a channel $\tC\in\transf{B}{C}$ such that \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{\Psi^\prime}& \poloFantasmaCn{\rA}\qw& \qw\\ \pureghost{\Psi^\prime}& \poloFantasmaCn{\rC}\qw& \qw\\ \end{aligned} \, = \, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{\Psi}& \qw& \poloFantasmaCn{\rA}\qw& \qw&\qw\\ \pureghost{\Psi}& \poloFantasmaCn{\rB}\qw& \gate{\tC}& \poloFantasmaCn{C}\qw& \qw\\ \end{aligned}\,. \end{equation} Moreover, channel $\tC$ has the form \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \poloFantasmaCn{\rB}\qw& \gate{\tC}& \poloFantasmaCn{\rC}\qw& \qw \end{aligned} \, = \, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \prepareC{\varphi_0}& \poloFantasmaCn{\rC}\qw& \multigate{1}{\tU}& \poloFantasmaCn{\rB}\qw& \measureD{e}\\ \poloFantasmaCn{\rB}\qw& \ghost{\tU}& \poloFantasmaCn{C}\qw& \qw\\ \end{aligned}\,. \end{equation} for some pure state $\varphi_0\in\st{C}$ and some reversible channel $\tU\in\transf{B}{C}$. Let $\eta$ and $\varphi_0$ be an arbitrary pure state of B and C, respectively. Then, due to Axiom <ref>, $\|\Psi^\prime)_{\rA\rC}\|\eta)_{\rB}$ and $\|\Psi)_{\rA\rB}\|\varphi_0)_{\rC}$ are still two pure states and so they are both purifications of $\rho$ with the same purifying system $\rB\rC$. Due to Axiom <ref>, we have \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{\Psi^\prime}& \poloFantasmaCn{\rA}\qw& \qw\\ \pureghost{\Psi^\prime}& \poloFantasmaCn{\rC}\qw& \qw\\ \prepareC{\eta}& \poloFantasmaCn{\rB}\qw& \qw \end{aligned} \, = \, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{\Psi}& \qw& \poloFantasmaCn{\rA}\qw& \qw&\qw\\ \pureghost{\Psi}& \poloFantasmaCn{\rB}\qw& \multigate{1}{\tU}& \poloFantasmaCn{\rC}\qw& \qw\\ \prepareC{\varphi_0}& \poloFantasmaCn{\rC}\qw& \ghost{\tU}& \poloFantasmaCn{B}\qw& \qw\\ \end{aligned}\,. \end{equation} Applying the deterministic effect $e_\rB$ on system $\rB$ we obtain the thesis, with $\tC=(e\|_\rB\;\tU\|\varphi_0)$. §.§ Reversible dilation of channels Before starting the proof of the impossibility of bit commitment it is in order to derive some useful results about reversible dilations of channels. Let $R\in\mathsf{St}(\rB\tilde{\rA})$ be a state such that \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{R}& \poloFantasmaCn{\rB}\qw& \measureD{e}\\ \pureghost{R}& \poloFantasmaCn{\tilde{\rA}}\qw& \qw \end{aligned} \, = \, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{\Psi^{(\rA)}}& \poloFantasmaCn{\rA}\qw& \measureD{e}\\ \pureghost{\Psi^{(\rA)}}& \poloFantasmaCn{\tilde{\rA}}\qw& \qw\\ \end{aligned}\,. \end{equation} where $\Psi^{(\rA)}\in\mathsf{St}(\rA\tilde{\rA})$ is a pure dynamically faithful state for system $\rA$. Then there exist a system $\rC$, a pure state $\varphi_0\in\mathsf{St}(\rB\rC)$, and a reversible channel $\tU\in\mathsf{Transf}(\rA\rB\rC)$ such that \begin{equation} \label{eq:revchannel} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{R}& \poloFantasmaCn{\rB}\qw& \qw\\ \pureghost{R}& \poloFantasmaCn{\tilde{\rA}}\qw& \qw \end{aligned} \, = \, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \prepareC{\varphi_0}& \poloFantasmaCn{\rB\rC}\qw& \multigate{1}{\tU}& \poloFantasmaCn{\rA\rC}\qw& \measureD{e}\\ \multiprepareC{1}{\Psi^{(\rA)}}& \poloFantasmaCn{\rA}\qw& \ghost{\tU}& \poloFantasmaCn{B}\qw& \qw\\ \pureghost{\Psi^{(\rA)}}& \qw& \poloFantasmaCn{\tilde{\rA}}\qw& \qw&\qw\\ \end{aligned}\,. \end{equation} Moreover the channel $\tV\in\mathsf{Transf}(\rA\rightarrow\rA\rB\rC)$ defined by $\tV:=\tU\|\varphi_0)_{\rB\rC}$ is unique up to reversible channels on $\rA\rC$. Take a purification of $R$, say $\Psi_R\in\mathsf{St}(\rC\rB\tilde{\rA})$ for some purifying system $\rC$ (existence of such a purification is granted by Axiom <ref>). One has \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{2}{\Psi_R}& \poloFantasmaCn{\rC}\qw& \measureD{e}\\ \pureghost{\Psi_R}& \poloFantasmaCn{\rB}\qw& \measureD{e}\\ \pureghost{\Psi_R}& \poloFantasmaCn{\tilde{\rA}}\qw& \qw \end{aligned} \, = \, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{R}& \poloFantasmaCn{\rB}\qw& \measureD{e}\\ \pureghost{R}& \poloFantasmaCn{\tilde{\rA}}\qw& \qw\\ \end{aligned} \, = \, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{\Psi^{(\rA)}}& \poloFantasmaCn{\rA}\qw& \measureD{e}\\ \pureghost{\Psi^{(\rA)}}& \poloFantasmaCn{\tilde{\rA}}\qw& \qw\\ \end{aligned} \end{equation} that is, the pure state $\Psi_R$ and $\Psi^{(\rA)}$ have the same marginal on system $\tilde{\rA}$. Applying the uniqueness of purification expressed by Lemma <ref> one then obtains \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{2}{\Psi_R}& \poloFantasmaCn{\rC}\qw& \qw\\ \pureghost{\Psi_R}& \poloFantasmaCn{\rB}\qw& \qw\\ \pureghost{\Psi_R}& \poloFantasmaCn{\tilde{\rA}}\qw& \qw \end{aligned} \, = \, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \prepareC{\varphi_0}& \poloFantasmaCn{\rB\rC}\qw& \multigate{1}{\tU}& \poloFantasmaCn{\rA}\qw& \measureD{e}\\ \multiprepareC{1}{\Psi^{(\rA)}}& \poloFantasmaCn{\rA}\qw& \ghost{\tU}& \poloFantasmaCn{\rB\rC}\qw& \qw\\ \pureghost{\Psi^{(\rA)}}& \qw& \poloFantasmaCn{\tilde{\rA}}\qw& \qw&\qw \end{aligned}\, . \end{equation} Applying the deterministic effect on system $\rC$ on both sides, one then proves Eq. (<ref>). Moreover, if $\tV^\prime\coloneqq\tU^\prime\|\varphi^\prime_0)_{\rB\rC}$ is channel such that Eq. (<ref>) holds, then the pure states $\tV\|\Psi^{(\rA)})_{\rA\tilde{\rA}}$ and $\tV^\prime\|\Psi^{(\rA)})_{\rA\tilde{\rA}}$ have the same marginal on system $\rB\tilde{\rA}$. Uniqueness of purification then implies \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { & & \amultigate{1}{\tV^\prime}& \poloFantasmaCn{\rA\rC}\qw& \qw\\ \multiprepareC{1}{\Psi^{(\rA)}}& \poloFantasmaCn{\rA}\qw& \ghost{\tV^\prime}& \poloFantasmaCn{\rB}\qw& \qw\\ \pureghost{\Psi^{(\rA)}}& \qw& \poloFantasmaCn{\tilde{\rA}}\qw& \qw&\qw \end{aligned} \, = \, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { & & \amultigate{1}{\tV}& \poloFantasmaCn{\rA\rC}\qw& \gate{\tW}& \poloFantasmaCn{\rA\rC}\qw& \qw\\ \multiprepareC{1}{\Psi^{(\rA)}}& \poloFantasmaCn{\rA}\qw& \ghost{\tV}& \poloFantasmaCn{\rB}\qw& \qw&\qw&\qw\\ \pureghost{\Psi^{(\rA)}}& \qw& \poloFantasmaCn{\tilde{\rA}}\qw& \qw&\qw&\qw&\qw \end{aligned} \end{equation} for some reversible channel $\tW\in\mathsf{Transf}(\rA\rC)$. Since $\Psi^{(\rA)}$ is dynamically faithful for $\rA$, this implies $\tV^\prime=\tW\tV$. We now give the definition of dilatation and reversible dilatation. A dilatation of channel $\tC\in\transf{A}{B}$ is a channel $\tV\in\transf{A}{BE}$ such that \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \poloFantasmaCn{\rA}\qw& \gate{\tC}& \poloFantasmaCn{\rB}\qw& \qw \end{aligned}\, = \, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { & & \amultigate{1}{\tV}& \poloFantasmaCn{\rE}\qw& \measureD{e}\\ \poloFantasmaCn{\rA}\qw& \ghost{\tV}& \poloFantasmaCn{\rB}\qw& \qw\\ \end{aligned}\, . \end{equation} We refer to system $\rE$ as to the environment. A dilatation $\tV\in\transf{A}{BE}$ is called reversible if there exists a system $\rE_0$ such that $\rA\rE_0\simeq\rB\rE$ and \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { & & \amultigate{1}{\tV}& \poloFantasmaCn{\rE}\qw& \qw\\ \poloFantasmaCn{\rA}\qw& \ghost{\tV}& \poloFantasmaCn{\rB}\qw& \qw\\ \end{aligned} \, = \, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \prepareC{\varphi_0}& \poloFantasmaCn{\rE_0}\qw& \multigate{1}{\tU}& \poloFantasmaCn{\rE}\qw& \qw\\ \poloFantasmaCn{\rA}\qw& \ghost{\tU}& \poloFantasmaCn{\rB}\qw& \qw\\ \end{aligned} \end{equation} for some pure state $\varphi_0\in\mathsf{St}(\rE_0)$ and some reversible channel $\tU\in\mathsf{Transf}(\rA\rE_0\rightarrow\rB\rE)$. According to the above definitions, we have the following dilatation theorem: Every channel $\tC\in\transf{A}{B}$ has a reversible dilatation $\tV\in\transf{A}{BE}$. If $\tV$, $\tV^\prime\in\transf{A}{BE}$ are two reversible dilatations of the same channel, then they are connected by a reversible transformation on the environment, namely \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { & & \amultigate{1}{\tV^\prime}& \poloFantasmaCn{\rE}\qw& \measureD{e}\\ \poloFantasmaCn{\rA}\qw& \ghost{\tV^\prime}& \poloFantasmaCn{\rB}\qw& \qw\\ \end{aligned} \, = \, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { & & \amultigate{1}{\tV}& \poloFantasmaCn{\rE}\qw& \measureD{e}\\ \poloFantasmaCn{\rA}\qw& \ghost{\tV}& \poloFantasmaCn{\rB}\qw& \qw\\ \end{aligned} \end{equation} \begin{equation} \Longrightarrow \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { & & \amultigate{1}{\tV^\prime}& \poloFantasmaCn{\rE}\qw& \qw\\ \poloFantasmaCn{\rA}\qw& \ghost{\tV^\prime}& \poloFantasmaCn{\rB}\qw& \qw\\ \end{aligned} \, = \, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { & & \amultigate{1}{\tV}& \poloFantasmaCn{\rE}\qw& \gate{\tW}& \poloFantasmaCn{\rE}\qw& \qw\\ \poloFantasmaCn{\rA}\qw& \ghost{\tV}& \qw& \poloFantasmaCn{\rB}\qw& \qw&\qw\\ \end{aligned} \end{equation} for some reversible channel $\tW\in\mathsf{Transf}(\rE)$. Let us store the channel $\tC$ in the faithful state $\Psi^{(\rA)}\in\mathsf{St}(\rA\tilde{\rA})$, thus getting the state $R_\tC$: \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{R_\tC}& \poloFantasmaCn{\rB}\qw& \qw\\ \pureghost{R_\tC}& \poloFantasmaCn{\tilde{\rA}}\qw& \qw\\ \end{aligned} \, \coloneqq \, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{\Psi}& \poloFantasmaCn{\rA}\qw& \gate{\tC}& \poloFantasmaCn{\rB}\qw& \qw\\ \pureghost{\Psi}& \qw& \poloFantasmaCn{\tilde{\rA}}\qw& \qw&\qw\\ \end{aligned}\, . \end{equation} Since $\tC$ is a channel, it satisfy the normalization condition \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \poloFantasmaCn{\rA}\qw& \gate{\tC}& \poloFantasmaCn{\rB}\qw& \measureD{e} \end{aligned} \, = \, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \poloFantasmaCn{\rA}\qw& \measureD{e} \end{aligned}\, , \end{equation} which implies \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{R_\tC}& \poloFantasmaCn{\rB}\qw& \measureD{e}\\ \pureghost{R_\tC}& \poloFantasmaCn{\tilde{\rA}}\qw& \qw\\ \end{aligned} \, = \, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{\Psi^{(\rA)}}& \poloFantasmaCn{\rA}\qw& \gate{\tC}& \poloFantasmaCn{\rB}\qw& \measureD{e}\\ \pureghost{\Psi^{(\rA)}}& \qw& \poloFantasmaCn{\tilde{\rA}}\qw& \qw&\qw\\ \end{aligned} \, = \, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{\Psi^{(\rA)}}& \poloFantasmaCn{\rA}\qw& \measureD{e}\\ \pureghost{\Psi^{(\rA)}}& \poloFantasmaCn{\tilde{\rA}}\qw& \qw\\ \end{aligned}\, . \end{equation} Now, applying Lemma <ref> we obtain \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{R_\tC}& \poloFantasmaCn{\rB}\qw& \qw\\ \pureghost{R_\tC}& \poloFantasmaCn{\tilde{\rA}}\qw& \qw\\ \end{aligned} \, = \, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \prepareC{\varphi_0}& \poloFantasmaCn{\rB\rC}\qw& \multigate{1}{\tU}& \poloFantasmaCn{\rA\rC}\qw& \measureD{e}\\ \multiprepareC{1}{\Psi^{(\rA)}}& \poloFantasmaCn{\rA}\qw& \ghost{\tU}& \poloFantasmaCn{\rB}\qw& \qw\\ \pureghost{\Psi^{(\rA)}}& \qw& \poloFantasmaCn{\tilde{\rA}}\qw& \qw&\qw\\ \end{aligned}\, . \end{equation} Since $\Psi^{(\rA)}$ is dynamically faithful for system $\rA$, this implies \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \poloFantasmaCn{\rA}\qw& \gate{\tC}& \poloFantasmaCn{\rB}\qw& \qw\\ \end{aligned} \, = \, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \prepareC{\varphi_0}& \poloFantasmaCn{\rB\rC}\qw& \multigate{1}{\tU}& \poloFantasmaCn{\rA\rC}\qw& \measureD{e}\\ \poloFantasmaCn{\rA}\qw& \ghost{\tU}& \poloFantasmaCn{\rB}\qw& \qw\\ \end{aligned}\, . \end{equation} Therefore, $\tV \coloneqq \tU\|\varphi_0)_{\rB\rC}$ is a reversible dilatation of $\tC$, with $\rE_0\coloneqq\rB\rC$ and $\rE\coloneqq\rA\rC$. Finally, the uniqueness clause in Lemma <ref> implies uniqueness of the dilatation. Moreover, two reversible dilatations of the same channel with different environments are related as follows. Let $\tV\in\transf{A}{BE}$ and $\tV^\prime\in\mathsf{Transf}(\rA\rightarrow\rB\rE^\prime)$ be two reversible dilatations of the same channel $\tC$, with generally different environment $\tE$ and $\rE^\prime$. Then there is a channel $\tL$ from $\rE$ to $\rE\rE^\prime$ such that \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { & & \amultigate{1}{\tV^\prime}& \poloFantasmaCn{\rE^\prime}\qw& \qw\\ \poloFantasmaCn{\rA}\qw& \ghost{\tV^\prime}& \poloFantasmaCn{\rB}\qw& \qw\\ \end{aligned} \, = \, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { & & & & \amultigate{1}{\tL}& \poloFantasmaCn{\rE}\qw& \measureD{e}\\ & & \amultigate{1}{\tV}& \poloFantasmaCn{\rE}\qw& \ghost{\tL}& \poloFantasmaCn{\rE^\prime}\qw& \qw\\ \poloFantasmaCn{\rA}\qw& \ghost{\tV}& \qw& \poloFantasmaCn{\rB}\qw& \qw&\qw\\ \end{aligned}\, . \end{equation} The channel $\tL$ has the form \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { & & \amultigate{1}{\tL}& \poloFantasmaCn{\rE}\qw& \qw\\ \poloFantasmaCn{\rE}\qw& \ghost{\tL}& \poloFantasmaCn{\rE^\prime}\qw& \qw\\ \end{aligned} \, = \, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \prepareC{\eta_0}& \poloFantasmaCn{\rE^\prime}\qw& \multigate{1}{\tU}& \poloFantasmaCn{\rE}\qw& \qw\\ \poloFantasmaCn{\rE}\qw& \ghost{\tU}& \poloFantasmaCn{\rE^\prime}\qw& \qw\\ \end{aligned} \end{equation} for some pure state $\eta_0$ and some reversible transformation $\tU\in\mathsf{Transf}(\rE\rE^\prime)$. Apply $\tV$ and $\tV^\prime$ to the faithful state $\Psi^{(\rA)}$ and then use the uniqueness of purification stated in Lemma <ref>. §.§ Casually ordered channels and channels with memory The last step before the proof of the theorem is the notion of casually ordered channels and channels with memory. A bipartite channel $\tC$ from $\rA_1\rA_2$ to $\rB_1\rB_2$ is casually ordered if there is a channel $\tD$ from $\rA_1$ to $\rB_1$ such that $(e\|_{\rB_2}\tC=\tD\otimes(e\|_{\rA_2}$. Diagrammatically, \begin{equation} \label{eq:casuallyorderedbipartitechannels} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \poloFantasmaCn{\rA_1}\qw& \multigate{1}{\tC}& \poloFantasmaCn{\rB_1}\qw& \qw\\ \poloFantasmaCn{\rA_2}\qw& \ghost{\tC}& \poloFantasmaCn{\rB_2}\qw& \measureD{e} \end{aligned} \, = \, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \poloFantasmaCn{\rA_1}\qw& \gate{\tD}& \poloFantasmaCn{\rB_1}\qw& \qw\\ \qw& \poloFantasmaCn{\rA_2}\qw& \qw& \measureD{e} \end{aligned}\,. \end{equation} Eq. (<ref>) means that the channel $\tC$ does not allow for signaling from the input system $\rA_2$ to the output system $\rB_1$. In a relativistic context, this can be interpreted as $\rB_1$ being outside the casual future of $\rA_2$. A bipartite channel $\tC$ from $\rA_1\rA_2$ to $\rB_1\rB_2$ can be realized as a sequence of two channels with memory if there exist two system $\rE_1,\,\rE_2$, called memory systems, and two channels $\tC_1\in\mathsf{Transf}(\rA_1\rightarrow\rB_1\rE_1)$ and $\tC_2\in\mathsf{Transf}(\rA_2\rE_1\rightarrow\rB_2\rE_2)$ such that $\tC=(e\|_{\rE_2}(\tC_2\otimes\tI_{\rB_1})(\tI_{\rA_2}\otimes\tC_1)$. Diagrammatically, \begin{equation} \label{eq:channelwithmemory} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \poloFantasmaCn{\rA_1}\qw& \multigate{1}{\tC}& \poloFantasmaCn{\rB_1}\qw& \qw\\ \poloFantasmaCn{\rA_2}\qw& \ghost{\tC}& \poloFantasmaCn{\rB_2}\qw& \qw\\ \end{aligned} \, = \, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \poloFantasmaCn{\rA_1}\qw& \multigate{1}{\tC_1}& \poloFantasmaCn{\rB_1}\qw& \qw & & \poloFantasmaCn{\rA_2}\qw& \multigate{1}{\tC_2}& \poloFantasmaCn{\rB_2}\qw& \qw\\ & & \aghost{\tC_1}& \qw& \poloFantasmaCn{\rE_1}\qw& \qw&\qw& \ghost{\tC_2}& \poloFantasmaCn{\rE_2}\qw& \measureD{e}\\ \end{aligned}\,. \end{equation} For causally ordered bipartite channels the dilatation theorem implies the following result: A bipartite channel $\tC$ from $\rA_1\rA_2$ to $\rB_1\rB_2$ is casually ordered if and only if it can be realized as a sequence of two channels with memory. Moreover, the channels $\tC_1,\,\tC_2$ in Eq. (<ref>) can be always chosen such that $\tC_2\tC_1$ is a reversible dilatation of $\tC$. If Eq. (<ref>) holds, the channel $\tC$ is clearly casually ordered, with the channel $\tD$ given by $\tD\coloneqq(e\|_{\rE_1}\tC_1$. Conversely, suppose that $\tC$ is casually ordered. Take a reversible dilatation of $\tC$, say $\tV\in\mathsf{Transf}(\rA_1\rA_2\rightarrow\rB_1\rB_2\rE)$, and a reversible dilatation of $\tD$, say $\tV_1\in\mathsf{Transf}(\rA_1\rightarrow\rB_1\rE_1)$. Now by the definition of casually ordered channels (Eq. (<ref>)) we have \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \poloFantasmaCn{\rA_1}\qw& \multigate{2}{\tV}& \poloFantasmaCn{\rB_1}\qw& \qw\\ \poloFantasmaCn{\rA_2}\qw& \ghost{\tV}& \poloFantasmaCn{\rB_2}\qw& \measureD{e}\\ & & \aghost{\tV}& \poloFantasmaCn{\rE}\qw& \measureD{e} \end{aligned} \, = \, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \poloFantasmaCn{\rA_1}\qw& \multigate{1}{\tV_1}& \poloFantasmaCn{\rB_1}\qw& \qw\\ & & \aghost{\tV_1}& \poloFantasmaCn{\rE_1}\qw& \measureD{e}\\ \qw& \poloFantasmaCn{\rA_2}\qw& \qw& \measureD{e} \end{aligned}\, . \end{equation} This means that $\tV$ and $\tV_1\otimes\tI_{\rA_2}$ are two reversible dilatations of the same channel. By the uniqueness of the reversible dilatation expressed by Lemma <ref> we then obtain \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \poloFantasmaCn{\rA_1}\qw& \multigate{2}{\tV}& \poloFantasmaCn{\rB_1}\qw& \qw\\ \poloFantasmaCn{\rA_2}\qw& \ghost{\tV}& \poloFantasmaCn{\rB_2}\qw& \qw\\ & & \aghost{\tV}& \poloFantasmaCn{\rE}\qw& \qw \end{aligned} \, = \, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \poloFantasmaCn{\rA_1}\qw& \multigate{1}{\tV_1}& \qw& \poloFantasmaCn{\rB_1}\qw& \qw& \qw\\ & & \aghost{\tV_1}& \poloFantasmaCn{\rE_1}\qw& \multigate{2}{\tL}& \poloFantasmaCn{\rB_2}\qw& \qw\\ \qw& \poloFantasmaCn{\rA_2}\qw& \qw& \ghost{\tL}& \poloFantasmaCn{\rE}\qw& \qw\\ & & & & \aghost{\tL}& \poloFantasmaCn{\rE_1\rA_2}\qw& \measureD{e} \end{aligned}\, . \end{equation} Once we have defined $\rE_2\coloneqq\rE\rE_1\rA_2$ it only remains to observe that the above diagram is nothing but the thesis, with $\tC_1=\tV_1$ and $\tC_2=\tL$. By construction, $\tC_2\tC_1$ is a reversible dilatation of $\tC$. The definition of casually ordered bipartite channel is easily extended to the multipartite case. Here we will only report the definition and the two main theorems of the theory. For their demonstrations we remand to the original article [2] since they are still valid also from our three Axiom as starting point. An N-partite channel $\tC^{(N)}$ from $\rA_1\dots\rA_N$ to $\rB_1\dots\rB_N$ is causally ordered if for every $k\le N$ there is a channel $\tC^{(k)}$ from $\rA_1\dots\rA_k$ to $\rB_1\dots\rB_k$ such that \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \poloFantasmaCn{\rA_1}\qw& \multigate{5}{\tC^{(N)}}& \poloFantasmaCn{\rB_1}\qw& \qw\\ & \vdots & \aghost{\tC^{(N)}}& \vdots& \\ \poloFantasmaCn{\rA_k}\qw& \ghost{\tC^{(N)}}& \poloFantasmaCn{\rB_k}\qw& \qw\\ \poloFantasmaCn{\rA_{k+1}}\qw& \ghost{\tC^{(N)}}& \poloFantasmaCn{\rB_{k+1}}\qw& \measureD{e}\\ & \vdots & \aghost{\tC^{(N)}}& \vdots& \\ \poloFantasmaCn{\rA_{N}}\qw& \ghost{\tC^{(N)}}& \poloFantasmaCn{\rB_{N}}\qw& \measureD{e}\\ \end{aligned} \, = \, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \poloFantasmaCn{\rA_1}\qw& \multigate{2}{\tC^{(k)}}& \poloFantasmaCn{\rB_1}\qw& \qw\\ & \vdots & \aghost{\tC^{(k)}}& \vdots& \\ \poloFantasmaCn{\rA_1}\qw& \ghost{\tC^{(k)}}& \poloFantasmaCn{\rB_1}\qw& \qw\\ \qw& \poloFantasmaCn{\rA_{k+1}}\qw& \qw& \measureD{e}\\ &\vdots& & \vdots&\\ \qw& \poloFantasmaCn{\rA_{N}}\qw& \qw& \measureD{e}\\ \end{aligned}\, . \end{equation} The definition means that the output systems $\rB_1\dots\rB_k$ are outside the casual future of any input system $\rA_l$ with $l>k$. Causally ordered channels can be characterized as follows. An N-partite channels $\tC^{(N)}$ from $\rA_1\dots\rA_N$ to $\rB_1\dots\rB_N$ is causally ordered if and only if there exist a sequence of memory systems $\{\rE_k\}_{k=0}^N$ with $\rE_0=\rI$ and a sequence of channels $\{\tV_k\}_{k=1}^N$, with $\tV_k\in\mathsf{Transf}(\rA_k\rE_{k-1}\rightarrow\rB_k\rE_k)$ such that \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \poloFantasmaCn{\rA_1}\qw& \multigate{2}{\tC^{(N)}}& \poloFantasmaCn{\rB_1}\qw& \qw\\ & \vdots & \aghost{\tC^{(N)}}& \vdots& \\ \poloFantasmaCn{\rA_N}\qw& \ghost{\tC^{(N)}}& \poloFantasmaCn{\rB_N}\qw& \qw \end{aligned} \, = \, \begin{aligned} \Qcircuit @C=1.1em @R=.7em @! R { \poloFantasmaCn{\rA_1}\qw& \multigate{1}{\tV_1}& \poloFantasmaCn{\tB_1}\qw& \qw& & \poloFantasmaCn{\rA_2}\qw& \multigate{1}{\tV_2}& \poloFantasmaCn{\rB_2}\qw& \qw& \cdots& & \poloFantasmaCn{\rA_N}\qw& \multigate{1}{\tV_N}& \poloFantasmaCn{\rB_N}\qw& \qw\\ & & \aghost{\tV_1}& \qw& \poloFantasmaCn{\rE_1}\qw& \qw& \qw& \ghost{\tV_2}& \poloFantasmaCn{\rE_2}\qw& \qw& \cdots& & \poloFantasmaCn{\rE_{N-1}}\qw& \ghost{\tV_N}& \poloFantasmaCn{\rE_N}\qw& \measureD{e}\\ \end{aligned}\, . \end{equation} Moreover, $\tV_N\dots\tV_1$ is a reversible dilation of $\tC^{(N)}$. We have also a uniqueness result: Let $\{\tV_k\}_{k=1}^N$, $\tV_k\in\mathsf{Transf}(\rA_k\rE_{k-1}\rightarrow\rB_k\rE_k)$ be a reversible realization of the casually ordered channel $\tC^{(N)}$ as a sequence of channels with memory, as in Theorem <ref>. Suppose that $\{\tV_k^\prime\}_{k=1}^N$, $\tV_k^\prime\in\mathsf{Transf}(\rA_k\rE_{k-1}^\prime\rightarrow\rB_k\rE_kì^\prime)$ is another reversible realization of $\tC^{(N)}$ as a sequence of channels with memory. Then there exist a channel $\tR$ from $\rE_N$ to $\rE_N^\prime$ such that \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \poloFantasmaCn{\rA_1}\qw& \multigate{1}{\tV_1^\prime}& \poloFantasmaCn{\tB_1}\qw& \qw& & \poloFantasmaCn{\rA_2}\qw& \multigate{1}{\tV_2^\prime}& \poloFantasmaCn{\rB_2}\qw& \qw& \cdots& & \poloFantasmaCn{\rA_N}\qw& \multigate{1}{\tV_N^\prime}& \poloFantasmaCn{\rB_N}\qw& \qw\\ & & \aghost{\tV_1^\prime}& \qw& \poloFantasmaCn{\rE_1^\prime}\qw& \qw& \qw& \ghost{\tV_2^\prime}& \poloFantasmaCn{\rE_2^\prime}\qw& \qw& \cdots& & \poloFantasmaCn{\rE_{N-1}^\prime}\qw& \ghost{\tV_N}& \poloFantasmaCn{\rE_N^\prime}\qw& \qw\\ \end{aligned}\, =\\ \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \poloFantasmaCn{\rA_1}\qw& \multigate{1}{\tV_1}& \poloFantasmaCn{\tB_1}\qw& \qw& & \poloFantasmaCn{\rA_2}\qw& \multigate{1}{\tV_2}& \poloFantasmaCn{\rB_2}\qw& \qw& \cdots& & \poloFantasmaCn{\rA_N}\qw& \multigate{1}{\tV_N}& \poloFantasmaCn{\rB_N}\qw& \qw&\qw&\qw\\ & & \aghost{\tV_1}& \qw& \poloFantasmaCn{\rE_1}\qw& \qw& \qw& \ghost{\tV_2}& \poloFantasmaCn{\rE_2}\qw& \qw& \cdots& & \poloFantasmaCn{\rE_{N-1}}\qw& \ghost{\tV_N}& \poloFantasmaCn{\rE_N}\qw& \gate{\tR}& \poloFantasmaCn{\rE_N^\prime}\qw& \qw\\ \end{aligned}\, . \end{aligned} \end{equation} §.§ No bit commitment The bit commitment protocol defined in Section <ref> is generally implemented with sequences of channels with memory, that can be used to describe sequences of moves of Alice or Bob. In this scenario, the memory systems are the private systems available to a party, while the other input-output systems are the systems exchanged in the communication with the other party. We recall that for every theory that satisfy the starting hypothesis, this proof has an absolutely general validity: for every kind of input states and every possible strategy adopted, including both atomic and non-atomic transformations. If a N-round protocol is perfectly concealing, then there is a perfect cheating. We first prove the impossibility for protocols that do not involve the exchange of classical information. Let $\tA_0,\,\tA_1\in\mathsf{Transf}(\rA_1...\rA_N\rightarrow\rB_1...\rB_N\rF_N)$ two causally ordered N-partite channels (here the last output system of the causally ordered channels is the bipartite system $\rB_N,\,F_N$), representing Alice's move to encode the bit value $b=0,\,1$, respectively. The system $\rF_N$ is the system sent from Alice to Bob at the opening phase in order to unveil the value of the bit. If the protocol is perfectly concealing, then the reduced channels before the opening phase must be indistinguishable, namely $(e\|_{\rF_N}\tA_0=(e\|_{\rF_N}\tA_1\coloneqq\tC$. Now, due to Theorem <ref>, there exist two reversible dilatations $\tV_0\in\mathsf{Transf}(\rA_1...\rA_N\rightarrow\rB_1...\rB_N\rF_N\rG_0)$ and $\tV_1\in\mathsf{Transf}(\rA_1...\rA_N\rightarrow\rB_1...\rB_N\rF_N\rG_1)$ for $\tA_0$ and $\tA_1$, respectively. Since $\tV_0$ and $\tV_1$ are also two dilatations of the channel $\tC$, due to Lemma <ref> there is a channel $\tR$ from $\rF_N\rG_0$ to $\rF_N\rG_1$ such that $\tV_1=\tR\tV_0$. Applying this channel to her private systems, Alice can switch from $\tV_0$ to $\tV_1$ just before the opening. Discarding the auxiliary system $\rG_1$, this yields channel $\tA_1$. The cheating is perfect, since Alice can play the strategy $\tV_0$ until the end of the commitment and decide the bit value before the opening without being detected by Bob. The above reasoning can be extended to N-round protocols involving the exchange of classical information. Indeed classical messages can be modeled by measure-and-prepare channels where the observation states are perfectly distinguishable. The fact that some systems can only be prepared in perfectly distinguishable states will be referred as to "communication interface" of the protocol. In this case, to construct Alice's cheating strategy we can first take the reversible dilatations $\tV_0$, $\tV_1$ and the channel $\tR$ such that $\tV_1=\tR\tV_0$. In order to comply with the communication interface protocol, one can compose $\tV_0$ and $\tV_1$ with classical channels on all system that must be "classical" before the opening, thus obtaining two channels $\tD_0$ and $\tD_1$ that are no longer reversible but still satisfy $\tD_1=\tR\tD_0$. Discarding the auxiliary system $\rG_1$ and, of required by the communication interface, applying a classical channels on $\rF_N$, Alice then obtains channel $\tA_1$. Again, this strategy allows Alice to decide the value of the bit just before the opening without being detected. CHAPTER: PR-BOXES In this Chapter we will analyse the probabilistic theory [21, 22, 23] corresponding to the popular PR-boxes model introduced in Ref. [7]. In particular, in the first Section we will retrace the crucial steps and the underlying reasons that led to the development of the PR-box model. Then, in the second Section, after the formalization of the PR-boxes model in the language of operational probabilistic theories, we will report our results: the general POVM that grants perfect discriminability between any two pure states in the bi-partite case, the existence and uniqueness of the purification exclusively for the maximally mixed state (again in the bi-partite scenario) and finally some consideration on the general case of N-partite boxes. § WHY PR-BOXES? One of the most striking feature of quantum theory is certentantly non-locality. In fact, since the very beginning of the theory the incompleteness of the Copenhagen interpretation of quantum mechanics in relation to the violation of local causality was one of the main discussed aspect. In 1935 Einstein Podolsky and Rosen published the famous article of the EPR paradox [24]. The thought experiment generated a great deal of interest in the following years. Their notion of a "complete description" was later formalized by the suggestion of hidden variables that determine the statistics of measurement results, but to which an observer does not have access. In 1964 John Bell proved that some predictions of quantum mechanics cannot be reproduced by any theory of local physical variables [3]. Although Bell worked within non-relativistic quantum theory, the definition of local variable is relativistic: a local variable can be influenced only by events in its backward light cone, not by events outside, and can influence events in its forward light cone only. Quantum mechanics, which does not allow us to transmit signals faster than light (super-luminal signalling), preserves relativistic causality. But quantum mechanics does not always allow us to consider distant systems as separate, as Einstein assumed. Now quantum non-locality has been experimentally verified under different physical assumptions. Any physical theory that aims at superseding or replacing quantum theory should account for such experiments and therefore must also be non-local in this sense; quantum non-locality is a property of the universe that is independent of our description of nature. So quantum non-locality is an essential feature of quantum theory but it often appear in a negative light. In 1994 Popescu and Rohrlich published a work [7] where they proposed to show quantum non-locality in a more positive light. They investigated the inversion of the logical approach to quantum mechanics, considering quantum non-locality as an axiom instead of as a theorem and wondering what non-locality together with relativistic causality would imply. They found that quantum theory is only one of a class of non-local theories consistent with causality, and not even the most non-local. In fact, in a certain sense, non-locality can be quantified. In 1969, John Clauser, Michael Horne, Abner Shimony, and Richard Holt reformulated Bell's inequality in a manner that best suites experimental testing, the homonymous CHSH inequality [5]. CHSH inequality, restricted to any classical theory, states that a particular algebraic combination of correlations lies between -2 and 2. This bound is obviously violated in quantum mechanics, where in fact the CHSH inequality allows a maximum value given by Cirel'son's theorem as $2\sqrt{2}$ [4]. However, Popescu and Rohrlich wrote down a set of correlations that return a value of 4 for the CHSH expression, the maximum value algebraically possible, and that yet are non(super-luminar)-signalling. A question that now rises spontaneously is why does quantum theory not allow these strongly non-local correlations. In the hope of making further progress with this question these correlations have been investigated in the context of a theory with well-defined dynamics. Abstractly this scenario may be described by introducing two observers that have access to a black box. Each observer selects an input from a range of possibilities and obtains an output. The box determines a joint probability for each output pair given each input pair. It is clear that a quantum state provides a particular example of such a box, with input corresponding to measurement choice and output to measurement outcome. More generally boxes can be divided into different types. Some will allow the observers to signal to one another via their choice of input, and correspond to two-way classical channels, as introduced by Shannon. Others will not allow signalling - it is well known, for example, that any box corresponding to an entangled quantum state will not. This is necessary for compatibility between quantum mechanics and special relativity. Among the non-signalling boxes, some will violate a Bell-type inequality, and we refer to any such a box as non-local. As we have described above, in terms of our boxes, there are some boxes that are non-signalling but are more non-local than any box allowed by quantum theory. § PR-BOXES AS OPT The model describes $N$ correlated boxes (in the original paper [7] it was $N=2$) in a casual context. Each box is represented by the same elementary system $\rA$, with the $N$ correlated boxes represented by the composite system $\rA^{\otimes N}$. We will consider the simplest situation where each box has both input and output as binary variables. On each elementary system $\rA_i$, $i=1,\ldots,N$ only two atomic binary observation tests are allowed, say $B^{(0)}\coloneqq\{b^{(0)},e_\rA-b^{(0)}\}$ or $B^{(1)}\coloneqq\{b^{(1)},e_\rA-b^{(1)}\}$, with $b^{(0)},b^{(1)}\in\eff{A}$ atomic effects, and $e_\rA$ the deterministic effect of system $\rA$. Notice that the since the model is causal, the deterministic effect $e_\rA$ is unique and it is $e_{\rA^{\otimes N}}=\otimes^N e_\rA$. The probabilistic model is typically presented in terms of the probability function $P:(b_1,b_2,\ldots, b_N\|B_1,B_2,\ldots B_N)\mapsto [0,1]$, with $B_i\in\{B^{(0)},B^{(1)}\}$ and $b_i\in\{b^{(0)},b^{(1)}\}$, for every $i=1,\ldots,N$, which returns the probability of the outcomes $b_1,b_2,\ldots,b_N$ given the observation tests $B_1,B_2,\ldots B_N$. The constraint imposed on the function $P$ is no-signalling, i.e. \begin{equation} \label{eq:no-signalling} \begin{aligned} \sum_{b_k=b^{(0)},e-b^{(0)}} &P(b_1,\ldots, b_N\|B_1,\ldots B_N)=\\ =&P(b_1,\ldots, b_{k-1},b_{k+1},\ldots,b_N\|B_1,\ldots B_{k-1},B_{k+1},\ldots, B_N). \end{aligned} \end{equation} §.§ States, effects and transformations We will start our analysis with the boxes of the elementary system $\rA$ that are necessary local. They are described in terms of the following probability distribution \begin{equation} \label{eq:prbox1} \begin{cases} 1& a=\alpha x\oplus \beta\\ 0 & \text{otherwise} \end{cases}, \end{equation} with $\alpha,\,\beta=0,1$. This four probability distributions correspond to the four pure states of system $\rA$. In fact the elementary system $\rA$ has dimension $\dim(\rA)=3$, namely its states are described by vectors in $\mathbb{R}^3$ ($\mathsf{St}_\mathbb{R}(\rA)=\mathsf{Eff}_\mathbb{R}(\rA)=\mathbb{R}^3$). The four pure normalized states (<ref>) can be represented by the following vectors \begin{equation} \label{eq:local-states} \begin{aligned} \omega_0 = \begin{pmatrix} \end{pmatrix},\: \omega_1 = \begin{pmatrix} \end{pmatrix},\: \omega_2 = \begin{pmatrix} \end{pmatrix},\: \omega_3 = \begin{pmatrix} \end{pmatrix}, \end{aligned} \end{equation} where the correspondence between $\omega_n$ and the probability rule $p_{\alpha\beta}$ is given by $\alpha\beta=$(binary form of $n$). The convex set of states normalized is then represented by a square (see the square in the plane $z=1$ in Fig. <ref>). (15, 44) $\omega_1$ (67, 54) $\omega_3$ (29, 54) $\omega_0$ (74, 44) $\omega_2$ (48, 27) $b_1$ (73, 33) $b_2$ (22, 33) $b_0$ (50, 38) $b_3$ (45, 53) $\bar e$ (72.5, 60) $0$ (52.5, 64) $1$ (93, 56) $-1$ (-2.5, 52.5) $1$ (-2, 36.5) $0$ (-4.5, 21) $-1$ (0, 56) $-1$ (29, 61) $0$ Elementary system of the PR-boxes theory. This picture depicts the “squit” elementary system often considered in generalized probabilistic theories (in analogy to the “bit” and the “qubit” which are elementary systems of classical and quantum theory respectively). The system is specified by its sets of states and effects here represented as vectors in $\mathbb{R}^3$. The convex set of normalized states is represented by the square at the top, while the convex set of effects corresponds to the truncated cone. The set of effects of system $\rA$ is defined as the set of vectors $b$ such that $0 \le \Tr[b^T \omega] \le 1$ for every state $\omega\in\st{A}$, with $p_{y\|x} = \Tr[{b^{(y)}}^T \omega_x]$ the rule providing the probability associated to an effect $b^{(y)}$ on a state $\omega_x$. This leads to the truncated cone of effects in Fig. <ref>, with extremal points given by \begin{align} b^{(0)} = \begin{pmatrix} \frac{1}{2}\\[2pt]\frac{1}{2}\\[2pt]\frac{1}{2} \end{pmatrix},\; b^{(1)} = \begin{pmatrix} \end{pmatrix},\; b^{(2)} = \begin{pmatrix} \end{pmatrix},\; b^{(3)} = \begin{pmatrix} \tfrac{1}{2}\\[2pt]-\tfrac{1}{2}\\[2pt]\tfrac{1}{2} \end{pmatrix}. \end{align} The deterministic effect (the effect $e_\rA$ such that $\Tr[e_\rA^T \omega]=1$ for every state $\omega\in\st{A}$) is the vector $e_\rA=(0,0,1)^T$ (when no ambiguity arises we will simply denote the deterministic effect with $e$). Our analysis now proceeds towards the composite system $\rA\otimes \rA$. In this case the probability function $p(a,b\|x,y)$ form a table with $2^4$ entries, although these are not all independent due to the constraints of Eq. (<ref>). The dimension of the set of boxes is found by subtracting the number of independent constraints from $2^4$, and turns out to be 8 (we notice that $8=\dim\mathsf{St}_1(\rA\otimes\rA)$, while $\dim\mathsf{St}_\mathbb{R}(\rA\otimes \rA)=\dim\mathsf{St}_\mathbb{R}(\rA)\dim\mathsf{St}_\mathbb{R}(\rA)=9$, so due to Theorem <ref> it satisfies local discriminability). In this case we will have no more a square like in Fig. <ref> but a polytope with 24 vertices. The vertices will be boxes that satisfy all of the constraints and saturates a sufficient number of the positivity constraints to be uniquely determined. These 24 bilocal pure states may be divided into two classes. The local boxes, given by the following 16 probability distributions \begin{equation} \label{eq:14} \begin{cases} 1& a=\alpha x\oplus\beta\\ 1& b=\gamma y\oplus\delta\\ 0& \text{otherwise} \end{cases}, \end{equation} with $\alpha,\beta,\gamma,\delta=0,1$, and the non-local boxes, given by the 8 probability distributions \begin{equation} \label{eq:15} \begin{cases} 1/2& a\oplus b=xy\oplus \alpha x\oplus \beta y\oplus\gamma\\ 0 & \text{otherwise} \end{cases} \end{equation} with $\alpha,\beta,\gamma=0,1$. For convenience we can represent states and effects as $3\times 3$ real matrices rather than as vectors in $\mathbb{R}^9$. The 16 local state of the bipartite system are nothing else that the factorized pure states \begin{align} \label{eq:local-bipartite-states} \Omega_{4i+j} \coloneqq \omega_i \otimes \omega_j^T, \end{align} where $i, j \in \{ 0, 1, 2, 3 \}$, and the 8 non-local states (playing the role of entangled states) are represented by the following matrices \begin{equation} \label{eq:non-local-bipartite-states} \begin{aligned} \Omega_{16} & := \frac12 \begin{pmatrix} -1 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 2 \end{pmatrix}, & \Omega_{17} & := \frac12 \begin{pmatrix} -1 & -1 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & 2 \end{pmatrix}, & \Omega_{18} := \frac12 \begin{pmatrix} 1 & -1 & 0 \\ -1 & -1 & 0 \\ 0 & 0 & 2 \end{pmatrix},\\ \Omega_{19} &:= \frac12 \begin{pmatrix} -1 & 1 & 0 \\ -1 & -1 & 0 \\ 0 & 0 & 2 \end{pmatrix}, & \Omega_{20} & := \frac12 \begin{pmatrix} -1 & -1 & 0 \\ 1 & -1 & 0 \\ 0 & 0 & 2 \end{pmatrix}, & \Omega_{21} := \frac12 \begin{pmatrix} 1 & -1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 2 \end{pmatrix}, \\ \Omega_{22} & := \frac12 \begin{pmatrix} 1 & 1 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & 2 \end{pmatrix}, & \Omega_{23} & := \frac12 \begin{pmatrix} 1 & 1 & 0 \\ 1 & -1 & 0 \\ 0 & 0 & 2 \end{pmatrix}.& \end{aligned} \end{equation} From a generic probability rule $p_{\alpha\beta\gamma}$ we can identify one of the above matrix $\Omega_n$ by the equation: $n=15+\left[(3+3\alpha+4\beta+6\gamma)\mod8\right]$. Again, the probability associated to a bipartite effect $B_y$ on the state $\Omega_x$ is given by $p_{y\|x} = \Tr[B_y^T \Omega_x]$. Accordingly, the set of bipartite effects is easily derived via the consistency condition $\Tr[B_j^T \Omega_i] \ge 0$ for every $j \in [0, 23]$. It follows that the only admissible extremal effects are the 16 factorized matrices \begin{align} B_{4i+j} := b^{(i)} \otimes {b^{(j)}}^T. \end{align} This is a relevant feature of PR-boxes model, whose strong correlation incapsulated in the eight non-local states $\Omega_x$, $x\in[16,23]$ are incompatible with any in principle admissible non-factorized measurements. This feature has been firstly noticed in Ref. [25] and later in Ref. [26] where all possible theories compatible with the squit local system have been classified (among these theories is the dual version of PR-boxes that only have factorized states but eight non-local effects). Finally, the deterministic effect for the bipartite system is $e\otimes e^T$. Analogously one defines the convex set of states and the convex set of effects for the arbitrary $N$-partite system $\rA^{\otimes N}$. However, while for effect is nothing but a trivial generalization, for the states the discussion is not so straight and some interesting features arise. We will dedicate a following Section to discuss some of the most immediate aspect about the $N$-partite system, we will start our discussion from the tripartite boxes. We can now turn our attention to the transformations of the theory. We focus here on the reversible transformations which are of interest for the present paper results. The set $\mathcal{U}(\rA)$ of reversible transformations of the system $\rA$ coincides with the finite group of symmetries of the square (the dihedral group of order eight $D_8$ containing four rotations and four reflections). In the chosen representation we have \begin{equation} \label{eq:single-system-unitaries} \begin{aligned} \mathcal{U}(\rA)=\{U_k^s: k=0,\ldots,3, s=\pm\}\\ U_k^s = \begin{pmatrix} \cos \frac{\pi k}2 & -s \sin \frac{\pi k}2 & 0 \\ \sin \frac{\pi k}2 & s \cos \frac{\pi k}2 & 0 \\ 0 & 0 & 1 \end{pmatrix}. \end{aligned} \end{equation} The matrices $U_k^{+}$ and $U_k^{-}$ representing the four rotations and the four reflections of the square respectively. When we apply them to the four pure states of Eq. (<ref>) we have \begin{align} \label{eq:reversible-mapping-on-local-states} \omega_{j+k} & = U_k^+\, \omega_{j}, & \omega_{k} & = U_k^-\, \omega_{j+k}\,, \end{align} for $j\in\left[0,3\right]$ and where the sum is$\mod 3$, i.e. if $j=2$ and $k=2$ then $j+k=0$. We notice that these transformations are atomic. In Ref. [23] all atomic transformations of the squit system have been classified. The set of reversible transformations $\mathcal{U}(A\otimes A)$ of the composite system $\rA\otimes\rA$ (which has been derived in Ref. [27]) is \begin{equation} \label{eq:reversible} \mathcal{U}(\rA\otimes \rA)=\{ W^i(U_j^{s_1} \otimes U_k^{s_2})\}\quad i=0,1,\; 0\leq j,k\leq 3,\; s_1,s_2=\pm, \end{equation} with $W$ the swap map, namely the map that exchanges the two subsystems. This means that any reversible map corresponds to the tensor product of single system reversible transformations with, possibly, the application of the swap. As noticed in Ref. [27] reversible transformations cannot create entanglement. This result has been extended in Ref. [28] to the composition of an arbitrary number $N$ of systems $\rA^{\otimes N}$, showing that also in that case the set of reversible multipartite transformations is generated by local reversible operations and permutations of systems. We notice that in the PR-boxes theory any non-local bipartite pure state can be reversibly mapped to any other non-local bipartite pure state. Moreover, this mapping can be done via the local application of a single system reversible map. For example, starting from the state $\Omega_{16}$, one has \begin{align} \label{eq:reversible-mapping} \Omega_{16+k} & = (U_k^+\otimes I)\Omega_{16}, & \Omega_{23-k} & = (U_k^-\otimes I) \Omega_{16}. \end{align} We finally remark that the set of reversible transformations of system $\rA$, $\tU(\rA)$, in terms of description by the probability distributions in Eqs. (<ref>), (<ref>), (<ref>) correspond to a local relabelling defined by the operations \begin{equation} \label{eq:local-relabelling} \begin{aligned} x&\rightarrow x\oplus 1,\qquad a\rightarrow a\oplus\alpha x\oplus \gamma,\\ y&\rightarrow y\oplus 1,\qquad b\rightarrow b\oplus\beta y\oplus \gamma. \end{aligned} \end{equation} §.§ Discriminability between PR-boxes An important question to address when dealing with the PR-boxes theory is the following: given two deterministic state of the theory is it possible to discriminate between them? We will address to this question only for the extremal points of the polytope, i.e. the pure states, in order to analyse a way to perfect discriminate between them. For local boxes it is easy to show that there exist POVMs that are able to perfect discriminate between every two of the four possible state in Eq. (<ref>) [23]. For what concern bipartite boxes, while it is trivial to find perfectly discriminable POVMs for each pair of the 16 local boxes (since they are simply the tensor product of the 4 local boxes, so also the discriminable POVMs are just the tensor product of the perfectly discriminable POVMs of the local boxes) it is a bit more elaborate to explore the discriminability between non-local boxes. To help in our analysis we introduce the following table regarding non-local boxes labelled by $(\alpha,\beta,\gamma)$ as described in Eq. (<ref>): $x$ $y$ $a\oplus b$ 0 0 $\gamma$ 0 1 $\beta\oplus\gamma$ 1 0 $\alpha\oplus\gamma$ 1 1 $1\oplus\alpha\oplus\beta\oplus\gamma$ Correlations for a generic $(\alpha,\beta,\gamma)$ non-local box for all the possible given input combinations of $x$ and $y$. It is now easy to verify that for every two different non-local boxes, i.e. for every choice of two different combinations of $(\alpha,\beta,\gamma)$: $c_1=(\alpha_1,\beta_1,\gamma_1)$ and $c_2=(\alpha_2,\beta_2,\gamma_2)$, there is always at least one input combination $(x,y)$, whose output relation is equal to 0 for $c_1$ and to 1 for $c_2$. If we denote with $e$ the deterministic effect of the bipartite system $\rA\otimes\rA$ and $a\coloneqq b^{(3(1-x))} \otimes b^{(3(1-y))}+b^{(1+x)} \otimes b^{(1+y)}$, the POVM $\{a,e-a\}$ is able to perfectly discriminate between the two chosen non-local boxes. Finally, the last remark regards the discrimination between one local and one non-local bipartite box. We can help ourselves with the two TABLEs of Appendix <ref> that follows the notation of Eqs. (<ref>),(<ref>). For every pair of bipartite boxes (one local and one non-local) there is always a combination of $(x,y)$ such that in one case the outcome relation $a\oplus b$ is equal to 0 for one box and 1 for the other. So we can construct a perfectly discriminating POVM following the same strategy of above. §.§ Purification in the PR-box (bipartite restriction) model Until now nothing has been said about purification in the PR-box model. The following discussion represent an important result of the theory, valid in the general context of $N$-partite boxes (with $N$ finite), but we will operate in the significant restriction of admitting no more than bipartite correlated boxes. For the rest of the subsection when we will refer to PR-box model/theory we will mean the model under this limitation. Let begin with some general definitions and preliminary considerations. The ability to transform any pure state into any other by means of reversible transformations will be called transitivity, meaning that the action of the set of reversible transformations is transitive on the set of pure states. Based on this definition, PR-box model clearly enjoys this property. In fact the transformations of Eq. (<ref>) is transitive on the set of pure states of Eq. (<ref>). Among the numerous consequences that transitivity implies one will be of our interest, the uniqueness of the maximally mixed state. So it is in order to properly define what a maximally mixed state is. If a state is invariant under the action of every reversible transformation, then it is a maximally mixed state. We will not report here the demonstration of the uniqueness of the maximally mixed state from transitivity, for it we refer to Ref. [11]. From the transformations of Eq. (<ref>) and the vector representation of the pure state in Eq. (<ref>) it is not difficult to write down the maximally mixed state: \begin{equation} \label{eq:maximally-mixed-state} \mu = \begin{pmatrix} \end{pmatrix}\,. \end{equation} We are now in position to state the main result of this Section. Given a system $\otimes^N\rA$, the maximally mixed state $\mu^{\otimes N}\in\st{\otimes^NA}$ of Eq. (<ref>) is the unique internal state that is purificable and its purification is unique up to a reversible transformation on the purifying systems. The proof is divided in two part. In the first one we will prove the thesis for $N=1$, then in the second part it is extended to an arbitrary number of systems. Given the system $\rA_1$ we will consider a system $\rA_2$ as the purifying system. The pure states of $\rA_1\otimes\rA_2$ are the 24 pure states $\Omega_{i}$, for $i=0,\ldots,23$ of Eqs. (<ref>), (<ref>). We know that 16 of them (namely the local ones, that are expressed in Eq. (<ref>)) are separable states. So if we consider the marginal state obtained by applying the deterministic effect $e_{\rA_2}$ on the purifying system $\rA_2$ on these separable states we get one of the local 4 state represented in Eq. (<ref>), that are pure. However, if we repeat the same procedure on the 8 non-local pure states (represented in Eq. (<ref>)) we get the same state for all of them: the maximally mixed state $\mu\in\mathsf{St}(\rA_1)$. So the unique internal state that can be purificated is the maximally mixed state and since the 8 non-local bipartite pure states are all mapped to any other non-local bipartite pure state by the application of a single system reversible map, as shown in Eq. (<ref>), the purification is unique up to a reversible transformation on the purifying system. We now deal with the $N$-partite scenario. The more general state $\Psi\in\mathsf{St}(\otimes^N\rA)$ has the form: \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.8em @! R { \multiprepareC{2}{\Psi}& \poloFantasmaCn{\rA_1}\qw& \qw\\ \pureghost{\Psi}& \vdots&\\ \pureghost{\Psi_i}& \poloFantasmaCn{\rA_N}\qw& \qw\\ \end{aligned} \: = \: \begin{aligned} \Qcircuit @C=1.2em @R=.6em @! R { \multiprepareC{1}{\omega_1}& \poloFantasmaCn{\rA_1}\qw& \qw\\ \pureghost{\omega_1}& \poloFantasmaCn{\rA_2}\qw& \qw\\ \vdots&&\\ \multiprepareC{1}{\omega_{M/2}}& \poloFantasmaCn{\rA_{M-1}}\qw& \qw\\ \pureghost{\omega_{M/2}}& \poloFantasmaCn{\rA_M}\qw& \qw\\ \prepareC{\omega_{M/2+1}}& \poloFantasmaCn{\rA_{M+1}}\qw& \qw\\ \vdots&&\\ \prepareC{\omega_{N-M/2}}& \poloFantasmaCn{\rA_{N}}\qw& \qw\\ \end{aligned}\, . \end{equation} In order to be purificable, every monopartite state has to be pure or purificable and every bipartite state to be pure (since we are admitting no more than bipartite correlations, a bipartite state can be purificable only if it is pure). But if one of the states $\omega_j$ for $j=1,\ldots,N-M/2$ is pure then $\Psi$ can not be internal. Since the only purificable internal state is $\mu$, we have that the only purificable internal $N$-partite state is $\mu\otimes\ldots\otimes\mu$ (thanks to local discriminability the parallel composition of internal states is still an internal state). Furthermore, for what we have seen before, the purification of $\mu\otimes\ldots\otimes\mu$ is unique up to a reversible transformation. This transformation is nothing else that the parallel composition of the maps $(U_k\otimes\tI)$ where $U_k$ are the maps of Eq. (<ref>). Remark: in the previous proof we noticed that the maximally mixed state of a system $\otimes^N\rA$ is not the unique mixed state that is purificable. In fact, the more general state $\Phi\in\mathsf{St}(\otimes^N\rA)$ that can be purificated is of the form: $\Phi=\phi_1\otimes\ldots\otimes\phi_N$ where \begin{equation} \phi_i= \begin{cases} &\omega_{j}\quad\text{for }j\in\left[0,3\right]\\ &\Omega_{k}\quad\text{for }j\in\left[16,23\right] \end{cases}\quad\text{for }i=1,\ldots,N\,, \end{equation} and its purification is still unique up to reversible transformations on the purifying systems. §.§ N-partite PR-boxes In the literature of PR-box theory a thorough and systematic study on $k$-partite correlated boxes, with $k\ge3$, has never been made. This represents an important absence within the model since it prevents the theory to be complete. In this Section we show some important consequences that emerge by just considering tripartite correlated boxes with some speculations about the complete $N$-partite model. In our analysis of PR-box model integrated with tripartite boxes we make use of the classification that has been made in Ref. [29]. In the article of Pironio et al., the no-signaling polytope is found to have 53856 extremal points, belonging to 46 inequivalent classes. The term inequivalent means that there not exist reversible local transformations that allow to move from a representative of one class to one of another class, while, inside the same class, all the extremal points are connected by local relabelling, see Eqs. (<ref>) (since we are dealing with three parties boxes, also permutation of the parties is a local relabelling, i.e. $x\rightarrow y$, $y\rightarrow z$, and $z\rightarrow x$ and so on for every possible permutation). Firstly, it is no more granted that the maximally mixed state is the unique internal state purificable in the theory. In fact it could happen that between the 53856 pure tripartite states, there will be one whose marginal state is an internal state different from the maximally mixed one. This leads to think that increasing the number of correlated systems that we are considering the number of internal states that are purificable will also increase. Even if there are not academic works in this matter, it is a very likely and reasonable possibility and we address to future studies to investigate in this direction. Secondly, even if the only internal purificable state would still be the maximally mixed one, the purification is no more unique. To see this it suffices to consider two states $\Psi_1,\Psi_2\in\mathsf{St}(\rA^{\otimes3})$ that have the following form: \begin{equation} \Psi_1 \, = \, \begin{aligned} \Qcircuit @C=1.2em @R=.8em @! R { \multiprepareC{1}{\Omega}& \poloFantasmaCn{\rA}\qw& \qw\\ \pureghost{\Omega}& \poloFantasmaCn{\rA}\qw& \qw\\ \prepareC{\omega}& \poloFantasmaCn{\rA}\qw& \qw\\ \end{aligned} \quad \text{and}\quad \Psi_2 \, = \, \begin{aligned} \Qcircuit @C=1.2em @R=.8em @! R { \multiprepareC{2}{\Phi^{(44)}}& \poloFantasmaCn{\rA}\qw& \qw\\ \pureghost{\Phi^{(44)}}& \poloFantasmaCn{\rA}\qw& \qw\\ \pureghost{\Phi^{(44)}}& \poloFantasmaCn{\rA}\qw& \qw\\ \end{aligned}\, , \end{equation} where $\omega\in\st{A}$ and $\Omega\in\mathsf{St}(\rA^{\otimes2})$ are two pure states and $\Phi^{(44)}$ is a pure state, representative of the $44^{th}$ class described in Ref. [29]. If $\Omega$ is one of the non-local bipartite states then they have the same marginal, namely \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.8em @! R { \multiprepareC{1}{\Omega}& \poloFantasmaCn{\rA}\qw& \qw\\ \pureghost{\Omega}& \poloFantasmaCn{\rA}\qw& \multimeasureD{1}{e}\\ \prepareC{\omega}& \poloFantasmaCn{\rA}\qw& \ghost{e}\\ \end{aligned} \, = \, \begin{aligned} \Qcircuit @C=1.2em @R=.8em @! R { \prepareC{\mu}& \poloFantasmaCn{\rA}\qw& \qw\\ \end{aligned} \, = \, \begin{aligned} \Qcircuit @C=1.2em @R=.8em @! R { \multiprepareC{2}{\Phi^{(44)}}& \poloFantasmaCn{\rA}\qw& \qw\\ \pureghost{\Phi^{(44)}}& \poloFantasmaCn{\rA}\qw& \multimeasureD{1}{e}\\ \pureghost{\Phi^{(44)}}& \poloFantasmaCn{\rA}\qw& \ghost{e}\\ \end{aligned}\, , \end{equation} where $\mu$ is the maximally mixed state of Eq. (<ref>) and the second equality derives straightforward once the probability rule of the $44^{th}$ class is written explicitly, as we will see in Eq. (<ref>). So we found that $\Psi_1$ and $\Psi_2$ are two purification of the same state but since every reversible transformation $U\in\tU(\rA^{\otimes2})$ is the composition of local reversible maps, "correlation" can not be created and so \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.8em @! R { \multiprepareC{2}{\Psi_1}& \poloFantasmaCn{\rA}\qw& \qw\\ \pureghost{\Psi_1}& \poloFantasmaCn{\rA}\qw& \qw\\ \pureghost{\Psi_1}& \poloFantasmaCn{\rA}\qw& \qw\\ \end{aligned} \, \ne \, \begin{aligned} \Qcircuit @C=1.2em @R=.8em @! R { \multiprepareC{2}{\Psi_2}& \qw& \poloFantasmaCn{\rA}\qw& \qw& \qw\\ \pureghost{\Psi_2}& \poloFantasmaCn{\rA}\qw& \multigate{1}{U}& \poloFantasmaCn{\rA}\qw& \qw\\ \pureghost{\Psi_2}& \poloFantasmaCn{\rA}\qw& \ghost{U}& \poloFantasmaCn{\rA}\qw& \qw\\ \end{aligned}\quad\forall\,U\in\tU(\rA^{\otimes2})\,. \end{equation} Furthermore, restricting our attention to the pure bipartite states, we have noticed that when we pick up two of these states that have the same marginal, than there is always a local reversible map from one to the other and vice-versa. This is the case for the 8 non-local bipartite boxes, that are all connected by local transformations in $\tU(\rA)$ of Eq. (<ref>). This mechanism will turn out to be exactly the one responsible for the impossibility of perfectly secure bit commitment, as we will see in detail in the next Chapter. Since in the tripartite scenario not all the tripartite non-local boxes are connected by local transformation (we remind that local relabelling is not enough to change from a class to another), it is reasonable to think that perfect bit commitment will be possible, or at least a completely new way of cheating has to be thought. With this purpose we will propose a scheme of bit commitment as the conclusive Section of the next Chapter. CHAPTER: NO BIT COMMITMENT IN PR-BOXES In the past years numerous protocols have been proposed to realize bit commitment using PR-boxes. However, as outlined by A.J. Short, N. Gisin, and S. Popescu in Ref. [9], “it is surprising that the possibility that non-local correlations which are stronger than those in quantum mechanics could be used for bit commitment, because it is the very existence of non-local correlations which in quantum mechanics prevents bit commitment”. In that article they particularly referred to the protocol prosed by S. Wolf and J. Wullschleger [8] and showed that it was erroneous by argument of causality. After that also Buhrman et al. [10] proposed a bit commitment protocol in PR-box theory that was claimed to be unconditionally secure and where the counter-proof of Short, Gisin and Popescu did not work anymore. From a OPT point of view, when dealing with PR-boxes, some issues arise since they still not have a complete and closed theory. In fact, $k$-partite boxes with $k\ge3$ have been studied only roughly and a coherent and comprehensive theory has not been proposed. As we have seen in Section <ref>, simplistic generalizations are not adequate since admitting more than bipartite correlations alters significantly the theory. Nevertheless, in the literature not only PR-box model is generally considered admitting no more than bipartite states, but also local transformations (that are admissible in the theory) are ignored. In this final Chapter we propose a proof of impossibility of perfectly secure bit commitment in PR-boxes (even if under two important limitations: pure input states and bipartite boxes, the proof includes almost all the protocols proposed in literature that make use of PR-boxes). Furthermore we will explicitly describe a cheating protocol, contextualized in OPTs, that confute both the scheme proposed by Wolf and Wullschleger and the one by Buhrman et al.. We will show that just admitting local atomic reversible transformations the protocols proposed in literature can be cheated. Our proof joins the work published by Barnum, Dahlsten, Leifer, and Toner [30] where, in the framework of probabilistic theories, they prove that in all theories that are locally non-classical but do not have entanglement, there exists a bit commitment protocol that is exponentially secure in the number of systems used. If the protocol of Buhrman et al. would have been turned out to be correct then it would have represented the first example of an unconditionally secure bit commitment protocol valid in a theory with entanglement. However the question if a theory with entanglement admits perfectly secure bit commitment is still open. Finally we will sketch at the end of the Chapter a bit commitment scheme that make use of tripartite non-local boxes that is not more cheatable by local transformations and it could satisfy perfectly secure bit commitment. But advancements in the theory are necessary in order to give a definitive answer. § NO-PERFECTLY SECURE BIT COMMITMENT In this Section we provide an explicit proof of the impossibility of perfectly secure bit commitment in PR-box theory. However, two important limitation will be adopted. Even if we will consider arbitrary $N$-partite systems (with $N$ finite), we will not admit more than bipartite correlations (it will be taken for granted for the rest of the Section). Furthermore $\Psi_0$ and $\Psi_1$ will always be selected between the pure states. This is due to the fact that discrimination has never been studied in PR-box theory and the only results we rely on are those of Section <ref> that refers to pure states. If it would turn out that the strategy exposed in Section <ref> is the unique one to grant perfect discriminability between two arbitrary bipartite states then the protocol could be easily extended also to $\Psi_0,\,\Psi_1$ as arbitrary mixed states. Actually we will prove our theorem in two different way. In the first proof, we will state the property of perfect bit commitment and we will show that inconsistencies arise. In the second, that we will call alternative proof, we will show that, with some shrewdness, the proof of Section <ref> can be used also in this context. §.§ First Proof In this Section we make use of the definition of the protocol given in Section <ref> and nomenclature of states and transformations given in Section <ref>. Before the main theorem a preliminary lemma is in order. If a bit commitment protocol is correct with probability one then the two input states $\Psi_0,\,\Psi_1\in\mathsf{St}(\rA^{\otimes N}\rB^{\otimes N})$ shared by Alice ($\rA^{\otimes N}$) and Bob ($\rB^{\otimes N}$) are of the form \begin{equation} \label{eq:input-states} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{4}{\Psi_i}& \poloFantasmaCn{\rA_1}\qw& \qw\\ \pureghost{\Psi_i}& \poloFantasmaCn{\rB_1}\qw& \qw\\ \pureghost{\Psi_i}& \vdots&\\ \pureghost{\Psi_i}& \poloFantasmaCn{\rA_N}\qw& \qw\\ \pureghost{\Psi_i}& \poloFantasmaCn{\rB_N}\qw& \qw \end{aligned} \: = \: \begin{aligned} \Qcircuit @C=1.2em @R=.6em @! R { \multiprepareC{1}{\Omega_{k_1(i)}}& \poloFantasmaCn{\rA_1}\qw& \qw\\ \pureghost{\Omega_{k_1(i)}}& \poloFantasmaCn{\rB_1}\qw& \qw\\ \vdots&&\\ \multiprepareC{1}{\Omega_{k_M(i)}}& \poloFantasmaCn{\rA_M}\qw& \qw\\ \pureghost{\Omega_{k_M(i)}}& \poloFantasmaCn{\rB_M}\qw& \qw\\ \prepareC{\omega_{k_{M+1}(i)}}& \poloFantasmaCn{\rA_{M+1}}\qw& \qw\\ \prepareC{\omega_{k_{M+2}(i)}}& \poloFantasmaCn{\rB_{M+1}}\qw& \qw\\ \vdots&&\\ \prepareC{\omega_{k_{2N-M-1}(i)}}& \poloFantasmaCn{\rA_{N}}\qw& \qw\\ \prepareC{\omega_{k_{2N-M}(i)}}& \poloFantasmaCn{\rB_{N}}\qw& \qw\\ \end{aligned} \quad \text{ for }i=0,\,1 \end{equation} where $k_j(i)\in\left[16,23\right]$ for $j=1,\ldots,M$ and $k_j(i)\in\left[0,3\right]$ for $j=M+1,\ldots,2N-M$. The thesis follows immediately from the two assumptions we made: no more than bipartite correlated boxes and pure input states. In fact, for every two pure states a perfectly discriminating procedure always exists, as analysed in Section <ref>. For discriminate between two parallel compositions of pure states the parallel composition of the discriminating POVMs for each pair of states is sufficient. In the previous lemma it would be possible that Alice and Bob have in control also non-factorized bipartite states, i.e. $\Omega_{k_{M+1}}\in\mathsf{St}(\rA_{M+1}\rA_{M+3})$ for $k_{M+1}\in\left[16,23\right]$ instead of $\omega_{k_{M+1}}\otimes\omega_{k_{M+3}}\in\mathsf{St}(\rA_{M+1}\rA_{M+3})$ for $k_{M+1},k_{M+3}\in\left[0,3\right]$, however this does not carry any modification in the proof and so, to not make the notation even more troublesome, we will refer to Eq. (<ref>) as the more general input states. Perfect bit commitment is impossible in PR-box theory. If a bit commitment is perfect it means that it should be correct with probability one, perfectly concealing and perfectly binding. If it is correct with probability one then, by the previous Lemma, the input states $\Psi_0,\,\Psi_1\in\mathsf{St}(\rA^{\otimes N}\rB^{\otimes N})$ must have the form of Eq. (<ref>). If it is also perfectly concealing, then we have to impose the condition of Eq. (<ref>), i.e. $(e\|_{\rA^{\otimes N}}\|\Psi_0)_{\rA^{\otimes N}\rB^{\otimes N}}=(e\|_{\rA^{\otimes N}}\|\Psi_1)_{\rA^{\otimes N}\rB^{\otimes N}}$: \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{\Psi_0}& \poloFantasmaCn{\rA^{\otimes N}}\qw& \measureD{e_{\rA^{\otimes N}}}\\ \pureghost{\Psi_0}& \poloFantasmaCn{\rB^{\otimes N}}\qw& \qw\\ \end{aligned} \: = \: \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{\Omega_{k_1(0)}}& \poloFantasmaCn{\rA_1}\qw& \measureD{e_\rA}\\ \pureghost{\Omega_{k_1(0)}}& \poloFantasmaCn{\rB_1}\qw& \qw\\ \vdots&&\\ \multiprepareC{1}{\Omega_{k_M(0)}}& \poloFantasmaCn{\rA_M}\qw& \measureD{e_\rA}\\ \pureghost{\Omega_{k_M(0)}}& \poloFantasmaCn{\rB_M}\qw& \qw\\ \prepareC{\omega_{k_{M+2}(0)}}& \poloFantasmaCn{\rB_{M+1}}\qw& \qw\\ \vdots&&\\ \prepareC{\omega_{k_{2N-M}(0)}}& \poloFantasmaCn{\rB_{N}}\qw& \qw\\ \end{aligned} \: = \: \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{\Omega_{k_1(1)}}& \poloFantasmaCn{\rA_1}\qw& \measureD{e_\rA}\\ \pureghost{\Omega_{k_1(1)}}& \poloFantasmaCn{\rB_1}\qw& \qw\\ \vdots&&\\ \multiprepareC{1}{\Omega_{k_M(1)}}& \poloFantasmaCn{\rA_M}\qw& \measureD{e_\rA}\\ \pureghost{\Omega_{k_M(1)}}& \poloFantasmaCn{\rB_M}\qw& \qw\\ \prepareC{\omega_{k_{M+2}(1)}}& \poloFantasmaCn{\rB_{M+1}}\qw& \qw\\ \vdots&&\\ \prepareC{\omega_{k_{2N-M}(1)}}& \poloFantasmaCn{\rB_{N}}\qw& \qw\\ \end{aligned} \: = \: \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{\Psi_1}& \poloFantasmaCn{\rA^{\otimes N}}\qw& \measureD{e_{\rA^{\otimes N}}}\\ \pureghost{\Psi_1}& \poloFantasmaCn{\rB^{\otimes N}}\qw& \qw\\ \end{aligned}\,. \end{equation} So, for the systems from $\rB_{M+1}$ to $\rB_{N}$ we have that $\omega_{k_{M+2s}(0)}=\omega_{k_{M+2s}(1)}$ for $s=1,\ldots,N-M$. For the systems from $\rB_{M+1}$ to $\rB_{N}$ we can not make any further deductions since these 8 pure states have all the same marginal. In any case if the protocol is correct with probability one and perfectly concealing it can not be perfectly binding. In fact, for the non-local bipartite states there will be one of the local transformations of Eq. (<ref>) such that \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{\Omega_{k_j(0)}}& \poloFantasmaCn{\rA}\qw& \gate{U_{k_j}}& \poloFantasmaCn{\rA}\qw& \qw\\ \pureghost{\Omega_{k_j(i)}}& \qw& \poloFantasmaCn{\rB}\qw& \qw&\qw\\ \end{aligned}\, \: = \: \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{\Omega_{k_j(1)}}& \poloFantasmaCn{\rA}\qw& \qw\\ \pureghost{\Omega_{k_j(1)}}& \poloFantasmaCn{\rB}\qw& \qw\\ \end{aligned}\,, \end{equation} as expressed by Eq. (<ref>), where $U_{k_j}\in\tU(\rA)$. Furthermore for all the other states, namely for the local states $\omega_{k_{M+2s+1}(i)}$ with $s=0,\ldots,N-M-1$, $i=0,1$, it is immediate that there will be local transformations $U_{k_{M+2s+1}}\in\tU(\rA)$, $s=0,\ldots,N-M-1$ chosen from the 4 of Eq. (<ref>) that will permit to Alice to switch unnoticed by Bob from $\Psi_0$ to $\Psi_1$ and vice-versa. §.§ Alternative Proof In Section <ref> we pointed out the three sufficient conditions that a theory has to satisfy in order to ensure the impossibility of perfectly secure bit commitment: causality, atomicity of composition and the one required in Axiom <ref>. Clearly PR-box theory is manifestly causal, since Eq. (<ref>) imposes exactly the no-signalling constraint. Furthermore, also atomicity of composition is fulfilled by PR-box theory. The parallel composition of atomic operations is still atomic due to local discriminability, see Ref. [31]. To verify that also the sequential composition of atomic operation is still atomic it only need to consider all the atomic operations in the theory, see Ref. [23], and straightforwardly compute their composition. For what concern Axiom <ref> some considerations are in order. We required the existence of at least one dynamically faithful pure state, $\Psi^{(\rA)}$, and the existence of a purification for every state $R$ that has the same marginal of $\Psi^{(\rA)}$. Actually the latter assumption is excessive. If we look at Theorem <ref>, where this assumption needs to work, it would be enough that every state $\|R)_{\rB\tilde{\rA}}$ that is obtained from $\|\Psi^{(\rA)})_{\rA\tilde{\rA}}$ by a local channel $\tC$, i.e. $\|R)_{\rB\tilde{\rA}}\coloneqq(\tC\otimes\tI_{\tilde{\rA}})\|\Psi^{(\rA)})_{\rA\tilde{\rA}}$, is purificable. In fact, if $R$ is so defined, it certainly has the same marginal of $\Psi^{(\rA)}$: \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{R}& \poloFantasmaCn{\rB}\qw& \measureD{e}\\ \pureghost{R}& \poloFantasmaCn{\tilde{\rA}}\qw& \qw\\ \end{aligned} \, = \, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{\Psi^{(\rA)}}& \poloFantasmaCn{\rA}\qw& \gate{\tC}& \poloFantasmaCn{\rB}\qw& \measureD{e}\\ \pureghost{\Psi^{(\rA)}}& \qw& \poloFantasmaCn{\tilde{\rA}}\qw& \qw&\qw\\ \end{aligned} \, = \, \begin{aligned} \Qcircuit @C=1.2em @R=.7em @! R { \multiprepareC{1}{\Psi^{(\rA)}}& \poloFantasmaCn{\rA}\qw& \measureD{e}\\ \pureghost{\Psi^{(\rA)}}& \poloFantasmaCn{\tilde{\rA}}\qw& \qw\\ \end{aligned}\, . \end{equation} Now if we limit to consider $\tC$ as a local reversible channel (since in the theory the reversible transformations are atomic), being $\Psi^{(\rA)}$ pure by hypothesis, due to atomicity of composition also $R$ is pure and hence trivially purificable. So, under this further constraint, we only need to check the existence of a dynamically faithful pure state within the theory. We can help ourselves by the fact that the purification of an internal state is dynamically faithful (and obviously pure). This result was published in Ref. [2] for theories that satisfy local discriminability and purification but it can easily extended to PR-box theory given the existence of at least one internal state that is purificable: $\mu$. Now the answer is immediate, all the non-local bipartite extremal point of the 8-dimensional polytope are dynamically faithful pure states. At this point it is easy to see that we can just choose one of the pure states in Eq. (<ref>), as the dynamically faithful pure state in Axiom <ref>. Finally, it is straightforward to verify that the purification is unique up to a local reversible transformation on the purifying system, see Eq. (<ref>). The further limitation that we imposed, on the atomicity of $\tC$, practically requires that for every commitment protocol the two encodings $\tA_0,\,\tA_1\in\mathsf{Transf}(\rA_1...\rA_N\rightarrow\rB_1...\rB_N\rF_N)$ have the marginal atomic: $(e\|_{\rF_N}\tA_0=(e\|_{\rF_N}\tA_1\coloneqq\tC$. But from Eq. (<ref>) we know that every element of the set of reversible transformation of the composite system $\rA\otimes\rA$ is nothing else than the tensor product of local reversible transformations (with eventually the swap map). So if we limit to consider only atomic local transformations the initial new constraint is satisfied. Finally, thanks to local discriminability, given two pure faithful states $\Psi^{(\rA)}\in\mathsf{St}(\rA\tilde{\rA})$ and $\Psi^{(\rB)}\in\mathsf{St}(\rB\tilde{\rB})$ for system $\rA$ and $\rB$, respectively, then also $\Psi^{(\rA)}\otimes\Psi^{(\rB)}\in\mathsf{St}(\rA\tilde{\rA}\rB\tilde{\rB})$ is a pure dynamically faithful state for the compound system $\rA\rB$ (a rigorous proof can be found in Ref. [2]) and so the previous discussion is easily generalizable to any $N$-partite system. In conclusion, the proof of impossibility of perfectly secure bit commitment of Section <ref> can be easily extended to include PR-box theory (limited to no more than bipartite correlations and only reversible transformations). § UNCONDITIONALLY SECURE BIT COMMITMENT As outlined before, in Ref. [10] Buhrman et al. proposed a bit commitment protocol that was claimed to be unconditionally secure. They would like to show that superstrong non-local correlations in the form of non-local boxes enable to solve cryptographic problems otherwise known to be impossible. In particular, their result would imply that the no-signaling principle and secure computation are compatible in principle. However, we now prove how it would be possible for Alice to perfectly cheat, i.e. with null probability of being detected by Bob, making use of the local reversible atomic transformations of Eq. (<ref>). In our analysis, we begin from dealing with only one bipartite PR-box and we suppose that Alice's input is the committed bit (even if this is not a well defined bit commitment protocol it is as well an instructive example in order to simplify the following analysis). Alice and Bob share the non-local bipartite PR-box $\Omega_{18}\in\mathsf{St}_\mathbb{R}(\rA\rB)$ (to which corresponds the probability rule $p_{000}$) but they have access only to system $\rA$ and $\rB$, respectively. We can summarise the protocol as follows. * Alice select her committed bit $x$, inputs it and obtain output bit $a$; * Bob inputs a random bit $y$ and obtains output bit $b$. * Alice sends $x$ and $a$ to Bob; * Bob checks to see if $a\oplus b=xy$. If this relation is true, Bob accepts $x$ as the revealed bit, otherwise he knows that Alice has cheated and rejects Alice’s revelation. It is easy to see that Alice has a probability to cheat successfully equal to $\frac{1}{2}$. With the help of local transformation Alice can make the probability of successful cheating equal to 1. In fact, it is sufficient to find $(\alpha,\beta,\gamma)$ of Eq. (<ref>) such that, for given $x$ and $y$ and for the output couple $(a,b)$ (i.e. such that $p_{\alpha\beta\gamma}(a,b\|x,y)\ne0$) exists a generic function $f:\{0,1\}\longrightarrow\{0,1\}$ such that, given $a^\prime=f(a)$ and $x^\prime=x\oplus1$, $p_{000}(a^\prime,b\|x^\prime,y)\ne0$. Mathematically, to find suitable $(\alpha,\beta,\gamma)$ it is sufficient to resolve \begin{equation}\label{eq:cheating} \begin{cases} & a\oplus b=xy\oplus \alpha x\oplus \beta y\oplus\gamma\\ & a^\prime\oplus b=x^\prime y \end{cases}. \end{equation} We find that $\beta=1$ and $f(a)=a\oplus\alpha x\oplus\gamma$, for every choice of $\alpha$ and $\gamma$. In conclusion, if Alice perform a local transformation (given by Eq. (<ref>)) such that $p_{000}\longrightarrow p_{\alpha1\gamma}$ and then inputs her bit $x$ and gets output $a$, she can reveal $x^\prime$ and $a^\prime=a\oplus\alpha x\oplus\gamma$ to Bob, who will accept with probability 1. We can now consider the unconditionally secure bit-commitment protocol proposed in Ref. [10] where $2n+1$ non-local bipartite PR-boxes in the state $\Omega_{18}\in\st{AB}$ are shared between Alice and Bob. The authors found that Alice probability of successfully cheating is at maximum equal to 1/2 but can be asymptotically reduced if the protocol is repeated $k$ times. However, using local transformation Alice can cheat without being detected with probability 1. We refer to the original article about the commitment protocol and we outline only the "cheating procedure". * Alice wants to commit to bit $c$ but to send to Bob bit $c^\prime=c\oplus1$. So she chooses $x\in\{0,1\}^{2n+1}$ by choosing the first $2n$ bits such that $\|x^\prime_1...x^\prime_{2n}\|_{11}$ is even where $x_i^\prime=x_i\oplus1$ for $i=1,2,...,2n$ (given a string of even length $x$, $\|x\|_{11}$ is the number of substring "11" in $x$ starting at an odd position) and then choosing $x_{2n+1}=c$ (analogously she can choose $x\in\{0,1\}^{2n+1}$ such that $\|x^\prime_1...x^\prime_{2n}\|_{11}$ is odd and $x_{2n+1}=c\oplus1$); * Alice, for each of the $2n+1$ shared boxes, apply a local transformation changing the probability law from $p_{000}\longrightarrow p_{\alpha1\gamma}$, then se puts the bits $x_1,x_2,...,x_{2n+1}$ into the boxes $1,2,...,2n+1$. Let $a_1,a_2,...,a_{2n+1}$ be Alice's output bits from the boxes; * Alice computes the parity of all the "cheated" output bits $A^\prime=\oplus_{i=0}^{2n+1}a^\prime_i$ and send $A^\prime$ to Bob, where $a^\prime_i=a_i\oplus\alpha x_i\oplus\gamma$ for $i=1,2,...,2n+1$; * Bob randomly chooses a string $y\in_\mathbb{R}\{0,1\}^{2n+1}$ and puts the bits $y_1,y_2,...,y_{2n+1}$ into his boxes. We call the output bits from his boxes $b_1,b_2,...,b_{2n+1}$. Then the REVEAL phase: * Alice sends $c^\prime$, her string $x^\prime$ (where $x_i^\prime=x_i\oplus1$ for $i=1,2,...,2n+1$) and all her $2n+1$ "cheated" outputs bits (i.e. $a^\prime_i$) to Bob; * Bob checks if Alice's data is consistent: $\forall i\in\{0,1\}^{2n+1}$, $x^\prime_i\cdot y_i=a^\prime_i\oplus b_i$ and $\|x^\prime_1...x^\prime_{2n}\|_{11}+x^\prime_{2n+1}+c^\prime$ is even. Since he finds no error, he accepts $c^\prime$ as the committed bit. In fact, according to Eq. (<ref>), every couple $(x^\prime_i,a_i^\prime)$ for $i=1,2,...,2n+1$ sent by Alice to Bob satisfies $x_i^\prime\cdot y=a_i^\prime\oplus b$ and by the proposed choices of $x\in\{0,1\}^{2n+1}$ also the parity constraints are § BIT COMMITMENT IN TRIPARTITE SCENARIO We have stressed that the proof of impossibility of perfectly secure bit commitment in PR-box theory was limited by considering no more than bipartite correlated boxes in $N$-partite systems. In fact, if we take in consideration just only tripartite boxes, the scenario changes considerably. In this Section we highlight that a scheme of bit commitment protocol like the one in Ref. [10] that would make use of tripartite non-local boxes would not subject to the cheating by local reversible transformations and it would represent a possible perfectly (or at least unconditionally) secure bit commitment protocol. In Section <ref> we used the classification done in Ref. [29] where the tripartite correlated boxes were divided in 46 non equivalent classes, i.e. 46 classes whose states are not connected by local reversible transformations. In order to see it directly, it is sufficient to write explicitly the probability rules for those classes. As an example we write out the representatives of the probability rule for three non-local tripartite classes (number 44, 45, and 46 in Ref. [29]): \begin{equation} \label{eq:tripartite-box-prob-rule} \begin{aligned} \text{Class 44: }& p(a,b,c\|x,y,z)= \begin{cases} 1/4& a\oplus b\oplus c=xyz\\ 0 & \text{otherwise} \end{cases}\\ \text{Class 45: }& p(a,b,c\|x,y,z)= \begin{cases} 1/4& a\oplus b\oplus c=xy\oplus xz\\ 0 & \text{otherwise} \end{cases}\\ \text{Class 46: }& p(a,b,c\|x,y,z)= \begin{cases} 1/4& a\oplus b\oplus c=xy\oplus xz\oplus yz\\ 0 & \text{otherwise} \end{cases}\\ \end{aligned}, \end{equation} and we note that the local relabelling of Eq. (<ref>) plus the permutations of the parties do not permit to move from every representative of one class to any one of any other class. Now to build our bit commitment protocol we decide to encode $b=0,1$ by choosing as input state $\Psi_0,\Psi_1\in\mathsf{St}(\rA^{\otimes3})$ one representative of the $44^{th}$ class and one of the $45^{th}$, say the ones in Eq. (<ref>) for the sake of simplicity. Following the strategy in Section <ref> we can construct a POVM that is able to discriminate between them, for example $\{a,e-a\}$ where $a\coloneqq b^{(0)} \otimes b^{(3)}\otimes b^{(0)} + b^{(0)}\otimes b^{(1)}\otimes b^{(2)}+ b^{(2)} \otimes b^{(1)}\otimes b^{(0)}+b^{(2)} \otimes b^{(3)}\otimes b^{(2)}$ and $e=e_\rA\otimes e_\rA\otimes e_\rA$. So the protocol is correct with probability one. Furthermore $\Psi_0$ and $\Psi_1$ are not connected by any local reversible transformation (as pointed out before) and so the protocol is also perfectly binding. At the same time the two input states have the same marginal on Bob system, and so it is also perfectly concealing. Namely, it is not difficult to derive from Eq. (<ref>) that \begin{equation} \begin{aligned} \Qcircuit @C=1.2em @R=.8em @! R { \multiprepareC{2}{\Psi_0}& \poloFantasmaCn{\rA}\qw& \qw\\ \pureghost{\Psi_0}& \poloFantasmaCn{\rA}\qw& \multimeasureD{1}{e}\\ \pureghost{\Psi_0}& \poloFantasmaCn{\rA}\qw& \ghost{e}\\ \end{aligned} \, = \, \begin{aligned} \Qcircuit @C=1.2em @R=.8em @! R { \prepareC{\mu}& \poloFantasmaCn{\rA}\qw& \qw\\ \end{aligned} \, = \, \begin{aligned} \Qcircuit @C=1.2em @R=.8em @! R { \multiprepareC{2}{\Psi_1}& \poloFantasmaCn{\rA}\qw& \qw\\ \pureghost{\Psi_1}& \poloFantasmaCn{\rA}\qw& \multimeasureD{1}{e}\\ \pureghost{\Psi_1}& \poloFantasmaCn{\rA}\qw& \ghost{e}\\ \end{aligned}\, . \end{equation} In conclusion, by simply choosing as our input states two suitable non-local tripartite boxes we build a protocol that is correct with probability 1, perfectly concealing and seems to be also perfectly binding (we are considering only reversible transformations). Naturally, since the theory is not complete, we would not like to fall in the same mistake of claiming our bit commitment protocol perfectly (or, when applied in Ref. [10], unconditionally) secure in PR-box theory but, less then unexpected turns in the theory, PR-box theory could really represent the first example of a theory with entanglement and bit commitment. CHAPTER: CONCLUSIONS In this thesis we have formalized the bit commitment protocol in the operational language to investigate its feasibility in a more general context than quantum theory. In this way we are willing to make the first step in the understanding of the relation that exists between bit commitment and the operational axioms of quantum theory. In particular we focused on the study of BC in PR-box theory, a theory that is more non-local than the quantum one but where the purification property does not hold. In performing this analysis we were forced to investigate new aspects of PR-box theory, since it is far away to be closed and complete. Some of the most remarkable results that we achieved are the following. Firstly, we described a strategy that grants to always find a POVM able to perfectly discriminate between any two bipartite pure states. In addition, we proved that the maximally mixed state $\mu$, is purificable. Furthermore, if the theory is limited to no more than bipartite correlated boxes, then $\mu$ is the unique internal state that is purificable and its purification is also unique up to reversible transformations on the purifying system. Then, we were able to show how simplistic generalizations to the arbitrary $N$-partite case are not appropriate. For example just admitting tripartite boxes we showed how the purification of the maximally mixed state is not more unique, and how it is not even granted that $\mu$ is still the unique internal state that is purificable. After the presentation of the PR-box theory in Chapter <ref>, in Chapter <ref> we presented the results about the impossibility of bit commitment in PR-boxes. In the literature of BC performed on non-local boxes, these have been rarely considered as part of a coherent theory and in fact tripartite correlated boxes or local reversible transformations (that are admissible in the theory) have always been neglected. By simply taking in consideration the reversible transformations we proposed a proof of impossibility of perfectly secure bit commitment in a PR-box theory limited by the following constraint: * no more than bipartite correlated boxes admitted; * only reversible transformations considered; * only pure input states. Even under these three important limitation our scenario is still enough general to include all the protocol proposed in literature. Furthermore we were also able to adapt the solid proof in Ref. [2] (in same context as above) to obtain an identical result for the impossibility of bit commitment in PR-boxes. In addition, even if we dealt only with perfectly secure bit commitment, we explicitly described a scheme in which Alice is able to cheat perfectly in the protocol proposed in Ref. [10], that was claimed to be unconditionally secure. Finally we relaxed the limitations that we imposed on the theory and we proposed a protocol that seems to be perfectly secure. However, since the theory is not complete, we address future studies to investigate this and the other questions that remain unanswered in this thesis. First of all, if the discriminating strategy could include not only pure states but all of them, then the proof of impossibility of perfectly secure BC could be extended to non-pure input states, too. Anyhow, the great unknown variable is still the integration of generic $N$-partite non-local correlated boxes in the theory. The consequences could be very surprising, for example it would even be possible that the number of internal states that are purificable asymptotically increases increasing $N$, and so that the purification principle could hold in the limit $N\rightarrow\infty$. In conclusion we presented some results on the impossibility of perfectly secure bit commitment in PR-boxes and we precisely pointed out their limit of validity, that, even if under considerable assumptions, still have an important comparison with the literature on the subject. However, as we have outlined many times in this work, to achieve definitive results, other progresses in the fundamental aspects of the theory are absolutely necessary. CHAPTER: TABLES ($\alpha\beta\gamma\delta$) $a\oplus b$ 0000 0 0001 1 0010 $y$ 0011 $y\oplus1$ 0100 1 0101 0 0110 $y\oplus1$ 0111 $y$ 1000 $x$ 1001 $x\oplus1$ 1010 $x\oplus y$ 1011 $x\oplus y\oplus1$ 1100 $x\oplus1$ 1101 $x$ 1110 $x\oplus y\oplus1$ 1111 $x\oplus y$ Outcome relations for the 16 bilocal-boxes. 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# Sparsistent filtering of comovement networks from high-dimensional data111ASC acknoweldges R&P grant from Indian Institute of Management Ahmedabad. We are grateful to Vikram Sarabhai Library for providing the data utilized in this paper. All remaining errors are ours. Arnab Chakrabarti Misra Centre for Financial Markets and Economy, Indian Institute of Management Ahmedabad, Gujarat 380015, India. Email: <EMAIL_ADDRESS>Anindya S. Chakrabarti (Corresponding author) Economics Area and Misra Centre for Financial Markets and Economy, Indian Institute of Management Ahmedabad, Gujarat 380015, India. Email<EMAIL_ADDRESS> ###### Abstract Network filtering is an important form of dimension reduction to isolate the core constituents of large and interconnected complex systems. We introduce a new technique to filter large dimensional networks arising out of dynamical behavior of the constituent nodes, exploiting their spectral properties. As opposed to the well known network filters that rely on preserving key topological properties of the realized network, our method treats the spectrum as the fundamental object and preserves spectral properties. Applying asymptotic theory for high dimensional data for the filter, we show that it can be tuned to interpolate between zero filtering to maximal filtering that induces sparsity and consistency while having the least spectral distance from a linear shrinkage estimator. We apply our proposed filter to covariance networks constructed from financial data, to extract the key subnetwork embedded in the full sample network. ## 1 Introduction Network representation of large dimensional complex systems has become a standard methodology to delineate the nature of linkages across a large number of constituent entities comprising the systems [33]. Examples range across systems varying widely in terms of nature and architecture: economic and financial networks [9, 5], social networks [44], biological networks like food webs [45], technological networks like world wide web [20] and transportation networks [40] among many others. Broadly speaking, there are two major strands of literature that starts from the analysis of the realized network. One strand of the literature utilizes networks to explore dynamics on it [32], using the realized network as the true representation of the linkages. The other literature goes backward to extract true linkages from the realized linkages [4, 36], maintaining the idea that some of the realized linkages in fact might be spurious. We are interested in the second stream of literature where the fundamental objective is to isolate and filter the key subnetwork out of a large dimensional realized network. In a complex dynamical system, the correlation matrix of time-varying responses of the constituent entities captures pairwise-linkages between the entities. A co-movement network is constructed by considering each response variable as a node of the graph and an undirected edge between two nodes exists if the corresponding correlation is nonzero. This kind of network construction out of observational multi-variate data has been very successful as a modeling paradigm in finance [31] and biology [12, 3] among others. However, such inference about existence of linkages from purely observational data has a problem. As the pairwise sample correlation is hardly equal to 0 (even when the true correlation is 0), the realized co-movement network will always be a complete graph. The size of the correlation matrix grows as square of the number of nodes. Therefore for real-life data, a complete graph constructed from a such a large correlation matrix might have edges carrying information that would be spurious in nature. Many of the edges, particularly the edges with very low correlation, contain very little information and a likely scenario is that they lead to false discovery of linkages. Hence, before carrying out the statistical analysis of a co-movement network, it is important to extract only the meaningful interactions or correlations. Prominent network filtering techniques, like _minimum spanning tree_ [28] (MST) or _planar maximally filtered Graph_ (PMFG) [43], aspire to do so by reducing the graph to a subgraph containing the maximum amount of information regarding the system’s collective behavior by preserving geometric properties of the realized network (connectivity in case of MST and closed loops with three or four nodes in case of PMFG). The second type of filtering emphasizes the statistical significance of edges [29]. The third type of filtering focuses on the spectral structure [19]. In this paper, we propose a new filtering technique for large networks constructed from high-dimensional data, utilizing the spectral properties. Drawing from statistical theory of high-dimensional covariance matrix estimation, we develop a flexible method to find out the sparse adjacency matrix that represents the key subnetwork of the full network. Theoretically, the filtered network retains maximum similarity with the true spectral structure. The method is quite flexible as it allows the tuning the degree of filtering within a range of zero to maximal permissible pruning of edges. Two important properties of the filter are as follows: first, the filter generates sparsity in the covariance matrix which makes the filtering possible, and secondly, the filter statistically consistently prunes spurious linkages leading to reduction on false discovery of linkages. The combination of these two properties lead to the sparsistence of the resultant filter. Fundamentally, our approach depends on the literature on large dimensional covariance matrix estimation. For a large interacting system, a comovement network ia a high-dimensional graph. Often the number of nodes is in the order of number of observations leading to a well recognized problem that the eigenvalues of the sample covariance matrix do not converge to their population counterpart [30]. This result dictates the fact that the sample covariance matrix is not an consistent estimate of the true covariance matrix [34, 14]. Therefore, efficient estimation of high-dimensional covariance matrix is a relevant problem in context to large network analysis. A broad class of well-conditioned shrinkage/ridge-type estimators were proposed to circumvent the problem [25]. Element-wise regularization methods were also proposed to achieve sparsity [7, 8, 37, 6]. Some of the methods require a natural ordering among the variables [39, 8]. Some of the proposed estimators fail to guaranty positive definiteness. Some form of _tapering_ matrix [17] and maximum likelihood estimator under positive definite and sparsity constraints [11] were proposed to ensure positive definite covariance estimator. We borrow the idea of consistent estimation of sparse covariance matrices from this literature. However, there are two ways of approaching the problem of inference of linkages. In this paper we deal with the graph implied by the covariance matrix and utilize a non-parametric approach. In particular, we leverage the properties of regularized covariance estimators in [25] and [38]. The complementary approach is through the application of graphical Lasso algorithm for Gaussian graphical model [10, 16]. However, graphical Lasso algorithm attempts to estimate the precision matrix and not the covariance matrix. Despite it’s popularity, graphical Lasso algorithm is a strongly parametric approach and applicable within a rather restricted class of models. Several attempts have been made to develop filtering methods while preserving large scale structure [18, 19]. [18] show that local filtering techniques can preserve network properties more that global filtering methods and propose a new sparsification technique that preserves edges leading to nodes of local hubs. [21] integrate spectral clustering and edge bundling for effective visual understanding. Under some distributional assumptions, statistical methods have been proposed to extract the backbone of the network [13, 29]. These works develop statistical tests for significance of edges. Some methods are proposed to find the irreducible backbone of a network from a sequence of temporal contacts between vertices [22]. Finally, we note that the proposed filter is more efficient than the filters based on random matrix theory, as those filters lead to shrinkage of all elements in the correlation matrices due to spectral decomposition without converting any of them to zero. Thus the resultant network is of the same dimension as the original network [41]. There are application of hard thresholding on the resultant network to reduce the size of the network by removing edges with low weights. However, such a technique is fundamentally ad-hoc as there is no intrinsic property that can fix the threshold [41]. In the present context, we avoid both problems by essentially targeting consistent covariance matrix estimator via sparsity and the distance between the eigenspectra of the target and the filtered matrix uniquely pins down the degree of thresholding and consequently, the degree of filtering. The rest of the paper is organized as follows: Section 2 introduces the essential notations used throughout and discusses necessary statistical background. Readers familiar with high-dimensional covariance matrix estimation problem can skip this part and can directly go to the next section 3. In section 4, we present some of the possible alternatives of the choices we made in the algorithm. We have presented applications of the filter to real-life data in section 5. Section 6 summarizes the paper and concludes. ## 2 Notations and Technical Background on Covariance Matrix Estimation Throughout this paper we maintain the following notations: * • $\mathcal{D}$: the $n\times p$ data matrix consisting of $p$ variables and $n$ independent observations. * • $\Sigma$: true (unobserved) covariance matrix ($p\times p$) of $p$ variables. * • $S$: sample covariance matrix of size $p\times p$ calculated from data matrix $\mathcal{D}$. * • $S_{LW}$: Ledoit-Wolf estimator of size $p\times p$ of covariance matrix $\Sigma$. * • $S_{\eta}$: Thresholded (sample) covariance matrix of size $p\times p$ corresponding to the threshold $\eta$. * • $S_{\eta^{}}$: Maximally filtered covariance matrix of size $p\times p$ for optimally chosen threshold parameter $\eta^{}$ with zero cost for filtering. * • $S_{\tilde{\eta}}$: Tuned filtering of covariance matrix of size $p\times p$ for optimally chosen threshold parameter $\tilde{\eta}$ for positive cost for filtering. * • $\Gamma(S_{\eta})$: Network corresponding to $S_{\eta}$. Following the above notation, the goal of our proposed methodology is to find an optimal threshold parameter $\eta^{}$ such that the corresponding filtered network $\Gamma(S_{\eta^{}})$ will have the sparsistence property. Below we define all concepts and discuss each of the steps in detail. With multivariate data, the population covariance matrix is estimated by its sample counterpart. The sample covariance matrix has unbiasedness and other useful large sample properties [2]. However, these properties are established under the assumption that the number of observations is large while the number of variables being constant. The difference between the multivariate statistical theory and high-dimensional statistics is that the latter considers the case where the number of variables ($p$) also grows with the number of observations ($n$). Under such assumption, the sample covariance matrix does not behave desirably and becomes inconsistent. When sample is drawn from a high-dimensional Gaussian distribution with true covariance matrix $I$, the difference between the true and sample spectra increases with the dimension to size ratio- as illustrated in Fig. 1. This fact is theoretically proved by _Marčenko-Pastur_ theorem and consequent developments [2]. For this reason, several attempts have been made to construct more efficient estimator of high-dimensional covariance matrix. Here, we will describe a few of these approaches which are relevant to this paper and used in section 3 to develop the algorithm. (a) p/n=0.1 (b) p/n=0.5 (c) p/n=1 (d) p/n=1.5 Figure 1: Plot of sorted eigenvalues corresponding to the true (dotted line) and sample covariance matrix (solid line) with $n$ draws from a $p$ dimensional normal distribution with covariance matrix $I$ (identity matrix). Since all eigenvalues of an identity matrix would be equal to one, the true spectrum is shown as a horizontal line at 1. Each panel represents a specific dimension to sample size ratio varying from 0.1 to 1.5. Higher $p/n$ ratio leads to larger deviation of the sample spectrum from the true spectrum [34], which necessitates spectral shrinkage as described in Sec. 2.1 and 2.2. ### 2.1 Stein’s approach Fig. 1 shows that under high-dimensional setup, the eigenvalues of the sample covariance matrix deviates considerably from their population counterparts. However, the problem itself suggests a possible way out. We can see (Fig. 1) that as the dimension to sample size ratio goes up, the sample spectra move further away from the true spectra. So shrinking the eigenvalues towards a central value may lead to a better estimator. Such a strategy was suggested by Stein [42] and the proposed covariance estimator takes the following form: $\hat{\Sigma}=\hat{\Sigma}(S)=P\psi(\Lambda)P^{\prime},$ (1) where the spectral decomposition of $S$ is given by $S=P\Lambda P^{\prime}$, with $\Lambda=\mathrm{diag}(\lambda_{1},\lambda_{2},...,\lambda_{p})$ being the diagonal matrix of eigenvalues of $S$ and $P$ being the matrix of eigenvectors; $\psi(\Lambda)=\mathrm{diag}(\psi(\lambda_{1}),\psi(\lambda_{2}),...,\psi(\lambda_{p}))$ is also a diagonal matrix. If $\psi(\lambda_{i})=\lambda_{i}\forall i$ then $\hat{\Sigma}$ is the usual sample covariance matrix $S$. $\psi$ shrinks the eigenvalues $\lambda$ and thus reduce the deviation from its true counterpart. Clearly, this approach only regularizes the eigenvalues and keep the eigen vectors of the sample covariance matrix unaltered. Due to this reason this type of estimators are also called _rotation equivariant_ covariance estimator. ### 2.2 Ledoit-Wolf estimator A problem with Stein’s original prescription is that it does not ensure monotonicity and nonnegativeness of the eigenvalues [34]. This problem had been addressed by Ref. [25] which formulated a general approach towards shrinkage by defining a rotation equivariant regularization based on the following minimization problem: $\underset{\Psi}{\text{min}}\\|P\Psi P^{\prime}-\Sigma\\|$ (2) where $\\|.\\|$ can be any matrix norm. Most widely considered norm is the Frobenius norm.222The Frobenius norm of an arbitrary matrix $A$ of order $r\times m$ is $\\|A\\|^{F}=\sqrt{\text{tr}(AA^{\prime})/r}$. A particularly useful solution for the optimal $\psi$ was proposed by Ledoit and Wolf [25] which is based on the observation that the sample covariance matrix is an unbiased estimator of the population covariance matrix. This fact remains true for high-dimensional data as well. But in high-dimensional setup, the sample covariance matrix becomes considerably unstable i.e. the deviation from the true covariance matrix can potentially be large. On the other hand if we use a structured covariance estimator- such as an identity matrix then the estimator, while being severely biased under misspecification of the structure, will have very little variability. They showed that a suitably chosen linear combination of these two types of estimators would outperform each of them where the coefficients/weights of linear combination is chosen to optimize the bias-variance trade off. Formally, the Ledoit-Wolf estimator $S_{LW}$ [25] is defined as $S_{LW}=\alpha_{1}I+\alpha_{2}S$ (3) where $I$ is a $p\times p$ identity matrix and $\alpha_{1}$ and $\alpha_{2}$ are chosen to minimize the risk corresponding to the loss function $p^{-1}\mathrm{tr}(S_{LW}-\Sigma)^{2}$. #### 2.2.1 Consistency of Ledoit-Wolf estimator Since $S$ is positive definite, $S_{LW}$ can also be shown to be positive- definite and consistency of such estimator depends on the growth rate of $p$, the moments and the association structure of the data [25]. More precisely, the $p(p+1)/2$ elements of the true covariance matrix can be consistently estimated if three conditions hold described below. In large dimensional covariance matrices, both $p$ and $n$ grows. Therefore it is a common practice to write $p$ as $p_{n}$ (function of $n$) and to consider $n$ in the limit. Let us define $Y=\mathcal{D}V$, where $\mathcal{D}$ is a $n\times p$ matrix and $V$ is the matrix whose columns are the normalized eigenvectors of $\mathcal{D}$. Denote the $i$th entry of any row by $Y_{i}$. Also, let us denote the set of all quadruples made of four distinct elements of $\\{1,~{}2,~{}3,~{}..,~{}p\\}$ as $Q_{n}$. The three conditions are the following: 1. C1: There exists a constant $K_{1}$ independent of $n$ such that $p_{n}/n\leq K_{1}$. 2. C2: There exists a constant $K_{2}$ independent of $n$ such that $\sum_{i=1}^{p_{n}}E(Y_{i})^{8}<K_{2}$. 3. C3: $\underset{n\rightarrow\infty}{\mathrm{lim}}\frac{p_{n}^{2}}{n^{2}}\times\frac{\sum_{(i,j,k,l)\in Q_{n}}(Cov(Y_{i}Y_{j},Y_{k}Y_{l}))^{2}}{\\#Q_{n}}=0,$ where $\\#Q$ denotes the cardinality of the set $Q$. C1 says that $p$ can either remain constant or grow with $n$. That means this method cannot be used (more specifically consistency cannot be achieved) for data for which $p>>n$. ### 2.3 Sparsity and threshold estimator Threshold estimator of high-dimensional covariance matrix regularizes both eigenvalues and eigenvectors as opposed to the Ledoit-Wolf estimator which only regularizes the eigenvalues of the sample covariance matrix. Threshold estimator is particularly useful when the true covariance matrix from the data generating process is _sparse_ , i.e. many of the non-diagonal entries of the covariance matrix are 0 or close to 0. This assumption is reasonable for a wide range of practical scenarios. Threshold estimator forces all the off- diagonal entries below a suitably chosen threshold to 0. Even if the corresponding entries of the true covariance matrix are nonzero, the threshold estimates of those entries entail only a bias but no dispersion as estimated by a fixed constant which is zero. The objective function would be the Frobenius norm (see footnote 2) of the difference between the thresholded matrix and empirical covariance matrix obtained from repeated sampling [7]. If the threshold is large, this method produces a sparse covariance matrix. We will denote the threshold matrix by $S_{\eta}$, where $\eta>0$ is the chosen threshold and $S$ is the usual sample covariance matrix; $\displaystyle S_{\eta}$ $\displaystyle\equiv$ $\displaystyle[s_{\eta}({i,j})]$ (4) $\displaystyle\equiv$ $\displaystyle[s_{\eta}(i,j)I(\|s(i,j)\|<\eta)]\quad~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ where $I(.)$ is the indicator function. So the entries of $S$ which are less than $\eta$ in magnitude are replaced by 0. The optimal threshold parameter $\eta$ can be chosen by cross validation. The resulting threshold estimator would be consistent under the assumption $\log(p)/n\rightarrow 0$, and it is shown to be uniform for a class of matrices satisfying a condition that captures a notion of “approximate sparsity”. One problem of such estimator is that it does not always preserves positive definiteness [7]. Threshold estimators can be further extended to a broader class of matrices called _generalized thresholding operators_ , which combines two regularization methods: thresholding and shrinkage [38]. When the true covariance is sparse, generalized thresholding estimators can identify the the true zero entries with probability 1. This property commonly called as _sparsistency_. The sufficient conditions required to achieve this are the following [34]: 1. C1: The data generating process is Gaussian. 2. C2: The variances are bounded above by a constant, i.e. $\sigma_{i,j}\leq C$ for a sufficiently large $C$. 3. C3: $C\sqrt{\frac{\text{log}~{}p}{n}}=o(1)$. ## 3 Sparsistent Filtering of Networks Fundamentally, our objective is to combine the feature of sparsity from the large dimensional covariance matrix estimation along with preservation of the underlying network topology. We elaborate on these two related but separate features below. Intuitively, the problem of filtering a complex network is equivalent to the problem of deleting a number of “less-important” edges from the original graph such that it become less complicated and reveals an underlying structure. Therefore sparsity is an essential property of the adjacency matrix of a filtered network. Evidently, this can be achieved by a threshold estimator with appropriately chosen threshold parameter. However, in the context of network filtering, the choice of a threshold is often ad-hoc and suffers from lack of robustness [41]. Therefore, the technical problem is what can be an efficient method for choosing the threshold that retains sparsity but is also statistically robust? One candidate would be cross-validation [35]. However, a threshold implied by cross-validation is purely numerical in nature and fully dependent on the realized covariance matrix where the realized covariance matrix is itself a random sample. As a direct implication, such a threshold does not allow inference on the underlying structure of the true covarince matrix (and therefore, the resulting network structure). Within a certain restricted class of data generating process, a threshold estimator can indeed be consistent [7]. However, the corresponding restrictions are too severe for direct applications to real-life data (e.g. the assumption of multivariate Gaussianity is often rejected in systems exhibiting large fluctuations, like in the case of stock market data [41]). Additionally, a cross-validated threshold estimator would exhibit theoretical consistency only when the true covariance matrix is sparse to begin with. However, a network filter should be flexible enough to consider a scenario where the linkages are of small strength, but non-zero. A further question arises here about the best way to capture the network topology. One can consider observable geometric properties as well as the spectral structure of a network. There are filters which focus on the geometric properties (like connectivity in case of minimum spanning tree or closed loops in case of planar maximally filtered graphs [43]). However, in the present context, the spectral structure of networks is the best candidate: one, the spectral structure by definition captures network topology, and two, it is amenable to asymptotic theories and links naturally with covariance matrix estimation. In sum, our goal is to find a way to retain the feature of sparsity in one hand (to make the estimator efficient) while preserving the network topology on the other. We require the filter should be flexible enough to interpolate between the two. We achieve it by combining the Ledoit-Wolf estimator as the target for retaining statistical consistency and imposing a threshold estimator that emulates the corresponding spectral structure. We show that the resulting filter inherits both sparsity as well as consistency. We explain the main idea in Fig. 2. The red thick line represents spectral distance of the candidate thresholded matrix from a target matrix by increasing the threshold from zero to a large enough value (we will provide the analytical details below). This distance represents a cost, indicating that a higher distance is less efficient. The distance as a function of the threshold is non-monotonic in nature. By increasing the threshold, initially the distance between the thresholded matrix from the target spectrum reduces and beyond a level, further increase in the threshold leads to an increase in the distance. The global minimum here corresponds to what we call maximal filtering, which is obtained by simply minimizing the spectral distance. However, we have to also consider a case where the true network is not necessarily sparse and there can be edges which can be of small magnitude. Thus in a general context, deleting them would entail a cost. We note that for a small threshold the edges being deleted would have small weights. But as the threshold is increased, we will filter edges with larger weights. This idea can be captured through a convex cost function, which indicates that as the threshold increases, the cost associated with deletion of edges with higher weights also increases. The blue dashed line shows a stylized cost curve. Therefore, the proper objective function is to minimize the total cost (adding spectral distance and the cost of edge deletion) indicated by the black dotted line. The final tuned filter would extract a threshold which is less than the threshold for maximal filtering as shown on the $x$-axis. Figure 2: Illustration of the distance function and cost function against the values of $y(\eta)$\- the number of deleted edges. The x-axis represents values of $y$, the black and red curves stand for $d(F^{S_{\eta}},F^{S_{LW}})$ (Eq. 5) and $C(\eta)$ (Eq. 9). The algorithm described below explains the steps. ### 3.1 Sequential steps of the algorithm Given the data matrix $\mathcal{D}$ of size $n\times p$, our algorithm goes through the following steps and return a thresholded matrix $S_{\tilde{\eta}}$ of size $p\times p$. 1. 1. Sample Covariance Matrix Construction: From the data $\mathcal{D}$, we calculate the sample covariance matrix $S$ of size $p\times p$. 2. 2. Construction of Ledoit-Wolf estimator: From the sample covariance matrix $S$, we calculate Ledoit-Wolf estimator $S_{LW}$ (following Eqn. 3).333For numerical implementation in R, we used the R package RiskPortfolios for Ledoit-Wolf matrix calculation (https://github.com/ArdiaD/RiskPortfolios) with type ‘oneparm’. For numerical implementation in Matlab, we have used the code titled ‘cov1para.m’ obtained from the code repository of Ledoit-Wolf estimator (https://www.econ.uzh.ch/en/people/faculty/wolf/publications.html#9). 3. 3. Finding the Spectrum of Ledoit-Wolf estimator: Eigenvalue decomposition of the covariation matrix444We choose to decompose the covariance matrix to obtain the spectrum. However one can certainly perform the identical analysis on the correlation matrix as well. Sometimes covariance can be too small and can exhibit some computational problem while implementation. Therefore we suggest it to be used on the correlation matrix. $S_{LW}$ gives us the spectrum denoted by $\lambda(S_{LW})$, which is a $p$ dimensional vector comprising $p$ eigenvalues. The empirical distribution function of $\lambda(S_{LW})$ is denoted by $F^{S_{LW}}$. 4. 4. Quantifying the Spectral Distance: The goal is to find a sparse thresholded matrix that is proximate to $S_{LW}$ in terms of spectrum. We define the spectral distance between two matrices $A$ and $B$ as $\text{d}(F^{A},F^{B})$ as the Euclidean distance between two spectra $\lambda(A)$ and $\lambda(B)$: $\text{d}(F^{A},F^{B})\equiv d(\lambda(A),\lambda(B))=\big{(}\sum_{i=1}^{p}\big{\|}\lambda_{i}^{A}-\lambda_{i}^{B}\big{\|}^{2}\big{)}^{1/2}.$ (5) 5. 5. Maximally Filtered Network: In this step, we obtain the “strong” or the “maximal” filter $S_{\eta}^{}$ corresponding to a threshold $\eta^{}$ that has the least spectral distance from $S_{LW}$. Formally, $S_{\eta^{}}$ can be obtained by minimizing the distance (Eqn. 5) of the resultant thresholded matrix from the Ledoit-Wolf matrix: $\eta^{}=\underset{\eta}{\text{argmin}}\\{d(F^{S_{\eta}},F^{S_{LW}})\\}.$ (6) Suppose, the number of edges being deleted in $S$ to reduce the matrix to $S_{\eta^{}}$ is $y^{}$.555$y^{}$ is a function of the threshold $\eta^{}$. A higher threshold $\eta^{}$ leads to deletion of a higher number of edges, implying that $y^{}$ would also be higher. Formally, we write $y^{}=\underset{i\neq j}{\sum}I(S_{\eta^{}}(i,j)=0)$ (7) where $I(.)$ is the identity function and $S_{\eta^{}}(i,j)$ is the $(i,j)$th entry of $S_{\eta^{}}$. 6. 6. Tuned Filtering with Costly Edge deletion: Now we impose a cost for deleting edges. The cost-adjusted optimal edge filtering leads to the following optimization: $\tilde{\eta}=\underset{\eta}{\text{argmin}}\\{d(F^{S_{\eta}},F^{S_{LW}})+C(\eta)\\}.$ (8) where $C(\eta)$ denotes the cost of deleting $y$ edges by implementing threshold $\eta$. For practical implementation, it is easier to work with the cost function on the edges to be deleted ($C(y)$) rather than the threshold ($C(\eta)$). A natural requirement for $C$ as a function of the number of edges to be filtered, is that it should be non-negative, continuous and potentially increasing in the first derivative leading to convexity. A typical candidate for a flexible functional form of $C$ is as follows666The parameters $\theta_{1}$ and $\theta_{2}$ in the cost function given by Eqn. 9 has to be specified by the user depending on the problem and the context of application.: $C(y)=\theta_{1}y^{\theta_{2}}\hskip 8.5359pt\text{where}\hskip 8.5359pt\theta_{1}\geq 0,~{}\theta_{2}>1~{}\text{and}~{}y\equiv y(\eta).$ (9) Eqn. 8 can be equivalently written in terms of deleted edges: $\tilde{y}=\underset{y}{\text{argmin}}\\{d(F^{S_{\eta}},F^{S_{LW}})+C(y)\\}.$ (10) The resulting tuned threshold is $\tilde{\eta}$, number of edges deleted is $\tilde{y}$ and finally, the thresholded matrix is $S_{\tilde{\eta}}$. 7. 7. Tuned Filtered Network with Sparsistence: We create the filtered network $\Gamma_{\tilde{\eta}}$ adjacency matrix from $S_{\tilde{\eta}}$. An edge is present if the corresponding element in $S_{\tilde{\eta}}$ is nonzero.777The covariances would not represent a metric since they can be negative. If we want to visualize the network in the metric space, we can convert covariances into correlations by dividing each covariance entry by the product of sample standard deviations of the corresponding pair of nodes, and these correlations can be transformed into a metric by using the transformation $\gamma_{ij}=\sqrt{2(1-\rho_{ij})}$ where $\rho_{ij}$ is the correlation between $i$ and $j$-th nodes [41] obtained from $S_{\eta}$ for any given $\eta$. By construction, the maximally filtered network $\Gamma(S_{\eta^{}})$ would be a subset of the tuned filtered network $\Gamma(S_{\tilde{\eta}})$. $\Gamma(S_{\tilde{\eta}})$ in turn would be a subset of the original network $\Gamma(S)$ corresponding to the sample covariance matrix $S$. In our empirical studies, we have seen that the threshold $\eta^{}$ is typically higher than the threshold chosen by the conventional threshold-estimator which is based on cross-validation (see Sec. 2.3). This implies that the sparsistency property (see Sec. 2.3 for sufficient conditions) will be maintained by maximal filtering because edges deleted for a threshold will also be deleted for all higher thresholds. In other words all spurious edges will be removed with probability one. ### 3.2 An illustrative example (a) True network: $\Gamma(\Sigma)$ (b) Realized network: $\Gamma(S)$ (c) Maximally filtered network: $\Gamma(S_{\eta^{}})$ (d) Network with tuned filtering: $\Gamma(S_{\tilde{\eta}})$ Figure 3: Illustration of the proposed algorithm. Panel (a): True sparse network $\Gamma(\Sigma)$ for simulation study with 15 edges and 10 nodes. Panel (b): The realized network $\Gamma(S)$ from simulated data corresponding to the sample covariance matrix $S$. Due to finite sampling ($n=50$), the network is fully connected with 45 edges (for $p$ = 10, the maximum number of edges $p(p-1)/2$ = 45). Panel (c): The maximally filtered network $\Gamma(S_{\eta^{}})$ when $\eta^{}=0.499$, with 6 edges. Panel (d): The tuned filtered network $\Gamma(S_{\tilde{\eta}})$ where the optimally chosen threshold $\tilde{\eta}$ is 0.170. $\Gamma(S_{\tilde{\eta}})$ has 20 edges. Here we present an example of our proposed filtering method to illustrate (1) how the filter produces a sparse network, (2) how different is this filtered network compared to the _true_ underlying network and (3) how the filtered network changes with the choice of the cost parameters. We choose the true data-generating process to be a $p(=10)$-dimensional Gaussian distribution with mean $\bm{0}$ and covariance matrix $\Sigma$. We illustrate the method for a particular choice of $\Sigma$ (given in Appendix A.1). $50$ sample observations are drawn from this $p$-dimensional distribution. Therefore $p/n$ ratio is 10/50=0.2. The sample covariance matrix ($S$) is calculated from the simulated data (see Appendix A.1). Applying the proposed algorithm, we get the filtered network. Fig. 3 shows the true comovement network for our chosen covariance matrix $\Sigma$. Although it has a moderately sparse structure (15 undirected edges), the sample correlation matrix obtained from the simulated data is not a sparse matrix. The network constructed from the sample covariance matrix $S$, is shown in Fig. 3 (45 undirected edges indicating a fully connected network). We plot two filtered networks corresponding to two choices of the threshold parameters. If we ignore the cost due to deletion of edges- i.e. if we only aim to reduce the distance between the spectral structure of the _threshold_ -network and the network induced by Ledoit-Wolf estimator- then we get the maximally filtered network shown in Fig. 3. We can see that this is not a connected network but it is able to preserve the stronger edges and the corresponding subnetworks of the true network. On the other hand, Fig. 3 represents the filtered network obtained by introducing a positive cost, which preserves the stronger edges along with the property of connectedness. Figure 4: Illustration of the distance function against number of deleted edges corresponding to a chosen threshold- denoted by $y(\eta)$. The distance is minimized at $y=39$ which means that the maximal filtering will remove 39 edges from $\Gamma(S)$ (the corresponding network is plotted in Fig. 3). ### 3.3 Filtering with known data generating process: Information loss and spuriousness If we know the true data generating process, then tuned filtering on a sample covariance matrix gives us two informational statistics related to information loss due to edge deletion and spuriousness of edges generated by finite sampling fluctuations. Fundamentally, these two statistics are related to finding false negatives (deleting edges that are actually informative) and false positives (retaining edges that appear in the sample covariance matrix due to sampling fluctuation, but are not there in the true covariance matrix). The first measure we define allows us to characterize true positives. We construct the measure by the proportion of _true_ edges which are retained in filtered network: $\text{P}_{t}(\eta)=\frac{\\#\\{E_{\Gamma(\Sigma)}\cap E_{\Gamma(S_{\eta})}\\}}{\\#\\{E_{\Gamma({\Sigma})}\\}}$ (11) where $\\#$ denotes the cardinality of a set, $E_{\Gamma(\Sigma)}$ and $E_{\Gamma(S_{\eta})}$ are the edge sets of the true and filtered networks. The numerator and denominator of the above equation only count the number of edges and do not take into account the relative importance of the edges. The next measure replace the _total number_ of edges by _total weight_ of the edges where weight is captured by absolute value of correlation: $\text{P}^{\prime}_{t}(\eta)=\frac{\underset{(i,j)\in\\{E_{\Gamma(\Sigma)}\cap E_{\Gamma(S_{\eta})}\\}}{\sum}w_{i,j}}{\underset{(i,j)\in\\{E_{\Gamma(\Sigma)}\\}}{\sum}w_{i,j}}.$ (12) Clearly, both $\text{P}_{t}(\eta)$ and $\text{P}_{t}^{\prime}(\eta)$ are bounded above by 1. As a consequence of being a complete graph, the sample covariance matrix has both the quantities equal to 1. The main challenge is to obtain a sparse graph with significantly high $\text{P}_{t}(\eta)$ and $\text{P}_{t}^{\prime}(\eta)$. However, high rate of edge retention might lead to retaining spurious edges. Therefore, it is important also to note how many edges the filtered network contains which are not part of the true network. The following proportion measures the same: $\text{P}_{f}(\eta)=\frac{\\#\\{E_{\Gamma(S_{\eta})}\cap E^{c}_{\Gamma(\Sigma)}\\}}{\\#\\{E_{\Gamma(S_{\eta})}\\}}$ (13) where $E^{c}_{\Gamma(\Sigma)}$ denotes the set of edges that do not exist in the true network (but might arise due to sampling). We report these three statistics in Table 1 for different threshold parameters $\eta$ for the data generating process discussed in Sec. 3.2. We see that when the threshold increases, $\text{P}_{t}(\eta)$ and $\text{P}^{\prime}_{t}(\eta)$ decrease on average. This is intuitive because a higher threshold leads to higher number of edges being deleted and therefore, the chances of deleting true edges also go up. On the contrary, a higher threshold simultaneously makes it more likely that spurious edges will be deleted. Therefore, the chances of having a false positive goes down. This is consistent with the column for $\text{P}_{f}(\eta)$ which shows that with higher threshold, the value of $\text{P}_{f}(\eta)$ decreases steadily. $\eta$ \| $\text{P}_{t}(\eta)$ \| $\text{P}^{\prime}_{t}(\eta)$ \| $\text{P}_{f}(\eta)$ ---\|---\|---\|--- 0.170 \| 0.742 ($\pm$ 0.09) \| 0.917 ($\pm$ 0.04) \| 0.445 ($\pm$ 0.07) 0.230 \| 0.698 ($\pm$ 0.09) \| 0.899 ($\pm$ 0.05) \| 0.351 ($\pm$ 0.09) 0.288 \| 0.568 ($\pm$ 0.07) \| 0.827 ($\pm$ 0.05) \| 0.157 ($\pm$ 0.10) 0.499 \| 0.454 ($\pm$ 0.14) \| 0.752 ($\pm$ 0.08) \| 0.072 ($\pm$ 0.14) Table 1: Measures $\text{P}_{t}(\eta)$, $\text{P}^{\prime}_{t}(\eta)$, $\text{P}_{f}(\eta)$ are shown for different threshold parameters $\eta$ for the data generating process discussed in Sec. 3.2. We have computed the measures based on 100 draw of sample covariance matrices of length $n=50$ and $p=10$. The average estimates over these 100 simulations appear in the table along with the corresponding standard deviation within the adjacent brackets. ## 4 Extensions and Robustness In the following, we discuss extensions and robustness of the proposed filtering algorithm. ### 4.1 Spectral similarity in terms of subset of eigenmodes In the algorithm presented in section 3, we have optimized on the threshold $\eta$ to minimize the distance between the spectrum of the Ledoit-Wolf estimator and the threshold estimator. However, one may not be interested in the full spectrum of covariance matrix. This is pertinent in the context of financial networks which is known to possess an eigenvalue distribution with wide heterogeneity. An array of statistical analysis (see e.g. [41]) shows that the highest eigenvalue captures the fluctuations due to the market mode, whereas sectoral fluctuations are associated with the deviating eigenvalues (except the largest one) from the bulk of the spectrum. The bulk of the spectrum on the other hand represents idiosyncratic fluctuations, which is modelled well by a Marčenko-Pastur distribution. Therefore in this context, only the deviating eigenvalues are informative. So one can argue for considering only these few eigenvalues and choose the threshold that minimizes the distance between two vectors of deviating eigenvalues. This represents evaluating the distance on a smaller set of eigenmodes as opposed to all eigenmodes, and the distance between eigenmodes of two matrices $A$ and $B$ to (Eqn. 5) to be modified as follows: $d(\lambda(A),\lambda(B))=\big{(}\sum_{i=p_{l}}^{p_{h}}\big{\|}\lambda_{i}^{A}-\lambda_{i}^{B}\big{\|}^{2}\big{)}^{1/2}$ (14) where $1\leq p_{l}\leq p_{h}\leq p$ and the choice of $p_{l}$ and $p_{h}$ can be chosen according to the specific system under analysis. Specifically, the upper bound of the Marčenko-Pastur distribution can provide such a natural cut-off for the choice of $p_{h}$ to capture the deviating eigenmodes and $p_{l}$ can be unity. ### 4.2 Non-linear shrinkage estimator All the rotation-equivariant estimators we have discussed so far are linear. They are linear combination of the sample covariance (or correlation) matrix and a suitable shrinkage target. This means that regardless of their ranks, all the sample eigenvalues are shrunk by same intensity. However our objective is to minimize Eqn. 2 and there is no guarantee that a linear shrinkage estimator would be our best choice. [24, 26] show that linear shrinkage is a first order approximation of a nonlinear problem whose utility depends very much on the situation- particularly on the limit of $p/n$. If this ratio is high then linear shrinkage will be a substantial improvement but not otherwise. Attempts have been made to find nonlinear solution to the problem which essentially results in individualized shrinkage intensity to every sample eigenvalue. First we describe the role of random matrix theory and why it is instrumental in finding the solution. Results from random matrix theory illustrate that for high dimensional set up the eigenvalues of sample covariance matrix do not converge to its population counterparts. However, random matrix theory attempts to establish a link between the two. First attempts in nonlinear shrinkage estimators harnessed the established relation between the limiting spectral distribution of the sample eigenvalues and that of the population eigenvalues. Once the spectral distribution of the population eigenvalues are obtained, it can be numerically inverted to calculate the population eigenvalues [24, 15]. Below, we describe one such solution. Let us introduce the following quantities: 1. 1. $p/n\rightarrow c~{}(>0)$. 2. 2. If $G$ be the cumulative distribution function of eigenvalues $\lambda$, then the Stieltjes transform of the $G$ is defined as below: $m_{G}(z)=\int_{-\infty}^{\infty}\frac{1}{\lambda-z}dG(\lambda)\quad\forall z\in\mathbb{C}^{+}.$ Stieltjes transform is an important tool in random matrix theory because of its one-one relationship with the distribution function (empirical spectral distribution in our context). Therefore, to determine the limiting spectral distribution one only needs to show the convergence of corresponding Stieltjes transform. 3. 3. The limiting empirical spectral distribution of the sample covariance matrix is denoted as $F$. 4. 4. The Stieltjes transform of the Marčenko-Pastur law [2] is denoted by $m_{F}(z)$. 5. 5. Define $\tilde{m}_{F}(\lambda)=\underset{z\in\mathbb{C}^{+}\rightarrow\lambda}{\text{lim}}m_{F}(z)\quad\forall\lambda\in\mathbb{R}-\\{0\\}$ 6. 6. Define $m_{LF}(z)=1+zm_{F}(z)\quad\forall z\in\mathbb{C}^{+}$. Under some general assumptions, [24] proposed nonlinear shrinkage intensities and the derived form of $\psi$ (see Eqn. 1) is the following: $\psi_{i}=\frac{\lambda_{i}}{\|1-c-c\lambda_{i}\tilde{m}_{F}(\lambda_{i})\|^{2}}.$ (15) Note that $\psi_{i}$ is dependent on $\lambda_{i}$ (unlike the linear shrinkage estimator). For more detailed discussion, see [15, 24, 26]. There are also some other methods of nonlinear shrinkage estimation. By exploiting the connection between nonlinear shrinkage and nonparametric estimation of Hilbert transform of the sample spectral density, an analytical formula for nonlinear shrinkage has been proposed recently [27]. [1] proposed a method called Nonparametric Eigenvalue-Regularized COvariance matrix estimator (NERCOME; [23]) that splits the sample into two parts. One part is used to estimate the eigenvectors of the covariance matrix and the other part of sample to estimate the eigenvalues associated with these eigenvectors. Averaging over a sufficiently large number of sample split results in reasonably good estimation. All these methods can be used as alternative to the linear shrinkage estimator considered in the algorithm proposed in Sec. 3. However, the difference in filtering via linear and nonlinear shrinkage estimators is often sample-dependent and exhibits large fluctuations. We attempted to filter financial networks via nonlinear shrinkage estimator (details given in Sec. 5). Through empirical analysis, we saw that when the filtering is moderate via linear estimator, then the nonlinear estimator does not produce radically different filtering. However, we have observed extreme cases where linear shrinkage estimator leads to complete filtering of all edges whereas nonlinear shrinkage estimator led to very minor filtering. Therefore, while nonlinear shrinkage estimator has more flexibility [26], the corresponding impact on the strength of filtering is sample-dependent. ### 4.3 Alternative choices of the cost function A convex cost function captures the idea that higher number of edges being deleted would entail a higher per unit cost.888A concave cost function would lead to maximal filtering, since more filtering leads to lower per unit cost. The underlying idea is that the first few edges being deleted would have the lowest weights. However, as we increase the threshold, the edges being filtered out would have larger and larger weights. This observation leads to the assumed convexity of the cost curve. The proposed functional form $C(y)=\theta_{1}y^{\theta_{2}}$ is useful for its simplicity and ease of manipulation. In principle, many other cost functions can be considered as long as they are convex in nature. A more general set of choices is presented below. Suppose the $(i,j)$th element of $S$ is denoted by $s_{i,j}$. We define the total edge weight as $W=\sum_{i\neq j}f(s_{i,j})$, for a suitable nonnegative function $f$. The total weight removed by filtering can be denote by $W_{\eta^{}}$, which is defined as follows: $W_{\eta}=\underset{(i,j)\in\bar{E}}{\sum}f(s_{i,j}),~{}\text{where}~{}\bar{E}=E_{\Gamma(S)}-E_{\Gamma(S_{\eta})}.$ (16) In principle, any convex function of $\frac{W_{\eta}}{W}$ is a valid choice for $C(\eta)$. As an example, if we choose $f(s_{i,j})=I(s_{i,j}\neq 0)/2$ then $W_{\eta^{}}$ becomes $y^{}$ as defined in Eqn. 7.999The cost function defined in Eqn. 10 is: $C(\eta)\equiv C(y(\eta))\equiv\theta_{1}W_{\eta}^{\theta_{2}}.$ (17) This choice of $f(.)$, although simple and easily understandable, does not directly incorporate the weight of each deleted edge. Therefore an exogenous cost function is needed to be defined and imposed (see Eqn. 9). An endogenous choice is the following: $f(s_{i,j})=\|s_{i,j}\|/2~{}\text{and}~{}C(\eta)\propto\frac{W_{\eta}}{W}.$ (18) Given the above discussion, we see that this function is convex. ### 4.4 Alternative choices of the distance function In the description of the algorithm, we have utilized Euclidean distance between the spectra (Eqn. 5). This is useful in terms of implementation as well as simplicity. In principle, one can look for alternative notions of distances as well to measure similarity or dissimilarity between two spectra. We consider them below and discuss the relative merits and demerits. Suppose $F(.)$ and $G(.)$ are two spectral distribution functions. For finite sample let us denote the vectors of ordered eigenvalues corresponding to two $p\times p$ matrices as $\lambda_{a}=(\lambda_{a,1},\lambda_{a,2},..,\lambda_{a,p})$ and $\lambda_{b}=(\lambda_{b,1},\lambda_{b,2},..,\lambda_{b,p})$. In this case, $F$ and $G$ are the discrete uniform distribution on $\lambda_{a}$ and $\lambda_{b}$. Three well known measures for distance are as follows: 1. 1. _Minkowski distance (for general $\kappa$)_: $d(F,G)=d(\lambda_{a},\lambda_{b})=\big{(}\sum_{i=1}^{p}\big{\|}\lambda_{a,i}-\lambda_{b,i}\big{\|}^{\kappa}\big{)}^{\nicefrac{{1}}{{\kappa}}}$, where $\kappa\geq 1$. 2. 2. _$L^{1}$ distance:_ $d(F,G)=d(\lambda_{a},\lambda_{b})=\sum_{i=1}^{p}\big{\|}\lambda_{a,i}-\lambda_{b,i}\big{\|}$. 3. 3. _$L^{\infty}$ distance:_ $d(F,G)=d(\lambda_{a},\lambda_{b})=\text{max}_{i}\big{\|}\lambda_{a,i}-\lambda_{b,i}\big{\|}$. Note that, Minkowski distance for $\kappa=2$ is Euclidean distance which we considered in the algorithm. The other two metrics ($L^{1}$ and $L^{\infty}$) are special cases of the Minkowski distance. While we can potentially evaluate the filter with $L^{1}$ or $L^{\infty}$, given the lack of smoothness in the derivatives, we consider the $L^{2}$ (i.e. Minkowski distance with $\kappa$ = 2) to be the most appropriate metric for ease of computation and exposition. ## 5 Real-life Data Analysis To demonstrate application of our proposed filter, we apply it on a real-life financial data set. Although the filter would be more useful for _big_ data sets with high number of variables, we use a data set with a moderate number of variables for the sake of visualization of the resulting network. Applicability of our method depends on three factors: 1) The data possess a high-dimensional covariance matrix. 2) There is a network representation based on the covariance matrix. 3) There is a _core_ underlying set of connections or _skeleton_ of the network that is captured by large enough covariances. ### 5.1 Application to financial networks We perform our method on historical NASDAQ data for 50 stocks with prices recorded over 70 consecutive days, from 2nd January, 2015 to 14th April, 2015. As is customary in the analysis of return comovement networks [41], we first construct the log return series for each of the stocks. If a stock’s price at time $t$ is $P_{t}$, then the log return at time $t$ is defined as the following: $r_{t}=\text{log}P_{t}-\text{log}P_{t-1}.$ (19) From the generated return series, the sample correlation matrix $S$ is calculated and the corresponding network $\Gamma(S)$ is generated (Fig. 5). As it is a complete graph we can see all possible edges are present (we have excluded the self-loops) as all pairs of stocks would exhibit non-zero covariance. Fig. 5 exhibits the maximally filtered network, which shows a drastic reduction in the number of edges(from 1225 to 256). We also observe that this filtered graph is not connected. In particular, the maximally filtered network produces 8 isolated vertices while the remaining 42 stocks create a giant component.101010In some sets of stocks, we have observed that the spectral distance between the true matrix and the Ledoit-Wolf analog is so large that the maximal filtering leads to fully diagonal matrix, i.e. all nodes become separate. In such cases, the spectral similarity is not an useful criterion for filtering. (a) Financial network (b) Maximally filtered network (c) Spectral distance Figure 5: Illustration for Financial network. Panel (a) shows the network ($\Gamma(S)$) obtained from sample correlation matrix. It is a complete graph with dense connectivity. Panel (b) shows the maximally filtered network with substantially fewer edges. The network splits into a giant connected component and unconnected peripheral nodes. Panel (c) shows the spectral distance function against number of deleted edges. The lowest distance corresponds to the maximal filtered network shown in panel (b). Finally, in Fig. 5 we plot the spectral distance between the Ledoit-Wolf estimator and the filtered matrices in an increasing number of edges being deleted (associated with increasing threshold). The global minimum for the distance function is reached at 1082 (=1225-143) number of edges being deleted. Therefore, the maximally filtered network would consist of 143 edges as shown in Fig. 5. Clearly, such strong filtering is associated with high thresholding and consequent loss of a large number of edges. For a less strong filtering, one can impose a positive cost of edge deletion (maximally filtering requires zero cost of edge deletion) following Eqn. 9. based on the choice of parameters, one can interpolate between zero filtering to maximal filtering. To complement the above analysis, we carry out the filtering on a large covariance matrix arising out of the largest $p$ = 300 stocks in NASDAQ in 2015 calendar year by varying the number of observations $n$ from 50, 200, 300 and 450. The values are chosen such that the $p/n$ ratio varies from a number smaller than one to larger than one. The resulting maximal thresholds are shown in Fig. 6 in the Appendix A.2. As can be seen, for large ratio of $p$ to $n$ (indicating very small number of observations for each stock) the filtering threshold is very high and a high fraction of edges get filtered. In the other extreme, when $p$ to $n$ ratio is small (indicating a large number of observations for each stock), then the filtering threshold is very low and therefore, very few edges are filtered. For the sake of completeness, we should mention that the filtering threshold is influenced by sampling fluctuations and therefore, such a monotonic relationship may not be found in all applications. However, in our numerical experiments we found that on an average a larger number of observations for each entity (stocks in this case) leads to smaller filtering threshold. ## 6 Summary and Conclusion Many large scale systems are best described as networks [3, 4, 5, 12, 20, 33, 31, 41]. A standard approach of network construction is to create covariance- based measures of interlinkages [41]. However, construction of the comovement network from an observed data set is a challenging problem because the resulting network is a complete graph and therefore resists any naive attempt to uncover the underlying network topology due to existence of spurious linkages. Statistically such networks suffer from false positives, i.e. false discovery of linkages. Therefore, a robust methodology is needed to identify and prune such non-informative linkages and isolate the key subnetwork embedded in the complete network. In this paper, we develop a filtering technique that attempts to resolve this problem utilizing spectral structure of the network. The existing filtering techniques have mainly two features. First, they are primarily based on some graph-theoretic constraints and not on explicit statistical motivation (e.g. minimum spanning tree or more general, hierarchical structures). Second, many of the filtering techniques are not tunable and often they lead to a drastic reduction in the number of edges (e.g. minimum spanning tree), which also makes the resulting network very unstable and sample-dependent. In this paper, we propose an new filter based on the properties of high-dimensional covariance estimators, utilizing the concept of sparsistence along with retaining flexibility for tuning the degree of filtering. We note that one can consider algorithms based on hypothesis testing of individual edge weights and prune statistically insignificant edges. However, this kind algorithms still suffer from the problem of false positives (i.e. false discovery of edges) as they do not account for joint hypothesis testing. We approach the problem in a new way by considering the spectral structure of the covariance network and sparsistent analogues of that, based on Ledoit-Wolf estimators which features predominantly in high dimensional covariance matrix estimation. Depending on the statistical properties of the Ledoit-Wolf estimator, we prescribe an endogenously thresholded covariance matrix estimator such that its spectrum is closest to that of the Ledoit-Wolf matrix. We complement the theoretical structure with numerical simulations along with applications to real world financial data. Our work is situated in the intersection of the literature on network filtering, covariance matrix estimation and large dimensional data. The proposed algorithm can be applied to any large dimensional data. We have demonstrated the usefulness of the filtering algorithm by applying it to financial stock return data. Further applications to various domains spanning biological, physical and technological comovement networks would lead to a more complete understanding of the corresponding topological structures and key linkages that contribute to the dynamics of the system. ## References * [1] Karim M Abadir, Walter Distaso, and Filip Žikeš. 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Nature, 404(6774):180–183, 2000. ## Appendix A Appendix ### A.1 True and sample covariance matrix for the illustrative example The true correlation matrix for the simulated illustrative example (Sec. 3.2) is the following: $\begin{bmatrix}1.00&0.80&0.00&0.00&0.00&0.30&0.00&0.00&0.00&0.00\\\ 0.80&1.00&0.00&0.00&0.00&0.09&0.00&0.00&0.00&0.00\\\ 0.00&0.00&1.00&0.30&0.09&0.09&0.00&0.09&0.00&0.00\\\ 0.00&0.00&0.03&1.00&0.00&0.00&0.30&0.00&0.00&0.00\\\ 0.00&0.00&0.09&0.00&1.00&0.80&0.00&0.00&0.00&0.00\\\ 0.30&0.09&0.09&0.00&0.80&1.00&0.00&0.00&0.00&0.00\\\ 0.00&0.00&0.00&0.30&0.00&0.00&1.00&0.30&0.80&0.80\\\ 0.00&0.00&0.09&0.00&0.00&0.00&0.30&1.00&0.09&0.30\end{bmatrix}$ After generating the sample, the sample covariance matrix is: $\begin{bmatrix}14.36&-2.12&2.36&-1.44&0.38&-4.01&0.64&2.65&-3.38&-2.13\\\ -2.12&7.02&-0.63&-0.01&-0.80&1.20&0.12&-3.29&1.87&-5.86\\\ 2.36&-0.63&11.12&5.55&-0.06&-1.31&5.04&4.63&-3.83&2.14\\\ -1.44&-0.01&5.55&9.73&-3.03&-1.69&7.45&1.15&-4.97&1.85\\\ 0.38&-0.80&-0.06&-3.03&6.76&3.57&-2.23&-3.82&-2.84&-0.96\\\ -4.01&1.20&-1.31&-1.69&3.57&5.99&-4.84&-2.91&-2.08&-1.12\\\ 0.64&0.12&5.04&7.45&-2.23&-4.84&13.54&-1.80&-3.02&0.33\\\ 2.65&-3.29&4.63&1.15&-3.82&-2.91&-1.80&9.56&1.38&4.99\end{bmatrix}$ We see that many entries of the true correlation matrix is 0 and therefore, the corresponding covariance matrix would be a sparse matrix. However, it is noteworthy that the sample covariance matrix does not contain any 0 due to sampling fluctuations. So the resulting network representation will be a fully connected network although the underlying network is sparsely connected. ### A.2 Application on large dimensional financial covariance matrix Figure 6: Application of the sparsistent filter on large dimensional financial network. We have considered the covariance matrices of $p=300$ largest stocks from NASDAQ stock exchange (in terms of market capitalization) with varying number of days. Each data set begins on 2nd January, 2015 and continues for $n$ days where $n$ varies from 50, 200, 300 and 450. Therefore, in Panel (a) $p/n$ = 2/3, Panel (b) $p/n$ = 1, Panel (c) $p/n$ = 3/2 and finally, Panel (d) $p/n$ = 6. A larger $p/n$ ratio leads to larger threshold for maximal filtering.
∎ aFEMclipboard 11institutetext: A. Chambolle 22institutetext: CEREMADE, CNRS & Université Paris Dauphine, PSL Research University, Paris 22email<EMAIL_ADDRESS>ORCID: 0000-0002-9465-4659 33institutetext: R. Tovey 44institutetext: MOKAPLAN, INRIA Paris, Paris 44email<EMAIL_ADDRESS>ORCID: 0000-0001-5411-2268 # “FISTA” in Banach spaces with adaptive discretisations Antonin Chambolle Robert Tovey ###### Abstract FISTA is a popular convex optimisation algorithm which is known to converge at an optimal rate whenever a minimiser is contained in a suitable Hilbert space. We propose a modified algorithm where each iteration is performed in a subset which is allowed to change at every iteration. Sufficient conditions are provided for guaranteed convergence, although at a reduced rate depending on the conditioning of the specific problem. These conditions have a natural interpretation when a minimiser exists in an underlying Banach space. Typical examples are L1-penalised reconstructions where we provide detailed theoretical and numerical analysis. ###### Keywords: Convex optimization Multiscale Multigrid Sparsity Lasso ††journal: Computational Optimization and Applications ## 1 Introduction The Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) was proposed by Beck and Teboulle Beck2009 as an extension of Nesterov’s fast gradient method Nesterov2004 and is now a very popular algorithm for minimising the sum of two convex functions. We write this as the problem of computing $\inf_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{H}}\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])\qquad\text{such that}\qquad\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])\coloneqq\operatorname{f}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])+\operatorname{g}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]),$ (1) for a Hilbert space $\mathds{H}$ where $\operatorname{f}\colon\mathds{H}\to\mathds{R}$ is a convex differentiable function with $L$-Lipschitz gradient and $\operatorname{g}\colon\mathds{H}\to\overline{\mathds{R}}$ is a “simple” convex function, whose “proximity operator” is easy to compute. Throughout this work we assume that $\operatorname{E}$ is bounded below so that the infimum is finite. The iterates of the FISTA algorithm will be denoted $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}\in\mathds{H}$. If, moreover the infimum is achieved, it has been shown that $\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})-\inf_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{H}}\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])$ converges at the optimum rate of $n^{-2}$ Beck2009 , and later (after a small modification) the convergence of the iterates was also shown in a general Hilbert space setting Chambolle2015 . Many further works have gone on to demonstrate faster practical convergence rates for slightly modified variants of FISTA Tao2016 ; Liang2017 ; Alamo2019 . In this work we address the case where the minimiser possibly fails to exist or lies in a larger space where $\mathds{H}$ is dense. There is much overlap between the techniques used in this work and those used in the literature of inexact optimisation, however, our interpretation is relatively novel. In particular, we emphasise the infinite-dimensional setting where errors come from “discretisation”, rather than random or decaying errors in $\mathds{H}$, which enables two new perspectives: * • Analytically, we prove new rates of convergence for FISTA when the minimum energy is not achieved (at least not in $\mathds{H}$). The exact rate can be computed by quantifying coercivity and regularity properties of $\operatorname{E}$. If there isn’t a minimiser in $\mathds{H}$, then this rate is strictly slower than $n^{-2}$. * • Numerically, we allow the optimisation domain to change on every iteration. This enables us to understand how FISTA behaves with adaptive discretisations. Adaptive finite-element methods are known to improve the efficiency of, for example, approximating the solutions of PDEs. Our analytical results show how to combine such tools with FISTA without reducing the guaranteed rate of convergence, and our numerical results confirm much improved time and computer memory efficiency in the Lasso example (Section 6). All the examples in this work, discussed from Section 5 onward, consider $\\{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{H}\operatorname{\;s.t.\;}\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])<\infty\\}$ to be contained in some ambient Banach space $\mathds{U}$. The idea is that FISTA provides a minimising sequence in $\mathds{H}\cap\mathds{U}$, but further properties like rate of convergence (of $\operatorname{E}$ or the iterates) must come from the topology of $\mathds{U}$. It will not be necessary for $\mathds{H}\hookrightarrow\mathds{U}$ to be a continuous embedding, nor in fact the full inclusion $\mathds{H}\subset\mathds{U}$. Some other works for FISTA-like algorithms include Jiang2012 ; Villa2013 . Of particular note, our stability estimate for FISTA in Theorem 4.1 is very similar to (Schmidt2011, , Prop 2) and (Aujol2015, , Prop 3.3). This is then used to analyse the convergence properties in our more general Banach space setting, but where all sources of inexactness come from subspace approximations. The ideas in Parpas2017 are similar although in application to the proximal gradient method with an additional smoothing on the functional $\operatorname{g}$. The permitted refinement steps are also more broad in our work. Very recent work in Yu2021 proposes a “Multilevel FISTA” algorithm which allows similar coarse-to-fine refinement strategies, although only a finite number. We also allow for non-uniform refinement with a posteriori strategies. ### 1.1 Outline This work is organised as follows. Section 2 defines notation and the generic form of our proposed refining FISTA algorithm, Algorithm 1. The main theoretical contribution of this work is the convergence analysis of Algorithm 1 which is split into two parts: first we outline the proof structure in Section 3, then we state the specific results in the case of FISTA in Section 4. The main results are Theorems 4.2/4.3 which extend the convergence of FISTA to cases with un-attained minima with uniform/adaptively chosen subspaces $\mathds{U}^{n}$ respectively. Section 5 presents some general results for the application of Algorithm 1 in Banach spaces and Section 6 gives a much more detailed discussion of adaptive refinement for Lasso minimisation. In particular, we describe how to choose efficient refining discretisations to approximate $\inf_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{H}}\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])$, estimate the convergence of $\operatorname{E}$, and identify the support of the minimiser. The numerical results in Section 7 demonstrate these techniques in four different models demonstrating the apparent sharpness of our convergence rates and the computational efficiency of adaptive discretisations. ## 2 Definitions and notation We consider optimisation of (1) over a Hilbert space $(\mathds{H},\left\langle\cdot,\cdot\right\rangle,{\left\lVert\cdot\right\rVert})$. In the more analytical section (Sections 3 and 4) it will be more convenient to use the translated energy $\operatorname{E}_{0}\colon\mathds{H}\to\mathds{R},\qquad\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])\coloneqq\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])-\inf_{\widetilde{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}\in\mathds{H}}\operatorname{E}(\widetilde{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]})$ (2) so that $\inf_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{H}}\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])=0$, although access to this function is not assumed for numerical examples. The proposed generalised FISTA algorithm is stated in Algorithm 1 for an arbitrary choice of closed convex subsets $\mathds{U}^{n}\subset\mathds{H}$ for $n\in\mathds{N}$. The only difference from standard FISTA is that on iteration $n$, all computations are performed in the subset $\mathds{U}^{n}$. If $\mathds{U}^{n}=\mathds{H}$, then we recover the original algorithm. More generally, the idea is that $\mathds{U}^{n}$ are “growing”, for example $\mathds{U}^{n}\subset\mathds{U}^{n+1}$, but this assumption is not necessary in most of the results. Without loss of generality we will assume $L=1$, i.e. $\nabla\operatorname{f}$ is 1-Lipschitz. To get the general statement of any of the results which follow, replace $\operatorname{E}$ with $\frac{\operatorname{E}}{L}$. In particular, ${\left\lVert\nabla\operatorname{f}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])-\nabla\operatorname{f}(\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1])\right\rVert}\leq{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]-\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]\right\rVert}$ (3) for all $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0],\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]\in\mathds{H}$ and $\operatorname{g}$ is called “simple” if it is proper, convex, weakly lower-semicontinuous, and $\operatorname{argmin}_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\widetilde{\mathds{U}}}\tfrac{1}{2}{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]-\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]\right\rVert}^{2}+\operatorname{g}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])$ (4) is exactly computable for all $\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]\in\mathds{H}$ and all $\widetilde{\mathds{U}}\in\\{\mathds{U}^{n}\\}_{n=0}^{\infty}$. Closed subsets of $\mathds{H}$ are locally weakly compact, therefore this argmin is always non-empty. One defining property of the FISTA algorithm is an appropriate choice of inertia, dictated by $t_{n}$. In particular, we will say that $(t_{n})_{n=0}^{\infty}$ is a _FISTA stepsize_ if $t_{0}=1,\qquad t_{n}\geq 1,\qquad\text{and}\qquad\rho_{n}\coloneqq t_{n}^{2}-t_{n+1}^{2}+t_{n+1}\geq 0\qquad\text{ for all }n=0,1,\ldots.$ (5) The precise constants associated to a given rate are given in the statements of the theorems but, for convenience, are otherwise omitted from the text. For sequences $(a_{n})_{n=0}^{\infty}$,$(b_{n})_{n=0}^{\infty}$ we will use the notation: $\displaystyle a_{n}\lesssim b_{n}\qquad$ $\displaystyle\iff\qquad\exists C,N>0\operatorname{\;s.t.\;}a_{n}\leq Cb_{n}\text{ for all }n>N,$ $\displaystyle a_{n}\simeq b_{n}\qquad$ $\displaystyle\iff\qquad a_{n}\lesssim b_{n}\lesssim a_{n}.$ For $n\in\mathds{N}$ we use the abbreviation $[n]=\\{1,2,\ldots,n\\}$. When the subdifferential of $\operatorname{E}$ is set-valued, we will use the short-hand ${\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\partial\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}\coloneqq\inf_{\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]\in\partial\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])}{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}$ (6) for any specified norm ${\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\cdot\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}$. Algorithm 1 Refining subset FISTA 1:Choose $(\mathds{U}^{n})_{n\in\mathds{N}}$, $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{0}\in\mathds{U}^{0}$ and some FISTA stepsize choice $(t_{n})_{n\in\mathds{N}}$ 2:$\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{0}\leftarrow\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{0},n\leftarrow 0$ 3:repeat 4: $\overline{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}_{n}\leftarrow(1-\tfrac{1}{t_{n}})\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}+\tfrac{1}{t_{n}}\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n}$ 5: $\displaystyle\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n+1}\leftarrow\operatorname{argmin}_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{U}^{n+1}}\tfrac{1}{2}{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]-\overline{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}_{n}+\nabla\operatorname{f}(\overline{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}_{n})\right\rVert}^{2}+\operatorname{g}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])$ $\triangleright$ Only modification, $\mathds{U}^{n+1}\subset\mathds{U}$ 6: $\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n+1}\leftarrow(1-t_{n})\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}+t_{n}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n+1}$ 7: $n\leftarrow n+1$ 8:until some stopping criterion is met ## 3 General proof recipe In this section we give an intuitive outline of the full proof for convergence of Algorithm 1 before giving formal theorems and proofs in the next section. First we recall the classical FISTA convergence guarantee given by (Chambolle2015, , Thm 3.1); if there exists $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}\in\operatorname{argmin}_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{H}}\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])$, then $t_{N}^{2}\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{N})+\sum^{N-1}_{n=1}\rho_{n}\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})+\tfrac{1}{2}{\left\lVert\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{N}-\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}\right\rVert}^{2}\leq\tfrac{1}{2}{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{0}-\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}\right\rVert}^{2}$ (7) for any FISTA stepsize choice $t_{N}\simeq N$ such that $\rho_{n}\geq 0$. ##### Step 1: Quantifying the stability The first step is to generalise (7) to account for the adapting subsets $\mathds{U}^{n}$. In the notation of Algorithm 1, Theorem 4.1 shows that $t_{N}^{2}\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{N})+\sum_{n=1}^{N-1}\rho_{n}\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})+\tfrac{1}{2}{\left\lVert\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{N}-\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{N}\right\rVert}^{2}\leq\tfrac{1}{2}{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{0}-\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{0}\right\rVert}^{2}+\tfrac{{\left\lVert\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{N}\right\rVert}^{2}-{\left\lVert\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{0}\right\rVert}^{2}}{2}+\sum^{N}_{n=1}t_{n}\operatorname{E}_{0}(\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n})+\left\langle\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n-1},\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n-1}-\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}\right\rangle$ (8) for any $\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}\in\mathds{U}^{n}$. The similarities to (7) are clear. If $\mathds{U}^{n}=\mathds{H}$, then we can choose $\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}=\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}$ and the two estimates agree. These extra terms in (8) quantify the robustness to changing of discretisation. ##### Step 2: Quantifying the scaling properties To show that the extra terms in (8) are small, we need to quantify the approximation properties of $\mathds{U}^{n}$. The idea is that there is a sequence $\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}\in\mathds{U}^{n}$, $n\in\mathds{N}$ such that ${\left\lVert\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}\right\rVert}$ grows slowly and $\operatorname{E}_{0}(\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n})$ decreases quickly. To quantify this balance, we introduce a secondary sequence $n_{0}<n_{1}<\ldots$ and constants $a_{\operatorname{U}},a_{\operatorname{E}}\geq 1$ such that for each $k\in\mathds{N}$ $n\leq n_{k}\implies{\left\lVert\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}\right\rVert}\lesssim a_{\operatorname{U}}^{k},\qquad n\geq n_{k}\implies\operatorname{E}_{0}(\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n})\lesssim a_{\operatorname{E}}^{-k}.$ (9) A canonical example would be $\mathds{U}^{n}=\\{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{H}\operatorname{\;s.t.\;}{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\rVert}\leq a_{\operatorname{U}}^{k}\\}$ for $n\in[n_{k},n_{k+1})$, then $a_{\operatorname{E}}$ reflects the smoothness of $\operatorname{E}_{0}$. The choice of exponential scaling is introduced to improve stability of Algorithm 1. It is natural if we consider the $\mathds{U}^{n}$ to be the subspace of functions discretised on a uniform mesh. If that mesh is sequentially refined, then the resolution of the mesh will be of order $h^{k}$ after $k$ refinements and for some $h<1$. The integer $n_{k}$ is then the time at which the mesh has refined $k$ times. The trade-off between $a_{\operatorname{E}}$ and $a_{\operatorname{U}}$ dictates the final convergence rate of the algorithm. If $a_{\operatorname{U}}>1$, then we cannot guarantee the original $n^{-2}$ rate of convergence. ##### Step 3: Generalising the convergence bound In this step we combine the FISTA stability estimate with the subset approximation guarantees to provide a sharper estimate of stability with respect to the parameters $a_{\operatorname{E}}$ and $a_{\operatorname{U}}$. For example, if for each $k\in\mathds{N}$ $\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}=\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n_{k}}\text{ for each }n=n_{k},n_{k}+1,\ldots,n_{k+1}-1,$ then many terms on the right-hand side of (8) telescope to 0. The result of this is presented in Lemma 3. The key idea is that the stability error in (8) has $K\ll N$ terms, rather than $N$. ##### Step 4: Sufficiently fast growth In Step 3 we develop a convergence bound, now we wish to show that it is only worse than the classical (7) by a constant factor. In particular, it is equivalent to either run Algorithm 1 for $N$ iterations, or the classical FISTA algorithm for $N$ iterations on the fixed subset $\mathds{U}^{N}$. The estimate from (7) provides the estimate $N^{2}\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{N})\lesssim{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{0}-\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{N}\right\rVert}^{2}=O(a_{\operatorname{U}}^{2K})$ for $N\leq n_{K}$. Lemma 4 shows that Algorithm 1 can achieve the same order of approximation, so long as $\mathds{U}^{n}$ grow sufficiently quickly (in particular $n_{k}^{2}\lesssim a_{\operatorname{E}}^{k}a_{\operatorname{U}}^{2k}$). ##### Step 5: Sufficiently slow growth The result of Step 3 is sufficient to prove convergence, but not yet a rate. If the subsets grow too quickly, then the influence of ${\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}\right\rVert}\to\infty$ will slow the rate of convergence. If $n_{k}$ is too large, then we overfit to the discrete problem, but if $n_{k}$ is too small, then FISTA converges slowly. Lemma 5 balances these two factors in an optimal way ($n_{k}^{2}\simeq a_{\operatorname{E}}^{k}a_{\operatorname{U}}^{2k}$) for Algorithm 1 resulting in a convergence rate of $\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{N})\lesssim\frac{a_{\operatorname{U}}^{2K}}{N^{2}}\lesssim\frac{N^{2\kappa}}{N^{2}}$ for all $N\in\mathds{N}$ and $\kappa=\frac{2\log a_{\operatorname{U}}}{\log a_{\operatorname{E}}+2\log a_{\operatorname{U}}}\in[0,1)$. In particular, if the minimum is attained in $\mathds{H}$, then we recover the classical rate with $\kappa=0$. ##### Step 6: Adaptivity Up to this point we have implicitly focused on the case where $\mathds{U}^{n}$ (and $n_{k}$) are chosen a priori. The main challenge for adaptive choice of $\mathds{U}^{n}$ is to guarantee (9) from Step 3 using a posteriori estimates. Combined with the partial telescoping requirement in Step 3, a natural choice is $\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}=\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n_{k}-1}$ for $n\in[n_{k},n_{k+1})$, i.e. the value of $n_{k}$ is chosen to be $n+1$ once the iterate $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}$ is observed. Theorem 4.3 shows that a sufficient condition is $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n_{k}-1}\in\mathds{U}^{n_{k}}\cap\mathds{U}^{n_{k}+1}\cap\ldots\cap\mathds{U}^{n_{k+1}-1},\qquad{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n_{k}-1}\right\rVert}\lesssim a_{\operatorname{U}}^{k},\quad\text{and}\quad\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n_{k}-1})\lesssim a_{\operatorname{E}}^{-k}.$ Convergence is most stable if the approximation spaces $\mathds{U}^{n}$ satisfy a monotone inclusion, breaking the monotonicity requires more care. The only non-trivial property to verify is the energy gap $\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})=\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})-\inf_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{H}}\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])$. Lemma 6 proposes some sufficient conditions to guarantee the same overall rate of convergence as in Step 3, $\min_{n\leq N}\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})\lesssim\frac{N^{2\kappa}}{N^{2}}$ for all $N\in\mathds{N}$, with the same $\kappa\in[0,1)$ from Step 3. The penalty for accelerating the change of discretisation is a potential loss of stability or monotonicity in $\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})$, although this behaviour has not been seen in numerical experiments. ## 4 Proof of convergence In this section we follow the recipe motivated in Section 3 to prove convergence of two variants of Algorithm 1. Each of the main theorems and lemmas will be stated with a sketch proof in this section. The details of the proofs are either trivial or very technical and are therefore placed in Section A to preserve the flow of the argument. ### 4.1 Computing the convergence bound For Step 3 of Section 3 we look to replicate the classical bound of the form in (7) for Algorithm 1. The proofs in this step follow the classical arguments Beck2009 ; Chambolle2015 very closely. Throughout this section we consider a sequence $(\mathds{U}^{n})_{n\in\mathds{N}}$ which generate the iterates $(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})_{n\in\mathds{N}}$ in Algorithm 1 such that $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{n}\in\mathds{U}^{n+1}\subset\mathds{H}\quad\text{where }\mathds{U}^{n}\text{ is a closed, convex subset for all }n\in\mathds{N}.$ (10) #### 4.1.1 Single iterations We first wish to understand a single iteration of Algorithm 1. This is done through the following two lemmas. ###### Lemma 1 (equivalent to (Chambolle2015, , Lemma 3.1)) thm: descent lemma Suppose $\nabla\operatorname{f}$ is 1-Lipschitz, for any $\overline{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}\in\mathds{H}$ define $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\coloneqq\operatorname{argmin}_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{U}^{n}}\tfrac{1}{2}{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]-\overline{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}+\nabla\operatorname{f}(\overline{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]})\right\rVert}^{2}+\operatorname{g}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]).$ Then, for all $\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]\in\mathds{U}^{n}$, we have $\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])+\tfrac{1}{2}{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]-\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]\right\rVert}^{2}\leq\operatorname{E}_{0}(\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2])+\tfrac{1}{2}{\left\lVert\overline{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}-\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]\right\rVert}^{2}.$ The proof is exactly the same as in Chambolle2015 on the subset $\mathds{U}^{n}$. Applying Lemma 1 to the iterates from Algorithm 1 gives a more explicit inequality. ###### Lemma 2 ((Chambolle2015, , (17)), (Beck2009, , Lemma 4.1)) thm: one step FISTA Let $\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}\in\mathds{U}^{n}$ be chosen arbitrarily and $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}$/$\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n}$ be generated by Algorithm 1 for all $n\in\mathds{N}$. For all $n>0$, it holds that $\Copy{thm:eq:onestepFISTA}{t_{n}^{2}(\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})-\operatorname{E}_{0}(\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}))-(t_{n}^{2}-t_{n})(\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n-1})-\operatorname{E}_{0}(\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}))\leq\tfrac{1}{2}\left[{\left\lVert\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n-1}\right\rVert}^{2}-{\left\lVert\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n}\right\rVert}^{2}\right]+\left\langle\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n}-\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n-1},\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}\right\rangle.}$ (11) The proof is given in Theorem A.1 and is a result of the convexity of $\operatorname{E}_{0}$ and $\mathds{U}_{n}$ for a well chosen $\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]$ in Lemma 1. #### 4.1.2 Generic convergence bound Lemma 2 gives us an understanding of a single iteration of Algorithm 1, summing over $n$ then gives our generic convergence bound for any variant of Algorithm 1. ###### Theorem 4.1 (analogous to (Chambolle2015, , Thm 3.2), (Beck2009, , Thm 4.1)) thm: mini FISTA convergence Fix a sequence of subsets $(\mathds{U}^{n})_{n\in\mathds{N}}$ satisfying (10), arbitrary $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{0}\in\mathds{U}^{0}$, and FISTA stepsize choice $(t_{n})_{n\in\mathds{N}}$. Let $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}$ and $\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n}$ be generated by Algorithm 1, then, for any choice of $\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}\in\mathds{U}^{n}$ and $N\in\mathds{N}$ we have $\Copy{thm:eq:miniFISTAconvergence}{t_{N}^{2}\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{N})+\sum_{n=1}^{N-1}\rho_{n}\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})+\frac{{\left\lVert\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{N}-\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{N}\right\rVert}^{2}}{2}\leq\frac{{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{0}-\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{0}\right\rVert}^{2}-{\left\lVert\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{0}\right\rVert}^{2}+{\left\lVert\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{N}\right\rVert}^{2}}{2}+\sum^{N}_{n=1}t_{n}\operatorname{E}_{0}(\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n})+\left\langle\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n-1},\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n-1}-\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}\right\rangle.}$ (12) The proof is given in Theorem A.2. This result is the key approximation for showing convergence of FISTA with changing subsets. In the classical setting, we have $\mathds{U}^{n}=\mathds{H}$, $\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}=\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{0}\in\operatorname{argmin}_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{H}}\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])$ and the extra terms on the right-hand side collapse to 0. If there exists a minimiser $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}\in\operatorname{argmin}_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{H}}\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])$, then the natural choice in (12) is $\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}=\mathsf{\Pi}_{n}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}$ for some projection $\mathsf{\Pi}_{n}\colon\mathds{H}\to\mathds{U}^{n}$, however, there are simple counter-examples which give $\operatorname{E}_{0}(\mathsf{\Pi}_{n}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{})=\infty$ and so this inequality becomes useless. For example, if $\operatorname{f}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])={\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\rVert}_{L^{2}([0,1])}^{2}$, $\operatorname{g}$ is the indicator on the set $\mathds{D}=\\{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in L^{1}([0,1])\operatorname{\;s.t.\;}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0](x)\geq x\\}$, and $\mathsf{\Pi}_{n}$ is the $L^{2}$ projection onto a set of piecewise constant functions, then $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}=x\mapsto x$. On the other hand, suppose one of the pixels of the discretisation is $[x_{0}-h,x_{0}+h]$, then $\mathsf{\Pi}_{n}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}\left(x_{0}+\tfrac{h}{2}\right)=\operatorname{argmin}_{\IfEqCase{5}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k5]\in\mathds{R}}\int_{x_{0}-h}^{x_{0}+h}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}(x)-\IfEqCase{5}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k5])^{2}\mathop{}\\!\mathrm{d}x=\operatorname{argmin}_{\IfEqCase{5}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k5]\in\mathds{R}}\int_{x_{0}-h}^{x_{0}+h}(x-\IfEqCase{5}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k5])^{2}\mathop{}\\!\mathrm{d}x=x_{0}<x_{0}+\tfrac{h}{2}.$ In particular $\mathsf{\Pi}_{n}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}\notin\mathds{D}$ therefore $\operatorname{E}_{0}(\mathsf{\Pi}_{n}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{})=\infty$. The choice $\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}=\operatorname{argmin}_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{U}^{n}}\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])$ is much more robust and allows us to apply Algorithm 1 more broadly. The penalty for this flexibility is a more complicated analysis; each time the subset changes, because $\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n}\in\mathds{U}_{n}$, the system receives a “shock” proportional to ${\left\lVert\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n}\right\rVert}{\left\lVert\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}-\mathsf{\Pi}_{n}\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n+1}\right\rVert}$. ### 4.2 Convergence bound with milestones In standard FISTA, the right-hand side of (12) is a constant. The following lemma minimises the growth of the “constant” as a function of $N$ by partially telescoping the sum on the right-hand side. Before progressing to the content of Step 3, we will first formalise the definition of the constants $a_{\operatorname{U}}$ and $a_{\operatorname{E}}$ introduced in Step 3. ###### Definition 1 Fix $a_{\operatorname{U}},a_{\operatorname{E}}\geq 1$ and a sequence $\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}\in\mathds{H}$. We say that $(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k})_{k\in\mathds{N}}$ is an _$(a_{\operatorname{U}},a_{\operatorname{E}})$ -minimising sequence of $\operatorname{E}$ _ if ${\left\lVert\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}\right\rVert}\lesssim a_{\operatorname{U}}^{k}\qquad\text{and}\qquad\operatorname{E}_{0}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k})\lesssim a_{\operatorname{E}}^{-k}$ for all $k\in\mathds{N}$. In this section we will simply assume that such sequences exist and in Section 5 we will give some more general examples. ###### Lemma 3 thm: mini exponential FISTA convergence Let $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}$, $\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n}$ be generated by Algorithm 1 with $(\mathds{U}^{n})_{n\in\mathds{N}}$ satisfying (10), $(n_{k}\in\mathds{N})_{k\in\mathds{N}}$ be a monotone increasing sequence, and choose $\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}\in\mathds{U}^{n_{k}}\cap\mathds{U}^{n_{k}+1}\cap\ldots\cap\mathds{U}^{n_{k+1}-1}$ for each $k\in\mathds{N}$. If such a sequence exists, then for all $K\in\mathds{N}$, $n_{K}\leq N<n_{K+1}$ we have $\Copy{thm:eq:miniexponentialFISTAconvergence}{t_{N}^{2}\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{N})+\sum_{n=1}^{N-1}\rho_{n}\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})+\frac{{\left\lVert\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{N}-\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{K}\right\rVert}^{2}}{2}\leq C+\frac{{\left\lVert\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{K}\right\rVert}^{2}}{2}+\frac{(N+1)^{2}-n_{K}^{2}}{2}\operatorname{E}_{0}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{K})\\\ +\sum_{k=1}^{K}\frac{n_{k}^{2}-n_{k-1}^{2}}{2}\operatorname{E}_{0}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k-1})+\left\langle\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n_{k}-1},\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k-1}-\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}\right\rangle}$ thm:end: mini exponential FISTA convergence where $C=\frac{{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{0}-\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{0}\right\rVert}^{2}-{\left\lVert\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{0}\right\rVert}^{2}}{2}$. The proof is given in Lemma 11. The introduction of $n_{k}$ has greatly compressed the expression of Theorem 4.1. On the right-hand side, we now only consider $\operatorname{E}_{0}$ evaluated on the sequence $\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}$ and there are $K$ elements to the sum rather than $N$. ### 4.3 Refinement without overfitting The aim of Step 3 is to show that $n$ iterations of Algorithm 1 is no slower (up to a constant factor) than $n$ iterations of classical FISTA on the space $\mathds{U}^{n}$. In other words, we would like to ensure that $\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})=\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})-\min_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{U}^{n}}\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])\lesssim\frac{{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{0}-\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}\right\rVert}^{2}}{n^{2}}$ (13) uniformly for $n\in[n_{k},n_{k+1})$. If this condition is not satisfied, then it indicates that computational effort has been wasted by a poor choice of subsets. This can be interpreted as an overfitting to the discretisation of $\operatorname{E}_{0}\|_{\mathds{U}^{n}}$ rather than the desired function $\operatorname{E}_{0}\|_{\mathds{H}}$. Combining the assumptions given by Definition 1 and the result of Lemma 3, the following lemma proves the convergence of Algorithm 1 provided that the refinement times $n_{k}$ are sufficiently small (i.e. $\mathds{U}^{n}$ grows sufficiently quickly). ###### Lemma 4 thm: sufficiently fast Suppose $\mathds{U}^{n},\ \IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n},\ \IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n}$ and $n_{k}$ satisfy the conditions of Lemma 3 and $(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k})_{k\in\mathds{N}}$ forms an $(a_{\operatorname{U}},a_{\operatorname{E}})$-minimising sequence of $\operatorname{E}$ with $\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}\in\mathds{U}^{n_{k}}\cap\mathds{U}^{n_{k}+1}\cap\ldots\cap\mathds{U}^{n_{k+1}-1}.$ If either: • $a_{\operatorname{U}}>1$ and $n_{k}^{2}\lesssim a_{\operatorname{E}}^{k}a_{\operatorname{U}}^{2k}$, * • or $a_{\operatorname{U}}=1$, $\sum_{k=1}^{\infty}n_{k}^{2}a_{\operatorname{E}}^{-k}<\infty$, and $\sum_{k=1}^{\infty}{\left\lVert\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}-\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k+1}\right\rVert}<\infty,$ then $\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{N})\lesssim\frac{a_{\operatorname{U}}^{2K}}{N^{2}}\qquad\text{for all}\qquad n_{K}\leq N<n_{K+1}.$ The proof is given in Lemma 12. We make two observations of the optimality of Lemma 4: * • The convergence guarantee for $N\in[n_{K},n_{K+1})$ iterations of classical FISTA in the space $\mathds{U}^{N}$ is $\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{N})\lesssim\frac{{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{0}-\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{K}\right\rVert}^{2}}{N^{2}}+\min_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{U}^{N}}\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])\lesssim\frac{a_{\operatorname{U}}^{2K}}{N^{2}}+a_{\operatorname{E}}^{-K}.$ This is equivalent to Lemma 4 after the assumptions on $n_{k}$. * • If $\mathds{H}$ is finite dimensional, then the condition $a_{\operatorname{U}}=1$ is almost trivially satisfied. Norms in finite dimensions are equivalent and any discretisation can be achieved with a finite number of refinements (i.e. the sums over $k$ are finite). ### 4.4 Convergence rate In Lemma 4 we show that $\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})$ converges at a rate depending on $k$ and $n$, so long as $k$ grows sufficiently quickly. On the other hand, as $k$ grows, the rate becomes worse and so we need to also put a lower limit on the growth of $n_{k}$. The following lemma completes Step 3 by computing the global convergence rate of $\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})$ when $k$ grows at the minimum rate which is consistent with Lemma 4. As a special case, note that if $a_{\operatorname{U}}=1$ then Lemma 4 already gives the optimal $O(N^{-2})$ convergence rate. This is in fact a special case of that shown in (Aujol2015, , Prop 3.3). If the minimum is achieved in $\mathds{H}$, then it is not possible to refine “too quickly” and the following lemma is not needed. ###### Lemma 5 thm: sufficiently slow Suppose $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}$ and $n_{k}$ are sequences satisfying $\forall N\in[n_{K},n_{K+1}),\ \operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{N})\lesssim\frac{a_{\operatorname{U}}^{2K}}{N^{2}}\qquad\text{where}\qquad n_{K}^{2}\gtrsim a_{\operatorname{E}}^{K}a_{\operatorname{U}}^{2K},$ then $\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{N})\lesssim\frac{1}{N^{2(1-\kappa)}}\qquad\text{ where }\qquad\kappa=\frac{\log a_{\operatorname{U}}^{2}}{\log a_{\operatorname{E}}+\log a_{\operatorname{U}}^{2}}.$ The proof is given in Lemma 13. #### 4.4.1 FISTA convergence with a priori discretisation We can summarise Lemmas 3 to 5 into a single theorem stating the convergence guarantees when $\mathds{U}^{n}$ and $n_{k}$ are chosen a priori. ###### Theorem 4.2 thm: exponential FISTA convergence Let $(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k})_{k\in\mathds{N}}$ be an $(a_{\operatorname{U}},a_{\operatorname{E}})$-minimising sequence of $\operatorname{E}$ and choose any $\mathds{U}^{n}$ satisfying (10) such that $\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}\in\mathds{U}^{n_{k}}\cap\mathds{U}^{n_{k}+1}\cap\ldots\cap\mathds{U}^{n_{k+1}-1}$ for all $k\in\mathds{N}$. Compute $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}$ and $\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n}$ by Algorithm 1. Suppose that either: * • $a_{\operatorname{U}}>1$ and $n_{k}^{2}\simeq a_{\operatorname{E}}^{k}a_{\operatorname{U}}^{2k}$, or * • $a_{\operatorname{U}}=1$, $\sum_{k=1}^{\infty}n_{k}^{2}a_{\operatorname{E}}^{-k}<\infty$ and $\sum_{k=1}^{\infty}{\left\lVert\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}-\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k+1}\right\rVert}<\infty$, then $\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{N})\lesssim\frac{1}{N^{2(1-\kappa)}}\qquad\text{ where }\qquad\kappa=\frac{\log a_{\operatorname{U}}^{2}}{\log a_{\operatorname{E}}+\log a_{\operatorname{U}}^{2}}\qquad\text{uniformly for }N\in\mathds{N}.$ Analytically, this theorem gives new rates of convergence for FISTA when the minimiser is not achieved in $\mathds{H}$. Indeed for the original algorithm ($\mathds{U}^{n}=\mathds{H}$), if $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{0}=0$ for simplicity and $(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k})_{k\in\mathds{N}}$ is _any_ $(a_{\operatorname{U}},a_{\operatorname{E}})$-minimising sequence of $\operatorname{E}$ exists, the result of Lemma 3 is $\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{N})\leq\inf_{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]\in\mathds{H}}\frac{{\left\lVert\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]\right\rVert}^{2}+N^{2}\operatorname{E}_{0}(\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2])}{2t_{N}^{2}}\leq\min_{k\in\mathds{N}}\frac{{\left\lVert\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}\right\rVert}^{2}+N^{2}\operatorname{E}_{0}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k})}{2t_{N}^{2}}\lesssim\min_{k\in\mathds{N}}\frac{a_{\operatorname{U}}^{2k}+N^{2}a_{\operatorname{E}}^{-k}}{N^{2}}\lesssim N^{-2(1-\kappa)}.$ (14) In this sense, we could say that $\operatorname{E}_{0}$ converges at the rate $N^{-2(1-\kappa)}$ if and only if such a sequence exists. Nothing is lost (or gained) analytically by choosing $\mathds{U}_{n}\subsetneq\mathds{H}$. Numerically, it is easy to implement the strategy of Theorem 4.2 and requires very little knowledge of how to estimate $\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})$. So long as $a_{\operatorname{U}}$ and $a_{\operatorname{E}}$ can be computed analytically, one can choose $\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}$ implicitly to be the discrete minimisers of some “uniform” discretisations (e.g. $\mathds{U}^{n}=\\{{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\rVert}\leq k\\}$ or finite element spaces with uniform mesh) to achieve the stated convergence rate. #### 4.4.2 FISTA convergence with adaptivity There are two properties of the sequence $(\mathds{U}^{n})_{n\in\mathds{N}}$ which we may wish to decide adaptively: the refinement times $n_{k}$ and the discretising spaces $\\{\mathds{U}^{n}\operatorname{\;s.t.\;}n_{k}\leq n<n_{k+1}\\}$. We will refer to these as temporal and spatial adaptivity respectively. Lemma 4 gives a sufficient condition on $n_{k}$ for converging at the rate $O(N^{2(\kappa-1)})$, but it is not necessary. Indeed for $n\leq n_{k}$ we have $\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})\geq\min_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{U}^{n}}\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])=O(a_{\operatorname{E}}^{-k})=O(n^{2(\kappa-1)}),$ which suggests that to converge faster than $n^{2(\kappa-1)}$ requires choosing smaller $n_{k}$. As an example, in Section 7.2 we will see Algorithm 1 can converge at a near-linear rate, although this is not possible without adaptive refinement times. On the other hand, choice of spatial adaptivity has no impact on rate but can impact computational efficiency. It will be permitted to use greedy discretisation techniques so long as it is sufficient to estimate $\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})$ accurately. Theorem 4.2 already allows for spatial adaptivity, so we focus on temporal adaptivity. Lemma 4 suggests that a good refinement time strategy is to choose $n_{k}$ to be the minimal integer such that $\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n_{k}-1})\lesssim a_{\operatorname{E}}^{-k}$. However, the value of $\operatorname{E}_{0}$ may be hard to estimate and so we retain a “backstop” condition which guarantees that convergence is no slower than the rate given by Theorem 4.2. In the non- classical case of $a_{\operatorname{U}}>1$, we provide the following theorem. ###### Theorem 4.3 thm: stronger exponential FISTA convergence Let $(\mathds{U}^{n}\subset\mathds{H})_{n\in\mathds{N}}$ be a sequence of subsets satisfying (10), compute $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}$ and $\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n}$ by Algorithm 1. Suppose that there exists a monotone increasing sequence $n_{k}\in\mathds{N}$ such that $\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}\coloneqq\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n_{k}-1}\in\mathds{U}^{n_{k}}\cap\mathds{U}^{n_{k}+1}\cap\ldots\cap\mathds{U}^{n_{k+1}-1}$ for all $k\in\mathds{N}$. If $(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k})_{k\in\mathds{N}}$ is an $(a_{\operatorname{U}},a_{\operatorname{E}})$-minimising sequence of $\operatorname{E}$ with $a_{\operatorname{U}}>1$ and $n_{k}^{2}\lesssim a_{\operatorname{E}}^{k}a_{\operatorname{U}}^{2k}$, then $\min_{n\leq N}\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})=\min_{n\leq N}\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})-\inf_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{H}}\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])\lesssim\frac{1}{N^{2(1-\kappa)}}\qquad\text{ where }\qquad\kappa=\frac{\log a_{\operatorname{U}}^{2}}{\log a_{\operatorname{E}}+\log a_{\operatorname{U}}^{2}}$ uniformly for $N\in\mathds{N}$. The proof is given in Theorem A.3. If we directly compare Theorems 4.2 and 4.3, both are a direct result of Lemma 4 assuming a specific choice of $n_{k}$ or $\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}$ respectively. We note that the convergence rate is the same in both theorems but the price for better adaptivity (i.e. only an upper bound on $n_{k}$) is a slightly weaker stability guarantee (now convergence of $\min_{n\leq N}\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})$). In Theorem 4.2, as in the original FISTA algorithm, the sequence $\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})$ is not monotone but the magnitude of oscillation is guaranteed to decay in time. This behaviour is lost in Theorem 4.3. Although we do not prove it here, it can be shown that the stronger condition $\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}\in\mathds{U}^{n_{k}}\cap\mathds{U}^{n_{k}+1}\cap\ldots\cap\mathds{U}^{n_{k+1}-1}\cap\ldots\cap\mathds{U}^{N}$ (15) is sufficient to restore the stronger last-iterate guarantee on $\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{N})$. Again, monotonicity of $\mathds{U}^{n}$ corresponds with improved stability of Algorithm 1. To enable a more practical implementation of Theorem 4.3, the following lemma describes several refinement strategies which provide sufficient condition for $\operatorname{E}_{0}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k})\lesssim a_{\operatorname{E}}^{-k}$. ###### Lemma 6 thm: practical refinement criteria Let $(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k})_{k\in\mathds{N}}$ be a sequence in $\mathds{H}$ with ${\left\lVert\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}\right\rVert}\lesssim a_{\operatorname{U}}^{k}$. Suppose $\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}\in\widetilde{\mathds{U}}^{k}\coloneqq\mathds{U}^{n_{k}}$ and denote $\operatorname{E}_{0}(\widetilde{\mathds{U}}^{k})\coloneqq\inf_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\widetilde{\mathds{U}}^{k}}\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])$. Any of the following conditions are sufficient to show that $\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}$ is an $(a_{\operatorname{U}},a_{\operatorname{E}})$-minimising sequence of $\operatorname{E}$: 1. 1. Small continuous gap refinement: $\operatorname{E}_{0}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k})\leq\IfEqCase{1}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k1]a_{\operatorname{E}}^{-k}$ for all $k\in\mathds{N}$, some $\IfEqCase{1}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k1]>0$. 2. 2. Small discrete gap refinement: $\operatorname{E}_{0}(\widetilde{\mathds{U}}^{k})\leq\IfEqCase{1}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k1]a_{\operatorname{E}}^{-k}$ and $\operatorname{E}_{0}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k})-\operatorname{E}_{0}(\widetilde{\mathds{U}}^{k-1})\leq\IfEqCase{1}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k1]a_{\operatorname{E}}^{-k}$ for all $k>0$, some $\IfEqCase{1}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k1]>0$. Otherwise, suppose there exists a Banach space $(\mathds{U},{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\cdot\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|})$ which contains each $\widetilde{\mathds{U}}^{k}$, $\sup_{k\in\mathds{N}}{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}<\infty$, and the sublevel sets of $\operatorname{E}$ are ${\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\cdot\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}$-bounded. With the subdifferential $\partial\operatorname{E}\colon\mathds{U}\rightrightarrows\mathds{U}^{}$, it is also sufficient if either: 1. 3. Small continuous gradient refinement: $\sup_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{U}}\inf_{\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]\in\partial\operatorname{E}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k})}\frac{\|\left\langle\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1],\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\rangle\|}{{\left\|\kern-0.75346pt\left\|\kern-0.75346pt\left\|\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\|\kern-0.75346pt\right\|\kern-0.75346pt\right\|}}\leq\IfEqCase{1}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k1]a_{\operatorname{E}}^{-k}$ for all $k\in\mathds{N}$, some $\IfEqCase{1}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k1]>0$. 2. 4. Small discrete gradient refinement: $\operatorname{E}_{0}(\widetilde{\mathds{U}}^{k})\leq\IfEqCase{1}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k1]a_{\operatorname{E}}^{-k}$ and $\sup_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0],\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}\in\widetilde{\mathds{U}}^{k}}\inf_{\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]\in\mathds{V}^{k}}\frac{\|\left\langle v,\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]-\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}\right\rangle\|}{{\left\|\kern-0.75346pt\left\|\kern-0.75346pt\left\|\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]-\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}\right\|\kern-0.75346pt\right\|\kern-0.75346pt\right\|}}\leq\IfEqCase{1}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k1]a_{\operatorname{E}}^{-k}$ for all $k\in\mathds{N}$, some $\IfEqCase{1}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k1]>0$, where $\mathds{V}^{k}\coloneqq\partial(\operatorname{E}\|_{\widetilde{\mathds{U}}^{k}})(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k})$. The proof is given in Lemma 14. The refinement criteria described by Lemma 6 can be split into two groups. Cases (1) and (3) justify that any choice of $\mathds{U}^{n_{k}}$ satisfies the required conditions, so long as $\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}\in\mathds{U}^{n_{k}}$. In cases (2) and (4), $\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}$ is sufficient to choose the refinement time $n_{k}$, but an apriori bound is required on $\operatorname{E}_{0}(\widetilde{\mathds{U}}^{k})$. In these cases one could, for example, choose $\widetilde{\mathds{U}}^{k}$ to be a uniform discretisation with a priori estimates. Another splitting of the criteria is into gap and gradient computations. Typically, gradient norms (in (4) and (5)) should be easier to estimate than function gaps because they only require local knowledge rather than global, i.e. $\partial\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})$ rather than an estimate of $\inf_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{H}}\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])$. Implicitly, the global information comes from an extra condition on $\operatorname{E}$ to assert that sublevel sets are bounded. ## 5 General examples We consider the main use of Algorithm 1 to be where there exists a Banach space $(\mathds{U},{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\cdot\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|})$ such that $\mathds{U}\supset\\{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{H}\operatorname{\;s.t.\;}\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])<\infty\\}$ and $\inf_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{H}}\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])=\min_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{U}}\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])=\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{})$ for some $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}\in\mathds{U}$. The cases where $\mathds{H}$ has finite dimension or is separable are more straightforward; if the total number of refinements is finite (i.e. $\mathds{U}^{n}=\mathds{U}^{N}$ for all $n\geq N$, some $N\in\mathds{N}$), then $a_{\operatorname{U}}=1$. This holds for most finite dimensional problems as well as the countable example discussed in detail in Section 6. In this section we give explicit computations of $a_{\operatorname{U}}$ and $a_{\operatorname{E}}$ in the setting where $\mathds{H}=L^{2}(\Omega)$ for some domain $\Omega\subset\mathds{R}^{d}$ and the subsets $\mathds{U}^{n}$ will be finite dimensional finite-element–like spaces, as defined below. ###### Definition 2 Suppose ${\left\lVert\cdot\right\rVert}_{q}\lesssim{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\cdot\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}$ (i.e. $\mathds{U}\subset L^{q}(\Omega)$) for some $q\in[1,\infty]$ and connected, bounded, measurable domain $\Omega\subset\mathds{R}^{d}$. We say that a collection $\mathds{M}$ is a _mesh_ if $\bigcup_{\omega\in\mathds{M}}\omega\supset\Omega\qquad\text{and}\qquad\|\omega\cap\omega^{\prime}\|=0\qquad\text{for all $\omega,\omega^{\prime}\in\mathds{M},\ \omega\neq\omega^{\prime}$.}$ Furthermore, we say a sequence of meshes $(\mathds{M}^{k})_{k\in\mathds{N}}$ is _consistent_ if there exists $\omega_{0}\subset\Omega$ such that $\forall\omega\in\mathds{M}^{k}\quad\exists(\IfEqCase{0}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k0]_{\omega},\vec{\IfEqCase{1}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k1]}_{\omega})\in\mathds{R}^{d\times d}\times\mathds{R}^{d}\quad\text{such that}\quad\vec{x}\in\omega_{0}\iff\IfEqCase{0}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k0]_{\omega}\vec{x}+\vec{\IfEqCase{1}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k1]}_{\omega}\in\omega.$ Fix $h\in(0,1)$, linear subspaces $\widetilde{\mathds{U}}^{k}\subset\mathds{H}$, and consistent meshes $\mathds{M}^{k}$. We say that the sequence $(\widetilde{\mathds{U}}^{k})_{k\in\mathds{N}}$ is an _$h$ -refining sequence of finite element spaces_ if there exists $c_{\IfEqCase{0}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k0]}>0$ such that: $\forall(\widetilde{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]},\omega)\in\widetilde{\mathds{U}}^{k}\times\mathds{M}^{k},\quad\operatorname{det}(\IfEqCase{0}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k0]_{\omega})\geq c_{\IfEqCase{0}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k0]}h^{kd}\quad\text{and}\quad\exists\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\widetilde{\mathds{U}}^{0}\quad\text{such that}\quad\forall\vec{x}\in\omega_{0},\ \IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0](\vec{x})=\widetilde{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}(\IfEqCase{0}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k0]_{\omega}\vec{x}+\vec{\IfEqCase{1}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k1]}_{\omega}).$ We say that $(\widetilde{\mathds{U}}^{k})_{k\in\mathds{N}}$ is _of order $p$_ if for any $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}\in\operatorname{argmin}_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{U}}\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])$ there exists a sequence $(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k})_{k\in\mathds{N}}$ such that $\forall k\in\mathds{N},\qquad\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}\in\widetilde{\mathds{U}}^{k}\quad\text{and}\quad{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}-\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}\lesssim_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}}h^{kp}.$ (16) We allow the implicit constant to have any dependence on $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}$ so long as it is finite. For example, in the case of Sobolev spaces we would expect an inequality of the form ${\left\lVert\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}-\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}\right\rVert}_{W^{0,2}}\lesssim h^{kp}{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}\right\rVert}_{W^{p,2}}$ Strang1972 . ###### Remark 1 To clarify this definition with an example, suppose we wish to approximate $L^{q}(\Omega)$ with piecewise linear finite elements with a triangulated mesh. Then, $\omega_{0}\subset\Omega$ is a single triangle of diameter $O(h)$ and all meshes $\mathds{M}^{k}$ must be triangulations of $\Omega$ with cell volumes scaling no faster than $O(h^{kd})$. The function $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]$ from the $h$-refining property is an arbitrary linear element, so that each $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\widetilde{\mathds{U}}^{k}$ is linear on each $\omega\in\mathds{M}^{k}$, which leads to an order $p=2$ if $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}\in W^{1,2}(\Omega)$. We note that any piecewise polynomial finite element (or spline) space can be used to form a $h$-refining sequence of subspaces. Wavelets with a compactly supported basis behave like a multi-resolution finite element space as there is always overlap in the supports of basis vectors. Similarly, a Fourier basis does satisfy the scaling properties, but each basis vector has global support. Both of these exceptions are important and could be accounted for with further analysis but we focus on the more standard finite element case. In order to align these discretisation properties with the assumptions of Theorems 4.2 and 4.3, we make the following observation. ###### Lemma 7 Fix $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}\in\operatorname{argmin}_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{U}}\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])$ and $p^{\prime},q^{\prime}>0$. If a sequence $\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}\in\mathds{H}$ satisfies ${\left\lVert\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}\right\rVert}\lesssim h^{-kq^{\prime}}\quad\text{and}\quad\operatorname{E}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k})-\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{})\lesssim h^{kp^{\prime}},$ then $(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k})_{k\in\mathds{N}}$ is an $(a_{\operatorname{U}},a_{\operatorname{E}})$-minimising sequence of $\operatorname{E}$ for $a_{\operatorname{U}}=h^{-q^{\prime}}$ and $a_{\operatorname{E}}=h^{-p^{\prime}}$. This is precisely rewriting the statement of Definition 1 into terms of resolution $h$. The following theorem links $p$ and $q$ from Definition 2 with $p^{\prime}$ and $q^{\prime}$ from Lemma 7. ###### Theorem 5.1 Suppose $\mathds{H}=L^{2}(\Omega)$ for some connected, bounded domain $\Omega\subset\mathds{R}^{d}$ and ${\left\lVert\cdot\right\rVert}_{q}\lesssim{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\cdot\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}$ for some $q\in[1,\infty]$. For $p\geq 0$ and $h\in(0,1)$, if $(\widetilde{\mathds{U}}^{k})_{k\in\mathds{N}}$ is an $h$-refining sequence of finite element spaces of order $p$, then $(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k})_{k\in\mathds{N}}$ is an $(a_{\operatorname{U}},a_{\operatorname{E}})$-minimising sequence of $\operatorname{E}$ for $\displaystyle a_{\operatorname{U}}$ $\displaystyle\leq\begin{cases}1&\text{if }q\geq 2,\\\ \sqrt{h^{-d}}&q<2\text{ and }\sup_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\widetilde{\mathds{U}}^{0}}\frac{{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\rVert}_{L^{\infty}(\omega_{0})}}{{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\rVert}_{L^{2}(\omega_{0})}}<\infty\end{cases},$ $\displaystyle a_{\operatorname{E}}$ $\displaystyle\geq\begin{cases}h^{-2p}&\text{if }\nabla\operatorname{E}\text{ is ${\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\cdot\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}$-Lipschitz at }\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{},\\\ h^{-p}\qquad&\text{if }\operatorname{E}\text{ is ${\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\cdot\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}$-Lipschitz at }\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{},\\\ 1&\text{otherwise.}\end{cases}$ The proof of this theorem is in Appendix B. Note that $\sup_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\widetilde{\mathds{U}}^{0}}{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\rVert}_{L^{\infty}(\omega_{0})}{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\rVert}_{L^{2}(\omega_{0})}^{-1}$ is finite whenever $\widetilde{\mathds{U}}^{0}\subset L^{\infty}(\Omega)$ is finite dimensional, so this is not a very strong assumption. The main take- home for this theorem is that the computation of $a_{\operatorname{U}}$ and $a_{\operatorname{E}}$ is typically very simple and clear given a particular choice of ${\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\cdot\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}$ and $\operatorname{E}$. We also briefly remark that the Lipschitz constants in this lemma do not need to be valid globally, only on the sequence $\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}$. The same result holds under a local-Lipschitz assumption, for example on the ball of radius $\sup_{k\in\mathds{N}}{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}$ which is finite whenever $p\geq 0$. ## 6 L1 penalised reconstruction The canonical example for FISTA is the LASSO problem with a quadratic data fidelity and L1 regularisation. In this section we develop the necessary analytical tools for the variant with general smooth fidelity term which will be used for numerical results in Section 7. We consider three forms which will be referred to as the continuous, countable, and discrete problem depending on whether the space $\mathds{U}$ is $\mathcal{M}([0,1]^{d})$, $\ell^{1}(\mathds{R})$, or $\mathds{R}^{M}$ respectively. We choose $\mathds{H}$ to be $L^{2}([0,1]^{d}),\ \ell^{2}(\mathds{R}),$ or $\mathds{R}^{M}$ correspondingly. Let $\mathsf{A}\colon\mathds{U}\cap\mathds{H}\to\mathds{R}^{m}$ be a linear operator represented by the kernels $\psi_{j}\in\mathds{H}$ such that $\forall\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{U}\cap\mathds{H},\ j=1,\ldots,m,\qquad(\mathsf{A}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])_{j}=\left\langle\psi_{j},\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\rangle.$ (17) In the continuous case we will assume the additional smoothness $\psi_{j}\in C^{1}([0,1]^{d})$. In Section 6.5 we will formally define and estimate several operator semi-norms for $\mathsf{A}$ of this form, for example Lemma 8 confirms that $\mathsf{A}$ is continuous on $\mathds{H}$ (without loss of generality ${\left\lVert\mathsf{A}\right\rVert}\leq 1$). In each case, the energy we consider is written as $\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])=\operatorname{f}(\mathsf{A}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]-\eta)+\mu{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}$ (18) for some $\mu>0$ where ${\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\cdot\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}={\left\lVert\cdot\right\rVert}_{1}$. We assume $\operatorname{f}\in C^{1}(\mathds{R}^{m})$ is convex, bounded from below, and $\nabla\operatorname{f}$ is 1-Lipschitz. Let $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}\in\operatorname{argmin}_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{U}}\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])$, which is non-empty so long as $\psi_{j}\in C([0,1]^{d})$, see the proof of (Bredies2013, , Prop. 3.1) when $\operatorname{f}$ is quadratic. The aim of this section is to develop all of the necessary tools for implementing Algorithm 1 on the energy (18) using the convergence guarantees of either Theorem 4.2 or Theorem 4.3. This includes computing the rates $a_{\operatorname{U}}$ and $a_{\operatorname{E}}$, estimating the continuous gap $\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})$, and developing an efficient refinement choice for $\mathds{U}^{n}$. Below we will just describe the form of $\widetilde{\mathds{U}}^{k}$ under the assumption that $\mathds{U}^{n}\subset\widetilde{\mathds{U}}^{k}$ is chosen adaptively for $n=n_{k-1}+1,\ldots,n_{k}$. The index $k$ refers to the scale or resolution and $n$ refers to the iteration number of the reconstruction algorithm. ### 6.1 Continuous case We start by estimating rates in the case $\mathds{U}=\mathcal{M}(\Omega)$ where $\Omega=[0,1]^{d}$. In this case we choose $\widetilde{\mathds{U}}^{k}$ to be the span of all piecewise constant functions on a mesh of squares with maximum side length $2^{-k}$ (i.e. $h=\tfrac{1}{2}$) and $\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}\coloneqq\sum_{\omega\in\mathds{M}^{k}}\frac{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}(\omega)}{\|\omega\|}\mathds{1}_{\omega}\quad\text{where}\quad\mathds{1}_{\omega}(\vec{x})=\begin{cases}1&\vec{x}\in\omega\\\ 0&\text{else}\end{cases}.$ By construction $\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}\in\widetilde{\mathds{U}}^{k}$, however note that for any $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in L^{1}(\Omega)$ and Dirac mass $\delta$ supported in $(0,1)^{d}$, ${\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]-\delta\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}=\sup_{\varphi\in C(\Omega),{\left\lVert\varphi\right\rVert}_{L^{\infty}}\leq 1}\left\langle\varphi,\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]-\delta\right\rangle={\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}+{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\delta\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}\geq 1=h^{0}.$ (19) Because of this, application of Theorem 5.1 with $p=0$ gives $a_{\operatorname{U}}=2^{\frac{d}{2}}$ but only $a_{\operatorname{E}}\geq 1$. To improve our estimate of $a_{\operatorname{E}}$ requires additional assumptions on $\mathsf{A}$. Note that ${\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}=\sum_{\omega\in\mathds{M}^{k}}\|\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}(\omega)\|\leq{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}$, therefore we have $\displaystyle\operatorname{E}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k})-\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{})$ $\displaystyle=\operatorname{f}(\mathsf{A}\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}-\eta)-\operatorname{f}(\mathsf{A}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}-\eta)+\mu\left({\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}-{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}\right)$ (20) $\displaystyle\leq\nabla\operatorname{f}(\mathsf{A}\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}-\eta)\vbox{\hbox{\leavevmode\resizebox{6.0pt}{}{$\cdot$}}}\mathsf{A}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}-\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{})$ (21) $\displaystyle\leq\left[{\left\lVert\nabla\operatorname{f}(\mathsf{A}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}-\eta)\right\rVert}_{\ell^{2}}+{\left\lVert\mathsf{A}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}-\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{})\right\rVert}_{\ell^{2}}\right]{\left\lVert\mathsf{A}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}-\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{})\right\rVert}_{\ell^{2}}.$ (22) as $\operatorname{f}$ is convex with 1-Lipschitz gradient. Clearly ${\left\lVert\nabla\operatorname{f}(\mathsf{A}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}-\eta)\right\rVert}_{\ell^{2}}$ is a constant. For the other term, for all $\vec{r}\in\mathds{R}^{m}$ denote $\varphi\coloneqq\mathsf{A}^{}\vec{r}$, then note that $\vec{r}\vbox{\hbox{\leavevmode\resizebox{6.0pt}{}{$\cdot$}}}\mathsf{A}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}-\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{})=\left\langle\varphi,\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}-\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}\right\rangle=\sum_{\omega\in\mathds{M}^{k}}\int_{\omega}\varphi(\vec{x})\mathop{}\\!\mathrm{d}[\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}-\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}]=\sum_{\omega\in\mathds{M}^{k}}\|\omega\|^{-1}\iint_{\omega^{2}}[\varphi(\vec{x})-\varphi(\vec{y})]\mathop{}\\!\mathrm{d}\vec{x}\mathop{}\\!\mathrm{d}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}(\vec{y}).$ (23) With the pointwise bound $\|\varphi(\vec{x})-\varphi(\vec{y})\|\leq\operatorname{diam}(\omega){\left\lVert\nabla\varphi\right\rVert}_{L^{\infty}}=\sqrt{d}2^{-k}{\left\lVert\nabla[\mathsf{A}^{}\vec{r}]\right\rVert}_{L^{\infty}}$, we deduce the estimate ${\left\lVert\mathsf{A}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}-\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{})\right\rVert}_{\ell^{2}}=\sup_{\vec{r}\in\mathds{R}^{m}}{\left\lVert\vec{r}\right\rVert}_{\ell^{2}}^{-1}\left\langle\mathsf{A}^{}\vec{r},\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}-\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}\right\rangle\leq\sqrt{d}2^{-k}{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}\sup_{\vec{r}\in\mathds{R}^{m}}{\left\lVert\vec{r}\right\rVert}_{\ell^{2}}^{-1}{\left\lVert\nabla[\mathsf{A}^{}\vec{r}]\right\rVert}_{L^{\infty}}.$ (24) In Lemma 9 we will show that this last term, which we denote the semi-norm $\|\mathsf{A}^{}\|_{\ell^{2}\to C^{1}}$, is bounded by $\sqrt{m}\max_{j\in[m]}{\left\lVert\nabla\Psi_{j}\right\rVert}_{\infty}$. We conclude that $\operatorname{E}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k})-\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{})\lesssim 2^{-k}$. In particular, this computation confirms two things. Firstly that the scaling constant is $a_{\operatorname{E}}=2$, and secondly that the required smoothness to achieve a good rate with Algorithm 1 is that $\mathsf{A}^{}\colon\mathds{R}^{m}\to C^{1}(\Omega)$ is a bounded operator. This accounts for using the weaker topology of $\mathcal{M}(\Omega)$ rather than $L^{1}(\Omega)$. Inserting the computed rates into Theorem 4.2 or Theorem 4.3 gives the guaranteed convergence rate $\kappa=\frac{\log a_{\operatorname{U}}^{2}}{\log a_{\operatorname{E}}+\log a_{\operatorname{U}}^{2}}=\frac{d}{1+d}\quad\implies\quad\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})-\inf_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{H}}\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])\lesssim n^{-2(1-\kappa)}=n^{-\frac{2}{1+d}}.$ (25) This rate can be used to infer the required resolution at each iteration, in particular on iteration $n$ with $n^{2}\simeq(a_{\operatorname{E}}a_{\operatorname{U}}^{2})^{k}$ we expect the resolution to be $2^{-k}=\left(a_{\operatorname{E}}a_{\operatorname{U}}^{2}\right)^{\frac{k}{1+d}}\simeq n^{-\frac{2}{1+d}}.$ (26) ### 6.2 Countable and discrete case We now extend the rate computations to the case when $\mathds{U}=\ell^{1}(\mathds{R})$, or a finite dimensional subspace. The key fact here is that, even when $\mathds{U}$ is infinite dimensional, it is known (e.g. (Unser2016, , Thm 6) and (Boyer2019, , Cor 3.8)) that there exists $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}\in\operatorname{argmin}_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{U}}\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])$ with at most $m$ non-zeros. If this is the case, then $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}\in\ell^{2}(\mathds{R})$, indeed ${\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}\right\rVert}_{\ell^{2}}\leq\sqrt{m}{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}\right\rVert}_{\ell^{1}}$. This makes the estimates of $a_{\operatorname{E}}/a_{\operatorname{U}}$ much simpler than in the continuous case as we can stay in the finite-dimensional Hilbert-space setting. For countable dimensions we consider discretisation subspaces of the form $\widetilde{\mathds{U}}^{k}=\\{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\ell^{1}(\mathds{R})\operatorname{\;s.t.\;}i\notin J_{k}\implies\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{i}=0\\}$ for some sets $J_{k}\subset\mathds{N}$, i.e. infinite vectors with finitely many non-zeros. The key change in analysis from the continuous case is ${\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}\right\rVert}<\infty$, so $a_{\operatorname{U}}=1$ and the expected rate of $n^{-2}$, independent of $a_{\operatorname{E}}$ or any additional properties of $\mathsf{A}$. The number of refinements will also be finite, therefore $n_{k}=\infty$ for some $k$, the remaining conditions of Theorems 4.2 and 4.3 hold trivially. ### 6.3 Refinement metrics Lemma 6 shows that adaptive refinement can be performed based on estimates of the function gap or the subdifferential. In this subsection we provide estimates for the forth case of Lemma 6 which can be easily computed. In this case we consider $\partial\operatorname{E}\colon\mathds{H}\rightrightarrows\mathds{H}$ so that subdifferentials are well behaved, for example for explicit computation assuming validity of the chain/sum rules for differentiation. #### 6.3.1 Bounds for discretised functionals We start by computing estimates for discretised energies. This covers the cases when either the continuous/countable energy is projected onto $\mathds{U}^{n}$, or $\mathds{U}$ is finite dimensional. For notation we will use the continuous case, to recover the other cases just replace continuous indexing with discrete (i.e. $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0](\vec{x})\leadsto\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{i}$). Let $\mathsf{\Pi}_{n}\colon\mathds{H}\to\mathds{U}^{n}$ denote the orthogonal projection. We consider the discretised function $\operatorname{E}\|_{\mathds{U}^{n}}\colon\mathds{U}^{n}\to\mathds{R}$ and its subdifferential $\partial_{n}\operatorname{E}(\cdot)=\mathsf{\Pi}_{n}\partial\operatorname{E}(\cdot)$ on $\mathds{U}^{n}$. In our case, the behaviour of $\operatorname{E}\|_{\mathds{U}^{n}}$ is equivalent to replacing $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]$ with $\mathsf{\Pi}_{n}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]$, and $\mathsf{A}^{}$ with $\mathsf{\Pi}_{n}\mathsf{A}^{}$. ##### Discrete gradient We can use $\mathsf{\Pi}_{n}$ to compute the discrete subdifferential at $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}\in\mathds{U}^{n}$: $\displaystyle\partial_{n}\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})(\vec{x})$ $\displaystyle=[\mathsf{\Pi}_{n}\mathsf{A}^{}\nabla\operatorname{f}(\mathsf{A}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}-\eta)](\vec{x})+\begin{cases}\\{+\mu\\}&\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}(\vec{x})>0\\\ [-\mu,\mu]&\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}(\vec{x})=0\\\ \\{-\mu\\}&\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}(\vec{x})<0\end{cases}$ (27) $\displaystyle\eqqcolon[\mathsf{\Pi}_{n}\mathsf{A}^{}\nabla\operatorname{f}(\mathsf{A}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}-\eta)](\vec{x})+\mu\mathsf{\Pi}_{n}\operatorname{sign}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}(\vec{x}))$ (28) where we define $s+\mu[-1,1]=[s-\mu,s+\mu]$ for all $s\in\mathds{R}$, $\mu\geq 0$. As ${\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\cdot\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}={\left\lVert\cdot\right\rVert}_{1}$, the natural metric for $\partial_{n}\operatorname{E}$ is ${\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\cdot\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}_{}={\left\lVert\cdot\right\rVert}_{\infty}$ which we can estimate $\displaystyle{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\partial_{n}\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}_{}$ $\displaystyle=\max_{\vec{x}\in\Omega}\min_{\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]}\left\\{\|\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]\|\operatorname{\;s.t.\;}\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]\in\mathsf{\Pi}_{n}\mathsf{A}^{}\nabla\operatorname{f}(\mathsf{A}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}-\eta)(\vec{x})+\mu\mathsf{\Pi}_{n}\operatorname{sign}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}(\vec{x}))\right\\}$ (29) $\displaystyle=\max_{\vec{x}\in\Omega}\begin{cases}\|[\mathsf{\Pi}_{n}\mathsf{A}^{}\nabla\operatorname{f}(\mathsf{A}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}-\eta)(\vec{x})+\mu\|&\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}(\vec{x})>0\\\ \|[\mathsf{\Pi}_{n}\mathsf{A}^{}\nabla\operatorname{f}(\mathsf{A}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}-\eta)(\vec{x})-\mu\|&\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}(\vec{x})<0\\\ \max\left(\|\mathsf{\Pi}_{n}\mathsf{A}^{}\nabla\operatorname{f}(\mathsf{A}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}-\eta)(\vec{x})\|-\mu,0\right)&\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}(\vec{x})=0\end{cases}$ (30) which can be used directly in Lemma 6. ##### Discrete gap We now move on to the discrete gap, $\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})-\min_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{U}^{n}}\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])$. This can be computed with a dual representation (e.g. Duval2017a ), $\displaystyle\min_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{U}^{n}}\operatorname{f}(\mathsf{A}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]-\eta)+\mu{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}$ $\displaystyle=\min_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{H}}\max_{\vec{\varphi}\in\mathds{R}^{m}}(\mathsf{A}\mathsf{\Pi}_{n}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]-\eta)\vbox{\hbox{\leavevmode\resizebox{6.0pt}{}{$\cdot$}}}\vec{\varphi}+\mu{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\mathsf{\Pi}_{n}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}-\operatorname{f}^{}(\vec{\varphi})$ (31) $\displaystyle=\max_{\vec{\varphi}\in\mathds{R}^{m}}\min_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{H}}(\mathsf{A}\mathsf{\Pi}_{n}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]-\eta)\vbox{\hbox{\leavevmode\resizebox{6.0pt}{}{$\cdot$}}}\vec{\varphi}+\mu{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\mathsf{\Pi}_{n}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}-\operatorname{f}^{}(\vec{\varphi})$ (32) $\displaystyle=\max_{\vec{\varphi}\in\mathds{R}^{m}}\begin{cases}-\eta\vbox{\hbox{\leavevmode\resizebox{6.0pt}{}{$\cdot$}}}\vec{\varphi}-\operatorname{f}^{}(\vec{\varphi})&\qquad{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\mathsf{\Pi}_{n}\mathsf{A}^{}\vec{\varphi}\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}_{}\leq\mu\\\ -\infty&\qquad\text{else}\end{cases}$ (33) $\displaystyle=-\min_{\vec{\varphi}\in\mathds{R}^{m}}\underbrace{\operatorname{f}^{}(\vec{\varphi})+\eta\vbox{\hbox{\leavevmode\resizebox{6.0pt}{}{$\cdot$}}}\vec{\varphi}}_{\eqqcolon\operatorname{E}^{\dagger}(\vec{\varphi})}+\chi({\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\mathsf{\Pi}_{n}\mathsf{A}^{}\vec{\varphi}\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}_{}\leq\mu).$ (34) In particular, $\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])-\min_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{U}^{n}}\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])=\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])+\min_{\vec{\varphi}\in\mathds{R}^{m}\operatorname{\;s.t.\;}{\left\|\kern-0.75346pt\left\|\kern-0.75346pt\left\|\mathsf{\Pi}_{n}\mathsf{A}^{}\vec{\varphi}\right\|\kern-0.75346pt\right\|\kern-0.75346pt\right\|}_{}\leq\mu}\operatorname{E}^{\dagger}(\vec{\varphi})\leq\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])+\operatorname{E}^{\dagger}(\vec{\varphi})$ (35) for any feasible $\vec{\varphi}\in\mathds{R}^{m}$. We further derive the criticality condition, if $(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{},\vec{\varphi}^{})$ is a saddle point, then $\mathsf{A}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}-\eta\in\partial\operatorname{f}^{}(\vec{\varphi}^{}),\qquad\text{or equivalently}\qquad\vec{\varphi}^{}=\nabla\operatorname{f}(\mathsf{A}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}-\eta).$ (36) We remark briefly that $\operatorname{E}^{\dagger}$ should be thought of as the dual of $\operatorname{E}$ but without the constraint. We choose to omit it here to highlight that it is only the constraint which changes between the discrete and continuous cases; the value of $\operatorname{E}^{\dagger}$ will remain the same. Given $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}\in\mathds{U}^{n}$, the optimality condition motivates a simple rule for choosing $\vec{\varphi}$: $\vec{\varphi}_{n}\coloneqq\nabla\operatorname{f}(\mathsf{A}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}-\eta),\qquad\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])-\min_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{\prime}\in\mathds{U}^{n}}\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{\prime})\leq\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])+\operatorname{E}^{\dagger}(\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]\vec{\varphi}_{n})$ (37) for some $0\leq\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]\leq\frac{\mu}{{\left\|\kern-0.75346pt\left\|\kern-0.75346pt\left\|\mathsf{\Pi}_{n}\mathsf{A}^{}\vec{\varphi}_{n}\right\|\kern-0.75346pt\right\|\kern-0.75346pt\right\|}_{}}$. In the case $\operatorname{f}(\cdot)=\frac{1}{2}{\left\lVert\cdot\right\rVert}_{\ell^{2}}^{2}$, one can use the optimal choice $\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]=\max\left(0,\min\left(\frac{-\eta\vbox{\hbox{\leavevmode\resizebox{6.0pt}{}{$\cdot$}}}\vec{\varphi}_{n}}{{\left\lVert\vec{\varphi}_{n}\right\rVert}_{\ell^{2}}^{2}},\frac{\mu}{{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\mathsf{\Pi}_{n}\mathsf{A}^{}\vec{\varphi}_{n}\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}_{}}\right)\right).$ (38) To apply Algorithm 1, we are assuming that both $\operatorname{f}(\mathsf{A}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}-\eta)$ and $\mathsf{\Pi}_{n}\nabla\operatorname{f}(\mathsf{A}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}-\eta)$ are easily computable, therefore $\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]$ and $\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})+\operatorname{E}^{\dagger}(\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]\vec{\varphi}_{n})$ are also easy to compute. #### 6.3.2 Bounds for countable functionals Extending the results of Section 6.3.1 to $\mathds{U}=\ell^{1}(\mathds{R})$ is analytically very simple but computationally relies heavily on the specific choice of $\mathsf{A}$. The computations of subdifferentials and gaps carry straight over replacing $\mathsf{\Pi}_{n}$ with the identity and adding the sets $J_{n}\subset\mathds{N}$ which define $\mathds{U}^{n}=\\{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\ell^{1}\operatorname{\;s.t.\;}i\notin J_{n}\implies\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{i}=0\\}$. Recall that ${\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\partial\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}_{}\coloneqq\inf_{s\in\operatorname{sign}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})}{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\mathsf{A}^{}\vec{\varphi}_{n}+\mu s\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}_{}$ where the $\operatorname{sign}$ function has the pointwise set-valued definition as indicated in (27)-(28). Where $[u_{n}]_{i}=0$, the choice $s_{i}=\min(1,\max(-1,-\mu^{-1}[\mathsf{A}^{}\vec{\varphi}_{n}]_{i}))$ achieves the minimal value $\displaystyle{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\partial\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}_{}$ $\displaystyle=\max_{i\in\mathds{N}}\begin{cases}\|[\mathsf{A}^{}\vec{\varphi}_{n}]_{i}+\mu\|&[\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}]_{i}>0\\\ \|[\mathsf{A}^{}\vec{\varphi}_{n}]_{i}-\mu\|&[\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}]_{i}<0\\\ \max\left(\|[\mathsf{A}^{}\vec{\varphi}_{n}]_{i}\|-\mu,0\right)&[\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}]_{i}=0\end{cases}$ (39) $\displaystyle\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})-\inf_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{H}}\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])$ $\displaystyle\leq\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})+\operatorname{E}^{\dagger}(\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]_{0}\vec{\varphi}_{n}),\qquad\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]_{0}\in\left[0,\frac{\mu}{{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\mathsf{A}^{}\vec{\varphi}_{n}\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}_{}}\right]$ (40) where $\vec{\varphi}_{n}=\nabla\operatorname{f}(\mathsf{A}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}-\eta)\in\mathds{R}^{m}$ is always exactly computable. In the countable case, the sets $J_{n}$ give a clear partition into known/unknown values in these definitions. For $i\in J_{n}$ the computation is the same as in Section 6.3.1, then for $i\notin J_{n}$ we know $[\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}]_{i}=0$ which simplifies the remaining computations. This leads to: $\displaystyle{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\partial\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}_{}$ $\displaystyle=\max\left(\max_{i\in J_{n}}\|[\partial\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})]_{i}\|,\ \sup_{i\notin J_{n}}\|[\partial\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})]_{i}\|\right)=\max\left({\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\partial_{n}\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}_{},\ \sup_{i\notin J_{n}}\|[\mathsf{A}^{}\vec{\varphi}_{n}]_{i}\|-\mu\right)$ (41) $\displaystyle{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\mathsf{A}^{}\vec{\varphi}_{n}\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}_{}$ $\displaystyle=\max\left(\max_{i\in J_{n}}\|[\mathsf{A}^{}\vec{\varphi}_{n}]_{i}\|,\ \sup_{i\notin J_{n}}\|[\mathsf{A}^{}\vec{\varphi}_{n}]_{i}\|\right)\hskip 19.0pt=\max\left({\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\mathsf{\Pi}_{n}\mathsf{A}^{}\vec{\varphi}_{n}\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}_{},\ \sup_{i\notin J_{n}}\|[\mathsf{A}^{}\vec{\varphi}_{n}]_{i}\|\right).$ (42) Both estimates only rely on an upper bound of $\max_{i\notin J_{n}}\|[\mathsf{A}^{}\vec{\varphi}_{n}]_{i}\|$. One example computing this value is seen in Section 7.2. #### 6.3.3 Bounds for continuous functionals Finally we extend the results of Section 6.3.1 to continuous problems. Similar to the countable case (39)-(40), the exact formulae can be written down immediately: $\displaystyle{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\partial\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}_{}$ $\displaystyle=\max_{\vec{x}\in\Omega}\begin{cases}\|[\mathsf{A}^{}\vec{\varphi}_{n}](\vec{x})+\mu\|&\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}(\vec{x})>0\\\ \|[\mathsf{A}^{}\vec{\varphi}_{n}](\vec{x})-\mu\|&\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}(\vec{x})<0\\\ \max\left(\|[\mathsf{A}^{}\vec{\varphi}_{n}](\vec{x})\|-\mu,0\right)&\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}(\vec{x})=0\end{cases}$ (43) $\displaystyle\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})-\inf_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{H}}\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])$ $\displaystyle\leq\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})+\operatorname{E}^{\dagger}(\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]_{0}\vec{\varphi}_{n}),\qquad\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]_{0}\in\left[0,\frac{\mu}{{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\mathsf{A}^{}\vec{\varphi}_{n}\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}_{}}\right]$ (44) with $\operatorname{E}^{\dagger}$ as defined in (34). Recall that there is a mesh $\mathds{M}^{n}$ corresponding to $\mathds{U}^{n}$ such that $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}$ is constant on each $\omega\in\mathds{M}^{n}$, so we can rewrite these bounds: $\displaystyle{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\partial\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}_{}$ $\displaystyle=\max_{\omega\in\mathds{M}^{n}}\begin{cases}{\left\lVert\mathsf{A}^{}\vec{\varphi}_{n}+\mu\right\rVert}_{L^{\infty}(\omega)}&\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}\|_{\omega}>0\\\ {\left\lVert\mathsf{A}^{}\vec{\varphi}_{n}-\mu\right\rVert}_{L^{\infty}(\omega)}&\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}\|_{\omega}<0\\\ \max(0,{\left\lVert\mathsf{A}^{}\vec{\varphi}_{n}\right\rVert}_{L^{\infty}(\omega)}-\mu)&\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}\|_{\omega}=0\end{cases}$ (45) $\displaystyle{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\mathsf{A}^{}\vec{\varphi}_{n}\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}_{}$ $\displaystyle=\max_{\omega\in\mathds{M}^{n}}{\left\lVert\mathsf{A}^{}\vec{\varphi}_{n}\right\rVert}_{L^{\infty}(\omega)}.$ (46) Now, both values can be estimated relying on pixel-wise supremum norms of $\mathsf{A}^{}\vec{\varphi}_{n}$ which we have assumed is sufficiently smooth. We will therefore use a pixel-wise Taylor expansion to provide a simple and accurate estimate. For instance, let $\vec{x}_{i}$ be the midpoint of the pixel $\omega$, then ${\left\lVert\mathsf{A}^{}\vec{\varphi}_{n}\right\rVert}_{L^{\infty}(\omega)}\leq\|[\mathsf{A}^{}\vec{\varphi}_{n}](\vec{x}_{i})\|+\frac{\operatorname{diam}(\omega)}{2}\|[\nabla\mathsf{A}^{}\vec{\varphi}_{n}](\vec{x}_{i})\|+\frac{\operatorname{diam}(\omega)^{2}}{8}\|\mathsf{A}^{}\vec{\varphi}_{n}\|_{C^{2}}.$ (47) In this work we chose a first order expansion because we are looking for extrema of $\mathsf{A}^{}\vec{\varphi}_{n}$, i.e. we are most interested in the squares $\omega$ such that $\|[\mathsf{A}^{}\vec{\varphi}_{n}](\vec{x}_{i})\|\approx\mu,\qquad\|[\nabla\mathsf{A}^{}\vec{\varphi}_{n}](\vec{x}_{i})\|\approx 0,\qquad[\nabla^{2}\mathsf{A}^{}\vec{\varphi}_{n}](\vec{x}_{i})\preceq 0.$ (48) A zeroth order expansion would be optimally inefficient (approximating $\|[\nabla\mathsf{A}^{}\vec{\varphi}_{n}](\vec{x}_{i})\|$ with $\|\mathsf{A}^{}\vec{\varphi}_{n}\|_{C^{1}}$) and a second order expansion would possibly be more elegant but harder to implement. We found that a first order expansion was simple and efficient. The bounds presented here for continuous problems emphasise the twinned properties required for adaptive mesh optimisation. The mesh should be refined greedily to the structures of $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}$, but also must be sufficiently uniform to provide a good estimate for $\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{})$. This is a classical exploitation/exploration trade-off; exploiting visible structure whilst searching for other structures which are not yet visible. ### 6.4 Support detection The main motivation for using L1 penalties in applications is because it recovers sparse signals, in the case of compressed sensing the support of $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}$ is also provably close to the “true” support Duval2017a ; Poon2018 . If $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}\approx\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}$ in the appropriate sense, then we should also be able to quantify the statement $\operatorname{supp}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})\approx\operatorname{supp}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{})$. Such methods are referred to as _safe screening_ rules ElGhaoui2010 which gradually identify the support and allow the optimisation algorithm to constrain parts of the reconstruction to 0. In this subsection we propose a new simple screening rule which is capable of generalising to our continuous subspace approximation setting. It is likely that more advanced methods Bonnefoy2015 ; Ndiaye2017 can also be adapted, although that is beyond the scope of this work. The key difference is the allowance of inexact computations resulting from estimates such as (47). The support of $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}$ has already been characterised very precisely Duval2017a ; Poon2018 . In particular, the support is at most $m$ distinct points and are a subset of $\\{\vec{x}\in\Omega\operatorname{\;s.t.\;}\|\mathsf{A}^{}\vec{\varphi}^{}\|(\vec{x})=\mu\\}$ (an equivalent statement holds for the countable case). Less formally, this can also be seen from the the subdifferential computations in Section 6.3, for all $\vec{x}\in\operatorname{supp}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{})$ we have $0\in\partial\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{})(\vec{x})=[\mathsf{A}^{}\vec{\varphi}^{}](\vec{x})+\mu\operatorname{sign}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}(\vec{x})).$ (49) Heuristically, we will use strong convexity of $\operatorname{E}^{\dagger}$ from (34) and smoothness of $\mathsf{A}^{}$ to quantify the statement: $\text{if}\quad\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})+\operatorname{E}^{\dagger}(\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]_{0}\vec{\varphi}_{n})\approx 0\quad\text{then}\quad\left\\{\vec{x}\operatorname{\;s.t.\;}\|[\mathsf{A}^{}\vec{\varphi}_{n}](\vec{x})\|\ll\mu\right\\}\subset\\{\vec{x}\operatorname{\;s.t.\;}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}(\vec{x})=0\\}.$ Recall that $\nabla\operatorname{f}$ is 1-Lipschitz if and only if $\operatorname{f}^{}$ is 1-strongly convex (Hiriart2013, , Chapter 10, Thm. 4.2.2). Therefore, if $\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]_{0}\vec{\varphi}_{n}$ and $\vec{\varphi}^{}$ are both dual-feasible, then $\tfrac{1}{2}{\left\lVert\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]_{0}\vec{\varphi}_{n}-\vec{\varphi}^{}\right\rVert}_{\ell^{2}}^{2}\leq\operatorname{E}^{\dagger}(\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]_{0}\vec{\varphi}_{n})-\operatorname{E}^{\dagger}(\vec{\varphi}^{})=\operatorname{E}^{\dagger}(\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]_{0}\vec{\varphi}_{n})+\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{})\leq\operatorname{E}^{\dagger}(\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]_{0}\vec{\varphi}_{n})+\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}),$ (50) which gives an easily computable bound on ${\left\lVert\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]_{0}\vec{\varphi}_{n}-\vec{\varphi}^{}\right\rVert}_{\ell^{2}}$. Now we estimate $\mathsf{A}^{}\vec{\varphi}_{n}$ on the support of $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}$: $\displaystyle\min_{\vec{x}\in\operatorname{supp}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{})}\|[\mathsf{\Pi}_{n}\mathsf{A}^{}\vec{\varphi}_{n}](\vec{x})\|$ $\displaystyle\geq\min_{\vec{x}\in\operatorname{supp}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{})}\|[\mathsf{A}^{}\vec{\varphi}_{n}](\vec{x})\|$ (51) $\displaystyle=\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]_{0}^{-1}\min_{\vec{x}\in\operatorname{supp}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{})}\|[\mathsf{A}^{}\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]_{0}\vec{\varphi}_{n}](\vec{x})\|$ (52) $\displaystyle\geq\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]_{0}^{-1}\min_{\vec{x}\in\operatorname{supp}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{})}\|[\mathsf{A}^{}\vec{\varphi}^{}](\vec{x})\|-\|[\mathsf{A}^{}\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]_{0}\vec{\varphi}_{n}-\mathsf{A}^{}\vec{\varphi}^{}](\vec{x})\|$ (53) $\displaystyle=\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]_{0}^{-1}\min_{\vec{x}\in\operatorname{supp}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{})}\mu-\|[\mathsf{A}^{}\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]_{0}\vec{\varphi}_{n}-\mathsf{A}^{}\vec{\varphi}^{}](\vec{x})\|$ (54) $\displaystyle\geq\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]_{0}^{-1}\left(\mu-\|\mathsf{A}^{}\|_{\ell^{2}\to L^{\infty}}{\left\lVert\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]_{0}\vec{\varphi}_{n}-\vec{\varphi}^{}\right\rVert}_{\ell^{2}}\right).$ (55) Therefore, $\|[\mathsf{\Pi}_{n}\mathsf{A}^{}\vec{\varphi}_{n}](\vec{x})\|<\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]_{0}^{-1}\left(\mu-\sqrt{2(\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})+\operatorname{E}^{\dagger}(\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]_{0}\vec{\varphi}_{n}))}\|\mathsf{A}^{}\|_{\ell^{2}\to L^{\infty}}\right)\qquad\implies\qquad\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}(\vec{x})=0.$ (56) This equation is valid when $\vec{x}$ is either a continuous or countable index, the only distinction is to switch to $\ell^{\infty}$ in the norm of $\mathsf{A}^{}$. To make the equivalent statement on the discretised problem, simply replace $\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]_{0}$ with $\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]$ and $\mathsf{A}^{}$ with $\mathsf{\Pi}_{n}\mathsf{A}^{}$. There are two short observations on this formula: • The convergence guarantee from Theorem 4.2 is for the primal gap $\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})-\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{})$, rather than the primal-dual gap $\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})+\operatorname{E}^{\dagger}(\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]_{0}\vec{\varphi}_{n})$ used here. Although there is no guaranteed rate for the primal-dual gap, it is much more easily computable than the primal gap. • In Section 6.1, $\|\mathsf{A}^{}\|_{\ell^{2}\to C^{1}}<\infty$ was required to compute a rate of convergence for $\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})$, but only $\|\mathsf{A}^{}\|_{\ell^{2}\to L^{\infty}}<\infty$ is needed to estimate the support. ### 6.5 Operator norms For numerical implementation of (18), we are required to accurately estimate several operator norms of $\mathsf{A}$ of the form in (17). In particular, there are kernels $\psi_{j}\in\mathds{H}$ such that $(\mathsf{A}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])_{j}=\left\langle\psi_{j},\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\rangle$ for each $j\in[m]$. Verifying that ${\left\lVert\mathsf{A}\right\rVert}\leq 1$ can be performed by computing $\|\mathsf{A}\mathsf{A}^{}\|_{\ell^{2}\to\ell^{2}}$, and the adaptivity described in Sections 6.1, 6.3.3, and 6.4 requires the values of $\|\mathsf{A}^{}\|_{\ell^{2}\to L^{\infty}}$, $\|\mathsf{A}^{}\|_{\ell^{2}\to C^{1}}$, and $\|\mathsf{A}^{}\|_{\ell^{2}\to C^{2}}$. The aim for this section is to provide estimates of these norms and seminorms for the numerical examples presented in Section 7. The following lemma allows for exact computation of the operator norm of $\mathsf{A}$. ###### Lemma 8 If $\mathsf{A}\colon\mathds{H}\to\mathds{R}^{m}$ has kernels $\psi_{j}\in\mathds{H}$ for $j\in[m]$, then $\mathsf{A}\mathsf{A}^{}\in\mathds{R}^{m\times m}$ has entries $(\mathsf{A}\mathsf{A}^{})_{i,j}=\left\langle\psi_{i},\psi_{j}\right\rangle$, so the spectral norm ${\left\lVert\mathsf{A}^{}\mathsf{A}\right\rVert}={\left\lVert\mathsf{A}\mathsf{A}^{}\right\rVert}$ can be computed efficiently. ###### Proof To compute the entries of $\mathsf{A}\mathsf{A}^{}\colon\mathds{R}^{m}\to\mathds{R}^{m}$, observe that for any $\vec{r}\in\mathds{R}^{m}$ $(\mathsf{A}\mathsf{A}^{}\vec{r})_{i}=\left\langle\psi_{i},\mathsf{A}^{}\vec{r}\right\rangle=\left\langle\psi_{i},\sum_{j=1}^{m}r_{j}\psi_{j}\right\rangle=\sum_{j=1}^{m}\left\langle\psi_{i},\psi_{j}\right\rangle r_{j}$ (57) as required. ∎ If ${\left\lVert\mathsf{A}^{}\mathsf{A}\right\rVert}$ is not analytically tractable, then Lemma 8 enables it to be computed using standard finite dimensional methods. The operator $\mathsf{A}\mathsf{A}^{}$ is always finite dimensional, and can be computed without discretisation error. In the continuous case, when $\mathds{H}=L^{2}(\Omega)$ we also need to estimate the smoothness properties of $\mathsf{A}^{}$. A generic result for this is given in the following lemma. ###### Lemma 9 If $\mathsf{A}\colon L^{2}([0,1]^{d})\to\mathds{R}^{m}$ has kernels $\psi_{j}\in L^{2}(\Omega)\cap C^{k}(\Omega)$ for $j\in[m]$, then for all $\frac{1}{q}+\frac{1}{q^{}}=1$, $q\in[1,\infty]$, we have $\displaystyle\|\mathsf{A}^{}\vec{r}\|_{C^{k}}$ $\displaystyle\coloneqq\sup_{\vec{x}\in\Omega}\|\nabla^{k}[\mathsf{A}^{}\vec{r}]\|(\vec{x})\leq\sup_{\vec{x}\in\Omega}{\left\lVert(\nabla^{k}\psi_{j}(\vec{x}))_{j=1}^{m}\right\rVert}_{\ell^{q^{}}}{\left\lVert\vec{r}\right\rVert}_{\ell^{q}},$ (58) $\displaystyle\|\mathsf{A}^{}\|_{\ell^{2}\to C^{k}}$ $\displaystyle\coloneqq\sup_{{\left\lVert\vec{r}\right\rVert}_{\ell^{2}}\leq 1}\|\mathsf{A}^{}\vec{r}\|_{C^{k}}\leq\sup_{\vec{x}\in\Omega}{\left\lVert(\nabla^{k}\psi_{j}(\vec{x}))_{j=1}^{m}\right\rVert}_{\ell^{q^{}}}\times\begin{cases}1&q\geq 2\\\ \sqrt{m^{2-q}}&q<2\end{cases}.$ (59) ###### Proof For the first inequality, we apply the Hölder inequality on $\mathds{R}^{m}$: $\|\nabla^{k}[\mathsf{A}^{}\vec{r}]\|(\vec{x})=\left\|\sum_{j=1}^{m}\nabla^{k}\psi_{j}(\vec{x})r_{j}\right\|\leq\left(\sum_{j=1}^{m}\|\nabla^{k}\psi_{j}(\vec{x})\|^{q^{}}\right)^{\frac{1}{q^{}}}{\left\lVert\vec{r}\right\rVert}_{\ell^{q}}={\left\lVert(\nabla^{k}\psi_{j}(\vec{x}))_{j}\right\rVert}_{\ell^{q^{}}}{\left\lVert\vec{r}\right\rVert}_{\ell^{q}}\;.$ For the second inequality, if $q\geq 2$ and $\sum_{j=1}^{m}r_{j}^{2}\leq 1$, then $\|r_{j}\|\leq 1$ for all $j$ and ${\left\lVert\vec{r}\right\rVert}_{\ell^{q}}^{q}\leq{\left\lVert\vec{r}\right\rVert}_{\ell^{2}}^{2}\leq 1$. If $q<2$ and ${\left\lVert\vec{r}\right\rVert}_{\ell^{2}}\leq 1$, then we again use Hölder’s inequality: $\sum_{j=1}^{m}r_{j}^{q}\leq\Big{(}\sum_{j=1}^{m}1^{Q^{}}\Big{)}^{\frac{1}{Q^{}}}\Big{(}\sum_{j=1}^{m}r_{j}^{qQ}\Big{)}^{\frac{1}{Q}}\leq m^{\frac{2-q}{2}}$ for $Q=\frac{2}{q}$. ∎ The examples in Section 7 require explicit computations of the expressions in Lemmas 8 and 9. These computations are provided in the appendix, Theorem C.1. ## 7 Numerical examples We present four numerical examples. The first two are in 1D to demonstrate the performance of different variants of Algorithm 1, both with and without adaptivity. In particular, we explore sparse Gaussian deconvolution and sparse signal recovery from Fourier data. We compare with the _continuous basis pursuit_ (CBP) discretisation Ekanadham2011 ; Duval2017b which is also designed to achieve super-resolution accuracy within a convex framework. More details of this method will be provided in Section 7.1. The next example is 2D reconstruction from Radon or X-ray data with wavelet- sparsity and a robust data fidelity. As the forward operator is not sufficiently smooth, we must optimise in $\ell^{1}(\mathds{R})$, which naturally leads to the choice of a wavelet basis. Finally, we process a dataset which represents a realistic application in biological microscopy, referred to as STORM microscopy. In essence, the task is to perform 2D Gaussian de-blurring/super-resolution and denoising to find the location of sparse spikes of signal. In this section, the main aim is to minimise $\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})=\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})-\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{})$, and so this will be our main metric for the success of an algorithm, referred to as the “continuous gap”. Lemma 6 only provides guarantees on the values of $\min_{n\leq N}\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})$ so it is this monotone estimate which is plotted. As $\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{})$ is not known exactly, we always use the estimate $\min_{n\leq N}\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})\approx\min_{n\leq N}\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})+\min_{n^{\prime}\leq n}\operatorname{E}^{\dagger}(\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]_{0}\vec{\varphi}_{n^{\prime}})$. Another quantity of interest is minimisation of the discrete energy $\min_{n\leq N}\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})+\min_{n^{\prime}\leq n}\operatorname{E}^{\dagger}(\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]\vec{\varphi}_{n^{\prime}})$ which will be referred to as the “discrete gap”. Note that for the adaptive schemes the discrete gap may not be monotonic as the discrete dual problem changes with $N$. The code to reproduce these examples can be found online111https://github.com/robtovey/2020SpatiallyAdaptiveFISTA. ### 7.1 1D continuous LASSO In this example we choose $\mathds{U}=\mathcal{M}([0,1])$, $\mathds{H}=L^{2}([0,1])$, $\operatorname{f}(\cdot)=\frac{1}{2}{\left\lVert\cdot\right\rVert}_{\ell^{2}}^{2}$ and $\mathsf{A}\colon\mathds{U}\to\mathds{R}^{30}$ with either random Fourier kernels: $(\mathsf{A}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])_{j}=\int_{0}^{1}\cos(\IfEqCase{3}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k3]_{j}x)\mathop{}\\!\mathrm{d}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0](x),\qquad\IfEqCase{3}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k3]_{j}\sim\operatorname{Uniform}[-100,100],\ j=1,2,\ldots,30,\ \mu=0.02,$ (60) or Gaussian kernels on a regular grid: $(\mathsf{A}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])_{j}=(2\pi\sigma^{2})^{-\frac{1}{2}}\int_{0}^{1}\exp\left(-\frac{(x-(j-1)\Delta)^{2}}{2\sigma^{2}}\right)\mathop{}\\!\mathrm{d}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0](x),\quad\sigma=0.12,\ \Delta=\tfrac{1}{29},\ j=1,2,\ldots,30,\ \mu=0.06.$ (61) Several variants of FISTA are compared for these examples but the key alternative shown here is the CBP discretisation. For this choice of $\operatorname{f}$, we call (18) the continuous LASSO problem, for which there are many numerical methods (c.f. Bredies2013 ; Castro2016 ; Boyd2017 ; Catala2019 ) however, most require the solution of a non-convex problem. We have focused on CBP because it approximates $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}$ through a convex discrete optimisation problem which is asymptotically exact in the limit $h\to 0$. It can also be optimised with FISTA which allows for direct comparison with the uniform and adaptive mesh approaches. The idea is that for a fixed mesh, the kernels of $\mathsf{A}$ are expanded to first order on each pixel and a particular first order basis is also chosen Ekanadham2011 ; Duval2017b . If $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}$ has only one Dirac spike in each pixel, then the zeroth order information should correspond to the mass of the spike, and additional first order information should determine the location. As shown in Section 6, in 1D we have $a_{\operatorname{U}}=a_{\operatorname{E}}=2$. The estimates given in (25) and (26) in dimension $d=1$ predict that the adaptive energy will decay at a rate of $\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})-\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{})\lesssim\frac{1}{n}$ so long as the pixel size also decreases at a rate of $h\sim\frac{1}{n}$. To achieve these rates, we implement a refinement criterion from Lemma 6 with guarantee of $\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n_{k}-1})-\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{})\lesssim 2^{-k}$ using the estimates made in Section 6.3. We choose subspaces $\mathds{U}^{n}$ to approximately enforce $\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})+\operatorname{E}^{\dagger}(\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]_{0}\vec{\varphi}_{n})\leq 2(\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})+\operatorname{E}^{\dagger}(\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]\vec{\varphi}_{n})),$ (62) i.e. the continuous gap is bounded by twice the discrete gap. In particular, note that for $\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]_{0}\approx\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]$, $\operatorname{E}^{\dagger}(\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]_{0}\vec{\varphi}_{n})=\tfrac{1}{2}{\left\lVert\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]_{0}\vec{\varphi}_{n}\right\rVert}^{2}+\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]_{0}\eta\vbox{\hbox{\leavevmode\resizebox{6.0pt}{}{$\cdot$}}}\vec{\varphi}_{n}=\frac{\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]_{0}}{\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]}\left(\frac{\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]_{0}}{\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]}\tfrac{1}{2}{\left\lVert\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]\vec{\varphi}_{n}\right\rVert}^{2}+\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]\eta\vbox{\hbox{\leavevmode\resizebox{6.0pt}{}{$\cdot$}}}\vec{\varphi}_{n}\right)\approx\frac{\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]_{0}}{\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]}\operatorname{E}^{\dagger}(\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]\vec{\varphi}_{n}).$ (63) Converting this into a spatial refinement criteria, recall $\frac{\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]_{0}}{\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]}\approx\frac{{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\mathsf{A}^{}\vec{\varphi}_{n}\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}_{}}{{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\mathsf{\Pi}_{n}\mathsf{A}^{}\vec{\varphi}_{n}\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}_{}}=\frac{\max_{\omega\in\mathds{M}^{n}}{\left\lVert\mathsf{A}^{}\vec{\varphi}_{n}\right\rVert}_{L^{\infty}(\omega)}}{\max_{\omega\in\mathds{M}^{n}}\|\mathsf{\Pi}_{n}\mathsf{A}^{}\vec{\varphi}_{n}(\omega)\|}\approx\max_{\omega\in\mathds{M}^{n}}\frac{{\left\lVert\mathsf{A}^{}\vec{\varphi}_{n}\right\rVert}_{L^{\infty}(\omega)}}{\|\mathsf{\Pi}_{n}\mathsf{A}^{}\vec{\varphi}_{n}(\omega)\|}$ (64) is the maximum ratio of second vs. zeroth order Taylor approximations of $\mathsf{A}^{}\vec{\varphi}_{n}$ on pixel $\omega$. This was found to be an efficient method of selecting pixels for refinement using quantities which had already been computed. Note briefly that this greedy strategy directly targets uncertainty, refinements also happen outside of the support of $u_{n}$ to guarantee that this is representative of $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}$. Such refinement is necessary to avoid discrete minimisers of $\operatorname{E}$ which are not global minimisers. Figure 1: Rates of continuous/discrete gap convergence for different LASSO algorithms with 128, 256, or 512 pixels. The “adaptive” method uses the proposed algorithm. Both “fixed” and “CBP” use standard FISTA with a uniform discretisation. Figure 2: Convergence plots for solving 1D problems with different algorithms. “Adaptive” methods use Algorithm 1 with fewer than 1024 pixels and the remaining methods use a uniform discretisation of 1024 pixels. Figure 3: Example reconstruction from the algorithms considered in Fig. 3. Pixel boundaries are indicated on the $x$-axis and the filtering method of Section 6.4 allows us to exclude the red shaded regions from $\operatorname{supp}(u^{})$. Values on the $y$-axis are normalised to units of mass, i.e. a Dirac mass would have height 1. ##### Comparison of discretisation methods In Fig. 3 we compare the three core approaches: fixed uniform discretisation, adaptive discretisation, and CBP. In particular, we wish to observe their convergence properties as the number of pixels is allowed to grow. In each case we use a FISTA stepsize of $t_{n}=\frac{n+19}{20}$. The adaptive discretisation is started with one pixel and limited to 128, 256, or 512 pixels while the fixed and CBP discretisations have uniform discretisations with the maximum number of pixels. The main observations are: * • The adaptive scheme is much more efficient, in both examples the adaptive scheme with 128 pixels is at least as good as both fixed discretisations with 512 pixels. In fact, only a maximum of 214 pixels were needed by the adaptive method in either example. * • With Fourier kernels the uniform piecewise constant discretisation is more efficient than CBP but in the Gaussian case this is reversed. This suggests that the performance of CBP depends on the smoothness of $\mathsf{A}$. * • The discrete gaps for non-adaptive optimisation behave as is common for FISTA, initial convergence is polynomial until a locally linear regime activates Tao2016 . CBP is always slower to converge than the piecewise constant discretisation. * • The adaptive refinement criterion succeeds in keeping the continuous/discrete gaps close for all $n$, i.e. (62). It is not completely fair to judge CBP with the continuous gap because, although it generates a continuous representation, this continuous representation is not necessarily consistent with the discrete gap being optimised, unlike when discretised with finite element methods. On the other hand, this is still the intended interpretation of the algorithm and we have no more appropriate metric for success in this case. ##### Comparison of FISTA variants Fig. 3 compares many methods with either fixed or adaptive discretisations. Each adaptive scheme is allowed up to 1024 pixels and each uniform discretisation uses exactly 1024. An example of each reconstruction method is shown in Fig. 3. The adaptive method better identifies the support of $u^{}$ and clearly localises pixels on that support. The reconstruction with uniform grid fails to provably identify the support of $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}$, despite having found a qualitatively accurate discrete minimiser. The “Greedy FISTA” implementation was proposed by in Liang2018 and we include the adaptive variant despite a lack of convergence proof. The remaining FISTA algorithms use a FISTA time step of $t_{n}=\frac{n+a-1}{a}$ for the given value of $a$, as proposed in Chambolle2015 . In this example CBP used the greedy FISTA implementation which gave faster observed convergence. Fig. 3 compares the discrete gaps because it is the accurate metric for fixed discretisations, and for the adaptive discretisation it should also be an accurate predictor of the continuous gap. The main observations are: * • Each algorithm displays very similar convergence properties. The main difference is that the reconstructions with fixed discretisations accelerate after $10^{4}$-$10^{5}$ iterations. * • During the initial “slow” phase, adaptive and fixed discretisations appear to achieve very similar (discrete) convergence rates. The coarse-to-fine adaptivity is not slower than fixed discretisations in this regime. * • Lemma 6 accurately predicts the $\frac{1}{n}$ rate of the adaptive methods, mirrored in the fixed discretisations. This suggests that high-resolution discretisations are also initially limited by this $\frac{1}{n}$ rate before entering the asymptotic regime, consistent with (14). * • The fastest FISTA stepsize choice is consistently the greedy variant, although $a=20$ is very comparable. * • While each adaptive algorithm is allowed to use up to 1024 pixels, in Fig. 3 the most used was 235. ##### Comparison of fixed and adaptive discretisation Motivated by the findings in Fig. 3, we now look more closely at the performance of the $a=20$ and the greedy FISTA schemes. We have convergence results for the former, but the latter typically performs the best for non- adaptive optimisation and is never worse than $a=20$ in the adaptive setting. The question is whether it is faster/more efficient to use the proposed adaptive scheme, or to use a classical scheme at sufficiently high uniform resolution. The fixed discretisations use 1024 pixels (i.e. constant pixel size of $2^{-10}$ in Fig. 6) and the adaptive discretisation starts with two pixels with an upper limit of 1024. As expected, the fixed discretisation starts with a smaller continuous gap before plateauing to a sub-optimal gap around $\operatorname{E}=\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{})+0.1$. Fig. 6 shows convergence of pixel size and continuous gap with respect to number of iterations. Fig. 6 shows the more practical attributes of continuous gap and number of pixels against execution time. We see that the adaptive discretisation is consistently capable of computing lower energies with fewer pixels and in less time than the uniform discretisation. The convergence behaviour is very consistent with respect to number of iterations. Suppose that the numerical aim is to find a function $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n}$ with $\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})-\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{})\leq 0.1$, all methods would converge after $O(10^{3})$ iterations, demonstrating some equivalence between the two FISTA algorithms. For $n\in[10^{3},10^{4}]$, in both problems, the adaptive schemes coincide with the fixed schemes in both energy and minimum pixel size. On the other hand, we also see that the adaptive scheme achieves this energy in almost an order of magnitude less time and fewer pixels. Figure 4: Continuous convergence of adaptive (coarse-to-fine pixel size) compared with uniform discretisation (constant pixel size) with respect to number of iterations. Figure 5: Continuous convergence of adaptive compared with uniform discretisation with respect to wall-clock time and total number of pixels (memory requirement). $\begin{aligned} J_{n}&=\\{(0,0),(0,1),(0,2),(1,2),(1,1)\\}\\\ \operatorname{leaf}(J_{n})&=\\{(0,2),(1,2),(1,1)\\}\\\ \mathds{M}^{n}&=\left\\{[0,\tfrac{1}{4}),[\tfrac{1}{4},\tfrac{1}{2}),[\tfrac{1}{2},1)\right\\}\end{aligned}$ $(0,0)$ $[0,1]$ $(0,1)$ $[0,\frac{1}{2}]$ $(0,2)$ $[0,\frac{1}{4}]$ $(1,2)$ $[\frac{1}{4},\frac{1}{2}]$ $(1,1)$ $[\frac{1}{2},1]$ Figure 6: Example tree representation of 1D wavelets. Left: nodes, leaves, and mesh of discretisation. Right: arrangement into a tree with index $(j,k)$ and corresponding support of wavelet $w_{j,k}$ underneath. ### 7.2 2D robust sparse wavelet reconstruction In this example we consider $\mathsf{A}$ to be a 2D Radon transform. In particular, the rows of $\mathsf{A}$ correspond to integrals over the sets $\mathds{X}^{I}_{i}$ where $\mathds{X}_{i}^{I}=\left\\{\vec{x}\in[-\tfrac{1}{2},\tfrac{1}{2}]^{2}\operatorname{\;s.t.\;}\vec{x}\vbox{\hbox{\leavevmode\resizebox{6.0pt}{}{$\cdot$}}}\begin{pmatrix}\cos\theta_{I}\\\ \sin\theta_{I}\end{pmatrix}\in\left[-\tfrac{1}{2}+\tfrac{i-1}{100},-\tfrac{1}{2}+\tfrac{i}{100}\right)\right\\},\quad\theta_{I}=\frac{180^{\circ}}{51}I$ (65) for $i=\in[100]$, $I\in[50]$. This is not exactly in the form analysed by Theorem C.1, only the sets $\\{\mathds{X}^{I}_{i}\operatorname{\;s.t.\;}i\in[100]\\}$ for each $I$ are disjoint, therefore we apply Theorem C.1 block-wise to estimate ${\left\lVert\mathsf{A}\right\rVert}_{L^{2}\to\ell^{2}}\leq\sqrt{\sum_{I\in[50]}\max_{i\in[100]}\|\mathds{X}^{I}_{i}\|}=\sqrt{\sum_{I\in[50]}\max_{i\in[100]}\int_{\mathds{X}^{I}_{i}}1\mathop{}\\!\mathrm{d}\vec{x}}=\sqrt{\sum_{I\in[50]}\max_{i\in[100]}\ (\mathsf{A}\mathds{1})_{i,I}}\;.$ (66) $\mathsf{A}$ is not smooth, therefore we can’t bound $\|\mathsf{A}^{}\|_{C^{k}}$ for $k>0$, and so we must look to minimise over $\ell^{1}$ rather than $L^{1}$. The natural choice is to promote sparsity in a wavelet basis which can be rearranged into the form of (18): $\min_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{U}}\operatorname{f}(\mathsf{A}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]-\eta)+\mu{\left\lVert\mathsf{W}^{-1}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\rVert}_{\ell^{1}}=\min_{\widehat{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}\in\ell^{1}(\mathds{R})}\operatorname{f}(\mathsf{A}\mathsf{W}\widehat{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}-\eta)+\mu{\left\lVert\widehat{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}\right\rVert}_{\ell^{1}}.$ (67) The minimisers are related by $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}=\mathsf{W}\widehat{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}^{}$ and, for wavelet bases, $\mathsf{W}$ is orthonormal so ${\left\lVert\mathsf{A}\mathsf{W}\right\rVert}_{\ell^{2}\to\ell^{2}}={\left\lVert\mathsf{A}\right\rVert}_{L^{2}\to\ell^{2}}$. In this example we consider the smoothed robust fidelity Rosset2007 $\operatorname{f}(\vec{\varphi})=\sum_{i=1}^{m}\begin{cases}10^{-4}\|\varphi_{i}\|&\|\varphi_{i}\|\geq 10^{-4}\\\ \tfrac{1}{2}\|\varphi_{i}\|^{2}+\tfrac{1}{2}10^{-8}&\text{else}\end{cases}\approx 10^{-4}{\left\lVert\vec{\varphi}\right\rVert}_{\ell^{1}}.$ (68) From Section 6.3 we know that to track convergence and perform adaptive refinement, it is sufficient to accurately bound $\|[\mathsf{W}^{\top}\mathsf{A}^{}\vec{\varphi}_{n}]_{j}\|$ for all $j\notin J_{n}$. If $\mathsf{W}$ is a wavelet transformation then its columns, $w_{j}\in L^{2}$, are simply the wavelets themselves and we can use the bound $\|\left\langle w_{j},\mathsf{A}^{}\vec{\varphi}_{n}\right\rangle\|=\left\|\left\langle w_{j},\mathds{1}_{\operatorname{supp}(w_{j})}\mathsf{A}^{}\vec{\varphi}_{n}\right\rangle\right\|\leq{\left\lVert\mathds{1}_{\operatorname{supp}(w_{j})}\mathsf{A}^{}\vec{\varphi}_{n}\right\rVert}_{L^{2}}\leq{\left\lVert\mathds{1}_{\mathds{X}}\mathsf{A}^{}\vec{\varphi}_{n}\right\rVert}_{L^{2}}$ (69) for all $\mathds{X}\supset\operatorname{supp}(w_{j})$. In the case of the Radon transform, we can compute the left-hand side explicitly for the finitely many $j\in J_{n}$, but we wish to use the right-hand side in a structured way to avoid computing the infinitely many $j\notin J_{n}$. To do this, we will take a geometrical perspective on the construction of wavelets to view them in a tree format. ##### Tree structure of wavelets Finite elements are constructed with a mesh which provided a useful tool for adaptive refinement in Section 6.3.3. For wavelets, we will associate a tree with every discretisation and the leaves of the tree correspond to a mesh. This perspective comes from the multi-resolution interpretation of wavelets. An example is seen in Fig. 6 for 1D Haar wavelets, $w_{j,k}(x)=\sqrt{2}^{k}\psi(2^{k}x-j)$ where $\psi=\mathds{1}_{[0,1)}-\mathds{1}_{[-1,0)}$. In higher dimensions, the only two things which change are the number of children ($2^{d}$ for non-leaves) and at each node you store the coefficients of $2^{d}-1$ wavelets. The support on each node is still a disjoint partition of unity consisting of regular cubes of side length $2^{-k}$ at level $k$. The only change in our own implementation is to translate the support to $[-\tfrac{1}{2},\tfrac{1}{2}]^{2}$. We briefly remark that the tree structuring of wavelets is not novel and appears more frequently in the Bayesian inverse problems literature Castillo2019 ; Kekkonen2021 . ##### Continuous gradient estimate In Section 7.1 we used the continuous gap as a measure for convergence, for wavelets we will use the continuous subdifferential. With the tree structure we can easily adapt the results of Section 6.3 to estimate subdifferentials (or function gaps). In particular, $\displaystyle{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\partial\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}_{}$ $\displaystyle=\max\left({\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\partial_{n}\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}_{},\max_{j\notin J_{n}}\|\left\langle w_{j},\mathsf{A}^{}\vec{\varphi}_{n}\right\rangle\|-\mu\right)$ (70) $\displaystyle\leq\max\left({\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\partial_{n}\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}_{},\max_{j\in\operatorname{leaf}(J_{n})}{\left\lVert\mathds{1}_{\operatorname{supp}(w_{j})}\mathsf{A}^{}\vec{\varphi}_{n}\right\rVert}_{L^{2}}-\mu\right).$ (71) ##### Numerical results We consider two phantoms where the ground-truth is either a binary disc or the Shepp-Logan phantom. Both examples are corrupted with $2\text{\,}\mathrm{\char 37\relax}$ Laplace distributed noise. This is visualised in Fig. 9. All optimisations shown are spatially adaptive using Haar wavelets and initialised with $\mathds{U}^{0}=\\{x\mapsto c\operatorname{\;s.t.\;}c\in\mathds{R}\\}$. The gradient metric shown throughout is the $\ell^{\infty}$ norm. Motivated by (71), the spatial adaptivity is chosen to refine nodes $j\in\operatorname{leaf}(J_{n})$ to ensure that ${\left\lVert\mathds{1}_{\operatorname{supp}(w_{j})}\mathsf{A}^{}\vec{\varphi}_{n}\right\rVert}_{L^{2}}-\mu\leq 10{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\partial_{n}\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}_{}$ for all $j$ and $n$ (i.e. so that the continuous gradient is less than 10 times the discrete gradient). We do not expect wavelet regularisation to have state-of-the-art performance in the examples of Fig. 9. What they demonstrate is the preference Haar wavelets have to align large discontinuities with a coarse grid, even when the discretisation is allowed to be as fine as necessary. There is an average of $2\cdot 10^{6}$ wavelet coefficients in each discretised reconstruction, although the higher frequencies have much smaller intensities. In limited data scenarios, wavelet regularisation automatically selects a local “resolution” which reflects the quality of data. Particularly in the Shepp-Logan reconstruction, we see that the outer ring is detected with a finer precision than the dark interior ellipses. The first numerical results shown in Fig. 9 compare the same adaptive FISTA variants as shown in Fig. 3. In these examples we see that the greedy FISTA and the $a=20$ algorithms achieve almost linear convergence while $a=2$ is significantly slower. Interestingly, in both examples the $a=20$ variant uses half as many wavelets as the Greedy variant, and therefore converges slightly faster in time. Figure 7: Phantoms, data and reconstructions for wavelet-sparse tomography optimisation. Both examples are corrupted with $2\text{\,}\mathrm{\char 37\relax}$ Laplace distributed noise. Figure 8: Convergence of different implementations of Algorithm 1 with an unlimited number of pixels for sparse wavelet optimisation. Figure 9: Example images from STORM dataset. ### 7.3 2D continuous LASSO Our final application is a super-resolution/de-blurring inverse problem from biological microscopy. In mathematical terms, the observed data is a large number of sparse images which are corrupted by blurring and a large amount of noise, examples are seen in Fig. 9. The task is to compute the centres of the spikes of signal in each image and then re-combine into a single super- resolved image, as in Fig. 11. This technique is referred to as _Single Molecule Localisation Microscopy_ (SMLM), of which we consider the specific example of _Stochastic Optical Reconstruction Microscopy_ (STORM). Readers are directed to the references Sage2015 ; Sage2019 ; Schermelleh2019 for further details. The LASSO formulation ($\operatorname{f}(\cdot)=\frac{1}{2}{\left\lVert\cdot\right\rVert}_{\ell^{2}}^{2}$) has previously been shown to be effective in the context of STORM Huang2017 ; Denoyelle2019 . Here we use a simulated dataset provided as part of the 2016 SMLM challenge222http://bigwww.epfl.ch/smlm/challenge2016/datasets/MT4.N2.HD/Data/data.html for benchmarking software in this application. The corresponding LASSO formulation is $(\mathsf{A}\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])_{i}=(2\pi\sigma^{2})^{-1}\int_{[0,6.4]^{2}}\exp\left(-\frac{1}{2\sigma^{2}}\left\|\vec{x}-\Delta\begin{pmatrix}i_{1}+\tfrac{1}{2}&i_{2}+\tfrac{1}{2}\end{pmatrix}^{\top}\right\|^{2}\right)\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0](\vec{x})\mathop{}\\!\mathrm{d}\vec{x},\qquad\sigma=0.2,\ \Delta=0.1$ (72) for $i_{1},i_{2}=1,2,\ldots,64$, $\mathds{U}=\mathcal{M}([0,6.4]^{2})$ and $\mathds{H}=L^{2}([0,6.4]^{2})$ with lengths in $\text{\,}\mathrm{\SIUnitSymbolMicro m}$. 3020 frames are provided, examples of which are shown in Fig. 9. To process this dataset, image intensities were normalised to $[0,1]$ then a constant was subtracted to approximate 0-mean noise. The greedy FISTA algorithm was used for optimisation with $\mu=0.15$, $10^{3}$ iterations, and a maximum of $10^{5}$ pixels per image. Finally, all the reconstructions were summed and the result shown in Fig. 11. The adaptive scheme used fewer than $10^{4}$ pixels per frame, a fixed discretisation with equivalent resolution of $1.3\text{\,}\mathrm{nm}$ would have required more than $3\cdot 10^{6}$ per frame. LASSO is compared with ThunderSTORM Ovesny2014 , a popular ImageJ plugin Schindelin2012 which finds the location of signal using Fourier filtering. The performance of ThunderSTORM was rated very highly in the initial SMLM challenge Sage2015 . Both methods compared here demonstrate the key structures of the reconstruction, however, both are sensitive to tuning parameters. In this examples, LASSO has possibly recovered too little signal and ThunderSTORM contains spurious signal. Fig. 11 shows various convergence metrics for the adaptive reconstructions. The magenta line in the first panel shows that the continuous gap converges slightly faster than the $n^{-2/3}$ predicted by (26) in dimension $d=2$. In this example we also implement the suggestion of Section 6.4 to remove pixels outside of the support of $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}$. From (56), any pixel $\omega\in\mathds{M}^{n}$ satisfying $\IfEqCase{2}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k2]_{0}{\left\lVert\mathsf{\Pi}_{n}\mathsf{A}^{}\vec{\varphi}_{n}\right\rVert}_{L^{\infty}(\omega)}\leq(1-\operatorname{threshold}_{n})\mu$ (73) guarantees that $\omega\cap\operatorname{supp}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{})=\emptyset$. This threshold is plotted in red in the first panel of Fig. 11. Once the value becomes less than 1, we can start reducing the number of pixels instead of continual refinement. We see that the resolution decreases steadily (second panel), but the total number of pixels (final panel) stops increasing after around 30 iterations. Figure 10: Convergence of adaptive FISTA for STORM dataset. Lines indicate the median value over 3020 STORM frames. Shaded regions indicate the $25\text{\,}\mathrm{\char 37\relax}75\text{\,}\mathrm{\char 37\relax}$ interquartile range. Pixel width is scaled $[0,1]$ rather than $[0,$6.4\text{\,}\mathrm{\SIUnitSymbolMicro m}$]$. Figure 11: Processed results of the STORM dataset. Top left: LASSO optimisation with Algorithm 1. Top right: Comparison with ThunderSTORM plugin. Bottom: Average data, no super-resolution or de-blurring. ## 8 Conclusions and outlook In this work we have proposed a new adaptive variant of FISTA and provided convergence analysis. This algorithm allows FISTA to be applied outside of the classical Hilbert space setting, still with a guaranteed rate of convergence. We have presented several numerical examples where convergence with the refining discretisation is at least as fast as a uniform discretisation, although more efficient with regards to both memory and computation time. In 1D we see good agreement with the theoretical rate. This rate also seems to be a good predictor for all variants of FISTA tested, although this is yet to be proven. Even the classical methods with a fixed discretisation are initially limited to the slower adaptive rate for small $n$. The results in 2D are similar, all tested FISTA methods converge at least at the guaranteed rate. The wavelet example was most impressive, achieving nearly linear convergence in energy. This is similar to the behaviour for classical FISTA although it is also yet to be formally proven. An interesting observation over all of the adaptive LASSO examples is that the standard oscillatory behaviour of FISTA has not occurred. With the monotone gaps plotted, oscillatory convergence should correspond to a piecewise constant descending gap. Either this behaviour only emerges for larger $n$, or the adaptivity provides a dampening effect for this oscillation. Moving forward, it would be interesting to see how far the analysis extends to other optimisation algorithms. Other variants of FISTA, such as the “greedy” implementation used here or the traditional Forward-Backward algorithm, should also be receptive to the analysis performed here. Furthermore, it would also interesting to attempt to replicate this refinement argument to extend the primal-dual algorithm Chambolle2011 or the Douglas-Rachford algorithm Douglas1956 . ###### Acknowledgements. R.T. acknowledges funding from EPSRC grant EP/L016516/1 for the Cambridge Centre for Analysis, and the ANR CIPRESSI project grant ANR-19-CE48-0017-01 of the French Agence Nationale de la Recherche. Most of this work was done while A.C. was still in CMAP, CNRS and Ecole Polytechnique, Institut Polytechnique de Paris, Palaiseau, France. 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Recall that the subsets $\mathds{U}^{n}\subset\mathds{H}$ satisfy (10). ### A.1 Proofs for Step 3 ###### Theorem A.1 (Lemma 2) thm: one step FISTA $t_{n}^{2}(\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})-\operatorname{E}(\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}))-(t_{n}^{2}-t_{n})(\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n-1})-\operatorname{E}(\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}))\leq\tfrac{1}{2}\left[{\left\lVert\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n-1}\right\rVert}^{2}-{\left\lVert\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n}\right\rVert}^{2}\right]+\left\langle\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n}-\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n-1},\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}\right\rangle.$ (74) ###### Proof Modifying (Chambolle2015, , Thm 3.2), for $n\geq 1$ we apply Lemma 1 with $\overline{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}=\overline{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}_{n-1}$ and $\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]=(1-\frac{1}{t_{n}})\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n-1}+\frac{1}{t_{n}}\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}$. By (10), $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n-1}\in\mathds{U}^{n}$ is convex so $\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]\in\mathds{U}^{n}$. This gives $\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})+\tfrac{1}{2}{\left\lVert\tfrac{1}{t_{n}}\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n}-\tfrac{1}{t_{n}}\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}\right\rVert}^{2}\leq\operatorname{E}\left((1-\tfrac{1}{t_{n}})\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n-1}+\tfrac{1}{t_{n}}\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}\right)+\tfrac{1}{2}{\left\lVert\tfrac{1}{t_{n}}\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n-1}-\tfrac{1}{t_{n}}\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}\right\rVert}^{2}.$ (75) By the convexity of $\operatorname{E}$, this reduces to $\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})-\operatorname{E}(\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n})-(1-\tfrac{1}{t_{n}})[\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n-1})-\operatorname{E}(\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n})]\leq\tfrac{1}{2t_{n}^{2}}{\left\lVert\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n-1}-\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}\right\rVert}^{2}-\tfrac{1}{2t_{n}^{2}}{\left\lVert\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n}-\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}\right\rVert}^{2}=\tfrac{1}{2t_{n}^{2}}\left[{\left\lVert\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n-1}\right\rVert}^{2}-{\left\lVert\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n}\right\rVert}^{2}\right]+\tfrac{1}{t_{n}^{2}}\left\langle\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n}-\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n-1},\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}\right\rangle.$ (76) Multiplying through by $t_{n}^{2}$ gives the desired inequality. ∎ ###### Theorem A.2 (Theorem 4.1) thm: mini FISTA convergence $\Paste{thm:eq:miniFISTAconvergence}$ (77) ###### Proof Theorem A.2 is just a summation of (74) over all $n=1,\ldots,N$. To see this: first add and subtract $\inf_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{H}}\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])$ to each term on the left-hand side to convert $\operatorname{E}$ to $\operatorname{E}_{0}$, then move $\operatorname{E}_{0}(\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n})$ to the right- hand side. Now (74) becomes $t_{n}^{2}\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})-(t_{n}^{2}-t_{n})\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n-1})\leq t_{n}\operatorname{E}_{0}(\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n})+\tfrac{1}{2}\left[{\left\lVert\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n-1}\right\rVert}^{2}-{\left\lVert\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n}\right\rVert}^{2}\right]+\left\langle\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n}-\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n-1},\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}\right\rangle.$ (78) Summing this inequality from $n=1$ to $n=N$ gives $t_{N}^{2}\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{N})+\sum_{n=1}^{N-1}(\underbrace{t_{n}^{2}-t_{n+1}^{2}+t_{n+1}}_{=\rho_{n}})\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{n})\leq\frac{{\left\lVert\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{0}\right\rVert}^{2}-{\left\lVert\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{N}\right\rVert}^{2}}{2}+\sum_{n=1}^{N}t_{n}\operatorname{E}_{0}(\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n})+\left\langle\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n}-\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n-1},\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}\right\rangle.$ (79) The final step is to flip the roles of $\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n}$/$\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}$ in the final inner product term. Re-writing the right-hand side gives $\sum_{n=1}^{N}\left\langle\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n}-\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n-1},\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}\right\rangle=\left\langle\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{N},\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{N}\right\rangle-\left\langle\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{0},\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{0}\right\rangle+\sum_{n=1}^{N}\left\langle\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n-1},\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n-1}-\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}\right\rangle.$ (80) Noting that $\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{0}=\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{0}$, the previous two equations combine to prove the statement of Theorem A.2. ∎ The following lemma is used to produce a sharper estimate on sequences $t_{n}$. ###### Lemma 10 If $\rho_{n}=t_{n}^{2}-t_{n+1}^{2}+t_{n+1}\geq 0$, $t_{n}\geq 1$ for all $n\in\mathds{N}$ then $t_{n}\leq n-1+t_{1}$. ###### Proof This is trivially true for $n=1$. Suppose true for $n-1$, the condition on $\rho_{n-1}$ gives $t_{n}^{2}-t_{n}\leq t_{n-1}^{2}\leq(n-2+t_{1})^{2}=(n-1+t_{1})^{2}-2(n-1+t_{1})+1.$ (81) Assuming the contradiction, if $t_{n}>n-1+t_{1}$ then the above equation simplifies to $n-1+t_{1}<1$. However, $t_{1}\geq 1$ implying that $n<1$ which completes the contradiction. ∎ ###### Lemma 11 (Lemma 3) thm: mini exponential FISTA convergence $\Paste{thm:eq:miniexponentialFISTAconvergence}$ (82) thm:end: mini exponential FISTA convergence ###### Proof This is just a telescoping of the right-hand side of (77) with the introduction of $n_{k}$ and simplification $\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}=\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}$, $\tfrac{1}{2}{\left\lVert\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{N}\right\rVert}^{2}+\sum^{N}_{n=1}t_{n}\operatorname{E}_{0}(\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n})+\left\langle\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n-1},\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n-1}-\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]_{n}\right\rangle=\tfrac{1}{2}{\left\lVert\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{K}\right\rVert}^{2}+\sum_{n=n_{K}}^{N}t_{n}\operatorname{E}_{0}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{K})+\sum_{k=1}^{K}\sum_{n=n_{k-1}}^{n_{k}-1}t_{n}\operatorname{E}_{0}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k-1})+\left\langle\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n_{k}-1},\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k-1}-\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}\right\rangle.$ (83) By Lemma 10, $t_{n}\leq n$ so we can further simplify $\sum_{n=\IfEqCase{3}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k3]}^{\IfEqCase{4}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k4]-1}t_{n}\leq\sum_{n=\IfEqCase{3}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k3]}^{\IfEqCase{4}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k4]-1}n=(\IfEqCase{4}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k4]-\IfEqCase{3}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k3])\frac{\IfEqCase{4}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k4]-1+\IfEqCase{3}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k3]}{2}\leq\frac{\IfEqCase{4}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k4]^{2}-\IfEqCase{3}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k3]^{2}}{2}$ to get the required bound. ∎ ### A.2 Proof for Step 4 ###### Lemma 12 (Lemma 4) thm: sufficiently fast ###### Proof Starting from Lemma 11 we have $\displaystyle t_{N}^{2}\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{N})+\tfrac{1}{2}{\left\lVert\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{N}-\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{K}\right\rVert}^{2}$ $\displaystyle\leq C+\frac{{\left\lVert\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{K}\right\rVert}^{2}}{2}+\frac{(N+1)^{2}-n_{K}^{2}}{2}\operatorname{E}_{0}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{K})$ $\displaystyle\hskip 70.0pt+\sum_{k=1}^{K}\frac{n_{k}^{2}-n_{k-1}^{2}}{2}\operatorname{E}_{0}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k-1})+\left\langle\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n_{k}-1},\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}-\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k+1}\right\rangle$ (84) $\displaystyle\leq C+\frac{{\left\lVert\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{K}\right\rVert}^{2}}{2}+\frac{n_{K+1}^{2}}{2}\operatorname{E}_{0}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{K})$ $\displaystyle\hskip 40.0pt+\sum_{k=1}^{K}\frac{n_{k}^{2}}{2}\operatorname{E}_{0}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k-1})+\left\langle\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n_{k}-1}-\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k-1}+\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k-1},{\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k-1}-\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}}\right\rangle.$ (85) The inductive step now depends on the value of $a_{\operatorname{U}}$. Case $a_{\operatorname{U}}>1$: We simplify the inequality $\displaystyle t_{N}^{2}\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{N})+\tfrac{1}{2}{\left\lVert\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{N}-\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{K}\right\rVert}^{2}$ $\displaystyle\lesssim a_{\operatorname{U}}^{2K}+n_{K+1}^{2}a_{\operatorname{E}}^{-K}+\sum_{k=1}^{K}n_{k}^{2}a_{\operatorname{E}}^{-k}+a_{\operatorname{U}}^{k}{\left\lVert\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n_{k}-1}-\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k-1}\right\rVert}+a_{\operatorname{U}}^{2k}$ (86) $\displaystyle\leq C_{1}\left[a_{\operatorname{U}}^{2K+2}+\sum_{k=1}^{K}a_{\operatorname{U}}^{2k}+a_{\operatorname{U}}^{k}{\left\lVert\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n_{k}-1}-\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k-1}\right\rVert}\right]$ (87) for some $C_{1}>C$. Choose $C_{2}\geq{\left\lVert\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n_{1}-1}-\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{1-1}\right\rVert}a_{\operatorname{U}}^{-1}$ such that $\frac{1}{2}C_{2}^{2}\geq\frac{C_{1}}{a_{\operatorname{U}}^{2}-1}(C_{2}+a_{\operatorname{U}}^{2}).$ (88) Assume ${\left\lVert\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n_{k}-1}-\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k-1}\right\rVert}\leq C_{2}a_{\operatorname{U}}^{k}$ for $1\leq k\leq K$ (trivially true for $K=1$), then for $N=n_{K+1}-1$ we have $\displaystyle\tfrac{1}{2}{\left\lVert\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n_{K+1}-1}-\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{K}\right\rVert}^{2}$ $\displaystyle\leq C_{1}\left[a_{\operatorname{U}}^{2K+2}+\sum_{k=1}^{K}a_{\operatorname{U}}^{2k}+a_{\operatorname{U}}^{k}{\left\lVert\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n_{k}-1}-\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k-1}\right\rVert}\right]$ (89) $\displaystyle\leq C_{1}\left[a_{\operatorname{U}}^{2K+2}+(1+C_{2})\frac{a_{\operatorname{U}}^{2K+2}}{a_{\operatorname{U}}^{2}-1}\right]$ (90) $\displaystyle\leq\frac{C_{1}a_{\operatorname{U}}^{2K+2}}{a_{\operatorname{U}}^{2}-1}\left(a_{\operatorname{U}}^{2}+C_{2}\right)\leq\tfrac{1}{2}(C_{2}a_{\operatorname{U}}^{K+1})^{2}.$ (91) Case $a_{\operatorname{U}}=1$: Denote $\IfEqCase{4}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k4]_{k}={\left\lVert\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}-\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k+1}\right\rVert}$ and note that ${\left\lVert\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k-1}\right\rVert}\leq{\left\lVert\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{0}\right\rVert}+\sum_{0}^{\infty}\IfEqCase{4}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k4]_{k}\lesssim 1$. We therefore bound $\displaystyle t_{N}^{2}\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{N})+\tfrac{1}{2}{\left\lVert\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{N}-\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{K}\right\rVert}^{2}$ $\displaystyle\lesssim 1+n_{K+1}^{2}a_{\operatorname{E}}^{-K}+\sum_{k=1}^{K}n_{k}^{2}a_{\operatorname{E}}^{-k}+({\left\lVert\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n_{k}-1}-\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k-1}\right\rVert}+1)b_{k}$ (92) $\displaystyle\leq C_{1}\left[1+\sum_{k=1}^{K}{\left\lVert\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n_{k}-1}-\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k-1}\right\rVert}\IfEqCase{4}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k4]_{k-1}\right]$ (93) for some $C_{1}>0$. Choose $C_{2}\geq\frac{{\left\lVert\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n_{1}-1}-\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{1-1}\right\rVert}}{\sum_{0}^{\infty}\IfEqCase{4}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k4]_{k}}$ such that $\frac{1}{2}C_{2}^{2}\geq C_{1}\left(1+C_{2}\sum_{0}^{\infty}\IfEqCase{4}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k4]_{k}\right).$ (94) Assume ${\left\lVert\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n_{k}-1}-\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k-1}\right\rVert}\leq C_{2}$ for $1\leq k\leq K$ (trivially true for $K=1$), then for $N=n_{K+1}-1$ we have $\tfrac{1}{2}{\left\lVert\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n_{K+1}-1}-\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{K}\right\rVert}^{2}\leq C_{1}\left[1+\sum_{k=1}^{K}{\left\lVert\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n_{k}-1}-\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k-1}\right\rVert}\IfEqCase{4}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k4]_{k-1}\right]\leq C_{1}\left(1+C_{2}\sum_{0}^{\infty}\IfEqCase{4}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k4]_{k}\right)\leq\frac{C_{2}^{2}}{2}$ (95) In both cases, the induction on ${\left\lVert\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]_{n_{K+1}-1}-\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{K}\right\rVert}$ holds for all $K$, and we have $t_{N}^{2}\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{N})\leq\frac{1}{2}C_{2}^{2}a_{\operatorname{U}}^{2K}$ for all $N<n_{K}-1$. ∎ ### A.3 Proof for Step 5 ###### Lemma 13 (Lemma 5) thm: sufficiently slow ###### Proof The proof is direct computation, note that $(a_{\operatorname{E}}a_{\operatorname{U}}^{2})^{\kappa}=\exp\left(\kappa\log(a_{\operatorname{E}}a_{\operatorname{U}}^{2})\right)=\exp(\log a_{\operatorname{U}}^{2})=a_{\operatorname{U}}^{2},$ (96) therefore $a_{\operatorname{U}}^{2K}=\left((a_{\operatorname{E}}a_{\operatorname{U}}^{2})^{K}\right)^{\kappa}\lesssim n_{K}^{2\kappa}\leq N^{2\kappa},$ (97) so $\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{N})\lesssim N^{-2(1-\kappa)}$ as required. ∎ ### A.4 Proofs for Step 6 ###### Theorem A.3 (Theorem 4.3) thm: stronger exponential FISTA convergence ###### Proof Let $C>0$ satisfy $n_{k}^{2}\leq Ca_{\operatorname{E}}^{k}a_{\operatorname{U}}^{2k}$ for each $k\in\mathds{N}$. Fix $N>C$ and choose $k$ such that $Ca_{\operatorname{E}}^{k-1}a_{\operatorname{U}}^{2k-2}\leq N<Ca_{\operatorname{E}}^{k}a_{\operatorname{U}}^{2k}$. By construction, and using the equality from (96), we have $\min_{n\leq N}\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]_{N})\leq\operatorname{E}_{0}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k-1})\lesssim a_{\operatorname{E}}^{-k}=(a_{\operatorname{E}}a_{\operatorname{U}}^{2})^{-k(1-\kappa)}<C^{\kappa-1}N^{-2(1-\kappa)}$ (98) as required. ∎ ###### Lemma 14 (Lemma 6) thm: practical refinement criteria ###### Proof The conditions for $a_{\operatorname{U}}$ in Definition 1 are already met, it remains to be shown that $\operatorname{E}_{0}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k})\leq Ca_{\operatorname{E}}^{-k}$ for some fixed $C>0$. For cases (3) and (4), fix $R>0$ such that both $\\{\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}\\}_{k\in\mathds{N}}$ and the sublevel set $\\{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{U}\operatorname{\;s.t.\;}\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])\leq 1+\IfEqCase{1}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k1]\\}$ are contained in the ball of radius $R$. Any minimising sequences of $\operatorname{E}$ in $\mathds{U}$ or $\widetilde{\mathds{U}}^{k}$ are contained in this ball. We can therefore compute $C$ in each case: * (1) $\operatorname{E}_{0}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k})\leq\IfEqCase{1}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k1]a_{\operatorname{E}}^{-k}$, so $C=\IfEqCase{1}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k1]$ suffices. * (2) $\operatorname{E}_{0}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k})\leq\operatorname{E}_{0}(\widetilde{\mathds{U}}^{k})+\IfEqCase{1}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k1]a_{\operatorname{E}}^{-k}\leq(a_{\operatorname{E}}+1)\beta a_{\operatorname{E}}^{-k}$, so $C=(a_{\operatorname{E}}+1)\beta$ suffices. * (3) $\operatorname{E}_{0}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k})-\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])\leq\inf_{\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]\in\partial\operatorname{E}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k})}\left\langle\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1],\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}-\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\rangle\leq 2R\IfEqCase{1}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k1]a_{\operatorname{E}}^{-k}$ for any $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{U}$ with ${\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}\leq R$. Maximising over $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]$ gives $C=2R\IfEqCase{1}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k1]$ * (4) $\operatorname{E}_{0}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k})-\operatorname{E}_{0}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])\leq\inf_{\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1]\in\partial\operatorname{E}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k})}\left\langle\IfEqCase{1}{{0}{u}{1}{v}{2}{w}}[u1],\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}-\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\rangle\leq 2R\IfEqCase{1}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k1]a_{\operatorname{E}}^{-k}$ for any $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\widetilde{\mathds{U}}^{k}$ with ${\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}\leq R$, so $\operatorname{E}_{0}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k})\leq\operatorname{E}_{0}(\widetilde{\mathds{U}}^{k})+2R\IfEqCase{1}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k1]a_{\operatorname{E}}^{-k}$ and $C=(1+2R)\IfEqCase{1}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k1]$. This completes the requirements of Definition 1. ∎ ## Appendix B Proof of Theorem 5.1 First we recall the setting of Definition 2, fix: $p\geq 0$, $q\in[1,\infty]$, $h\in(0,1)$, $N\in\mathds{N}$, connected and bounded domain $\Omega\subset\mathds{R}^{d}$, and $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}\in\operatorname{argmin}_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\mathds{U}}\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0])$. We assume that $\mathds{H}=L^{2}(\Omega)$, ${\left\lVert\cdot\right\rVert}_{q}\lesssim{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\cdot\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}$, and there exist spaces $(\widetilde{\mathds{U}}^{k})_{k\in\mathds{N}}$ with $\widetilde{\mathds{U}}^{k}\subset\mathds{U}$ containing a sequence $(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}\in\widetilde{\mathds{U}}^{k})_{k\in\mathds{N}}$ such that ${\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}-\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}\lesssim h^{kp}$, c.f. (16). Furthermore, there exists constant $c_{\alpha}>0$ and meshes $\mathds{M}^{k}$ such that: $\displaystyle\exists\omega_{0}\subset\Omega\quad\text{such that}\quad\forall\omega\in\mathds{M}^{k}\quad\exists(\IfEqCase{0}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k0]_{\omega},\vec{\IfEqCase{1}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k1]}_{\omega})\in\mathds{R}^{d\times d}\times\mathds{R}^{d}\quad\text{such that}\quad\vec{x}\in\omega_{0}\iff\IfEqCase{0}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k0]_{\omega}\vec{x}+\vec{\IfEqCase{1}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k1]}_{\omega}\in\omega,\qquad\text{ and}$ (99) $\displaystyle\forall(\widetilde{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]},\omega)\in\widetilde{\mathds{U}}^{k}\times\mathds{M}^{k},\quad\exists\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\widetilde{\mathds{U}}^{0}\quad\text{such that}\quad\operatorname{det}(\alpha_{\omega})\geq c_{\alpha}h^{kd}\quad\text{and}\quad\forall\vec{x}\in\omega_{0},\ \IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0](\vec{x})=\widetilde{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}(\alpha_{\omega}\vec{x}+\vec{\beta}_{\omega}).$ (100) In this section, these assumptions will be summarised simply by saying that $\mathds{H}$ and $(\widetilde{\mathds{U}}^{k})_{k\in\mathds{N}}$ satisfy Definition 2. We prove Theorem 5.1 as a consequence of Lemma 7, namely we compute exponents $p^{\prime},q^{\prime}$ with $a_{\operatorname{U}}=h^{-q^{\prime}}$ and $a_{\operatorname{E}}=h^{-p^{\prime}}$. These values are computed as the result of the following three lemmas. The first, Lemma 15, is a quantification of the equivalence between $L^{q}$ and $L^{2}$ norms on general sub-spaces. Lemma 16 applies this result to finite-element spaces to compute the value of $q^{\prime}$. Finally, Lemma 17 then performs the computations for $p^{\prime}$ depending on the smoothness properties of $\operatorname{E}$. ###### Lemma 15 (Equivalence of norms for fixed $k$) Suppose $\mathds{H}=L^{2}(\Omega)$ for some connected, bounded domain $\Omega\subset\mathds{R}^{d}$ and ${\left\lVert\cdot\right\rVert}_{q}\leq C{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\cdot\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}$ for some $q\in[1,\infty]$, $C>0$. For any linear subspace $\widetilde{\mathds{U}}\subset\mathds{U}$ and $\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}\in\widetilde{\mathds{U}}$, ${\left\lVert\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}\right\rVert}\leq\sup_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0],\widetilde{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}\in\widetilde{\mathds{U}}}\frac{\left\langle\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0],\widetilde{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}\right\rangle}{{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\rVert}\,{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\widetilde{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}}{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}\quad\text{where}\quad\sup_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0],\widetilde{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}\in\widetilde{\mathds{U}}}\frac{\left\langle\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0],\widetilde{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}\right\rangle}{{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\rVert}\,{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\widetilde{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}}\leq C^{-1}\begin{cases}\|\Omega\|^{\frac{1}{2}-\frac{1}{q}}&\text{ if }q\geq 2\text{, otherwise}\\\ \|\Omega\|^{1-\frac{1}{q}}\sup_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\widetilde{\mathds{U}}}{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\rVert}_{\infty}/{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\rVert}\qquad&\text{ if }q\in[1,2).\end{cases}$ (101) ###### Proof The first statement of the result is by definition, for each $\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}\in\widetilde{\mathds{U}}\subset L^{\infty}(\Omega)\subset\mathds{H}$ we have ${\left\lVert\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}\right\rVert}=\frac{\left\langle\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]},\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}\right\rangle}{{\left\lVert\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}\right\rVert}}\leq\sup_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\widetilde{\mathds{U}}}\frac{\left\langle\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0],\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}\right\rangle}{{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\rVert}\,{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}}{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}\leq\sup_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0],\widetilde{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}\in\widetilde{\mathds{U}}}\frac{\left\langle\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0],\widetilde{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}\right\rangle}{{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\rVert}\,{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\widetilde{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}}{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}.$ Recall ${\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\cdot\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}\geq C^{-1}{\left\lVert\cdot\right\rVert}_{q}$. To go further we use Hölder’s inequality. If $\frac{1}{q}+\frac{1}{q^{}}=1$, then for any $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0],\widetilde{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}\in\widetilde{\mathds{U}}$ $\frac{\left\langle\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0],\widetilde{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}\right\rangle}{{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\rVert}\,{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\widetilde{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}}\leq C^{-1}\frac{\left\langle\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0],\widetilde{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}\right\rangle}{{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\rVert}\,{\left\lVert\widetilde{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}\right\rVert}_{q}}\leq C^{-1}\frac{{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\rVert}_{q^{}}}{{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\rVert}}.$ (102) If $q\geq 2$ we use Hölder’s inequality a second time: $\int_{\Omega}\|\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0](\vec{x})\|^{q^{}}\mathop{}\\!\mathrm{d}\vec{x}\leq\left(\int_{\Omega}1\mathop{}\\!\mathrm{d}\vec{x}\right)^{1-q^{}/2}\left(\int_{\Omega}\|\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0](\vec{x})\|^{2}\mathop{}\\!\mathrm{d}\vec{x}\right)^{q^{}/2}=\left(\|\Omega\|^{\frac{1}{2}-\frac{1}{q}}{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\rVert}\right)^{q^{}}.$ (103) This confirms the inequality when $q\geq 2$. If $q<2$, we can simply upper bound ${\left\lVert\cdot\right\rVert}_{q^{}}\leq\|\Omega\|^{\frac{1}{q^{}}}{\left\lVert\cdot\right\rVert}_{\infty}$ as required. ∎ ###### Lemma 16 Suppose $\mathds{H}$ and $(\widetilde{\mathds{U}}^{k})_{k\in\mathds{N}}$ satisfy Definition 2, then 1. 1. If $q\geq 2$, then ${\left\lVert\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}\right\rVert}\lesssim 1$ (i.e. $q^{\prime}=0$). 2. 2. If $q<2$ and $\sup_{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\widetilde{\mathds{U}}^{0}}\frac{{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\rVert}_{L^{\infty}(\omega_{0})}}{{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\rVert}_{L^{2}(\omega_{0})}}<\infty$, then ${\left\lVert\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}\right\rVert}\lesssim h^{-\frac{kd}{2}}$ (i.e. $q^{\prime}=-\frac{d}{2}$). ###### Proof Most of the conditions of Lemma 15 are already satisfied. Furthermore observe that ${\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}\lesssim{\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}+h^{kp}\lesssim 1$. For the $q\geq 2$ case, this is already sufficient to conclude ${\left\lVert\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}\right\rVert}\lesssim 1$ from Lemma 15, as required. For the case $q<2$, from Lemma 15 recall that we are required to bound $\sup_{\widetilde{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}\in\widetilde{\mathds{U}}^{k}}\frac{{\left\lVert\widetilde{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}\right\rVert}_{\infty}}{{\left\lVert\widetilde{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}\right\rVert}}=\sup_{\widetilde{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}\in\widetilde{\mathds{U}}^{k}}\sup_{\omega\in\mathds{M}^{k}}\frac{{\left\lVert\widetilde{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}\right\rVert}_{L^{\infty}(\omega)}}{{\left\lVert\widetilde{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}\right\rVert}_{L^{2}(\Omega)}}\leq\sup_{\widetilde{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}\in\widetilde{\mathds{U}}^{k}}\sup_{\omega\in\mathds{M}^{k}}\frac{{\left\lVert\widetilde{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}\right\rVert}_{L^{\infty}(\omega)}}{{\left\lVert\widetilde{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}\right\rVert}_{L^{2}(\omega)}}.$ (104) However, due to the decomposition property (100), for each $\omega\in\mathds{M}^{k}$ and $\widetilde{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}\in\widetilde{\mathds{U}}^{k}$ there exists $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\in\widetilde{\mathds{U}}^{0}$ such that ${\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\rVert}_{L^{\infty}(\omega_{0})}={\left\lVert\widetilde{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}\right\rVert}_{L^{\infty}(\omega)},\qquad{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\rVert}_{L^{2}(\omega_{0})}^{2}=\int_{\omega_{0}}\|\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0](\vec{x})\|^{2}\mathop{}\\!\mathrm{d}\vec{x}=\int_{\omega_{0}}\|\widetilde{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}(\alpha\vec{x}+\vec{\beta})\|^{2}\mathop{}\\!\mathrm{d}\vec{x}=\operatorname{det}(\alpha)^{-1}{\left\lVert\widetilde{\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]}\right\rVert}_{L^{2}(\omega)}^{2}.$ (105) Combining these two equations with the assumed bound on $\frac{{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\rVert}_{L^{\infty}(\omega_{0})}}{{\left\lVert\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]\right\rVert}_{L^{2}(\omega_{0})}}$ confirms ${\left\lVert\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}\right\rVert}\lesssim\sqrt{\operatorname{det}(\alpha)^{-1}}\leq c_{\alpha}^{-\frac{1}{2}}h^{-\frac{kd}{2}}$ as required. ∎ ###### Lemma 17 Suppose $\mathds{H}$ and $(\widetilde{\mathds{U}}^{k})_{k\in\mathds{N}}$ satisfy Definition 2 and $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}$ is the minimiser of $E$ such that ${\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}-\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}\lesssim h^{kp}$. 1. 1. If $\operatorname{E}$ is ${\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\cdot\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}$-Lipschitz at $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}$, then $\operatorname{E}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k})-\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{})\lesssim h^{kp}$ (i.e. $p^{\prime}=p$). 2. 2. If $\nabla\operatorname{E}$ is ${\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\cdot\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}$-Lipschitz at $\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}$, then $\operatorname{E}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k})-\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{})\lesssim h^{2kp}$ (i.e. $p^{\prime}=2p$). ###### Proof Both statements are direct by definition, observe $\displaystyle\operatorname{E}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k})-\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{})\leq\operatorname{Lip}(\operatorname{E}){\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}-\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|},$ (106) $\displaystyle\operatorname{E}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k})-\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{})\leq\left\langle\nabla\operatorname{E}(\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]),\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}-\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}\right\rangle=\left\langle\nabla\operatorname{E}(\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k})-\nabla\operatorname{E}(\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}),\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}-\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}\right\rangle\leq\operatorname{Lip}(\nabla\operatorname{E}){\left\|\kern-1.07639pt\left\|\kern-1.07639pt\left\|\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}-\IfEqCase{0}{{0}{u}{1}{v}{2}{w}}[u0]^{}\right\|\kern-1.07639pt\right\|\kern-1.07639pt\right\|}^{2}.$ (107) The proof is concluded by using the approximation bounds of $\widetilde{\IfEqCase{2}{{0}{u}{1}{v}{2}{w}}[u2]}_{k}$ in Definition 2. ∎ ## Appendix C Operator norms for numerical examples ###### Theorem C.1 Suppose $\mathsf{A}\colon\mathds{H}\to\mathds{R}^{m}$ has kernels $\psi_{j}\in L^{\infty}([0,1]^{d})$ for $j\in[m]$. 1. Case 1: If $\psi_{j}(\vec{x})=\begin{cases}1&\vec{x}\in\mathds{X}_{j}\\\ 0&\text{ else}\end{cases}$ for some collection $\mathds{X}_{j}\subset\Omega$ such that $\mathds{X}_{i}\cap\mathds{X}_{j}=\emptyset$ for all $i\neq j$, then ${\left\lVert\mathsf{A}\right\rVert}_{L^{2}\to\ell^{2}}=\max_{j\in[m]}\sqrt{\|\mathds{X}_{j}\|}.$ 2. Case 2: If $\psi_{j}(\vec{x})=\cos(\vec{\IfEqCase{3}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k3]}_{j}\vbox{\hbox{\leavevmode\resizebox{6.0pt}{}{$\cdot$}}}\vec{x})$ for some frequencies $\vec{\IfEqCase{3}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k3]}_{j}\in\mathds{R}^{d}$ with $\|\vec{\IfEqCase{3}{{0}{\alpha}{1}{\beta}{2}{\gamma}{3}{a}{4}{b}{5}{c}}[k3]}_{j}\|\leq A$, then ${\left\lVert\mathsf{A}\right\rVert}_{L^{2}\to\ell^{2}}\leq\sqrt{m},\qquad\|\mathsf{A}^{}\vec{r}\|_{C^{k}}\leq m^{1-\frac{1}{q}}A^{k}{\left\lVert\vec{r}\right\rVert}_{q},\quad\text{and}\quad\|\mathsf{A}^{}\|_{\ell^{2}\to C^{k}}\leq\sqrt{m}A^{k}$ for all $\vec{r}\in\mathds{R}^{m}$ and $q\in[1,\infty]$. 3. Case 3: Suppose $\psi_{j}(\vec{x})=(2\pi\sigma^{2})^{-\frac{d}{2}}\exp\left(-\frac{\|\vec{x}-\vec{x}_{j}\|^{2}}{2\sigma^{2}}\right)$ for some regular mesh $\vec{x}_{j}\in[0,1]^{d}$ and separation $\Delta$. i.e. $\\{\vec{x}_{j}\operatorname{\;s.t.\;}j\in[m]\\}=\\{\vec{x}_{0}+(j_{1}\Delta,\ldots,j_{d}\Delta)\operatorname{\;s.t.\;}j_{i}\in[\widehat{m}]\\}$ for some $\vec{x}_{0}\in\mathds{R}^{d}$, $\widehat{m}\coloneqq\sqrt[d]{m}$. For all $\frac{1}{q}+\frac{1}{q^{}}=1$, $q\in(1,\infty]$, we have $\displaystyle{\left\lVert\mathsf{A}\right\rVert}_{L^{2}\to\ell^{2}}$ $\displaystyle\leq\bigg{(}(4\pi\sigma^{2})^{-\frac{1}{2}}\sum_{j=-2\widehat{m},\ldots,2\widehat{m}}\exp(-\tfrac{\Delta^{2}}{4\sigma^{2}}j^{2})\bigg{)}^{d},$ (108) $\displaystyle\|\mathsf{A}^{}\vec{r}\|_{C^{0}}$ $\displaystyle\leq(2\pi\sigma^{2})^{-\frac{d}{2}}\bigg{(}\sum_{\vec{j}\in J}\exp\left(-\tfrac{q^{}\Delta^{2}}{2\sigma^{2}}\max(0,\|\vec{j}\|-\delta)^{2}\right)\bigg{)}^{\frac{1}{q^{}}}{\left\lVert\vec{r}\right\rVert}_{q},$ (109) $\displaystyle\|\mathsf{A}^{}\vec{r}\|_{C^{1}}$ $\displaystyle\leq\frac{(2\pi\sigma^{2})^{-\frac{d}{2}}}{\sigma}\frac{\Delta}{\sigma}\bigg{(}\sum_{\vec{j}\in J}(\|\vec{j}\|+\delta)^{q^{}}\exp\left(-\tfrac{q^{}\Delta^{2}}{2\sigma^{2}}\max(0,\|\vec{j}\|-\delta)^{2}\right)\bigg{)}^{\frac{1}{q^{}}}{\left\lVert\vec{r}\right\rVert}_{q},$ (110) $\displaystyle\|\mathsf{A}^{}\vec{r}\|_{C^{2}}$ $\displaystyle\leq\frac{(2\pi\sigma^{2})^{-\frac{d}{2}}}{\sigma^{2}}\bigg{(}\sum_{\vec{j}\in J}\left(1+\tfrac{\Delta^{2}}{\sigma^{2}}(\|\vec{j}\|+\delta)^{2}\right)^{q^{}}\exp\left(-\tfrac{q^{}\Delta^{2}}{2\sigma^{2}}\max(0,\|\vec{j}\|-\delta)^{2}\right)\bigg{)}^{\frac{1}{q^{}}}{\left\lVert\vec{r}\right\rVert}_{q},$ (111) where $\delta=\frac{\sqrt{d}}{2}$ and $J=\\{\vec{j}\in\mathds{Z}^{d}\operatorname{\;s.t.\;}{\left\lVert\vec{j}\right\rVert}_{\ell^{\infty}}\leq 2\widehat{m}\\}$. The case for $q=1$ can be inferred from the standard limit of ${\left\lVert\cdot\right\rVert}_{{q^{}}}\to{\left\lVert\cdot\right\rVert}_{\infty}$ for $q^{}\to\infty$. ###### Proof (Case 1.) From Lemma 8 we have $(\mathsf{A}\mathsf{A}^{})_{i,j}=\left\langle\mathds{1}_{\mathds{X}_{i}},\mathds{1}_{\mathds{X}_{j}}\right\rangle=\|\mathds{X}_{i}\cap\mathds{X}_{j}\|=\begin{cases}\|\mathds{X}_{i}\|&i=j\\\ 0&i\neq j\end{cases}.$ (112) Therefore, $\mathsf{A}\mathsf{A}^{}$ is a diagonal matrix and ${\left\lVert\mathsf{A}\mathsf{A}^{}\right\rVert}_{\ell^{2}\to\ell^{2}}=\max_{j\in[m]}\|\mathds{X}_{j}\|$ completes the result. ∎ ###### Proof (Case 2.) $\psi_{j}$ are not necessarily orthogonal however $\|\left\langle\psi_{i},\psi_{j}\right\rangle\|\leq 1$ therefore we can estimate ${\left\lVert\mathsf{A}\mathsf{A}^{}\right\rVert}_{\ell^{2}\to\ell^{2}}\leq{\left\lVert\mathsf{A}\mathsf{A}^{}\right\rVert}_{\ell^{\infty}\to\ell^{\infty}}\leq m.$ (113) Now looking to apply Lemma 9, note ${\left\lVert\nabla^{k}\psi_{j}\right\rVert}_{\infty}\leq A^{k}$, therefore $\|\mathsf{A}^{}\vec{r}\|_{C^{k}}\leq A^{k}m^{\frac{1}{q^{}}}{\left\lVert\vec{r}\right\rVert}_{q}=A^{k}m^{1-\frac{1}{q}}{\left\lVert\vec{r}\right\rVert}_{q}\quad\text{and}\quad\|\mathsf{A}^{}\|_{\ell^{2}\to C^{k}}\leq A^{k}\min_{q\in[1,\infty]}m^{1-\frac{1}{q}}\sqrt{m}^{\max(0,2-q)}=\sqrt{m}A^{k}.$ (114) ∎ ###### Proof (Case 3.) In the Gaussian case, we build our approximations around the idea that sums of Gaussians should converge very quickly. The first example can be used to approximate the operator norm. Computing the inner products gives $\left\langle\psi_{i},\psi_{j}\right\rangle=(2\pi\sigma^{2})^{-d}\int_{[0,1]^{d}}\exp\left(-\tfrac{\|\vec{x}-\vec{x}_{i}\|^{2}}{2\sigma^{2}}-\tfrac{\|\vec{x}-\vec{x}_{j}\|^{2}}{2\sigma^{2}}\right)\mathop{}\\!\mathrm{d}\vec{x}\leq(2\pi\sigma^{2})^{-d}(\pi\sigma^{2})^{\frac{d}{2}}\exp\left(-\tfrac{\|\vec{x}_{i}-\vec{x}_{j}\|^{2}}{4\sigma^{2}}\right).$ (115) Estimating the operator norm, $\displaystyle{\left\lVert\mathsf{A}\mathsf{A}^{}\right\rVert}_{\ell^{2}\to\ell^{2}}$ $\displaystyle\leq{\left\lVert\mathsf{A}\mathsf{A}^{}\right\rVert}_{\ell^{\infty}\to\ell^{\infty}}=\max_{i\in[m]}\sum_{j=1}^{m}\|\left\langle\psi_{i},\psi_{j}\right\rangle\|$ (116) $\displaystyle=\max_{i\in[m]}(4\pi\sigma^{2})^{-\frac{d}{2}}\sum_{j_{1},\ldots,j_{d}\in[\widehat{m}]}\exp\left(-\frac{(j_{1}\Delta- i_{1}\Delta)^{2}+\ldots+(j_{d}\Delta-i_{d}\Delta)^{2}}{4\sigma^{2}}\right)$ (117) $\displaystyle\leq(4\pi\sigma^{2})^{-\frac{d}{2}}\sum_{\vec{j}\in\mathds{Z}^{d}\cap[-\widehat{m},\widehat{m}]^{d}}\exp\left(-\frac{(j_{1}\Delta)^{2}+\ldots+(j_{d}\Delta)^{2}}{4\sigma^{2}}\right)=\left[(4\pi\sigma^{2})^{-\frac{1}{2}}\sum_{j=-\widehat{m}}^{\widehat{m}}\exp\left(-\frac{\Delta^{2}j^{2}}{4\sigma^{2}}\right)\right]^{d}.$ (118) This is a nice approximation because it factorises simply over dimensions. Applying the results from Lemma 9, note $\begin{array}[]{rll}\displaystyle\|\psi_{j}(\vec{x})\|&\displaystyle=\left\|\psi_{j}(\vec{x})\right\|&\displaystyle=(2\pi\sigma^{2})^{-\frac{d}{2}}\exp\left(-\frac{\|\vec{x}-\vec{x}_{j}\|^{2}}{2\sigma^{2}}\right),\\\ \displaystyle\|\nabla\psi_{j}(\vec{x})\|&\displaystyle=\left\|\frac{\vec{x}-\vec{x}_{j}}{\sigma^{2}}\psi_{j}(\vec{x})\right\|&\displaystyle=\frac{(2\pi\sigma^{2})^{-\frac{d}{2}}}{\sigma}\frac{\|\vec{x}-\vec{x}_{j}\|}{\sigma}\exp\left(-\frac{\|\vec{x}-\vec{x}_{j}\|^{2}}{2\sigma^{2}}\right),\\\ \|\nabla^{2}\psi_{j}(\vec{x})\|&\displaystyle=\left\|\frac{1}{\sigma^{2}}+\frac{(\vec{x}-\vec{x}_{j})(\vec{x}-\vec{x}_{j})^{\top}}{\sigma^{4}}\right\|\psi_{j}(\vec{x})&\displaystyle=\frac{(2\pi\sigma^{2})^{-\frac{d}{2}}}{\sigma^{2}}\left(1+\frac{\|\vec{x}-\vec{x}_{j}\|^{2}}{\sigma^{2}}\right)\exp\left(-\frac{\|\vec{x}-\vec{x}_{j}\|^{2}}{2\sigma^{2}}\right).\end{array}$ We now wish to sum over $j=1,\ldots,m$ and produce an upper bound on these, independent of $t$. To do so we will use the following lemma. ###### Lemma 18 Suppose $q>0$. If the polynomial $p(\|\vec{x}\|)=\sum p_{k}\|\vec{x}\|^{k}$ has non-negative coefficients and $\vec{x}\in[-m,m]^{d}$, then $\sum_{{\left\lVert\vec{j}\right\rVert}_{\ell^{\infty}}\leq m}p(\|\vec{j}-\vec{x}\|)\exp\left(-\tfrac{q\|\vec{j}-\vec{x}\|^{2}}{2}\right)\leq\sum_{{\left\lVert\vec{j}\right\rVert}_{\ell^{\infty}}\leq 2m}p(\|\vec{j}\|+\delta)\exp\left(-\frac{q\max(0,\|\vec{j}\|-\delta)^{2}}{2}\right)$ where $\delta\coloneqq\frac{\sqrt{d}}{2}$ and $\vec{j}\in\mathds{Z}^{d}$. ###### Proof There exists $\widehat{\vec{x}}\in[-\tfrac{1}{2},\tfrac{1}{2}]^{d}$ such that $\vec{x}+\widehat{\vec{x}}\in\mathds{Z}^{d}$, therefore $\displaystyle\sum_{{\left\lVert\vec{j}\right\rVert}_{\ell^{\infty}}\leq m}p(\|\vec{j}-\vec{x}\|)\exp\left(-\tfrac{q\|\vec{j}-\vec{x}\|^{2}}{2}\right)$ $\displaystyle=\sum_{{\left\lVert\vec{j}\right\rVert}_{\ell^{\infty}}\leq m}p(\|\vec{j}-(\vec{x}+\widehat{\vec{x}})+\widehat{\vec{x}}\|)\exp\left(-\tfrac{q\|\vec{j}-(\vec{x}+\widehat{\vec{x}})+\widehat{\vec{x}}\|^{2}}{2}\right)$ $\displaystyle\leq\sum_{{\left\lVert\vec{j}\right\rVert}_{\ell^{\infty}}\leq 2m}p(\|\vec{j}+\widehat{\vec{x}}\|)\exp\left(-\tfrac{q\|\vec{j}+\widehat{\vec{x}}\|^{2}}{2}\right)$ $\displaystyle\leq\sum_{\begin{subarray}{c}\vec{j}\in\mathds{Z}^{d}\\\ {\left\lVert\vec{j}\right\rVert}_{\ell^{\infty}}\leq 2m\end{subarray}}p(\|\vec{j}\|+\delta)\exp\left(-\tfrac{q\max(0,\|\vec{j}\|-\delta)^{2}}{2}\right)$ as $\|\widehat{\vec{x}}\|\leq\delta$ and $p$ has non-negative coefficients. ∎ Now, continuing the proof of Theorem C.1, for $\widehat{m}=\sqrt[d]{m}$, $\delta=\frac{\sqrt{d}}{2}$ and $J=\\{\vec{j}\in\mathds{Z}^{d}\operatorname{\;s.t.\;}{\left\lVert\vec{j}\right\rVert}_{\ell^{\infty}}\leq 2\widehat{m}\\}$, Lemma 18 bounds $\displaystyle\sum_{j=1}^{m}\|\psi_{j}(\vec{x})\|^{q^{}}$ $\displaystyle\leq(2\pi\sigma^{2})^{-\frac{dq^{}}{2}}\left[\sum_{\vec{j}\in J}\exp\left(-\frac{q^{}\Delta^{2}}{2\sigma^{2}}\max(0,\|\vec{j}\|-\delta)^{2}\right)\right]$ $\displaystyle\sum_{j=1}^{m}\|\nabla\psi_{j}(\vec{x})\|^{q^{}}$ $\displaystyle\leq\frac{(2\pi\sigma^{2})^{-\frac{dq^{}}{2}}}{\sigma^{q^{}}}\frac{\Delta^{q^{}}}{\sigma^{q^{}}}\left[\sum_{\vec{j}\in J}(\|\vec{j}\|+\delta)^{q^{}}\exp\left(-\frac{q^{}\Delta^{2}}{2\sigma^{2}}\max(0,\|\vec{j}\|-\delta)^{2}\right)\right]$ $\displaystyle\sum_{j=1}^{m}\|\nabla^{2}\psi_{j}(\vec{x})\|^{q^{}}$ $\displaystyle\leq\frac{(2\pi\sigma^{2})^{-\frac{dq^{}}{2}}}{\sigma^{2q^{}}}\left[\sum_{\vec{j}\in J}\left(1+\frac{\Delta^{2}}{\sigma^{2}}(\|\vec{j}\|+\delta)^{2}\right)^{q^{}}\exp\left(-\frac{q^{}\Delta^{2}}{2\sigma^{2}}\max(0,\|\vec{j}\|-\delta)^{2}\right)\right]$ for all $\vec{x}\in\Omega$. In a worst case, this is $O(2^{d}m)$ time complexity however the summands all decay faster than exponentially and so should converge very quickly. ∎
# X-ray Scatter Estimation Using Deep Splines Philipp Roser, Annette Birkhold, Alexander Preuhs, Christopher Syben, Lina Felsner, Elisabeth Hoppe, Norbert Strobel, Markus Korwarschik, Rebecca Fahrig, Andreas Maier P. Roser, A. Preuhs, C. Syben, L. Felsner, E. Hoppe, and A. Maier are with the Pattern Recognition Lab, Department of Computer Science, Friedrich-Alexander Universität Erlangen-Nürnberg, Erlangen, Germany. P. Roser is funded by the Erlangen Graduate School in Advanced Optical Technologies (SAOT), Friedrich-Alexander Universität Erlangen-Nürnberg, Erlangen, Germany. A. Maier is principal investigator at the SAOT. A. Birkhold, M. Kowarschik, and R. Fahrig are employees of Siemens Healthcare GmbH, 91301 Forchheim, Germany. N. Strobel is with the Institute of Medical Engineering Schweinfurt, University of Applied Sciences Würzburg-Schweinfurt, 97421 Schweinfurt, Germany. ###### Abstract Algorithmic X-ray scatter compensation is a desirable technique in flat-panel X-ray imaging and cone-beam computed tomography. State-of-the-art U-net based image translation approaches yielded promising results. As there are no physics constraints applied to the output of the U-Net, it cannot be ruled out that it yields spurious results. Unfortunately, those may be misleading in the context of medical imaging. To overcome this problem, we propose to embed B-splines as a known operator into neural networks. This inherently limits their predictions to well-behaved and smooth functions. In a study using synthetic head and thorax data as well as real thorax phantom data, we found that our approach performed on par with U-net when comparing both algorithms based on quantitative performance metrics. However, our approach not only reduces runtime and parameter complexity, but we also found it much more robust to unseen noise levels. While the U-net responded with visible artifacts, our approach preserved the X-ray signal’s frequency characteristics. ###### Index Terms: Approximation, B-spline, Neural network, X-ray scatter. ## I Introduction Xray transmission imaging enables advanced diagnostic imaging or facilitates instrument guidance during minimally-invasive interventions. Unfortunately, primary photons scattered by the patient’s body impair image quality at the detector. Especially cone-beam X-ray systems using flat-panel detectors suffer from this phenomenon due to the large X-ray field and the resulting high scatter-to-primary ratio. Besides contrast degradation in X-ray projections, artifacts in cone-beam computed tomography (CBCT) deteriorate the image quality in the reconstructed volume. Since the advent of CBCT, various methods to compensate for scatter have been developed [1, 2]. These can be categorized into physical and algorithmic scatter compensation methods. ### I-A Physical Scatter Compensation Physical scatter compensation refers to the direct manipulation of the X-ray field to either suppress scattered photons from reaching the detector, or modulate them to enable differentiation into scattered and primary photons. A simple yet effective approach is to increase the patient to detector distance, the so-called air gap [3, 4, 5, 6]. Since the number of scattered photons depends on the irradiated volume, the scatter-to-primary ratio can be reduced by shaping the X-ray field to a small volume of interest. In slit scanning, the stitching of these smaller volumes of interest yields a normal-sized reconstruction [7, 8]. Although slit scanning reduces patient dose and scattered radiation, the prolonged acquisition time complicates motion compensation and X-ray tube heat management [9]. Since relying on large air gaps or strict collimation limits the flexibility of the imaging system, clinical systems are equipped with anti-scatter grids that physically block scattered photons [5, 10, 11, 12, 13, 14]. Although state-of-the-art for clinical CBCT, anti-scatter grids have disadvantages. For example, they can increase radiation exposure [15] and lead to Moiré and grid line artifacts [16]. Overall, the efficiency and effectiveness of anti-scatter grids depends on multiple factors, e.g., air gap, photon energy, source-to-detector distance, and detector resolution [2, 17, 18]. In contrast to detector-side grids, primary modulation follows a different operation principle [19, 20, 21]. In the shadow of this modulator grid, almost no primary radiation is measured. Instead, information of the scatter properties of the irradiated volume is encoded. Using demodulation algorithms, the X-ray projection and the associated scatter distribution can be distinguished. Due to its complex structure, primary modulation, especially with C-arm systems, has not made its way into the clinical practice. In conclusion, hardware-based solutions induce additional manufacturing costs and limit flexibility. Also, since most approaches need algorithmic support or look-up tables, software-only solutions are highly desirable [1, 2]. ### I-B Algorithmic Scatter Compensation Algorithmic scatter compensation approaches typically try to estimate the scatter signal from the acquired X-ray projections first. The estimated scatter image is then subtracted to obtain the primary signal. For CBCT, using Monte Carlo (MC) methods or Boltzmann transport solvers are common approaches [22, 23, 24]. Both require an initial reconstruction. Their computational complexity can partially be coped with using dedicated hardware and variance reduction techniques [25] or by using a coarse simulation to derive the scattering model online [26]. Trading off effectiveness for efficiency, model- based approaches are typically preferred for clinical applications. Such approaches rely on either simplified physical or analytical models [27, 28, 29] or convolution kernels [30, 31, 32] in combination with an iterative correction scheme. Although being fast, their ability to generalize to different imaging settings is limited. Recently, learning-based methods found their way to modeling physical processes [33, 34, 35]. Especially the application of the U-net [36] to the task of scatter estimation, referred to as deep scatter estimation, yields superior results compared to kernel-based or MC methods, respectively [37]. The U-net based method has the potential to become the new gold standard in the domain of scatter correction. However, there are several drawbacks to employing a deep U-net. First, deep convolutional neural networks are challenging to comprehend in their operating principle. Given the high parameter complexity of such a U-net, large amounts of training samples are mandatory to arrive at a robust solution. Second, to maintain a fast performance, the U-net requires a dedicated top tier graphics processing unit (GPU), which might not be universally available. Third, the U-net is a universal function approximator without including any known physical characteristics of scattered radiation. To come up with an alternative, it appears attractive to build on the rich prior knowledge about X-ray scatter. Also, it has been shown that the incorporation of prior knowledge into neural networks reduces error bounds [38] and simplifies the analysis of the deployed network [39]. Although subject to photon Poisson noise, X-ray scatter is mainly a low-frequency signal in the diagnostic energy regime [40, 41]. In a previous proof-of- concept study, we exploited this property by approximating X-ray scatter using a smooth bivariate B-spline and directly inferred spline coefficients using a lean convolutional encoder architecture [42]. This (a) allowed us to reduce the number of parameters and computational complexity tremendously, (b) ensured that no high-frequency details of diagnostic value can get manipulated, while (c) still meeting the high accuracy of the U-net. ### I-C Contributions In this work, we extend the idea of approximating X-ray scatter as bivariate B-spline by integrating its evaluation into the computational graph of a neural network. We analyze our approach in depth and compare it to several U-net realizations in a nested cross-validation study using synthetic head and thorax X-ray image data. Besides benchmarking the parameter and computational complexity, we investigate all networks in terms of their frequency response, the power spectral density of the co-domain, and the response to unseen noise levels. Finally, we demonstrate how our method performs when applied to real data in an anthropomorphic thorax phantom study. ## II Materials and Methods ### II-A X-ray Projection Formation using B-splines In general, in both techniques, X-ray fluoroscopy and CBCT it is assumed that primary photons either reach the detector along a straight line or are absorbed completely. Therefore, the observed X-ray projection is typically described in terms of its primary component $\boldsymbol{I}_{\text{p}}\in\mathbb{R}^{w\times h}$ with image width $w$ and height $h$ in pixels. The intensity at pixel $(u,v)$ is given by the Beer- Lambert law ${I}_{\text{p}}\left(u,v\right)=\int_{E}{I}_{0}\left(u,v,E\right)e^{-\int_{\lambda}\mu\left(\boldsymbol{s}+\lambda\cdot\boldsymbol{r}\left(u,v\right),E\right)d\lambda}dE\enspace.$ (1) The polychromatic flat-field projection or X-ray spectrum $\boldsymbol{I}_{0}$ as well as the linear attenuation coefficient $\mu\left(\boldsymbol{l},E\right)$ depend on the photon energy $E$. Note that, due to the cone-beam geometry, $\boldsymbol{I}_{0}$ decreases towards the borders and thus depends on the pixel position $(u,v)$. The linear photon attenuation further depends on the media along the straight line $\boldsymbol{l}:\mathbb{R}\mapsto\mathbb{R}^{3}$ from the X-ray source $\boldsymbol{s}\in\mathbb{R}^{3}$ in direction $\boldsymbol{r}\in\mathbb{R}^{3}$ to the pixel $(u,v)$. Unfortunately, in reality, besides being photoelectrically absorbed, X-ray photons also undergo Compton scattering, Rayleigh scattering, or multiple occurrences of both effects. Therefore, a more realistic formation model $\boldsymbol{I}$ adds scattered photons $\boldsymbol{I}_{\text{s}}$ leading to ${I}\left(u,v\right)={I}_{\text{p}}\left(u,v\right)+{I}_{\text{s}}\left(u,v\right)\enspace.$ (2) Since, for CBCT, we are mostly interested in the low-frequency components of the scatter, we neglect photon shot noise and detector noise below. In a recent study [42], we experimentally showed that the main scatter components, which are low-frequency for diagnostic X-rays, can be well approximated by sparse bivariate B-splines with error rates below $1\text{\,}\mathrm{\char 37\relax}$. The domain and co-domain of a B-spline of degree $k$ and order $n=k+1$ are fully characterized by the knot vectors $\boldsymbol{t}_{u}=(t_{u,1},t_{u,2},\dots t_{u,w_{c}-n})$ and $\boldsymbol{t}_{v}=(t_{v,1},t_{v,2},\dots t_{v,h_{c}-n})$, as well as the coefficient matrix $\boldsymbol{C}\in\mathbb{R}^{w_{\text{c}}\times h_{\text{c}}}$ with width $w_{\text{c}}$ and height $h_{\text{c}}$, respectively. Based on this, we can approximate the spatial scatter distribution as a tensor product B-spline $\tilde{\boldsymbol{I}}_{\text{s},n}$ $\tilde{I}_{\text{s},n}\left(u,v\right)=\sum_{i=1}^{w_{\text{c}}}\sum_{j=1}^{h_{\text{c}}}C\left(i,j\right)B_{i,n,\boldsymbol{t}_{u}}\left(u\right)B_{j,n,\boldsymbol{t}_{v}}\left(v\right)\,,$ (3) with the basis splines $B$, which are zero for knots that do not affect the spline at the pixel $(u,v)$. The basis splines are recursively defined by $\displaystyle B_{i,n,\boldsymbol{t}}(x)$ $\displaystyle={}$ $\displaystyle\frac{x-t_{i}}{t_{i+n}-t_{i}}$ $\displaystyle\cdot B_{i,n-1,\boldsymbol{t}}(x)$ (4) $\displaystyle+{}$ $\displaystyle\frac{t_{i+n+1}-x}{t_{i+n+1}-t_{i+1}}$ $\displaystyle\cdot B_{i+1,n-1,\boldsymbol{t}}(x)\enspace,$ with $B_{i,0,\boldsymbol{t}}(x)=\begin{cases}1,&\text{if }t_{i}\leq x<t_{i+1}\\\ 0,&\text{otherwise}\end{cases}\enspace.$ (5) Since the basis functions $B$ are equal to zero for coefficients that do not contribute to a pixel $(u,v)$, only a $n\times n$ sub-grid of $\boldsymbol{C}$ needs to be considered for each pixel. Thus, using matrix notation, the tensor product can be reformulated to $\tilde{I}_{\text{s},n}\left(u,v\right)=\boldsymbol{u}^{\text{T}}\cdot\left[\boldsymbol{M}_{n,\boldsymbol{t}_{u}}\left(u\right)\right]\cdot\boldsymbol{C}_{\text{uv}}\cdot\left[\boldsymbol{M}_{n,\boldsymbol{t}_{v}}\left(v\right)\right]^{\text{T}}\cdot\boldsymbol{v}\enspace,$ (6) with the coefficient patch $\boldsymbol{C}_{\text{uv}}\in\mathbb{R}^{n\times n}$ impacting the pixel $(u,v)$, the general matrix representation of a univariate B-spline $\boldsymbol{M}_{n,\boldsymbol{t}}\left(\cdot\right)\in\mathbb{R}^{n\times n}$ [43], and the vectors $\boldsymbol{u}=\begin{pmatrix}u^{0}&u^{1}&\cdots&u^{k}\end{pmatrix}^{\text{T}}$ and $\boldsymbol{v}=\begin{pmatrix}v^{0}&v^{1}&\cdots&v^{k}\end{pmatrix}^{\text{T}}$. The general matrix representation of B-splines allows for arbitrarily spaced knot grids, i.e., in our case, it allows for endpoint interpolation, which exposes a more convenient behavior towards the borders of the evaluation grid. Besides the borders of the grid, we currently limit our approach to uniform knot grids and cubic B-splines ($k=3$). Thus, for most pixels $(u,v)$, the B-spline is evaluated using $\boldsymbol{M}_{4}=\frac{1}{6}\begin{pmatrix}1&4&1&0\\\ -3&0&3&0\\\ 3&-6&3&0\\\ 1&3&-3&1\end{pmatrix}\enspace.$ (7) With using a fixed knot grid and evaluation grid, we can pre-calculate both $\boldsymbol{u}^{\text{T}}\cdot\left[\boldsymbol{M}_{n,\boldsymbol{t}_{u}}\left(u\right)\right]$ and $\boldsymbol{v}^{\text{T}}\cdot\left[\boldsymbol{M}_{n,\boldsymbol{t}_{v}}\left(v\right)\right]$ for each pixel $(u,v)$. Padding the resulting vectors with zeros to account for all coefficients in $\boldsymbol{C}$ and stacking them row-wisely, we obtain the evaluation matrices $\boldsymbol{U}_{n}\in\mathbb{R}^{w\times w_{\text{c}}}$ and $\boldsymbol{V}_{n}\in\mathbb{R}^{h\times h_{\text{c}}}$, respectively. Thus, we can calculate the cubic ($n=4$) scatter distribution given the B-spline coefficients $\boldsymbol{C}$ via $\tilde{\boldsymbol{I}}_{\text{s},4}=\boldsymbol{U}_{4}\cdot\boldsymbol{C}\cdot\boldsymbol{V}_{4}^{\text{T}}\enspace.$ (8) Since neural networks can extract scatter distributions from a measured X-ray projection either directly [37] or indirectly via B-spline coefficients [42] we aim to combine deep learning with the matrix evaluation scheme. Using the Kronecker product ‘$\otimes$’, the derivative $\frac{\partial\tilde{\boldsymbol{I}}_{\text{s},4}}{\partial\boldsymbol{C}}$ is $\frac{\partial\tilde{\boldsymbol{I}}_{\text{s},4}}{\partial\boldsymbol{C}}=\boldsymbol{V}_{4}\otimes\boldsymbol{U}_{4}\enspace.$ (9) Thus, we can straightforwardly embed the B-spline evaluation into a computational graph $f_{\boldsymbol{\theta}}:\mathbb{R}^{w\times h}\mapsto\mathbb{R}^{w\times h}$ with parameters to train $\boldsymbol{\theta}$ without breaking differentiability and thus back- propagation. ### II-B Network Architecture Figure 1: The general architecture of our proposed approach. First, a (convolutional) encoder extracts the latent variables $\boldsymbol{Z}$ from the input X-ray projection $\boldsymbol{I}$. Second, a bottleneck network maps the latent space to bivariate spline coefficients $\boldsymbol{C}$. The different choices for both networks are discussed later in more detail. Third, we obtain the scatter estimate $\tilde{\boldsymbol{I}}_{\text{s},n}$ by evaluating the spline coefficients $\boldsymbol{C}$ using the pre-calculated evaluation grid defined by the pre-computed matrices $\boldsymbol{U}$ and $\boldsymbol{V}$. The top (green) path refers to the forward pass, and the bottom (orange) path to the backward pass, respectively. In the following, we describe the devised neural network. The overall architecture comprises a generic convolutional encoder followed by a bottleneck network to infer spline coefficients from measured X-ray projections, as proposed in our previous study [42]. This architecture is depicted in Fig. 1. We employ a lean convolutional encoder $f_{\boldsymbol{\theta}}:\mathbb{R}^{w\times h}\mapsto\mathbb{R}^{w_{\text{c}}\times h_{\text{c}}}$ that consists of $d$ convolutional blocks. A block comprises two convolutional layers with $c$ feature channels and $3\times 3$ kernels. Each convolutional layer is followed by a rectified linear unit (ReLU) [44] activation, and between two blocks, $2\times 2$ average pooling is applied. As X-ray scatter is low-frequency and to potentially limit the computational complexity of our model, we allow for additional pooling layers before the first convolutional block. The encoder $f_{\boldsymbol{\theta}}$ is completed by a $1\times 1$ convolution to build the weighted sum of all channels $c$. Overall, based on convolutions only, $f_{\boldsymbol{\theta}}$ only encodes a local scatter representation $\boldsymbol{Z}$, since its receptive field does not necessarily cover the whole input image. To establish a global context, we employ different types of bottleneck networks $g_{\boldsymbol{\phi}}$ to map $\boldsymbol{Z}$ to spline coefficients $\boldsymbol{C}$: (1) a constrained weighting matrix $\boldsymbol{W}$ with $W_{i,j}>0$, (2) an unconstrained fully-connected layer with ReLU activation (which merely relates to an unconstrained weighting matrix with enforced non-negativity constraint), (3) two fully-connected layers with ReLU activation, or (4) two additional convolutional blocks followed by a fully-connected layer. ### II-C Synthetic Dataset Acquiring raw scatter-free X-ray projections and their scatter-contaminated counterparts is tedious and time-consuming, especially at the scale to carry out deep learning. As a solution, we leveraged the X-ray transport code MC-GPU [25] to generate artificial pairs of scatter-contaminated and scatter-free X-ray projections. We used openly available CT scans from The Cancer Imaging Archive (TCIA) [45] as inputs to the simulation. In order to keep the simulation time manageable, we selected 20 head scans from the HNSCC-3DCT-RT dataset [46] and 15 thorax scans from the CT Lymph Nodes dataset [47], respectively. To prepare the phantoms for MC simulation, we employed a basic pre-processing pipeline [48] based on tissue and density estimation [49], and connected component labeling [50]. For each CT scan, we simulated a stack of 260 X-ray projections ($w=1152$, $h=768$) over an angular range of $200\text{\,}\mathrm{\SIUnitSymbolDegree}$. The source-to-isocenter and source-to-detector distances are $785\text{\,}\mathrm{mm}$ and $1300\text{\,}\mathrm{mm}$, respectively. Per X-ray projection, we simulated $5\text{\times}{10}^{10}$ photons sampled from an $85\text{\,}\mathrm{kV}$ peak voltage tungsten spectrum. All projections were flat-field normalized. Note that we used the data as is, without registering similar anatomies in a common reference frame. To be more comparable to the related work [37], speed- up training times, and suppress simulation noise, we down-sampled the projections to $384\times 256$ pixels. Before the down-sampling, we applied Gaussian filtering to the primary and scatter projections independently ($\sigma_{\text{p}}=2$, $\sigma_{\text{s}}=30$). Corresponding cross-sectional slices were reconstructed on an isotropic ${256}^{3}$ grid with $1\text{\,}{\mathrm{mm}}^{3}$ voxels. ### II-D Real Dataset To evaluate the proposed method on real data, we scanned the thorax of an anthropomorphic phantom (PBU-60, Kyoto Kagaku Co., Ltd., Kyoto, Japan) using a C-arm CBCT system (ARTIS icono floor, Siemens Healthineers AG, Forchheim, Germany). In total, we acquired 12 datasets, three short scans for each full view grid (referred to as full), and maximum collimation (referred to as slit), both with and without an anti-scatter grid. Each short scan consists of 397 projections ($648\text{\times}472$ pixels, $616\text{\,}\mathrm{\SIUnitSymbolMicro m}$ isotropic spacing) over an angular range of $197.5\text{\,}\mathrm{\SIUnitSymbolDegree}$ using $85\text{\,}\mathrm{kV}$ peak tube voltage. The source-to-isocenter and source-to-detector distances are $750\text{\,}\mathrm{mm}$ and $1200\text{\,}\mathrm{mm}$, respectively. We reconstructed $512\text{\times}512$ slices on an $484\text{\,}\mathrm{\SIUnitSymbolMicro m}$ isotropic voxel grid using an in-house reconstruction pipeline. We regard the slit scan in conjunction with the anti-scatter grid as ground truth. ## III Experiments In the following, we describe the experiments carried out to evaluate our proposed method. Since the U-net based approach outperformed other computational scatter estimation methods by far [37], we considered different configurations of the U-net as a baseline method. To account for our relatively small training corpora, we evaluated our method and the baseline using a nested cross-validation approach. For the head dataset (20 subjects), we used a $4^{}3$-fold cross-validation approach and, for the thorax dataset (15 subjects), we used a $5^{}4$-fold cross-validation approach. The real dataset was only used for testing. To keep the training procedure manageable, we divided the evaluation into meta-parameter search based on the scatter mean absolute percentage error (MAPE), and further in-depth analysis. ### III-A Meta-Parameter Search Since clinic CBCT systems usually have fine-tuned acquisition protocols for each anatomic region, we evaluated each network architecture for head and thorax data separately. To fix the meta-parameters, we only used the synthetic head dataset and the scatter MAPE as metric. We validated both our approach and the U-net for different combinations of depths $d$ and feature channels $c$ using Glorot initialization [51], which was also used by the baseline [37]. Furthermore, we distinguished between a deep U-net (DU-net) and a shallow U-net (SU-net), in which the number of feature maps is not doubled at each level. For all experiments in this section, we performed a $4^{}3$-fold cross-validation and trained all networks using the adaptive moments optimizer (Adam) [52] with an initial learning rate of ${10}^{-4}$ for 100 epochs. In total, we trained 12 networks for each configuration and used the averaged MAPE to assess their quality. To reduce the total computation time, we stopped each training procedure when no significant performance increase was observed for 20 epochs. In a first step, we scanned different parametrizations of the network architectures to find the most promising ones for in-depth comparison. Overall, we tested both U-nets with $c=16$ and $d\in\\{4,5,6,7\\}$, and our approach with $c=16$, $d\in\\{4,5,6\\}$, and additional pre-pooling $p\in\\{0,1,2\\}$. In addition, to decouple the architecture of our convolutional encoder $f_{\boldsymbol{\theta}}$ and the spline coefficient dimensionality, we investigated the four bottleneck architectures $g_{\boldsymbol{\phi}}$ as described in Sec. II-B. ### III-B Qualitative and Quantitative Results Based on the findings of the previous experiments, we adapted the learning rate to ${10}^{-5}$ but kept the overall training routine. We separately performed 4∗3-fold and 5∗4-fold cross-validations for the head and thorax datasets, respectively. In addition to the scatter MAPE, we included the structural similary index (SSIM) of scatter-compensated reconstructions with respect to the ground truth in our evaluation. ### III-C In-Depth Analysis #### III-C1 Spectral Analysis For clinical applications, data integrity is of utmost importance to ensure that automated systems do not alter diagnostically relevant content. While the predictions of neural networks may appear reasonable at first glance, unrealistic perturbations can be unveiled by investigating the spectral properties of (a) the predicted images [53] or (b) the neural network itself. Therefore, we first investigate all networks’ performance concerning the predicted scatter distributions’ power spectral density. Second, from scatter estimation theory, we know that the scatter distribution can be recovered from the measured signal by the convolution with a so-called scatter kernel [30]. This allows us to interpret a neural network for a specific pair of scatter distribution and X-ray signal as a filtering operation and assess its frequency response. #### III-C2 Noise Analysis Assessing a network’s robustness is inherently difficult given small training corpora. Therefore, adversarial attacks are often used to expose weaknesses deliberately. Since we trained all networks on noise-free data, testing them on data with different unseen noise levels appeared appropriate. To this end, we applied Poisson noise associated with different photon counts ranging from ${10}^{3}{10}^{5}$ to our head dataset and investigated the accuracy of the networks’ predictions. #### III-C3 Runtime Analysis For interventional applications, the fast execution speed of computer programs is essential. Therefore, we benchmarked the average execution time for all networks for different batch sizes using a 12-core CPU (Intel(R) Xeon(R) Silver 4116 CPU 2.10GHz). #### III-C4 Real Data Analysis To confirm the generalizability to real data, we tested the networks, which performed best on our synthetic thorax dataset, on the real thorax dataset. For reference, we considered various different scatter suppression techniques, namely using an anti-scatter grid or slit scanning. As an almost scatter free baseline, we considered the configuration to use both, an anti-scatter grid and slit scanning. Note that we neither performed an intensity or a geometry calibration in between the scans to account for the missing anti-scatter grid. ## IV Results ### IV-A Meta-Parameter Search #### IV-A1 Network Parametrization $\displaystyle 10^{5}$$\displaystyle 10^{6}$$\displaystyle 10^{7}$$\displaystyle 10^{8}$No. parameters$\displaystyle 6$$\displaystyle 7$$\displaystyle 8$$\displaystyle 9$MAPE [$\displaystyle\%$]DU-netSU-netOurs Figure 2: Distribution of different network configurations for our approach and the U-net in terms of absolute percentage errors averaged over all folds and patients. Note that, due to visualization purposes, the standard deviation encoded by the circular margins does not correspond to absolute values but rather indicates the relative spread between the networks. The standard deviation is in the range of $1\text{\,}\mathrm{\char 37\relax}5\text{\,}\mathrm{\char 37\relax}$. The x-axis refers to the number of parameters in the convolutional layers. Figure 2 establishes the relationship between the number of convolutional parameters to train for each network to the averaged error rates of all folds and patients. All networks’ error rates range between $6\text{\,}\mathrm{\char 37\relax}9\text{\,}\mathrm{\char 37\relax}$, and we observed that overall the more compact networks outperform the DU-nets. Based on these findings, we selected two spline networks ($c=16$, $(d,p)\in\\{(4,1),(5,0)\\}$) and four U-nets (deep and shallow, $c=16$, $d\in\\{6,7\\}$) for further investigations. #### IV-A2 Bottleneck 1234Fold05101520MAPE [$\displaystyle\%$]constrainedunconstrainedfc-netconv-net Figure 3: Boxplots of the mean absolute percentage error (MAPE) of our proposed method for different bottleneck architectures averaged over the validation folds for each training fold. The horizontal lines indicate the mean value over all test and validation folds. $\displaystyle 0$$\displaystyle 200$$\displaystyle 0$$\displaystyle 50$constrained$\displaystyle 0$$\displaystyle 200$$\displaystyle 0$$\displaystyle 50$unconstrained Figure 4: Normalized constrained (left) and unconstrained (right) weighting matrices used to map the latent variables $\boldsymbol{Z}$ to spline coefficients $\boldsymbol{C}$. Figure 3 shows the results for the different bottleneck architectures. Our proposed constrained weighting matrix, homogeneously initialized, achieves the best results on average, even surpassing the convolutional bottleneck followed by a fully-connected layer. The unconstrained fully-connected layer architectures yield considerably worse results. Figure 4, which shows the weighting matrix for the constrained and unconstrained case, substantiates this finding. While the constrained matrix converges to a block circulant matrix, which merely relates to a convolution, the unconstrained one hardly resembles a sensible operation and is overall noisy. ### IV-B Qualitative and Quantitative Results Head data \| Thorax data ---\|--- 1234Fold05101520MAPE [$\displaystyle\%$] \| 12345Fold05101520MAPE [$\displaystyle\%$] 1234Fold0.950.960.970.991.00SSIM \| 12345Fold0.950.960.970.991.00SSIM Figure 5: Quantitative boxplots for our synthetic datasets. The top row shows the mean absolute percentage errors (MAPE) with respect to the scatter ground truth for each test fold. The bottom row shows structural similarity indices (SSIM) between reconstructed volumes from the simulated ideal primary signal and the scatter-corrected ones using neural networks. The horizontal lines indicate the overall average across all folds. For the sake of clarity, we only depict the best performing networks in each category. For the head dataset, we depict the DU-net with $d=6$, SU-net with $d=7$, and our method with $d=4$ and $p=1$. For the thorax dataset, we depict the DU-net with $d=7$, SU-net with $d=6$, and our method with $d=5$ and $p=0$. Figure 6: Selected subjects of both datasets. The results of the learning- based approaches are given in terms of error maps using the absolute percentage error (APE) for scatter and the absolute error (AE) for the reconstructed slices. For convenience, the associated mean APE (MAPE) and mean AE (MAE) are also provided below each output. Figure IV-B shows fold-wise boxplots for the two synthetic datasets. We observe similar error rates of approximately $5\text{\,}\mathrm{\char 37\relax}$ across all folds and network configurations regarding the predicted scatter distributions for the head dataset. In general, all networks achieve high SSIM values above 0.99 when comparing the reconstructed volumes from the simulated ideal primary signal to their counterparts obtained using the scatter-corrected projections. Note that for the head data, our approach performed equally well for all folds, whereas the results obtained with the U-nets varied more widely. Processing the thorax datasets, on the other hand, was more challenging. Again, all networks performed comparably well with error rates of approximately $7.5\text{\,}\mathrm{\char 37\relax}$. In comparison to the head dataset, larger error margins and outliers were found. This can also be seen with the SSIM, which is, on average, just below 0.98. Again, we find that our proposed method is on par with U-net-like structures from a quantitative point of view. Looking at selected subjects of both datasets in Fig. 6 reveals the potential advantages of the proposed approach. Modeling the predicted scatter as a B-spline intrinsically limits the network output to smooth surfaces. In contrast, both U-nets preserve some details of the input, especially the shallow architecture. However, the reconstructed slices show no systematic trend, and all methods can adequately recover the desired signal. ### IV-C In-Depth Analysis #### IV-C1 Spectral Analysis $\displaystyle 10^{-2}$$\displaystyle 10^{-1}$$\displaystyle 10^{0}$Normalized frequency$\displaystyle 10^{-2}$$\displaystyle 10^{-1}$$\displaystyle 10^{0}$Normalized powerGTDU-netSU-netOurs Figure 7: Power spectral density plots for the X-ray scatter distributions of the ground truth (black solid), ours (light orange dashed), and the U-net (light purple dotted) averaged over all test folds. Note that the graphs associated with both U-nets are almost identical. We calculated the power spectral density by azimuthally averaging the magnitudes of the 2D Fourier transform of X-ray scatter distributions. Also, we averaged all power spectral densities of all projection, patients, and test folds. The resulting densities plotted in Fig. 7 support our previous findings. While the U-nets yield numerically sound predictions, they systematically boost the high frequencies in the scatter distributions. Our approach, however, preserves the real power spectral density over the whole frequency spectrum. AmplitudeGTDU-netSU- netOurs$\displaystyle-3$$\displaystyle-2$$\displaystyle-1$$\displaystyle 0$Phase$\displaystyle 0$$\displaystyle 1$$\displaystyle 2$ Figure 8: Frequency responses of the systems for one test patient averaged over all validation folds. From top to bottom: Normalized $\log$-amplitude, phase in radians. From left to right: ideal system (GT), our spline-net (Ours), deep U-net (DU-net), shallow U-net (SU-net). As mentioned above, conventional convolutional neural networks can be interpreted for a single input image in terms of a filtering operation. Thus, we divided the Fourier transform of the output by the Fourier transform of the input to obtain the respective frequency responses. Figure 8 shows the amplitude and phase of an ideal system’s frequency responses, for our method and both U-nets, averaged over all projections for one patient. Our spline- net’s frequency response was closer to the ideal frequency response in both, amplitude and phase. However, we observed a noticeable intensity shift in our method. The U-nets, in contrast, both exhibited larger deviations in the patterns of amplitude and phase, indicating that their represented operation is less predictable. #### IV-C2 Noise Analysis $\displaystyle 10^{3}$$\displaystyle 10^{4}$$\displaystyle 10^{5}$Photon count$\displaystyle 5.0$$\displaystyle 7.5$$\displaystyle 10.0$$\displaystyle 12.5$$\displaystyle 15.0$$\displaystyle 17.5$MAPE [$\displaystyle\%$]DU-netSU- netOurs Figure 9: Error rates of predicted scatter distributions for different noise levels averaged over all validation and test fold networks. All networks have been trained with noise-free data. Note that a lower photon count relates to a higher noise level. Figure 9 shows plots of the networks’ error rates when confronted with noise. We could confirm that the U-net is very sensitive to unseen noise levels in both configurations, whereas our approach performs more robustly. #### IV-C3 Runtime Analysis 12481632Batch size$\displaystyle 0$$\displaystyle 25$$\displaystyle 50$$\displaystyle 75$$\displaystyle 100$$\displaystyle 125$$\displaystyle 150$Runtime [ms]DU-net 6DU-net 7SU-net 6SU-net 7Ours 4\1Ours 5\0 Figure 10: Runtimes per projection in $\mathrm{ms}$ for different architectures (specified by the depth and optional pre-pooling, $d$ \ $p$) and batch sizes. Figure 10 compiles the inference speed of all considered networks. As expected from the parameter complexity, our approach is the fastest with $4\text{\,}\mathrm{ms}30\text{\,}\mathrm{ms}$ and therefore $1.78.5$ times faster than the SU-net with $34\text{\,}\mathrm{ms}50\text{\,}\mathrm{ms}$. The DU-net is the slowest with $89\text{\,}\mathrm{ms}144\text{\,}\mathrm{ms}$. #### IV-C4 Real Data Analysis ReconstructionGrid+SlitGrid+FullSlitFullDU-netSU- netOursDifference$\displaystyle 39.31\pm 0.21$$\displaystyle 58.46\pm 0.09$$\displaystyle 123.84\pm 0.32$$\displaystyle 62.86\pm 0.25$$\displaystyle 64.52\pm 0.23$$\displaystyle 63.97\pm 0.19$ Figure 11: Central slices of reconstructed volumes for different scatter compensation strategies and errors taken with respect to the reconstruction result resulting from the grid + slit data acquisition. Grid refers to employing a conventional anti-scatter grid. Slit refers to the most narrow collimator setting available on the X-ray imaging system. Full refers to using no collimation at all. Both, the reconstructed slices as well as the difference images are shown using a gray level window $[-1000,1000]$ HU. For convenience the absolute average HU errors and the associated standard deviations with respect to three consecutive measurements are provided. As shown in Fig. 11, all methods were able to compensate for scatter artifacts well. However, employing an anti-scatter grid still yielded the lowest overall error in Hounsfield units (HU) as compared to using slit collimation in addition. The learning-based approaches performed about as well as the slit scanning without an anti-scatter grid, which is in accordance to previous findings [37]. Overall, the networks achieved similar error rates, and no systematic trend was observable. ## V Discussion X-ray scatter is a major source of artifacts in interventional CBCT. Deep- learning-based approaches have shown the potential to outperform conventional physical or algorithmic approaches to scatter compensation. Without incorporating prior information, scatter distributions can be inferred from the measured X-ray signal [37]. However, data integrity and robustness are critical aspects of clinical imaging, which can be violated by deep neural networks [54]. To ensure sound scatter estimates, we proposed to embed bivariate B-splines in neural networks to constrain their co-domain to smooth results [42]. By reformulating the spline evaluation in terms of matrix multiplications, we were able to integrate B-splines with neural networks without further ado such that end-to-end training was feasible. In an extensive cross-validation using synthetic data, we showed that our proposed lean convolutional encoder using B-spline evaluation performs on par with several U-net based architectures. We substantiated this finding in a first phantom study. There, our approach performed basically as well as the U-net and the slit scanning technique (without anti-scatter grid), which was used as a baseline before [37]. Note, however, that the proposed method offers several advantages not present in the U-net architectures. First, we considerably lowered the parameter and runtime complexity, rendering our method suitable for a variety of hardware. More importantly, we verified that our spline-based approach ensures data integrity concerning the power spectral density of scatter estimates and the overall network’s frequency response. This property ensures that no high- frequency details, which relate to anatomic structures or pathologies, are altered. In comparison, the U-net considerably changes the concerning spectral contents, which was already shown for neural networks containing up- convolutions [53]. As our spline network corresponds to a low-pass filtering operation, it is robust towards noise even when trained on noise-free data. While we implemented the U-net baselines to the best of our knowledge, our error rates are higher than previously reported [37]. Potential reasons include but are not limited to (a) different simulation codes, (b) our smaller training corpora, and (c) the heterogeneity of our data. Our simulation setup currently assumes an ideal detector, and we do not consider the domain shift between synthetic and real data. For future work, we indicate several research directions for either method or data and experiments. First, since our preferred bottleneck weighting matrix converges to a block circulant matrix, a fixed representation is desirable to further reduce the number of trainable parameters and increase the plausibility of our approach. For instance, training both the encoder and the bottleneck separately or introducing additional constraints are promising approaches to do so. Second, replacing the bottleneck fully-connected layer with a small U-net reduces the number of parameters while still covering the entire latent space with its receptive field. Third, since our network is end- to-end trainable, it appears reasonable to include the reconstruction into the computational graph to calculate the loss function in the CBCT domain [55, 56]. Last but not least, we believe our approach applies to low-frequency signal estimation and correction in general, e.g., bias field correction in magnetic resonance imaging, ultrasound imaging, or microscopy techniques. ## VI Conclusion Embedding B-splines in neural networks ensures data integrity for low- frequency signals. This reduces the number of network parameters needed to arrive at physically sensible results, and thanks to the reduced parameter set, network inference can be made faster. ## Disclaimer The concepts and information presented in this article are based on research and are not commercially available. ## References [1] E.-P. Rührnschopf and K. Klingenbeck, “A general framework and review of scatter correction methods in x-ray cone-beam computerized tomography. 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# On the Negativity of Wigner Function as a measure of entanglement under quantum polarization converter devices Mustapha<EMAIL_ADDRESS>and Morad El <EMAIL_ADDRESS> Université Mohammed V, Faculté des Sciences Equipe Sciences de la Matière et du Rayonnement Av. Ibn Battouta, B.P. 1014, Agdal, Rabat, Morocco ###### Abstract We study the behaviour of the Negativity of Wigner Function (NWF) as a measure of entanglement in non-Gaussian states under quantum polarisation converter devices. We analyze comparatively this quantity with other measures of entanglement in a system prepared in a superposition of two-mode coherent states. We show that the (WF) can be identified as a quantifier of non- Gaussian entanglement. ## Keywords Wigner function, Q-function, Negativity, Non-Gaussian state, Non-gaussianity, Coherent states, Quantum polarization. ## Introduction Quantum entanglement, is recognised as one of quantum resources for several applications in quantum information theory including quantum computing, quantum communications and quantum cryptography. However, quantifying quantum correlations particularly, entanglement is one of the most relevant challenges in quantum information theory. On the other hand polarization play a central role in a large number of optical phenomena. It appear in several paradigmatic application including remote sensing and light scattering [., 3 abdo] In recent years, a considerable attention has been paid to the polarization of quantum light due to its application in quantum information protocols since the light can be extensively used in the quantum information-coding. There is a wide consensus that coherent states are the most classical quantum states. It has been shown that these states minimize the Heisenberg uncertainty relation for position and momentums operators. In addition the dynamic of their expectation values has the same form as their classical counterpart. These properties, make coherent states least quantum states. For this reason, they are called quasi-classical states. These states describe coherent optical light and can be generated in the laboratory. In this way, the quantum polarization formalism is extended from that of classical light polarization through replacing the Stokes parameters by the associated stocks operators. In addition, the degree of polarization of classical light does not depend on its intensity. However, the degree of polarization of coherent states decreases with decreasing values of optical power. Furthermore, the quantum Stokes parameters $\hat{S}_{1},\hat{S}_{2}$ and $\hat{S}_{3}$ do not commute with each other. Then, it is not possible to know the values of any two of them simultaneously without uncertainties. This fact has been used in [1] in order to create a continuous variable quantum key distribution system. Basically, polarization of coherent states has extensively been used in quantum information processing, precisely in quantum key distribution protocols with continuous variables. [1, 2, 3, 4]. Recently, one of the most important applications of the Wigner function in quantum information theory is the classification of classical and non- classical states based on its non-Gaussianity and its negativity [5, 6]. In fact, a Gaussian state is associated with a Gaussian Wigner function in phase space of one mode or multimode pure state in continuous variables systems [7]. On the other hand, a non-Gaussian state with negative Wigner function reveals that the system possesses non-classical correlations [8]. Recent works showed that the negativity of the Wigner function can detect the entanglement and it is not sensitive to all kinds of quantum correlations and can be a best quantifier of genuine entanglement in tripartite systems [9, 10]. In this work, we study the strength of the NWF as a quantifier of entanglement under a quantum polarization converter devices. In this direction we describe in section (1) the Stokes parameters in order to analyse the quantum polarization of the superposition of two bi-mode coherent states; As well in section (3), we use the entanglement of formation and the NWF to analyse the entanglement behavior before and after the polarization converter devices; Finally, in section (4) we discuss our results and provide conclusions. ## 1 Review of quantum polarization In classical optics the polarization of light beams is determined by computing the four Stokes parameters [11]. Similarly, In quantum optics the degree of polarization of states of light is quantified by the calculation of the mean values of the associated Stocks operators[12, 13]. For a monochromatic plane wave propagating in the $z$-direction, whose electric field lines in the $xy$ plane. In terms of the annihilation and creation operators of horizontally and vertically polarized modes noted by $\hat{a}_{H}$ and $\hat{a}_{V}$, respectively, the Stokes operators can be expressed as[12, 13] $\begin{matrix}\hat{S}_{0}&=&\hat{a}_{H}^{+}\hat{a}_{H}+\hat{a}_{V}^{+}\hat{a}_{V},\\\ \hat{S}_{1}&=&\hat{a}_{H}^{+}\hat{a}_{H}-\hat{a}_{V}^{+}\hat{a}_{V},\\\ \hat{S}_{2}&=&\hat{a}_{H}^{+}\hat{a}_{V}+\hat{a}_{V}^{+}\hat{a}_{H},\\\ \hat{S}_{3}&=&i\left(\hat{a}_{V}^{+}\hat{a}_{H}+\hat{a}_{H}^{+}\hat{a}_{V}\right)\end{matrix}$ (1) and the Stokes parameters are given by calculation of the corresponding average values $<\hat{S}_{k}>$. Using the bosonic commutation relations $\left[\hat{a}_{i},\hat{a}_{j}^{+}\right]=\hat{\textbf{1}}\delta_{ij}\quad;\quad\\{i,j\\}\in\\{H,V\\}.$ (2) It has been shown that, the operators $\hat{S}_{1}$, $\hat{S}_{2}$ and $\hat{S}_{3}$ all commute with $\hat{S}_{0}$ and satisfy the corresponding commutation relations, $\left[\hat{S}_{k},\hat{S}_{l}\right]=2i\hat{S}_{m}\quad;\quad\\{k,l,m\\}\in\\{1,2,3\\}.$ (3) The Stokes operators $\hat{S}_{1}$, $\hat{S}_{2}$ and $\hat{S}_{3}$ thus from $SU\left(2\right)$ algebra and generate all transformations from this group; * • $\hat{S}_{2}$ is the infinitesimal generator of geometric rotations around the direction of propagation, * • $\hat{S}_{3}$ is the differential phase shifts between the two modes. For a quasi-classical two-mode coherent state $\lvert\alpha,\beta\rangle$ defined as $\lvert\alpha,\beta\rangle=\mathrm{e}^{-\dfrac{\lvert\alpha\rvert+\lvert\beta\rvert}{2}}\sum_{n,m}\dfrac{\left(\alpha\right)^{n}}{\sqrt{n!}}\dfrac{\left(\beta\right)^{m}}{\sqrt{m!}}\lvert n,m\rangle,$ (4) the mean values $<\hat{S}_{i}>$ of the three Stokes operators and the variances $V_{i}$ are expressed as a function of the field amplitudes: $\begin{matrix}<\hat{S}_{1}>=&\lvert\alpha\rvert^{2}-\lvert\beta\rvert^{2},&<\hat{S}_{1}^{2}>=&\left(\left(\lvert\alpha\rvert\right)^{2}-\left(\lvert\beta\rvert\right)^{2}\right)^{2}+\left(\lvert\alpha\rvert\right)^{2}+\left(\lvert\beta\rvert\right)^{2},\\\ <\hat{S}_{2}>=&\alpha^{}\beta-\alpha\beta^{},\quad&<\hat{S}_{2}^{2}>=&\left(\alpha^{}\beta\right)^{2}+\left(\alpha\beta^{}\right)^{2}+\lvert\alpha\rvert^{2}+\lvert\beta\rvert^{2}+2\lvert\alpha\rvert^{2}\lvert\beta\rvert^{2},\\\ <\hat{S}_{3}>=&i\left(\alpha\beta^{}-\alpha^{}\beta\right),\quad&<\hat{S}_{3}^{2}>=&-\left(\alpha^{}\beta\right)^{2}-\left(\alpha\beta^{}\right)^{2}+\lvert\alpha\rvert^{2}+\lvert\beta\rvert^{2}+2\lvert\alpha\rvert^{2}\lvert\beta\rvert^{2},\\\ \end{matrix}$ (5) and $V_{1}=V_{2}=V_{3}=\lvert\alpha\rvert^{2}+\lvert\beta\rvert^{2}.$ The average values of the quantum Stokes parameters of coherent state are equal to of classical light Stokes parameters values and the variance of the three parameters increase while the optical power increases. ## 2 Quantum degree of polarization of superposition of two mode coherent states Classically, the light is considered unpolarized if it’s Stokes parameters vanish. In quantum mechanics, this is condition necessary but not sufficient. A quantum light beam can be considered unpolarized if its observable do not change after a geometric rotation and/or a phase shift between the components. These condition are mathematically described by [14]: $\left[\hat{\rho},\hat{S}_{1}\right]=\left[\hat{\rho},\hat{S}_{3}\right]=0,$ (6) where $\hat{\rho}$ is the density matrix of the quantum state. By analogy with the classical definition, many measures of the quantum polarization degree have been proposed [13, 12]. Here we consider a measure based on Q-function [13]: $P=\dfrac{D}{1+D}$ (7) with $D=4\pi\int_{0}^{2\pi}d\Omega\int_{0}^{\pi}\left[Q\left(\theta,\phi\right)-\dfrac{1}{4\pi}\right]^{2}\sin\left(\theta\right)d\theta d\phi$ where $d\Omega=\sin\left(\theta\right)d\theta d\phi$ is the differential of solid angle and $Q\left(\theta,\phi\right)$ is the Q-function of the light. For the two-mode coherent state $\lvert\alpha e^{i\phi_{\alpha}},\beta e^{i\phi_{\beta}}\rangle$ the Q-function reads [15]: $Q\left(\theta,\phi\right)=\dfrac{e^{-\left(\lvert\alpha\rvert^{2}+\lvert\beta\rvert^{2}\right)}}{4\pi}\left(1+z\right)e^{2}$ (8) where $\displaystyle z$ $\displaystyle=\left[\lvert\alpha\rvert\cos\left(\dfrac{\theta}{2}\right)\cos\left(\phi_{\alpha}+\phi\right)+\beta\sin\left(\dfrac{\theta}{2}\right)\cos\left(\phi_{\beta}\right)\right]^{2}$ $\displaystyle+\left[\lvert\alpha\rvert\cos\left(\dfrac{\theta}{2}\right)\sin\left(\phi_{\alpha}+\phi\right)+\beta\sin\left(\dfrac{\theta}{2}\right)\sin\left(\phi_{\beta}\right)\right]^{2}.$ Using (7) and (8), the quantum polarization degree of the quantum state $\lvert\alpha,0\rangle$ is $P=1-\dfrac{4\lvert\alpha\rvert^{2}}{1+2\lvert\alpha\rvert}.$ (9) When $\rvert\alpha\lvert^{2}\gg 1$, equation (9) can be approximated by $P\simeq 1-\dfrac{2}{\rvert\alpha\lvert^{2}}$ (10) Showing that the quantum degree of polarization of two-mode coherent state increase with increasing of the light power. In Figure (2) we show the quantum degree of polarization of the states $\left\lvert 0,\pm\alpha\right\rangle,$ $\left\lvert\pm\alpha,0\right\rangle,$ $\left\lvert\pm\alpha,\mp\alpha\right\rangle$ and $\left\lvert\pm\alpha,\pm\alpha\right\rangle$ that are equivalent to vertical polarization $\left\lvert V\right\rangle$, horizontal polarization $\left\lvert H\right\rangle$, anti-diagonal polarization $\left\lvert-\dfrac{\pi}{4}\right\rangle$ and diagonal polarization $\left\lvert\dfrac{\pi}{4}\right\rangle$, respectively. In Figure (2) the diagonal states have a larger quantum degree of polarization, compared with horizontal and vertical states, because they have a larger mean photon number (in total). In order to discuss the quantum degree of polarization, we consider the quantum Stokes parameters of quantum state composed by the superposition of bimodal coherent states defined by $\left\lvert\psi_{\pm}\right\rangle=N\left(\left\lvert\alpha,\beta\right\rangle\pm\left\lvert\gamma,\lambda\right\rangle\right)$ (11) where $\rvert N\lvert^{2}=\left\\{2+\left(\zeta+\zeta^{}\right)\emph{Exp}\left[-\left(\lvert\alpha\rvert^{2}+\lvert\beta\rvert^{2}+\lvert\gamma\rvert^{2}+\lvert\lambda\rvert^{2}\right)/2\right]\right\\}^{-1}$ and $\zeta=\emph{Exp}\left(\alpha^{}\gamma+\beta^{}\lambda\right)$. The average of the quantum Stokes parameters and of their squared values and the Q-function of the state (11) are given in Appendix (A). For the raison of simplification, we choose to consider the following particular cases of the state (11), $\displaystyle\left\lvert\psi_{1}\right\rangle=$ $\displaystyle N_{1}\left(\left\lvert\alpha,\beta\right\rangle+\left\lvert\beta,\alpha\right\rangle\right)$ (12a) $\displaystyle\left\lvert\psi_{2}\right\rangle=$ $\displaystyle N_{2}\left(\left\lvert-\alpha,-\alpha\right\rangle+\left\lvert\alpha,\alpha\right\rangle\right)$ (12b) $\displaystyle\left\lvert\psi_{3}\right\rangle=$ $\displaystyle N_{3}\left(\left\lvert\alpha,0\right\rangle+\left\lvert 0,\alpha\right\rangle\right)$ (12c) with $N_{1}=\left\\{2\left[1+\textnormal{Exp}\left(2\alpha\beta-\lvert\alpha\rvert^{2}-\lvert\beta\rvert^{2}\right)\right]\right\\}^{-1/2}$, $N_{2}=\left\\{2\left[1+\textnormal{Exp}\left(-4\lvert\alpha\rvert^{2}\right)\right]\right\\}^{-1/2}$ and $N_{3}=\left\\{2\left[1+\textnormal{Exp}\left(-\lvert\alpha\rvert^{2}\right)\right]\right\\}^{-1/2}$. The averages and the covariances of the quantum Stokes parameters of the states $\left\lvert\psi_{1}\right\rangle$, $\left\lvert\psi_{2}\right\rangle$ and $\left\lvert\psi_{3}\right\rangle$ are calculated from their general expressions of superposed bi-mode coherent state and given in Appendix (A). we found that $<\hat{S}_{1}>$ and $<\hat{S}_{3}>$ vanish for the states (12a), (12b) and (12c). That is consequence of the fact that the average of optical powers in horizontal and vertical polarizations are equal for these cases. In the separable bi-mode state given in (4), the Stockes parameters and the variances expressed propationally to optical power contrary to the superposed entangled state defined in (11). The entangled state gives an interesting behavior of the variances of the Stokes parameters. The variances can increase and decrease when the total mean photon number increases, as are shown in Figure (1). Figure 1: Variances $V_{1}$, $V_{2}$, and $V_{3}$ of $\hat{S}_{1}$, $\hat{S}_{2}$ and $\hat{S}_{3}$ (respectivly) versus $\lvert\alpha\rvert^{2}$ for $\left\lvert\psi_{1}\right\rangle$ having $\lvert\beta\rvert^{2}=4$. Figure 2: Quantum degree of polarization for the states $\left\lvert 0,\pm\alpha\right\rangle$ (Solidligne) and $\left\lvert\pm\alpha,\mp\alpha\right\rangle$ (Dashedligne). ## 3 Wigner function and entanglement under polarization converter devices Wigner function is an important tool in physics, especially in quantum physics to detect the non-classical behavior of light by studying its negativity. Recently, in quantum information theory the NWF is used as a measure of entanglement[9]. In this section we study the NWF and the entanglement behavior of the superposition of two-mode coherent states before and after pass by a polarization converter devices, in order to testing the strength of the NWF as a measure of entanglement. For a single quantum system described by a density matrix $\hat{\rho}$, the associated Wigner function is defined by $\mathcal{W}\left(q,p\right)=\dfrac{1}{2\pi}\int\exp{(\dfrac{-ipy}{\hbar})}\left\langle q+\dfrac{y}{2}\right\rvert\hat{\rho}\left\lvert q-\dfrac{y}{2}\right\rangle dy,$ (13) where $\lvert q\pm\dfrac{y}{2}\rangle$ are the eigenkets of the position operator. If the state in question is a pure state $\hat{\rho}=\lvert\psi\rangle\langle\psi\rvert$ then $\mathcal{W}\left(q,p\right)=\dfrac{1}{2\pi\hbar}\int\psi^{}\left(q-\dfrac{y}{2}\right)\psi\left(q+\dfrac{y}{2}\right)\exp{(\dfrac{-ipy}{\hbar})}dy.$ (14) Hence, the doubled volume of the integrated negative part of the Wigner function may be written as [5] $\delta\left(\rho\right)=\iint\left\|\mathcal{W}\left(q,p\right)\right\|\,dqdp-1.$ (15) (a) Wigner function of $\psi_{1}$ for $\left\|\alpha\right\|=\left\|\beta\right\|=1$. (b) Wigner function of $\psi_{+}$ for $\left\|\alpha\right\|=\left\|\beta\right\|=2$. Figure 3: Wigner function of superposed coherent state ((12a)). It is clear from the plot in Figure (3) that the Wigner function of the bi- mode superposed state (12a) is not positive on the all phase space. The volume of the negative part of the Wigner function is plotted in Figure (4) versus $\lvert\alpha\lvert^{2}$ for $\lvert\beta\lvert=2.$ The quantum entanglement of the same state ((11)) can be measured by the concurrence [16, 17] $C=\dfrac{\sqrt{\left(1-\lvert\langle\alpha\|\gamma\rangle\rvert^{2}\right)\left(1-\lvert\langle\lambda\|\beta\rangle\rvert^{2}\right)}}{1+Re\left(\langle\alpha\|\gamma\rangle\langle\beta\|\lambda\rangle\right)}.$ (16) The concurrence $C_{\psi_{1}}\left(\alpha\right)$ of the particular state $\left\lvert\psi_{1}\right\rangle$ defined in (12a) is showen in Figure (5) . Figure 4: The NWF of the state $\left\lvert\psi_{1}\right\rangle$ versus $\lvert\alpha\rvert$ for $\lvert\beta\rvert=2$. Figure 5: The concurrence of the state $\left\lvert\psi_{1}\right\rangle$ versus $\lvert\alpha\rvert$ and $\lvert\beta\rvert$. Now let as assume that, the state (12a) pass by $\textbf{C}\left(\phi_{2}\right)\textbf{R}\left(\theta\right)\textbf{C}\left(\phi_{1}\right)$ device (Compensator-Rotationer-Compensator device), where $\textbf{C}\left(\phi\right)=\textnormal{Exp}\left(i\dfrac{\phi}{2}\hat{S}_{1}\right)$ is the application of phase shift $\phi$ between the horizontal and vertical modes and $\textbf{R}\left(\theta\right)=\textnormal{Exp}\left(i\dfrac{\theta}{2}\hat{S}_{3}\right)$ is a geometric rotation by angle $\theta$ in the polarization. The quantum state at the output is given by $\displaystyle\left\lvert\psi_{out}\right\rangle=N_{1}$ $\displaystyle\biggl{(}\left\lvert\beta\sin\left(\theta\right)e^{i\left(\phi_{2}-\phi_{1}\right)/2}+\alpha\cos\left(\theta\right)e^{i\left(\phi_{2}+\phi_{1}\right)/2}\right\rangle\left\lvert\beta\cos\left(\theta\right)e^{-i\left(\phi_{2}+\phi_{1}\right)/2}-\alpha\sin\left(\theta\right)e^{-i\left(\phi_{2}-\phi_{1}\right)/2}\right\rangle$ $\displaystyle+\left\lvert\alpha\sin\left(\theta\right)e^{i\left(\phi_{2}-\phi_{1}\right)/2}+\beta\cos\left(\theta\right)e^{i\left(\phi_{2}+\phi_{1}\right)/2}\right\rangle\left\lvert\alpha\cos\left(\theta\right)e^{-i\left(\phi_{2}+\phi_{1}\right)/2}-\beta\sin\left(\theta\right)e^{-i\left(\phi_{2}-\phi_{1}\right)/2}\right\rangle\biggr{)}.$ (17) The NWF and the concurrence of the output state (3) are plotted in the Figure (6) versus $\lvert\alpha\rvert^{2}$ for $\lvert\beta\lvert^{2}=2$. (a) (b) Figure 6: The concurrence (a) and the NWF (b) versus rotator’s angle $\theta$ for $\phi_{1}\in\\{0,\dfrac{\pi}{8},\dfrac{\pi}{6},\dfrac{\pi}{4}\\}$ and $\phi_{2}=0$. ## 4 Discussion and conclusion In this section, we will discuss comparatively the behavior of the NWF as a measure of entanglement and the concurrence under a polarization converter device, $\textbf{C}\left(\phi_{2}\right)\textbf{R}\left(\theta\right)\textbf{C}\left(\phi_{1}\right)$. For this purpose we chose to plot the NWF and the concurrence for the input and output states $\left\lvert\psi_{1}\right\rangle$ and $\left\lvert\psi_{out}\right\rangle$ ((12a) and (3)). Figure (6) shows the entanglement and the NWF in the input state (12a) that are dependent to the parameters $\theta$ and $\phi$ according to $\lvert\alpha\rvert$ and $\lvert\beta\rvert$. We see that the entanglement increases with increasing values of the parameter $\alpha$ to reach its maximum when $\lvert\alpha\rvert^{2}\geq 1.5$ that is a result of the fact that at the limit of large values of the parameter $\lvert\alpha-\beta\rvert$ the coherent states $\left\lvert\alpha\right\rangle$ and $\left\lvert\beta\right\rangle$ become orthogonal. Thus the behavior of the bi-mode superposed coherent state is, as expected, exactly that of the Bell state. After the state passed through the CRC device, we show in Figure (6) its entanglement as a function of $\theta$ for different values of $\phi_{1}$ fixing $\lvert\alpha-\beta\rvert^{2}=4$. It is interesting to say that the rotation of $\dfrac{\pi}{4}$ applied on the input state $\left\lvert\psi_{1}\right\rangle$ destroys completely the entanglement. This implies that, the $\textbf{R}\left(\theta\right)$ device can be a perfect entangler$/$disentangler gate. In figure (6), the NWF is plotted as a function of $\theta$ for $\lvert\alpha-\beta\rvert^{2}=4$ for different values of $\phi_{1}$. For a specific values of $\phi_{1}$, we see that, the NWF decrease with increasing values of $\theta$ to reach its minimum and vanish for $\phi_{1}=\dfrac{\pi}{4}$. Then, it increases again with increasing values of $\theta$. This allows to show that the NWF and the concurrence behave identically and they have the same inflection points which does confirm that the NWF is a true measure of entanglement in non-Gaussian states. As conclusion, in this paper we have studied the behavior of the entanglement and the polarization degree in superposition of two-mode coherent states. We have confirm that the NWF can be used as a good quantifier of entanglement in non-Gaussian systems. As matter of fact, it turn out that the volume of the negative part of Wigner function is in fact a best quantifier of bipartite entanglement in non- Gaussian systems. This work allows as to describe the Wigner function and the polarization of superposition of two-mode coherent states and the important use of the WF to study the entanglement in non-Gaussian systems. Consequently, the NWF can be considered as a measure of entanglement in non-Gaussian systems. We believe that this result will be efficienct in quantum information theory, mostly in quantum computing [18], because the Wigner function can be measured experimentally, [19, 20], including the measurements of its negative values [5]. The interest point on such experiments has triggered a search for operational definitions of the Wigner functions, based on experimental setup [21, 22]. It does represent a major step forward in the detection and the quantification of non-Gaussian entanglement in bipartite systems. ## References * [1] A. Vidiella-Barranco and L. 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Cambridge university press, 1997. ## Appendix A Average values and variances of quantum Stokes parameters and Q-function: For each Quantum Stokes parameter $\hat{S}_{i}$ the variance is defined by $V_{i}=\left\langle\hat{S}_{i}^{2}\right\rangle-\left\langle\hat{S}_{i}\right\rangle^{2}$, where the averages of the quantum Stokes parameters and of their squared values in the state $\left\lvert\psi_{\pm}\right\rangle$ defined in (11) are: $\left\langle\hat{S}_{1}\right\rangle=\lvert N\rvert^{2}\Big{(}\left(\lvert\alpha\rvert^{2}-\lvert\beta\rvert^{2}\right)+\left(\lvert\gamma\rvert^{2}-\lvert\lambda\rvert^{2}\right)+\big{[}\left(\alpha^{}\gamma-\beta\lambda\right)+\left(\alpha\gamma^{}-\beta\lambda^{}\right)\big{]}\delta\Big{)}$ (A.1) $\left\langle\hat{S}_{2}\right\rangle=\lvert N\rvert^{2}\Big{(}\left(\alpha^{}\beta+\alpha\beta^{}\right)+\left(\gamma^{}\lambda+\gamma\lambda^{}\right)+\big{[}\left(\alpha^{}\lambda+\gamma\beta^{}\right)+\left(\gamma^{}\beta+\alpha\gamma^{}\right)\big{]}\delta\Big{)}$ (A.2) $\left\langle\hat{S}_{3}\right\rangle=\lvert N\rvert^{2}\Big{(}\left(\alpha\beta^{}-\alpha^{}\beta\right)+\left(\gamma\lambda^{}-\gamma^{}\lambda\right)+\big{[}\left(\gamma\beta^{}-\alpha^{}\lambda\right)+\left(\alpha\lambda^{}-\beta\gamma^{}\right)\big{]}\delta\Big{)}$ (A.3) $\left\langle\hat{S}_{1}^{2}\right\rangle=\lvert N\rvert^{2}\left\\{\begin{array}[]{cc}\lvert\alpha\rvert^{2}+\lvert\beta\rvert^{2}+\lvert\gamma\rvert^{2}+\lvert\lambda\rvert^{2}+\left(\lvert\alpha\rvert^{2}-\lvert\beta\rvert^{2}\right)^{2}+\left(\lvert\gamma\rvert^{2}-\lvert\lambda\rvert^{2}\right)^{2}+\\\ \big{[}\alpha^{}\gamma+\beta^{}\lambda+\alpha\gamma^{}+\beta\lambda^{}+\left(\alpha^{}\gamma-\beta^{}\lambda\right)^{2}+\left(\alpha\gamma^{}-\beta\lambda^{}\right)^{2}\big{]}\delta\end{array}\right\\}$ (A.4) $\left\langle\hat{S}_{2}^{2}\right\rangle=\lvert N\rvert^{2}\left\\{\begin{array}[]{cc}\lvert\alpha\rvert^{2}+\lvert\beta\rvert^{2}+\lvert\gamma\rvert^{2}+\lvert\lambda\rvert^{2}+2\left(\lvert\alpha\rvert^{2}\lvert\beta\rvert^{2}+\lvert\gamma\rvert^{2}\lvert\lambda\rvert^{2}\right)+\left(\alpha^{}\beta\right)^{2}+\left(\alpha\beta^{}\right)^{2}\left(\gamma^{}\lambda\right)^{2}+\\\ \left(\gamma\lambda^{}\right)^{2}+\big{[}\alpha^{}\gamma+\beta^{}\lambda+\alpha\gamma^{}+\beta\lambda^{}+\left(\alpha^{}\lambda+\beta^{}\gamma\right)^{2}+\left(\gamma^{}\beta+\alpha\lambda^{}\right)^{2}\big{]}\delta\par\end{array}\right\\}$ (A.5) $\left\langle\hat{S}_{3}^{2}\right\rangle=\lvert N\rvert^{2}\left\\{\begin{array}[]{cc}\lvert\alpha\rvert^{2}+\lvert\beta\rvert^{2}+\lvert\gamma\rvert^{2}+\lvert\lambda\rvert^{2}+2\left(\lvert\alpha\rvert^{2}\lvert\beta\rvert^{2}-\lvert\gamma\rvert^{2}\lvert\lambda\rvert^{2}\right)-\left(\alpha^{}\beta\right)^{2}-\left(\alpha\beta^{}\right)^{2}\left(\gamma^{}\lambda\right)^{2}-\\\ \left(\gamma\lambda^{}\right)^{2}+\big{[}\alpha^{}\gamma+\beta^{}\lambda+\alpha\gamma^{}+\beta\lambda^{}-\left(\alpha^{}\lambda-\beta^{}\gamma\right)^{2}-\left(\gamma^{}\beta-\alpha\lambda^{}\right)^{2}\big{]}\delta\end{array}\right\\}$ (A.6) where $\delta=\textnormal{exp}\big{[}\alpha^{}\gamma+\beta^{}\lambda-\left(\lvert\alpha\rvert^{2}+\lvert\beta\rvert^{2}+\lvert\gamma\rvert^{2}+\lvert\lambda\rvert^{2}\right)/2\big{]}$ and the Q-function of the same state ((11)) is $Q\left(\theta,\phi\right)=\dfrac{\lvert N\rvert^{2}}{4\pi}\left\\{\begin{array}[]{cc}e^{-\left(\lvert\alpha\rvert^{2}+\lvert\beta\rvert^{2}\right)}\Big{(}1+z_{1}\Big{)}e^{z_{1}}+e^{-\left(\lvert\gamma\rvert^{2}+\lvert\lambda\rvert^{2}\right)}\Big{(}1+z_{2}\Big{)}e^{z_{2}}+\\\ e^{-\left(\lvert\alpha\rvert^{2}+\lvert\beta\rvert^{2}+\lvert\gamma\rvert^{2}+\lvert\lambda\rvert^{2}\right)/2}\left[\Big{(}1+z_{12}\Big{)}e^{z_{12}}+\Big{(}1+z_{12}^{}\Big{)}e^{z_{12}^{}}\right]\end{array}\right\\}$ (A.7)
# The extended Bregman divergence and parametric estimation Sancharee Basak Ayanendranath Basu (Interdisciplinary Statistical Research Unit Indian Statistical Institute, Kolkata, INDIA ) ###### Abstract Minimization of suitable statistical distances (between the data and model densities) has proved to be a very useful technique in the field of robust inference. Apart from the class of $\phi$-divergences of [1] and [2], the Bregman divergence ([3]) has been extensively used for this purpose. However, since the data density must have a linear presence in the cross product term of the Bregman divergence involving both the data and model densities, several useful divergences cannot be captured by the usual Bregman form. In this respect, we provide an extension of the ordinary Bregman divergence by considering an exponent of the density function as the argument rather than the density function itself. We demonstrate that many useful divergence families, which are not ordinarily Bregman divergences, can be accommodated within this extended description. Using this formulation, one can develop many new families of divergences which may be useful in robust inference. In particular, through an application of this extension, we propose the new class of the GSB divergence family. We explore the applicability of the minimum GSB divergence estimator in discrete parametric models. Simulation studies as well as conforming real data examples are given to demonstrate the performance of the estimator and to substantiate the theory developed. Keywords— Bregman divergence; $S$-divergence; B-exponential divergence; GSB divergence; discrete model; robustness ## 1 Introduction In the domain of statistical inference, there is, generally, an inherent trade-off between the model efficiency and the robustness of the procedure. Often we compromise the efficiency of the procedure to a certain allowable extent for achieving better robustness properties. In the present age of big data, the robustness angle of statistical inference has to be dealt with greater care than ever before. Such methods do exist in the literature which allow full asymptotic efficiency simultaneously with strong robustness properties; see, e.g., [4], [5] and [6]. In practice, however, one would be hard-pressed to find a procedure which matches the likelihood-based methods in terms of efficiency in small to moderate samples without inheriting any of the robustness limitations of the latter. Many of these trade-off issues are discussed in the canonical texts on robustness such as [7], [8] and [9]; for a minimum divergence view of this issue, see [6]. There are several types of divergences which are used in minimum distance inference. Most of them are not mathematical metrics. They may not satisfy the triangle inequality or may not even be symmetric in their arguments. The only properties we demand of these measures are that they are non-negative and are equal to zero if and only if the two arguments are identically equal. Sometimes we will refer to these divergence measures as ‘statistical distances’ or, loosely, as ‘distances’ without any claim to metric properties. Most of the density-based divergences in the literature belong to either the class of chi-square type distances (formally called $\phi$-divergences, $f$-divergences or disparities) or Bregman divergences. See [1], [2] and [5] for a description of the divergences of the chi-square type and [3] for Bregman divergences. Although [3] introduced the Bregman divergence in order to be used in convex programming, it has, because of its flexible characteristics, been used in many branches of natural science as well as in areas like information theory and computational geometry. The class of chi- square type distances between two densities $g$ and $f$ includes, for example, the likelihood disparity (LD), the Kullback-Leibler divergence (KLD) and the (twice, squared) Hellinger distance (HD), given by ${\rm LD}(g,f)=\int g\log\left(\frac{g}{f}\right),~{}{\rm KLD}(g,f)=\int f\log\left(\frac{f}{g}\right),~{}{\rm HD}(g,f)=\frac{1}{2}\int(f^{1/2}-g^{1/2})^{2},$ (1) respectively. Representative members of the class of Bregman divergences include the LD and the squared $L_{2}$ distance, where $L_{2}(g,f)=\int(g-f)^{2}.$ (2) The LD is the only common member between the class of chi-square type distances and Bregman divergences. In the parametric estimation scheme that we consider, the estimator corresponds to the parameter of the model density which is closest to the observed data density in terms of the given divergence, the observed data density being a non-parametric representative of the true unknown density based on the given sample. In case of chi-square type distances, the construction of the data density inevitably requires the use of an appropriate non-parametric smoothing technique, like kernel density estimation, in continuous models (the LD is the only exception). This makes the derivation of the asymptotic properties far more involved, and complicates the computational aspect of this estimation. On the other hand, all the minimum Bregman divergence estimators are M-estimators and, hence, they avoid this density estimation component. In this paper, our primary aim is to extend the scope of the Bregman divergence by utilizing the powers of densities as arguments, rather than the densities themselves; this leads to the generalized class of the extended Bregman divergences which can then be used to generate new divergences which could provide more refined tools for minimum divergence inference compared to the current state of the art. This is the key idea of this work. Note that the use of the Bregman divergence in statistics is relatively recent; the class of density power divergences by [10], defined in Section 2, is a prominent example of Bregman divergences having significant applications in statistical inference. Many minimum divergence procedures have natural robustness properties against data contamination and outliers. The extended Bregman divergence allows us to express several existing divergence families as special cases of it, which is not possible through the ordinary Bregman divergence. Consequently, the extended Bregman idea can be used to generate large super-families of divergences containing, together with the existing divergences, many new and useful divergence families as special cases. In Section 2, we propose the extended Bregman divergence family. We demonstrate how it allows us to capture well known divergences that are not within the ordinary Bregman class and give a potential route for constructing new divergences. In Section 3, by considering a specific form of the convex function along with a particular choice of exponent of densities, we construct a large super-family of divergences within the extended Bregman family. Several known divergence families are obtained as special cases of this super- family. Section 4 introduces the corresponding minimum distance estimator, while Section 5 studies its asymptotic properties. Section 6 explores the robustness properties of the estimator based on its influence function. A large scale simulation study is taken up in Section 7, and a tuning parameter selection strategy is discussed in Section 8. The final section has some concluding remarks. Before concluding this section, we summarize what we believe to be the main achievements in this paper. 1. 1. We provide a simple extension of the Bregman divergence by considering powers of densities (instead of the densitites themselves) as arguments. Many divergence families (which are ordinarily not members of the class of Bregman divergences) can now be looked upon as members of this extended family, and the properties of the corresponding minimum distance estimators may be obtained from the general properties common to all the members of the extended family. 2. 2. Ordinarily, all minimum Bregman divergence estimators are also M-estimators. But, through this extension, several minimum distance estimators which are not M-estimators also become a part of this extended family. 3. 3. Consideration of exponentiated arguments with a specific choice of the convex function introduces a generalized super-family which we refer to as the GSB divergence family. The power divergence family of [11], the density power divergence (DPD) family of [10], the Bregman exponential divergence (BED) of [12] and the $S$-divergence family of [13] can all be brought under the umbrella of this super-family. 4. 4. This GSB divergence family consists of three tuning parameters $\alpha$, $\beta$ and $\lambda$. By simultaneously varying all these three parameters, we can generate new divergences (and hence new minimum divergence estimators) which are outside the union of the BED and S-divergence family, but can potentially provide improved performance over both of these classes. ## 2 The extended Bregman divergence and special cases Being motivated by the problem of convex programming, [3] introduced the Bregman divergence, a measure of dissimilarity between any two vectors in the Euclidean space. In $\mathbb{R}^{p}$, it has the form $\displaystyle D_{\psi}\left(\boldsymbol{x},\boldsymbol{y}\right)=\Bigg{\\{}\psi\left(\boldsymbol{x}\right)-\psi\left(\boldsymbol{y}\right)-\langle\nabla\psi\left(\boldsymbol{y}\right),\boldsymbol{x}-\boldsymbol{y}\rangle\Bigg{\\}},$ (3) for any strictly convex function $\psi:\mathcal{S}\rightarrow\mathbb{R}$ and for any two $p$-dimensional vectors $\boldsymbol{x},\boldsymbol{y}\in\mathcal{S}$, where $\mathcal{S}$ is a convex subset of $\mathbb{R}^{p}$. Here, $\nabla\psi\left(\boldsymbol{y}\right)$ denotes the gradient of $\psi$ with respect to its argument at $\boldsymbol{y}=\left(y_{1},y_{2},\ldots,y_{p}\right)^{T}$. It is evident that only the convexity criterion of the function $\psi\left(\cdot\right)$ is necessary for the non-negativity property of the divergence $D_{\psi}\left(\boldsymbol{x},\boldsymbol{y}\right)$ to hold. One could, therefore, consider other quantities as the arguments rather than the points themselves in this measure. Hence, as long as $\psi$ remains convex, any set of arguments whose equivalence translates to the equivalence of $\boldsymbol{x}$ and $\boldsymbol{y}$ can be used in the distance expression. This observation may be used to extend the Bregman divergence to have the form $\displaystyle D_{\psi}\left(\boldsymbol{x},\boldsymbol{y}\right)=\Bigg{\\{}\psi\left(\boldsymbol{x}^{k}\right)-\psi\left(\boldsymbol{y}^{k}\right)-\langle\nabla\psi\left(\boldsymbol{y}^{k}\right),\boldsymbol{x}^{k}-\boldsymbol{y}^{k}\rangle\Bigg{\\}}.$ (4) Here, $\nabla\psi\left(\boldsymbol{y}^{k}\right)$ denotes the gradient of $\psi$ with respect to its argument, evaluated at $\boldsymbol{y}^{k}=\left(y_{1}^{k},y_{2}^{k},\ldots,y_{d}^{k}\right)^{T}$ and $\psi$ is a strictly convex function, mapping $\mathcal{S}$ to $\mathbb{R}$, $\mathcal{S}$ being a convex subset of $\mathbb{R^{+}}^{p}$. Since our main purpose is to utilize this extension in the field of statistics where the arguments, being probability density functions, are inherently non-negative, restricting the domain of $\psi$ to $\mathbb{R^{+}}^{p}$ does not cause any difficulty. It is also not difficult to see that many of the properties of the Bregman divergence in Equation (3), as described by [14], are retained by the extended version in Equation (4). However, we will not make use of these properties in this paper, so we do not discuss them here any further. The Bregman divergence has significant applications in the domain of statistical inference for both discrete and continuous models. Given two densities $g$ and $f$, the Bregman divergence between these densities (associated with the convex function $\psi$) is given by $\displaystyle D_{\psi}\left(g,f\right)=\displaystyle\int\Bigg{\\{}\psi\left(g\left(x\right)\right)-\psi\left(f\left(x\right)\right)-\left(g\left(x\right)-f\left(x\right)\right)\nabla\psi\left(f\left(x\right)\right)\Bigg{\\}}dx.$ (5) By the strict convexity of the function $\psi$, the integrand in the above Equation (5) is non-negative and, therefore, so is the integral. It is also clear that the divergence equals zero if and only if the arguments $g$ and $f$ are identically equal. Well-known examples include the LD and the (squared) $L_{2}$ distance which correspond to $\psi\left(x\right)=x\log x$ and $\psi\left(x\right)=x^{2}$ respectively. In a real scenario, one uses $g$ as the true data generating density and $f=f_{\theta}$ as the parametric model density. In [10], the important class of density power divergences (DPDs) has been proposed, which is a subfamily of the class of Bregman divergences. This family is generated by the function $\psi\left(x\right)=\frac{x^{\alpha+1}-x}{\alpha}$, indexed by a non-negative tuning parameter $\alpha$. As a function of $\alpha$, the density power divergence may be expressed as $\displaystyle DPD_{\alpha}\left(g,f\right)=\displaystyle\int\left\\{f^{\alpha+1}\left(x\right)-\left(1+\frac{1}{\alpha}\right)g\left(x\right)f^{\alpha}\left(x\right)+\frac{1}{\alpha}g^{\alpha+1}\left(x\right)\right\\}dx.$ (6) It is a simple matter to check that for $\alpha=1$, the above reduces to the (squared) $L_{2}$ distance between $g$ and $f$, whereas when $\alpha\rightarrow 0$, one recovers the likelihood disparity defined in Equation (1). The DPD class should not be confused with the power divergence (PD) class of divergences (see [11]) which has the form $\displaystyle PD_{\lambda}\left(g,f\right)=\displaystyle\frac{1}{\lambda\left(\lambda+1\right)}\int\left\\{g\left(x\right)\left(\frac{g\left(x\right)}{f\left(x\right)}\right)^{\lambda}-1\right\\}dx,\lambda\in\mathbb{R}.$ (7) The PD class is a subfamily of chi-square type distances. The latter class of divergences has the form $\rho\left(g,f\right)=\displaystyle\int C\left(\delta\left(x\right)\right)f\left(x\right)dx,$ (8) where $C$ is a strictly convex function and $\delta\left(x\right)=\frac{g\left(x\right)}{f\left(x\right)}-1$. The power divergence corresponds to the specific convex function $C\left(\delta\right)=\frac{\left(\delta+1\right)^{\lambda+1}-\left(\delta+1\right)}{\lambda\left(\lambda+1\right)}-\frac{\delta}{\lambda+1}.$ (9) Important special cases of the PD class include the LD (obtained in the limit as $\lambda\rightarrow 0$) and the twice, squared HD (obtained for $\lambda=-\frac{1}{2}$). The LD is the only common member between the PD and the DPD classes. The Bregman exponential divergence (BED) class ([12]), on the other hand, has the form $\displaystyle BED_{\beta}\left(g,f\right)=\displaystyle\frac{2}{\beta}\displaystyle\int\left\\{e^{\beta f\left(x\right)}\left(f\left(x\right)-\frac{1}{\beta}\right)-e^{\beta f\left(x\right)}g\left(x\right)+\frac{e^{\beta g\left(x\right)}}{\beta}\right\\}dx.$ (10) The defining convex function is $\psi\left(x\right)=\frac{2\left(e^{\beta x}-\beta x-1\right)}{\beta^{2}}$ which is indexed by the real parameter $\beta$. This family generates the (squared) $L_{2}$ distance in the limit $\beta\rightarrow 0$. A list of some Bregman divergences useful in the context of statistical inference is presented in Table 1. Table 1: Different divergences as special cases of the Bregman divergence Choice of convex function \| Divergences ---\|--- $B\left(x\right)=\displaystyle x^{2}$ \| (squared) $L_{2}$ Distance $B\left(x\right)=\displaystyle x\log\left(x\right)$ \| Likelihood Disparity $B\left(x\right)=\displaystyle\frac{x^{1+\alpha}-x}{\alpha}$ \| Density Power Divergence (DPD) $B\left(x\right)=\displaystyle-\frac{\log\left(x\right)}{2\pi}$ \| Itakura-Saito Distance $B\left(x\right)=\frac{2\left(e^{\beta x}-\beta x-1\right)}{\beta^{2}}$ \| Bregman Exponential Divergence Consider the standard set up of parametric estimation where $G$ is the true data generating distribution which is modeled by the parametric family ${\cal F}=\\{F_{\theta}:\theta\in\Theta\subset{\mathbb{R}}^{p}\\}$. Let $g$ and $f_{\theta}$ be the corresponding density functions. Further we assume that both $G$ and $F_{\theta}$ belong to $\mathcal{G}$, the class of all cumulative distribution functions having densities with respect to some appropriate dominating measure. Our aim is to estimate the unknown parameter $\theta$ by choosing the model density closest to the true density in the Bregman sense. The definition of ordinary Bregman divergences as given in Equation (5), useful as it is, does not include many well-known and popular divergences which are extensively used in the literature for different purposes including parameter estimation. The PD family is a prominent example. An inspection of the Bregman form in Equation (5) indicates that the term which involves both densities $g$ and $f$ is of the form $\int g\left(x\right)\nabla\psi\left(f\left(x\right)\right)dx.$ (11) Here, the density $g$ is present only as a linear term having exponent one. Given a random sample $X_{1},X_{2},\ldots,X_{n}$ from the true distribution $G$, the term in Equation (11) can be empirically estimated by $\frac{1}{n}\sum\nabla\psi(f_{\theta}(X_{i}))$ (with $f=f_{\theta}$ under the parametric model) so that one can construct an empirical version of the divergence without any non-parametric density estimation. On the other hand, this restricts the class of divergences that are expressible in the Bregman form. Using an extension in the spirit of Equation (4) may allow the construction of richer classes of divergences. With this aim, we define the extended Bregman divergence between two densities $g$ and $f$ as $\displaystyle D^{(k)}_{\psi}\left(g,f\right)=\displaystyle\int\left\\{\psi\left(g^{k}\left(x\right)\right)-\psi\left(f^{k}\left(x\right)\right)-\left(g^{k}\left(x\right)-f^{k}\left(x\right)\right)\nabla\psi\left(f^{k}\left(x\right)\right)\right\\}dx.$ Apart from the requirement of strict convexity of the function $\psi$, this formulation also depends on a positive index $k$ with which the density is exponentiated. For the rest of the paper, the notation $D^{(k)}_{\psi}(\cdot,\cdot)$ will refer to this general form in Equation (LABEL:ss1), of which the divergence in Equation (5) is a special case for $k=1$. Evidently, $D^{(k)}_{\psi}\left(g,f\right)\geq 0$ for any choices of densities $f$ and $g$ with respect to the same measure. Moreover, the fact that $D^{(k)}_{\psi}\left(g,f\right)=0$ if and only if $g=f$, holds true in this case due to non-negativity property of a density as well as the consideration of strict convexity of the function $\psi\left(\cdot\right)$. In the following, we present some special cases of extended Bregman divergence. 1. 1. $S$-Hellinger family ([13]) If we take $\psi\left(x\right)=\frac{2e^{\beta x}}{\beta^{2}}$ with $k=\frac{1+\alpha}{2}$, $\alpha\in\left(0,1\right)$ in Equation (LABEL:ss1), it will generate an extension of the BED family having the form $BED^{(k)}_{\beta}\left(g,f\right)=\frac{2}{\beta}\int\left\\{e^{\beta f^{k}\left(x\right)}\left(f^{k}\left(x\right)-\frac{1}{\beta}\right)-e^{\beta f^{k}\left(x\right)}g^{k}\left(x\right)+\frac{e^{\beta g^{k}\left(x\right)}}{\beta}\right\\}dx.$ (13) It can be easily shown that, as $\beta\rightarrow 0$ and $k=\frac{1+\alpha}{2}$, $\alpha\in\left(0,1\right)$, the application of L’Hospital’s rule leads to the S-Hellinger Distance (SHD) family with the form $SHD_{\alpha}\left(g,f\right)=\frac{2}{1+\alpha}\int\left(g^{\frac{1+\alpha}{2}}\left(x\right)-f^{\frac{1+\alpha}{2}}\left(x\right)\right)^{2}dx.$ (14) This was introduced by [13] as a special case of the $S$-divergence family. This family cannot be expressed through the normal expression of the Bregman divergence, but through this extension, we can express this member of the $S$-divergence family as a (limiting) member of the extended BED class. 2. 2. PD family ([11]) If we take $\psi\left(x\right)=\frac{x^{1+\frac{B}{A}}}{B}$, $A=1+\lambda$, $B=-\lambda$ and $\lambda\in\mathbb{R}$, with $k=A$ in Equation (LABEL:ss1), we get the PD family introduced in Equation (7). 3. 3. $S$-divergence family ([13]) If we take $\psi\left(x\right)=\frac{x^{1+\frac{B}{A}}}{B}$, $A=1+\lambda\left(1-\alpha\right)$, $B=\alpha-\lambda\left(1-\alpha\right)$, $A+B=1+\alpha$, $\alpha\geq 0$, $\lambda\in\mathbb{R}$ and $k=A$ in Equation (LABEL:ss1), we get the $S$-divergence having the following form $SD_{(\alpha,\lambda)}\left(g,f\right)=\int\left\\{\frac{1}{B}\left(g^{A+B}\left(x\right)-f^{A+B}\left(x\right)\right)-\left(g^{A}\left(x\right)-f^{A}\left(x\right)\right)\frac{A+B}{AB}f^{B}\left(x\right)\right\\}dx.$ (15) This is one of the most useful divergence families in the domain of robust inference due to its capacity to generate much more robust estimator(s) than the DPD and PD families can generate. Through this extension of Bregman divergence, it is now possible to express this divergence as a special case of the extended family. Note that, for $k\neq 1$, $f^{k}$ and $g^{k}$ will generally no longer represent probability densities, and by extending the divergence idea to general positive measures (beyond probability measures), [15] has suggested certain constructions where the power divergence has been exhibited in the Bregman divergence form for general measures. Also, see the discussion in [16]. We differ from the interpretation in these papers in the sense that we still view the family of divergences presented here in Equation (LABEL:ss1) as divergences between valid probability densities. Given any two probability densities, the expression in Equation (LABEL:ss1) is non-negative, and equals zero if and only if the densities $g$ and $f$ are identical, irrespective of the value of $k$. For the ordinary Bregman divergence, the term in Equation (11), with $f=f_{\theta}$, may be approximated by $\frac{1}{n}\sum_{i=1}^{n}\nabla\psi\left(f_{\theta}\left(X_{i}\right)\right),$ (16) through the replacement of $dG\left(x\right)$ by $dG_{n}\left(x\right)$, where $G_{n}$ is the empirical distribution function obtained from the random sample $X_{1},X_{2},\ldots,X_{n}$. It is evident that the minimizer of the empirical version of the Bregman divergence is an M-estimator. But there are several useful divergences (or divergence families) where the empirical representation of the term involving $f$ and $g$ is not possible using the above trick, and such divergences generate estimators beyond the M-estimator class. See [17] for more discussion on this issue. In the above we have given several examples where the extended Bregman class contains such divergences which are not covered by the ordinary Bregman form. Thus the structure of the extended class allows us to extend the scope much beyond that of the ordinary Bregman divergence. ## 3 Introducing a new divergence family Our aim here is to exploit the extended Bregman idea and generate rich new super families of divergences by choosing a suitable convex generating function and a suitable exponent. In particular, we use the convex function $\psi(x)=e^{\beta x}+\frac{x^{1+\frac{B}{A}}}{B}$, $A=1+\lambda\left(1-\alpha\right)$, $B=\alpha-\lambda\left(1-\alpha\right)$, $A+B=1+\alpha$, $\alpha\geq-1$, $\beta,\lambda\in\mathbb{R}$, which, together with the exponent $k=A$, generates the divergence $D^{}\left(g,f\right)=\int\left\\{e^{\beta f^{A}}\left(\beta f^{A}-\beta g^{A}-1\right)+e^{\beta g^{A}}+\frac{1}{B}\left(g^{A+B}-f^{A+B}\right)-\left(g^{A}-f^{A}\right)\frac{A+B}{AB}f^{B}\right\\}dx,$ (17) which we refer to as the GSB divergence (being the abbreviated form of generalized S-Bregman divergence). The divergence measure $D^{}$ is also a function of $\alpha,\lambda$ and $\beta$, which we suppress for brevity. If we put $A+B=0$ in the above expression with $A\neq 0$ and $B\neq 0$, we will get the extended BED family with parameter $\beta$ and exponent $k=A$. Moreover, if $A=1$, i.e., $\lambda=0$ then it will lead to the ordinary BED family with parameter $\beta$. On the contrary, if we put $\beta=0$, it will lead to the $S$-divergence family with parameters $\alpha$ and $\lambda$ (in terms of $A$ and $B$). More specifically, when $\alpha=0$ and $\beta=0$, it leads to the power divergence (PD) family. On the other hand, $\beta=0$ and $\lambda=0$ lead us to the density power divergence (DPD) family. Thus, it acts as a connector between the BED and the $S$-divergence family. ### 3.1 Special cases We will get several well-known divergences or divergence families from the general form of GSB for particular choices of the three tuning parameters $\alpha$, $\lambda$ and $\beta$. Some such choices are given in Table 2. Table 2: Different divergences as special cases of GSB divergence $\alpha$ \| $\lambda$ \| $\beta$ \| Divergences ---\|---\|---\|--- $\alpha=-1$ \| $\lambda=0$ \| $\beta\in\mathbb{R}$ \| Bregman Exponential Divergence1 $\alpha=0$ \| $\lambda\in\mathbb{R}$ \| $\beta=0$ \| Power Divergence $\alpha\in\mathbb{R}$ \| $\lambda=0$ \| $\beta=0$ \| Density Power Divergence $\alpha\in\mathbb{R}$ \| $\lambda\in\mathbb{R}$ \| $\beta=0$ \| $S$-Divergence $\alpha=0$ \| $\lambda=-1$ \| $\beta=0$ \| Kullback-Liebler Divergence $\alpha=0$ \| $\lambda=0$ \| $\beta=0$ \| Likelihood Disparity $\alpha=0$ \| $\lambda=-.5$ \| $\beta=0$ \| Hellinger Distance $\alpha\in\mathbb{R}$ \| $\lambda=-.5$ \| $\beta=0$ \| S-Hellinger Distance $\alpha=0$ \| $\lambda=1$ \| $\beta=0$ \| Pearson’s Chi-square Divergence $\alpha=0$ \| $\lambda=-2$ \| $\beta=0$ \| Neyman’s Chi-square Divergence $\alpha=1$ \| $\lambda\in\mathbb{R}$ \| $\beta=0$ \| (squared) $L_{2}$ Distance * 1 This is a constant time B-exponential divergence. It basically generates all the members of BED family corresponding to the same $\beta$ except (squared) $L_{2}$ distance, which occurs when $\beta\rightarrow 0$. However, as seen above, the (squared) $L_{2}$ distance remains a member of the GSB class for other choices of the tuning parameters. ## 4 The minimum GSB divergence estimator Under the parametric set-up described in Section 2, we would like to identify the best fitting parameter $\theta^{g}$ by choosing the element of the model family of distributions which provides the closest match to the true density $g$ in terms of the given divergence. The minimum GSB divergence functional $T_{\alpha,\lambda,\beta}:\mathcal{G}\rightarrow\Theta$ is defined by the relation $D^{}\left(g,f_{T_{\alpha,\lambda,\beta}}\right)=\min\\{D^{}\left(g,f_{\theta}\right):\theta\in\Theta\\},$ provided the minimum exists. If the parametric model family is identifiable, it follows from the definition of the divergence that $D^{}\left(g,f_{\theta}\right)=0$, if and only if $g=f_{\theta}$. Thus, $T_{\alpha,\lambda,\beta}\left(F_{\theta}\right)=\theta$, uniquely. Hence, the functional $T_{\alpha,\lambda,\beta}$ is Fisher consistent. Given the density $g$, a straightforward differentiation of the GSB divergence of Equation (17) leads to the estimating equation $\displaystyle\displaystyle\int\left\\{A^{2}\beta^{2}e^{\beta f_{\theta}^{A}\left(x\right)}f_{\theta}^{A}\left(x\right)+\left(A+B\right)f_{\theta}^{B}\left(x\right)\right\\}\left(f_{\theta}^{A}\left(x\right)-g^{A}\left(x\right)\right)u_{\theta}\left(x\right)dx=0.$ (18) In practice, the true density $g$ is unknown, so one has to use a suitable non-parametric density estimator $\hat{g}$ for $g$, depending on the situation. Under a discrete parametric set-up, the natural choice for $\hat{g}$ is the vector of relative frequencies as obtained from the sample data. Thus, assuming a discrete parametric model, and assuming, without loss of generality, that the support of the random variable is $\\{0,1,2,\ldots,\\}$, the estimating equation becomes $\displaystyle\displaystyle\sum_{x=0}^{\infty}A^{2}\beta^{2}e^{\beta f_{\theta}^{A}\left(x\right)}f_{\theta}^{2A}\left(x\right)u_{\theta}\left(x\right)+\sum_{x=0}^{\infty}\left(A+B\right)f_{\theta}^{A+B}\left(x\right)u_{\theta}\left(x\right)$ (19) $\displaystyle=$ $\displaystyle\displaystyle\sum_{x=0}^{\infty}A^{2}\beta^{2}e^{\beta f_{\theta}^{A}\left(x\right)}f_{\theta}^{A}\left(x\right)\hat{g}^{A}\left(x\right)u_{\theta}\left(x\right)+\displaystyle\sum_{x=0}^{\infty}\left(A+B\right)f_{\theta}^{B}\left(x\right)\hat{g}^{A}\left(x\right)u_{\theta}\left(x\right).$ For continuous models, on the other hand, some suitable non-parametric smoothing technique such as kernel density estimation is inevitable unless the exponent $A$ equals 1. In the latter case, $g$ appears as a linear term in the estimating equation (18). In that case, we can use $G_{n}$, the empirical distribution function, as an estimator of $G$. Hence, for $A=1$, Equation (19) can be reduced as $\displaystyle\displaystyle\sum_{x=0}^{\infty}\beta^{2}e^{\beta f_{\theta}\left(x\right)}f_{\theta}^{2}\left(x\right)u_{\theta}\left(x\right)+\sum_{x=0}^{\infty}\left(1+B\right)f_{\theta}^{1+B}\left(x\right)u_{\theta}\left(x\right)$ (20) $\displaystyle=$ $\displaystyle\frac{1}{n}\displaystyle\sum_{i=1}^{n}\beta^{2}e^{\beta f_{\theta}\left(X_{i}\right)}f_{\theta}\left(X_{i}\right)u_{\theta}\left(X_{i}\right)+\frac{1}{n}\sum_{i=1}^{n}\left(1+B\right)f_{\theta}^{B}\left(X_{i}\right)u_{\theta}\left(X_{i}\right).$ Since the left hand side of the above equation is non-random and the right hand side is a sum of independent and identically distributed terms, it is of the form $\sum_{i=1}^{n}\psi\left(X_{i},\theta\right)=0$ and the corresponding estimator belongs to the M-estimator class. In accordance with the information on the first three rows of Table 2, we will refer to the parameters $\alpha$, $\lambda$ and $\beta$ as the DPD parameter, the PD parameter and the BED parameter, respectively. ## 5 Asymptotic properties of the GSB divergence In this section, we concentrate on the asymptotic properties of our proposed minimum divergence estimator. As mentioned, we will focus on the discrete set- up throughout the rest of the paper. Let $X_{1},X_{2},\ldots,X_{n}$ be independent and identically distributed observations from an unknown distribution $G$ with support $\chi=\\{0,1,2,3,\ldots,\\}$. On the other hand, we consider a parametric family of distributions $\mathcal{F}=\\{F_{\theta}:\theta\in\Theta\subseteq\mathbb{R}^{p}\\}$, also supported on $\chi$, to model the true data generating distribution $G$. In this set-up, we assume both $G$ and $\mathcal{F}$ to have densities $g$ and $f_{\theta}$ with respect to the counting measure. Let $\theta^{g}=T_{\alpha,\beta,\lambda}(G)$ be the best fitting parameter. Since $G$ is unknown, we are going to use the vector of relative frequencies, obtained from the data, as an estimate of $g$ throughout the rest of this paper. Let $r_{n}(x)$ be the relative frequency of the value $x$ in the sample. The minimum GSB divergence estimator is obtained as a root of the estimating equation $\displaystyle\displaystyle\sum_{x=0}^{\infty}\left\\{A^{2}\beta^{2}e^{\beta f_{\theta}^{A}\left(x\right)}f_{\theta}^{A}\left(x\right)+\left(A+B\right)f_{\theta}^{B}\left(x\right)\right\\}\left(f_{\theta}^{A}\left(x\right)-\hat{g}^{A}\left(x\right)\right)u_{\theta}\left(x\right)=0$ (21) $\displaystyle\Rightarrow$ $\displaystyle\displaystyle\sum_{x=0}^{\infty}\left\\{A^{2}\beta^{2}e^{\beta f_{\theta}^{A}\left(x\right)}f_{\theta}^{2A}\left(x\right)+\left(A+B\right)f_{\theta}^{A+B}\left(x\right)\right\\}\frac{\left(1-\frac{\hat{g}^{A}\left(x\right)}{f_{\theta}^{A}\left(x\right)}\right)}{A}u_{\theta}\left(x\right)=0$ $\displaystyle\Rightarrow$ $\displaystyle\displaystyle\sum_{x=0}^{\infty}K\left(\delta\left(x\right)\right)\left(A^{2}\beta^{2}e^{\beta f_{\theta}^{A}\left(x\right)}f_{\theta}^{2A}\left(x\right)+\left(A+B\right)f_{\theta}^{A+B}\left(x\right)\right)u_{\theta}\left(x\right)=0,$ where, $\delta\left(x\right)=\delta_{n}\left(x\right)=\frac{\hat{g}\left(x\right)}{f_{\theta}\left(x\right)}-1=\frac{r_{n}\left(x\right)}{f_{\theta}\left(x\right)}-1$, $K\left(\delta\right)=\frac{\left(\delta+1\right)^{A}-1}{A}$ and $u_{\theta}\left(x\right)$ is the likelihood score function at $x$. We denote the minimum GSB divergence estimator obtained as a solution of the above equation as $\hat{\theta}$. Let $\displaystyle J_{g}$ $\displaystyle=$ $\displaystyle J_{\alpha,\beta,A}\left(g\right)$ $\displaystyle=$ $\displaystyle\displaystyle E_{g}\left(u_{\theta^{g}}\left(X\right)u^{T}_{\theta^{g}}\left(X\right)K^{\prime}\left(\delta_{g}^{g}\left(X\right)\right)\left(\left(A+B\right)f_{\theta^{g}}^{\alpha}\left(X\right)+A^{2}\beta^{2}e^{\beta f_{\theta^{g}}^{A}\left(X\right)}f_{\theta^{g}}^{2A-1}\left(X\right)\right)\right)$ $\displaystyle+$ $\displaystyle\sum_{x=0}^{\infty}K\left(\delta_{g}^{g}\left(x\right)\right)\left(\left(A+B\right)f_{\theta^{g}}^{1+\alpha}\left(x\right)+A^{2}\beta^{2}e^{\beta f_{\theta^{g}}^{A}\left(x\right)}f_{\theta^{g}}^{2A}\left(x\right)\right)i_{\theta^{g}}\left(x\right)$ $\displaystyle-$ $\displaystyle\sum_{x=0}^{\infty}K\left(\delta_{g}^{g}\left(x\right)\right)\left(\left(A+B\right)^{2}f_{\theta^{g}}^{1+\alpha}\left(x\right)+A^{3}\beta^{2}e^{\beta f_{\theta^{g}}^{A}\left(x\right)}f_{\theta^{g}}^{2A}\left(x\right)\left(2+\beta f_{\theta^{g}}^{A}\left(x\right)\right)\right)u_{\theta^{g}}\left(x\right)u^{T}_{\theta^{g}}\left(x\right)$ $\displaystyle V_{g}$ $\displaystyle=$ $\displaystyle\displaystyle Var_{g}\left(u_{\theta^{g}}\left(X\right)K^{\prime}\left(\delta_{g}^{g}\left(X\right)\right)\left(\left(A+B\right)f_{\theta^{g}}^{\alpha}\left(X\right)+A^{2}\beta^{2}e^{\beta f_{\theta^{g}}^{A}\left(X\right)}f_{\theta^{g}}^{2A-1}\left(X\right)\right)\right),$ (22) where, $X$ is a random variable having density $g$, $Var_{g}$ represents variance under the density $g$, $\delta_{g}\left(x\right)=\frac{g\left(x\right)}{f_{\theta}\left(x\right)}-1$, $K^{\prime}\left(\cdot\right)$ is the derivative of $K\left(\cdot\right)$ with respect to its argument, $\delta_{g}^{g}\left(x\right)=\frac{g\left(x\right)}{f_{\theta}^{g}\left(x\right)}-1$ and $i_{\theta}\left(x\right)=-u^{\prime}_{\theta}\left(x\right)$, the negative of the derivative of the score function with respect to the parameter. ###### Theorem 1. Under the above-mentioned set-up and certain regularity assumptions given in the Online Supplement, there exists a consistent sequence of roots $\hat{\theta}_{n}$ of the estimating equation (18). Moreover, the asymptotic distribution of $\sqrt{n}\left(\hat{\theta}_{n}-\theta^{g}\right)$ is p-dimensional normal with mean $0$ and $J_{g}^{-1}V_{g}J_{g}^{-1}$. ###### Corollary 5.1. When $g=f_{\theta}$ for some $\theta\in\Theta$, then $\sqrt{n}\left(\theta_{n}-\theta\right)\sim N\left(0,J^{-1}VJ^{-1}\right)$ asymptotically, where, $\displaystyle J$ $\displaystyle=$ $\displaystyle E_{f_{\theta}}\left\\{u_{\theta}\left(X\right)u^{T}_{\theta}\left(X\right)\left(\left(A+B\right)f^{\alpha}\left(x\right)+A^{2}\beta^{2}e^{\beta f_{\theta}^{A}\left(X\right)}f_{\theta}^{2A-1}\left(X\right)\right)\right\\}$ $\displaystyle=$ $\displaystyle\sum_{x=0}^{\infty}\left\\{u_{\theta}\left(x\right)u^{T}_{\theta}\left(x\right)\left(\left(A+B\right)f^{\alpha}\left(x\right)+A^{2}\beta^{2}e^{\beta f_{\theta}^{A}\left(x\right)}f_{\theta}^{2A-1}\left(x\right)\right)\right\\}f_{\theta}\left(x\right),$ $\displaystyle V$ $\displaystyle=$ $\displaystyle V_{f_{\theta}}\left\\{u_{\theta}\left(X\right)\left(\left(A+B\right)f_{\theta}^{\alpha}\left(X\right)+A^{2}\beta^{2}e^{\beta f_{\theta}^{A}\left(X\right)}f_{\theta}^{2A-1}\left(X\right)\right)\right\\}$ (24) $\displaystyle=$ $\displaystyle\left(A+B\right)^{2}\sum_{x=0}^{\infty}f^{1+2\alpha}\left(x\right)u_{\theta}\left(x\right)u^{T}_{\theta}\left(x\right)$ $\displaystyle+$ $\displaystyle A^{4}\beta^{4}\sum_{x=0}^{\infty}e^{2\beta f_{\theta}^{A}\left(x\right)}f_{\theta}^{4A-1}\left(x\right)u_{\theta}\left(x\right)u^{T}_{\theta}\left(x\right)$ $\displaystyle+$ $\displaystyle 2\left(A+B\right)A^{2}\beta^{2}\sum_{x=0}^{\infty}e^{\beta f_{\theta}^{A}\left(x\right)}f_{\theta}^{2A+\alpha}\left(x\right)u_{\theta}\left(x\right)u^{T}_{\theta}\left(x\right)-\zeta\zeta^{\prime},$ where, $\zeta=\displaystyle\sum_{x=0}^{\infty}u_{\theta}\left(x\right)\left(\left(A+B\right)f^{A+B}\left(x\right)+A^{2}\beta^{2}e^{\beta f_{\theta}^{A}\left(x\right)}f_{\theta}^{2A}\left(x\right)\right).$ ## 6 Influence analysis of the minimum GSB estimator Here we study the stability of our proposed class of estimators on the basis of the influence function (IF), which measures the effect of adding an infinitesimal mass to the distribution and is one of the most important heuristic tools of robustness. A simple differentiation of a contaminated version of the estimating equation (18) leads to the expression $IF\left(y,G,T_{\alpha,\lambda,\beta}\right)=J_{G}^{-1}N_{G}\left(y\right),~{}~{}\rm{where},$ (25) $\displaystyle N_{G}\left(y\right)$ $\displaystyle=$ $\displaystyle\displaystyle\left(A^{2}\beta^{2}e^{\beta f^{A}_{\theta^{g}}\left(y\right)}f^{A}_{\theta^{g}}\left(y\right)+(A+B)f^{B}_{\theta^{g}}\left(y\right)\right)g^{A-1}\left(y\right)u_{\theta^{g}}\left(y\right)$ $\displaystyle-$ $\displaystyle\sum_{x=0}^{\infty}\left(A^{2}\beta^{2}e^{\beta f^{A}_{\theta^{g}}\left(x\right)}f^{A}_{\theta^{g}}\left(x\right)+(A+B)f^{B}_{\theta^{g}}\left(x\right)\right)g^{A}\left(x\right)u_{\theta^{g}}\left(x\right),$ $\displaystyle J_{G}$ $\displaystyle=$ $\displaystyle\displaystyle A^{2}\beta^{2}\sum_{x=0}^{\infty}e^{\beta f^{A}_{\theta^{g}}\left(x\right)}f^{A}_{\theta^{g}}\left(x\right)\left(2f^{A}_{\theta^{g}}\left(x\right)-g^{A}\left(x\right)\right)u_{\theta^{g}}\left(x\right)u^{T}_{\theta^{g}}\left(x\right)$ $\displaystyle+$ $\displaystyle A^{3}\beta^{3}\sum_{x=0}^{\infty}e^{\beta f^{A}_{\theta^{g}}\left(x\right)}f^{2A}_{\theta^{g}}\left(x\right)\left(f^{A}_{\theta^{g}}\left(x\right)-g^{A}\left(x\right)\right)u_{\theta^{g}}\left(x\right)u^{T}_{\theta^{g}}\left(x\right)$ $\displaystyle+$ $\displaystyle(A+B)\sum_{x=0}^{\infty}f^{B}_{\theta^{g}}\left(x\right)\left(\left(A+B\right)f^{A}_{\theta^{g}}\left(x\right)-Bg^{A}\left(x\right)\right)u_{\theta^{g}}\left(x\right)u^{T}_{\theta^{g}}\left(x\right)$ $\displaystyle+$ $\displaystyle A^{2}\beta^{2}\sum_{x=0}^{\infty}e^{\beta f^{A}_{\theta^{g}}\left(x\right)}f^{A}_{\theta^{g}}\left(x\right)\left(g^{A}\left(x\right)-f^{A}_{\theta^{g}}\left(x\right)\right)i_{\theta^{g}}\left(x\right)$ $\displaystyle-$ $\displaystyle(A+B)\sum_{x=0}^{\infty}f^{B}_{\theta^{g}}\left(f^{A}_{\theta^{g}}\left(x\right)-g^{A}\left(x\right)\right)i_{\theta^{g}}\left(x\right).$ If the distribution $G$ belongs to the model family $\mathcal{F}$ with $g=f_{\theta}$, then the influence function reduces to, $\displaystyle IF\left(y,F_{\theta},T_{\alpha,\lambda,\beta}\right)$ $\displaystyle=$ $\displaystyle J_{F_{\theta}}^{-1}N_{F_{\theta}}\left(y\right),\rm{where},$ (26) $\displaystyle J_{F_{\theta}}$ $\displaystyle=$ $\displaystyle\displaystyle\sum_{x=0}^{\infty}\left(A^{2}\beta^{2}e^{\beta f^{A}_{\theta}\left(x\right)}f^{2A}_{\theta}\left(x\right)+(A+B)f^{A+B}_{\theta}\left(x\right)\right)u_{\theta}\left(x\right)u^{T}_{\theta}\left(x\right),$ $\displaystyle N_{F_{\theta}}\left(y\right)$ $\displaystyle=$ $\displaystyle\displaystyle A^{2}\beta^{2}e^{\beta f^{A}_{\theta}\left(y\right)}f^{2A-1}_{\theta}\left(y\right)u_{\theta}\left(y\right)+(A+B)f^{A+B-1}_{\theta}\left(y\right)u_{\theta}\left(y\right)$ $\displaystyle-$ $\displaystyle\sum_{x=0}^{\infty}A^{2}\beta^{2}e^{\beta f^{A}_{\theta}\left(x\right)}f^{2A}_{\theta}\left(x\right)u_{\theta}\left(x\right)-\sum_{x=0}^{\infty}(A+B)f^{A+B}_{\theta}\left(x\right)u_{\theta}\left(x\right).$ Evidently, the influence function is dependent on all the three tuning parameters. Whenever the matrix $J_{F_{\theta}}$ is non singular, the boundedness of the influence function depends on the ability of the coefficients to control the score function $u_{\theta}(y)$ in the first two terms of the numerator. In most parametric models including all exponential family models, $f_{\theta}^{\tau}(y)u_{\theta}(y)$ remains bounded for any $\tau>0$; in the case $\tau=0$, however the expression equals $u_{\theta}(y)$ and there is no control over it to keep it bounded. For the second term of the numerator in Equation (6), this is achieved when $A+B>1$, i.e., when $\alpha>0$. The first term of the numerator contains an additional exponential term. However, given that $f_{\theta}(y)\leq 1$ for any value $y$ in the support of a discrete random variable, the first term of the numerator is easily seen to be bounded for any fixed non-zero real $\beta$ when $2A-1>0$, i.e., $A>1/2$. We now list the different possible cases for boundedness of the influence function as follows: 1. 1. $\beta=0$; here the first and third terms of the numerator vanish, and the only other condition necessary is $A+B>1$, i.e., $\alpha>0$. This is essentially the $S$-divergence case, and shows that all minimum $S$-divergence functionals with $\alpha>0$ have bounded influence (irrespective of the value of $\lambda$). In this case the allowable region for the triplet $(\alpha,\lambda,\beta)$ for bounded influence is ${\mathbb{S}}_{1}=\left(\alpha>0,\lambda\in{\mathbb{R}},\beta=0\right)$. 2. 2. $\beta\neq 0$, $A=0$. In this case also the first and third terms of the numerator drop out and the additional required condition is $\alpha>0$. However, since $A=1+\lambda(1-\alpha)=0$, this implies $\lambda=-\frac{1}{1-\alpha}$. In this case the influence function is independent of $\beta$. Now the relevant region for the triplet is ${\mathbb{S}}_{2}=\left(\alpha>0,\lambda=-\frac{1}{1-\alpha},\beta\neq 0\right)$. 3. 3. Now suppose $A+B=0$, without the components being individually zero. In this case the second and fourth terms get eliminated and we have $\alpha=-1$. In this case the condition $2A-1>0$ translates to $\lambda>-\frac{1}{4}$. Here the corresponding region for the triplet is ${\mathbb{S}}_{3}=\left(\alpha=-1,\lambda\geq-\frac{1}{4},\beta\neq 0\right)$. 4. 4. Now we allow all the terms $\beta$, $A$ and $A+B$ to be non-zero. In this case all the four terms of the numerator are non-vanishing. Then, beyond the condition on $\beta$, the required conditions are $\alpha>0$ and $\lambda\left(1-\alpha\right)>-\frac{1}{2}$. The region here is ${\mathbb{S}}_{4}=\left(\alpha>0,\lambda(1-\alpha)>-\frac{1}{2},\beta\neq 0\right)$. Combining all the cases, we see that the IF will be bounded if the triplet $\left(\alpha,\lambda,\beta\right)\in\mathbb{S}={\mathbb{S}}_{1}\cup{\mathbb{S}}_{2}\cup{\mathbb{S}}_{3}\cup{\mathbb{S}}_{4}$. It is easily seen that the four constituent subregions are disjoint. For illustration, we present some plots for bounded and unbounded influence functions for the minimum GSB functional under the Poisson($\theta$) model in Figure 1, where the true data distribution is Poisson(3). In the four rows of the right panel we give examples of triplets belonging to the four disjoint components of ${\mathbb{S}}$. In the first two rows of the right panel, the $\alpha$ value alone determines the shape of the curve. On the $i$-th row of the left panel, on the other hand, the triplets are slightly different from the triplets of $i$-th row on the right, but far enough to be pushed out of ${\mathbb{S}}_{i}$. Accordingly, all the plots on the left correspond to unbounded influence functions. Generally, it may also be observed that for increasing $\beta$ the curves get flatter in each plot, where IF varies over different $\beta$. We will provide further illustration of the bounded influence region of the triplet through three-dimensional graphs at the end of the simulation section. ## 7 Simulation results In the simulation section our aim is to demonstrate that by choosing non-zero values of the parameter $\beta$, we may be able to generate procedures which, in a suitable sense, improve upon the estimators that are provided by the existing standard, the class of $S$-divergences. We consider the Poisson ($\theta$) model, and choose samples of size 50 from the $(1-\epsilon){\rm Poisson}(3)+\epsilon{\rm Poisson}(10)$ mixture, where the second component is the contaminant and $\epsilon\in[0,1)$ is the contaminating proportion. The values 0, 0.05, 0.1 and 0.2 are considered for $\epsilon$, and at each contamination level, the samples are replicated 1000 times. The Poisson parameter is estimated in each of the 1000 replications, for each contamination level, and at each of several $(\alpha,\lambda,\beta)$ triplets considered in our study. Subsequently we construct the empirical mean square error (MSE) against the target value of 3, for each tuning parameter triplet and each contamination level over the 1000 replications. In case of the minimum $S$-divergence estimator, [13] have empirically identified a subset of $\left(\alpha,\lambda\right)$ collections which represent good choices. According to them, the zone of “best” estimators correspond to an elliptical subset of the tuning parameter space, with $\alpha\in[0.1,0.6]$ and $\lambda\in[-1,-0.3]$. We hope to show that for most of the $(\alpha,\lambda)$ combinations (including the best ones) there is a corresponding better or competitive $\left(\alpha,\lambda,\beta\right)$ combination with a non-zero $\beta$, thus providing an option which appears to perform better, at least to the extent of the findings in these simulations. We begin with an exploration of the $S$-divergence, since this is the basis for comparison. The MSEs are presented in Table 3 over a cross-classified grid with $\alpha$ values in {0.1, 0.25, 0.4, 0.5, 0.6, 0.8, 1} and $\lambda$ values in {-1, -0.7, -0.5, -0.3, 0, 0.2, 0.5, 0.8, 1}, a total of 63 cells. In each cell the empirical MSEs for $\epsilon=0,0.05,0.1$ and $0.2$ are presented in a column of four elements, in that order, followed by the corresponding combination of tuning parameters $(\alpha,\lambda,\beta=0)$. We have carried the $\beta=0$ parameter in each triplet of parameters, to indicate that the $S$-divergence is indeed a special case of the GSB divergence. It may be noted that between all the cells, there is no unique $(\alpha,\lambda)$ combination which produces an overall best result (in terms of smallest MSE) over all the four columns (levels of contamination). We now expand the exploration by considering, in addition, a grid of possible non-zero $\beta$ values at each $(\alpha,\lambda)$ combination to see if the results can be improved. To be conservative about our definition of improvement, we declare the existence of a “better” triplet in the GSB sense if all the four mean square errors corresponding to a ${(\alpha,\lambda)}$ combination within the $S$-divergence family in Table 3 are improved (reduced) by a suitable member of the GSB divergence class which is strictly outside the $S$-divergence family (corresponding to a non-zero $\beta$). \| ---\|--- \| \| \| Figure 1: Examples of unbounded influence functions (left panel) and bounded influence functions (right panel) corresponding to $\left(\alpha,\lambda,\beta\right)\in$ each disjoint subsets contained in $\mathbb{S}$. Our exploration indicates that in a large majority of the 63 cells there is a member of the GSB divergence with a non-zero $\beta$ parameter which improves (over all the four cells) the performance of the corresponding $S$-divergence estimator with the same $(\alpha,\lambda)$ combination. Interestingly it turns out that in practically all the cases where an improvement is observed it happens for a negative value of $\beta$ (it is observed to be zero in rare cases, but is never positive). A more detailed inspection indicates that in many of these cases, the improvement occurs at the value $\beta=-4$. In order to summarize the findings of this rather large exploration (presented in Table 4) in a meaningful manner, we first note the following different cases, 1. 1. (First Case) These are the cells where all the four mean square errors for the $S$-divergence case are reduced by the minimum GSB divergence estimator with the same values of $(\alpha,\lambda)$ and $\beta=-4$. These cells are highlighted with the blue colour in Table 4. (There are 18 such cells). 2. 2. (Second Case) These are the cells where all the four MSEs for the $S$-divergence case are reduced by a minimum GSB divergence estimator with $\beta=-4$ but with a different $(\alpha,\lambda)$ combination than that for the corresponding cell. These cells are highlighted in red in Table 4. (There are 39 such cells). 3. 3. (Third Case) These are the cells where all the four MSEs are reduced by a minimum GSB divergence estimator outside the $S$-divergence family, but with $\beta\neq-4$, and not necessarily the same $(\alpha,\lambda)$. These cells are highlighted in orange in Table 4. (There is one such cell). 4. 4. (Fourth Case) These are the cells where some triplet within the minimum GSB divergence class can improve upon the three MSEs under contamination ($\epsilon=0.05,0.1,0.2$) but not all the four MSEs simultaneously. While these are not “better” triplets in the sense described earlier in the section, the pure data MSEs (not reported here) for these triplets are close to those of the S-Divergence MSEs for these cells; in this sense these triplets are at least competitive. These cells are highlighted in green in Table 4. (There are three such cases). 5. 5. (Fifth Case) These are the cells where no $(\alpha,\lambda,\beta)$ provides an improvement over the S-divergence results in the sense of any of the previous four cases (although there are competitive alternatives). These cells remain in black in Table 4. There are 2 such cells. On the whole, therefore, it turns out that we observe improvements in 57 out of the 63 cells in all four rows of the column of MSEs in that cell by choosing $\beta=-4$ together with the $S$-divergence parameters. Even in the handful of cases (cells) where we do not have an improvement in all the rows of the column, there generally are competitive (although not strictly better) options within the minimum GSB divergence class with a negative value of $\beta$. In Table 4, in each cell, we also present the particular $(\alpha,\lambda,\beta)$ combination which generates the mean square errors (improved over Table 3 in most cases, as we have seen) reported in that cell. In Figure 2, we provide a three-dimensional plot (as described in that section) in the three-dimensional $(\alpha,\lambda,\beta)$ plane, where the region ${\cal S}$ has been expressed as a union of several colour-coded subregions representing the individual components. The triplets corresponding to the improved MSE solutions reported in the cells of Table 4 all belong to the blue subregion of this figure, indicating that all improved solutions are provided by bounded influence estimators. Table 3: MSEs of the minimum divergence estimators within the $S$-divergence family for pure and contaminated data 0.1968 \| 0.0836 \| 0.0704 \| 0.0708 \| 0.0733 \| 0.0802 \| 0.0876 ---\|---\|---\|---\|---\|---\|--- 0.1974 \| 0.0981 \| 0.0855 \| 0.0852 \| 0.0869 \| 0.0926 \| 0.0994 0.1753 \| 0.1063 \| 0.1012 \| 0.1028 \| 0.1054 \| 0.1116 \| 0.1118 0.3099 \| 0.2245 \| 0.2119 \| 0.2113 \| 0.2130 \| 0.2200 \| 0.2298 (0.1, $-1$, 0) \| (0.25, -1, 0) \| (0.4, -1, 0) \| (0.5, -1, 0) \| (0.6, -1, 0) \| (0.8, -1, 0) \| (1, -1, 0) 0.0751 \| 0.0666 \| 0.0673 \| 0.0698 \| 0.0729 \| 0.0800 \| 0.0876 0.0893 \| 0.0830 \| 0.0831 \| 0.0847 \| 0.0869 \| 0.0927 \| 0.0994 0.1081 \| 0.1044 \| 0.1045 \| 0.1056 \| 0.1073 \| 0.1121 \| 0.1118 0.2830 \| 0.2505 \| 0.2328 \| 0.2264 \| 0.2231 \| 0.2233 \| 0.2298 (0.1, -0.7, 0) \| (0.25, -0.7, 0) \| (0.4, -0.7, 0) \| (0.5, -0.7, 0) \| (0.6, -0.7, 0) \| (0.8, -0.7, 0) \| (1, -0.7, 0) 0.0638 \| 0.0635 \| 0.0665 \| 0.0694 \| 0.0727 \| 0.0799 \| 0.0876 0.0836 \| 0.0821 \| 0.0832 \| 0.0849 \| 0.0871 \| 0.0927 \| 0.0994 0.1203 \| 0.1120 \| 0.1087 \| 0.1083 \| 0.1089 \| 0.1125 \| 0.1118 0.3715 \| 0.2958 \| 0.2559 \| 0.2408 \| 0.2319 \| 0.2258 \| 0.2298 (0.1, -0.5, 0) \| (0.25, -0.5, 0) \| (0.4, -0.5, 0) \| (0.5, -0.5, 0) \| (0.6, -0.5, 0) \| (0.8, -0.5, 0) \| (1, -0.5, 0) 0.0600 \| 0.0622 \| 0.0660 \| 0.0691 \| 0.0725 \| 0.0798 \| 0.0876 0.0895 \| 0.0846 \| 0.0843 \| 0.0856 \| 0.0875 \| 0.0928 \| 0.0994 0.1554 \| 0.1264 \| 0.1149 \| 0.1118 \| 0.1109 \| 0.1129 \| 0.1118 0.5669 \| 0.3709 \| 0.2904 \| 0.2605 \| 0.2424 \| 0.2286 \| 0.2298 (0.1, -0.3, 0) \| (0.25, -0.3, 0) \| (0.4, -0.3, 0) \| (0.5, -0.3, 0) \| (0.6, -0.3, 0) \| (0.8, -0.3, 0) \| (1, -0.3, 0) 0.0592 \| 0.0617 \| 0.0657 \| 0.0688 \| 0.0721 \| 0.0796 \| 0.0876 0.1415 \| 0.0971 \| 0.0880 \| 0.0873 \| 0.0883 \| 0.0929 \| 0.0994 0.3491 \| 0.1774 \| 0.1308 \| 0.1196 \| 0.1147 \| 0.1136 \| 0.1118 1.3860 \| 0.6555 \| 0.3832 \| 0.3061 \| 0.2655 \| 0.2333 \| 0.2298 (0.1, 0, 0) \| (0.25, 0, 0) \| (0.4, 0, 0) \| (0.5, 0, 0) \| (0.6, 0, 0) \| (0.8, 0, 0) \| (1, 0, 0) 0.0608 \| 0.0621 \| 0.0657 \| 0.0687 \| 0.0721 \| 0.0794 \| 0.0876 0.3302 \| 0.1231 \| 0.0930 \| 0.0892 \| 0.0890 \| 0.0930 \| 0.0994 0.8565 \| 0.2745 \| 0.1508 \| 0.1276 \| 0.1181 \| 0.1141 \| 0.1118 2.5938 \| 1.0853 \| 0.5550 \| 0.3553 \| 0.2867 \| 0.2370 \| 0.2298 (0.1, 0.2, 0) \| (0.25, 0.2, 0) \| (0.4, 0.2, 0) \| (0.5, 0.2, 0) \| (0.6, 0.2, 0) \| (0.8, 0.2, 0) \| (1, 0.2, 0) 0.0671 \| 0.0638 \| 0.0658 \| 0.0685 \| 0.0718 \| 0.0792 \| 0.0876 1.1434 \| 0.3251 \| 0.1115 \| 0.0943 \| 0.0907 \| 0.0931 \| 0.0994 2.3829 \| 0.8165 \| 0.2234 \| 0.1489 \| 0.1255 \| 0.1151 \| 0.1118 4.8817 \| 2.4261 \| 0.8641 \| 0.4847 \| 0.3338 \| 0.2434 \| 0.2298 (0.1, 0.5, 0) \| (0.25, 0.5, 0) \| (0.4, 0.5, 0) \| (0.5, 0.5, 0) \| (0.6, 0.5, 0) \| (0.8, 0.5, 0) \| (1, 0.5, 0) 0.0778 \| 0.0676 \| 0.0665 \| 0.0685 \| 0.0716 \| 0.0790 \| 0.0876 1.9928 \| 0.9339 \| 0.1951 \| 0.1068 \| 0.0936 \| 0.0933 \| 0.0994 3.7890 \| 1.9909 \| 0.4869 \| 0.1994 \| 0.1378 \| 0.1162 \| 0.1118 6.6731 \| 4.2592 \| 1.6784 \| 0.7520 \| 0.4130 \| 0.2511 \| 0.2298 (0.1, 0.8, 0) \| (0.25, 0.8, 0) \| (0.4, 0.8, 0) \| (0.5, 0.8, 0) \| (0.6, 0.8, 0) \| (0.8, 0.8, 0) \| (1, 0.8, 0) 0.0863 \| 0.0717 \| 0.0673 \| $0.0686$ \| 0.0714 \| 0.0789 \| 0.0876 2.4449 \| 1.3987 \| 0.3803 \| $0.1283$ \| 0.0967 \| 0.0934 \| 0.0994 4.5000 \| 2.7992 \| 0.9117 \| $0.2793$ \| 0.1514 \| 0.1171 \| 0.1118 7.5320 \| 5.3554 \| 2.5215 \| $1.0745$ \| 0.4969 \| 0.2572 \| 0.2298 (0.1, 1, 0) \| (0.25, 1, 0) \| (0.4, 1, 0) \| (0.5, 1, 0) \| (0.6, 1, 0) \| (0.8, 1, 0) \| (1, 1, 0) Table 4: MSEs of the minimum GSB divergence estimators under pure and contaminated data 0.0623 \| 0.0696 \| 0.0704 \| 0.0708 \| 0.0687 \| 0.0720 \| 0.0763 ---\|---\|---\|---\|---\|---\|--- 0.0816 \| 0.0843 \| 0.0855 \| 0.0852 \| 0.0833 \| 0.0859 \| 0.0892 0.1115 \| 0.1056 \| 0.1012 \| 0.1028 \| 0.1042 \| 0.1060 \| 0.1077 0.2831 \| 0.2207 \| 0.2119 \| 0.2113 \| 0.2110 \| 0.2162 \| 0.2115 (0.4, $-0.4$, -4) \| (0.8, -0.5, -4) \| (0.4, -1, 0) \| (0.5, -1, 0) \| (0.8, 0, -7.5) \| (0.8, -0.3, -4) \| (1, -1, -4) 0.0642 \| 0.0681 \| 0.0681 \| 0.0696 \| 0.0696 \| 0.0720 \| 0.0763 0.0816 \| 0.0826 \| 0.0826 \| 0.0843 \| 0.0843 \| 0.0859 \| 0.0892 0.1076 \| 0.1043 \| 0.1043 \| 0.1055 \| 0.1055 \| 0.1060 \| 0.1077 0.2514 \| 0.2135 \| 0.2135 \| 0.2207 \| 0.2207 \| 0.2162 \| 0.2115 (0.6, -0.5, -4) \| (0.8, 0, -8) \| (0.8, 0, -8) \| (0.8, -0.5, -4) \| (0.8, -0.5, -4) \| (0.8, -0.3, -4) \| (1, -0.7, -4) 0.0623 \| 0.0623 \| 0.0659 \| 0.0678 \| 0.0678 \| 0.0696 \| 0.0763 0.0816 \| 0.0816 \| 0.0825 \| 0.0834 \| 0.0834 \| 0.0843 \| 0.0892 0.1115 \| 0.1115 \| 0.1071 \| 0.1061 \| 0.1061 \| 0.1055 \| 0.1077 0.2831 \| 0.2831 \| 0.2417 \| 0.2295 \| 0.2295 \| 0.2207 \| 0.2115 (0.4, -0.4, -4) \| (0.4, -0.4, -4) \| (0.5, -0.3, -4) \| (0.6, -0.3, -4) \| (0.6, -0.3, -4) \| (0.8, -0.5, -4) \| (1, -0.5, -4) 0.0600 \| 0.0619 \| 0.0642 \| 0.0659 \| 0.0678 \| 0.0720 \| 0.0763 0.0845 \| 0.0822 \| 0.0819 \| 0.0825 \| 0.0834 \| 0.0859 \| 0.0892 0.1294 \| 0.1154 \| 0.1091 \| 0.1071 \| 0.1061 \| 0.1060 \| 0.1077 0.4049 \| 0.3080 \| 0.2602 \| 0.2417 \| 0.2295 \| 0.2162 \| 0.2115 (0.1, -0.3, -4) \| (0.25, -0.3, -4) \| (0.4, -0.3, -4) \| (0.5, -0.3, -4) \| (0.6, -0.3, -4) \| (0.8, -0.3, -4) \| (1, -0.3, -4) 0.0600 \| 0.0600 \| 0.0644 \| 0.0644 \| 0.0644 \| 0.0755 \| 0.0763 0.0845 \| 0.0845 \| 0.0817 \| 0.0817 \| 0.0817 \| 0.0887 \| 0.0892 0.1294 \| 0.1294 \| 0.1073 \| 0.1073 \| 0.1073 \| 0.1077 \| 0.1077 0.4049 \| 0.4049 \| 0.2492 \| 0.2492 \| 0.2492 \| 0.2139 \| 0.2115 (0.1, -0.3, -4) \| (0.1, -0.3, -4) \| (0.8, -1, -4) \| (0.8, -1, -4) \| (0.8, -1, -4) \| (0.8, 0, -4) \| (1, 0, -4) 0.0600 \| 0.0600 \| 0.0644 \| 0.0644 \| 0.0644 \| 0.0779 \| 0.0763 0.0845 \| 0.0845 \| 0.0817 \| 0.0817 \| 0.0817 \| 0.0907 \| 0.0892 0.1294 \| 0.1294 \| 0.1073 \| 0.1073 \| 0.1073 \| 0.1092 \| 0.1077 0.4049 \| 0.4049 \| 0.2492 \| 0.2492 \| 0.2492 \| 0.2146 \| 0.2115 (0.1, -0.3, -4) \| (0.1, -0.3, -4) \| (0.8, -1, -4) \| (0.8, -1, -4) \| (0.8, -1, -4) \| (0.8, 0.2, -4) \| (1, 0.2, -4) 0.0600 \| 0.0600 \| 0.0644 \| 0.0644 \| 0.0644 \| 0.0696 \| 0.0763 0.0845 \| 0.0845 \| 0.0817 \| 0.0817 \| 0.0817 \| 0.0843 \| 0.0892 0.1294 \| 0.1294 \| 0.1073 \| 0.1073 \| 0.1073 \| 0.1055 \| 0.1077 0.4049 \| 0.4049 \| 0.2492 \| 0.2492 \| 0.2492 \| 0.2207 \| 0.2115 (0.1, -0.3, -4) \| (0.1, -0.3, -4) \| (0.8, -1, -4) \| (0.8, -1, -4) \| (0.8, -1, -4) \| (0.8, -0.5, -4) \| (1, 0.5, -4) 0.0600 \| 0.0600 \| 0.0644 \| 0.0644 \| 0.0644 \| 0.0696 \| 0.0763 0.0845 \| 0.0845 \| 0.0817 \| 0.0817 \| 0.0817 \| 0.0843 \| 0.0892 0.1294 \| 0.1294 \| 0.1073 \| 0.1073 \| 0.1073 \| 0.1055 \| 0.1077 0.4049 \| 0.4049 \| 0.2492 \| 0.2492 \| 0.2492 \| 0.2207 \| 0.2115 (0.1, -0.3, -4) \| (0.1, -0.3, -4) \| (0.8, -1, -4) \| (0.8, -1, -4) \| (0.8, -1, -4) \| (0.8, -0.5, -4) \| (1, 0.8, -4) 0.0600 \| 0.0600 \| 0.0644 \| 0.0644 \| 0.0644 \| 0.0696 \| 0.0763 0.0845 \| 0.0845 \| 0.0817 \| 0.0817 \| 0.0817 \| 0.0843 \| 0.0892 0.1294 \| 0.1294 \| 0.1073 \| 0.1073 \| 0.1073 \| 0.1055 \| 0.1077 0.4049 \| 0.4049 \| 0.2492 \| 0.2492 \| 0.2492 \| 0.2207 \| 0.2115 (0.1, -0.3, -4) \| (0.1, -0.3, -4) \| (0.8, -1, -4) \| (0.8, -1, -4) \| (0.8, -1, -4) \| (0.8, -0.5, -4) \| (1, 1, -4) --- --- Figure 2: The figure shows the region needed for bounded IF. Here, the grey, the orange, the green and the blue planes represent the boundaries of the sets ${\mathbb{S}}_{1}$, ${\mathbb{S}}_{2}$, ${\mathbb{S}}_{3}$ and ${\mathbb{S}}_{4}$, respectively. ## 8 Selection of tuning parameters Our simulations in the previous section seem to suggest that the minimum divergence estimators within the GSB class with $\beta=-4$ often provides good options for data analysis. To take full advantage of this observation, this subclass of the GSB family should be explored further. However, we want to fully exploit the flexibility of the three parameter system, and noting that in some cases the optimal is outside the $\beta=-4$ subclass, including some which generate the most competitive solutions in the full system, we want to use an overall data-based tuning parameter selection rule in which all the three parameters are allowed to vary over reasonable supports. The aim is to select the “best” tuning parameter combination depending on the amount of anomaly in the data. Thus datasets which show very close compatibility to the model should be analyzed by a triplet providing an efficient solution, while a more anomalous one should have a more robust member of the GSB class to deal with it. The current literature contains some suggestions for choosing data-driven tuning parameters in specific situations which we can make use of. The works of [18], [19] and [20] are relevant in this connection. Different algorithms for generating the “optimal” tuning parameter in case of the DPD have been described in these papers, which we will denote as the HK (for Hong and Kim), the OWJ (for one-step Warwick-Jones) and the IWJ (for the iterated Warwick- Jones) algorithms, respectively. The essential idea here is to construct an empirical approximation to the mean square error as a function of the tuning parameters (and a pilot estimator) and minimize it over the tuning parameter. The IWJ algorithm of [20] refines the OWJ algorithm of [19] by choosing the solution at a particular stage as the pilot for the next stage and going through an iterative process, thus eliminating the dependence on the pilot estimator as long as one has a good robust estimator to start with. The Hong and Kim algorithm does not consider the bias part of the mean square error and occasionally throws up highly non-robust solutions. See [20] for a full description and a comparative discussion of all the three algorithms. We will implement the same algorithms here, but for the GSB parameters rather than just the DPD parameter. In the following we have taken up two real data examples and considered the problem of selecting the “optimal” tuning parameters in each case. The OWJ algorithm considered here uses the minimum $L_{2}$ distance estimator as the pilot. Although the IWJ algorithm is pilot independent, for computational purposes it needs to commence from some suitable robust pilot for which also we utilize the minimum $L_{2}$ distance estimator. While the IWJ algorithm is our preferred method, we demonstrate the use of all the three algorithms in the following data sets. ###### Example 8.1. (Peritonitis Data): This example involves the incidence of peritonitis in $390$ kidney patients. The data are available in Chapter 2 of [6], and are also presented in the Online Supplement. The observations at 10 and 12 may be regarded as mild outliers. A geometric model with success probability $\theta$ has been fitted to these frequency data. Here, the IWJ solution coincides with the HK solution where the estimate of success probability is 0.5110 corresponding to $\left(\alpha,\lambda,\beta\right)=\left(0.41,-0.84,-3.5\right)$. The OWJ solution gives a slightly different success probability of 0.5105 corresponding to $\left(\alpha,\lambda,\beta\right)=\left(0.17,-0.60,-3\right)$. In case of clean data these IWJ, OWJ and HK estimates will be 0.5044, 0.5061 and 0.5029 corresponding to $\left(\alpha,\lambda,\beta\right)=\left(0.47,-1,-2\right)$, $\left(0.29,-1,-1\right)$ and $\left(0.55,-1,-3\right)$, respectively, being slightly different from each other. On the contrary, the MLEs for the full dataset and the (two) outlier deleted dataset are $0.4962$ and $0.5092$, respectively. Now we consider a more recent dataset for the implementation of our new proposal. ###### Example 8.2. (Stolen Bases Data): In “Major League Baseball (MLB) Player Batting Stats” for the 2019 MLB Regular Season, obtained from the ESPN.com website, one variable of interest is the number of Stolen Bases (SB) awarded to the top 40 Home Run (HR) scorers of the American League (AL). This dataset, containing three extreme and six moderate outliers, could be well-modelled by the Poisson distribution if not for the outliers. We are interested to estimate $\theta$, the average number of Stolen Bases (SB) awarded to the MLB batters of the AL throughout the whole regular season. The “optimal” estimates, derived from the implementation of the three algorithms under the Poisson model, are presented in Table 5. The fitted curves corresponding to some of these optimal estimates are given in Figure 3. It is clear that except for the full data MLE, all the other estimators primarily model the main model conforming part of the data and sacrifice the outliers. Table 5: Optimal estimates in different cases for the Stolen Bases Data data \| method \| optimal $\hat{\theta}$ \| optimal $\left(\alpha,\lambda,\beta\right)$ ---\|---\|---\|--- Full data (with outliers) \| IWJ \| $2.6270$ \| $\left(0.65,-0.98,-8\right)$ \| OWJ \| $2.5086$ \| $\left(0.73,-1,-8\right)$ \| HK \| $2.6409$ \| $\left(0.65,-1,-8\right)$ \| MLE \| $4.875$ \| $\left(0,0,0\right)$ excluding 9 outliers \| IWJ \| $2.3949$ \| $\left(0.01,1.00,0\right)$ \| OWJ \| $2.3229$ \| $\left(0.25,1.00,0\right)$ \| HK \| $2.6918$ \| $\left(0.45,-1,-8\right)$ \| MLE \| $2.3871$ \| $\left(0,0,0\right)$ --- Figure 3: Some significant fits for the Stolen Bases Data under the Poisson model. Here “clean data” refer to the modified data after removing all 9 outliers. ## 9 Concluding remarks In this paper, we have provided an extension of the ordinary Bregman divergence which has direct applications to developing new classes of divergence measures, and, in turn, in providing more options for minimum distance inference with better mixes of model efficiency and robustness. In the second part of the paper, we have made use of the suggested approach in generating a particular super-family of divergences which seems to work very well in practice and provides new minimum divergence techniques which appear to improve the performance of the $S$-divergence based procedures in many cases. Since the results presented here are based on a single study, more research will certainly be needed to decide to what extent the observed advantages of the procedures considered here can be generalized, but clearly there appears to be enough evidence to suggest such explorations are warranted. Even apart from the search for other divergences, several possible follow ups of this research immediately present themselves. This paper is restricted to discrete parametric models. An obvious follow up step is to suitably handle the case of continuous models, where the construction of the density and the divergence are more difficult. Another obvious extension will be to extend the procedures to more complicated data structures beyond the simple independently and identically distributed data scenarios. Yet another extension would be to apply this and other similarly developed super-divergences in the area of robust testing of hypothesis. We hope to take up all of these extension in our future work. The subfamily of the class of GSB divergences with $\beta=-4$ also needs some attention, and we hope to take it up in the future. For the time being we have presented the results for our real data examples for the $\beta=-4$ subfamily of GSB in the Online Supplement. ## Disclosure statement No potential conflict of interest was reported by the authors. ## References [1] Csiszár I. Eine informations theoretische ungleichung und ihre anwendung auf den beweis der ergodizitat von Markoffschen ketten. Publ. Math. Inst. Hungar. Acad. Sci. 1963;3:85–107. * [2] Ali SM, Silvey SD. A general class of coefficients of divergence of one distribution from another. Journal of the Royal Statistical Society B (Methodological). 1966;28:131–142. * [3] Bregman LM. The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Computational Mathematics and Mathematical Physics. 1967;7:200–217. * [4] Beran R. Minimum Hellinger distance estimates for parametric models. The Annals of Statistics. 1977;5:445–463. * [5] Lindsay BG. Efficiency versus robustness: the case for minimum Hellinger distance and related methods. The Annals of Statistics. 1994;22:1081–1114. * [6] Basu A, Shioya H, Park C. Statistical inference: the minimum distance approach. Boca Raton (FL): CRC Press; 2011. * [7] Hampel FR, Ronchetti EM, Rousseeuw PJ, et al. Robust statistics: the approach based on influence functions. New York (NY): John Wiley and Sons; 1986. * [8] Huber PJ, Ronchetti EM. Robust statistics. Hoboken (NJ): John Wiley and Sons; 2009. * [9] Maronna RA, Martin RD, Yohai VJ. Robust statistics. Chichester: John Wiley and Sons; 2006. * [10] Basu A, Harris IR, Hjort NL, et al. Robust and efficient estimation by minimising a density power divergence. Biometrika. 1998;85:549–559. * [11] Cressie N, Read TRC. Multinomial goodness‐of‐fit tests. Journal of the Royal Statistical Society B (Methodological). 1984;46:440–464. * [12] Mukherjee T, Mandal A, Basu A. The B-exponential divergence and its generalizations with applications to parametric estimation. Statistical Methods and Applications. 2019;28:241–257. * [13] Ghosh A, Harris IR, Maji A, et al. A generalized divergence for statistical inference. Bernoulli. 2017;23:2746–2783. * [14] Banerjee A, Merugu S, Dhillon IS, et al. Clustering with Bregman divergences. Journal of Machine Learning Research. 2005;6:1705–1749. * [15] Amari S. Alpha-divergence is unique, belonging to both $f$-divergence and Bregman divergence classes. IEEE Transactions on Information Theory. 2009;55:4925–4931. * [16] Gutmann M, Hirayama J. Bregman divergence as general framework to estimate unnormalized statistical models. arXiv preprint arXiv:1202.3727. 2012. * [17] Jana S, Basu A. A characterization of all single-integral, non-kernel divergence estimators. IEEE Transactions on Information Theory. 2019;65:7976–7984. * [18] Hong C, Kim Y. Automatic selection of the tuning parameter in the minimum density power divergence estimation. J. Korean Stat. Soc. 2001;30:453–465. * [19] Warwick J, Jones MC. Choosing a robustness tuning parameter. J. Stat. Comput. Simul. 2005;75:581–588. * [20] Basak S, Basu A, Jones MC. On the ‘optimal’ density power divergence tuning parameter. Journal of Applied Statistics. 2020. doi:10.1080/02664763.2020.1736524.
# Comment on “Strong Quantum Darwinism and Strong Independence are Equivalent to Spectrum Broadcast Structure” Alexandre Feller1 Benjamin Roussel1 Irénée Frérot2,3 Pascal Degiovanni4 (1) European Space Agency - Advanced Concepts Team, ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands. (2) ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Av. Carl Friedrich Gauss 3, 08860 Barcelona, Spain (3) Max-Planck-Institut für Quantenoptik, D-85748 Garching, Germany (4) Univ Lyon, Ens de Lyon, Université Claude Bernard Lyon 1, CNRS, Laboratoire de Physique, F-69342 Lyon, France ###### Abstract In a recent Letter [Phys. Rev. Lett. 122, 010403 (2019)], an equivalence is proposed between the so-called Spectrum Broadcast Structure for a system- multienvironment quantum state, and the conjunction of two information-theory notions: (a) Strong Quantum Darwinism; and (b) Strong Independence. Here, we show that the mathematical formulation of condition (b) by the authors (namely, the pairwise independence of the fragments of the environment, conditioned on the system), is necessary but not sufficient to ensure the equivalence. We propose a simple counter-example, together with a strengthened formulation of condition (b), ensuring the equivalence proposed by the authors. In their paper Castro-2019 , the authors introduce the notions of: (a) “strong quantum Darwinism”; and (b) “strong independence”. As the main result of the paper, it is proposed that, taken together, conditions (a) and (b) are equivalent to the so-called “spectrum broadcast structure” (SBS) for the system–environment quantum state $\rho_{{\cal S}{\cal E}}$ (throughout this Comment, we follow the notations and definitions used in the paper). By definition, if $\rho_{{\cal S}{\cal E}}$ has a SBS, the state of ${\cal E}={\cal E}_{1}\cdots{\cal E}_{F}$ relative to ${\cal S}$ is fully factorized, and has no correlations whatsoever; and the authors aimed at capturing this fully-factorized structure by an information-theory criterion of “strong independence”. In this Comment, we show that condition (b), while ensuring pairwise independence of the fragments ${\cal E}_{i}$ when conditioning on the system (a necessary condition for having the SBS), does not rule out the existence of higher-order correlations, and as such is insufficient to imply a SBS. We first construct a (completely classical) counter-example of a state with no SBS yet satisfying conditions (a) and (b). We then propose a stronger condition (b’), ensuring the equivalence of conditions (a) and (b’) with SBS. At a conceptual level, the main conclusion of the paper, namely the equivalence between, on the one hand, the SBS, and on the other, the conjunction of “strong quantum Darwinism” (as defined by the authors) and “strong independence” (as now defined by (b’)), remains therefore unaltered. Counter-example. We consider a qubit system ${\cal S}:\\{\|0\rangle,\|1\rangle\\}$, and the environment ${\cal E}={\cal E}_{1}{\cal E}_{2}{\cal E}_{3}$, with three qutrits fragments ${\cal E}_{i}:\\{\|0_{i}\rangle,\|1_{i}\rangle,\|2_{i}\rangle\\}$ ($i\in\\{1,2,3\\}$). We define the projectors $\Pi_{a}^{({\cal S})}=\|a\rangle\langle a\|$ and $\Pi_{a}^{(i)}=\|a_{i}\rangle\langle a_{i}\|$. Finally, we define the projectors $\Pi_{abc}=\Pi_{a}^{(1)}\otimes\Pi_{b}^{(2)}\otimes\Pi_{c}^{(3)}$. Our counter-example is the state ($0\leq p\leq 1$): $\rho_{{\cal S}{\cal E}}=(1-p)~{}\Pi_{0}^{({\cal S})}\otimes\rho_{{\cal E}}^{(0)}+p~{}\Pi_{1}^{({\cal S})}\otimes\rho_{{\cal E}}^{(1)}~{},$ (1) with the relative states: $\displaystyle\rho_{{\cal E}}^{(0)}$ $\displaystyle=$ $\displaystyle\Pi_{000}$ (2) $\displaystyle\rho_{{\cal E}}^{(1)}$ $\displaystyle=$ $\displaystyle\frac{1}{4}[\Pi_{111}+\Pi_{122}+\Pi_{212}+\Pi_{221}]$ (3) $\displaystyle\rho_{{\cal E}_{i}{\cal E}_{j}}^{(1)}$ $\displaystyle=$ $\displaystyle\frac{1}{2}[\Pi_{1}^{(i)}+\Pi_{2}^{(i)}]\otimes\frac{1}{2}[\Pi_{1}^{(j)}+\Pi_{2}^{(j)}]$ (4) $\displaystyle\rho_{{\cal E}_{i}}^{(1)}$ $\displaystyle=$ $\displaystyle\frac{1}{2}[\Pi_{1}^{(i)}+\Pi_{2}^{(i)}]$ (5) As the relative states $\rho_{{\cal E}_{i}}^{(0)}=\Pi_{0}^{(i)}$ and $\rho_{{\cal E}_{i}}^{(1)}=[\Pi_{1}^{(i)}+\Pi_{2}^{(i)}]/2$ have disjoint supports, it is clear that each fragment has full access to the system’s state. Therefore: $I({\cal S}:{\cal E}_{i})=I_{\rm acc}({\cal S}:{\cal E}_{i})=\chi({\cal S}^{\Pi}:{\cal E}_{i})=H({\cal S})$ [condition (a)]. The state also fulfills condition (b) (pairwise-independence conditioned on ${\cal S}$). Indeed, the relative states $\rho_{{\cal E}_{i}{\cal E}_{j}}^{(0)}=\Pi_{0}^{(i)}\otimes\Pi_{0}^{(j)}$ and $\rho_{{\cal E}_{i}{\cal E}_{j}}^{(1)}$ [Eq. (4)] are product states, which implies that $I({\cal E}_{i}:{\cal E}_{j}\|{\cal S})=0$ (as can be checked by direct computation on $\rho_{{\cal S}{\cal E}_{i}{\cal E}_{j}}$). Yet, the state $\rho_{{\cal S}{\cal E}_{1}{\cal E}_{2}{\cal E}_{3}}$ does not admit a SBS. Indeed, the relative state $\rho_{{\cal E}}^{(1)}$ [Eq. (3)] does not factorize (while pairs of fragments are uncorrelated, the parity of the total number of fragments in state $\|1_{i}\rangle$ is odd, yielding genuine 3-partite correlations). This concludes our proof that the conditions (a) and (b) are insufficient to imply a SBS. Proposed correct formulation of the theorem. The above counter-example suggests to introduce a stronger condition (b’) ensuring the complete factorization of the fragments when conditioning on the system, which is a defining property of the SBS. Noticing that a multipartite state is factorized, $\rho_{{\cal E}_{1},\dots\cal E_{F}}=\otimes_{i=1}^{F}\rho_{{\cal E}_{i}}$, iff the multipartite mutual information vanishesYang-2009 : $I({\cal E}_{1},\dots\cal E_{F}):=\sum_{i=1}^{F}H(\rho_{{\cal E}_{i}})-H(\rho_{{\cal E}_{1},\dots\cal E_{F}})=0$, we are led to propose the following: Theorem. The state $\rho_{{\cal S}{\cal E}}$ has the SBS iff: * (a1) $I({\cal S}:{\cal E})=\chi({\cal S}^{\Pi},{\cal E})$ (classical–quantum state), * (a2) $I_{\rm acc}({\cal S}:{\cal E}_{i})=H({\cal S})$ for all $i$ (the information about ${\cal S}$ can be fully recovered by measuring any fragment), * (b’) $I({\cal E}_{1},\cdots{\cal E}_{F}\|{\cal S})=0$ (totally-factorized relative states). ###### Proof. If the state has the SBS, it is clear that conditions (a1), (a2) and (b’) are fulfilled. Conversely, condition (a1) (vanishing discord) implies that the system–environment state is of the form $\rho_{{\cal S}{\cal E}}=\sum_{s}p_{s}\Pi_{s}^{({\cal S})}\otimes\rho_{\cal E}^{(s)}$ with $\\{\Pi_{s}^{({\cal S})}\\}$ forming mutually orthogonal projectors for the system. Condition (b’) then implies that the relative states factorize: $\rho_{\cal E}^{(s)}=\otimes_{i=1}^{F}\rho_{{\cal E}_{i}}^{(s)}$. Finally, condition (a2) implies that for each $i$, the states $\rho_{{\cal E}_{i}}^{(s)}$ are pairwise orthogonal. ∎ ###### Acknowledgements. We thank Thao P. Le and Alexandra Olaya-Castro for constructive feedback on our manuscript. This work has been supported by the European Space Agency (Ariadna study 1912-01). IF acknowledges support from the Government of Spain (FIS2020-TRANQI and Severo Ochoa CEX2019-000910-S), Fundació Cellex and Fundació Mir-Puig through an ICFO-MPQ Postdoctoral Fellowship, Generalitat de Catalunya (CERCA, AGAUR SGR 1381 and QuantumCAT). ## References * (1) Thao P. Le and Alexandra Olaya-Castro, _Strong quantum darwinism and strong independence are equivalent to spectrum broadcast structure_ , Phys. Rev. Lett. 122 (2019), 010403. * (2) Dong Yang, Karol Horodecki, Michal Horodecki, Pawel Horodecki, Jonathan Oppenheim, and Wei Song, _Squashed entanglement for multipartite states and entanglement measures based on the mixed convex roof_ , IEEE Transactions on Information Theory 55 (2009), no. 7, 3375–3387.
# Third-harmonic light polarization control in magnetically-resonant silicon metasurfaces Andrea Tognazzi 1,2 Kirill I. Okhlopkov 3 Attilio Zilli 4 Davide Rocco 1,2 Luca Fagiani 4,5 Erfan Mafakheri 5 Monica Bollani 5 Marco Finazzi 4 Michele Celebrano 4 Maxim R. Shcherbakov 3 Andrey A. Fedyanin 3 and Costantino De Angelis1,2 1Department of Information Engineering, University of Brescia, Via Branze 38, 25123 Brescia, Italy 2CNR-INO (National Institute of Optics), Via Branze 45, 25123 Brescia, Italy 3Faculty of Physics, Lomonosov Moscow State University, Moscow 119991, Russia 4Department of Physics, Politecnico di Milano, Piazza Leonardo Da Vinci 32, 20133 Milano, Italy 5CNR-IFN, LNESS laboratory, Via Anzani 42, 22100 Como, Italy <EMAIL_ADDRESS> ###### Abstract Nonlinear metasurfaces have become prominent tools for controlling and engineering light at the nanoscale. Usually, the polarization of the total generated third harmonic is studied. However, diffraction orders may present different polarizations. Here, we design an high quality factor silicon metasurface for third harmonic generation and perform back focal plane imaging of the diffraction orders, which present a rich variety of polarization states. Our results demonstrate the possibility of tailoring the polarization of the generated nonlinear diffraction orders paving the way to a higher degree of wavefront control. ††journal: oe ## 1 Introduction Manipulation of light is of paramount importance in many fields such as opto- electronics, image processing, sensing and cryptography[1, 2]. The 2D nature of metasurfaces, which are composed by an array of resonators, makes them suitable candidates for compact photonic devices[3, 4]. The optical properties of such structures can be tailored by tuning the geometrical parameters of each resonator in the periodic array or by changing the material[5, 6, 7]. Thanks to the improved accuracy of nanofabrication techniques, it is nowadays possible to obtain high quality nano-objects consisting of metals or dielectric materials to manipulate light in the visible and near-infrared regimes[8]. The applications of metasurfaces include beam steering [9, 10], light focusing[11, 12], holography[13], and sensing[14]. Implementing nonlinear optics at the nanoscale is very challenging because one cannot exploit phase matching, which can be achieved only over mesoscopic scales. In this frame, the added value of metasurfaces consists in the possibility of exploiting collective modes stemming from the interactions between neighboring nanoresonators to enhance the local electric field, improve the conversion efficiency, and tailor the emitted radiation[15, 16, 17, 18]. The low losses make dielectrics more suitable than metals for second- and third-harmonic generation (THG), all-optical switching and modulation of visible and near-infrared light[19, 20, 21, 22, 23, 24, 25, 26]. In the past few years, high-refractive index dielectric materials were employed to build nanoresonators to improve nonlinear frequency conversion [27, 28] and manipulate light emission [29, 30]. One of the most attractive materials for nanophotonics applications is silicon due to its well-established fabrication technology, high-refractive index and technological relevance[31, 32]. Previously, nonlinear beam deflection has been achieved by inducing a phase shift using different building blocks[33]. However, the polarization of the diffraction orders is usually an overlooked property when studying nonlinear gratings [34]. In this article, we report the design and fabrication of high quality factor ($Q$-factor) metasurfaces and we propose a simple electromagnetic model to explain the polarization of the third harmonic (TH) diffraction orders. Orthogonal polarizations are measured for different diffraction orders depending on the dominant multipolar component at resonance. Our results pave the way to the realization of a higher degree of polarization-controlled nonlinear diffractive metasurfaces. Figure 1: (a) Sketch of the metasurface and of the electric field distribution at normal incidence showing the magnetic quadrupole behaviour of the fundamental frequency. The near field distribution can be used to predict the diffraction orders polarization. The unitary cell is constituted by a silicon cuboid laying on a SiO2 substrate. (b) SEM planar view image of a dielectric metasurface image. Figure 2: Simulated reflectance for $p$ (a) and $s$ (b) incident polarization as a function of the wavelength and the incidence angle $\vartheta$. The dashed white lines delimit the FWHM bandwidth ($1554\pm 8$ $\mathrm{n}\mathrm{m}$) of the pump used in the experiments. For incident p-polarization, the high quality factor is preserved when $\vartheta$ increases and the resonance blue shifts. For incident s-polarization, the high quality factor resonance fades away when $\vartheta$ increases and a broader resonance appears at shorter wavelengths. ## 2 Design and fabrication We employ a commercial finite element solver (Comsol Multiphysics) to optimize the design of high-$Q$ metasurfaces made of silicon cuboids arranged in a periodic rectangular lattice (see Fig. 1a). We created a waveguide-like system with a channel coupling the light and the structure to obtain a metasurface with different quality factors depending on the excitation geometry. In the experiments, we achieve the resonant condition by changing the angle of incidence, this allows us to excite two different modes under incident p\- and s-polarization. The metasurfaces are realized on a silicon-on-insulator (SOI) substrate with a device layer of $H=125$ $\mathrm{n}\mathrm{m}$ on 2 $\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ of buried oxide (see Fig. 1b). Arrays of rectangles (width $W=428$ $\mathrm{n}\mathrm{m}$, length $L=942$ $\mathrm{n}\mathrm{m}$ and periodicity $P_{x}$ and $P_{y}$ 1065–1060 $\mathrm{n}\mathrm{m}$, respectively) aligned along the [110] direction are patterned by means of e-beam lithography (EBL) and reactive ion etching (RIE). The resist is spin-coated on the SOI substrate and then exposed to the electron beam of a converted scanning electron microscope (SEM) along the designed pattern (acceleration voltage of 30 $\mathrm{k}\mathrm{V}$). A double layer of PMMA diluted in chlorobenzene, respectively, at 3.5% and 1.5% is employed. The dose used for the structures is 350 $\mathrm{\SIUnitSymbolMicro}\mathrm{C}\mathrm{/}\mathrm{c}\mathrm{m}\textsuperscript{2}$. After exposure, PMMA is developed in a solution of methyl isobutyl ketone (MIBK) and isopropanol (IPA) in a 1:3 ratio; MIBK is diluted in order to obtain well-defined profiles. The sample is immersed in this solution and agitated manually for 90 $\mathrm{s}$; a pure IPA solution is used for 1 minute to stop the development of the resist. Then, the pattern is transferred to the thin Si film by RIE in a CF4 plasma, using 80 $\mathrm{W}$ of radio frequency power and a total gas pressure of 5.4 $\mathrm{m}\mathrm{T}\mathrm{o}\mathrm{r}\mathrm{r}$. Finally, the resist is removed using acetone and the sample surface is exposed to O2 plasma in order to remove any residual resist. In the simulations, we model the silicon refractive index as reported in [35] and we assume a wavelength-independent refractive index (n=1.45) for the SiO2 substrate. The spatial period $P_{x}=P_{y}=P$ of the through-notches has been chosen to satisfy the matching condition resulting from momentum conservation between a normal incident plane wave with in-plane modes, i.e. $2\pi/P=\beta(\omega)$ where $\beta(\omega)$ is the propagation wavevector of the mode[15]. Possible deviations of $\beta(\omega)$ in the fabricated device from the simulated value can be matched by tuning the wavelength $\lambda_{0}$ or the angle $\theta$ of the incident plane wave. Fig. 2 shows sketches of the incident polarization and the reflectance (R) at the fundamental frequency (FF) as a function of $\lambda_{0}$ and $\theta$ for p and s polarized excitation with $\vec{E}_{0}(\omega)\perp\hat{x}$. When $\vec{E}_{0}(\omega)$ is p polarized the metasurface shows a sharp resonance ($Q=399$) which blue-shifts when $\vartheta$ is varied, albeit maintaining a narrow spectral width. When the impinging wave is s polarized, it excites a magnetic dipole mode with a lower $Q$ factor ($Q=29$). We then simulate the TH field $\vec{E}(3\omega)$ by evaluating in a second step of the computation the nonlinear current generated within the structures by the fundamental field $\vec{E}_{0}(\omega)$ through the third-order susceptibility as reported in [36]. The diffraction orders are calculated by performing the Fourier transform of the near field simulated at the TH frequency. ## 3 Experiments We employ a pulsed laser (160 fs) centered at 1554 $\mathrm{n}\mathrm{m}$ (FWHM=17 $\mathrm{n}\mathrm{m}$) and focus the beam in the back focal plane (BFP) of a 60x objective (Nikon, CFI Plan Fluor 60XC, NA=0.85) to obtain a loosely focused beam on the sample. We shift the pump beam in the BFP plane of the objective to change the angle of incidence. We collect the emitted TH through the same objective used for the excitation, and chromatically filter it. A Bertrand lens in the detection path focuses the TH beam in the BFP of the objective to image the TH diffraction orders, while a polarizer is used to analyze the polarization of the TH. A cooled CCD camera is used to acquire BFP images such as the ones in Fig. 3, where $\vartheta$ is the angle of incidence and $\phi$ is the polarizer angle with respect to the x-axis. ## 4 Results In Figs. 4(a,b) the intensity of TH light as function of the analyzer angle of different diffraction orders is reported for experiments (dashed) and simulations (continuous) for s and p-polarized fundamental wavelength light illuminating the sample at the incidence angle leading to maximum THG. Normalized experimental data in Fig. 4a refer to p-polarized light impinging on the sample at 41∘, corresponding to the maximum THG signal, while simulations corresponds to 32°. All the THG diffraction orders are polarized along the $y$ axis as predicted by the simulations. THG is maximum for incident s-polarization at 14∘ in the experiment and 22∘ in the simulations. The discrepancies in the resonant angles and the systematic tilt of the (-1,0) diffraction order polarization may be due to the uncertainty in the experimental pump beam angle of incidence and fabrication defects. In order to have a better insight on the polarization of diffraction orders we performed a cartesian multipolar decomposition of the TH field (see Fig. 4c,d). For incident p-polarization, the main multipolar component is always a magnetic quadrupole, $Q_{xz}$, whose amplitude is maximum at the resonant angle, leading to no variations of the diffraction order polarization when the angle of incidence is changed . For s-polarization, the main multipolar component changes at resonance and becomes a magnetic dipole along the z-axis, with a spatially non uniform far field polarization (see Fig.4d inset). This corresponds to a variation of the polarization of all the diffraction orders $(m,n)$ with $n\neq 1$. The polarization of the diffraction orders can be described by a simple formula, which takes into account the electromagnetic field distribution of each scatterer and the periodic structure. In the far- field region the total electric field radiated ($E_{t}$) by the metasurface is proportional to the far field radiated by the single array element ($E_{s}$) through the array factor ($AF$): $\vec{E}_{t}^{3\omega}=\vec{E}_{s}^{3\omega}AF(P,3\omega),$ (1) where the $AF(P,3\omega)$ is a function that depends only on the periodicity of the array and the TH frequency. Here, each cuboid can be envisioned as an antenna whose emission pattern is determined by the superposition of all the multipolar components and the diffraction orders are determined according to the $AF$ as described in [37]. This formalism enables one to tailor the polarization state of the nonlinear diffraction orders by engineering the main multipolar components at the TH frequency describing the single antenna behaviour. It is worth noting that this approach can be applied also for closely packed unit-cells once the multipolar decomposition is completely resolved for the meta-units that form the metasurface under test. Figure 3: (a,c) Experimental BFP images under normal incidence excitation. The red circle represents the numerical aperture of the objective (NA=0.85). (b,d) BFP images with tilted illumination at 41° for p and 14° for s polarized light. When the angle of incidence is increased, the (0,0) order is not emitted perpendicularly to the metasurface due to the in-plane components. For p-polarization, the diffraction orders move along $k_{y}$ since the incident beam wavevector lies in the yz-plane, while, for s-polarization, they move along $k_{x}$ since the incident wavevector is in the xz-plane. As a consequence, at certain angles, some diffraction orders disappear and others fall in the NA view window. Figure 4: Experimental (dashed) and simulated (continuous) polarization-resolved TH power excited at the angle with maximum THG for $p$ (a) and $s$ (b) polarization, respectively. The vertical double arrows represent the incident pump polarization. (c,d) Cartesian multipolar decomposition for incident $p$ and $s$ polarization, respectively. The insets in (c,d) represent the far field polarization of the magnetic quadrupole and of the magnetic dipole, respectively. ## 5 Conclusions We showed, both experimentally and numerically, a complex behaviour of the polarization of the TH diffraction orders as a function of the incidence angle of the fundamental pump beam. We applied a cartesian multipolar decomposition and a simple formula to describe the polarization of the diffraction orders and provide a method to tailor the far field properties of the metasurface. Our results demonstrate that the polarization of the diffraction orders is solely influenced by the near field distribution within each single element, which can be modelled by considering the multipolar decomposition, while the far field is determined by the period of the metasurface relative to the exciting wavelength. 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11institutetext: Anadijiban Das 22institutetext: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, V5A 1S6, Canada 22email<EMAIL_ADDRESS>33institutetext: Rupak Chatterjee 44institutetext: Center for Quantum Science and Engineering Department of Physics, Stevens Institute of Technology, Hoboken, NJ 07030, USA 44email<EMAIL_ADDRESS> # Discrete phase space and continuous time relativistic quantum mechanics II: Peano circles, hyper-tori phase cells, and fibre bundles Anadijiban Das Rupak Chatterjee (Received: date / Accepted: date) ###### Abstract The discrete phase space and continuous time representation of relativistic quantum mechanics is further investigated here as a continuation of paper I DasRCI . The main mathematical construct used here will be that of an area- filling Peano curve. We show that the limit of a sequence of a class of Peano curves is a Peano circle denoted as $\bar{S}^{1}_{n}$, a circle of radius $\sqrt{2n+1}$ where $n\in\\{0,1,\cdots\\}$. We interpret this two-dimensional Peano circle in our framework as a phase cell inside our two-dimensional discrete phase plane. We postulate that a first quantized Planck oscillator, being very light, and small beyond current experimental detection, occupies this phase cell $\bar{S}^{1}_{n}$. The time evolution of this Peano circle sweeps out a two-dimensional vertical cylinder analogous to the world-sheet of string theory. Extending this to three dimensional space, we introduce a $(2+2+2)$-dimensional phase space hyper-tori $\bar{S}^{1}_{n^{1}}\times\bar{S}^{1}_{n^{2}}\times\bar{S}^{1}_{n^{3}}$ as the appropriate phase cell in the physical dimensional discrete phase space. A geometric interpretation of this structure in state space is given in terms of product fibre bundles. We also study free scalar Bosons in the background $[(2+2+2)+1]$-dimensional discrete phase space and continuous time state space using the relativistic partial difference-differential Klein-Gordon equation. The second quantized field quantas of this system can cohabit with the tiny Planck oscillators inside the $\bar{S}^{1}_{n^{1}}\times\bar{S}^{1}_{n^{2}}\times\bar{S}^{1}_{n^{3}}$ phase cells for eternity. Finally, a generalized free second quantized Klein-Gordon equation in a higher $[(2+2+2)N+1]$-dimensional discrete state space is explored. The resulting discrete phase space dimension is compared to the significant spatial dimensions of some of the popular models of string theory. ###### Keywords: Discrete phase space Peano curves Partial difference-differential equations Fibre bundles ###### pacs: 11.10Ef 11.15Ha 02.70Bf 03.65Fd ## 1 Introduction We begin in section 2 by introducing the concept of a Peano curve Clark ; Gelbaum . In 1890, G. Peano startled the mathematical world with the introduction of an area-filling curve. After that, many area-filling curves have been discovered including those by David Hilbert. To our knowledge, very few papers in the mathematical physics arena have used this concept DasHaldar . We begin by discussing the original Peano curve that fills a square $\bar{D}$ of unit area inside $\mathbb{R}^{2}$. In section 3, we derive other Peano curves filling a sequence $\\{\bar{D}_{M}\\}^{\infty}_{M=1}$ of unit areas each in the shape of a rectangle. Next, we introduce a double sequence $\\{\bar{D}_{Mn}\\}^{\infty}_{M=1}$ for $n\in\\{0,1,2,...\\}$. Each of closed regions $\bar{D}_{Mn}$ for a fixed $n$ is endowed with a unit area and a rectangular shape. Moreover, each of these regions $\bar{D}_{Mn}$ is covered by a Peano curve. Next, we discuss another double sequence $\\{\bar{A}_{Mn}\\}$ of closed regions where each region $\bar{A}_{Mn}$ is annular in shape and possesses a unit area. This annular region $\bar{A}_{Mn}$ is also covered by a Peano curve. Finally, in the limiting process of $M\rightarrow\infty$ for a fixed $n$, the annular region $\bar{A}_{Mn}$ collapses into a circle $\bar{S}^{1}_{n}$ of unit area. Since this cirlce contains a ’squashed’ Peano curve, one calls $\bar{S}^{1}_{n}$ a Peano circle. Section 4 details more analysis on this Peano circle. The simplified Klein-Gordon equation in the background of a $[(1+1)+1]$-dimensional discrete phase space and continuous time is discussed in section 5. Some physically relevant results are also derived. In section 6, we introduce the abstract concept of fibre bundles Choquet and discuss their applications to certain physical problems. We apply this concept in section 7 to the physically important $[(2+2+2)+1]$-dimensional phase space. This discrete phase space and continuous time arena has been investigated with respect to the second quantization of many free relativistic field theories DasI ; DasII . The corresponding formulations of interacting relativistic second quantized fields yielding an $S^{\\#}$-matrix series were given in DasIII . Second order expansion terms of this $S^{\\#}$-matrix involving Møller scattering of quantum electrodynamics Jauch ; Peskin produced a non-singular Coulomb potential in DasBen ; DasRC . In section 7, we define a new set-theoretic mapping Goldberg by the direct or Cartesian product Lightstone as: $\begin{array}[]{c}\bar{S}^{1}_{n^{1}}\times\bar{S}^{1}_{n^{2}}\times\bar{S}^{1}_{n^{3}}=\displaystyle{\lim_{M\to\infty}}\\\ \\\ \left([g^{M}_{n^{1}}\times h^{M}_{n^{1}}\times h^{M}_{1}]\times[g^{M}_{n^{2}}\times h^{M}_{n^{2}}\times h^{M}_{2}]\times[g^{M}_{n^{3}}\times h^{M}_{n^{3}}\times h^{M}_{3}]\right)\left(\bar{D}_{1}\times\bar{D}_{2}\times\bar{D}_{3}\right)\\\ \\\ \subset\mathbb{R}^{2}\times\mathbb{R}^{2}\times\mathbb{R}^{2}.\end{array}$ (1) The left hand side of this equation ($\bar{S}^{1}_{n^{1}}\times\bar{S}^{1}_{n^{2}}\times\bar{S}^{1}_{n^{3}}$) is a hyper-torus Massey of physical dimension $\left[\dfrac{ML^{2}}{T}\right]^{3}$. It will be shown that these hyper-tori can represent a phase cell in the usual $(2+2+2)$-dimensional discrete phase space. In section 8, the second quantization of the scalar field $\Phi(n^{1},n^{2},n^{3};t)=:\Phi(\mathbf{n};t)$ is discussed in the arena of $[(2+2+2)+1]$-dimensional discrete phase space and continuous time using the relativistic Klein-Gordon operator equation. A partial difference-differential version of the Klein-Gordon equation is solved for a special class of solutions involving Fourier-Hermite transforms DasII . The total energy $\mathcal{H}$, momentum component $\mathcal{P}_{j}$ and electric charge $\mathcal{Q}$ are worked out for the second quantized Klein-Gordon scalar field operator $\Phi(\mathbf{n};t)$. Finally, in section 9, we study discrete $(2+2+2)N$-dimensional phase space many-particle systems and the corresponding Peano hyper-tori $\bar{S}^{1}_{n^{1}}\times\bar{S}^{1}_{n^{2}}\times\cdots\bar{S}^{1}_{n^{N}}$. Each of these Peano hyper-tori has the physical dimension of $\left[\dfrac{ML^{2}}{T}\right]^{N}$ and represent appropriate phase cells inside the discrete $(2+2+2)N$-dimensional phase space. The generalized Klein- Gordon equation within this discrete $[(2+2+2)N+1]$-dimensional phase space and continuous time arena is also discussed. As our $2N$-dimensional Peano hyper-tori is a phase-cell inside a discrete $6N$-dimensional phase space, we compare our structure with that of the spatial dimension of string theory Green ; PolchinskiI ; PolchinskiII . ## 2 The area filling curve of Peano Consider a continuous, piece-wise linear, oriented, parametrized curve Clark $f^{1}(u)$ inside a $(1+1)$-dimensional plane $\mathbb{R}^{2}$ as depicted in figure 1. Figure 1: The graph of the parametrized curve $f^{1}(u)$. Note that the curve $f^{1}(u)$ is defined over nine closed intervals $\left[\dfrac{k-1}{9},\dfrac{k}{9}\right]$ for $k\in\\{1,2,\cdots,9\\}\subset\mathbb{R}$. The image of the curve of figure 1 is a continuous, piece-wise zigzag, oriented, parametrized curve over a closed square $\bar{D}\subset\mathbb{R}$ of unit area inside the $x-y$ plane $\mathbb{R}^{2}$. The image of the function $f^{1}(u)$ in figure 1 has for the closed domain$[0,1/9]$ the closed range marked as the first linear segment with an arrow and denoted with a ”1”. Similarly, the closed domain $[1/9,2/9]$ is mapped to the linear segment denoted ”2”. The ten corners of the zig-zag curve are situated at $f^{1}(k/9),k=0,1,...,9$. Figure 2: The graph of the parametrized curve $f^{2}(u)$. The image of the next parametrized curve $f^{2}(u)$, depicted in figure 2, is continuous, oriented, and piece-wise linear in the $x-y$ plane $\mathbb{R}^{2}$. It has $9^{2}=81$ linear segments inside the unit square $\bar{D}$ within $\mathbb{R}^{2}$. This figure is constructed by replicating figure 1 nine times within the unit square area resulting in $81$ oriented linear pieces. Let us define a distance function $d(f^{1}(u),f^{2}(u))$ abstractly as $d(f^{1}(u),f^{2}(u)):=\displaystyle{\textit{sup}_{u\in[0,1]}}\|\|f^{1}(u)-f^{2}(u)\|\|\leq\dfrac{\sqrt{2}}{3}.$ (2) Here, the number $\dfrac{\sqrt{2}}{3}$ is the diameter of a square with each side being $\dfrac{1}{3}$. The sequence of pre-Peano parametrized curves $\\{f^{j}(u)\\}^{\infty}_{j=1}$ is obtained by replicating the above process over and over again. Therefore, one may obtain the result that $d(f^{j}(u),f^{j+1}(u))\leq\dfrac{\sqrt{2}}{3^{j}}.$ (3) It follows by the uniform Cauchy criterion Clark that for each $\epsilon>0$, there exists an integer $N>0$ such that $d(f^{j}(u),f^{m}(u))<\epsilon$ for all $j$ and $m\geq N$. Therefore, there exists a uniformly continuous function $f(u)$ such that $\displaystyle{\lim_{j\rightarrow\infty}}f^{j}(u)=f(u)$. Moreover, on can prove that this parametrized curve $f(u)$ passes through every point of the closed unit square $\bar{D}$. This is the first example of a Peano curve filling in the area of $\bar{D}$ Clark . ## 3 Other Peano curves filling a $(1+1)$-dimensional physical phase space We can interpret physically the areas $\bar{D}$ in both figures 1 and 2 as phase cells inside a $(1+1)$-dimensional phase plane. Furthermore, the uncertainty of quantum physics Dirac $\|\Delta x\|\|\Delta y\|\geq 1$ $(i.e.\,\|\Delta q\|\|\Delta p\|\geq 1,\hbar=1)$ automatically holds inside a phase cell $\bar{D}$ for a quantized particle occupying $\bar{D}$ (here, $\Delta$ stands for an increment and not a finite difference operator). Any possible movement of the quantized particle along the Peano curve $f(u)$ covering $\bar{D}$ is physically unobservable. Now, we shall define an infinite sequence of functions $\\{h^{M}\\}^{\infty}_{M=1}$ from the region $\bar{D}\subset\mathbb{R}^{2}$ into regions $\\{\bar{D}_{M}\\}^{\infty}_{M=1}$ as depicted in figure 3. Figure 3: The graph of the function $h^{M}$ for fixed $M$. The function $h^{M}$ is defined for a specific $M$ as the following linear transformation: $\begin{array}[]{c}\rho=\left(\dfrac{1}{2M\pi}\right)x+\dfrac{1}{2},\\\ \\\ \alpha=(2M\pi)y-M\pi,\\\ \\\ M\geq 1.\end{array}$ (4) The Jacobian of these transformations is $\dfrac{\partial(\rho,\alpha)}{\partial(x,y)}=1,$ (5) making these transformations canonical transformations of Hamiltonian mechanics Lanczos ; Goldstein . Therefore, the area of each of the regions $\\{\bar{D}_{M}\\}$ is furnished by the double integral of the following differential 2-form Spivak ; DasTA $\begin{array}[]{c}Area(\bar{D}^{M}):=\displaystyle{{\int\int}_{\bar{D}^{M}}}d\rho\wedge d\alpha=\int_{\frac{1}{2}}^{\frac{1}{2M\pi}+\frac{1}{2}}\int_{-M\pi}^{M\pi}d\rho d\alpha=1,\\\ \\\ M\in\\{1,2,3,...\\}.\end{array}$ (6) Remarks: (i) In the limiting process $M\rightarrow\infty$, the sequence of regions $\\{\bar{D}^{M}\\}^{\infty}_{M=1}$ collapses into the open vertical line given by $\\{(\rho,\alpha)\in\mathbb{R}^{2}:\rho=1/2,\alpha\in\mathbb{R}\\}$. (ii) This infinite vertical line is an open string-like phase cell of unit area as seen by using (6) above. (iii) The infinite vertical line in remark (i) is consistent with the uncertainty principle $\|\Delta\rho\|\|\Delta\alpha\|\geq 1$. (iv) The string-like phase cell may or may not contain any quanta of a second quantized relativistic wave field (see section 5). (v) Moreover, unlike in conventional string theory Green ; PolchinskiI ; PolchinskiII , in our approach of string-like phase cells, an open string-like phase cell cannot be of finite length. Consider another sequence of transformations $\\{h^{M}_{n}\\}^{\infty}_{M=1}$ where $n$ is chosen to be a fixed non-negative integer. A typical mapping $h^{M}_{n}$ maps the closed domain $\bar{D}_{M}$ into the closed region $\bar{D}_{Mn}$ by the linear transformation $\begin{array}[]{c}r=\rho+n,\\\ \\\ \theta=\alpha,\\\ \\\ \dfrac{\partial(r,\theta)}{\partial(\rho,\alpha)}=1.\end{array}$ (7) This transformation is exhibited in figure 4. Figure 4: The mapping $h^{M}_{n}$ from $\bar{D}_{M}$ to $\bar{D}_{Mn}$. The area of each domain $\\{\bar{D}_{Mn}\\}$ is given by $\begin{array}[]{c}Area(\bar{D}_{Mn}):=\displaystyle{{\int\int}_{\bar{D}_{Mn}}}dr\wedge d\theta=\int_{n+\frac{1}{2}}^{n+\frac{1}{2M\pi}+\frac{1}{2}}\int_{-M\pi}^{M\pi}drd\theta=1,\\\ \\\ M\in\\{1,2,3,...\\},\,\,\,\ n\in\\{0,1,2,....\\}.\end{array}$ (8) Now, consider a third sequence of canonical transformations $\\{g^{M}_{n}\\}^{\infty}_{M=1}$ for a fixed $n$ as follows, $\begin{array}[]{c}q=\sqrt{2\rho}\cos\theta,\\\ \\\ p=\sqrt{2\rho}\sin\theta,\\\ \\\ \dfrac{\partial(q,p)}{\partial(\rho,\theta)}=1,\\\ \\\ q^{2}+p^{2}=2r>0,\\\ \\\ \left(\dfrac{p}{q}\right)=\tan\theta,\,\,\,q\neq 0.\end{array}$ (9) The closed domain of each of the mappings of $\\{g^{M}_{n}\\}^{\infty}_{M=1}$ is furnished by $\bar{D}_{Mn}:=\left\\{(r,\theta)\in\mathbb{R}^{2}:\left(n+\frac{1}{2}\right)\leq r\leq\left(n+\frac{1}{2M\pi}+\frac{1}{2}\right),-M\pi\leq\theta\leq M\pi\right\\}.$ (10) The corresponding co-domain $\bar{A}_{Mn}$ in the $q-p$ phase plane is provided by the annular region: $\begin{array}[]{c}\bar{A}_{Mn}:=\left\\{(q,p)\in\mathbb{R}^{2}:2n+1\leq q^{2}+p^{2}\leq\left(2n+1+\frac{1}{M\pi}\right)\right\\},\\\ \\\ Area(\bar{A}_{Mn}):=\displaystyle{\int\int_{\bar{A}_{Mn}}}dp\wedge dq=1.\end{array}$ (11) The mappings $\\{g^{M}_{n}\\}^{\infty}_{M=1}$ are depicted in the figure 5. Figure 5: The mapping $g^{M}_{n}$ with topological identifications of the two horizontal sides of $\bar{D}_{Mn}$. Note that the annular region $\bar{A}_{Mn}$ is not a simply connected region of the circular disk $q^{2}+p^{2}=2n+1+\frac{1}{M\pi}$. Furthermore, the winding number of each boundary of the annular region $\bar{A}_{Mn}$ is exactly $M$. ## 4 The Peano circle $\bar{S}^{1}_{n}$ in the $(1+1)$-dimensional phase plane We shall now investigate the closed domain $\bar{A}_{Mn}$ of the mapping $g^{M}_{n}$ for a fixed $n$ as exhibited in figure 5. Using (9) and (11), the inner and outer boundaries of the closed co-domain $\bar{A}_{Mn}$ are found to be $\begin{array}[]{c}\partial_{-}[\bar{A}_{Mn}]:=\\{(q,p)\in\mathbb{R}:q^{2}+p^{2}=2n+1\\},\\\ \\\ \partial_{+}[\bar{A}_{Mn}]:=\left\\{(q,p)\in\mathbb{R}:q^{2}+p^{2}=2n+1+\frac{1}{M\pi}\right\\}.\end{array}$ (12) Consider the limiting map $\displaystyle{\lim_{M\rightarrow\infty}}\\{g^{M}_{n}\\}_{M=1}^{\infty}$. Clearly, one has for this limiting map $\displaystyle{\lim_{M\rightarrow\infty}}\left(\left\\{(q,p)\in\mathbb{R}:q^{2}+p^{2}=2n+1+\frac{1}{M\pi}\right\\}\right)=\partial_{-}[\bar{A}_{Mn}]$ (13) which is depicted in figure 6. Figure 6: The limiting map $\displaystyle{\lim_{M\rightarrow\infty}}\\{g^{M}_{n}\\}$ and the closed range as the Peano circle $\bar{S}^{1}_{n}$. Remarks: (i) Note that by (11), $Area(\bar{A}_{Mn})=1$ for all $M\in\\{1,2,3,...\\}$. (ii) Since the closed range of the mapping $\displaystyle{\lim_{M\rightarrow\infty}}\\{g^{M}_{n}\\}=\bar{S}^{1}_{n}$, the Peano circle $\bar{S}^{1}_{n}$ itself is endowed with the unit $(1+1)$-dimensional area. (iii) The Peano circle can act as a string-like Green ; PolchinskiI ; PolchinskiII phase-cell. (iv) The quantum mechanical uncertainty principle $\|\Delta q\|\|\Delta p\|\geq 1$ holds for each quanta inside the phase cell $\bar{S}^{1}_{n}$. (v) The original Peano-curve $\displaystyle{\lim_{j\rightarrow\infty}}f^{j}(u)=f(u)$ is squashed inside the Peano circle $\bar{S}^{1}_{n}$. (vi) The first quantized oscillator introduced in part I of this work DasRCI follows a zig-zag motion or zitter-bewengung Greiner along the squashed Peano curve inside $\bar{S}^{1}_{n}$. (vii) Unlike the Peano circle $\bar{S}^{1}_{n}$ here, the one-dimensional circle of figure 2 in part I DasRCI cannot act as a two-dimensional phase cell within the $q-p$ phase plane. Consider now the infinite $[(1+1)+1]$-dimensional hyper-circular cylinder $\bar{S}^{1}_{n}\times\mathbb{R}$ inside the $[(1+1)+1]$-dimensional discrete phase plane plus continuous time $t\in\mathbb{R}$ called the state space Lanczos . Within this state space is a world-sheet like object Green ; PolchinskiI ; PolchinskiII depicted in figure 7. Figure 7: The $[(1+1)+1]$-dimensional infinite hyper-cylinder $\bar{S}^{1}_{n}\times\mathbb{R}$ inside the $[(1+1)+1]$-dimensional discrete phase plane plus continuous time. The $[(1+1)+1]$-dimensional circular cylinder $S^{1}_{n}\times\mathbb{R}$ inside the state space in figure 4 in part I DasRCI cannot represent an evolving phase cell unlike the hyper-circular cylinder $\bar{S}^{1}_{n}\times\mathbb{R}$ depicted in figure 7. Let us summarize what we have achieved so far. Figures 1 and 2 have described the area filling curve of Peano, which completely fills the area of a unit square. We identify this unit area as a possible phase cell within a $(p,q)$-phase plane. The area filling Peano curve is identified as the trajectory of one quanta of certain quantized simple harmonic oscillators introduced in DasRCI . The trajectory of the Peano curve is not observable due to the uncertainty principle of quantum mechanics Greiner ; Bethe . Equations (4), (5), (7), and (9) indicate sequences of area preserving mathematical mappings physically representing canonical transformations of Hamiltonian mechanics Lanczos ; Goldstein . Therefore, in figure 5, the original unit square has yielded a sequence of circular, annular domains each of unit area. In figure 6, this sequence of circular annular domains collapses into a Peano circle $\bar{S}^{1}_{n}$ of unit area. The quanta of the Planck oscillator DasRCI still goes around the Peano circle with constant energy $\sqrt{2n+1}$ in a zig-zag pattern but is undetectable by external observations. Note that a Peano circle $\bar{S}^{1}_{n}$ of unit area in the $(1+1)$-phase plane is analogous to a closed string of prototypical string theory Green ; PolchinskiI ; PolchinskiII . Figure 7 depicts the $[(1+1)+1]$-dimensional hyper-circular-cylinder inside the three-dimensional state space that is analogous to the two-dimensional world-sheet evolution of closed strings. ## 5 The Klein-Gordon equation in a $[(1+1)+1]$-dimensional discrete phase- plane and continuous time Now, we investigate the second quantization of the Klein-Gordon equation in a $[(1+1)+1]$-dimensional discrete phase-plane and continuous time scenario. The relativistic Klein-Gordon equation in a $[(2+2+2)+1]$-dimensional discrete phase-plane and continuous time was discussed in section 7 of part I of this paper DasRCI . The second quantized scalar field linear operator, denoted by $\Phi(n,t)$, acts on a Hilbert space bundle Choquet ; DasTA . The corresponding Klein-Gordon equation is $\begin{array}[]{c}[\mathbf{P}\cdot\mathbf{P}-(\mathbf{P_{t}})^{2}+m^{2}\mathbf{I}]\overrightarrow{\mathbf{\Psi}}=\overrightarrow{\mathbf{0}},\\\ or\\\ (\Delta^{\\#})^{2}\Phi(n,t)-(\partial_{t})^{2}\Phi(n,t)-m^{2}\Phi(n,t)=0.\end{array}$ (14) Consider the complex valued functions $\xi_{n}(k)$ involving Hermite polynomials $H_{n}(k)$ Olver , $\begin{array}[]{c}\xi_{n}(k):=\dfrac{(i)^{n}e^{-k^{2}/2}H_{n}(k)}{\pi^{1/4}2^{n/2}\sqrt{n!}},\\\ \\\ \xi_{n}(-k)=\overline{\xi_{n}(k)},\\\ \xi_{2n+1}(0)=0,\\\ \displaystyle{\int_{\mathbb{R}}}\overline{\xi_{m}(k)}\xi_{n}(k)dk=\delta_{mn},\\\ \displaystyle{\sum_{n=0}^{\infty}}\overline{\xi_{n}(k)}\xi_{n}(\hat{k})=\delta(k-\hat{k}),\\\ \Delta^{\\#}\xi_{n}(k)=ik\xi_{n}(k).\end{array}$ (15) Here, $\delta(k-\hat{k})$ indicates the Dirac delta distribution function Zemanian . A special class of exact solutions to the partial difference- differential equations of (14) are given by $\begin{array}[]{c}\Phi^{-}(n,t):=\displaystyle{\int_{\mathbb{R}}}\dfrac{1}{\sqrt{2\omega(k)}}\left[A(k)\xi_{n}(k)e^{-i\omega t}\right]dk,\\\ \\\ \Phi^{+}(n,t):=\displaystyle{\int_{\mathbb{R}}}\dfrac{1}{\sqrt{2\omega(k)}}\left[B^{\dagger}(k)\overline{\xi_{n}(k)}e^{i\omega t}\right]dk,\\\ \\\ \Phi(n,t):=\Phi^{-}(n,t)+\Phi^{+}(n,t),\\\ \\\ \omega=\omega(k):=+\sqrt{k^{2}+m^{2}}>0.\end{array}$ (16) The Fourier-Hermite integrals in (16) are supposed to be uniformly convergent Buck . Moreover, the linear operators $A(k),A^{\dagger}(k),B(k)$ and $B^{\dagger}(k)$ (creation and annihilation operators in momentum space) acting linearly on a Hilbert space bundle must satisfy the following commutation relations: $\begin{array}[]{c}[A(k),A^{\dagger}(\hat{k})]=[B(k),B^{\dagger}(\hat{k})]=\delta(k-\hat{k})I(k),\\\ \\\ \,[A(k),A(\hat{k})]=[A^{\dagger}(k),A^{\dagger}(\hat{k})]=[B(k),B(\hat{k})]=[B^{\dagger}(k),B^{\dagger}(\hat{k})]=0,\\\ \\\ N^{+}(k):=A^{\dagger}(k)A(k),\\\ \\\ N^{-}(k):=B^{\dagger}(k)B(k),\end{array}$ (17) where the eigenvalues of the number operators $N^{\pm}(k)$ are the set $\\{0,1,2,...\\}$. Physically speaking, the scalar field operator $\Phi(n,t)$ represents a collection of massive, spin-less, electrically charged, second quantized scalar particle excitations. One can show the following relations for total energy, total momentum and total electric charge respectively DasI ; DasII , $\begin{array}[]{c}\mathcal{H}:=\displaystyle{\int_{\mathbb{R}}}[N^{+}(k)+N^{-}(k)+\delta(0)I(k)]\omega(k)dk,\\\ \\\ \mathcal{P}:=\displaystyle{\int_{\mathbb{R}}}[N^{+}(k)+N^{-}(k)]kdk,\\\ \\\ \mathcal{Q}:=e\displaystyle{\int_{\mathbb{R}}}[N^{+}(k)-N^{-}(k)]dk.\end{array}$ (18) The divergent null point energy term $\delta(0)I(k)$ may be ignored for physical interpretations but cannot be rectified directly. ## 6 Fibre Bundles Let $M$ and $M^{\\#}$ be two topological manifolds Choquet ; DasTA . The Cartesian product $M\times M^{\\#}$ and the projection mapping $\Pi$ from $M\times M^{\\#}$ into $M$ constitute a product or trivial bundle $(M\times M^{\\#},M,\Pi)$ depicted in figure 8. The vertical linear segment inside $M\times M^{\\#}$ is called a fibre Choquet ; DasTA . Figure 8: The trivial or product bundle $(M\times M^{\\#},M,\Pi)$. Figure 9 provides an explicit example of a product bundle using our Peano cirle $\bar{S}^{1}_{n}$ to create a vertical circular cylinder $\bar{S}^{1}_{n}\times I_{t}$, Figure 9: The product bundle $(\bar{S}^{1}_{n}\times I_{t},\bar{S}^{1}_{n},\Pi_{t})$. The closed interval $I_{t}:=[0,T]\subset\mathbb{R}$ and $(\chi,[-\pi,\pi])$ is a coordinate chart DasTA in figure 9. The second quantized scalar field operators defined in (16) are restricted for fixed numbers $n$ and $T$ of the product bundle in figure 9. Another product bundle associated with the linear operators (16) is created by using our Peano circle $\bar{S}^{1}_{n}$ and the closed linear line segment $I_{k}:=[K_{1},K_{2}]\subset\mathbb{R}$ as in figure 10. Figure 10: The product bundle $(\bar{S}^{1}_{n}\times I_{k},\bar{S}^{1}_{n},\Pi_{k})$. Remarks: (i) Figure 9 represents geometrically the occupation of a tiny first quantized Planck oscillator of figures 6 and 7 in the interval $[0,T]\subset\mathbb{R}$ (the full actual physics takes place in the time interval $-\infty<t<\infty$). (ii) Moreover, in figure 9, the second quantized scalar field $\Phi(n,t)$ quanta can cohabit with a tiny first quantized Planck oscillator during the time interval $I_{t}:=[0,T]\subset\mathbb{R}$. (iii) Furthermore, in figure 10, the second quantized scalar field quanta with a range of momentum $K_{1}<k<K_{2}$ can cohabit with a tiny first quantized Planck oscillator. (iv) The complete mathematical treatment of this phenomena involves the product bundles $(\bar{S}^{1}_{n}\times I_{t},\bar{S}^{1}_{n},\Pi_{t})\times(\bar{S}^{1}_{n}\times I_{k},\bar{S}^{1}_{n},\Pi_{k})$ (which is difficult to depict). ## 7 Discrete phase space and hyper-tori like phase cells Here, we shall extend the various mappings of figures 3,4, and 5 to the $(2+2+2)$-dimensional discrete phase space arena. Choosing the fixed indices $a\in\\{1,2,3\\}$ and $M\in\\{1,2,3,\cdots\\}$, the relevant mappings are illustrated in figure 11. It depicts the composite map $g^{M}_{n^{a}}\circ h^{M}_{n^{a}}\circ h^{M}_{a}$ for fixed indices $a$ and $M$. Figure 11: The discrete phase space mappings $h^{M}_{a},h^{M}_{n^{a}}$, and $g^{M}_{n^{a}}$. The closed domains and the closed range of the composite map $g^{M}_{n^{a}}\circ h^{M}_{n^{a}}\circ h^{M}_{a}$ are provided by $\begin{array}[]{c}closed\,\,\ domain\,\left[g^{M}_{n^{a}}\circ h^{M}_{n^{a}}\circ h^{M}_{a}\right]=\bar{D}_{a}\subset\mathbb{R}^{2},\\\ \\\ closed\,\,\ range\,\left[g^{M}_{n^{a}}\circ h^{M}_{n^{a}}\circ h^{M}_{a}\right]=\bar{A}_{Mn^{a}}\subset\mathbb{R}^{2},\\\ or\\\ \bar{A}_{Mn^{a}}=\left[g^{M}_{n^{a}}\circ h^{M}_{n^{a}}\circ h^{M}_{a}\right](\bar{D}_{a}).\end{array}$ (19) where the last relation illustrates a set theoretic mapping Goldberg . Consider the sequence of composite mappings $\left\\{g^{M}_{n^{a}}\circ h^{M}_{n^{a}}\circ h^{M}_{a}\right\\}_{M=1}^{\infty}$ for a fixed $a$ and $n^{a}$. The limiting set-theoretic mapping from figure 6 and the last relation of (19) is given by $\bar{S}^{1}_{n}=\displaystyle{\lim_{M\rightarrow\infty}}\left\\{\left[g^{M}_{n^{a}}\circ h^{M}_{n^{a}}\circ h^{M}_{a}\right](\bar{D}_{a})\right\\}$ (20) From this, one can derive the set-theoretic Cartesian product mapping Goldberg ; Lightstone : $\begin{array}[]{c}\bar{S}^{1}_{n^{1}}\times\bar{S}^{1}_{n^{2}}\times\bar{S}^{1}_{n^{3}}=\displaystyle{\lim_{M\rightarrow\infty}}\left(\left\\{\bar{A}_{Mn^{1}}\right\\}\times\left\\{\bar{A}_{Mn^{2}}\right\\}\times\left\\{\bar{A}_{Mn^{3}}\right\\}\right),\\\ or\\\ \bar{S}^{1}_{n^{1}}\times\bar{S}^{1}_{n^{2}}\times\bar{S}^{1}_{n^{3}}=\displaystyle{\lim_{M\rightarrow\infty}}\left(\left\\{\left[g^{M}_{n^{1}}\circ h^{M}_{n^{1}}\circ h^{M}_{1}\right](\bar{D}_{1})\right\\}\times\left\\{\left[g^{M}_{n^{2}}\circ h^{M}_{n^{2}}\circ h^{M}_{2}\right](\bar{D}_{2})\right\\}\right.\\\ \left.\times\left\\{\left[g^{M}_{n^{3}}\circ h^{M}_{n^{3}}\circ h^{M}_{3}\right](\bar{D}_{3})\right\\}\right).\end{array}$ (21) Furthermore, one has $\begin{array}[]{c}\left[g^{M}_{n^{1}}\times h^{M}_{n^{1}}\times h^{M}_{1}\right]\times\left[g^{M}_{n^{2}}\times h^{M}_{n^{2}}\times h^{M}_{2}\right]\times\left[g^{M}_{n^{3}}\times h^{M}_{n^{3}}\times h^{M}_{3}\right]\left(\bar{D}_{1}\times\bar{D}_{2}\times\bar{D}_{3}\right):=\\\ \\\ \left\\{\left[g^{M}_{n^{1}}\circ h^{M}_{n^{1}}\circ h^{M}_{1}\right](\bar{D}_{1})\right\\}\times\left\\{\left[g^{M}_{n^{2}}\circ h^{M}_{n^{2}}\circ h^{M}_{2}\right](\bar{D}_{2})\right\\}\times\left\\{\left[g^{M}_{n^{3}}\circ h^{M}_{n^{3}}\circ h^{M}_{3}\right](\bar{D}_{3})\right\\}\end{array}$ (22) From the above two relations, a new set-theoretic mapping is furnished by $\begin{array}[]{c}\bar{S}^{1}_{n^{1}}\times\bar{S}^{1}_{n^{2}}\times\bar{S}^{1}_{n^{3}}=\displaystyle{\lim_{M\rightarrow\infty}}\left\\{\left[g^{M}_{n^{1}}\times h^{M}_{n^{1}}\times h^{M}_{1}\right]\times\left[g^{M}_{n^{2}}\times h^{M}_{n^{2}}\times h^{M}_{2}\right]\times\right.\\\ \\\ \left.\left[g^{M}_{n^{3}}\times h^{M}_{n^{3}}\times h^{M}_{3}\right]\left(\bar{D}_{1}\times\bar{D}_{2}\times\bar{D}_{3}\right)\right\\}\subset\mathbb{R}^{2}\times\mathbb{R}^{2}\times\mathbb{R}^{2}.\end{array}$ (23) Remarks: (i) The $(2+2+2)$-dimensional hyper-sphere or hyper-torus Massey $\bar{S}^{1}_{n^{1}}\times\bar{S}^{1}_{n^{2}}\times\bar{S}^{1}_{n^{3}}$ for three fixed integers $(n^{1},n^{2},n^{3})$ denotes a region where a tiny first quantized Planck oscillator inhabits this region for all time. (ii) The whole of the $(2+2+2)$-dimensional discrete phase space contains a denumerably infinite number of concentric hyper-tori each containing a single tiny first quantized Planck oscillator. (iii) Each $\bar{S}^{1}_{n^{1}}\times\bar{S}^{1}_{n^{2}}\times\bar{S}^{1}_{n^{3}}$ constitutes a phase cell of physical dimension $\left[\dfrac{ML^{2}}{T}\right]^{3}$. Therefore, in the fundamental physical units, each of these phase cells is endowed with a hyper-volume of $1\left[\dfrac{ML^{2}}{T}\right]^{3}$. (iv) Each of these phase-cells bears a resemblance to a $D$-dimensional hyper- torus in standard string theory Green ; PolchinskiI ; PolchinskiII . (v) In the arena of $[(2+2+2)+1]$-dimensional discrete phase space and continuous time, the hyper-torus $\bar{S}^{1}_{n^{1}}\times\bar{S}^{1}_{n^{2}}\times\bar{S}^{1}_{n^{3}}$ sweeps out a world-sheet like vertical, circular hyper-cylinder. Such a vertical, circular cylinder always contains just one first quantized tiny Planck oscillator for all time. (vi) Finally, one or more second quantized scalar field $\Phi(\mathbf{n};t)$ quanta may cohabit the tiny first quantized Planck oscillator temporarily or forever. ## 8 Second quantized, relativistic Klein-Gordon field equations in $[(2+2+2)+1]$-dimensional discrete phase space and continuous time From part I of this work DasRCI , the second quantized, relativistic Klein- Gordon field equations are given by $\begin{array}[]{c}a,b\in\\{1,2,3\\},\\\ \\\ n^{a}\in\\{0,1,2,...\\},\\\ \\\ \mathbf{n}:=(n^{1},n^{2},n^{3}),\\\ \\\ \left[(\delta^{ab}\Delta^{\\#}_{a}\Delta^{\\#}_{b})-(\partial_{t})^{2}-m^{2}\right]\Phi(\mathbf{n};t)=\mathbf{0}.\end{array}$ (24) The three dimensional extensions of the functions $\xi_{n}(k)$ in (15) imply that Olver $\begin{array}[]{c}\xi_{n^{a}}(k_{a}):=\dfrac{(i)^{n^{a}}e^{-k_{a}^{2}/2}H_{n^{a}}(k_{a})}{\pi^{1/4}2^{n^{a}/2}\sqrt{n^{a}!}},\\\ \\\ \mathbf{k}:=(k_{1},k_{2},k_{3}),\\\ \\\ \Delta^{\\#}_{a}\xi_{n^{a}}(k_{a})=ik_{a}\xi_{n^{a}}(k_{a}),\\\ \\\ \displaystyle{\sum_{n^{1}=0}^{\infty}\sum_{n^{2}=0}^{\infty}\sum_{n^{3}=0}^{\infty}}\left[\overline{\xi_{n^{1}}}(k_{1})\xi_{n^{1}}(\hat{k}_{1})\right]\left[\overline{\xi_{n^{2}}}(k_{2})\xi_{n^{2}}(\hat{k}_{2})\right]\left[\overline{\xi_{n^{3}}}(k_{3})\xi_{n^{3}}(\hat{k}_{3})\right]\\\ =\delta({k_{1}}-{\hat{k}_{1}})\delta({k_{2}}-{\hat{k}_{2}})\delta({k_{3}}-{\hat{k}_{3}})=:\delta^{3}(\mathbf{k}-\mathbf{\hat{k}}).\end{array}$ (25) A special class of exact solutions for these relativistic partial difference- differential operator equations (24) are given by $\begin{array}[]{c}\Phi^{-}(\mathbf{n};t):=\displaystyle{\int_{\mathbb{R}^{3}}}\dfrac{1}{\sqrt{2\omega(\mathbf{k})}}\left[A(\mathbf{k})\xi_{n^{1}}(k_{1})\xi_{n^{2}}(k_{2})\xi_{n^{3}}(k_{3})e^{-i\omega t}\right]dk_{1}dk_{2}dk_{3},\\\ \\\ \Phi^{+}(\mathbf{n};t):=\displaystyle{\int_{\mathbb{R}^{3}}}\dfrac{1}{\sqrt{2\omega(\mathbf{k})}}\left[B^{\dagger}(\mathbf{k})\overline{\xi_{n^{1}}}(k_{1})\overline{\xi_{n^{2}}}(k_{2})\overline{\xi_{n^{3}}}(k_{3})e^{i\omega t}\right]dk_{1}dk_{2}dk_{3},\\\ \\\ \Phi(\mathbf{n};t):=\Phi^{-}(\mathbf{n};t)+\Phi^{+}(\mathbf{n};t),\\\ \\\ \omega=\omega(\mathbf{k}):=+\sqrt{k_{1}^{2}+k_{3}^{2}+k_{3}^{2}+m^{2}}>0.\end{array}$ (26) The Fourier-Hermite integrals in (26) are supposed to be uniformly convergent Buck . Moreover, the linear operators $A(\mathbf{k}),A^{\dagger}(\mathbf{k}),B(\mathbf{k})$ and $B^{\dagger}(\mathbf{k})$ (creation and annihilation operators in momentum space) acting linearly on a Hilbert space bundle Choquet ; DasTA must satisfy the following commutation relations: $\begin{array}[]{c}[A(\mathbf{k}),A^{\dagger}(\hat{\mathbf{k}})]=[B(\mathbf{k}),B^{\dagger}(\hat{\mathbf{k}})]=\delta(\mathbf{k}-\hat{\mathbf{k}})\mathbf{I}(\mathbf{k}),\\\ \\\ \,[A(\mathbf{k}),A(\hat{\mathbf{k}})]=[A^{\dagger}(\mathbf{k}),A^{\dagger}(\hat{\mathbf{k}})]=[B(\mathbf{k}),B(\hat{\mathbf{k}})]=[B^{\dagger}(\mathbf{k}),B^{\dagger}(\hat{\mathbf{k}})]=\mathbf{0},\\\ \\\ N^{+}(\hat{\mathbf{k}}):=A^{\dagger}(\mathbf{k})A(\mathbf{k}),\\\ \\\ N^{-}(\hat{\mathbf{k}}):=B^{\dagger}(\mathbf{k})B(\mathbf{k}),\end{array}$ (27) where the eigenvalues of the number operators $N^{\pm}(\hat{\mathbf{k}})$ are the set $\\{0,1,2,...\\}$. The second quantized scalar field operator $\Phi(\mathbf{n};t)$ represents a collection of massive, spin-less, electrically charged, physical particles or quantas. One can show the following relations for total energy, total momentum and total electric charge respectively DasI ; DasII , $\begin{array}[]{c}\mathcal{H}:=\displaystyle{\int_{\mathbb{R}^{3}}}[N^{+}(\mathbf{k})+N^{-}(\mathbf{k})+\delta^{3}(\mathbf{0})\mathbf{I}(\mathbf{k})]\omega(\mathbf{k})dk_{1}dk_{2}dk_{3},\\\ \\\ \mathcal{P}_{j}:=\displaystyle{\int_{\mathbb{R}^{3}}}[N^{+}(\mathbf{k})+N^{-}(\mathbf{k})]k_{j}dk_{1}dk_{2}dk_{3},\\\ \\\ \mathcal{Q}:=e\displaystyle{\int_{\mathbb{R}^{3}}}[N^{+}(\mathbf{k})-N^{-}(\mathbf{k})]dk_{1}dk_{2}dk_{3}.\end{array}$ (28) The divergent null point energy term $\delta^{3}(\mathbf{0})\mathbf{I}(\mathbf{k})$ may be ignored for physical interpretations but cannot be directly remedied. ## 9 Many-particle systems, $(2+2+2)N$-dimensional phase space and $(2+2+2)N$-dimensional hyper-tori as phase cells Recall Hamilton’s canonical equations of motion for non-relativistic classical mechanics Lanczos ; Goldstein $\begin{array}[]{c}A\in\\{1,2,3,...,N\\},\\\ \\\ \dot{q}^{A}:=\dfrac{d\mathcal{Q}^{A}(t)}{dt},\\\ \\\ \dot{p}_{A}:=\dfrac{d\mathcal{P}_{A}(t)}{dt},\\\ \\\ \dot{q}^{A}:=\dfrac{\partial H}{\partial p_{A}}(q^{1},\cdots,q^{3N};p_{1},\cdots,p_{3N};t),\\\ \\\ \dot{p}_{A}:=-\dfrac{\partial H}{\partial q^{A}}(q^{1},\cdots,q^{3N};p_{1},\cdots,p_{3N};t).\end{array}$ (29) The corresponding Schrödinger wave equation for the first quantized physical system is furnished by $\begin{array}[]{c}i\dfrac{\partial}{\partial_{t}}\psi\left(q^{1},\cdots,q^{3N};t\right)=\\\ H\left(q^{1},\cdots,q^{3N};-i\frac{\partial}{\partial q^{1}},\cdots,-i\frac{\partial}{\partial q^{3N}};t\right)\psi(q^{1},\cdots,q^{3N};t),\end{array}$ (30) which has become ubiquitous with standard quantum mechanical systems being verified experimentally for a multitude of specific systems. From part I of this paper DasRCI , the $N=1$ relativistic Klein-Gordon equation in four-dimensional space is given by ($\eta_{\mu\nu}:=[1,1,1,-1]$) $\begin{array}[]{c}\eta^{\mu\nu}\dfrac{\partial^{2}}{\partial q^{\mu}\partial q^{\mu}}\psi\left(q^{1},q^{2},q^{3},q^{4}\right)-m^{2}\psi\left(q^{1},q^{2},q^{3},q^{4}\right)=0,\\\ or\\\ \delta^{ab}\dfrac{\partial^{2}}{\partial q^{a}\partial q^{b}}\psi\left(\mathbf{q};t\right)-(\partial_{t})^{2}\psi\left(\mathbf{q};t\right)-m^{2}\psi\left(\mathbf{q};t\right)=0.\\\ \end{array}$ (31) The relativistic wave equation for the case of two spin-$\frac{1}{2}$ particles $q_{(1)}^{\mu}$ and $q_{(2)}^{\nu}$ is furnished by the Bethe- Salpeter equation Bethe $\begin{array}[]{c}\left\\{\gamma_{(1)}^{\mu}\left[\left(\dfrac{m_{1}}{m_{1}+m_{2}}P_{\mu}+P_{\mu}\right)-im_{1}\right]\right\\}\left\\{\gamma_{(2)}^{\nu}\left[\left(\dfrac{m_{1}}{m_{1}+m_{2}}P_{\nu}+P_{\nu}\right)-im_{2}\right]\right\\}\\\ \\\ \cdot\psi\left(q_{(1)}^{1}-q_{(2)}^{1},q_{(1)}^{2}-q_{(2)}^{2},q_{(1)}^{3}-q_{(2)}^{3},q_{(1)}^{4}-q_{(2)}^{4}\right)=\\\ \\\ i\bar{G}\left(q_{(1)}^{1}-q_{(2)}^{1},q_{(1)}^{2}-q_{(2)}^{2},q_{(1)}^{3}-q_{(2)}^{3},q_{(1)}^{4}-q_{(2)}^{4}\right)\psi\left(q_{(1)}^{1}-q_{(2)}^{1},\cdots\right)\end{array}$ (32) where $\bar{G}$ is the appropriate Green’s function. In $N+1$-dimensional state space, the generalization to the relativistic Klein-Gordon wave equation is $\begin{array}[]{c}\left[\delta^{AB}\dfrac{\partial^{2}}{\partial q^{A}\partial q^{B}}-(\partial_{t})^{2}-m^{2}\right]\psi\left(q^{1},\cdots,q^{N};t\right)=0,\\\ A,B\in\\{1,\cdots,N\\}.\\\ \end{array}$ (33) with a group invariance of $\mathcal{I}[O(N,1)]_{+}^{+}$. Similarly, we can express within a $[(2+2+2)N+1]$-dimensional discrete phase space and continuous time arena, the first quantized partial differential- difference Klein-Gordon equation as $\left[\delta^{AB}\Delta^{\\#}_{A}\Delta^{\\#}_{B}-(\partial_{t})^{2}-m^{2}\right]\phi(n^{1},\cdots,n^{N};t)=0.$ (34) The second quantized version of this generalized Klein-Gordon equation is given by $\left[\delta^{AB}\Delta^{\\#}_{A}\Delta^{\\#}_{B}-(\partial_{t})^{2}-m^{2}\right]\Phi(n^{1},\cdots,n^{N};t)=\mathbf{0}$ (35) The group invariance of the above equation is provided by $\mathcal{I}[O(N,1)]_{+}^{+}$ (see DasRCI ). A special class of exact solutions for these relativistic partial difference- differential operator equations (35) are given by $\begin{array}[]{c}\Phi^{-}(n^{1},\cdots,n^{N};t):=\displaystyle{\int_{\mathbb{R}^{N}}}\dfrac{1}{\sqrt{2\omega(\mathbf{k})}}\left[A(\mathbf{k})\xi_{n^{1}}(k_{1})\cdots\xi_{n^{N}}(k_{N})e^{-i\omega t}\right]dk_{1}\cdots dk_{N},\\\ \\\ \Phi^{+}(n^{1},\cdots,n^{N};t):=\displaystyle{\int_{\mathbb{R}^{N}}}\dfrac{1}{\sqrt{2\omega(\mathbf{k})}}\left[B^{\dagger}(\mathbf{k})\overline{\xi_{n^{1}}}(k_{1})\cdots\overline{\xi_{n^{N}}}(k_{N})e^{i\omega t}\right]dk_{1}\cdots dk_{N},\\\ \\\ \Phi(n^{1},\cdots,n^{N};t):=\Phi^{-}(n^{1},\cdots,n^{N};t)+\Phi^{+}(n^{1},\cdots,n^{N};t),\\\ \\\ \omega=\omega(k_{1},\cdots,k_{N}):=+\sqrt{\delta^{AB}k_{A}k_{B}+m^{2}}>0.\end{array}$ (36) The second quantized linear operators $A(k_{1},\cdots,k_{N}),B^{\dagger}(k_{1},\cdots,k_{N})$, etc. (creation and annihilation operators in momentum space) acting linearly on a Hilbert space bundle must satisfy very similar commutation relations as those of (27). The linear operator $\Phi(n^{1},\cdots,n^{N};t)$ is defined over the geometric configuration $\bar{S}_{n^{1}}^{1}\times\bar{S}_{n^{2}}^{1}\times\cdots\times\bar{S}_{n^{N}}^{1}\times\mathbb{R}$ where $\bar{S}_{n^{A}}^{1}$ is a Peano circle of physical dimension $\left[\dfrac{ML^{2}}{T}\right]$. Table 1: Discrete Phase Space and Popular String Theory Dimension Comparison Finally, we would like to make a final comparison with our method to two popular approaches to quantizing gravity: string theory and loop quantum gravity. Table 1 consists of a higher dimensional comparison between our discrete phase space theory put forward here and that of various popular string theories Green ; PolchinskiI ; PolchinskiII . There appears to be a striking similarity between the spatial or gauge group dimension of string theory and the dimension of our discrete phase space. One clear advantage of our method is that we have gone further in understanding the scattering matrix of our theory, the $S^{\\#}$-matrix, as delineated in DasI ; DasII ; DasIII . In DasRC , we have explicitly calculated new Feynman rules for computations of the $S^{\\#}$-matrix elements and have derived an exact singularity free Coulomb-type potential within our discrete phase space approach. See Green ; PolchinskiI ; PolchinskiII for further information on the actual calculation abilities of string theory. Loop quantum gravity Rovelli , and its spin network structure of space-time composed of extremely fine but finite loops has a similarity to our Peano circle $\bar{S}^{1}_{n}$ of unit area in the $(1+1)$ dimensional phase plane. The area filling Peano curve is identified as the trajectory of one quanta as shown here and in DasRCI . It would be interesting to further investigate this analogy to possibly find a quantum theory of gravity within our framework or find Peano type motion within the framework of loop quantum gravity. ## 10 Concluding Remarks In this paper, we have shown how an area filling Peano curve represents a possible particle trajectory in the unit phase cell of a discrete phase space and continuous time relativistic quantum mechanical system. This is one of the first uses of this fascinating mathematical structure as a physical construct. Both first quantized Planck oscillators, first explored in part I of this work DasRCI , and second quantized Klein-Gordon excitations were explored with respect to the Peano curve formalism. Furthermore, the state space evolution of our Peano circle was shown to be analogous to the world-sheet evolution of closed strings. The geometric framework of this evolution was interpreted in terms of a product fibre bundle structure. Finally, extensions of our model to higher dimensions and striking similarities to popular dimensions used in traditional string theory were explored. ## Acknowledgements A.D. thanks Dr. Jack Gegenberg for some informal discussions. ## References * (1) A. Das and R. Chatterjee, Discrete phase space and continuous time relativistic quantum mechanics I: Planck oscillators and closed string-like circular orbits, arXiv:2012.14256, (2020). * (2) C. Clark, The Theoretical Side of Calculus, Wadsworth Publ. Co., Belmont, (1972). * (3) B.R. Gelbaum and J.M.H. Olmstead Counterexamples in Analysis, Holden-Day, Inc., San Francisco, (1964). * (4) A. Das and S. Haldar, Physical Science International Journal 17(2), 1 (2018). * (5) Y. Choquet-Bruhat, C. Dewitt-Morette, and M. Dillard-Bleck Analysis, Manifolds, and Physics North-Holland, Amsterdam, (1978). * (6) A. Das, Can. J. Phys. 88, 73 (2010). * (7) A. Das, Can. J. Phys. 88, 93 (2010). * (8) A. Das, Can. J. Phys. 88, 111 (2010). * (9) J.M. Jauch and F. Rohrlich, The Theory of Photons and Electrons, Addison-Wesley, Cambridge, MA, (1955). * (10) M.E. Peskin and D.V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, Cambridge, MA, (1995). * (11) A. Das and A. DeBenedictis, Scientific Voyage, 1, 45 (2015). * (12) A. Das, R. Chatterjee, and T. Yu, Mod. Phys. Lett. A 35, 24 (2020). * (13) R.R. Goldberg Methods of Real Analysis John-Wiley & Sons, New York, (1976). * (14) A.H. Lightstone Symbolic Logic and the Real Number System Harper & Row, New York, (1965). * (15) W.S. Massey Algebraic Topology: An Introduction Harcourt, Brace & World, , New York, (1967). * (16) M.B. Green, J.H. Schwarz, and E. Witten, Superstring Theory Vol 1 & 2, Cambridge University Pres, Cambridge, (1987). * (17) J. Polchinski, String Theory Vol 1, Cambridge University Pres, Cambridge, (1998). * (18) J. Polchinski, String Theory Vol 2, Cambridge University Pres, Cambridge, (1998). * (19) P.A.M. Dirac, The Principles of Quantum Mechanics, 4th Edition Oxford University Press, London, (1967). * (20) C. Lanczos, The Variational Principles of Mechanics, University of Toronto Press, Toronto, (1970). * (21) H. Goldstein, Classical Mechanics, 2nd, Addison-Wesley, Reading, Mass., (1980). * (22) M. Spivak, Calculus on Manifolds, Benjamin-Cummings, Melno Park, CA, (1965). * (23) A. Das, Tensors:The Mathematics of Relativity Theory and Continuum Mechanics, Springer-Verlag, Berlin, (2007). * (24) W. Greiner, Relativistic Quantum Mechanics, 3rd Edition Springer-Verlag, Berlin, (2000). * (25) F.W.J. Olver, Introduction to Asymptotics and Special Functions Academic Press, New York, (1974). * (26) A.H.. Zemanian, Distribution Theory and Transform Analysis Dover Publications, New York, (1965). * (27) R.C. Buck and E.F. Buck, Advanced Calculus McGraw-Hill, New York, (1965). * (28) H.A. Bethe and E.E. Salpeter, Quantum Mechanics of One and Two Electron Atoms Springer-Verlag, Berlin, (1957). * (29) C. Rovelli, Quantum Gravity Cambridge University Press, Cambridge, (2007).
]www.polyphys.mat.ethz.ch # Coordinate conditions and field equations for pure composite gravity Hans Christian Öttinger<EMAIL_ADDRESS>[ ETH Zürich, Department of Materials, CH-8093 Zürich, Switzerland ###### Abstract Whenever an alternative theory of gravity is formulated in a background Minkowski space, the conditions characterizing admissible coordinate systems, in which the alternative theory of gravity may be applied, play an important role. We here propose Lorentz covariant coordinate conditions for the composite theory of pure gravity developed from the Yang-Mills theory based on the Lorentz group, thereby completing this previously proposed higher derivative theory of gravity. The physically relevant static isotropic solutions are determined by various methods, the high-precision predictions of general relativity are reproduced, and an exact black-hole solution with mildly singular behavior is found. ## I Introduction Only two years after the discovery of Yang-Mills theories Yang and Mills (1954), it has been recognized that that there is a striking formal relationship between the Riemann curvature tensor of general relativity and the field tensor of the Yang-Mills theory based on the Lorentz group Utiyama (1956). However, developing this particular Yang-Mills theory into a consistent and convincing theory of gravity is not at all straightforward. The ideas of Utiyama (1956) have been found to be “unnatural” by Yang (see footnote 5 of Yang (1974)), whose work has later been criticized massively in Chap. 19 of Blagojević and Hehl (2013). Nevertheless, the pioneering work Utiyama (1956) may be considered as the origin of what is now known as gauge gravitation theory Capozziello and De Laurentis (2011); Ivanenko and Sardanashvily (1983). An obvious problem with the Yang-Mills theory based on the Lorentz group is that it has the large number of $48$ degrees of freedom, half of which are physically relevant. One is faced with six four-vector fields satisfying second-order evolution equations. For the pure field theories, the physical degrees of freedom are essentially given by the two transverse components of the four-vector fields, like in electrodynamics with its single vector potential. In view of this enormous number of degrees of freedom we need an almost equally large number of constraints to keep only a few degrees of freedom in a theory of gravity. In other words, we need a structured principle for selecting just a few ones among all the solutions of the Yang-Mills theory based on the Lorentz group. A powerful selection principle can be implemented by means of the tool of composite theories Öttinger (2018a, 2019). The basic idea is to write the gauge vector fields of the Yang-Mills theory in terms of fewer, more fundamental variables and their derivatives. The admission of derivatives in this so-called composition rule implies that the composite theory involves higher than second derivatives. The power of the tool of composite theories results from the fact that, in their Hamiltonian formulations Öttinger (2018a, 2019), the structure of the constraints providing the selection principle is highly transparent. As the composite theory of gravity Öttinger (2020a), just like the underlying Yang-Mills theory, is formulated in a background Minkowski space, the question arises how to characterize the “good” coordinate systems in which the theory may be applied. This characterization should be Lorentz invariant, but not invariant under more general coordinate transformations, that is, it shares the formal properties of coordinate conditions in general relativity. However, the unique solutions obtained from Einstein’s field equations only after specifying coordinate conditions are all physically equivalent, whereas the coordinate conditions in composite gravity characterize physically preferred systems. From a historical perspective, it is remarkable that Einstein in 1914 still believed that the metric should be completely determined by the field equations and, therefore, a generally covariant theory of gravity was not desirable (see Giovanelli (2020) for a detailed discussion). The important task of characterizing the preferred systems in composite gravity is addressed in the present paper. Once it is solved, we can provide a canonical Hamiltonian formulation of composite theory of gravity beyond the weak-field approximation Öttinger (2020b) and we obtain the static isotropic black-hole solution in a proper coordinate system. The structure of the paper is as follows. As a preparatory step, we present the various variables and relations between them (Sec. II) and discuss their gauge transformation behavior (Sec. III). A cornerstone of the development is the close relationship between the covariant derivatives associated (i) with a connection with torsion and (ii) with the Yang-Mills theory based on the Lorentz group. The core of the composite theory of gravity consists of the field equations presented for several sets of variables (Sec. IV) and the coordinate conditions characterizing the admissible coordinate systems (Sec. V). As an application, we determine the static isotropic solutions and provide the results for the high-precision tests of gravity as well as an exact black- hole solution (Sec. VI). We finally offer a detailed summary of our results and draw a number of conclusions (Sec. VII). A number of detailed results and arguments are provided in six appendices. ## II Various variables and relations between them For the understanding of composite theories, it is important to introduce different kinds of variables and to clarify the relations between them. On the one hand, we have the metric tensors, tetrad variables, connections and curvature tensors familiar from general relativity and other theories of gravity. On the other hand, we have the gauge vectors and field tensors of the Yang-Mills theory based on the Lorentz group. An important step is the decomposition of metric tensors in terms of tetrad or _vierbein_ variables, $g_{\mu\nu}=\eta_{\kappa\lambda}\,{b^{\kappa}}_{\mu}{b^{\lambda}}_{\nu}={b^{\kappa}}_{\mu}\,b_{\kappa\nu},$ (1) where $\eta_{\kappa\lambda}=\eta^{\kappa\lambda}$ is the Minkowski metric with signature $(-,+,+,+)$. Throughout this paper, the Minkowski metric is used for raising or lowering space-time indices. For the inverses of the metric and the tetrad variables we introduce the components $\bar{g}^{\mu\nu}$ and $\mbox{$\bar{b}^{\mu}$}_{\kappa}$. Note that they are not obtained by raising or lowering indices of $g_{\mu\nu}$ and ${b^{\kappa}}_{\mu}$, respectively. Equation (1) may be regarded as the characterization of metric tensors by symmetry and definiteness properties. A general metric tensor may also be regarded as the result of transforming the Minkowski metric. The decomposition of a metric $g_{\mu\nu}$ into tetrad variables ${b^{\kappa}}_{\mu}$ is not unique. If we multiply ${b^{\kappa}}_{\mu}$ from the left with any Lorentz transformation, the invariance of the Minkowski metric under Lorentz transformations implies that we obtain another valid decomposition. This observation reveals the origin of the underlying gauge symmetry of the composite theory of gravity. The key role of the metric tensor in the present theory is the characterization of the momentum-velocity relation, so that it can be interpreted as an indication of tensorial properties of mass. While this is also the case in general relativity, Einstein’s theory of gravity goes much further in the geometric interpretation of the metric by assuming that it characterizes the underlying space-time. In contrast, the present theory is developed in an underlying Minkowski space, which is the standard situation for Yang-Mills theories. As a next step, we introduce the vector fields $A_{(\kappa\lambda)\rho}$ in terms of the tetrad variables (the pair $(\kappa,\lambda)$ of space-time indices should be considered as a label associated with the Lorentz group, $\rho$ as a four-vector index), $\displaystyle{b^{\kappa}}_{\mu}{b^{\lambda}}_{\nu}\,A_{(\kappa\lambda)\rho}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\left(\frac{\partial g_{\nu\rho}}{\partial x^{\mu}}-\frac{\partial g_{\mu\rho}}{\partial x^{\nu}}\right)$ (2) $\displaystyle+$ $\displaystyle\frac{1}{2\tilde{g}}\left({b^{\kappa}}_{\mu}\,\frac{\partial b_{\kappa\nu}}{\partial x^{\rho}}-\frac{\partial{b^{\kappa}}_{\mu}}{\partial x^{\rho}}\,b_{\kappa\nu}\right).\qquad$ From the Yang-Mills perspective, $\tilde{g}$ is the coupling constant. From a metric viewpoint, $\tilde{g}\neq 1$ implies torsion (see Eq. (6) below). The antisymmetry of the right-hand side of Eq. (2) in $\mu$ and $\nu$ leads, after resolving for $A_{(\kappa\lambda)\rho}$, to antisymmetry in $\kappa$ and $\lambda$. We have thus introduced six vector fields associated with six pairs $(\kappa,\lambda)$, or with a label $a$ taking the values from $1$ to $6$ according to Table 1. The pairs $(0,1)$, $(0,2)$, $(0,3)$ correspond to Lorentz boosts in the respective directions (involving also time) and the pairs $(2,3)$, $(3,1)$, $(1,2)$ correspond to rotations in the respective planes, as can be recognized by analyzing the gauge transformation behavior of the fields $A_{(\kappa\lambda)\rho}$ resulting from the freedom of acting with Lorentz transformations on ${b^{\kappa}}_{\mu}$ (see Sec. III for details). $a$ \| $1$ \| $2$ \| $3$ \| $4$ \| $5$ \| $6$ ---\|---\|---\|---\|---\|---\|--- $(\kappa,\lambda)$ \| $(0,1)$ \| $(0,2)$ \| $(0,3)$ \| $(2,3)$ \| $(3,1)$ \| $(1,2)$ Table 1: Correspondence between the label $a$ for the base vectors of the six- dimensional Lie algebra ${\rm so}(1,3)$ and ordered pairs $(\kappa,\lambda)$ of space-time indices. Following standard procedures for Yang-Mills theories (see, e.g., Sect. 15.2 of Peskin and Schroeder (1995), Chap. 15 of Weinberg (2005), or Öttinger (2018b)), we can introduce a field tensor in terms of the vector fields, $F_{a\mu\nu}=\frac{\partial A_{a\nu}}{\partial x^{\mu}}-\frac{\partial A_{a\mu}}{\partial x^{\nu}}+\tilde{g}f^{bc}_{a}A_{b\mu}A_{c\nu},$ (3) where $f^{bc}_{a}$ are the structure constants of the Lorentz group. A Lie algebra label, say $a$, can be raised or lowered by raising or lowering the indices in the pairs associated with $a$ according to Table 1. The structure constants can then be specified as follows: $f^{abc}$ is $1$ ($-1$) if $(a,b,c)$ is an even (odd) permutation of $(4,5,6)$, $(1,3,5)$, $(1,6,2)$ or $(2,4,3)$ and $0$ otherwise (see also Eq. (51)). The definition (2) suggests the following general passage from quantities labeled by a Lie algebra index to a quantity with space-time indices, $\tilde{X}_{\mu\nu}={b^{\kappa}}_{\mu}{b^{\lambda}}_{\nu}\,X_{(\kappa\lambda)}.$ (4) One then gets a deep relation between covariant derivatives associated with metrics and connections on the one hand and covariant derivatives associated with a Yang-Mills theory based on the Lorentz group on the other hand (for a proof of this fundamental relation based on the structure of the Lorentz group, see Appendix A), $\displaystyle\frac{\partial\tilde{X}_{\mu\nu}}{\partial x^{\rho}}-\Gamma^{\sigma}_{\rho\mu}\tilde{X}_{\sigma\nu}-\Gamma^{\sigma}_{\rho\nu}\tilde{X}_{\mu\sigma}$ $\displaystyle=$ (5) $\displaystyle{b^{\kappa}}_{\mu}{b^{\lambda}}_{\nu}\left[\frac{\partial X_{(\kappa\lambda)}}{\partial x^{\rho}}+\tilde{g}\,f_{(\kappa\lambda)}^{bc}A_{b\rho}X_{c}\right],$ where the connection $\Gamma^{\rho}_{\mu\nu}$ is given by $\Gamma^{\rho}_{\mu\nu}=\frac{1}{2}\,\bar{g}^{\rho\sigma}\left[\frac{\partial g_{\sigma\nu}}{\partial x^{\mu}}+\tilde{g}\left(\frac{\partial g_{\mu\sigma}}{\partial x^{\nu}}-\frac{\partial g_{\mu\nu}}{\partial x^{\sigma}}\right)\right]=\bar{g}^{\rho\sigma}\,\bar{\Gamma}_{\sigma\mu\nu}.$ (6) Unlike the Christoffel symbols obtained for $\tilde{g}=1$, $\Gamma^{\rho}_{\mu\nu}$ is not symmetric in $\mu$ and $\nu$ for $\tilde{g}\neq 1$. This lack of symmetry indicates the presence of torsion. Note, however, that the connection is metric-compatible for all $\tilde{g}$ Jiménez _et al._ (2019), that is, $\frac{\partial g_{\mu\nu}}{\partial x^{\rho}}-\Gamma^{\sigma}_{\rho\mu}g_{\sigma\nu}-\Gamma^{\sigma}_{\rho\nu}g_{\mu\sigma}=0,$ (7) which can be recast in the convenient form $\frac{\partial g_{\mu\nu}}{\partial x^{\rho}}=\bar{\Gamma}_{\mu\rho\nu}+\bar{\Gamma}_{\nu\rho\mu}.$ (8) From the connection $\Gamma^{\rho}_{\mu\nu}$, we can further construct the Riemann curvature tensor (see, e.g. Jiménez _et al._ (2019) or Weinberg (1972)) ${R^{\mu}}_{\nu\mu^{\prime}\nu^{\prime}}=\frac{\partial\Gamma^{\mu}_{\mu^{\prime}\nu}}{\partial x^{\nu^{\prime}}}-\frac{\partial\Gamma^{\mu}_{\nu^{\prime}\nu}}{\partial x^{\mu^{\prime}}}+\Gamma^{\sigma}_{\mu^{\prime}\nu}\Gamma^{\mu}_{\nu^{\prime}\sigma}-\Gamma^{\sigma}_{\nu^{\prime}\nu}\Gamma^{\mu}_{\mu^{\prime}\sigma}.$ (9) In Appendix B, it is shown that the field tensor (3) can be written in the alternative form $\displaystyle\tilde{F}_{\mu\nu\mu^{\prime}\nu^{\prime}}$ $\displaystyle=$ (10) $\displaystyle\frac{1}{2}\left(\frac{\partial^{2}g_{\nu\nu^{\prime}}}{\partial x^{\mu}\partial x^{\mu^{\prime}}}-\frac{\partial^{2}g_{\nu\mu^{\prime}}}{\partial x^{\mu}\partial x^{\nu^{\prime}}}-\frac{\partial^{2}g_{\mu\nu^{\prime}}}{\partial x^{\nu}\partial x^{\mu^{\prime}}}+\frac{\partial^{2}g_{\mu\mu^{\prime}}}{\partial x^{\nu}\partial x^{\nu^{\prime}}}\right)$ $\displaystyle+\,\frac{1}{\tilde{g}}\,\bar{g}^{\rho\sigma}(\bar{\Gamma}_{\rho\mu^{\prime}\mu}\bar{\Gamma}_{\sigma\nu^{\prime}\nu}-\bar{\Gamma}_{\rho\nu^{\prime}\mu}\bar{\Gamma}_{\sigma\mu^{\prime}\nu}).$ This explicit expression for $\tilde{F}_{\mu\nu\mu^{\prime}\nu^{\prime}}$ reveals its symmetry properties: antisymmetry under $\mu\leftrightarrow\nu$ and $\mu^{\prime}\leftrightarrow\nu^{\prime}$ and, more surprisingly, symmetry under $(\mu\nu)\leftrightarrow(\mu^{\prime}\nu^{\prime})$. A comparison between the expressions (9) and (10) yields a remarkable relationship between the Riemann curvature tensor and the field tensor of the Yang-Mills theory based on the Lorentz group, $\tilde{g}\,\bar{g}^{\mu\rho}\tilde{F}_{\rho\nu\mu^{\prime}\nu^{\prime}}={R^{\mu}}_{\nu\mu^{\prime}\nu^{\prime}},$ (11) which holds for all values of the coupling constant $\tilde{g}$. ## III Gauge transformation behavior As a consequence of the decomposition (1), there exists the gauge freedom of acting with a Lorentz transformation from the left on the tetrad variables ${b^{\kappa}}_{\mu}$. In its infinitesimal version, this possibility corresponds to the transformation $\delta b_{\kappa\mu}=\tilde{g}\,\Lambda_{(\kappa\lambda)}{b^{\lambda}}_{\mu},$ (12) where $\Lambda_{(\kappa\lambda)}$ is antisymmetric in $\kappa$ and $\lambda$ and can hence be understood as $\Lambda_{a}$ according to Table 1. For $\kappa=0$, time is mixed with a spatial dependence in one of the coordinate directions so that we deal with the respective Lorentz boosts. If both $\kappa=k$ and $\lambda=l$ are both spatial indices, the antisymmetric matrix $\Lambda_{(\kappa\lambda)}$ describes rotations in the corresponding $(k,l)$ plane. For the inverse of ${b^{\kappa}}_{\mu}$, Eq. (12) implies $\delta\mbox{$\bar{b}^{\mu}$}_{\kappa}=-\tilde{g}\,\Lambda_{(\kappa\lambda)}\bar{b}^{\mu\lambda}.$ (13) By using Eq. (12) in the composition rule (2), we obtain $\delta A_{(\kappa\lambda)\rho}-\tilde{g}\,\eta^{\kappa^{\prime}\lambda^{\prime}}\Big{[}A_{(\kappa^{\prime}\lambda)\rho}\Lambda_{(\kappa\lambda^{\prime})}-\Lambda_{(\kappa^{\prime}\lambda)}A_{(\kappa\lambda^{\prime})\rho}\Big{]}=\frac{\partial\Lambda_{(\kappa\lambda)}}{\partial x^{\rho}},$ (14) which, by means of Eq. (49), can be written as $\delta A_{a\rho}=\frac{\partial\Lambda_{a}}{\partial x^{\rho}}+\tilde{g}f^{bc}_{a}\,A_{b\rho}\,\Lambda_{c}.$ (15) This result demonstrates that the six vector fields $A_{a\rho}$ indeed possess the proper gauge transformation behavior for the vector fields of the Yang- Mills theory based on the Lorentz group. By means of the Jacobi identity for the structure constants, $f^{sb}_{a}f^{cd}_{s}+f^{sc}_{a}f^{db}_{s}+f^{sd}_{a}f^{bc}_{s}=0,$ (16) we further obtain the gauge transformation behavior of the field tensor, $\delta F_{a\mu\nu}=\tilde{g}f^{bc}_{a}\,F_{b\mu\nu}\,\Lambda_{c}.$ (17) Finally, we look at the gauge transformation behavior obtained for the Yang- Mills variables transformed according to Eq. (4). From Eqs. (12) and (14) we obtain $\delta\tilde{A}_{\mu\nu\rho}={b^{\kappa}}_{\mu}{b^{\lambda}}_{\nu}\,\frac{\partial\Lambda_{(\kappa\lambda)}}{\partial x^{\rho}}.$ (18) As the metric is gauge invariant (gauge degrees of freedom result only from its decomposition), the representations (6) and (10) imply the gauge invariance properties $\delta\Gamma^{\rho}_{\mu\nu}=\delta\bar{\Gamma}_{\sigma\mu\nu}=0,$ (19) and $\delta\tilde{F}_{\mu\nu\mu^{\prime}\nu^{\prime}}=0.$ (20) ## IV Field equations With the help of Eq. (5), the standard field equations for our Yang-Mills theory based on the Lorentz group (see, e.g., Sect. 15.2 of Peskin and Schroeder (1995), Chap. 15 of Weinberg (2005), or Öttinger (2018b)) can be written in the manifestly gauge invariant form $\eta^{\mu^{\prime}\mu^{\prime\prime}}\left(\frac{\partial\tilde{F}_{\mu\nu\mu^{\prime\prime}\nu^{\prime}}}{\partial x^{\mu^{\prime}}}-\Gamma^{\sigma}_{\mu^{\prime}\mu}\tilde{F}_{\sigma\nu\mu^{\prime\prime}\nu^{\prime}}-\Gamma^{\sigma}_{\mu^{\prime}\nu}\tilde{F}_{\mu\sigma\mu^{\prime\prime}\nu^{\prime}}\right)=0.$ (21) By means of Eq. (11), these field equations can be rewritten in terms of the Riemann curvature tensor, $\eta^{\rho\nu^{\prime}}\left(\frac{\partial{R^{\mu}}_{\nu\mu^{\prime}\nu^{\prime}}}{\partial x^{\rho}}+\Gamma^{\mu}_{\rho\sigma}{R^{\sigma}}_{\nu\mu^{\prime}\nu^{\prime}}-\Gamma^{\sigma}_{\rho\nu}{R^{\mu}}_{\sigma\mu^{\prime}\nu^{\prime}}\right)=0.$ (22) In view of Eq. (9), this latter equation is entirely in terms of the variables $\Gamma^{\rho}_{\mu\nu}$. The explicit form of the resulting equation is given in Appendix C. This observation offers the option of the following two-step procedure: one first determines the most general solution of the second-order differential equations (LABEL:compactGameq) for $\Gamma^{\rho}_{\mu\nu}$ and then, in a post-processing step, one obtains the metric by solving the first- order differential equations (6). The post-processing step selects those solutions $\Gamma^{\rho}_{\mu\nu}$ that can actually be expressed in terms of the metric. Finally, we write the field equations directly as third-order differential equations for the metric. As the solutions of these third-order equations can be understood in terms of selected solutions of the Yang-Mills theory found by post-processing, there is no reason to be concerned about the potential instabilities resulting from higher-order differential equations, known as Ostrogradsky instabilities Ostrogradsky (1850); Woodard (2015). Avoiding such instabilities is an important topic, in particular, in alternative theories of gravity j. Chen _et al._ (2013); Raidal and Veermäe (2017); Stelle (1977, 1978); Krasnikov (1987); Grosse-Knetter (1994); Becker _et al._ (2017); Salvio (2019). We write all the third and second derivatives of the metric explicitly, whereas the first derivatives are conveniently combined into connection variables. The result is the following set of equations for the composite theory of gravity obtained by expressing the gauge vector fields of the Yang-Mills theory based on the Lorentz group in terms of the tetrad variables obtained by decomposing a metric, $\displaystyle\Xi_{\mu\nu\mu^{\prime}}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\frac{\partial}{\partial x^{\mu}}\square g_{\mu^{\prime}\nu}-\frac{1}{2}\frac{\partial^{2}}{\partial x^{\mu}\partial x^{\mu^{\prime}}}\frac{\partial g_{\nu\rho}}{\partial x_{\rho}}$ (23) $\displaystyle-\,\frac{1}{2}\Gamma^{\sigma}_{\mu^{\prime}\mu}\bigg{(}\frac{1}{\tilde{g}}\square g_{\sigma\nu}+\frac{\partial}{\partial x^{\nu}}\frac{\partial g_{\sigma\rho}}{\partial x_{\rho}}-\frac{\partial}{\partial x^{\sigma}}\frac{\partial g_{\nu\rho}}{\partial x_{\rho}}\bigg{)}$ $\displaystyle+\,\frac{\eta^{\rho\rho^{\prime}}}{2}\Gamma^{\sigma}_{\rho\nu}\bigg{(}\frac{\partial^{2}g_{\sigma\rho^{\prime}}}{\partial x^{\mu}\partial x^{\mu^{\prime}}}-\frac{\partial^{2}g_{\mu\rho^{\prime}}}{\partial x^{\sigma}\partial x^{\mu^{\prime}}}$ $\displaystyle\hskip 5.0pt+\,2\frac{\partial^{2}g_{\mu\mu^{\prime}}}{\partial x^{\sigma}\partial x^{\rho^{\prime}}}-2\frac{\partial^{2}g_{\sigma\mu^{\prime}}}{\partial x^{\mu}\partial x^{\rho^{\prime}}}-\frac{1}{\tilde{g}}\frac{\partial^{2}g_{\sigma\mu}}{\partial x^{\mu^{\prime}}\partial x^{\rho^{\prime}}}\bigg{)}$ $\displaystyle+\,\frac{\eta^{\rho\rho^{\prime}}}{\tilde{g}}\bigg{[}\Gamma^{\alpha}_{\mu^{\prime}\mu}\bigg{(}2\bar{\Gamma}_{\alpha\rho^{\prime}\beta}+\bar{\Gamma}_{\beta\rho^{\prime}\alpha}\bigg{)}-\Gamma^{\alpha}_{\rho^{\prime}\mu}\bar{\Gamma}_{\alpha\mu^{\prime}\beta}\bigg{]}\Gamma^{\beta}_{\rho\nu}$ $\displaystyle\hskip 80.00012pt-\;\boxed{\mu\leftrightarrow\nu}=0.$ In view of the antisymmetry of $\Xi_{\mu\nu\mu^{\prime}}$ implied by the last line of the above equation, we can assume $\mu<\nu$ so that Eq. (23) provides a total of $24$ equations for the ten components of the symmetric matrix $g_{\mu\nu}$. If we wish to determine the time evolution of the metric from the third-order differential equations (23), we need $30$ initial conditions for the matrix elements $g_{\mu\nu}$ and their first and second time derivatives as well as expressions for the third time derivatives. Closer inspection of the third-order terms in Eq. (23) reveals that the six equations $\Xi_{0mn}=0$ for $m\leq n$ provide the derivatives $\partial^{3}g_{mn}/\partial t^{3}$, but that the remaining equations do not contain any information about $\partial^{3}g_{0\mu}/\partial t^{3}$. Therefore, the remaining $18$ equations constitute constraints for the initial conditions, and we are faced with two tasks: (i) find equations for the time evolution of $g_{0\mu}$, and (ii) show that the constraints are satisfied at all times if they hold initially (or count the additional constraints that need to be satisfied otherwise). It is not at all trivial to find the number of further constraints arising from the dynamic invariance of the constraints contained in Eq. (23). A controlled handling of constraints is more straightforward in a Hamiltonian setting. As the canonical Hamiltonian formulation has been elaborated only in the weak-field approximation Öttinger (2020b), we sketch the generalizations required for the full, nonlinear theory of composite pure gravity in Appendix E. As a conclusion, we expect (at least) four physical degrees of freedom remaining in the field equations (23) for $g_{\mu\nu}$. Note that the Hamiltonian approach also provides the natural starting point for a generalization to dissipative systems. In particular, this approach allows us to formulate quantum master equations Breuer and Petruccione (2002); Weiss (2008); Öttinger (2011); Taj and Öttinger (2015) and to make composite gravity accessible to the robust framework of dissipative quantum field theory Öttinger (2017); Oldofredi and Öttinger (2021). The issue of missing evolution equations is addressed in the subsequent section. As in the weak-field approximation, coordinate conditions characterizing those coordinate systems in which the composite theory of gravity can be applied provide the missing evolution equations. ## V Coordinate conditions As we have assumed an underlying Minkowski space for developing composite gravity, we need to characterize those coordinate systems in which the theory actually holds. These characteristic coordinate conditions should clearly be Lorentz covariant. Furthermore, the coordinate conditions should provide evolution equations for $g_{0\mu}$ because the field equations (23) determine the third-order time derivatives of $g_{mn}$, but not of $g_{0\mu}$. Therefore, the formulation of appropriate coordinate conditions is an important task. The status of coordinate conditions in composite theory is very different from their status in general relativity, where they have no influence on the physical predictions. The coordinate conditions should be a set of four Lorentz covariant equations. An appealing form is given by $\frac{\partial g_{\mu\rho}}{\partial x_{\rho}}=\frac{\partial\phi}{\partial x^{\mu}},$ (24) where the potential $\phi$ is often assumed to be proportional to the trace of the metric. To eliminate the need of specifying a potential, we can write the second-order integrability conditions $\frac{\partial}{\partial x^{\nu}}\frac{\partial g_{\mu\rho}}{\partial x_{\rho}}=\frac{\partial}{\partial x^{\mu}}\frac{\partial g_{\nu\rho}}{\partial x_{\rho}}.$ (25) After taking the derivatives with respect to $x_{\nu}$ and summing over $\nu$, we arrive at the four Lorentz covariant coordinate conditions $\square\frac{\partial g_{\mu\rho}}{\partial x_{\rho}}=K\frac{\partial}{\partial x^{\mu}}\frac{\partial^{2}g_{\rho\sigma}}{\partial x_{\rho}\partial x_{\sigma}},$ (26) actually with $K=1$. Note that it is very appealing to use third-order equations as coordinate conditions because we actually need only expressions for the third time derivatives of $g_{0\mu}$ (stronger, first-order conditions are needed for the Hamiltonian formulation; see Appendix E). For $K=1$, we would obtain such equations for $g_{0m}$, but not for $g_{00}$. This is the reason why we have introduced the factor $K$ in Eq. (26). For any $K\neq 1$, we obtain the desired four evolution equations for $g_{0\mu}$. Formally, we could stick to the first-order conditions (24), but then the potential $\phi$ would be described by the second-order differential equations $\square\phi=K\frac{\partial^{2}g_{\rho\sigma}}{\partial x_{\rho}\partial x_{\sigma}},$ (27) where suitable space-time boundary conditions would be required. Note, however, that for $K\neq 1$, Eqs. (24) and (27) imply $\frac{\partial^{2}g_{\rho\sigma}}{\partial x_{\rho}\partial x_{\sigma}}=0,$ (28) whereas Eq. (26) implies the weaker requirement $\square\frac{\partial^{2}g_{\rho\sigma}}{\partial x_{\rho}\partial x_{\sigma}}=0.$ (29) The coordinate conditions (26) are an essential new ingredient into the composite theory of gravity. Of course, these coordinate conditions take a particularly simple form for $K=0$, which is a possible choice. Alternatively, we could choose $K=\tilde{g}/(1+\tilde{g})$ because we can then express the coordinate conditions as $\frac{\partial^{2}\bar{\Gamma}_{\mu\rho\sigma}}{\partial x_{\rho}\partial x_{\sigma}}=0.$ (30) In the following, we leave the particular choice of $K\neq 1$ open. From a structural point of view, the coordinate conditions (26) have the important advantage that they can be implemented in exactly the same way as the gauge conditions in Yang-Mills theories: one can add a term to the Lagrangian that does not lead to any modification of the field equations, provided that the desired (coordinate or gauge) conditions are imposed as constraints. For the coordinate conditions (26), the additional contribution to the Lagrangian is given in Appendix D. ## VI Static isotropic solution The study of static isotropic solutions of composite gravity is of great importance because these solutions provide the predictions for the high- precision tests of general relativity (deflection of light by the sun, anomalous precession of the perihelion of Mercury, gravitational redshift of spectral lines from white dwarf stars, travel time delay for radar signals reflecting off other planets) and the properties of black holes. Therefore, we here discuss these solutions in great detail. We assume that the static isotropic solutions are of the general form, $g_{\mu\nu}=\left(\begin{matrix}-\beta&0\\\ 0&\alpha\,\delta_{mn}+\xi\,\frac{x_{m}x_{n}}{r^{2}}\end{matrix}\right),$ (31) with inverse $\bar{g}^{\mu\nu}=\left(\begin{matrix}-\frac{1}{\beta}&0\\\ 0&\frac{\delta_{mn}}{\alpha}-\frac{\xi}{\alpha(\alpha+\xi)}\,\frac{x_{m}x_{n}}{r^{2}}\end{matrix}\right),$ (32) where $\alpha$, $\beta$ and $\xi$ are functions of the single variable $r=(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})^{1/2}$. The static isotropic metric (31) is given in terms of the three real-valued functions $\alpha$, $\beta$ and $\xi$. In the original work on the composite theory of gravity (see Sec. V of Öttinger (2020a)), we had parametrized these three functions in terms of only two functions $A$ and $B$: $\alpha=1$, $\beta=B$, and $\xi=A-1$. This particular parametrization corresponds to standard quasi-Minkowskian coordinates. A problem with these quasi-Minkowskian coordinates is that it is unclear how they can be generalized to full coordinate conditions for general metrics. The more general form (31) of the metric is consistent with the coordinate conditions (26). In particular, we do not need to introduce a further function for characterizing the components $g_{0m}$. In general relativity, the form (31) of the metric (with $g_{0m}=0$) can be achieved by shifting time by a function depending on $r$ (see Sec. 8.1 of Weinberg (1972)). Nonzero $g_{0m}$ arise by Lorentz transformation of the metric (31) so that the form (31) belongs to a particularly simple solution of coordinate conditions and field equations. The field equations (23) provide two third-order ordinary differential equations involving all three functions $\alpha$, $\beta$ and $\xi$. For $K\neq 1$, the coordinate conditions (26) lead to another third-order differential equation relating $\alpha$ and $\xi$, which is actually independent of $K$; only for $K=1$, no further condition arises. In the remainder of this section, we solve the three differential equations for our three unknown functions for $K\neq 1$ by various methods. ### VI.1 Robertson expansion The high-precision tests for theories of gravity depend on the behavior of the static isotropic solutions at large distances. We therefore construct the so- called Robertson expansion in terms of $1/r$. One obtains the following results, $\alpha=1+\alpha_{1}\frac{r_{0}}{r}+\alpha_{3}\frac{r_{0}^{3}}{r^{3}}+\ldots,$ (33) $\xi=\xi_{1}\frac{r_{0}}{r}+\ldots,$ (34) and $\beta=1-2\frac{r_{0}}{r}+\big{[}2+(\tilde{g}-1)(\alpha_{1}+\xi_{1})\big{]}\,\frac{r_{0}^{2}}{2r^{2}}+\ldots,$ (35) where all higher terms indicated by $\ldots$ in these Robertson expansions are uniquely determined by the dimensionless parameters $\alpha_{1}$, $\alpha_{3}$, $\xi_{1}$ and the coupling constant $\tilde{g}$. However, $\alpha_{1}$ and $\xi_{1}$ are not independent but rather related by a cubic algebraic equation with a single real solution establishing a one-to-one relation between $\alpha_{1}$ and $\xi_{1}$ (see Appendix F). The parameter $r_{0}$ with dimension of length is determined by the mass at the center creating the static isotropic field, as can be shown by reproducing the limit of Newtonian gravity (see, e.g., Sec. 3.4 of Weinberg (1972)). An obvious strategy for finding the dimensionless parameters is to make sure that the high-precision predictions of general relativity are reproduced. This is achieved by choosing $\alpha_{1}+\xi_{1}=2,\qquad\alpha_{1}=\tilde{g}.$ (36) Imposing a further relation between $\alpha_{1}$ and $\xi_{1}$ is subtle as we have already established the cubic relationship between these parameters given explicitly in Eq. (69). This implies that the first part of Eq. (36) can be satisfied only for particular values of the coupling constant $\tilde{g}$. By using Eq. (36) for eliminating $\alpha_{1}$ and $\xi_{1}$ from Eq. (69), we obtain the following equation for $\tilde{g}$, $(4+4\tilde{g}-\tilde{g}^{2}-5\tilde{g}^{3})(2-\tilde{g})=0.$ (37) Two of the roots of this polynomial equation of degree four are real. In addition to the obvious root $2$, implying $\alpha_{1}=2$ and $\xi_{1}=0$, one finds the further real-valued root $\frac{1}{15}\big{[}(1259+30\sqrt{1509})^{1/3}+(1259-30\sqrt{1509})^{1/3}-1\big{]},$ which is approximately equal to $1.13164$; although closer to unity, this irrational number seems to be less appealing than the integer $2$. Only for these two values of $\tilde{g}$ composite gravity with the coordinate conditions (26) for $K\neq 1$ can reproduce the high-precision predictions of general relativity. Note that the parameter $\alpha_{3}$ in the expansions (33)–(35) remains undetermined as it is the only term among the listed ones that does not affect the high-precision tests of gravity. ### VI.2 Short-distance singularity We next focus on singular behavior at small distances, which we expect to describe black holes. A glance at the field equations (23) reveals that any fixed multiple of a solution is another solution of the field equations. For the “equidimensional” third-order differential equations determining the functions $\alpha$, $\beta$ and $\xi$ of $r$, we assume the following form, $\alpha=\frac{c_{\alpha}}{r^{x}},\qquad\beta=\frac{c_{\beta}}{r^{x}},\qquad\xi=\frac{c_{\xi}}{r^{x}},$ (38) with constants $c_{\alpha}$, $c_{\beta}$, $c_{\xi}$ and an exponent $x$. We further assume that $c_{\alpha}$, $c_{\beta}$, and $x$ are different from zero. For $\tilde{g}=2$, we then find that the field equations and coordinate conditions are equivalent to $c_{\xi}=0$ and $x=1$. For general $\tilde{g}$, one can verify that the values $x=\frac{2}{\tilde{g}},\qquad c_{\xi}=0,$ (39) lead to a static isotropic solution of both field equations and coordinate conditions. Of course, this solution is physically unacceptable as a global solution because it does not converge to the Minkowski metric at large distances. It does, however, characterize the asymptotic singular behavior of physical solutions at short distances. The exponent $x$ given in Eq. (39) speaks strongly in favor of choosing $\tilde{g}=2$ (rather than an irrational value). We then obtain a solution decaying according to a $1/r$ power law, the spatial part of which is a multiple of the three-dimensional unit matrix. ### VI.3 Numerical solution After discussing the static isotropic solutions at large and small distances from the center, we would now like to consider their behavior over the entire range of $r$. In particular, we are interested in the influence of the so far undetermined parameter $\alpha_{3}$ in Eq. (33) on the behavior of the solutions. To explore the full solutions, we solve the field equations and coordinate conditions by numerical integration, starting from a large initial distance $r_{\rm i}$ and then proceeding to smaller values of $r$. Assuming $\tilde{g}=2$, the initial conditions at $r_{\rm i}$ are given by the truncated third-order expansions $\alpha=1+2\frac{r_{0}}{r}+\alpha_{3}\frac{r_{0}^{3}}{r^{3}},$ (40) $\xi=-3\alpha_{3}\frac{r_{0}^{3}}{r^{3}},$ (41) $\beta=1-2\frac{r_{0}}{r}+2\frac{r_{0}^{2}}{r^{2}}-2\frac{r_{0}^{3}}{r^{3}}.$ (42) These expressions do not only provide the values of the coefficient functions at $r_{\rm i}$, but also their first and second derivatives required for solving the third-order differential equations for the functions $\alpha$, $\beta$ and $\xi$ of $r$. The actual numerical solution is performed with an implicit Runge-Kutta scheme of Mathematica. Figure 1: The functions $\beta$ (dashed line) and $\xi$ (continuous lines) characterizing the temporal and off-diagonal components of the isotropic metric (31) obtained from the composite theory for gravity for $\tilde{g}=2$ and $\alpha_{3}=\pm 0.25$. Positive and negative values of $\xi$ correspond to $\alpha_{3}=-0.25$ and $\alpha_{3}=0.25$, respectively. If $r_{\rm i}$ is sufficiently large, that is, in the range of validity of the asymptotic solutions (40)-(42), the numerical solutions are expected to be independent of the choice of $r_{\rm i}$. This expectation is scrutinized in Figure 1. This figure displays the functions $\beta$ and $\xi$ for the values $\alpha_{3}=\pm 0.25$ in the conditions (40), (41). The numerical solutions have been calculated for $r_{\rm i}=50$ and $r_{\rm i}=500$, so that each curve for $\xi$ actually consists of two overlapping curves and the anticipated independence of the results of $r_{\rm i}$ is confirmed. The result for $\beta$ actually consists of four curves, which implies that $\xi$ has remarkably little influence on the function $\beta$ until it touches the $r$ axis. Figure 1 suggests that $\xi$ diverges around the value $r$ at which $\beta$ touches the $r$ axis (and numerical difficulties arise). According to Eqs. (38), (39), $\xi$ must go to zero for small $r$. The real function $\xi$ might actually end in a cusp singularity and develop a complex branch at smaller $r$ that reaches zero at $r=0$ (see Sec. V C of Öttinger (2020a)). Alternatively, $\xi$ might jump from $+\infty$ to $-\infty$, or vice versa, to return as a real function to zero at $r=0$, where it started at large $r$ (this kind of behavior is found for the Schwarzschild solution of general relativity; see Sec. VI.5). To avoid singularities at finite $r$ we from now on assume $\alpha_{3}=0$, for which $\xi(r)$ is found to be identically zero. Note that singularities would be much more alarming in the composite theory of gravity than in general relativity because they cannot be considered as artifacts (“coordinate singularities”) removable by general coordinate transformations. ### VI.4 An exact solution As we have by now fixed the values of the coupling constant ($\tilde{g}=2$) and all the free parameters in the Robertson expansions (33)-(35) ($\alpha_{1}=2$, $\xi_{1}=0$, $\alpha_{3}=0$), there should be a unique static isotropic solution, which is the counterpart of the Schwarzschild solution in general relativity. The Robertson expansions suggest that all higher coefficients $\alpha_{n}$, $\xi_{n}$ for $n\geq 2$ vanish, so that $\alpha$ consists of only two terms and $\xi$ vanishes identically, as already noted in the numerical solutions. Then, a closed-form expression for $\beta$ can be found from the field equation $4r_{0}^{2}\,\beta=r^{4}\left(1+2\frac{r_{0}}{r}\right){\beta^{\prime}}^{2},$ (43) so that we arrive at the complete solution $\alpha=1+2\frac{r_{0}}{r},\quad\xi=0,\quad\beta=\left(2-\sqrt{1+2\frac{r_{0}}{r}}\right)^{2}.$ (44) These functions $\alpha$ and $\beta$ are shown in Figure 2. The present results are qualitatively similar to what was found in previous work on the composite theory of gravity for different coordinate conditions (see Fig. 1 of Öttinger (2020a)). Figure 2: The exact solutions (44) for the functions $\alpha$ and $\beta$ characterizing the diagonal components of the isotropic metric (31) in the composite theory for gravity. Note that $\beta$ is non-negative, vanishes at $r=(2/3)r_{0}$, and that $\sqrt{\alpha}\pm\sqrt{\beta}=2$, where the $+$ sign holds for $r\geq(2/3)r_{0}$ and the $-$ sign for $r\leq(2/3)r_{0}$. The only singularities occur at the origin, and they are of the Newtonian $1/r$ type. The most remarkable feature is that $\beta$ reaches a local minimum at $r=(2/3)r_{0}$, where $\beta$ becomes zero. The observation that the proper time stands still at this distance from the origin is the essence of black- hole behavior in the composite theory of gravity. An interesting consequence of $\beta=0$ is revealed by considering the curvature scalar $R=\bar{g}^{\nu\nu^{\prime}}{R^{\mu}}_{\nu\mu\nu^{\prime}}=\tilde{g}\,\bar{g}^{\mu\mu^{\prime}}\bar{g}^{\nu\nu^{\prime}}\tilde{F}_{\mu\nu\mu^{\prime}\nu^{\prime}}=\tilde{g}\,\mbox{$\bar{b}^{\mu}$}_{\kappa}\mbox{$\bar{b}^{\nu}$}_{\lambda}{F^{(\kappa\lambda)}}_{\mu\nu}.$ (45) For the isotropic solution given in Eq. (44), we find $R=\frac{16r_{0}^{2}}{r^{4}\left(1+2\frac{r_{0}}{r}\right)^{3}\left(\sqrt{1+2\frac{r_{0}}{r}}-2\right)},$ (46) which implies infinite curvature at $r=(2/3)r_{0}$ where $\beta$ vanishes, and a change of sign at that point. This is an important insight because, in the weak-field approximation, the curvature scalar and tensor have been explored for the coupling of gravitational field and matter Öttinger (2020b). If we want to keep geodesic motion of a mass point in a gravitational field, however, the coupling should be done in terms of a scalar or tensor quantity that is given in terms of second-derivatives of the metric and vanishes, at least for the static isotropic metric. In this context, the scalar identity (28) holding for the static isotropic solution might be useful. A tensorial coupling could be based on the following identity for the static isotropic solution, $\frac{\partial^{2}g_{\mu\nu}}{\partial x^{\rho}\partial x_{\rho}}+\frac{1}{2}\bar{g}^{\rho\rho^{\prime}}\,\frac{\partial g_{\mu\nu}}{\partial x^{\rho}}\,\frac{\partial g_{\rho^{\prime}\sigma}}{\partial x_{\sigma}}=\frac{2r_{0}^{2}}{(r+2r_{0})r^{3}}\,\eta_{\mu\nu},$ (47) which implies that the trace-free part of the tensor on the left-hand side vanishes. ### VI.5 Comparison to Schwarzschild solution The Schwarzschild solution of general relativity in harmonic coordinates is given by (see, e.g., Eq. (8.2.15) of Weinberg (1972)) $\alpha=\left(1+\frac{r_{0}}{r}\right)^{2},\quad\xi=\frac{r+r_{0}}{r-r_{0}}\,\frac{r_{0}^{2}}{r^{2}},\quad\beta=\frac{r-r_{0}}{r+r_{0}}.$ (48) We here compare this solution to the static isotropic solution (44) of composite gravity. The functions $\alpha$ in Eqs. (44) and (48) differ by the term $r_{0}^{2}/r^{2}$. This term does not matter for the high-precision tests. Whereas the singularity at the origin is $1/r$ for composite gravity, it is $1/r^{2}$ for general relativity. This observation goes nicely with the exponent $x$ in Eqs. (38), (39), for $\tilde{g}=2$ and $\tilde{g}=1$, respectively, where general relativity corresponds to the torsion-free case $\tilde{g}=1$. Whereas $\xi$ vanishes in composite gravity, it has a singularity at $r=r_{0}$ for the Schwarzschild solution, with a jump from $+\infty$ for $r=r_{0}^{+}$ to $-\infty$ for $r=r_{0}^{-}$. While this may be considered as a coordinate singularity in general relativity, this would not be possible for a theory in Minkowski space. For the high-precision tests, the absence of a $1/r$ contribution to $\xi$ is crucial. Also $\beta$ is remarkably different for the two solutions. Whereas $\beta$ is non-negative in composite gravity, it changes sign at $r_{0}$ for the Schwarzschild solution. Although the two solutions look so different, their truncated third-order expansions (42) coincide. The coincidence of these expansions to order $1/r^{2}$ is crucial for satisfying the high-precision tests. ## VII Summary and conclusions Yang-Mills theories are formulated on a background Minkowski space, and so is the composite theory of gravity that selects a small subset of solutions from the Yang-Mills theory based on the Lorentz group. Such theories are covariant under Lorentz transformations but, unlike general relativity, not under general coordinate transformations. Therefore, it is important to characterize the coordinate systems, in which the composite theory of gravity should be valid, by coordinate conditions. We here propose the Lorentz covariant third- order equations (26) for the metric as appealing coordinate conditions that nicely supplement the third-oder differential equations for the composite theory of gravity. Their alternative formulation in Eqs. (24) and (27) shows that we essentially introduce a potential for the divergence of the metric, where the potential itself satisfies a second-order differential equation. In the original work on composite gravity Öttinger (2020a), no general coordinate conditions were given. The static isotropic solution was determined for quasi-Minkowskian coordinates, which are defined only for solutions of this particular type and do not satisfy the new coordinate conditions. Also the coordinate conditions previously used in the complete Hamiltonian formulation of the linearized theory, or weak-field approximation, of composite gravity Öttinger (2020b) differ from the present proposal. Therefore, previous results are qualitatively similar but quantitatively different from our previous results. The coordinate conditions (26) complete the nonlinear theory of pure composite gravity proposed in Öttinger (2020a). The field equations for pure composite gravity can be expressed in a number of different ways. One option is to solve the field equations of the Yang-Mills theory based on the Lorentz group and, in a post-processing step, select those solutions that can be properly expressed in terms of the derivatives of the tetrad variables obtained by decomposing the metric. Alternatively, one can introduce a gauge-invariant connection with torsion and formulate second-order differential equations entirely in terms of those. One is then interested in the solutions for the connection that can be properly expressed in terms of first derivatives of the metric. A final possibility is to write third-order evolution equations directly for the metric. In the various formulations of the field equations, it is difficult to count the number of degrees of freedom of composite gravity. This difficulty is a consequence of the primary constraints arising from the composition rule of composite theories and serving as a selection principle for the relevant solutions of the underlying Yang-Mills theory. A canonical Hamiltonian formulation on the combined spaces of tetrad and Yang-Mills variables provides the most structured form of both field equations and coordinate conditions. This formulation suggests that composite gravity has four degrees of freedom (whereas the Yang-Mills theory based on the Lorentz group has $24$ degrees of freedom). The Hamiltonian formulation suggests that we deal with two types of constraints: (i) constraints resulting from the composition rule and (ii) gauge constraints. As the former can be handled by Dirac brackets Dirac (1950, 1958a, 1958b) and the latter by the BRST methodology (the acronym derives from the names of the authors of the original papers Becchi _et al._ (1976); Tyutin (1975); see also Nemeschansky _et al._ (1988); Öttinger (2018b)), the path to quantization of composite gravity is clear. This is a major advantage of an approach starting from the class of Yang-Mills theories, which so successfully describe electro-weak and strong interactions and for which quantization is perfectly understood, and imposing Dirac-type constraints. In addition, this background reveals why composite theories, although they are higher derivative theories, are not prone to Ostrogradsky instabilities. The fact that just a few degrees of freedom of the Yang-Mills theory based on the Lorentz group survive in the composite theory of gravity is also reflected in its static isotropic solutions. Its Robertson expansion has two free dimensionless parameters in addition to the Yang-Mills coupling constant. For reproducing the high-precison predictios of general relativity, one of the free parameters and the coupling constant ($\tilde{g}=2$) need to be fixed. The remaining dimensionless parameter can be chosen to avoid singularities at finite distances from the origin. A closed-form solution for the static isotropic metric, which plays the same role in composite gravity as the Schwarzschild solution in general relativity, has been found. The solution displays a $1/r$ singularity at the origin but remains finite at all finite values of $r$. The only remarkable feature is $g_{00}=0$ at a particular distance from the origin, which is of the order of the Schwarzschild radius; for all other values of $r$, we have $g_{00}<0$. This paper develops only the pure theory of gravity. The coupling to matter still needs to be elaborated. For the linearized composite theory of gravity, we had proposed scalar and tensorial coupling mechanisms Öttinger (2020b). As the curvature tensor for the static isotropic metric no longer vanishes for the nonlinear theory, which would lead to a deviation from geodesic motion for a coupling based on the curvature tensor, an alternative scalar [see, e.g., Eq. (28)] or tensor [see, e.g., Eq. (47)] must be identified for the coupling of the gravitational field to the energy-momentum tensor of matter. ###### Acknowledgements. I am grateful for the opportunity to do this work during my sabbatical at the _Collegium Helveticum_ in Zürich. ## Appendix A Relation between covariant derivatives The reformulation of equations for the Yang-Mills theory based on the Lorentz group in the metric language is based on the identity $f_{(\kappa\lambda)}^{bc}B_{b}C_{c}=\eta^{\kappa^{\prime}\lambda^{\prime}}\Big{[}B_{(\kappa^{\prime}\lambda)}C_{(\kappa\lambda^{\prime})}-C_{(\kappa^{\prime}\lambda)}B_{(\kappa\lambda^{\prime})}\Big{]},$ (49) which, in view of the definition (4), can be rewritten in the alternative form ${b^{\kappa}}_{\mu}{b^{\lambda}}_{\nu}f_{(\kappa\lambda)}^{bc}B_{b}C_{c}=\bar{g}^{\rho\sigma}\Big{(}\tilde{B}_{\rho\mu}\tilde{C}_{\sigma\nu}+\tilde{B}_{\rho\nu}\tilde{C}_{\mu\sigma}\Big{)}.$ (50) These remarkably simple identities follow from the form of the structure constants of the Lorentz group. After writing the structure constants in the following explicit form (see Table 1 for the index conventions), $\displaystyle f^{abc}$ $\displaystyle=$ $\displaystyle\eta^{\kappa_{a}\lambda_{c}}\eta^{\kappa_{b}\lambda_{a}}\eta^{\kappa_{c}\lambda_{b}}-\eta^{\kappa_{a}\lambda_{b}}\eta^{\kappa_{b}\lambda_{c}}\eta^{\kappa_{c}\lambda_{a}}$ (51) $\displaystyle+$ $\displaystyle\eta^{\kappa_{a}\kappa_{b}}\big{(}\eta^{\kappa_{c}\lambda_{a}}\eta^{\lambda_{b}\lambda_{c}}-\eta^{\kappa_{c}\lambda_{b}}\eta^{\lambda_{a}\lambda_{c}}\big{)}$ $\displaystyle+$ $\displaystyle\eta^{\kappa_{a}\kappa_{c}}\big{(}\eta^{\kappa_{b}\lambda_{c}}\eta^{\lambda_{a}\lambda_{b}}-\eta^{\kappa_{b}\lambda_{a}}\eta^{\lambda_{b}\lambda_{c}}\big{)}$ $\displaystyle+$ $\displaystyle\eta^{\kappa_{b}\kappa_{c}}\big{(}\eta^{\kappa_{a}\lambda_{b}}\eta^{\lambda_{a}\lambda_{c}}-\eta^{\kappa_{a}\lambda_{c}}\eta^{\lambda_{a}\lambda_{b}}\big{)},$ the result (49) is obtained by straightforward calculation. We can now use Eq. (50) to evaluate the right-hand side of Eq. (5), $\displaystyle{b^{\kappa}}_{\mu}{b^{\lambda}}_{\nu}\left[\frac{\partial X_{(\kappa\lambda)}}{\partial x^{\rho^{\prime}}}+\tilde{g}\,f_{(\kappa\lambda)}^{bc}A_{b\rho^{\prime}}X_{c}\right]$ $\displaystyle=$ $\displaystyle\frac{\partial\tilde{X}_{\mu\nu}}{\partial x^{\rho^{\prime}}}$ (52) $\displaystyle+\,\bar{g}^{\rho\sigma}\bigg{[}\left(\tilde{g}\tilde{A}_{\rho\mu\rho^{\prime}}-{b^{\kappa}}_{\rho}\,\frac{\partial b_{\kappa\mu}}{\partial x^{\rho^{\prime}}}\right)\tilde{X}_{\sigma\nu}$ $\displaystyle+\,\left(\tilde{g}\tilde{A}_{\rho\nu\rho^{\prime}}-{b^{\kappa}}_{\rho}\,\frac{\partial b_{\kappa\nu}}{\partial x^{\rho^{\prime}}}\right)\tilde{X}_{\mu\sigma}\bigg{]}.\qquad$ By using the composition rule (2) we recover the fundamental relationship (5) with the definition (6) of the connection following from $\bar{\Gamma}_{\mu\rho\nu}={b^{\kappa}}_{\mu}\,\frac{\partial b_{\kappa\nu}}{\partial x^{\rho}}-\tilde{g}\tilde{A}_{\mu\nu\rho}.$ (53) ## Appendix B Alternative expression for field tensor From the definitions (3) and (4) and the fundamental relations (5) and (50), we obtain $\displaystyle\tilde{F}_{\mu\nu\mu^{\prime}\nu^{\prime}}$ $\displaystyle=$ $\displaystyle\frac{\partial\tilde{A}_{\mu\nu\nu^{\prime}}}{\partial x^{\mu^{\prime}}}-\Gamma^{\sigma}_{\mu^{\prime}\mu}\tilde{A}_{\sigma\nu\nu^{\prime}}+\Gamma^{\sigma}_{\mu^{\prime}\nu}\tilde{A}_{\sigma\mu\nu^{\prime}}$ (54) $\displaystyle-$ $\displaystyle\frac{\partial\tilde{A}_{\mu\nu\mu^{\prime}}}{\partial x^{\nu^{\prime}}}+\Gamma^{\sigma}_{\nu^{\prime}\mu}\tilde{A}_{\sigma\nu\mu^{\prime}}-\Gamma^{\sigma}_{\nu^{\prime}\nu}\tilde{A}_{\sigma\mu\mu^{\prime}}$ $\displaystyle-$ $\displaystyle\tilde{g}\,\bar{g}^{\rho\sigma}\Big{(}\tilde{A}_{\rho\mu\mu^{\prime}}\tilde{A}_{\sigma\nu\nu^{\prime}}-\tilde{A}_{\rho\nu\mu^{\prime}}\tilde{A}_{\sigma\mu\nu^{\prime}}\Big{)}.\qquad$ By means of Eq. (53), we obtain $\displaystyle\tilde{g}\left(\frac{\partial\tilde{A}_{\mu\nu\nu^{\prime}}}{\partial x^{\mu^{\prime}}}-\frac{\partial\tilde{A}_{\mu\nu\mu^{\prime}}}{\partial x^{\nu^{\prime}}}\right)$ $\displaystyle=$ $\displaystyle\frac{\partial\bar{\Gamma}_{\mu\mu^{\prime}\nu}}{\partial x^{\nu^{\prime}}}-\frac{\partial\bar{\Gamma}_{\mu\nu^{\prime}\nu}}{\partial x^{\mu^{\prime}}}$ (55) $\displaystyle+\,\frac{\partial{b^{\kappa}}_{\mu}}{\partial x^{\mu^{\prime}}}\frac{\partial b_{\kappa\nu}}{\partial x^{\nu^{\prime}}}-\frac{\partial{b^{\kappa}}_{\mu}}{\partial x^{\nu^{\prime}}}\frac{\partial b_{\kappa\nu}}{\partial x^{\mu^{\prime}}},$ and, again Eq. (53), gives $\frac{\partial{b^{\kappa}}_{\mu}}{\partial x^{\mu^{\prime}}}\frac{\partial b_{\kappa\nu}}{\partial x^{\nu^{\prime}}}=\bar{g}^{\rho\sigma}(\bar{\Gamma}_{\rho\mu^{\prime}\mu}+\tilde{g}\tilde{A}_{\rho\mu\mu^{\prime}})(\bar{\Gamma}_{\sigma\nu^{\prime}\nu}+\tilde{g}\tilde{A}_{\sigma\nu\nu^{\prime}}).$ (56) By combining Eqs. (54)–(56), we finally arrive at $\tilde{F}_{\mu\nu\mu^{\prime}\nu^{\prime}}=\frac{1}{\tilde{g}}\bigg{(}\frac{\partial\bar{\Gamma}_{\mu\mu^{\prime}\nu}}{\partial x^{\nu^{\prime}}}-\frac{\partial\bar{\Gamma}_{\mu\nu^{\prime}\nu}}{\partial x^{\mu^{\prime}}}+\bar{\Gamma}_{\sigma\mu^{\prime}\mu}\Gamma^{\sigma}_{\nu^{\prime}\nu}-\bar{\Gamma}_{\sigma\nu^{\prime}\mu}\Gamma^{\sigma}_{\mu^{\prime}\nu}\bigg{)}.$ (57) This expression for the field tensor coincides with the one given in Eq. (10) when the definition (6) of the connection is used. ## Appendix C Field equation for connection By inserting the expression (9) for the Riemann curvature tensor in terms of the connection, the field equation (22) for the composite theory of gravity can be written as a second-order differential equation for the connection, $\displaystyle\frac{\partial^{2}\Gamma^{\mu}_{\mu^{\prime}\nu}}{\partial x_{\rho}\partial x^{\rho}}$ $\displaystyle-$ $\displaystyle\frac{\partial^{2}\Gamma^{\mu}_{\rho\nu}}{\partial x_{\rho}\partial x^{\mu^{\prime}}}+\eta^{\rho\rho^{\prime}}\bigg{[}\Gamma^{\sigma}_{\mu^{\prime}\nu}\frac{\partial\Gamma^{\mu}_{\rho\sigma}}{\partial x^{\rho^{\prime}}}-\Gamma^{\mu}_{\mu^{\prime}\sigma}\frac{\partial\Gamma^{\sigma}_{\rho\nu}}{\partial x^{\rho^{\prime}}}$ $\displaystyle+\,\Gamma^{\mu}_{\rho\sigma}\bigg{(}2\frac{\partial\Gamma^{\sigma}_{\mu^{\prime}\nu}}{\partial x^{\rho^{\prime}}}-\frac{\partial\Gamma^{\sigma}_{\rho^{\prime}\nu}}{\partial x^{\mu^{\prime}}}\bigg{)}-\Gamma^{\sigma}_{\rho\nu}\bigg{(}2\frac{\partial\Gamma^{\mu}_{\mu^{\prime}\sigma}}{\partial x^{\rho^{\prime}}}-\frac{\partial\Gamma^{\mu}_{\rho^{\prime}\sigma}}{\partial x^{\mu^{\prime}}}\bigg{)}\quad\;\;$ $\displaystyle+\,\Gamma^{\mu}_{\rho^{\prime}\sigma}\Gamma^{\sigma}_{\rho\sigma^{\prime}}\Gamma^{\sigma^{\prime}}_{\mu^{\prime}\nu}+\Gamma^{\mu}_{\mu^{\prime}\sigma}\Gamma^{\sigma}_{\rho\sigma^{\prime}}\Gamma^{\sigma^{\prime}}_{\rho^{\prime}\nu}-2\Gamma^{\mu}_{\rho\sigma}\Gamma^{\sigma}_{\mu^{\prime}\sigma^{\prime}}\Gamma^{\sigma^{\prime}}_{\rho^{\prime}\nu}\bigg{]}=0.$ Note that $\eta^{\rho\rho^{\prime}}$ occurs rather than $\bar{g}^{\rho\rho^{\prime}}$, so that there is no need to know the metric for solving this equation. ## Appendix D Modified Lagrangian The Lagrangian for a pure Yang-Mills theory, including a covariant but gauge breaking term for removing degeneracies associated with gauge invariance (the particular form corresponds to the convenient Feynman gauge), is given by $L=-\int\left(\frac{1}{4}F^{a}_{\mu\nu}F_{a}^{\mu\nu}+\frac{1}{2}\frac{\partial A^{a}_{\mu}}{\partial x_{\mu}}\frac{\partial A_{a}^{\nu}}{\partial x^{\nu}}\right)d^{3}x.$ (59) We propose to add the further term $L_{\rm cc}=\frac{1}{2}\int\left(\frac{\partial^{2}g_{\mu\nu}}{\partial x_{\mu}\partial x^{\sigma}}\frac{\partial^{2}{g_{\rho}}^{\nu}}{\partial x_{\rho}\partial x_{\sigma}}-K\frac{\partial^{2}g_{\mu\nu}}{\partial x_{\mu}\partial x_{\nu}}\frac{\partial^{2}g_{\rho\sigma}}{\partial x_{\rho}\partial x_{\sigma}}\right)d^{3}x,$ (60) implying the functional derivative $\frac{\delta L_{\rm cc}}{\delta g_{\mu\nu}}=\frac{\partial}{\partial x_{\mu}}\left(\square\frac{\partial{g_{\rho}}^{\nu}}{\partial x_{\rho}}-K\frac{\partial}{\partial x_{\nu}}\frac{\partial^{2}g_{\rho\sigma}}{\partial x_{\rho}\partial x_{\sigma}}\right),$ (61) which vanishes upon imposing the coordinate conditions (26) as constraints. If the gauge conditions and the coordinate conditions are imposed as constraints, the above modifications of the Lagrangian for the pure Yang-Mills theory have no effect on the field equations. ## Appendix E Hamiltonian formulation For the weak-field approximation of composite gravity, a canonical Hamiltonian formulation with a detailed analysis of all constraints has been given in Öttinger (2020b). We here sketch how that approach can be generalized to a full, nonlinear theory of pure gravity selected from the Yang-Mills theory based on the Lorentz group. The underlying space of the Hamiltonian formulation consists of the tetrad variables ${b^{\kappa}}_{\mu}$ and the gauge vector fields $A_{a\nu}$ associated with the Lorentz group as configurational variables, together with their conjugate momenta ${p_{\kappa}}^{\mu}$ and $E^{a\nu}$ (where $E_{aj}=F_{aj0}$ and $E_{a0}=\partial A_{a\mu}/\partial x_{\mu}$) Öttinger (2019, 2020b). This space consists of $80$ fields, but massive constraints arise from the composition rule and gauge invariance so that, in the end, the composite theory of pure gravity turns out to possess only four degrees of freedom. The generalization of the Hamiltonian (25)–(27) of Öttinger (2020b) is obtained by introducing the Hamiltonian for the full, nonlinear version of Yang-Mills theory, $\displaystyle H_{\rm pure}$ $\displaystyle=$ $\displaystyle\int\bigg{[}\frac{1}{2}E^{a\mu}E_{a\mu}+\frac{1}{4}F_{aij}F^{aij}-E^{a0}\frac{\partial A_{aj}}{\partial x_{j}}$ (62) $\displaystyle-\,E^{aj}\left(\frac{\partial A_{a0}}{\partial x^{j}}+\tilde{g}f_{a}^{bc}A_{bj}A_{c0}\right)+\mbox{$\dot{b}^{\kappa}$}_{\mu}\,{p_{\kappa}}^{\mu}\bigg{]}d^{3}x,\qquad$ where the functional form of the $16$ time derivatives $\mbox{$\dot{b}^{\kappa}$}_{\mu}$ in terms of the configurational variables ${b^{\kappa}}_{\mu}$ and $A_{a\nu}$ is obtained from $12$ components of the composition rule (2) and the four coordinate conditions (24) (the potential $\phi$ is assumed to be a functional of $g_{\mu\nu}$). For pure gravity without external sources, we can impose the $16$ constraints ${p_{\kappa}}^{\mu}=0$ so that the composite theory consists of selected solutions of the Yang-Mills theory based on the Lorentz group Öttinger (2019, 2020b). The terms involving $E^{a0}$ in the Hamiltonian (62) are associated with the gauge breaking term in the Lagrangian (59). Of course, this Hamiltonian implies the canonical evolution equations for the entire set of $80$ fields. The generalization of the weak-field approximation becomes particularly simple if we introduce the following variables eliminating the nonlinear effects of the coupling constant, $\breve{A}_{\mu\nu\rho}=\tilde{A}_{\mu\nu\rho}-\frac{1}{2\tilde{g}}\,\Omega_{\mu\nu/\rho},$ (63) with $\Omega_{\mu\nu/\rho}={b^{\kappa}}_{\mu}\,\frac{\partial b_{\kappa\nu}}{\partial x^{\rho}}-\frac{\partial{b^{\kappa}}_{\mu}}{\partial x^{\rho}}\,b_{\kappa\nu},$ (64) and $\breve{E}_{\mu\nu 0}=\tilde{E}_{\mu\nu 0}-\frac{\eta^{\rho\rho^{\prime}}}{2\tilde{g}}\Big{(}\Gamma^{\sigma}_{\rho\nu}\bar{\Gamma}_{\mu\rho^{\prime}\sigma}-\Gamma^{\sigma}_{\rho\mu}\bar{\Gamma}_{\nu\rho^{\prime}\sigma}\Big{)},$ (65) $\breve{E}_{\mu\nu j}=\tilde{E}_{\mu\nu j}-\frac{1}{\tilde{g}}\bigg{(}\Gamma^{\sigma}_{j\mu}\bar{\Gamma}_{\sigma 0\nu}-\Gamma^{\sigma}_{j\nu}\bar{\Gamma}_{\sigma 0\mu}\bigg{)},$ (66) as further modifications of the variables $\tilde{A}_{\mu\nu\rho}$ and $\tilde{E}_{\mu\nu\rho}$ defined in Eq. (4). For example, the composition rule (2) takes the linear form $\breve{A}_{\mu\nu\rho}=\frac{1}{2}\left(\frac{\partial g_{\nu\rho}}{\partial x^{\mu}}-\frac{\partial g_{\mu\rho}}{\partial x^{\nu}}\right),$ (67) which corresponds to Eq. (7) of Öttinger (2020b) in the symmetric gauge and includes $12$ primary constraints. Also the evolution equations for $\breve{A}_{\mu\nu\rho}$ and hence also the $12$ secondary constraints keep the same form as in the linearized theory (cf. Eqs. (39), (40) and (46), (47) of Öttinger (2020b)). The $12$ tertiary constraints can be obtained by acting with the operator $\square$ on the primary constraints. The invariance of the tertiary constraints follows from ${p_{\kappa}}^{\mu}=0$. In order to verify the above statements, one needs the identity $\frac{\partial\Omega_{\mu\nu/\rho}}{\partial x_{\rho}}-\eta^{\rho\rho^{\prime}}\left(\Gamma^{\sigma}_{\rho\mu}\Omega_{\sigma\nu/\rho^{\prime}}+\Gamma^{\sigma}_{\rho\nu}\Omega_{\mu\sigma/\rho^{\prime}}\right)=0,$ (68) which is the counterpart of Eq. (16) of Öttinger (2020b) and can be inferred from the gauge invariance of the left-hand side of Eq. (68). Finally, the $24$ evolution equations for $\breve{E}_{\mu\nu\rho}$ correspond to the field equations given in various forms in Sec. IV. As the structure of the Hamiltonian and the constraints for the full, nonlinear theory is so similar (mostly even formally identical) to the case of the linear weak-field approximation, we expect the same count of $24+3\cdot 12+16=76$ constraints for $2\cdot(16+24)=80$ variables. Half of the $24$ constraints associated with gauge invariance result from the gauge conditions $E_{a0}=\partial A_{a\mu}/\partial x_{\mu}=0$, which establish a relationship between the (unphysical) temporal and longitudinal modes of the four-vector potentials. The above arguments suggest that pure composite gravity possesses (at least) four physical degrees of freedom, just as in the thoroughly elaborated special case of the weak-field approximation Öttinger (2020b). ## Appendix F A cubic equation The coefficients $\alpha_{1}$ and $\xi_{1}$ in the Robertson expansions (33), (34) are related by the following cubic equation, $\displaystyle 10\xi_{1}^{3}$ $\displaystyle+$ $\displaystyle 10\tilde{g}\xi_{1}(4\alpha_{1}^{2}+5\alpha_{1}\xi_{1}+2\xi_{1}^{2})$ (69) $\displaystyle-$ $\displaystyle 5\tilde{g}^{2}\big{[}4\xi_{1}-(\alpha_{1}+\xi_{1})(8\alpha_{1}^{2}+9\alpha_{1}\xi_{1}-\xi_{1}^{2})\big{]}$ $\displaystyle-$ $\displaystyle 5\tilde{g}^{3}\big{[}4+3(\alpha_{1}+\xi_{1})^{2}\big{]}(3\alpha_{1}+2\xi_{1})$ $\displaystyle+$ $\displaystyle\tilde{g}^{4}(\alpha_{1}+\xi_{1})(36+11(\alpha_{1}+\xi_{1})^{2})=0.$ Its only real solution for $\alpha_{1}$ in terms of $\xi_{1}$ is given by $\displaystyle\alpha_{1}$ $\displaystyle=$ $\displaystyle\Big{[}\Big{(}w_{3}+\sqrt{w_{3}^{2}-w_{2}^{3}}\Big{)}^{1/3}+w_{2}\Big{(}w_{3}+\sqrt{w_{3}^{2}-w_{2}^{3}}\Big{)}^{-1/3}$ $\displaystyle-\,\xi_{1}(40+85\tilde{g}-120\tilde{g}^{2}+33\tilde{g}^{3})\Big{]}/\big{[}3\tilde{g}(40-45\tilde{g}+11\tilde{g}^{2})\big{]},$ with $\displaystyle w_{2}$ $\displaystyle=$ $\displaystyle 36\tilde{g}^{3}\big{(}200-345\tilde{g}+190\tilde{g}^{2}-33\tilde{g}^{3}\big{)}$ (71) $\displaystyle+$ $\displaystyle 5\big{(}320+160\tilde{g}-85\tilde{g}^{2}-282\tilde{g}^{3}+111\tilde{g}^{4}\big{)}\xi_{1}^{2},\qquad$ and $\displaystyle w_{3}$ $\displaystyle=$ $\displaystyle-5\xi_{1}\Big{[}108\tilde{g}^{4}\big{(}520-985\tilde{g}+633\tilde{g}^{2}-155\tilde{g}^{3}+11\tilde{g}^{4})$ (72) $\displaystyle+$ $\displaystyle\big{(}12800+52800\tilde{g}-82200\tilde{g}^{2}-10735\tilde{g}^{3}$ $\displaystyle+63045\tilde{g}^{4}-33273\tilde{g}^{5}+5427\tilde{g}^{6}\big{)}\xi_{1}^{2}\Big{]}.$ ## References * Yang and Mills (1954) C. 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See Tesi_RossiP-twoside-frn.pdf ## Abstract Equivariant localization theory is a powerful tool that has been extensively used in the past thirty years to elegantly obtain exact integration formulas, in both mathematics and physics. These integration formulas are proved within the mathematical formalism of equivariant cohomology, a variant of standard cohomology theory that incorporates the presence of a symmetry group acting on the space at hand. A suitable infinite-dimensional generalization of this formalism is applicable to a certain class of Quantum Field Theories (QFT) endowed with _supersymmetry_. In this thesis we review the formalism of equivariant localization and some of its applications in Quantum Mechanics (QM) and QFT. We start from the mathematical description of equivariant cohomology and related localization theorems of finite-dimensional integrals in the case of an Abelian group action, and then we discuss their formal application to infinite-dimensional path integrals in QFT. We summarize some examples from the literature of computations of partition functions and expectation values of supersymmetric operators in various dimensions. For 1-dimensional QFT, that is QM, we review the application of the localization principle to the derivation of the Atiyah- Singer index theorem applied to the Dirac operator on a twisted spinor bundle. In 3 and 4 dimensions, we examine the computation of expectation values of certain Wilson loops in supersymmetric gauge theories and their relation to 0-dimensional theories described by “matrix models”. Finally, we review the formalism of non-Abelian localization applied to 2-dimensional Yang-Mills theory and its application in the mapping between the standard “physical” theory and a related “cohomological” formulation. ## Acknowledgments First and foremost, I would like to express my sincere gratitude to professor Diego Trancanelli, who supervised me during the draft of this thesis. I thank him for his constant availability, his valuable advice and his sincere interest for my learning process, as well as his great patience in meticulously reviewing my work step by step. It is also a pleasure for me to thank professor Olindo Corradini, who initiated me to the wonderful world of QFT with two brilliant courses, and was always available to discuss and answer my questions. I would like to thank all the friends and colleagues that grew up with me in this journey. Even though our paths and interests separated, they have always been a precious source of inspiration to me. A heartfelt thanks goes to Giulia, whose presence alone makes everything easier. Finally I would like to thank my family, for the love and support, no matter what, during all these years. ###### Contents 1. Abstract 2. 1 Introduction 3. 2 Equivariant cohomology 1. 2.1 A brief review of standard cohomology theory 2. 2.2 Group actions and equivariant cohomology 3. 2.3 The Weil model and equivariant de Rham’s theorem 4. 2.4 The Cartan model 5. 2.5 The BRST model 4. 3 Localization theorems in finite-dimensional geometry 1. 3.1 Equivariant localization principle 2. 3.2 The ABBV localization formula for Abelian actions 3. 3.3 Equivariant cohomology on symplectic manifolds 1. 3.3.1 Pills of symplectic geometry 2. 3.3.2 Equivariant cohomology for Hamiltonian systems 5. 4 Supergeometry and supersymmetry 1. 4.1 Gradings and superspaces 1. 4.1.1 Definitions 2. 4.1.2 Integration 2. 4.2 Supergeometric proof of ABBV formula for a circle action 3. 4.3 Introduction to Poincaré-supersymmetry 1. 4.3.1 Super-Poincaré algebra and superspace 2. 4.3.2 Chiral superspace and superfields 3. 4.3.3 Supersymmetric actions and component field expansion 4. 4.3.4 R-symmetry 5. 4.3.5 Supersymmetry multiplets 6. 4.3.6 Euclidean 3d N=2 supersymmetric gauge theories 7. 4.3.7 Euclidean 4d N=4,2,2* gauge theories 4. 4.4 From flat to curved space 1. 4.4.1 Coupling to background SUGRA 2. 4.4.2 Supercurrent multiplets and metric multiplets 3. 4.4.3 N=2 gauge theories on the round 3-sphere 4. 4.4.4 N=4,2,2* gauge theories on the round 4-sphere 5. 4.4.5 Trial and error method 5. 4.5 BRST cohomology and equivariant cohomology 6. 5 Localization for circle actions in supersymmetric QFT 1. 5.1 Localization principle in Hamiltonian QM 2. 5.2 Localization and index theorems 3. 5.3 Equivariant structure of supersymmetric QFT and supersymmetric localization principle 4. 5.4 Localization of N=4,2,2* gauge theory on the 4-sphere 1. 5.4.1 The action and the supersymmetric Wilson loop 2. 5.4.2 Quick localization argument 3. 5.4.3 The equivariant model 4. 5.4.4 Localization formulas 5. 5.4.5 The Matrix Model for N=4 SYM 5. 5.5 Localization of N=2 Chern-Simons theory on the 3-sphere 1. 5.5.1 Matter-coupled N=2 Euclidean SCS theory on the 3-sphere 2. 5.5.2 The supersymmetric Wilson loop 3. 5.5.3 Localization: gauge sector 4. 5.5.4 Localization: matter sector 5. 5.5.5 The ABJM matrix model 7. 6 Non-Abelian localization and 2d YM theory 1. 6.1 Prelude: moment maps and YM theory 2. 6.2 A localization formula for non-Abelian actions 3. 6.3 “Cohomological” and “physical” YM theory 4. 6.4 Localization of 2-dimensional YM theory 8. 7 Conclusion 9. A Some differential geometry 1. A.1 Principal bundles, basic forms and connections 2. A.2 Spinors in curved spacetime 10. B Mathematical background on equivariant cohomology 1. B.1 Equivariant vector bundles and equivariant characteristic classes 2. B.2 Universal bundles and equivariant cohomology 3. B.3 Fixed point sets and Borel localization 4. B.4 Equivariant integration and Stokes’ theorem ## Chapter 1 Introduction Quantum Field Theory (QFT) is the framework in which modern theoretical physics describes fundamental interactions between elementary particles and it is also central in the study of condensed matter physics and statistical mechanics. QFT has made the most precise predictions ever in the history of science and it has been tested against a huge amount of experimental data. Nowadays, the most useful formulation of QFT is made in terms of _path integrals_ , which are integrals over the space of all possible field configurations. A “field” mathematically speaking can be thought roughly as a function over spacetime, so these integrals are computed over functional spaces, that are infinite-dimensional. This makes the exact computation of such objects a complicated task, except for some very special cases, and in fact their precise mathematical formulation is still an open problem. Despite these formal difficulties, many interesting results can be extracted from these objects, that can describe _partition functions_ and _expectation values_ of physical observables in QFT. The favorite approach to deal with such computations is perturbation theory, applied in the case in which the QFT is weakly coupled. In this regime, one can compute approximately the expectation values in a perturbative expansion, order by order in the coupling constant. This method, applied to the computation of the partition function, is the infinite-dimensional analogous of a “saddle-point” or “stationary- phase” approximation. Intuitively it represents a semi-classical approach to the quantum dynamics. There are however many cases in which perturbation theory is not applicable, mainly when the QFT is strongly coupled, i.e. the coupling constant is of order 1. This is not a very rare situation. For example we know that one of the fundamental interactions of the Standard Model of particle physics, the strong nuclear force, is well described by “quantum chromodynamics” (QCD), a QFT that is strongly coupled at low energies (so in the “phenomenological” regime). Understanding the behavior of QFT in the strong coupling regime is then a major problem from the physical point of view, and one is lead to develop techniques that permit to study the path integral in a non- perturbative approach. In this thesis we describe features of one of these techniques, that has been exploited in the last few decades for a class of special QFTs, those who exhibit some kind of _supersymmetry_. This technique is called _supersymmetric localization_ , or _equivariant localization_ , or simply localization. Its name derives from the fact that, when one is able to use this method, the path integral of the QFT at hand simplifies (so “localizes”) to an integral over a smaller domain, sometimes even a finite-dimensional integral over constant field configurations. This result can be viewed as an exact stationary-phase approximation, so that the full quantum spectrum of the localized theory is completely determined by its semi-classical limit. Without entering in the technical details of this localization phenomenon, we just point out that this method fundamentally relies on the presence of a large amount of symmetry of the theory, that can be described by the presence of a _group action_ on the space of fields. When the space of fields is graded, i.e. there is a distinction between “bosonic” and “fermionic” degrees of freedom, the symmetry group action can exchange these two types of fields and in this case it is called _super_ symmetry. This situation arises mainly in BRST-fixed and topological field theories, where the grading is regarded as the _ghost number_ , and in Poincaré-supersymmetric QFT, where the grading distinguishes between bosons and fermions in the standard sense of particle physics. In both cases, the action of a supersymmetry transformation “squares” to a canonical (bosonic) one, i.e. a gauge transformation or a Poincaré transformation. It is reasonable that such a huge amount of symmetry can simplify the dynamics of the theory, but it can be non-trivial a priori how to translate this in a simplification of the path integral. At this point, one wishes to understand if there is a theoretical framework that allows to systematically understand why and when such a drastic simplification of the path integral can occur, and at which level this is related to (super-)symmetries in QFT. To be mathematically more rigorous, we can think in terms of integration over finite-dimensional spaces, and then try to extrapolate and generalize the important results to the infinite- dimensional case. Integrals of differential forms over manifolds are built up technically from the smooth (so local) structure of the space, but it is a well-known fact that their result can describe and is regulated by topological (so global) properties. It is a consequence of de Rham’s theorem and Stokes’ theorem that they really depend on the _cohomology class_ of the integrand and not on the particular differential form that represents the class. It is thus reasonable that a theory of integration that embeds the presence of a symmetry group action should arise from a topological construction. Indeed, the mathematical framework in which the localization formulas were firstly derived is a suitable modification of the standard cohomology theory. This is called _equivariant cohomology_. As de Rham’s theorem relates the usual cohomology to differential forms on a smooth manifold, equivariant cohomology can be associated to a modification of them, called _equivariant differential forms_. Equivariant cohomology theory was initiated in the mathematical literature by Cartan, Borel and others during the 50’s [1, 2, 3, 4], but the first instance of a localization formula was presented by Duistermaat and Heckman in 1982 [5]. In this paper, they proved the exactness of a stationary-phase approximation in the context of symplectic geometry and Hamiltonian Abelian group actions. Subsequently, Atiyah and Bott realized that the Duistermaat- Heckman localization formula can be viewed as a special case of a more general theorem, that they proved in the topological language of equivariant cohomology [6]. Almost at the same time, Berline and Vergne derived an analogous localization formula valid for Killing vectors on general compact Riemannian manifolds [7]. Roughly, the Atiyah-Bott-Berline-Vergne (ABBV) formula says that the integral over a manifold that is acted upon by an Abelian group localizes as a sum of contributions arising only from the fixed points of the group action. The first infinite-dimensional generalization of this localization formula was given soon after by Atiyah and Witten in 1985 [8], applied to supersymmetric Quantum Mechanics (QM). This turns out to be an example of topological theory, since their localization formula relates the partition function to the index of a Dirac operator. Many generalizations of this approach followed, first mainly in the context of topological field theories. Here localization allows to get closed formulas relating partition functions of physical QFTs to topological invariants of the spaces where they live. In all these cases, the BRST cohomology is interpreted as the equivariant structure of the theory, and is responsible for the localization of the path integral. In recent years, the formal application of the ABBV localization formula for Abelian symmetry actions was employed in the context of Poincaré- supersymmetric theories, whose explicit supersymmetry is an extension of the spacetime symmetry that is common to all QFTs. In this case, the equivariant structure is generated by the cohomology of a _supersymmetry charge_ , that acts as a differential on the space of Poincaré symmetric field configurations. This formal structure of supersymmetric field theories was firstly realized by Niemi, Palo and Morozov in the 90’s [9, 10]. Starting from the work of Pestun in 2007 [11], localization has been applied to the computation of partition functions and expectation values of supersymmetric operators on curved compact manifolds [12]. In many of these cases, the partition function localizes to a finite-dimensional integral over matrices, a so-called _matrix model_. To carry out this procedure, a number of technical difficulties have to be overcome, the most urgent being to understand how to define Poincaré-supersymmetric theory on curved spaces. Nowadays there is a well-defined and well-understood procedure that allows to do that, essentially deforming an original theory defined in flat space through the coupling to a non-trivial “rigid” supersymmetric background. This in general reduces the degree of supersymmetry of the original theory, but if some of it is preserved on the new background then one is able in principle to perform localization. Beside the general and abstract motivation of understanding strongly coupled QFT, the importance of these computations can be viewed in a string theory perspective, and in particular as possible tests of the so called _AdS/CFT correspondence_ [13]. Some generalizations of the ABBV formula for non-Abelian group actions have been proposed both in the mathematical and physical literature. The first generalization of the Duistermaat-Heckman theorem to non-Abelian group actions was presented by Guillemin and Prato [14], restricting the localization principle to the action of the maximal Abelian subgroup. An infinite- dimensional generalization of the localization principle was proposed by Witten in 1992 [15], applied to the study of 2-dimensional Yang-Mills theory, and a more rigorous proof appeared in the mathematical literature in 1995 by Jeffrey and Kirwan [16]. Of course these localization formulas find many interesting applications not only in physics but also in pure mathematics, but this side of the story is far from the purpose of this work. ### An exact saddle-point approximation To give a feeling of what we mean by “exact saddle-point approximation”, we present a very simple but instructive example here, in the finite-dimensional setting. The saddle-point (or stationary-phase) method is applied to oscillatory integrals of the form $I(t)=\int_{-\infty}^{+\infty}dx\ e^{itf(x)}g(x),$ (1.1) when one is interested in the asymptotic behavior of $I(t)$ at positive large values of the real parameter $t$. In this limit, the integral is dominated by the critical points of $f(x)$, where its first derivative vanishes and it can be expanded in Taylor series as $f(x)=f(x_{0})+\frac{1}{2}f^{\prime\prime}(x_{0})(x-x_{0})^{2}+\cdots.$ (1.2) If $F\subset\mathbb{R}$ is the set of critical points, that for simplicity we assume to be discrete, the leading contribution to (1.1) is then given by a Gaussian integral, $\displaystyle I(t)$ $\displaystyle\approx\sum_{x_{0}\in F}g(x_{0})e^{itf(x_{0})}\int_{-\infty}^{+\infty}dx\ e^{\frac{it}{2}f^{\prime\prime}(x_{0})(x-x_{0})^{2}}$ (1.3) $\displaystyle=\sum_{x_{0}\in F}g(x_{0})e^{itf(x_{0})}e^{\frac{i\pi}{4}\mathrm{sign}(f^{\prime\prime}(x_{0}))}\sqrt{\frac{2\pi}{t\|f^{\prime\prime}(x_{0})\|}}.$ If the integral is performed over $\mathbb{R}^{n}$, this last formula generalizes easily to $I(t)\approx\left(\frac{2\pi}{t}\right)^{n/2}\sum_{x_{0}\in F}g(x_{0})e^{itf(x_{0})}\frac{e^{\frac{i\pi}{4}\sigma(x_{0})}}{\|\det(\mathrm{Hess}_{f}(x_{0}))\|^{1/2}},$ (1.4) where $\mathrm{Hess}_{f}(x)$ is the matrix of second derivatives of $f$ at $x\in\mathbb{R}^{n}$, and $\sigma(x)$ denotes the sum of the signs of its eigenvalues. Of course, there is no reason for the RHS of (1.4) to be the exact answer for $I(t)$, but the claimed property of localization is that in some cases this turns out to be true! To see this, let us consider the integration over the 2-sphere $\mathbb{S}^{2}$, defined by its embedding in $\mathbb{R}^{3}$ as the set of points whose distance from the origin is 1. For this example we chose $f(x,y,z)=z$, the “height function”, and $g(x,y,z)=1$. The resulting oscillatory integral is then $I(t)=\int_{\mathbb{S}^{2}}dA\ e^{itz},$ (1.5) where $dA$ is the volume form on the sphere, normalized such that $\int_{\mathbb{S}^{2}}dA=4\pi$. The critical points of the height function are the North and the South poles, where $z\approx\pm\left(1-\frac{1}{2}(x^{2}+y^{2})\right).$ (1.6) The volume form at the poles is just $dA=dxdy$, so if we apply the saddle- point approximation to (1.5) we get $\displaystyle\int_{\mathbb{S}^{2}}dA\ e^{itz}$ $\displaystyle\approx e^{it}\int dxdy\ e^{-\frac{it}{2}(x^{2}+y^{2})}+e^{-it}\int dxdy\ e^{\frac{it}{2}(x^{2}+y^{2})}$ (1.7) $\displaystyle=\frac{2\pi}{it}e^{it}-\frac{2\pi}{it}e^{-it}$ $\displaystyle=4\pi\frac{\sin(t)}{t}.$ Now, since this integral is rather easy, in this case we can actually compare the result of the saddle-point approximation with its exact value. Using spherical coordinates, $I(t)=\int_{-1}^{+1}d\cos(\theta)\int_{0}^{2\pi}d\varphi\ e^{it\cos(\theta)}=4\pi\frac{\sin(t)}{t}.$ (1.8) As promised, the result coincides with the stationary-phase result (1.7). This is the simplest example of equivariant localization! The scope of the first chapters of the thesis is to describe in general the structure underlying this result, how it can be related to symmetry properties of the specific function and space under consideration, and the localization theorems that generalize this specific computation to possibly more complicated examples. In the remaining part, we will deal instead with examples of the infinite-dimensional analog of this exact stationary-phase approximation. ### Structure of the thesis The aim of this project is to summarize some results in the context of equivariant localization applied to physics. We will draw a line from the first mathematical results concerning the theory of equivariant cohomology and associated powerful localization theorems of finite-dimensional integrals, to the generalization to path integration in quantum mechanical models and finally to applications in quantum field theories. One of the purposes of this work is to provide a suitable reference for other students in theoretical and mathematical physics with a background at the level of a master degree, who are interested in approaching the subject. In this spirit, we will try to expose the material in a pedagogical order, being as self-contained as possible, and otherwise giving explicit references to background material. In Chapter 2, we will review the basics of equivariant cohomology theory, starting from its construction in algebraic topology. After having recalled some notions of basic homology and cohomology, we will introduce group actions, and define equivariant cohomology with the so-called _Borel construction_. Then, we will describe the most common algebraic models that generalize de Rham’s theorem in the equivariant setup, the _Weil_ , _Cartan_ and _BRST models_. These give a description of equivariant cohomology in terms of a suitable modification of the complex of differential forms. In Chapter 3, we will describe the common rationale behind the localization property of equivariant integrals in finite-dimensional geometry, the so- called _equivariant localization principle_. Then we will state and explain the Abelian localization formula derived by Atiyah-Bott and Berline-Vergne. In the final part of the chapter, we will connect the discussion with the context of symplectic geometry that, as we will see, can be rephrased in terms of equivariant cohomology. We will review the basic notions of symplectic manifolds, symmetries and Hamiltonian systems, and then state the Duistermaat- Heckman localization formula as a special case of the ABBV theorem. Chapter 4 can be viewed as a long technical aside. Here we will review some notions about supergeometry that are needed to understand a proof of the ABBV theorem, and then specialize the discussion to Poincaré-supersymmetric theories. We will discuss their construction from the perspective of superspace, give some practical examples, and then generalize their description over general curved backgrounds. This is achieved by coupling the given theory to the supersymmetric version of Einstein gravity, _supergravity_ , and then requiring the gravitational sector of the resulting theory to decouple from the rest in a “rigid limit”, analogous to $G_{N}\to 0$. This method will bring up to the notion of _Killing spinors_ , a special type of spinorial fields whose existence ensures the preservation of some supersymmetry on the curved background. Finally, we will comment about a possible “super-interpretation” of the models of equivariant cohomology described in Chapter 2, that connects them to the usual BRST formalism for quantization of constrained Hamiltonian systems. In Chapter 5 we will discuss examples of Abelian supersymmetric localization of path integrals in the infinite-dimensional setting of QFT. The first case we will report is 1-dimensional, i.e. QM. In an Hamiltonian formulation on phase space, we will describe how it is possible to give a supersymmetric (so equivariant) interpretation to the path integral in a model-independent way, the supersymmetry arising as a “hidden” BRST symmetry which is linked to the Hamiltonian dynamics. This results in a localization of the path integral over the space of classical field configurations or over constant field configurations, that applied to supersymmetric QM gives an alternative proof the Atiyah-Singer index theorem for the Dirac operator over a twisted spinor bundle. Next, we will review two modern applications of the localization principle to the computation of expectation values of _Wilson loop operators_ in supersymmetric gauge theories. The first application computes the expectation value in $\mathcal{N}=4,2,2^{}$ Super Yang-Mills theory on the 4-sphere $\mathbb{S}^{4}$, the second one in $\mathcal{N}=2$ Super Chern- Simons theory on the 3-sphere $\mathbb{S}^{3}$. In both cases, the partition function and the Wilson loop expectation value can be reduced to finite- dimensional integrals over the Lie algebra of the gauge group of the theory. This makes the supersymmetric theories in exam equivalent to a suitable “matrix model”, whose path integral can be computed exactly with some special regularization. We will give an example of computations of such a matrix model, since this class of objects arises in many important areas of modern theoretical physics. In Chapter 6 we will introduce Witten’s non-Abelian localization formula, and briefly describe its possible application in the study of 2-dimensional Yang- Mills theory. This more general formalism is able to show the mapping between the standard “physical” version of Yang-Mills theory and its “cohomological” (i.e. topological, in some sense) formulation, and is at the base of the localization of the 2-dimensional Yang-Mills partition function. Some technical asides are relegated to the appendices. Appendix A is devoted to some background in differential geometry, concerning principal bundles and the definition of spinors in curved spacetime. In Appendix B we report more details about equivariant cohomology, equivariant vector bundles and characteristic classes. This can be seen as a completion of the discussion of Chapter 2, from a more mathematical point of view. ## Chapter 2 Equivariant cohomology In this chapter we review the theory of _equivariant cohomology_ , as a modification of the standard cohomology theory applied to spaces that are equipped with the action of a _symmetry group_ $G$ on them, the so-called $G$-manifolds. First, we will review the basic notions about cohomology and homology, from their algebraic definition to the application in topology and differential geometry. The main result that we need to care about, and extend to the equivariant case, is _de Rham’s theorem_ [17], that gives an _algebraic model_ for the cohomology of a smooth manifold in terms of the complex of its differential forms. This and Stokes’ theorem relate the theory of cohomology classes to the integration on smooth manifolds. Next, we will extend this to the equivariant setting, giving a topological definition of equivariant cohomology, and then discussing, in the smooth case, an equivariant version of de Rham’s theorem. This, analogously to the standard case, will give an equivalence between the topological definition of equivariant cohomology and the cohomology of some suitable differential complex built from the smooth structure of the space at hand. There are different, but equivalent, possibilities of such algebraic models for the equivariant cohomology of a $G$-manifold: we will see the _Weil model_ , the _Cartan model_ and the _BRST model_ , and discuss how they are related one to each other, since at the end they have to describe the same equivariant cohomology. The purpose of all this, from the physics point of view, is that with equivariant cohomology we can describe a theory of cohomology and integration over manifolds that are acted upon by a symmetry group, the standard setup of classical mechanics and QFT. In the next chapter we will review one of the climaxes of this theory applied to the problem of integration over $G$-manifolds: the famous _localization formulas_ of Berline-Vergne [7] and Atiyah-Bott [6], that permit to highly simplify a large class of integrals thanks to the equivariant structure of the underlying manifold. The aim and the core of this thesis will be then the description of some generalizations and application of those theorems to the context of QM and QFT, where the integrals of interest are the infinite-dimensional _path integrals_ describing partition functions or expectation values of operators. For this introductory chapter, we follow mainly [18, 19, 20, 21]. Another classical reference is [22]. Some background tools from differential geometry that are needed can be found in Appendix A. ### 2.1 A brief review of standard cohomology theory In this section we will review some of the basic facts about standard homology and cohomology theory, and in particular its application to topological spaces with the definition of _singular_ homology and cohomology groups. Since this is after all standard material, we refer to any book of topology/geometry/algebra (for example [23, 24, 25, 26]) for the various proofs, while we will give some intuitive examples to help making concrete the various abstract definitions. The main result that we aim to recall is _de Rham’s theorem_ , that relates the cohomology theory to differential forms over smooth manifolds, and that will be extended in the next sections to the modified equivariant setup. We start with the abstract definition of homology and cohomology as algebraic constructions. From this point of view, (co)homology groups are defined in relation to _(co)chain complexes_ (or _differential complexes_). ###### Definition 2.1.1. Given a ring $R$, a _chain complex_ is an ordered sequence $A=(A_{p},d_{p})_{p\in\mathbb{N}}$ of $R$-modules $A_{p}$ and homomorphisms $d_{p}:A_{p}\to A_{p-1}$ such that $d_{p-1}\circ d_{p}=0$. A _cochain complex_ has the same structure but with homomorphisms $d_{p}:A_{p}\to A_{p+1}$, and $d_{p+1}\circ d_{p}=0$. Chain complex $\displaystyle:\qquad\cdots A_{p-1}\xleftarrow{d_{p}}A_{p}\xleftarrow{d_{p+1}}A_{p+1}\cdots$ Cochain complex $\displaystyle:\qquad\cdots A_{p-1}\xrightarrow{d_{p-1}}A_{p}\xrightarrow{d_{p}}A_{p+1}\cdots$ An element $\alpha$ of a (co)chain complex is called _(co)cycle_ or _closed_ if $\alpha\in\mathrm{Ker}(d_{p})$ for some $p$. It is instead called _(co)boundary_ or _exact_ if $\alpha\in\mathrm{Im}(d_{p})$ for some $p$. By definition $\mathrm{Im}(d_{p})\subseteq\mathrm{Ker}(d_{p\pm 1}),$ where the $-$ is for chain and the $+$ for cochain complexes, so the quotient sets $\faktor{\mathrm{Ker}}{Im}$ of $p$-(co)cycles modulo $p$-(co)boundries are well defined. ###### Definition 2.1.2. Given a (co)chain complex $A$, the _$p^{th}$ (co)homology group_ of $A$ is $\left(H^{p}(A):=\faktor{\mathrm{Ker}(d_{p})}{\mathrm{Im}(d_{p+1})}\right)\qquad H_{p}(A):=\faktor{\mathrm{Ker}(d_{p})}{\mathrm{Im}(d_{p-1})}.$ (Co)homology groups are called like that because they inherit a natural Abelian group structure (or equivalently $\mathbb{Z}$-module structure) from the sum in the original chain complex.111Notice that $[\alpha]+[\beta]:=[\alpha+\beta]$ is well defined. As usual, once we have a definition of a class of mathematical objects, a prime interest lies in the study of structure preserving maps between them. A morphism between (co)chain complexes $A$ and $B$ is then a sequence of homomorphisms $\left(f_{p}:A_{p}\to B_{p}\right)_{p\in\mathbb{N}}$ such that, schematically, $f\circ d^{(A)}=d^{(B)}\circ f$. It is easy to see that every such a morphism induces an homomorphism of (co)homology groups, since for example $f^{}:H^{p}(A)\to H^{p}(B)$ such that $f^{}([\alpha]):=[f(\alpha)]$ is well defined. Notice that, in many applications one considers (co)chain complexes defined by _graded_ modules or algebras with a suitable _differential_. An example of this type is the complex of differential forms $(\Omega(M),d)$ over a smooth manifold, that we will recover later on. For a general (co)chain complex $(A_{p},d_{p})_{p\in\mathbb{N}}$, we can always see $A:=\bigoplus_{p}A_{p}$ as a _graded_ $R$-module, whose elements as $\alpha\in A_{p}\subset A$ are said to have pure _degree_ $\textrm{deg}(\alpha):=p$. A generic element will be a sum of elements of pure degree. ###### Definition 2.1.3. A _differential graded algebra_ (_dg-algebra_ for short) over $R$ is then an $R$-algebra with the decomposition (grading) $A=\bigoplus_{p}A_{p}$, the product satisfying $A_{p}A_{q}\subseteq A_{p+q}$, and a _differential_ $d:A\to A$ such that 1. (i) it has degree $\textrm{deg}(d)=\pm 1$, meaning that for every $\alpha$ of degree $p$, $\textrm{deg}(d\alpha)=p\pm 1$; 2. (ii) it is nilpotent, $d^{2}=0$; 3. (iii) it satisfies the graded Leibniz rule, $d(\alpha\beta)=(d\alpha)\beta+(-1)^{\textrm{deg}(\alpha)}\alpha(d\beta)$. Every such an algebra clearly defines an underlying complex (cochain if $\textrm{deg}(d)=1$, chain if $\textrm{deg}(d)=-1$) and thus has associated (co)homology groups. Even if the algebra structure (the product and the Leibniz rule) are not needed to define the complex, we included it in the definition because this is the kind of structure that arises in physics or in differential geometry. Morphisms of dg-algebras are naturally defined as structure preserving maps between them, analogously to the above discussion. Beside the abstract algebraic definitions of above, one of the most important applications of homology and cohomology groups is in the study and classification of topological spaces. In order to define these groups in a topological setup, the complexes one takes into consideration are the _simplicial complexes_ , that intuitively represents a formal way of constructing “polyhedra” over $\mathbb{R}^{n}$, and that can be used in turn to study properties of topological spaces. Given $\mathbb{R}^{\infty}$ with the standard basis $\\{e_{i}\\}_{i=0,1,\cdots}$ ($e_{0}=0$), a _standard q-simplex_ is $\Delta_{q}:=\left\\{\left.x=\sum_{i=0}^{q}\lambda_{i}e_{i}\right\|\sum_{i=0}^{q}\lambda_{i}=1,\lambda_{i}\in[0,1]\ \forall i=0,\cdots,q\right\\}.$ (2.1) Although this definition takes into account any possible dimensionality, we can embed these simplices in in finite-dimensional Euclidean spaces, giving them a more practical interpretation. Given $q+1$ points $v_{0},\cdots,v_{q}\in\mathbb{R}^{n}$, the associated _affine singular q-simplex_ in $\mathbb{R}^{n}$ is the map $\displaystyle\left[v_{0}\cdots v_{q}\right]:\Delta_{q}$ $\displaystyle\to\mathbb{R}^{n}$ (2.2) $\displaystyle\sum_{i=0}^{q}\lambda_{i}e_{i}$ $\displaystyle\mapsto\sum_{i=0}^{q}\lambda_{i}v_{i}.$ This is the convex hull in $\mathbb{R}^{n}$ generated by the _vertices_ $(v_{i})$. Geometrically, the 0-simplex is just the point $0\in\mathbb{R}^{n}$, 1-simplices are line segments, 2-simplices are triangles and so on. Notice that $\Delta_{q-1}\subset\Delta_{q}$, and its image through $[v_{0}\cdots v_{q}]$ is a “face” of the resulting polygon. More precisely, $[v_{0}\cdots\hat{v_{i}}\cdots v_{q}]:\Delta_{q-1}\to\Delta_{q}$ (the hat means we take away that point from the list) is regarded as the _$i^{th}$ face map_, denoted concisely as $F^{(i)}_{q}$. (a) (b) Figure 2.1: (a) First standard simplices. (b) An oriented affine 2-simplex, its face maps and a singular 2-simplex $\sigma_{2}$ on a 2-dimensional topological space. The same idea can be used to embed the simplices in a generic topological space $M$, changing the codomain of the simplex map. A _singular q-simplex_ in $M$ is then a continuous map222The standard topology on $\mathbb{R}^{q}$ is induced on $\Delta_{q}$. $\sigma_{q}:\Delta_{q}\to M$ (2.3) where now $\\{\sigma_{q}(e_{0}),\cdots,\sigma_{q}(e_{q})\\}$ are the _vertices_ of $\sigma_{q}$. Two simplices are said to have the same/opposite _orientation_ if the vertex sets are respectively even/odd permutations of each other. The word “singular” is there because only continuity is required, thus from a “smooth” point of view these simplices can present singularities. With this setup, we can construct chain complexes on topological spaces in terms of singular simplices. In fact, defining the sum of two singular simplices $\sigma_{q},\rho_{p}$ as $\displaystyle(\sigma_{q}+\rho_{p}):\Delta_{q}\sqcup\Delta_{p}$ $\displaystyle\to M$ (2.4) $\displaystyle\lambda$ $\displaystyle\mapsto\begin{cases}\sigma_{q}(\lambda)&\text{if}\ \lambda\in\Delta_{q}\\\ \rho_{q}(\lambda)&\text{if}\ \lambda\in\Delta_{p},\end{cases}$ whose image is the (disjoint) union in $M$ of the images of the two starting simplices. Since the “+” is clearly commutative, $\mathcal{C}_{q}(M):=C^{0}(\Delta_{q},M)$ is an Abelian group, called the _(singular) q-chain group_ of $M$. We can define a _boundary operator_ as a group homomorphism $\displaystyle\partial:\mathcal{C}_{q}(M)$ $\displaystyle\to\mathcal{C}_{q-1}(M)$ (2.5) $\displaystyle\sigma$ $\displaystyle\mapsto\partial\sigma:=\sum_{i=0}^{q}(-1)^{i}\left(\sigma\circ F^{(i)}_{q}\right),$ that restricts to the (oriented) sum of faces of a given simplex, and happens to satisfy the nilpotency condition $\partial\circ\partial=0$. This means that $\mathcal{C}(M):=\bigoplus_{q=0}^{\infty}\mathcal{C}_{q}(M)$ with the operator $\partial$ defines a dg-module over $\mathbb{Z}$, and an associated chain complex, that we use to define the homolgy groups of $M$. ###### Definition 2.1.4. The _singular $q^{th}$ homology group_ of $M$ is $H_{q}(M;\mathbb{Z}):=\faktor{\mathrm{Ker}(\partial_{q})}{\mathrm{Im}(\partial_{q+1})}.$ It is often useful to work with homology groups _with coefficients_ in some $\mathbb{Z}$-module $A$ (like the real numbers), that is considering $H_{q}(M;A)$ as defined from the simplicial complex $\mathcal{C}(M)\otimes A$. ###### Example 2.1.1 (Homology of spheres). In practice, the strategy to get $H_{}(M)$ is the so-called _triangulation_ of $M$, i.e. constructing a suitable simplicial complex $K$ in $\mathbb{R}^{\dim(M)}$ as a set of standard simplices, whose union gives a polyhedron that is homeomorphic to $M$. Then, one can count and classify all the cycles and the boundaries in $K$, and then get $H_{}(K)\cong H_{}(M)$. Some examples of this rigorous approach can be found in [23]. We can still give some examples, less rigorously, by looking directly at simple topological spaces, just to help building some intuition. Remember that a $q$-cycle $\sigma$ on $M$ is a boundary-less singular $q$-simplex _up to continuous deformations_ , and it is also a boundary if it can be seen as the border of a $(q+1)$-simplex. 1. $(\mathbb{S}^{1})$ On the circle there is no place for simplices of dimension higher than 1, so we look at the 1-simplices. There are two inequivalent ways of deforming the standard 1-simplex onto the circle: it can join at the end points covering all $\mathbb{S}^{1}$ or not. In the first case, that we call $\sigma_{1}$, we have $\partial\sigma_{1}=0$ since $\mathbb{S}^{1}$ has no boundaries, in the second the boundaries are the end points of the singular 1-simplex. The first boundary-less case cannot be seen as a boundary of something else, by dimensionality, so the Abelian group $H_{1}(\mathbb{S}^{1};\mathbb{Z})=\mathrm{Ker}(\partial_{1})/\mathrm{Im}(\partial_{2})$ is generated by a single element, $[\sigma_{1}]$. In other words, $H_{1}(\mathbb{S}^{1};\mathbb{Z})\cong\mathrm{span}_{\mathbb{Z}}\\{[\sigma_{1}]\\}\cong\mathbb{Z}$. The case $q=0$ is trivial, since we have only one way of drawing a point on the circle, and every point is boundary-less. We have just said that the boundary of a 1-simplex is either zero or two points, so a single point is never a boundary. Thus the Abelian group $H_{0}(\mathbb{S}^{1};\mathbb{Z})\cong\mathbb{Z}$ since it is generated by only one element. Generalizing a little, we can already see from this example that the homology group in 0-degree will always follow this trend for _connected_ topological spaces. If the space has $n$ connected components, there will be $n$ inequivalent ways of drawing a point on it, so $n$ generators for the homology group, giving $H_{0}(M_{(n)};\mathbb{Z})\cong\mathbb{Z}\oplus\cdots\oplus\mathbb{Z}$ ($n$ factors). 2. $(\mathbb{S}^{2})$ The 2-sphere does not necessitate of much more work, at least with this level of rigor. Again, by dimensionality the homology groups in degrees higher than $\dim(\mathbb{S}^{2})=2$ are empty. For $q=2$, the only way we can construct a boundary-less figure on the sphere from $\Delta_{2}$ is joining the vertices and the edges together and cover the whole sphere. All other singular 2-simplices have boundaries, and the 2-sphere cannot be seen as a boundary of something else by dimensionality, so analogously to the previous case $H_{2}(\mathbb{S}^{2};\mathbb{Z})\cong\mathbb{Z}$. For the 1-simplices, we notice that the only two ways of drawing a segment on the sphere (up to continuous deformations) is to close it or not at the end points. In the first case, the 1-simplex has no boundary, but can be seen as the boundary of its internal area, so it is in fact exact. In the second case, the 1-simplex has boundaries so it is outside $\mathrm{Ker}(\partial)$. This means that every 1-cycle is also a boundary, and thus $H_{1}(\mathbb{S}^{2};\mathbb{Z})\cong 0$. In 0-degree we can argue in the same way as for the circle that $H_{0}(\mathbb{S}^{2};\mathbb{Z})\cong\mathbb{Z}$. 3. $(\mathbb{S}^{n})$ It turns out that all spheres follow this trend, giving homology groups $H_{q}(\mathbb{S}^{n};\mathbb{Z})\cong\begin{cases}\mathbb{Z}&q=0,n\\\ 0&\text{otherwise}.\end{cases}$ If interested in the case with real coefficient, the homology of spheres are again very simple, since $\mathbb{Z}\otimes\mathbb{R}\cong\mathbb{R}$. Now we can turn to the construction of singular cohomology groups on topological spaces. This is done considering the dual spaces $\text{Hom}(\mathcal{C}_{q}(M),A)$ with values in a $\mathbb{Z}$-module $A$. The simplest choice is of course $A=\mathbb{Z}$. Notice that $\text{Hom}(\mathcal{C}_{q}(M),A)$ itself is a $\mathbb{Z}$-module. The _coboundary operator_ in this case is defined as the $\mathbb{Z}$-module homomorphism $\displaystyle\delta:\text{Hom}(\mathcal{C}_{q}(M),A)$ $\displaystyle\to\text{Hom}(\mathcal{C}_{q+1}(M),A)$ (2.6) $\displaystyle f$ $\displaystyle\mapsto\delta f\quad s.t.\quad\delta f(\sigma_{q+1}):=f(\partial\sigma_{q+1})$ and from the nilpotency $\partial^{2}=0$ we get easily $\delta^{2}=0$. ###### Definition 2.1.5. The _singular $q^{th}$ cohomology group_ of $M$, _with coefficients in $A$_, is $H^{q}(M;A):=\faktor{\mathrm{Ker}(\delta_{q})}{\mathrm{Im}(\delta_{q-1})}.$ Note that for a commutative ring $A$ (as for example $\mathbb{R}$), the cohomology groups are naturally $A$-modules. Although the definition is less practical than the one for homology groups, there is an important theorem that allows to relate the two, so that homology computations can be used to infer the structure of singular cohomology groups. This is the so-called _universal coefficient theorem_ [24]. Since for applications to smooth manifolds we will be primarily interested in cohomology groups with real coefficients (as it will become clearer later) we take $A=\mathbb{R}$. For this special case, the theorem says $H^{q}(M;\mathbb{R})\cong H^{q}(M;\mathbb{Z})\otimes\mathbb{R}\cong H_{q}(M;\mathbb{R})^{},$ (2.7) so that the cohomolgy groups are exactly the dual spaces of the homology groups. Notice that, from the example above, $H^{q}(\mathbb{S}^{n};\mathbb{R})\cong\mathbb{R}$ in degree $q=0,n$. The above construction relates the topological properties of the space $M$ to the algebraic concept of (co)homology groups. In general we said that morphisms of complexes induce morphisms of associated (co)homologies, and this extends to the present topological case: if we consider a continuous map (morphism of topological spaces) $F:M\to N$ between the topological spaces $M,N$, we can lift it to $F_{\\#}:\mathcal{C}_{q}(M)\to\mathcal{C}_{q}(N)$ such that $F_{\\#}(\sigma):=F\circ\sigma$, that is a morphism of dg-modules. This in turn induces the morphism of homology groups as descrbed above. For the cohomology groups we have, analogously, the lifted map in the opposite direction $F^{\\#}:\text{Hom}(\mathcal{C}_{q}(N),A)\to\text{Hom}(\mathcal{C}_{q}(M),A)$ such that $F^{\\#}(f):=f\circ F_{\\#}$, giving the morphism of cochain complexes. This induces $F^{}:H^{q}(N)\to H^{q}(M)$ such that $F^{}([f]):=[F^{\\#}(f)]$.333In the context of smooth manifolds and de Rham cohomology, this is analogous to the _pull-back_ of differential forms. Also, we notice that if we have two continuous maps $F,G$ between the topological spaces, then444In category theory language, we can summarize these properties saying that singular homology $H_{}(\cdot)$ is a _covariant functor_ between the categories Top of topological spaces and Ab of Abelian groups, and singular cohomology $H^{}(\cdot)$ is a _contravariant functor_ between Top and Ab. Anyway, we will not need such a terminology for what follows. See for example [27], Appendix A, for a quick introduction to the subject. $\displaystyle(F\circ G)_{\\#}=F_{\\#}\circ G_{\\#}\quad\Rightarrow\quad(F\circ G)_{}=F_{}\circ G_{}$ (2.8) $\displaystyle(F\circ G)^{\\#}=G^{\\#}\circ F^{\\#}\quad\Rightarrow\quad(F\circ G)_{}=G_{}\circ F_{}.$ An important fact that permits to use homology and cohomology groups to classify and characterize topological spaces, is that these objects are _topological invariants_ , meaning that isomorphic spaces have the same (co)homology groups. Moreover, a stricter result holds: two homotopy- equivalent topological spaces have the same cohomology and homology groups. We recall that two continuous maps $F,G:M\to N$ between topological spaces are _homotopic_ if it exists a continuous map $H:[0,1]\times M\to N$ that deforms continuously $F$ in $G$, i.e. $H(0,x)=F(x)$ and $H(1,x)=G(x)$ for every $x\in M$. Homotopy of maps is an equivalent relation, and we denote it by $F\sim G$. Two topological spaces $M,N$ are said to be _homotopy-equivalent_ , or of the same homotopy type, if there exist two maps $F:M\to N$ and $G:N\to M$ such that $(G\circ F)\sim id_{M}$. Homotopy-equivalence is also an equivalence relation, that we denote also as $M\sim N$. The result stated above is then, for cohomologies $M\sim N\Rightarrow H^{}(M)\cong H^{}(N).$ (2.9) ###### Example 2.1.2 (More singular homologies). 1. $(pt)$ We can consider the very trivial case of $M$ being just a point. In this case, the informal discussion of Example 2.1.1 can be carried out just for the 0-dimensional simplices: $H_{}(pt;\mathbb{Z})\cong\mathbb{Z}$ in degree 0. It follows by definition and by the universal coefficient theorem that for any commutative ring $A$ (as $\mathbb{R}$), $H^{0}(pt;A)\cong H_{0}(pt;A)\cong A$ and $H^{q}(pt;A)\cong H_{q}(pt;A)\cong 0$ for $q>0$. By the homotopy- invariance property discussed above, any contractible space will have the same trivial cohomology and homology as the point! 2. $(\mathcal{C})$ Let us look at another simple case, the cylinder $\mathcal{C}=\mathbb{S}^{1}\times[0,1]$. To compute its homology groups we could follow the intuitive discussion of Example 2.1.1, or we can just notice that since the interval $[0,1]$ is contractible, $\mathbb{S}^{1}\times[0,1]\sim\mathbb{S}^{1}.$ This means that $H^{q}(\mathbb{S}^{1}\times[0,1];\mathbb{R})\cong H^{q}(\mathbb{S}^{1};\mathbb{R})\cong H_{q}(\mathbb{S}^{1};\mathbb{R})$. 3. $(T)$ A less trivial example is the 2-torus $T=\mathbb{S}^{1}\times\mathbb{S}^{1}$. In this case no one of the factors is contractible, so we cannot use the homotopy invariance to get the result from a simpler space. We can anyway get the answer using the same method of Example 2.1.1. Starting from the top- degree homology group, we notice that the only boundary-less surface on the torus is the torus itself. Thus analogously to all the other cases, $H_{2}(T;\mathbb{Z})\cong\mathbb{Z}$ or $H_{2}(T;\mathbb{R})\cong\mathbb{R}$. Since the torus is connected, in degree 0 we get trivially $H_{0}(T;\mathbb{R})\cong\mathbb{R}$. In degree 1 we see the difference with the other cases. On the torus there are two inequivalent ways of drawing a closed line that is not a boundary of any 2-dimensional surface, following essentially the two factors of $\mathbb{S}^{1}$ (see figure 2.2). This means that the $1^{st}$ homology group is generated by two elements, and thus $H_{1}(T;\mathbb{Z})\cong\mathbb{Z}\oplus\mathbb{Z}$. The same reasoning can be applied to higher genus surfaces $\Sigma_{g}$, giving $H_{1}(\Sigma_{g};\mathbb{Z})\cong(\mathbb{Z})^{\oplus 2g}$. (a) (b) Figure 2.2: (a) The 2 inequivalent non-exact 1-cycles $\sigma_{1}$ and $\sigma_{1}^{\prime}$ on the 2-torus. (b) The genus-$g$ surface $\Sigma_{g}$ has $2g$ inequivalent non-exact 1-cycles. Figures adapted from [23]. We leave now the purely topological setup, since in physics we are mostly interested in studying local properties, i.e. from the differential geometry point of view. We assume to work in the smooth setting and consider $M$ to be a $d$-dimensional $C^{\infty}$-manifold. With $TM$ we denote its _tangent bundle_ , and with $T^{}M$ its _cotangent bundle_.555Sections of any bundle $E\to M$ over $M$ will be denoted in the following with $\Gamma(M,E)$, or $\Gamma(E)$ when the base space is clear from the context. For example, vector fields are elements of $\Gamma(TM)$. At every point $p\in M$, $T_{p}M$ and $T^{}_{p}M$ are the dual vector spaces of tangent vectors and 1-forms at $p$, respectively. We consider the _exterior algebra_ $\bigwedge(T^{}_{p}M)$, with the wedge product $\wedge$ making it in a graded-commutative algebra, and the exterior derivative $d:\bigwedge(T^{}_{p}M)\to\bigwedge(T^{}_{p}M)$ acting as a graded derivation of $\textrm{deg}(d)=+1$. Extending these operations point-wise for every point $p\in M$, we have the bundle of _differential forms_ over $M$, $\Omega(M):=\bigoplus_{k=0}^{d}\Omega^{k}(M)\qquad\text{with}\ \Omega^{k}(M):=\bigsqcup_{p\in M}\bigwedge^{k}(T^{}_{p}M).$ (2.10) $(\Omega(M),\wedge,d)$ is thus a dg-algebra over the commutative ring $C^{\infty}(M)$, and naturally defines a cochain complex called the _de Rham complex_. The associated cohomology groups are the _de Rham cohomology groups_ , constituting the graded-commutative ring666Analogously to the singular cohomology, in category theory language the de Rham cohomology $H_{dR}(\cdot)$ is a contravariant functor between the categories Man of smooth manifolds and Ab of Abelian groups. $H_{dR}(M)=\bigoplus_{k=0}^{d}H_{dR}^{k}(M)\qquad\text{with}\quad H_{dR}^{k}(M):=H^{k}(\Omega(M),d)=\faktor{\mathrm{Ker}(d_{k})}{\mathrm{Im}(d_{k-1})}.$ (2.11) The ring structure of $H_{dR}(M)$ is naturally inherited from the wedge product of differential forms, that lifts at the level of cohomology classes. In fact, for two closed forms $\omega,\eta\in\Omega(M)$, $[\omega]\wedge[\eta]:=[\omega\wedge\eta]$ (2.12) is well-defined.777This result can be seen also in the topological setup for singular cohomology groups, as it should be by de Rham’s theorem. The operation that corresponds to the wedge product between singular cohomology classes is called _cup-product_ [24]. It is important to remember that cohomology in general has a ring structure. The final important result that we state, and that will be crucial to extend to the equivariant setting in the following section, is the so called _de Rham’s theorem_ : ###### Theorem 2.1.1 (de Rham). The de Rham cohomology of the smooth manifold $M$ is isomorphic to its singular cohomology with real coefficients: $H_{dR}(M)\cong H^{}(M;\mathbb{R}).$ The power of this theorem is that it allows to study topological properties of the manifold (recall that $H^{}(M;\mathbb{R})$ are homotopy-invariants) using differential geometric (so local) objects, the differential forms. We say that the de Rham complex $(\Omega(M),d)$ constitute an _algebraic model_ for the singular cohomology of $M$. Notice that, by dimensionality reasons, we get trivially also in this case that the cohomology groups $H_{dR}^{q}(M)$ for $q>\dim(M)$ are automatically zero. Another important property that is intuitively very clear from the de Rham complex is the _Poincaré duality_. For a closed connected manifold $M$ this states that, as vector spaces $H^{k}_{dR}(M)\cong H^{\dim(M)-k}_{dR}(M).$ (2.13) A crucial tool for the proof of de Rham’s theorem is the so-called _Stokes’ theorem_ , that relates the integral of an exact $d$-form over a $d$-dimensional manifold to the integral of its primitive over the $(d-1)$-dimensional boundary, $\int_{M}d\omega=\int_{\partial M}\omega.$ (2.14) Notice that integration over $M$ when $\partial M=\emptyset$ can be regarded as a function on the $d^{th}$ de Rham cohomology $\int:H^{d}_{dR}(M)\to\mathbb{R}$. ###### Example 2.1.3 (Cohomology rings). With the help of de Rham’s theorem, we can compute some of the previous example directly at the level of cohomology using differential forms and integration.888Another powerful tool to practically compute cohomology groups and rings, at the topological level, goes by the name of _spectral sequences_. See for example [18]. Let us consider the case of the tours $T=(\mathbb{S}^{1})^{2}$. We can parametrize it with coordinates $(x,y)$ taking values in $[0,1)^{2}\subset\mathbb{R}^{2}$. If we call $\alpha:=dx$ and $\beta:=dy$ in $\Omega^{1}(T)$, a natural choice of volume form is $\omega:=\alpha\wedge\beta$, that gives $\mathrm{vol}(T)=1$. The volume form is of course closed by dimensionality, but it cannot be exact since otherwise by Stokes’ theorem the volume of the torus would be 0, so it defines a non- trivial cohomology class $[\alpha\wedge\beta]$. Any other 2-form is of the type $\omega^{\prime}=f\omega$ for some $f\in C^{\infty}(T)$, but closed forms must satisfy $df=0$, so $f\in\mathbb{R}$ constant. We conclude that any other independent closed 2-form has to be “cohomologous” to $[\alpha\wedge\beta]$, so that in top-degree $H^{2}_{dR}(T)\cong\mathrm{span}\\{[\alpha\wedge\beta]\\}\cong\mathbb{R}$. In degree 1, any closed form must be a combination of $\alpha$ and $\beta$ with real coefficients (since again $d(f\alpha)=0\ \Leftrightarrow\ df=0$), so they are the only independent closed 1-forms (they correspond to the volume forms for the two $\mathbb{S}^{1}$ factors). To see whether or not they are exact, we can use Stokes’ theorem: if they are, then their integral over _any_ closed curve on $T$ must be zero. But we can take the two curves $\sigma_{1}(t)=(t,0)$ and $\sigma_{1}^{\prime}(t)=(0,t)$ of Figure 2.2 and see that $\int_{\sigma_{1}}\alpha=1=\int_{\sigma_{1}^{\prime}}\beta,$ so they define two independent cohomology classes $[\alpha]$ and $[\beta]$. This means that $H^{1}_{dR}(T)\cong\mathrm{span}\\{[\alpha],[\beta]\\}\cong\mathbb{R}\oplus\mathbb{R}$. Since the torus is connected, the only closed 0-form is a constant number, that we can chose to be $1$. Thus, $H^{0}_{dR}\cong\mathbb{R}$. We can further easily get the ring structure of $H_{dR}(T)$ by looking at the multiplication rules between the generators. If we call $a:=[\alpha]$ and $b:=[\beta]$, the wedge product of differential forms gives the following rules $a^{2}\sim 0,\qquad b^{2}\sim 0,\qquad ab+ba\sim 0.$ Thus we can rewrite the cohomology ring as a polynomial ring over the indeterminates $(a,b)$, taken in degree 1, that satisfy the above rules: $H^{}(T;\mathbb{R})\cong\faktor{\mathbb{R}[a,b]}{(a^{2},b^{2},ab+ba)},$ where $(a^{2},b^{2},ab+ba)$ denotes the quotient by the ideal generated by the corresponding expressions. In the same fashion we can rewrite the cohomology rings of the other examples that we gave above for the $n$-sphere. Introducing an indeterminate $u$ of degree $n$, and the multiplication rule $u^{2}\sim 0$, its cohomology ring can be expressed as $H^{}(\mathbb{S}^{n};\mathbb{R})\cong\faktor{\mathbb{R}[u]}{u^{2}}.$ We quote another example, that will enter in the case of equivariant cohomology with respect to a circle action by $U(1)\cong\mathbb{S}^{1}$. For the complex projective plane $\mathbb{C}P^{n}$, it turns out that $H^{}(\mathbb{C}P^{n};\mathbb{R})\cong\faktor{\mathbb{R}[u]}{u^{(n+1)}}$ where $\mathrm{deg}(u):=2$. In the limiting case $n\to\infty$, one has thus $H^{}(\mathbb{C}P^{\infty};\mathbb{R})\cong\mathbb{R}[u]$, the polynomials in $u$. ### 2.2 Group actions and equivariant cohomology As already mentioned, equivariant cohomology is an extension of the standard cohomology theory, partly reviewed in the last section, to the cases in which the space $M$ is acted upon by some group $G$. This is the common setup in physics, from the finite-dimensional cases of classical Lagrangian or Hamiltonian mechanics to the infinite dimensional case of Quantum Field Theory, where $M$ can be the configuration space, the phase space, or the space of fields, and $G$ is a Lie group representing a _symmetry_ of the physical system. In gauge theory for example, we want to identify those physical configurations that are equivalent modulo a gauge transformations, so the moduli space of gauge orbits $M/G$. In Poincaré-supersymmetric theories, the group $G$ is actually the Poincaré group of spacetime symmetries. In all these cases we are interested in the cohomology of $M$ modulo these symmetry transformations, since many primary objects of study (partition functions, expectation values…) are usually given in terms of integrals over $M$. Before moving to the technical definition of $G$-equivariant cohomology of $M$, we recall some terminology about group actions. ###### Definition 2.2.1. 1. (i) Given a group $G$ and a topological space $M$,999We are going to work practically always with smooth manifolds and (compact) Lie groups, but for the moment we do not need this level of structure on $M$ and $G$. a _$G$ -action_ on $M$ is given by a group homomorphism (_left_ action) or anti-homomorphism (_right_ action) $\rho:G\to\mbox{Homeo}(M)\quad(\text{or Diff}(M)\text{ for smooth manifolds}).$ If $m\in M,g\in G$, the left action of $g$ on $m$ can be denoted $\rho(g)m\equiv g\cdot m$, and the right action $m\cdot g$, if this causes no confusion. $M$ is said to be a (left or right) _$G$ -space_. 2. (ii) If $M,N$ are two $G$-spaces, on the product $M\times N$ it is canonically defined the _diagonal $G$-action_ $\rho^{M\times N}(g)(m,n):=(\rho^{M}(g)m,\rho^{N}(g)n)\quad\text{for }m\in M,n\in N,g\in G.$ 3. (iii) Given a point $m\in M$, the _orbit_ of $m$ is the subset of $M$ of all points that are reached from $m$ by the action of $G$. The _orbit space_ with respect to the $G$-action is $M/G$.101010It is easy to check that $m\sim m^{\prime}\Leftrightarrow m^{\prime}=g\cdot m$ for some $g\in G$ is an equivalence relation. 4. (iv) The _stabilizer_ (or _isotropy group_ , or _little group_) of $m$ is the subgroup of $G$ of all elements that act trivially on $m$, i.e. $g\cdot m=m$. The $G$-action is called _free_ if the stabilizer of every point in $M$ is given by the identity of $G$. The $G$-action is called _locally free_ if the stabilizer of every point is _discrete_. The _fixed point set_ $F\subseteq M$, is the set of all points that are stabilized by the entire $G$. 5. (v) Morphisms of $G$-spaces are called _$G$ -equivariant_ functions. $f:M\to N$ is $G$-equivariant if $f(g\cdot m)=g\cdot f(m)$, for every $m\in M,g\in G$. In the following we will not care much about distinguishing between left and right actions, and assume all $G$-actions are from the left, unless otherwise stated. For the first part of the discussion it is not needed, but we are going to assume $M$ and $G$ to be at least topological manifolds, and then specialize to the case of smooth manifolds, since these are the most common structures arising in physics. Since, as we said above, we are interested in identifying those elements in $M$ that are equivalent up to a “symmetry” transformation by $G$, the first candidate for the $G$-equivariant cohomology of $M$ could be simply the cohomology of the orbit space $M/G$, $H_{G}^{}(M):=H^{}\left(\faktor{M}{G}\right).$ (2.15) This definition has the problem that, if the $G$-action is not free and has fixed points on $M$, the orbit space is singular: in the neighborhood of those fixed points there is no well-defined notion of dimensionality. This kind of singular quotient spaces are called _orbifolds_. ###### Example 2.2.1 (Some group actions and orbit spaces). 1. (i) Let us consider the Euclidean space $\mathbb{R}^{n}$. $O(n)$ rotations (and reflections) act naturally on it, with the only fixed point being the origin. Any point but the origin identifies a direction in the Euclidean space, and thus is stabilized by the subgroup of $n-1$ rotations, $O(n-1)$. For example we see that, without considering parity transformations, the standard $SO(2)$-action is free on $\mathbb{R}^{2}\setminus\\{0\\}$. If we bring translations into the game, considering the Euclidean space as an affine space acted upon by $ISO(n)=\mathbb{R}^{n}\rtimes O(n)$, then the stabilizer of any point is the entire $O(n)$, since any point can be considered an origin after translation. So the action of $ISO(n)$ is neither free nor locally free, but has no fixed points on the entire $\mathbb{R}^{n}$. Considering only rotations, the orbit space $\mathbb{R}^{n}/O(n)$ is the space of points identified up to their angular coordinates, that is an half-line starting from the origin, $\mathbb{R}^{n}/O(n)\cong[0,\infty)$. This is not a manifold, since the interval is closed on the left, giving a “singularity” on the original fixed point of the action. Notice that, by embedding $\mathbb{S}^{n-1}$ in $\mathbb{R}^{n}$, an $O(n)$-action descends on it, and the orbit of any point of $\mathbb{S}^{n-1}$ is the sphere itself. The orbit of any point can be seen as the quotient of $O(n)$ by the stabilizer of that point, so that one has $\faktor{O(n)}{O(n-1)}\cong\mathbb{S}^{n-1}.$ This quotient describes the common situation of spontaneous symmetry braking inside $O(n)$-models, in Statistical Mechanics. 2. (ii) One can always consider circle actions on the spheres $\mathbb{S}^{n}$. Starting with $n=1$, and considering the circle as embedded in the $\mathbb{C}$-plane, $U(1)$ acts on itself by multiplication: $e^{i\varphi}\mapsto e^{ia}e^{i\varphi}$ for some $a\in[0,2\pi)$. This action has clearly no fixed points, and it is also free. Thus the quotient is well defined, giving simply $\mathbb{S}^{1}/U(1)\cong pt$. The $U(1)$-action on the 2-sphere is already more interesting. Rotations around a given axis fix two points on $\mathbb{S}^{2}$, that we identify with the North and the South poles. If we exclude the poles the resulting space is homeomorphic to a cylinder, the $U(1)$-action becomes free and indeed we have that $\mathbb{S}^{1}\times(0,1)\to(0,1)$ is a trivial principal $U(1)$-bundle. But considering the poles, the quotient space is singular since $\mathbb{S}^{2}/U(1)\cong[0,1]$. This is an elementary example of an orbifold. Figure 2.3: The circle acting on the 2-sphere, the orbits being the parallels. The orbit space $\mathbb{S}^{2}/U(1)$ is a meridian, homeomorphic to the interval $[0,1]$. Let us consider also the case $n=3$. The 3-sphere $\mathbb{S}^{3}\cong SO(2)$ can be parametrized by a pair of complex numbers $(z_{1},z_{2})$ such that $\|z_{1}\|^{2}+\|z_{2}\|^{2}=1$. The circle then acts naturally by diagonal multiplication: $(z_{1},z_{2})\mapsto(e^{ia}z_{1},e^{ia}z_{2})$ for some $e^{ia}\in U(1)$. This action is clearly free, since the two coordinates $z_{i}$ cannot be simultaneously zero on the sphere, thus the quotient is well defined, giving the 3-sphere the structure of a principal $U(1)$-bundle known as the _Hopf bundle_. The equivalence classes $[z_{1},z_{2}]\in\mathbb{S}^{3}/U(1)$ describe, by definition, points on the complex projective line $\mathbb{C}P^{1}\cong\mathbb{S}^{2}$, that is isomorphic to the Riemann 2-sphere. The Hopf bundle can be thus seen as $\mathbb{S}^{3}\to\mathbb{S}^{2}$, with typical fiber $\mathbb{S}^{1}$. Clearly this is not a trivial bundle, since $\mathbb{S}^{3}\neq\mathbb{S}^{2}\times\mathbb{S}^{1}$. The above case generalizes to any odd-dimensional sphere $\mathbb{S}^{2n+1}$, since they all can be embedded in complex spaces $\mathbb{C}^{n+1}$. The circle acts always by diagonal multiplication, and the resulting action is free. The bundles $\mathbb{S}^{2n+1}\to\mathbb{S}^{2n+1}/U(1)\cong\mathbb{C}P^{n}$ are all principal $U(1)$-bundles over the complex projective spaces $\mathbb{C}P^{n}$. 3. (iii) The last case we mention is the possible $U(1)$-action on a torus $T=(\mathbb{S}^{1})^{2}$, by rotations along one of the two factors. This is the only possible free action on a closed surface, giving the well defined quotient $T/U(1)\cong\mathbb{S}^{1}$. More examples can be found in [28]. The example above showed that also in very simple cases singularities can appear in quotient spaces, so that one cannot define cohomology in a smooth way using the powerful de Rham theorem. It is thus more convenient to set up a definition of equivariant cohomology that automatically avoids this problem. This more clever definition is given by the _Borel construction_ for the $G$-space $M$. ###### Definition 2.2.2. Considering a $G$-space $M$, its associated _Borel construction_ , or _homotopy quotient_ , is $M_{G}:=\faktor{(M\times EG)}{G}\equiv M\times_{G}EG$ where $EG$ is some contractible space on which $G$ acts freely, called the _universal bundle of $G$_ (see Appendix B.2 for the precise definition). The _$G$ -equivariant cohomology_ of $M$ is then defined as111111From now on we always consider cohomologies with coefficients in $\mathbb{R}$, unless otherwise stated. $H_{G}^{}(M):=H^{}(M_{G}).$ We assume the action on the product $M\times EG$ to be the diagonal action. Notice that, since $G$ acts freely on $EG$, the action on the product is automatically free. Indeed, if in the worst case $p\in M$ is a fixed point, for every point $e\in EG$, $g\cdot(p,e)=(p,g\cdot e)\neq(p,e)$. This means that the homotopy quotient defines a smooth manifold, and we can hope for a generalization of de Rham’s theorem, allowing to study this topological definition from its smooth structure in terms of something analogous to the differential forms on $M$. We will discuss this result in the next section. Since the space $EG$ does not need to either exist or be unique a priori, one could think that the above definition contains some degree of arbitrariness, so a natural question is: is equivariant cohomology well defined? The answer is of course yes, and the crucial fact allowing this stands in the contractibility of the space $EG$. The arguments that lead to this conclusion are summarized in Appendix B.2, together with some examples of universal bundles. The important property that one has to keep in mind is that, intuitively, to get it acted freely by $G$ and being contractible, one has to define it “so big” that for _any_ other principal $G$-bundle $P$, there is a copy of $P$ sitting inside $EG$. This is why it is called “universal”. Here we just notice that, if we assume a contractible free $G$-space $EG$ to exist, we have an homotopy equivalence $M\times EG\sim M$, that descends also to the homotopy quotient $\faktor{(M\times EG)}{G}\sim\faktor{M}{G},$ (2.16) since one can show that $M\times_{G}EG\to M/G$ is a fiber bundle with typical fiber $EG$ [18]. From homotopy invariance of cohomology, we see that at least in the case in which $G$ acts freely on $M$ and $M/G$ is well defined, the equivariant cohomology reduces to the naive definition above, $H_{G}^{}(M)=H^{}\left(M\times_{G}EG\right)\cong H^{}\left(\faktor{M}{G}\right).$ (2.17) Notice that the contractible space $EG$ alone has a very simple cohomology. Indeed, for what we pointed out in Example 2.1.2, it must be $H^{}(EG)\cong H^{}(pt)\cong\mathbb{R}$ in degree zero. When we take the quotient, the base space $BG:=EG/G$ can have a less trivial cohomology. This space is called _classifying space_ of the Lie group $G$. When, for example, the $G$-action on $M$ is trivial (all points are fixed points), the homotopy quotient is just $M\times_{G}EG\cong M\times(EG/G)=M\times BG$, and in this case we have121212This is an application of the so-called _Künneth theorem_ [24]. $H^{}_{G}(M)=H^{}(M\times BG)\cong H^{}(M)\otimes H^{}(BG),$ (2.18) so that the homotopy quotient by a trivial action does not bring any further information to the cohomology of $M$ but for tensoring it with the cohomology of the classifying space. In Section 2.4 we will see that the latter can be described in general by a very simple algebraic model, while here we carry on the example of the case $G=U(1)$. ###### Example 2.2.2 (A few $U(1)$-equivariant cohomologies). To search for a suitable principal $U(1)$-bundle whose total space is contractible, we can first notice from Example 2.2.1 that we already described a class of principal $U(1)$-bundles, $\mathbb{S}^{2n+1}\to\mathbb{C}P^{n}$, whose total spaces are the odd-dimensional spheres. The bad news is that any of these total spaces are contractible, but this problem can be solved considering the limiting case $n\to\infty$, since it turns out that $\mathbb{S}^{\infty}=\bigcup_{n}S^{2n+1}\equiv\bigcup_{n}S^{n}$ is contractible [18]!131313Notice that any sphere $\mathbb{S}^{n}$ can be embedded as the equator of $\mathbb{S}^{n+1}$. Thus there is a sequence of inclusions $\mathbb{S}^{1}\subset\mathbb{S}^{3}\subset\cdots$ as well as $\mathbb{C}P^{1}\subset\mathbb{C}P^{3}\cdots$, and the circle action is compatible with the inclusion. Thus, in the limit, a free circle action induces on $\mathbb{S}^{\infty}$. Thus the universal bundle for $U(1)$ can be chosen to be $EU(1)=\mathbb{S}^{\infty}$, and the classifying space $BU(1)=\mathbb{C}P^{\infty}$. Being infinite-dimensional, they are strictly speaking not manifolds, but $\mathbb{S}^{\infty}\to\mathbb{C}P^{\infty}$ is still a topological bundle, and this is enough for the definition of an homotopy quotient. 1. (i) In Example 2.1.2 we quoted the resulting cohomology ring of the complex projective planes, $H^{}(\mathbb{C}P^{n})\cong\mathbb{R}[\phi]/\phi^{n+1}$ and $H^{}(\mathbb{C}P^{\infty})=H^{}(BU(1))\cong\mathbb{R}[\phi]$, with $\phi$ in degree 2. Thus the $U(1)$-equivariant cohomology of any space $M$ on which $U(1)$ acts trivially is, from (2.18), $H^{}_{U(1)}(M)=H^{}(M)\otimes\mathbb{R}[\phi].$ In particular if $M$ is contractible, $H^{}_{U(1)}(M)=H^{}_{U(1)}(pt)=\mathbb{R}[\phi]$. 2. (ii) The opposite case is the one of a free action, for example the circle acting on itself. As we pointed out above, $\mathbb{S}^{1}/U(1)\cong pt$, so simply $H^{}_{U(1)}(\mathbb{S}^{1})\cong\mathbb{R}$. 3. (iii) The last example that we mention is the case of $U(1)$ acting on $\mathbb{S}^{2}$. The equivariant cohomology $H^{}_{U(1)}(\mathbb{S}^{2})$ is non-trivial a priori, as we remarked above, and it can be calculated easily for example using spectral sequences. We will not enter in the detail of the calculation but only describe the result. Consider first the standard cohomology of the 2-sphere that we already saw in various examples, being $H^{}(\mathbb{S}^{2})\cong\mathbb{R}\oplus\mathbb{R}y$, where we explicitly wrote a generator $y$ for the term in degree 2, that can be identified in the de Rham model by a volume form $y=[\omega],\omega\in\Omega^{2}(\mathbb{S}^{2})$. It turns out that its equivariant version can be obtained simply by tensoring with the polynomial ring $H^{}(BU(1))=\mathbb{R}[\phi]$, $H^{}_{U(1)}(\mathbb{S}^{2})\cong\mathbb{R}[\phi]\oplus\mathbb{R}[\phi]y,$ although the generator $y$ has now a different interpretation, that we will give in terms of an equivariant version of the de Rham model in the next sections.141414In terms of differential forms, $\omega$ will have to be _equivariantly extended_ in the Cartan model, as discussed at the end of Section 2.4. This equivariant cohomology actually can be given a ring structure, defining the multiplication $y\cdot y=a\phi y+b\phi^{2}$ for some constants $a,b$. It turns out [18] that the correct constants are $a=1,b=0$, making $\faktor{\mathbb{R}[y,\phi]}{(y^{2}-\phi^{2})}\to H_{U(1)}^{}(\mathbb{S}^{2})$ into a ring isomorphism, where the denominator stands for the ideal generated by the expression $(y^{2}-\phi^{2})$ in $\mathbb{R}[y,\phi]$. Notice that the cohomology groups $H^{n}_{U(1)}(\mathbb{S}^{2})$ are now non-empty in every even-degree (while in odd-degree they are all trivial), even when $n$ is bigger than the dimension of the sphere! This intuitively matches the fact that the quotient $\mathbb{S}^{2}/U(1)$ is singular, and thus simple dimensionality arguments do not make sense anymore at the fixed points. ### 2.3 The Weil model and equivariant de Rham’s theorem From now on, we specialize the equivariant cohomological theory to $G$ being a Lie group with Lie algebra $\mathfrak{g}$,151515Some of what follows is only rigorous if $G$ is compact, but the formal discussion can be applied generically. and $M$ being a smooth $G$-manifold. We saw that de Rham’s theorem provides an _algebraic model_ for the singular cohomology (with real coefficients) of the smooth manifold $M$, through the complex of differential forms. We now describe a way to obtain an algebraic model for the homotopy quotient $(M\times EG)/G$, the so-called _Weil model_ for the $G$-equivariant cohomology of $M$. From the discussion of the last sections, it is already imaginable that this will contain in some way the de Rham complex of $M$, but modifying it through a somewhat “trivial” extension, in the sense of the triviality of the cohomology of $EG$. This is thus the most natural model that is connected to the topological definition of the last section, but we will see that it is also overly complicated. In fact, in the next section we will describe a simpler but equivalent way to obtain the same equivariant cohomology, the _Cartan model_ , that is more intuitive from the differential geometry point of view, and that we will use to generalize the theory of integration to the equivariant setting. This is what we often use in physics for practical calculations. Before defining the Weil model, we notice that, in presence of a $G$-action, the de Rham complex $(\Omega(M),d)$ of differential forms on $M$ has more structure than being a dg algebra. In fact, if $\rho:G\to\text{Diff}(M)$ is the $G$-action, this induces an infinitesimal action of the Lie algebra $\mathfrak{g}$ on any tensor space via the Lie algebra homomorphism161616Usually, in physics conventions, in the action of the exponential map one collects a factor of $i$ at the exponent, in order to consider the Lie algebra element Hermitian for the most commonly considered group actions. Left and right actions should be taken with different signs at the exponent. $\displaystyle\mathfrak{g}$ $\displaystyle\to\Gamma(TM)$ (2.19) $\displaystyle X$ $\displaystyle\mapsto\underline{X}:=\left.\frac{d}{dt}\right\|_{t=0}\rho\left(e^{-tX}\right)^{}$ that defines for any $X\in\mathfrak{g}$ the corresponding _fundamental vector field_ $\underline{X}\in\Gamma(TM)$. Then $\mathfrak{g}$ acts infinitesimally on $\Omega(M)$ via the _Lie derivative_ and the _interior multiplication_ with respect to the fundamental vector fields, $\begin{aligned} \mathcal{L}_{X}(\alpha)&:=\mathcal{L}_{\underline{X}}(\alpha)\\\ \iota_{X}\alpha&:=\iota_{\underline{X}}\alpha\end{aligned}\qquad\text{for}\ X\in\mathfrak{g},\alpha\in\Omega(M),$ (2.20) with the additional property (_Cartan’s magic formula_) $\mathcal{L}_{X}=d\circ\iota_{X}+\iota_{X}\circ d.$ (2.21) This makes $\Omega(M)$ into a so-called _$\mathfrak{g}$ -gd algebra_. In general, a $\mathfrak{g}$-gd algebra is defined as a differential graded algebra (cf. definition 2.1.3) with two actions of $\mathfrak{g}$, denoted by analogy as $\iota$ and $\mathcal{L}$, such that for any $X\in\mathfrak{g}$ 1. (i) $\iota_{X}$ acts as an _antiderivation_ of degree $-1$, satisfying $(\iota_{X})^{2}=0$; 2. (ii) $\mathcal{L}_{X}$ acts as a derivation (of degree 0); 3. (iii) the Cartan’s magic formula holds: $\mathcal{L}_{X}=d\circ\iota_{X}+\iota_{X}\circ d$. Morphisms of $\mathfrak{g}$-dg algebras are naturally defined as maps between $\mathfrak{g}$-dg algebras that commute with all the above stated operations. It is not difficult to show that $G$-equivariant maps of $G$-manifolds induce pull-backs of differential forms that preserve the $\mathfrak{g}$-dg algebra structure. Now we can define the Weil model via an extension of $(\Omega(M),d)$ that preserves this new structure. We want this extension to be an algebraic analog of $EG$, so its cohomology must be trivial, but carrying information about $\mathfrak{g}$. To do this, we associate it to the characteristic differential structure of a generic principal $G$-bundle $P$ (remember that any principal $G$-bundle sits inside $EG$): its connection 1-form $A\in\Omega^{1}(P)\otimes\mathfrak{g}$ and the associated curvature $F=dA+\frac{1}{2}[A\stackrel{{\scriptstyle\wedge}}{{,}}A]\in\Omega^{2}(P)\otimes\mathfrak{g}$, satisfying the Bianchi identity $dF=[F,A]$. We first notice that the connection 1-form and the curvature 2-form can be seen as linear maps $\begin{aligned} A:\mathfrak{g}^{}&\to\Omega^{1}(P)\\\ \eta&\mapsto A(\eta):=(\eta\circ A),\end{aligned}\qquad\begin{aligned} F:\mathfrak{g}^{}&\to\Omega^{2}(P)\\\ \eta&\mapsto F(\eta):=(\eta\circ F).\end{aligned}$ (2.22) These maps can be extended multi-linearly to the whole $\Omega(P)$, if we start from algebras constructed by $\mathfrak{g}^{}$ that respect the commutativity of 1- and 2-forms, respectively. This means that $A$ has to “eat” an element of the (anticommutative) exterior algebra $\bigwedge(\mathfrak{g}^{})$, while $F$ has to “eat” an element of the (commutative) symmetric algebra $S(\mathfrak{g}^{})$: $\begin{aligned} A:\bigwedge(\mathfrak{g}^{})&\to\Omega(P)\\\ \eta_{1}\wedge\cdots\wedge\eta_{k}&\mapsto A(\eta_{1})\wedge\cdots\wedge A(\eta_{k}),\end{aligned}\qquad\begin{aligned} F:S(\mathfrak{g}^{})&\to\Omega(P)\\\ \eta_{1}\cdots\eta_{k}&\mapsto F(\eta_{1})\wedge\cdots\wedge F(\eta_{k}).\end{aligned}$ (2.23) We can combine the two maps in the homomorphism of graded-algebras $\displaystyle f:S(\mathfrak{g}^{})\otimes\bigwedge(\mathfrak{g}^{})$ $\displaystyle\to\Omega(P)$ (2.24) $\displaystyle\eta\otimes\xi$ $\displaystyle\mapsto F(\eta)\wedge A(\xi).$ This captures the fact that a connection of $P$ could be _defined_ as a map $S(\mathfrak{g}^{})\otimes\bigwedge(\mathfrak{g}^{})\to\Omega(P)$, and motivates the following definition. ###### Definition 2.3.1. The _Weil algebra_ of $\mathfrak{g}$ is the graded algebra $W(\mathfrak{g}):=S(\mathfrak{g}^{})\otimes\bigwedge(\mathfrak{g}^{})$ and the map $f:W(\mathfrak{g})\to\Omega(P)$ is called the _Weil map_. We define the graded structure of $W(\mathfrak{g})$ by assigning to the generators $\\{\phi^{a}\\}$ of $S(\mathfrak{g}^{})$ degree $\textrm{deg}(\phi^{a})=2$, and to the generators $\\{\theta^{a}\\}$ of $\bigwedge(\mathfrak{g}^{})$ degree $\textrm{deg}(\theta^{a})=1$. The generators $\\{\phi^{a},\theta^{a}\\}$ are two copies of a basis set for $\mathfrak{g}^{}$, but taken in different degrees. With respect to this _graded_ basis, the Weil algebra can also be written as $W(\mathfrak{g})=\bigwedge\left(\mathbb{R}[\phi^{1},\cdots,\phi^{\dim\mathfrak{g}}]\oplus\mathbb{R}[\theta^{1},\cdots,\theta^{\dim{\mathfrak{g}}}]\right),$ (2.25) and a generic element will be expanded as $\alpha=\alpha_{0}+\alpha_{a}\theta^{a}+\frac{1}{2}\alpha_{ab}\theta^{a}\theta^{b}+\cdots+\alpha^{(top)}\theta^{1}\theta^{2}\cdots\theta^{\dim\mathfrak{g}}\quad\text{with}\quad\alpha_{I}\in\mathbb{R}[\phi^{1},\cdots,\phi^{\dim\mathfrak{g}}],$ (2.26) since higher order terms vanish by the anticommutativity of the $\theta$’s. Here we suppressed tensor and wedge products to simplify the notation, as we will often do in the following. On this basis, the Weil map projects simply the connection and the curvature on the given Lie algebra components, $f(\theta^{a})=\theta^{a}\circ A=A^{a},\qquad f(\phi^{a})=\phi^{a}\circ\Omega=\Omega^{a}.$ (2.27) The Weil algebra is the central object to define an algebraic model for $EG$. We need to define a $\mathfrak{g}$-dg algebra structure on it to properly take its cohomology, but this is naturally done requiring the Weil map to be a morphism of $\mathfrak{g}$-dg algebras. This means introducing a differential $d_{W}:W(\mathfrak{g})\to W(\mathfrak{g})$ and two $\mathfrak{g}$-actions $\mathcal{L},\iota$ such that the following diagram commutes for all the three operations separately, ${W(\mathfrak{g})}$${\Omega(P)}$${W(\mathfrak{g})}$${\Omega(P).}$$\scriptstyle{d_{W}\ \iota\ \mathcal{L}}$$\scriptstyle{f}$$\scriptstyle{d\ \iota\ \mathcal{L}}$$\scriptstyle{f}$ (2.28) One can check that, defining the _Weil differential_ $d_{W}$ on the generators as171717In the second column we introduced $\theta:=\theta^{i}\otimes T_{i}$ and $\phi:=\phi^{i}\otimes T_{i}$ in $W(\mathfrak{g})\otimes\mathfrak{g}$, where $\\{T_{i}\\}$ is a basis of $\mathfrak{g}$ dual to the generators. Notice that this notation make the formulas independent on a choice of basis. Also, these are the objects that are really correspondent to the connection $A$ and the curvature $F$ on $P$, respectively. $\begin{aligned} d_{W}\theta^{a}&=\phi^{a}-\frac{1}{2}f^{a}_{bc}\theta^{b}\theta^{c}\\\ d_{W}\phi^{a}&=f^{a}_{bc}\phi^{b}\theta^{c}\end{aligned}\quad\text{or}\quad\begin{aligned} d_{W}\theta&=\phi-\frac{1}{2}[\theta\stackrel{{\scriptstyle\wedge}}{{,}}\theta]\\\ d_{W}\phi&=[\phi\stackrel{{\scriptstyle\wedge}}{{,}}\theta]\end{aligned}$ (2.29) where $f_{ab}^{c}$ are the structure constants of $\mathfrak{g}$, it commutes with $f$ giving correctly the definition of curvature and the Bianchi identity. Moreover, extending the differential on $W(\mathfrak{g})$ as an antiderivation of degree $+1$, it gives $d_{W}^{2}=0$ (since $d_{W}^{2}$ is a derivation, it is enough to check it on the generators). To be compatible with the properties of the connection and the curvature $\iota_{X}A=A(\underline{X})=X,\qquad\iota_{X}F=0\qquad\forall X\in\mathfrak{g},$ (2.30) the interior multiplication must be defined as $\begin{aligned} \iota_{X}\theta^{a}&:=\theta^{a}(X)=X^{a}\\\ \iota_{X}\phi^{a}&:=0\end{aligned}\quad\text{or}\quad\begin{aligned} \iota_{X}\theta&:=\theta(X)=X\\\ \iota_{X}\phi&:=0\end{aligned}$ (2.31) and extended as an antiderivation of degree $-1$. Then the Lie derivative is simply defined via Cartan’s magic formula. We finally have defined the Weil algebra as a $\mathfrak{g}$-dg algebra. ###### Theorem 2.3.1. The cohomology of the Weil algebra is $H^{0}(W(\mathfrak{g}),d_{W})\cong\mathbb{R},\qquad H^{k}(W(\mathfrak{g}),d_{W})\cong 0\ \text{for}\ k>0.$ ###### Proof. The full proof can be found in [18]. Schematically it follows the proof of the Poincaré lemma: one has to find an _cochain homotopy_ , i.e. a map $K:W(\mathfrak{g})\to W(\mathfrak{g})$ of degree -1, such that $[K,d_{W}]_{+}=id$. Then any cocycle ($d_{W}\alpha=0$) is also a coboundary, since $\alpha=[K,d_{W}]_{+}\alpha=d_{W}(K\alpha)$. This can be found for any degree $k>0$. In degree zero $W^{0}(\mathfrak{g})\cong\mathbb{R}$ by definition, so every element is a cocycle, and no one is a coboundary for degree reasons. ∎ ###### Example 2.3.1 (Weil model for torus and circle actions). Consider the case of a compact Abelian group, i.e. a torus $T=U(1)^{l}$ for some $l$. Remember that a possible purpose of the Weil algebra is to describe the connection and the curvature of any principal $T$-bundle, so we are somewhat analyzing the structure of $l$ electromagnetic fields, from the point of view of the Lie algebra $\mathfrak{t}$. Since the structure constants are all zero, the Weil differential (2.29) and the $\mathfrak{t}$-actions (2.31) on the generators $(\theta^{a},\phi^{a})$ simplify as $\begin{array}[]{ll}d_{W}\theta^{a}=\phi^{a},&d_{W}\phi^{a}=0,\\\ \iota_{b}\theta^{a}=\delta^{a}_{b},&\iota_{b}\phi^{a}=0,\\\ \mathcal{L}_{b}\theta^{a}=0,&\mathcal{L}_{b}\phi^{a}=0,\end{array}$ where we denoted $\iota_{a}\equiv\iota_{T_{a}}$ and $\mathcal{L}_{a}\equiv\mathcal{L}_{T_{a}}$, with $\\{T_{a}\\}$ the basis of $\mathfrak{t}$ dual to the generators of $W(\mathfrak{t})$. We jump ahead a little and notice that the first line really resembles the structure of a “supersymmetry” transformation, with $\phi^{a}$ being the “bosonic partner” of $\theta^{a}$. The remaining non-Abelian piece of the generic case can be viewed as the action of a Chevalley-Eilemberg differential, so that $d_{W}=d_{susy}+d_{CE}$.181818Remember that the C-E differential is the one that appears in BRST quantization of gauge theories. We will return to this point in Section 4.5, after having introduced some technology about supergeometry and supersymmetry. Let us simplify again and prove theorem 2.3.1 for $l=1$. In the case of $U(1)$, the Lie algebra has only one generator $T\cong i$, and the symmetric algebra is the algebra of polynomials in the indeterminate $\phi\in\mathfrak{g}^{}$, $S(\mathfrak{g}^{})=\mathbb{R}[\phi]$, while the exterior algebra reduces to $\bigwedge(\theta)=\mathbb{R}\oplus\mathbb{R}\theta$ by anticommutativity. The Weil algebra is thus $W(\mathfrak{u}(1))=\mathbb{R}[\phi]\otimes\left(\mathbb{R}\oplus\mathbb{R}\theta\right)=\mathbb{R}[\phi]\oplus\mathbb{R}[\phi]\theta.$ The cohomology of $W(\mathfrak{u}(1))^{0}=\mathbb{R}$ in degree zero is as always trivial, since all constant numbers are closed, and none of them is exact, giving $H^{0}(W(\mathfrak{u}(1)),d_{W})\cong\mathbb{R}$. In degree 1 we have $W(\mathfrak{u}(1))^{1}=\mathbb{R}\theta$, thus no one element (but zero) is closed. This extends to any odd-degree, since $W(\mathfrak{u}(1))^{2n+1}=\mathbb{R}\phi^{n}\theta$, and $d_{W}(\phi^{n}\theta)=\phi^{n+1}\neq 0$. This means that $H^{2n+1}(W(\mathfrak{u}(1)),d_{W})\cong 0$. In degree 2, we have $W(\mathfrak{u}(1))^{2}=\mathbb{R}\phi$, so that any element is closed but also exact, since $\phi=d_{W}\theta$. This extends to any even-degree, since $W(\mathfrak{u}(1))^{2n}=\mathbb{R}\phi^{n}$, and $\phi^{n}=d_{W}\theta\phi^{n-1}=d_{W}(\theta\phi^{n-1})$. Thus we have also $H^{2n}(W(\mathfrak{u}(1)),d_{W})\cong 0$, showing the triviality of the Weil algebra in the simplest case of a circle action. Almost the same direct computation can be carried out in the $l$-dimensional case. Theorem 2.3.1 shows that we are in business: the Weil algebra is exactly an algebraic analog of the universal bundle $EG$. Since the de Rham model for $M$ is just $\Omega(M)$, the product $M\times EG$ can be modeled by the complex $W(\mathfrak{g})\otimes\Omega(M)$, since by the Künneth formula [24] and de Rham’s theorem $H^{}(EG\times M)=H^{}(EG)\otimes H^{}(M)=H^{}(W(\mathfrak{g}^{}),d_{W})\otimes H^{}(\Omega(M),d).$ (2.32) The differential and the $\mathfrak{g}$-actions are extended naturally on this complex as graded derivations, making it into a $\mathfrak{g}$-dg algebra too. Explicitly, $\displaystyle d_{T}$ $\displaystyle:=d_{W}\otimes 1+1\otimes d,$ (2.33) $\displaystyle\iota$ $\displaystyle\equiv\iota\otimes 1+1\otimes\iota,$ $\displaystyle\mathcal{L}$ $\displaystyle\equiv\mathcal{L}\otimes 1+1\otimes\mathcal{L}.$ A model for the homotopy quotient $M_{G}$ can be guessed by the following argument. Since $M_{G}$ is the base of the principal bundle $EG\times G\to M_{G}$, differential forms on $M_{G}$ identify the _basic forms_ on $EG\times M$ (see Appendix A), i.e. those that are both _$G$ -invariant_ and _horizontal_. It is thus reasonable that the homotopy quotient can be modeled by the _basic subcomplex_ of the Weil model. Since the differential closes on the basic subcomplex, we are allowed to take its cohomology, giving the $G$-equivariant cohomology of $M$. This is exactly the content of the _equivariant de Rham’s theorem_. A recent original proof of it can be found in [18]. ###### Theorem 2.3.2 (equivariant de Rham). If $G$ is a connected Lie group, and $M$ is a $G$-manifold, $\boxed{H^{}_{G}(M)\cong H^{}\left(\left(W(\mathfrak{g})\otimes\Omega(M)\right)_{bas},d_{T}\right)}.$ The equivariant de Rham’s theorem is telling us that the “correct” differential complex that encodes the topology of the $G$-action on $M$ is _not_ anymore the complex $\Omega(M)$ of differential forms, but a modification of it through the presence of the Weil algebra. Remember always that, via the Weil map, $W(\mathfrak{g})$ can be thought as in correspondence with the presence of a connection and a curvature on some principal $G$-bundle. This means that the right extension of the de Rham complex in presence of a $G$-action embeds somewhat the presence of a connection and a curvature with respect to $G$. We can analyze as an example the simplest case of a $U(1)$-action. The unrestricted Weil model is (from Example 2.3.1) $W(\mathfrak{u}(1))\otimes\Omega(M)=\Omega(M)[\phi]\oplus\Omega(M)[\phi]\theta,$ (2.34) thus any element can be written as $\alpha=\alpha^{(0)}+\alpha^{(1)}\theta,$ (2.35) where $\alpha^{(0)},\alpha^{(1)}\in\Omega(M)[\phi]$ are polynomials in $\phi$ with differential forms as coefficients. The subcomplex of basic forms consists of those elements that satisfy both $\iota_{T}\alpha=0$ and $\mathcal{L}_{T}\alpha=0$, imposing the conditions $\alpha^{(1)}=-\iota_{T}\alpha^{(0)},\qquad\mathcal{L}_{T}\alpha^{(0)}=0.$ (2.36) Thus any basic element can be written as $\alpha=(1-\theta\iota_{T})\sum_{i}(\phi)^{i}\alpha^{(0)}_{i}$, where all the differential forms $\alpha^{(0)}_{i}\in\Omega(M)^{G}$ must be $G$-invariant, and the basic subcomplex can be identified with the polynomials in $\phi$ with invariant differential forms as coefficients. In the next section we will argue that this is not a special case, and that the Weil model can be simplified in general, producing another model for the same equivariant cohomology. ### 2.4 The Cartan model As we said at the beginning of the last section, the cohomology of the Weil complex is not the unique algebraic model for the $G$-equivariant cohomology of the $G$-manifold $M$. Moreover, although very transparent, the Weil model seems overly complicated for differential geometric applications. In fact, the extreme simplicity of the basic subcomplex of the Weil algebra $W(\mathfrak{g})_{bas}$ suggests that a simpler model for equivariant cohomology can be obtained simplifying this one. To see this, we analyze this basic subcomplex first. As we recalled at the end of the last section (for further details see Appendix A), a basic element $\alpha\in W(\mathfrak{g})_{bas}$ is both _horizontal_ and _invariant_ , i.e. $\iota_{X}\alpha=0=\mathcal{L}_{X}\alpha.$ (2.37) The horizontal condition means that we pick only the symmetric algebra inside $W(\mathfrak{g})$, since by definition $\iota_{X}\phi^{a}=0$. Imposing also the $G$-invariance we have $W(\mathfrak{g})_{bas}\cong S(\mathfrak{g}^{})^{G},$ (2.38) i.e. the basic subcomplex is the algebra of Casimir invariants. It is easy to check that on this subcomplex $d_{W}\cong 0$, so that $H^{}(W(\mathfrak{g})_{bas},d_{W})=H^{}(S(\mathfrak{g}^{})^{G},d_{W})=S(\mathfrak{g}^{})^{G}\quad\text{in degree 0,}$ (2.39) since every element is closed, and no one element can be exact. Moreover, from the equivariant de Rham’s theorem $H_{G}^{}(pt)\cong H^{}(W(\mathfrak{g})_{bas},d_{W})$, so the Casimir invariants are precisely the cohomology of the classifying space, $H^{}(BG)\cong S(\mathfrak{g}^{})^{G}.$ (2.40) Motivated by the above simplification, we can turn now to analyze the complete Weil model $(W(\mathfrak{g})\otimes\Omega(M))_{bas}$. Let us see concretely what it means to restrict the attention to a basic element $\alpha\in(W(\mathfrak{g})\otimes\Omega(M))_{bas}$, starting from its expansion on a basis (2.26). Imposing the horizontality condition means, using the multi-index notation $I=(a_{1},\cdots,a_{\|I\|})$, $0=\iota_{X}\alpha=\iota_{X}\left(\alpha_{0}+\frac{1}{\|I\|!}\alpha_{I}\theta^{I}\right)\quad\Rightarrow\quad 0=\iota_{X}\alpha_{0}+\frac{1}{\|I\|!}(\iota_{X}\alpha_{I})\theta^{I}+\frac{1}{\|I\|!}\alpha_{I}(\iota_{X}\theta^{I}).$ (2.41) Equating the terms of the same degree and taking $X$ to be a basis element, one arrives at the condition on the various components, $\alpha_{a_{1}\cdots a_{\|I\|}}=(-1)^{\|I\|}\iota_{a_{1}}\cdots\iota_{a_{\|I\|}}\alpha_{0}$ (2.42) where we denoted $\iota_{k}:=\iota_{T_{k}}$, meaning that a horizontal element is fully determined by its first component $\alpha_{0}\in S(\mathfrak{g}^{})\otimes\Omega(M)$, and it can be expressed as $\alpha=\left(\prod_{k=1}^{\dim\mathfrak{g}}(1-\theta^{k}\iota_{k})\right)\alpha_{0}.$ (2.43) This comment, with some more checks (see again [18] for a complete proof), proves the following theorem, and extends the above discussion to the complete Weil model. ###### Theorem 2.4.1 (Mathai-Quillen isomorphism). There is an isomorphism of $\mathfrak{g}$-dg algebras, called the _Mathai- Quillen isomorphism_ [29] (or _Cartan-Weil_ in [18]), $\displaystyle\varphi:(W(\mathfrak{g})\otimes\Omega(M))_{hor}$ $\displaystyle\to S(\mathfrak{g}^{})\otimes\Omega(M)$ $\displaystyle\alpha=\alpha_{0}+\frac{1}{\|I\|!}\alpha_{I}\theta^{I}$ $\displaystyle\mapsto\alpha_{0}$ $\displaystyle\left(\prod_{k=1}^{\dim\mathfrak{g}}(1-\theta^{k}\iota_{k})\right)\alpha_{0}$ $\displaystyle\mathrel{\reflectbox{$\mapsto$}}\alpha_{0}.$ The RHS of the isomorphism above inherits the $\mathfrak{g}$-actions and the differential from the Weil model on the LHS, making commutative the following diagram, similarly to (2.28), ${(W(\mathfrak{g})\otimes\Omega(M))_{hor}}$${S(\mathfrak{g}^{})\otimes\Omega(M)}$${(W(\mathfrak{g})\otimes\Omega(M))_{hor}}$${S(\mathfrak{g}^{})\otimes\Omega(M).}$$\scriptstyle{d_{T}\ \iota\ \mathcal{L}}$$\scriptstyle{\varphi}$$\scriptstyle{d_{C}\ \iota\ \mathcal{L}}$$\scriptstyle{\varphi}$ (2.44) In particular, the new differential is called _Cartan differential_ , defined such that $d_{C}:=\varphi\circ d_{T}\circ\varphi^{-1}.$ (2.45) It is not difficult to get, directly from this definition, that it can be expressed more simply as $d_{C}=1\otimes d-\phi^{k}\otimes\iota_{k}$ (2.46) where $d$ is the de Rham differential on $\Omega(M)$. The two $\mathfrak{g}$-actions commute with $\varphi$ without modification, so they agree with their behavior in the Weil model, $\mathcal{L}_{X}\phi^{a}=f^{a}_{bc}\phi^{b}X^{c},\qquad\iota_{X}\phi^{a}=0.$ (2.47) Using the nilpotency of $d$ and $\iota$, and Cartan’s magic formula, wee see that the Cartan differential on the horizontal subcomplex squares to a Lie derivative (an infinitesimal symmetry transformation) $d_{C}^{2}=-\phi^{k}\otimes\mathcal{L}_{k},$ (2.48) so when we restrict to the $G$-invariant subspace, $d_{C}^{2}\cong 0$ on $\left(S(\mathfrak{g}^{})\otimes\Omega(M)\right)^{G}$, as it should. Using the Mathai-Quillen isomorphism and the equivariant de Rham theorem we then have the fundamental result, $\boxed{H_{G}^{}(M)\cong H^{}\left(\left(S(\mathfrak{g}^{})\otimes\Omega(M)\right)^{G},d_{C}\right)}$ (2.49) that simplifies the algebraic model for the $G$-equivariant cohomology of $M$. ###### Definition 2.4.1. The $\mathfrak{g}$-dg algebra $\Omega_{G}(M):=\left(S(\mathfrak{g}^{})\otimes\Omega(M)\right)^{G}$ with the Cartan differential $d_{C}$ is called the _Cartan model_ for the equivariant cohomology of $M$. Elements of $\Omega_{G}(M)$ are called _equivariant differential forms_ on $M$. The degree of an equivariant form is the total degree with respect to the generators of $\Omega(M)$ (in degree 1), and the generators of $S(\mathfrak{g^{}})$ (in degree 2). Equivariant forms will be in the next chapters the the principal object of study. In the physical applications we are interested in, we will always search for an interpretation of the space of interest as a Cartan model with respect to the action of a symmetry group $G$. The Cartan differential will be some object that squares to an infinitesimal symmetry, and on the subspace of $G$-invariant forms (or “fields”, in the following) it will define a $G$-equivariant cohomology. Cartan differentials arise in Field Theory as _supersymmetry transformations_ , that we will contextualize in Chapter 4 and relate to equivariant cohomology in Chapter 5. This interpretation will be crucial in treating some of the most important objects in QM and QFT that arise as (infinite-dimensional) path integrals over the space of fields. In fact, we will see in the next chapter that integration of equivariant forms leads to powerful _localization thorems_ , that formally extended to the infinite-dimensional case greatly simplifying those integrals. ###### Example 2.4.1 (Cartan model for $U(1)$-equivariant cohomology). Until Chapter 6, we will actually deal with the equivariant cohomology with respect to a circle action of $G=U(1)$, or at most a torus action of $U(1)^{n}$ for some $n$. As we saw also for the Weil model, this greatly simplifies the problem, so we carry on that example also in the Cartan model for a $U(1)$-action. We recall from Example 2.3.1 that in the Weil model $\iota_{T}\phi=\mathcal{L}_{T}\phi=d_{W}\phi=0,$ i.e. $\phi$ is automatically also $U(1)$-invariant. The equivariant forms are thus $\Omega_{U(1)}(M)=(\mathbb{R}[\phi]\otimes\Omega(M))^{U(1)}\cong\Omega(M)^{U(1)}[\phi],$ so polynomials in $\phi$ with $U(1)$-invariant forms as coefficients. The Cartan differential is, suppressing tensor products, $d_{C}=d-\phi\ \iota_{T}.$ In this 1-dimensional case, the indeterminate $\phi$ is just a spectator, and serves only to properly count the equivariant form-degree. This is important of course, but for many purposes it creates no confusion to suppress its presence. More precisely, we often _localize_ the algebra $\Omega_{G}(M)$, substituting the _indeterminate_ $\phi$ with a _variable_ , and setting it for example to $\phi=-1$,191919This could seem harmless, but it is definitely a non-trivial move. We are really able to do this without spoiling the resulting equivariant cohomology because (algebraic) localization commutes with taking cohomology. More details on this are reported in Appendix B.3. so that $d_{C}=d+\iota_{T}.$ This differential squares to an infinitesimal symmetry generated by $\underline{T}$, $d_{C}^{2}=\mathcal{L}_{T}$. Equivariant differential forms after this localization are just $U(1)$-invariant forms. Often it is useful to generate equivariant forms from invariant differential forms in $\Omega(M)$, for the purpose of integration for example. If $\alpha\in\Omega^{2n}(M)$, an _equivariant extension_ of $\alpha$ is $\tilde{\alpha}\in\Omega_{U(1)}(M)$ such that $\tilde{\alpha}=\alpha+f_{(2n-2)}\phi+f_{(2n-4)}\phi^{2}+\cdots$ where any coefficient is an invariant form in $\Omega(M)$. As an example, we can take the circle acting on the 2-sphere $\mathbb{S}^{2}$, via rotations around a chosen axis. If $\theta$ is the polar coordinate and $\varphi$ is the azimutal coordinate, $(\theta,\varphi)\left(e^{it}\cdot p\right):=(\theta(p),\varphi(p)+t)\qquad\text{for}\ p\in\mathbb{S}^{2},e^{it}\in U(1),$ so that the fundamental vector field is $\underline{T}=\frac{\partial}{\partial\varphi}$.202020Recall that this action has two fixed points, at the North and the South pole. Consider the canonical volume-form $\omega=d\cos{(\theta)}\wedge d\varphi$. It is obviously closed, and also $U(1)$-invariant, since $\mathcal{L}_{T}\omega=0$. Aiming to the extension of $\Omega(M)$ to $\Omega_{U(1)}(M)$, we can find an _equivariantly closed extension_ of the volume form $\tilde{\omega}=\omega+f\phi$, with $f\in C^{\infty}(M)$ such that $d_{C}\tilde{\omega}=0$, so that it is closed in the “correct” complex. This imposes the equation $df=\iota_{T}\omega$,212121$\omega$ is a _symplectic_ form on $\mathbb{S}^{2}$, and the $df=\iota_{T}\omega$ means that $H:=-f$ is the _Hamiltonian function_ with respect to the $U(1)$-action on the sphere. We will deepen this point of view in the next chapter. and so $f=-\cos(\theta)$: $\tilde{\omega}=\omega-\cos(\theta)\phi.$ ### 2.5 The BRST model In this section we mention the last popular model for equivariant cohomology: the so-called _BRST model_ , or sometimes _intermediate model_. It is worth to mention it because we will see in the next chapter that it is (as the first name suggests) intimately related to the BRST method for gauge-fixing in the Hamiltonian formalism. Moreover, its complex is the one that arises naturally in Topological Field Theories (TFT), as we will mention in Chapter 6. It is also important because it provides (as the second name suggests) an “interpolation” between the Weil and the Cartan models that we saw in the last sections, relating the latter more “physical”(or differential geometric) point of view with the former more “topological” one. As an algebra, the (unrestricted) complex of the BRST model is identical to that of the Weil model, $B:=W(\mathfrak{g})\otimes\Omega(M),$ (2.50) but with the new differential (compare to (2.29) and (2.33)) $d_{B}=d_{W}\otimes 1+1\otimes d+\theta^{a}\otimes\mathcal{L}_{a}-\phi^{a}\otimes\iota_{a}$ (2.51) that satisfies $d_{B}^{2}=0$ on $B$, and has the same trivial cohomology of the unrestricted Weil model. The idea that brought to the construction of this model in [30], was essentially to prove along the line we did in the last section the equivalence of the models, but from a slightly different point of view. In fact one can construct $d_{B}$ using an algebra automorphism that carries the Weil model $(B,d_{W})$ into the BRST model $(B,d_{B})$, at the level of the unrestricted algebras. The restriction to the basic subcomplex gives then automatically the Cartan model. The automorphism is given by the map $\varphi:=e^{\theta^{a}\iota_{a}}\equiv\prod_{a}(1+\theta^{a}\otimes\iota_{a}),$ (2.52) that looks very similar to the Mathai-Quillen isomorphism of theorem 2.4.1, but now is applied to the whole algebra and not only on the horizontal part. Analogously to the definition of the Cartan differential, $d_{B}$ is got as (2.51) from the commutativity of the diagram ${B}$${B}$${B}$${B,}$$\scriptstyle{d_{W}}$$\scriptstyle{\varphi}$$\scriptstyle{d_{B}}$$\scriptstyle{\varphi}$ (2.53) so that $d_{B}=\varphi^{-1}\circ d_{W}\circ\varphi$, as well as the two $\mathfrak{g}$-actions. In particular, it results $\displaystyle\iota^{(B)}$ $\displaystyle=\iota\otimes 1\neq\iota^{(W)},$ (2.54) $\displaystyle\mathcal{L}^{(B)}$ $\displaystyle=\mathcal{L}\otimes 1+1\otimes\mathcal{L}=\mathcal{L}^{(W)},$ where we called $\iota^{(W)},\mathcal{L}^{(W)}$ the one defined in (2.33). Thus the BRST differential carries the same information of the Weil differential, giving the same trivial cohomology of the unrestricted Weil model, $H^{}(B,d_{B})\cong H^{}(B,d_{W})\cong H_{dR}(M)$ (2.55) where the last equivalence follows from the triviality of the cohomology of the Weil algebra $W(\mathfrak{g}^{})$. Of course, we have to restrict the the action of $d_{B}$ to the basic subcomplex, i.e. to the intersection with the kernels of $\iota^{(B)}$ and $\mathcal{L}^{(B)}$, to get a meaningful $G$-equivariant cohomology. This reproduces again the Cartan model, as expected. This result shows that there is in fact a whole continuous family of $\mathfrak{g}$-dg algebras that give equivalent models for the $G$-equivariant cohomology of $M$, because we can conjugate the Weil differential through the modified automorphism $\varphi_{t}:=e^{t\theta^{a}\iota_{a}}\qquad\text{with}\ t\in\mathbb{R}.$ (2.56) This produces, by conjugation, the family of differentials and $\mathfrak{g}$-actions on $B$, $\displaystyle d^{(t)}$ $\displaystyle=d_{W}\otimes 1+1\otimes d+t\theta^{a}\otimes\mathcal{L}_{a}-t\phi^{a}\otimes\iota_{a}+\frac{1}{2}t(1-t)f_{ab}^{c}\theta^{a}\theta^{b}\otimes\iota_{c},$ (2.57) $\displaystyle\iota^{(t)}$ $\displaystyle=\iota\otimes 1+(1-t)1\otimes\iota,$ $\displaystyle\mathcal{L}^{(t)}$ $\displaystyle=\mathcal{L}^{(W)}\quad\forall t.$ We see that for $t=0$ we recover the Weil model, while for $t=1$ we get the BRST model, as special cases. When restricted to the basic subcomplex, they all give the same equivariant cohomology. ## Chapter 3 Localization theorems in finite-dimensional geometry In this chapter we are going to introduce one of the most important results of the equivariant cohomology theory: the _Atiyah-Bott-Berline-Vergne (ABBV) localization formula_ for torus actions, discovered independently by Berline and Vergne [7], and by Atiyah and Bott [6]. For the most applications to QM and QFT, we will focus on the case of a circle action, and higher-dimensional generalizations will be postponed to Chapter 6. This formula can be viewed as a generalization of an analogous result of Duistermaat and Heckman [5], that treats the special case in which the torus action is Hamiltonian on a symplectic manifold. We will expand on this point of view in the second part of the chapter, since this is the situation we are more commonly interested in when we treat dynamical systems in physics, at least at the classical level. The formal generalization of these formulas in the infinite-dimensional setting of QFT will be discussed in Chapter 5. Since we are going to deal with integration of equivariant forms, we consider $U(1)$-equivariant cohomologies from the point of view of the Cartan model. The definition and notational conventions for integration of equivariant forms on a smooth $G$-manifold are reported in Appendix B.4, as well as an equivariant version of the Stokes’ theorem, needed for the proof of the localization formulas that are presented in the following. ### 3.1 Equivariant localization principle Let $U(1)$ act (smoothly) on a compact oriented $n$-dimensional manifold $M$ without boundaries,111If not specified a manifold is always “without boundaries” since, strictly speaking, manifolds with boundaries have to be defined in an appropriate separated way. In particular, near points at the boundary the manifold is locally homeomorphic not to an open set in $\mathbb{R}^{n}$, but to an _half-open_ disk in $\mathbb{R}^{n}$. with fixed point set $F\subseteq M$, and consider the integral of a generic $U(1)$-invariant top-form $\int_{M}\alpha\qquad\quad\text{with}\ \alpha\in\Omega^{n}(M)^{U(1)}.$ (3.1) As we saw in Example 2.4.1, in some cases we can find an _equivariantly closed extension_ $\tilde{\alpha}\in\Omega_{U(1)}(M)$ such that $d_{C}\alpha=0$, with $d_{C}=d+\iota_{T}$ (3.2) and $T\cong i$ being the generator of $U(1)$.222Notice that we have localized the Cartan model and set $\phi=-1$, as discussed in Example 2.4.1. This will be our standard convention up to Chapter 6. Then we can deform the integral without changing its value, $I[\tilde{\alpha}]:=\int_{M}\tilde{\alpha}=\int_{M}\alpha$ (3.3) since only the top-degree component $\alpha$ is selected by integration. We are going to argue now that such integration of an equivariantly closed form is completely captured by its values at the fixed point locus $F$, using two different arguments. The first is cleaner, the second less explicit but more common especially in the physics literature. We are going to need in both cases some preliminary facts, that we collect in the following lemma. ###### Lemma 3.1.1. 1. (i) If $G$ is a compact Lie group, any smooth $G$-manifold $M$ admits a $G$-invariant Riemannian metric. In other words, $G$ acts via isometry on $M$, and the fundamental vector field $\underline{T}$ is a Killing vector field,333This follows from two facts: if a $G$-action on $M$ is smooth and _proper_ , then $M$ admits a $G$-invariant Riemannian structure [31]; also, it is easy to prove that any smooth action of a _compact_ Lie group is proper. $\mathcal{L}_{T}g=0.$ 2. (ii) If $G$ is a connected Lie group, then the fixed point locus is the zero locus of all the fundamental vector fields:444This is just reasonable, see [18] for a proof. Connectedness is required because we passed from the action of $G$ to the action of $\mathfrak{g}$ by the exponential map. $F\cong\\{\left.p\in M\right\|\underline{A}_{p}=0\quad\forall A\in\mathfrak{g}\\}.$ 3. (iii) For any point $p\in M$, the stabilizer of $p$ under the action of a Lie group $G$ is a closed subgroup of $G$.555By continuity of the action, every sequence inside the stabilizer of $p$ converges inside the stabilizer. ##### $1^{st}$ argument: Poincaré lemma For simplicity, suppose that $F$ contains only isolated fixed points. From lemma (i), we can pick any $U(1)$-invariant metric on $M$, and define through it _open balls_ of radius $\epsilon$ $B(p,\epsilon)$ around any fixed point $p\in F$. Then $U(1)$ acts without fixed points on the complement $\tilde{M}(\epsilon):=M\setminus\bigcup_{p\in F}B(p,\epsilon),$ (3.4) that is a manifold _with boundaries_ , them being the union of the surfaces of the balls at every fixed point (oriented in the opposite direction to the usual one). From lemma (iii), the stabilizer of any point in $\tilde{M}$ is a closed subgroup of $U(1)$, but it cannot be $U(1)$ since we excluded the fixed points, so it is discrete.666The closed subgroups of $U(1)$ are $U(1)$ and the finite cyclic groups $\\{1\\},\mathbb{Z}/n$ with $n\in\mathbb{Z}$. This means that the $U(1)$-action on $\tilde{M}$ is locally free. We would like to find an equivariant version of the Poincaré lemma on $\tilde{M}$, where the action is locally free. This means finding a map $K:\Omega(\tilde{M})^{U(1)}\to\Omega(\tilde{M})^{U(1)}$ of odd-degree such that $[d_{C},K]_{+}=id$. If we are able to find such a map, then any equivariantly closed form $\eta\in\Omega(\tilde{M})^{U(1)}$ is also equivariantly exact, $\eta=(Kd_{C}+d_{C}K)\eta=K(d_{C}\eta)+d_{C}(K\eta)=d_{C}(K\eta).$ (3.5) We can define the map $K$ by multiplication with respect to an equivariant form $\xi\in\Omega(\tilde{M})^{U(1)}$ of pure odd-degree such that $d_{C}\xi=1$, since $[\xi,d_{C}]_{+}=\xi d_{C}+(d_{C}\xi)+(-1)^{\textrm{deg}(\xi)}\xi d_{C}=1.$ (3.6) This form can be defined using again a $U(1)$-invariant metric on $M$, that we call $g$. We define the following 1-form away from the fixed point set, where $\underline{T}=0$, $\beta:=\frac{1}{g(\underline{T},\underline{T})}g(\underline{T},\cdot)$ (3.7) and notice that it is $U(1)$-invariant by invariance of $g$, and $\iota_{T}\beta=1$, so that the action of the Cartan differential on it gives $d_{C}\beta=d\beta+1$. Then the odd-degree form $\xi$ can be defined as $\xi:=\beta(d_{C}\beta)^{-1}=\beta\left(1+d\beta\right)^{-1}=\beta\sum_{i=0}^{n-1}(-1)^{i}(d\beta)^{i}.$ (3.8) The inverse of the form $(1+d\beta)$ can be guessed pretending that $d\beta$ is a number, and using the Taylor expansion $(1+z)^{-1}=\sum_{i=0}^{\infty}(-1)^{i}z^{i}.$ In the case of forms, the sum at the RHS stops at finite order, since by degree reasons $(d\beta)^{i}=0$ for $i>(n/2)$. It is easy to check that $(d_{C}\beta)^{-1}(d_{C}\beta)=1$, $d_{C}\xi=1$, and $\textrm{deg}(\xi)$ is odd. Now we know that any equivariantly closed form in $\Omega(\tilde{M})^{U(1)}$ is also equivariantly exact, so we can simplify the integral $I[\alpha]$ of an equivariantly closed form $\alpha$ using an equivariant version of Stokes’ theorem (see Appendix B.4): $\int_{\tilde{M}}\alpha=\int_{\tilde{M}}d_{C}(\xi\alpha)=\int_{\partial\tilde{M}}\xi\alpha.$ (3.9) Taking the limit $\epsilon\to 0$, the domain of integration on the LHS covers all $M$, and the integral over the boundary on the RHS reduces to a sum of integrals over the boundaries of $n$-spheres centered at each fixed point $p\in F$ (since $\partial M=\emptyset$). Thus the integral of an equivariantly closed form “localizes” as a sum over the fixed points of the $U(1)$-action, $I[\alpha]=\int_{M}\alpha=\lim_{\epsilon\to 0}\int_{\tilde{M}(\epsilon)}\alpha=\sum_{p\in F}\lim_{\epsilon\to 0}\left(-\int_{S^{n-2}_{\epsilon}(p)}(\xi\alpha)\right)=\sum_{p\in F}c_{p}$ (3.10) for some contributions $c_{p}$ at each fixed point. The precise form of these contributions will be discussed in the next section. ##### $2^{nd}$ argument: localization principle The second argument for the localization of the equivariant integral is less explicit, but more direct. Also, it is closer to the approach we will use in the infinite-dimensional context of supersymmetric QFT. Again, we start from the integral $I[\alpha]$ of an equivariantly closed form $\alpha\in\Omega(M)^{U(1)}$. The basic idea is to take advantage of the equivariant cohomological nature of the integral over $M$: this depends really on the cohomology class of the integrand, not on the particular representative. So we can deform the integral staying in the same class in a way that simplifies its evaluation, without changing the final result. To do this, we pick a positive definite $U(1)$-invariant 1-form $\beta$ on $M$, and define the new integral $I_{t}[\alpha]:=\int_{M}\alpha e^{-td_{C}\beta}$ (3.11) with $t\in\mathbb{R}$. It is again an integral of an equivariantly closed form, $d_{C}\left(\alpha e^{-td_{C}\beta}\right)=(d_{C}\alpha)e^{-td_{C}\beta}-t\alpha(d_{C}^{2}\beta)e^{-td_{C}\beta}=0,$ (3.12) since $d_{C}^{2}=\mathcal{L}_{T}$ and $\beta$ is $U(1)$-invariant. To show that this integral is equivalent to $I[\alpha]$, we show that it is independent on the parameter $t$: $\displaystyle\frac{d}{dt}I_{t}[\alpha]$ $\displaystyle=\int_{M}\alpha(-d_{C}\beta)e^{-td_{C}\beta}$ (3.13) $\displaystyle=-\int_{M}d_{C}\left(\alpha\beta e^{-td_{C}\beta}\right)\qquad$ (integration by parts) $\displaystyle=0\qquad$ $\displaystyle\text{(equivariant Stokes' theorem)}.$ Noticing that $I[\alpha]=I_{t=0}[\alpha]$, from the $t$-independence it follows that $I[\alpha]=I_{t}[\alpha]$ for every value of the parameter. We showed that the deformation via the exponential $e^{-td_{C}\beta}$ does not change the equivariant cohomology class of the integrand, so we are free to compute the integral for any value of the parameter. In particular, in the limit $t\to\infty$, we see that the only contributions come from the zero locus of the exponential. This gives the “localization formula” $\int_{M}\alpha=\lim_{t\to\infty}\int_{M}\alpha e^{-td_{C}\beta},$ (3.14) that will be the starting point for all the applications of the equivariant localization principle of the next chapters, also in the infinite-dimensional case in which $M$ describes generically the “space of fields” of a given QFT. The 1-form $\beta$ is usually called “localization 1-form”. Notice that choosing different localization 1-forms produces different practical localization schemes, but at the end of the computation they must all agree on the final result! In particular, by lemma (i) we can pick a $U(1)$-invariant Riemannian metric $g$, and choose the 1-form as $\beta:=g(\underline{T},\cdot).$ (3.15) This makes it positive definite and produces the same localization scheme of the first argument, since its zeros coincide with the zeros of the fundamental vector field $\underline{T}$ and thus with the fixed point locus $F$ of the circle action, by lemma (ii). ### 3.2 The ABBV localization formula for Abelian actions Here we state the celebrated result by Atiyah-Bott and Berline-Vergne, about the localization formulas for circle and torus actions. The rationale of the last section showed that the equivariant cohomology of the manifold $M$ is encoded in the fixed point set $F$ of the symmetry action, but left us with the evaluation of an integral over the fixed point set. We show the result of this integration here, and we are going to give an argument for the proof in the next chapter, with some tools from supergeometry. That proof is different from the original ones in [6, 7], but will introduce a method that can be easily generalized to functional integrals. To warm up, we consider first the simple case of isolated fixed point set $F\subseteq M$, and a $U(1)$-action. Notice that, at any fixed point $p\in F$, the circle action gives a representation of $U(1)$ on the tangent space, since for any $\psi\in U(1)$ $(\psi\cdot)_{}:T_{p}M\to T_{\psi\cdot p}M\equiv T_{p}M,$ (3.16) so $(\psi\cdot)_{}\in GL(T_{p}M)$. Since $T_{p}M$ is finite dimensional, it can be decomposed in irreducible representations of $U(1)$, $T_{p}M\cong V_{1}\oplus\cdots\oplus V_{n}.$ (3.17) The circle has to act faithfully on $T_{p}M$, since if there was $v\in T_{p}M$ such that $(\psi\cdot)_{}v=v$, then the whole curve $\exp(tv)=\exp(t(\psi\cdot)_{}v)=\psi\cdot\exp(tv)$ would be fixed by $U(1)$, thus $p$ would not be isolated. Recall that the irreducible representations of $U(1)$ are complex 1-dimensional, and are labeled by integers, $\psi=e^{ia}\in U(1),\qquad\rho_{m}(\psi):=e^{ima}\quad\text{with}\ m\in\mathbb{Z}.$ (3.18) This means that the irreducible representations in (3.17) are all non-trivial (of real dimension 2), and that $\dim(M)=2n$. In other words, if a circle action on a manifold $M$ has isolated fixed points, $M$ must be even- dimensional. Excluding the trivial representation with $m=0$, the tangent spaces at the fixed points are thus labeled by a set of integers, $T_{p}M\cong V_{m_{1}}\oplus\cdots\oplus V_{m_{n}}$ (3.19) where $(m_{1},\cdots,m_{n})\in\mathbb{Z}^{n}$ are called the _exponents_ of the circle action at $p\in F$.777In terms of the Lie algebra representation, every exponent $m$ coincide with the _weight_ of the single generator of $U(1)$ in the fundamental representation. They can be regarded as maps $m_{i}:F\to\mathbb{Z}$. We formulate now a simplified version of the localization theorem in term of this local data. The proof of this can be found in [18]. ###### Theorem 3.2.1 (Localization for circle actions). Let $U(1)$ act on a compact oriented manifold $M$ of dimension $\dim(M)=2n$, with isolated fixed point locus $F$. If $m_{1},\cdots,m_{n}:F\to\mathbb{Z}$ are the exponents of the circle action, and $\alpha=\alpha^{(2n)}+\alpha^{(2n-2)}\phi+\alpha^{(2n-4)}\phi^{4}+\cdots+\alpha^{(0)}$ is an equivariant top-form in $\Omega_{U(1)}(M)$ such that $d_{C}\alpha=0$, then $\boxed{\int_{M}\alpha^{(2n)}=\int_{M}\alpha=(2\pi)^{n}\sum_{p\in F}\frac{\alpha^{(0)}(p)}{m_{1}(p)\cdots m_{n}(p)}}$ where the last component $\alpha^{(0)}\in C^{\infty}(M)$. ###### Example 3.2.1 (Localization on the 2-sphere). Let us consider again the case of the height function $H:\mathbb{S}^{2}\to\mathbb{R}$ such that, in spherical coordinates $(\theta,\varphi)$, $H(\theta,\varphi):=\cos(\theta)$. In Example 2.4.1 we related this function to the equivariantly closed extension of the volume form on the 2-sphere, $\tilde{\omega}=\omega+H.$ We can use the last localization theorem to compute integrals involving this “Hamiltonian” function on $\mathbb{S}^{2}$. The 2-sphere has two isolated fixed points at the poles, and only one exponent $m:F\to\mathbb{Z}$. It is not difficult to see that the exponent of the action at the fixed points is $m(N)=1$ at the North pole, and $m(S)=-1$ at the South pole (the sign comes from the orientation of the charts). We can check the theorem with two instructive integrals. The first is simply the area of the sphere, i.e. the integral of $\omega$. Using the theorem we easily get the correct result, $\int_{\mathbb{S}^{2}}\omega=\int_{\mathbb{S}^{2}}(\omega+H)=2\pi\sum_{p\in\\{N,S\\}}\frac{H(p)}{m(p)}=2\pi\left(\frac{\cos(0)}{1}+\frac{\cos(\pi)}{-1}\right)=4\pi.$ The second integral is the “partition function”on the sphere, $Z(t):=\int_{\mathbb{S}^{2}}\omega e^{itH}=\frac{1}{it}\int_{\mathbb{S}^{2}}e^{it(H+\omega)}$ where the second equality comes from degree arguments. This is the integral of an equivariantly closed form, since $d_{C}e^{it(H+\omega)}\propto d_{C}\tilde{\omega}=0$, whose $C^{\infty}(\mathbb{S}^{2})$ component is given by $e^{itH}$. Using the localization theorem we get $Z=\frac{1}{it}2\pi\left(\frac{e^{it\cos(0)}}{1}+\frac{e^{it\cos(\pi)}}{-1}\right)=4\pi\frac{\sin(t)}{t}$ matching the result from the “semiclassical” saddle-point approximation (1.7). We now get to the main theorem, considering a more generic torus action with higher dimensional fixed point locus on $M$. ###### Theorem 3.2.2 (Atiyah-Bott [6], Berline-Vergne [7]). Let the torus $T=U(1)^{l}$ of dimension $l$ act on a compact oriented $d$-dimensional manifold $M$, with fixed point locus $F$. If $\alpha\in\Omega_{T}(M)$ is an equivariantly closed form, i.e. $d_{C}\alpha=0$, and $i:F\hookrightarrow M$ is the inclusion map, then $\boxed{\int_{M}\alpha=\int_{F}\frac{i^{}\alpha}{\left.e_{T}(R)\right\|_{N}}}$ where $\left.e_{T}(R)\right\|_{N}$ is the _T-equivariant Euler class_ of the normal bundle of $F$ in $M$. This is the localization formula as originally presented for a torus action and fixed point locus $F$, that is generically an embedded (regular) submanifold of $M$. The _normal bundle_ to $F$ can be regarded as $TN=\faktor{TM}{i_{}TF},$ (3.20) where the quotient is taken pointwise at any $p\in F$, so that the tangent bundle of $M$ is split as $TM=i_{}TF\oplus TN$. The finite sum is replaced by an integral over $F$, and the zero-degree component of $\alpha$ is replaced by the component with the correct dimensionality, that matches $\dim(F)$, by pulling-back $\alpha$ on $F$. The product of the exponents at the denominator is represented in general by the equivariant Euler class of the normal bundle, $\left.e_{T}(R)\right\|_{N}=\mathrm{Pf}_{N}\left(\frac{R^{T}}{2\pi}\right)=\mathrm{Pf}_{N}\left(\frac{R+\mu}{2\pi}\right),$ (3.21) where the pfaffian is taken over the coordinates that span the normal bundle $TN$, $R$ is the curvature of an invariant Riemannian metric on $M$, $\mu:\mathfrak{t}\to\Omega^{0}(M;\mathfrak{gl}(d))$ is the “moment map” that makes $R^{T}$ an equivariant extension of the Riemannian curvature in the Cartan model (see Appendix B.1). As an example, let us apply the ABBV localization formula in the case of a discrete fixed point set $F$, so that we can recover at least the more readable version of theorem 3.2.1. The normal bundle in this case is the whole tangent bundle and, since $F$ is 0-dimensional, the restriction of the equivariant curvature $R^{T}$ to $F$ makes only its $\Omega^{0}$ component contribute, so $\mathrm{Pf}(R^{T})=\phi^{a}\otimes\mathrm{Pf}(\mu_{a})$. At an isolated fixed point $p$, as we said before, the tangent space $T_{p}$ is a representation space for the torus action. Since the torus is Abelian, analogously to the above discussion this representation can be decomposed as the sum of 2-dimensional _weight spaces_ [32, 22], $T_{p}M\cong\bigoplus_{i=1}^{d/2}V_{v_{i}}.$ (3.22) In Section 4.2 we will see that the moment map at an isolated fixed point encodes exactly these weights, being the representation $\mu(p):\mathfrak{t}\to\mathfrak{gl}(d)\cong\mathrm{End}(T_{p}M)$. The equivariant Euler class computes exactly the product of the weights, $e_{T}(R)_{p}=\frac{1}{(2\pi)^{d/2}}\prod_{i}v_{i}=\frac{1}{(2\pi)^{d/2}}\prod_{i}\phi^{a}\otimes v_{i}(T_{a}),$ (3.23) where $T_{a}$ are the generators of $T$. This recovers the formula for the circle action in theorem 3.2.1, where the exponents play the role of the weights for the single generator of $\mathbb{S}^{1}$. Notice that, as it is clear from the above example, in the generic $l$-dimensional case it is not so convenient to forget about the generators $\\{\phi^{a}\\}$ of $\mathfrak{t}^{}$, and the ABBV localization formula should be thought as an equivalence of elements in $H^{}_{T}(pt)=\mathbb{R}[\phi^{1},\cdots,\phi^{l}]$. The LHS is clearly polynomial in $\phi^{a}$, so has to be the RHS. Since in the latter both the numerator and the denominator are polynomials in $\phi^{a}$, some simplification has to occur in the rational expression to give a polynomial as the final answer. ###### Remark. We anticipate that in QFT the pfaffian in the definition of the Euler class is usually realized in terms of a Gaussian integral over Grassmann (anticommuting) variables, as we will see in detail in Section 4.2. These “fermionic” Gaussian integrals arise naturally as “1-loop determinants” from some saddle-point (semi-classical) approximation technique to the partition function of the theory, for example. In general, the differential form $\alpha$ will be an “observable” of the QFT, and the equivariantly closeness condition will be interpreted as it being “supersymmetric”. The localization locus $F$ will be then the fixed point set of a symmetry group that is the “square” of this supersymmetry (as $d_{C}^{2}\propto\mathcal{L}_{T}$ schematically), so a Poincaré symmetry or a gauge symmetry. The integral then localizes onto the “moduli space” of gauge-invariant (or BPS) field configurations. In the context of Hamiltonian mechanics, the gauge symmetry can be one generated by the dynamics of the theory itself, and in this case the path integral localizes onto the classical solutions of the equations of motion. The ABBV formula thus gives a systematic way to understand in which cases the semi-classical approximation results to be exact. We will reexamine this point of view in the next section in the context of finite-dimensional Hamiltonian mechanics, while in Chapter 5 we will describe the infinite- dimensional case of QM and QFT, giving some examples of the ABBV localization formula at work. ### 3.3 Equivariant cohomology on symplectic manifolds As we remarked at the beginning of the chapter, the localization formulas of the last section can be seen as generalizing a similar result showed by Duistermaat and Heckman [5] in the context of Hamiltonian group actions on symplectic manifolds. This special case is of fundamental importance in physics, because this is the context in which classical Hamiltonian mechanics is constructed. In some special cases also the quantum theory can be formally given such a structure, and thus some results from symplectic geometry can be extended to QM and QFT in general. We begin this section by quickly recalling some basic concepts about symplectic and Hamiltonian geometry, then we will describe how this can be seen as a special case of equivariant cohomology theory from the point of view of the localization formulas. #### 3.3.1 Pills of symplectic geometry The notion of _phase space_ can be constructed in a basis-independent way in differential geometry through the definition of _symplectic manifold_. We suggest for example [33, 28, 34] for a complete introduction to the subject. ###### Definition 3.3.1. A _symplectic manifold_ is a pair $(M,\omega)$, where $M$ is a $2n$-dimensional smooth manifold, and $\omega$ is a _symplectic form_ on $M$: 1. (i) $\omega\in\Omega^{2}(M)$; 2. (ii) $d\omega=0$; 3. (iii) $\omega$ is non-degenerate. The fact that $M$ is even-dimensional is not really a requirement but a consequence of its symplectic structure. This is because any skew-symmetric bilinear map on a $d$-dimensional vector space can be represented in a suitable basis by the matrix $\left(\begin{array}[]{c\|cc}\mathbf{0}_{k}&\mathbf{0}&\mathbf{0}\\\ \hline\cr\mathbf{0}&\mathbf{0}&-\mathds{1}_{n}\\\ \mathbf{0}&\mathds{1}_{n}&\mathbf{0}\end{array}\right)$ (3.24) with $2n+k=d$. To be non degenerate, it must be $k=0$. The symplectic form is a skew-symmetric bilinear form on $T_{p}M$ at any point $p\in M$, so the even- dimensionality of $M$ follows from its non-degeneracy. On manifolds, a stronger result than the above one holds: the so-called _Darboux theorem_. It states that, for every point $p\in M$, there exists an entire open neighborhood $U_{p}\subseteq M$ and a coordinate system $x:U_{p}\to\mathbb{R}^{2n}$ with respect to which $\omega_{\mu\nu}=\omega(\partial_{\mu},\partial_{\nu})$ has the canonical form (3.24), with $k=0$. The coordinates $x$ are called _Darboux coordinates_.888This means that all symplectic manifolds look locally as the prototype $\mathbb{R}^{2n}$ with $\omega=\sum_{i=1}^{n}dx^{i}\wedge dx^{i+n}$. This is a very strong property, compared for example with the Riemannian case. Notice that from the non-degeneracy of $\omega$ we have a canonical choice for the volume form on $M$, the so-called _Liouville volume form_ $\mbox{vol}:=\frac{\omega^{n}}{n!}=\mathrm{Pf}\|\|\omega^{(x)}_{\mu\nu}\|\|d^{2n}x=dp_{1}\wedge dp_{2}\wedge\cdots dp_{n}\wedge dq^{1}\wedge dq^{2}\wedge\cdots\wedge dq^{n},$ (3.25) where $(q^{\mu},p_{\mu})_{\mu=1,\cdots,n}$ are Darboux coordinates. The closeness of $\omega$ implies that in some cases there can be a 1-form $\theta\in\Omega^{1}(M)$ such that $d\theta=\omega.$ (3.26) Such a 1-form, if it exists, is called _symplectic potential_. In practice, sometimes it is useful to _locally_ define a symplectic potential even if $\omega$ is not globally integrable. Isomorphisms of symplectic manifolds are called _symplectomorphisms_ or _canonical transformations_ , defined as diffeomorphisms that preserve the symplectic structure via pull-back. The standard example of a symplectic manifold is exactly the _phase space_ associated to some $n$-dimensional _configuration space_ $Q$, i.e. its cotangent bundle $M:=T^{}Q$. A point $p\in Q$ represents the “generalized position” of the system with coordinates $q(p)=(q^{\mu}(p))$ with $\mu=1,\cdots,n$, and a point $p\in T^{}Q$ represents the “generalized momentum”, with coordinates $\xi(p):=(q^{\mu}\circ\pi(p),\iota_{\mu}(p))\equiv(q^{\mu},p_{\mu})$, where $\pi:TQ\to Q$ is the projection and $\iota_{\mu}\equiv\iota_{\partial/\partial q^{\mu}}$. The cotangent bundle has a canonical integrable symplectic form. In fact, the symplectic potential is the so-called _tautological 1-form_ given by the pull-back of the projection map, $\theta:=\pi^{}\in\Omega^{1}(T^{}Q)$. In Darboux coordinates, at a point $p\in T^{}Q$, $\theta_{p}=\pi^{}(p)=p_{\mu}dq^{\mu}$ (3.27) where we denoted $dq^{\mu}\equiv d(q\circ\pi)^{\mu}=d\xi^{\mu}$ with $\mu=1\cdots,n$, as 1-forms on the cotangent bundle. The canonical symplectic form is then just $\omega=d\theta$, and in Darboux coordinates $\omega=dp_{\mu}\wedge dq^{\mu}$ (3.28) where again we simplified the notation setting $dp_{\mu}\equiv d\xi^{\mu}$ for $\mu=n+1,\cdots,2n$. Thus the canonical coordinates on the cotangent bundle are Darboux coordinates. One can show that canonical symplectic structures over diffeomorphic manifolds are “canonically compatible”, i.e. if $\phi:Q_{1}\to Q_{2}$ is a diffeomorphism, there is a lift of it as a symplectomorphism between $(T^{}Q_{1},\omega_{1})$ and $(T^{}Q_{2},\omega_{2})$. If we take $Q_{1}=Q_{2}$, this means that there is a group homomorphism $\text{Diff}(Q)\to\text{Symp}(T^{}Q,\omega).$ (3.29) This example showed that symplectic manifolds are the right generalization of the concept of phase space in a fully covariant setting. It is thus common to call functions on a symplectic manifold _observables_. Let us return to a generic symplectic manifold $(M,\omega)$. Giving to it some additional structure, it is possible to define on it _dynamics_ and _symmetries_ in the sense of classical mechanics. Naturally, we call symmetry of $(M,\omega)$ a diffeomorphism $\phi:M\to M$ that preserves the symplectic structure, $\phi^{}\omega=\omega$, that is a symplectomorphism. At the infinitesimal level, a diffeomorphism can be generated by the flow of a vector field $X\in\Gamma(M)$, and the symmetry condition is rephrased to $\mathcal{L}_{X}\omega=0.$ (3.30) Such a vector field is called _symplectic vector field_. It is easy to realize that a vector field is symplectic if and only if $\iota_{X}\omega=\omega(X,\cdot)$ is closed, by Cartan’s magic formula. More special vector fields are those for which $\iota_{X}\omega$ is exact, so that it exists an observable $f\in C^{\infty}(M)$ such that $df=-\iota_{X}\omega,$ (3.31) where the minus sign is conventional. The vector field $X$ is called _Hamiltonian vector field_ associated to the observable $f$. In components, $\partial_{\mu}f=\omega_{\mu\nu}X^{\nu}\qquad\text{or}\qquad X^{\mu}=\omega^{\mu\nu}\partial_{\nu}f,$ (3.32) where $\omega^{\mu\nu}$ is the “inverse” of the symplectic form. Of course Hamiltonian vector fields are symplectic, and the flow of the Hamiltonian vector field $X$ preserves the value of the Hamiltonian function $f$, since $\mathcal{L}_{X}(f)=X(f)=df(X)=\omega(X,X)=0$. The flow of the Hamiltonian vector field is regarded as the “time-evolution” over the generalized phase space $M$, generated by the observable $f$. ###### Definition 3.3.2. An _Hamiltonian (or dynamical) system_ is a tuple $(M,\omega,H)$, where $(M,\omega)$ is a symplectic manifold and $H\in C^{\infty}(M)$ an observable called _Hamiltonian_. The _time-evolution_ of points $p\in M$ is defined by the flow of the Hamiltonian vector field $X_{H}$ of $H$, $p(t):=\gamma^{H}_{p}(t)$ where $\gamma^{H}_{p}$ is the integral curve of $X_{H}$ with $\gamma^{H}_{p}(0)=p$. In particular, the evolution of an observable $f\in C^{\infty}(M)$ is regulated by the _equation of motion_ $\dot{f}(p):=(f\circ\gamma^{H}_{p})^{\prime}(0)\equiv\left.\mathcal{L}_{X_{H}}(f)\right\|_{p}.$ The equation of motion can be rewritten in a more usual way introducing the _Poisson brackets_ $\\{\cdot,\cdot\\}:C^{\infty}(M)\times C^{\infty}(M)\to C^{\infty}(M)$ such that $\\{f,g\\}:=\omega(X_{g},X_{f})$, where $X_{f},X_{g}$ are the Hamiltonian vector fields of $f$ and $g$, respectively. In a chart and with respect to Darboux coordinates $(q^{\mu},p_{\mu})$ on $M$, by the Darboux theorem the Poisson brackets take the usual form $\\{f,g\\}=\frac{\partial f}{\partial q^{\mu}}\frac{\partial g}{\partial p_{\mu}}-\frac{\partial g}{\partial q^{\mu}}\frac{\partial f}{\partial p_{\mu}}.$ (3.33) With this definition we can write $\dot{f}=-\\{H,f\\},\qquad\dot{q}^{\mu}=\frac{\partial H}{\partial p_{\mu}},\qquad\dot{p}_{\mu}=-\frac{\partial H}{\partial q^{\mu}},$ (3.34) recovering the Hamilton’s equations for the Darboux coordinates. The Poisson brackets are anti-symmetric and satisfy the Jacobi identity, so this turns $(C^{\infty}(M),\\{\cdot,\cdot\\})$ into a Lie algebra,999In fact this is a _Poisson algebra_ , i.e. a Lie algebra whose brackets act as a derivation. and one can check that there is a Lie algebra homomorphism $\displaystyle(C^{\infty}(M),\\{\cdot,\cdot\\})$ $\displaystyle\to(\text{Hamiltonian v.f.},[\cdot,\cdot])$ (3.35) $\displaystyle f$ $\displaystyle\mapsto X_{f},$ where we also already used the fact that Hamiltonian vector fields form a Lie subalgebra with respect to the standard commutator on $\Gamma(TM)$. We just reviewed that the concept of symmetry in symplectic geometry is correlated with the concept of dynamics on the symplectic manifold. The next fact that we need is to connect this formalism to the equivariant cohomology one, identifying these symmetries as generated by a _group action_ on $M$. In particular, we would like to identify the Lie subalgebra of Hamiltonian vector fields as the Lie algebra of a Lie group that acts on the symplectic manifold. We can start thus the discussion of symmetry by declaring that $M$ is a $G$-manifold with respect to a Lie group $G$ of Lie algebra $\mathfrak{g}$. Denoting the $G$-action as $\rho$, this is called _symplectic_ if it makes $G$ act by symplectomorphisms on $(M,\omega)$, i.e. $\rho:G\to\text{Symp}(M,\omega).$ (3.36) We can characterize again infinitesimally this action by saying that $\mathfrak{g}$ acts on $\Omega(M)$ via symplectic vector fields: if $A\in\mathfrak{g}$, the corresponding fundamental vector field preserves the symplectic structure, $\mathcal{L}_{A}\omega=0$. We are interested in the special case analogous to the one above, in which not only a fundamental vector field is symplectic, but it is also Hamiltonian. This forces a generalization of the concept of Hamiltonian function, because now there are more than one independent fundamental vector fields to take into account, if $\dim(\mathfrak{g})>1$. ###### Definition 3.3.3. The $G$-action $\rho:G\to\text{Symp}(M,\omega)$ on the symplectic manifold $(M,\omega)$ is said to be an _Hamiltonian action_ if every fundamental vector field is Hamiltonian. In particular, there exists a $\mathfrak{g}^{}$-valued function $\mu\in\mathfrak{g}^{}\otimes C^{\infty}(M)$ such that: 1. (i) For every $A\in\mathfrak{g}$, $\mu(A)\equiv\mu_{A}\in C^{\infty}(M)$ is the Hamiltonian function with respect to $\underline{A}$, $d\mu_{A}=-\iota_{A}\omega=\omega(\cdot,\underline{A}).$ 2. (ii) It is $G$-equivariant with respect to the canonical (co)adjoint action of $G$ on $\mathfrak{g}\ (\mathfrak{g}^{})$,101010If, for every $g\in G$, $Ad_{g}:G\to G$ is the action by conjugation, the adjoint action $Ad_{g}$ on $\mathfrak{g}$ is the push-forward of $Ad_{g}$, while the coadjoint action $Ad^{}_{g}$ on $\mathfrak{g}^{}$ is the pull-back of $Ad_{g^{-1}}$. If $g=\exp(tA)$ for some $A\in\mathfrak{g}$, differentiating one gets the infinitesimal actions of $\mathfrak{g}$ on $\mathfrak{g}$ and $\mathfrak{g}^{}$, $ad_{A}(B)=[A,B]$ and $ad^{}_{A}(\eta):=\eta([\cdot,A])$. so for any $g\in G$ $\mu\circ Ad_{g}=\rho_{g}^{}\circ\mu\qquad\text{or}\qquad Ad^{}_{g}\circ\mu=\mu\circ\rho_{g},$ where in the first equation $\mu$ is considered as $\mathfrak{g}\xrightarrow{\mu}C^{\infty}(M)$, in the second one as $M\xrightarrow{\mu}\mathfrak{g}^{}$. If $G$ is connected, this is equivalent to requiring $\mu:\mathfrak{g}\to C^{\infty}(M)$ to be a Lie algebra anti- homomorphism with respect to the Poisson brackets, $\mu_{[A,B]}=\\{\mu_{B},\mu_{A}\\}\qquad\forall A,B\in\mathfrak{g}.$ The map $\mu$ is called _moment map_ , and $(M,\omega,G,\mu)$ is called _Hamiltonian $G$-space_. In general the job of the moment map is to collect all the “Hamiltonians” with respect to which the system can flow. There are $\dim(G)$ independent of them, one for every generator. In the 1-dimensional case, where $G=U(1)$ (or its non-compact counterpart $G=\mathbb{R}$), the moment map produces only one independent Hamiltonian, $\mu_{T}\equiv H$, and the above definition reduces to the Hamiltonian system $(M,\omega,H)$ of definition 3.3.2. Notice that for any Hamiltonian structure we build on $(M,\omega)$, its flow preserves the symplectic form and thus the canonical Liouville volume form $\omega^{n}/n!$. This is the content of the so-called _Liouville theorem_. #### 3.3.2 Equivariant cohomology for Hamiltonian systems We can first generalize what we noticed in examples 2.4.1 and 3.2.1, in the case of a circle action on a symplectic manifold $(M,\omega)$. In the above examples the manifold was the 2-sphere $\mathbb{S}^{2}$ and the symplectic form was the canonical volume form. Rephrased in terms of symplectic geometry, the existence of an _equivariantly closed extension_ $\tilde{\omega}$ of the symplectic form is the condition of $U(1)$ acting in an Hamiltonian way, since $d_{C}\tilde{\omega}=d_{C}(\omega+H)=\iota_{T}\omega+dH=0$ (3.37) is satisfied if and only if $dH=-\iota_{T}\omega$. This is readily generalizable to the multidimentional case, so that we can describe the Hamiltonian $G$-space $(M,\omega,G,\mu)$ and its classical mechanics in equivariant cohomological terms. In fact, we can always find an equivariantly closed extension of the symplectic form in $\Omega_{G}(M)$, $\tilde{\omega}:=1\otimes\omega-\phi^{a}\otimes\mu_{a}$ (3.38) where $\mu_{a}\equiv\mu_{T_{a}}\in C^{\infty}(M)$ and $T_{a}$ are the dual basis elements with respect to the generators $\phi^{a}$ of $S(\mathfrak{g}^{})$. It is straightforward to check that $\tilde{\omega}\in\Omega_{G}(M)$ is indeed $G$-invariant, and closed with respect to $d_{C}$ thanks to the Hamiltonian property of the $G$-action, $d\mu_{a}=-\iota_{a}\omega$. In the language of $G$-equivariant bundles (see Appendix B.1), the symplectic structure on $M$ can be seen as the presence of a principal $U(1)$-bundle $P\to M$ whose connection 1-form is the symplectic potential $\theta$ (that has not always a global trivialization on $M$), and whose curvature is the symplectic 2-form $\omega=d\theta$ (that instead transforms covariantly on $M$). $G$ acts symplectically if also $\theta$ is $G$-invariant, $\mathcal{L}_{X}\theta=0\quad\Rightarrow\quad\mathcal{L}_{X}\omega=0\qquad\forall X\in\mathfrak{g},$ (3.39) so that $P\to M$ is a $G$-equivariant bundle. Thus, this equivariant extension to the curvature $\omega$ is the same as in [35, 7]. We return for a moment to the symplectic geometric interpretation, to describe the results of Duistermaat and Heckman related to the localization formulas that we described in the last section. In [5] they proved an important property of the Liouville measure in the presence of an Hamiltonian action by a torus $T$ on $(M,\omega)$. Namely, defining a measure on $\mathfrak{g}^{}$ as the _push-forward_ of the Liouville measure, $\mu_{}\left(\frac{\omega^{n}}{n!}\right)(U)=\int_{\mu^{-1}(U)}\frac{\omega^{n}}{n!}\qquad\forall U\subseteq M\ \text{measurable},$ (3.40) they proved that $\mu_{}\left(\omega^{n}/n!\right)$ is a _piecewise polynomial function_.111111To be more precise, denoting $\mu_{}\left(\omega^{n}/n!\right)=fd\xi$ with $d\xi$ the standard Lebesgue measure on $\mathfrak{g}^{}\cong\mathbb{R}^{\dim\mathfrak{g}}$, the function $f$ is piecewise polynomial. This, and an application of the stationary phase approximation showed a localization formula for the oscillatory integral $\int_{M}\frac{\omega^{n}}{n!}\exp{\left(i\mu_{X}\right)}$ (3.41) for every $X\in\mathfrak{t}:=Lie(T)$ with non-null weight at every fixed point of the $T$-action. This can be viewed as the Fourier transform of the Liouville measure, or as the _partition function_ of a 0-dimensional QFT with target space $M$. Let the fixed point locus $F$ be the union of compact connected symplectic manifolds $M_{k}\hookrightarrow M$ of even codimension $2n_{k}$, and denote $(m_{kl})_{l=1,\cdots,n_{k}}$ the weights of the $T$-action at a tangent space of a fixed point $p\in M_{k}$.121212The components of the fixed point set being symplectic is not an assumption, but a consequence of the Hamiltonian action. See [28], proposition IV.1.3. Then the _Duistermaat-Heckman (DH) localization formula_ is $\boxed{\int_{M}\frac{\omega^{n}}{n!}\exp{\left(i\mu_{X}\right)}=\sum_{k}\frac{\mbox{vol}(M_{k})\exp{(i\mu_{X}(M_{k}))}}{\prod_{l}^{n_{k}}(m_{kl}(X)/2\pi)}}$ (3.42) where $\mu_{X}(M_{k})$ denotes the common value of $\mu_{X}$ at every point in $M_{k}$. In equivariant cohomological terms, we can see the above result as a localization formula for the integral of an equivariantly closed form. In fact, if we fix the $U(1)$ symmetry subgroup generated by $X\in\mathfrak{t}$, and consider the Cartan model defined by the differential $d_{C}=d+i\iota_{X}$, the LHS can be rewritten as $\int_{M}\frac{\omega^{n}}{n!}\exp(i\mu_{X})=\int_{M}\exp(\omega+i\mu_{X}),$ (3.43) analogously to what we did in Example 3.2.1, and this is clearly the integral of an equivariantly closed form with respect to the differential $d_{C}$. To see the correspondence with the ABBV formula, let us examine the case of a circle action and discrete fixed point locus $F$. In this case we have only one Hamiltonian function $H:=\mu_{X}$, the weights are just the exponents $m_{1},\cdots,m_{n}$ of the circle action, and the sum over $k$ runs over the isolated fixed points. The DH formula thus recovers exactly the localization formula of theorem 3.2.1: $\int_{M}\exp(\omega+iH)=(2\pi)^{n}\sum_{p\in F}\frac{e^{iH(p)}}{m_{1}(p)\cdots m_{n}(p)}.$ (3.44) As we remarked in (3.23), the denominator can be expressed as the equivariant Euler class of the normal bundle to $F$ (that is just the tangent bundle since $F$ is 0-dimensional), recovering the DH formula as a special case of the ABBV localization formula for torus actions. See also [19] for an explicit correspondence between the two. We wish only to remark again that, especially in the context of Hamiltonian mechanics, this localization formula can be seen as the result of an “exact” saddle-point approximation on the partition function (3.43). This is the point of view we are going to take in the next chapters, when we are going to discuss the generalization of this formula to higher-dimensional QFT, where the integral of the partition function is turned into an infinite-dimensional _path integral_. To see the correspondence with the saddle-point approximation, we recall that the isolated fixed points of the $U(1)$-action are those in which $\underline{X}=0$, so $dH=0$, and thus they are the critical points of the Hamiltonian. We need to assume that the function $H$ is _Morse_ , so that these fixed points are non-degenerate, i.e. the Hessian $\mathrm{Hess}_{p_{0}}(H)_{\mu\nu}=\partial_{\mu}\partial_{\nu}H(p_{0})$ at a given $p_{0}\in F$ has non-null determinant.131313This subject was in fact firstly connected with Morse theory by Witten in [36], where localization is applied in the context of supersymmetric QM to prove Morse inequalities. We do not need to deepen this point of view for what follows, but a discussion about Morse theory and its connection with the DH formula can be found in [19], and references therein. This Hessian can be expressed in terms of the exponents $m_{k}(p_{0})$ via an equivariant version of the Darboux theorem [37, 28]: at any fixed point we can choose Darboux coordinates in which the symplectic form takes its canonical form (3.28), and moreover the action of the fundamental vector field at that tangent space decomposes as in (3.19). The latter can then be expressed as $n$ canonical rotations of the type $\underline{X}=\sum_{\mu=1}^{n}im_{\mu}\left(q^{\mu}\frac{\partial}{\partial p_{\mu}}-p_{\mu}\frac{\partial}{\partial q^{\mu}}\right)$ (3.45) with different weights $m_{k}$. By the general form of the Hamilton’s equations (3.32), this means that the Hamiltonian near the isolated fixed point $p_{0}\in F$ can be expanded as $H(x)=H(p_{0})+\frac{1}{2}\sum_{\mu=1}^{n}im_{\mu}(p_{0})\left(p_{\mu}(x)^{2}+q^{\mu}(x)^{2}\right)+\cdots$ (3.46) Plugging this expansion into the oscillatory integral, we get the saddle-point approximation $\displaystyle\int_{M}d^{n}pd^{n}q\ e^{iH(p,q)}$ $\displaystyle\approx\sum_{p_{0}\in F}e^{iH(p_{0})}\prod_{\mu=1}^{n}\left(\int dp\ e^{-\frac{m_{\mu}(p_{0})}{2}p^{2}}\int dq\ e^{-\frac{m_{\mu}(p_{0})}{2}q^{2}}\right)$ (3.47) $\displaystyle\approx\sum_{p_{0}\in F}e^{iH(p_{0})}\frac{(2\pi)^{n}}{\prod_{\mu}m_{\mu}(p_{0})}$ that, again, is exactly the result of the localization formula above. This motivates in the context of Hamiltonian mechanics, and generalization to infinite-dimensional case, that the denominators appearing in these formulas are exactly the “1-loop determinants” of a would-be semiclassical approximation to the partition function. More aspects of the equivariant theory in contact with symplectic geometry can be found in [22]. ## Chapter 4 Supergeometry and supersymmetry ### 4.1 Gradings and superspaces We give some definitions concerning _graded_ spaces and _super_ -spaces, that are useful for many applications of the localization theorems in physics. In particular, we will see how to translate the problem of integration of differential forms in the context of supergeometry, and how this is useful to prove the ABBV localization formula for circle actions. Also, in the next chapter we will apply this theorem to path integrals in QM and QFT, where the coordinates over which one integrates are those of a “field space” over a given manifold. To construct a suitable Cartan model over this kind of spaces, it is necessary to introduce a graded structure, that physically means to distinguish between _bosonic_ and _fermionic_ fields, and some operation that acts as a “Cartan differential” transforming one type of field into the other. These structures arise in the context of _supersymmetric field theories_ , or in the context of _topological field theories_ , and the differentials here are called _supersymmetry_ transformations or _BRST_ transformations. The precise mathematics behind this is a great subject and we do not seek to be complete here, we just give some of the basic concepts that are necessary to understand what follows. For a more extensive review of the subject, we suggest for example [38]. #### 4.1.1 Definitions To understand the concept of a supermanifold, we need first to recall the linearized case. We already introduced a _graded module_ or _graded algebra_ $V$ over a ring $R$, that is a collection of $R$-modules $\\{V_{n}\\}_{n\in\mathbb{Z}}$ such that $V=\bigoplus_{n\in\mathbb{Z}}V_{n}$. If $V$ is an algebra, it must also satisfy $V_{n}V_{m}\subseteq V_{n+m}$. An element $a\in V_{n}$ for some $n$ is called _homogeneous_ of _degree_ $\mathrm{deg}(a)\equiv\|a\|:=n$. We now can specialize to the case of _super_ \- vector spaces (or modules) and _super_ -algebras. ###### Definition 4.1.1. A _super vector space_ is a $\mathbb{Z}_{2}$-graded vector space $V=V_{0}\oplus V_{1}$ where $V_{0},V_{1}$ are vector spaces. Its _dimension_ as a super vector space is defined as $\dim{V}:=\left(\dim{V_{0}}\|\dim{V_{1}}\right)$. A _superalgebra_ is a super vector space $V$ with the product satisfying $V_{0}V_{0}\subseteq V_{0}\ ;\quad V_{0}V_{1}\subseteq V_{1}\ ;\quad V_{1}V_{1}\subseteq V_{0}.$ A _Lie superalgebra_ is a superalgebra where the product $[\cdot,\cdot]:V\times V\to V$, called _Lie superbracket_ , satisfies also $\displaystyle[a,b]=-(-1)^{\|a\|\|b\|}[b,a]$ $\displaystyle(\mathrm{supercommutativity}),$ $\displaystyle(-1)^{\|a\|\|c\|}[a,[b,c]]+(-1)^{\|b\|\|c\|}[c,[a,b]]+(-1)^{\|a\|\|b\|}[b,[c,a]]=0$ $\displaystyle(\mathrm{super\ Jacobi\ identity}).$ ###### Definition 4.1.2. The _k-shift_ of a graded vector space $V$ is the graded vector space $V[k]$ such that $(V[k])_{n}=V_{n+k}$ $\forall n\in\mathbb{Z}$. A few remarks are in order. First, it is clear that every graded vector space has naturally also the structure a super vector space, if we split its grading according to “parity”: $V_{even}:=\bigoplus_{n\in 2\mathbb{Z}}V_{n}\ ,\qquad V_{odd}:=\bigoplus_{n\in 2\mathbb{Z}+1}V_{n}.$ (4.1) In physics, the $\mathbb{Z}$-grading occurs on the “field space” as the so- called _ghost number_ , while the $\mathbb{Z}_{2}$-grading with respect to parity distinguish between _bosonic_ and _fermionic_ coordinates. Second, we notice that every vector space $V$ can be considered as a (trivial) super vector space, if we think of it as $V=V\oplus 0$ in even degree or $V[1]=0\oplus V$ in odd degree. Notice that the even/odd parts of a super vector space can be considered as eigenspaces of an automorphism $P:V\to V$ such that $P^{2}=id_{V}$. In this sense, a super vector space is a pair $(V,P)$ made by a vector space and the given automorphism $P$. Morphisms of graded vector spaces are graded linear maps, i.e. grading preserving maps: ###### Definition 4.1.3. A _graded linear map_ $f$ between graded vector spaces $V$ and $W$ is a collection of linear maps $\\{f_{k}:V_{k}\to W_{k}\\}_{k\in\mathbb{Z}}$. A linear map of _k-degree_ is a graded linear map $f:V\to W[k]$. Now we can turn to the non-linear case and consider _supermanifolds_.111Historically two (apparently) different concepts of _supermanifolds_ and _graded manifolds_ were firstly developed. They both aimed to generalizing the mathematics of manifolds to a non-commutative setting, following different approaches. Eventually it was proven in [39] that their definitions are equivalent. Locally, they can be thought as extensions of a manifold via “anticommuting coordinates”: if we take an open set $U\subset\mathbb{R}^{n}$ and a set of coordinates $\\{x^{\mu}:U\to\mathbb{R}\\}_{\mu=1,\cdots,m}$, we can consider a set of additional coordinates $\\{\theta^{i}\\}_{i=1,\cdots,n}$ with the algebraic properties $x^{\mu}\theta^{i}=\theta^{i}x^{\mu}\ ,\qquad\theta^{i}\theta^{j}=-\theta^{j}\theta^{i}.$ (4.2) The anticommuting $\\{\theta^{i}\\}$ can be thought as generators of $\bigwedge(V^{})$ for some vector space $V$, and the product between them and coordinates of $C^{\infty}(U)$ is then interpreted as a tensor product in $C^{\infty}(U)\otimes\bigwedge(V^{})=:C^{\infty}(U\times V[1])$.222Being generators of an exterior algebra, $\theta^{i}$ are called “Grassmann-odd” coordinates, while $x^{\mu}$ are called “Grassmann-even” consequently. This terminology is commonly inherited by every graded object (vector fields, forms, etc.) on the supermanifold. If we then patch together different open sets we get globally a manifold structure, with a modified atlas made by a _graded_ ring of local functions $C^{\infty}(U\times V[1])$. More formally, we define: ###### Definition 4.1.4. A (smooth) _supermanifold_ $SM$ of dimension $(m\|n)$ is a pair $(M,\mathcal{A})$, where $M$ is a $C^{\infty}$-manifold of dimension $m$, and $\mathcal{A}$ is a sheaf of $\mathbb{R}$-superalgebras that makes $SM$ locally isomorphic to $\left(U,C^{\infty}(U)\otimes\bigwedge(V^{})\right)$ for some $U\subseteq\mathbb{R}^{m}$ open and some vector space $V$ of finite dimension $\dim{V}=n$. $M$ is called the _body_ of $SM$ and $\mathcal{A}$ is called the _structure sheaf_ (or, sometimes, “soul”) of $SM$.333Note that also a regular d-dimensional smooth manifold can be viewed as a pair $(M,\mathcal{O}_{M})$ composed by a topological space $M$ (Hausdorff and paracompact) with a structure sheaf of local functions $\mathcal{O}_{M}:\ \mathcal{O}_{M}(U)=C^{\infty}(U)$ for every $U\subseteq M$ open, such that locally every $U$ is isomorphic to a subset of $\mathbb{R}^{d}$. We just associated to a real manifold $M$ a graded-commutative algebra $C^{\infty}(SM)$ of functions over $SM$. Locally in a patch $U\subseteq M$, this matches the idea above of having coordinate systems as tuples $(x^{\mu},\theta^{i})_{\stackrel{{\scriptstyle\mu=1,\cdots,m}}{{i=1,\cdots,n}}}$ with the property (4.2). In particular, any local function $\Phi\in\mathcal{A}(U)$ can be trivialized with respect to the graded basis of $\bigwedge(V^{})$: $\Phi(x,\theta)=\Phi^{(0)}(x)+\Phi_{i}^{(1)}(x)\theta^{i}+\Phi_{ij}^{(2)}\theta^{i}\wedge\theta^{j}+\cdots+\Phi^{(n)}_{i_{1},\cdots,i_{n}}\varepsilon^{i_{1}\cdots i_{n}}\theta^{1}\wedge\cdots\wedge\theta^{n}$ (4.3) where $\Phi^{(l)}_{i_{1}\cdots i_{l}}\in C^{\infty}(U)$ $\forall l\in\\{0,\cdots,n\\}$. The restriction to the zero-th degree $\epsilon:\mathcal{A}\to C^{\infty}_{M}$ such that $\epsilon(\Phi):=\Phi^{(0)}$ is usually called the _evaluation map_. ###### Example 4.1.1. To every super vector space $V=V_{0}\oplus V_{1}$ we can associate the supermanifold $\hat{V}=\left(V_{0},C^{\infty}(V_{0})\otimes\Lambda(V_{1}^{})\right)\cong\mathbb{R}^{\dim(V_{0})\|\dim(V_{1})}.$ More generally, to every vector bundle $E\to M$ with sections $\Gamma(E)$ we can associate the _odd vector bundle_ , denoted $\Pi E$ or $E[1]$, that is the supermanifold with body $M$ and structure sheaf $\mathcal{A}=\Gamma\left(\bigwedge E^{}\right)$. The _odd tangent bundle_ $\Pi TM$ is the supermanifold with $\mathcal{A}=\Gamma\left(\bigwedge T^{}M\right)$, i.e. globally the functions here are the differential forms on $M$, $C^{\infty}(\Pi TM)=\Omega(M)$. Coordinates on $\Pi TM$ are just $(x^{\mu},dx^{\mu})_{\mu=1\cdots,m}$, exactly as the coordinates on the tangent bundle $TM$, but now we consider them as generators of a graded algebra. Morphisms of supermanifolds can be given in terms of local morphisms of superalgebras, that respect compatibility between different patches. In particular, a morphism $(f,f^{\\#}):SM\to SN$ is a pair such that $f:M\to N$ is a diffeomorphism, and for every $U\subseteq M$ there is a morphism of superalgebras $f_{U}^{\\#}:\mathcal{A}_{M}(U)\to\mathcal{A}_{N}(f(U))$ that respects $f^{\\#}_{V}\circ\mathrm{res}_{U,V}=\mathrm{res}_{f(U),f(V)}\circ f^{\\#}_{U}$, where $\mathrm{res}_{U,V}$ is the restriction to a subset $V\subseteq U$. In less fancy words, if $\dim{SM}=(m\|p)$ and $\dim{SN}=(n\|q)$, a local coordinate system $(x,\theta)$ in $SM$ is mapped through $n$ functions $y^{\nu}=y^{\nu}(x,\theta)$ and $q$ functions $\varphi^{j}=\varphi^{j}(x,\theta)$ to a coordinate system $(y,\varphi)$ of $SN$. A vector field $X$ on a supermanifold $SM$, or _supervector field_ , is a derivation on $C^{\infty}(SM)$. Locally, considering $U\subseteq M$ open and $\mathcal{A}(U)=C^{\infty}(U)\otimes\bigwedge(V^{})$, it can be expressed with respect to a coordinate system $(x,\theta)$ as $X=X^{\mu}(x,\theta)\frac{\partial}{\partial x^{\mu}}+X^{i}(x,\theta)\frac{\partial}{\partial\theta^{i}},$ (4.4) where $(\partial/\partial x^{\mu})$ acts as the corresponding vector field in $\Gamma(TM)$ on the $C^{\infty}(U)$ components and acts trivially on the odd coordinates $\theta^{i}$; $(\partial/\partial\theta^{i})$ acts trivially on $C^{\infty}(U)$, and as an _interior multiplication_ by the dual basis vector $u_{i}\in V$: $\frac{\partial}{\partial\theta^{i}}\theta^{j}:=\theta^{j}(u_{i})=\delta^{j}_{i}$. $X^{\mu},X^{i}$ are local sections in $\mathcal{A}(U)$. Supervectors on $SM$ form the tangent bundle $TSM$. Notice that, in particular $(\partial/\partial x^{\mu})$ preserves the grading of an homogeneous function, i.e. it is a derivation of degree 0, while $(\partial/\partial\theta^{i})$ shifts the grading by -1. ###### Definition 4.1.5. A _graded vector field_ of degree $k$ on $SM$ is a graded linear map $X:C^{\infty}(SM)\to C^{\infty}(SM)[k]$ that satisfies the graded Leibniz rule: $X(\phi\psi)=X(\phi)\psi+(-1)^{k\|\phi\|}\phi X(\psi)$ for any $\phi,\psi\in\mathcal{A}$ of pure degree. The _graded commutator_ between graded vector fields $X,Y$ is defined as $[X,Y]:=X\circ Y-(-1)^{\|X\|\|Y\|}Y\circ X.$ From what we said above, partial derivatives $(\partial/\partial x^{\mu})$ with respect to even coordinates commute between each other, while $(\partial/\partial\theta^{i})$ anticommute, being respectively graded vector fields of degree 0 and -1. Then from (4.4) and (4.3) we see that any supervector field can be decomposed with respect to the $\mathbb{Z}_{2}$-grading given by the parity, as the sum $X=X_{(0)}+X_{(1)}$ of an even (_bosonic_) and an odd (_fermionic_) vector field. This makes $\left(\Gamma(TSM),[\cdot,\cdot]\right)$ into a Lie superalgebra. The _value_ at a point $p\in M$ of a supervector field $X\in\Gamma(TSM)$ is defined through the evaluation map: $X_{p}(\Phi):=\epsilon_{p}(X(\Phi))=\left[X^{\mu}(x,\theta)\frac{\partial\Phi}{\partial x^{\mu}}+X^{i}(x,\theta)\frac{\partial\Phi}{\partial\theta^{i}}\right]_{\stackrel{{\scriptstyle x=x(p)}}{{\theta=0}}}.$ (4.5) Clearly a super vector field $X$ is not determined by its values at points, since the evaluation map throws away all the dependence on the Grassmann-odd coordinates $\theta^{i}$ in the coefficient functions $X^{\mu},X^{i}$. This means that at every point $p\in M$, $T_{p}SM$ is a super vector space generated by the symbols $\partial/\partial x^{\mu},\partial/\partial\theta^{i}$ of opposite degrees, with _real_ coefficients. We collect this result in the following proposition. ###### Proposition 4.1.1. Let $SM=(M,\mathcal{A})$ be a supermanifold such that for every chart $U\subseteq M$ $\mathcal{A}=C^{\infty}(U)\otimes\Lambda(V^{})$. Then at every point $p\in M$, $T_{p}SM\cong T_{p}M\oplus V[1]$ as real super vector spaces. For an odd vector bundle $\Pi E$ this specializes as $T_{p}\Pi E\cong T_{p}M\oplus E_{p}[1]$. Notice that one has always really both a $\mathbb{Z}$ and a $\mathbb{Z}_{2}$ grading of functions and supervector fields, analogously to the remark (4.1). As already mentioned, in field theory and in particular in the BRST formalism, the first one is called _ghost number_ , while the second one is the distinction between _bosonic_ and _fermionic_ degrees of freedom in the theory. The physical (i.e. gauge-invariant) combinations are those of ghost number zero. From the point of view of equivariant cohomology, it is very useful to relate the algebraic models we saw in Chapter 2 to these graded manifold structures. In particular, in field theory we interpret the graded complex of fields as a Cartan model, with a suitable graded equivariant differential given in terms of supersymmetry or BRST transformations. Generically, a _differential_ in supergeometry can be interpreted as a special supervector field on a supermanifold: ###### Definition 4.1.6. A _cohomological vector field_ $Q$ on a supermanifold $SM$ is a graded supervector field of degree +1 satisfying $[Q,Q]=0.$ It is immediate that any cohomological vector field corresponds to a _differential_ on the algebra of functions $C^{\infty}(SM)$, since being it of degree +1, $Q\circ Q=(1/2)[Q,Q]=0$. For example, consider the de Rham differential $d:\Omega(M)\to\Omega(M)$ on a regular smooth manifold $M$. It corresponds to the cohomological vector field on the odd tangent bundle $\Pi TM$ given in local coordinates by $d=\theta^{\mu}\frac{\partial}{\partial x^{\mu}},$ (4.6) where now $\theta^{\mu}\equiv dx^{\mu}$ are odd coordinate functions on $\Pi TM$. Similarly, the _interior multiplication_ $\iota_{X}:\Omega(M)\to\Omega(M)$ with respect to some vector field $X\in\Gamma(TM)$ is a nilpotent supervector field on $\Pi TM$ of degree -1, $\iota_{X}=X^{\mu}\frac{\partial}{\partial\theta^{\mu}}.$ (4.7) #### 4.1.2 Integration In the following we will use this graded machinery to translate the problem of integration of differential forms $\Omega(M)$ on a smooth manifold $M$, to an integration over the related odd tangent bundle $\Pi TM$. For this, we need to consider differential forms on a supermanifold $SM$. If $SM$ has local coordinates $(x^{\mu},\theta^{i})$, we can locally form an algebra generated by the 1-forms $dx^{\mu},d\theta^{i}$, where now $d$ is the de Rham differential on $SM$, acting as a cohomological vector field on $\Pi TSM$: $d=dx^{\mu}\partial_{x^{\mu}}+d\theta^{i}\partial_{\theta^{i}}.$ (4.8) The odd tangent bundle $\Pi TSM$ has thus coordinates $(x^{\mu},\theta^{i},dx^{\mu},d\theta^{i})$, where now $dx^{\mu}$ is odd whereas $d\theta^{i}$ is even.444To be more precise, the algebra of functions locally generated by $(x^{\mu},\theta^{i},dx^{\mu},d\theta^{i})$ on $\Pi TSM$ has _bi-grading_ , _i.e._ it inherits a $\mathbb{Z}_{2}$ grading from the original supermanifold $SM$ and a $\mathbb{Z}$ grading from the action of the de Rham differential (the form-degree). The coordinates have thus bi-degrees $x^{\mu}:(\text{even},0),\qquad\theta^{i}:(\text{odd},0),\qquad dx^{\mu}:(\text{even},1),\qquad d\theta^{i}:(\text{odd},1),$ which result in a total even degree for $d\theta^{i}$ and a total odd degree for $dx^{\mu}$. From this point of view, the de Rham differential acts as a “supersymmetry” transformation: $d:\left\\{\begin{aligned} x^{\mu}&\mapsto dx^{\mu}\\\ \theta^{i}&\mapsto d\theta^{i}\end{aligned}\right.$ (4.9) exchanging bosonic coordinates with fermionic coordinates. Since the odd 1-forms $d\theta^{i}$ are commuting elements, it is not possible to construct, at least in the usual sense, a form of “top degree”on $SM$. We will thus interpret integration over the odd coordinates by the purely algebraic rules of Berezin integration for Grassmann variables: $\int d\theta^{i}\theta^{i}=1,\qquad\int d\theta^{i}1=0,$ (4.10) and such that Fubini’s theorem holds for multidimensional integrals. We see that symbolically $\int d\theta^{i}\leftrightarrow\frac{\partial}{\partial\theta^{i}},$ (4.11) and in particular $\int d\theta^{i}\frac{\partial}{\partial\theta^{i}}\Phi(\theta^{i})=0$ always holds. We will use the important property of the Berezin integral: $\int d^{n}\theta\ e^{-\theta^{i}A_{ij}\theta^{j}}=\mathrm{Pf}(A)$ (4.12) where $d^{n}\theta\equiv d\theta^{1}d\theta^{2}\cdots d\theta^{n}$, to be compared with the usual Gaussian integral for real variables $\int d^{n}x\ e^{-x^{i}A_{ij}x^{j}}=\frac{\pi^{n/2}}{\sqrt{\det(A)}}.$ (4.13) Notice that under an homogeneous change of coordinates $\varphi^{i}=B^{i}_{j}\theta^{j}$, the “measure” shifts as $\int d^{n}\theta\to\det{B}\int d^{n}\varphi$, such that the Gaussian integral (4.12) is invariant under similarity transformations. For a mathematically refined theory of superintegration, we suggest looking at [40]. Concerning the integration of functions on the odd tangent bundle $\Pi TM$, we notice how, making use of the Berezin rules (4.10), this is nothing but a reinterpretation of the usual integrals of differential forms on $M$. If $\dim{M}=d$, the integral of the form $\omega=\sum_{i}\omega^{(i)}$, where $\omega^{(i)}\in\Omega^{i}(M)$ is $\int_{M}\omega=\int_{M}\omega^{(d)}=\int_{M}d^{d}x\ \omega^{(d)}(x),$ (4.14) selecting the top-form of degree $d$. If we consider the same form $\omega\in C^{\infty}(\Pi TM)$, its trivialization in coordinates $(x,\theta)$ is $\omega(x,\theta)=\sum_{i}\omega^{(i)}_{\mu_{1}\cdots\mu_{i}}(x)\theta^{\mu_{1}}\cdots\theta^{\mu_{i}}.$ (4.15) Now the Berezin integration over $d^{d}\theta$ selects just the term with the right number of $\theta$’s, giving $\int_{\Pi TM}d^{d}xd^{d}\theta\ \omega(x,\theta)=\int_{M}d^{d}x\ \omega^{(d)}(x)\int d^{d}\theta\ \theta^{d}\theta^{d-1}\cdots\theta^{1}=\int_{M}d^{d}x\ \omega^{(d)}(x).$ (4.16) ### 4.2 Supergeometric proof of ABBV formula for a circle action We give now a proof of the ABBV integration formula for a $U(1)$-action, starting from the expression (3.14). The “localization 1-form” is chosen as $\beta:=g(\underline{T},\cdot)$ (4.17) where $g$ is a $U(1)$-invariant metric and $\underline{T}$ is the fundamental vector field corresponding to the generator $T\in\mathfrak{u}(1)$. In local coordinates, the action of the Cartan differential $d_{C}=d+\iota_{T}$ on this 1-form can be written as $\begin{split}d_{C}\beta&=B_{\mu\nu}(x)dx^{\mu}dx^{\nu}+g_{\mu\nu}(x)T^{\mu}(x)T^{\nu}(x)\\\ B_{\mu\nu}&=(\nabla_{\mu}T)_{\nu}-(\nabla_{\nu}T)_{\mu}\end{split}$ (4.18) where, again, we suppressed the $S(\mathfrak{u}(1)^{})$ generator setting $\phi=-1$. We use the result of the last section to rewrite (3.14) as an integral over the odd tangent bundle $\Pi TM$, identifying the odd coordinates $\theta^{\mu}\equiv dx^{\mu}$: $\begin{split}I[\alpha]&=\int_{M}\alpha=\lim_{t\to\infty}\int_{M}\alpha e^{-td_{C}\beta}\\\ &=\lim_{t\to\infty}\int_{\Pi TM}d^{d}xd^{d}\theta\ \alpha(x,\theta)\exp{\left\\{-tB_{\mu\nu}(x)\theta^{\mu}\theta^{\nu}-tg_{\mu\nu}(x)T^{\mu}(x)T^{\nu}(x)\right\\}},\end{split}$ (4.19) where the equivariantly closed form $\alpha$ is the sum of $U(1)$-invariant differential forms in $\Omega(M)^{U(1)}$ suppressing the $S(\mathfrak{u}(1)^{})$ generator, $\alpha(x,\theta)=\sum_{i}\alpha^{(i)}_{\mu_{1}\cdots\mu_{i}}(x)\theta^{\mu_{1}}\cdots\theta^{\mu_{i}},$ (4.20) such that $d_{C}\alpha=\left(\theta^{\mu}\partial_{x^{\mu}}+T^{\mu}\partial_{\theta^{\mu}}\right)\alpha=0$. Using the Gaussian integrals (4.12) and (4.13), we have the following delta- function representations for Grassmann-even and Grassmann-odd variables $\displaystyle\delta^{(n)}(y)$ $\displaystyle=\lim_{t\to\infty}\left(\frac{t}{\pi}\right)^{n/2}\sqrt{\det{A}}\ e^{-ty^{\mu}A_{\mu\nu}y^{\nu}}$ (4.21) $\displaystyle\delta^{(n)}(\eta)$ $\displaystyle=\lim_{t\to\infty}t^{-n/2}\frac{1}{\mathrm{Pf}A}\ e^{-tA_{\mu\nu}\eta^{\mu}\eta^{\nu}}$ where the limits are understood in the weak sense. Multiplying and dividing by $(t^{n/2})$ in (4.19), and using the delta-representation we rewrite the integral as $I[\alpha]=\pi^{d/2}\int_{\Pi TM}d^{d}xd^{d}\theta\ \alpha(x,\theta)\frac{\mathrm{Pf}B(x)}{\sqrt{\det{g}(x)}}\delta^{(d)}(T(x))\delta^{(d)}(\theta).$ (4.22) The delta function on the odd coordinates simply puts $\theta^{\mu}=0$, that is analogous to selecting the top-degree form in (4.19), so that it remains $\alpha(x,0)\equiv\alpha^{(0)}(x)\in C^{\infty}(M)$. The delta function on the even coordinates instead selects the values at which $T^{\mu}(x)=0$, that corresponds to the fixed point set $F\hookrightarrow M$ of the $U(1)$-action. Suppose this fixed point set to be of dimension 0, i.e. composed by isolated points in $M$. If this is the case, we can simply separate the integral $\int d^{d}x$ in a sum of integrals, each of which domain $\mathcal{D}(p):p\in F$ contains one and only one of those fixed points, and in each of them apply the delta function $\begin{split}I[\alpha]&=\pi^{d/2}\sum_{p\in F}\int_{\mathcal{D}(p)}d^{d}x\ \alpha^{(0)}(x)\frac{\mathrm{Pf}B(x)}{\sqrt{\det{g}(x)}}\delta^{(d)}(T(x))\\\ &=\pi^{d/2}\sum_{p\in F}\frac{\alpha^{(0)}(p)}{\|\det{dT}\|(p)}\frac{\mathrm{Pf}B(p)}{\sqrt{\det{g}(p)}}.\end{split}$ (4.23) Here the factor $\|\det{dT}\|(p)$ is the Jacobian from the change of variable $y^{\mu}:=T^{\mu}(x)$. At any point $p\in F$ we have $(\nabla_{\mu}T)_{\nu}(p)=\partial_{\mu}T_{\nu}(p)=\partial_{\mu}T^{\rho}(p)g_{\rho\mu}(p)$ since $T^{\mu}(p)=0$, so the pfaffian in the numerator becomes $\mathrm{Pf}B(p)=\sqrt{2^{d}}\ \mathrm{Pf}{dT}(p)\sqrt{\det{g}(p)},$ (4.24) and we get the result $I[\alpha]=(2\pi)^{d/2}\sum_{p\in F}\frac{\alpha^{(0)}(p)}{\mathrm{Pf}dT(p)}.$ (4.25) Notice that, at $p\in F$, the operator $dT(p)=\partial_{\mu}T^{\nu}\theta^{\mu}\otimes\partial_{\nu}=-[\underline{T},\cdot]$ coincide up to a sign with the infinitesimal action $\mathcal{L}_{T}$ of $T\in\mathfrak{u}(1)$ on the tangent space $T_{p}M$, just because here $T^{\mu}(p)=0$. If we consider a continuous fixed point set, so that $F$ is a regular submanifold of $M$, it is not possible to use the delta functions like in (4.23), but we can consider the decomposition as a disjoint union $M=F\sqcup N$. Points far away from $F$ give a zero contribution to $I[\alpha]$ in the limit $t\to\infty$, so we can consider a neighborhood of $F$ and split here the tangent bundle as $TM\cong i_{}TF\oplus TN$ where $TN$ is the _normal bundle_ to $F$ in $M$ ($i$ is the inclusion map). Consequently, in this neighborhood we can split the coordinates $(x^{\mu},\theta^{\mu})$ on the odd tangent bundle in tangent and normal to $F$, and rescale the normal components as $1/\sqrt{t}$, $x^{\mu}=x^{\mu}_{0}+\frac{x^{\mu}_{\perp}}{\sqrt{t}},\qquad\theta^{\mu}=\theta_{0}^{\mu}+\frac{\theta_{\perp}^{\mu}}{\sqrt{t}}.$ (4.26) The measure simply splits as $d^{d}xd^{d}\theta=d^{n}x_{0}d^{\hat{n}}x_{\perp}d^{n}\theta_{0}d^{\hat{n}}\theta_{\perp}$, where $n+\hat{n}=d$, thanks to the Berezin integration rules (4.10). Expanding $B_{\mu\nu}(x),g_{\mu\nu}(x),T^{\mu}(x)$ around the $x_{0}$ components, and taking the limit $t\to\infty$, the integral becomes [41] $\displaystyle I[\alpha]=\int d^{n}x_{0}d^{n}\theta_{0}\ \alpha(x_{0},\theta_{0})$ $\displaystyle\int d^{\hat{n}}x_{\perp}d^{\hat{n}}\theta_{\perp}\ $ (4.27) $\displaystyle\exp{\left\\{-B_{\mu\sigma}(x_{0})\left(B_{\nu}^{\sigma}(x_{0})+R^{\sigma}_{\nu\lambda\rho}(x_{0})\theta_{0}^{\lambda}\theta_{0}^{\rho}\right)x_{\perp}^{\mu}x_{\perp}^{\nu}-B_{\mu\nu}(x_{0})\theta_{\perp}^{\mu}\theta_{\perp}^{\nu}\right\\}}$ where $R^{\sigma}_{\nu\lambda\rho}(x_{0})$ is the curvature relative to the metric $g$. The integrals over the normal coordinates are Gaussian, giving the exact “saddle point” contribution $\frac{1}{\mathrm{Pf}_{N}\left(\frac{R+B}{2\pi}\right)(x_{0})}=\frac{1}{\left.e_{T}(R)\right\|_{N}},$ (4.28) where $\left.e_{T}(R)\right\|_{N}$ is the $U(1)$-_equivariant Euler class_ of the normal bundle to $F$. We notice that this matches the definition in Appendix B.1, since $B:=\nabla T$, seen as an element of the adjoint bundle $\Omega^{0}(M;\mathfrak{gl}(n))$, is a moment map for the Riemannian curvature $R$ satisfying (as can be checked by direct computation) $\nabla B^{\sigma}_{\nu}=-\iota_{T}R^{\sigma}_{\nu}=R^{\sigma}_{\nu}(\cdot,\underline{T}),$ (4.29) where the covariant derivative acts as in the adjoint bundle with respect to the Levi-Civita connection, $\nabla=d+[\Gamma,\cdot]$. As the final piece, the Berezin integration selects the component of $\alpha$ of degree $\dim{F}$ evaluated on $F$, that is just the pull-back $i^{}\alpha$ along the inclusion map. Summarizing, we are left with $I[\alpha]=\int_{F}\frac{i^{}\alpha}{\left.e_{T}(R)\right\|_{N}}$ (4.30) as presented in Section 3.2, for the case of a circle action. Notice that in Section 3.2 we called the moment map $B^{\sigma}_{\nu}(p)=\mu_{T}(p)^{\sigma}_{\mu}\in\mathrm{End}(T_{p}M)$ at a fixed point $p$. If the fixed point is isolated, the tangent space $T_{p}M$ splits, as in (3.22), as the direct sum of the weight spaces of the $U(1)$-representation, and the vector field $\underline{T}$ acts as a rotation in any subspace. On a suitable coordinate basis $\underline{T}=\sum_{i}v_{i}\left(x^{i}\frac{\partial}{\partial y^{i}}-y^{i}\frac{\partial}{\partial x^{i}}\right),$ (4.31) so that the moment map block-diagonalizes as $B^{\sigma}_{\nu}=\partial_{\nu}T^{\sigma}=\left(\begin{array}[]{ccccc}0&-v_{1}&\cdots&&\\\ v_{1}&0&&&\\\ \cdots&&0&-v_{2}&\\\ &&v_{2}&0&\\\ &&&&\ddots\end{array}\right).$ (4.32) Taking the pfaffian then one gets exactly the product of the weights (or the “exponents”) $v_{i}$ of the circle action, so that the Euler class results $e_{T}(R)=(2\pi)^{-\dim(M)/2}\prod_{i}v_{i}.$ (4.33) This recovers (3.23) for the case of a circle action. See [6] for the result in presence of a torus action. ### 4.3 Introduction to Poincaré-supersymmetry We have seen in the last section how it is useful to translate the integration problem of a differential form on the manifold $M$, into an integration over the supermanifold $\Pi TM$. Here the differential forms $\Omega(M)$ are seen as the _graded_ ring of functions over $\Pi TM$, and the differential $d_{C}=d+\iota_{T}$ being the sum of two graded derivations555Both $d$ and $\iota_{T}$ are nilpotent supervector fields on $\Pi TM$. Without suppressing the degree-2 generator $\phi$ of $S(\mathfrak{u(1)}^{})$, it is apparent that their sum $d_{C}$ is a cohomological vector field, i.e. a good differential of degree +1, on the subspace of $U(1)$-invariant forms. of degree $\pm 1$, can be viewed as an infinitesimal _supersymmetry_ transformation mapping odd-degree (_fermionic_) forms to even-degree (_bosonic_) forms. In field theory, we already mentioned that the presence of a supersymmetry on the relevant complex of fields often arises in two different ways: • The differential $d_{C}$ is represented in the physical model by a BRST-like supercharge, introduced because of some gauge freedom. In this case, the complex of fields (analogously to the graded ring of functions $\Omega(M)$) is the BRST complex, and the grading is referred to as _ghost number_. Physical states of the quantum field theory are then created by fields of 0-degree, i.e. the functions on $M$. We could refer to this type of supersymmetry as a “hidden” one, coming from the original internal gauge symmetry of the model. In Hamiltonian systems, as we shall see in the next chapter, this gauge freedom can be simply associated to the Hamiltonian flow. * • The original theory could also be explicitly endowed with a supersymmetry. In this case the base space has a supermanifold structure, and the grading of the field complex follows. This is the case of QFT with Poincaré supersymmetry, where the action of a _supercharge_ as generator of the super-Poincaré algebra can be interpreted as an equivariant differential. In this and the next sections we will review the geometric setup of Poincaré supersymmetry and its generalization to curved spacetimes, interesting case for practical applications of the localization technique in QFT. As QFT (with global Poincaré symmetry) is formulated on Minkowski spacetime666We are really interested in both the Lorentzian and the Euclidean case, so we will express both the Minkowski and Euclidean spaces as $\mathbb{R}^{d}$ without stressing of the signature in the notation. The choice of metric will be clear from the context. $\mathbb{R}^{d}\cong ISO(\mathbb{R}^{d})/O(d),$ (4.34) we can formulate a Poincaré-supersymmetric theory on a super-extension of this space, coming from a given super-extension of the Poincaré group $ISO(\mathbb{R}^{d})$. We then first introduce the super-Poincaré groups starting from the super-extension of their algebras. #### 4.3.1 Super-Poincaré algebra and superspace ###### Definition 4.3.1. A super-Poincaré algebra is the extension of the Poincaré algebra $\mathfrak{iso}(d)\cong\mathbb{R}^{d}\oplus\mathfrak{so}(d)$ as a Lie superalgebra, via a given _real_ spin representation space $S$ of $Spin(d)$ taken in odd-degree: $\mathfrak{siso}_{S}(d)\cong\mathfrak{iso}(d)\oplus S[1].$ (4.35) The super Lie bracket are extended on $S[1]$ through the _symmetric_ and Spin- equivariant bilinear form $\Gamma:S\times S\to\mathbb{R}^{d}$: $[\Psi,\Phi]:=2\Gamma(\Psi,\Phi)=2(\overline{\Psi}\gamma^{\mu}\Phi)P_{\mu}=2(\Psi^{T}C\gamma^{\mu}\Phi)P_{\mu}$ (4.36) where $P_{\mu}$ are generators of the translation algebra $\mathbb{R}^{d}$, $C$ is the charge conjugation matrix, $\overline{\Psi}\in S^{}$ is the Dirac adjoint of $\Psi$,777In Lorentzian signature $\overline{\Psi}=\Psi^{\dagger}\beta$ with $\beta_{ab}\equiv(\gamma^{0})^{a}_{\ b}$, in Euclidean signature $\overline{\Psi}=\Psi^{\dagger}$. This is due to hermitianity of the generators of the Euclidean algebra, unlike the Lorentzian case. and $\gamma^{\mu}$ are the generators of the Clifford algebra acting on $S$. In a real representation, the Majorana condition $\Psi^{T}C\stackrel{{\scriptstyle!}}{{=}}\overline{\Psi}$ is satisfied. The other brackets involving $S$ are defined by the natural action of $\mathfrak{so}(d)$ on it, and by the trivial action on $\mathbb{R}^{d}$: $\displaystyle\lambda\in\mathfrak{so}(d):\quad$ $\displaystyle[\lambda,\Psi]:=\frac{i}{2}\lambda_{\mu\nu}\Sigma^{\mu\nu}(\Psi),$ (4.37) $\displaystyle[\Psi,\lambda]=-[\lambda,\Psi],$ (4.38) $\displaystyle a\in\mathbb{R}^{d}:\quad$ $\displaystyle[a,\Psi]:=0,$ (4.39) where $\Sigma^{\mu\nu}=\frac{i}{2}\gamma^{[\mu}\gamma^{\nu]}$ are the generators of the rotation algebra in the Spin representation $S$. If the charge conjugation matrix is symmetric in the given representation, we can further enlarge this superalgebra via a “central extension”, considering $\mathfrak{siso}_{S}(d)\oplus\mathbb{R}$ with the extended brackets $\displaystyle\Psi,\Phi\in S[1]:\quad$ $\displaystyle[\Psi,\Phi]:=\Gamma(\Psi,\Phi)+(\Psi^{T}C\Phi),$ (4.40) $\displaystyle x\in\mathbb{R},A\in\mathfrak{siso}_{S}(d):\quad$ $\displaystyle[x,A]:=0.$ (4.41) The super Jacobi identity is satisfied thanks to the Spin-equivariance of the spinor bilinear form: $\Gamma\left(e^{R^{(s)}}\Psi,e^{R^{(s)}}\Phi\right)=e^{R^{(v)}}\Gamma(\Psi,\Phi)$ (4.42) where $R^{(s)},R^{(v)}$ are the same element $R\in\mathfrak{so}(d)$ in the spin and vector representations, respectively. Often the bracket structure of this superalgebra is given in terms of the generators. Regarding the odd part and picking a basis $\left\\{Q_{a}\right\\}$ of $S[1]$, their brackets are888We conventionally raise and lower spinor indices with the charge conjugation matrix: $(\gamma^{\mu})_{ab}:=C_{ac}(\gamma^{\mu})^{c}_{\ b},\qquad(\gamma^{\mu})_{ab}=(\gamma^{\mu})_{ba}.$ Notice that the matrix $C$ represents an inner product on $S$, while the Clifford algebra generators $\\{\gamma^{\mu}\\}$ act on $S$ as endomorphisms, so the index structure follows. A review of classification of Clifford algebras, Spin groups and Majorana spinors can be found in [42]. $[Q_{a},Q_{b}]=2(\gamma^{\mu})_{ab}P_{\mu}+C_{ab}.$ (4.43) The generators $\left\\{Q_{a}\right\\}$ are referred to as _supercharges_. If the real spin representation on $S$ is irreducible as a representation of the corresponding Clifford algebra, we have the minimal amount of supersymmetry and we refer to $\mathfrak{siso}_{S}(d)$ as to an $\mathcal{N}=1$ supersymmetry algebra. If instead the representation is reducible, then $S=\bigoplus_{I=1}^{\mathcal{N}}S^{(I)}$ and we can split the basis of supercharges as $\left\\{Q^{I}_{a}\right\\}$. This case is referred to as _extended_ supersymmetry. In this basis the gamma matrices block-diagonalize as $\gamma^{\mu}\otimes\mathbb{I}$, with $\gamma^{\mu}$ the (minimal) gamma matrices in every $S^{(I)}$, and the central extension part separates as $C\otimes Z$, with $C$ being the (minimal) charge conjugation matrix in every $S^{(I)}$ and $Z$ a matrix of so-called _central charges_. The odd part of the superalgebra then looks like $[Q_{a}^{I},Q_{b}^{J}]=2(\gamma^{\mu})_{ab}\delta^{IJ}P_{\mu}+C_{ab}Z^{IJ}.$ (4.44) The matrix $Z$ must be (anti)symmetric if $C$ is (anti)symmetric. ###### Definition 4.3.2. The subspace $\mathfrak{st}_{S}(d):=\mathbb{R}^{d}\oplus S[1]$ is a Lie superalgebra itself (if there is no central extension), and can be referred to as the _super-translation algebra_. Even if $\mathfrak{st}_{S}(d)$ it is not Abelian, it has the property $[a,[b,c]]=0\qquad\forall a,b,c\in\mathfrak{st}_{S}(d)$ (4.45) that can be easily checked by the definition. This means that the elements of the corresponding _super-translation group_ can be computed exactly using the exponential map and the BHC formula. This space can be identified with the _super-spacetime_. ###### Definition 4.3.3. We can define the full super-Poncaré group as $SISO_{S}(d)=\exp{(\mathfrak{st}_{S}(d))}\rtimes Spin(d),$ (4.46) so that we can identify the superspacetime with respect to the spin representation $S$, analogously to (4.34), as $S\mathbb{R}_{S}^{d}=SISO_{S}(d)/Spin(d)\cong\exp{(\mathfrak{st}_{S}(d))}.$ (4.47) As a supermanifold of dimension $(d\|\dim(S))$, $S\mathbb{R}_{S}^{d}$ is characterized by its sheaf of functions, $\mathcal{A}=C^{\infty}(\mathbb{R}^{d})\otimes\bigwedge(S^{}).$ (4.48) So, $S\mathbb{R}_{S}^{d}$ is the odd vector bundle associated to the spinor bundle over $\mathbb{R}^{d}$ of typical fiber $S$. In particular, on $S\mathbb{R}_{S}^{d}$ we have respectively even and odd coordinates $(x^{\mu},\theta^{a})$, with $\mu=1,\cdots,d$ and $a=1,\cdots,\dim(S)$. As a Lie group, we can get the group operation (the sum by supertranslation) from the exponentiation of its Lie superalgebra. Technically, to use the BHC formula $e^{A}e^{B}=e^{A+B+\frac{1}{2}[A,B]+\cdots}$ (4.49) we would like to deal with a _Lie algebra_ , so we consider the group operation on coordinates functions instead of points on $S\mathbb{R}_{S}^{d}$, taking the space $\left(\mathcal{A}\otimes\mathfrak{st}_{S}(d)\right)_{(0)}=\left(\mathcal{A}_{(0)}\otimes\mathbb{R}^{d}\right)\oplus\left(\mathcal{A}_{(1)}\otimes S[1]\right).$ (4.50) The Lie brackets on this space are inherited from those on $\mathfrak{st}_{S}(d)$ and the (graded) multiplication of functions in $\mathcal{A}$. The only non-zero ones come from couples of elements of $\mathcal{A}_{(1)}\otimes S[1]$: $[f_{1}\otimes\epsilon_{1},f_{2}\otimes\epsilon_{2}]=-2\Gamma(\epsilon_{1},\epsilon_{2})f_{1}f_{2}$ (4.51) where the sign rule has been used since both $f_{1},f_{2}$ and $\epsilon_{1},\epsilon_{2}$ are odd.999We are being a little informal here, but this can be made more rigorous with the help of a construction called _functor of points_. The important thing for us is that in this approach one can work with coordinate functions $x,\theta$ instead of some would-be “points” on the supermanifold (a misleading concept since we know from the last section that a supermanifold is not a set). See [43] for more details. We will use the combination (suppressing tensor products) $[\theta^{a}Q_{a},\varphi^{b}Q_{b}]=-2\Gamma(Q_{a},Q_{b})\theta^{a}\varphi^{b}=-2(\gamma^{\mu})_{ab}\theta^{a}\varphi^{b}P_{\mu}\equiv-2(\theta\gamma^{\mu}\varphi)P_{\mu}.$ This makes $\left(\mathcal{A}\otimes\mathfrak{st}_{S}(d)\right)_{(0)}$ into a Lie algebra, by the antisymmetry of the product between odd functions. Representing via the exponential map $(x,\theta)$ as $\exp\left\\{i(xP+\theta Q)\right\\}$, where we suppressed also index contractions, we can finally use the BHC formula on this algebra to get the group operation on coordinates: $\displaystyle(x,\theta)\,(y,\varphi)$ $\displaystyle=\exp{\left\\{i(xP+\theta Q)\right\\}}\exp{\left\\{i(yP+\varphi Q)\right\\}}$ (4.52) $\displaystyle=\exp{\left\\{i\left(xP+\theta Q+yP+\varphi Q+\frac{i}{2}[\theta Q,\varphi Q]\right)\right\\}}$ $\displaystyle=\exp{\left\\{i\left(x+y+i\theta\gamma\varphi\right)P+i\left(\theta+\varphi\right)Q\right\\}}$ $\displaystyle=\left(x+y-i\theta\gamma\varphi,\theta+\varphi\right).$ If the odd dimension $\dim{(S)}$ is zero, this reduces to a standard translation in $\mathbb{R}^{d}$. The infinitesimal action of the superalgebra $\mathfrak{st}_{S}(d)$ is defined as a Lie derivative with respect to the fundamental vector field representing a given element of the supertranslation algebra. For $\Phi\in\mathcal{A}$, $\epsilon=\epsilon^{a}Q_{a}\in(\mathcal{A}_{(1)}\otimes S[1])$ $\delta_{\epsilon}\Phi(x,\theta):=\mathcal{L}_{\underline{\epsilon}}(\Phi(x,\theta))=\underline{\epsilon}(\Phi)=\epsilon^{a}\underline{Q}_{a}(\Phi(x,\theta)),$ (4.53) where the odd vector field $\underline{Q}_{a}$ is associated to the supercharge $Q_{a}$ through the _left_ translation (4.52): $\displaystyle\underline{Q}_{a}\Phi(x,\theta)$ $\displaystyle=\left.\frac{\partial}{\partial\varphi^{a}}\left(e^{-i\varphi Q}\right)^{}\Phi(x,\theta)\right\|_{\varphi=0}=\left.\frac{\partial}{\partial\varphi^{a}}\Phi\big{(}(0,-\varphi)(x,\theta)\big{)}\right\|_{\varphi=0}=$ (4.54) $\displaystyle=\partial_{\mu}\Phi(x,\theta)(i\gamma^{\mu}\theta)_{a}+\partial_{b}\Phi(x,\theta)\delta^{b}_{a}.$ We recognize then $\underline{Q}_{a}=-\frac{\partial}{\partial\theta^{a}}+i(\theta\gamma^{\mu})_{a}\frac{\partial}{\partial x^{\mu}},$ (4.55) and one can check that $[\underline{Q}_{a},\underline{Q}_{b}]=2(\gamma^{\mu})_{ab}\underline{P}_{\mu}$ with $\underline{P}_{\mu}=-i\partial/\partial x^{\mu}$ is associated to the momentum generator. The rest of the super-Poincaré group and its algebra acts naturally on superspace following the same type of arguments. In particular, the spin generators act in the vector and spin representations on the even and odd sectors, respectively. It is useful to introduce also the fundamental vector fields with respect to _right_ supertranslations on $S\mathbb{R}^{d}_{S}$. These are called _superderivatives_ and are easily obtained from the law (4.52) as101010Remember that left and right actions correspond to opposite signs at the exponent in the definition of the fundamental vector fields. $D_{a}:=\frac{\partial}{\partial\theta^{a}}+i(\theta\gamma^{\mu})_{a}\frac{\partial}{\partial x^{\mu}}.$ (4.56) One can check that indeed they satisfy $[\underline{Q}_{a},D_{b}]=0$ and $[D_{a},D_{b}]=-2(\gamma^{\mu})_{ab}P_{\mu}$, since right invariant and left invariant vector fields form anti-isomorphic algebras. #### 4.3.2 Chiral superspace and superfields It is customary to construct supersymmetric field theories starting not from a _real_ spin representation, but from a _complex_ Dirac representation $S$ endowed with a _real structure_ , i.e. an antilinear map $J:S\to S$ which is an involution ($J^{2}=id_{S}$).111111$J$ is the generalization of the “complex conjugation” operation on a $\mathbb{C}$-vector space. In this case, diagonalizing $J$ the representation splits as $S\cong S_{\mathbb{R}}\otimes\mathbb{C}\cong S^{(+)}\oplus S^{(-)}$ where $S_{\mathbb{R}}\cong S^{(+)}\cong iS^{(-)}$ are a real vector spaces. Choosing some basis, the matrix representing the real structure is $J=C(\gamma^{0})^{T}$ in Lorenzian signature, or $J=C$ in Euclidean signature. The _Majorana spinors_ are the elements of $S_{\mathbb{R}}$, that satisfy $J(\Psi)=\Psi$, or in matrix notation $\overline{\Psi}=\Psi^{T}C$. The subspace of Majorana spinors, taken as a real representation, would then give the super-extension of the last paragraph. If instead we are interested in working with the whole complex representation $S$, we are forced to introduce _complexified_ supersymmetry algebra and superspace, and then impose constraints on the resulting objects to properly reduce their degrees of freedom a posteriori. In particular, complex spinors from $S$ generating the supertranslations are taken satisfying the Majorana condition. This is always the case in QFT. As a paradigmatic example, we can take $\mathcal{N}=1$ supersymmetry in (3+1)-dimensions, where $\Psi$ is a Dirac spinor in $S=\mathbb{C}^{4}\cong\mathbb{R}^{4}\otimes\mathbb{C}$. On the complex representation, we can chose a to work in the _chiral basis_ $\\{Q_{a},\tilde{Q}_{\dot{a}}\\}$, splitted in left- and right-handed Weyl spinors. The dotted and undotted indices now run between $1,2$. Here the symmetric pairing $\Gamma:S\times S\to\mathbb{C}^{4}$ is non-zero only on $S^{(L/R)}\times S^{(R/L)}$, and and the restriction to the symmetrized subspace $\Gamma:S^{L}\odot S^{R}\to\mathbb{C}^{4}$ is actually an isomorphism, so the relevant non-zero brackets are $[Q_{a},\tilde{Q}_{\dot{b}}]=2(\gamma^{\mu})_{a\dot{b}}P_{\mu}.$ (4.57) On the chiral basis, $\gamma^{\mu}=\begin{pmatrix}0&\sigma^{\mu}\\\ \bar{\sigma}^{\mu}&0\end{pmatrix},\qquad C=\begin{pmatrix}\varepsilon&0\\\ 0&-\varepsilon\end{pmatrix},\qquad\sigma^{\mu}=(\mathbf{1},\sigma^{i}),\quad\overline{\sigma}^{\mu}=(\mathbf{1},-\sigma^{i}),$ where $(\sigma^{i})_{i=1,2,3}$ are the Pauli matrices and $\varepsilon_{ab}=\varepsilon_{\dot{a}\dot{b}}$ is the totally antisymmetric tensor. The supercharges are represented by the odd vector fields $\underline{Q}_{a}=-\frac{\partial}{\partial\theta^{a}}+i\tilde{\theta}^{\dot{b}}(\gamma^{\mu})_{\dot{b}a}\frac{\partial}{\partial x^{\mu}},\qquad\underline{\tilde{Q}}_{\dot{a}}=-\frac{\partial}{\partial\tilde{\theta}^{\dot{a}}}+i\theta^{b}(\gamma^{\mu})_{b\dot{a}}\frac{\partial}{\partial\overline{x^{\mu}}},$ (4.58) and the supersymmetry action on a superfield is $\delta_{\epsilon}\Phi=(\epsilon^{a}\underline{Q}_{a}+\tilde{\epsilon}^{\dot{a}}\underline{\tilde{Q}}_{\dot{a}})\Phi$ (4.59) where $\overline{\epsilon^{a}}=\tilde{\epsilon}^{\dot{a}}$, being a Majorana spinor. The superderivatives are $D_{a}=\frac{\partial}{\partial\theta^{a}}+i\tilde{\theta}^{\dot{b}}(\gamma^{\mu})_{\dot{b}a}\frac{\partial}{\partial x^{\mu}},\qquad\tilde{D}_{\dot{a}}=\frac{\partial}{\partial\tilde{\theta}^{\dot{a}}}+i\theta^{b}(\gamma^{\mu})_{b\dot{a}}\frac{\partial}{\partial\overline{x^{\mu}}}.$ (4.60) Here we split the odd coordinates $\theta^{a},\tilde{\theta}^{\dot{a}}$ according to the split of the supercharges. The reality constraint on the coordinates then reads $\overline{\theta^{a}}=\tilde{\theta}^{\dot{a}},\qquad\overline{x^{\mu}}=x^{\mu}.$ (4.61) Since we are operating over $\mathbb{C}$, the subspaces of the supertranslation algebra $\mathfrak{st}^{(L/R)}:=\mathbb{C}^{4}\oplus S^{(L/R)}[1]$ (4.62) are both Lie superalgebras over $\mathbb{C}$, and determines the corresponding complex Lie supergroups $S\mathbb{C}^{(L/R)}$. Moreover these subalgebras are _Abelian_ , since $\Gamma$ vanishes on $S^{(L/R)}\times S^{(L/R)}$. ###### Definition 4.3.4. $S\mathbb{C}^{(L)}$ is called _chiral_ superspace and $S\mathbb{C}^{(R)}$ _anti-chiral_ superspace. We can write the complexified superspace as $S\mathbb{C}^{4}_{S}\cong S\mathbb{C}^{(L)}\times_{\mathbb{C}^{4}}S\mathbb{C}^{(R)}$ (4.63) where the $\times_{\mathbb{C}^{4}}$ here denotes the fiber product with respect to the base $\mathbb{C}^{4}$.121212This is analogous to the pull-back bundle, not to be confused with an homotopy quotient. The chiral and anti- chiral superspaces are those identified by the flows of the corresponding superderivatives, since they generates the chiral and anti-chiral part of the supertranslation algebra on $\Gamma(TS\mathbb{C}_{S}^{4})$.131313Here $\Gamma$ denotes the space of sections on the tangent bundle, i.e. the vector fields on $S\mathbb{C}_{S}^{4}$, not the spinor pairing. We can easily find sets of “holomorphic-like” coordinates $y^{\mu}_{(\pm)}:=x^{\mu}\pm i\tilde{\theta}^{\dot{a}}(\gamma^{\mu})_{\dot{a}b}\theta^{b},\qquad\varphi^{a}:=\theta^{a},\qquad\tilde{\varphi}^{\dot{a}}:=\tilde{\theta}^{\dot{a}},$ (4.64) where the superderivatives simplify as $D_{a}=\left\\{\begin{aligned} &\frac{\partial}{\partial\varphi^{a}}\\\ &\frac{\partial}{\partial\varphi^{a}}+2i(\tilde{\varphi}\gamma^{\mu})_{a}\frac{\partial}{\partial y^{\mu}_{(-)}}\end{aligned}\right.,\qquad\tilde{D}_{\dot{a}}=\left\\{\begin{aligned} &\frac{\partial}{\partial\tilde{\varphi}^{\dot{a}}}+2i(\varphi\gamma^{\mu})_{\dot{a}}\frac{\partial}{\partial y^{\mu}_{(+)}}\\\ &\frac{\partial}{\partial\tilde{\varphi}^{\dot{a}}}\end{aligned}\right..$ (4.65) Complexifying the superspace we doubled its real-dimension, and that leads to a sort of “reducibility” of the relevant physical objects, _i.e._ the fields on superspace, or _superfields_. The simplest kind of superfields in the complexified setting are complex _even_ maps $\Phi:S\mathbb{C}^{4}\to\mathbb{C}$, that is sections of a trivial $\mathbb{C}$-line bundle over $S\mathbb{C}^{4}$. We can impose now some constraints on them in order to restore the correct number of degrees of freedom. This is usually done in two different ways: asking the superfields to depend only on the chiral (or anti-chiral) sector of the complexified superspace, or imposing a reality condition. ###### Definition 4.3.5. 1. (i) A _chiral (anti-chiral) superfield_ is a superfield $\Phi$ such that $\tilde{D}_{\dot{a}}\Phi=0\qquad\left(D_{a}\Phi=0\right).$ 2. (ii) A _vector superfield_ is a superfield $V$ such that $V=V^{\dagger}.$ Notice how the first one is a sort of (anti)holomorphicity condition with respect to the chiral/anti-chiral sectors of $S\mathbb{C}^{4}_{S}$, spanned by the coordinates $\varphi^{a},\tilde{\varphi}^{\dot{a}}$. Moreover, we can see that the complex conjugate $\Phi^{\dagger}$ of a chiral superfield $\Phi$ is antichiral. Next we will see how these conditions reflect on the various component fields of the coordinate expansions of $\Phi$ and $V$. It is important to stress that in this particular case of $\mathcal{N}=1$ in (3+1)-dimensions, the complex representation $S\cong\mathbb{C}^{4}$ allows for a real structure and the presence of Majorana spinors, and also for a chiral decomposition into left- and right-handed parts. It does not exist though a common basis for the two decompositions, i.e. $S^{(\pm)}\neq S^{(L/R)}$. In other words, it is not possible to require _both_ the chirality and the Majorana conditions on spinors in 4-dimension, since Majorana spinors contain both left- and right- handed components. This means that in a theory with some supersymmetry in 4-dimensions there will be the same number of left-handed and right-handed degrees of freedom. In $d=2\text{mod}8$ dimensions (in Lorentzian signature), instead, the minimal complexified spin representation $S$ can be decomposed into Majorana-Weyl subrepresentations $S=S_{\mathbb{R}}^{(+)}\oplus S_{\mathbb{R}}^{(-)}$, so that one can choose to work only with real left- handed spinors. In the case of extended supersymmetry, we can thus have in general a different number of left-handed and right-handed real supercharges, and $S=(S_{\mathbb{R}})^{\mathcal{N}_{+}}\oplus(S_{\mathbb{R}})^{\mathcal{N}_{-}}$. This is denoted with $\mathcal{N}=(\mathcal{N}_{+},\mathcal{N}_{-})$. When $\mathcal{N}_{+}$ or $\mathcal{N}_{-}$ is zero, the supersymmetry is called _chiral_. For a review on spinors in different dimensions, Majorana and chirality conditions we refer to [42]. #### 4.3.3 Supersymmetric actions and component field expansion Actions for supersymmetric field theories are constructed integrating over superspace combinations of superfields and their derivatives. For this purpose, it is useful to write the superfields in a so-called “component expansion”, with respect to the generators of $\bigwedge(S^{})$. In this paragraph we continue with the example of $\mathcal{N}=1$ in (3+1)-dimensions, but the construction is immediatly generalizable to other cases. Consider a complex superfield $\Phi:S\mathbb{C}^{4}\to\mathbb{C}$, its trivialization on the set of coordinates $(x,\theta,\tilde{\theta})$ being $\displaystyle\Phi(x,\theta,\tilde{\theta})=\phi(x)+v_{\mu}(x)(\tilde{\theta}\gamma^{\mu}\theta)+\psi_{a}(x)\theta^{a}+\tilde{\psi}_{\dot{a}}\tilde{\theta}^{\dot{a}}+F(x)\theta^{(2)}+\tilde{F}(x)\tilde{\theta}^{(2)}+$ (4.66) $\displaystyle+\xi_{a}(x)\theta^{a}\tilde{\theta}^{(2)}+\tilde{\xi}_{\dot{a}}(x)\tilde{\theta}^{\dot{a}}\theta^{(2)}+D(x)\theta^{(2)}\tilde{\theta}^{(2)}$ where the the wedge product between the $\theta$’s has been suppressed, and we agree they are anticommuting, $\theta^{(2)}:=\theta^{2}\theta^{1}$ and $\tilde{\theta}^{(2)}:=\tilde{\theta}^{\dot{2}}\tilde{\theta}^{\dot{1}}$. We used the isomorphism $\Gamma:S^{L}\odot S^{R}\to\mathbb{C}^{4}$ to represent the component of degree (1,1) as $v_{a\dot{a}}\mapsto v_{\mu}(\gamma^{\mu})_{a\dot{a}}$, the reason for this will become clear shortly. The expansion stops at top-degree (here 4) for the anticommuting property of the exterior product. In order for $\Phi$ to be an even (scalar) field, we must take the functions $\phi,v_{\mu},F,\tilde{F},D$ to be even (commuting), and the functions $\psi_{a},\tilde{\psi}_{\dot{a}},\xi_{a},\tilde{\xi}_{\dot{a}}$ carrying a spinor index to be odd (anticommuting). These are called _component fields_ of the superfield $\Phi$.141414The possibility of writing down an expansion similar to (4.66) with these properties could again be justified more rigorously thanks to the concept of _functor of point_. If we impose the chirality condition $\tilde{D}_{\dot{a}}\Phi=0$, when expressed in the coordinates $(y_{(-)},\varphi,\tilde{\varphi})$ this simply requires the independence on $\tilde{\varphi}$, so on these coordinates a chiral superfield can be written as $\Phi(y_{(-)},\varphi,\tilde{\varphi})=\phi(y_{(-)})+\psi_{a}(y_{(-)})\varphi^{a}+F(y_{(-)})\varphi^{(2)}.$ (4.67) Taylor-expanding in the old coordinates, this is equivalent to $\Phi(x,\theta,\tilde{\theta})=\phi(x)+\psi_{a}(x)\theta^{a}+F(x)\theta^{(2)}-i\partial_{\mu}\phi(x)(\tilde{\theta}\gamma^{\mu}\theta)+i(\tilde{\theta}\gamma^{\mu}\partial_{\mu}\psi(x))\theta^{(2)}-\partial^{2}\phi(x)\theta^{(2)}\tilde{\theta}^{(2)}$ (4.68) where higher order terms are again automatically zero for degree reasons.151515Here we used $\theta^{a}\theta^{b}=\varepsilon^{ab}\theta^{(2)}$, and $(\tilde{\theta}\gamma^{\mu}\theta)(\tilde{\theta}\gamma^{\mu}\theta)=2\eta^{\mu\nu}\theta^{(2)}\tilde{\theta}^{(2)}$ with a “mostly plus” signature. The “irreducible” chiral superfield has three non-zero field components: a complex scalar field $\phi$, a left-handed Weyl spinor field $\psi$ and another complex scalar field $F$. One can work out the supersymmetry transformations of these component fields from the general rule (4.59), and the result is $\displaystyle\delta_{\epsilon}\phi=\epsilon\psi$ (4.69) $\displaystyle\delta_{\epsilon}\psi=2i(\tilde{\epsilon}\gamma^{\mu})\partial_{\mu}\phi+\epsilon F$ $\displaystyle\delta_{\epsilon}F=-2i\tilde{\epsilon}\gamma^{\mu}\partial_{\mu}\psi$ where spinor contractions are implied, and spinor indices are lowered/raised via the charge conjugation matrix as usual. To construct a minimal Lagrangian density from the superfield $\Phi$, we can look at its mass dimension: this is equal to its lowest component $\phi$, that being a scalar field is $(d-2)/2=1$ in $d=4$ dimensions. Since $\psi$ is a spinor, it has dimension $(d-1)/2=3/2$, so the odd coordinates have always dimension -1/2. The highest component of any superfield has thus dimension two more than the superfield. This means that to construct a Lagrangian we have to take a quadratic expression in $\Phi$. Since the action should be real, the simplest choice is $\Phi^{\dagger}\Phi$. Its top-degree component is $\int d^{2}\theta d^{2}\tilde{\theta}\ \Phi^{\dagger}\Phi=4\left\|\partial\phi\right\|^{2}+i\overline{\Psi}\not{\partial}\Psi+\|F\|^{2}+\partial_{\mu}(\cdots)^{\mu}$ (4.70) where $\Psi$ is a Majorana 4-spinor, whose left-handed component is $\psi$. Up to a total derivative, this is the Lagrangian of the free, massless _Wess- Zumino model_ : $S_{WZ,free}[\Phi]=\int_{S\mathbb{R}^{4}_{\mathbb{C}^{4}}}d^{4}xd^{2}\theta d^{2}\tilde{\theta}\ \Phi^{\dagger}\Phi=\int_{\mathbb{R}^{4}}d^{4}x\left(4\left\|\partial\phi\right\|^{2}+i\overline{\Psi}\not{\partial}\Psi+\|F\|^{2}\right).$ (4.71) Since the field $F$ appears without derivatives, its equation of motion is an algebraic equation. For this reason, it is called an _auxiliary field_ , and it is customary to substitute its on-shell value in the action. For this simple model, this means putting $F=0$. This procedure makes in general the action to be supersymmetric only if the equations of motion (EoM) are imposed, and is often called _on-shell_ supersymmetry. The physical field content of an $\mathcal{N}=1$ chiral superfield is thus the supersymmetry _doublet_ $(\phi,\psi)$. By CPT invariance, the theory must contain both the chiral field $\Phi$ and its antichiral conjugate $\Phi^{\dagger}$, so that the physical field content of a meaningful theory constructed from it is made by two real scalar fields $\mathrm{Re}(\phi),\mathrm{Im}(\phi)$ and the Majorana spinor $\Psi$. If we now start from a vector superfield $V$, satisfying the reality condition $V=V^{\dagger}$, we reach a different physical field content and supersymmetric Lagrangian. The component expansion in a chart $(x,\theta,\tilde{\theta})$ is the following: $\displaystyle V(x,\theta,\tilde{\theta})=C(x)+\xi_{a}(x)\theta^{a}+\xi^{\dagger}_{\dot{a}}\tilde{\theta}^{\dot{a}}+v_{\mu}(x)(\tilde{\theta}\gamma^{\mu}\theta)+G(x)\theta^{(2)}+G^{\dagger}(x)\tilde{\theta}^{(2)}+$ (4.72) $\displaystyle+\eta_{a}(x)\theta^{a}\tilde{\theta}^{(2)}+\eta^{\dagger}_{\dot{a}}(x)\tilde{\theta}^{\dot{a}}\theta^{(2)}+E(x)\theta^{(2)}\tilde{\theta}^{(2)}$ where $C,v_{\mu},E$ are real fields. It is clear that this is the right type of superfield needed to describe (Abelian) gauge boson fields, represented here by $v_{\mu}$. This is why $V$ is called _vector_ superfield. Notice that the real part of a chiral superfield is a special kind of vector superfield. In particular, its vector component is a derivative: if $\Lambda$ is chiral, from (4.68) $\Lambda+\Lambda^{\dagger}\supset i\partial_{\mu}(\phi-\phi^{\dagger})(\tilde{\theta}\gamma^{\mu}\theta).$ (4.73) This suggest to interpret the transformation $V\mapsto V+(\Lambda+\Lambda^{\dagger})$ (4.74) as the action of a $U(1)$ internal gauge symmetry on superfields. In terms of component fields this gauge transformation reads $\begin{array}[]{lll}C\mapsto C+(\phi+\phi^{\dagger})&\xi_{a}\mapsto\xi_{a}+\psi_{a}&G\mapsto G+F\\\ v_{\mu}\mapsto v_{\mu}-i\partial_{\mu}(\phi-\phi^{\dagger})&\eta_{a}\mapsto\eta_{a}-i(\partial_{\mu}\psi^{\dagger}\gamma^{\mu})_{a}&E\mapsto E-\partial^{2}(\phi+\phi^{\dagger}).\end{array}$ (4.75) We can notice two main things. The first is that the combinations $\displaystyle\lambda_{a}:=\eta_{a}+i(\gamma^{\mu}\partial_{\mu}\xi^{\dagger})_{a}$ (4.76) $\displaystyle D:=E+\partial^{2}C$ are gauge invariants. The second is that, since $C,G,\xi_{a}$ transform as shifts, we can chose a special gauge in which they vanish. This is called the _Wess-Zumino (WZ) gauge_. Chosing a gauge of course breaks explicitly supersymmetry, but it is convenient for most of the calculations. In the WZ gauge, the vector superfield looks like $V=v_{\mu}(\tilde{\theta}\gamma^{\mu}\theta)+\lambda_{a}\theta^{a}\tilde{\theta}^{(2)}+\lambda^{\dagger}_{\dot{a}}\tilde{\theta}^{\dot{a}}\theta^{(2)}+D\theta^{(2)}\tilde{\theta}^{(2)}.$ (4.77) Gauge transformations with immaginary scalar component ($\phi+\phi^{\dagger}=0$) preserve the Wess-Zumino gauge and moreover induce on $v_{\mu}$ the usual $U(1)$ transformation of Abelian vector bosons. Indeed, if $\alpha:=2\mathrm{Im}(\phi)$, $v_{\mu}\mapsto v_{\mu}+\partial_{\mu}\alpha.$ (4.78) As for $F$ in the case of chiral superfields, $D$ is the top component field of the vector superfield $V$. It will have a purely algebraic equation of motion, so it can be considered as an auxiliary field. The physical field content of an $\mathcal{N}=1$ vector superfield is thus the supersymmetry _doublet_ $(v_{\mu},\lambda)$ composed by an Abelian gauge boson and a Majorana spinor, called the _gaugino_. A gauge-invariant supersymmetric action for the Abelian vector superfield can be given in terms of the spinorial superfields defined as $W_{a}:=\frac{1}{2}\tilde{D}^{2}D_{a}V,\qquad\tilde{W}_{\dot{a}}:=\frac{1}{2}D^{2}\tilde{D}_{\dot{a}}V.$ (4.79) $W_{a}$ ($\tilde{W}_{\dot{a}}$) is both chiral (antichiral) and gauge invariant, and moreover it satisfies the “reality” condition $D^{a}W_{a}=\tilde{D}^{\dot{a}}\tilde{W}_{\dot{a}}$. Expanding the component fields in coordinates $(y_{(\pm)},\theta,\tilde{\theta})$, we have $\displaystyle W_{a}=\lambda_{a}+if_{\mu\nu}(\gamma^{\mu}\gamma^{\nu})_{ab}\theta^{b}+D\varepsilon_{ab}\theta^{b}+2i(\gamma^{\mu}\partial_{\mu}\lambda^{\dagger})_{a}\theta^{(2)}$ (4.80) $\displaystyle\tilde{W}_{\dot{a}}=\lambda^{\dagger}_{\dot{a}}-if_{\mu\nu}(\gamma^{\mu}\gamma^{\nu})_{\dot{a}\dot{b}}\tilde{\theta}^{\dot{b}}+D\varepsilon_{\dot{a}\dot{b}}\tilde{\theta}^{\dot{b}}-2i(\gamma^{\mu}\partial_{\mu}\lambda)_{\dot{a}}\tilde{\theta}^{(2)}$ where $f_{\mu\nu}:=(\partial_{\mu}v_{\nu}-\partial_{\nu}v_{\nu})$ is the gauge invariant Abelian field-strength of $v_{\mu}$. A gauge invariant action is then $\displaystyle S[V]$ $\displaystyle=\int d^{4}x\frac{1}{8}\left(\int d^{2}\theta\ W^{a}W_{a}+\int d^{2}\tilde{\theta}\ \tilde{W}^{\dot{a}}\tilde{W}_{\dot{a}}\right)$ (4.81) $\displaystyle=\int d^{4}x\left\\{f_{\mu\nu}f^{\mu\nu}+i\lambda\gamma^{\mu}\partial_{\mu}\lambda^{\dagger}+2D^{2}\right\\}$ that is an $\mathcal{N}=1$ supersymmetric extension of the Abelian Yang-Mills theory in 4-dimensions. The supersymmetry transformations of the component field, under which (4.81) is invariant can be obtained applying the supertranslation on $V$ in WZ gauge. The result will not be in this gauge anymore, but can be translated back in WZ gauge applying an appropriate gauge transformation as (4.75). The result is $\displaystyle\delta_{\epsilon}v_{\mu}=\frac{1}{2}\left(\tilde{\epsilon}\gamma_{\mu}\lambda-\lambda^{\dagger}\gamma_{\mu}\epsilon\right)$ (4.82) $\displaystyle\delta_{\epsilon}\lambda=\frac{i}{2}f_{\mu\nu}\epsilon\gamma^{\mu}\gamma^{\nu}+D\epsilon$ $\displaystyle\delta_{\epsilon}D=-i\left(\tilde{\epsilon}\gamma^{\mu}\partial_{\mu}\lambda+\epsilon\gamma^{\mu}\partial_{\mu}\lambda^{\dagger}\right).$ Notice that in this case the Super Yang-Mills (SYM) action remains supersymmetric even if we impose the EoM on the auxiliary field $D$, setting $D=0$ in both (4.81) and (4.82). This is a special result, that holds in 4, 6 and 10 dimensions [44]. In a generic gauge theory with gauge group $G$, we consider a $G-$valued chiral multiplet $\Phi$ which transforms under a gauge transformations as $\Phi\mapsto e^{\Lambda}\Phi$ (4.83) where $\Lambda$ is a $\mathfrak{g}$-valued chiral superfield. Now the combination $\Phi^{\dagger}\Phi$ is not gauge invariant, so we introduce a $\mathfrak{g}-$valued vector superfield $V$, transforming as $e^{V}\mapsto e^{\Lambda^{\dagger}}e^{V}e^{\Lambda}$ (4.84) that reduces to the previous case (4.74) for Abelian $G=U(1)$. The exponential of a superfield can be defined through its component field expansion, that stops at finite order for degree reasons: $e^{V}=1+v_{\mu}(\tilde{\theta}\gamma^{\mu}\theta)+\lambda_{a}\theta^{a}\tilde{\theta}^{(2)}+\lambda^{\dagger}_{\dot{a}}\tilde{\theta}^{\dot{a}}\theta^{(2)}+(D+2v_{\mu}v^{\mu})\theta^{(2)}\tilde{\theta}^{(2)}.$ (4.85) The kinetic term for the chiral superfield can be rewritten as a gauge invariant combination: $\int d^{4}xd^{2}\theta d^{2}\tilde{\theta}\ \Phi^{\dagger}e^{V}\Phi.$ (4.86) Generalizing the supersymmetric field-strength $W_{a}$ to the non-Abelian case as $W_{a}=\frac{1}{2}\tilde{D}^{2}e^{-V}D_{a}e^{V}$ (4.87) we can write the full matter-coupled gauge theory action: $S[V,\Phi]=\int d^{4}x\left\\{\int d^{2}\theta d^{2}\tilde{\theta}\ \Phi^{\dagger}e^{V}\Phi+\left[\int d^{2}\theta\ \left(\frac{1}{4}\mathrm{Tr}W^{a}W_{a}+W(\Phi)\right)+c.c.\right]\right\\}$ (4.88) where $W(\Phi)$ is a holomorphic function of $\Phi$ called _superpotential_. The expansion in terms of component fields and the supersymmetry variations can be calculated with the same procedure we did in the other cases.161616A more detailed treatment can be found in [45], or [46]. Notice that whenever the center of the Lie algebra $\mathfrak{g}$ is non- trivial, i.e. when there is a $U(1)$ factor in $G$, we could add another supersymmetric and gauge-invariant term to the action (4.88). This is the so- called _Fayet-Iliopoulos term_ : $\int d^{4}xd^{2}\theta d^{2}\tilde{\theta}\ \xi(V)=\int d^{4}x\ \xi_{A}D^{A}$ (4.89) where $\xi=\xi_{A}\tilde{T}^{A}$ is a constant element in the dual of the center of $\mathfrak{g}$. #### 4.3.4 R-symmetry The subgroup of (outer) automorphisms of the supersymmetry group which fixes the underlying Poincaré (Euclidean) group is called _R-symmetry group_. At the level of the algebra, these are linear transformations that act only on the spin representation $S$, leaving the brackets of two spinors unchanged. In the complexified case, when different chiral sectors are present, the R-symmetry acts differently on any sector. For example, in the case of $\mathcal{N}=1$ in (3+1)-dimensions, there is a $U(1)_{R}$ R-symmetry group acting as $Q_{a}\mapsto e^{-i\alpha}Q_{a},\qquad\tilde{Q}_{\dot{a}}\mapsto e^{i\alpha}\tilde{Q}_{\dot{a}},$ (4.90) with $\alpha\in\mathbb{R}$. This clearly leaves the brackets $[Q_{a},\tilde{Q}_{\dot{a}}]$ invariant. The odd coordinates $\theta^{a}$ on superspacetime, being elements of $S^{}$ transform as $\begin{array}[]{lcl}\theta^{a}\mapsto e^{i\alpha}\theta^{a}&&\tilde{\theta}^{\dot{a}}\mapsto e^{-i\alpha}\tilde{\theta}^{\dot{a}}\\\ d^{2}\theta\mapsto e^{-2i\alpha}d^{2}\theta&&d^{2}\tilde{\theta}\mapsto e^{2i\alpha}d^{2}\tilde{\theta},\end{array}$ (4.91) so that the volume element $d^{4}\theta=d^{2}\theta d^{2}\tilde{\theta}$ is invariant under R-symmetry. This fixes the _R-charge_ of the superpotential $W(\Phi)$ to be 2, if we want the action to be invariant under R-symmetry: $W(\Phi)\mapsto e^{2i\alpha}W(\Phi).$ (4.92) In principle we can chose the chiral superfield $\Phi$ to have any R-charge $r$, since the combination $\Phi^{\dagger}\Phi$ is R-invariant. This, combined with (4.91) means that the different field components in the chiral multiplet transform differently with respect to R-symmetry: $\phi\mapsto e^{ir\alpha}\phi,\quad\psi\mapsto e^{i(r-1)\alpha}\psi,\quad F\mapsto e^{i(r-2)\alpha}F.$ (4.93) The vector superfield, being real is acted upon trivially by $U(1)_{R}$. Its component fields are then forced to transform as $v_{\mu}\mapsto v_{\mu},\quad\lambda\mapsto e^{i\alpha}\lambda,\quad D\mapsto D,$ (4.94) thus the gauge-invariant supersymmetric field-strength $W_{a}$ has R-charge $1$. In general, if the spin representation is reducible and we have extended supersymmetry, the R-group is always compact. For $S=(S_{0})^{\mathcal{N}}$, where $S_{0}$ is a real representation, it is of the type $U(\mathcal{N})$, while for $S=(S^{(+)})^{\mathcal{N}_{+}}\oplus(S^{(-)})^{\mathcal{N}_{-}}$, where $S^{(\pm)}$ are the two real representations of different chirality, it is of the type $U(\mathcal{N}_{+})\times U(\mathcal{N}_{-})$ [43]. Notice the isomorphism $U(n)\cong(SU(n)\times U(1))/\mathbb{Z}_{n}$ (4.95) i.e. $U(n)$ is an n-fold cover of $SU(n)\times U(1)$. In particular, their Lie algebras are isomorphic. In terms of infinitesimal transformations then, the R-symmetry generators can be decomposed in one R-charge plus $\mathcal{N}^{2}-1$ rotation generators. The supercharges are rotated into one another by $Q_{a}^{I}\mapsto e^{-i\alpha}\mathcal{U}^{I}_{J}Q_{a}^{J},\quad\tilde{Q}_{\dot{a}}^{I}\mapsto e^{i\alpha}\mathcal{(U^{\dagger})}^{I}_{J}\tilde{Q}_{\dot{a}}^{J}.$ (4.96) In QFT, R-symmetry may or may not be present as a symmetry of the theory, and in many cases part of this symmetry may be broken by anomaly at quantum level. #### 4.3.5 Supersymmetry multiplets A geometric analysis as the one carried out in the last subsections allows one to find the physical field content of a supersymmetric theory in every dimensions and for any degree of reducibility of the spin representation $S$ that is used to extend the Poincaré algebra. Another systematic way to obtain the same result, from a more algebraic point of view, is to study the representation of the supersymmetry algebra $\mathfrak{siso}_{S}(d)$, in analogy with the Wigner analysis of massive and massless representations of the Poincaré algebra. As the cases encountered above, this study leads to the presence of different _supersymmetry multiplets_ for different choices of spin and $\mathcal{N}$. We will not present this here but refer for example to [47] for a comprehensive review, and list here some results for the multiplets at various $\mathcal{N}$. For $1\leq\mathcal{N}\leq 4$ with spin less or equal to 1, the supersymmetry particle representations simply consists of spin 1 vector particles, spin 1/2 fermions and spin 0 scalars. In the supergeometric approach, these fields are interpreted as components of the same superfield, and thus transform one into another under the supersymmetry algebra. Let $G$ be the gauge group, and $\mathfrak{g}$ its Lie algebra. We are interested mainly in two types of multiplets. The first is the (massless) _vector_ or _gauge multiplet_ , which transforms under the adjoint representation of $\mathfrak{g}$. For $\mathcal{N}=3,4$, this is the only possible multiplet. It turns out that quantum field theories with $\mathcal{N}=3$ supersymmetries coincide with those with $\mathcal{N}=4$ in view of CPT invariance, thus we shall limit our discussion to the $\mathcal{N}=4$ theories.171717To be more precise, it is possible to construct theories with genuine $\mathcal{N}=3$ supersymmetry, but they lack of a Lagrangian description in terms of component fields. For $\mathcal{N}=1,2$, we also have (possibly massive) matter multiplets: for $\mathcal{N}=1$, this is the _chiral multiplet_ , and for $\mathcal{N}=2$ this is the _hypermultiplet_ , both of which may transform under an arbitrary (unitary, and possibly reducible) representation of $G$. In (3+1)-dimensions, the on-shell field content of these multiplets is: • $\mathcal{N}=1$ _gauge multiplet_ $(A_{\mu},\lambda)$: a gauge boson and a Majorana fermion, the gaugino. * • $\mathcal{N}=1$ _chiral multiplet_ $(\phi,\psi)$: a complex scalar and a left- handed Weyl fermion. * • $\mathcal{N}=2$ _gauge multiplet_ $(A_{\mu},\lambda_{\pm},\phi)$: $\lambda_{\pm}$ form a Dirac spinor, and $\phi$ is a complex _gauge scalar_. Under the $SU(2)_{R}$ symmetry, $A_{\mu}$ and $\phi$ are singlets, while $\lambda_{+},\lambda_{-}$ transform as a doublet. * • $\mathcal{N}=2$ _hypermultiplet_ $(\psi_{+},H,\psi_{-})$: $\psi_{\pm}$ form a Dirac spinor and $H_{\pm}$ are complex scalars. Under the $SU(2)_{R}$ symmetry, $\psi_{+}$ and $\psi_{-}$ transform as singlets, while $H_{+},H_{-}$ transform as a doublet. * • $\mathcal{N}=4$ _gauge multiplet_ $(A_{\mu},\lambda^{i},\Phi_{A})$: $\lambda^{i}$ , $i=1,2,3,4$ are Weyl fermions (equivalents to two Dirac fermions), and $\Phi_{A}$, $A=1,\cdots,6$ are real scalars (equivalents to three complex scalars). Under the $SU(4)_{R}$ symmetry181818The R-symmetry group is actually $SU(2)\times SU(2)\times U(1)$, as we will see in a practical application in the following. the gauge field $A_{\mu}$ is a singlet, the fermions $\lambda^{i}$ transform in the fundamental representation $\mathbf{4}$, the scalars $\Phi_{A}$ transform in the rank-two antisymmetric representation $\mathbf{6}$. Even though in this thesis we do not work explicitly with gravity theories, we will see in the next section that the introduction of _off-shell_ supergravity is necessary in a possible approach to construct globally supersymmetric theories on curved base-spaces. For this purpose, it is useful to remind also the content of massless supersymmetry particle representations with helicity between 1 and 2. These are the _gravitino multiplet_ and the _graviton multiplet_ (or _supergravity multiplet_ , or _metric multiplet_). In general the gravitino multiplet contains degrees of freedom with helicity less or equal than 3/2. Since in a theory without gravity one cannot accept particles with helicity greater than one,191919This comes from the so-called _Weinberg- Witten theorem_ [48]. that multiplet cannot appear in a supersymmetric theory if also a graviton, with helicity 2, does not appear. In (3+1)-dimensions, the field content of the relevant multiplets are: * • $\mathcal{N}=1$ _gravitino multiplet_ $(\Phi_{\mu},B_{\mu})$: a helicity 3/2 fermionic particle and a vector boson. * • $\mathcal{N}=1$ _graviton multiplet_ $(h_{\mu\nu},\Psi_{\mu})$: the _graviton_ , with helicity 2, and its supersymmetric partner the _gravitino_ , of helicity 3/2. * • $\mathcal{N}=2$ _gravitino multiplet_ : a spin 3/2 particle, two vectors and one Weyl fermion. * • $\mathcal{N}=2$ _graviton multiplet_ : graviton, two gravitinos and a vector boson. For $\mathcal{N}>4$ it is not possible to avoid gravity since there do not exist representations with helicity smaller than 3/2. Hence, theories with $\mathcal{N}>4$ are all supergravity theories. #### 4.3.6 Euclidean 3d $\mathcal{N}=2$ supersymmetric gauge theories As an example, which will be used in some applications of the localization principle in the next chapter, we can look at $\mathcal{N}=2$ Euclidean supersymmetry in 3-dimensions. First, notice that the rotation algebra for 3d Euclidean space is $\mathfrak{so}(3)$. The corresponding spin group is thus $SU(2)$, whose fundamental representation $\mathbf{2}$ does not admit a real structure. In fact here the charge conjugation can be taken as the totally antisymmetric symbol $C_{ab}=\varepsilon_{ab}$, and the Majorana condition would be inconsistent: $\psi^{T}C=\psi^{\dagger}\Leftrightarrow\psi=0.$ (4.97) Thus we cannot construct an $\mathcal{N}=1$ Euclidean supersymmetry algebra in 3 dimensions, in the sense of definition (4.3.1). The problem can be cured considering a reducible spin representation $S$, where the spinors and the charge conjugation matrix can be split as $\Psi=(\psi^{a}_{I})^{a=1,2}_{I=1,\cdots,\mathcal{N}},\qquad\mathcal{C}=(\Omega_{IJ}C^{ab})^{a,b=1,2}_{I,J=1,\cdots,\mathcal{N}},$ (4.98) and the same reality condition $\Psi^{\dagger}=\Psi^{T}\mathcal{C}$ now is consistent if also the matrix $\Omega$ squares to $-\mathds{1}$ and is anti- orthogonal: $\Omega=-\Omega^{T}=-\Omega^{-1}.$ (4.99) If we now fix $\mathcal{N}=2$, the resulting spinor representation is analogous to the one of $\mathcal{N}=1$ in 4-dimensions, but now the two Weyl sectors are independent since they generates the two supersymmetries. To see this corrispondence, we can change basis of $S=\mathbf{2}^{(1)}\oplus\mathbf{2}^{(2)}$ from the natural one in terms of the generators $\\{Q^{1}_{a},Q^{2}_{a}\\}$ to $Q_{a}:=\frac{1}{\sqrt{2}}(Q_{a}^{1}+iQ_{a}^{2}),\qquad\tilde{Q}_{a}:=\frac{1}{\sqrt{2}}(Q_{a}^{1}-iQ_{a}^{2}).$ (4.100) In this basis, using (4.44) the super Lie brackets become $\begin{array}[]{lr}\lx@intercol[Q_{a},\tilde{Q}_{b}]=2(\gamma^{\mu})_{ab}P_{\mu}+Z\varepsilon_{ab}\hfil\lx@intercol\\\ \left[Q_{a},Q_{b}\right]=0&[\tilde{Q}_{a},\tilde{Q}_{b}]=0\end{array}$ (4.101) where $Z$ is a constant central charge, and the gamma matrices in this representation can be chosen to be the Pauli matrices $\gamma^{\mu}=\sigma^{\mu}$ for $\mu=1,2,3$.202020There is also another inequivalent representation of the Clifford algebra, as in any odd dimensions, in which $\gamma^{3}=-\sigma^{3}$. We chose the former one. Note that the 4-dimensional $Spin(3,1)$ Lorentz group breaks to $SU(2)\times SU(2)_{R}$, where $Spin(3)\cong SU(2)$ is the 3-dimensional Lorentz group, and the remaining $SU(2)_{R}$ is an R-symmetry acting on the $\mathcal{N}=2$ algebra. The generators $Q_{a}$ and $\tilde{Q}_{a}$ are represented in superspace by odd vector fields whose expressions are formally the same as in (4.58), and the $\mathcal{N}=2$ supersymmetry variation of a superfield $\Phi$ is $\delta_{\epsilon,\eta}\Phi=(\epsilon^{a}Q_{a}+\eta^{a}\tilde{Q}_{a})\Phi$ (4.102) where now, as said before, $\epsilon$ and $\eta$ are two independent complex spinors. If we want to construct a supersymmetric gauge theory in 3-dimensions, we consider the vector superfield, now expressed in WZ gauge as $V(x,\theta,\tilde{\theta})=A_{\mu}(\tilde{\theta}\gamma^{\mu}\theta)+i\sigma\theta\tilde{\theta}+\lambda_{a}\theta^{a}\tilde{\theta}^{(2)}+\lambda^{\dagger}_{a}\tilde{\theta}^{a}\theta^{(2)}+D\theta^{(2)}\tilde{\theta}^{(2)}.$ (4.103) The off-shell $\mathcal{N}=2$ gauge multiplet is then composed by a gauge field $A_{\mu}$, two real scalars $\sigma,D$ and a 2-component complex spinor $\lambda$. Notice that this is just the _dimensional reduction_ of the $\mathcal{N}=1$ multiplet in 4-dimensions, with $\sigma$ coming from the zero- th component of the gauge field in higher dimensions. The only difference with the 4-dimensional vector multiplet is that this zero-th component has been considered purely immaginary, _i.e._ $A_{0}=i\sigma$ with real $\sigma$. This ensures the kinetic term for $\sigma$ to be positive definite and the path integral to converge, matching the would-be dimensional reduction from an Euclidean 4-dimensional theory. If the gauge group is $G$, all fields are valued in its Lie algebra $\mathfrak{g}$. For what we are going to discuss in the next chapter, we now adopt the convention of [49, 50] for the supersymmetry variations of the vector superfield and the supersymmetric actions. Under a proper rescaling of the component fields and of the supercharges, one can work them out in an analogous way to which we did in the last sections, and get $\displaystyle\delta_{\epsilon,\eta}A_{\mu}=\frac{i}{2}(\eta^{\dagger}\gamma_{\mu}\lambda-\lambda^{\dagger}\gamma_{\mu}\epsilon)$ (4.104) $\displaystyle\delta_{\epsilon,\eta}\sigma=\frac{1}{2}(\eta^{\dagger}\lambda-\lambda^{\dagger}\epsilon)$ $\displaystyle\delta_{\epsilon,\eta}D=\frac{i}{2}\left(\eta^{\dagger}\gamma^{\mu}D_{\mu}\lambda-(D_{\mu}\lambda^{\dagger})\gamma^{\mu}\epsilon\right)-\frac{i}{2}\left(\eta^{\dagger}[\lambda,\sigma]-[\lambda^{\dagger},\sigma]\epsilon\right)$ $\displaystyle\delta_{\epsilon,\eta}\lambda=\left(-\frac{1}{2}\gamma^{\mu\nu}F_{\mu\nu}-D+i\gamma^{\mu}D_{\mu}\sigma\right)\epsilon$ $\displaystyle\delta_{\epsilon,\eta}\lambda^{\dagger}=\eta^{\dagger}\left(-\frac{1}{2}\gamma^{\mu\nu}F_{\mu\nu}+D-i\gamma^{\mu}D_{\mu}\sigma\right)$ where $D_{\mu}=\partial_{\mu}+[A_{\mu},\cdot]$ is the gauge-covariant derivative and $\gamma^{\mu\nu}:=\frac{1}{2}[\gamma^{\mu},\gamma^{\nu}]$. Up to some prefactors, they can be seen as a dimensional reduction of (4.82). We can consider two types of gauge supersymmetric actions constructed from the vector multiplet in 3 Euclidean dimensions: the Super Yang-Mills theory, that is a reduction of (4.88), and the Super Chern-Simons (SCS) theory. In superspace, the former one is constructed in the same way as the 4-dimensional case from the spinorial superfield $W_{a}$, while the SCS term is constructed as $S_{CS}=\int d^{3}xd^{2}\theta d^{2}\tilde{\theta}\ \frac{k}{4\pi}\left(\int_{0}^{1}dt\ \mathrm{Tr}\left\\{V\tilde{D}^{a}e^{-tV}D_{a}e^{tV}\right\\}\right).$ (4.105) Integrating out the odd coordinates in superspace, these are given by [49]: $\displaystyle S_{YM}=\int d^{3}x\ \mathrm{Tr}\left\\{\frac{i}{2}\lambda^{\dagger}\gamma^{\mu}D_{\mu}\lambda+\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}D_{\mu}\sigma D^{\mu}\sigma+\frac{i}{2}\lambda^{\dagger}[\sigma,\lambda]+\frac{1}{2}D^{2}\right\\},$ (4.106) $\displaystyle S_{CS}=\frac{k}{4\pi}\int d^{3}x\ \mathrm{Tr}\left\\{\varepsilon^{\mu\nu\rho}\left(A_{\mu}\partial_{\nu}A_{\rho}+\frac{2i}{3}A_{\mu}A_{\nu}A_{\rho}\right)-\lambda^{\dagger}\lambda+2\sigma D\right\\}.$ (4.107) #### 4.3.7 Euclidean 4d $\mathcal{N}=4,2,2^{}$ supersymmetric gauge theories We describe here another example that will be useful in the next chapter, when we will apply the localization principle to supersymmetric QFT. The $\mathcal{N}=4$ SYM theory on flat space can be derived via dimensional reduction of $\mathcal{N}=1$ SYM in $(9+1)$ dimensions.212121For convergence of the partition function, it would be nicer to start from the $(10,0)$ Euclidean signature. We follow the convention of [11] and start from the $(9,1)$ one, Wick rotating a posteriori the path integral when needed, to match the would-be reduction from the $(10,0)$ theory. The $\mathcal{N}=2$ and $\mathcal{N}=2^{}$ theories can be derived as modification of the $\mathcal{N}=4$ theory, as we will see later. We start recalling the structure of the 10-dimensional Clifford algebra following the conventions of [11]. This is independent from the choice of the signature, $Cl(9,1)\cong Cl(1,9)\cong\mathrm{Mat}_{32}(\mathbb{R})$, it is real, and generated by the gamma matrices $(\gamma^{M})_{M=0,1,\cdots,9}$ such that $\\{\gamma^{M},\gamma^{N}\\}=2\eta^{MN}$ where $\eta$ is the 10-dimensional Minkowski metric, that we take with signature $(-,+,\cdots,+)$. The fundamental representation of the spin group $Spin(9,1)\hookrightarrow Cl(9,1)$ is then Majorana, and it is moreover reducible under chirality [42] $\gamma^{c}:=-i\gamma^{0}\gamma^{1}\cdots\gamma^{9}$ as $Spin(9,1)=S^{+}\oplus S^{-}\cong\mathrm{Mat}_{16}(\mathbb{R})\oplus\mathrm{Mat}_{16}(\mathbb{R})$. Thus fundamental spinors are Majorana-Weyl, and have 16 real components. In the chiral basis we denote $\displaystyle\gamma^{M}=\left(\begin{array}[]{cc}0&\tilde{\Gamma}^{M}\\\ \Gamma^{M}&0\end{array}\right)\qquad\tilde{\Gamma}^{M},\Gamma^{M}:S^{\pm}\to S^{\mp}$ (4.108) $\displaystyle\gamma^{MN}=\left(\begin{array}[]{cc}\tilde{\Gamma}^{[M}\Gamma^{N]}&0\\\ 0&\Gamma^{[M}\tilde{\Gamma}^{N]}\end{array}\right)=:\left(\begin{array}[]{cc}\Gamma^{MN}&0\\\ 0&\tilde{\Gamma}^{MN}\end{array}\right)$ where $\Gamma^{M},\tilde{\Gamma}^{M}$ act on the Majorana-Weyl subspaces, exchanging chirality, and are taken to be symmetric.222222In the Euclidean signature, we would use $\Gamma^{M}_{E}=\\{\Gamma^{1},\cdots,\Gamma^{9},i\Gamma^{0}\\}$. Let the gauge group $G$ be a compact Lie group, and $\mathfrak{g}$ its Lie algebra. The (on-shell) component field content of the gauge multiplet in 10 dimensions is of a gauge field, locally represented as $A\in\Omega^{1}(\mathbb{R}^{9,1},\mathfrak{g})$, and a gaugino, a Mayorana- Weyl spinor $\Psi:\mathbb{R}^{9,1}\to S^{+}\otimes\mathfrak{g}$ with values in the Lie algebra $\mathfrak{g}$. The field strength of the gauge field is locally represented by $F=dA+[A,A]$, and the associated gauge-covariant derivative on $\mathbb{R}^{9,1}$ is $D_{M}=\partial_{M}+A_{M}$. The supersymmetry variations under the action of the 10-dimensional super-Poincaré algebra are $\displaystyle\delta_{\epsilon}A_{M}$ $\displaystyle=\epsilon\Gamma_{M}\Psi$ (4.109) $\displaystyle\delta_{\epsilon}\Psi$ $\displaystyle=\frac{1}{2}\Gamma^{MN}F_{MN}\epsilon$ where $\epsilon$ is a Majorana-Weyl spinor, analogously to the on-shell version of (4.82) up to the chirality projection and conventional prefactors. The action functional for the $\mathcal{N}=1$ 10-dimensional theory is $S_{10d}=\int d^{10}x\ \mathcal{L}$, with Lagrangian $\mathcal{L}=\frac{1}{g_{YM}^{2}}\mathrm{Tr}\left(\frac{1}{2}F_{MN}F^{MN}-\Psi\Gamma^{M}D_{M}\Psi\right)$ (4.110) where $\mathrm{Tr}$ denotes a symmetric bilinear pairing in $\mathfrak{g}$,232323For semisimple $\mathfrak{g}$, this is the Killing form as usual. and $g_{YM}$ is the Yang-Mills coupling constant. As we remarked in Section 4.3.3, this action is exactly supersymmetric under (4.109) without the addition of auxiliary fields. To get the Euclidean 4-dimensional theory, we perform dimensional reduction along the directions $x^{0},x^{5},\cdots,x^{9}$, assuming independence of the fields on these coordinates. The fields split as $\displaystyle A_{M}$ $\displaystyle\to\left((A_{\mu})_{\mu=1,\cdots,4},(\Phi_{A})_{A=5,\cdots,9,0}\right)$ (4.111) $\displaystyle\Psi$ $\displaystyle\to\left(\psi^{L}\ \chi^{R}\ \psi^{R}\ \chi^{L}\right)^{T}$ where $\psi^{L/R},\chi^{L/R}$ are four-component real chiral spinors. The spacetime symmetry group $Spin(9,1)$ is broken to $Spin(4)\times Spin(5,1)^{\mathcal{R}}\hookrightarrow Spin(9,1)$, where $Spin(4)\cong SU(2)_{L}\times SU(2)_{R}$ acts on the $x^{1},\cdots,x^{4}$ directions, and the R-symmetry group $Spin(5,1)^{\mathcal{R}}$ rotates the other ones. It is often convenient to further break the R-symmetry group to $Spin(4)^{\mathcal{R}}\times SO(1,1)^{\mathcal{R}}\hookrightarrow Spin(5,1)^{\mathcal{R}}$, where the first piece $Spin(4)^{\mathcal{R}}\cong SU(2)_{L}^{\mathcal{R}}\times SU(2)_{R}^{\mathcal{R}}$ rotates the $x^{5},\cdots,x^{8}$ directions, and $SO(1,1)^{\mathcal{R}}$ acts on the $x^{9},x^{0}$ ones. We thus consider the symmetry group $SU(2)_{L}\times SU(2)_{R}\times SU(2)_{L}^{\mathcal{R}}\times SU(2)_{R}^{\mathcal{R}}\times SO(1,1)^{\mathcal{R}}$ (4.112) under which the fields behave as * • $A_{\mu}$: vector of $SU(2)_{L}\times SU(2)_{R}$, scalar under R-symmetry; * • $(\Phi_{I})_{I=4,\cdots,8}$: 4 scalars under $SU(2)_{L}\times SU(2)_{R}$, vector of $SU(2)_{L}^{\mathcal{R}}\times SU(2)_{R}^{\mathcal{R}}$, scalars under $SO(1,1)^{\mathcal{R}}$; * • $\Phi_{9},\Phi_{0}$: scalars under $SU(2)_{L}\times SU(2)_{R}\times SU(2)_{L}^{\mathcal{R}}\times SU(2)_{R}^{\mathcal{R}}$, vector of $SO(1,1)^{\mathcal{R}}$; * • $\psi^{L/R}$: $\left(\frac{1}{2},0\right)/\left(0,\frac{1}{2}\right)$ of $SU(2)_{L}\times SU(2)_{R}$, $\left(\frac{1}{2},0\right)$ of $SU(2)_{L}^{\mathcal{R}}\times SU(2)_{R}^{\mathcal{R}}$, $+/-$ of $SO(1,1)^{\mathcal{R}}$; * • $\chi^{L/R}$: $\left(\frac{1}{2},0\right)/\left(0,\frac{1}{2}\right)$ of $SU(2)_{L}\times SU(2)_{R}$, $\left(0,\frac{1}{2}\right)$ of $SU(2)_{L}^{\mathcal{R}}\times SU(2)_{R}^{\mathcal{R}}$, $-/+$ of $SO(1,1)^{\mathcal{R}}$; here we denoted $+,-$ the inequivalent Majorana-Weyl representations of $SO(1,1)^{\mathcal{R}}$, seen as a subgroup of $Cl(1,1)\cong\mathrm{Mat}_{2}(\mathbb{R})$. The above decomposition of $Spin(9,1)$ into four subrepresentations of $Spin(4)$, rotated into each other by the R-symmetry group, gives the $\mathcal{N}=4$ supersymmetry algebra on $\mathbb{R}^{4}$. The supersymmetry variations of the reduced component fields are given by (4.109), read in terms of the splitting (4.111), $\displaystyle\delta_{\epsilon}A_{\mu}$ $\displaystyle=\epsilon\Gamma_{\mu}\Psi$ (4.113) $\displaystyle\delta_{\epsilon}\Phi_{A}$ $\displaystyle=\epsilon\Gamma_{A}\Psi$ $\displaystyle\delta_{\epsilon}\Psi$ $\displaystyle=\frac{1}{2}\left(\Gamma^{\mu\nu}F_{\mu\nu}+\Gamma^{AB}[\Phi_{A},\Phi_{B}]+\Gamma^{\mu A}D_{\mu}\Phi_{A}\right)\epsilon.$ The action of the 4d $\mathcal{N}=4$ SYM theory is $S^{\mathcal{N}=4}=\int d^{4}x\ \mathcal{L}$ with the Lagrangian obtained by the reduction of (4.110). More explicitly, $S^{\mathcal{N}=4}=\int d^{4}x\frac{1}{g_{YM}^{2}}\mathrm{Tr}\left(\frac{1}{2}F_{\mu\nu}F^{\mu\nu}+(D_{\mu}\Phi_{A})^{2}-\Psi\Gamma^{\mu}D_{\mu}\Psi+\frac{1}{2}[\Phi_{A},\Phi_{B}]^{2}-\Psi\Gamma^{A}[\Phi_{A},\Psi]\right).$ (4.114) Notice that, since contractions of $A,B$ indices are done with a reduced Minkowski metric, upon dimensional reduction from the Lorentzian theory the scalar $\Phi_{0}$ has a negative kinetic term. Analogously to the last section, we consider it to be purely immaginary, i.e. $\Phi_{0}=:i\Phi_{0}^{E}$ with $\Phi_{0}^{E}$ real. This makes the path integral match with the would-be reduction from the Euclidean $(10,0)$-dimensional theory. The $\mathcal{N}=4$ algebra closes _on-shell_. In fact, it can be obtained from (4.113) that $\delta_{\epsilon}^{2}=\frac{1}{2}[\delta_{\epsilon},\delta_{\epsilon}]=-\mathcal{L}_{v}-G_{\Phi}$ (4.115) up to the imposition of the EoM for $\Psi$, $\Gamma^{M}D_{M}\Psi=0$. Here $v^{M}:=\epsilon\Gamma^{M}\epsilon$, $\mathcal{L}_{v}$ is the Lie derivative (the action of the translation algebra) with respect to $v\sim v^{\mu}\partial_{\mu}$, and $G_{\Phi}$ is an infinitesimal gauge transformation with respect to $\Phi:=A_{M}v^{M}$. A famous non- renormalization theorem by Seiberg [51] states that the $\mathcal{N}=4$ theory is actually _superconformal_ , i.e. it has a larger supersymmetry algebra that squares to the _conformal algebra_ , whose generators are the Poincaré generators plus the generators of dilatations and special conformal transformations.242424More precisely, the theorem states that the beta function of $g_{YM}$ is zero non-perturbatively. This means that the theory is fully scale invariant at quantum level. In fact, one can see that $S^{\mathcal{N}=4}$ is classically invariant under supersymmetry variations with respect to the non-constant spinor $\epsilon=\hat{\epsilon}_{s}+x^{\mu}\Gamma_{\mu}\hat{\epsilon}_{c}$ (4.116) where $\hat{\epsilon}_{s},\hat{\epsilon}_{c}$ are constant spinors parametrizing supertranslations and superconformal transformations. This enlarged supersymmetry algebra closes now on the superconformal algebra, $\delta_{\epsilon}^{2}=-\mathcal{L}_{v}-G_{\Phi}-R-\Omega$ (4.117) where $R$ is a $Spin(5,1)^{\mathcal{R}}$ rotation, acting on scalars as $(R\cdot\Phi)_{A}=R_{A}^{B}\Phi_{B}$, and on spinors as $(R\cdot\Psi)=\frac{1}{4}R_{AB}\Gamma^{AB}\Psi$, where $R_{AB}=2\epsilon\tilde{\Gamma}_{AB}\tilde{\epsilon}$. $\Omega$ is an infinitesimal dilatation with respect to the parameter $2(\tilde{\epsilon}\epsilon)$, acting on the gauge field trivially, on scalars as $\Omega\cdot\Phi=-2(\tilde{\epsilon}\epsilon)\Phi$ and on spinors as $\Omega\cdot\Psi=-3(\tilde{\epsilon}\epsilon)\Psi$. This new bosonic transformations are clearly symmetries of $S^{\mathcal{N}=4}$. Now we can restrict the attention to an $\mathcal{N}=2$ subalgebra, considering the variations with respect to Majorana-Weyl spinors of the form $\epsilon=\left(\epsilon^{L}\ 0\ \epsilon^{R}\ 0\right)^{T}$ (4.118) so in the subrepresentation $\left(\left(\frac{1}{2},0\right)\oplus\left(0,\frac{1}{2}\right)\right)\oplus\left(\frac{1}{2},0\right)^{\mathcal{R}}\oplus(+\oplus-)^{\mathcal{R}}$, the eigenspace of $\Gamma^{5678}$ with eigenvalue +1. With respect to these supersymmetry variations, the gauge multiplet further splits in * • $(A_{\mu},\Phi_{9},\Phi_{0},\psi^{L},\psi^{R})$: the $\mathcal{N}=2$ vector multiplet; * • $(\Phi_{I},\chi^{L},\chi^{R})$: the $\mathcal{N}=2$ hypermultiplet, with value in the adjoint representation of $G$. These two multiplets are completely disentangled in the free theory limit $g_{YM}^{2}\to 0$.252525Working out the restricted supersymmetry variations taking into account the splitting of the gaugino, some non-linear term, coupling the fermionic sectors of the two multiplets, survives because of the gauge interaction. In the free theory limit, after the rescaling $A_{M}\mapsto g_{YM}A_{M},\Psi\mapsto g_{YM}\Psi$, these terms go to zero. The same Lagrangian thus equivalently describes an $\mathcal{N}=2$ matter-coupled gauge theory. It is also possible to insert a mass for the hypermultiplet, breaking explicitly the conformal invariance, and obtain the so-called $\mathcal{N}=2^{}$ theory. Since the fields of the vector multiplet are all scalars under $SU(2)_{R}^{\mathcal{R}}$, and the hypermultiplet fields are all in the $\frac{1}{2}$ representation, these mass terms can at most rotate the hypermultiplet content with an $SU(2)_{R}^{\mathcal{R}}$ transformation. Thus replacing $D_{0}\Phi_{I}\mapsto[\Phi_{0},\Phi_{I}]+M_{I}^{J}\Phi_{J}$ and $D_{0}\Psi\mapsto[\Phi_{0},\Psi]+\frac{1}{4}M_{IJ}\Gamma^{IJ}\Psi$, where $(M^{I}_{J})$ represents an $SU(2)_{R}^{\mathcal{R}}$ rotation in the vector representation, one obtains the mass terms for the $\Phi_{I}$ and $\chi$ fields. Notice that $\delta_{\epsilon}^{2}$ gets a contribution from the Lie derivative with respect to $v^{0}\partial_{0}\cong 0\mapsto v^{0}M$, so that in the $2^{}$ theory $\displaystyle\delta_{\epsilon}^{2}\Phi_{I}$ $\displaystyle\mapsto(\delta_{\epsilon}\Phi_{I})_{\mathcal{N}=2}-v^{0}M_{I}^{J}\Phi_{J}$ (4.119) $\displaystyle\delta_{\epsilon}^{2}\chi$ $\displaystyle\mapsto(\delta_{\epsilon}\chi)_{\mathcal{N}=2}-\frac{1}{4}v^{0}M_{IJ}\Gamma^{IJ}\chi.$ In the limits of infinite or zero mass, the pure $\mathcal{N}=2$ or $\mathcal{N}=4$ theory is recovered. Notice that, since we argued that $\Phi_{0}$ should be integrated over purely immaginary values for the convergence of the path integral, also $(M_{IJ})$ should be taken purely immaginary. ### 4.4 From flat to curved space Recently, localization theory has been extensively used in the framework of quantum field theories with rigid super-Poincaré symmetry, to compute exactly partition functions or expectation values of certain supersymmetric observables, when the theory is formulated on a _curved_ compact manifold. This cures the corresponding partition functions from infrared divergences making the path integral better defined, and is consistent with the requirement of periodic boundary conditions on the fields, that allows to generalize properly the Cartan model on the infinite dimensional field space. We will come back to this last point in the next chapter, when we will study circle localization of path integrals, while we close this chapter reviewing the idea behind some common approaches used to formulate rigid supersymmetry on curved space. Following the approach of the last section, we would have to understand what does it mean to have supersymmetry on a generic metric manifold $(\mathcal{M},g)$ (Riemannian or pseudo-Riemannian) of dimension $\dim{\mathcal{M}}=d$ from a geometric point of view. The supersymmetry of flat space was constructed as a super-extension of Minkowski (or Euclidean) space $\mathbb{R}^{d}$, starting from a super-extension of the Lie algebra of its isometry group, the Poincaré group. Now in general the Poincaré group is not an isometry group for $\mathcal{M}$, so the super Poincaré algebra $\mathfrak{siso}_{S}(d)$ with respect to some (real or Majorana) spin representation $S$ cannot be fully interpreted as a “supersymmetry” algebra for the space at hand. We can nonetheless associate in some way this algebra to a suitable super-extension of $\mathcal{M}$, and then ask what part of it can be preserved as a supersymmetry of this supermanifold. We follow [52] for this geometric introduction. Since we want to work with spinors, we assume that $\mathcal{M}$ admits a spin-structure. In particular, it exists a (real) spinor bundle $S\to\mathcal{M}$ associated to the spin-structure, with structure sheaf $\mathcal{S}:\mathcal{S}(U)=\Gamma(U,S),\forall U\subset\mathcal{M}$ open. Analogously to the flat superspace of the last section, we make now a super- extension of $\mathcal{M}$ through this spinor bundle considering the _odd spinor bundle_ $S\mathcal{M}_{S}:=\Pi S$, with body $\mathcal{M}$ and structure sheaf $\bigwedge\mathcal{S}^{}:\bigwedge\mathcal{S}^{}(U)=C^{\infty}(\mathcal{U})\otimes\bigwedge(S_{0}^{}),\forall U\subset\mathcal{M}$ open, where $S_{0}$ is the typical fiber of $S$. From proposition 4.1.1, for any $p\in\mathcal{M}$, there is an isomorphism of $\mathbb{Z}_{2}$-graded vector spaces $T_{p}S\mathcal{M}_{S}\cong T_{p}\mathcal{M}\oplus S_{p}[1]$. Now, the vector bundle $V:=T\mathcal{M}\oplus S$ over $\mathcal{M}$ carries the canonical spin-connection induced by the Levi-Civita connection of the manifold $(\mathcal{M},g)$. Assume that we can pick a parallel non-degenerate $Spin(d)$-invariant bilinear form $\beta$ on $S$ with respect to this connection.262626This is always true if $\mathcal{M}$ is simply-connected. We can think of the $Spin(d)$-invariant bilinear form $\tilde{g}=g+\beta$ as a (pseudo-)Riemannian metric on the supermanifold $S\mathcal{M}_{S}$. Moreover, associated to the bilinear form $\beta$ we have the map $\Gamma:S^{2}\to T\mathcal{M}$, that is a point-wise generalization of the usual symmetric and Spin-equivariant bilinear form for a Spin representation $S_{0}\cong S_{p},\forall p\in\mathcal{M}$. This means that we can consider the bundle $\mathfrak{p}(V):=\mathfrak{spin}(d)\oplus V$ (4.120) as a _bundle of super Poincaré algebras_ over $\mathcal{M}$, with the bracket structure extended through $\Gamma$. Having found how to (point-wise) set up the super Poincaré algebra on top of the supermanifold $S\mathcal{M}_{S}$ constructed from $(\mathcal{M},g)$, we wish to establish which section of the super Poincaré bundle $\mathfrak{p}(V)$ produces a suitable generalization of “super-isometry” for $S\mathcal{M}_{S}$. In particular, we pay attention to which sections of $S$, as the odd subbundle of $\mathfrak{p}(V)$, generates “supersymmetries” of the generalized metric $\tilde{g}$. This problem was analyzed in [52], and connected to the problem of finding solution to the so called _Killing spinor equation_ for a section $\psi$ of $S\to\mathcal{M}$. ###### Definition 4.4.1. A section $\psi$ of the spinor bundle $S\to\mathcal{M}$ is called a _twistor spinor_ (or _conformal Killing spinor_) if it exists another section $\phi$ such that, for any vector field $X\in\Gamma(T\mathcal{M})$, $\nabla_{X}\psi=X\cdot\phi$ (4.121) where $X\cdot\phi=X^{\mu}\gamma_{\mu}\phi$ is the _Clifford multiplication_. If in particular $\phi=\lambda\psi$, for some constant $\lambda$, the spinor $\psi$ is called _Killing spinor_. The equation (4.121) is called _twistor_ or _Killing spinor equation_. Note that (4.121) directly implies $\phi=\pm(1/\dim(\mathcal{M}))\not{\nabla}\psi$, where $\not{\nabla}:=\gamma^{\mu}\nabla_{\mu}$ is the Dirac operator, and the sign depends on conventions. The twistor spinor equation is thus equivalently written as $\nabla_{X}\psi=\pm\frac{1}{\dim{(\mathcal{M})}}X\cdot\not{\nabla}\psi.$ (4.122) This characterizes the Killing spinors as those twistor spinors that satisfies also the Dirac equation $\not{\nabla}\psi=m\psi$ for some constant $m$. The main result proved in [52] is stated in the following theorem. ###### Theorem 4.4.1. Consider the supermanifold $S\mathcal{M}_{S}$ with the bilinear form $\tilde{g}=g+\beta$, and a section $\psi$ of $S$. The odd vector field $X_{\psi}$ associated to $\psi$ is a Killing vector field of $(S\mathcal{M}_{S},\tilde{g})$ if and only if $\psi$ is a twistor spinor. Here the Killing vector condition on the supermanifold is a conceptually straightforward generalization of the usual concept of Killing vector fields on a smooth manifold. It can be natually stated in terms of superframe fields. We refer to the above cited article for the details. Notice that, in particular, Killing spinors generate infinitesimal isometries of the supermanifold $S\mathcal{M}_{S}$, and thus are good candidates to describe the “preserved” supersymmetries of the odd part of the super Poincaré algebra, when this is associated to the generic curved manifold $\mathcal{M}$ in the way we saw above. See also [53] for a review on Killing spinors in (pseudo-)Riemannian geometry. From the QFT point of view, it is possible to derive a (generalized) Killing spinor equation, describing the preserved supercharges on the curved space, from a dynamical approach. This idea is based on a procedure also valid in the non-supersymmetric setting, when one aims to deform a certain QFT to redefine it on a generic curved manifold. In this case, one couples the theory to _background_ gravity, letting the metric fluctuate.272727Since the metric can fluctuate and the field theory is defined locally, there is no harm in principle in considering different topologies of the base manifold, like requiring it to be compact. Then the gravitational sector is decoupled from the rest of the theory taking the gravitational constant $G_{N}\to 0$, while the metric is linearized around the chosen _off-shell_ configuration $g=\eta+h$ and the higher order corrections disappear in the limit of weak gravitational interaction. It is important that we do not constrain the gravitational field to satisfy the equation of motion, since it is considered as a background (classical) field. The same idea applies when the theory is defined in the supersymmetric setting: in this case to preserve supersymmetry one has to couple to background _supergravity (SUGRA)_. The resulting field theory will contain then more fields belonging to the so-called _supergravity multiplet_. This time, taking the limit $G_{N}\to 0$ we fix all the background supergravity multiplet to an allowed off-shell configuration. Note that in particular, the auxiliary fields are not eliminated in terms of the other fields using their equations of motion. If then there are supergravity transformations that leave the given background invariant, we say that the corresponding rigid supercharges are _preserved_ on this background. This procedure was systematically introduced in [54], then many cases and classification were made in different dimensions and with different degree of supersymmetriy (see for example [55, 56, 57, 58]). #### 4.4.1 Coupling to background SUGRA Suppose we have a supersymmetric field theory formulated on flat space specified by its Lagrangian $\mathcal{L}^{(0)}$, whose variation under supersymmetry is a total derivative:282828For simplicity, we consider now the formulation on the even space $\mathbb{R}^{d}$ or $\mathcal{M}$, at the level of component fields of the given supersymmetry multiplets. The supersymmetry variation of these component fields are those coming from the action of the odd supertranslations in superspace. $\delta\mathcal{L}^{(0)}=d(\cdots)=\partial_{\mu}(\cdots)^{\mu}$ (4.123) We can introduce supergravity by requiring the action of the super-Poincaré group to be _local_ , employing the usual gauge principle and minimal coupling or Noether procedure. In the non-supersymmetric setting, this would mean to introduce a gauge symmetry under local coordinate transformations, realized via diffeomorphisms on $\mathcal{M}$. The Noether current associated to such infinitesimal transformations is the _energy-momentum tensor_ $T^{\mu\nu}$, that we take to be symmetric.292929In general this will not be a symmetric tensor, but there always exists a suitable modification that makes it symmetric, and moreover equivalent to the Hilbert definition of energy momentum tensor as a source of gravitational field. This is the _Belinfante–Rosenfeld tensor_ $\tilde{T}^{\mu\nu}:=T^{\mu\nu}+\frac{1}{2}\nabla_{\lambda}(S^{\mu\nu\lambda}+S^{\nu\mu\lambda}-S^{\lambda\nu\mu})$ (4.124) where $S^{\mu}_{\nu\lambda}$ is the spin part of the Lorentz generators in a given spin representation satisfying $\nabla_{\mu}S^{\mu}_{\nu\lambda}=T_{\nu\mu}-T_{\mu\nu}$, and $\nabla$ is an appropriate torsion-free spin-covariant derivative induced from the metric (see for example [59]). The minimal coupling procedure then requires to modify the Lagrangian, $\mathcal{L}^{\prime}=\mathcal{L}^{(0)}+h_{\mu\nu}T^{\mu\nu}+O(h^{2})$ (4.125) where $h_{\mu\nu}$ is regarded as a variation of the metric from the flat space values $\eta_{\mu\nu}$, and $O(h^{2})$ are seagull non-linear terms that can be fixed requiring the gauge invariance of $\mathcal{L}^{\prime}$. The resulting non-linear coupling is obtainable substituting in the original theory $d\mapsto\nabla,\qquad\eta\mapsto g=\eta+h,$ (4.126) where $\nabla$ is the gauge-covariant derivative with respect to a connection $\Gamma$, that we take as the Levi-Civita connection. The theory is now coupled to a gravitational (classical) _background_. If we want to make the graviton field $h$ dynamical, we can add an Hilbert-Einstein term to $\mathcal{L}^{(0)}$, $\mathcal{L}_{HE}=-\frac{\sqrt{\|\det{g}\|}}{2\kappa^{2}}Ric_{g}$ (4.127) where $Ric_{g}$ is the Ricci scalar associated to $g$, and $\kappa:=1/M_{p}=\sqrt{8\pi G_{N}}$. In the supersymmetric setting, gauging the super-Poincaré group leads to the introduction of more fields into the theory, since as we know they can be interpreted as components of superfields in superspace, and thus belong to supersymmetry multiplets. In particular, we have to introduce the _graviton multiplet_ composed by the metric $g$, the gravitino $\Psi$ and other (maybe auxiliary) fields. The particular field content depends on the number $\mathcal{N}$ of supersymmetries, the dimensionality of the theory, the presence or absence of an R-symmetry and whether the theory is or not superconformal. Consequently, also the energy-momentum tensor will belong to a multiplet, the so-called _supercurrent multiplet_ , composed by $T$, a _supercurrent_ $J$ associated to the local invariance under (odd) supertranslations, and other fields. We can schematically perform the first steps of the Noether procedure to see how the components of these multiplets arise naturally. We start from the odd part of the super-Poincaré algebra $\mathfrak{iso}(\mathbb{R}^{d})\oplus S[1]$, writing the infinitesimal variation of the Lagrangian in terms of the supercurrent: $\delta_{\epsilon}\mathcal{L}^{(0)}\equiv\epsilon\cdot\mathcal{L}^{(0)}=(\partial_{\mu}J^{\mu})\epsilon$ (4.128) where $\epsilon$ is now a Majorana spinor field, i.e. a section of the spinor bundle with fiber $S$. The supercurrent $J$ is an $S$-valued vector field, and the spinor contraction is done via the usual charge conjugation matrix. We couple this current to a gauge field $\Psi$, to be identified with the gravitino, that is locally an $S$-valued 1-form such that, at linearized level, $\delta_{\epsilon}\Psi_{\mu}=\frac{1}{\kappa}\partial_{\mu}\epsilon$ (4.129) where the constant $\kappa$ is introduced for dimensional reasons.303030If we canonically take mass dimensions of scalars to be $(d-2)/2$ and of spinors to be $(d-1)/2$, and since schematically $\delta_{\epsilon}(boson)=(fermion)\epsilon$ than $[\epsilon]=-1/2$, so $\kappa$ must be a dimensionfull parameter of mass dimension $[\kappa]=(2-d)/d$. We can so identify this constant as the gravitational constant previously defined. Then we add a term to the Lagrangian: $\mathcal{L}^{\prime}=\mathcal{L}^{(0)}+\kappa\Psi_{\mu}J^{\mu}.$ (4.130) Now the variation of $\mathcal{L}^{\prime}$ is proportional to the variation of the current $\delta_{\epsilon}J$. Since the supercurrent is a supersymmetry variation of the original Lagrangian, its variation will be proportional to the action of the translation generators $P_{\mu}$: $\displaystyle\delta_{\epsilon}^{2}\mathcal{L}^{(0)}$ $\displaystyle=\epsilon\cdot(\epsilon\cdot\mathcal{L}^{(0)})=\partial_{\mu}(\delta_{\epsilon}J^{\mu})\epsilon$ (4.131) $\displaystyle=(1/2)[\epsilon,\epsilon]\cdot\mathcal{L}^{(0)}=\overline{\epsilon}\gamma^{\nu}\epsilon(P_{\nu}\cdot\mathcal{L}^{(0)})=(\overline{\epsilon}\gamma_{\nu}\epsilon)\partial_{\mu}T^{\mu\nu}$ $\displaystyle\Rightarrow\delta_{\epsilon}J^{\mu}$ $\displaystyle=\overline{\epsilon}\gamma_{\nu}T^{\nu\mu}$ where in the second line we wrote the variation under the action of the translation generators in terms of the energy-momentum tensor $T$. We try to restore the gauge-invariance of the Lagrangian by minimally coupling this new current to a new gauge field $h$, that we identify as a metric variation, the _graviton_ $\mathcal{L}^{\prime\prime}=\mathcal{L}^{(0)}+\kappa\Psi_{\mu}J^{\mu}+h_{\mu\nu}T^{\mu\nu},$ (4.132) and naturally requiring the supersymmetry variation of the graviton to be $\delta_{\epsilon}h_{\mu\nu}=\kappa\overline{\epsilon}\gamma_{(\mu}\Psi_{\nu)},$ (4.133) making it the superpartner of the gravitino $\Psi$. The Lagrangian $\mathcal{L}^{\prime\prime}$ is again not supersymmetric, since the variation $\delta_{\epsilon}T^{\mu\nu}\neq 0$ in general, so the Noether procedure is not terminated yet. It is not easy to complete this procedure in this way, but in principle repeating this passages we would introduce more linearly coupled currents and gauge fields that, motivated by supersymmetry, we expect to come from the SUGRA supermultiplets mentioned above. To ensure the supersymmetry of the full Lagrangian at non-linear level, as in non supersymmetric gauge theories, non-linear couplings could have to be introduced as well as non-linear terms in the supersymmetry variations. Summarizing, we expect the fully coupled Lagrangian to be schematically of the form $\mathcal{L}=\mathcal{L}^{(0)}+\kappa J^{\mu}\Psi_{\mu}+h_{\mu\nu}T^{\mu\nu}+\sum_{i}\mathcal{B}^{i}\cdot\mathcal{J}^{i}+(\mathrm{seagull\ terms})$ (4.134) where $\mathcal{B}$ is the multiplet of background gauge fields $(h,\Psi,\cdots)$, $\mathcal{J}$ the supercurrent multiplet $(T,J,\cdots)$, and we referred to possible higher-order terms in the background fields as seagull terms. As already said, the particular field content of these multiplets is not unique, so we remain generic for the moment and refer to the next subsections for some examples. We can absorb the terms proportional to $h$ as in the non-supersymmetric case, by making the substitutions $d\mapsto\nabla$ and $\eta\mapsto g=\eta+h$. If we want to have a full gravitational theory, we can add a kinetic term for the source fields, and complete their supersymmetry variations with possible non-linear terms to ensure gauge invariance. Regarding the metric and the gravitino, the kinetic terms are given by the Hilbert-Einstein action (4.127) and the _Rarita- Schwinger_ action $\mathcal{L}_{RS}=-\frac{\sqrt{\|\det{g}\|}}{2}\overline{\Psi}_{\mu}\gamma^{\mu\nu\rho}(\nabla_{\nu}\Psi)_{\rho},$ (4.135) where $\gamma^{\mu\nu\rho}:=\gamma^{[\mu}\gamma^{\nu}\gamma^{\rho]}$, and $\nabla$ acts on spinors via the spin connection, $(\nabla_{\mu}\Psi)_{\nu}=\partial_{\mu}\Psi_{\nu}+\frac{1}{4}\omega_{\mu}^{ab}\gamma_{ab}\Psi_{\nu}-\Gamma_{\mu\nu}^{\rho}\Psi_{\rho}$. The supersymmetry variations will be generically $\displaystyle\delta_{\epsilon}h_{\mu\nu}=\kappa\overline{\epsilon}\left\\{\gamma_{(\mu}\Psi_{\nu)}+(\cdots)^{F}\right\\}$ (4.136) $\displaystyle\delta_{\epsilon}\Psi_{\mu}=\frac{1}{\kappa}\left\\{\nabla_{\mu}+(\cdots)_{\mu}^{B}\right\\}\epsilon+O(\kappa\Psi^{2}\epsilon)$ where we stressed that non-linear higher-oreder terms for the gravitino are $\kappa$-suppressed, and the ellipses in both cases collect contributions from the other (fermionic or bosonic, respectively) fields of the supergravity multiplet. Notice that also the supersymmetry variations of the field content of the original $\mathcal{L}^{(0)}$ get modified with respect to their flat- space version. Once one has the full supergravity theory, their transformation rules follow from the corresponding formulas in the appropriate matter-coupled off-shell supergravity. We now consider the _rigid limit_ $G_{N}\to 0$ (or $\kappa\to 0$, or $M_{P}\to\infty$) together with the choice of a given background gravitational multiplet $\mathcal{B}$ compatible with the original request $(\mathcal{M},g)$.313131We stress that a rigid supersymmetric background is characterized by a full set of supergravity background fields, i.e. specifying only the metric does not determine the background. In particular, there are distinct backgrounds that have the same metric but lead to different partition functions. Since we think at this classical configuration as a VEV, we require all the fermion fields in the supergravity multiplet to vanish on this background. We also look for those supergravity transformations that leave this background invariant.323232This can be interpreted as a superisometry requirement with respect to the graviton multiplet. These requirements produce the following effects: • Fermionic gravitational fields as well as the kinetic term for the bosonic gravitational sector do not contribute to the lagrangian: $\mathcal{L}=\left.\mathcal{L}^{(0)}\right\|_{\begin{subarray}{c}d\to\nabla\\\ \eta\to g\end{subarray}}+\sum_{i}\mathcal{B}_{B}^{i}\cdot\mathcal{J}_{B}^{i}+(\mathrm{seagull\ terms})_{B}$ (4.137) At the same time, supersymmetry variations of the bosonic gravitational fields automatically vanish. * • Requiring the supersymmetry of the background is then equivalent to $\delta_{\epsilon}B_{F}=0.$ (4.138) In particular, this condition on the gravitino generates the _generalized Killing spinor equation_ $\delta_{\epsilon}\Psi_{\mu}=0\quad\Leftrightarrow\quad\nabla_{\mu}\epsilon=(\cdots)_{\mu}\epsilon$ (4.139) where, again, ellipses stand for terms proportional to bosonic fields in the graviton multiplet. The solutions to this equation determine which sections of the spinor bundle $S\to\mathcal{M}$ generates the preserved supersymmetry transformations on $\mathcal{M}$. #### 4.4.2 Supercurrent multiplets and metric multiplets As we wrote before, the field content of supercurrent multiplets depends on the general properties of the theory at hand. In [60] it was given a definition from basic general requirements starting from superfields in superspace, that allows a classification by specializing to the various particular cases. It is shown that the most general supercurrent is a real superfield $\mathcal{S}_{a\dot{a}}$ satisfying $\begin{array}[]{ll}\lx@intercol\tilde{D}^{\dot{a}}\mathcal{S}_{a\dot{a}}=\chi_{a}+\mathcal{Y}_{a}\hfil\lx@intercol\\\ \tilde{D}^{\dot{a}}\chi_{a}=0&\tilde{D}^{\dot{a}}\chi^{\dagger}_{\dot{a}}=D^{a}\chi_{a}\\\ D_{(a}\mathcal{Y}_{b)}=0&\tilde{D}^{2}\mathcal{Y}_{a}=0.\end{array}$ (4.140) Every supersymmetric theory has such an $\mathcal{S}$-multiplet, containing the stress energy tensor $T$ and the supercurrent $S$. They are the only component fields with spin larger than one, since they couple to the graviton and the gravitino in the metric multiplet, that respectively are the only component fields with spin higher than one in this multiplet. We report here special examples in 4 and 3 dimensions, that can be derived solving the constraints (4.140) in cases where additional conditions on the superfields $\chi_{a}$ and $\mathcal{Y}_{a}$ hold. For $\mathcal{N}=1$ in 4-dimensions we have three possible interesting special cases: 1. 1. The majority of theories admit a reduction of the $\mathcal{S}$-multiplet into the so-called _Ferrara-Zumino (FZ) multiplet_ $\mathcal{J}^{FZ}_{\mu}\to\left(j_{\mu},(S_{\mu})_{a},x,T_{\mu\nu}\right)$ (4.141) where $j_{\mu}$ is a vector field and $x$ a complex scalar field. 2. 2. If the theory has a $U(1)_{R}$ symmetry, the $\mathcal{S}$-multiplet reduces to the so-called _$\mathcal{R}$ -multiplet_, whose lower degree component is the conserved R-current $j_{\mu}^{(R)}$: $\mathcal{R}_{\mu}\to\left(j_{\mu}^{(R)},(S_{\mu})_{a},T_{\mu\nu},C_{\mu\nu}\right)$ (4.142) where $C_{\mu\nu}$ are the components of a conserved 2-form current, the so- called _brane current_.333333In curved space and in presence of topological defects as strings (1-brane) or domain walls (2-brane), the supersymmetry algebra (4.57) can be modified by the presence of _brane charges_ , $\displaystyle\left[Q_{a},\tilde{Q}_{\dot{b}}\right]$ $\displaystyle=2(\gamma^{\mu})_{a\dot{b}}(P_{\mu}+Z_{\mu})$ $\displaystyle[Q_{a},Q_{b}]$ $\displaystyle=(\gamma^{\mu\nu})_{ab}\tilde{Z}_{\mu\nu}$ where $Z_{\mu},\tilde{Z}_{\mu\nu}$ are non-zero for strings and domain walls, respectively. The corresponding tensor currents are the _brane currents_ $C_{\mu\nu},\tilde{C}_{\mu\nu\rho}$, that are topologically conserved. See [61, 62, 60] for more details. 3. 3. For a superconformal theory, the $\mathcal{S}$-multiplet decomposes into the smaller supercurrent $\mathcal{J}_{\mu}\to\left(j_{\mu}^{(R)},(S_{\mu})_{a},T_{\mu\nu}\right)$ (4.143) where $j_{\mu}^{(R)}$ is a conserved superconformal $U(1)_{R}$-current. Both the FZ multiplet and the $\mathcal{R}$-multiplet contain 12+12 real degrees of freedom out of the initial 16+16 of the general $\mathcal{S}$-multiplet, while the superconformal multiplet is reduced to 8+8 real degrees of freedom. The FZ multiplet can be coupled to the so-called “old minimal supergravity multiplet” [63]: $\mathcal{H}_{\mu}\to\left(b_{\mu},(\Psi_{\mu})_{a},M,h_{\mu\nu}\right)$ (4.144) where $b_{\mu}$ is a genuine 1-form field (i.e. non gauge), $M$ is a complex scalar, $(\Psi_{\mu})_{a}$ is the gravitino and $h_{\mu\nu}$ is the graviton. The variation of the gravitino in this case is given by [54] $\delta_{\epsilon}\Psi_{\mu}=-2\nabla_{\mu}\epsilon+\frac{i}{3}\left(M\gamma_{\mu}+2b_{\mu}+2b^{\nu}\gamma_{\mu\nu}\right)\epsilon$ (4.145) that implies a generalized Killing spinor equation of the form $\nabla_{\mu}\epsilon=\frac{i}{6}\left(M\gamma_{\mu}+2b_{\mu}+2b^{\nu}\gamma_{\mu\nu}\right)\epsilon$ (4.146) in the Majorana spinor $\epsilon$, given a background multiplet. In theories with an R-symmetry, one can couple the $\mathcal{R}$-multiplet to the “new minimal supergravity multiplet”[64]: $\mathcal{H}^{(new)}_{\mu}\to\left(A^{(R)}_{\mu},(\Psi_{\mu})_{a},h_{\mu\nu},B_{\mu\nu}\right)$ (4.147) where $A^{(R)}_{\mu}$ is the Abelian gauge field associated to the $U(1)_{R}$ symmetry, and $B_{\mu\nu}$ is a 2-form gauge field that is often treated through its Hodge dual $V^{\mu}:=i(\star B)^{\mu}=(i/2)\varepsilon^{\mu\nu\rho\sigma}\partial_{\nu}B_{\rho\sigma}$. The variation of the gravitino in this case gives rise to the following Killing spinor equation, that in 2-component notation is [54] $\displaystyle\left(\nabla_{\mu}-iA^{(R)}_{\mu}\right)\epsilon_{a}=-iV_{\mu}\epsilon_{a}-iV^{\nu}(\gamma_{\mu\nu}\epsilon)_{a}$ (4.148) $\displaystyle\left(\nabla_{\mu}+iA^{(R)}_{\mu}\right)\tilde{\epsilon}_{\dot{a}}=iV_{\mu}\tilde{\epsilon}_{\dot{a}}+iV^{\nu}(\gamma_{\mu\nu}\tilde{\epsilon})_{\dot{a}}$ where in the parenthesis on the LHS there is a gauge-covariant derivative with respect to the $U(1)_{R}$ symmetry, that acts with opposite charge on the two chiral sector of the spin representation, see (4.90). The $\mathcal{N}=2$ case in 3 Euclidean dimensions can be derived in superspace by dimensional reduction from the four dimensional case: the supercurrent is reduced to a three dimensional $\mathcal{S}$-multiplet with 12+12 real DoF, plus a real scalar superfield $\hat{\mathcal{J}}=\mathcal{S}_{0}\equiv\mathcal{S}_{a\dot{a}}(\sigma_{0})^{a\dot{a}}$, that contains 4+4 real DoF. Again, there are special cases analogue of those above: a FZ multiplet, an $\mathcal{R}$-multiplet, and a superconformal multiplet. For example, the $\mathcal{N}=2$ $\mathcal{R}$-multiplet in 3 dimensions has the field content $\mathcal{R}_{\mu}\to\left(j_{\mu}^{(R)},j_{\mu}^{(Z)},J,(S_{\mu})_{a},(\tilde{S}_{\mu})_{a},T_{\mu\nu}\right)$ (4.149) where $j_{\mu}^{(R)}$ is the conserved R-current, $j_{\mu}^{(Z)}$ is the conserved central charge current and $J$ is a scalar operator, that with the conserved supercurrents and enery-momentum tensor sum up to 8+8 real DoF. This multiplet couples to the tree dimensional $\mathcal{N}=2$ new minimal supergravity multiplet $\mathcal{H}^{(new)}_{\mu}\to\left(A^{(R)}_{\mu},C_{\mu},H,(\psi_{\mu})_{a},(\tilde{\psi}_{\mu})_{a},h_{\mu\nu}\right)$ (4.150) with the graviton, two gravitini, two gauge 1-forms $A^{(R)}$ and $C$, and a scalar $H$. The 1-form $C$ is often treated in terms of the vector field $V^{\mu}:=i(\star dC)^{\mu}=i\varepsilon^{\mu\nu\rho}\partial_{\nu}C_{\rho}$ that is Hodge dual to its field strength. Putting to zero the gravitini and their variations leads to the generalized Killing spinor equations [56] $\displaystyle\left(\nabla_{\mu}-iA^{(R)}_{\mu}\right)\epsilon=-\left(\frac{1}{2}H\gamma_{\mu}+iV_{\mu}+\frac{1}{2}\varepsilon_{\mu\nu\rho}V^{\nu}\gamma^{\rho}\right)\epsilon$ (4.151) $\displaystyle\left(\nabla_{\mu}+iA^{(R)}_{\mu}\right)\tilde{\epsilon}=-\left(\frac{1}{2}H\gamma_{\mu}-iV_{\mu}-\frac{1}{2}\varepsilon_{\mu\nu\rho}V^{\nu}\gamma^{\rho}\right)\tilde{\epsilon}$ where in this case the two spinors $\epsilon,\tilde{\epsilon}$ have to be treated as independent. Notice that both equations (4.148) and (4.151) are linear in the 4 spinor components, so their solutions (if exist) span a vector space of dimension less or equal than 4. #### 4.4.3 $\mathcal{N}=2$ gauge theories on the round 3-sphere It was shown that, in general, solutions of the Killing condition (4.148) in four dimensions exist if $(\mathcal{M},g)$ is an Hermitian manifold, i.e. $\mathcal{M}$ has an integrable complex structure and $g$ is a compatible Hermitian metric. Analogously, the existence of a solution to (4.151) in three dimensions was shown to be equivalent to the manifold admitting a _transversally holomorphic fibration_.343434This is an odd-dimensional analogue to a complex structure. It means, roughly speaking, that $\mathcal{M}$ is locally isomorphic to $\mathbb{R}\times\mathbb{C}$, and its transition functions are holomorphic in the $\mathbb{C}$-sector. If one is interested in the case of maximal number of Killing spinor solutions, a suitable integrability condition (see [56]) gives $\displaystyle H=\text{const},\qquad d(A^{(R)}-V)=0,\qquad g(V,V)=\text{const},$ (4.152) $\displaystyle(\nabla_{\mu}V)_{\nu}=-iH\varepsilon_{\mu\nu\rho}V^{\rho},$ $\displaystyle R_{\mu\nu}=-V_{\mu}V_{\nu}+g_{\mu\nu}(g(V,V)+2H^{2}).$ In particular, if we take $A^{(R)}=V=0$, then $\mathcal{M}$ is of Einstein type and so it has constant sectional curvature. $H^{2}$ is then interpreted as a cosmological constant, and $\mathcal{M}$ can be $\mathbb{S}^{3},\mathbb{T}^{3}$ or $\mathbb{H}^{3}$ if $H$ is purely immaginary, zero or real. All of them are examples of maximally supersymmetric backgrounds in $\mathcal{N}=2$, thus we have 2 solutions for $\epsilon$ and 2 solutions for $\tilde{\epsilon}$ to the Killing equations $\nabla_{\mu}\epsilon=-\frac{H}{2}\gamma_{\mu}\epsilon\ ;\qquad\nabla_{\mu}\tilde{\epsilon}=-\frac{H}{2}\gamma_{\mu}\tilde{\epsilon}.$ (4.153) In particular, if we take $H=-(i/l)$, this solutions are consistent with the $\mathbb{S}^{3}$ round metric $g=l^{2}\left(d\varphi_{1}\otimes d\varphi_{1}+\sin^{2}\varphi_{1}\ d\varphi_{2}\otimes d\varphi_{2}+\sin^{2}\varphi_{1}\sin^{2}\varphi_{2}\ d\varphi_{3}\otimes d\varphi_{3}\right).$ (4.154) The action of the supersymmetry algebra on the curved manifold can be derived by taking the rigid limit of the appropriate algebra of supergravity transformations. In the 3-dimensional case, it can be derived by a “twisted” reduction of the $\mathcal{N}=1$ supergravity in 4 dimensions. The 4-dimensional supersymmetry algebra realizes on the curved manifold as $[\delta_{\epsilon},\delta_{\epsilon}]\phi_{(r)}=[\epsilon,\epsilon]\cdot\phi_{(r)}=2(\overline{\epsilon}\gamma^{\mu}\epsilon)P_{\mu}\cdot\phi_{(r)}$ (4.155) where $\epsilon$ is a Majorana Killing spinor, $\phi_{(r)}$ is a generic field of R-charge $r$, and the local action of the momentum operator is through the fully covariant derivative $P_{\mu}\to-\left(i\nabla_{\mu}+rA^{(R)}_{\mu}\right),$ (4.156) so that (4.155) can be written as $[\delta_{\epsilon},\delta_{\epsilon}]\phi_{(r)}=-2i\left(\mathcal{L}_{v}-irA^{(R)}(v)\right)\phi_{(r)}$ (4.157) where $v^{\mu}:=(\overline{\epsilon}\gamma^{\mu}\epsilon)$ is a Killing vector field thanks to $\epsilon$ being a Killing spinor field. This is reduced to the 3-dimensional case, taking $\epsilon,\tilde{\epsilon}$ now as independent 2-component Killing spinors and $v^{\mu}:=\tilde{\epsilon}\gamma^{\mu}\epsilon$ in 3 dimensions, as (see again [56]) $\displaystyle[\delta_{\tilde{\epsilon}},\delta_{\epsilon}]\phi_{(r,z)}=-2i\left[\mathcal{L}_{v}-iv^{\mu}\left(r(A^{(R)}_{\mu}-\frac{1}{2}V_{\mu})+zC_{\mu}\right)+\tilde{\epsilon}\epsilon(z-rH)\right]\phi_{(r,z)},$ (4.158) $\displaystyle[\delta_{\tilde{\epsilon}},\delta_{\tilde{\epsilon}}]\phi_{(r,z)}=0,\qquad[\delta_{\epsilon},\delta_{\epsilon}]\phi_{(r,z)}=0,$ where $z$ is the charge associated to the action of the central charge $Z$ in (4.101). For the 3-sphere of radius $l=1$, this is simplified to $\displaystyle[\delta_{\tilde{\epsilon}},\delta_{\epsilon}]\phi_{(r,z)}=-2i\left[\mathcal{L}_{v}+\tilde{\epsilon}\epsilon(z+ir)\right]\phi_{(r,z)},$ (4.159) $\displaystyle[\delta_{\tilde{\epsilon}},\delta_{\tilde{\epsilon}}]\phi_{(r,z)}=0,\qquad[\delta_{\epsilon},\delta_{\epsilon}]\phi_{(r,z)}=0.$ We report the resulting supersymmetry variation for the 3-dimensional $\mathcal{N}=2$ vector multiplet $(A_{\mu},\sigma$, $\lambda_{a},\tilde{\lambda}_{a},D)$, with respect to two Killing spinors $\tilde{\epsilon},\epsilon$. This multiplet is uncharged under the action of R-symmetry and of the central charge $Z$. Following conventions of [50] and [49], $\displaystyle\delta A_{\mu}=\frac{i}{2}(\tilde{\epsilon}\gamma_{\mu}\lambda-\tilde{\lambda}\gamma_{\mu}\epsilon)$ (4.160) $\displaystyle\delta\sigma=\frac{1}{2}(\tilde{\epsilon}\lambda-\tilde{\lambda}\epsilon)$ $\displaystyle\delta\lambda=\left(-\frac{1}{2}F_{\mu\nu}\gamma^{\mu\nu}-D+i(D_{\mu}\sigma)\gamma^{\mu}+\frac{2i}{3}\sigma\gamma^{\mu}D_{\mu}\right)\epsilon$ $\displaystyle\delta\tilde{\lambda}=\left(-\frac{1}{2}F_{\mu\nu}\gamma^{\mu\nu}+D-i(D_{\mu}\sigma)\gamma^{\mu}-\frac{2i}{3}\sigma\gamma^{\mu}D_{\mu}\right)\tilde{\epsilon}$ $\displaystyle\delta D=-\frac{i}{2}\left(\tilde{\epsilon}\gamma^{\mu}D_{\mu}\lambda-(D_{\mu}\tilde{\lambda})\gamma^{\mu}\epsilon\right)+\frac{i}{2}\left([\tilde{\epsilon}\lambda,\sigma]-[\tilde{\lambda}\epsilon,\sigma]\right)-\frac{i}{6}\left(\tilde{\lambda}\gamma^{\mu}D_{\mu}\epsilon+(D_{\mu}\tilde{\epsilon})\gamma^{\mu}\lambda\right)$ where now $D_{\mu}=\nabla_{\mu}-iA_{\mu}$ is the gauge-covariant derivative. On the 3-sphere, the actions in (4.106) and (4.107) acquire a factor $\sqrt{g}$ in the measure,353535The pure Chern-Simons term $\left(A\wedge dA+\frac{2i}{3}A^{3}\right)$ is actually unmodified, being already a 3-form. and the Super Yang-Mills Lagrangian gets modified to $\mathcal{L}_{YM}=\mathrm{Tr}\left\\{\frac{i}{2}\tilde{\lambda}\gamma^{\mu}D_{\mu}\lambda+\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}D_{\mu}\sigma D^{\mu}\sigma+\frac{i}{2}\tilde{\lambda}[\sigma,\lambda]+\frac{1}{2}\left(D+\frac{\sigma}{l}\right)^{2}-\frac{1}{4l}\tilde{\lambda}\lambda\right\\}\\\ $ (4.161) where we reinstated the radius $l$, to see that indeed in the limit $l\to\infty$ this becomes the standard Euclidean SYM theory in 3 dimensions. We note an important feature of this Lagrangian, that will be important for the application of the localization principle: this can be written as a supersymmetry variation, i.e. $\tilde{\epsilon}\epsilon\mathcal{L}_{YM}=\delta_{\tilde{\epsilon}}\delta_{\epsilon}\mathrm{Tr}\left\\{\frac{1}{2}\tilde{\lambda}\lambda-2D\sigma\right\\}.$ (4.162) The SCS Lagrangian does not get modified on curved space, since the term depending on the gauge field is topological, and the other ones do not contain derivatives. It is important to remark that, in general, unbroken supersymmetry is consistent only with Anti-de Sitter geometry (or, in Euclidean signature, hyperbolic geometry) [54]. An exception to this is given by those theories that possess a larger group of symmetries, the _superconformal_ group. This is an extension of the super-Poincaré group, to include also conformal transformations of the metric. In this case, supersymmetry can be consistent also on conformally flat backgrounds with positive scalar curvature, of which the $n$-spheres are an example. The $\mathcal{N}=2$ SCS theory of above is an example of superconformal theory. #### 4.4.4 $\mathcal{N}=4,2,2^{}$ gauge theories on the round 4-sphere We continue also the example of the $\mathcal{N}=4$ 4-dimensional theory, understanding how it can be realized on a different background compatible with $\mathbb{S}^{4}$, and what part of the supersymmetry algebra can be preserved on this background. As in Section 4.3.7, the $\mathcal{N}=2$ and $\mathcal{N}=2^{}$ cases follow from modifications of the $\mathcal{N}=4$ theory. Using stereographic coordinates $x^{1},\cdots,x^{4}$, such that the North pole is located at $x^{\mu}=0$, the round metric of the 4-sphere of radius $r$ looks explicitly as a conformal transformation of the flat Euclidean metric, $g_{\mu\nu}^{(x)}=\delta_{\mu\nu}e^{2\Omega(x)},\quad\mathrm{where}\ e^{2\Omega(x)}=\frac{1}{\left(1+\frac{x^{2}}{4r^{2}}\right)^{2}}$ (4.163) where $x^{2}=\sum_{\mu=1}^{4}(x^{\mu})^{2}$. As remarked at the end of the last section, the conformal flatness of $\mathbb{S}^{4}$ allows us to deform the superconformal YM theory on it, provided we preserve the conformal symmetry. In order to do this, we modify the kinetic term of the scalars $(\Phi_{A})_{A=5,\cdots,9,0}$ adding a conformal coupling to the curvature: $(\partial\Phi_{A})^{2}\to(\partial\Phi_{A})^{2}+\frac{R}{6}(\Phi_{A})^{2}$ where $R=\frac{12}{r^{2}}$ is the scalar curvature of the metric $g$.363636In $d$-dimensions, the conformal coupling to the curvature scalar is made adding a term $\xi R(\Phi)^{2}$ for the scalar field of canonical mass dimension $\frac{d-2}{2}$, with $\xi=(d-2)/4(d-1)$ (see [65], Appendix D). The scalar curvature of the $d$-sphere of radius $r$ is $R=d(d-1)/r^{2}$. This ensures conformal invariance of the action on the 4-sphere, $S^{\mathcal{N}=4}_{\mathbb{S}^{4}}=\int_{\mathbb{S}^{4}}d^{4}x\sqrt{g}\ \frac{1}{g_{YM}^{2}}\mathrm{Tr}\left(\frac{1}{2}F_{MN}F^{MN}-\Psi\Gamma^{M}D_{M}\Psi+\frac{2}{r^{2}}\Phi_{A}\Phi^{A}\right)$ (4.164) where the derivatives have been promoted to covariant derivatives with respect to the Levi-Civita connection of $g$. Now we have to understand which supersymmetries of the $\mathcal{N}=4$ algebra can be preserved on the new curved background. From theorem 4.4.1, we know that a necessary condition for a section $\epsilon$ of the Majorana-Weyl spinor bundle on $\mathbb{S}^{4}$, to produce a superisometry for the new background, is that it satisfies the _twistor spinor equation_ , or _conformal Killing equation_ $\nabla_{\mu}\epsilon=\tilde{\Gamma}_{\mu}\tilde{\epsilon}$ (4.165) for some other section $\tilde{\epsilon}$. It can be checked that, to ensure supersymmetry of (4.164), $\tilde{\epsilon}$ must be also a twistor spinor satisfying $\nabla_{\mu}\tilde{\epsilon}=-\frac{1}{4r^{2}}\Gamma_{\mu}\epsilon,$ (4.166) and the variations (4.109) have to be modified as the superconformal transformations $\displaystyle\delta_{\epsilon}A_{M}$ $\displaystyle=\epsilon\Gamma_{M}\Psi$ (4.167) $\displaystyle\delta_{\epsilon}\Psi$ $\displaystyle=\frac{1}{2}\Gamma^{MN}F_{MN}\epsilon+\frac{1}{2}\Gamma^{\mu A}\Psi_{A}\nabla_{\mu}\epsilon.$ Since $\mathbb{S}^{4}$ is conformally flat, the number of solutions to (4.165) is maximal and equal to $2\dim{(S^{\pm})}=32$ [53], so the whole $\mathcal{N}=4$ superconformal algebra is preserved.373737The number of generators of the $\mathcal{N}=4$ super-Euclidean algebra is $\dim{(S^{\pm})}=16$. The other 16 are the remaining generators of the superconformal algebra. If one restricts the attention to the $\mathcal{N}=2$ subalgebra, then half of the generators are preserved. If instead the $\mathcal{N}=2^{}$ theory is considered, the conformal symmetry is broken and only 8 supercharges are preserved on $\mathbb{S}^{4}$. With the above modifications, the $\mathcal{N}=4$ superconformal algebra closes again only on-shell: imposing the EoM for $\Psi$, one gets (see Appendix of [11] for the details of the computation) $\delta_{\epsilon}^{2}=-\mathcal{L}_{v}-G_{\Phi}-R-\Omega$ (4.168) as (4.117). In order to prepare the ground for the exploitation of the localization principle on supersymmetric gauge theories, we remark that, if we want to define correctly an equivariant structure with respect to (at least a $U(1)$ subgroup of) the Poincaré group, we need at least an $\mathcal{N}=1$ supersymmetry subalgebra to close properly (i.e. off-shell). If this is the case, we can use the corresponding variation $\delta_{\epsilon}$ as a Cartan differential with respect to this equivariant cohomology (we are going to justify better this in the next chapter). It is not possible to close off- shell the full $\mathcal{N}=2$ algebra on the hypermultiplet, but fixing a conformal Killing spinor $\epsilon$ satisfying (4.165) and (4.166), it is possible to close the subalgebra generated by $\delta_{\epsilon}$ only. To do this, one has to add auxiliary fields to match the number of off-shell bosonic/fermionic degrees of freedom of the theory [66], analogously to what happens for example to the $\mathcal{N}=1$ vector multiplet in 4-dimensions. In 10-dimensions, we have 16 real fermionic components, and $(10-1)$ real physical bosonic components, so we have to add 7 bosonic (scalar) fields $(K_{i})_{i=1,\cdots,7}$. The modified action $S^{\mathcal{N}=4}_{\mathbb{S}^{4}}=\int_{\mathbb{S}^{4}}d^{4}x\sqrt{g}\ \frac{1}{g_{YM}^{2}}\mathrm{Tr}\left(F_{MN}F^{MN}-\Psi\Gamma^{M}D_{M}\Psi+\frac{2}{r^{2}}\Phi_{A}\Phi^{A}-\sum_{i=1}^{7}K_{i}K_{i}\right)$ (4.169) is supersymmetric under the modified $\mathcal{N}=4$ superconformal transformations $\displaystyle\delta_{\epsilon}A_{M}$ $\displaystyle=\epsilon\Gamma_{M}\Psi$ (4.170) $\displaystyle\delta_{\epsilon}\Psi$ $\displaystyle=\frac{1}{2}\Gamma^{MN}F_{MN}\epsilon+\frac{1}{2}\Gamma^{\mu A}\Psi_{A}\nabla_{\mu}\epsilon+\sum_{i=1}^{7}K_{i}\nu_{i}$ $\displaystyle\delta_{\epsilon}K_{i}$ $\displaystyle=-\nu_{i}\Gamma^{M}D_{M}\Psi.$ Here $\epsilon$ is a fixed conformal Killing spinor, and $(\nu_{i})$ are seven spinors satisfying $\displaystyle\epsilon\Gamma^{M}\nu_{i}=0$ (4.171) $\displaystyle(\epsilon\Gamma_{M}\epsilon)\tilde{\Gamma}^{M}_{ab}=2\left(\sum_{i}(\nu_{i})_{a}(\nu_{i})_{b}+\epsilon_{a}\epsilon_{b}\right)$ $\displaystyle\nu_{i}\Gamma^{M}\nu_{j}=\delta_{ij}\epsilon\Gamma^{M}\epsilon.$ To ensure convergence of the path integral, as we did with the scalar field $\Phi_{0}$, we path integrate the new auxiliary scalars on purely immaginary values, i.e. $K_{j}=:iK_{j}^{E}$ with $K_{j}^{E}$ real. For every fixed non- zero $\epsilon$, there exist seven linearly independent $\nu_{i}$ satisfying these constraints, up to an $SO(7)$ internal rotation, ensuring the closure (4.117) off-shell. Although, if we want $\delta_{\epsilon}$ to describe the equivariant cohomology of a subgroup of the Poincaré group (not the conformal one), we should restrict to those $\epsilon$ that generates only translations and R-symmetries at most (up to unphysical gauge transformations). Thus the dilatation term in (4.117) must vanish, imposing the condition $(\epsilon\tilde{\epsilon})=0$ on the conformal Killing spinors. We describe now which modifications to the above discussion have to be made in order to describe the $\mathcal{N}=2$ and $\mathcal{N}=2^{}$ theories. The pure $\mathcal{N}=2$ is classically obtained by restricting to the $\mathcal{N}=2$ supersymmetry algebra generated by (4.118) and putting all the fields of the hypermultiplet to zero. At quantum level, this theory breaks in general the conformal invariance, so it is equivalent to consider the $\mathcal{N}=2^{}$ with hypermultiplet masses introduced as at the end of Section 4.3.7, by $D_{0}\mapsto D_{0}+M$ where $M$ is an $SU(2)_{R}^{\mathcal{R}}$ mass matrix. The mass terms for the fermions break the $SO(1,1)^{\mathcal{R}}$ R-symmetry, so we must restrict the superconformal algebra further to those $\epsilon$ for which the corresponding piece of the R-symmetry in (4.117) vanish. This imposes $(\tilde{\epsilon}\Gamma^{09}\epsilon)=0$. Moreover, this deformed theory is not invariant under the $\mathcal{N}=2$ supersymmetry, because of the non- triviality of the conformal Killing spinor. In fact, using the conformal Killing equations it results that $\delta_{\epsilon}\left(\frac{1}{2}F_{MN}F^{MN}-\Psi\Gamma^{M}D_{M}\Psi+\frac{2}{r^{2}}\Phi_{A}\Phi^{A}\right)=-4\Psi\Gamma^{i}\tilde{\Gamma}^{0}\tilde{\epsilon}M_{i}^{j}\Phi_{j}$ (4.172) where $i,j=5,\cdots,8$, up to a total derivative. If $\epsilon,\tilde{\epsilon}$ are restricted to the the $+1$ eigenspace of $\Gamma^{5678}$, we write $\tilde{\epsilon}=\frac{1}{2r}\Lambda\epsilon$, where $\Lambda$ is a generator of $SU(2)_{L}^{\mathcal{R}}$. Explicitly $\Lambda=\frac{1}{4}\Gamma^{ij}R_{ij}$, with components $(R_{ij})$ normalized such that $R_{ij}R^{ij}=4$. Then, after some gamma matrix technology, (4.172) gives $\delta_{\epsilon}(\cdots)=\frac{1}{2r}(\Psi\Gamma_{i}\epsilon)R^{ik}M_{k}^{j}\Phi_{j}=\frac{1}{2r}(\delta_{\epsilon}\Phi_{i})R^{ik}M_{k}^{j}\Phi_{j}.$ (4.173) Hence, we can modify further the mass-deformed action to get invariance with respect to this subalgebra of the original superconformal algebra on $\mathbb{S}^{4}$, adding the new term $-\frac{1}{4r}(R^{ki}M_{k}^{j})\Phi_{i}\Phi_{j}.$ (4.174) Finally, the action $\displaystyle S^{\mathcal{N}=2^{}}_{\mathbb{S}^{4}}=\int_{\mathbb{S}^{4}}d^{4}x\sqrt{g}\ \frac{1}{g_{YM}^{2}}\mathrm{Tr}\Biggl{(}F_{MN}F^{MN}-\Psi\Gamma^{M}D_{M}\Psi$ $\displaystyle+\frac{2}{r^{2}}\Phi_{A}\Phi^{A}-$ (4.175) $\displaystyle-\frac{1}{4r}(R^{ki}M_{k}^{j})\Phi_{i}\Phi_{j}-\sum_{i=1}^{7}K_{i}K_{i}\Biggr{)}$ where $D_{0}\Phi_{i}\mapsto[\Phi_{0},\Phi_{i}]+M_{i}^{j}\Phi_{j}$ and $D_{0}\Psi\mapsto[\Phi_{0},\Psi]+\frac{1}{4}M_{ij}\Gamma^{ij}\Psi$, is invariant under the subalgebra generated by a fixed conformal Killing spinor satisfying the conditions $\Gamma^{5678}\epsilon=\epsilon,\qquad\nabla_{\mu}\epsilon=\frac{1}{8r}\Gamma_{\mu}\Gamma^{ij}R_{ij}\epsilon.$ (4.176) #### 4.4.5 Trial and error method Another method that was extensively used in the physics literature to promote a supersymmetric theory on curved spaces is based on a trial and error procedure [67]. Suppose to have a supersymmetric QFT formulated in terms of component fields on flat Minkowski (or Euclidean) space $\mathbb{R}^{d}$, specified by the Lagrangian density $\mathcal{L}^{(0)}$ invariant under the supersymmetry variation $\delta^{(0)}$. The starting point of this approach is to simply “covariantize” the original theory, replacing the flat metric $\eta$ to the desired metric $g$ defined on $\mathcal{M}$ and every derivative $\partial_{\mu}$ with the appropriate Levi-Civita or spin covariant derivative $\nabla_{\mu}$ corresponding to $g$. The problem is that in general this does define the theory on the curved space, but it is not guaranteed that the supersymmetry survives: $\left[\delta^{(0)}\mathcal{L}^{(0)}\right]_{(\eta,d)\to(g,\nabla)}\neq\nabla_{\mu}(\cdots)^{\mu}.$ (4.177) The idea then is to correct the action of the supersymmetry and the Lagrangian with an expansion in powers of $1/r$, where $r$ is a characteristic length of $\mathcal{M}$,383838Being $\mathcal{M}$ compact, we can take it as an embedding in $\mathbb{R}^{n}$ for some $n$, and scale the metric according to some characteristic length $r$. $\displaystyle\delta$ $\displaystyle=\delta^{(0)}+\sum_{i\geq 1}\frac{1}{r^{i}}\delta^{(i)}$ (4.178) $\displaystyle\mathcal{L}$ $\displaystyle=\mathcal{L}^{(0)}+\sum_{i\geq 1}\frac{1}{r^{i}}\mathcal{L}^{(i)}$ requiring order by order the symmetry of the Lagrangian _and_ the closure of the super-algebra. This “trial and error” terminates if one is able to ensure both conditions at some finite order in $1/r$, even though a priori the series contains an infinite number of terms. This procedure has the quality to be simple and operational in principle, but can be very cumbersome in practice to apply. ### 4.5 BRST cohomology and equivariant cohomology In gauge theories, BRST cohomology is a useful device to provide an algebraic description of the path integral quantization procedure, and the renormalizability of non-Abelian Yang-Mills theory in 4 dimensions. This formalism makes use of Lie algebra cohomology, while the BRST model of Section 2.5 describes equivariant cohomology, that is what we use in topological or supersymmetric field theories. It is natural to ask whether there is a relation between these two cohomology theories, and in fact there is. It turns out that equivariant cohomology of a Lie algebra $\mathfrak{g}$ is the same as a “supersymmetrized” Lie algebra cohomology of a corresponding graded Lie algebra $\mathfrak{g[\epsilon]}:=\mathfrak{g}\otimes\bigwedge\epsilon$ [20]. Let us first see how the Weil model $W(\mathfrak{g})=S(\mathfrak{g}^{})\otimes\bigwedge(\mathfrak{g}^{})$ (4.179) for the equivariant cohomology of $\mathfrak{g}$ can be seen in more supergeometric terms. Notice that the space $S(\mathfrak{g}^{})$ may be identified with the (commutative) algebra of functions on the Lie algebra $\mathfrak{g}$, and thus us we can see the Weil algebra $W(\mathfrak{g}^{})$ as the space of functions on a supermanifold built from the tangent bundle of $\mathfrak{g}$, that is exactly the odd tangent bundle $\Pi T\mathfrak{g}\equiv\Pi\mathfrak{g}$.393939Notice that since $\mathfrak{g}$ is a vector space, $T^{}\mathfrak{g}\cong\mathfrak{g}^{}$. Denoting $\\{\tilde{c}^{i}\\}$ and $\\{c^{i}\\}$ the generators of $S(\mathfrak{g}^{})$ and $\bigwedge(\mathfrak{g}^{})$ respectively, indeed a function on this superspace is trivialized as $\Phi=\Phi^{(0)}(\tilde{c})+\Phi_{j}^{(1)}(\tilde{c})c^{j}+\Phi_{jk}^{(2)}(\tilde{c})c^{j}c^{k}+\cdots$ (4.180) Introducing generators $\\{\tilde{b}_{i}\\}$ and $\\{b_{i}\\}$ of $\mathfrak{g}[1]$ and $\mathfrak{g}$, such that404040Sometimes the action of this generators is denoted as a graded bracket structure, like $[b_{i},c^{j}]=[\tilde{b}_{i},\tilde{c}^{j}]_{+}=\delta^{j}_{i}$. $b_{i}(c^{j}):=c^{j}(b_{i})=\delta^{j}_{i},\qquad\tilde{b}_{i}(\tilde{c}^{j}):=c^{j}(b_{i})=\delta^{j}_{i},$ (4.181) the Weil differential (2.29) can be written as $d_{W}=\tilde{c}^{i}b_{i}+f^{i}_{jk}c^{j}\tilde{c}^{k}b_{i}-\frac{1}{2}f^{i}_{jk}c^{j}c^{k}\tilde{b}_{i},$ (4.182) that is very reminiscent of the form of a “BRST operator”. In Lie algebra cohomology, the Chevalley-Eilenberg differential on $\bigwedge(\mathfrak{g}^{})$ is defined on 1-forms $\alpha\in\mathfrak{g}^{}$ as $\delta\alpha=-\alpha\left([\cdot,\cdot]\right)=\alpha_{i}\left(-\frac{1}{2}f^{i}_{jk}c^{j}c^{k}\right),$ (4.183) and then extended as an antiderivation on the whole complex. If we consider a $\mathfrak{g}$-module $V$, such as the target space of a given field theory of gauge group $G$, with a representation $\rho:\mathfrak{g}\to\mathrm{End}(V)$,414141If $V$ is a field space $C^{\infty}(\mathcal{M})$ over some (super)manifold $\mathcal{M}$, $\mathfrak{g}$ acts as usual as a Lie derivative with respect to the fundamental vector field, $\rho(X)=\mathcal{L}_{X}$. then the CE differential is extended to $\bigwedge(\mathfrak{g}^{})\otimes V$ as $\displaystyle\delta v(X):=\rho(X)v\qquad\forall v\in V,X\in\mathfrak{g}$ (4.184) $\displaystyle\delta(\alpha\otimes v)=\delta\alpha\otimes v+(-1)^{k}\alpha\otimes\delta v\qquad\forall v\in V,\alpha\in\bigwedge\nolimits^{\\!k}(\mathfrak{g}^{}).$ This, expressed with respect to a basis $\\{b_{i}\\}$ of $\mathfrak{g}$, coincide with the BRST operator $\delta=c^{i}\rho(b_{i})-\frac{1}{2}f^{i}_{jk}c^{j}c^{k}b_{i}$ (4.185) that satisfies $\delta^{2}=0$. The $c^{i}$ are _ghosts_ , while the $b_{i}$ are _anti-ghosts_. The zero-th cohomology group of the complex $\bigwedge(\mathfrak{g}^{})\otimes V$ with respect to the differential (4.185) contains those states that have _ghost number_ 0 and are $\mathfrak{g}$-invariant, $H^{0}(\mathfrak{g},V)\cong V^{\mathfrak{g}}$ (4.186) so the interesting “physical” states. We see that there is a difference between the BRST operator (4.185) and the Weil differential (4.182), but we can connect these differentials as follows. To the Lie algebra $\mathfrak{g}$ we can associate the _differential graded Lie algebra_ $\mathfrak{g}[\epsilon]:=\mathfrak{g}\otimes\bigwedge\epsilon$. Here $\epsilon$ is a single generator taken in odd degree, $\mathrm{deg}(\epsilon):=-1$, while $\mathrm{deg}(\mathfrak{g}):=0$. A differential $\partial:\mathfrak{g}[\epsilon]\to\mathfrak{g}[\epsilon]$ is defined as $\partial\epsilon:=1\in\mathfrak{g},\qquad\qquad\partial X:=0\quad\forall X\in\mathfrak{g}.$ (4.187) This superalgebra has generators $b_{i}:=b_{i}\otimes 1$ and $\tilde{b}_{i}:=b_{i}\otimes\epsilon$, and the superbracket structure coming from the Lie brackets on $\mathfrak{g}$ and the (trivial) wedge product on $\bigwedge\epsilon$: $\displaystyle[b_{i},b_{j}]=f^{k}_{ij}b_{k}$ (4.188) $\displaystyle[b_{i},\tilde{b}_{j}]=f^{k}_{ij}\tilde{b}_{k}$ $\displaystyle[\tilde{b}_{i},\tilde{b}_{j}]=0$ making it into a Lie superalgebra. The differential on the generators is rewritten as $\partial b_{i}=0,\qquad\quad\partial\tilde{b}_{i}=b_{i}.$ (4.189) To this “supersymmetrized” algebra we can associate the Lie algebra cohomology with respect to the complex $\bigwedge(\mathfrak{g}[\epsilon]^{})$, that is generated by $\\{c^{i},\tilde{c}^{i}\\}$ of degrees $\mathrm{deg}(c^{i})=1,\mathrm{deg}(\tilde{c}^{i})=2$, such that $c^{i}(b_{j})=\tilde{c}^{i}(\tilde{b}_{j})=\delta^{i}_{j}.$ (4.190) The BRST differential for the $\mathfrak{g}[\epsilon]$-Lie algebra cohomology is naturally defined analogously to before as $Q\alpha:=-\alpha\left([\cdot,\cdot]\right)$ (4.191) on 1-forms $\alpha\in\mathfrak{g}[\epsilon]^{}$. But now, because of the Lie algebra extension (4.188), its expression in terms of the generators results $Q=-\frac{1}{2}f^{k}_{ij}c^{i}c^{j}b_{k}+f^{k}_{ij}c^{i}\tilde{c}^{j}\tilde{b}_{k}.$ (4.192) Moreover, the dual $\partial^{}$ acts as $\partial^{}=\tilde{c}^{i}b_{i}$ (4.193) with $b_{i}$ acting as in (4.181). Then the total differential on this complex coincides with the Weil differential, $d_{W}\cong\partial^{}+Q$ (4.194) and we identify an isomorphism of dg algebras $\left(\bigwedge(\mathfrak{g}[\epsilon]^{}),\partial^{}+Q\right)\cong\left(W(\mathfrak{g}),d_{W}\right).$ (4.195) If we bring into the game the $\mathfrak{g}$-module $V$ as before, we can work with $\Omega(V)$ as a $\mathfrak{g}[\epsilon]$-dg algebra, defining the $\mathfrak{g}[\epsilon]$ action as424242Again, if $V=C^{\infty}(\mathcal{M})$, then $\Omega(\mathcal{M})$ is the space we considered when we constructed the equivariant cohomology of a $G$-manifold. $(X\otimes 1)\to\mathcal{L}_{X},\qquad(X\otimes\epsilon)\to\iota_{X}.$ (4.196) On the complex $\bigwedge(\mathfrak{g}[\epsilon]^{})\otimes\Omega(V)$, the total differential inherited from the Weil differential and the $\mathfrak{g}[\epsilon]$-action is $d_{B}:=c^{k}\otimes\mathcal{L}_{k}+\tilde{c}^{k}\otimes\iota_{k}+Q\otimes 1+\partial^{}\otimes 1+1\otimes d$ (4.197) and it coincides with the one of the BRST model of equivariant cohomology (2.51)! This demonstrates how the BRST quantization formalism and equivariant cohomology are intimately related, and suggests that indeed BRST symmetry operators are good candidates to represent equivariant differentials in QFT, that can be used to employ the localization principle in these kind of physical systems. ## Chapter 5 Localization for circle actions in supersymmetric QFT In this chapter we describe how the equivariant localization principle can be carried out in the infinite dimensional case of path integrals in QM or QFT. In this setting, the first object of interest is the _partition function_ $Z=\int_{\mathcal{F}}D\phi\ e^{iS[\phi]}$ (5.1) where $\mathcal{F}=\Gamma(M,E)$ is the space of fields, i.e. sections of some fiber bundle $E\to M$ with typical fiber (the _target space_) $V$ over the (Lorentzian) $n$-dimensional spacetime $M$, and $S\in C^{\infty}(\mathcal{F})$ is the action functional.111In the (common) case of a trivial bundle, this is equivalent to considering $\mathcal{F}=C^{\infty}(M,V)$, i.e. $V$-valued functions over $M$. Often $V$ is a vector space, otherwise the theory describes a so-called _non-linear $\sigma$-model_. In supersymmetric field theories, $V=\bigwedge(S^{})$ for some vector space $S$, and the field space acquires a natural graded structure. The fields are supposed to satisfy some prescribed boundary conditions on $\partial M$. In the Riemannian case, the corresponding object has the form $Z=\int_{\mathcal{F}}D\phi\ e^{-S_{E}[\phi]}$ (5.2) where we denoted $S_{E}$ the Euclidean action. If the spacetime has the form $M=\mathbb{R}_{t}\times\Sigma$, this last expression can be reached from the Lorentzian theory via _Wick rotation_ of the time direction $t\mapsto\tau:=it$. If the $\tau$ direction is compactified to a circle of length $T$, we can interpret the Euclidean path integral as the canonical ensemble partition function describing the original QFT at a finite temperature $1/T$. If needed, we are always free to set the length $T$ of the circle to be very large, and find the zero temperature limit when $T\to\infty$. Given an observable $\mathcal{O}\in C^{\infty}(M)$, its _expectation value_ is given by $\left\langle\mathcal{O}\right\rangle=\frac{1}{Z}\int_{\mathcal{F}}D\phi\ \mathcal{O}[\phi]e^{iS[\phi]}\quad\mathrm{or}\quad\left\langle\mathcal{O}\right\rangle_{E}=\frac{1}{Z}\int_{\mathcal{F}}D\phi\ \mathcal{O}[\phi]e^{-S_{E}[\phi]}.$ (5.3) The path integral measure $D\phi$ on the infinite dimensional space $\mathcal{F}$ is not rigorously defined,222In fact, it does not exist in general. but it is usually introduced as $\int D\phi=\mathcal{N}\prod_{x\in M}\int_{V}d\phi(x)$ (5.4) where $\mathcal{N}$ is some (possibly infinite) multiplicative factor, and at every point $x\in M$ we have a standard integral over the fiber $V$. Notice that the infinite factors $\mathcal{N}$ cancel in ratios in the computations of expectation values, so we can still make sense of such objects and formally manipulate them to obtain physical information. Another convergence issue comes with the prescription of boundary conditions in computing the action $S[\phi]$. If $M$ is non compact, this often requires a specific regularization,333For example, one can first assume that the spacetime just extends up to some large but finite typical lenght $r_{0}$, and then send this value to infinity at the end of the calculations. while taking compact spacetimes ensure better convergence properties. Very few quantum systems have an exactly solvable path integral. When this functional integral method was introduced, the only examples where (5.1) could be directly evaluated were the free particle and the harmonic oscillator.444Later on, ad hoc methods for other particular systems were developed, like the solution of the Hydrogen atom by Duru and Kleinert [68], and others. Both these theories are quadratic in the fields and their derivatives, thus the partition function can be computed using the formal functional analog of the classical Gaussian integration formula $\int_{-\infty}^{\infty}d^{n}x\ e^{-\frac{1}{2}x^{k}M_{kl}x^{l}+A_{k}x^{k}}=\frac{(2\pi)^{n/2}}{\sqrt{\det{M}}}e^{\frac{1}{2}A_{i}(M^{-1})^{ij}A_{j}}$ (5.5) where $M=[M_{ij}]$ is an $n\times n$ non singular matrix. The analogue in field theory has $n\to\infty$ and a functional determinant at the denominator, that must be properly regularized in order to be a meaningful convergent quantity (see any standard QFT book, like [69, 70]). In perturbative QFT, one almost never has to explicitly perform such a functional integration. Suppose that the action has the generic form $S=S_{0}+S_{int}$ (5.6) where $S_{0}$ is the free term containing up to quadratic powers of the fields and their derivatives, and the rest is collected in $S_{int}$. Then the expectation value of an observable $\mathcal{O}$, expressible as a combination of local fields, is computed expanding the exponential of the interacting part in Taylor series, and exploiting _Wick’s theorem_ 555Again, see any standard QFT book. for the vacuum expectation values in the free theory: $\left\langle\mathcal{O}\right\rangle=\sum_{k}\frac{1}{k!}\left\langle(iS_{int})^{k}\mathcal{O}\right\rangle_{0}.$ (5.7) Another perturbative approach, especially useful to compute the effective action in a given theory, is the so-called _background field method_ , where the action $S$ is expanded around a classical “background”element $\overline{\phi}\in\mathcal{F}$, $S[\overline{\phi}+\eta]=S[\overline{\phi}]+\int_{M}d^{n}x\left(\frac{\delta S}{\delta\phi(x)}\right)_{\overline{\phi}}\eta(x)+\frac{1}{2}\int_{M}d^{n}xd^{n}y\left(\frac{\delta^{(2)}S}{\delta\phi(x)\delta\phi(y)}\right)_{\overline{\phi}}\eta(x)\eta(y)+\cdots$ (5.8) If we chose $\overline{\phi}$ to be a solution of the classical equation of motion $\left(\frac{\delta S}{\delta\phi(x)}\right)_{\overline{\phi}}=0$, the first order term disappears from the expansion. If we also neglect the terms of order higher than quadratic in $\eta$, and substitute the resulting expression in (5.1) or (5.2), we get the equivalent of the saddle point approximation, or “one-loop approximation” of the partition function $Z\approx e^{-S[\overline{\phi}]}Z_{1-loop}[\overline{\phi}],$ (5.9) where $\displaystyle Z_{1-loop}[\phi]$ $\displaystyle:=\int_{\mathcal{F}}D\eta\ e^{-\frac{1}{2}\eta\cdot\Delta[\phi]\cdot\eta}\equiv\left[\det{\left(\Delta[\phi]\right)}\right]^{-1/2}$ (5.10) $\displaystyle\Delta[\phi](x,y)$ $\displaystyle:=\left(\frac{\delta^{(2)}S}{\delta\phi(x)\delta\phi(y)}\right)_{\phi},$ and we denoted convolution products over $M$ with $(\cdot)$ for brevity. We are interested in those cases in which such a “semiclassical” approximation of the partition function turns out to give the exact result for the path integral in the full quantum theory. This is possible if some symmetry of the field theory, i.e. acting on the space $\mathcal{F}$, allows us to formally employ the equivariant localization principle and reduce the path integration domain from $\mathcal{F}$ to a (possibly finite-dimensional) subspace. In the next part of the chapter we will see some cases in which it is possible to interpret $\mathcal{F}$, or a suitable extension of it, as a Cartan model with some (super)symmetry operator acting as the Cartan differential. As we already anticipated, this is possible if $\mathcal{F}$ has a graded structure that can both arise from the supersymmetry of the underlying spacetime, or can be introduced via an extension analogous to what happens in the BRST formalism. We will first describe the application of the Duistermaat-Heckman theorem in the case of Hamiltonian QM, where the equivariant structure can be constructed from the symplectic structure of the underlying theory. Then we will be concerned with more general applications of the localization principle in supersymmetric QFT, where the super-Poincaré group action allows for an equivariant cohomological interpretation. In both frameworks, we present examples of localization under the action of a single supersymmetry, whose “square” generates a bosonic $U(1)$ symmetry. Supersymmetric localization was recently applied to many cases of QFT on curved spacetimes, so we must consider those curved background that preserve at least one supersymmetry, as discussed in Section 4.4. ### 5.1 Localization principle in Hamiltonian QM We consider now, following [19] and refernces therein, the path integral quantization of an Hamiltonian system $(M,\omega,H)$, of the $2n$-dimensional phase space $M$ with symplectic form $\omega$, and an Hamiltonian function $H\in C^{\infty}(M)$. This is simply QM viewed as a (0+1)-dimensional QFT over the base “spacetime” $\mathbb{R}$ or $\mathbb{S}^{1}$, that now is only “time”, and with target space $M$, that physically represents the phase space of the system. The fields of the theory are the paths $\gamma:\mathbb{R}(\mathbb{S}^{1})\to M$, that means we consider a trivial total space $E=\mathbb{R}(\mathbb{S}^{1})\times M$. In principle the time axis can be chosen to be the real line (or an interval with some prescribed boundary conditions), or the circle (that corresponds to periodic boundary conditions), but we will soon see that it is much convenient technically to chose the latter possibility, so consider the “loop space” $\mathcal{F}=C^{\infty}(\mathbb{S}^{1},M)$. A field for us is so a closed curve $\gamma:[0,T]\to M$ such that $\gamma(0)=\gamma(T)$. Since we make the periodicity explicit in $t$, we interpret this parameter as an “Euclidean” Wick-rotated time, so that $T$ is the inverse temperature of the canonical ensemble. We set up now some differential geometric concept on the loop space that we are going to use in the following. If $\\{x^{\mu}\\}$ are coordinates on $M$, on the loop space we can choose an infinite set of coordinates $\\{x^{\mu}(t)\\}$ for $\mu\in{1,\cdots,2n}$ and $t\in[0,T]$, such that for any $\gamma\in\mathcal{F}$ $x^{\mu}(t)[\gamma]:=x^{\mu}(\gamma(t)).$ Using the standard rules of functional derivation, a vector field in $\Gamma(T\mathcal{F})$ can be thus expressed locally with respect to these coordinates as $X=\int_{0}^{T}dt\ X^{\mu}(t)\frac{\delta}{\delta x^{\mu}(t)}$ (5.11) where $X^{\mu}(t)$ are functions over $\mathcal{F}$, and $(\delta/\delta x^{\mu}(t))_{\gamma}$ is a basis element of the tangent space $T_{\gamma}\mathcal{F}$ at $\gamma$. Many other geometric objects can be lifted from $M$ to $\mathcal{F}$ following this philosophy. For example, for any function in $C^{\infty}(M)$ as the Hamiltonian $H$, we can define $H(t)\in C^{\infty}(\mathcal{F})$ at a given time $t$, such that $H(t)[\gamma]:=H(\gamma(t))$. The action functional instead is the function over the loop space such that $\displaystyle S[\gamma]$ $\displaystyle=\int_{0}^{T}dt\ \left[\dot{q}^{a}(t)p_{a}(t)-H(p(t),q(t))\right]$ (5.12) $\displaystyle=\int_{0}^{T}dt\ \left[\theta_{\gamma(t)}(\dot{\gamma})-H(\gamma(t))\right]$ where in the first line we expressed $\gamma$ through its trivialization in Darboux coordinates $\\{q^{a},p_{a}\\}$ with $a\in\\{1,\cdots,n\\}$, and in the second line we expressed the same thing more covariantly using the (local) symplectic potential $\theta$ of $\omega$ and the velocity vector field $\dot{\gamma}$ of the curve. Concerning differential forms, if we consider the basis set $\\{\eta^{\mu}(t):=dx^{\mu}(t)\\}$, a $k$-degree element of $\Omega(\mathcal{F})$ can be expressed locally as $\alpha=\int_{0}^{T}dt_{1}\cdots\int_{0}^{T}dt_{k}\ \frac{1}{k!}\alpha_{\mu_{1}\cdots\mu_{k}}(t_{1},\cdots,t_{k})\eta^{\mu_{1}}(t_{1})\wedge\cdots\wedge\eta^{\mu_{k}}(t_{k})$ (5.13) and we recall that $\Omega(\mathcal{F})$ can be considered as the space of functions over the _super-loop space_ $\Pi T\mathcal{F}$, of coordinates $\\{x^{\mu}(t),\eta^{\mu}(t)\\}$. The de Rham differential on the loop space can be expressed as the cohomological vector field on $\Pi T\mathcal{F}$ $d_{\mathcal{F}}=\int_{0}^{T}dt\ \eta^{\mu}(t)\frac{\delta}{\delta x^{\mu}(t)}.$ (5.14) Finally, it is natural to lift on the loop space the symplectic structure of $M$, as well as a choice of Riemannian metric, as $\displaystyle\Omega$ $\displaystyle=\int_{0}^{T}dt\ \frac{1}{2}\omega_{\mu\nu}(t)\eta^{\mu}(t)\wedge\eta^{\nu}(t)$ (5.15) $\displaystyle G$ $\displaystyle=\int_{0}^{T}dt\ g_{\mu\nu}(t)\eta^{\mu}(t)\otimes\eta^{\nu}(t),$ i.e. $\Omega_{\mu\nu}(t,t^{\prime}):=\omega_{\mu\nu}(t)\delta(t-t^{\prime})$ and $G_{\mu\nu}:=g_{\mu\nu}(t)\delta(t-t^{\prime})$. $\Omega$ is closed under the loop space differential $d_{\mathcal{F}}$.666Strictly speaking, the 2-form $\Omega$ should be called “pre-symplectic”, since although it is certainly closed, it is not necessarily non-degenerate on the loop space. Considering the standard Liouville measure on $M$ $\frac{\omega^{n}}{n!}=d^{2n}x\ \mathrm{Pf}(\omega(x))=d^{n}qd^{n}p,$ (5.16) we can write now the path integral measure for QM on the loop space as an infinite product of the Liouville one for any time $t\in[0,T]$, and get $\int_{\mathcal{F}}\frac{\Omega^{n}}{n!}=\int_{\mathcal{F}}D^{2n}x\ \mathrm{Pf}(\Omega[x])=\int_{\Pi T\mathcal{F}}D^{2n}xD^{2n}\eta\ \frac{\Omega^{n}}{n!}.$ (5.17) Here in the last equality we rewrote the integral over $\mathcal{F}$ as an integral over the super-loop space, analogously to (4.16). The path integral for the quantum partition function is thus $\displaystyle Z(T)$ $\displaystyle=\int_{\mathcal{F}}D^{2n}x\ \mathrm{Pf}(\Omega[x])e^{-S[x]}$ (5.18) $\displaystyle=\int_{\Pi T\mathcal{F}}D^{2n}xD^{2n}\eta\ \frac{\Omega^{n}[x,\eta]}{n!}e^{-S[x]}$ $\displaystyle=\int_{\Pi T\mathcal{F}}D^{2n}xD^{2n}\eta\ e^{-(S[x]+\Omega[x,\eta])}$ where in the last line we exponentiated the loop space symplectic form, making explicit the formal analogy with the Duistermaat-Heckman setup. In particular, we associated to the “classical” Hamiltonian system $(M,\omega,H)$ an Hamiltonian system $(\mathcal{F},\Omega,S)$ on the loop space. Here the loop space Hamiltonian function $S$ generates an Hamiltonian $U(1)$-action on $\mathcal{F}$, that can be used to formally apply the same equivariant localization principle as in the finite-dimensional case. As for the proof of the ABBV formula, we introduced a graded structure on field space formally rewriting the path integration on the super-loop space $\Pi T\mathcal{F}$. This graded structure is now simply given by the form-degree on the extended field space $\Omega(\mathcal{F})$. Let now $X_{S}$ be the Hamiltonian vector field associated to $S\in C^{\infty}(\mathcal{F})$, such that $d_{\mathcal{F}}S=-\iota_{X_{S}}\Omega$, or equivalently $X_{S}=\Omega(\cdot,d_{\mathcal{F}}S)$. Explicitly, in coordinates $\\{x^{\mu}(t)\\}$ $\begin{split}X_{S}^{\mu}(t)&=\int_{0}^{T}dt^{\prime}\ \Omega^{\mu\nu}(t,t^{\prime})\frac{\delta S}{\delta x^{\nu}(t)}\\\ &=\omega^{\mu\nu}(x(t))\bigl{(}\omega_{\nu\rho}(x(t))\dot{x}^{\rho}(t)-\partial_{\nu}H(x(t))\bigr{)}\\\ &=\dot{x}^{\mu}(t)-X_{H}^{\mu}(x(t))\end{split}$ (5.19) where $\dot{x}(t)$ is the vector field on $\mathcal{F}$ with components such that $\dot{x}^{\mu}(t)[\gamma]:=(x^{\mu}\circ\gamma)^{\prime}(t)\equiv\dot{\gamma}^{\mu}(t)$. The flow of $X_{S}$ defines the Hamiltonian $U(1)$-action on $\mathcal{F}$ and the infinitesimal action of the Lie algebra $\mathfrak{u}(1)$ on $\Omega(\mathcal{F})$ through the Lie derivative $\mathcal{L}_{X_{S}}$. The Cartan model for the $U(1)$-equivariant cohomology of $\mathcal{F}$ is then defined by the space of equivariant differential forms $\Omega_{S}(\mathcal{F}):=\left(\mathbb{R}[\phi]\otimes\Omega(\mathcal{F})\right)^{U(1)}\cong\Omega(\mathcal{F})^{U(1)}[\phi]$ (5.20) and the equivariant differential $\displaystyle Q_{S}$ $\displaystyle:=\mathds{1}\otimes d_{\mathcal{F}}-\phi\otimes\iota_{X_{S}}\equiv d_{\mathcal{F}}+\iota_{X_{S}}$ (5.21) $\displaystyle=\int_{0}^{T}dt\bigl{(}\eta^{\mu}(t)+\dot{x}^{\mu}(t)-X^{\mu}_{H}(x(t))\bigr{)}\frac{\delta}{\delta x^{\mu}(t)},$ where as usual we localized algebraically setting $\phi=-1$ to ease the notation. The square of this operator gives, after some simplifications $Q_{S}^{2}=\int_{0}^{T}dt\left(\frac{d}{dt}-\left.\mathcal{L}_{X_{H}}\right\|_{x(t)}\right)$ (5.22) where the second term is the Lie derivative on $M$ with respect to $X_{H}$, lifted on $\mathcal{F}$ at every value of $t$. The first term, when evaluated on a field, gives only contributions from the values at $t=0,T$, and so it vanishes thanks to the fact that we chose periodic boundary conditions! This means that, with this choice, the Cartan model on field space is completely determined by the lift of the $U(1)$-invariant forms on $M$, for which $\mathcal{L}_{X_{H}}\alpha=0$. Consistently, if we restrict to this subspace of $\Omega(M)$ where the energy is preserved, $Q_{S}\equiv Q_{\dot{x}}=d_{\mathcal{F}}+\iota_{\dot{x}}$ acts as the _supersymmetry_ operator generating time-translations on the base $\mathbb{S}^{1}$: $Q_{\dot{x}}^{2}=\frac{1}{2}[Q_{\dot{x}},Q_{\dot{x}}]=\int_{0}^{T}dt\frac{d}{dt}$ (5.23) resembling the supersymmetry algebra (4.44) for $\mathcal{N}=1$ and 1-dimensional spacetime. We will see in the next section that this is not just a coincidence, but we can relate this model to a supersymmetric version of QM. This restricted differential acts on coordinates of the super-loop space as $Q_{\dot{x}}x^{\mu}(t)=\eta^{\mu}(t),\qquad Q_{\dot{x}}\eta^{\mu}(t)=\dot{x}^{\mu}(t),$ (5.24) while the full equivariant differential acts as $Q_{S}x^{\mu}(t)=\eta^{\mu}(t),\qquad Q_{S}\eta^{\mu}(t)=X_{S}^{\mu}(t),$ (5.25) both exchanging “bosonic” with “fermionic” degrees of freedom. We remark that we started from a standard (non supersymmetric) Hamiltonian theory on $\mathcal{F}$, and from this we constructed a supersymmetric theory on $\Pi T\mathcal{F}$, whose supersymmetry is encoded in the Hamiltonian symmetry (so in the symplectic structure) of the original theory. This “hidden” supersymmetry is thus interpretable, in the spirit of Topological Field Theory, as a BRST symmetry, and the differential $Q_{S}$ as a “BRST charge” under which the augmented action $(S+\Omega)$ is supersymmetric: $Q_{S}(S+\Omega)=d_{\mathcal{F}}S+d_{\mathcal{F}}\Omega+\iota_{X_{S}}\Omega=d_{\mathcal{F}}S+0-d_{\mathcal{F}}S=0.$ (5.26) In other words, $(S+\Omega)$ is an _equivariantly closed extension_ of the symplectic 2-form $\Omega$, analogously to the finite-dimensional Hamiltonian geometry discussed in Chapter 3.3. The same argument cannot be straightforwardly applied to any QFT, since in general we do not have a symplectic structure on the field space,777A symplectic structure can be induced from the action principle on the subspace of solutions of the classical EoM, but it does not lift on the whole field space in general. but in the presence of a gauge symmetry we know that a BRST supersymmetry can be used to define the equivariant cohomology on the field space and exploit the localization principle. We will expand this a bit in the next chapter. It is now possible to mimic the procedure of Section 4.2 to localize the supersymmetric path integral $Z(T)=\int_{\Pi T\mathcal{F}}D^{2n}xD^{2n}\eta\ e^{-(S[x]+\Omega[x,\eta])}$ (5.27) seen as an integral of an equivariantly closed form. We modify the integral introducing the “localizing action” $S_{loc}[x,\eta]:=Q_{S}\Psi[x,\eta]$, with localization 1-form $\Psi\in\Omega^{1}(\mathcal{F})^{U(1)}$, the so-called “gauge-fixing fermion”: $Z_{T}(\lambda)=\int_{\Pi T\mathcal{F}}D^{2n}xD^{2n}\eta\ e^{-(S[x]+\Omega[x,\eta]-\lambda Q_{S}\Psi[x,\eta])}$ (5.28) where $\lambda\in\mathbb{R}$ is a parameter. The resulting integrand is again explicitly equivariantly closed, and we can check that this path integral is formally independent on the parameter $\lambda$. Indeed, after a shift $\lambda\mapsto\lambda+\delta\lambda$, (5.28) becomes $Z_{T}(\lambda+\delta\lambda)=\int_{\Pi T\mathcal{F}}D^{2n}xD^{2n}\eta\ e^{-(S[x]+\Omega[x,\eta]-\lambda Q_{S}\Psi[x,\eta]-\delta\lambda Q_{S}\Psi[x,\eta])},$ (5.29) and we can make a change of integration variables to absorb the resulting shift at the exponential. Since the exponential is supersymmetric, we change variables using a supersymmetry transformation: $\begin{split}&x^{\mu}(t)\mapsto x^{\prime\mu}(t):=x^{\mu}(t)+\delta\lambda\Psi Q_{S}x^{\mu}(t)=x^{\mu}(t)+\delta\lambda\eta^{\mu}(t)\\\ &\eta^{\mu}(t)\mapsto\eta^{\prime\mu}(t):=\eta^{\mu}(t)+\delta\lambda\Psi Q_{S}\eta^{\mu}(t)=\eta^{\mu}(t)+\delta\lambda X_{S}^{\mu}(t)\end{split}$ (5.30) so that the only change in the integral comes from the integration measure, $\displaystyle D^{2n}xD^{2n}\eta\mapsto D^{2n}x^{\prime}D^{2n}\eta^{\prime}$ $\displaystyle=\mathrm{Sdet}\left[\begin{array}[]{cc}\partial x^{\prime}/\partial x&\partial x^{\prime}/\partial\eta\\\ \partial\eta^{\prime}/\partial x&\partial\eta^{\prime}/\partial\eta\end{array}\right]D^{2n}xD^{2n}\eta$ (5.31) $\displaystyle=e^{-\delta\lambda Q_{S}\Psi}D^{2n}xD^{2n}\eta.$ Putting all together, $\displaystyle Z_{T}(\lambda+\delta\lambda)$ $\displaystyle=\int_{\Pi T\mathcal{F}}D^{2n}x^{\prime}D^{2n}\eta^{\prime}\ e^{-(S[x^{\prime}]+\Omega[x^{\prime},\eta^{\prime}]-\lambda Q_{S}\Psi[x^{\prime},\eta^{\prime}]-\delta\lambda Q_{S}\Psi[x^{\prime},\eta^{\prime}])}$ (5.32) $\displaystyle=\int_{\Pi T\mathcal{F}}D^{2n}xD^{2n}\eta\ e^{-(S[x]+\Omega[x,\eta]-\lambda Q_{S}\Psi[x,\eta])}=Z_{T}(\lambda).$ The same property can be seen less rigorously by exploiting some sort of (arguable) infinite-dimensional version of Stokes’ theorem: $\displaystyle\frac{d}{d\lambda}Z_{T}(\lambda)$ $\displaystyle=\int_{\Pi T\mathcal{F}}D^{2n}xD^{2n}\eta\ (Q_{S}\Psi)e^{-(S[x]+\Omega[x,\eta]-\lambda Q_{S}\Psi[x,\eta])}$ (5.33) $\displaystyle=\int_{\Pi T\mathcal{F}}D^{2n}xD^{2n}\eta\left(Q_{S}\Psi e^{-(S[x]+\Omega[x,\eta]-\lambda Q_{S}\Psi[x,\eta])}\right)=0,$ that holds if we assume the path integration measure to be non-anomalous under $Q_{S}$. Since the path integral (5.28) is independent on the parameter, we can take the limit $\lambda\to\infty$ and obtain the localization formula $Z(T)=\lim_{\lambda\to\infty}\int_{\Pi T\mathcal{F}}D^{2n}xD^{2n}\eta\ e^{-(S[x]+\Omega[x,\eta]-\lambda Q_{S}\Psi[x,\eta])}$ (5.34) that “localizes” $Z(T)$ onto the zero locus of $S_{loc}$. Of course different choices of gauge-fixing fermion induce different final localization formulas for the path integral, but at the end they should all give the same result. We now present two different localization formulas derived from (5.34) with different choices of localizing term. The fist canonical choice of gauge fixing fermion we can make mimics the same procedure we used in the finite-dimensional case. Under the same assumptions we made in Section 3.1, we pick a $U(1)_{H}$-invariant metric $g$ on $M$, and lift it to $\mathcal{F}$ using (5.15), so that the resulting $G$ is $U(1)_{S}$-invariant: $\mathcal{L}_{S}G=Q_{S}^{2}G=0$. Then the localization 1-form is taken to be $\Psi[x,\eta]:=G(X_{S},\cdot)=\int_{0}^{T}dt\ g_{\mu\nu}(x(t))\Bigl{(}\dot{x}^{\mu}(t)-X^{\mu}_{H}(x(t))\Bigr{)}\eta^{\nu}(t),$ (5.35) so that the localization locus is the subspace where $X_{S}=0$, i.e. the moduli space of classical trajectories [41]: $\mathcal{F}_{S}=\left\\{\gamma\in\mathcal{F}:\left(\frac{\delta S}{\delta x^{\mu}(t)}\right)_{\gamma}=0\right\\}.$ (5.36) If this space consists of isolated, non-degenerate trajectories, we can apply the non-degenerate version of the ABBV formula for a circle action, and get $Z(T)=\sum_{\gamma\in\mathcal{F_{S}}}\frac{\mathrm{Pf}\left[\omega(\gamma(t))\right]}{\sqrt{\det{[dX_{S}[\gamma]/2\pi]}}}e^{-S[\gamma]}$ (5.37) where the pfaffian and the determinant are understood in the functional sense, spanning both the phase space indices $\mu\in\\{1,\cdots,2n\\}$ and the time continuous index $t\in[0,T]$. In general, for non isolated classical trajectories we have to decompose any $\gamma\in\mathcal{F}$ near to the fixed point set, splitting the classical component and normal fluctuations, as we did in Section 4.2. Then, rescaling the normal fluctuations by $\sqrt{\lambda}$ and thanks to the Berezin integration rules on the super-loop space, the same argument of the finite-dimensional case leads to $Z(T)=\int_{\mathcal{F_{S}}}D^{2n}x\ \frac{\mathrm{Pf}\left[\omega(x(t))\right]}{\left.\mathrm{Pf}\left[\delta^{\mu}_{\nu}\partial_{t}-(B+R)^{\mu}_{\nu}(x(t))\right]\right\|_{N\mathcal{F}_{S}}}e^{-S[x]}$ (5.38) where $B^{\mu}_{\nu}=g^{\mu\rho}(\nabla_{[\rho}X_{H})_{\nu]}$, while $\nabla$ and $R$ are the connection and curvature of the metric $g$ on $M$, evaluated on $\mathcal{F}$ time-wise as usual. Notice that this localization scheme makes the contribution from the classical configurations explicit, resembling the exactness of the saddle point approximation (5.9), with 1-loop determinant given by the pfaffian at the denominator. However, even if the integration domain has been reduced, one must still perform a difficult infinite- dimensional path integration to get the final answer, whose $T$-dependence for example looks definitely non-trivial from (5.38). We can consider another choice of localizing term to simplify the final result, setting the gauge-fixing fermion to $\Psi[x,\eta]:=G(\dot{x},\cdot)=\int_{0}^{T}dt\ g_{\mu\nu}(x(t))\dot{x}^{\mu}(t)\eta^{\nu}(t).$ (5.39) With this choice, the gauge-fixed action reads $\displaystyle S$ $\displaystyle[x]+\Omega[x,\eta]+\lambda Q_{S}\Psi[x,\eta]=$ (5.40) $\displaystyle=\int_{0}^{T}dt\biggl{(}\dot{x}^{\mu}\theta_{\mu}-H+\frac{1}{2}\omega_{\mu\nu}\eta^{\mu}\eta^{n}u+\lambda\left(g_{\mu\nu,\sigma}\dot{x}^{\mu}\eta^{\sigma}\eta^{\nu}+\eta^{\mu}\partial_{t}(g_{\mu\nu}\eta^{\nu})+g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}-g_{\mu\nu}\dot{x}^{\mu}X_{H}^{\nu}\right)\biggr{)}$ $\displaystyle=\int_{0}^{T}dt\biggl{(}\lambda g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}+\lambda\eta^{\mu}\nabla_{t}\eta^{\nu}+\dot{x}^{\mu}\theta_{\mu}+\frac{1}{2}\omega_{\mu\nu}\eta^{\mu}\eta^{\nu}-H-\lambda g_{\mu\nu}\dot{x}^{\mu}X_{H}^{\nu}\biggr{)}$ where in the second line the time-covariant derivative acts as $\nabla_{t}\eta^{\nu}=\partial_{t}\eta^{\nu}+\Gamma^{\nu}_{\rho\sigma}\dot{x}^{\rho}\eta^{\sigma}$, and the localization locus is the subset of constant loops $\mathcal{F}_{0}:=\left\\{\gamma\in\mathcal{F}:\dot{\gamma}=0\right\\}\cong M$ (5.41) that is, points in $M$. Splitting again the loops near this subspace in constant modes plus fluctuations, and rescaling the latter as we did before, the path integral is reduced to a finite-dimensional integral over $M$, the Niemi-Tirkkonen localization formula [71] $Z(T)=\int_{\Pi TM}\sqrt{g}d^{2n}xd^{2n}\eta\ \frac{e^{-T(H(x)-\omega(x,\eta))}}{\sqrt{\det^{\prime}{\left[g_{\mu\nu}\partial_{t}-(B_{\mu\nu}+R_{\mu\nu})\right]}}}$ (5.42) where the prime on the determinant means it is taken over the normal fluctuation modes, excluding the constant ones, giving exactly the equivariant Euler form of the normal bundle to $\mathcal{F}_{0}$. This formula is much more appealing since it contains no refernce to $T$-dependent submanifolds of $\mathcal{F}$, and the evaluation of the action on constant modes simply gives the Hamiltonian at those points multiplied by $T$. The functional determinant at the denominator requires a specific regularization: using the $\zeta$-function method, it can be simplified to $\frac{1}{\sqrt{\det^{\prime}{\left[g_{\mu\nu}\partial_{t}-(B_{\mu\nu}+R_{\mu\nu})\right]}}}=\sqrt{\det{\left[\frac{\frac{T}{2}(B+R)_{\mu\nu}}{\sinh{\left(\frac{T}{2}(B+R)_{\mu\nu}\right)}}\right]}}=\hat{A}_{H}(TR)$ (5.43) where in the last equality we rewrote, by definition, the determinant as the _$U(1)_{H}$ -equivariant Dirac $\hat{A}$-genus_ of the curvature $R$ up to a constant $T$, that is the Dirac $\hat{A}$-genus of the equivariant extension of the curvature, $R+B$ [72]. Note that the exponential can be rewritten as the $U(1)_{H}$-equivariant Chern character of the symplectic form, $e^{-H+\omega}=\mathrm{ch}_{H}(\omega)$ (5.44) and so the partition function can be nicely rewritten as $Z(T)=\int_{M}\mathrm{ch}_{H}(T\omega)\wedge\hat{A}_{H}(TR)$ (5.45) in terms of equivariant characteristic classes of the phase space $M$ with respect to the $U(1)_{H}$ Hamiltonian group action, that are determined by the initial classical system. The only remnant of the quantum theory is in the, now very explicit, dependence on the inverse temperature $T$. This form of the partition function emphasizes the fact that if we put $H=0$, we end up with a _topological_ theory. In this case there are no propagating physical degrees of freedom, and the partition function only describes topological properties of the underlying phase space. In the next section we will report a non- trivial example of this kind. ### 5.2 Localization and index theorems A famous and important application of the localization principle to loop space path integrals is the supersymmetric derivation of the _Atiyah-Singer index theorem_ [73] [74] [75]. The theorem relates the _analytical index_ of an elliptic differential operator on a compact manifold to a topological invariant, connecting the local data associated to solutions of partial differential equations to global properties of the manifold. Supersymmetric localization allowed to prove this statement, in a new way with respect to the original proof, for different examples of classical differential operators. We describe here the application to the index of the Dirac operator acting on the spinor bundle $S$ on an even-dimensional compact manifold $M$, in presence of a gravitational and electromagnetic background, specified by the metric $g$ and a $U(1)$ connection 1-form $A$ on a $\mathbb{C}$-line bundle $L_{\mathbb{C}}$ over $M$.888This can be extended to non-Abelian gauge groups as well, but for simplicity we report the Abelian case. The Dirac operator is defined as the fiber-wise Clifford action of the covariant derivative on the twisted spinor bundle $TM\otimes S\otimes L_{\mathbb{C}}$ over $M$: $i\not{\nabla}=i\gamma^{\mu}\left(\partial_{\mu}+\frac{1}{8}\omega_{\mu ij}[\gamma^{i},\gamma^{j}]+iA_{\mu}\right)$ (5.46) where $\\{\gamma^{\mu}\\}$ are the gamma-matrices generating the Clifford algebra in the given spin representation, satisfying the anticommutation relation $\\{\gamma^{\mu},\gamma^{\nu}\\}=2g^{\mu\nu},$ (5.47) and $\omega$ is the spin connection related to the metric.999See Appendix A.2. The “curved” and “flat” indices are related through the vielbein $e^{i}_{\mu}(x)$, $g_{\mu\nu}(x)=e^{i}_{\mu}(x)e^{j}_{\nu}(x)\eta_{ij},\qquad\gamma^{i}=e^{i}_{\mu}(x)\gamma^{\mu}(x),$ (5.48) with $\eta$ the flat metric. The analytical index of the Dirac operator is defined as [76] $\mbox{index}(i\not{\nabla}):=\dim\mathrm{Ker}(i\not{\nabla})-\dim\mathrm{coKer}(i\not{\nabla})=\dim\mathrm{Ker}(i\not{\nabla})-\dim\mathrm{Ker}(i\not{\nabla}^{\dagger}).$ (5.49) We have thus to find the number of zero-energy solutions of the Dirac equation $i\not{\nabla}\Psi=E\Psi,$ (5.50) where $\Psi$ is a Dirac spinor. In even dimensions, we can decompose the problem in the chiral basis of the spin representation, where $\gamma^{i}=\left(\begin{array}[]{cc}0&\sigma^{i}\\\ \overline{\sigma}^{i}&0\end{array}\right),\quad\gamma^{c}=\left(\begin{array}[]{cc}\mathds{1}&0\\\ 0&-\mathds{1}\end{array}\right),\quad i\not{\nabla}=\left(\begin{array}[]{cc}0&D\\\ D^{\dagger}&0\end{array}\right),\quad\Psi=\left(\begin{array}[]{c}\psi_{-}\\\ \psi_{+}\end{array}\right),$ (5.51) and the chirality matrix is denoted by $\gamma^{c}$. In this representation, we see that the index counts the number of zero-energy modes with positive chirality minus the number of zero-energy modes of negative chirality, $\mbox{index}(i\not{\nabla})=\dim\mathrm{Ker}(D)-\dim\mathrm{Ker}(D^{\dagger}).$ (5.52) It is possible to give a path integral representation of this index, a key ingredient to apply the localization principle. In order to do that, we first prove that it can be rewritten as a _Witten index_ , $\mbox{index}(i\not{\nabla})=\mathrm{Tr}_{\mathcal{H}}\left(\gamma^{c}e^{-T\Delta}\right)$ (5.53) where $\Delta:=(i\not{\nabla})^{2}$ is the Shröedinger operator (the covariant Laplacian) and the parameter $T>0$ is a regulator for the operator trace, taken over the space $\mathcal{H}$ of Dirac spinors, sections of the twisted spinor bundle over $M$. ###### Proof of (5.53). First, we notice that $(i\not{\nabla})$ is symmetric and elliptic, and since $M$ is compact it is also essentially self-adjoint [77]. Thus, it has a well defined spectrum $\\{\Psi^{E}\\}$ that forms a basis of the function space at hand. The same modes diagonalize also the Schrödinger operator, $\Delta\Psi^{E}=E^{2}\Psi^{E}$ (5.54) so we can shift the attention to solutions of the Schrödinger equation with eigenvalue satisfying $E^{2}=0$. It is easy to see that the Dirac operator anticommutes with the chirality matrix $\gamma^{c}$, and so the Schrödinger operator commutes with it. Thus we can split the field space in complementary subspaces $\mathcal{S}^{E}_{\pm}:=\left\\{\Psi\in\mathcal{H}:\Delta\Psi=E^{2}\Psi,\gamma^{c}\Psi=\pm\Psi\right\\}$, for every eigenvalue $E^{2}$ and chirality $(\pm)$. For every non-zero energy, we establish an isomorphism $\mathcal{S}^{E}_{+}\cong\mathcal{S}^{E}_{-}$: using the fact that $(i\not{\nabla})$ and $\gamma^{c}$ anticommute, starting from a solution with eigenvalue $E^{2}$ and definite chirality $\Psi_{\pm}$ we can construct another one with opposite chirality $(i\not{\nabla}\Psi_{\pm})$, $\displaystyle\Delta(i\not{\nabla}\Psi_{\pm})=i\not{\nabla}i\not{\nabla}i\not{\nabla}\Psi_{\pm}=E^{2}(i\not{\nabla}\Psi_{\pm})$ (5.55) $\displaystyle\gamma^{c}(i\not{\nabla}\Psi_{\pm})=-i\not{\nabla}\gamma^{c}\Psi_{\pm}=-(\pm)(i\not{\nabla}\Psi_{\pm}).$ Thus the maps $\phi_{\pm}:\mathcal{S}^{E}_{\pm}\to\mathcal{S}^{E}_{\mp}$ such that $\phi_{\pm}(\Psi):=\frac{i\not{\nabla}}{\|E\|}\Psi$ (5.56) are both well defined and are right and left inverse of each other, giving the bijective correspondence for every $\|E\|\neq 0$. This is the well known fact that particles and antiparticles are created in pairs with opposite energy and chirality. If we take the trace over the field space $\mathcal{H}$, this is splitted into the sum of the traces over every subspace $\mathcal{S}^{E}$ of definite energy squared. In every one of them the restricted Witten index gives $\mathrm{Tr}_{\mathcal{S}^{E}}\left(\gamma^{c}e^{-T\Delta}\right)=e^{-TE^{2}}\left(\mathrm{Tr}_{\mathcal{S}^{E}_{+}}(\mathds{1})-\mathrm{Tr}_{\mathcal{S}^{E}_{-}}(\mathds{1})\right)=0$ (5.57) for every $E\neq 0$. So the whole trace is formally independent on $T$, resolving on the subspace of zero-energy, where the number of chirality + and - eigenstates is different in general: $\displaystyle\mathrm{Tr}_{\mathcal{H}}\left(\gamma^{c}e^{-T\Delta}\right)$ $\displaystyle=\mathrm{Tr}_{E=0}(\gamma^{c})$ (5.58) $\displaystyle=\\#^{E=0}(\mathrm{chirality\ (+)\ modes})-\\#^{E=0}(\mathrm{chirality\ (-)\ modes}).$ ∎ The Witten index representation (5.53) and the chirality decomposition of the operators of interest (5.51) permit to see the current problem as a $\mathcal{N}=1$ supersymmetric QM on the manifold $M$, identifying chirality +(-) spinors with bosonic(fermionic) states. Here, the supersymmetry algebra (4.57) is simply101010Often this is called $\mathcal{N}=1/2$ supersymmetry, leaving the name $\mathcal{N}=1$ for the complexified algebra with generators $Q,\tilde{Q}$, and imposition of Majorana condition. (choosing an appropriate normalization) $[Q,Q]=2H,$ (5.59) and corresponds to the Schrödinger operator above, if we make the following identifications: $\displaystyle i\not{\nabla}$ $\displaystyle\leftrightarrow\quad Q$ (5.60) $\displaystyle\Delta=(i\not{\nabla})^{2}$ $\displaystyle\leftrightarrow\quad H=Q^{2}.$ The chirality matrix $\gamma^{c}$ is identified with the operator $(-1)^{F}$, where $F$ is the fermion number operator, that assigns eigenvalue $+1$ to bosonic states and $-1$ to fermionic states. The Witten index representation in the quantum system is thus $\displaystyle\mbox{index}(i\not{\nabla})$ $\displaystyle=\mathrm{Tr}\left((-1)^{F}e^{-TH}\right)$ (5.61) $\displaystyle=n^{E=0}(\mathrm{bosons})-n^{E=0}(\mathrm{fermions}).$ The proof above, translated in terms of the quantum system, shows that in supersymmetric QM eigenstates of the Hamiltonian have non-negative energy, and are present in fermion-boson pairs for every non-zero energy. Since $Q^{2}$ is a positive-definite Hermitian operator, the zero modes $\|0\rangle$ of $H$ are supersymmetric, $Q\|0\rangle=0$ (they do not have supersymmetric partners). Thus the non-vanishing of the Witten index (5.61) is a sufficient condition to ensure that there is at least one supersymmetric vacuum state available, whereas its vanishing is a necessary condition for spontaneous braking of supersymmetry by the vacuum. The Witten index has a super-loop space path integral representation [78], so that we can rewrite (5.61) as $\mbox{index}(i\not{\nabla})=\int D^{2n}\phi D^{2n}\psi\ e^{-TS[\phi,\psi]}$ (5.62) where $S$ is the Euclidean action corresponding to the Hamiltonian $H$ and the fields are defined on the unit circle. The appropriate supersymmetric theory which describes a spinning particle on a gravitational background is the 1-dimensional _supersymmetric non-linear $\sigma$ model_. The superspace formulation of this model considers the base space as a super extension of the 1-dimensional spacetime with coordinates $(t,\theta)$, and $M$ as the target space with covariant derivative given by the Dirac operator. A superfield, trivialized with respect to coordinates $(t,\theta)$ and $(x^{\mu})$ on $M$ is then $\Phi^{\mu}(t,\theta)\equiv(x^{\mu}\circ\Phi)(t,\theta)=\phi^{\mu}(t)+\psi^{\mu}(t)\theta,$ (5.63) and supersymmetry transformations are given by the action of the odd vector field $\underline{Q}=\partial_{\theta}+\theta\partial_{t}$, that in terms of component fields reads $\delta\phi^{\mu}=\psi^{\mu},\qquad\delta\psi^{\mu}=\dot{\phi}^{\mu}.$ (5.64) Denoting the superderivative as $D=-\partial_{\theta}+\theta\partial_{t}$, the action of the non-linear $\sigma$ model coupled to the gauge field $A$ can be given as $S[\Phi]=\int dt\int d\theta\ \frac{1}{2}g_{\Phi(t,\theta)}\left(D\Phi,\dot{\Phi}\right)+A(D\Phi)$ (5.65) where $D\Phi$ and $\dot{\Phi}$ are thought as (super)vector fields on $M$ such that, for any function $f\in C^{\infty}(M)$, $D\Phi(f):=D(f\circ\Phi)$ and $\dot{\Phi}(f):=\partial_{t}(f\circ\Phi)$. Inserting the trivialization for the metric components $g_{\mu\nu}(\Phi(t,\theta))=g_{\mu\nu}(\phi)+\theta\psi^{\sigma}(t)g_{\mu\nu,\sigma}(\phi)$, and the component expansion for $\Phi$, the action is simplified to $S[\phi,\psi]=\int dt\left(\frac{1}{2}g_{\mu\nu}(\phi)\dot{\phi}^{\mu}\dot{\phi}^{\nu}+\frac{1}{2}g_{\mu\nu}(\phi)\psi^{\mu}(\nabla_{t}\psi)^{\nu}+A_{\mu}(\phi)\dot{\phi}^{\mu}-\frac{1}{2}\psi^{\mu}F_{\mu\nu}\psi^{\nu}\right)$ (5.66) where we suppressed the $t$-dependence, $F_{\mu\nu}=\partial_{[\mu}A_{\nu]}$ are the components of the electromagnetic field strength, and the time- covariant derivative $\nabla_{t}$ acts as $(\nabla_{t}V)^{\sigma}(\phi(t))=\dot{V}^{\sigma}(\phi(t))+\Gamma^{\sigma}_{\mu\nu}\dot{\phi}^{\mu}(t)V^{\nu}(\phi(t)).$ The action (5.66) is formally equivalent to the model-independent action (5.40) of the last section, with $T$ behaving like the localization parameter $\lambda$, if we identify $\theta$ with the electromagnetic potential $A$ and $\omega$ with the field strength $F$, and if we set the Hamiltonian and its associated vector field $H,X_{H}$ to zero. This means that we can give to it an equivariant cohomological interpretation on the super-loop space $\Pi T\mathcal{F}$ over $M$, with coordinates identified with $(\phi^{\mu},\psi^{\mu})$, as was pointed out first by Atiyah and Witten [8]. Moreover, since the Hamiltonian vanishes, this action describes no propagating physical degrees of freedom, and thus the model is topological. Indeed, its value has to give the index of the Dirac operator, expected to be a topological quantity. To emphasize the equivariant cohomological nature of the model, we notice that the action functional can be split in the loop space (pre-)symplectic 2-form $\Omega[\phi,\psi]:=\int dt\ \frac{1}{2}\psi^{\mu}\left(g_{\mu\nu}\nabla_{t}-F_{\mu\nu}\right)\psi^{\nu}$ (5.67) and the loop space Hamiltonian $\mathcal{H}=\int dt\ \left(\frac{1}{2}g_{\mu\nu}\dot{\phi}^{\mu}\dot{\phi}^{\nu}+A_{\mu}\dot{\phi}^{\mu}\right).$ (5.68) They satisfy $d_{\mathcal{F}}\mathcal{H}=-\iota_{\dot{\phi}}\Omega$, so the supersymmetry transformation (5.64) is rewritten in terms of the Cartan differential $Q_{\dot{\phi}}=d_{\mathcal{F}}+\iota_{\dot{\phi}}$, and $Q_{\dot{\phi}}S=Q_{\dot{\phi}}(\mathcal{H}+\Omega)=0$. Moreover, we can find a loop space symplectic potential $\Sigma$ such that $S$ is equivariantly exact: $\displaystyle S[\phi,\psi]$ $\displaystyle=Q_{\dot{x}}\Sigma[\phi,\psi]$ (5.69) $\displaystyle\mathrm{where}\quad\Sigma[\phi,\psi]$ $\displaystyle:=\int dt\left(g_{\mu\nu}(\phi)\dot{\phi}^{\mu}+A_{\nu}(\phi)\right)\psi^{\nu}.$ Notice that, since the Hamiltonian $H$ vanishes, the localizing $U(1)$ symmetry is the one generated by time-translation with respect to the base space $\mathbb{S}^{1}$, an intrinsic property of the geometric structure that underlies the model. We can now apply the Niemi-Tirkkonen formula, localizing the path integral (5.62) into the moduli space of constant loops, i.e. as an integral over $M$. The result is the same as equation (5.45), but now since the Hamiltonian vanishes, the Chern class and the Dirac $\hat{A}$-genus (see Appendix B.1) are not equivariantly extended by the presence of an Hamiltonian vector field, giving the topological formula $\mbox{index}(i\not{\nabla})=\int_{M}\mbox{ch}(F)\wedge\hat{A}(R).$ (5.70) This is the result of the Atiyah-Singer index theorem for the Dirac operator on the twisted spinor bundle over $M$. Similar applications of the localization principle to variations of the non-linear $\sigma$ model give correct results for other classical complexes as well (de Rham, Dolbeault for example), in terms of different topological invariants [74]. ### 5.3 Equivariant structure of supersymmetric QFT and supersymmetric localization principle In the last section we saw how to give an equivariant cohomological interpretation to a model exhibiting Poincaré-supersymmetry, in terms of the super-loop space symplectic structure introduced before. In the following we are interested in applying the same kind of supersymmetric localization principle to higher dimensional QFT on a, possibly curved, $n$-dimensional spacetime $M$, where there is some preserved supersymmetry. In [9][10] it is argued that any generic quantum field theory with at least an $\mathcal{N}=1$ Poincaré supersymmetry admits a field space Hamiltonian (symplectic) structure and a corresponding $U(1)$-equivariant cohomology responsible for localization of the supersymmetric path integral. The key feature is an appropriate off-shell component field redefinition which defines a splitting of the fields into loop space “coordinates” and their associated “differentials”. In general, unlike the simplest case of the last section where bosonic fields were identified with coordinates and fermionic fields with 1-forms, loop space coordinates and 1-forms involve both bosonic and fermionic fields. It is proven that, within this field redefinition on the super-loop space, any supersymmetry charge $Q$ can be identified with a Cartan differential $Q=d_{\mathcal{F}}+\iota_{X_{+}}$ (5.71) analogously to (5.21), whose square generates translations in a given “light- cone” direction $x_{+}$, $Q^{2}=\mathcal{L}_{X_{+}}\sim\int_{M}\frac{\partial}{\partial x^{+}}$ (5.72) that corresponds to the $U(1)$ symmetry that can be used to exploit the localization principle. Taking the base spacetime to be compact in the light- cone direction, the periodic boundary conditions ensure $Q^{2}=0$, analogously to the loop space assumption of the one dimensional case. Also, it is argued that the supersymmetric action can be generally split into the sum of a loop space scalar function and a (pre-)symplectic 2-form, $S_{susy}=\mathcal{H}+\Omega$ (5.73) related by $d_{\mathcal{F}}\mathcal{H}=-\iota_{X_{+}}\Omega$. Thus, the supersymmetry of the action can be seen in general as the $U(1)$-equivariant closeness required to the application of the localization principle, and the path integral localizes onto the locus of constant loops, i.e. zero-modes of the fields. Even without entering in the details of this construction in terms of auxiliary fields redefinition, we feel now allowed to translate in full generality the circle localization principle in the framework of Poincaré- supersymmetric QFT. In the component-field description, we consider a rigid supersymmetric background over the given compact111111This ensures the loop space interpretation of above. spacetime $M$ and a graded field space $\mathcal{F}$ that plays the role of the super-loop space over $M$, whose even-degree forms are bosonic fields and the odd-degree forms are fermionic fields. The (infinitesimal) supersymmetry action of a preserved supercharge $Q$ plays the role of the Cartan differential $d_{\mathcal{F}}$, squaring to a bosonic symmetry that corresponds to the (infinitesimal) action of a $U(1)$ symmetry group, $Q^{2}\sim\mathcal{L}_{X}$ with $X$ an _even_ vector field. The Cartan model for the $U(1)$-equivariant cohomology of $\mathcal{F}$ is defined by the subcomplex of $Q$-invariant (or supersymmetric, or “BPS”) observables, where the supercharge squares to zero. We consider a supersymmetric model specified by the (Euclidean) action functional $S\in C^{\infty}(\mathcal{F})$ such that $\delta_{Q}S=0$, and a BPS observable $\mathcal{O}$ such that $\delta_{Q}\mathcal{O}=0$. Now the partition function (5.2) and the expectation value (5.3) are seen as integrations of equivariantly closed forms with respect to the differential $Q$. The supersymmetric localization principle then tells us that we can modify the respective integrals adding an equivariantly exact localizing term to the action, $\lambda S_{loc}[\Phi]:=\lambda\delta_{Q}\mathcal{V}[\Phi]$ (5.74) where $\mathcal{V}\in\Omega^{1}(\mathcal{F})^{U(1)}$ is a $Q^{2}$-invariant fermionic functional, the “gauge-fixing fermion” of Section 5.1, and $\lambda\in\mathbb{R}$ is a parameter. Assuming the supersymmetry $\delta_{Q}$ to be non anomalous, the partition function and the expectation value are not changed by this modification, i.e. the the new integrand lies in the same equivariant cohomology class, $\displaystyle\frac{d}{d\lambda}Z(\lambda)=\int_{\mathcal{F}}D\Phi\ (-\delta_{Q}\mathcal{V}[\Phi])e^{-(S+\lambda\delta_{Q}\mathcal{V})[\Phi]}=-\int_{\mathcal{F}}D\Phi\ \delta_{Q}\left(\mathcal{V}[\Phi]e^{-(S+\lambda\delta_{Q}\mathcal{V})[\Phi]}\right)=0$ (5.75) $\displaystyle\frac{d}{d\lambda}\langle\mathcal{O}\rangle_{\lambda}=-\frac{1}{Z(\lambda)}\left(\frac{d}{d\lambda}Z(\lambda)\right)\langle\mathcal{O}\rangle_{\lambda}+\frac{1}{Z(\lambda)}\int_{\mathcal{F}}D\Phi\ \delta_{Q}\left(\mathcal{V}[\Phi]\mathcal{O}[\Phi]e^{-(S+\lambda\delta_{Q}\mathcal{V})[\Phi]}\right)=0$ by the same argument of Section 5.1. Assuming the bosonic part of $\delta_{Q}\mathcal{V}$ to be positive-semidefinite, and using the $\lambda$-independence of the path integral, we can evaluate the partition function or the expectation value in the limit $\lambda\to+\infty$, getting the localization formulas $Z=\lim_{\lambda\to\infty}\int_{\mathcal{F}}D\Phi\ e^{-(S+\lambda S_{loc})[\Phi]},\qquad\langle\mathcal{O}\rangle=\frac{1}{Z}\lim_{\lambda\to\infty}\int_{\mathcal{F}}D\Phi\ \mathcal{O}[\Phi]e^{-(S+\lambda S_{loc})[\Phi]}.$ (5.76) The path integrals localize then onto the locus $\mathcal{F}_{0}$ of saddle points of $S_{loc}$. Following again the same argument of Section 4.2 we can in fact expand the fields about these saddle point configurations, rescale the normal fluctuations as $\Phi=\Phi_{0}+\frac{1}{\sqrt{\lambda}}\tilde{\Phi},$ (5.77) and the augmented action functional as $(S+\lambda S_{loc})[\Phi]=S[\Phi_{0}]+\frac{1}{2}\int_{M}d^{n}x\int_{M}d^{n}y\left(\frac{\delta^{2}S_{loc}}{\delta\Phi(x)\delta\Phi(y)}\right)_{\Phi_{0}}\tilde{\Phi}(x)\tilde{\Phi}(y)+O(\lambda^{-1/2}).$ (5.78) The functional measure on the normal sector $D\tilde{\Phi}$ is not affected by the rescaling, since the supersymmetric model contains the same number of bosonic and fermionic physical component fields,121212Note that this has to be true _off-shell_ , i.e. without imposing any EoM. and the corresponding Jacobians cancel by Berezin integration rules. The integral over this fluctuations is Gaussian and can be performed, giving the “1-loop determinant” analogous to the equivariant Euler class that appeared in Theorem 3.2.2. The leftover integral corresponds to the saddle point formula (5.9), but as an exact equality: $\displaystyle Z$ $\displaystyle=\int_{\mathcal{F}_{0}}D\Phi_{0}\ e^{-S[\Phi_{0}]}Z_{1-loop}[\Phi_{0}]$ (5.79) $\displaystyle\langle\mathcal{O}\rangle$ $\displaystyle=\frac{1}{Z}\int_{\mathcal{F}_{0}}D\Phi_{0}\ \mathcal{O}[\Phi_{0}]e^{-S[\Phi_{0}]}Z_{1-loop}[\Phi_{0}]$ where $Z_{1-loop}[\Phi_{0}]:=\left(\mathrm{Sdet}\left[\frac{\delta^{2}S_{loc}}{\delta\Phi(x)\delta\Phi(y)}[\Phi_{0}]\right]\right)^{-1}$ (5.80) and the super-determinant denotes collectively the result of Gaussian intergrations over bosonic or fermionic fields. Although the choice of localizing term $\mathcal{V}$ is arbitrary, and different choices give in principle different localization loci, the final result must be the same for every choice. At the end of Section 4.4 we remarked that the $\mathcal{N}=2$ supersymmetric Yang-Mills Lagrangian on a 3-dimensional maximally supersymmetric background is $Q$-exact, and thus can be used as a localizing term for supersymmetric gauge theories on this type of 3-dimensional spacetimes. It turns out that also the $\mathcal{N}=2$ matter (chiral) Lagrangian is $Q$-exact in three dimensions [79]. A canonical choice of localizing action can be, schematically [67] $S_{loc}[\Phi]:=\int_{M}\delta_{Q}\sum_{f}\left((\delta_{Q}\Phi_{f})^{\dagger}\Phi_{f}+\Phi_{f}^{\dagger}(\delta_{Q}\Phi_{f}^{\dagger})^{\dagger}\right)$ (5.81) where the sum runs over the fermionic fields of the theory. Its bosonic part is $\left.S_{loc}[\Phi]\right\|_{bos}=\sum_{f}\left(\|\delta_{Q}\Phi_{f}\|^{2}+\|\delta_{Q}\Phi_{f}^{\dagger}\|^{2}\right),$ (5.82) that is indeed positive semidefinite. The corresponding localization locus is the subcomplex of BPS configurations, $[\mathrm{fermions}]=0,\qquad\delta_{Q}[\mathrm{fermions}]=0.$ (5.83) Concluding this general and schematic discussion, there are a couple of remarks we wish to point out. Firstly, in the above formulas we always considered generic BPS observables that are expressed through (local or non- local) combinations of the fields. In other words, their quantum expectation values are defined as insertions in the path integral of corresponding (classical) functionals on the field space. Examples of local objects of this kind are correlation functions of fundamental fields. A famous class of non- local quantum operators that are expressible as classical functionals are the so-called _Wilson loops_. In gauge theory with gauge group $G$ and local gauge field $A$, a Wilson loop in the representation $R$ of $Lie(G)$ over the closed curve $C:\mathbb{S}^{1}\to M$ is defined by $W_{R}(C):=\frac{1}{\dim R}\mathrm{Tr}_{R}\left(\mathcal{P}\exp{i\oint_{\mathbb{S}^{1}}C^{}(A)}\right)$ (5.84) where the trace $\mathrm{Tr}_{R}$ is taken in the given representation.131313In the adjoint representation, this denotes an invariant inner product in $\mathfrak{g}$, for example the Killing form for a semisimple Lie algebra. This is gauge invariant, and represents physically the phase acquired by a charged probe particle in the representation $R$ after a tour on the curve $C$, in presence of the gauge potential $A$. Mathematically, if $R$ is the adjoint representation, the Wilson loop represents the parallel transport map between the fibers of the principal $G$-bundle defining the gauge theory. These operators have many interesting applications in physics: depending on the chosen curve $C$ their expectation value can be interpreted as an order parameter for the confinement/deconfinement phase transitions in QCD or the Bremsstrahlung function for an accelerated particle [80, 81, 82]. In the case of 3-dimensional Chern-Simons theory, they can be used to study topological invariants in knot theory [83]. In supersymmetric theories, they are particularly relevant for tests of the AdS/CFT correspondence [84]. In the next sections we will review some interesting cases in which expectation values of this type of operators can be evaluated exactly using the supersymmetric localization principle. There exists another class of interesting operators in the quantum theory, that cannot be expressed as classical functionals on the field space. These are the so-called _disorder operators_ , and their expectation values are defined by a restriction of the path integral to those field configurations which have prescribed boundary conditions around some artificial singularity introduced in spacetime. An example of these are the _’t Hooft operators_ , which introduce a Dirac monopole singularity along a path in a 4-dimensional space [85]. These kind of operators can be also studied non-perturbatively with the help of localization techniques [86]. For a great review of different examples of localization computations in supersymmetric QFT, see [12]. The second remark we wish to make is that, in presence of a gauge symmetry, the action functionals in the above formulas have to be understood as the quantum (i.e. gauge-fixed) action in order to give meaning to the corresponding partition function. That is, one has to introduce Faddeev-Popov ghost fields in the theory and the associated BRST transformations $\delta_{BRST}$. We have seen in Section 4.5 that it is always possible to see the BRST complex in terms of equivariant cohomology on the field space, so this supersymmetry transformation have to be incorporated in the equivariant structure of the supersymmetric theory. In this case, the field space acquires a $\mathbb{Z}$-grading corresponding to the ghost number, on top of the $\mathbb{Z}_{2}$ one from supersymmetry, and the appropriate Cartan differential with respect to which the equivariant cohomological structure is defined is then the total supersymmetry variation $Q=\delta_{susy}+\delta_{BRST}$. ### 5.4 Localization of $\mathcal{N}=4,2,2^{}$ gauge theory on the 4-sphere In this section we review, following the seminal work of Pestun [11], how to exploit the supersymmetric localization principle in $\mathcal{N}=4$ Euclidean Super Yang-Mills theory on the four-sphere $\mathbb{S}^{4}$. The $\mathcal{N}=2$ and $\mathcal{N}=2^{}$ theories can be also treated with the same technique. In particular, it was possible to solve exactly the partition function of the theory and the expectation value of the Wilson loop defined by $W_{R}(C):=\frac{1}{\dim R}\mathrm{Tr}_{R}\left(\mathcal{P}\exp{i\oint_{C(\mathbb{S}^{1})}(A_{\mu}\dot{C}^{\mu}+\|\dot{C}\|\Phi_{0})dt}\right)$ (5.85) where $C$ is a closed equatorial curve on $\mathbb{S}^{4}$ of tangent vector $\dot{C}$, and the scalar field $\Phi_{0}$ is required by supersymmetry, as will become clear later. The localization procedure makes the path integral reduce to a finite-dimensional integral over the Lie algebra of the gauge group, a so-called “matrix model”. #### 5.4.1 The action and the supersymmetric Wilson loop We will consider the theories revisited in Sections 4.3.7 and 4.4.4. We report the action of the $\mathcal{N}=2^{}$ theory on the 4-sphere, $\displaystyle S^{\mathcal{N}=2^{}}_{\mathbb{S}^{4}}=\int_{\mathbb{S}^{4}}d^{4}x\sqrt{g}\ \frac{1}{g_{YM}^{2}}\mathrm{Tr}\left(F_{MN}F^{MN}-\Psi\Gamma^{M}D_{M}\Psi\right.$ $\displaystyle+\frac{2}{r^{2}}\Phi_{A}\Phi^{A}-$ (5.86) $\displaystyle\left.-\frac{1}{4r}(R^{ki}M_{k}^{j})\Phi_{i}\Phi_{j}-\sum_{i=1}^{7}K_{i}K_{i}\right)$ where $D_{0}\Phi_{i}\mapsto[\Phi_{0},\Phi_{i}]+M_{i}^{j}\Phi_{j}$ and $D_{0}\Psi\mapsto[\Phi_{0},\Psi]+\frac{1}{4}M_{ij}\Gamma^{ij}\Psi$, $i,j=5,\cdots,8$. In the limit of zero mass $M$ we get the $\mathcal{N}=4$ YM theory, while in the limit of infinite mass the $\mathcal{N}=2$ hypermultiplet decouples and the pure $\mathcal{N}=2$ YM theory is recovered. This is invariant under the superconformal transformations (4.170) that we report here, $\displaystyle\delta_{\epsilon}A_{M}$ $\displaystyle=\epsilon\Gamma_{M}\Psi$ (5.87) $\displaystyle\delta_{\epsilon}\Psi$ $\displaystyle=\frac{1}{2}\Gamma^{MN}F_{MN}\epsilon+\frac{1}{2}\Gamma^{\mu A}\Psi_{A}\nabla_{\mu}\epsilon+\sum_{i=1}^{7}K_{i}\nu_{i}$ $\displaystyle\delta_{\epsilon}K_{i}$ $\displaystyle=-\nu_{i}\Gamma^{M}D_{M}\Psi$ with $(\nu_{i})_{i=1,\cdots,7}$ satisfying (4.171), and $\epsilon$ being a conformal Killing spinor satisfying (4.165) and (4.166). When the mass is non- zero, the Killing condition is restricted to (4.176), or equivalently $\tilde{\epsilon}=\frac{1}{2r}\Lambda\epsilon$ (5.88) where $\Lambda$ is an $SU(2)_{L}^{R}$ generator. The superconformal algebra closes schematically as $\delta_{\epsilon}^{2}=-\mathcal{L}_{v}-G_{\Phi}-(R+M)-\Omega.$ (5.89) To obtain a Poincaré-equivariant differential interpretation of this variation, we want $\delta_{\epsilon}$ to generate rigid supersymmetry, i.e. square only to the Poincaré algebra (plus R-symmetry, up to gauge transformations). Thus, to eliminate the dilatation contribution, we impose also the condition $\epsilon\tilde{\epsilon}=0.$ (5.90) If the mass is non-zero, the $SU(1,1)^{\mathcal{R}}$ is broken, so also its contribution should be eliminated, imposing further the condition $\tilde{\epsilon}\Gamma^{09}\epsilon=0.$ (5.91) Solutions to (4.165) and (4.166) are easy to compute in the flat space limit $r\to\infty$: here $\nabla_{\mu}=\partial_{\mu}$, and $\partial_{\mu}\tilde{\epsilon}=0$ imposes $\tilde{\epsilon}(x)=\hat{\epsilon}_{c}$ constant. Thus, the conformal Killing spinor in flat space is just the one considered in (4.116), $\epsilon(x)=\hat{\epsilon}_{s}+x^{\mu}\Gamma_{\mu}\hat{\epsilon}_{c}$ (5.92) where the first constant term generates supertranslations, while the term linear in $x$ generates superconformal transformations. The constant spinors $\hat{\epsilon}_{s},\hat{\epsilon}_{c}$ parametrize in general the space of solutions of the conformal Killing spinor equation. For a finite radius $r$, using stereographic coordinates and the round metric (4.163), the covariant derivative acts as $\nabla_{\mu}\epsilon=\left(\partial_{\mu}+\frac{1}{4}\omega_{ij\mu}\Gamma^{ij}\right)\epsilon$, where $\omega$ is the spin connection $\omega^{i}_{j\mu}=\left(e^{i}_{\mu}e^{\nu}_{j}-e_{j\mu}e^{i\nu}\right)\partial_{\nu}\Omega$ (5.93) and $e$ is the vielbein corresponding to the metric.141414Here we use latin indices as “flat” indices and greek indices as “curved” indices, so that as $g_{\mu\nu}=e^{i}_{\mu}e^{j}_{\nu}\delta_{ij}$. The general solution in this coordinate system is $\epsilon(x)=\frac{1}{\sqrt{1+\frac{x^{2}}{4r^{2}}}}\left(\hat{\epsilon}_{s}+x^{\mu}\Gamma_{\mu}\hat{\epsilon}_{c}\right)\qquad\tilde{\epsilon}(x)=\frac{1}{\sqrt{1+\frac{x^{2}}{4r^{2}}}}\left(\hat{\epsilon}_{c}-\frac{x^{\mu}\Gamma_{\mu}}{4r^{2}}\hat{\epsilon}_{s}\right)$ (5.94) that indeed simplifies to (5.92) in the limit of infinite radius. The conditions (5.90), (5.91), (5.88) are rewritten in terms of the constant spinors as $\hat{\epsilon}_{s}\hat{\epsilon}_{c}=\hat{\epsilon}_{s}\Gamma^{09}\hat{\epsilon}_{c}=0\qquad\hat{\epsilon}_{s}\Gamma^{\mu}\hat{\epsilon}_{s}=\frac{1}{4r^{2}}\hat{\epsilon}_{c}\Gamma^{M}\hat{\epsilon}_{c}\qquad\hat{\epsilon}_{c}=\frac{1}{2r}\Lambda\hat{\epsilon}_{s}.$ (5.95) The second condition is solved if the two constant spinors are taken to be chiral with respect to the 4-dimensional chirality operator $\Gamma^{1234}$, so that both terms vanish automatically. In Pestun’s conventions, they are chosen to have the same definite chirality and orthogonal to each other (to satisfy the first condition), so that $\epsilon$ is chiral only at the North and South poles, where $x^{2}=0,\infty$.151515There cannot be chiral spinor fields on $\mathbb{S}^{4}$ without zeros, because a chiral spinor defines an almost complex structure at each point, but $\mathbb{S}^{4}$ has no almost complex structure. Since $\mathbb{S}^{4}$ has constant scalar curvature, it can be proved that the conformal Killing condition on $\epsilon$ actually implies that $\epsilon$ is also a Killing spinor, $\nabla_{\mu}\epsilon=\mu\Gamma_{\mu}\epsilon$ for some constant $\mu$. This condition implies that the spinor is never zero, since it has constant norm. Thus $\epsilon$ cannot be chiral. [55] The Wilson loop under consideration is of the type considered in [87, 88], $W_{R}(C):=\frac{1}{\dim R}\mathrm{Tr}_{R}\left(\mathcal{P}\exp{i\oint_{C}dt(A_{\mu}\dot{C}^{\mu}+\|\dot{C}\|\Phi_{0})}\right)$ (5.96) where $C:[0,1]\to\mathbb{S}^{4}$ is an equatorial closed curve, parametrized in stereographic coordinates as $(x\circ C)(t)=2r(\cos{(t)},\sin{(t)},0,0)$, spanning a great circle of radius $r$. Its tangent vector is $\dot{C}(t)=2r(-\sin(t),\cos(t),0,0)$, and the normalization $\|\dot{C}\|=2r$ in front of $\Phi_{0}$ is needed for the reparametrization invariance of the line integral. We argue now that this Wilson loop preserves some supersymmetry under the action of $\delta_{\epsilon}$. In fact, its variation is proportional to $\delta_{\epsilon}W_{R}(C)\propto\epsilon\left(\Gamma_{\mu}\dot{C}^{\mu}+2r\Gamma_{0}\right)\Psi$ (5.97) and for this to vanish for every value of the gaugino $\Psi$, it must be that $\displaystyle 0=$ $\displaystyle\epsilon\left(\Gamma_{\mu}\dot{C}^{\mu}+2r\Gamma_{0}\right)\propto\left(\hat{\epsilon}_{s}+C^{\mu}\Gamma_{\mu}\hat{\epsilon}_{c}\right)\left(\Gamma_{\mu}\dot{C}^{\mu}+2r\Gamma_{0}\right)$ (5.98) $\displaystyle\Leftrightarrow 0=$ $\displaystyle\sin(t)\left(-\hat{\epsilon}_{s}\Gamma_{1}+2r\hat{\epsilon}_{c}\Gamma_{2}\Gamma_{0}\right)+\cos(t)\left(\hat{\epsilon}_{s}\Gamma_{2}+2r\hat{\epsilon}_{c}\Gamma_{1}\Gamma_{0}\right)+\left(2r\hat{\epsilon}_{c}\Gamma_{1}\Gamma_{2}+\hat{\epsilon}_{s}\Gamma_{0}\right)$ where we inserted the values for $C^{\mu},\dot{C}^{\mu}$ and simplified some trivial terms. For this to vanish at all $t$, the three parentheses have to vanish separately, giving the condition $\hat{\epsilon}_{c}=\frac{1}{2r}\Gamma_{0}\Gamma_{1}\Gamma_{2}\hat{\epsilon}_{s}.$ (5.99) This condition halves the number of spinors that preserve the Wilson loop under supersymmetry, so this is called a _1/2-BPS_ operator.161616This is just common terminology, that does not refer to any BPS condition between mass and central charges in the supersymmetry algebra (see [47]). It only means that the observable under consideration preserves half of the supercharges. In the $\mathcal{N}=4$ case, it preserves 16 supercharges. If the mass of the hypermultiplet is turned on, the third condition in (5.95) has a non-zero solution for $\hat{\epsilon}_{s}$ if $\det(\Lambda-\Gamma_{0}\Gamma_{1}\Gamma_{2})=0$, that fixes $\Lambda$ up to a sign. #### 5.4.2 Quick localization argument Without considering the unphysical redundancy in field space given by the gauge symmetry of the theory, we can give a quick argument for the localization of the $\mathcal{N}=4$ SYM, using the procedure outlined in Section 5.3. We consider the $U(1)$-equivariant cohomology generated by the action of a fixed supersymmetry $\delta_{\epsilon}$.171717 $\delta_{\epsilon}$ squares to the Poincaré algebra up to an gauge transformation, so we are really considering an $(U(1)\rtimes G)$-equivariant cohomology, because $\delta_{\epsilon}$-closed equivariant forms are supersymmetric and gauge invariant observables. Ignoring the gauge fixing procedure, we are really not considering the complete field space, since the BRST procedure teaches us that in presence of a gauge symmetry this is automatically extended to include ghosts, that may contribute to the localization locus. It turns out that ghosts contribution is trivial, so the rough argument already gives the correct localization locus. Here we continue with this simplified procedure, and in the next section we are going to argue the above claim. Since $\delta_{\epsilon}S^{\mathcal{N}=4}_{\mathbb{S}^{4}}=0$ is equivariantly closed (off-shell) with respect to the variations (5.87), we can perform the usual trick and add the localizing term $\displaystyle\lambda S_{loc}$ $\displaystyle:=\lambda\delta_{\epsilon}\mathcal{V}$ (5.100) $\displaystyle\mathrm{where}\quad\mathcal{V}$ $\displaystyle:=\mathrm{Tr}\left(\Psi\overline{\delta_{\epsilon}\Psi}\right)$ where $\overline{\delta_{\epsilon}\Psi}$ is defined by complex conjugation in the Euclidean signature, $\overline{\delta_{\epsilon}\Psi}=\frac{1}{2}\tilde{\Gamma}^{MN}F_{MN}\epsilon+\frac{1}{2}\Gamma^{\mu A}\Psi_{A}\nabla_{\mu}\epsilon-\sum_{i=1}^{7}K_{i}\nu_{i}.$ (5.101) The bosonic part of the localizing action is $\left.S_{loc}\right\|_{bos}=\mathrm{Tr}\left(\delta_{\epsilon}\Psi\overline{\delta_{\epsilon}\Psi}\right)$ (5.102) that is positive semi-definite. The localization locus is then the subspace of fields such that $[\mathrm{fermions}]=0;\qquad\delta_{\epsilon}[\mathrm{fermions}]\ s.t.\ \left.S_{loc}\right\|_{bos}=0.$ (5.103) The solution to (5.103) is found by inserting the relevant supersymmetry variation in $\left.S_{loc}\right\|_{bos}$, collecting the terms as a sum of positive semi-definite contributions and requiring them to vanish separately. Under the assumption of smooth gauge field, this is given by the field configurations such that, up to a gauge transformation (see [11] for the details) $\left\\{\begin{array}[]{ll}A_{\mu}=0&\mu=1,\cdots,4\\\ \Phi_{i}=0&i=5,\cdots,9\\\ \Phi^{E}_{0}=a\in\mathfrak{g}&\mathrm{constant}\\\ K_{i}^{E}=-2(\nu_{i}\tilde{\epsilon})a&i=5,6,7\\\ K_{I}=0&I=1,\cdots,4\end{array}\right..$ (5.104) So the physical sector of the theory localizes onto the zero-modes of $\Phi_{0}$. If also singular gauge field configurations are allowed, (5.103) receives contributions from instanton solutions, where $F_{\mu\nu}=0$ everywhere except from the North or the South pole. These configurations can contribute non-trivially to the partition function. Computing the action on the smooth solutions one gets $S^{\mathcal{N}=4}_{\mathbb{S}^{4}}[a]=\frac{1}{g_{YM}^{2}}\int_{\mathbb{S}^{4}}d^{4}x\sqrt{g}\mathrm{Tr}\left(\frac{2}{r^{2}}(\Phi^{E}_{0})^{2}+(K_{i}^{E})^{2}\right)=\frac{1}{g_{YM}^{2}}\mbox{vol}(\mathbb{S}^{4})\frac{3}{r^{2}}\mathrm{Tr}\left(a^{2}\right)=\frac{8\pi^{2}r^{2}}{g_{YM}^{2}}\mathrm{Tr}(a^{2})$ (5.105) where we used $\mbox{vol}(\mathbb{S}^{4})=\frac{8}{3}\pi^{2}r^{4}$ and $(\nu_{i}\tilde{\epsilon})^{2}=\frac{1}{4r^{2}}$. This last equation can be derived from the conditions (4.171) and the form of the conformal Killing spinor $\epsilon$. The action is given by constant field contributions, so the path integral is expected to be reduced to a finite-dimensional integral over the Lie algebra $\mathfrak{g}$, of the form $Z\sim\int_{\mathfrak{g}}da\ e^{-\frac{8\pi^{2}r^{2}}{g_{YM}^{2}}\mathrm{Tr}(a^{2})}\|Z_{inst}[a]\|^{2}Z_{1-loop}[a].$ (5.106) Here $Z_{inst}$ is the instanton partition function, coming from the singular gauge field contributions of above. Since $G$ is often considered to be a matrix group, this partition function is said to describe a _matrix model_. The Wilson loop (5.96), evaluated on this locus is given by $W_{R}(C)=\frac{1}{\dim R}\mathrm{Tr}_{R}e^{2\pi ra}.$ (5.107) This type of matrix models can be approached by reducing the integration over $\mathfrak{g}$ to an integration over its Cartan subalgebra $\mathfrak{h}$ (we will discuss this better later, in Section 5.4.5).181818For finite-dimensional semi-simple complex Lie algebras, this is the maximal Abelian subalgebra. In general it is the maximal Lie subalgebra such that there exists a basis extension $\mathfrak{h}\oplus span(e_{\alpha})\cong\mathfrak{g}$, and it holds the eigenvalue equation $[h,e_{\alpha}]=\rho_{\alpha}(h)e_{\alpha}$ for any $h\in\mathfrak{h}$ and a certain eigenvalue $\rho_{\alpha}(h)$. $\rho_{\alpha}:\mathfrak{h}\to\mathbb{C}$ are called _roots_ of $\mathfrak{g}$. Assuming the zero-mode $a\in\mathfrak{h}$, we can conveniently rewrite the trace in the representation $R$ as the sum of all the _weights_ $\rho(a)$ of $a$ in $R$,191919Analogously to the definition of root in the adjoint representation of $\mathfrak{g}$, if the Lie algebra acts on the representation $R$, the _weight space_ $R_{\rho}$ of weight $\rho:\mathfrak{h}\to\mathbb{C}$ is defined as the subspace of elements $A\in R$ such that $h\cdot A=\rho(h)A$. $W_{R}(C)=\frac{1}{\dim R}\sum_{\rho\in\Omega(R)}n(\rho)e^{2\pi r\rho(a)},$ (5.108) where $n(\rho)$ is the multiplicity of the weight $\rho$, and $\Omega(R)$ is the set of all the weights in the representation $R$. We finally notice that the same result for the localization locus works also for the $\mathcal{N}=2$ and the $\mathcal{N}=2^{}$ theories, and both theories localize to the same matrix model. If the mass term for the hypermultiplet is considered, this can of course give a non-trivial contribution to the 1-loop determinant. #### 5.4.3 The equivariant model As remarked at the end of Section 5.3, in presence of a gauge symmetry the path integral has to be defined with respect to a gauge-fixed action. To do so, one has to enlarge the field space to include the appropriate Faddeev- Popov ghosts in a BRST complex with the differential $\delta_{B}$. We consider then the total differential $Q=\delta_{\epsilon}+\delta_{B}$ (5.109) where $\epsilon$ is a fixed conformal Killing spinor that closes off-shell the superconformal algebra, so that the gauge-invariant SYM action is $Q$-closed. From the equivariant cohomology point of view, this operator is an equivariant differential with respect to the $U(1)_{\epsilon}\rtimes G$ symmetry group acting on the enlarged field space. To gauge fix the path integral, following the BRST procedure with respect to the differential $Q$, the action has to be extended as $S_{phys}[A,\Psi,K,ghosts]=S_{SYM}[A,\Psi,K]+Q\mathcal{O}_{g.f.}[A,ghosts]$ (5.110) with a gauge-fixing fermion $\mathcal{O}[A,ghosts]$. Upon path integration over ghosts, this new term has to give the gauge-fixing action and Fadee-Popov determinant. The localization principle is then exploited augmenting again the action with a $Q$-exact term, $\lambda Q\mathcal{V}$ with again $\mathcal{V}:=\mathrm{Tr}(\Psi\overline{\delta_{\epsilon}\Psi}).$ (5.111) This effectively gives the same localization term of the previous paragraph, since $\mathcal{V}$ is gauge-invariant. The BRST-like complex considered in [11] is given by the following ghost and auxiliary field extension. The ghost $c$, anti-ghost $\tilde{c}$ and standard Lagrange multiplier for the $R_{\xi}$-gauges $b$ (“Nakanishi-Lautrup” field) are introduced, respectively odd, odd and even with respect to Grassmann parity. Since the path integral is expected to localize on zero-modes, constant fields $c_{0},\tilde{c}_{0}$ (odd) and $a_{0},\tilde{a}_{0},b_{0}$ (even) are also introduced. On the original fields of the SYM theory, the BRST differential acts as a gauge transformation parametrized by $c$. On the gauge field $A_{\mu}$ $\delta_{B}A_{\mu}=-[c,D_{\mu}].$ (5.112) On ghosts and zero-modes the BRST transformation is defined by $\begin{array}[]{llll}\delta_{B}c=-a_{0}-\frac{1}{2}[c,c]&\delta_{B}\tilde{c}=b&\delta_{B}\tilde{a}_{0}=\tilde{c}_{0}&\delta_{B}b_{0}=c_{0}\\\ \delta_{B}a_{0}=0&\delta_{B}b=[a_{0},\tilde{c}]&\delta_{B}\tilde{c}_{0}=[a_{0},\tilde{a}_{0}]&\delta_{B}c_{0}=[a_{0},b_{0}]\end{array}$ (5.113) and its square generates a gauge transformation with respect to the (bosonic) constant field $a_{0}$, $\delta_{B}^{2}=[a_{0},\cdot].$ (5.114) The supersymmetry complex, constituted by the original fields, is reparametrized with respect to the basis $\\{\Gamma^{M}\epsilon,\nu^{i}\\}$, with $M=1,\cdots,9;i=1,\cdots,7$, of the 10-dimensional Majorana-Weyl bundle over $\mathbb{S}^{4}$. Expanding $\Psi$ over such a basis we have $\Psi=\sum_{M=1}^{9}\Psi_{M}(\Gamma^{M}\epsilon)+\sum_{i=1}^{7}\Upsilon_{i}\nu^{i}$ (5.115) and the superconformal transformations are rewritten as $\displaystyle\delta_{\epsilon}A_{M}=\Psi_{M}$ (5.116) $\displaystyle\delta_{\epsilon}\Psi_{M}=-(\mathcal{L}_{v}+R+G_{\Phi})A_{M}$ $\displaystyle\delta_{\epsilon}\Upsilon_{i}=H_{i}$ $\displaystyle\delta_{\epsilon}H_{i}=-(\mathcal{L}_{v}+R+G_{\Phi})\Upsilon_{i},$ where $H_{i}:=K_{i}+2(\nu_{i}\tilde{\epsilon})\Phi_{0}+\frac{1}{2}F_{MN}\nu_{i}\Gamma^{MN}\epsilon+\frac{1}{2}\Phi_{A}\nu_{i}\Gamma^{\mu A}\nabla_{\mu}\epsilon.$ (5.117) With this field redefinition, we see that the supersymmetry transformations can be schematized in the form $\delta_{\epsilon}X=X^{\prime}\qquad\delta_{\epsilon}X^{\prime}=[\phi+\epsilon,X]\qquad\left(\delta_{\epsilon}\phi=0\right)$ (5.118) where $\phi:=-\Phi=v^{M}A_{M}$, $[\phi,X^{\prime}]:=-G_{\Phi}X^{\prime}$ denotes a gauge transformation, $[\epsilon,X^{\prime}]:=-(\mathcal{L}_{v}+R)X^{\prime}$ denotes a Lorentz transformation. Here $X=(A_{M}(x),\Upsilon_{i}(x))$, $X^{\prime}=(\Psi_{M}(x),H_{i}(x))$ are the coordinates in the super-loop space interpretation of the supersymmetric model, of opposite statistics. As we pointed out in Section 5.3, we espect every Poincaré-supersymmetric theory to have such super-loop equivariant structure, and this is an example of the fact that in higher dimensional QFT the reparametrization of the fields necessary to make this apparent can be non-trivial. Indeed, the loop space coordinates $X$ and the corresponding 1-forms $X^{\prime}$ mix the bosonic/fermionic field components of the original parametrization! Combining the two complexes, and giving supersymmetry transformation properties to the ghost sector, the equivariant differential $Q$ is taken to act as $\begin{array}[]{lll}QX=X^{\prime}-[c,X]&Qc=\phi- a_{0}-\frac{1}{2}[c,c]&Q\tilde{c}=b\\\ QX^{\prime}=[\phi+\epsilon,X]-[c,X^{\prime}]&Q\phi=-[c,\phi+\epsilon]&Qb=[a_{0}+\epsilon,\tilde{c}]\\\ Q\tilde{a}_{0}=\tilde{c}_{0}&Qb_{0}=c_{0}&\\\ Q\tilde{c}_{0}=[a_{0},\tilde{c}_{0}]&Qc_{0}=[a_{0},b_{0}].&\end{array}$ (5.119) Moreover, $Qa_{0}=Q\epsilon=0$. This differential squares to a constant gauge transformation generated by $a_{0}$ and the Lorentz transformation generated by $\epsilon$, $Q^{2}=[a_{0}+\epsilon,\cdot].$ (5.120) Notice that to make explicit the super-loop structure when the combined complex is taken into account, one needs another non-trivial reparametrization of the fields, $\tilde{X}^{\prime}:=X^{\prime}-[c,X]\qquad\tilde{\phi}:=\phi- a_{0}-\frac{1}{2}[c,c].$ (5.121) This makes the tranformations look like $Q(\mbox{field})=\mbox{field}^{\prime}\qquad Q(\mbox{field}^{\prime})=[a_{0}+\epsilon,\mbox{field}]$ (5.122) and the new pairs of coordinate/1-form in the extended super-loop space are $(c,\tilde{\phi})$, $(\tilde{c},b)$, $(\tilde{a}_{0},\tilde{c}_{0})$, $(b_{0},c_{0})$. The gauge-fixing term considered for the extended quantum action is, schematically $S_{g.f.}=\int_{\mathbb{S}^{4}}Q\left(\tilde{c}\left(\nabla^{\mu}A_{\mu}+\frac{\xi_{1}}{2}b+b_{0}\right)-c\left(\tilde{a}_{0}-\frac{\xi_{2}}{2}a_{0}\right)\right)$ (5.123) where the bilinear product in $\mathfrak{g}$ is suppressed in the notation, assuming contraction of Lie algebra indices. Upon integration of the auxiliary field, this term produces the usual gauge fixing term for the Lorentz gauge $\nabla^{\mu}A_{\mu}=0$, and the ghost term of the action. Moreover, the path integral is independent of the parameters $\xi_{1},\xi_{2}$ (we refer to [11] for the proof). We finally claim that the localization principle for the gauge-fixed theory remains the same, with the additional condition of vanishing ghosts in the localization locus, and identifying the zero-mode of $\Phi_{0}$ with $a_{0}$. In fact from the gauge-fixing term, $S_{g.f.}\supset-\int_{\mathbb{S}^{4}}\left(\phi- a_{0}-\frac{1}{2}[c,c]\right)\tilde{a}_{0}$ (5.124) and integrating over $\tilde{a}_{0}$, we have the condition $\phi=a_{0}+\frac{1}{2}[c,c]$, that in the localization locus where $c=0$ and $\phi=-v^{M}A_{M}=\Phi_{0}$, becomes precisely $\Phi_{0}=a_{0}$. #### 5.4.4 Localization formulas We stated that the path integral localizes (apart from instanton corrections) to the zero-modes of the bosonic constant $a_{0}\in\mathfrak{g}$, that correspond to the zero-modes of $\Phi_{0}$. For the same reason of the scalar field corresponding to the reduced time-direction of the (9,1)-theory, we integrate over immaginary $a_{0}=ia_{0}^{E}$, where $a_{0}^{E}$ is real. The application of the localization principle is now straightforward in principle, although very cumbersome in practice. In particular, integrating out the Gaussian fluctuations around the localization locus, the arising one-loop determinant in the partition function results of the form $Z_{1-loop}=\left(\frac{\det{K_{f}}}{\det{K_{b}}}\right)^{1/2}$ (5.125) where $K_{f},K_{b}$ are the kinetic operators acting on the fermionic and bosonic fluctuation modes after the usual expansion of $Q\mathcal{V}$. This factor requires in general a regularization, and it has been computed for the $\mathcal{N}=2,\mathcal{N}=2^{}$ and $\mathcal{N}=4$ theory, using an appropriate generalization of the Atiyah-Singer theorem seen in Section 5.2 applied to transversally elliptic operators. The instanton partition functions have been also simplified for the theories under consideration. We refer to [89, 11, 90] for the explicit form of instanton contributions in the cases of $\mathcal{N}=2,2^{}$. For the maximally supersymmetric $\mathcal{N}=4$ SYM theory, the results for the 1-loop determinant and the instanton partition function are of the very simple form $Z_{1-loop}^{\mathcal{N}=4}=1,\qquad Z_{inst}^{\mathcal{N}=4}=1,$ (5.126) so that the resulting localization formulas for the partition function and the expectation value of the supersymmetric Wilson loop presented before become $\displaystyle Z_{\mathbb{S}^{4}}$ $\displaystyle=\frac{1}{\mbox{vol}(G)}\int_{\mathfrak{g}}da\ e^{-\frac{8\pi^{2}r^{2}}{g_{YM}^{2}}\mathrm{Tr}(a^{2})},$ (5.127) $\displaystyle\langle W_{R}(C)\rangle$ $\displaystyle=\frac{1}{\dim R}\frac{1}{Z\ \mbox{vol}(G)}\int_{\mathfrak{g}}da\ e^{-\frac{8\pi^{2}r^{2}}{g_{YM}^{2}}\mathrm{Tr}(a^{2})}\mathrm{Tr}_{R}\left(e^{2\pi ra}\right).$ This result proved a previous conjecture, based on a perturbative analysis by Erickson-Semenoff-Zarembo [91]. Their calculation for $\langle W_{R}(C)\rangle$ with $G=U(N)$ showed that the Feynman diagrams with internal vertices cancel up to order $g^{4}N^{2}$, and that the sum of all ladder diagrams (planar diagrams with no internal vertices) exponentiate to a matrix model. The result of this exponentiation gives an expectation value that coincides with the strong-coupling prediction of the AdS/CFT correspondence for $\mathcal{N}=4$ SYM,202020This “correspondence” conjectures a duality between the $\mathcal{N}=4$ SYM in 4 dimensions and type IIB superstring theory in an $AdS_{5}\times\mathbb{S}^{5}$ background. In particular, when the parameters of the gauge theory are taken to be such that $N\to\infty$ and $g_{YM}^{2}N\to\infty$ (namely, in the planar and strong ’t Hooft coupling limit), $\mathcal{N}=4$ SYM is dual to classical type IIB supergravity on $AdS_{5}\times\mathbb{S}^{5}$ and the computation of the Wilson loop in this limit is mapped to the evaluation of a minimal surface in this space [80, 92]. thus they conjectured that the diagrams with vertices have to vanish at all orders. Later this conjecture was supported by Drukker-Gross [93], and finally proven with the exact localization technique described above. We quote now the results for the 1-loop determinants in the $\mathcal{N}=2,2^{}$ theories. For this, it is useful to introduce the notation $\displaystyle\mbox{det}_{R}f(a)$ $\displaystyle:=\prod_{\rho}f(\rho(a))$ (5.128) $\displaystyle H(z)$ $\displaystyle:=e^{-(1+\gamma)z^{2}}\prod_{n=1}^{\infty}\left(1-\frac{z^{2}}{n^{2}}\right)^{n}\prod_{n=1}^{\infty}e^{z^{2}/n}$ with $\rho$ running over the weights of $R$ (if $R=Ad$, the weights are the roots of $\mathfrak{g}$), $\gamma$ being the Euler-Mascheroni constant. Let also be $m^{2}:=\frac{1}{4}M_{ij}M^{ij}$, and recall that $m$ (as well as $a_{0}$) should take immaginary values. From [11] we have $\displaystyle Z_{1-loop}^{\mathcal{N}=2^{}}[a_{0};M]$ $\displaystyle=\exp{\left(-r^{2}m^{2}\left((1+\gamma)-\sum_{n=1}^{\infty}\frac{1}{n}\right)\right)}\mbox{det}_{Ad}\left[\frac{H(ra_{0})}{\left[H(r(a_{0}+m))H(r(a_{0}-m))\right]^{-1/2}}\right],$ (5.129) $\displaystyle Z_{1-loop}^{\mathcal{N}=2,pure}[a_{0}]$ $\displaystyle=\mbox{det}_{Ad}H(ra_{0}),$ (5.130) $\displaystyle Z_{1-loop}^{\mathcal{N}=2,W}[a_{0}]$ $\displaystyle=\frac{\det_{Ad}H(ra_{0})}{\det_{W}H(ra_{0})},$ (5.131) where the first result is for the massive $\mathcal{N}=2^{}$ theory, the second one is derived putting $m=0$ in the first line, and describes the pure $\mathcal{N}=2$ SYM, the third one is for the matter-coupled theory to a massles hypermultiplet in the representation $W$. Notice that the exponential prefactor in the first line diverges, but is independent of $a_{0}$, and thus simplifies in ratios during the computation of expectation values. Also, the third line holds literally if the ($a_{0}$-independent) divergent factors are the same for the vector and the hypermultiplet. #### 5.4.5 The Matrix Model for $\mathcal{N}=4$ SYM As an example, we include here an explicit computation for the Gaussian matrix model (5.127) in the case of $\mathcal{N}=4$ SYM [85, 94, 93]. We will take in particular the case of the compact matrix group $G=U(N)$ with the Wilson loop in the fundamental representation $R=\mathbf{N}$, but first we analyze generically how to simplify such an integration over the Lie algebra $\mathfrak{g}$. We normalize the invariant volume element $da$ on $\mathfrak{g}$ such that $\int_{\mathfrak{g}}da\ e^{-\frac{2}{\xi^{2}}\mathrm{Tr}(a^{2})}=\left(\frac{\xi^{2}\pi}{2}\right)^{\dim(G)/2}$ (5.132) for any parameter $\xi$. In the $U(N)$ case, $\mathfrak{g}=\mathfrak{u}(N)=\\{\text{Hermitian}\ N\times N\ \text{matrices}\\}$, so this means taking $da=2^{N(N-1)/2}\prod_{i=1}^{N}da_{ii}\prod_{1\leq j<i\leq N}d\text{Re}(a_{ij})d\text{Im}(a_{ij}).$ (5.133) Setting $\xi^{2}:=g_{YM}^{2}/(4\pi^{2}r^{2})$ we have $Z_{\mathbb{S}^{4}}=\frac{1}{\mbox{vol}(G)}\left(\frac{g_{YM}^{2}}{8\pi r^{2}}\right)^{\dim(G)/2}.$ (5.134) To simplify the integration of the Wilson loop expectation value, we notice that the matrix model has a leftover gauge symmetry under constant gauge transformations, since both the measure and the traces are invariant under the adjoint action of $G$. Thus we can “gauge-fix” the integrand to depend only on the Cartan subalgebra $\mathfrak{h}\subset\mathfrak{g}$, setting $a=Ad_{g}(X)$ (5.135) for some $g\in G/T$ and $X\in\mathfrak{h}$, with $T$ being the maximal torus in $G$ generated by $\mathfrak{h}$. There is more than one $X$ related to $a$ by conjugation, but they are related via the action of the _Weyl group_ of $G$, that we call $\mathcal{W}$. Taking this into account, after the gauge- fixing we can perform the integral over the orbits obtaining a volume factor $\frac{\mbox{vol}(G/T)}{\|\mathcal{W}\|}.$ (5.136) The gauge-fixing can be done with the usual Faddeev-Popov (FP) procedure, that is inserting the unity decomposition $1=\int dg\ \Delta^{2}(X)\delta(F(a^{(g)})),$ (5.137) where the delta-function fixes the condition (5.135), and the FP determinant is given by $\Delta(X)^{2}=\prod_{\alpha}\|\alpha(X)\|=\prod_{\alpha>0}\alpha(X),$ (5.138) where $\alpha:\mathfrak{h}\to\mathbb{C}$ are the roots of $\mathfrak{g}$, and in the second equality we used that roots come in pairs $(\alpha,-\alpha)$. We can rewrite the expectation value of the circular Wilson loop as $\displaystyle\langle W_{R}(C)\rangle$ $\displaystyle=\left(\frac{\xi^{2}\pi}{2}\right)^{\dim(G)/2}\frac{\mbox{vol}(G/T)}{\|\mathcal{W}\|\dim R}\int_{\mathfrak{h}}dX\ \Delta(X)^{2}e^{-\frac{2}{\xi^{2}}\mathrm{Tr}(X^{2})}\mathrm{Tr}_{R}\left(e^{2\pi rX}\right)$ (5.139) $\displaystyle=\left(\frac{\xi^{2}\pi}{2}\right)^{\dim(G)/2}\frac{\mbox{vol}(G/T)}{\|\mathcal{W}\|\dim R}\sum_{\rho\in\Omega(R)}n(\rho)\int_{\mathfrak{h}}dX\ \Delta(X)^{2}e^{-\frac{2}{\xi^{2}}\mathrm{Tr}(X^{2})}e^{2\pi r\rho(X)}$ where in the second line we used (5.108). We specialize now to the case $G=U(N)$, $\mathfrak{h}=\\{X=\mathrm{diag}(\lambda_{1},\cdots,\lambda_{N})\|\lambda_{i}\in\mathbb{R}\\}$, and we take $r=1$. The adjoint action of $U(N)$ is the conjugation $a\mapsto gag^{\dagger}$, so the FP determinant is defined by $1=\int dg\ \Delta(X)^{2}\prod_{ij}\delta((gXg^{\dagger})_{ij}),$ (5.140) that imposes the off-diagonal terms to vanish in the given gauge. Expressing $g=e^{M}$ with $M\in\mathfrak{u}(N)$, $\Delta(X)^{2}=\prod_{ij}\det_{kl}\left\|\frac{\delta(e^{M}Xe^{-M})_{ij}}{\delta M_{kl}}\right\|=\prod_{ij}\det_{kl}\left\|\delta_{ki}\delta_{lj}(\lambda_{j}-\lambda_{i})\right\|=\prod_{i>j}(\lambda_{i}-\lambda_{j})^{2}$ (5.141) is the so called _Vandermonde determinant_ , that can be related to the following matrix $\Delta(\lambda)=\det\|\|\lambda_{i}^{j-1}\|\|=\det\left(\begin{array}[]{ccccc}1&\lambda_{1}&\lambda_{1}^{2}&\cdots&\lambda_{1}^{N-1}\\\ 1&\lambda_{2}&\lambda_{2}^{2}&\cdots&\lambda_{2}^{N-1}\\\ &&\vdots&&\\\ 1&\lambda_{N}&\lambda_{N}^{2}&\cdots&\lambda_{N}^{N-1}\\\ \end{array}\right).$ (5.142) The partition function can thus be expressed as $Z_{\mathbb{S}^{4}}=\frac{1}{N!}\frac{1}{(2\pi)^{N}}\int\left(\prod_{i}d\lambda_{i}\right)\left(\prod_{i>j}(\lambda_{i}-\lambda_{j})^{2}\right)e^{-\frac{2}{\xi^{2}}\sum_{i}\lambda_{i}^{2}},$ (5.143) where $N!$ is the order of the Weyl group $\mathcal{W}=S_{N}$ and $(2\pi)^{N}$ is the volume of the $N$-torus $U(1)^{N}$, while the Wilson loop in the fundamental representation inserts in the path integral a factor $\frac{1}{N}\sum_{j=1}^{N}e^{2\pi\lambda_{j}}.$ (5.144) There are two main approaches to the evaluation of this matrix model and the computation of the Wilson loop expectation value, at least in the limit $N\to\infty$. ##### $1^{st}$ method: saddle-point The first method that we present is based on a suitable saddle-point approximation in the large-$N$ limit. To see the possibility for this interpretation, we rewrite the partition function as $\displaystyle Z_{\mathbb{S}^{4}}$ $\displaystyle=\frac{1}{N!}\int\prod_{i}\frac{d\lambda_{i}}{2\pi}\ e^{-N^{2}S_{eff}(\lambda)}$ (5.145) $\displaystyle\text{with}\quad S_{eff}(\lambda)$ $\displaystyle:=\frac{8\pi^{2}}{tN}\sum_{i=1}^{N}\lambda_{i}^{2}-\frac{2}{N^{2}}\sum_{i>j}\log\|\lambda_{i}-\lambda_{j}\|,$ where $t:=g_{YM}^{2}N$ is the ’t Hooft coupling constant. This can be viewed as an effective action of a zero-dimensional QFT describing $N$ sites (the eigenvalues $\lambda_{i}$), where the first piece is a “one-body” harmonic potential, and the second one is a repulsive “two-body” interaction. Notice that every sum is roughly of order $\sim N$, so $S_{eff}\sim O(1)$ in $N$. The limit $N\to\infty$, with $t$ fixed, can be regarded as a semi-classical approximation (we could compare it to “$1/\sqrt{\hbar}\to\infty$”), and in that limit we can solve the integral using a saddle-point approximation. The saddle points are those values of $\lambda_{i}$ that solve the classical EoM $0=\frac{\delta S_{eff}}{\delta\lambda_{i}}\qquad\Rightarrow\qquad 0=\frac{16\pi^{2}}{tN}\lambda_{i}-\frac{2}{N^{2}}\sum_{j\neq i}\frac{1}{\lambda_{i}-\lambda_{j}}.$ (5.146) In the large-$N$ limit we can study this equation in the continuum approximation, assuming the eigenvalues $\lambda_{i}$ to take values in a compact interval $I=[a,b]$, so that the (normalized) _eigenvalue distribution_ $\rho(\lambda)=\frac{1}{N}\sum_{i=1}^{N}\delta(\lambda-\lambda_{i})$ (5.147) is regarded as a continuous function of compact support on $I$. Then every sum can be replaced by an integration over the reals, $\frac{1}{N}\sum_{i=1}^{N}f(\lambda_{i})\to\int d\lambda\ f(\lambda)\rho(\lambda),$ (5.148) and (5.146) becomes $\frac{8\pi^{2}}{t}\lambda=\mathcal{P}\int\frac{\rho(\lambda^{\prime})d\lambda^{\prime}}{\lambda-\lambda^{\prime}},$ (5.149) where we took the principal value of the integral to avoid the pole at $\lambda_{i}=\lambda_{j}$. This is an integral equation in $\rho(\lambda)$, whose solution gives the distribution of the eigenvalues at the saddle-point locus of the partition function. It is useful to introduce an auxiliary function on the complex plane, the “resolvent” $\omega(z):=\int\frac{\rho(\lambda)d\lambda}{z-\lambda},$ (5.150) that has three important properties for our purposes: 1. (i) it is analytic on $\mathbb{C}\setminus I$, since there are poles for $z=\lambda$ when $z\in I$; 2. (ii) thanks to the normalization of $\rho$, asymptotically for $\|z\|\to\infty$ it goes as $\omega(z)\sim\frac{1}{z}$; 3. (iii) using the residue theorem and the delta-function representation $\frac{\epsilon}{z^{2}+\epsilon^{2}}\xrightarrow{\epsilon\to 0^{+}}\pi\delta(z),$ (5.151) it relates to the eigenvalue distribution by the discontinuity equation $\rho(\lambda)=-\frac{1}{2\pi i}\lim_{\epsilon\to 0^{+}}\left[\omega(\lambda+i\epsilon)-\omega(\lambda-i\epsilon)\right].$ (5.152) Knowing the resolvent we can easily compute the eigenvalue distribution by this last property, so we rewrite the saddle-point equation in terms of it. To compute $\omega$, we can start again from (5.149), multiply by $1/(\lambda-z)$ and integrate over $\lambda$ with the usual measure $\rho(\lambda)d\lambda$: $\frac{8\pi^{2}}{t}\int d\lambda\ \rho(\lambda)\frac{\lambda}{\lambda-z}=\int d\lambda\frac{\rho(\lambda)}{\lambda-z}\ \mathcal{P}\int d\lambda^{\prime}\frac{\rho(\lambda^{\prime})}{\lambda-\lambda^{\prime}}.$ (5.153) We can add $\pm z$ at the numerator of the LHS, and use the formula (_Sokhotski–Plemelj theorem_) $\mathcal{P}\int\frac{f(z)}{z}dz=\lim_{\epsilon\to 0^{+}}\frac{1}{2}\left(\int\frac{f(z)}{z+i\epsilon}dz+\int\frac{f(z)}{z-i\epsilon}dz\right)$ (5.154) to break the principal value on the RHS. Inserting the definition of the resolvent and using the residue theorem, this gives $\frac{8\pi^{2}}{t}-\frac{8\pi^{2}}{t}\lambda\omega(\lambda)=-\frac{1}{2}\omega(\lambda)^{2},$ (5.155) that is solved for $\omega(\lambda)=\frac{8\pi^{2}}{t}\left(\lambda\pm\sqrt{\lambda^{2}-\frac{t}{4\pi^{2}}}\right).$ (5.156) In order to match the right asymptotic behavior $\omega(z\to\infty)\sim 1/z$, we have to chose the minus sign. With this choice, we can compute the saddle- point eigenvalue distribution using the discontinuity equation (5.152), $\displaystyle\rho(\lambda)$ $\displaystyle=-\frac{1}{2\pi i}\frac{8\pi^{2}}{t}\lim_{\epsilon\to 0^{+}}\left[\omega(\lambda+i\epsilon)-\omega(\lambda-i\epsilon)\right]$ (5.157) $\displaystyle=\frac{4\pi}{it}\lim_{\epsilon\to 0^{+}}\left[\sqrt{\lambda^{2}-\frac{t}{4\pi^{2}}+2i\epsilon\lambda}-\sqrt{\lambda^{2}-\frac{t}{4\pi^{2}}-2i\epsilon\lambda}\right]$ $\displaystyle=\frac{4\pi}{it}\left(2\sqrt{\lambda^{2}-\frac{t}{4\pi^{2}}}\right)$ $\displaystyle=\frac{8\pi}{t}\sqrt{\frac{t}{4\pi^{2}}-\lambda^{2}}$ where we used that the principal square root has a branch cut on the real line. This function is called _Wigner semi-circle distribution_ , it has support on the interval $I=[-\sqrt{t}/2\pi,\sqrt{t}/2\pi]$, and here it is correctly normalized to 1. Now that we have the saddle-point locus in terms of the eigenvalue distribution, we can compute the expectation value for the circular Wilson loop in the fundamental representation. Since the exponential factor (5.144) is of order $\sim N^{0}$, this does not contribute to the saddle-point equation in the $N\to\infty$ limit. We can thus still use the Wigner distribution at zero-order in $1/N^{2}$, and insert in the path integral the trace in the continuum limit, $\displaystyle\langle W_{\mathbf{N}}(C)\rangle$ $\displaystyle=\int d\lambda\ \langle\rho(\lambda)\rangle e^{2\pi\lambda}$ (5.158) $\displaystyle=\frac{8\pi}{t}\int_{-\sqrt{t}/2\pi}^{\sqrt{t}/2\pi}d\lambda\ e^{2\pi\lambda}\sqrt{\frac{t}{4\pi^{2}}-\lambda^{2}}+O\left(1/N^{2}\right)$ $\displaystyle=\frac{2}{\sqrt{t}}I_{1}\left(\sqrt{t}\right)+O\left(1/N^{2}\right)$ where $I_{1}(z)$ is a modified Bessel function of the first kind. In the weak and strong coupling limits $t\gg,\ll 1$ the expectation value gives $\displaystyle t\ll 1:\qquad\langle W_{\mathbf{N}}(C)\rangle$ $\displaystyle\sim 1+\frac{t^{2}}{8}+\frac{t^{4}}{192}+\cdots$ (5.159) $\displaystyle t\gg 1:\qquad\langle W_{\mathbf{N}}(C)\rangle$ $\displaystyle\sim\sqrt{\frac{2}{\pi}}t^{-3/4}e^{\sqrt{t}},$ (5.160) so it explodes in the strong coupling limit, with an _essential_ singularity.212121Interestingly, the strong coupling limit can be checked independently using holography, where Wilson loops are given by minimal surfaces in AdS [80, 92]. ##### $2^{nd}$ method: orthogonal polynomials Another technique to solve matrix models involve the use of orthogonal polynomials [93]. Our starting point is again the partition function, $Z=\frac{1}{N!}\int\prod_{i=1}^{N}\left(\frac{d\lambda_{i}}{2\pi}e^{-\frac{8\pi^{2}N}{t}\lambda_{i}^{2}}\right)\Delta(\lambda)^{2}.$ (5.161) Introducing the $L^{2}(\mathbb{R})$ measure $d\mu(x):=dx\ e^{-\frac{8\pi^{2}N}{t}x^{2}},$ (5.162) we can write the partition function as $Z=\frac{1}{N!}\int\prod_{i=1}^{N}d\mu(\lambda_{i})\Delta(\lambda)^{2}.$ (5.163) Recalling that the Vandermonde determinant is evaluated from the matrix (5.142), expressed in terms of the polynomials $\\{1,x,x^{2},\cdots\\}$, we notice that we can equivalently express it in terms of another set of _monic_ polynomials, $p_{k}(x)=x^{k}+\sum_{j=0}^{k-1}a_{j}^{(k)}x^{j}$ (5.164) since by elementary row operations $\Delta(\lambda)=\det\|\|\lambda_{i}^{j-1}\|\|=\det\|\|p_{j-1}(\lambda_{i})\|\|.$ (5.165) It is useful to chose the set $\\{p_{k}\\}_{k\geq 0}$ to be _orthogonal_ with respect to the matrix model measure, $\int d\mu(\lambda)p_{n}(\lambda)p_{m}(\lambda)=h_{n}\delta_{nm}$ (5.166) since the knowledge of this set, and in particular of the normalization constants $h_{n}$, allows to compute the partition function. Writing the determinant as $\Delta(\lambda)=\sum_{\sigma\in S_{N}}(-1)^{\mbox{sign}(\sigma)}\prod_{k=1}^{N}p_{\sigma(k)-1}(\lambda_{k}),$ then (5.163) reduces to $Z=\prod_{k=0}^{N-1}h_{k}.$ (5.167) In our case the matrix model is Gaussian, and the corresponding set of orthogonal polynomials are the _Hermite polynomials_ , $H_{n}(x):=e^{x^{2}}\left(-\frac{d}{dx}\right)^{n}e^{-x^{2}},\qquad\int_{-\infty}^{+\infty}dx\ e^{-x^{2}}H_{n}(x)H_{m}(x)=\delta_{nm}2^{n}n!\sqrt{\pi}$ (5.168) so, normalizing $h_{n}=1$ and inserting the correct prefactors, we consider the set of _orthonormal_ polynomials with respect to the measure $d\mu(\lambda)$ $P_{n}(\lambda):=\sqrt{\sqrt{\frac{8\pi N}{t}}\frac{1}{2^{n}n!}}\ H_{n}\left(\frac{\sqrt{8\pi^{2}N}}{t}\lambda\right).$ (5.169) The expectation value of any observable of the type $\mathrm{Tr}(f(X))=\sum_{k}f(\lambda_{k})$ can be simplified as $\displaystyle\langle\mathrm{Tr}f(X)\rangle$ $\displaystyle=\frac{1}{N!Z}\int\left(\prod_{i=1}^{N}d\mu(\lambda_{i})\right)\Delta(\lambda)^{2}\sum_{k=1}^{N}f(\lambda_{k})$ (5.170) $\displaystyle\begin{aligned} =\frac{1}{N!}\sum_{k}\sum_{\sigma\in S_{N}}\int d\mu(\lambda_{1})\ P_{\sigma(1)-1}(\lambda_{1})^{2}\cdots\int d\mu(\lambda_{k})\ P_{\sigma(k)-1}(\lambda_{k})^{2}f(\lambda_{k})\cdots&\\\ \cdots\int d\mu(\lambda_{N})\ P_{\sigma(N)-1}(\lambda_{N})^{2}&\end{aligned}$ $\displaystyle=\sum_{j=0}^{N-1}\int d\mu(\lambda)\ P_{j}(\lambda)^{2}f(\lambda).$ Applying this formula to the expectation value of the circular Wilson loop in the fundamental representation we have $\displaystyle\langle W_{\mathbf{N}}(C)\rangle$ $\displaystyle=\frac{1}{N}\left\langle\mathrm{Tr}\exp(2\pi X)\right\rangle$ (5.171) $\displaystyle=\frac{1}{N}\sum_{j=0}^{N-1}\int d\lambda\ P_{j}(\lambda)^{2}e^{-\frac{8\pi^{2}N}{t}\lambda^{2}+2\pi\lambda}.$ A useful formula to simplify this integral is $\int_{-\infty}^{+\infty}dx\ H_{n}(x)^{2}e^{-(x-c)^{2}}=2^{n}n!\sqrt{\pi}L_{n}(-2c^{2})$ (5.172) where $c$ is a constant and $L_{n}(x)$ are the _Laguerre polynomials_ , satisfying the properties $\displaystyle L_{n}^{(m)}(x)$ $\displaystyle=\frac{1}{n!}e^{x}x^{m}\left(\frac{d}{dx}\right)^{n}\left(e^{-x}x^{n+m}\right),$ (5.173) $\displaystyle L_{n}(x)$ $\displaystyle\equiv L_{n}^{(0)}(x),$ (5.174) $\displaystyle L_{n}^{(m+1)}(x)$ $\displaystyle=\sum_{j=0}^{n}L_{j}^{(m)}(x),$ (5.175) $\displaystyle L_{n}^{(m)}(x)$ $\displaystyle=\sum_{k=0}^{n}\binom{n+m}{n-k}\frac{(-x)^{k}}{k!}.$ (5.176) Substituting (5.172) in (5.171), and expanding in series we have $\displaystyle\langle W_{\mathbf{N}}(C)\rangle$ $\displaystyle=\frac{1}{N}e^{c^{2}}L_{N-1}^{(1)}(-2c^{2})\qquad\text{with}\ c:=\sqrt{\frac{t}{8N}}$ (5.177) $\displaystyle\begin{aligned} =&\frac{1}{N}\sum_{k=0}^{\infty}\frac{1}{k!}\left(\frac{t}{8N}\right)^{k}\sum_{j=0}^{N-1}\frac{N!}{(j+1)!(N-1-j)!}\frac{1}{j!}\left(\frac{t}{4N}\right)^{j}-\\\ &-\frac{1}{N}\sum_{j=1}^{N-1}\frac{1}{j!(j+1)!}\left(\frac{t}{4}\right)^{j}\frac{j(j+1)}{2}+\frac{1}{N}\sum_{j=0}^{N-1}\frac{1}{j!(j+1)!}\left(\frac{t}{4}\right)^{j}\frac{1}{2}+O(1/N^{2})\end{aligned}$ $\displaystyle=\sum_{j=0}^{N-1}\frac{1}{j!(j+1)!}\left(\frac{t}{4}\right)^{j}+O\left(1/N^{2}\right)$ where we expanded the first terms with respect to powers of $1/N$, and already noticed that for $N\gg 1$ the odd-power terms cancel. We can thus examine the large-$N$ limit, and inserting the definition of the modified Bessel function $I_{n}(2x)=\sum_{k=0}^{\infty}\frac{x^{n+2k}}{k!(n+k)!}$ the expectation value gives $\langle W_{\mathbf{N}}(C)\rangle=\frac{2}{\sqrt{t}}I_{1}\left(\sqrt{t}\right)+O(1/N^{2}),$ (5.178) matching the result obtained with the saddle point technique in (5.158). In general, the expansion is in powers of $1/N^{2}$ rather than $1/N$, as expected from the analogy “$N^{2}\leftrightarrow 1/\hbar$” that we noticed in (5.145). Solutions to the matrix model for higher representations have also been found, see [85, 95, 96]. ### 5.5 Localization of $\mathcal{N}=2$ Chern-Simons theory on the 3-sphere In this section we review another example of supersymmetric localization applied to the computation of Wilson loop expectation values, in an $\mathcal{N}=2$ matter-coupled Euclidean Super Chern-Simons (SCS) theory on the 3-sphere $\mathbb{S}^{3}$. We follow the derivation of Kapustin-Willet- Yaakov [49], and Mariño [50], inspired in part by the work discussed in the previous section. We consider a generic compact Lie group $G$ as the gauge group, with Lie algebra $\mathfrak{g}$. #### 5.5.1 Matter-coupled $\mathcal{N}=2$ Euclidean SCS theory on $\mathbb{S}^{3}$ The case of $\mathcal{N}=2$ Euclidean supersymmetry on $\mathbb{S}^{3}$ was discussed as an example in Sections 4.3.6 and 4.4.3 for the gauge sector. We report the action for the SCS theory $S_{CS}=\frac{k}{4\pi}\int_{\mathbb{S}^{3}}d^{3}x\sqrt{g}\ \mathrm{Tr}\left\\{\frac{\varepsilon^{\mu\nu\rho}}{\sqrt{g}}\left(A_{\mu}\partial_{\nu}A_{\rho}+\frac{2i}{3}A_{\mu}A_{\nu}A_{\rho}\right)-\tilde{\lambda}\lambda+2\sigma D\right\\}$ (5.179) and the supersymmetry variations, already considered in curved space $\displaystyle\delta A_{\mu}=\frac{i}{2}(\tilde{\epsilon}\gamma_{\mu}\lambda-\tilde{\lambda}\gamma_{\mu}\epsilon)$ (5.180) $\displaystyle\delta\sigma=\frac{1}{2}(\tilde{\epsilon}\lambda-\tilde{\lambda}\epsilon)$ $\displaystyle\delta\lambda=\left(-\frac{1}{2}F_{\mu\nu}\gamma^{\mu\nu}-D+i(D_{\mu}\sigma)\gamma^{\mu}+\frac{2i}{3}\sigma\gamma^{\mu}D_{\mu}\right)\epsilon$ $\displaystyle\delta\tilde{\lambda}=\left(-\frac{1}{2}F_{\mu\nu}\gamma^{\mu\nu}+D-i(D_{\mu}\sigma)\gamma^{\mu}-\frac{2i}{3}\sigma\gamma^{\mu}D_{\mu}\right)\tilde{\epsilon}$ $\displaystyle\delta D=-\frac{i}{2}\left(\tilde{\epsilon}\gamma^{\mu}D_{\mu}\lambda-(D_{\mu}\tilde{\lambda})\gamma^{\mu}\epsilon\right)+\frac{i}{2}\left([\tilde{\epsilon}\lambda,\sigma]-[\tilde{\lambda}\epsilon,\sigma]\right)-\frac{i}{6}\left(\tilde{\lambda}\gamma^{\mu}D_{\mu}\epsilon+(D_{\mu}\tilde{\epsilon})\gamma^{\mu}\lambda\right).$ where the $D_{\mu}$ are gauge-covariant derivatives with respect to the metric and spin connection induced by the round metric (4.154), that in stereographic coordinates $x^{\mu=1,2,3}$ is given by $g_{\mu\nu}=e^{2\Omega(x)}\delta_{\mu\nu}\qquad e^{2\Omega(x)}=\left(1+\frac{x^{2}}{4r^{2}}\right)^{-2}$ (5.181) with $r$ being the radius of the embedding $\mathbb{S}^{3}\hookrightarrow\mathbb{R}^{4}$. We remark again that this supersymmetric action is actually superconformal, thus can preserve supersymmetry on this conformally flat background, even with positive scalar curvature. The new background preserves all the original $\mathcal{N}=2$ algebra, generated by conformal Killing spinors $\epsilon,\tilde{\epsilon}$, taken to satisfy $\nabla_{\mu}\epsilon=\frac{i}{2r}\gamma_{\mu}\epsilon,\qquad\nabla_{\mu}\tilde{\epsilon}=\frac{i}{2r}\gamma_{\mu},\tilde{\epsilon}$ (5.182) where every equation has two possible solutions. We consider also coupling the theory to matter fields, adding them in chiral multiplets in a representation $R$ of the gauge group, to preserve supersymmetry. The 3-dimensional $\mathcal{N}=2$ chiral multiplet (or hypermultiplet) is, as for the gauge multiplet, given by dimensional reduction of the $\mathcal{N}=1$ chiral multiplet in 4 dimensions: a complex scalar $\phi$, a 2-component Dirac spinor222222Recall that $Spin(3)=SU(2)$ has no Majorana spinors. We consider the reduced 4-dimensional Majorana spinor $\psi$ as a 3-dimensional Dirac (complex) spinor, since they have the same number of real components. $\psi$ and an auxiliary complex scalar $F$. Every field comes with its complex conjugate from the corresponding anti-chiral multiplet. The supersymmetric action for the matter multiplet coupled to the gauge multiplet is given by $S_{m}=\int_{\mathbb{S}^{3}}d^{3}x\sqrt{g}\ \left(D_{\mu}\tilde{\phi}D^{\mu}\phi+\frac{3}{4r^{2}}\tilde{\phi}\phi+i\tilde{\psi}\not{D}\psi+\tilde{F}F+\tilde{\phi}\sigma^{2}\phi+i\tilde{\phi}D\phi+i\tilde{\psi}\sigma\psi+i\tilde{\phi}\tilde{\lambda}\psi-i\tilde{\psi}\lambda\phi\right)$ (5.183) where the $\mathfrak{g}$-valued fields in the gauge multiplets act on the chiral multiplet in the representation $R$. This is the “covariantization” of the flat space action for the matter multiplet (see for example [97]), with the addition of the conformal coupling of the scalar field to the curvature, $\frac{3}{4r^{2}}\tilde{\phi}\phi$. The supersymmetry transformations for the chiral multiplet, with respect to the conformal Killing spinors $\epsilon,\tilde{\epsilon}$, are $\displaystyle\delta\phi$ $\displaystyle=\tilde{\epsilon}\psi\qquad\delta\tilde{\phi}=\tilde{\psi}\epsilon$ (5.184) $\displaystyle\delta\psi$ $\displaystyle=(-i\gamma^{\mu}D_{\mu}\phi-i\sigma\phi)\epsilon-\frac{i}{3}\gamma^{\mu}(\nabla_{\mu}\epsilon)\phi+\tilde{\epsilon}F$ $\displaystyle\delta\tilde{\psi}$ $\displaystyle=\tilde{\epsilon}(i\gamma^{\mu}D_{\mu}\tilde{\phi}+i\sigma\tilde{\phi})+\frac{i}{3}(\nabla_{\mu}\tilde{\epsilon})\gamma^{\mu}\tilde{\phi}+\epsilon\tilde{F}$ $\displaystyle\delta F$ $\displaystyle=\epsilon(-i\gamma^{\mu}D_{\mu}\psi+i\lambda\phi+i\sigma\psi)$ $\displaystyle\delta\tilde{F}$ $\displaystyle=(iD_{\mu}\tilde{\psi}\gamma^{\mu}-i\tilde{\lambda}\tilde{\phi}+i\sigma\tilde{\psi})\tilde{\epsilon}.$ The above variations generates a superconformal algebra that closes off-shell: $[\delta_{\epsilon},\delta_{\tilde{\epsilon}}]=-i(\mathcal{L}_{v}+G_{\Lambda}+R_{\alpha}+\Omega_{f})$ (5.185) where $\mathcal{L}_{v}$ is the Lie derivative (translation) along the Killing vector field $v=(\tilde{\epsilon}\gamma^{\mu}\epsilon)\partial_{\mu}$, acting on one forms as $\mathcal{L}_{v}(A)_{\mu}=v^{\nu}\partial_{\nu}A_{\mu}+A_{\nu}\partial_{\mu}v^{\nu}$, and on spinors as $\mathcal{L}_{v}\psi=\nabla_{\nu}\psi-\frac{1}{4}(\nabla_{\mu}v_{\nu})\gamma^{\mu\nu}\psi$. $G_{\Lambda}$ is a gauge transformation with respect to the parameter $\Lambda:=A(v)+\sigma(\tilde{\epsilon}\epsilon)$. $R_{\alpha}$ is a $U(1)^{\mathcal{R}}$ R-symmetry transformation, and $\Omega_{f}$ is a dilatation [50]. The matter coupled action $S_{CS}+S_{m}$ is known to be superconformal at quantum level, but one could also add a superpotential for the matter multiplet. This choice is restricted by the condition of unbroken superconformal symmetry both at classical and at quantum level, since the localization principle works only if the supersymmetry algebra closes off- shell. It turns out that the localization locus is at trivial configurations of the matter sector, thus the precise choice of superpotential does not influence the computation. #### 5.5.2 The supersymmetric Wilson loop The Wilson loop under consideration, in the representation $R$ of the gauge group, is defined as [97] $W_{R}(C)=\frac{1}{\dim{R}}\mathrm{Tr}_{R}\left(\mathcal{P}\exp{\oint_{C}dt\ (iA_{\mu}\dot{C}^{\mu}+\sigma)}\right)$ (5.186) with $C:\mathbb{S}^{1}\to\mathbb{S}^{3}$ a closed curve of tangent vector $\dot{C}$, normalized such that $\|\dot{C}\|=1$. In order to localize its expectation value, we have to consider those curves such that this operator preserves some supersymmetry on the 3-sphere. Its variation under (5.180) is proportional to $\delta W_{R}(C)\propto-\tilde{\epsilon}(\gamma_{\mu}\dot{C}^{\mu}+1)\lambda+\tilde{\lambda}(\gamma_{\mu}\dot{C}^{\mu}-1)\epsilon.$ (5.187) Imposing the vanishing of this expression for all gauginos, we get the following conditions on the conformal Killing spinors, $\tilde{\epsilon}(\gamma_{\mu}\dot{C}^{\mu}+1)=0,\qquad(\gamma_{\mu}\dot{C}^{\mu}-1)\epsilon=0.$ (5.188) We have two more conditions on the conformal Killing spinors, thus the maximum number of solutions is reduced by half. The Wilson loop can at most be invariant under two of the four possible supersymmetry variations, and for that it is called _1/2-BPS_. We can find explicitly one family of supersymmetric Wilson loops and one supersymmetry variation with respect to which we are going to perform the localization procedure. In order to solve the conformal Killing equations and the conditions (5.188), we chose explicitly an orthonormal basis and a corresponding vielbein on $\mathbb{S}^{3}$. Since as a manifold $\mathbb{S}^{3}\cong SU(2)$, we can use Lie theory to describe the geometry on the 3-sphere. In particular, the vielbein can be chosen proportional to the Maureer-Cartan form $\Theta\in T^{}(SU(2))\otimes\mathfrak{su}(2)$,232323Again, we use Roman letters as “flat” indices, and Greek letters as “curved” indices. $e^{i}_{\mu}:=\frac{r}{2}e^{i}(\Theta(\partial_{\mu}))$ (5.189) where $\\{e^{i}\\}$ is a basis of $\mathfrak{su}(2)^{}$, dual to a basis $\\{T_{i}\\}$ of $\mathfrak{su}(2)$.242424Say, the standard basis given by the Pauli matrices, $T_{i}:=\sigma_{i}/\sqrt{2}$. One can check that this vielbein is consistent with the round metric, giving $g_{\mu\nu}=e_{\mu}^{i}e_{\nu}^{j}\delta_{ij}$ (see [50]). Using this orthonormal basis, the spin connection components are $(\omega_{\mu})_{ij}=\frac{1}{r}e^{k}_{\mu}\varepsilon_{ijk}$ (5.190) where $\varepsilon_{ijk}$ is the Levi-Civita symbol. In this basis the conformal Killing spinor equation for $\epsilon$ looks particularly simple, $\left(\partial_{\mu}+\frac{1}{8}(\omega_{\mu})_{ij}[\gamma^{i},\gamma^{j}]\right)\epsilon=\frac{i}{2r}\gamma_{\mu}\epsilon\quad\Leftrightarrow\quad\partial_{\mu}\epsilon=0$ (5.191) where we used the commutator $[\gamma^{i},\gamma^{j}]=2i\left.\varepsilon^{ij}\right._{k}\gamma^{k}$. We see that the components of $\epsilon$ are constants. The corresponding condition for the supersymmetry of the Wilson loop then requires $\gamma_{\mu}\dot{C}^{\mu}$ to be constant too, as the components of the vector field $\dot{C}^{i}$ in the orthonormal frame. This means that the Wilson loop has to describe grat circles on $\mathbb{S}^{3}$. Following [49], we take $\dot{C}$ parallel to one of the $e^{i}$, say $e^{3}$, and the conformal Killing spinor to satisfy $(\gamma_{3}-1)\epsilon=0.$ (5.192) We will consider the one dimensional subalgebra generated by such restricted spinor, and put $\tilde{\epsilon}=0$. #### 5.5.3 Localization: gauge sector We focus now on the localization of the Chern-Simons path integral, without coupling to the matter multiplet. Ignoring the issue of gauge fixing, we would add to the action the localizing term $tS_{loc}=t\delta\mathcal{V}$, with $t\in\mathbb{R}^{+}$ a parameter, $\delta$ being the supersymmetry transformation generated by the conformal Killing spinor $\epsilon$ described in the last section, and $\mathcal{V}$ some fermionic functional whose bosonic part is positive semi-definite. At the end of Section 4.4, we pointed out that the Super Yang-Mills Lagrangian is an example of $\delta$-exact term, so we put $\displaystyle S_{loc}:=2S_{YM}=\int_{\mathbb{S}^{3}}d^{3}x\sqrt{g}\ \mathrm{Tr}\left(i\tilde{\lambda}\gamma^{\mu}D_{\mu}\lambda+\frac{1}{2}F_{\mu\nu}F^{\mu\nu}+D_{\mu}\sigma D^{\mu}\sigma+i\tilde{\lambda}[\sigma,\lambda]+\right.$ (5.193) $\displaystyle+\left.\left(D+\frac{\sigma}{r}\right)^{2}-\frac{1}{2r}\tilde{\lambda}\lambda\right)$ whose bosonic part is indeed positive semi-definite. This localizing term can be derived also from the functional [49] $\mathcal{V}=\int_{\mathbb{S}^{3}}d^{3}x\sqrt{g}\ \mathrm{Tr}\left((\delta\tilde{\lambda})\lambda\right)$ (5.194) analogously to the one used in the previous chapter for the gauge multiplet. $S_{YM}$ being supersymmetric means that $\delta^{2}=0$ on $\mathcal{V}$, making the localization principle applicable. As usual, the limit $t\to\infty$ localizes the path integral on the configurations that make this term vanish: the terms involving bosonic fields are separately non-negative, while the gaugino and its conjugate have to vanish identically. Summarizing, the localization locus is given by $\left\\{\begin{aligned} &\lambda=\tilde{\lambda}=0\\\ &F=0\Rightarrow A=0\ \text{(up to a gauge transformation)}\\\ &\sigma=a\in\mathfrak{g}\ (\mathrm{constant})\\\ &D=-\frac{1}{r}a\end{aligned}\right.$ (5.195) Keeping into account the gauge-fixing procedure (as we should), the ghost $c$, anti-ghost $\tilde{c}$ and Lagrange multiplier $b$ are added to the theory, taking value in the Lie algebra $\mathfrak{g}$, together with the BRST differential $\delta_{B}$ that acts as $\delta_{B}X=-[c,X]\qquad\delta_{B}c=-\frac{1}{2}[c,c]\qquad\delta_{B}\tilde{c}=b\qquad\delta_{B}b=0$ (5.196) where $X$ is any field in the original theory, acted by a gauge transformation parametrized by $c$. The BRST differential is nilpotent, $\delta_{B}^{2}=0$. The total differential $Q:=\delta_{\epsilon}+\delta_{B}$ (5.197) acts now as the equivariant differential for the $(U(1)\rtimes G)$-equivariant cohomology in the BRST-augmented field space. The original CS action is automatically $Q$-closed since it is gauge invariant, so we can combine the localization principle with the gauge-fixing procedure adding to the Lagrangian the term $Q\left((\delta\tilde{\lambda})\lambda-\tilde{c}\left(\frac{\xi}{2}b-\nabla^{\mu}A_{\mu}\right)\right)$ (5.198) where we suppressed the Lie algebra bilinear $\mathrm{Tr}$ for notational convenience. Since the first term is gauge invariant, $\delta_{B}\left((\delta\tilde{\lambda})\lambda\right)=0$, this gives the same localization term as before. If $\delta[\mbox{ghosts}]=0$ on the gauge-fixing subcomplex, the second term gives $Q\left(\tilde{c}\left(\frac{\xi}{2}b-\nabla^{\mu}A_{\mu}\right)\right)=\frac{\xi}{2}b^{2}-b\nabla^{\mu}A_{\mu}+\tilde{c}\nabla^{\mu}D_{\mu}c+\tilde{c}\nabla^{\mu}\delta A_{\mu}.$ (5.199) The first two terms give, upon path integration over $b$, the usual gauge- fixing Lagrangian in the $R_{\xi}$-gauge; the third term is the ghost Lagrangian. The fourth term $\propto\left(\tilde{c}\nabla^{\mu}\tilde{\lambda}\gamma_{\mu}\right)$ does not change the partition function: if we see this term as a perturbation of the gauge-fixed action, all diagrams with insertion of $\left(\tilde{c}\nabla^{\mu}\tilde{\lambda}\gamma_{\mu}\right)$ will vanish, since $\tilde{c}$ is coupled only to $c$ via the propagator but there are no vertices containing $c$. In other words, the fermionic determinant arising from the path integration over ghosts is not changed by this term. The modified localizing term (5.199) is $Q$-closed: the old localizing term because of gauge invariance and supersymmetry, while the gauge-fixing and ghost terms follows by $Q^{2}A_{\mu}=0$ that is easy to check. After path integration over the auxiliary $b$ the limit $t\to\infty$ finally localizes the theory to the same locus (5.195), with ghosts put to zero. Evaluating the classical action at the saddle point configuration, we get $S_{CS}[a]=\frac{k}{4\pi}\int_{\mathbb{S}^{3}}d^{3}x\sqrt{g}\ \mathrm{Tr}\left(-\frac{2}{r}a^{2}\right)=-k\pi r^{2}\mathrm{Tr}(a^{2})$ (5.200) where we used $\mbox{vol}(\mathbb{S}^{3})=2\pi^{2}r^{3}$. The supersymmetric Wilson loop observable (5.186) localizes to $W_{R}(C)=\frac{1}{\dim{R}}\mathrm{Tr}_{R}\left(e^{2\pi ra}\right)$ (5.201) since the curve $C$ is a great circle of radius $r$. Integrating as usual the rescaled fluctuations above the localization configuration, and taking the limit $t\to\infty$ as in (5.76), the partition function and the Wilson loop expectation value are thus given by a finite-dimensional integral over $\mathfrak{g}$ with Gaussian measure, the “matrix model” $\displaystyle Z$ $\displaystyle=\int_{\mathfrak{g}}da\ e^{-k\pi r^{2}\mathrm{Tr}(a^{2})}Z_{1-loop}^{g}[a]$ (5.202) $\displaystyle\langle W_{R}(C)\rangle$ $\displaystyle=\frac{1}{Z\dim{R}}\int_{\mathfrak{g}}da\ e^{-k\pi r^{2}\mathrm{Tr}(a^{2})}Z_{1-loop}^{g}[a]\mathrm{Tr}_{R}\left(e^{2\pi ra}\right).$ As we pointed out in the last section, the integration over the Lie algebra $\mathfrak{g}$ can be reduced over its Cartan subalgebra $\mathfrak{h}$, exploiting the gauge invariance of the matrix model under the adjoint action of $\mathfrak{g}$ itself. This for example means, in the case of a matrix gauge group, that we integrate over the diagonalized matrices “fixing the gauge” of the matrix model. The corresponding Faddeev-Popov determinant is also called _Vandermonde determinant_ , $\prod_{\alpha}\left(\rho_{\alpha}(a)\right)$ (5.203) where the product runs over the roots of $\mathfrak{g}$. There is left an overcounting given by the possible permutations of the roots, the action of the _Weyl group_ $\mathcal{W}$ of $\mathfrak{g}$, cured dividing by its order $\|\mathcal{W}\|$. The path integrals are thus rewritten as $\displaystyle Z$ $\displaystyle=\frac{1}{\|\mathcal{W}\|}\int_{\mathfrak{\mathfrak{h}}}da\ \prod_{\alpha}\left(\rho_{\alpha}(a)\right)e^{-k\pi r^{2}\mathrm{Tr}(a^{2})}Z_{1-loop}^{g}[a]$ (5.204) $\displaystyle\langle W_{R}(C)\rangle$ $\displaystyle=\frac{1}{Z\|\mathcal{W}\|\dim{R}}\int_{\mathfrak{h}}da\ \prod_{\alpha}\left(\rho_{\alpha}(a)\right)e^{-k\pi r^{2}\mathrm{Tr}(a^{2})}Z_{1-loop}^{g}[a]\mathrm{Tr}_{R}\left(e^{2\pi ra}\right).$ Here we summarize the computation of the 1-loop determinant from [49]. For convenience, we put $r=1$ and $\xi=1$. Inserting the contribution of ghosts, the Lagrangian for the localizing term is given by (suppressing the $\mathrm{Tr}$) $\mathcal{L}_{loc}=\frac{1}{2}F_{\mu\nu}F^{\mu\nu}+D_{\mu}\sigma D^{\mu}\sigma+\left(D+\sigma\right)^{2}+i\tilde{\lambda}\not{D}\lambda+i[\tilde{\lambda},\sigma]\lambda-\frac{1}{2}\tilde{\lambda}\lambda+\partial_{\mu}\tilde{c}D^{\mu}c-\frac{1}{2}b^{2}+b\nabla^{\mu}A_{\mu}.$ (5.205) Considering the limit $t\to\infty$, we rescale as usual the fields around the configuration (5.195): $\sigma=a+\sigma^{\prime}/\sqrt{t},\qquad D=-a+D^{\prime}/\sqrt{t},\qquad X=X^{\prime}/\sqrt{t},$ (5.206) where $X$ are all the fields without zero modes, and then rename $\sigma^{\prime}\to\sigma$, $D^{\prime}\to D$, $X^{\prime}\to X$. In the limit, only quadratic terms in the fluctuations survive, $\mathcal{L}_{loc}\sim\frac{1}{2}\partial_{[\mu}A_{\nu]}\partial^{[\mu}A^{\nu]}-[A_{\mu},a]^{2}+(\partial\sigma)^{2}+(D+\sigma)^{2}+i\tilde{\lambda}\not{\nabla}\lambda+i[\tilde{\lambda},a]\lambda-\frac{1}{2}\tilde{\lambda}\lambda+\|\partial\tilde{c}\|^{2}-\frac{1}{2}b^{2}+b\nabla^{\mu}A_{\mu}.$ (5.207) The resulting theory is free, and we can integrate it giving the corresponding 1-loop determinant. We will neglect all overall normalization constant from the Gaussian integrations. The integral over the auxiliary field $b$ gives the gauge fixing term $-\frac{1}{2}(\nabla^{\mu}A_{\mu})^{2}$. The contribution from $D$ is purely Gaussian and can be integrated out removing the corresponding term. The integration over $\sigma$ gives a determinant $\det{(\nabla^{2})}^{-1/2}$, and the (Grassman) integral over the ghosts gives $\det{(\nabla^{2})}$. It is useful to separate the gauge field as (Helmolz- Hodge decomposition) $A_{\mu}=B_{\mu}+\partial_{\mu}\phi$ with $\phi$ scalar and $B_{\mu}$ divergenceless, $\nabla^{\mu}B_{\mu}=0$. With this decomposition, the Lorentz gauge condition becomes $\nabla^{2}\phi=0$, and we can integrate $\phi$ giving a determinant $\det{(\nabla^{2})}^{-1/2}$, that cancels the above two other contributions. We are left with $-B_{\mu}\Delta B^{\mu}-[a,B_{\mu}]^{2}+i\tilde{\lambda}\not{\nabla}\lambda+i[\tilde{\lambda},a]\lambda-\frac{1}{2}\tilde{\lambda}\lambda$ (5.208) where $\Delta$ is the vector Laplacian. Now we use the fact that the path integral can be reduced over the Cartan subalgebra of $\mathfrak{g}$, considering $a\in\mathfrak{h}$, and $B_{\mu}=B_{\mu}^{(\mathfrak{h})}+B_{\mu}^{\alpha}e_{\alpha}$ (5.209) where $B_{\mu}^{(\mathfrak{h})}$ is the component of $B_{\mu}$ along $\mathfrak{h}$, and similarly for the gaugino. This component does not enter in the Lie brackets with $a$, so its contribution to the path integral is independent of $a$, and we drop it. The remaining interesting terms are $\sum_{\alpha}\left(B_{\mu}^{-\alpha}(-\Delta+\rho_{\alpha}(a)^{2})B_{\mu}^{\alpha}+\tilde{\lambda}^{-\alpha}\left(i\not{\nabla}+i\rho_{\alpha}(a)-\frac{1}{2}\right)\lambda^{\alpha}\right)$ (5.210) where the $a$-dependent kinetic terms are clearly identified, and the component fields appearing are real or complex valued scalars and spinors. The Gaussian integration over these fields lead to the determinant factors $Z^{g}_{1-loop}[a]=\prod_{\alpha}\frac{\det\left(i\not{\nabla}+i\rho_{\alpha}(a)-\frac{1}{2}\right)}{\det{\left(-\Delta+\rho_{\alpha}(a)^{2}\right)}^{1/2}}.$ (5.211) Now, using the fact that the eigenvalues of the Laplacian on divergenceless vectors are $(l+1)^{2}$ with degeneracy $2l(l+2)$, and the eigenvalues of $i\not{\nabla}$ are $\pm\left(l+\frac{1}{2}\right)$ with degeneracy $l(l+1)$, where $l\in\mathbb{Z}^{+}$, the corresponding determinants can be written as infinite products $\prod_{\alpha}\prod_{l=1}^{\infty}\frac{(l+i\rho_{\alpha}(a))^{l(l+1)}(-l-1+i\rho_{\alpha}(a))^{l(l+1)}}{((l+1)^{2}+\rho_{\alpha}(a)^{2})^{l(l+2)}}=\prod_{\alpha}\prod_{l=1}^{\infty}\frac{(l+i\rho_{\alpha}(a))^{(l+1)}}{(l-i\rho_{\alpha}(a))^{(l-1)}}$ (5.212) where the equality follows after some simplifications. Since roots come in pairs $(\rho_{\alpha},-\rho_{\alpha})$, taking the square of this one gets $\left(Z^{g}_{1-loop}[a]\right)^{2}=\prod_{\alpha}\prod_{l=1}^{\infty}\frac{(l^{2}+\rho_{\alpha}(a)^{2})^{(l+1)}}{(l^{2}+\rho_{\alpha}(a)^{2})^{(l-1)}}=\prod_{\alpha}\prod_{l=1}^{\infty}\left(l^{2}+\rho_{\alpha}(a)^{2}\right)^{2}.$ (5.213) Collecting a factor $l^{4}$ the product splits in the factorization formula for the hyperbolic sine, $\frac{\sinh(\pi z)}{\pi z}=\prod_{l=1}^{\infty}\left(1+\frac{z^{2}}{l^{2}}\right)$ (5.214) and an $a$-independent divergent part that can be regularized with the zeta- function method, $\prod_{l=1}^{\infty}l^{4}=e^{4\sum_{l=1}^{\infty}\log(l)}=e^{-4\zeta^{\prime}(0)}=e^{2\log(2\pi)}.$ (5.215) Up to an overall normalization constant, the $a$-dependence of the 1-loop determinant is finally given by $Z_{1-loop}^{g}[a]=\prod_{\alpha}\left(\frac{2\sinh(\pi\rho_{\alpha}(a))}{\pi\rho_{\alpha}(a)}\right)$ (5.216) where we see cancellation between the denominator and the Vandermonde determinant (5.203). Collecting the above results, the localization formulas for the partition function and the expectation value of the supersymmetric Wilson loop in the pure CS theory are $\displaystyle Z$ $\displaystyle\sim\frac{1}{\|\mathcal{W}\|}\int_{\mathfrak{\mathfrak{h}}}da\ e^{-k\pi\mathrm{Tr}(a^{2})}\prod_{\alpha}\left(2\sinh(\pi\rho_{\alpha}(a))\right)$ (5.217) $\displaystyle\langle W_{R}(C)\rangle$ $\displaystyle=\frac{1}{Z\|\mathcal{W}\|\dim{R}}\int_{\mathfrak{h}}da\ e^{-k\pi\mathrm{Tr}(a^{2})}\mathrm{Tr}_{R}\left(e^{2\pi a}\right)\prod_{\alpha}\left(2\sinh(\pi\rho_{\alpha}(a))\right)$ $\displaystyle=\frac{1}{\dim{R}}\frac{\int_{\mathfrak{h}}da\ e^{-k\pi\mathrm{Tr}(a^{2})}\mathrm{Tr}_{R}\left(e^{2\pi a}\right)\prod_{\alpha}\left(2\sinh(\pi\rho_{\alpha}(a))\right)}{\int_{\mathfrak{\mathfrak{h}}}da\ e^{-k\pi\mathrm{Tr}(a^{2})}\prod_{\alpha}\left(2\sinh(\pi\rho_{\alpha}(a))\right)}.$ These general localization formulas can be tested comparing their results for specific choices of $G$ to perturbative calculations, for example. In the case of $U(N)$ gauge group, the integral over the Cartan subalgebra is an integral over diagonal matrices $a=\mathrm{diag}(\lambda_{1},\cdots,\lambda_{N})$, and the roots are given by $\rho_{ij}(a)=\lambda_{i}-\lambda_{j}$ for $i\neq j$. The Weyl group is $S_{N}$, thus $\|\mathcal{W}\|=N!$. If we take the Wilson loop in the fundamental representation, from (5.217) we get $\displaystyle Z$ $\displaystyle\sim\frac{1}{N!}\int\left(\prod_{i}d\lambda_{i}\ e^{-k\pi\lambda_{i}^{2}}\right)\prod_{i\neq j}2\sinh(\pi(\lambda_{i}-\lambda_{j})),$ (5.218) $\displaystyle\langle W_{\mathbf{N}}(C)\rangle$ $\displaystyle=\frac{1}{ZN!N}\int\left(\prod_{i}d\lambda_{i}\ e^{-k\pi\lambda_{i}^{2}}\right)\left(e^{2\pi\lambda_{1}}+\cdots+e^{2\pi\lambda_{N}}\right)\prod_{i\neq j}2\sinh(\pi(\lambda_{i}-\lambda_{j})),$ that are sums of Gaussian integrals, and can be computed exactly. The result for the Wilson loop expectation value is $\langle W_{\mathbf{N}}(C)\rangle=\frac{1}{N}e^{-Ni\pi/k}\frac{\sin\left(\frac{\pi N}{k}\right)}{\sin\left(\frac{\pi}{k}\right)},$ (5.219) which is known as the exact result [83], up to the overall phase factor $e^{-Ni\pi/k}$. This kind of phase factors arise in perturbative calculations in the so-called _framing_ of the Wilson loop. A perturbative calculation of the Wilson loop involves computations of correlators of the type $\langle A_{\mu_{1}}(x_{1})A_{\mu_{2}}(x_{2})\cdots\rangle$, where $x_{1},x_{2},\cdots$ are coordinates of points on the image of the curve $C$. This contribution diverges when $x_{1}=x_{2}$, so it is necessary to choose some regularization scheme to perform the computations. For example, considering the 2-point function $\langle A_{\mu_{1}}(x)A_{\mu_{2}}(y)\rangle$, this clashing of points can be avoided requiring that $y$ is integrated over a shifted curve $C_{f}$ such that $C^{\mu}_{f}(\tau)=C^{\mu}(\tau)+\alpha\ n^{\mu}(\tau)$ (5.220) where $n$ is orthogonal to $\dot{C}$. The choice of such an orthogonal component (frame) at every point on the curve is called framing. Even if at the end of the calculation one takes $\alpha\to 0$, this procedure leaves a deformation-dependent term, that in pure $U(N)$ CS is $e^{\frac{i\pi N}{k}\chi(C,C_{f})}$ (5.221) where $\chi(C,C_{f})$ is a topological invariant that takes integer values corresponding to the number of times the path $C_{f}$ winds around $C$. We see that localization produces an expectation value at framing -1 (see also [98] for a detailed discussion about framing). #### 5.5.4 Localization: matter sector We turn now to the result for the localization of the matter-coupled theory. This is of course gauge invariant, so the equivariant differential acts effectively as $Q\sim\delta$, since the ghost sector has been already considered in the previous paragraph. This means that, following the localization principle, we have to extend the matter action with a $\delta$-exact term. We are free to consider the canonical choice (5.81) as in [49], or using the fact that [50] the matter action (5.183) is actually given by a supersymmetry variation, as the case of the YM action. This means that we can consider the localizing terms $tS_{m}+tS_{YM+ghosts}$ or, schematically $t\int\delta\left((\delta\psi)^{\dagger}\psi+\tilde{\psi}(\delta\tilde{\psi})^{\dagger}\right)+tS_{YM+ghosts}$ that have positive semi-definite bosonic parts. The second term in both choices is the one analyzed in the previous paragraph, and gives the same localization locus for the gauge and ghost sector, while both the first terms vanishes for the field configurations $\psi=0,\qquad\phi=0,\qquad F=0.$ (5.222) This means that the classical action of the matter sector does not contribute to the partition function, but only in the 1-loop determinant. Expanding the fields around this configuration and scaling the fluctuations with the usual $1/\sqrt{t}$ factor, we see that there are no couplings to the gauge sector fluctuations that survive in the $t\to\infty$ limit, but only to the zero mode $a$ of $\sigma$. Thus the determinant factorizes as $Z_{1-loop}[a]=Z_{1-loop}^{g}[a]Z_{1-loop}^{m}[a].$ (5.223) If matter is present in different copies of chiral multiplets, in maybe different representations of the gauge group, the determinant factorizes in the same way for each multiplet. The determinant for the matter sector can be computed diagonalizing the the kinetic operators acting on the scalar and the fermion field, after having integrated out the auxiliary $F$, and considering the path integration over the Cartan subalgebra with $a\in\mathfrak{h}$. In particular, the relevant kinetic operators that have to be diagonalized are $K_{b}^{(\rho)}=\left(-\nabla^{2}+\rho(a)^{2}-i\rho(a)+\frac{3}{4}\right),\qquad K_{f}^{(\rho)}=\left(i\not{\nabla}+i\rho(a)\right),$ (5.224) for the (complex) bosonic and fermionic parts, where $a$ is regarded as acting on the representation $R$ with weights $\\{\rho\\}$. The eigenvalues of $-\nabla^{2}$ are $4j(j+1)$ with $j=0,\frac{1}{2},\cdots$ with degeneracy $(2j+1)^{2}$, that we can rewrite as $l(l+2)$ with degeneracy $(l+1)^{2}$ and $l=0,1,\cdots$. The eigenvalues of $i\not{\nabla}$ are $\pm\left(l+\frac{1}{2}\right)$ with degeneracy $l(l+1)$, with $l=1,2,\cdots$. Thus the one loop determinant results, after a change of dummy index and some simplifications $\displaystyle Z_{1-loop}^{m}[a]$ $\displaystyle=\prod_{\rho}\frac{\det(K_{f})}{\det(K_{b})}$ (5.225) $\displaystyle=\prod_{\rho}\prod_{l=1}^{\infty}\frac{\left(l+\frac{1}{2}+i\rho(a)\right)^{l(l+1)}\left(l+\frac{1}{2}-i\rho(a)\right)^{l(l+1)}}{\left(l+\frac{1}{2}+i\rho(a)\right)^{l^{2}}\left(l-\frac{1}{2}-i\rho(a)\right)^{l^{2}}}$ $\displaystyle=\prod_{\rho}\prod_{l=1}^{\infty}\left(\frac{l+\frac{1}{2}+i\rho(a)}{l-\frac{1}{2}-i\rho(a)}\right)^{l}$ This product can be regularized using the zeta-function. We refer to [50] for the details of the computation, and report here the result in the case the fields take value in a self-conjugate representation $R$ of the gauge group:252525For example, if $R=S\oplus S^{}$. $Z_{1-loop}^{m}[a]=\prod_{\rho}\left(2\cosh{(\pi\rho(a))}\right)^{-1/2}$ (5.226) where now $a\in R(\mathfrak{h})$ and $\rho(a)$ is the weight of the Cartan element in the representation $R$. Summarizing, we have seen that the application of the supersymmetric localization principle to the matter-coupled SCS theory on $\mathbb{S}^{3}$ reduces the path integral to a finite-dimensional integral describing a matrix model over the Lie algebra of the theory. Using the notation (5.128), the localization formulas for the partition function and the supersymmetric Wilson loop expectation value, with matter multiplets coming in self-conjugate representations $R_{1}\oplus R_{1}^{},R_{2}\oplus R_{2}^{},\cdots$ are $\displaystyle Z$ $\displaystyle=\frac{1}{\|\mathcal{W}\|}\int_{\mathfrak{\mathfrak{h}}}da\ e^{-k\pi\mathrm{Tr}(a^{2})}\frac{\det_{ad}2\sinh(\pi a)}{\left(\det_{R_{1}}2\cosh{(\pi a)}\right)\left(\det_{R_{2}}2\cosh{(\pi a)}\right)\cdots}$ (5.227) $\displaystyle\langle W_{R}(C)\rangle$ $\displaystyle=\frac{1}{Z\|\mathcal{W}\|\dim{R}}\int_{\mathfrak{h}}da\ e^{-k\pi\mathrm{Tr}(a^{2})}\mathrm{Tr}_{R}\left(e^{2\pi a}\right)\frac{\det_{ad}2\sinh(\pi a)}{\left(\det_{R_{1}}2\cosh{(\pi a)}\right)\left(\det_{R_{2}}2\cosh{(\pi a)}\right)\cdots}$ #### 5.5.5 The ABJM matrix model ABJM theory is a special type of matter-coupled SCS theory in 3-dimensions constructed in [99], that has the interesting property to be dual under the AdS/CFT conjecture to a certain orbifold background in M-theory. It consists of two copies of $\mathcal{N}=2$ SCS theory, each one with gauge group $U(N)$, and opposite levels $k,-k$. In addition, the are four matter (chiral and anti- chiral) supermultiplets $\Phi_{i},\tilde{\Phi}_{i}$, with $i=1,2$, in the bi- fundamental representation of $U(N)\times U(N)$, $(\mathbf{N},\bar{\mathbf{N}})$ and $(\bar{\mathbf{N}},\mathbf{N})$. This field content can be represented as the _quiver_ in Fig. 5.1. Figure 5.1: The quiver for ABJM theory. The two nodes represent the gauge multiplets, with the convention of specifying the level of the CS term. The oriented links represent the matter multiplets in the bi-fundamental and anti- bi-fundamental representations. The superpotential for the matter part is given by $W=\frac{4\pi}{k}(\Phi_{1}\tilde{\Phi}_{1}\Phi_{2}\tilde{\Phi}_{2}-\Phi_{1}\tilde{\Phi}_{2}\Phi_{2}\tilde{\Phi}_{1}),$ (5.228) and this structure actually enhance the supersymmetry of the resulting theory to $\mathcal{N}=6$.262626This is not apparent from the original action, but can be realized noticing that the superpotential has an $SU(2)\times SU(2)$ symmetry that rotates separately the $\Phi_{i}$ and the $\tilde{\Phi}_{i}$. This, combined with the original $SU(2)^{\mathcal{R}}$ symmetry of the theory, gives an $SU(4)\cong Spin(6)$ symmetry that acts non-trivially on the supercharges. Thus the final theory has to have an enhanced $\mathcal{N}=6$ supersymmetry. If now $a=\mathrm{diag}(\lambda_{1},\cdots,\lambda_{N},\hat{\lambda}_{1},\cdots,\hat{\lambda}_{N})$, the weights in the bi-fundamental representations are $\rho_{i,j}^{(N,\bar{N})}(a)=\lambda_{i}-\hat{\lambda}_{j},\qquad\rho_{i,j}^{(\bar{N},N)}(a)=\hat{\lambda}_{j}-\lambda_{i}.$ (5.229) Plugging this information into (5.227), the partition function in this case localizes to the following matrix model, $Z\sim\frac{1}{N!N!}\int\left(\prod_{i}d\lambda_{i}d\hat{\lambda}_{i}\ e^{-k\pi(\lambda_{i}^{2}-\hat{\lambda}_{i}^{2})}\right)\frac{\prod_{i\neq j}\left(2\sinh(\pi(\lambda_{i}-\lambda_{j}))2\sinh(\pi(\hat{\lambda}_{i}-\hat{\lambda}_{j}))\right)}{\prod_{i,j}\left(2\cosh(\pi(\lambda_{i}-\hat{\lambda}_{j}))\right)}.$ (5.230) The circular Wilson loop under consideration can be called now _1/6 BPS_ with respect to the enhanced supersymmetry of the model. Its expectation value in the fundamental representation is obtained by plugging a factor $(1/N)\sum_{i}e^{2\pi\lambda_{i}}$ as before. This matrix model cannot be solved exactly as in the case of the pure CS discussed above, but can be studied in the $N\to\infty$ limit with the saddle-point technique showed in Section 5.4.5 [50, 94]. We also mention that, in this particular theory with enhanced $\mathcal{N}=6$ supersymmetry, it was possible to construct a _1/2 BPS_ Wilson loop (so invariant under half of the $\mathcal{N}=6$ supersymmetry algebra). The latter can be solved applying the same localization scheme that brings to the matrix model describing the 1/6 BPS Wilson loop presented above [100, 101]. A compact review introducing the state of the art on recent results about supersymmetric Wilson loops in ABJM and related theories can be found in [84]. ## Chapter 6 Non-Abelian localization and 2d YM theory In this chapter we are going to summarize the result obtained mainly in [15] by Witten. This was the first attempt in the physics literature of extending the equivariant localization formalism to possibly non-Abelian group actions. In that work, a modified definition of equivariant integration was defined, and this allowed for an extension of the same procedure discussed in Chapter 3 to show the localization property of integrals computed over spaces with generic symmetry group $G$. This new formalism was applied to the study of 2-dimensional Yang-Mills (YM) theory over a Riemann surface, a relatively simple model from the physical point of view, but with a very rich underlying mathematical structure. In the following, we are going first to review the geometry of this special model, in connection with the symplectic geometry introduced in Section 3.3, as a motivation for the more mathematical discussion about the Witten’s equivariant integration and non-Abelian localization principle that will follow. Next, we will review the ideas underlying the application of this new localization principle to the YM theory, and how this application results in a “mapping” between this model and a suitable topological theory, establishing the topological nature of the YM theory in the weak coupling limit. In the final section, we will summarize the interpretation given by the localization framework to the already existing solution for the partition function of this model. As we pointed out in the Introduction, other generalizations of the Duistermaat-Heckman theorem to non-Abelian Hamiltonian systems also appeared in the mathematical literature, as the result obtained by Jeffrey and Kirwan in [16]. Other applications of this extended formalism followed, and Witten’s approach was used for example more recently to describe Chern-Simons theories over a special class of 3-manifolds in [102]. ### 6.1 Prelude: moment maps and YM theory In the next section we are going to review Witten’s extension of the equivariant localization principle to possibly non-Abelian group actions, and a generalization of the DH formula in this direction. In [15] this was applied to reinterpret the weak coupling limit of pure YM theory on a Riemann surface. This theory is exactly solvable, in the sense that its partition function can be expressed in closed form, and its zero-coupling limit is known to describe a topological field theory. These features make 2-dimensional YM theory very appealing from the mathematical structure it carries, and make it possible to compare results or interpretations obtained via this “new” localization method with already existing solutions of the problem. We are going to discuss more about the topological interpretation of 2d YM theory later, while in this section we review some results introduced by Atiyah and Bott [103] about the symplectic structure underlying this special QFT. This can be useful to contextualize the generic discussion of the next section, and it prepares the ground for the formal application of the non- Abelian localization principle. We start by considering the partition function of YM theory on a compact orientable Riemannian manifold $\Sigma$ of arbitrary dimension, $\displaystyle Z(\epsilon)$ $\displaystyle=\frac{1}{\mathrm{vol}(\mathcal{G}(P))}\left(\frac{1}{2\pi\epsilon}\right)^{\dim(\mathcal{G})/2}\int_{\mathcal{A}(P)}DA\ e^{-S[A]},$ (6.1) $\displaystyle S[A]$ $\displaystyle=-\frac{1}{2\epsilon}\int_{\Sigma}\mathrm{Tr}(F^{A}\wedge\star F^{A}).$ Here $\epsilon:=g_{YM}^{2}$ is the square of the YM coupling constant. To describe the rest of the ingredients, let us recall the geometry underlying the gauge theory (to fill some of the details, see Appendix A.1). The dynamical field here is the _connection_ $A\in\Omega(P;\mathfrak{g})$ on a principal $G$-bundle $P\xrightarrow{\pi}\Sigma$, where $G$ is a compact connected Lie group with Lie algebra $\mathfrak{g}$. The path integral is thus taken over the space $\mathcal{A}(P)$ of $G$-equivariant vertical 1-forms with values in $\mathfrak{g}$, that is naturally an _affine space_ modeled on the infinite-dimensional vector space $\mathfrak{a}$ of $G$-equivariant horizontal 1-forms with values in $\mathfrak{g}$. This gives to $\mathcal{A}(P)$ the structure of an infinite-dimensional manifold, whose tangent spaces are $T_{A}\mathcal{A}(P)\cong\mathfrak{a}\cong\Omega^{1}(\Sigma;\mathrm{ad}(P))$, where we identified horizontal forms over $P$ with forms over the base $\Sigma$.111Recall that horizontality means essentially to have components only in the “directions” of the base space, and the $G$-equivariance ensures the right transformation behavior as forms valued in the _adjoint bundle_ $\mathrm{ad}(P)$, the associated bundle to $P$ that has $\mathfrak{g}$ as typical fiber. Thus $\mathfrak{a}\cong\Omega^{1}(\Sigma;\mathrm{ad}(P))$. In other words, any vector field $\alpha\in\Gamma(T\mathcal{A}(P))$ can be expanded locally as $\alpha=\alpha_{\mu}^{a}T_{a}\otimes dx^{\mu},\qquad\alpha_{\mu}^{a}\in C^{\infty}(\Sigma\times\mathcal{A}(P)),$ (6.2) with coefficients that depend on the point $A\in\mathcal{A}(P)$ and $p\in\Sigma$. The curvature $F^{A}=dA+\frac{1}{2}[A\stackrel{{\scriptstyle\wedge}}{{,}}A]$ of the connection $A$ is a horizontal 2-form over $P$, so we can identify it as a 2-form on the adjoint bundle without loss of information, $F^{A}\in\Omega^{2}(\Sigma;\mathrm{ad}(P))$. As such, it can be integrated as a differential form over $\Sigma$. In the action $S[A]$, “$\mathrm{Tr}$” represents a (negative definite) invariant inner product on $\mathfrak{g}$, and $\star$ is the Hodge dual operation, that is identified by the presence of a metric on $\Sigma$.222The definition of the Hodge star is, implicitly, $\alpha\wedge\star\beta=g^{-1}(\alpha,\beta)\omega$ for any $\alpha,\beta\in\Omega^{k}(\Sigma)$. Here $g^{-1}$ is the “inverse” metric on $\Sigma$, that extends multi-linearly its action on every tangent space as $g^{-1}(\alpha,\beta)=g^{\mu_{1}\nu_{1}}\cdots g^{\mu_{k}\nu_{k}}\alpha_{\mu_{1}\cdots\mu_{k}}\beta_{\nu_{1}\cdots\nu_{k}}$. $\omega$ is a volume form (that can be induced by $g$, for example). The Hodge star satisfies the property $\star^{2}\alpha=(-1)^{k(\dim(\Sigma)-k)}\alpha$. $\mathcal{G}(P)\cong\Omega^{0}(\Sigma;\mathrm{Ad}(P)$ is the group of gauge transformations, that is locally equivalent to the space of $G$-valued functions over $\Sigma$, and acts naturally on $\mathcal{A}(P)$. If $\phi\in Lie(\mathcal{G}(P))\cong\Omega^{0}(\Sigma;\mathrm{ad}(P))$ is an element of the Lie algebra of infinitesimal gauge transformations, its associated fundamental vector field at the point $A\in\mathcal{A}(P)$ is $\underline{\phi}_{A}\equiv\delta_{\phi}A=\nabla^{A}\phi=d\phi+[A,\phi].$ (6.3) The path integral measure $DA$ can be defined formally as the Riemannian measure induced by a metric on the affine space $\mathcal{A}(P)$. The latter can be induced by the metrics on $\Sigma$ and on $\mathfrak{g}$, and defined pointwise in $\mathcal{A}(P)$ as $(\alpha,\beta)_{A}:=-\int_{\Sigma}\mathrm{Tr}(\alpha^{A}\wedge\star\beta^{A})$ (6.4) for every $\alpha^{A},\beta^{A}\in\Omega^{1}(\Sigma;\mathrm{ad}(P))$. With this definition, the YM action can be rewritten as $S[A]=\frac{1}{2\epsilon}(F,F)_{A}.$ (6.5) We can now specialize the discussion to the case in which $\dim(\Sigma)=2$, i.e. the base space is a Riemann surface. It is a well-known fact in geometry that any Riemann surface is a _Kähler manifold_ : it admits a Riemannian metric $g$, a symplectic form $\omega$ (that can be a choice of volume form), and a complex structure $J$ such that the compatibility condition $g(\cdot,\cdot)=\omega(\cdot,J(\cdot))$ is satisfied.333A complex structure on a vector space $V$ is an isomorphism $J:V\to V$ such that $J^{2}=-id_{V}$. It intuitively plays the role of “multiplication by $i$” when one considers the complexified $V^{\mathbb{C}}:=V\otimes\mathbb{C}$, allowing for a decomposition of $V^{\mathbb{C}}$ in a holomorphic subspace (generated by the eigenvectors with eigenvalue $+i$) and anti-holomorphic subspace (generated by the eigenvectors with eigenvalue $-i$). A manifold $M$ has _almost complex structure_ if there is a tensor $J\in\Gamma(T^{1}_{1}M)$ that acts as a complex structure in every tangent space. If the holomorphic decomposition can be extended on an entire neighborhood of every point by a suitable choice of coordinates, $M$ has _complex structure_ , and admits an atlas of holomorphic coordinates. Riemann surfaces can thus be thought as 2-dimensional real manifolds, or 1-dimensional complex manifolds. This special property holds also for $\mathcal{A}(P)$, since in addition to the metric (6.4) we can define the symplectic form $\Omega\in\Omega^{2}(\mathcal{A}(P))$ such that $\Omega_{A}(\alpha,\beta):=-\int_{\Sigma}\mathrm{Tr}(\alpha^{A}\wedge\beta^{A}),$ (6.6) and the complex structure on $T\mathcal{A}(P)$ is provided by the Hodge duality, $\star:\Omega^{1}(\Sigma;\mathrm{ad}(P))\to\Omega^{1}(\Sigma;\mathrm{ad}(P))$ such that $\star^{2}=-1$. Then the compatibility condition is immediately satisfied, since $(\cdot,\cdot)=\Omega(\cdot,\star(\cdot))$. The fact that $\Omega$ is symplectic can be seen by noticing that, in any basis, it has constant components (i.e. independent from $A\in\mathcal{A}(P)$): $\Omega_{ab}^{\mu\nu}(A)=\Omega_{A}(T_{a}\otimes dx^{\mu},T_{b}\otimes dx^{\nu})=-\mathrm{Tr}(T_{a}T_{b})\varepsilon^{\mu\nu}\left(\int_{\Sigma}dx^{1}dx^{2}\right)\quad\in\mathbb{R}.$ (6.7) The non-degeneracy follows from the non-degeneracy of $\mathrm{Tr}$ and of $\int_{\Sigma}$, and the skew-symmetry is obvious from the definition. Thus $\mathcal{A}(P)$ is Kähler. For our applications, we focus on the fact that $\mathcal{A}(P)$ has now a canonical symplectic structure. It is natural to wonder if it possible to extend all the machinery that we introduced in Section 3.3 also to this case, and in particular if the $\mathcal{G}(P)$-action on $\mathcal{A}(P)$ results to be symplectic or Hamiltonian with respect to $\Omega$. The answer was given by in [103], and we state it in the following theorem. ###### Theorem 6.1.1 (Atiyah-Bott). In 2-dimensions, the group $\mathcal{G}(P)$ of gauge transformations acts in an Hamiltonian way on $\mathcal{A}(P)$, with a moment map identified by the curvature $F$. ###### Proof. To see this, let us introduce the moment map as $\mu:Lie(\mathcal{G}(P))\to C^{\infty}(\mathcal{A}(P))$ such that $\mu_{\phi}(A):=\langle F^{A},\phi\rangle=-\int_{\Sigma}\mathrm{Tr}(F^{A}\phi),$ (6.8) and check that the Hamiltonian property is satisfied. For every $\alpha\in\Gamma(T\mathcal{A}(P))$ and $\phi\in Lie(\mathcal{G}(P))$, we compute $\displaystyle(\iota_{\phi}\Omega_{A})(\alpha)$ $\displaystyle=\Omega_{A}(\underline{\phi},\alpha)=-\int_{\Sigma}\mathrm{Tr}(\nabla^{A}\phi\wedge\alpha^{A})=\int_{\Sigma}\mathrm{Tr}(\phi\nabla^{A}\alpha^{A}),$ (6.9) $\displaystyle\left.\delta\mu_{\phi}\right\|_{A}(\alpha)$ $\displaystyle=-\int_{\Sigma}\mathrm{Tr}\left(F^{A+\alpha}\phi-F^{A}\phi\right)=-\int_{\Sigma}\mathrm{Tr}\left(\phi\nabla^{A}\alpha^{A}\right),$ (6.10) where $\delta$ is the de Rham differential on $\mathcal{A}(P)$, that acts in the usual sense of variational calculus. We see that $\iota_{\phi}\Omega=-\delta\mu_{\phi}$, thus $\mu$ provides a correct moment map for the $\mathcal{G}(P)$-action. If we identify $Lie(\mathcal{G}(P))$ with $Lie(\mathcal{G}(P))^{}$ through the pairing $\langle\cdot,\cdot\rangle$ introduced above, and regard the curvature $F:\mathcal{A}(P)\to\Omega^{2}(\Sigma;\mathrm{ad}(P))$ as an element of $C^{\infty}(\mathcal{A}(P))\otimes Lie(\mathcal{G}(P))^{}$, we can simply write that $\mu\equiv F$. ∎ Another corollary of $\mathcal{A}(P)$ being Kähler is that the path integral measure $DA$ is formally equivalent to the Liouville measure induced from $\Omega$, since by compatibility of the structures the latter is equivalent to the Riemannian measure induced by $(\cdot,\cdot)$. Since we are working on an infinite-dimensional space, we can write this measure formally as $DA=\exp(\Omega),$ (6.11) as we did in (3.43) but with $n=\infty$. With this identification, we see that the path integral of the 2-dimensional YM theory acquires the very suggestive form $Z(\epsilon)\propto\int_{\mathcal{A}(P)}\exp\left(\Omega-\frac{1}{2\epsilon}(\mu,\mu)\right).$ (6.12) This path integral resembles very much an infinite-dimensional version of the type of integrals we treated when discussing the Duistermaat-Heckman localization formula in Section 3.3, but with the fundamental difference that now the exponent of the integrand is not the moment map, but its square. We will return to this point in the next section. Here we notice that in the weak coupling limit $\epsilon\to 0$ the path integral will receive contributions from the saddle points of the action $S=\frac{1}{2\epsilon}(\mu,\mu)$, that is the space of solutions of the classical equations of motion $\nabla^{A}\star F^{A}=0$. Every one of these contributions brings roughly a term that decays as $\sim\exp\left(-1/\epsilon\right)$ to the partition function, the main one being determined by the absolute minimum at $\mu=0$, the subspace of flat connections $\mu^{-1}(0)\subset\mathcal{A}(P)$. Eliminating the redundancy from the gauge freedom of the theory, the most interesting piece of the _physical_ field space, especially in the weak coupling limit, is thus determined by the quotient $\mathcal{A}_{0}:=\faktor{\mu^{-1}(0)}{\mathcal{G}(P)},$ (6.13) or in other words, when computing the path integral one is interested in the $\mathcal{G}(P)$-equivariant cohomology of $\mu^{-1}(0)$, $H_{\mathcal{G}}^{}(\mu^{-1}(0))\cong H^{}(\mathcal{A}_{0})$. The quotient $\mathcal{A}_{0}$ is the _moduli space of flat connections_. It turns out that this space has a nice interpretation in symplectic geometry in terms of _symplectic reduction_. A theorem by Marsden-Weinstein-Meyer (MWM) [104, 105, 34] in fact states that, in a generic Hamiltonian $G$-space $(M,\omega,G,\mu)$, if the zero-section of the moment map $\mu^{-1}(0)\subset M$ is acted on _freely_ by $G$, then the base space $M_{0}:=\mu^{-1}(0)/G$ of the principal $G$-bundle $\mu^{-1}(0)\xrightarrow{\pi}M_{0}$ is a symplectic manifold, with symplectic form $\omega_{0}\in\Omega^{2}(M_{0})$ satisfying $\left.\omega\right\|_{\mu^{-1}(0)}=\pi^{}(\omega_{0}).$ (6.14) In other words, the restriction of $\omega$ to $\mu^{-1}(0)$ is a basic form, completely determined by a symplectic form $\omega_{0}$ on the base space. The space $(M_{0},\omega_{0})$ is called Marsden–Weinstein quotient, symplectic quotient or symplectic reduction of $M$ by $G$.444Symplectic reduction in classical mechanics on $M=\mathbb{R}^{2n}$ occurs when one of the momenta is an integral of motion, $0=\dot{p}_{n}=-\partial_{n}H$. In that case, one can solve the system in the reduced coordinates $(q^{1},\cdots,q^{n-1},p_{1},\cdots,p_{n-1})$ and then solve for the $n^{th}$ coordinate separately. The MWM theorem essentially generalizes this process in a fully covariant setting. Returning to the case of YM theory, this means that in the limit $\epsilon\to 0$, when the path integral is reduced to $\mathcal{A}_{0}$ by gauge fixing, the symplectic form can be reduced without loss of information on this base space. In the next section we will see that, applying localization, this is extended to the whole exponential. ### 6.2 A localization formula for non-Abelian actions In the last section we found an Hamiltonian interpretation of the system $(\mathcal{A}(P),\Omega,\mathcal{G}(P),\mu\equiv F)$ for YM theory on a 2-dimensional Riemann surface $\Sigma$. Here we would like to make contact with the DH formula, that we described for analogous systems in finite- dimensional geometry. We notice that the main differences with the case treated in Section 3.3 are essentially two: $\mathcal{G}(P)$ is non-Abelian in general for non-Abelian gauge groups $G$, and the path integral is not in the form of an oscillatory integral of the DH type. Indeed, schematically we have $\text{DH}\to\int\exp{\left(\omega+i\mu\right)},\qquad\quad\text{YM}\to\int\exp{\left(\Omega-\frac{1}{2}\|\mu\|^{2}\right)}.$ (6.15) In the following, we will describe the solution proposed in [15] to generalize the DH formula to the non-Abelian case starting from the first integral in (6.15), and how this procedure can be used to recover the second one, of the YM type. We consider a generic Hamiltonian system $(M,\omega,G,\mu)$ with compact semisimple Lie group $G$ of dimension $\dim(G)=s$, and the associated Cartan model defined by the space $\Omega_{G}(M)=(S(\mathfrak{g}^{})\otimes\Omega(M))^{G}$ of equivariant forms, on which we defined the action of the extended operators555We adopt Witten’s conventions and substitute $\phi^{a}\mapsto i\phi^{a}$ in the definition of the Cartan differential, analogously to the DH case of Section 3.3. $\displaystyle d_{C}$ $\displaystyle=1\otimes d-i\phi^{a}\otimes\iota_{a},$ (6.16) $\displaystyle\mathcal{L}_{a}$ $\displaystyle\equiv 1\otimes\mathcal{L}_{a}+\mathcal{L}_{a}\otimes 1\qquad\text{with}\quad\mathcal{L}_{a}\phi^{b}=f^{b}_{ac}\phi^{c},$ $\displaystyle\iota_{a}$ $\displaystyle\equiv 1\otimes\iota_{a}.$ An element $\alpha\in\Omega_{G}(M)$ is an invariant polynomial in the generators $\phi^{a}$ of $S(\mathfrak{g}^{})$, with differential forms on $M$ as coefficients. This means that integration over $M$ provides a map in equivariant cohomology of the type $\int_{M}:H^{}_{G}(M)\to S(\mathfrak{g}^{})^{G},$ (6.17) or in other words that the integral of an equivariant form is in general a polynomial in the $\phi^{a}$. This is not quite satisfactory, as we would like an integration that generalizes the standard de Rham case, giving a map $H^{}_{G}(M)\to\mathbb{R}$ (or $\mathbb{C}$). In the case of $G=U(1)$ we often solved this problem by setting the unique generator $\phi=-1$ (or $\phi=i$ in this conventions), thus constructing a map in the localized cohomology, $\int_{M}:H_{U(1)}^{}(M)_{\phi}\to\mathbb{R}$. Here in the non- Abelian case, the trick of algebraic localization is not so trivial in practice, and we avoid it. An alternative and fruitful idea to saturate the $\phi$-dependence is to make them dynamical variables, and integrate over them too. Since $\phi^{a}$ can be regarded as an Euclidean coordinate over $\mathfrak{g}$, this means defining an integration over $M\times\mathfrak{g}$. As a vector space, the Lie algebra has a natural measure $d^{s}\phi$ that is unique up to a multiplicative factor. We fix that factor by choosing a (positive-definite) inner product $(\cdot,\cdot)$ on $\mathfrak{g}^{}$ and setting666The inner product on $\mathfrak{g}^{}$ is induced from an inner product on $\mathfrak{g}$, and when we write $(\phi,\phi)$ we really mean $\sum_{a}(\phi^{a},\phi^{a})$. This can be stated more formally defining $\phi:=\phi^{a}\otimes T_{a}\in\mathfrak{g}^{}\otimes\mathfrak{g}$, and then letting act the inner- product on the $T_{a}$’s, normalized in order to produce a Kronecker delta. We avoid this cumbersome notation, since the action of the various inner products is always clear from the context. $\int_{\mathfrak{g}}d^{s}\phi\ e^{-\frac{\epsilon}{2}(\phi,\phi)}=\left(\frac{2\pi}{\epsilon}\right)^{s/2},$ (6.18) essentially as we did in (5.132). Since our goal is to integrate equivariant forms that have polynomial dependence on the $\phi^{a}$, or at most expressions of the form $\exp(\phi^{a}\otimes\mu_{a})$ that have exponential dependence, integrating over $\mathfrak{g}$ with the bare measure $d^{s}\phi$ would produce possible divergences. To ensure convergence of these class of functions, the _equivariant integration_ is defined as [15] $\boxed{\int_{M\times\mathfrak{g}}\alpha:=\frac{1}{\mathrm{vol}(G)}\int_{M}\int_{\mathfrak{g}}\frac{d^{s}\phi}{(2\pi)^{s}}e^{-\frac{\epsilon}{2}(\phi,\phi)}\alpha}$ (6.19) where $\epsilon$ is inserted as a regulator. Notice that in general the limit $\epsilon\to 0$ is not well-defined, for what we said above. With this enhanced definition of equivariant integration of elements of $\Omega_{G}(M)$, we can apply the equivariant localization principle to the present case. Let $\alpha\in\Omega_{G}(M)$ be an equivariantly closed form, so that $d_{C}\alpha=0$, and choose an equivariant 1-form $\beta\in\Omega_{G}^{1}(M)=\Omega^{1}(M)^{G}$. The latter is independent on $\phi^{a}$, and plays the role of the localization 1-form. By the same arguments of Section 3.1, $\alpha$ and $\alpha e^{d_{C}\beta}$ are representatives of the same equivariant cohomology class in $H^{}_{G}(M)$, and we can deform the integral of $\alpha$ as $I[\alpha;\epsilon]:=\int_{M\times\mathfrak{g}}\alpha=\int_{M\times\mathfrak{g}}\alpha e^{td_{C}\beta}\qquad\forall t\in\mathbb{R}.$ (6.20) In particular, taking the limit $t\to\infty$, this integral localizes on the critical point set of the localization 1-form $\beta$. This can be seen simply by expanding the definition of equivariant integration from (6.20), $\displaystyle I[\alpha;\epsilon]$ $\displaystyle=\frac{1}{\mathrm{vol}(G)}\int_{M}\int_{\mathfrak{g}}\frac{d^{s}\phi}{(2\pi)^{s}}\ \alpha\exp\left(-\frac{\epsilon}{2}(\phi,\phi)-it\phi^{a}(\iota_{a}\beta)+td\beta\right)$ (6.21) $\displaystyle=\frac{1}{\mathrm{vol}(G)}\int_{M}\int_{\mathfrak{g}}\frac{d^{s}\phi}{(2\pi)^{s}}\ \alpha\exp\left(-\frac{\epsilon}{2}(\phi,\phi)-\frac{t^{2}}{2\epsilon}\sum_{a}(\iota_{a}\beta)^{2}+td\beta\right),$ where in the second line we completed the square and shifted variable in the $\phi$-integral. Since the term $td\beta$ gives a polynomial dependence on $t$ (by degree reasons, it is expanded up to a finite order), the limit $t\to\infty$ converges and makes the integral localize on the critical points of $\iota_{a}\beta$. This shows the localization property of equivariant integrals in the non-Abelian setting. If for example we suppose $\alpha$ to be independent on the $\phi^{a}$, we can perform the Gaussian integration to further simplify $I[\alpha;\epsilon]$, $I[\alpha;\epsilon]=\frac{1}{\mathrm{vol}(G)(2\pi\epsilon)^{s/2}}\int_{M}\alpha\exp\left(-\frac{t^{2}}{2\epsilon}\sum_{a}(\iota_{a}\beta)^{2}+td\beta\right).$ (6.22) We now apply the above non-Abelian localization principle to the special case in which $\alpha=\exp(\omega-i\phi^{a}\otimes\mu_{a})$, i.e. generalizing the DH formula of Section 3.3. We will suppress tensor products in the following, for notational convenience. First of all, it is straightforward to see that this form is equivariantly closed, $d_{C}e^{\omega-i\phi^{a}\mu_{a}}\propto d_{C}(\omega-i\phi^{a}\mu_{a})=-i\phi^{a}\iota_{a}\omega-i\phi^{a}d\mu_{a}=-i\phi^{a}\iota_{a}\omega+i\phi^{a}\iota_{a}\omega=0.$ (6.23) The DH oscillatory integral becomes, following the same steps of (6.20) and (6.21), $\displaystyle Z(\epsilon)$ $\displaystyle=\int_{M\times\mathfrak{g}}\frac{\omega^{n}}{n!}e^{-i\phi^{a}\mu_{a}}$ (6.24) $\displaystyle=\frac{1}{\mathrm{vol}(G)}\int_{M}\int_{\mathfrak{g}}\frac{d^{s}\phi}{(2\pi)^{s}}\exp\left(\omega-i\phi^{a}\mu_{a}-\frac{\epsilon}{2}(\phi,\phi)+t(d\beta-i\phi^{a}(\iota_{a}\beta))\right)$ $\displaystyle=\frac{1}{\mathrm{vol}(G)(2\pi\epsilon)^{s/2}}\int_{M}\exp\left(\omega-\frac{1}{2\epsilon}(\mu,\mu)-\frac{t^{2}}{2\epsilon}\sum_{a}(\iota_{a}\beta)^{2}+\frac{it}{\epsilon}\sum_{a}\mu_{a}(\iota_{a}\beta)\right),$ and it is independent of $t$. Specializing to the case $t=0$, we get $\boxed{\int_{M\times\mathfrak{g}}\frac{\omega^{n}}{n!}e^{-i\phi^{a}\mu_{a}}=\frac{1}{\mathrm{vol}(G)(2\pi\epsilon)^{s/2}}\int_{M}\exp\left(\omega-\frac{1}{2\epsilon}(\mu,\mu)\right)},$ (6.25) that shows the equivalence of the YM type partition function and the equivariant integral of the DH type! If instead we take the limit $t\to\infty$, we see that the integral localizes on the critical points of $(\iota_{a}\beta)$. With a smart choice of localization 1-form, we can show that this localization locus coincides with the critical point set of the function $S:=\frac{1}{2}(\mu,\mu)$. ###### Proof. Since $(M,\omega)$ is symplectic, it admits an almost complex structure $J\in\Gamma(T^{1}_{1}M)$ and a Riemannian metric $G\in\Gamma(T^{0}_{2}M)$ such that $\omega(\cdot,J(\cdot))=G(\cdot,\cdot)$ (see [34], proposition 12.6).777In the case of 2-dimensional YM theory, we recall that $J\equiv\star$ is the Hodge duality operator. We pick the localization 1-form $\beta:=dS\circ J=J^{\sigma}_{\nu}\partial_{\sigma}S\ dx^{\nu}=J^{\sigma}_{\nu}\sum_{a}\mu_{a}(\partial_{\sigma}\mu_{a})\ dx^{\nu},$ and the localization condition $\iota_{a}\beta=0$. Now we use the compatible metric $G$, that has components $G_{\mu\nu}=\omega(\partial_{\mu},J(\partial_{n}u))=\omega_{\mu\sigma}J^{\sigma}_{\nu}$. We consider its “inverse” $G^{-1}$ acting on $T^{}M$ with components $G^{\mu\nu}=J^{\mu}_{\sigma}\omega^{\nu\sigma}$, where $\omega^{\nu\sigma}$ are the components of the “inverse” symplectic form, and compute the norm of the 1-form $dS$, $G^{-1}(dS,dS)=(\partial_{\mu}SJ^{\mu}_{\sigma})\omega^{\nu\sigma}\partial_{\nu}S=\beta_{\sigma}\sum_{a}\mu_{a}(\omega^{\nu\sigma}\partial_{\nu}\mu_{a})=-\sum_{a}\mu_{a}\beta_{\sigma}(T_{a})^{\sigma}=-\sum_{a}\mu_{a}(\iota_{a}\beta)=0,$ where we used the Hamiltonian equation $d\mu_{a}=-\iota_{a}\omega$ and the localization condition $\iota_{a}\beta=0$. By the non-degeneracy of $G$, this condition is equivalent to $dS=0$, that precisely identifies the critical points of $S$. ∎ Rephrasing the above result in the language of the last section, we just showed in general terms that the 2-dimensional YM partition function localizes on the moduli space of solutions of the EoM, meaning that this theory is essentially classical. We remark again that this localization locus consists of two qualitatively different types of points: those that minimize absolutely $S$, that is $\mu^{-1}(0)\subset M$, and the higher extrema with $\mu\neq 0$. The former ones in the gauge theory are the flat connections, and they give the dominant contribution to the partition function. The latter ones decays exponentially in the limit $\epsilon\to 0$ as $\sim\exp(-S/\epsilon)$. In general thus the partition function can be written as a sum of terms coming from all these disconnected regions of $M$, $Z(\epsilon)=\sum_{n}Z_{n}(\epsilon).$ (6.26) Let us consider the dominant piece $Z_{0}(\epsilon)$ coming from $\mu^{-1}(0)$, that we interpret in the gauge theory as the rough answer in the weak coupling limit, and that we can select by restricting the integration over $M$ to a suitable neighborhood $N$ of $\mu^{-1}(0)$, $Z_{0}(\epsilon)=\frac{1}{\mathrm{vol}(G)}\int_{N}\int_{\mathfrak{g}}\frac{d^{s}\phi}{(2\pi)^{s}}\exp\left(\omega-i\phi^{a}\mu_{a}-\frac{\epsilon}{2}(\phi,\phi)+td_{C}(dS\circ J)\right),$ (6.27) where we inserted the localization 1-form such that the $t\to\infty$ limit identifies the critical locus $\mu^{-1}(0)$. Cohomological arguments show that, if $G$ acts freely on $\mu^{-1}(0)$, this integral retracts on the symplectic quotient $M_{0}:=\mu^{-1}(0)/G$, giving $Z_{0}(\epsilon)=\int_{M_{0}}\exp\left(\omega_{0}+\epsilon\Theta\right)$ (6.28) for some 4-form $\Theta\in\Omega^{4}(M_{0})$. In particular, we see that in the weak coupling limit $\epsilon\to 0$, $Z_{0}(0)$ gives the volume of the symplectic quotient $M_{0}$. ###### Argument for (6.28). The precise proof is technical and it can be found in [15], we only sketch the main instructive ideas here. The neighborhood $N$ is chosen small enough to be preserved by the $G$-action, and represents the split with respect to the normal bundle we used in Section 4.2. Thus it retracts equivariantly onto $\mu^{-1}(0)$, meaning that it is homotopic to $\mu^{-1}(0)$ and that the homotopy commutes with the $G$-action. First of all we recall what we noticed at the end of the last section: if $G$ acts freely on $\mu^{-1}(0)$ the MWM theorem tells us that the symplectic form retracts on the symplectic quotient $M_{0}:=\mu^{-1}(0)/G$, so it does not contribute to the integration over the “normal directions” to $M_{0}$ in $N$. Here we are not considering a simple integration over $N$, but an equivariant integration, that provides a map $H^{}_{G}(N)\to\mathbb{C}$. So in this case we consider the equivariantly closed extension $\tilde{\omega}=\omega-i\phi^{a}\mu_{a}$ as representative of a cohomology class $[\tilde{\omega}]\in H^{2}_{G}(M)$. When restricted over $N$, this class is the pull-back of a cohomology class $[\omega_{0}]\in H^{2}(M_{0})$, since $H^{}_{G}(N)\cong H^{}_{G}(\mu^{-1}(0))\cong H^{}(M_{0})$, the first equivalence following from the retraction of $N$ onto $\mu^{-1}(0)$ and the second from the fact that the $G$-action is free on $\mu^{-1}(0)$ (these properties were explained in Chapter 2). Thus we can substitute in the integral $\tilde{\omega}\mapsto\omega_{0}$ without changing the final result. The same kind of argument works for the term $\frac{1}{2}(\phi,\phi)$. It is easy to check that this is both $G$-invariant and equivariantly closed, so it represents an element $\left[\frac{1}{2}(\phi,\phi)\right]\in H^{4}_{G}(M)$. When we restrict it to $N$, as above, this class is the pull-back of some class $[\Theta]\in H^{4}(M_{0})$, and we can make the substitution $\frac{1}{2}(\phi,\phi)\mapsto\Theta$ in the integral without changing the final result. Since both $\omega_{0}$ and $\Theta$ are standard differential forms over $M_{0}$ and thus independent of $\phi$, the integration over $\mathfrak{g}$ goes along only with the remaining term $d_{C}(dS\circ J)$. We already know that on $\mu^{-1}(0)\subset N$ this term is zero, so one has to show that its integral over the normal directions to $\mu^{-1}(0)$ in $N$ produces a trivial factor of 1. In [15] it is proven that $\frac{1}{\mathrm{vol}(G)}\int_{F}\int_{\mathfrak{g}}\frac{d^{s}\phi}{(2\pi)^{s}}\exp\left(td_{C}(dS\circ J)\right)=1,$ where $F$ is any fiber of the normal bundle to $\mu^{-1}(0)$ in $N$. From this (6.28) follows. ∎ ###### Example 6.2.1 (The height function on the 2-sphere, again). Beside the main application of Witten’s localization principle to non-Abelian gauge theories, we try now to apply this new formalism to the old and simple example of the height function on the 2-sphere, to compare it with the results obtained in Chapter 3. Setting $G=U(1)$, $M=\mathbb{S}^{2}$, $\mu=\cos(\theta)$ and $\omega=d\cos(\theta)\wedge d\varphi$, the equivariant integration (6.19) of the DH oscillatory integral gives $\displaystyle Z(\epsilon)$ $\displaystyle=\int_{\mathbb{S}^{2}\times\mathfrak{g}}\omega e^{-i\phi\mu}=\frac{1}{2\pi}\int_{0}^{2\pi}d\varphi\int_{-1}^{+1}d\cos\theta\int_{-\infty}^{+\infty}\frac{d\phi}{2\pi}\exp\left(-i\phi\cos\theta-\frac{\epsilon}{2}\phi^{2}\right)$ $\displaystyle=\frac{1}{\sqrt{2\pi\epsilon}}\int_{-1}^{+1}dx\ e^{-\frac{x^{2}}{2\epsilon}}=1-2I(\epsilon),$ where $I(\epsilon):=\int_{1}^{\infty}\frac{dx}{\sqrt{2\pi\epsilon}}\exp(-x^{2}/2\epsilon)$ is a trascendental error function. The three terms in the final result for $Z(\epsilon)$ (two of which are equal to $-I(\epsilon)$) correspond to the contributions of the extrema of $(\cos\theta)^{2}$: the two maxima at $\theta=0,\pi$ contribute with $-I(\epsilon)$ and the minimum at $\theta=\pi/2$ contributes with $1$. The latter is the dominant piece when $\epsilon\to 0$, since $I(0)=0$. We see that in general the modified equivariant integration of this new formalism gives an incredibly complicated answer, when compared to the simple result of Example 3.2.1 obtained via the usual equivariant localization principle. ###### Remark. We notice that we could have expressed equivalently the whole dissertation above in supergeometric language, since we discussed in Section 4.1 that integration over $M$ is equivalent to integration over $\Pi TM$. In these terms, maybe more common in QFT, we can introduce coordinates $(x^{\mu},\psi^{\mu},\phi^{a})$ over $\Pi TM\times\mathfrak{g}$, where $\psi^{\mu}:=dx^{\mu}$ are Grassmann-odd, and interpret elements of $\Omega_{G}(M)$ as elements of $C^{\infty}(\Pi TM\times\mathfrak{g})^{G}$. An equivariant form $\alpha$ is thus (locally) a $G$-invariant function of $(x,\psi,\phi)$. For example, the Cartan differential and the definition of equivariant integration become $\displaystyle d_{C}$ $\displaystyle=\psi^{\mu}\frac{\partial}{\partial x^{\mu}}-i\phi^{a}T_{a}^{\mu}\frac{\partial}{\partial\psi^{\mu}},$ (6.29) $\displaystyle\int_{M\times\mathfrak{g}}\alpha$ $\displaystyle:=\frac{1}{\mathrm{vol}(G)}\int d^{2n}xd^{2n}\psi\frac{d^{s}\phi}{(2\pi)^{s}}\ \alpha(x,\psi,\phi)e^{-\frac{\epsilon}{2}(\phi,\phi)}.$ ### 6.3 “Cohomological” and “physical” YM theory In this section we are going to review the relation between 2-dimensional YM theory that we described in Section 6.1 and a topological field theory (TFT) that can be viewed as its “cohomological” counterpart. We can translate almost verbatim the general principles that we discussed in the last section, setting $M\mapsto\mathcal{A}(P),\quad\omega\mapsto\Omega,\quad G\mapsto\mathcal{G}(P),\quad\mu\mapsto F,$ (6.30) while we regard $G$ as a compact connected Lie group that acts as the gauge group on the principal bundle $P\to\Sigma$ over a Riemann surface $\Sigma$. The moment map is formally equivalent to the curvature $F$ if we identify $Lie(\mathcal{G}(P))\cong Lie(\mathcal{G}(P))^{}$ through an inner product on $\mathfrak{g}$, as in (6.8). The non-Abelian localization principle of the last section, if used in reverse, already showed that an equivalent way to express the standard YM theory is through a “first-order formulation” $S[A,\phi]=-\int_{\Sigma}\mathrm{Tr}\left(i\phi F^{A}+\frac{\epsilon}{2}\phi\star\phi\right)$ (6.31) where we consider $\phi\in\Omega^{0}(\Sigma;\mathrm{ad}(P))\cong Lie(\mathcal{G}(P))$, and $F^{A}\in\Omega^{2}(\Sigma;\mathrm{ad}(P))$ is the curvature of $A$.888More precisely, we should say that $A^{a}_{\mu}$ and $\phi^{a}$ are _coordinates_ functions on $\mathcal{A}(P)\times Lie(\mathcal{G}(P))$, so they effectively are elements of $C^{\infty}(\mathcal{A}(P))\otimes Lie(\mathcal{G}(P))^{}$. This caveat will be logically important in the following, and it goes along with the _functor of points_ approach we used in the supergeometric discussion of Chapter 4. This is essentially what is written in (6.25), where on the LHS we have the first-order action (the $\epsilon$ dependence is contained in the equivariant integration), and on the RHS we have the standard YM action $S[A]=\frac{1}{2\epsilon}(F,F)_{A}$. The first-order formulation has the quality of showing very clearly the weak coupling limit behavior when $\epsilon\to 0$, that is less obvious in the standard formulation. In this limit, the theory becomes topological, in the sense that the action does not depend on the metric anymore (the metric appears in the Hodge duality $\star$), $S_{\epsilon\to 0}[A,\phi]=-\int_{\Sigma}\mathrm{Tr}(i\phi F^{A}).$ (6.32) This theory is called “BF model”, and it is the prototype of a TFT of Schwarz- type. The YM theory can thus be seen as a “regulated version” of a truly topological field theory.999Notice that, although YM theory is clearly dependent on the metric of $\Sigma$, in 2 dimensions it shows a very “weak” dependence to it. In fact, in this dimensionality the action can be simplified as (suppressing the constants and the Lie algebra inner product) $\displaystyle\int F\wedge\star F$ $\displaystyle=\int g^{\mu\rho}g^{\nu\sigma}F_{\mu\nu}F_{\rho\sigma}\sqrt{\|g\|}d^{2}x=\int(F_{01})^{2}g^{\mu\rho}g^{\nu\sigma}\varepsilon_{\mu\nu}\varepsilon_{\rho\sigma}\sqrt{\|g\|}d^{2}x=$ $\displaystyle=\int(F_{01})^{2}\|g^{-1}\|\sqrt{\|g\|}d^{2}x=\int(F_{01})^{2}\|g\|^{-1/2}d^{2}x,$ so the metric does not appear through its components, but only in the invariant quantity $\det(g)$. At least classically, it is intuitive from the EoM with respect to $\phi$ that the only contribution to the classical solutions comes from the moduli space of flat connections, where $F^{A}=0$ up to gauge transformations. It is not trivial, though, to infer that this is all the theory has to offer also at the quantum level, that is essentially the result we showed in general terms in the last section, via the localization principle applied to the path integral $Z(\epsilon\to 0)$. In this section we discuss, following [15], how the localization principle can be translated in the language of TFT, in order to give a more physical interpretation of the abstract mathematical results that we discussed in finite dimensions. In particular, we will see that the BF model partition function can be recovered as an expectation value in a TFT, and that this ensures its localization properties onto the moduli space of flat connections. The regulated version at $\epsilon\neq 0$ will not follow precisely this behavior, as we already know that higher extrema of the YM action contribute to the partition function $Z(\epsilon)$, but these will have a nice interpretation in terms of the moduli space. #### Intermezzo: TFT This is a good moment to explain briefly in more general terms what one usually means by TFT, and how localization enters in this subject. Traditionally, TFT borrows the language of BRST formalism for quantization of gauge theories, as many examples of topological theories arise from that context. Recall that the standard BRST quantization procedure is based on the definition of a differential, the “BRST charge” $Q$, that acts on an extended graded field space, whose grading counts the “ghost number” (in other words, $Q$ is an operator of degree $\mathrm{gh}(Q)=+1$). The BRST charge represents an infinitesimal supersymmetry transformation, that squares to zero in the gauge-fixed theory. On the Hilbert space, physical states are those of ghost number zero, and that are annihilated by the BRST charge, $Q\|\text{phys}\rangle=0,\quad\mathrm{gh}\|\text{phys}\rangle=0.$ (6.33) The action of $Q$ on the field space is often denoted with a Poisson bracket- like notation, $\Phi\mapsto\delta_{Q}\Phi:=-\\{Q,\Phi\\}$ for any field $\Phi$. By gauge invariance of the vacuum, $Q\|0\rangle=0$, and so for any operator $\mathcal{O}$ one has that $\langle 0\|\\{Q,\mathcal{O}\\}\|0\rangle=0$. For a QFT being “topological”, in physics one usually means that all its quantum properties are independent from a choice of a metric on the base space $M$.101010Here the word “topological” is somewhat overused. Mathematically, a topological space consists of a set $M$ and a _topology_ $\mathcal{O}_{M}$, that is roughly the set of all “open neighborhoods” in $M$. In QFT one almost always works on base spaces that have the structure of a _manifold_ of some kind (smooth, complex, …), so that it allows for the presence of an _atlas_ $\mathcal{A}_{M}$ of charts that identifies it locally as $\mathbb{R}^{n}$ for some (constant) $n$. A manifold is thus a triple $(M,\mathcal{O}_{M},\mathcal{A}_{M})$, and the choice of a metric is only on top of this structure. So metric-independence does not generically mean that the QFT describes only the topology of $M$, but it can depend on the choice of a smooth (or complex, …) structure on $M$. This is rephrased in the requirement that the partition function of the theory should be metric- independent. Assuming the path integral measure to be metric-independent and $Q$-invariant (so that the BRST symmetry is non anomalous), the variation with respect to the metric of the partition function is $\delta_{g}Z\propto\int_{\mathcal{F}}[D\Phi]e^{-S[\Phi]}\delta_{g}S[\Phi]=\int_{\mathcal{F}}[D\Phi]e^{-S[\Phi]}\left(\int_{M}d^{n}x\sqrt{\|g\|}\delta g^{\mu\nu}T_{\mu\nu}\right)\propto\langle 0\|T_{\mu\nu}\|0\rangle,$ (6.34) so a suitable definition of TFT is the one that requires the energy-momentum tensor $T_{\mu\nu}$ to be a BRST variation, $T_{\mu\nu}=\\{Q,V_{\mu\nu}\\}$ for some operator $V_{\mu\nu}$. This would ensure $\delta_{g}Z=0$ for what we said above. Collecting the above remarks, we can give the following “working definition” [106]. A _Topological Field Theory_ is a QFT defined over a $\mathbb{Z}$-graded field space $\mathcal{F}$, with a nilpotent operator $Q$ (i.e. a _cohomological vector field_ on $\mathcal{F}$), and a $Q$-exact energy-momentum tensor $T_{\mu\nu}=\\{Q,V_{\mu\nu}\\}$, for some $V_{\mu\nu}\in C^{\infty}(\mathcal{F})$. _Physical states_ are defined to be elements of the $Q$-cohomology of $\mathcal{F}$ in degree zero, $\|\text{phys}\rangle\in H^{0}(\mathcal{F},Q)$.111111We stress that, if we see $\mathcal{F}$ as a graded extension of an original field space $\mathcal{F}_{0}$ acted upon by gauge transformations, the $Q$-cohomology of $\mathcal{F}$ is exactly the analogous of the gauge-equivariant cohomology $H_{gauge}^{}(\mathcal{F}_{0})$, computed in the Cartan model with Cartan differential $Q$. ###### Remark. • $Q$ is called “BRST charge” or “operator”, but in general it can be every supersymmetry charge (as it is a cohomological vector field). We saw examples in the last chapter where $Q=\delta_{susy}+\delta_{BRST}$, where $\delta_{BRST}$ is the actual gauge-supersymmetry, and $\delta_{susy}$ is a Poincaré-supersymmetry. * • The one above is a good “working definition” for most examples, but it is not completely adequate in all cases. Indeed, there are examples of QFT where $T_{\mu\nu}$ fails to be BRST-exact, but nonetheless one can still establish the topological nature of the model. We do not need to treat any example of this kind, so we refer to [106] for more details. * • We already encountered an example of TFT in Section 5.2, i.e. supersymmetric QM. There, we called it topological because its partition function was determined by topological invariants of the base space, here we point out that it indeed fits into this general discussion. In fact, in (5.69) we saw that its action can be expressed as $S=\\{Q,\Sigma\\}$, where $Q\equiv Q_{\dot{x}}$ was the $U(1)$-Cartan differential on the loop space, and $\Sigma$ some other loop space functional. The $Q$-exactness of the energy-momentum tensor follows from this. * • If the energy-momentum tensor is $Q$-exact, in particular the Hamiltonian satisfies $\langle\text{phys}\|H\|\text{phys}\rangle=\langle\text{phys}\|T_{00}\|\text{phys}\rangle=0.$ (6.35) This means that the energy of any physical state is zero, and thus the TFT does not contain propagating degrees of freedom. It can only describe “topological” properties of the base space. TFTs fall in two main broad categories. The first one is constituted by the so-called TFT of _Witten-type_ (or also _cohomological TFT_). Their defining property is to have a $Q$-exact quantum action, $S_{q}=\\{Q,V\\}$ (6.36) for some operator $V$. As the case of supersymmetric QM, the energy-momentum tensor is automatically $Q$-exact too, $T_{\mu\nu}=\frac{2}{\sqrt{\|g\|}}\left\\{Q,\delta V/\delta g_{\mu\nu}\right\\}.$ (6.37) From the equivariant point of view, these theories have a very simple localization property. The action is representative of the trivial $Q$-equivariant cohomology class, so the partition function can be in fact written as $Z\propto\int_{\mathcal{F}}[D\Phi]e^{-tS_{q}[\Phi]}$ (6.38) for any $t\in\mathbb{R}$, by the standard argument of supersymmetric localization. In this case $\mathcal{F}$ is non-compact and so simply taking $t=0$ is not really allowed, but the limit $t\to\infty$ is perfectly defined, producing the path integral localization onto the space of solutions of the classical EoM of $S_{q}$. This means that all TFT of Witten-type are completely determined by their semiclassical approximation! The second main class of TFT is called of _Schwarz-type_ (or also _quantum TFT_). In this case one starts with a classical action $S$ that is metric- independent, so that the classical energy-momentum tensor is zero. The usual BRST quantization of this action produces a quantum action of the type $S_{q}=S+\\{Q,V\\}$ for some $V$, and again $T_{\mu\nu}=\frac{2}{\sqrt{\|g\|}}\left\\{Q,\delta V/\delta g_{\mu\nu}\right\\},$ (6.39) since the classical piece does not contribute. Chern-Simons theory and BF theory are of this type. Regarding the second remark above, we mention that some Schwarz-type TFTs fail to respect the $Q$-exactness property of $T_{\mu\nu}$, for example in the case of non-Abelian BF theories in dimension $d>3$ (this happens essentially because one can ensure nilpotency of the BRST charge only on-shell, and moreover $Q$ has to be defined in a metric-dependent way). As we anticipated before we are interested in BF theories, an example of Schwarz-type TFT, and in particular in their 2-dimensional realization. On a base space $\Sigma$ of generic dimension $d$, the classical action of the BF theory is $S_{BF}[A,B]:=-\int_{\Sigma}\mathrm{Tr}(B\wedge F^{A}),$ (6.40) where $F^{A}$ is the curvature 2-form of a connection $A$, and $B$ is a $(d-2)$-form with values in the adjoint bundle. The BRST charge is the usual gauge-supersymmetry. In the 2-dimensional case, $B\equiv i\phi\in\Omega^{0}(\Sigma;\mathrm{ad}(P))$, and the naive BRST quantization of this theory has no problems (as we mentioned, in more than 3 dimensions the definition of $Q$ have to be modified and things complicate). The quantum action is $\displaystyle S_{q}[A,B,\pi,b,c]$ $\displaystyle=-\int_{\Sigma}\mathrm{Tr}\left(BF^{A}+\pi(\nabla^{A}\star A)+b(\nabla^{A}\star\nabla^{A})c\right)$ (6.41) $\displaystyle=S_{BF}[A,B]+\\{Q,V\\},$ $\displaystyle\text{with}\quad V$ $\displaystyle:=-\int_{\Sigma}\mathrm{Tr}\left(b(\nabla^{A}\star A)\right),$ where we introduced a ghost $c$, an anti-ghost $b$ and an auxiliary field $\pi$ with the BRST transformations properties $\delta_{Q}A=\nabla^{A}c,\quad\delta_{Q}c=-\frac{1}{2}[c,c],\quad\delta_{Q}b=\pi,\quad\delta_{Q}\pi=0,\quad\delta_{Q}B=-[B,c].$ (6.42) Even though this theory is not of Witten-type, so it has no direct localization onto the subspace of classical solutions, it turns out that this is still the case. Traditionally, this is shown finding a suitable redefinition of coordinates in field space that trivializes the bosonic sector of the action (meaning that there are no derivatives acting on bosonic fields), and whose Jacobian cancels in the path integral with the 1-loop determinant obtained by integrating out the fermions (in this case, the ghosts fields). This is called _Nicolai map_ , and for the 2-dimensional BF model is given by the redefinitions [106, 107] $\xi(A):=F^{A},\qquad\eta(A):=\nabla^{A_{c}}\star A_{q},$ (6.43) where $A:=A_{c}+A_{q}$ is the expansion of the gauge field around a classical (on-shell) solution $A_{c}$. Assuming the fermions being integrated out, the path integrals over $B$ and $\pi$ exactly identify the space of zeros of $\xi,\eta$, that is the moduli space of solutions to $F^{A}=0$ up to gauge transformations, $\mathcal{A}_{0}$. We do not pursue this direction further, but summarize the approach taken in [15], more related to localization. #### Cohomological approach Thinking in equivariant cohomological terms, one can expect to show the localization of the BF model (and of its “regulated” version, _i.e._ YM theory) finding a suitable “localization 1-form” $V^{\prime}$. This has to be such that the deformation of the action given by $t\\{Q,V^{\prime}\\}$ induces the path integral to localize in this subspace when $t\to\infty$, analogously to what we did for example in Section 5.1, and also to the discussion of the last section in the finite-dimensional case. This is exactly what is shown in [15], where the partition function of the model of interest is found as an expectation value inside a cohomological TFT, proving automatically its localization behavior. This cohomological TFT is constructed in a way such that the action of the BRST operator $Q$ coincides with the Cartan differential $d_{C}$ arising in the symplectic formulation of 2-dimensional YM theory. Using the more common supergeometric language of Sections 5.1, 5.2 and recalled in (6.29) (in finite dimensions), we move from the field space $\mathcal{A}(P)\times Lie(\mathcal{G}(P))$ to $\mathcal{F}:=\Pi T\mathcal{A}(P)\times Lie(\mathcal{G}(P))$, introducing the graded coordinates $(A_{\mu}^{a},\psi_{\mu}^{a},\phi^{a})$ and regarding $d_{C}$ as a supersymmetry (BRST) transformation, $d_{C}\equiv-\\{Q,\cdot\\}=\int_{\Sigma}d\Sigma\left(\psi_{\mu}^{a}\frac{\delta}{\delta A^{a}_{\mu}}-i(\underline{\phi}^{a})^{\mu}\frac{\delta}{\delta\psi_{\mu}^{a}}\right),$ (6.44) where we recall from (6.3) that the fundamental vector field associated to the action of a Lie algebra element $\phi\in Lie(\mathcal{G}(P))$ is $\underline{\phi}=\nabla\phi$. The BRST transformation of every field follows the rule $\delta_{Q}\Phi=-\\{Q,\Phi\\}\equiv d_{C}\Phi$, and on the coordinates we have $\delta_{Q}A=\psi,\qquad\delta_{Q}\psi=-i\nabla\phi,\qquad\delta_{Q}\phi=0.$ (6.45) The ghost numbers of the elementary fields are $\mathrm{gh}(A,\psi,\phi)=(0,1,2)$. Other multiplets could be added as well, but this is the basic one we start with. Before describing in more detail the particular cohomological theory and its relation with the physical YM theory, we summarize the strategy that has been followed. One starts with a suitable cohomological TFT with action $S_{c}=\\{Q,V\\},$ (6.46) for some operator $V$, that ensures the localization onto the moduli space $\mathcal{A}_{0}$ of flat connections. Then the mapping to the physical theory is done by the common equivariant localization procedure. The TFT is deformed adding a cohomologically trivial localizing action, $S(t)=S_{c}+t\\{Q,V^{\prime}\\}=\\{Q,V+tV^{\prime}\\},$ (6.47) for some gauge invariant operator $V^{\prime}$ that forces only the interesting YM multiplet $(A,\psi,\phi)$ to survive. Since the field space here is non-compact, some additional care must be taken in claiming the $t$-independence of the deformed theory. In particular, the new term must not introduce new fixed points of the $Q$-symmetry, that would contribute to the localization locus of the resulting theory. These fixed points, if presents in the theory at $t\neq 0$, can be interpreted as “flowing from infinity” in the moduli space (since this is, as just remarked, non-compact).121212This will be exactly the case in going to the YM theory with $\epsilon\neq 0$, where the new fixed points are just the higher extrema of the action $S=\frac{1}{2}(F,F)$. If this is not the case, one can infer properties of the “physical” theory at $t=\infty$ by making computations in the cohomological one at $t=0$.131313Notice that this is logically the opposite of what one does usually in using localization. Here the “easy” theory is the cohomological one at $t=0$, while the more difficult (but more interesting) is the one at $t\neq 0$. ##### The cohomological theory The cohomological theory considered in [15] makes use of the additional multiplets $(\lambda,\eta)$ and $(\chi,H)$ with the transformation properties $\delta_{Q}\lambda=\eta,\qquad\delta_{Q}\eta=-i[\phi,\lambda],\qquad\delta_{Q}\chi=-iH,\qquad\delta_{Q}H=[\phi,\chi].$ (6.48) The extended field space and ghost numbers are $\begin{array}[]{rccccc}\mathcal{F}=&\underbrace{\Pi T\mathcal{A}(P)\times Lie(\mathcal{G}(P))}&\times&\underbrace{\left(\Omega^{0}(\Sigma;\mathrm{ad}(P))\right)^{2}}&\times&\underbrace{\left(\Omega^{0}(\Sigma;\mathrm{ad}(P))\right)^{2}}\\\ &(A,\psi,\phi)&&(\lambda,\eta)&&(\chi,H)\\\ \mathrm{gh}=&(0,1,2)&&(-2,-1)&&(-1,0).\end{array}$ (6.49) Of course the definition of the BRST operator will be extended from (6.44), but we do not need it explicitly. The operator $V$ defining the TFT is chosen to be $V=\frac{1}{h^{2}}\int_{\Sigma}d\Sigma\ \mathrm{Tr}\left(\frac{1}{2}\chi(H-2(\star F^{A}))+g^{\mu\nu}(\nabla^{A}_{\mu}\lambda)\psi_{\nu}\right),$ (6.50) where $h\in\mathbb{R}$ is a parameter from which the theory is completely independent (it is analogous to $1/t$ appearing in (6.38)), that can be interpreted as the “coupling constant” of the TFT. Computing the cohomological action $S_{c}$ one sees that the $H$ field plays an auxiliary role, and can be eliminated setting $H=\star F$. Analyzing then the theory in the $h\to 0$ limit (its semiclassical “exact” approximation), the localization locus is identified by the BRST-fixed point (we refer to [15] for the details) $\delta_{Q}\chi=0\quad(\Rightarrow F=0),\qquad\delta_{Q}\psi=0\quad(\Rightarrow\nabla\phi=0).$ (6.51) This is analogous to the usual situation in Poincaré-supersymmetric theories, as discussed in Chapter 5, where the localization locus was always identified by the subcomplex of BPS configurations, given by the vanishing of the variation of the fermions. Here the “fermions” are the fields with odd ghost number. This means that the final moduli space contains $\mathcal{A}_{0}$, plus maybe some contributions from the zero-modes of the other bosonic (even ghost number) fields, $\lambda$ and $\phi$. ##### $t\neq 0$ deformation The deformation (6.47) can be made in order to reduce effectively the field content of the theory to the YM multiplet only, $(A,\psi,\phi)$. In particular, to eliminate the non-trivial presence of the field $\lambda$ from the contributions to the localization locus one can consider $V^{\prime}=-\frac{1}{h^{2}}\int_{\Sigma}d\Sigma\ \mathrm{Tr}\left(\chi\lambda\right).$ (6.52) Computing the deformed action $S(t)$, one sees that for $t\neq 0$ all the additional fields $H,\chi,\lambda,\eta$ can be integrated out (again, some more details on the technical passages can be found in [15]). In particular, the EoM for $H$ and $\lambda$ are $H=0,\qquad\lambda=-\frac{1}{t}(\star F).$ (6.53) This is already the sign that the localizing term $V^{\prime}$ qualitatively changed the localization property of the theory. In fact, we see that for $t=0$ we do not have an algebraic equation for $\lambda$, but (6.53) reduces to the solution $F=0,H=0$ of the cohomological theory. The theories defined by $S(t)$ and $S_{c}$ may be thus different, but the failure of their equivalence can only come from new components of the moduli space that flow in from infinity for $t\neq 0$; the contribution of the “old” component must be independent of $t$. Taking the limit $t\gg 1$, the dominant contribution to the deformed action is (suppressing the $A$-dependence) $\displaystyle S(t\gg 1)$ $\displaystyle=-\frac{1}{t}\left\\{Q,\int_{\Sigma}d\Sigma\ \mathrm{Tr}\left(\psi^{\mu}\nabla_{\mu}(\star F)\right)\right\\}$ (6.54) $\displaystyle=\frac{1}{t}\int_{\Sigma}d\Sigma\ \mathrm{Tr}\Bigl{(}i\nabla_{\mu}\phi\nabla^{\mu}(\star F)-(\star F)[\psi_{\mu},\psi^{\mu}]+\nabla_{\mu}\psi^{\mu}\epsilon^{\nu\sigma}\nabla_{\nu}\psi_{\sigma}\Bigr{)}.$ The main point is that the $\phi$-EoM is actually equivalent to the YM equation $\nabla\star F=0$. This means that the moduli space of the deformed theory contains the moduli space of the standard YM theory, and indeed includes all the higher extrema corresponding to non-flat connections. These solutions with $F\neq 0$ have $\lambda\sim-1/t$, and thus their contribution to the path integral goes roughly as $\exp(-1/t)$, as expected. When $t=0$, the cohomological theory is recovered and the only contribution to the moduli space is given by the flat connections. ##### Connection with 2-dimensional YM theory We already argued that the deformed TFT gained all the YM spectrum “flowing from infinity in the moduli space”, but it remains to see how one can get practically the YM (and BF) partition function from the theory defined by $S(t)$. This is simply obtained by another deformation of the exponential in the partition function: we notice that the YM action is gauge invariant, so it is meaningful to compute the expectation value of $e^{S_{YM}}$ in the TFT. Thus we consider an exponential operator of the form $\begin{gathered}\exp\left(\omega_{0}+\epsilon\Theta\right)\\\ \text{with}\quad\omega_{0}:=\int_{\Sigma}\mathrm{Tr}\left(i\phi F+\frac{1}{2}\psi\wedge\psi\right),\qquad\Theta:=\frac{1}{2}\int_{\Sigma}\mathrm{Tr}(\phi\star\phi).\end{gathered}$ (6.55) Since the quantity $\left\langle\exp\left(\omega_{0}+\epsilon\Theta\right)\right\rangle_{t}\propto\int_{\Pi T\mathcal{A}(P)\times Lie(\mathcal{G}(P))}DAD\psi D\phi\ \exp\left(\omega_{0}+\epsilon\Theta-S(t)\right)$ (6.56) is well defined for $t\to\infty$, we can actually take $t=\infty$ and drop $S(\infty)=0$ (recall that the path integral is independent on the actual value of $t$), getting exactly the YM partition function $\left\langle\exp\left(\omega_{0}+\epsilon\Theta\right)\right\rangle_{t}\propto\int_{\Pi T\mathcal{A}(P)\times Lie(\mathcal{G}(P))}DAD\psi D\phi\ \exp\left(\omega_{0}+\epsilon\Theta\right)\propto Z(\epsilon),$ (6.57) up to some normalization constant. In the limit $\epsilon\to 0$, when only the BF model survives, the path integral over $\phi$ produces the constraint $\delta(F)$, so localizing the expectation value onto the space of flat connections. This means that, although we started from different theories $S_{c}\cong S(0)$ and $S(t)$, this particular expectation value satisfies $\left\langle\exp\left(\omega_{0}\right)\right\rangle_{t}=\left\langle\exp\left(\omega_{0}\right)\right\rangle_{t=0}$ (6.58) and the BF model is recovered as an expectation value in the cohomological theory. This gives another interpretation to the topological behavior of the BF model, and a measure of the failure of 2-dimensional YM theory in being topological. Concluding, we only point out that the the operators $\omega_{0}$ and $\Theta$ are precisely the infinite-dimensional realization in this example of the general expressions in (6.28). In fact, the symplectic 2-form on $\mathcal{A}(P)$ $\Omega=\int_{\Sigma}\mathrm{Tr}(\psi\wedge\psi)$ (6.59) only serves to have a formal interpretation of the measure $DAD\psi e^{\Omega}$, since the field $\psi$ is really a spectator in the action $S[A,\psi,\phi]=-\int_{\Sigma}\mathrm{Tr}\left(i\phi F+\frac{\epsilon}{2}\phi\star\phi+\frac{1}{2}\psi\wedge\psi\right).$ (6.60) ### 6.4 Localization of 2-dimensional YM theory As we said at the beginning of the chapter, 2-dimensional YM theory is an exactly solvable theory, whose partition function can be expressed in closed form, for example by group characters expansion methods [20, 15]. This makes it possible to compare results from the localization formalism, and obtain a new geometric interpretation of the already present solution of the theory. In general, its partition function on a Riemann surface $\Sigma$ of genus $g$, with a simply-connected gauge group $G$, is given as $Z(\epsilon)=(\mathrm{vol}(G))^{2g-2}\sum_{R}\frac{1}{\dim(R)^{2g-2}}e^{-\epsilon\tilde{C}_{2}(R)},$ (6.61) where the sum runs over the representations $R$ of $G$, and $\tilde{C}_{2}(R)$ is related to the quadratic Casimir $C_{2}(R):=\sum_{a}\mathrm{Tr}_{R}(T_{a}T_{a})$ of the representation $R$ by some normalization constant. For not simply-connected $G$ this formula has to be slightly modified (see [15]).141414Simply-connectedness implies that the principal $G$-bundle $P\to\Sigma$ has to be trivial. When one drops this condition, the triviality is not ensured and contributions to the formula appear due to singular points in $\Sigma$ for the connection. We are not interested in reviewing the proof of (6.61) in general, but we present a quick argument for a simple example, that already contains the logic behind it. ###### Example 6.4.1 (YM theory with genus $g=1$). We quickly motivate the result for the YM partition function on a genus 1 surface. We can think of this surface as a disk with boundary, that is homeomorphic to a sphere with one hole. The radial direction is identified with the interval $[0,T]$, and the angular coordinate as $[0,L]$ with the edges identified. It is more natural to compute the partition function of the theory in the Hamiltonian formulation, $Z=\mathrm{Tr}_{\mathcal{H}}\mathcal{P}\exp\left(-\int_{0}^{T}dt\ H(t)\right)$ (6.62) up to possible normalization factors, where $\mathcal{H}$ is the Hilbert space of the system. To this end, let us consider the canonical quantization of the YM action. To make sense of the partition function we must fix a gauge, and we do this setting $A_{t}=0$ (temporal gauge). In this gauge the action simplifies as $S[A]=-\frac{1}{2\epsilon}\int dxdt\ \mathrm{Tr}(F_{01}^{2})=\frac{1}{2\epsilon}\int dxdt\ (\partial_{t}A^{a}_{x})^{2},$ (6.63) where we expanded $A_{\mu}=A^{a}_{\mu}T_{a}$ with respect to the generators of $\mathfrak{g}$, and suppressed the inner product implicitly summing over the Lie algebra indices. We see that the only non-zero canonical momentum is $\Pi^{a}_{x}=(1/\epsilon)\partial_{t}A^{a}_{x}$, acting on the Hilbert space as $\Pi^{a}_{x}(t,x)\mapsto\frac{\delta}{\delta A_{x}^{a}(t,x)}$. The canonical Hamiltonian in temporal gauge is thus $H(t)=\epsilon\int_{0}^{L}dx\ (\Pi^{a}_{x}(t,x))^{2}\mapsto\epsilon\int_{0}^{L}dx\ \frac{\delta}{\delta A_{x}^{a}(t,x)}\frac{\delta}{\delta A_{x}^{a}(t,x)}.$ (6.64) The Hilbert space $\mathcal{H}$ can be considered to consist of gauge invariant functions $\Psi(A)$. The only gauge invariant data obtained from the gauge field at any point $p\in\Sigma$ is its holonomy, $U_{p}[A]:=\mathcal{P}\exp\left(\oint_{C(p)}A\right)\quad\in G,$ (6.65) where $C(p)$ is a loop about $p$. In the case of the one holed-sphere, all loops are homotopic to the one on the boundary, so $\Psi\in\mathcal{H}$ must be an invariant function of $U=\mathcal{P}\exp\int_{0}^{L}dxA_{1},$ (6.66) and independent of $t\in[0,T]$. Any invariant function must be expandible in characters of representations of $G$, so $\Psi(U)=\sum_{R}\Psi_{R}\mathrm{Tr}_{R}(U)$, where $\Psi_{R}\in\mathbb{C}$ and $\mathrm{Tr}_{R}(U)$ is the Wilson loop in the representation $R$ of $G$. We notice that the basis functions $\chi_{R}(U):=\mathrm{Tr}_{R}(U)$ diagonalize the Hamiltonian, since $H\chi_{R}(U)=\epsilon\mathrm{Tr}_{R}\int_{0}^{1}dx\ T_{a}T_{a}\ \mathcal{P}\exp\int_{0}^{L}dxA_{1}=\epsilon LC_{2}(R)\chi_{R}(U),$ (6.67) where $C_{2}(R):=\mathrm{Tr}_{R}(T_{a}T_{a})$ is the quadratic Casimir in the representation $R$, a time-independent eigenvalue of $H$. Via this diagonalization the partition function is easily computed, $Z=\sum_{R}e^{-TL\epsilon c(R)},$ (6.68) matching (6.61) for $g=1$. All the geometric information about $\Sigma$ that enters in $Z$ is its total area $TL$, and any other local property. Notice that in the topological limit $\epsilon\to 0$, the Hamiltonian vanishes (as the theory has no propagating degrees of freedom) and the partition function simplifies further. From the localization formalism discussed in the last sections, we expect the partition function to be of the type $Z(\epsilon)=Z_{0}(\epsilon)+\sum_{n}Z_{n}(\epsilon),$ (6.69) with $Z_{0}(\epsilon)$ representing the contribution from the moduli space $\mathcal{A}_{0}$ of flat connections, such that $Z_{0}(0)\sim\mathrm{vol}(\mathcal{A}_{0})$, and the other $Z_{n}(\epsilon)$ coming from contributions of the higher extrema of the YM action, such that $Z_{n}(\epsilon)\sim\exp(-1/\epsilon)$ in the weak coupling limit $\epsilon\ll 1$. Using cohomological arguments, in [15] (also nicely reviewed in [102]) it was shown how to recover the general features of (6.61), and in particular how to interpret it in terms of an $\epsilon$-expansion at weak coupling, in relation to the expected form (6.69). The detailed derivation is cumbersome and requires some more technical background, so we refer to the article for it, but the logic is essentially the same as for the discussion at the end of Section 6.2. The strategy is the following. Any solution to the YM EoM identifies a disconnected region $\mathcal{S}_{n}\subset\mathcal{A}(P)$. For every such region, one fixes a small neighborhood $N_{n}$ around $\mathcal{S}_{n}$ that equivariantly retracts onto it. The technically difficult passage is to perform the integral over the “normal directions” to $\mathcal{S}_{n}$ in $N_{n}$, and then reduce it on the moduli space $\mathcal{A}_{n}:=\mathcal{S}_{n}/\mathcal{G}(P)$. The main difficulty is that in general the MWM theorem (or its equivariant counterpart) does not work, since the action of $\mathcal{G}(P)$ is not generally free on $\mathcal{S}_{n}$ (also for $n=0$), as we assumed in writing down (6.28) for the $\mathcal{S}_{0}\equiv\mu^{-1}(0)$ component. For the higher extrema, this is readily seen by the fact that the equation $\nabla(\star F)=0\qquad\text{with}\ F\neq 0$ (6.70) identifies a vacuum $f:=(\star F)$ as a preferred element of $\mathfrak{g}$ (being it covariantly constant over $\Sigma$), and thus the gauge group is spontaneously broken to a subgroup $G_{f}\subseteq G$. The action of the whole gauge group thus cannot be free on this subspace, and the quotient $\mathcal{S}_{n}/\mathcal{G}(P)$ is singular. Via a suitable choice of localization 1-form one is still able to extract information by this integral over the normal directions, and in particular to compute the $\epsilon$-dependence of the higher extrema contributions. We limit ourselves now to the comparison of the exact result (6.61) applied to the case $G=SU(2)$, with the expectation (6.69) obtained by cohomological arguments. For this gauge group, the character expansion of the partition function results $Z(\epsilon)=\frac{1}{(2\pi^{2})^{g-1}}\sum_{n=1}^{\infty}\frac{\exp(-\epsilon\pi^{2}n^{2})}{n^{2g-2}}.$ (6.71) Simply taking $\epsilon=0$, we see that this is finite and proportional to a Riemann zeta-function, but to explore better the $\epsilon$-dependence it is convenient to consider $\frac{\partial^{g-1}Z}{\partial\epsilon^{g-1}}=\left(-\frac{1}{2}\right)^{g-1}\sum_{n=1}^{\infty}\exp\left(-\epsilon\pi^{2}n^{2}\right)=\frac{(-1)^{g-1}}{2^{g}}\left(-1+\sum_{n\in\mathbb{Z}}\exp\left(-\epsilon\pi^{2}n^{2}\right)\right).$ (6.72) This is not quite in the expected form, since the exponentials in the sum go to zero for $\epsilon\to 0$ but not as $\exp(-1/\epsilon)$. We can bring this expression closer to the desired result using the Poisson summation formula $\sum_{n\in\mathbb{Z}}f(n)=\sum_{k\in\mathbb{Z}}\hat{f}(k)\qquad\text{with}\ \hat{f}(k):=\int_{-\infty}^{+\infty}f(x)e^{-2\pi ikx},$ (6.73) where $f$ is a function and $\hat{f}$ its Fourier transform, and rewriting the sum of exponentials in (6.72) as $\frac{\partial^{g-1}Z}{\partial\epsilon^{g-1}}=\frac{(-1)^{g-1}}{2^{g}}\left(-1+\sqrt{\frac{1}{\pi\epsilon}}\sum_{k\in\mathbb{Z}}\exp\left(-\frac{k^{2}}{\epsilon}\right)\right).$ (6.74) This is exactly the result that could be obtained via integration over normal coordinates in the localization framework (see [102], eq. (4.102)), but fundamentally differs from our expectation, since for $\epsilon\to 0$ the contribution from the flat connections (with $k=0$) is singular for the presence of the square root. This means that the partition function is not really a polynomial in $\epsilon$ for small couplings, but an expression of the form $Z(\epsilon)=\sum_{m=0}^{g-2}a_{m}\epsilon^{m}+a_{g-3/2}\epsilon^{g-3/2}+\text{exponentially small terms}.$ (6.75) The singularity in $Z(\epsilon\to 0)$ arises because, for gauge group $SU(2)$, the subspace $\mu^{-1}(0)$ is singular and the MWM theorem does not apply. A simpler situation would occur considering the gauge group $SO(3)$ (which is not simply connected) and a non-trivial principal bundle over $\Sigma$. In this case, the character expansion of the partition function requires some modifications with respect to (6.61), the result being $Z(\epsilon)=\frac{1}{(8\pi^{2})^{g-1}}\sum_{n=1}^{\infty}\frac{(-1)^{n+1}\exp(-\pi^{2}\epsilon n^{2})}{n^{2g-2}}.$ (6.76) Following the same idea as above, we look at the $(g-1)^{th}$ derivative $\frac{\partial^{g-1}Z}{\partial\epsilon^{g-1}}=\frac{(-1)^{g}}{8^{g-1}}\sum_{n=1}^{\infty}(-1)^{n}\exp(-\pi^{2}\epsilon n^{2})=\frac{(-1)^{g}}{8^{g-1}}\frac{1}{2}\left(-1+\sum_{n\in\mathbb{Z}}(-1)^{n}\exp(-\pi^{2}\epsilon n^{2})\right),$ (6.77) and we rewrite the sum using the Poisson summation formula, getting $\frac{\partial^{g-1}Z}{\partial\epsilon^{g-1}}=\frac{(-1)^{g}}{2\cdot 8^{g-1}}\left(-1+\sqrt{\frac{1}{\pi\epsilon}}\sum_{k\in\mathbb{Z}}\exp\left(-\frac{\left(k+1/2\right)^{2}}{\epsilon}\right)\right).$ (6.78) This time we see that the contribution for $k=0$ from the moduli space of flat connections is finite for $\epsilon\to 0$, and the whole $\partial^{g-1}Z/\partial\epsilon^{g-1}$ is constant up to exponentially small terms. This means that the partition function at weak coupling $Z(\epsilon\to 0)$ is a regular polynomial of degree $g-1$ in $\epsilon$, up to exponentially decaying terms, $Z(\epsilon)=\sum_{m=0}^{g-2}a_{m}\epsilon^{m}+O(\epsilon^{g-1}),$ (6.79) and it reflects the fact that, for a non-trivial $SO(3)$-bundle, $\mu^{-1}(0)$ is smooth and acted on freely by $G$. These two quick examples capture the way this localization framework can give a very geometric interpretation to the $\epsilon$-expansion of the partition function, and its dependence on the classical geometry of the moduli space. Finally, we point out that an analogous treatment was done more recently in [102] to analyze in this cohomological framework Chern-Simons theory on 3-dimensional _Seifert manifolds_. A Seifert manifold is a smooth object that can be described as an $\mathbb{S}^{1}$-bundle over a 2-dimensional orbifold, and this feature makes it possible to dimensionally reduce the Chern-Simons theory along the direction of the circle $\mathbb{S}^{1}$ to a 2-dimensional YM theory over a singular base space. It turns out that the localization locus of the resulting theory receives contributions only from the flat connections over the total space. This is in accordance with the fact that Chern-Simons theories are by themselves TFT (of Schwarz-type). 2-dimensional YM theories have been studied extensively in the past years, and many interesting results were obtained thanks to their non-perturbative solvability. For example, exact results for Wilson loops expectation values and their relation with higher dimensional supersymmetric theories were studied in [108, 109, 110]. A relation with certain topological string theories and supersymmetric black hole entropy computations were analyzed in [111]. A duality between higher- dimensional supersymmetric gauge theories and deformations of 2-dimensional YM theory was revisited in [112, 113]. Localization techniques play an important role in all those cases. ## Chapter 7 Conclusion In this thesis we reviewed and summarized the main features of the formalism of equivariant cohomology, the powerful localization theorems first introduced by Atiyah-Bott and Berline-Vergne, and the principles that allow to formally apply these integration formulas to QFT. From the physical point of view, the equivariant (or supersymmetric) localization principle gives a systematic approach to understand when the “semiclassical” approximation of the path integral, describing the partition function or an expectation value in QFT, can give an exact result for the full quantum dynamics. We discussed the applicability of these techniques in the context of _supersymmetric_ theories. These are characterized by a space of fields that is endowed with a graded structure and the presence of some symmetry operator whose “square” gives a standard “bosonic” symmetry of the action functional. This supersymmetry operator is interpreted as a differential acting on the subspace of symmetric configurations in field space, and its cohomology describes the field theoretical analog of the $G$-equivariant cohomology of a $G$-manifold. After having introduced the general features of the mathematical theory of equivariant cohomology and equivariant localization, we reviewed the concepts in supergeometry that allow for the construction of supersymmetric QFT, and that constitute the correct framework to translate the mathematical theory in the common physical language. Since many recent applications of the localization principle aimed at the computations of path integrals in supersymmetric QFT on curved spaces, we included a discussion of the main tools needed to define supersymmetry in such instances. Then we collected some examples from the literature of application of the supersymmetric localization principle to path integrals in QFT of diverse dimensions. The common feature of these examples is that, via a suitable “cohomological” deformation of the action functional, it is possible to reduce the infinite-dimensional path integral to a finite-dimensional one that represents its semiclassical limit, as stressed above. We described cases in which this reduction relates the partition function to topological invariants of the geometric structure underlying the theory, namely the cases of supersymmetric QM (a 1-dimensional QFT) and the weak coupling limit of 2-dimensional Yang-Mills theory (its “topological” limit). We also reviewed the more recent applications to the computations of the expectation values of supersymmetric Wilson loops in 3- and 4-dimensional gauge theories, namely Supersymmetric Chern-Simons theory and Supersymmetric Yang-Mills theory defined on the 3- and the 4-sphere. In these cases, the path integral results to be equivalently described by some 0-dimensional QFT with a Lie algebra as target space, called “matrix model”. In the last few decades, the literature concerning the applications of supersymmetric localization has grown exponentially, and many other advanced examples of its use in the physics context have been found. From the point of view of supersymmetric QFT, a consistent slice of the state-of-the-art on the subject can be found in [12], including computations analogue to the one we showed for Wilson loop expectation values or topological invariants over more complicated geometries. From the point of view of Quantum (Super)Gravity, these techniques have found applications in the computations of the Black Hole quantum entropy [114, 115]. In many circumstances, localization allows for the analysis of properties of QFT at strong coupling, an otherwise prohibited region of study with conventional perturbative techniques. This feature can be used also to test a class of conjectural dualities between some types of gauge theories and string theories, the so-called AdS/CFT correspondences [96]. Concerning the subject of Wilson loops in 3-dimensional Chern-Simons theories and their relations to matrix models and holography, for which localization has played an important role, a recent reference that concisely reviews the state-of-the-art is [84]. ## Appendix A Some differential geometry ### A.1 Principal bundles, basic forms and connections Here we recall some notions about principal bundles that can be useful to follow the discussion, especially of the first chapters of this thesis. Principal bundles are the geometric construction behind the concepts of _covariant derivatives_ and _connections_ in gauge theory or General Relativity, for example. If $G$ is a Lie group, a principal $G$-bundle is a smooth bundle $P\xrightarrow{\pi}M$ such that 1. (i) $P$ is a (right) $G$-manifold; 2. (ii) the $G$-action on $P$ is _free_ ; 3. (iii) as a bundle, $P\to M$ is isomorphic to $P\to P/G$, where the projection map is canonically defined as $p\mapsto[p]$. Notice that since the $G$-action is free, a principal $G$-bundle is a fiber bundle with typical fiber $G$, and by the third property it is at least _locally trivial_ , i.e. over every open set $U\subseteq M$ it looks like $G\times U$. Morphisms of principal bundles are naturally defined as maps between bundles that preserve the $G$-structure, so $G$-equivariant maps. A principal bundle is _trivial_ if it is isomorphic through a principal bundle isomorphism to the trivial product bundle $G\times M$. A useful fact is that the triviality of a principal bundle is completely captured by the existence of a _global_ section $\sigma:M\to P$ such that $\pi\circ\sigma=id_{M}$. Since every principal bundle is locally trivial, than local sections can always be chosen and they constitute a so-called _local trivialization of the bundle_. The main example of principal bundle that occurs in the geometric construction of spacetime is the _frame bundle_ $LM$ over some $n$-dimensional smooth manifold $M$. At every point $p\in M$, the elements of the fiber $L_{p}M$ are the _frames_ at $p$, i.e. all the possible bases $e=(e_{1},\cdots,e_{n})$ for the tangent space $T_{p}M$. $LM$ has a natural $GL(n,\mathbb{R})$ right action that corresponds to the rotation of the basis, $e\cdot g:=(e_{k}g^{k}_{1},\cdots,e_{k}g^{k}_{n})$ for $g\in GL(n,\mathbb{R})$. In gauge theories, the _structure group_ (or sometimes _gauge group_) $G$ of the theory is the Lie group acting on the right on a principal $G$-bundle. Since the fibers of the principal bundle are essentially the Lie group $G$, tangent vectors on $P$ can come from its Lie algebra $\mathfrak{g}$. This leads to the following definition. ###### Definition A.1.1. The _vertical sub-bundle_ $VP$ of the tangent bundle $TP$ is the disjoint union $VP:=\bigsqcup_{p\in P}V_{p}P,\qquad\text{with}\ V_{p}P:=\mathrm{Ker}(\pi_{p})=\\{X\in T_{p}P\|\pi_{}(X)=0\\}\subset T_{p}P.$ Analogously but for differential forms, the _basic forms_ inside $\Omega(P)$ are those forms $\omega\in\mathrm{Im}(\pi^{})$, so that it exists an $\alpha\in\Omega(M)$ such that $\omega=\pi^{}\alpha$. The space of basic forms is denoted $\Omega(P)_{bas}$. The vertical vectors in every $V_{p}P$ are in one to one correspondence with the Lie algebra elements in $\mathfrak{g}$, through the Lie algebra homomorphism $X\in\mathfrak{g}\mapsto\underline{X}:=\left.\frac{d}{dt}\right\|_{t=0}\left(\cdot e^{tX}\right)^{}$ (A.1) that maps Lie algebra elements to the corresponding _fundamental vector fields_.111The choice of the sign at the exponential differs from the one in (2.19) because here we are considering a right action. Fundamental vector fields satisfy the following properties: 1. (i) $[\underline{X},\underline{Y}]=\underline{[X,Y]}$; 2. (ii) the integral curve of $\underline{X}$ through $p\in M$ is $\displaystyle\gamma_{p}:\mathbb{R}$ $\displaystyle\to M$ (A.2) $\displaystyle t$ $\displaystyle\mapsto\gamma_{p}(t)=p\cdot e^{tX};$ 3. (iii) denoting with $r_{g}$ the right action of $g\in G$, $\left(r_{g}\right)_{}(\underline{X}_{p})=\left(\underline{Ad_{g^{-1}}(X)}\right)_{r_{g}(p)}.$ (A.3) As for the vertical vector fields being encoded in the Lie algebra $\mathfrak{g}$, also the basic forms can be characterized in terms on the (infinitesimal) action of $\mathfrak{g}$ on $\Omega(P)$. This can be seen introducing the following definitions. ###### Definition A.1.2. A differential form $\omega\in\Omega(P)$ is said to be _$G$ -invariant_ if it is preserved by the $G$ action: $\omega=(r_{g})^{}\omega\qquad\forall g\in G.$ The space of $G$-invariant forms is commonly denoted $\Omega(P)^{G}$. A differential form is called _horizontal_ if it is annihilated by vertical vector fields, $\iota_{X}\omega=0\qquad\forall X\in\Gamma(VP).$ The properties of being invariant and horizontal can be also stated infinitesimally with respect to the action of the Lie algebra $\mathfrak{g}$. If we define $\mathcal{L}_{X}:=\mathcal{L}_{\underline{X}},\qquad\iota_{X}:=\iota_{\underline{X}},\qquad\forall X\in\mathfrak{g},$ (A.4) then an invariant form is characterized by $\mathcal{L}_{X}\omega=0$ for every $X\in\mathfrak{g}$, and a horizontal form by $\iota_{X}\omega=0$ for every $X\in\mathfrak{g}$. This makes the concepts of invariant and horizontal elements independent from the principal bundle structure, so that they can be defined by this characterization for every $\mathfrak{g}$-dg algebra, as in Section 2.3. Also basic forms can be defined for every $\mathfrak{g}$-dg algebra, combining the definitions of invariant and horizontal forms, thanks to the following theorem: ###### Theorem A.1.1 (Characterization of basic forms). Schematically, $\text{invariant}+\text{horizontal}\Leftrightarrow\text{basic}.$ ###### Proof. For notational convenience only, let us consider 1-forms. * $(\Leftarrow)$ If $\omega=\pi^{}\alpha$ is basic, then $(r_{g})^{}\omega=(r_{g})^{}\pi^{}\alpha=(\pi\circ r_{g})^{}\alpha=\omega,$ since the principal bundle is locally trivial. So $\omega$ is also invariant. For a vertical vector $X\in VP$, $\iota_{X}\omega=(\pi^{}\alpha)(X)=\alpha(\pi_{}(X))=0.$ So $\omega$ is also horizontal. $(\Rightarrow)$ Let $\omega\in\Omega^{1}(P)$ be horizontal and invariant. Since $\pi$ is surjective, for every vector $X\in T_{p}P$ there exists $Y=\pi_{}(X)\in T_{\pi(p)}M$. We can define $\alpha\in\Omega^{1}(M)$ such that, at every $x\in M$ $\alpha_{x}(Y):=\omega_{p}(X)\qquad\text{for}\ p\in\pi^{-1}(x),$ and thanks to the horizontality and invariance of $\omega$ we can check that this form is well defined, i.e. independent from the choice of point $p$ in the fiber $\pi^{-1}(x)$ and from the choice of vector $X$ such that $\pi_{}(X)=Y$. In fact, if $X^{\prime}\in T_{p}P$ is another vector such that $\pi_{}(X^{\prime})=Y$, then $\pi_{}(X-X^{\prime})=0$ so $(X-X^{\prime})\in V_{p}P$. By horizontality, $\omega(X-X^{\prime})=0\Rightarrow\omega(X)=\omega(X^{\prime})$, so $\alpha$ is independent from the choice of vector. Moreover, if $p^{\prime}\in\pi^{-1}(x)$ is another point in the fiber, there exists a $g\in G$ such that $r_{g}(p)=p^{\prime}$, so by $G$-invariance $\omega_{p^{\prime}}=\omega_{p}$ and thus $\alpha$ is independent from the choice of point in the fiber. ∎ ###### Proposition A.1.1. The differential $d$ closes on the subspace $\Omega(P)_{bas}$ of basic forms, defining a proper subcomplex. This extends to any $\mathfrak{g}$-dg algebra. ###### Proof. Consider $\omega\in\Omega(P)_{bas}$, and its differential $d\omega$. We characterize basic forms by being horizontal and $G$-invariant. By Cartan’s magic formula the Lie derivative commutes with the differential, so for every $X\in\mathfrak{g}$, $\mathcal{L}_{X}d\omega=d(\mathcal{L}_{X}\omega)=0$. Thus $d\omega$ is still $G$-invariant. Also, $\iota_{X}d\omega=\mathcal{L}_{X}\omega-d\iota_{X}\omega=0$. Thus $d\omega$ is still horizontal, and so basic. ∎ We recall now the definition of connection and curvature on principal bundles, from which one inherits covariant derivatives on associated vector bundles. ###### Definition A.1.3. An _(Ehresmann) connection_ on a principal $G$-bundle $P\to M$ is an _horizontal distribution_ $HP$, i.e. a smooth choice at every point $p\in P$ of vector subspaces $H_{p}P\subset T_{p}P$ such that 1. (i) $T_{p}P=V_{p}P\oplus H_{p}P$; 2. (ii) $(r_{g})_{}(H_{p}P)=H_{p\cdot g}P$ ($G$-equivariance of the horizontal projection). Given a horizontal distribution $HP$, every vector $X\in T_{p}P$ decomposes into an _horizontal_ and a _vertical_ part, $X=hor(X)+ver(X).$ A _connection 1-form_ on $P\to M$ is Lie algebra-valued 1-form $A\in\Omega^{1}(P)\otimes\mathfrak{g})$ such that 1. (i) for any $X\in\mathfrak{g}$, $\iota_{X}A=A(\underline{X})=X$ (_vertical_ 1-form); 2. (ii) for any $g\in G$, $(r_{g})^{}A=(Ad_{g^{-1}}\circ A)$ ($G$-equivariance). The choice of a horizontal distribution is equivalent to the choice of a connection 1-form on $P$, since at every $p\in P$ one can use $A$ as a projection onto the vertical subspace $V_{p}P\cong\mathfrak{g}$, and $\pi_{}$ as a projection onto the horizontal subspace, identifying $V_{p}P:=\mathrm{Ker}(\pi_{p})$ and $H_{p}P:=\mathrm{Ker}(A_{p})$. This choice is smooth and $G$-equivariant since $A$ is, by definition. Notice that the splitting $HP\oplus VP$ induces a splitting $\Omega^{1}(P)=\Omega^{1}_{hor}(P)\oplus\Omega^{1}_{ver}(P)$, and that we can identify the “space of connection 1-forms” as $\mathcal{A}(P):=\\{A\in(\Omega^{1}_{ver}(P)\otimes\mathfrak{g})\|A\ \text{is }G\text{-equivariant}\\}.$ (A.5) It is easy to see that for every $A,A^{\prime}\in\mathcal{A}(P)$, their difference is not a connection, and in fact it is an _horizontal_ $\mathfrak{g}$-valued 1-form, $(A-A^{\prime})\in\mathfrak{a}:=\\{a\in(\Omega^{1}_{hor}(P)\otimes\mathfrak{g})\|a\ \text{is }G\text{-equivariant}\\}.$ (A.6) This means that every connection $A$ can be written as another connection $A^{\prime}$ plus a horizontal form, or in other words that $(\mathcal{A}(P),\mathfrak{a})$ can be seen as a natural _affine space_ , modeled on the infinite-dimensional vector space $\mathfrak{a}$. As for any affine space, one can think of the space of connections as an infinite- dimensional smooth manifold, with tangent spaces $T_{A}\mathcal{A}(P)\cong\mathfrak{a}$ at every $A\in\mathcal{A}(P)$. ###### Definition A.1.4. The _covariant exterior derivative_ on $\Omega(P)$ is $D:=d\circ hor^{}$. The _curvature_ of a connection 1-form $A$ is $F:=DA=dA(hor(\cdot),hor(\cdot))\in\Omega^{2}(P)\otimes\mathfrak{g}.$ The curvature $F$ satisfies the following properties: 1. (i) by definition, $F$ is horizontal: $\iota_{X}F=0$ for every $X\in\mathfrak{g}$; 2. (ii) by $G$-equivariance of $A$, $F$ is $G$-equivariant too; 3. (iii) it obeys the structural equation $F=dA+\frac{1}{2}[A\stackrel{{\scriptstyle\wedge}}{{,}}A]$ (A.7) where $[A\stackrel{{\scriptstyle\wedge}}{{,}}A]=f^{a}_{bc}A^{b}\wedge A^{c}\otimes T_{a}$ with respect to a basis $\\{T_{a}\\}$ of $\mathfrak{g}$ and the structure constants $f^{a}_{bc}$; 4. (iv) it obeys the second Bianchi identity, $DF=0\qquad\text{or}\qquad dF=[F\stackrel{{\scriptstyle\wedge}}{{,}}A].$ (A.8) One can consider the very trivial construction of a principal $G$-bundle as $G\to pt$, where $P=G\times pt\cong G$. Here the right $G$-action is simply the diagonal action (trivial on $pt$, induced by the natural action on $G$). On this bundle there is a canonical choice of connection 1-form, the _Maureer- Cartan (MC) form_ $\Theta\in\Omega^{1}(G)\otimes\mathfrak{g}$. For every vector $X\in T_{g}G$ at some $g\in G$, there is a Lie algebra element $A\in T_{e}G\cong\mathfrak{g}$ such that $X=l_{g}(A)$, and the MC form is defined by $\Theta_{g}(X):=l_{g^{-1}}(X)=A.$ (A.9) One can check that this form is indeed $G$-equivariant, and it is obviously vertical, giving a connection 1-form. Moreover it satisfies the _Maurer-Cartan equation_ $d\Theta+\frac{1}{2}[\Theta\stackrel{{\scriptstyle\wedge}}{{,}}\Theta]=0,$ (A.10) so that by (A.7) we see that its curvature is zero. On the trivial principal $G$-bundle $G\times M\to M$ one can always define a connection 1-form by pulling back the MC connection along the projection $\pi_{1}:G\times M\to G$. In the general case, the principal bundle $P$ is locally trivial, so in any local patch $G\times U_{\alpha}\to U_{\alpha}$ one can pull back the MC connection and use a suitable partition of unity to glue together the local pieces to a global connection 1-form on $P$. This shows that any principal bundle allows for a connection. The curvature $F$ of the chosen connection $A$ measures, in a sense, the deviation of $A$ from being the Maurer-Cartan connection. As said before, a connection on a principal $G$-bundle allows for the definition of a covariant derivative on _associated vector bundles_. An associated vector bundle to the principal $G$-bundle $P\xrightarrow{\pi}M$ is a vector bundle constructed over $M$ with some typical fiber $V$ (a vector space) that has a (left) $G$-action compatible with the one on $P$. Precisely, the associated bundle is $P_{V}\xrightarrow{\pi_{V}}M$, where $\displaystyle P_{V}:=\faktor{(P\times V)}{\sim_{G}}\qquad\text{with}\ (p,v)\sim_{G}(p\cdot g,g^{-1}\cdot v)\ \forall(p,v)\in P\times V,g\in G,$ (A.11) $\displaystyle\pi_{V}([p,v]):=\pi(p),$ and it has indeed typical fiber $V$. In the case of the frame bundle $P=LM$, one can construct the tangent bundle $TM$, the cotangent bundle $T^{}M$ and all the tensor bundles as associated to $LM$. In fact, for the tangent bundle for example, the typical fiber is $V:=\mathbb{R}^{n}$ and the $GL(n,\mathbb{R})$-action is $(g\cdot v)^{k}=g^{k}_{j}v^{j}$. This encodes the change of basis rule if we see vectors as elements $[e,v]\in LM_{\mathbb{R}^{n}}$, $[e,v]\equiv e_{k}v^{k}\sim_{GL}[e\cdot g,g^{-1}\cdot v]\equiv e_{j}g^{j}_{k}(g^{-1})^{k}_{j}v^{j}=e_{k}v^{k}.$ (A.12) On the associated vector bundle, a _field_ (in physics terms) is a (local, at least) section $\phi:M\to P_{V}$, that can be always seen locally as a $V$-valued function on every $U\subseteq M$, $\tilde{\phi}:U\to V$, so that $\phi(x)=[p,\tilde{\phi}(x)]$ for some chosen $p\in\pi^{-1}(x),x\in U$. Another example of this concept that came up in Chapter 2 is the _homotopy quotient_ $M_{G}:=(M\times EG)/G$ of a $G$-manifold $M$. This is precisely the associated bundle with fiber $V=M$ to the principal $G$-bundle $EG\to BG$ (however, this is not an associated _vector_ bundle, since $M$ is not a vector space in general). As we said at the beginning of this appendix, every principal bundle is locally trivial, so that it exists a set of _local trivializations_ $\\{U_{\alpha},\varphi_{\alpha}:\pi^{-1}(U_{\alpha})\to U_{\alpha}\times G\\}$, where $\\{U_{\alpha}\\}$ covers $M$, and $\varphi_{\alpha}$ is $G$-equivariant. This means that $\varphi_{\alpha}(p)=(\pi(p),g_{\alpha}(p))$ for some $G$-equivariant map $g_{\alpha}:\pi^{-1}(U_{\alpha})\to G$,222In this case $G$-equivariance means $g_{\alpha}(p\cdot h)=g_{\alpha}(p)\cdot h$. that makes every fiber diffeomorphic to $G$. To this local trivialization, one can canonically associate a family of local sections $\\{\sigma_{\alpha}:U_{\alpha}\to\pi^{-1}(U_{\alpha})\\}$, determined by the maps $\varphi_{\alpha}$ so that for every $m\in U_{\alpha}$, $\varphi_{\alpha}(\sigma_{\alpha}(m))=(m,e)$, where $e\in G$ is the identity element. In other words, $g_{\alpha}\circ\sigma_{\alpha}:U_{\alpha}\to G$ is the constant function over the local patch $U_{\alpha}\subseteq M$ that maps every point to the identity. Conversely, a local section $\sigma_{\alpha}$ allows us to identify the fiber over $m$ with $G$. Indeed, given any $p\in\pi^{-1}(m)$, there is a unique group element $g_{\alpha}(p)\in G$ such that $p=\sigma_{\alpha}(m)\cdot g_{\alpha}(p)$. Using these canonical local data, the connection $A$ and the curvature $F$ can be pulled back on $M$ giving the local _gauge field_ $A^{(\alpha)}:=\sigma_{\alpha}^{}(A)$ and _field strength_ $F^{(\alpha)}:=\sigma_{\alpha}^{}(F)$. The _covariant derivative_ along the tangent vector $X\in TM$ of a local $V$-valued function $\tilde{\phi}:U_{\alpha}\to V$ is defined as $\nabla_{X}\tilde{\phi}:=d\tilde{\phi}(X)+A^{(\alpha)}(X)\cdot\tilde{\phi},$ (A.13) where the second term denotes the action of the Lie algebra on $V$, that for matrix groups coincides with the action of $G$. We denote schematically the covariant derivative as $\nabla=d+A$ on a generic associated vector bundle. When $V=\mathfrak{g}$ we have the so-called _adjoint bundle_ , often denoted $\text{ad}(P)$, that is in one-to-one correspondence with the space $\mathfrak{a}$ above, of horizontal and $G$-equivariant Lie algebra-valued forms on $P$. On this special associated bundle, the covariant derivative acts with the infinitesimal adjoint action of $\mathfrak{g}$, $\nabla=d+[A,\cdot].$ (A.14) By the horizontal property of the curvature, we see that $F$ can be regarded as a 2-form on $M$ with values in $\text{ad}(P)$.333Being horizontal means that pulling it back on the base space, we do not lose information on the 2-form. In fact, the local representation of the curvature $F^{(\alpha)}=\sigma_{\alpha}^{}(F)$ still transforms covariantly also as a $\mathfrak{g}$-valued 2-form over $M$. Strictly speaking, the covariant derivative on the adjoint bundle acts on this local representative. Notice that the gauge field $A^{(\alpha)}$ instead looks only locally as an element of the adjoint bundle, but globally it does not respect the “right” transformation property, and indeed it comes from a global 1-form on $P$ that is not horizontal, but vertical. Then the Bianchi identity can be rewritten in terms of the covariant derivative, $\nabla F=dF+[A,F]=[F,A]+[A,F]=0.$ (A.15) As the last piece of information, we recall the meaning of _gauge transformations_ from the perspective of the principal bundle. Locally, we can think of them as local actions of the gauge group $G$, so that a gauge transformation is a map that associates to every point $x\in M$ an element $g(x)\in G$, acting on the local field strength in the adjoint representation. At the level of the principal bundle, this can be viewed more formally defining the group $\mathcal{G}(P)\subset\mathrm{Diff}(P)$ of principal bundle maps of the type444As a principal bundle map it is by definition $G$-equivariant, $\Psi(p\cdot g)=\Psi(p)\cdot g$, and it commutes with the projection, $\pi(\Psi(p))=\pi(p)$, for every $p\in P$. ${P}$${P}$${M.}$$\scriptstyle{\Psi}$$\scriptstyle{\pi}$$\scriptstyle{\pi}$ (A.16) We notice right-away that, from the local point of view, this can indeed be identified with the space of sections $\Omega^{0}\left(M;\mathrm{Ad}(P)\right)$ of the bundle $\mathrm{Ad}(P)$, associated to $P$ with typical fiber $G$ and $G$-action defined by conjugation (the adjoint representation of $G$ on itself).555Notice that this looks like $P\to\Sigma$ as a fiber bundle, since both are locally trivial with fiber $G$. It is only the $G$ action that distinguishes them. On $P$ we have a right action, on $\mathrm{Ad}(P)$ we have the left action on the fibers $g\cdot f:=gfg^{-1}$. ###### Proof of $\mathcal{G}(P)\cong\Omega^{0}(M;\mathrm{Ad}(P))$. We can see that associated to every element $\Psi\in\mathcal{G}(P)$ there is a unique class of local sections $\\{\psi_{\alpha}:U_{\alpha}\to G\\}$ that transforms in the adjoint representation, and vice versa. * $(\Rightarrow)$ In every local patch $U_{\alpha}$, let us define the map $\tilde{\psi}_{\alpha}:\pi^{-1}(U_{\alpha})\to G$ such that $\tilde{\psi}_{\alpha}(p):=g_{\alpha}(\Psi(p))\ g_{\alpha}(p)^{-1},$ where $g_{\alpha}$ is the trivialization map inside $U_{\alpha}$. By equivariance of $\Psi$ and $g_{\alpha}$, $\tilde{\psi}_{\alpha}$ is $G$-invariant, so it depends only on the base point $\pi(p)$. Thus we can define $\psi_{\alpha}:U_{\alpha}\to G$ such that $\psi_{\alpha}(x):=\tilde{\psi}_{\alpha}(p)\qquad\text{for some}\ p\in\pi^{-1}(x).$ In changing local patch, this transforms in the adjoint representation. In fact, if $x\in U_{\alpha}\cap U_{\beta}$ $\displaystyle\psi_{\beta}(x)$ $\displaystyle=g_{\beta}(\Psi(p))\ g_{\beta}(p)^{-1}$ $\displaystyle=\left[g_{\beta}(\Psi(p))g_{\alpha}(\Psi(p))^{-1}\right]g_{\alpha}(\Psi(p))g_{\alpha}(p)\left[g_{\alpha}(p)^{-1}g_{\beta}(p)\right]$ $\displaystyle=g_{\alpha\beta}(x)\psi_{\alpha}(x)g_{\alpha\beta}(x)^{-1},$ where in the last passage we recognized that $g_{\beta}(p)g_{\alpha}(p)^{-1}$ is $G$-invariant and thus can be written as a map $g_{\alpha\beta}:U_{\alpha}\cap U_{\beta}\to G$ that depends only on the base point $\pi(p)$, and we used that $\pi\circ\Psi=\pi$. * $(\Leftarrow)$ Starting from a class of local sections $\\{\psi_{\alpha}\\}$, we define the $G$-invariant maps $\tilde{\psi}_{\alpha}:=\psi_{\alpha}\circ\pi$. Then we can obtain $\Psi$ by “inverting” the above definition in every patch and gluing them together, $\Psi(p):=\sigma_{\alpha}(p)\cdot\left(\tilde{\psi}_{\alpha}(p)g_{\alpha}(p)\right).$ ∎ The group of gauge transformations $\mathcal{G}(P)$ acts naturally on the space $\mathcal{A}(P)$ via pull-back, $\Psi\cdot A:=\Psi^{}(A)\qquad\forall A\in\mathcal{A}(P),\Psi\in\mathcal{G}(P).$ (A.17) If we consider the local gauge field $A^{(\alpha)}$, one can prove that the trivialization of the gauge-transformed connection follows the usual rule $(\Psi\cdot A)^{(\alpha)}=\psi_{\alpha}^{-1}A^{(\alpha)}\psi_{\alpha}+\psi^{-1}_{\alpha}(d\psi_{\alpha}).$ (A.18) From this local expression, it is easy to find the representation of $Lie(\mathcal{G}(P))$ on $T\mathcal{A}(P)$. In fact, writing $\Psi$ as $\exp(X)$ for some $X\in Lie(\mathcal{G}(P))\cong\Omega^{0}(M;\mathrm{ad}(P))$, we can recognize the associated fundamental vector field as $\underline{X}_{A}=\left.\frac{d}{dt}\right\|_{t=0}e^{-tX}\cdot A=dX+[A,X]=\nabla^{A}X.$ (A.19) This make us see the usual “infintesimal variation” $\delta_{X}A$ as a tangent vector $\delta_{X}A\equiv\underline{X}_{A}\in T_{A}\mathcal{A}(P)$ at the point $A\in\mathcal{A}(P)$. ### A.2 Spinors in curved spacetime In QFT, _fermionic_ particles are described geometrically by _spinors_ , i.e. fields that transform under the Lorentz algebra in representations whose angular momentum is _half-integer_. At the level of Lie groups, they transform thus in representations of the double-cover of the rotation group of spacetime, $SO(1,d-1)$ (or $SO(d)$ in the Euclidean case), $SO(1,d-1)\cong\faktor{Spin(1,d-1)}{\mathbb{Z}_{2}}.$ (A.20) For simplicity, let us denote the dimension by $d$ for the rest of the section, since the discussion is valid both for the Euclidean and the Lorentzian signature. In Minkowski spacetime $(\mathbb{R}^{d},\eta)$, there exists a preferred class of _global_ coordinate systems, the global “inertial frames”, where the metric is diagonal $\eta_{\mu\nu}=diag(-1,+1,\cdots,+1)$ (A.21) and that are preserved by the Lorentz transformations. Working only with such special type of coordinate systems, one can introduce and work with spinors as living in double-valued representations of the Lorentz algebra, and transforming as $\displaystyle\text{vector fields}:\qquad V^{\mu}\mapsto\Lambda^{\mu}_{\nu}V^{\nu},$ (A.22) $\displaystyle\text{spinor fields}:\qquad\Psi_{\alpha}\mapsto S(\Lambda)^{\beta}_{\alpha}\Psi_{\beta},$ where, if $\Lambda=\exp(i\omega_{\mu\nu}M^{\mu\nu})$, $S(\Lambda)=\exp(i\omega_{\mu\nu}\Sigma^{\mu\nu})$. When we move on to the description of a generically curved spacetime $M$, there is a priori no such choice of “preferred” coordinate systems, and a general coordinate transformation (GCT) is generated by a diffeomorphism $M\to M$, reflecting on the tangent spaces as $GL(d,\mathbb{R})$ basis changes. $SO(d)$ injects as a subgroup of the General Linear group, but $Spin(d)$ does not, since it is a double cover, so it is not clear a priori how GCTs act on spinor fields. Tensor fields are naturally present in the fully covariant formalism as fields over the manifold $M$, but to define spinors one has to introduce further structure. The solution to this puzzle is really to (try to) mimic the same idea applied the the Minkowski case, and employ the presence of a (pseudo-)Riemannian metric $g$ on $M$. On the metric manifold $(M,g)$ all the tangent bundles arise as associated bundles to the _frame bundle_ $LM$, that is a principal $GL(d,\mathbb{R})$-bundle over $M$. Using the presence of a metric on $M$, one can restrict the frame bundle to a principal $SO(d)$-bundle, by considering only those frames $e=(e_{1},\cdots,e_{d})$ such that, at a given point $g(e_{i},e_{j})=\eta_{ij},$ (A.23) where $\eta$ is the “flat” Minkowski (or Euclidean) metric. This reduction defines the so called _orthonormal frame bundle_ $LM^{(SO)}\xrightarrow{\pi}M$. A section of this bundle is an orthonormal frame, or _tetrad_. It is customary to denote with Latin indices the expansion of every vector field with respect to an orthonormal frame, and with Greek indices the expansion with respect to a generic (for example chart-induced) frame:666Latin indices are sometimes called “flat”, and Greek ones “curved”. If one needs to raise and lower indices, flat indices are understood to be multiplied by the diagonalized metric $\eta_{ij}$, curved indices by $g_{\mu\nu}$. $V=V^{i}e_{i}=V^{\mu}\frac{\partial}{\partial x^{\mu}}\qquad\text{for}\ V\in\Gamma(TM).$ (A.24) The choice of an orthonormal frame is encoded in the choice of a _vielbein_ , or _solder form_ on $M$, that is a linear identification of the tangent bundle with the typical fiber $\mathbb{R}^{d}$: $\displaystyle E:TM$ $\displaystyle\to\mathbb{R}^{d}$ (A.25) $\displaystyle V$ $\displaystyle\mapsto E(V):=(\tilde{e}^{i}(V))_{i=1,\cdots,d},$ where $(\tilde{e}^{i})$ is the dual frame to a chosen orthonormal frame $(e_{i})$. Notice that the choice of a metric is in one to one correspondence with the choice of a vielbein, since $g(\cdot,\cdot)=\langle E(\cdot),E(\cdot)\rangle,$ (A.26) where $\langle\cdot,\cdot\rangle$ is the canonical inner product on $\mathbb{R}^{d}$ with the chosen signature. In a chart-induced basis, $g_{\mu\nu}=e^{i}_{\mu}e^{j}_{\nu}\eta_{ij}$, where we denoted the components of the vielbein $(E(\partial_{\mu}))^{i}\equiv e^{i}_{\mu}$. The “inverse vielbein” at any point is the matrix $e^{\mu}_{i}$ such that $e^{i}_{\mu}e^{\mu}_{j}=\delta^{i}_{j}$. Once this orthonormal reduction is made, one can define spinor bundles as associated bundles to a principal $Spin(d)$-bundle, that must be compatible with the orthonormal frame bundle. This is made precise by defining the presence of a spin-structure on $M$. ###### Definition A.2.1. A _spin-structure_ on $(M,g)$ is a principal $Spin(d)$-bundle $Spin(M)\xrightarrow{\pi_{S}}M$, together with a principal bundle map777Recall that a principal bundle map by definition commutes with the projections, $\pi(\Phi(p))=\pi_{S}(p)$. ${Spin(M)}$${LM^{(SO)}}$${M}$$\scriptstyle{\Phi}$$\scriptstyle{\pi_{S}}$$\scriptstyle{\pi}$ with respect to the double-cover map $\varphi:Spin(d)\to SO(d)$. This means that the equivariance condition is $\Phi(s\cdot g)=\Phi(s)\cdot\varphi(g)\qquad\forall s\in Spin(M),g\in Spin(d).$ A section of $Spin(M)\to M$ is called _spin-frame_. We notice that the equivariance condition in this definition is just the formal requirement that spinors and tensors transform all together with compatible rotations by the action of the respective groups. Although the above restriction of the frame bundle to the orthonormal frame bundle can always be done in presence of a metric on $M$, a spin-structure does not necessarily exist, and if it does it is not necessarily unique. There can be topological obstructions to this process that can be characterized in terms of the cohomology of $M$.888In particular, it turns out that a spin-structure exists if and only if the second Stiefel–Whitney class of $M$ vanishes [27]. By this construction, and from the canonical Levi-Civita covariant derivative $\nabla$ on $(M,g)$, we can induce a connection 1-form on the orthonormal frame bundle and on the spin-frame bundle, and thus have a compatible covariant derivative on associated spinor bundles. Let us recall that the Levi-Civita connection on $(M,g)$ is the unique metric-compatible and torsion free connection, i.e. $\begin{array}[]{lcl}\nabla_{X}g=0&\Leftrightarrow&X(g(Y,Z))=g(\nabla_{X}Y,Z)+g(Y,\nabla_{X}Z),\\\ T=0&\Leftrightarrow&\nabla_{X}Y-\nabla_{Y}X=[X,Y].\end{array}$ (A.27) This covariant derivative is associated to the gauge field $\Gamma\in\Omega^{1}(M)\otimes\mathfrak{gl}(n,\mathbb{R})$ such that $\Gamma^{\rho}_{\mu\nu}:=(\nabla_{\mu}(\partial_{\nu}))^{\rho}$. Simply restricting to orthonormal frames, one can induce a connection 1-form on $LM^{(SO)}$, $\omega\in\Omega^{1}(LM^{(SO)})\otimes\mathfrak{so}(d)$ such that in any trivialization induced by a local frame $(U\subset M,e:U\to LM^{(SO)})$ the gauge field has components $\omega(X)^{i}_{j}:=(\nabla_{X}(e_{i}))^{j}=X^{\mu}e^{j}_{\nu}(\nabla_{\mu}e_{i})^{\nu}=X^{\mu}e^{j}_{\nu}\left(\partial_{\mu}e^{\nu}_{i}+\Gamma^{\nu}_{\mu\sigma}e^{\sigma}_{i}\right)\quad\text{or}\quad\omega(X)_{ij}=g(\nabla_{X}e_{i},e_{j}),$ (A.28) and it can be written as $\omega^{(U)}:=e^{}\omega=\frac{1}{2}\omega_{ij}M^{ij}$, where $M^{ij}$ are the generators of $\mathfrak{so}(d)$. Given a spin-structure as in the above definition, we can induce a _compatible spin-connection_ $\tilde{\omega}\in\Omega^{1}(Spin(M))\otimes\mathfrak{so}(d)$ by pulling back $\omega$, $\tilde{\omega}:=\Phi^{}\omega$.999Notice that $Lie(Spin(d))\cong Lie(SO(d))\cong\mathfrak{so}(d)$. In a given patch $U\subset M$, if $s:U\to Spin(M)$ is a local spin-frame and $e:=\Phi\circ s$ is the associated tangent frame, the local gauge fields representing the spin-connection and the Levi- Civita connection coincide, $\tilde{\omega}^{(U)}:=s^{}\tilde{\omega}=(\Phi\circ s)^{}\omega=\omega^{(U)},$ (A.29) so in particular the local components of the compatible spin-connection are defined as $\tilde{\omega}(X)^{i}_{j}=(\nabla_{X}(e_{i}))^{j}.$ (A.30) The covariant derivative on an associated spinor bundle is defined as usual. Let $V$ be the typical fiber, acted upon by the representation $\rho:Spin(d)\to GL(V)$. Then for every local $V$-valued function $\psi:U\to V$, $\nabla_{X}\psi=d\psi(X)+\frac{1}{2}\omega(X)_{ij}\rho(M^{ij})\cdot\psi.$ (A.31) If in particular we take the fundamental representation of $Spin(d)$, i.e. $\psi$ is a _Dirac spinor_ , the generators are $\rho(M^{ij})=\Sigma^{ij}:=\frac{1}{4}[\gamma^{i},\gamma^{j}]$, where $\gamma^{i}$ are the Dirac matrices. Thus, $\nabla_{\mu}\psi=\partial_{\mu}\psi+\frac{1}{8}\omega_{\mu ij}[\gamma^{i},\gamma^{j}]\cdot\psi.$ (A.32) We quote the fact that, in general, one is not forced to consider a spin- connection that is compatible with the Levi-Civita connection.101010For example in SUGRA it is sometimes convenient to work with torsion-full spin connections. However, in this work we always implicitly define covariant derivatives on spinors via a compatible spin-connections. A discussion about spinors in curved spacetime can be found also in [65]. ## Appendix B Mathematical background on equivariant cohomology ### B.1 Equivariant vector bundles and equivariant characteristic classes We recall the definitions of characteristic classes on principal bundles [116] and then their equivariant version when the bundle supports a $G$-action for some Lie group $G$. Consider a principal $H$-bundle $P\xrightarrow{\pi}M$ with connection 1-form $A$, and curvature $F$. Both are forms on $P$ with values in the Lie algebra $\mathfrak{h}$. A _polynomial_ on $\mathfrak{h}$ is an element $f\in S(\mathfrak{h}^{})$, and it is called _invariant polynomial_ if it is invariant with respect to the adjoint action of $H$ on $\mathfrak{h}$, $f(Ad_{h}X)=f(X)\qquad\forall X\in\mathfrak{h},h\in H.$ (B.1) For example, if $H$ is a matrix group, the adjoint action is simply $Ad_{h}X=hXh^{-1}$. If $f$ is an invariant polynomial of degree $k$, then $f(F)$ is an element of $\Omega^{2k}(P)$. Explicitly, with respect to a basis $(T_{a})_{a=1,\cdots,\dim\mathfrak{h}}$ of $\mathfrak{h}$ and the dual basis $(\alpha^{a})_{a=1,\cdots,\dim\mathfrak{h}}$ of $\mathfrak{h}^{}$, if $F=F^{a}T_{a}$ and $f=f_{a_{1}\cdots a_{k}}\alpha^{a_{1}}\cdots\alpha^{a_{k}}$, then $f(F)=f_{a_{1}\cdots a_{k}}F^{a_{1}}\wedge\cdots\wedge F^{a_{k}}.$ (B.2) The above form has three remarkable properties: 1. (i) $f(F)$ is a basic form on $P$, i.e. it exists a $2k$-form $\Lambda\in\Omega^{2k}(M)$ such that $f(F)=\pi^{}\Lambda$; 2. (ii) $d\Lambda=0$, or equivalently $df(F)=0$; 3. (iii) the cohomology class $[\Lambda]\in H^{2k}(M)$ is independent on the connection $F$. The cohomology class $[\Lambda]$ on $M$ is called _characteristic class_ of $P$ associated to the invariant polynomial $f$. Denoting $\text{Inv}(\mathfrak{h})\subseteq S(\mathfrak{h}^{})$ the algebra of invariant polynomials on $\mathfrak{h}$, the map $\displaystyle w:\text{Inv}(\mathfrak{h})$ $\displaystyle\to H^{}(M)$ (B.3) $\displaystyle f$ $\displaystyle\mapsto[\Lambda]$ is called _Chern-Weil homomorphism_. If one is considering a vector bundle $E\to M$ associated to the principal $H$-bundle $P\to M$, here the connection 1-form $A$ and the curvature $F$ are represented only locally via $\mathfrak{h}$-valued forms on $M$. Under a change of trivialization the local connection does not transform covariantly, but the local curvature does (by conjugation), so the invariant polynomial $f(F)$ is independent on the frame and it defines a global form on $M$. The definition of characteristic classes could be thus given in terms of the local curvature of a vector bundle, without changing the result. We need mainly three examples of characteristic classes, associated to the invariant polynomials $\mathrm{Tr},\det$ and $\mathrm{Pf}$, that corresponds for matrix groups to the standard trace, determinant and pfaffian. These are the _Chern character_ $\mbox{ch}(F):=\mathrm{Tr}\left(e^{F}\right),$ (B.4) the _Euler class_ $e(F):=\mathrm{Pf}\left(\frac{F}{2\pi}\right),$ (B.5) and the _Dirac $\hat{A}$-genus_ $\hat{A}(F):=\sqrt{\det{\left[\frac{\frac{1}{2}F}{\sinh\left(\frac{1}{2}F\right)}\right]}}.$ (B.6) Now we turn the discussion to the case of $G$-equivariant bundles [35, 7, 19]. ###### Definition B.1.1. A _$G$ -equivariant vector bundle_ is a vector bundle $E\xrightarrow{\pi}M$, such that: 1. (i) both $E$ and $M$ are $G$-spaces and $\pi$ is $G$-equivariant; 2. (ii) $G$ acts linearly on the fibers. A principal $H$-bundle $P\xrightarrow{\pi}M$ is $G$-equivariant if 1. (i) both $E$ and $M$ are $G$-spaces and $\pi$ is $G$-equivariant; 2. (ii) the $G$-action commutes with the $H$-action on $P$. Usually, a connection $A$ on a $G$-equivariant principal bundle is required to be _$G$ -invariant_, that is $\mathcal{L}_{X}A=0$ for every $X\in\mathfrak{g}$. If $G$ is compact, this choice is always possible by averaging any connection over $G$ to obtain a $G$-invariant one [35]. Since the $G$\- and the $H$-actions commute, a principal $H$-bundle $P\xrightarrow{\pi}M$ induces another principal $H$-bundle $P_{G}\xrightarrow{\pi_{G}}M_{G}$ over the homotopy quotient $M_{G}$. Topologically, the _equivariant characteristic classes_ of $P\xrightarrow{\pi}M$ are the ordinary characteristic classes of $P_{G}\xrightarrow{\pi_{G}}M_{G}$, thus defining elements in the $G$-equivariant cohomology $H_{G}^{}(M)$. From the differential geometric point of view, they can be derived as _equivariantly closed extensions_ of the ordinary characteristic classes in the Cartan model. In particular, in [35, 7] it was shown that the equivariant characteristic class associated to an invariant polynomial $f$ is represented by $f(F^{\mathfrak{g}})$, where $F^{\mathfrak{g}}=1\otimes F+\phi^{a}\otimes\mu_{a}$ (B.7) is the equivariant extension of the curvature $F$ on the principal $H$-bundle. $\phi^{a=1,\cdots,\dim\mathfrak{g}}$ are the generators of $S(\mathfrak{g}^{})$ in the Cartan model, and the map $\mu:\mathfrak{g}\to\Omega(P;\mathfrak{h})$ such that $\mu_{X}:=-\iota_{X}A=-A(\underline{X})$ (B.8) is called _moment map_ , with analogy to the symplectic case. We denoted $\mu_{a}\equiv\mu_{T_{a}}$ with $T_{a=1,\cdots,\dim\mathfrak{g}}$ the basis of $\mathfrak{g}$ dual to $\phi^{a}$. If we define $\nabla=d+A$ the covariant derivative, that in the adjoint bundle acts as $\nabla\omega=d\omega+[A\stackrel{{\scriptstyle\wedge}}{{,}}\omega]$, we notice that we can obtain the above equivariant curvature in the Cartan model from the _equivariant covariant derivative_ $\nabla^{\mathfrak{g}}:=1\otimes\nabla-\phi^{a}\otimes\iota_{a}$ (B.9) that is completely analogous to the definition of the Cartan differential (2.46). With this definition, the equivariant curvature can be expressed as $F^{\mathfrak{g}}=(\nabla^{\mathfrak{g}})^{2}+\phi^{a}\otimes\mathcal{L}_{a},$ (B.10) where the last piece takes care of the non-nilpotency of the Cartan differential on generic differential forms, and moreover it satisfies an equivariant variation of the Bianchi identity $(\nabla^{\mathfrak{g}}F^{\mathfrak{g}})=0.$ (B.11) Notice that, if we assume the connection $A$ to be $G$-invariant, the moment map $\mu$ indeed satisfies a moment map equation with respect to the curvature $F$ (see Section 3.3.2), $\nabla\mu_{X}=-\iota_{X}F\qquad\forall X\in\mathfrak{g}.$ (B.12) Once a suitable equivariant extension of the curvature $F^{\mathfrak{g}}$ is known, the particular equivariant characteristic classes are simply a modification of the old ones, so the equivariant version of the above Chern character, Euler class and Dirac $\hat{A}$-genus are given by $\mathrm{ch}_{G}(F):=\mathrm{Tr}\left(e^{F^{\mathfrak{g}}}\right),\quad e_{G}(F):=\mathrm{Pf}\left(\frac{F^{\mathfrak{g}}}{2\pi}\right),\quad\hat{A}_{G}(F):=\sqrt{\det{\left[\frac{\frac{1}{2}F^{\mathfrak{g}}}{\sinh\left(\frac{1}{2}F^{\mathfrak{g}}\right)}\right]}},$ (B.13) respectively. ### B.2 Universal bundles and equivariant cohomology In this section we motivate the well-definiteness of equivariant cohomology of Section 2.2, starting from the definition of the space $EG$. Proofs for the various propositions we are going to state informally and/or without proof can be found for example in [18, 24, 21]. We should mention that the mathematically correct approach to this subject works considering only _CW complexes_. These are special types of topological spaces that can be constructed by “attaching deformed disks” to each other [24]. We only quote that any smooth manifold can be given the structure of a CW complex, so that in the smooth setting we do not need to bother with this subtlety.111This is a result of Morse theory, see [18] and references therein. ###### Definition B.2.1. A principal $G$-bundle $\pi:EG\rightarrow BG$ is called _universal G-bundle_ if: 1. (i) for any principal $G$-bundle $P\rightarrow X$, there exists a map $h:X\rightarrow BG$ such that $P\cong h^{}(EG)$ (the pull-back bundle of $EG$ through $h$); 2. (ii) if $h_{0},h_{1}:X\rightarrow BG$ are such that $h_{0}^{}(EG)\cong h_{1}^{}(EG)$, then the two maps are homotopic. The base space $BG$ is called _classifying space_. The classifying property (i) required of $EG$ means that for every principal $G$-bundle there is a copy of it sitting inside $EG\rightarrow BG$. The important fact is that existence can be proven for a large class of interesting cases, the argument going as follows. First recall that _homotopic maps pull back to isomorhic bundles_ , i.e. if $E\rightarrow B$ is a vector bundle, $X$ a paracompact space, then $g,h:X\rightarrow B\ \text{homotopic maps}\ \Rightarrow g^{}(E)\cong h^{}(E).$ (B.14) Then the property (ii) in the definition above states that if $E\rightarrow B$ is a universal bundle, $\Rightarrow$ is replaced by $\Leftrightarrow$. Now let, for any paracompact space $X$, $P_{G}(X):=\left\\{\text{isomorphism classes of principal G-bundles over X}\right\\}$ (B.15) and for some space $BG$ (to be identified with the classifying space), $[X,BG]:=\left\\{\text{homotopy classes of maps }X\rightarrow BG\right\\}.$ (B.16) Notice that the definition of $P_{G}(X)$ is totally independent from the notion of universal $G$-bundle. Considering then the map $\displaystyle\phi:[X,BG]$ $\displaystyle\rightarrow P_{G}(X)$ (B.17) $\displaystyle[h:X\rightarrow BG]$ $\displaystyle\mapsto h^{}(EG),$ by (B.14) we have that it is well-defined (independent from the representatives). The conditions (i) and (ii) are equivalent to surjectivity and injectivity of $\phi$, so finally $P_{G}(X)\cong[X,BG]$. Since $P_{G}(X)$ exists, this proves the existence of the classifying space $BG$ and of the universal bundle $EG\rightarrow BG$.222In the language of category theory, we could say that $P_{G}(\cdot)$ is a contravariant functor, _representable_ through $[\cdot,BG]$. We can now motivate the well-definiteness of the Borel construction for equivariant cohomology. A fundamental result for this is that _a principal $G$-bundle is a universal bundle if and only if its total space is (weakly) contractible_.333A _weakly contractible_ space is a topological space whose homotopy groups are all trivial. Clearly any contractible space is weakly contractible. It is a fact that every CW complex that is weakly contractible is also contractible [24], so for our purposes the two concepts coincide. The contractibility of $EG$ makes its cohomology trivial, so that, since $(M\times EG)\sim M$, we have $H^{}(M\times EG)\cong H^{}(M)$. When we take the homotopy quotient, the product by $EG$ acts as a “regulator” of the resulting cohomology. In fact, if the action of $G$ on $M$ is free, such that $M\to M/G$ is a principal $G$-bundle, one can prove that for any (weakly) contractible $G$-space $E$ $\faktor{(M\times E)}{G}\sim\faktor{M}{G},$ (B.18) where $\sim$ here stands for “weakly homotopic”.444Two spaces are _weakly homotopic_ if they have the same homotopy groups. Again, homotopy equivalence implies weak homotopy equivalence, and for CW complexes these two concepts coincide. In general, even if the $G$-action is not free, two homotopy quotients with respect to different (weakly) contractible $G$-spaces $E$ and $E^{\prime}$ are (weakly) homotopy equivalent, $\faktor{(M\times E)}{G}\sim\faktor{(M\times E^{\prime})}{G}.$ (B.19) Another known fact is that _weakly homotopic spaces have the same (co)homology groups, for all coefficients_ , generalizing (2.9). Putting together these properties, we have that $H^{}\left(\faktor{(M\times E)}{G}\right)\cong H^{}\left(\faktor{(M\times E^{\prime})}{G}\right),$ (B.20) so that the resulting cohomology is independent of the choice of contractible principal $G$-bundle. The homotopy quotient thus well-defines the $G$-equivariant cohomology of $M$, producing an “homotopically correct” version of its orbit space. As pointed out in Section 2.2, when the $G$-action is free on $M$ this reproduces the naive definition of cohomology of the quotient space $M/G$. #### Every compact Lie group has a universal bundle In Section 2.2 we gave the example of the universal bundle for the circle, $EU(1)=\mathbb{S}^{\infty}$ and $BU(1)=\mathbb{C}P^{\infty}$. One can generalize this construction to concretely define a universal bundle for any compact Lie group $G$. This is because any such Lie group embeds into $U(n)$ or $O(n)$ (the maximal compact subgroups of $GL(n,\mathbb{C})$ and $GL(n,\mathbb{R})$), for some $n$, and for them one can construct universal bundles explicitly. As a subgroup, $G$ will act freely on the given universal bundle. Then one can take this to be its universal bundle too. A class of principal $O(n)$\- or $U(n)$-bundles is given by the so-called _Stiefel manifolds_. A Stiefel manifold $V_{k}(\mathbb{F}^{n})$ is the set of all _orthonormal $k$-frames_ in $\mathbb{F}^{n}$, where $\mathbb{F}=\mathbb{R},\mathbb{C}$, and the orthonormality is defined with respect to the canonical Euclidean or sesquilinear inner products. A _$k$ -frame_ is an ordered set $(v_{1},\cdots,v_{k})$ of $k$ linearly independent vectors in $\mathbb{F}^{n}$. Notice that when $k=1$, $V_{1}(\mathbb{C}^{n})$ is the set of all unit vectors in $\mathbb{C}^{n}\cong\mathbb{R}^{2n}$, i.e. the $(2n-1)$-sphere. The latter is acted freely by the group $U(1)$ by diagonal multiplication, and analogously the Stiefel manifold $V_{k}(\mathbb{C}^{n})$ is acted freely by $U(k)$, that essentially rotates the vectors of the $k$-frames. Analogously, $V_{k}(\mathbb{R}^{n})$ is acted freely by $O(k)$. Thus we have the generalization of the sequence of principal $U(k)$\- and $O(k)$-bundles, that in the limit $n\to\infty$ produces the contractible universal bundles $EU(k)=V_{k}(\mathbb{C}^{\infty})$ and $EO(k)=V_{k}(\mathbb{R}^{\infty})$. The Stiefel manifolds can thus be seen as a “higher dimensional versions” of the spheres, in the sense of the following consideration: $V_{k}(\mathbb{R}^{n})\cong\faktor{O(n)}{O(n-k)}.$ (B.21) Comparing with Example 2.2.1, where we remarked that $\mathbb{S}^{n-1}\cong O(n)/O(n-1)$, we see that indeed $\mathbb{S}^{n-1}\equiv V_{1}(\mathbb{R}^{n})$. The $(n-1)$-sphere is just the set of unit vectors in $\mathbb{R}^{n}$, so orthonormal 1-frames. The base spaces $G_{k}(\mathbb{C}^{n})=V_{k}(\mathbb{C}^{n})/U(k)$ and $G_{k}(\mathbb{R}^{n})=V_{k}(\mathbb{R}^{n})/O(k)$ are the sets of equivalence classes of $k$-frames, that identify $k$_-hyperplanes_ through the origin inside $\mathbb{C}^{n}$ or $\mathbb{R}^{n}$. These manifolds are called _Grassmannians_. The _infinite Stiefel manifold_ $V_{k}(\mathbb{F}^{\infty})$ and the _infinite Grassmannian_ $G_{k}(\mathbb{F}^{\infty})$ are thus the total space of the universal bundle and the classifying space for the unitary and orthogonal groups $U(k)$ and $O(k)$, and generalize the universal bundle $\mathbb{S}^{\infty}\to\mathbb{C}P^{\infty}$ of the circle. As recalled above, any compact Lie group $G$ can be embedded as a closed subgroup of an orthogonal group (or a unitary group). This means that $G$ also acts freely on $V_{k}(\mathbb{F}^{\infty})$ for some $k$, and in turn $V_{k}(\mathbb{F}^{\infty})\to V_{k}(\mathbb{F}^{\infty})/G$ is a principal $G$-bundle, whose total space is a contractible space. This gives the universal bundle for any compact Lie group $G$. #### Module structure of equivariant cohomology We end this section with a more algebraic comment about the construction of equivariant cohomology. Notice first that $pt_{G}=\faktor{(pt\times EG)}{G}\cong BG\quad\Rightarrow\quad H_{G}^{}(pt)\cong H^{}(BG),$ (B.22) so the equivariant cohomology of a point is the standard cohomology of the classifying space $BG$, generalizing Example 2.2.2. Thus the equivariant cohomology $H_{G}^{}(\cdot)$ inherits analogous functorial properties to the standard (singular) cohomology of the last section, with respect to the ring $H^{}(BG)$ instead of the coefficient ring $A\cong H^{}(pt;A)$. To see this, let us first notice that a $G$-equivariant function $f:M\to N$ between the two $G$-spaces $M,N$ induces a well-defined map between the two homotopy quotients, $\displaystyle f_{G}:M_{G}$ $\displaystyle\rightarrow N_{G}$ (B.23) $\displaystyle[m,e]$ $\displaystyle\mapsto[f(m),e].$ This induced map inherits many properties from $f$: 1. (i) if $f$ is injective (surjective), then $f_{G}$ is injective (surjective); 2. (ii) if $id:M\rightarrow M$ is the identity, then $id_{G}:M_{G}\rightarrow M_{G}$ is the identity; 3. (iii) $(h\circ f)_{G}=h_{G}\circ f_{G}$; 4. (iv) if $f:M\rightarrow N$ is a fiber bundle with fiber $F$, then $f_{G}:M_{G}\rightarrow N_{G}$ is also a fiber bundle with fiber $F$. As pointed out in Section 2.1, a map between two topological spaces induces a map (in the opposite direction) between the associated singular cohomologies, so $f_{G}^{}:\left(H^{}(N_{G})\equiv H^{}_{G}(N)\right)\to\left(H^{}(M_{G})\equiv H^{}_{G}(M)\right).$ (B.24) Defining thus a trivial map $\phi:M\to pt$, we see from (B.22) that the induced homomorphism $\phi^{}_{G}:H^{}(BG)\to H^{}_{G}(M)$ makes the equivariant cohomology $H^{}_{G}(M)$ naturally into a $H^{}(BG)$-module!555Recall that singular cohomology has a ring structure. Also, in general $f_{G}^{}:H^{}_{G}(N)\to H^{}_{G}(M)$ is a $H^{}(BG)$-module homomorphism.666In category theory terminology, we could say that the Borel construction $(\cdot)_{G}$ is a covariant functor from the category of $G$-spaces to Top (or Man), and $H_{G}^{}(\cdot)$ is a contravariant functor between Top (or Man) and the category of $H^{}(BG)$-modules. Notice that the cohomology of the classifying space $BG$ is usually very simple, as we pointed out in Section 2.4 via its associated Weil model. There is a curious difference between standard cohomology and equivariant cohomology regarding the associated coefficient rings. In the former case, it is clear from the various examples in Section 2.1 that the coefficient ring $\mathbb{R}\cong H^{}(pt)$ always embeds into the cohomology $H^{}(M)$ (also for other commutative rings). In the case of equivariant cohomology, on the other hand, the coefficient ring $H_{G}^{}(pt)=H^{}(BG)=S(\mathfrak{g}^{})^{G}$ does not, since the map $\phi_{G}^{}$ above is not injective in general, as it is clear also from the example of $H^{}_{U(1)}(\mathbb{S}^{1})=\mathbb{R}$. It turns out that the condition for $H^{}(BG)$ to embed in $H^{}_{G}(M)$ is that $G$ acts on $M$ _with fixed points_. We can argue briefly why this is the case. Let $p\in M$ be a fixed point. The inclusion $i:\\{p\\}\to M$ is $G$-equivariant since the action on $p$ is trivial, so there is a well-defined map $i_{G}:pt_{G}=BG\to M_{G}$. This is easily checked to be a section of the bundle $M_{G}\xrightarrow{\pi}BG$, with respect to the projection map $\pi([m,e]):=[e]\in BG$. The identity $\pi\circ i_{G}=id_{M_{G}}$ lifts to the pull-backs in the opposite direction: $i_{G}^{}\circ\pi^{}=id$ on $H_{G}^{}(pt)=H^{}(BG)$. This means that the map $\pi^{}:H^{}(BG)\to H_{G}^{}(M)$ has a left-inverse, and thus it is injective. This property can be seen in the example of the $U(1)$-equivariant cohomology of the 2-sphere. In this case there are two fixed points, and indeed $H^{}(BU(1))=\mathbb{R}[\phi]$ embeds in $H^{}_{U(1)}(\mathbb{S}^{2})=\mathbb{R}[\phi]\oplus\mathbb{R}[\phi]y$, where $y$ can be identified in the Cartan model with the equivariantly closed extension of the volume form, $y\equiv[\tilde{\omega}]$. ### B.3 Fixed point sets and Borel localization We now spend a few words about a procedure that we used many times without many worries, that is to “algebraically localize” the space of equivariant differential forms $\Omega(M)^{U(1)}[\phi]$ with respect to the indeterminate $\phi$, setting it to $\phi=-1$. This localization was useful to simplify the notation in many occasions, but it really has a non-trivial deeper meaning. In fact, it allows to show in a more algebraic way that the $G$-equivariant cohomology of the smooth $G$-manifold $M$ is encoded in the fixed point set $F$ of the $G$-action, at least when $G$ is a torus. The fundamental theorem concerning this point is the so-called _Borel localization theorem_ , that sometimes allows to obtain the ring structure of the equivariant cohomology of the manifold from that of its fixed point set. We consider the case of a circle action here. First, let us recall what localization in algebra means. If $R$ is a commutative ring, the _localization_ of $R$ with respect to a closed subset $S\subseteq R$ is a way to formally introduce a multiplicative inverse for every element of $S$ in $R$, so to introduce _fractions_ in $R$, analogously to what one does in the construction of the rational numbers $\mathbb{Q}$ from the integers $\mathbb{Z}$. This procedure makes the former commutative ring into a field (in the algebraic sense). Since we are interested in $U(1)$-equivariant cohomologies, let us consider an $\mathbb{R}[\phi]$-module $N$, and practically define the _localization of N with respect to $\phi$_ as $N_{\phi}\cong\left\\{\left.\frac{x}{\phi^{n}}\right\|x\in N,n\in\mathbb{N}\right\\},$ (B.25) identifying elements in $N_{\phi}$ as $\frac{x}{\phi^{n}}\sim\frac{y}{\phi^{m}}\Leftrightarrow\exists k\in\mathbb{N}:\phi^{k}(\phi^{m}x-\phi^{n}y)=0\ \text{in}\ N.$ (B.26) The simplest example of such a localized module is just $\mathbb{R}[\phi]_{\phi}\cong\mathbb{R}[\phi^{-1},\phi]$, i.e. the Laurent polynomials in $\phi$. Notice that there is always an $\mathbb{R}[\phi]$-module homomorphism that makes $N$ inject into $N_{\phi}$, $i:N\to N_{\phi}$ such that $i(x):=x/\phi^{0}$. If $f:N\to M$ is an $\mathbb{R}[\phi]$-module homomorphism, then there is a well-defined induced homomorphism between the localized modules $f_{\phi}:N_{\phi}\to M_{\phi}$ such that $f(x/\phi^{n}):=f(x)/\phi^{n}$. The important algebraic property of localization for what concerns this discussion is that _it commutes with cohomology_ : if $(A,d)$ is a differential complex, $A^{(0)}\xrightarrow{d}A^{(1)}\xrightarrow{d}\cdots,\qquad d^{2}=0,$ (B.27) where $A^{(i)}$ are $\mathbb{R}[\phi]$-modules, then also $(A_{\phi},d_{\phi})$ is a differential complex, and $H^{}(A,d)_{\phi}\cong H^{}(A_{\phi},d_{\phi}).$ (B.28) Quite analogously, from Example 2.4.1 onward we substitute the _indeterminate_ $\phi\in S(\mathfrak{u}(1)^{})$ with a _variable_ , and then set it to the value $\phi=-1$ for notational convenience. Stated more formally, we start from the Cartan model of $U(1)$-equivariant differential forms $\Omega(M)^{U(1)}[\phi]$, that has clearly an $\mathbb{R}[\phi]$-module structure, and localize it to $\Omega(M)^{U(1)}[\phi]_{\phi}$, so introducing $\phi$ also at the denominator. This puts $\phi$ on the same footing as a real variable, so that we are allowed to fix it to some value, for convenience only. Notice that operations like (3.8), where we “invert” an equivariant form, are allowed only in the localized module $\Omega(M)^{U(1)}[\phi]_{\phi}$, where the division by $\phi$ is meaningful. From the result (B.28), we understand that this localization of the Cartan model does not spoil the resulting equivariant cohomology $H_{U(1)}^{}(M)$, because the two operations commute.777Notice that $H_{U(1)}^{}(M)$ has generically an $\mathbb{R}[\phi]$-module structure, by the discussion in Appendix B.2 and the application of the Weil model (see Section 2.4) $H^{}(BG)\cong S(\mathfrak{u}(1)^{})\cong\mathbb{R}[\phi]$. The Borel localization theorem relates really the localized equivariant cohomologies of the $U(1)$-manifold $M$ and of its fixed point locus $F$. To understand what this has to say about the actual equivariant cohomology of $M$, we recall first some other algebraic facts. A _torsion_ element in a module $N$ over a ring $R$, is an element $x\in N$ such that $\exists r\neq 0\in R:rx=0$. If $N$ is an $\mathbb{R}[\phi]$ module, the element $x$ is said to be $\phi$-_torsion_ if it exists some power of $\phi$ that annihilates it: $\phi^{k}x=0$ for some $k\in\mathbb{N}$. The module $N$ is $\phi$-torsion if every one of its elements is $\phi$-torsion. It is easy to see that888Just consider that in the localized module $x\sim\frac{\phi^{k}}{\phi^{k}}x$, so if $x$ is $\phi$-torsion it is equivalent to 0 in $N_{\phi}$. $N\ \text{is }\phi\text{-torsion}\quad\Leftrightarrow\quad N_{\phi}=0.$ (B.29) Applying this to the case of $N=H^{}_{U(1)}(M)$, we can see that the equivariant cohomology in the case of a free $U(1)$-action on $M$ is $\phi$-torsion. In fact, if the action is free, we can easily compute $H_{U(1)}^{}(M)=H^{}(M/U(1))$, so that $H_{U(1)}^{k}(M)=0$ in some degree $k>\dim(M/U(1))$. This means that $\phi^{k}\cdot H_{U(1)}^{}(M)=0$ for some $k$ high enough. The first argument in Section 3.1 in fact is the proof that more is true: $H^{}_{U(1)}(M)$ is $\phi$-torsion if the $U(1)$-action is _locally free_ on $M$, since we found essentially $(H^{}_{U(1)}(M))_{\phi}=0$ as the Poincaré lemma, after having introduced $\phi$ at the denominator.999In Section 3.1 $M$ is the manifold without its fixed point set, there called $\tilde{M}$. This motivates the following theorem, that states that, _up to torsion_ , the $U(1)$-equivariant cohomology of $M$ is concentrated on its fixed point set. A proof can be found in [22, 18]. ###### Theorem B.3.1 (Borel localization). Let $U(1)$ act smoothly on the manifold $M$, with compact fixed point set $F$. The inclusion $i:F\hookrightarrow M$ induces an isomorphism of algebras over $\mathbb{R}[\phi]$, $i^{}_{\phi}:H^{}_{U(1)}(M)_{\phi}\to H^{}_{U(1)}(F)_{\phi}.$ This theorem is an “abstract version” of the localization theorems described in Chapter 3, and intuitively gives another way to see that they have to be true, without having to travel through all the smooth algebraic models and the integration theory that we described in due time. It shows that localization is something present at a very low level of structure, originating just from the topological nature of equivariant cohomology. ### B.4 Equivariant integration and Stokes’ theorem In this section we define what it means to integrate a $G$-equivariant differential form $\omega\in\Omega_{G}(M)$ over a smooth, oriented $G$-manifold $M$ of dimension $dim(M)=n$ and we report an extended version of Stokes’ theorem that applies in the equivariant setup. Let $G$ be a connected Lie group acting (smoothly) on the left on $M$, being $\\{\phi^{a}\\}_{a=1,\cdots,dim(\mathfrak{g})}$ a basis for $\mathfrak{g}^{}:=Lie(G)^{}$. If the equivariant form $\omega$ is of degree $k$, we can express it as $\omega=\omega^{(k)}+\omega^{(k-2)}_{a}\phi^{a}+\omega^{(k-4)}_{ab}\phi^{a}\phi^{b}+\cdots=\sum_{p\geq 0}\omega^{(k-2p)}_{a_{1}\cdots a_{p}}\phi^{a_{1}}\cdots\phi^{a_{p}}$ (B.30) where the coefficients are differential forms on $M$, tensor products have been suppressed and we require $\omega$ to be $G$-invariant. The natural way to define integration of such objects is obtained just making the integral $\int_{M}$ act on the coefficients $\omega^{(k-2p)}_{a_{1}\cdots a_{p}}$ of the $\phi$-expansion of $\omega$. In this way, one obtains a map $\int_{M}:\Omega_{G}(M)\to S(\mathfrak{g}^{})\equiv\mathbb{R}[\phi^{a}].$ (B.31) Thanks to the equivariant Stokes’ theorem (to be stated later), this descends also in equivariant cohomology, $\int_{M}:H^{}_{G}(M)\to S(\mathfrak{g}^{})$, analogously to the standard (non-equivariant) case. ###### Definition B.4.1. The integral on $M$ of the $G$-equivariant form $\omega$ of $\textrm{deg}(\omega)=k$ is defined as $\int_{M}\omega:=\sum_{p\geq 0}\left(\int_{M}\omega^{(k-2p)}_{a_{1}\cdots a_{p}}\right)\phi^{a_{1}}\cdots\phi^{a_{p}}.$ Notice that if $n$ and $k$ are of different parity, the integral is automatically zero. If instead $k=n+2m$ for some $m\in\mathbb{Z}$, then $\int_{M}\omega=\begin{cases}\left(\int_{M}\omega^{(n)}_{a_{1}\cdots a_{m}}\right)\phi^{a_{1}}\cdots\phi^{a_{m}}&k\geqslant n\\\ 0&k<n.\end{cases}$ (B.32) In particular, if we have a top form $\omega\in\Omega(M)$ on M and $\tilde{\omega}$ is any equivariant extension of $\omega$ in $\Omega_{G}(M)$, then we can deform the integral $\int_{M}\omega=\int_{M}\tilde{\omega}$ (B.33) without changing its value. We can then prove the equivariant version of the Stokes’ theorem. ###### Theorem B.4.1. Let $G$ be a connected Lie group acting (smoothly) on the left on a smooth manifold $M$ with boundary $\partial M$. If $\omega\in\Omega_{G}(M)$ of $\textrm{deg}(\omega)=k$, then $\int_{M}d_{C}\omega=\int_{\partial M}\omega$ where $d_{C}=1\otimes d+\phi^{a}\otimes\iota_{a}$ is the Cartan differential and $\iota_{a}\equiv\iota_{T_{a}}$, with $\\{T_{a}\\}$ a basis of $\mathfrak{g}^{}$ dual to $\\{\phi^{a}\\}$. ###### Proof. The proof follows from the direct evaluation and the standard Stokes’ theorem. If the integral is not zero, selecting the component of top-degree, $\left.(d_{C}\omega)\right\|_{(n)}=d\omega^{(n-1)}_{I}\phi^{I}+(\iota_{\\{a}\omega^{(n+1)}_{J\\}})\phi^{a}\phi^{J},$ where $I=(a_{1},\cdots,a_{(k-n)/2})$ and $J=(a_{1},\cdots,a_{(k-n+1)/2})$. The second term vanishes since $\omega^{(n+1)}=0$ by dimensionality. So the integral of $d_{C}\omega$ is the integral of the first term, on which we can use the standard version of Stokes’ theorem, and getting the statement of the theorem. ∎ ## References [1] H. Cartan “Notions d’algèbre différentielle; application aux groupes de Lie et aux variétés où opère un groupe de Lie” In _Colloque de topologie (espaces fibrés)_ , 1951 * [2] H. Cartan “La transgression dans un groupe de Lie et dans un espace fibré principal” In _Colloque de topologie (espaces fibrés)_ , 1951 * [3] A. 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# Some applications of transversality for infinite dimensional manifolds Kaveh Eftekharinasab Topology lab. Institute of Mathematics of National Academy of Sciences of Ukraine, Tereshchenkivska st. 3, Kyiv, 01601 Ukraine <EMAIL_ADDRESS> ###### Abstract. We present some transversality results for a category of Fréchet manifolds, the so-called $MC^{k}$-Fréchet manifolds. In this context, we apply the obtained transversality results to construct the degree of nonlinear Fredholm mappings by virtue of which we prove a rank theorem, an invariance of domain theorem and a Bursuk-Ulam type theorem. ###### Key words and phrases: Transversality, degree of nonlinear Fredholm mappings, Fréchet manifolds ###### 2020 Mathematics Subject Classification: 57N75, 58B15, 47H11. This paper is devoted to the development of transversality and its applications to degree theory of nonlinear Fredholm mappings for non- Banachable Fréchet manifolds. The elaboration is mostly, but not entirely, routine; we shall discuss the related issues. In attempting to develop transversality to Fréchet manifolds we face the following drawbacks which are related to lack of a suitable topology on a space of continuous linear maps: 1. (1) In general, the set of isomorphisms between Fréchet spaces is not open in the space of continuous linear mappings. 2. (2) In general, the set of Fredholm operators between Fréchet spaces is not open in the space of continuous linear mappings. Also, a key point in the proof of an infinite dimensional version of Sard’s theorem is that a Fredholm mapping $\varphi$ near origin has a local representation of the form $\varphi(u,v)=(u,\eta(u,v))$ for some smooth mapping $\eta$; indeed, this is a consequence of an inverse function theorem. To obtain a version of Sard’s theorem for Fréchet manifolds, based on the ideas of Müller [4], it was proposed by the author ([1]) to consider Fredholm operators which are Lipschitz on their domains. There is an appropriate metrizable topology on a space of Lipschitz linear mappings so that if we employ this space instead of a space of continuous linear mappings, the mentioned openness issues and the problem of stability of Fredholm mappings under small perturbation can be resolved. Furthermore, for mappings belong to a class of differentiability, bounded or $MC^{k}$-differentiability which is introduced in [4], a suitable version of an inverse function theorem is available, [4, Theorem 4.7]. An example of Lipschitz-Fredholm mapping of class $MC^{k}$ can be found in [3], where the Sard’s theorem [1, Theorem 4.3] is applied to classify all the holomorphic functions locally definable; this gives the additional motivation to study further applications of Sard’s theorem. In this paper, first we improve the transversality theorem [2, Theorem 4.2] by considering all mappings of class $MC^{k}$, then use it to prove the parametric transversality theorem. Then, for Lipschitz-Fredholm mappings of class $MC^{k}$ we apply the transversality theorem to construct the degree (due to Cacciappoli, Shvarts and Smale), which is defined as the group of non- oriented cobordism class of $\varphi^{-1}(q)$ for some regular value $q$. We then prove a rank theorem for Lipschitz-Fredholm mappings of class $MC^{k}$ , and use it to prove an invariance of domain theorem and a Fredholm alternative theorem. Also, using the parametric transversality theorem we obtain a Bursuk-Ulam type theorem. ## 1\. Bounded Fréchet manifolds In this section, we shall briefly recall the basics of $MC^{k}$-Fréchet manifolds for the convenience of readers, which also allows us to establish our notations for the rest of the paper. For more studies, we refer to [1, 2]. Throughout the paper we assume that $E,F$ are Fréchet spaces and $CL(E,F)$ is the space of all continuous linear mappings from $E$ to $F$ topologized by the compact-open topology. If $T$ is a topological space by $U\mathrel{\mathchoice{\ooalign{$\displaystyle\subseteq$\cr\raise 1.67915pt\hbox{$\displaystyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\textstyle\subseteq$\cr\raise 1.67915pt\hbox{$\textstyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptstyle\subseteq$\cr\raise 1.18399pt\hbox{$\scriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptscriptstyle\subseteq$\cr\raise 0.6458pt\hbox{$\scriptscriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}}T$ we mean $U$ is open in $T$. Let $\varphi:U\mathrel{\mathchoice{\ooalign{$\displaystyle\subseteq$\cr\raise 1.67915pt\hbox{$\displaystyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\textstyle\subseteq$\cr\raise 1.67915pt\hbox{$\textstyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptstyle\subseteq$\cr\raise 1.18399pt\hbox{$\scriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptscriptstyle\subseteq$\cr\raise 0.6458pt\hbox{$\scriptscriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}}E\to F$ be a continuous map. If the directional (Gâteaux) derivatives $\operatorname{D}\varphi(x)h=\lim_{t\to 0}\dfrac{\varphi(x+th)-\varphi(x)}{t}$ exist for all $x\in U$ and all $h\in E$, and the induced map $\operatorname{D}\varphi(x):U\rightarrow CL(E,F)$ is continuous for all $x\in U$, then we say that $\varphi$ is a Keller’s differentiable map of class $C^{1}_{c}$. The higher directional derivatives and $C^{k}_{c}$-mappings, $k\geq 2$, are defined in the obvious inductive fashion. To define bounded or $MC^{k}$-differentiability, we endow a Fréchet space $F$ with a translation invariant metric $\varrho$ defining its topology, and then introduce the metric concepts which strongly depend on the choice of $\varrho$. We consider only metrics of the following form $\varrho(x,y)=\sup_{n\in\mathbb{N}}\dfrac{1}{2^{n}}\dfrac{\left\lVert x-y\right\rVert_{F,n}}{1+\left\lVert x-y\right\rVert_{F,n}},$ where $\left\lVert\cdot\right\rVert_{F,n}$ is a collection of seminorms generating the topology of $F$. Let $\sigma$ be a metric that defines the topology of a Fréchet space $E$. Let $\mathbb{L}_{\sigma,\varrho}(E,F)$ be the set of all linear mappings $L:E\rightarrow F$ which are (globally) Lipschitz continuous as mappings between metric spaces $E$ and $F$, that is $\mathpzc{Lip}(L)\,\coloneq\displaystyle\sup_{x\in E\setminus\\{0_{F}\\}}\dfrac{\varrho(L(x),0_{F})}{\sigma(x,0_{F})}<\infty,$ where $\mathpzc{Lip}(L)$ is the (minimal) Lipschitz constant of $L$. The translation invariant metric $\mathbbm{d}_{\sigma,\varrho}:\mathbb{L}_{\sigma,\varrho}(E,F)\times\mathbb{L}_{\sigma,\varrho}(E,F)\longrightarrow[0,\infty),\,\,(L,H)\mapsto\mathpzc{Lip}(L-H)_{\sigma,\varrho}\,,$ (1.1) on $\mathbb{L}_{\sigma,\varrho}(E,F)$ turns it into an Abelian topological group. We always topologize the space $\mathbb{L}_{\sigma,\varrho}(E,F)$ by the metric (1.1). Let $\varphi:U\mathrel{\mathchoice{\ooalign{$\displaystyle\subseteq$\cr\raise 1.67915pt\hbox{$\displaystyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\textstyle\subseteq$\cr\raise 1.67915pt\hbox{$\textstyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptstyle\subseteq$\cr\raise 1.18399pt\hbox{$\scriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptscriptstyle\subseteq$\cr\raise 0.6458pt\hbox{$\scriptscriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}}E\rightarrow F$ be a continuous map. If $\varphi$ is Keller’s differentiable, $\operatorname{D}\varphi(x)\in\mathbb{L}_{\sigma,\varrho}(E,F)$ for all $x\in U$ and the induced map $\operatorname{D}\varphi(x):U\rightarrow\mathbb{L}_{\sigma,\varrho}(E,F)$ is continuous, then $\varphi$ is called bounded differentiable or $MC^{1}$ and we write $\varphi^{(1)}=\varphi^{\prime}$. We define for $k>1$ mappings of class $MC^{k}$, recursively. An $MC^{k}$-Fréchet manifold is a Hausdorff second countable topological space modeled on a Fréchet space with an atlas of coordinate charts such that the coordinate transition functions are all $MC^{k}$-mappings. We define $MC^{k}$-mappings between Fréchet manifolds as usual. Henceforth, we assume that $M$ and $N$ are connected $MC^{k}$-Fréchet manifolds modeled on Fréchet spaces $(F,\varrho)$ and $(E,\sigma)$, respectively. A mapping $\varphi\in\mathbb{L}_{\sigma,\varrho}(E,F)$ is called Lipschitz- Fredholm operator if its kernel has finite dimension and its image has finite co-dimension. The index of $\varphi$ is defined by $\operatorname{Ind}\varphi=\dim\ker\varphi-\operatorname{codim}\operatorname{Img}\varphi.$ We denote by $\mathcal{LF}(E,F)$ the set of all Lipschitz-Fredholm operators, and by $\mathcal{LF}_{l}(E,F)$ the subset of $\mathcal{LF}(E,F)$ consisting of those operators of index $l$. An $MC^{k}$-Lipschitz-Fredholm mapping $\varphi:M\rightarrow N,\,k\geq 1$, is a mapping such that for each $x\in M$ the derivative $\operatorname{D}\varphi(x):T_{x}M\longrightarrow T_{f(x)}N$ is a Lipschitz- Fredholm operator. The index of $\varphi$, denoted by $\operatorname{Ind}{\varphi}$, is defined to be the index of $\operatorname{D}\varphi(x)$ for some $x$ which does not depend on the choice of $x$, see [1, Definition 3.2 ]. Let $\varphi:M\rightarrow N$ $(k\geq 1)$ be an $MC^{k}$-mapping. We denote by $T_{x}\varphi:T_{x}M\rightarrow T_{\varphi(x)}N$ the tangent map of $f$ at $x\in M$ from the tangent space $T_{x}M$ to the tangent space $T_{\varphi(x)}N$. We say that $\varphi$ is an immersion (resp. submersion) provided $T_{x}\varphi$ is injective (resp. surjective) and the range $\operatorname{Img}(T_{x}\varphi)$ (resp. the kernel $\ker(T_{x}\varphi)$) splits in $T_{\varphi(x)}N$ (resp. $T_{x}M$) for any $x\in M$. An injective immersion $f:M\rightarrow N$ which gives an isomorphism onto a submanifold of $N$ is called an embedding. A point $x\in M$ is called a regular point if $\operatorname{D}f(x):T_{x}M\longrightarrow T_{f(x)}N$ is surjective. The corresponding value $f(x)$ is a regular value. Points and values other than regular are called critical points and values, respectively. Let $\varphi:M\to N$ be an $MC^{k}$-mapping, $k\geq 1$. We say that $\varphi$ is transversal to a submanifold $S\subseteq N$ and write $\varphi\pitchfork S$ if either $\varphi^{-1}(S)=\emptyset$, or if for each $x\in\varphi^{-1}(S)$ 1. (1) $(T_{x}\varphi)(T_{x}M)+T_{\varphi(x)}S=T_{\varphi(x)}N$, and 2. (2) $(T_{x}\varphi)^{-1}(T_{\varphi(x)}S)$ splits in $T_{x}M$. In terms of charts, $\varphi\pitchfork S$ when $x\in\varphi^{-1}(S)$ there exist charts $(\phi,\mathcal{U})$ around $x$ and $(\psi,\mathcal{V})$ around $\varphi(x)$ such that $\psi:\mathcal{V}\to\mathcal{V}_{1}\times\mathcal{V}_{2}$ is an $MC^{k}$-isomorphism on a product, with $\psi(\varphi(x))=(0_{E},0_{E})\,\quad\varphi(S\cap\mathcal{V})=\mathcal{V}_{1}\times\left\\{0_{E}\right\\}.$ Then the composite mapping $\mathcal{U}\xrightarrow{\varphi}\mathcal{V}\xrightarrow{\psi}\mathcal{V}_{1}\times\mathcal{V}_{2}\xrightarrow{\mathrm{Pr}_{V_{2}}}\mathcal{V}_{2}.$ is an $MC^{k}$-submersion, where $\mathrm{Pr}_{V_{2}}$ is the projection onto $\mathcal{V}_{2}$. ## 2\. Transversality theorems We generalize [2, Theorem 4.2] and [2, Corollary 4.1] for not necessarily Lipschitz-Fredholm mappings and finite dimensional submanifolds. We shall need the following version of the inverse function theorem for $MC^{k}$-mappings. ###### Theorem 2.1. [4, Theorem 4.7] Let $\mathcal{U}\mathrel{\mathchoice{\ooalign{$\displaystyle\subseteq$\cr\raise 1.67915pt\hbox{$\displaystyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\textstyle\subseteq$\cr\raise 1.67915pt\hbox{$\textstyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptstyle\subseteq$\cr\raise 1.18399pt\hbox{$\scriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptscriptstyle\subseteq$\cr\raise 0.6458pt\hbox{$\scriptscriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}}E$, $u_{0}\in\mathcal{U}$ and $\varphi:\mathcal{U}\rightarrow E$ an $MC^{k}$-mapping, $k\geq 1$. If $\varphi^{\prime}(u_{0})\in\operatorname{Aut}{(E)}$, then there exists $\mathcal{V}\mathrel{\mathchoice{\ooalign{$\displaystyle\subseteq$\cr\raise 1.67915pt\hbox{$\displaystyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\textstyle\subseteq$\cr\raise 1.67915pt\hbox{$\textstyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptstyle\subseteq$\cr\raise 1.18399pt\hbox{$\scriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptscriptstyle\subseteq$\cr\raise 0.6458pt\hbox{$\scriptscriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}}\mathcal{U}$ of $u_{0}$ such that $\varphi(\mathcal{V})$ is open in $E$ and $\varphi\|_{\mathcal{V}}:\mathcal{V}\to\varphi(\mathcal{V})$ is an $MC^{k}$\- diffeomorphism. ###### Proposition 2.1. Let $\varphi:M\to N$ be an $MC^{k}$-mapping, $S\subset N$ an $MC^{k}$-submanifold and $x\in\varphi^{-1}(S)$. Then $\varphi\pitchfork S$ if and only if there are charts $(\mathcal{U},\phi)$ around $x$ with $\phi(x)=0_{E}$ and $(\mathcal{V},\psi)$ around $y=\varphi(x)$ in $S$ with $\psi(y)=0_{F}$ such that the following hold: 1. (1) There are subspaces $\bf E_{1}$ and $\bf E_{2}$ of $E$, and $\bf F_{1}$ and $\bf F_{2}$ of $F$ such that $E=\bf E_{1}\oplus\bf E_{2}$ and $F=\bf F_{1}\oplus F_{2}$. Moreover, $\psi(S\cap\mathcal{V})=F_{1}$ and $\displaystyle\phi(\mathcal{U})=E_{1}+E_{2}\subseteq{\bf E_{1}}\oplus{\bf E_{2}}$ $\displaystyle\psi(\mathcal{V})=F_{1}+F_{2}\subseteq{\bf F_{1}}\oplus{\bf F_{2}},$ where $0_{E}\in E_{i}\mathrel{\mathchoice{\ooalign{$\displaystyle\subseteq$\cr\raise 1.67915pt\hbox{$\displaystyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\textstyle\subseteq$\cr\raise 1.67915pt\hbox{$\textstyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptstyle\subseteq$\cr\raise 1.18399pt\hbox{$\scriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptscriptstyle\subseteq$\cr\raise 0.6458pt\hbox{$\scriptscriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}}{\bf E_{i}}$ and $0_{F}\in F_{i}\mathrel{\mathchoice{\ooalign{$\displaystyle\subseteq$\cr\raise 1.67915pt\hbox{$\displaystyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\textstyle\subseteq$\cr\raise 1.67915pt\hbox{$\textstyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptstyle\subseteq$\cr\raise 1.18399pt\hbox{$\scriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptscriptstyle\subseteq$\cr\raise 0.6458pt\hbox{$\scriptscriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}}{\bf F_{i}}$, $i=1,2$. 2. (2) In the charts the local representative of $\varphi$ has the form $\varphi_{\phi\psi}=\overline{\varphi}+\hat{\varphi}\circ\mathrm{Pr}_{E_{2}},$ (2.1) where $\overline{\varphi}:E_{1}+E_{2}\to F_{1}$ is an $MC^{k}$-mapping, $\hat{\varphi}$ is an $MC^{k}$-isomorphism of $\bf E_{2}$ onto $\bf F_{2}$ and $\mathrm{Pr}_{E_{2}}:E\to{\bf E_{2}}$ is the projection. ###### Proof. Sufficiency: Let $(\mathcal{U},\phi)$ and $(\mathcal{V},\psi)$ be charts that satisfy the assumptions we will prove $\varphi\pitchfork S.$ In the charts, by using the identifications $T_{x}M\simeq E,\,T_{y}\simeq F$, the tangent map $T_{x}\varphi:T_{x}M\to T_{y}N$ has the representation $T_{x}\varphi=\varphi^{\prime}_{\phi\psi}(0_{E}):E\to F.$ Also, we have the identification $T_{y}S\simeq{\bf F_{1}}$. Let $\mathrm{Pr}_{F_{2}}:F\to{\bf F_{2}}$ be the projection onto ${\bf F_{2}}$. Since $\varphi^{\prime}_{\phi\psi}(0_{E})=(\overline{\varphi})^{\prime}+\hat{\varphi}\circ\mathrm{Pr}_{E_{2}}$ and $(\overline{\varphi})^{\prime}(0):E\to{\bf F_{1}}$, it follows that for all $e\in E,\,e=e_{1}+e_{2}\in{\bf E_{1}}\oplus{\bf E_{2}}$ $\varphi^{\prime}_{\phi\psi}(0_{E})e=(\overline{\varphi})^{\prime}(0_{E})e+\hat{\varphi}\circ\mathrm{Pr}_{E_{2}}.$ Thus, $\mathrm{Pr}_{F_{2}}\circ\varphi^{\prime}_{\phi\psi}(0_{E})e=\mathrm{Pr}_{F_{2}}\circ\hat{\varphi}\circ\,\mathrm{Pr}_{F_{2}}(e)$ which means $\mathrm{Pr}_{F_{2}}\circ\varphi^{\prime}_{\phi\psi}(0_{E})=\mathrm{Pr}_{F_{2}}\circ\hat{\varphi}\circ\,\mathrm{Pr}_{F_{2}}$ (2.2) it is a surjective mapping of $E$ onto ${\bf F_{2}}$. Moreover, we have $\displaystyle\ker(\mathrm{Pr}_{F_{2}}\circ\varphi^{\prime}_{\phi\psi}(0_{E}))$ $\displaystyle=\varphi^{\prime}_{\phi\psi}(0_{E})^{-1}(\mathrm{Pr}_{F_{2}}(0_{E}))=\varphi^{\prime}_{\phi\psi}(0_{E})^{-1}(F_{1})$ $\displaystyle=\left\\{e=e_{1}+e_{2}\in{\bf E_{1}}\oplus{\bf E_{2}}\mid(\overline{\varphi})^{\prime}(0_{E})e+\hat{\varphi}(e_{2})\in\mathbf{F}_{1}\right\\}$ $\displaystyle=\left\\{e=e_{1}+e_{2}\in{\bf E_{1}}\oplus{\bf E_{2}}\mid\hat{\varphi}(e_{2})=0_{F}\right\\}$ $\displaystyle=\left\\{e=e_{1}+e_{2}\in{\bf E_{1}}\oplus{\bf E_{2}}\mid e_{2}=0_{E}\right\\}$ $\displaystyle=\mathbf{E}_{1}.$ Which is an $MC^{k}$\- splitting in $E$ with a component $E_{2}$. From (3) it follows that $\mathrm{Pr}_{F_{2}}\circ\varphi^{\prime}_{\phi\psi}(0_{E})=\hat{\varphi}:\mathrm{Pr}_{E_{2}}\to\mathrm{Pr}_{F_{2}}$ which is an $MC^{k}$-isomorphism. Necessity: Suppose $\varphi\pitchfork S$. Since $S$ is an $MC^{k}$-submanifold of $N$ and $y=\varphi(x)\in S$, there is a chart $(\mathcal{W},\mathbbmss{w})$ around $y$ having the submanifold property for $S$ in $N$: $\displaystyle\mathbbmss{w}(\mathcal{W})=W_{1}+W_{2}\subset\mathbf{F_{1}}\oplus\mathbf{F_{2}}=F,$ $\displaystyle\mathbbmss{w}(S\cap\mathcal{W})=W_{1}\subset F,\quad\mathbbmss{w}(y)=0_{E}.$ Also, there is a chart $(\mathcal{X},\mathbbmss{x})$ around $x$ such that $\mathbbmss{x}(x)=0_{F},\,\varphi(\mathcal{X})\subset\mathcal{W}$ and $\varphi_{\mathbbmss{x}\mathbbmss{w}}:\mathbbmss{x}(\mathcal{X})\mathrel{\mathchoice{\ooalign{$\displaystyle\subseteq$\cr\raise 1.67915pt\hbox{$\displaystyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\textstyle\subseteq$\cr\raise 1.67915pt\hbox{$\textstyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptstyle\subseteq$\cr\raise 1.18399pt\hbox{$\scriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptscriptstyle\subseteq$\cr\raise 0.6458pt\hbox{$\scriptscriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}}E\to\mathbbmss{w}(\mathcal{W})\mathrel{\mathchoice{\ooalign{$\displaystyle\subseteq$\cr\raise 1.67915pt\hbox{$\displaystyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\textstyle\subseteq$\cr\raise 1.67915pt\hbox{$\textstyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptstyle\subseteq$\cr\raise 1.18399pt\hbox{$\scriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptscriptstyle\subseteq$\cr\raise 0.6458pt\hbox{$\scriptscriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}}F$ is of class $MC^{k}$. It follows that $\varphi_{\mathbbmss{x}\mathbbmss{w}}(0_{E})\circ\mathrm{Pr}_{F_{2}}:E\to\mathbf{F_{2}}$ is an $MC^{k}$-submersion as $\varphi\pitchfork S$. That is, $\varphi_{\mathbbmss{x}\mathbbmss{w}}\circ\mathrm{Pr}_{F_{2}}$ and $\mathbf{E_{1}}\coloneq\varphi_{\mathbbmss{x}\mathbbmss{w}}^{\prime}(0_{E})^{-1}(\mathbf{F_{1}})$ splits in $E$ with the complement $\mathbf{E_{2}}$ such that $\eta\coloneq\mathrm{Pr}_{F_{2}}\circ\varphi_{\mathbbmss{x}\mathbbmss{w}}(0_{E})\mid_{\mathbf{E_{2}}}:\mathbf{E_{2}}\to\mathbf{F_{2}}$ is an $MC^{k}$-isomorphism. Set $\tau\coloneq(\mathrm{Pr}_{E_{1}}+\eta^{-1}\circ\mathrm{Pr}_{F_{2}}\circ\varphi_{\mathbbmss{x}\mathbbmss{w}}):\mathbbmss{x}(\mathcal{X})\to\mathbbmss{w}(\mathcal{W})$, then $\tau$ is an $MC^{k}$-mapping and $\tau(0_{E})=0_{E}$ and $\tau^{\prime}(0_{E})=(\mathrm{Pr}_{E_{1}}+\eta^{-1}\circ\mathrm{Pr}_{F_{2}}\circ\varphi_{\mathbbmss{x}\mathbbmss{w}})^{\prime}(0_{E})=\mathrm{Pr}_{E_{1}}+\mathrm{Pr}_{E_{2}}=\mathrm{Id}_{E}$. Because, for all $e=e_{1}+e_{2}\in\mathbf{E_{1}}\oplus\mathbf{E_{2}}$ we have $(\varphi_{\mathbbmss{x}\mathbbmss{w}})^{\prime}(0_{E})e=(\varphi_{\mathbbmss{x}\mathbbmss{w}})^{\prime}(0_{E})e_{1}+(\varphi_{\mathbbmss{x}\mathbbmss{w}})^{\prime}(0_{E})e_{2}$. Whence, $\mathrm{Pr}_{F_{2}}\circ(\varphi_{\mathbbmss{x}\mathbbmss{w}})^{\prime}(0_{E})e=\mathrm{Pr}_{F_{2}}\circ(\varphi_{\mathbbmss{x}\mathbbmss{w}})^{\prime}(0_{E})e_{2}=\tau(e_{2})$, hence, $\tau^{-1}\circ\mathrm{Pr}_{F_{2}}\circ(\varphi_{\mathbbmss{x}\mathbbmss{w}})^{\prime}(0_{E})e=\tau^{-1}(\tau(e_{2}))=\mathrm{Pr}_{E_{2}}.$ By the inverse mapping theorem 2.1, $\tau$ is a local $MC^{k}$-diffeomorphism. Assume $0_{E}\in\mathcal{X}_{1}\mathrel{\mathchoice{\ooalign{$\displaystyle\subseteq$\cr\raise 1.67915pt\hbox{$\displaystyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\textstyle\subseteq$\cr\raise 1.67915pt\hbox{$\textstyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptstyle\subseteq$\cr\raise 1.18399pt\hbox{$\scriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptscriptstyle\subseteq$\cr\raise 0.6458pt\hbox{$\scriptscriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}}\mathbbmss{x}(\mathcal{X})$ is small enough. Let $\mathbbmss{x}_{1}:\mathcal{X}_{1}\to 0_{E}\in\mathcal{X}_{2}\mathrel{\mathchoice{\ooalign{$\displaystyle\subseteq$\cr\raise 1.67915pt\hbox{$\displaystyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\textstyle\subseteq$\cr\raise 1.67915pt\hbox{$\textstyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptstyle\subseteq$\cr\raise 1.18399pt\hbox{$\scriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptscriptstyle\subseteq$\cr\raise 0.6458pt\hbox{$\scriptscriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}}E$ be an $MC^{k}$-diffeomorphism such that $\tau\circ\mathbbmss{x}_{1}^{-1}=\mathrm{Id}_{F}.$ (2.3) Thus, $\mathrm{Pr}_{F_{2}}\circ\tau\circ\varphi_{\mathbbmss{x}\mathbbmss{w}}=\eta\circ\mathrm{Pr}_{E_{2}}.$ (2.4) If, $e=e_{1}+e_{2}\in\mathbbmss{x}_{1}(\mathcal{X}_{1})$ and $\mathbbmss{x}_{1}^{-1}(e)=\bar{e_{1}}+\bar{e_{2}}$, then by (2.3) and (2.4) we obtain $\displaystyle\tau\circ\mathbbmss{x}_{1}^{-1}(e)$ $\displaystyle=\tau(\bar{e_{1}}+\bar{e_{2}})$ $\displaystyle=\bar{e_{1}}+\eta^{-1}\circ\mathrm{Pr}_{F_{2}}\circ\varphi_{\mathbbmss{x}\mathbbmss{w}}(\bar{e_{1}}+\bar{e_{2}})$ $\displaystyle=e_{1}+e_{2}.$ Therefore, $\bar{e_{1}}=e_{1}$ and $\tau^{-1}\circ\mathrm{Pr}_{F_{2}}\circ\varphi_{\mathbbmss{x}\mathbbmss{w}}=e_{2}$ and so $\mathrm{Pr}_{F_{2}}\circ\varphi_{\mathbbmss{x}\mathbbmss{w}}(\bar{e_{1}}+\bar{e_{2}})=\tau(e_{2})=\tau\circ\mathrm{Pr}_{E_{2}}(e_{1}+e_{2}).$ This means, $\mathrm{Pr}_{E_{2}}\circ\varphi_{\mathbbmss{x}\mathbbmss{w}}\circ\mathbbmss{x}_{1}^{-1}(e)=\eta\circ\mathrm{Pr}_{E_{2}}(e)$ for all $e\in\mathbbmss{x}_{1}(\mathcal{X}_{1})$. Now, define $\displaystyle\phi\coloneq\mathbbmss{x}_{1}\circ\mathbbmss{x},\quad\mathcal{U}\coloneq\mathbbmss{x}^{-1}(\mathcal{X}),$ $\displaystyle\psi\coloneq\mathbbmss{w},\quad\mathcal{V}\coloneq\mathrm{small\,enough\,neighborhood\,of}\,y\,\mathrm{in}\mathcal{W}.$ Then, $(\phi,\mathcal{U})$ and $(\psi,\mathcal{V})$ are the desired charts. Indeed, $\displaystyle\varphi_{\phi\psi}$ $\displaystyle=\mathbbmss{w}\circ\varphi\circ(\mathbbmss{x}_{1}\circ\mathbbmss{x})$ $\displaystyle=\mathbbmss{w}\circ\varphi\circ\mathbbmss{x}^{-1}\circ\mathbbmss{x}^{-1}=\varphi_{\mathbbmss{x}\mathbbmss{w}}\circ\mathbbmss{x}_{1}^{-1}.$ Thus, $\displaystyle\varphi_{\phi\psi}$ $\displaystyle=\mathrm{Pr}_{F_{1}}\circ\varphi_{\mathbbmss{x}\mathbbmss{w}}\circ\mathbbmss{x}_{1}^{-1}+\mathrm{Pr}_{F_{2}}\circ\varphi_{\mathbbmss{x}\mathbbmss{w}}\circ\mathbbmss{x}_{1}^{-1}$ $\displaystyle=\overline{\varphi}+\hat{\varphi}\circ\mathrm{Pr}_{E_{2}},$ if we set $\hat{\varphi}\coloneq\eta$ and $\overline{\varphi}\coloneq\mathrm{Pr}_{E_{1}}\circ\varphi_{\mathbbmss{x}\mathbbmss{w}}\circ\mathbbmss{x}_{1}^{-1}$. ∎ ###### Theorem 2.2 (Transversality Theorem). Let $\varphi:M\to N$ be an $MC^{k}$-mapping, $k\geq 1$, $S\subset N$ an $MC^{k}$-submanifold and $\varphi\pitchfork S$. Then, $\varphi^{-1}(S)$ is either empty of $MC^{k}$-submanifold of $M$ with $(T_{x}\varphi)^{-1}(T_{y}S)=T_{x}(\varphi^{-1}(S)),\,x\in\varphi^{-1}(S),\,y=\varphi(x).$ If $S$ has finite co-dimension in $N$, then $\operatorname{codim}(\varphi^{-1}(S))=\operatorname{codim}S$. Moreover, if $\dim S=m<\infty$ and $\varphi$ is an $MC^{k}$-Lipschitz-Fredholm mapping of index $l$, then $\dim\varphi^{-1}(S)=l+m$. ###### Proof. Let $x\in\varphi^{-1}(S)$, then by Proposition 2.1 there are chart $(\phi,\mathcal{U})$ around $x$ and $(\psi,\mathcal{V})$ around $y=\varphi(x)$ such that $\displaystyle\phi(\mathcal{U})=E_{1}+E_{2}\subseteq{\bf E_{1}}\oplus{\bf E_{2}},$ $\displaystyle\psi(\mathcal{V})=F_{1}+F_{2}\subseteq{\bf F_{1}}\oplus{\bf F_{2}},$ $\displaystyle\varphi_{\phi\psi}=\overline{\varphi}+\hat{\varphi}\circ\mathrm{Pr}_{E_{2}},$ (2.5) where $\overline{\varphi}:E_{1}+E_{2}\to F_{1}$ is an $MC^{k}$-mapping, $\hat{\varphi}$ is an $MC^{k}$-isomorphism of $\bf E_{2}$ onto $\bf F_{2}$ and $\mathrm{Pr}_{E_{2}}:E\to{\bf E_{2}}$ is the projection. Let $\hat{e}\in\varphi^{-1}(S)\cap\mathcal{U}$, then $\hat{f}=\varphi(\hat{e})\in S\cap\mathcal{V}$ and $\psi(\varphi(\hat{e}))\in F_{1}\subset\mathbf{F_{1}}$. By (2), if $\phi(\hat{e})=e_{1}+e_{2}\in E_{1}+E_{2}$ we have $\displaystyle\varphi_{\phi\psi}(\phi(\hat{e}))$ $\displaystyle=\varphi_{\phi\psi}(e_{1}+e_{2})$ $\displaystyle=\overline{\varphi}(e_{1}+e_{2})+\hat{\varphi}(e_{2})\in F_{1}\subset\mathbf{F_{1}}.$ It follows $\hat{\varphi}(e_{2})=0_{E},e_{2}=0_{E},$ since $\hat{\varphi}(e_{2})\in\mathbf{F_{2}}$ and $\mathbf{F_{1}}\cap\mathbf{F_{2}}=\left\\{0_{F}\right\\}$. Thus, $\phi(\hat{e})\in F_{1}$ for all $\hat{e}\in\varphi^{-1}(S)\cap\mathcal{U}$. Therefore, $E_{1}\subset\phi(\varphi^{-1}(S)\cap\mathcal{U})$, since for each $e_{1}\in E_{1}$ we have $\varphi_{\phi\psi}(e_{1})=\overline{\varphi}(e_{1})+\hat{\varphi}\circ\mathrm{Pr}_{E_{2}}(e_{1})=\overline{\varphi}(e_{1})\in F_{1}.$ Hence, $\psi\circ\varphi\circ\phi^{-1}(e_{1})\in F_{1}$ implies that $\varphi\circ\phi^{-1}(e_{1})\in\psi^{-1}(F_{1})=S\cap\mathcal{V}$ and so $\varphi\circ\phi^{-1}(e_{1})$ which means $\phi^{-1}(e_{1})\in\varphi(S)\cap\mathcal{V}$ that yields $e_{1}\in\psi(\varphi^{-1}(S)\cap\mathcal{V})$. Therefore, for $x\in\varphi^{-1}(S)$ there is a chart $(\phi,\mathcal{U})$ with $\phi(\mathcal{U})=E_{1}+E_{2}\subset\mathbf{E_{1}}\oplus\mathbf{E_{2}}$ and $\phi(x)=0_{E},\,\phi(\varphi^{-1}(S)\cap\mathcal{V})=E_{1},$ which means $\varphi^{-1}(S)$ is an $MC^{k}$-submanifold in $M$. In the charts, we have $T_{x}\simeq E,\,T_{y}N\simeq F,\,T_{x}(\varphi^{-1}(S))\simeq\mathbf{E_{1}}$ and $T_{y}S\simeq\mathbf{F_{1}}$. From the proof of Proposition 2.1 we have $\varphi_{\phi\psi}^{\prime}(0_{E})^{-1}(\mathbf{F_{1}})=\mathbf{E_{1}}$ which yields $(T_{x}\varphi)^{-1}(T_{y}S)=T_{x}(\varphi^{-1}(S))$. If $S$ has finite co-dimension then $\mathbf{F_{2}}$ has finite dimension and thus by Proposition 2.1, $\operatorname{codim}(\varphi^{-1}(S))=\operatorname{codim}\varphi^{-1}(S\cap\mathcal{V})=\dim(\mathbf{F_{2}})=\operatorname{codim}(S).$ The proof of the last statement is standard. ∎ As an immediate consequence we have: ###### Corollary 2.1. Let $\varphi:M\rightarrow N$ be an $MC^{k}$-mapping, $k\geq 1$. If $q$ is a regular value of $\varphi$, then the level set $\varphi^{-1}(q)$ is a submanifold of $M$ and its tangent space at $p=\varphi(q)$ is $\ker T_{p}\varphi$. Moreover, if $q$ is a regular value of $\varphi$ and $\varphi$ is an $MC^{k}$-Lipschitz-Fredholm mapping of index $l$, then $\dim\varphi^{-1}(S)=l$. To prove the parametric transversality theorem we apply the following Sard’s theorem. ###### Theorem 2.3. [2, Theorem 3.2] If $\varphi:M\rightarrow N$ is an $MC^{k}$-Lipschitz-Fredholm map with $k>\max\\{{\operatorname{Ind}\varphi,0}\\}$. Then, the set of regular values of $\varphi$ is residual in $N$. ###### Theorem 2.4 (The Parametric Transversality Theorem). Let $A$ be a manifold of dimension $n$, $S\subset N$ a submanifold of finite co-dimension $m$. Let $\varphi:M\times A\to N$ be an $MC^{k}$-mapping, $k\geq\left\\{1,n-m\right\\}$. If $\varphi\pitchfork S$, then the set of all points $x\in M$ such that the mappings $\varphi_{x}:A\to N,\,(\varphi_{x}(\cdot)\coloneq\varphi(x,\cdot))$ are transversal to $S$, is residual $M$. ###### Proof. Let $\mathbf{S}=\varphi^{-1}(S)$, $\mathrm{Pr}_{M}:M\times A\to M$ the projection onto $M$ and $\mathrm{Pr}_{\mathbf{S}}$ be its restriction to $\mathbf{S}$. First, we prove that $\mathrm{Pr}_{\mathbf{S}}$ is an $MC^{k}$-Fredholm-Lipschitz mapping of index $n-m$, i.e., $T_{(m,a)}\mathrm{Pr}_{\mathbf{S}}:T_{(m,a)}\mathbf{S}\to T_{m}M$ is a Lipschitz-Fredholm operator of index $n-m$. By Theorem 2.2 the inverse image $\mathbf{S}$ is an $MC^{k}$-submanifold of $M\times A$, with model space $\mathbb{S}$, so that $\mathrm{Pr}_{\mathbf{S}}$ is an $MC^{k}$-mapping. Let $\pi_{M}$ and $\pi_{\mathbf{S}}$ be the local representatives of $\mathrm{Pr}_{M}$ and $\mathrm{Pr}_{\mathbf{S}}$, respectively. We show that $\pi_{M}$ and consequently $\pi_{\mathbf{S}}$ are Lipschitz-Fredholm operators of index $n-m$. Finite dimensionality of $\mathbb{R}^{n}$ and closedness of $\mathbb{S}$ implies that $K\coloneq\mathbb{S}+(\\{0\\}\times\mathbb{R}^{n})$ is closed in $E\times\mathbb{R}^{n}$. Also, $\operatorname{codim}K$ is finite because it contains the finite co-dimensional subspace $\mathbb{S}$. Therefore $K$ has a finite-dimensional complement $K_{1}\subset E\times\left\\{0\right\\}$, that is $E\times\mathbb{R}^{n}=K\oplus K_{1}$. Let $K_{2}\coloneq\mathbb{S}\cap\left\\{0\right\\}\times\mathbb{R}^{n}$. Since $K_{2}\subset\mathbb{R}^{n}$ we can choose closed subspaces $\mathbb{S}_{1}\subset\mathbb{R}^{n}$ and $\mathbb{R}_{0}\subset\left\\{0\right\\}\times\mathbb{R}^{n}$ such that $\mathbb{S}=\mathbb{S}_{1}\oplus K_{1}$ and $\left\\{0\right\\}\times\mathbb{R}^{n}=K_{1}\oplus\mathbb{R}_{0}$. Whence, $K=\mathbb{S}_{1}\oplus K_{1}\oplus\mathbb{R}_{0}$ and $E\times\mathbb{R}^{n}=\mathbb{S}_{1}\oplus K_{1}\oplus\mathbb{R}_{0}\oplus K_{2}$. The mapping $\pi_{\mathbf{S}}\mid_{\mathbb{S}_{1}\oplus K_{2}}:\mathbb{S}_{1}\oplus K_{2}\to E$ is an isomorphism, $K_{1}=\ker\pi_{\mathbf{S}}$, and $\pi_{M}(K_{2})$ is a finite dimensional complement to $\pi_{M}(\mathbb{S})$ in $\mathbb{R}^{n}$. Thus, $\pi_{M}$ is a Lipschitz-Fredholm operator and we have $\displaystyle\operatorname{Ind}\pi_{M}$ $\displaystyle=\dim K_{1}-\dim K_{2}$ $\displaystyle=\dim(K_{1}\oplus\mathbb{R}_{0})-\dim(\mathbb{R}_{0}\oplus K_{2}).$ Since, $K_{1}\oplus\mathbb{R}_{0}=\left\\{0\right\\}\times\mathbb{R}^{n}$ and $\mathbb{R}_{0}\oplus K_{2}$ is a complement to $\mathbb{S}$ in $E\times\mathbb{R}^{n}$ and therefore its dimension is $n$, so the index of $\pi_{M}$ is $n-m$. Now, we prove that if $x$ is a regular value of $\mathrm{Pr}_{\mathbf{S}}$ if and only if $\varphi_{x}\pitchfork S$. From the definition of $\varphi\pitchfork$ we have $\forall(x,a)\in\mathbf{S}$ $(T_{(x,a)}\varphi)(T_{x}M\times T_{a}A)+T_{\varphi(x,a)}S=T_{\varphi(x,a)}N,$ (2.6) and $(T_{(x,a)}\varphi)^{-1}(T_{\varphi_{(x,a)}}S)\,\mathrm{splits\,in}\,T_{x}M\times T_{a}A.$ (2.7) Since $A$ has finite dimension, it follows that the mapping $a\in A\mapsto\varphi{(x,a)}$ for a fixed $x\in M$ is transversal to $S$ if and only if $\forall(x,a)\in\mathbf{S},T_{a}\varphi_{x}(T_{a}A)+T_{\varphi(x,a)}S=T_{\varphi(x,a)}S.$ (2.8) Since $\mathrm{Pr}_{\mathbf{S}}$ is a Lipschitz-Fredholm mapping, $\ker T\mathrm{Pr}_{\mathbf{S}}$ splits at any point as its dimension is finite. Then $x$ is a regular value of $\mathrm{Pr}_{\mathbf{S}}$ if and only if $\forall(x,a)\in\mathbf{S},\forall v\in T_{x}M,\exists u\in T_{a}A\colon T_{(v,u)}\varphi(v,u)\in T_{(x,a)}S.$ (2.9) Pick $x\in M$ and $a\in A$ such that $(x,a)\in\mathbf{S}$ and let $w\in T_{(x,a)}S$. By (2.6) and (2.7) we obtain that there exist $v\in T_{a}A,\,x_{1}\in T_{x}M,\,y_{1}\in T_{(x,a)}S$ such that $T_{(x,a)}\varphi(v,x_{1})+y_{1}=w.$ (2.10) Then, there exists $x_{2}\in T_{x}M$ such that $T_{(x,a)}\varphi(v,x_{2})\in T_{\varphi_{(x,a)}}S$. Hence, $\displaystyle w$ $\displaystyle=T_{(x,a)}\varphi(v,x_{1})-T_{(x,a)}\varphi(v,x_{2})+T_{(x,a)}\varphi(v,x_{2})+y_{1}$ $\displaystyle=T_{(x,a)}\varphi(0,x_{1}-x_{2})+T_{(x,a)}\varphi(v,x_{2})+y_{1}$ $\displaystyle=T_{(x,a)}\varphi(0,u)+T_{\varphi(x,a)}S+y_{2}\in T_{a}\varphi_{x}(T_{a}A),$ where $u=x_{1}-x_{2}$ and $y_{2}=T_{(x,a)}\varphi(v,x_{2})+y_{1}\in T_{\varphi_{(x,a)}}S$. Thus, (2.8) holds. Now we show that (2.8) implies (2.9). Pick $a\in A,\,x\in M$ such that $(x,a)\in\mathbf{S}$. Let $v\in T_{x}M,\,a_{1}\in T_{a}A,\,y_{1}\in T_{\varphi_{(x,a)}}S$ and set $w\coloneq T_{(x,a)}\varphi_{(v,x_{1})}+y_{1}$. By (2.8) there exist $a_{2}\in T_{a}A$ and $y_{2}\in T_{\varphi_{(x,a)}}S$ such that $w=T_{a}\varphi_{x}(a_{2})+y_{2}$. Then, $0_{E}=T_{(x,a)}\varphi(v,a_{1})-T_{a}\varphi_{x}(a_{2})+y_{1}-y_{2}=T_{(x,a)}\varphi(v,a_{1}-a_{2})+y_{1}-y_{2},$ so $T_{(x,a)}\varphi(v,a_{1}-a_{2})=y_{2}-y_{1}\in T_{\varphi_{(x,a)}}S$ so (2.9) holds. Thus, we showed that if $x$ is a regular value of $\mathrm{Pr}_{\mathbf{S}}$ if and only if $\varphi_{x}\pitchfork S$. Since $\mathrm{Pr}_{\mathbf{S}}:\mathbf{S}\to M$ is a Lipschitz-Fredholm of class $MC^{k}$with the index $n-m$ and $\operatorname{codim}\mathbf{S}=\operatorname{codim}S=m$ and $k>\left\\{0,n-m\right\\}$, the Sard’s theorem 2.3 concludes the theorem. ∎ ## 3\. The degree of Lipschitz-Fredholm mappings In this section we construct the degree of $MC^{k}$-Lipschitz-Fredholm mappings and apply it to prove an invariance of domain theorem, a rank theorem and a Bursuk-Ulam type theorem. The construction of the degree relies on the following transversality result. ###### Theorem 3.1. [2, Theorem 3.3] Let $\varphi:M\to N$ be an $MC^{k}$-Lipschitz-Fredholm mapping, $k\geq 1$. Let $\imath:\mathcal{A}\to N$ be an $MC^{1}$-embedding of a finite dimension manifold $\mathcal{A}$ with $k>\max\\{\operatorname{Ind}\varphi+\dim\mathcal{A},0\\}$. Then there exists an $MC^{1}$ fine approximation $\mathbf{g}$ of $\imath$ such that $\mathbf{g}$ is embedding and $\varphi\pitchfork\mathbf{g}$. Moreover, suppose $S$ is a closed subset of $\mathcal{A}$ and $\varphi\pitchfork\imath(S)$, then $\mathbf{g}$ can be chosen so that $\imath=\mathbf{g}$ on $S$. We shall need the following theorem that gives the connection between proper and closed mappings. ###### Theorem 3.2. [5, Theorem 1.1] Let $A,B$ be Hausdorff manifolds, where $A$ is a connected infinite dimensional Fréchet manifold, and $B$ satisfies the first countability axiom, and let $\varphi:A\to B$ be a continuous closed non- constant map. Then $\varphi$ is proper. Let $\varphi:M\to N$ be a non-constant closed Lipschitz-Fredholm mapping with index $l\geq 0$ of class $MC^{k}$such that $k>l+1$. If $q$ is a regular value of $\varphi$, then by Theorem 3.2 and Corollary 2.1 the preimage $\varphi^{-1}(q)$ is a compact submanifold of dimension $l.$ Let $\imath:[0,1]\hookrightarrow N$ be an $MC^{1}$-embedding that connects two distinct regular values $q_{1}$ and $q_{2}$. By Theorem 3.1 we may suppose $\imath$ is transversal to $\varphi$. Thus, by Theorem 2.2 the preimage ${\bf M}\coloneq\varphi^{-1}(\imath([0,1]))$ is a compact $(l+1)$-dimensional submanifold of $M$ such that its boundary, $\partial{\bf M}$, is the disjoint union of $\varphi^{-1}(q_{1})$ and $\varphi^{-1}(q_{2})$, $\partial{\bf M}=\varphi^{-1}(q_{1})\amalg\varphi^{-1}(q_{2})$. Therefore, $\varphi^{-1}(q_{1})$ and $\varphi^{-1}(q_{2})$ are non-oriented cobordant which gives the invariance of the mapping. Following Smale [6] we associate to $\varphi$ a degree, denoted by $\deg\varphi$, defined as the non-oriented cobordism class of $\varphi^{-1}(q)$ for some regular value $q$. If $l=0$, then $\deg\varphi\in\mathbb{Z}_{2}$ is the number modulo 2 of preimage of a regular value. Let $\mathcal{O}\mathrel{\mathchoice{\ooalign{$\displaystyle\subseteq$\cr\raise 1.67915pt\hbox{$\displaystyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\textstyle\subseteq$\cr\raise 1.67915pt\hbox{$\textstyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptstyle\subseteq$\cr\raise 1.18399pt\hbox{$\scriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptscriptstyle\subseteq$\cr\raise 0.6458pt\hbox{$\scriptscriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}}M$. Suppose $\varphi:\overline{\mathcal{O}}\to N$ is a non- constant closed continuous mapping such that its restriction to $\mathcal{O}$ is an $MC^{k+1}$-Lipschitz-Fredholm mapping of index $k$, $k\geq 0$. Let $p\in N\setminus\varphi(\partial\overline{\mathcal{O}})$ and let $\mathbf{p}$ a regular value of $\varphi$ in the connected component of $N\setminus\varphi(\partial\overline{\mathcal{O}})$ containing $p$, the existence of such regular value follows from Sard’s theorem 2.3. Again, we associate to $\varphi$ a degree, $\deg(\varphi,p)$, defined as non-oriented class of $k$-dimensional compact manifold $\varphi^{-1}(\mathbf{p})$. This degree does not depend on the choice of $\mathbf{p}$. The following theorem which presents the local representation of $MC^{k}$-mappings is crucial for the rest of the paper. ###### Theorem 3.3. [1, Theorem 4.2] Let $\varphi:\mathcal{U}\mathrel{\mathchoice{\ooalign{$\displaystyle\subseteq$\cr\raise 1.67915pt\hbox{$\displaystyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\textstyle\subseteq$\cr\raise 1.67915pt\hbox{$\textstyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptstyle\subseteq$\cr\raise 1.18399pt\hbox{$\scriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptscriptstyle\subseteq$\cr\raise 0.6458pt\hbox{$\scriptscriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}}E\rightarrow F$ be an $MC^{k}$-mapping, $k\geq 1$, $u_{0}\in\mathcal{U}$. Suppose that $\operatorname{D}\varphi(u_{0})$ has closed split image $\mathbf{F_{1}}$ with closed topological complement $\mathbf{F_{2}}$ and split kernel $\mathbf{E_{2}}$ with closed topological complement $\mathbf{E_{1}}$. Then, there are two open sets $\mathcal{U}_{1}\mathrel{\mathchoice{\ooalign{$\displaystyle\subseteq$\cr\raise 1.67915pt\hbox{$\displaystyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\textstyle\subseteq$\cr\raise 1.67915pt\hbox{$\textstyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptstyle\subseteq$\cr\raise 1.18399pt\hbox{$\scriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptscriptstyle\subseteq$\cr\raise 0.6458pt\hbox{$\scriptscriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}}\mathcal{U}$ and $\mathcal{V}\mathrel{\mathchoice{\ooalign{$\displaystyle\subseteq$\cr\raise 1.67915pt\hbox{$\displaystyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\textstyle\subseteq$\cr\raise 1.67915pt\hbox{$\textstyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptstyle\subseteq$\cr\raise 1.18399pt\hbox{$\scriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptscriptstyle\subseteq$\cr\raise 0.6458pt\hbox{$\scriptscriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}}\mathbf{F_{1}}\oplus\mathbf{E_{2}}$ and an $MC^{k}$-diffeomorphism $\Psi:\mathcal{V}\rightarrow\mathcal{U}_{1}$, such that $(\varphi\circ\Psi)(f,e)=(f,\eta(f,e))$ for all $(f,e)\in\mathcal{V}$, where $\eta:\mathcal{V}\to\mathbf{E_{2}}$ is an $MC^{k}$\- mapping. ###### Theorem 3.4 (Rank theorem for $MC^{k}$-mappings). Let $\varphi:\mathcal{U}\mathrel{\mathchoice{\ooalign{$\displaystyle\subseteq$\cr\raise 1.67915pt\hbox{$\displaystyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\textstyle\subseteq$\cr\raise 1.67915pt\hbox{$\textstyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptstyle\subseteq$\cr\raise 1.18399pt\hbox{$\scriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptscriptstyle\subseteq$\cr\raise 0.6458pt\hbox{$\scriptscriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}}E\to F$ be an $MC^{k}$-mapping, $k\geq 1$. Suppose $u_{0}\in\mathcal{U}$ and $\operatorname{D}\varphi(u_{0})$ has closed split image $\mathbf{F_{1}}$ with closed complement $\mathbf{F_{2}}$ and split kernel $\mathbf{E_{2}}$ with closed complement $\mathbf{E_{1}}$. Also, assume $\operatorname{D}\varphi(\mathcal{U})(E)$ is closed in F and $\operatorname{D}\varphi(u)\|_{\mathbf{E_{1}}}:\mathbf{E_{1}}\to\operatorname{D}\varphi(u)(E)$ is an $MC^{k}$-isomorphism for each $u\in\mathcal{U}$. Then, there exist open sets $\mathcal{U}_{1}\mathrel{\mathchoice{\ooalign{$\displaystyle\subseteq$\cr\raise 1.67915pt\hbox{$\displaystyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\textstyle\subseteq$\cr\raise 1.67915pt\hbox{$\textstyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptstyle\subseteq$\cr\raise 1.18399pt\hbox{$\scriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptscriptstyle\subseteq$\cr\raise 0.6458pt\hbox{$\scriptscriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}}\mathbf{F_{1}}\oplus\mathbf{E_{2}},\,\mathcal{U}_{2}\mathrel{\mathchoice{\ooalign{$\displaystyle\subseteq$\cr\raise 1.67915pt\hbox{$\displaystyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\textstyle\subseteq$\cr\raise 1.67915pt\hbox{$\textstyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptstyle\subseteq$\cr\raise 1.18399pt\hbox{$\scriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptscriptstyle\subseteq$\cr\raise 0.6458pt\hbox{$\scriptscriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}}E,\,\mathcal{V}_{1}\mathrel{\mathchoice{\ooalign{$\displaystyle\subseteq$\cr\raise 1.67915pt\hbox{$\displaystyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\textstyle\subseteq$\cr\raise 1.67915pt\hbox{$\textstyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptstyle\subseteq$\cr\raise 1.18399pt\hbox{$\scriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptscriptstyle\subseteq$\cr\raise 0.6458pt\hbox{$\scriptscriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}}F$, and $\mathcal{V}_{2}\mathrel{\mathchoice{\ooalign{$\displaystyle\subseteq$\cr\raise 1.67915pt\hbox{$\displaystyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\textstyle\subseteq$\cr\raise 1.67915pt\hbox{$\textstyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptstyle\subseteq$\cr\raise 1.18399pt\hbox{$\scriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptscriptstyle\subseteq$\cr\raise 0.6458pt\hbox{$\scriptscriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}}F$ and there are $MC^{k}$-diffeomorphisms $\phi:\mathcal{V}_{1}\to\mathcal{V}_{2}$ and $\psi:\mathcal{U}_{1}\to\mathcal{U}_{2}$ such that $(\phi\circ\varphi\circ\psi)(f,e)=(f,0),\quad\forall(f,e)\in\mathcal{U}_{1}.$ ###### Proof. By Theorem 3.3 there exits an $MC^{k}$-diffeomorphism $\psi:\mathcal{U}_{1}\mathrel{\mathchoice{\ooalign{$\displaystyle\subseteq$\cr\raise 1.67915pt\hbox{$\displaystyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\textstyle\subseteq$\cr\raise 1.67915pt\hbox{$\textstyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptstyle\subseteq$\cr\raise 1.18399pt\hbox{$\scriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptscriptstyle\subseteq$\cr\raise 0.6458pt\hbox{$\scriptscriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}}\mathbf{F_{1}}\oplus\mathbf{E_{2}}\to\mathcal{U}_{2}\mathrel{\mathchoice{\ooalign{$\displaystyle\subseteq$\cr\raise 1.67915pt\hbox{$\displaystyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\textstyle\subseteq$\cr\raise 1.67915pt\hbox{$\textstyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptstyle\subseteq$\cr\raise 1.18399pt\hbox{$\scriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptscriptstyle\subseteq$\cr\raise 0.6458pt\hbox{$\scriptscriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}}E$ such that $\varphi(f,e)=(\varphi\circ\psi)(f,e)=(f,\eta(f,e)),$ where $\eta:\mathcal{V}\to\mathbf{E_{2}}$ is an $MC^{k}$\- mapping. Let $\mathrm{Pr}_{1}:F\to\mathbf{F_{1}}$ be the projection. We obtain $\mathrm{Pr}_{1}\circ\operatorname{D}\varphi(f,e)(w,v)=(w,0)$, for $w\in\mathbf{F_{1}}$ and $v\in\mathbf{E_{2}}$ because $\operatorname{D}\varphi(f,e)(w,v)=(w,\operatorname{D}\eta(f,e)(w,v)).$ Hence, $\mathrm{Pr}_{1}\circ\operatorname{D}\varphi(f,e)\|_{\mathbf{F_{1}}\times\left\\{0\right\\}}$ is the identity mapping, $\mathrm{Id}_{\mathbf{F_{1}}}$, on $\mathbf{F_{1}}$. Thereby, $\operatorname{D}\varphi(f,e)\|_{\mathbf{F_{1}}\times\left\\{0\right\\}}:\mathbf{F_{1}}\times\left\\{0\right\\}\to\operatorname{D}\varphi(f,e)(\mathbf{F_{1}}\oplus\mathbf{E_{2}})$ is one-to-one and therefore by our assumption $\operatorname{D}\varphi(f,e)\circ\mathrm{Pr}_{1}\|_{\operatorname{D}\varphi(f,e)(\mathbf{F_{1}}\oplus\mathbf{E_{2}})}$ is the identity mapping. Suppose $(w,\operatorname{D}\eta(f,e)(w,v))\in\operatorname{D}\varphi(f,e)(\mathbf{F_{1}}\oplus\mathbf{E_{2}})$, we obtain $\operatorname{D}\eta(f,e)v=0$ for all $v\in\mathbf{E_{2}}$, which means $\operatorname{D}_{2}\eta(f,e)=0$, since $\displaystyle(\operatorname{D}\varphi(f,e)\circ\mathrm{Pr}_{1})(w,\operatorname{D}\eta(f,e)(w,v))=$ $\displaystyle\operatorname{D}\varphi(f,e)(w,0)$ $\displaystyle=$ $\displaystyle(w,\operatorname{D}\eta(f,e)(w,0))$ $\displaystyle=$ $\displaystyle(w,\operatorname{D}_{1}(f,e)w).$ We have $\operatorname{D}^{2}\varphi(f,e)v=(0,\operatorname{D}_{2}\eta(f,e)v)$, i.e., $\operatorname{D}^{2}\varphi(f,e)=0$ which means $\varphi$ does not depend on the variable $y\in\mathbf{E_{2}}$. Let $\mathrm{Pr}_{2}:\mathbf{F_{1}}\oplus\mathbf{E_{2}}\to\mathbf{F_{1}}$ be the projection and $\varphi_{f}\coloneq\varphi(f,e)=(\varphi\circ\mathrm{Pr}_{2})(f,e),$ so that $\varphi_{f}:\mathrm{Pr}_{2}(\mathcal{U}_{1})\subset\mathbf{F_{1}}\to F$. Let $\mathbf{v_{0}}\coloneq(\mathrm{Pr}_{2}\circ\psi^{-1})(u_{0})$ and $\mathcal{V}\mathrel{\mathchoice{\ooalign{$\displaystyle\subseteq$\cr\raise 1.67915pt\hbox{$\displaystyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\textstyle\subseteq$\cr\raise 1.67915pt\hbox{$\textstyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptstyle\subseteq$\cr\raise 1.18399pt\hbox{$\scriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptscriptstyle\subseteq$\cr\raise 0.6458pt\hbox{$\scriptscriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}}\mathcal{U}$ be an open neighborhood of $\mathbf{v_{0}}$. Define the mapping $\begin{array}[]{cccc}\Phi:\mathcal{V}\times\mathbf{F_{2}}\to\mathbf{F_{1}}\oplus\mathbf{F_{2}}\\\ \Phi(f,e)=\varphi(f)+(0,e).\end{array}$ By the open mapping theorem $\operatorname{D}\Phi(\mathbf{v_{0}},0)=(\operatorname{D}\varphi(\mathbf{v_{0}}),\mathrm{Id}_{\mathbf{F_{2}}}):E\oplus\mathbf{F_{2}}\to F$ is a linear $MC^{k}$-isomorphism, where $\mathrm{Id}_{\mathbf{F_{2}}}$ is the identity mapping of $\mathbf{F_{2}}$. Now $\Phi$ satisfies the inverse function theorem 2.1 at $\mathbf{v_{0}}$, therefore, there exist $\mathcal{V}_{1},\mathcal{V}_{2}\mathrel{\mathchoice{\ooalign{$\displaystyle\subseteq$\cr\raise 1.67915pt\hbox{$\displaystyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\textstyle\subseteq$\cr\raise 1.67915pt\hbox{$\textstyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptstyle\subseteq$\cr\raise 1.18399pt\hbox{$\scriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptscriptstyle\subseteq$\cr\raise 0.6458pt\hbox{$\scriptscriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}}F$ such that $(\mathbf{v_{0}},0)\in\mathcal{V}_{2}$ and $\Phi(\mathbf{v_{0}},0)=\varphi_{\mathbf{v_{0}}}(\mathbf{v_{0}})\in\mathcal{V}_{1}$ and an $MC^{k}$-diffeomorphism $\phi:\mathcal{V}_{1}\to\mathcal{V}_{2}$ such that $\phi^{-1}=\Phi\|_{\mathcal{V}_{1}}$. Thus, for $(f,0)\in\mathcal{V}_{2}$ we have $(\phi\circ\varphi)(f)=(\varphi\circ\Phi)(f,0)=(f,0),$ and therefore, $(\phi\circ\varphi\circ\psi)(f,e)=(f,0),\quad\forall(f,e)\in\mathcal{U}_{1}.$ ∎ As an immediate consequence we have the following: ###### Corollary 3.1 (Rank theorem for Lipschitz-Fredholm mappings). Let $\varphi:M\to N$ be an $MC^{\infty}$-Lipschitz-Fredholm mapping of index $k$ and $\dim\ker\operatorname{D}\varphi(x)=m$, $\forall x\in M$. Let $\complement_{1},\complement_{2}$ be topological complements of $\mathbb{R}^{m}$ in $E$ and $\mathbb{R}^{m-k}$ in $F$, respectively. Then, there exist charts $\phi:\mathcal{U}\mathrel{\mathchoice{\ooalign{$\displaystyle\subseteq$\cr\raise 1.67915pt\hbox{$\displaystyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\textstyle\subseteq$\cr\raise 1.67915pt\hbox{$\textstyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptstyle\subseteq$\cr\raise 1.18399pt\hbox{$\scriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptscriptstyle\subseteq$\cr\raise 0.6458pt\hbox{$\scriptscriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}}M\to E=\mathbb{R}^{m}\oplus\complement_{1}$ with $\phi(x)=0_{E}$ and $\psi:\mathcal{V}\mathrel{\mathchoice{\ooalign{$\displaystyle\subseteq$\cr\raise 1.67915pt\hbox{$\displaystyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\textstyle\subseteq$\cr\raise 1.67915pt\hbox{$\textstyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptstyle\subseteq$\cr\raise 1.18399pt\hbox{$\scriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptscriptstyle\subseteq$\cr\raise 0.6458pt\hbox{$\scriptscriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}}N\to F=\mathbb{R}^{m-k}\oplus\complement_{2}$ with $\psi(x)=0_{F}$ such that $\psi\circ\varphi\circ\phi^{-1}(f,0)=(f,0).$ The following theorem gives the openness property of the set of Lipschitz- Fredholm mappings. ###### Theorem 3.5. [1, Theorem 3.2] The set $\mathcal{LF}(E,F)$ is open in $\mathcal{L}_{d,g}(E,F)$ with respect to the topology defined by the metric (1.1). Furthermore, the function $T\rightarrow\operatorname{Ind}T$ is continuous on $\mathcal{LF}(E,F)$, hence constant on connected components of $\mathcal{LF}(E,F)$. The proof of the following theorem is a minor modification of [7, Theorem 2]. ###### Theorem 3.6. Let $\varphi:M\to N$ be a Lipschitz-Fredholm mapping of class $MC^{k}$, $k\geq 1$. Then, the set $\mathsf{Sing}(\varphi)\coloneq\left\\{m\mid\operatorname{D}\varphi(m)\mathrm{is\,not\,injective}\right\\}$ is nowhere dense in $M$. ###### Proof. This is a local problem so we assume $M$ is an open set in $E$ and $N$ is an open set in $F$. Let $s\in\mathsf{Sing}(\varphi)$ be arbitrary and $\mathcal{U}$ an open neighborhood of $s$ in $\mathsf{Sing}(\varphi)$. For each $n\in\mathbb{N}\cup\left\\{0\right\\}$ define $S_{n}\coloneq\left\\{m\in M\mid\dim\operatorname{D}\varphi(m)\geq n\right\\}.$ Then, $M=M_{0}\supset M_{1}\supset\cdots,$ therefore, is a unique $n_{0}$ such that $M=M_{n_{0}}\neq M_{n_{0}+1}$. Let $m_{0}\in M_{n_{0}}\setminus M_{n_{0}+1}$ such that $\dim\ker\operatorname{D}\varphi(m_{0})=n_{0}$. By Theorem 3.5, there exists an open neighborhood $\mathcal{V}$ of $m_{0}$ in $\mathcal{U}$ such that for all $v\in\mathcal{V}$ we have $\dim\ker\operatorname{D}\varphi(v)\leq n_{0}$ and hence $\dim\operatorname{D}\varphi(v)=n_{0}\geq 1$. By Corollary 3.1, there is a local representative $\varphi$ around zero such that $\psi\circ\varphi\circ\phi^{-1}(f,e)=(f,0)$ for $(f,e)\in\complement_{1}\oplus\mathbb{R}^{n_{0}}$ which contradicts the injectivity of $\varphi$, therefore, $\mathsf{Sing}(\varphi)$ contains a nonempty open set. The closedness of $\mathsf{Sing}(\varphi)$ is obvious in virtue of Theorem 3.5. ∎ ###### Theorem 3.7 (Invariance of domain for Lipschitz-Fredholm mappings). Let $\varphi:M\to N$ be an $MC^{k}$-Lipschitz-Fredholm mapping of index zero, $k>1$. If $\varphi$ is locally injective, then $\varphi$ is open. ###### Proof. Let $p\in U\mathrel{\mathchoice{\ooalign{$\displaystyle\subseteq$\cr\raise 1.67915pt\hbox{$\displaystyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\textstyle\subseteq$\cr\raise 1.67915pt\hbox{$\textstyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptstyle\subseteq$\cr\raise 1.18399pt\hbox{$\scriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptscriptstyle\subseteq$\cr\raise 0.6458pt\hbox{$\scriptscriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}}M$ and $q=\varphi(p)$. The point $p$ has a connected open neighborhood $\,\mathcal{U}\mathrel{\mathchoice{\ooalign{$\displaystyle\subseteq$\cr\raise 1.67915pt\hbox{$\displaystyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\textstyle\subseteq$\cr\raise 1.67915pt\hbox{$\textstyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptstyle\subseteq$\cr\raise 1.18399pt\hbox{$\scriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptscriptstyle\subseteq$\cr\raise 0.6458pt\hbox{$\scriptscriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}}M$ such that $\varphi\mid_{\overline{\mathcal{U}}}:\overline{\mathcal{U}}\to N$ is proper and injective. Whence $q\notin\varphi(\partial\mathcal{U})$ and $\varphi(\partial\mathcal{U})$ is closed in $N$. Let $\mathcal{V}$ be a connected component of $N\setminus\varphi(\partial\mathcal{U})$ containing $q$ which is its open neighborhood. Since $\mathcal{U}$ is connected it implies that $\varphi(\mathcal{U})\subset\mathcal{V}$. It follows from $\varphi(\partial\mathcal{U})\cap\mathcal{V}=\emptyset$ that $\overline{\mathcal{U}}\cap\varphi^{-1}(\mathcal{V})=\mathcal{U}$ and so $\varphi\mid_{\mathcal{U}}:\mathcal{U}\to N$ is proper and injective. By Theorem 3.6 there is a point $x\in M$ such that the tangent map $T_{x}\varphi$ is injective and since $\operatorname{Ind}\varphi=0$ it is surjective too. Therefor, $y=\varphi(x)$ is a regular value with $\varphi^{-1}(y)=\left\\{x\right\\}$ and so $\deg\varphi=1$. It follows that $\varphi$ is surjective, because if it is not , then any point in $N\setminus\varphi(M)$ is regular and $\deg\varphi=0$ which is contradiction. Then, $\mathcal{V}=\varphi(\mathcal{U})$ is the open neighborhood of $q$. ∎ ###### Corollary 3.2 (Nonlinear Fredholm alternative). Let $\varphi:M\to N$ be an $MC^{k}$-Lipschitz-Fredholm mapping of index zero, $k>1$. If $N$ is connected and $\varphi$ is locally injective, then $\varphi$ is surjective and finite covering mapping. If $M$ is connected and $N$ is simply connected, then $\varphi$ is a homeomorphism. The following theorem is a generalization of the Bursuk-Ulam theorem, the proof is slight modification of the Banach case. ###### Theorem 3.8. Let $\varphi:\overline{\mathcal{U}}\to F$ be a non-constant closed Lipschitz- Fredholom mapping of class $MC^{2}$ with index zero, where $U\mathrel{\mathchoice{\ooalign{$\displaystyle\subseteq$\cr\raise 1.67915pt\hbox{$\displaystyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\textstyle\subseteq$\cr\raise 1.67915pt\hbox{$\textstyle{\scriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptstyle\subseteq$\cr\raise 1.18399pt\hbox{$\scriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}{\ooalign{$\scriptscriptstyle\subseteq$\cr\raise 0.6458pt\hbox{$\scriptscriptstyle{\scriptscriptstyle\circ}\mkern 2.0mu$}\cr}}}F$ is symmetric. If $\varphi$ is odd and for $u_{0}\in\overline{U}$ we have $u_{0}\notin\varphi(\partial\overline{\mathcal{U}})$. Then $\deg(\varphi,u_{0})\equiv 1\mod 2.$ ###### Proof. Since $\operatorname{D}\varphi(u_{0})$ is a Lipschitz-Fredholm mapping with index zero $F=\mathbf{F_{1}}\oplus\ker\varphi=\mathbf{F_{2}}\oplus\operatorname{Img}\varphi$ and $\dim\mathbf{F_{2}}=\dim\ker\varphi$. The image $\varphi(\partial\mathcal{U})$ is closed as $\varphi$ is closed, hence $\mathbf{a}=\varrho(\varphi(\mathcal{U}),u_{0})>0$ because $u_{0}\notin\varphi(\partial\mathcal{U})$. Let $\phi:F\to F$ be a global Lipschitz-compact linear operator with $\mathpzc{Lip}(\phi)<\mathbf{b}$ for some $\mathbf{b}>0$. Define the mapping $\Phi_{\phi}:\overline{U}\to F$ by $\Phi_{\phi}(u)=\varphi(u)+\phi(u)$. Then $\Phi_{\phi}$ is a Lipschitz-Fredholm mapping of index zero. Suppose $\mathbf{b}<\mathbf{a}/\mathbf{k}$ for some $\mathbf{k}>1$, then $\varrho(\Phi_{\phi}(u),u_{0})\geq\varrho(\varphi(e),u_{0})-\mathpzc{Lip}(\phi)\varrho(U,u_{0})>\mathbf{a}-\mathbf{bk}>0,\quad\forall u\in\partial\mathcal{U}.$ Therefore, $u_{0}\notin\Phi_{\phi}(\partial\mathcal{U})$. We obtain $\deg(\varphi,u_{0})=\deg(\Phi_{\phi},u_{0})$ as the mapping $\psi:[0,1]\times\overline{U}\to F$ defined by $(t,u)\to\varphi(u)+t\phi(u)$ is proper and $u_{0}\notin\psi(\partial U)$ for all $t$. Considering the fact that $\psi(-u)=-\psi(u)$, we may use the perturbation by compact operators to find the degree of $\varphi$. Let $\mathsf{C}$ be a set of global Lipschitz-compact linear operators $\phi:F\to F$ with $\mathpzc{Lip}(\phi)<\mathbf{b}<\mathbf{a}/\mathbf{k}$. Let $\widehat{\phi}\in\mathsf{C}$ be such that its restriction to $\mathbf{F_{1}}$ equals $u_{0}$ and $\widehat{\phi}\mid_{\ker\operatorname{D}\varphi(u_{0})}:\ker\operatorname{D}\varphi(u_{0})\to\mathbf{F_{2}}$ is an $MC^{1}$-isomorphism. Therefore, $\operatorname{D}\varphi(u_{0})+\widehat{\phi}$ and consequently $\operatorname{D}\varphi(u_{0})$ is an $MC^{1}$-isomorphism. Now define the mapping $\Psi:\mathcal{U}\times\mathsf{C}\to F$ by $(u,\phi)=\Phi_{\phi}(u)$. For sufficiently small $\mathbf{b}$ the differential $\operatorname{D}\Psi(u,\phi)(v,\psi)=(\operatorname{D}\varphi(u)+\phi)v+\psi(u)$ is surjective at $u_{0}$ as $\operatorname{D}\varphi(u_{0})$ is an $MC^{1}$-isomorphism. Also, it is clear that it is surjective at the other points. Then, the mapping $\Psi$ satisfies the assumption of Theorem 2.4, therefore, $\Psi^{-1}(u_{0})$ is a submanifold and the mapping $\Pi:\Psi^{-1}(u_{0})\to\mathsf{C}$ induced by the projection onto the second order is Lipschitz-Fredholm of index zero. By employing the local version of Sard’s theorem we may find a regular point $\overline{\phi}$ of $\Pi$, and from the proof of the Theorem 2.4 it follows that $u_{0}$ is a regular value of $\Phi_{\overline{\phi}}$ and consequently $u_{0}$ is a regular value of $\varphi$. Thus, properness and $\varphi(-u)=-\varphi(u)$ imply that $\varphi^{-1}(u_{0})=\left\\{u_{0},f_{1},-f_{1},\cdots f_{m},-f_{m}\right\\}$ and therefore $\deg(\varphi,u_{0})\equiv 1\mod 2$. ∎ ## References * [1] Eftekharinasab, K., Sard’s theorem for mappings between Fréchet manifolds, Ukr. Math. J., Vol. 62, No. 11 (2011) 1896–1905. doi: 10.1007/s11253-011-0478-z. * [2] Eftekharinasab, K., Transversality and Lipschitz-Fredholm maps, Zb. Pr. Inst. Mat. NAN Ukr. Vol. 12, No. 6 (2015), 89–104. * [3] Jones, G., Kirby, J., Le Gal, O. and Servi, T., On Local definability of holomorphic functions, The Quarterly J. Math., Vol. 70, No. 4 (2019) 1305–1326. doi: 10.1093/qmath/haz015 * [4] O. Müller, A metric approach to Fréchet geometry, Journal of Geometry and physics, Vol. 58, No.11 (2008), 1477-1500. doi: 10.1016/j.geomphys.2008. * [5] R. S. Sadyrkhanov, On infinite dimensional features of proper and closed mappings, Proceedings of the AMS, Vol. 98, No. 4 (1986) 643–648. doi:10.2307/2045743. * [6] Smale S. An infinite dimensional version of Sard’s theorem, Amr. J. Math., Vol. 87, No. 4 (1965) 861–866. doi: 10.2307/2373250. * [7] Tromba, A. J., Some theorems on Fredholm maps, Proceedings of AMS., Vol. 34, No. 2 (1972) 578–585. doi:10.2307/2038410.
# QCD Critical Point and High Baryon Density Matter ††thanks: Presented at workshop on ”Criticality in QCD and the Hadron Resonance Gas”, Wroclaw (online), July 29-31, 2020 B. Mohanty1,2 and N. Xu2,3 1School of Physical Sciences, National Institute of Science Education and Research, HBNI, Jatni 752050, India, 2Institute of Modern Physics, 509 Nanchang Road, Lanzhou 730000, China and 3Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA ###### Abstract We report the latest results on the search for the QCD critical point in the QCD phase diagram through high energy heavy-ion collisions. The measurements discussed are based on the higher moments of the net-proton multiplicity distributions in heavy-ion collisions. A non-monotonic variation in the product of kurtosis times the variance of the net-proton distribution is observed as a function of the collision energy with 3$\sigma$ significance. We also discuss the results of the thermal model in explaining the measured particle yield ratios in heavy-ion collisions and comparison of the different variants of hardon resonance gas model calculation to the data on higher moments of net-proton distributions. We end with a note that the upcoming programs in high baryon density regime at various experimental facilities will complete the search for the QCD critical point through heavy-ion collisions. 25.75.-q,25.75.Nq, 12.38.Mh, 12.38.-t,25.75.Gz ## 1 Introduction Figure 1: Conjectured QCD phase diagram of temperature ($T$) versus baryonic chemical potential ($\mu_{\mathrm{B}}$). See text for details. Relativistic heavy-ion collisions at varying center of mass energy ($\sqrt{s_{NN}}$) allows for the study of the phase diagram of nuclear matter [1]. The underlying theory is the one that governs the strong interactions - Quantum Chromodynamics (QCD). The conjectured phase diagram of QCD is shown in Fig. 1. The current status of the phase diagram is as follows. There are two distinct phases in the phase structure: de-confined state of quarks and gluons called the quark gluon plasma (QGP) and the confined state of gas of hadrons and resonances (HRG). The phase boundary (shown as a solid line in Fig. 1) between the hadronic gas phase and the high-temperature quark-gluon phase is a first-order phase transition line, which begins at large baryon chemical potential ($\mu_{B}$) and small temperature ($T$) and curves towards smaller $\mu_{B}$ and larger $T$. This line ends at the QCD critical point whose conjectured position, indicated by a square, is uncertain both theoretically and experimentally. At smaller $\mu_{B}$ there is a cross over indicated by a dashed line. The region of $\mu_{\mathrm{B}}$/$T$ $\leq$ 2 is shown as dot- dashed line. A comparison between RHIC data and lattice QCD (LQCD) calculations disfavours the possible QCD critical point being located at $\mu_{\mathrm{B}}$/$T$ $\leq$ 2 [2, 3]. The red-yellow dotted line corresponds to the chemical freeze-out obtained from the fits of particle yields in heavy- ion collisions using a thermal model. The liquid-gas transition region features a second order critical point (red-circle) and a first-order transition line (yellow line) that connect the critical point to the ground state of nuclear matter ($T$ $\sim$ 0 and $\mu_{\mathrm{B}}$ $\sim$ 925 MeV) [4]. The regions of the phase diagram accessed by past (AGS and SPS), ongoing (LHC, RHIC, SPS and RHIC operating in fixed target mode), and future (FAIR and NICA) experimental facilities are also indicated. In this proceeding, we discuss the success and tests of the hadron resonance gas model using the particle ratios and fluctuations in net-proton number produced in heavy-ion collisions. We also discuss the status of the search for the QCD critical point and future experimental directions in this connection at the upcoming facilities. ## 2 Particle ratio and thermal model Thermal models, assuming approximate local thermal equilibrium, have been successfully applied to matter produced in heavy-ion collisions. Most popular variant of such a model employs Grand Canonical Ensemble (GCE), hence uses chemical potentials to account for conservation of quantum numbers on an average [5]. For systems created via elementary collisions (small system) or via low energy heavy-ion collisions, the Canonical Ensemble (CE) approach is used. In the large volume limit, the GCE and the CE formalisms should be equivalent. In heavy-ion collisions at energies spanning from few GeV to few TeV it may be worthwhile to ask at what collision energy a transition from GCE to CE occurs [6] ? ### 2.1 Success of thermal model Figure 2: (1) Ratio of yields of kaon to pion ($K^{+}/\pi^{+}$ (circles) and $K^{-}/\pi^{-}$ (triangles) produced in central heavy-ion collisions at mid- rapidity as a function of $\sqrt{s_{NN}}$. Thermal fits are also shown as bands (yellow band for $K^{+}$/$\pi^{+}$ and green band for $K^{-}$/$\pi^{-}$) in the plot. Dot-dashed line represents the net-baryon density at the chemical freeze-out. The dot-dashed line represents the net-baryon density at the Chemical Freeze-out as a function of collision energy, calculated from the thermal model [13]. (2) Ratio of yields of $\phi$-meson to kaon ($\phi/K^{-}$) produced in central heavy-ion collisions at mid-rapidity as a function of $\sqrt{s_{NN}}$. The various bands shows the thermal model expectation from grand canonical ensemble (GCE) and canonical ensemble (CE) formulations in the HRG model. Figure 2 (1) in the upper panel shows the energy dependence of $K$/$\pi$ particle yield ratio produced in heavy-ion collisions at AGS [7, 8, 9], SPS [10, 11] and RHIC [12]. The thermal model calculation explains the $K$/$\pi$ ratios that reflect the strangeness content relative to entropy of the system formed in heavy-ion collisions. This can be treated as a success of the application of thermal model to heavy-ion collisions. A peak in the energy dependence of $K^{+}$/$\pi^{+}$ could be due to associated production dominance at lower energies as the baryon stopping is large. The peak is consistent with the calculated net baryon density reaching a maximum [13] has been suggested to be a signature of a change in degrees of freedom (baryon to meson [14] or hadrons to QGP [15]) while going from lower to higher energies. The $K^{-}$/$\pi^{-}$ ratio seems unaffected by the changes in the net-baryon density with collision energy and shows a smooth increasing trend. ### 2.2 Transition from grand canonical to canonical ensemble Figure 2 (2) in the lower panel shows the energy dependence of $\phi$/$K^{-}$ yield ratio measured in heavy-ion collisions [16, 17, 18]. As one moves from higher to lower collision energy, the $\phi$/$K^{-}$ ratio changes rapidly from a constant value to larger values. The transition happens below the collision energy where the freeze-out net-baryon density peaks (see upper panel). Thermal model calculations with GCE explains the measurements up to collision energy of $5$ GeV. At lower energies the GCE model expectation is that the $\phi$/$K^{-}$ ratio should decrease in contrast to that observed in experiments. On the other hand, the increase in $\phi$/$K^{-}$ at lower energies is explained by thermal model with CE framework for strangeness production. The results are also sensitive to the choice of the additional control parameter, $r_{\rm sc}$, in CE framework, which decides the typical spatial size of $s\bar{s}$ correlations. Hence, we find that a high statistics and systematic measurement of $\phi$/$K^{-}$ yield ratio can be used to test the transition of GCE to CE in thermal models. As the size of the $s\bar{s}$ correlations depends on the medium properties, such studies will provide valuable data for estimation of the volume in which open strangeness is produced. ## 3 Net-proton number fluctuations and QCD critical point The QCD critical point is a landmark on the QCD phase diagram. Experimental signatures for critical point is enhanced fluctuations coupled to the critical modes. In this respect the baryon number fluctuations are sensitive to the criticality [19]. At the critical point, generally, the correlation length takes large values, and that leads to non-Gaussian fluctuations [20]. Higher- order fluctuations are more sensitive to the criticality, the third order ($S\sigma$) and the fourth order ($\kappa\sigma^{2}$) are common measures for the QCD critical point search, where $\sigma$, $S$ and $\kappa$ are called the standard deviation, skewness and the kurtosis of the distribution, respectively. Experimentally, net-proton distribution is considered as a proxy for net-baryon distributions. ### 3.1 Net-proton number fluctuations Figure 3: (1) $S\sigma$ and (2) $\kappa\sigma^{2}$ of net-proton distributions for 70-80% peripheral (open squares) and 0-5% central (filled-circles) Au+Au collisions as a function of $\sqrt{s_{NN}}$ [21]. Projected statistical uncertainty for the second phase of the RHIC BES program is shown by the green-band and the blue arrow shows the region of $\sqrt{s_{NN}}$ to be covered by the STAR experiments fixed-target program. Results of calculations are shown for different variants (Ideal GCE [23], excluded volume [24] and CE [25]) of HRG model and transport model (UrQMD). The solid red and the dashed blue line in (2) is a schematic representation of expectation from a QCD based model calculation in presence of a critical point. Figure 3 shows the most relevant measurements over the widest range in $\mu_{B}$ ($20-450$ MeV) to date for the critical point search [21]. As we go from observables involving lower order moments ($S\sigma$) to higher order moments ($\kappa\sigma^{2}$), deviations between central and peripheral collisions for the measured values increases. Central collisions $\kappa\sigma^{2}$ data show a non-monotonic variation with collision energy with respect to the statistical baseline of $\kappa\sigma^{2}$ = 1 at a significance of $\sim$ 3$\sigma$ [21]. The deviations of $\kappa\sigma^{2}$ below the baseline are qualitatively consistent with theoretical considerations including a critical point [22]. In addition, experimental data show deviation from heavy-ion collision models without a critical point. This can be seen from the table 1 which shows values of a $\chi^{2}$ test between the experimental data and various models. In all cases, within 7.7 $<$ $\sqrt{s_{NN}}$ (GeV) $<$ 27, the $\chi^{2}$ tests return $p$-values that are less than 0.05. This implies that the monotonic energy dependence from all of the models are statistically inconsistent with the data. Although a non- monotonic variation of the experimental data with collision energy looks promising for the QCD critical point search, a more robust conclusion can be derived when the uncertainties get reduced and significance above $5\sigma$ is reached. This is the plan for the RHIC Beam Energy Scan Phase-II program. Table 1: The $p$ values of a $\chi^{2}$ test between data and various models for the $\sqrt{s_{NN}}$ dependence of $\it{S}\sigma$ and $\kappa\sigma^{2}$ values of net-proton distributions in 0-5% central Au+Au collisions. The results are for the $\sqrt{s_{NN}}$ range 7.7 to 27 GeV [21] which is the relevant region for the physics analysis presented here. Moments \| HRG GCE \| HRG EV \| HRG CE \| UrQMD ---\|---\|---\|---\|--- \| \| (r = 0.5 fm) \| \| $\it{S}\sigma$ \| $<$ 0.001 \| $<$ 0.001 \| 0.0754 \| $<$ 0.001 $\kappa\sigma^{2}$ \| 0.00553 \| 0.0145 \| 0.0450 \| 0.0221 ### 3.2 Comparison to Lattice QCD inspired fits In the previous sub-section we have seen that the data deviates from the expectations based on UrQMD and HRG models. Figures 4 and 5 show that several features of the data are qualitatively consistent with LQCD calculations of net baryon-number fluctuations up to NLO in $\mu_{B}/T$ [2]. Specifically, (a) $M/\sigma^{2}>S\sigma$, where $M$ is the mean of the net-proton distribution; $C_{3}/C_{1}$ is smaller than unity and tending to decrease with increasing $M/\sigma^{2}$; and with increasing $M/\sigma^{2}$, the cumulant ratio $C_{4}/C_{2}$ departs further away from unity than the ratio $C_{3}/C_{1}$ for $\sqrt{s_{{}_{NN}}}\geq 19.6$ GeV. The LQCD inspired fits are of the form: $C_{3}/C_{1}$ = $p_{0}$ \+ $p_{1}$ $(C_{1}/C_{2})^{2}$; $C_{4}/C_{2}$ = $p_{2}$ \+ $p_{3}$ $(C_{1}/C_{2})^{2}$ and $C_{3}/C_{2}$ = $p_{0}$ $C_{1}/C_{2}$\+ $p_{1}$ $(C_{1}/C_{2})^{3}$. Where $p_{0}$, $p_{1}$, $p_{2}$, and $p_{3}$ are fit parameters and we have used the equivalence between product of the moments and ratios of cumulants as $C_{1}/C_{2}$ = $M/\sigma^{2}$; $C_{3}/C_{1}$ = $S\sigma^{3}/M$ and $C_{4}/C_{2}$ = $\kappa\sigma^{2}$. The good agreement between data and LQCD inspired fits for $\sqrt{s_{NN}}$ range between 200 to 19.6 GeV, suggests that the heavy-ion collisions have produced a strongly interacting QCD matter. Figure 4: Net-proton cumulant ratios as a function of $M/\sigma^{2}$. Also shown are the expectations from different variants of HRG model (lines), UrQMD (yellow band) and LQCD inspired fits (green bands) [2]. Figure 5: $S\sigma$ versus the $M/\sigma^{2}$ of net-proton distribution in high energy heavy-ion collisions. Also shown are the expectation from HRG, UrQMD and LQCD inspired fits [2]. ## 4 Experimental programs for high baryon density Figure 6: Interaction rates (in Hz) for high-energy nuclear collision facilities as a function of $\sqrt{s_{NN}}$ [26]. Accelerators in collider mode are shown by blue symbols (ALICE, sPHENIX, RHIC BES-II and NICA) and those operating in fixed target mode by red symbols (STAR fixed traget (FXT), FAIR (CBM, SIS), HADES, and HIAF). As seen from the measurements discussed in previous section, to complete the critical point search program a high statistics phase - II of the beam energy scan program at RHIC is needed. In addition, future new experiments, which are all designed with high rates, large acceptance, and the state-of-the-art particle identification, at the energy region where baryon density is high, i.e., 500 MeV $<\mu_{B}<$ 800 MeV, see Fig. 6, will be needed. The new facilities for studying high baryon density matter includes (a) Nuclotron- based Ion Collider fAcility (NICA) at the Joint Institute for Nuclear Research (JINR), Dubna, Russia [27], (b) Compressed Baryonic Matter (CBM) at Facility for Antiproton and Ion Research (FAIR), Darmstadt, Germany [28], and (c) CSR External-target Experiment (CEE) at High Intensity heavy-ion Accelerator Facility (HIAF), Huizhou, China [29]. ## 5 Summary and Outlook The workshop dealt with two topics: Criticality and hadron resonance gas models. Criticality: A robust and vibrant research program is now established both experimentally (several facilities) and theoretically to study the QCD phase structure [30] and seeking for the QCD critical point in the phase diagram. The observables are well established and the results from a first systematic measurements are promising. Thermal models: Another success story has been use of hadron resonance gas models to extract freeze-out dynamics, provide evidences for local thermalisation in heavy-ion collisions and act as baseline for several measurements in heavy-ion collisions. This can be extended further to test the details of the model, like GCE vs. CE, and applications to higher order fluctuations to probe true thermal nature of the system formed in heavy-ion collisions [31]. High baryon density: Gradual shift of attention of the heavy-ion community is expected towards a return to the low energy collisions, where state-of-art accelerator facility with large luminosity and much advances detector systems with excellent particle identification will allow us to unravel the physics of a rotating high baryon density QCD matter subjected to magnetic field, similar to the neutron stars. A̱cknowledgments F. Karsch, V. Koch, A. Pandav, and K. Redlich for exciting discussions. We also thank the colleagues from STAR and ALICE collaborations. B.M. was supported in part by the Chinese Academy of Sciences President’s International Fellowship Initiative and J C Bose Fellowship from Department of Science of Technology, Government of India. N.X. was supported in part by the Chinese NSF grant No.11927901 and the US DOE grant No.KB0201022. ## References * [1] STAR Internal Note - SN0493, 2009. * [2] Bazavov, A._et al._ , _Phys. 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# On a “Wonderful" Bruhat-Tits group scheme Vikraman Balaji Chennai Mathematical Institute, Plot number H1, Sipcot IT Park, Siruseri, Chennai, India<EMAIL_ADDRESS>and Yashonidhi Pandey Indian Institute of Science Education and Research, Mohali Knowledge city, Sector 81, SAS Nagar, Manauli PO 140306, India<EMAIL_ADDRESS><EMAIL_ADDRESS> ###### Abstract. In this paper we make a universal construction of Bruhat-Tits group scheme on wonderful embeddings of adjoint groups in the absolute and relative settings. We make a similar construction for the wonderful embeddings of adjoint Kac- Moody groups. These have natural classifying properties reflecting the orbit structure on the wonderful embeddings. ###### Key words and phrases: Bruhat-Tits group scheme, parahoric group, Loop groups, Wonderful compactification ###### 2000 Mathematics Subject Classification: 14L15,14M27,14D20 The support of Science and Engineering Research Board under Mathematical Research Impact Centric Support File number: MTR/2017/000229 is gratefully acknowledged. ###### Contents 1. 1 Introduction 1. 1.1 Group embeddings and buildings 2. 1.2 Statement of main results 1. 1.2.1 The case of Tits building 2. 1.2.2 The relative case of the Bruhat-Tits buiding 2. 2 Preliminaries 1. 2.0.1 Lie-data of $G/k$ 2. 2.0.2 Apartment data 3. 2.0.3 Loop groups and their parahoric subgroups 4. 2.0.4 Bruhat-Tits group scheme 5. 2.0.5 Standard Parahoric subgroups 3. 3 A Bruhat-Tits group scheme on the wonderful compactification 1. 3.0.1 The structure of ${\bf X}$ 2. 3.0.2 The local toric structure of ${\bf X}$ 3. 3.0.3 Construction of a Lie-algebra bundle $\mathcal{R}$ on ${\bf Y}_{{}_{0}}$ 4. 3.0.4 Construction of a Lie-algebra bundle $\mathcal{R}$ on ${\bf Y}$ 5. 3.0.5 Weil restrictions and Lie algebras 6. 3.0.6 Restriction of $\mathcal{R}$ to infinitesimal standard curves $U_{I}$ 7. 3.0.7 The Lie algebra bundle $\mathcal{R}$ on $\bf X$ 8. 3.0.8 Bruhat-Tits group scheme ${\mathcal{G}}_{{}_{\bf X}}^{{}^{\varpi}}$ on $\bf X$ 4. 4 The Weyl alcove and apartment case 1. 4.0.1 On the torus-embedding ${\bf Y}_{{}_{0}}^{{}^{\text{aff}}}$ 2. 4.0.2 Construction of a finite-dimensional Lie algebra bundle $J$ on ${\bf Y}_{{}_{0}}^{{}^{\text{aff}}}$ together with parabolic structures 3. 4.0.3 The parahoric group scheme on the torus embedding ${{\bf Y}^{{}^{\text{aff}}}}$ 5. 5 The Bruhat-Tits group scheme on ${\bf X}^{{}^{{aff}}}$ 6. 6 The Bruhat-Tits group scheme on ${\bf X}^{{}^{{poly}}}$ 7. 7 An analogue of a construction of Mumford 8. 8 Appendix on parabolic and equivariant bundles 1. 8.0.1 The group scheme situation 2. 8.0.2 Kawamata Coverings ## 1\. Introduction Let $G$ be an almost simple, simply-connected group over an algebraically closed field $k$ of characteristic zero and let $G_{{}_{{}_{\text{ad}}}}:=G/Z(G)$. The aim of this note is to construct certain universal group schemes on * • the De Concini-Procesi wonderful compactification [7] $\bf X$ of $G_{{}_{{}_{\text{ad}}}}$, * • the loop “wonderful embedding" ${\bf X}^{{}^{\text{aff}}}$ of the adjoint affine Kac-Moody group $G^{{}^{\text{aff}}}_{{}_{\text{ad}}}$, constructed by Solis [16], * • certain toroidal embeddings $\bar{G}_{{}_{{}_{\text{ad,A}}}}$ of the relative group scheme $G_{{}_{{}_{\text{ad,A}}}}$ modeled after [13]. The group schemes we construct are sufficiently universal to be called “wonderful". A new point of view, playing a central role in this work, is that parabolic vector bundle on logarithmic schemes can be used as a tool to make geometric constructions. Hitherto, they have been objects of study occuring as points in certain moduli spaces. Let us briefly motivate the theme of this note. ### 1.1. Group embeddings and buildings Let us recall that Tits’ buildings are basically of two types. The first one is the “absolute" Tits building or spherical building which is attached to a semi-simple group over a general field. This simplicial complex is built from simplices which correspond to parabolic subgroups. The apartments of the building correspond to parabolic subgroups containing a fixed maximal torus. This is built up out of Euclidean spaces decomposed by the usual Weyl chambers. The second one is the Bruhat-Tits building which is the “relative" building attached to a semi-simple group over a complete-valued field. This is based on its parahoric subgroups and built up out of Euclidean spaces decomposed into affine Weyl chambers. The two types of buildings can also be seen from an algebro-geometric perspective. In the absolute case, we work with a semisimple group $G_{{}_{\text{ad}}}$ of adjoint type. In this setting one has the wonderful embedding $G_{{}_{\text{ad}}}\subset{\bf X}$ where $G_{{}_{\text{ad}}}$ sits as an open dense subset of $\bf X$. The complement ${\bf X}\setminus G_{{}_{\text{ad}}}$ is stratified by subsets $Y$ and there is a bijection $Y\mapsto\\{P_{{}_{Y}}\|B\subset P_{{}_{Y}}\\}$ from these strata to parabolic subgroups $P_{{}_{Y}}\subset G$ containing a fixed Borel subgroup $B$. Furthermore, this bijection extends to an isomorphism between the Tits building and the canonical complex associated with the toroidal embedding $G_{{}_{\text{ad}}}\subset{\bf X}$ (see Mumford [13, Page 178]). A second perspective is when the ground field is endowed with a complete non- archimedean discrete valuation. Let $A=k\llbracket z\rrbracket$ be a complete discrete valuation ring, $K=k(\\!(z)\\!)$ its quotient field. In this setting our basic model was constructed by Mumford [13]. He constructs a toroidal embedding $G_{{}_{\text{ad, A}}}\subset\bar{G}_{{}_{\text{ad, A}}}$ of the split group scheme $G_{{}_{\text{ad, A}}}=G_{{}_{\text{ad, A}}}\times{\rm Spec}\,A$. The strata of $\bar{G}_{{}_{\text{ad, A}}}\setminus G_{{}_{\text{ad, A}}}$ correspond bijectively to parahoric subgroups of $G(K)$ in a way that naturally extends to an isomorphism of the graph of the embedding $G_{{}_{\text{ad, A}}}\subset\bar{G}_{{}_{\text{ad, A}}}$ with the Bruhat-Tits building of $G\times{\rm Spec}\,A$ over $A$. ### 1.2. Statement of main results Classical Bruhat-Tits theory associates, to each facet $\Sigma$ of the Bruhat- Tits building, a smooth group scheme $\mathcal{G}_{{}_{\Sigma}}$ on ${\rm Spec}\,~{}A$, with connected fibres and whose generic fibre is $G\times_{{}_{{\rm Spec}\,k}}{\rm Spec}\,~{}K$. We call $\mathcal{G}_{{}_{\Sigma}}$ a Bruhat-Tits group scheme on ${\rm Spec}\,A$. The $A$-valued points $\mathcal{G}_{{}_{\Sigma}}(A)\subset G(K)$ are precisely the parahoric subgroups of $G(K)$. In this paper we construct universal analogues of the Bruhat-Tits group scheme. #### 1.2.1. The case of Tits building In the first setting, namely in the case of the Tits building, we construct an affine group scheme $\mathcal{G}_{{}_{\bf X}}$ over $\bf X$ whose restriction along each curve transversal to a strata of ${\bf X}\setminus G_{{}_{\text{ad}}}$ corresponds to the Bruhat-Tits group scheme associated to the parabolic subgroup under the bijection $Y\mapsto\\{P_{{}_{Y}}\|B\subset P_{{}_{Y}}\\}$ mentioned above. To state our theorem, we introduce some relevant notations and notions. Let ${\bf X}:=\overline{G_{{}_{{}_{\text{ad}}}}}$ be the wonderful compactification of $G_{{}_{{}_{\text{ad}}}}$. We construct a locally free sheaf of Lie algebras on $\bf X$. This construction is essentially toric in the sense that it is first constructed on the toric varieties based on the negative Weyl chambers and then the ones on the bigger spaces is deduced from this construction. We endow the locally free sheaf $\mathcal{R}$ with a canonical parabolic structure at the generic points of the divisor ${\bf X}\setminus G_{{}_{{}_{\text{ad}}}}$ together with a compatible loop Lie algebra structure. We fix data $(T,B,G)$ of $G$. Let $S$ denote the set of simple roots of $G$ and $\mathbb{S}=S\cup\\{\alpha_{0}\\}$ denote the set of affine simple roots. For $\emptyset\neq\mathbb{I}\subset\mathbb{S}$ let $\mathcal{G}_{\mathbb{I}}$ denote the associated Bruhat-Tits group scheme on a dvr and for any $I\subset S$ let $\mathcal{G}^{{}^{st}}_{I}=\mathcal{G}_{\mathbb{I}}$ where $\mathbb{I}=I\cup\\{\alpha_{0}\\}$. For ${I}\subset S$, let $Z_{I}$ denote the corresponding strata of $\bf X$ and for any point $z_{{}_{I}}\in Z_{{}_{I}}$, let $C_{{}_{I}}\subset{\bf X}$ be a smooth curve with generic point in $G_{{}_{\text{ad}}}$ and closed point $z_{{}_{I}}$. Let $U_{{}_{z}}\subset C_{{}_{I}}$ be a formal neighbourhood of $z_{{}_{I}}$. We these notations, Theorem 3.10 is as follows. ###### Theorem 1.1. There exists an affine “wonderful" Bruhat-Tits group scheme ${\mathcal{G}}_{{}_{\bf X}}^{{}^{\varpi}}$ on $\bf X$ satisfying the following classifying properties. 1. (1) There is an identification of the Lie-algebra bundles $\text{Lie}({\mathcal{G}}_{{}_{\bf X}}^{{}^{\varpi}})\simeq\mathcal{R}$. 2. (2) For $\emptyset\neq I\subset S$ the restriction of ${\mathcal{G}}_{{}_{{\bf X}}}^{{}^{\varpi}}$ to the formal neighbourhood $U_{{}_{z_{{}_{I}}}}$ of $z_{{}_{I}}$ in $C_{{}_{I}}$ §(3.0.3) is isomorphic to the standard Bruhat- Tits group scheme $\mathcal{G}^{{}^{st}}_{{}_{I}}$ §(2.0.5). In (3.11), we have indicated generalizations to fields of positive characteristics. #### 1.2.2. The relative case of the Bruhat-Tits buiding In the second scenario we work in the setting of loop groups and construct an affine group scheme over a “wonderful" embedding ${\bf X}^{{}^{\text{aff}}}$ constructed by Solis [16]. In the relative case the group scheme is obtained by “integrating" a locally free sheaf of Lie algebras $\bf R$ on ${\bf X}^{{}^{\text{aff}}}$. Its construction is achieved by constructing a locally free sheaf of Lie-algebras $J$ on a finite dimensional scheme ${\bf Y}^{{}^{\text{aff}}}$ which is the closure of a torus-embedding and whose translates build up the ind-scheme ${\bf X}^{{}^{\text{aff}}}$. The sheaf $J$ comes equipped with a canonical parabolic structure on a normal crossing divisor. The sheaf $J$ plays the role analogous to $\mathcal{R}$ (on $\bf X$) once we view ${\bf Y}^{{}^{\text{aff}}}$ as built out of affine Weyl group translates of the affine space $\mathbb{A}^{\ell+1}$ whose standard coordinate hyperplanes play the role of strata of largest dimension of $\bf X$. Using this perspective, we then briefly consider a toroidal embedding $\bar{G}_{{}_{ad,A}}$ the structure of which is modeled after Mumford’s construction in [13]. We then define an affine group scheme over $\bar{G}_{{}_{ad,A}}$ which has properties analogous to those of $\mathcal{G}_{{}_{\bf X}}$. We first introduce some notation to state our theorem more precisely. Let $LG$ denote the loop group of $G$. Let $L^{\ltimes}G=\mathbb{G}_{m}\ltimes LG$ where the rotational torus $\mathbb{G}_{m}$ acts on $LG$ by acting on the uniformizer (cf. §4). Let $G^{{}^{\text{aff}}}$ denote the Kac-Moody group associated to the affine Dynkin diagram of $G$. Recall that $G^{{}^{\text{aff}}}$ is given by a central extension of $L^{\ltimes}G$ by $\mathbb{G}_{m}$. Let $T_{{}_{\text{ad}}}:=T/Z(G)$ and let us denote by $T^{\ltimes}_{{}_{\text{ad}}}$ the torus $\mathbb{G}_{m}\times T_{{}_{\text{ad}}}\subset G^{{}^{\text{aff}}}_{{}_{\text{ad}}}$. In ${\bf X}^{{}^{\text{aff}}}$, the closure ${\bf Y}^{{}^{\text{aff}}}:=\overline{T^{\ltimes}_{{}_{\text{ad}}}}$ gives a torus- embedding. The complement $Z:={\bf Y}^{{}^{\text{aff}}}\setminus T^{\ltimes}_{{}_{\text{ad}}}$ is a union $\cup_{{}_{\alpha\in\mathbb{S}}}H_{{}_{\alpha}}$ of $\ell+1$ smooth divisors meeting at normal crossings. For $\alpha\in\mathbb{S}$, let $\xi_{{}_{\alpha}}\in H_{{}_{\alpha}}$ denote the generic points of the divisors $H_{{}_{\alpha}}$’s. Let $A_{{}_{\alpha}}=\mathcal{O}_{{}_{{\bf Y}^{{}^{\text{aff}}},\xi_{{}_{\alpha}}}}$ be the dvr’s obtained by localizing at the height $1$-primes given by the $\xi_{{}_{\alpha}}$’s. Let $Y_{{}_{\alpha}}:={\rm Spec}\,(A_{{}_{\alpha}})$. In Theorem 4.1 we show the existence of a finite dimensional Lie-algebra bundle $J$ on ${\bf Y}^{{}^{\text{aff}}}$ which extends the trivial bundle with fiber $\mathfrak{g}$ on the open dense subset ${T^{\ltimes}_{{}_{\text{ad}}}}\cap{\bf Y}^{{}^{\text{aff}}}\subset{\bf Y}^{{}^{\text{aff}}}$ and for each $\alpha\in\mathbb{S}$ we have $L^{+}(J_{{}_{Y_{{}_{\alpha}}}})\simeq L^{+}({\text{Lie}}(\mathcal{G}_{{}_{\alpha}}))$. Then in Proposition 5.1 we show the existence of a finite dimensional Lie-algebra bundle $\bf R$ on ${\bf X}^{{}^{\text{aff}}}$ which extends the trivial Lie algebra bundle $G_{{}_{\text{ad}}}^{{}^{\text{aff}}}\times\mathfrak{g}$ on the open dense subset $G_{{}_{\text{ad}}}^{{}^{\text{aff}}}\subset{\bf X}^{{}^{\text{aff}}}$ and whose restriction to ${\bf Y}^{{}^{\text{aff}}}$ is $J$. The ind-scheme ${\bf X}^{{}^{\text{aff}}}$ has divisors $D_{\alpha}$ for $\alpha\in\mathbb{S}$ such that the complement of their union is ${\bf X}^{{}^{\text{aff}}}\setminus G_{ad}^{aff}$. With these notations, let us state Theorem 5.2. ###### Theorem 1.2. There exists an affine “wonderful" Bruhat-Tits group scheme ${\mathcal{G}}_{{}_{{\bf X}^{{}^{\text{aff}}}}}^{{}^{\varpi}}$ on ${\bf X}^{{}^{\text{aff}}}$ together with a canonical isomorphism $\text{Lie}({\mathcal{G}}_{{}_{\bf X^{aff}}}^{{}^{\varpi}})\simeq{\bf R}$. It further satisfies the following classifying property: For $h\in{\bf X}^{{}^{\text{aff}}}\setminus G_{ad}^{aff}$, let $\mathbb{I}\subset\mathbb{S}$ be the subset such that $h\in\cap_{\alpha\in I}D_{\alpha}$. Let $C_{{}_{\mathbb{I}}}\subset{\bf X}^{{}^{\text{aff}}}$ be a smooth curve with generic point in $G^{aff}_{{}_{\text{ad}}}$ and closed point $h$. Let $U_{{}_{h}}\subset C_{{}_{\mathbb{I}}}$ be a formal neighbourhood of the closed point $h$. Then, the restriction ${\mathcal{G}}_{{}_{{\bf X}^{{}^{\text{aff}}}}}^{{}^{\varpi}}\|_{{}_{U_{{}_{h}}}}$ is isomorphic to the Bruhat-Tits group scheme $\mathcal{G}_{{}_{\mathbb{I}}}$ §(2.0.4) on $U_{{}_{h}}$. ###### Acknowledgments . We thank J. Heinloth and M.Brion for their questions and comments. They have helped us very much in expressing our results with greater precision. ## 2\. Preliminaries #### 2.0.1. Lie-data of $G/k$ Let $G$ be an almost simple, simply-connected group over $k$ (see (3.11)) with the data $(T,B,G)$. Let $X(T)\,=\,\text{Hom}(T,\mathbb{G}_{{}_{m}})$ be the group of characters of $T$ and $Y(T)\,=\,\text{Hom}(\mathbb{G}_{m},T)$ be the group of all one–parameter subgroups of $T$. Let $G_{{}_{{}_{\text{ad}}}}:=G/Z(G)$ and let $\mathfrak{g}$ denote the Lie- algebra. We denote $\Phi^{+},\Phi^{-}\subset\Phi$ the set of positive and negative roots with respect to $B$. Let $S=\\{\alpha_{{}_{1}},\ldots,\alpha_{{}_{\ell}}\\}$ denote the set of simple roots of $G$, where $\ell$ is the rank of $G$. Let $\alpha^{\vee}$ denote the coroot corresponding to $\alpha\in S$ . #### 2.0.2. Apartment data Let $\mathbb{S}=S\cup\\{\alpha_{0}\\}$ denote the set of affine simple roots. Let $\mathcal{A}_{T}$ denote the affine apartment corresponding to $T$. It can be identified with the affine space ${\mathbb{E}}=Y(T)\otimes_{\mathbb{Z}}\mathbb{R}$ together with its origin $0$. Let $\mathbf{a}_{0}$ be the unique Weyl alcove of $G$ whose closure contains $0$ and which is contained in the dominant Weyl chamber corresponding to $B$. Under the natural pairing between $Y(T)\otimes_{\mathbb{Z}}\mathbb{Q}$ and $X(T)\otimes_{\mathbb{Z}}\mathbb{Q}$, the integral basis elements dual to $S$ are called the fundamental co-weights $\\{\omega^{\vee}_{\alpha}\|\alpha\in S\\}$. Let $c_{\alpha}$ be the coefficient of $\alpha$ in the highest root. The vertices of the Weyl alcove $\mathbf{a}_{0}$ are indexed by $0$ and $\displaystyle\theta_{{}_{\alpha}}:={{\omega^{\vee}_{\alpha}}\over{c_{\alpha}}},\alpha\in S.$ (2.0.1) Indeed, any rational point $\theta\in\mathbf{a}_{0}$ can be expressed as $\theta_{{}_{\lambda}}$ so that there is a unique pair $(d,\lambda)\in\mathbb{N}\times Y(T)$ defined by the condition that $d$ is the least positive integer such that $\lambda=d.\theta_{{}_{\lambda}}\in Y(T).$ (2.0.2) Thus, if $e_{\alpha}$ is the order of $\omega^{\vee}_{\alpha}$ in the quotient of the co-weight lattice by $Y(T)$, it follows that for $\theta_{{}_{\alpha}}$, the number $d$ is: $d_{{}_{\alpha}}:=e_{\alpha}.c_{\alpha}.$ (2.0.3) An affine simple root $\alpha\in\mathbb{S}$ may be viewed as an affine functional on $\mathcal{A}_{T}$. Any non-empty subset $\mathbb{I}\subset\mathbb{S}$ defines the facet $\Sigma_{\mathbb{I}}\subset\overline{\mathbf{a}_{0}}$ where exactly the $\alpha$’s not lying in $\mathbb{I}$ vanish. So $\mathbb{S}$ corresponds to the interior of the alcove and the vertices of the alcove correspond to $\alpha\in\mathbb{S}$. Conversely any facet $\Sigma\subset\overline{\mathbf{a}_{0}}$ defines non-empty subset $\mathbb{I}\subset\mathbb{S}$. For $\emptyset\neq\mathbb{I}\subset\mathbb{S}$, the barycenter of $\Sigma_{\mathbb{I}}$ is given by $\theta_{\mathbb{I}}:=\frac{1}{\|\mathbb{I}\|}\sum_{\alpha\in\mathbb{I}}\theta_{{}_{\alpha}}.$ (2.0.4) #### 2.0.3. Loop groups and their parahoric subgroups Let $k=\mathbb{C}$, $A=k\llbracket z\rrbracket$ and let $R$ denote an arbitrary $k$-algebra. We denote the field of Laurent polynomials by $K=k(\\!(z)\\!)=k\llbracket z\rrbracket[z^{-1}]$. The loop group $LG$ on a $k$-algebra $R$ is given by $G(R(\\!(t)\\!))$. Similarly the loop Lie algebra $L\mathfrak{g}$ is given by $L\mathfrak{g}(R)=\mathfrak{g}(R(\\!(t)\\!))$. We can similarly define the positive loops (also called jet groups in the literature) $L^{+}(G)$ to be the subfunctor of $LG$ defined by $L^{+}(G)(R):=G(R\llbracket z\rrbracket)$. The positive loop construction extends more generally to any group scheme $\mathcal{G}\rightarrow{\rm Spec}\,(A)$ and any vector bundle $\mathfrak{P}\rightarrow{\rm Spec}\,(A)$ whose sheaf of sections carries a Lie-bracket. Let $L^{\ltimes}G=\mathbb{G}_{m}\ltimes LG$ where the rotational torus $\mathbb{G}_{m}$ acts on $LG$ by acting on the uniformizer via the loop rotation action as follows: $u\in\mathbb{G}_{m}(R)$ acts on $\gamma(z)\in LG(R)=G(R(\\!(t)\\!))$ by $u\gamma(z)u^{-1}=\gamma(uz)$. A maximal torus of $L^{\ltimes}G$ is $T^{\ltimes}=\mathbb{G}_{m}\times T$. A $\eta\in Hom(\mathbb{G}_{m},T^{\ltimes})\otimes_{\mathbb{Z}}\mathbb{Q}$ over $R(s)$ can be viewed as a rational $1$-PS, i.e., a $1$-PS $\mathbb{G}_{m}\rightarrow T^{\ltimes}$ over $R(w)$ where $w^{n}=s$ for some $n\geq 1$. In this case for $\gamma(z)\in L^{\ltimes}G(R)$ we will view $\eta(s)\gamma(z)\eta(s)^{-1}$ as an element in $L^{\ltimes}G(R(w))$. Hence, by the condition on $\gamma(z)\in L^{\ltimes}G(R)$ “$\lim_{{}_{s\rightarrow 0}}\eta(s)\gamma(z)\eta(s)^{-1}$ exists in $L^{\ltimes}G(R)$", we mean that there exists a $n\geq 1$ such that for $w^{n}=s$, we have $\eta(s)\gamma(z)\eta(s)^{-1}\in L^{\ltimes}G(R\llbracket w\rrbracket).$ (2.0.5) If $\eta=(a,\theta)$ for $a\in\mathbb{Q}$ and $\theta$ a rational $1$-PS of $T$, then we have $\eta(s)\gamma(z)\eta(s)^{-1}=\theta(s)\gamma(s^{a}z)\theta(s)^{-1}.$ (2.0.6) We note further that for any $0<d\in\mathbb{N}$, setting $\eta=(\frac{a}{d},\frac{\theta}{d})$ we have $\eta(s)\gamma(z)\eta(s)^{-1}=\theta(s^{\frac{1}{d}})\gamma(s^{\frac{a}{d}}z)\theta(s^{\frac{1}{d}})^{-1}.$ (2.0.7) So for $\eta=\frac{1}{d}(1,\theta)$ observe that the statement "$\lim_{s\rightarrow 0}\eta(s)\gamma(z)\eta(s)^{-1}$ exists " is a condition which is equivalent to $\theta(s)\gamma(s)\theta(s)^{-1}\in L^{\ltimes}G(R\llbracket w\rrbracket).$ (2.0.8) In other words, the condition of existence of limits is independent of $d$, and we may further set $z=s$ in $\gamma(z)$. More generally, if $a>0$ the condition for $(a,\theta)$ and $(1,\frac{\theta}{a})$ are equivalent. We may write this condition on $\gamma(s)\in L^{\ltimes}G(R)$ or $LG(R)$ as $"\lim_{s\rightarrow 0}\theta(s)\gamma(s)\theta(s)^{-1}\quad\text{ exists in}\quad L^{\ltimes}G(R)\quad\text{or}\quad LG(R)\quad\text{if}\quad\gamma(z)\in LG(R)"$ (2.0.9) For $r\in\Phi$, let $u_{r}:\mathbb{G}_{a}\rightarrow G$ denote the root subgroup. If for some $b\in\mathbb{Z}$ and $t(z)\in R\llbracket z\rrbracket$ we take $\gamma(z):=u_{r}(z^{b}t(z))\in Lu_{r}(R)$ , and $\eta:=(1,\theta)$, then $\eta(s)u_{r}(z^{b}t(z))\eta(s)^{-1}=\theta(s)u_{r}((sz)^{b}t(sz))\theta(s)^{-1}=u_{r}(s^{r(\theta)}(sz)^{b}t(sz)).$ (2.0.10) So for $\eta=(1,\theta)$ the condition that the limit exists is equivalent to $r(\theta)+b\geq 0\iff\lfloor r(\theta)+b\rfloor=\lfloor r(\theta)\rfloor+b\geq 0\iff b\geq-\lfloor r(\theta)\rfloor.$ (2.0.11) We note the independence of the above implications on the number $d$ occurring in the equation $\eta:=\frac{1}{d}(1,\theta)$. Let $\pi_{{}_{1}}:T^{\ltimes}\rightarrow\mathbb{G}_{m}$ be the first projection. For any rational $1$-PS $\eta:\mathbb{G}_{m}\rightarrow T^{\ltimes}$ we say $\eta$ is positive if $\pi_{{}_{1}}\circ\eta>0$, negative if $\pi_{{}_{1}}\circ\eta<0$ and non-zero if it is either positive or negative. Any non-zero $\eta=(a,\theta)$ defines the following positive-loop functors from the category of $k$-algebras to the category of groups and Lie-algebras: $\displaystyle{\mathcal{P}}_{{}_{\eta}}^{\ltimes}(R):=\\{\gamma\in L^{\ltimes}G(R)\|\quad\lim_{s\rightarrow 0}\eta(s)\gamma(z)\eta(s)^{-1}\quad\text{exists in}\quad L^{\ltimes}G(R)\\},$ (2.0.12) $\displaystyle{\mathfrak{P}}_{{}_{\eta}}^{\ltimes}(R):=\\{h\in L^{\ltimes}\mathfrak{g}(R)\|\quad\lim_{s\rightarrow 0}{Ad}(\eta(s))(h(z))\quad\text{exists in}\quad L^{\ltimes}\mathfrak{g}(R)\\},$ (2.0.13) $\displaystyle{\mathcal{P}}_{{}_{\eta}}(R):=\\{\gamma\in LG(R)\|\quad\lim_{s\rightarrow 0}\eta(s)\gamma(z)\eta(s)^{-1}\quad\text{exists in}\quad LG(R)\\},$ (2.0.14) $\displaystyle{\mathfrak{P}}_{{}_{\eta}}(R)=\\{h\in L\mathfrak{g}(R)\|\quad\lim_{s\rightarrow 0}Ad(\eta(s))(h(z))\quad\text{exists in}\quad L\mathfrak{g}(R)\\},$ (2.0.15) $\displaystyle\text{Thus},~{}~{}\mathcal{P}_{{}_{\eta}}:=\mathcal{P}_{{}_{\eta}}^{\ltimes}\cap(1\times LG)\quad\text{and}\quad{\mathfrak{P}}_{{}_{\eta}}:={\mathfrak{P}}_{{}_{\eta}}^{\ltimes}\cap(0\oplus L\mathfrak{g}).$ (2.0.16) A parahoric subgroup of $L^{\ltimes}G$ (resp. $LG$) is a subgroup that is conjugate to $\mathcal{P}^{\ltimes}_{{}_{\eta}}$ (resp. $\mathcal{P}_{{}_{\eta}}$) for some $\eta$. In this paper, we will mostly be using only the case when $\eta=\frac{1}{d}(1,\theta)$. In this case, letting $L\mathfrak{g}(R)=\mathfrak{g}(R(\\!(s)\\!))$ as in (2.0.9), we may reformulate (2.0.16) as ${\mathfrak{P}}_{{}_{\eta}}(R)=\\{h\in L\mathfrak{g}(R)\|\quad\lim_{s\rightarrow 0}Ad(\theta(s))(h(s))\quad\text{exists in}\quad L\mathfrak{g}(R)\\}.$ (2.0.17) By the conditions (2.0.12) and (2.0.16), using (2.0.11), we may express $\mathcal{P}_{{}_{\eta}}$ in terms of generators as follows: $\mathcal{P}_{{}_{\eta}}(R)=<T(R\llbracket z\rrbracket),u_{r}(z^{-\lfloor(r,\theta)\rfloor}R\llbracket z\rrbracket),r\in\Phi>,$ (2.0.18) Let $\mathfrak{t}$ denote the Cartan sub-algebra of $T$ and $\mathfrak{u}_{r}$ denote the root algebras associated to $u_{r}$. Then the Lie-algebra functor of $\mathcal{P}_{{}_{\eta}}$ in terms of generators is given by $Lie(\mathcal{P}_{{}_{\eta}})(R)=<\mathfrak{t}(R\llbracket z\rrbracket),\mathfrak{u}_{r}(z^{-\lfloor(r,\theta)\rfloor}R\llbracket z\rrbracket),r\in\Phi>.$ (2.0.19) Thus, for any $\eta$ of the type $(1,\theta)$, we get the equality $\text{Lie}(\mathcal{P}_{{}_{\eta}})={\mathfrak{P}}_{{}_{\eta}}.$ (2.0.20) This can be seen by (2.0.13) and (2.0.16) and then replacing the conjugation in (2.0.6) by $Ad(\eta(s))$. For a rational $1$-PS $\theta$ of $T$, with $\eta=(1,\theta)$ we will have the notation: ${\mathfrak{P}}_{{}_{\theta}}:={\mathfrak{P}}_{{}_{\eta}}.$ (2.0.21) In particular, for $r\in\Phi$ let $\mathfrak{g}_{{}_{r}}\subset\mathfrak{g}$ be the root subspace. Let $m_{{}_{r}}(\theta):=-\lfloor{(r,\theta)}\rfloor$. The parahoric Lie algebra ${\mathfrak{P}}_{{}_{\theta}}(k)\subset\mathfrak{g}(K)$ has $T$-weight space decomposition as: $\displaystyle{\mathfrak{P}}_{{}_{\theta}}(k)=\mathfrak{t}(A)\bigoplus z^{{}^{m_{{}_{r}}(\theta)}}\mathfrak{g}_{{}_{r}}(A).$ (2.0.22) #### 2.0.4. Bruhat-Tits group scheme To each facet $\Sigma_{\mathbb{I}}\subset\overline{\mathbf{a}_{0}}$, Bruhat- Tits theory associates a smooth group scheme $\mathcal{G}_{{}_{\mathbb{I}}}$ on ${\rm Spec}\,~{}A$, with connected fibres and whose generic fibre is $G\times_{{}_{{\rm Spec}\,k}}{\rm Spec}\,~{}K$. We call $\mathcal{G}_{{}_{\mathbb{I}}}$ a Bruhat-Tits group scheme on ${\rm Spec}\,A$. To $\mathcal{G}_{{}_{\mathbb{I}}}$, we can associate a pro-algebraic group $L^{+}(\mathcal{G}_{{}_{\mathbb{I}}})$ over $k$ as follows: $L^{+}(\mathcal{G}_{{}_{\mathbb{I}}})(R):=\mathcal{G}_{{}_{\mathbb{I}}}(R\llbracket z\rrbracket).$ (2.0.23) The $L^{+}(\mathcal{G}_{{}_{\mathbb{I}}})$ also characterise $\mathcal{G}_{{}_{\mathbb{I}}}$. We thus have the identifications: $L^{+}(\mathcal{G}_{{}_{\mathbb{I}}})=\mathcal{P}_{{}_{\eta}}\quad\text{and}\quad\text{Lie}(L^{+}(\mathcal{G}_{{}_{\mathbb{I}}}))={\mathfrak{P}}_{{}_{\eta}}$ (2.0.24) for $\eta=(1,\theta)$ where $\theta$ is any rational $1$-PS lying in $\Sigma_{\mathbb{I}}$. #### 2.0.5. Standard Parahoric subgroups The standard parahoric subgroups of $G(K)$ are parahoric subgroups of the distinguished hyperspecial parahoric subgroup $G(A)$. These are realized as inverse images under the evaluation map $ev:G(A)\to G(k)$ of standard parabolic subgroups of $G$. In particular, the standard Iwahori subgroup ${\mathfrak{I}}$ is a standard parahoric and indeed, ${\mathfrak{I}}=ev^{-1}(B)$. Denoting by $Q_{{}_{I}}\subset G$ the parabolic subgroup associated to the subset $I\subset S$ we will denote by $\mathcal{G}^{{}^{st}}_{{}_{{I}}}$ the standard parahoric subgroups of $G(A)$ determined by $L^{+}(\mathcal{G}^{{}^{st}}_{{}_{{I}}})(k):=ev^{-1}(Q_{{}_{I}}).$ (2.0.25) Thus, for ${I}\subset S$, setting $\mathbb{I}:=I\cup\\{\alpha_{0}\\}$ we have $L^{+}(\mathcal{G}^{{}^{st}}_{{}_{{I}}})(k)=L^{+}(\mathcal{G}_{{}_{\mathbb{I}}})(k)\quad\text{and}\quad L^{+}(\mathfrak{P}_{{}_{I}}^{{}^{st}})=L^{+}(\mathfrak{P}_{{}_{\mathbb{I}}}).$ (2.0.26) ## 3\. A Bruhat-Tits group scheme on the wonderful compactification #### 3.0.1. The structure of ${\bf X}$ Let ${\bf X}:=\overline{G_{{}_{{}_{\text{ad}}}}}$ be the wonderful compactification of $G_{{}_{{}_{\text{ad}}}}$. Let $\\{D_{\alpha}\|\alpha\in S\\}$ denote the irreducible smooth divisors of ${\bf X}$. Let $D:=\cup_{{\alpha\in S}}D_{\alpha}$. Then $X\setminus G_{{}_{{}_{\text{ad}}}}=D$. The pair $({\bf X},D)$ is the primary example of a $(G_{{}_{{}_{\text{ad}}}}\times G_{{}_{{}_{\text{ad}}}})$-homogenous pair [5, Brion]. Let $P_{{}_{I}}$ be the standard parabolic subgroup defined by subsets $I\subset S$ , the notation being such that the Levi subgroup $L_{{}_{I,ad}}$ containing $T_{{}_{\text{ad}}}$ has root system with basis $S\setminus I$. Recall that the $(G_{{}_{{}_{\text{ad}}}}\times G_{{}_{{}_{\text{ad}}}})$-orbits in $\bf X$ are indexed by subsets $I\subset S$ and have the following description: $Z_{{}_{I}}=(G_{{}_{{}_{\text{ad}}}}\times G_{{}_{{}_{\text{ad}}}})\times_{{}_{P_{{}_{I}}\times P^{-}_{{}_{I}}}}L_{{}_{I,ad}}.$ Then by [4, Proposition A1], each $Z_{{}_{I}}$ contains a unique base point $z_{{}_{I}}$ such that $(B\times B^{-}).z_{{}_{I}}$ is dense in $Z_{{}_{I}}$ and there is a $1$-PS $\lambda$ of $T$ satisfying $P_{{}_{I}}=P(\lambda)$ and $\lim_{{}_{t\to 0}}\lambda(t)=z_{{}_{I}}$. The closures of these $\lambda$ define curves $C_{{}_{I}}\subset{\bf X}$ which meet the strata $Z_{{}_{I}}$ transversally at $z_{{}_{I}}$. In particular, if $I=\\{\alpha\\}$ a singleton, then the divisor $D_{{}_{\alpha}}$ is the orbit closure $\bar{Z}_{{}_{I}}$ and the $1$-PS can be taken to be the fundamental co-weight $\omega^{\vee}_{\alpha}$. The closure of the $1$-PS $\omega^{\vee}_{\alpha}:\mathbb{G}_{{}_{m}}\to G_{{}_{\text{ad}}}$ defines the curve $C_{{}_{\alpha}}\subset\bf X$ transversal to the divisor $D_{{}_{\alpha}}$ at the point $z_{{}_{\alpha}}$. #### 3.0.2. The local toric structure of ${\bf X}$ Let ${\bf Y}:=\overline{T_{{}_{\text{ad}}}}$ be the closure of $T_{{}_{\text{ad}}}$ in ${\bf X}$. Recall that ${\bf Y}$ is a projective toric variety associated to the fan of Weyl chambers. In what follows, we will mostly work with the affine toric embedding ${\bf Y}_{{}_{0}}:=\overline{T_{{}_{ad,0}}}\simeq{\mathbb{A}}^{{}^{\ell}}$ which is the toric variety associated to the negative Weyl chamber. Let $U$ (resp $U^{{}^{-}}$) be the unipotent radical (resp. its opposite) of the Borel subgroup $B\subset G$. We also recall that one may identify $U\times U^{{}^{-}}\times{\bf Y}_{{}_{0}}$ with an open subset of $\bf X$ and moreover the $G\times G$-translates of $U\times U^{{}^{-}}\times{\bf Y}_{{}_{0}}$ covers the whole of $\bf X$. #### 3.0.3. Construction of a Lie-algebra bundle $\mathcal{R}$ on ${\bf Y}_{{}_{0}}$ The aim of this section is to construct a Lie algebra bundle on ${\bf Y}_{{}_{0}}$ (see (3.3)), which may be termed “parahoric". We close by making a similar construction on $\bf Y$ and $\bf X$. The construction is motivated by the structure of the kernel of the Tits fibration in [5]. We begin with a couple of elementary lemmas which should be well known. ###### Lemma 3.1. Let $E$ be a locally free sheaf on an irreducible smooth scheme $X$. Let $\xi\in X$ be the generic point and let $W\subset E_{{}_{\xi}}$ be an $\mathcal{O}_{{}_{\xi}}$-submodule. Then there exists a unique coherent subsheaf $F\subset E$ such that $F_{{}_{\xi}}=W$ and $Q:=Coker(F\hookrightarrow E)$ is torsion-free. Moreover $F$ is a reflexive sheaf. ###### Proof. Define $\tilde{F}$ on the affine open $U\subset X$ by $\tilde{F}(U):=E(U)\cap W$. Then it is easily seen that $\tilde{F}$ defines a coherent subsheaf $F\subset E$ and it is the maximal coherent subsheaf of $E$ whose fiber over $\xi$ is $W$. To check $Q$ is torsion-free, let $T$ be the torsion submodule of $Q$. Let $K:=Ker(E\to Q/T)$. Then since $\xi\notin Supp(T)$, so $K_{{}_{\xi}}=W$ and hence, by the maximality of $F$ we have $K=F$. Since $E$ is locally free, we have $F^{\vee\vee}/F\hookrightarrow E/F$. But since $F^{\vee\vee}/F$ is only torsion, and $E/F=Q$ is torsion-free, it follows that $F$ is automatically a reflexive sheaf.∎ ###### Lemma 3.2. Let $X$ be as above and $i:U\hookrightarrow X$ an open subset such that $X\setminus U$ has codimension $\geq 2$ in $X$. Let $F_{{}_{U}}$ be a reflexive sheaf on $U$. Then $i_{{}_{}}(F_{{}_{U}})$ is a reflexive sheaf on $X$ which extends $F_{{}_{U}}$. ###### Proof. . By [11, Corollaire 9.4.8], there exists a coherent $\mathcal{O}_{{}_{X}}$-module $F_{1}$ such that $F_{1}\|_{{}_{U}}\simeq F_{{}_{U}}$. Set $F:=F_{1}^{{}^{}}$ to be the double dual. Then $F$ is reflexive and also since $F_{{}_{U}}$ is reflexive, $F\|_{{}_{U}}\simeq F_{{}_{U}}$. Hence, we have $i_{{}_{}}(F_{{}_{U}})=F$ [12, Proposition 1.6]. ∎ Let $\lambda=\sum_{\alpha\in S}k_{{}_{\alpha}}\omega_{{}_{\alpha}}^{{\vee}}$ be a dominant $1$-PS of $T_{{}_{\text{ad}}}$. It defines the curve $C_{{}_{\lambda}}\subset\bf Y$. When the $k_{{}_{\alpha}}$ are constrained to be in $\\{0,1\\}$, then these curves are the standard curves $C_{{}_{I}},I\subset S$ considered in §(3.0.1), which meet the strata transversally at the points $z_{{}_{I}}$. We call these dominant $\lambda$’s as standard. In particular, the curve defined by $\omega_{{}_{\alpha}}^{{\vee}}$ meets the divisor $H_{{}_{\alpha}}$ transversally at $z_{{}_{\alpha}}$. The formal neighbourhood of the closed point $z_{{}_{\lambda}}$ in $C_{{}_{\lambda}}$ identifies with the spectrum $U_{{}_{\lambda}}:={\rm Spec}\,(A_{{}_{\lambda}})$ of $A_{{}_{\lambda}}=k\llbracket s\rrbracket$ with quotient field $K_{{}_{\lambda}}=k(\\!(s)\\!)$. We set $\theta_{\lambda}:=\sum_{\alpha\in S}k_{{}_{\alpha}}\frac{\theta_{{}_{\alpha}}}{\ell+1}.$ (3.0.1) When $k_{\alpha}\in\\{0,1\\}$, then $\theta_{\lambda}$ does not lie on the far wall of $\mathbf{a}_{0}$. So $\mathfrak{P}^{{}^{st}}_{{}_{\theta_{\lambda}}}=\mathfrak{P}_{{}_{\theta_{\lambda}}}$ §(2.0.5). ###### Theorem 3.3. There exists a canonical Lie-algebra bundle $\mathcal{R}$ on ${\bf Y}_{{}_{0}}$ which extends the trivial bundle with fiber $\mathfrak{g}$ on ${T_{{}_{\text{ad}}}}\subset{\bf Y}_{{}_{0}}$ and such that for $K_{{}_{\alpha}}\in\\{0,1\\}$ we have the identification of functors from the category of $k$-algebras to $k$-Lie-algebras: $L^{+}(\mathfrak{P}^{{}^{st}}_{{}_{\theta_{\lambda}}})=L^{+}(\mathcal{R}\mid_{{}_{U_{{}_{\lambda}}}}).$ (3.0.2) ###### Proof. With notations as in (2.0.3), consider the inclusion of lattices: $\bigoplus_{{}_{\alpha\in S}}{\mathbb{Z}}.\omega_{{}_{\alpha}}^{{\vee}}\hookrightarrow\bigoplus_{{}_{\alpha\in S}}{\mathbb{Z}}.\frac{{\theta_{{}_{\alpha}}}}{e_{{}_{\alpha}}.(\ell+1)}.$ (3.0.3) This induces an inclusion of $k$-algebras $B_{{}_{0}}\subset B$, where $B:=k[y_{\alpha},y_{\alpha}^{-1}]_{{}_{\alpha\in S}},~{}~{}\\\ B_{{}_{0}}:=k[x_{\alpha},x_{\alpha}^{-1}]_{{}_{\alpha\in S}}$ (3.0.4) and $B_{{}_{0}}\subset B$ is given by the equations: $\\{{y_{{}_{\alpha}}^{{}^{(\ell+1)d_{{}_{\alpha}}}}=x_{{}_{\alpha}}}\\}_{{}_{\alpha\in S}}$. Let ${\bf T}:={\rm Spec}\,(B),\text{and}T_{{}_{\text{ad}}}={\rm Spec}\,(B_{{}_{0}})$ and let $p:{\bf T}\to T_{{}_{\text{ad}}}$ be the natural morphism. We define the “roots" map: $\displaystyle\mathfrak{r}:{\bf T}\to T_{{}_{\text{ad}}},~{}~{}~{}\text{as}$ (3.0.5) $\displaystyle{\mathfrak{r}}^{\\#}\big{(}\big{(}x_{\alpha}\big{)}\big{)}:=\big{(}y_{\alpha}\big{)}.$ (3.0.6) Note that as a map between tori, $\mathfrak{r}$ is an isomorphism. We consider the map $\displaystyle Ad\circ\mathfrak{r}:{\bf T}\times\mathfrak{g}\rightarrow{\bf T}\times\mathfrak{g}$ (3.0.7) $\displaystyle({\bf t},x)\mapsto\big{(}{{\bf t}},Ad\big{(}\mathfrak{r}({\bf t})\big{)}(x)\big{)}.$ (3.0.8) We define the embedding of modules $j:\mathfrak{g}(B_{{}_{0}})\hookrightarrow\mathfrak{g}(B)\stackrel{{\scriptstyle Ad\circ\mathfrak{r}}}{{\rightarrow}}\mathfrak{g}(B)$ (3.0.9) where the second map is the one induced by (3.0.7) on sections. When $\mathfrak{g}(B_{0})$ is viewed as sections of the trivial bundle on $T_{{}_{\text{ad}}}$ with fibers $\mathfrak{g}$, then it also has a $T_{{}_{\text{ad}}}$-weight space decomposition. Let $B^{+}=k[y_{{}_{\alpha}}]_{{}_{\alpha\in S}}$ and $B_{{}_{0}}^{+}:=k[x_{{}_{\alpha}}]_{{}_{\alpha\in S}}$. Taking intersection as Lie submodules of $\mathfrak{g}(B)$ we define the following $B_{{}_{0}}^{+}$-module $\mathcal{R}(B_{{}_{0}}^{+}):=j(\mathfrak{g}(B_{{}_{0}}))\cap\mathfrak{g}(B^{+}).$ (3.0.10) Observe that $\mathcal{R}$ is a reflexive sheaf (3.1) on the affine embedding $T_{{}_{\text{ad}}}\hookrightarrow{\bf Y}_{{}_{0}}:={\rm Spec}\,(B_{{}_{0}}^{+})(\simeq{\mathbb{A}}^{{}^{\ell}})$. Further, this is a sheaf of Lie algebras with its Lie bracket induced from $\mathfrak{g}(B)$. We now check that $\mathcal{R}$ is locally-free on ${\bf Y}_{{}_{0}}$. Observe that the intersection (3.0.10) also respects $T_{{}_{\text{ad}}}$-weight space decomposition on the sections. More precisely, for a root $r\in\Phi$, since by definition we have the identification $\mathcal{R}(B_{{}_{0}}^{+})_{r}=j(\mathfrak{g}_{r}(B_{{}_{0}}))\cap\mathfrak{g}_{r}(B^{+})$, we get the following equalities: $j(\mathfrak{g}(B_{{}_{0}}))\cap\mathfrak{g}_{r}(B^{+})=j(\mathfrak{g}_{r}(B_{{}_{0}}))\cap\mathfrak{g}(B^{+})=\mathcal{R}(B_{{}_{0}}^{+})_{r}.$ (3.0.11) Since $\mathcal{R}$ is reflexive, so there is an open subset $U\subset{\bf Y}_{{}_{0}}$, whose complement is of codimension at least two such that the restriction $\mathcal{R}^{\prime}:=\mathcal{R}\|_{{}_{U}}$ is locally free. Clearly $U$ contains $T_{{}_{\text{ad}}}$ and the generic points $\zeta_{{}_{\alpha}}$ of the divisors $H_{{}_{\alpha}}$. This gives a decomposition on $\mathcal{R}^{\prime}$ obtained by restriction from $\mathcal{R}$. The locally free sheaf $\mathcal{R}^{\prime}$ is a direct sum of the trivial bundle $\text{Lie}(T_{{}_{\text{ad}}})\times U$ (coming from the $0$-weight space) and the invertible sheaves coming from the root decomposition. Now since invertible sheaves extend across codimension $\geq 2$, the reflexivity of $\mathcal{R}$ implies that this direct sum decomposition of $\mathcal{R}^{\prime}$ extends to ${\bf Y}_{{}_{0}}$ (see [12, Proposition 1.6, page 126]). Whence $\mathcal{R}$ is locally free. Let $\lambda:\mathbb{G}_{{}_{m}}\to T_{{}_{\text{ad}}}$ be a $1$-PS. This defines a rational $1$-PS $\theta_{{}_{\lambda}}:\mathbb{G}_{{}_{m}}\to T_{{}_{\text{ad}}}$. The map $\mathfrak{r}:{\bf T}\to T_{{}_{\text{ad}}}$ is abstractly an isomorphism of tori and we can therefore consider the rational $1$-PS $\boldsymbol{\theta_{{}_{\lambda}}}:\mathbb{G}_{{}_{m}}\to{\bf T}$ defined by $\boldsymbol{\theta_{{}_{\lambda}}}:=\mathfrak{r}^{{}^{-1}}\circ\theta_{{}_{\lambda}}$. Let $p:{\bf T}\to T_{{}_{\text{ad}}}$ be the canonical map induced by $B_{{}_{0}}\subset B$. We observe the following: 1. (1) $p\circ\boldsymbol{\theta_{{}_{\lambda}}}=\lambda$, 2. (2) $\mathfrak{r}\circ\boldsymbol{\theta_{{}_{\lambda}}}=\theta_{{}_{\lambda}}.$ Let $\lambda=\sum k_{{}_{\alpha}}.\omega_{{}_{\alpha}}^{{\vee}}$ be a standard dominant $1$-PS of $T_{{}_{\text{ad}}}$. For example, the case $\lambda=\omega_{{}_{\alpha}}^{{\vee}}$, viewed as a $1$-PS, may be expressed in $\ell$-many coordinates as follows: $\omega_{{}_{\alpha}}^{{\vee}}(s)=(1,\ldots,s,\ldots 1)$ (3.0.12) with $s$ at the coordinate corresponding to $\alpha\in S$. Thus, we may express $\lambda$ as: $\lambda(s)=\prod_{\alpha\in S}\omega_{{}_{\alpha}}^{{\vee}}(s^{{}^{k_{{}_{\alpha}}}}).$ (3.0.13) Set $f_{{}_{\alpha}}:=\frac{k_{{}_{\alpha}}}{d_{{}_{\alpha}}.(\ell+1)}$. In this case we have the expression: $\boldsymbol{\theta_{{}_{\lambda}}}(s)=\prod_{\alpha\in S}\omega_{{}_{\alpha}}^{{\vee}}(s^{{}^{f_{{}_{\alpha}}}}).$ (3.0.14) We now check the isomorphism $L^{+}({\mathcal{R}}\|_{{}_{U_{{}_{\lambda}}}})\simeq L^{+}(\mathfrak{P}_{{}_{\theta_{\lambda}}}^{{}^{st}})$ by first evaluating at $k$-valued points. By (3.0.10), a section s of $\mathcal{R}$ over $U_{{}_{\lambda}}={\rm Spec}\,A_{{}_{\lambda}}$, is firstly given by a local section $\text{\cursive s}_{{}_{K}}$ over the generic point of $A_{{}_{\lambda}}=k\llbracket s\rrbracket$. This is firstly an element in $j(\mathfrak{g}(K))$. Observe that this element may be written as $\text{\cursive s}_{{}_{K}}=Ad\big{(}\mathfrak{r}({\boldsymbol{\theta_{{}_{\lambda}}}(s)}\big{)}(x_{{}_{K}}))=Ad\big{(}\theta_{{}_{\lambda}}(s)\big{)}(x_{{}_{K}})).$ Let $d$ be any positive integer such that $d.\theta_{\lambda}$ becomes a $1$-PS of $T_{{}_{\text{ad}}}$. Let $(\tilde{B},w,L)$ be defined by taking a $d$-th root of the uniformizer $s$ of $A_{{}_{\lambda}}$ and let $\tilde{U}_{{}_{\lambda}}={\rm Spec}\,(\tilde{B})$. By (3.0.14), we may express $\boldsymbol{\theta_{{}_{\lambda}}}$ in terms of $w$ as $\boldsymbol{\theta_{{}_{\lambda}}}(s)=\prod_{\alpha\in S}\omega_{{}_{\alpha}}^{{\vee}}(w^{{}^{d.f_{{}_{\alpha}}}}).$ In other words, $\theta_{\lambda}(s)=\theta_{\lambda}(w^{d})=(d\theta_{{}_{\lambda}})(w)$ has become integral in $w$. Therefore, the membership of s in $\mathfrak{g}(B^{+})$ (3.0.10) gets interpreted as follows: $Ad(\theta_{\lambda}(s))(x_{K})\in L\mathfrak{g}(\tilde{B}).$ (3.0.15) But this is exactly the Lie-algebra version of the condition "$\lim_{s\rightarrow 0}$" condition for $\eta=(1,\theta)$ in the observation (2.0.8). In our situation we have assumed $R=k$ and so (3.0.15) is equivalent to $\text{\cursive s}\in\mathfrak{P}_{{}_{\theta_{{}_{\lambda}}}}(k).$ (3.0.16) But by (2.0.26), $\mathfrak{P}_{{}_{\theta_{{}_{\lambda}}}}(k)=\mathfrak{P}_{{}_{\theta_{{}_{\lambda}}}}^{{}^{st}}(k)$ because $\theta_{{}_{\lambda}}$ lies in the alcove $\bf a_{{}_{0}}$ but not on the far wall. Thus, we get the equality $\mathcal{R}\|_{{}_{U_{{}_{\lambda}}}}(A_{{}_{\lambda}})=\mathfrak{P}_{{}_{\theta_{{}_{\lambda}}}}^{{}^{st}}(k).$ (3.0.17) This proves the assertion for $k$-valued points. The above proof goes through for all $k$-algebras because the Lie-bracket is already defined on $k$, and so is the underlying module structure. ∎ ###### Remark 3.4. Let $\lambda$ be an arbitrary dominant $1$-PS of $T_{{}_{\text{ad}}}$. In general, the coefficients $k_{{}_{\alpha}}$ in its expression in terms of the $\omega_{{}_{\alpha}}^{\vee}$ need not be such that $\sum k_{{}_{\alpha}}/(\ell+1)<1$. Then we get the identification $L^{+}(\mathfrak{P}_{{}_{\theta_{\lambda}}})=L^{+}(\mathcal{R}\mid_{{}_{U_{{}_{\lambda}}}})$ without the standardness of the parahoric. ###### Remark 3.5. In the setting of Theorem 3.3, we may view $p:{\rm Spec}\,(B^{{}^{+}})\to{\rm Spec}\,(B_{{}_{0}}^{{}^{+}})$ as a ramified covering space of affine toric varieties induced by the inclusion (3.0.3) of lattices. The Galois group for this covering is the dual of the quotient of lattices. The computation in 3.3 can be seen in the light of [1], in the sense that via an "invariant direct image" process, one is able to recover the complete data of all standard parahoric Lie algebras from this explicit Kawamata cover. #### 3.0.4. Construction of a Lie-algebra bundle $\mathcal{R}$ on ${\bf Y}$ In this subsection, we deduce the existence of Lie algebra bundles on the projective non-singular toric variety $\bf Y$ with the classifying properties. The variety $\bf Y$ is the toric variety for ${T_{{}_{\text{ad}}}}$ with fan consisting of the Weyl chambers. The complement ${\bf Y}\setminus{T_{{}_{\text{ad}}}}$ is a union of translates of the hyperplanes $H_{{}_{\alpha}}\subset{\bf Y}_{{}_{0}}$ by the Weyl group $W$. Thus, for each $w\in W$, we can take the locally free sheaf $\mathcal{R}$ of Lie-algebras on ${\bf Y}_{{}_{0}}$ and its $w$-translate $w.\mathcal{R}$ on ${\bf Y}_{{}_{w}}:=w{\bf Y}_{{}_{0}}$ in $\bf Y$. Let $Y^{\prime\prime}\subset{\bf Y}_{{}_{0}}$ be the open subset consisting of the open orbit $T_{{}_{\text{ad}}}$ and of the orbits of codimension 1. In other words, $Y^{\prime\prime}$ is obtained from ${\bf Y}_{{}_{0}}$ by removing all the $T_{{}_{\text{ad}}}$-orbits of codimension at least 2. The data above defines a locally free sheaf of Lie algebras on $\cup_{{}_{w\in W}}w.Y^{\prime\prime}$ because $w.Y^{\prime\prime}\cap Y^{\prime\prime}$ is either $Y^{\prime\prime}$ or $T_{{}_{\text{ad}}}$. Using (3.2), we get a reflexive sheaf which by abuse of notation can be called $\mathcal{R}$ on the whole of ${\bf Y}$. That $\mathcal{R}$ on $\bf Y$ is locally free is immediate since its restrictions to the open cover given by ${\bf Y}_{{}_{0}}$ and its $W$-translates is locally free. The Lie algebra structure also extends since it is already there on an open subset with complement of codimension $\geq 2$. Thus, we conclude: ###### Corollary 3.6. There exists a canonical Lie-algebra bundle $\mathcal{R}$ on ${\bf Y}$ which extends the trivial bundle with fiber $\mathfrak{g}$ on ${T_{{}_{\text{ad}}}}\subset{\bf Y}$ with properties as in (3.3) at the translates of the hyperplanes $H_{{}_{\alpha}}$. #### 3.0.5. Weil restrictions and Lie algebras Let $X$ be an arbitrary $k$-scheme. For an affine (or possibly ind-affine) group scheme $\mathcal{H}\rightarrow X$, we denote $\text{Lie}(\mathcal{H})$ the sheaf of Lie-algebras on $X$ whose sections on $U\rightarrow X$ are given by $\text{Lie}(\mathcal{H})(U)=\text{ker}(\mathcal{H}(U\times k[\epsilon])\rightarrow\mathcal{H}(U)).$ (3.0.18) ###### Lemma 3.7. Let $p:\tilde{X}\rightarrow X$ be a finite flat map of noetherian schemes. Let $Res_{\tilde{X}/X}$ denote the “Weil restriction of scalars" functor. Let $\mathcal{H}\rightarrow\tilde{X}$ be an affine group scheme. Then we have a natural isomorphism ${\text{Lie}}(Res_{\tilde{X}/X}\mathcal{H})\simeq Res_{\tilde{X}/X}{\text{Lie}}(\mathcal{H}).$ (3.0.19) When $p$ is also Galois with Galois group $\Gamma$, then in characteristic $0$, we have ${\text{Lie}}((Res_{\tilde{X}/X}\mathcal{H})^{\Gamma})\simeq(Res_{\tilde{X}/X}{\text{Lie}}(\mathcal{H}))^{\Gamma}.$ (3.0.20) ###### Proof. See [9, Page 533] and [10, 3.1, page 293] respectively. ∎ #### 3.0.6. Restriction of $\mathcal{R}$ to infinitesimal standard curves $U_{I}$ Let $U_{{}_{I}}$ be the formal neighbourhood of the standard curve $C_{{}_{I}}$ §(3.0.3) at its closed point. We call the corresponding dominant $1$-PS $\lambda$ of $T_{{}_{\text{ad}}}$ to be standard. Let $\theta_{{}_{\lambda}}$ be the point in $\mathbf{a}_{0}$ (3.0.1). Recall that by [1, Proposition 5.1.2] for each $\theta_{{}_{\lambda}}$ there exists a ramified cover $q_{{}_{\lambda}}:U^{\prime}_{{}_{\lambda}}\to U_{{}_{\lambda}}$ (of ramification index $d$ as in (2.0.2)) together with a $\Gamma_{{}_{\lambda}}$-equivariant $G$-torsor $E_{{}_{\lambda}}$ such that the adjoint group scheme $\mathcal{H}_{{}_{\lambda}}=E_{{}_{\lambda}}(G)$ has simply-connected fibers isomorphic to $G$ and we have the identification of $U_{{}_{\lambda}}$-group schemes: $\displaystyle\big{(}\text{Res}_{{}_{U^{\prime}/U}}(\mathcal{H}_{{}_{\lambda}})\big{)}^{{}^{\Gamma_{{}_{\lambda}}}}\simeq\mathcal{G}^{{}^{st}}_{{}_{\theta_{\lambda}}}.$ (3.0.21) We therefore get the following useful corollary to (3.3). ###### Corollary 3.8. For any standard dominant $1$-PS $\lambda$ of $T_{{}_{ad}}$, with notations as above we have an isomorphism as sheaves of Lie algebras: $\mathcal{R}\|_{{}_{U_{{}_{\lambda}}}}\simeq q^{{}^{\Gamma}}_{{}_{\lambda,}}(E_{{}_{\lambda}}(\mathfrak{g})).$ (3.0.22) ###### Proof. This is an immediate consequence of (3.3) and [1, Proposition 5.1.2]. ∎ All results in this subsection generalize suitably to non-standard curves corresponding to dominant $1$-PS in $T_{{}_{\text{ad}}}$ by (3.4). #### 3.0.7. The Lie algebra bundle $\mathcal{R}$ on $\bf X$ Let $\\{D_{\alpha}\|\alpha\in S\\}$ denote the irreducible smooth boundary divisors of ${\bf X}$. Set $H_{{}_{\alpha}}:=D_{{}_{\alpha}}\cap{\bf Y}_{{}_{0}}$ for each $\alpha$. Recall that $Z:={\bf Y}_{{}_{0}}\setminus T_{{}_{\text{ad}}}$ is a union $\cup_{{}_{\alpha\in S}}H_{{}_{\alpha}}$ which are $\ell$ smooth hyperplanes meeting at simple normal crossings. For $\alpha\in S$, let $\zeta_{{}_{\alpha}}$’s denote the generic points of the divisors $H_{{}_{\alpha}}$’s. Let $A_{{}_{\alpha}}=\mathcal{O}_{{}_{{{\bf Y}_{{}_{0}}},\zeta_{{}_{\alpha}}}}$ (3.0.23) be the dvr’s obtained by localizing at the height $1$-primes given by the $\zeta_{{}_{\alpha}}$’s and let $Y_{{}_{\alpha}}:={\rm Spec}\,(A_{{}_{\alpha}})$. Base changing by the local morphism $Y_{{}_{\alpha}}\to\bf Y_{{}_{0}}$, we have a Lie algebra bundle $\mathcal{R}\|_{{}_{Y_{{}_{\alpha}}}}$ for each $\alpha$. Moreover, the Lie algebra bundle $\mathcal{R}$ on ${\bf Y}_{{}_{0}}$ gives canonical gluing data to glue $\mathcal{R}\|_{{}_{Y_{{}_{\alpha}}}}$ with the trivial bundle $\mathfrak{g}\times T_{{}_{\text{ad}}}$. Let $\xi_{{}_{\alpha}}\in D_{{}_{\alpha}},\alpha\in S$ denote the generic points of the divisors $D_{{}_{\alpha}}$’s and let $B_{{}_{\alpha}}:=\mathcal{O}_{{}_{{\bf X},\xi_{{}_{\alpha}}}}$ (3.0.24) be the dvr’s obtained by localizing at the height $1$-primes given by the $\xi_{{}_{\alpha}}$’s. By a transport of structures, for each $\alpha$ the gluing datum of $\mathcal{R}\|_{{}_{Y_{{}_{\alpha}}\cap T_{{}_{\text{ad}}}}}$, gives a Lie-algebra bundle gluing datum on ${\rm Spec}\,(B_{{}_{\alpha}})\cap G_{{}_{\text{ad}}}$. Thus, the gluing data of the bundle $\mathcal{R}$ on ${\bf Y}_{{}_{0}}$ at the the $\zeta_{{}_{\alpha}}$’s can be now used to extend the trivial bundle $\mathfrak{g}\times G_{{}_{\text{ad}}}$ to the $\xi_{{}_{\alpha}}$’s as a locally free sheaf of Lie algebras. The rest of the proof is as for $\bf Y$, together with the observation that the $G\times G$-translates of the open subset $U\times U^{{}^{-}}\times{\bf Y}_{{}_{0}}$ cover $\bf X$, where the bundle is simply the pull-back from ${\bf Y}_{{}_{0}}$. Thus we have: ###### Corollary 3.9. There exists a canonical Lie-algebra bundle $\mathcal{R}_{{}_{\bf X}}$ on ${\bf X}$ which extends the trivial bundle with fiber $\mathfrak{g}$ on ${G_{{}_{\text{ad}}}}\subset{\bf X}$ with properties as in (3.3). #### 3.0.8. Bruhat-Tits group scheme ${\mathcal{G}}_{{}_{\bf X}}^{{}^{\varpi}}$ on $\bf X$ Let $L_{{}_{\alpha}}$ be the quotient field of $B_{{}_{\alpha}}$. Let $X_{{}_{\alpha}}:={\rm Spec}\,(B_{{}_{\alpha}})$. By (3.8), for each $\alpha$ there exist a $\Gamma_{{}_{\alpha}}$-equivariant $G$-torsor $E_{\alpha}$ on a ramified cover $q_{{}_{\alpha}}:X^{\prime}_{{}_{\alpha}}\to X_{{}_{\alpha}}$ such that the adjoint group scheme $\mathcal{H}_{{}_{\alpha}}=E_{\alpha}(G)$ has simply-connected fibers isomorphic to $G$ and we have the identification of $B_{{}_{\alpha}}$-group schemes: $\displaystyle\big{(}\text{Res}_{{}_{X^{\prime}_{{}_{\alpha}}/X_{{}_{\alpha}}}}(\mathcal{H}_{{}_{\alpha}})\big{)}^{{}^{\Gamma_{{}_{\alpha}}}}\simeq\mathcal{G}^{{}^{st}}_{{}_{\theta{{}_{{}_{\alpha}}}}}.$ (3.0.25) Before stating the main result of this section, we make a few remarks which might help the reader. The basic underlying principle in these constructions is that the combinatorial data encoded in the triple consisting of the Weyl chamber, the fan of Weyl chambers, and the Tits building, is geometrically replicated by the inclusion ${\bf Y_{{}_{0}}}\subset{\bf Y}\subset\bf X$. The "wonderful" Bruhat-Tits group scheme which arises on $\bf X$ has its local Weyl-chamber model on the affine toric variety $\bf Y_{{}_{0}}$. Indeed, in this case, the Kawamata cover is even explicit (3.5). In particular, the theorem below, can be executed for ${\bf Y}_{{}_{0}}$ but this will give the group scheme associated to the data coming from the Weyl chamber alone. ###### Theorem 3.10. There exists an affine “wonderful" Bruhat-Tits group scheme ${\mathcal{G}}_{{}_{\bf X}}^{{}^{\varpi}}$ on $\bf X$ satisfying the following classifying properties. 1. (1) There is an identification of the Lie-algebra bundles $\text{Lie}({\mathcal{G}}_{{}_{\bf X}}^{{}^{\varpi}})\simeq\mathcal{R}_{{}_{\bf X}}$. 2. (2) For $\emptyset\neq I\subset S$ the restriction of ${\mathcal{G}}_{{}_{{\bf X}}}^{{}^{\varpi}}$ to the formal neighbourhood $U_{{}_{z_{{}_{I}}}}$ of $z_{{}_{I}}$ in $C_{{}_{I}}$ §(3.0.3) is isomorphic to the standard Bruhat- Tits group scheme $\mathcal{G}^{{}^{st}}_{{}_{I}}$ §(2.0.5). ###### Proof. Let $X_{{}_{\alpha}}$ be as above. Note that we can identify the open subset ${\rm Spec}\,(L_{{}_{\alpha}})$ with ${G_{{}_{\text{ad}}}}\cap X_{{}_{\alpha}}$. Set $X^{\prime}:=G_{{}_{\text{ad}}}\cup_{\alpha\in S}X_{{}_{\alpha}}.$ (3.0.26) Hence ${\bf X}\setminus X^{\prime}$ is a colimit of closed subschemes of $\bf X$ of codimension at least $2$. Consider the fpqc morphism: $G_{{}_{{}_{\text{ad}}}}\bigsqcup_{\alpha\in S}X_{{}_{\alpha}}\to{X^{\prime}}.$ (3.0.27) We restrict $\mathcal{R}=\mathcal{R}_{{}_{\bf X}}$ further to $X^{\prime}$. By (3.9) , over the open subset $G_{{}_{\text{ad}}}\subset X^{\prime}$, we have $\mathcal{R}\simeq G_{ad}\times\mathfrak{g}$. Further, the transition functions of $\mathcal{R}$ on the intersections ${\rm Spec}\,(L_{{}_{\alpha}}):=G_{{}_{{}_{\text{ad}}}}\cap{\rm Spec}\,(B_{{}_{\alpha}})$ take values in $\text{Aut}(\mathfrak{g})$. Since $\text{Aut}(\mathfrak{g})=\text{Aut}(G)$, it follows that the effective descent datum provided by pulling back $\mathcal{R}$ to $G_{{}_{{}_{\text{ad}}}}\bigsqcup_{\alpha\in S}X_{{}_{\alpha}}$ gives a descent datum to glue the trivial group scheme $G\times G_{{}_{{}_{\text{ad}}}}$ on $G_{{}_{{}_{\text{ad}}}}$ with $\mathcal{G}^{{}^{st}}_{{}_{\alpha}}$ on ${\rm Spec}\,(B_{{}_{\alpha}})$ along ${\rm Spec}\,(L_{{}_{\alpha}})$. Since the group schemes are affine, this descent datum is also effective. In other words, we get the group scheme $\mathcal{G}^{\circ}\rightarrow X^{\prime}.$ (3.0.28) This can also be seen without “descent" theory as follows. Since the group schemes $\mathcal{G}^{{}^{st}}_{{}_{\alpha}}$ on ${\rm Spec}\,(B_{{}_{\alpha}})$ are of finite type, they can be extended to an affine subscheme $\bf X_{{}_{f}}$. By a further shrinking of this neighbourhood of $\xi_{{}_{\alpha}}$ one can glue it to $G\times G_{{}_{{}_{\text{ad}}}}$ along the intersection. So we can think of $X^{\prime}$ as an honest open subset of $X$ with complement of codimension at least $2$. By (3.8), the Lie algebra bundle $\mathcal{R}$ gets canonical parabolic structures at the generic points $\xi_{{}_{\alpha}}$ of the divisors $\\{D_{{}_{\alpha}}\\}_{{}_{\alpha\in S}}$. For a parabolic vector bundle with prescribed rational weights such as $\mathcal{R}$, by [2] (see §8) we get the following data: • a global Kawamata cover (8.0.2) $p:Z\rightarrow{\bf X}$ ramified over $D$ with ramification prescribed by the weights $\\{d_{{}_{\alpha}}\\}$ (2.0.3), with Galois group $\Gamma$ which “realizes the local ramified covers $q_{{}_{\alpha}}$ at the points $\xi_{{}_{\alpha}}$" i.e. the isotropy subgroup of $\Gamma$ at $\xi_{{}_{\alpha}}$ is $\Gamma_{{}_{\alpha}}$ and * • an equivariant vector bundle $V$ on $Z$ such that $p_{{}_{}}^{\Gamma}(V)\simeq\mathcal{R}.$ (3.0.29) Let $Z^{\prime}:=p^{-1}(X^{\prime})$ and $V^{\prime}:=V\|_{{}_{Z^{\prime}}}$. Gluing the trivial $G$-torsor with the $\\{E_{\alpha}\\}$ by the transition functions of $V^{\prime}$, we make a $\Gamma$-equivariant principal $\text{Aut}(G)$-torsor $\mathcal{E}^{\circ}$. Its associated group scheme $\mathcal{E}^{\circ}\times^{{}^{\text{Aut}(G)}}G=\mathcal{H}^{\circ}$ is a now group scheme on $Z^{\prime}$ with simply-connected fibers $G$, albeit with transition functions in $\text{Aut}(G)$. Moreover, we have $\big{(}\text{Res}_{{}_{Z^{\prime}/X^{\prime}}}(\mathcal{H}^{\circ})\big{)}^{{}^{\Gamma}}\simeq\mathcal{G}^{\circ}.$ (3.0.30) Further, as locally-free sheaves, we also have $\displaystyle\text{Lie}(\mathcal{H}^{\circ})=V^{\prime}.$ (3.0.31) We transport the structure of Lie-bracket from $\text{Lie}(\mathcal{H}^{\circ})$ to $V^{\prime}$ on $Z^{\prime}$. This is non-degenerate everywhere on $Z^{\prime}$ since $\mathcal{H}^{\circ}$ is the adjoint group scheme of $\mathcal{E}^{\circ}$. Hence $V^{\prime}$ has fiber type $\mathfrak{g}$. Since $\text{codim}(Z\setminus Z^{\prime})\geq 2$, the Lie bracket $[.,.]$ on $V^{\prime}$ extends to a Lie bracket on the locally free sheaf $V$ with the Killing form being non-degenerate on the whole of $Z$. In other words $V$ is now a locally free sheaf of Lie algebras on the whole of $Z$ with semisimple fibres; these fibres are isomorphic to the Lie algebra $\mathfrak{g}$ by the rigidity of semisimple Lie algebras. We now wish to extend $\mathcal{E}^{\circ}$ as an equivariant $\text{Aut}(G)$-torsor $\mathcal{E}$ on the whole of $Z$ such that its associated Lie-algebra bundle $\mathcal{E}(\mathfrak{g}):=\mathcal{E}\times^{{}^{\text{Aut}(G)}}\mathfrak{g}$ becomes isomorphic $V$. Since $G$ is simply-connected, making the identification $\text{Aut}(\mathfrak{g})=\text{Aut}(G)$, we see that the transition functions of $V$ give the gluing for defining $\mathcal{E}$ on $Z$ satisfying our desired requirements. Let $\mathcal{H}:=\mathcal{E}\times^{{}^{\text{Aut}(G)}}G$ denote the group scheme associated to the adjoint group scheme of $\mathcal{E}$. This is an equivariant group scheme on $Z$ and we define $\displaystyle{\mathcal{G}}_{{}_{\bf X}}^{{}^{\varpi}}:=\big{(}\text{Res}_{{}_{Z/{\bf X}}}(\mathcal{H})\big{)}^{{}^{\Gamma}}.$ (3.0.32) Then ${\mathcal{G}}_{{}_{\bf X}}^{{}^{\varpi}}\|_{{X^{\prime}}}=\mathcal{G}^{\circ}$. Since by Lemma (3.7), the functor “invariant direct image" commutes with taking Lie algebras, we moreover get isomorphisms of locally-free sheaves of Lie algebras $\text{Lie}({\mathcal{G}}_{{}_{\bf X}}^{{}^{\varpi}})\simeq p_{{}_{}}^{\Gamma}(\mathcal{E}(\mathfrak{g}))\simeq\mathcal{R}.$ (3.0.33) This proves the first claim in the theorem. Finally, let us verify the classifying property of the group scheme $\mathcal{H}$. For $\alpha\in S$, at the closed points $z_{{}_{\alpha}}$ of the curves ${\rm Spec}\,(A_{{}_{\alpha}})$ we have $\mathcal{H}\|{{}_{{\rm Spec}\,(A_{{}_{\alpha}})}}\simeq\mathcal{H}_{{}_{\alpha}}$. The classifying property is tautologically valid here because this was designed expressly in (3.0.25). For the closed points $z_{{}_{\lambda}}$ of $C_{{}_{\lambda}}$ corresponding to strata of lower dimension, we proceed as follows. Consider the base change of the Kawamata cover $p:Z\rightarrow{\bf X}$ to the curve $C_{{}_{\lambda}}\subset{\bf X}$ and further to the formal neighbourhood $U_{{}_{z_{{}_{\lambda}}}}\subset C_{{}_{\lambda}}$ of $z_{{}_{\lambda}}$. Let $p_{{}_{z}}:W_{{}_{z}}\to U_{{}_{z}}$ be the restriction of $p$ to a connected component of $Z\times_{{}_{U_{{}_{z}}}}{\bf X}$. Then, $p_{{}_{z}}$ gives a Galois cover with Galois group some cyclic group $\mu_{{}_{d}}\subset\Gamma$ of order $d$. By (3.0.32), the restriction ${\mathcal{G}}_{{}_{{\bf Y}}}^{{}^{\varpi}}\|_{{}_{U_{{}_{z}}}}$ is the “invariant direct image" of $\mathcal{H}\|_{{}_{W_{{}_{z}}}}$. Also, we have the isomorphism $\mathcal{R}\|_{{}_{U_{{}_{z}}}}\simeq p_{{}_{}}^{\mu_{{}_{d}}}(\mathcal{E}(\mathfrak{g})_{{}_{W_{{}_{z}}}})$. Further, by (3.8), $L^{+}(\mathcal{R}\|_{{}_{U_{{}_{z_{{}_{\lambda}}}}}})\simeq L^{+}(\mathfrak{P}^{{}^{std}}_{{}_{\theta_{{}_{\lambda}}}})$. By (3.8), we have another ramified cover $q_{{}_{\lambda}}:U^{\prime}_{{}_{z}}\to U_{{}_{z}}$, but now with Galois group $\Gamma_{{}_{\lambda}}$ and an equivariant $(\Gamma_{{}_{\lambda}},G)$-torsor $E_{{}_{\lambda}}$ on $U^{\prime}_{{}_{z}}$, such that $q^{{}^{\Gamma_{{}_{\lambda}}}}_{{}_{}}(E_{{}_{\lambda}}(G))\simeq\mathcal{G}^{{}^{st}}_{{}_{\theta_{{}_{\lambda}}}}$ and $q^{{}^{\Gamma_{{}_{\lambda}}}}_{{}_{}}(E_{{}_{\lambda}}(\mathfrak{g}))\simeq\mathcal{R}\|_{{}_{U_{{}_{z_{{}_{\lambda}}}}}}$. To finish the proof, we need to show the following isomorphism of group schemes: ${\mathcal{G}}_{{}_{{\bf X}}}^{{}^{\varpi}}\|_{{}_{U_{{}_{z}}}}\simeq\mathcal{G}^{{}^{st}}_{{}_{\theta_{{}_{\lambda}}}}.$ (3.0.34) Over a common cover of $U^{\prime}_{{}_{z}}$ and $W_{{}_{z}}$ (which will continue to be a Kawamata cover of $U_{{}_{z}}$), we can identify the pull- back of Lie sheaves $E_{{}_{\lambda}}(\mathfrak{g})$ and $\mathcal{E}(\mathfrak{g})\|_{{}_{W_{{}_{z}}}}$ as equivariant Lie sheaves since both give invariant direct images isomorphic to $\mathcal{R}\|_{{}_{U_{{}_{z_{{}_{\lambda}}}}}}$. Therefore, on the same common cover we have an identification of pull-backs of $E_{{}_{\lambda}}\times^{G}\mathfrak{g}$ with $\mathcal{E}(\mathfrak{g})\|_{{}_{W_{{}_{z}}}}=\mathcal{E}\times^{Aut(G)}\mathfrak{g}\|_{{}_{W_{{}_{z}}}}$. Hence we have an identification of pull-backs of $E_{{}_{\lambda}}\times^{G}G$ with $\mathcal{E}\times^{Aut(G)}G\|_{{}_{W_{{}_{z}}}}$ as equivariant group schemes with simply-connected fibers. Now the invariant direct image of the first group scheme is a standard parahoric group scheme because the curve $C_{{}_{\lambda}}$ is standard. On the other hand, the second group scheme is $\mathcal{H}\|_{{}_{W_{{}_{z}}}}$ whose invariant direct image by construction is ${\mathcal{G}}_{{}_{{\bf X}}}^{{}^{\varpi}}\|_{{}_{U_{{}_{z}}}}$. Thus, we have proven (3.0.34). ∎ ###### Remark 3.11. The above construction goes through without change over an algebraically closed field $k$ of characteristic $p$ coprime to the $d_{{}_{\alpha}}$’s (2.0.3). The existence of $\bf X$ is known from the works of Strickland [17] and De Concini-Springer [8] and Kawamata covering works under the above conditions on characteristics. ## 4\. The Weyl alcove and apartment case We continue to use the notations as in previous sections. Let $G^{{}^{\text{aff}}}$ denote the Kac-Moody group associated to the affine Dynkin diagram of $G$. Recall that $G^{{}^{\text{aff}}}$ is given by a central extension of $L^{\ltimes}G$ by $\mathbb{G}_{m}$. Analogous to the wonderful compactification of $G_{{}_{{}_{\text{ad}}}}$, P. Solis in [16] has constructed a wonderful embedding ${\bf X}^{{}^{\text{aff}}}$ for $G^{{}^{\text{aff}}}_{{}_{\text{ad}}}:=G^{{}^{\text{aff}}}/Z(G^{{}^{\text{aff}}})=\mathbb{G}_{m}\ltimes LG/Z(G)$. It is an ind-scheme containing $G^{{}^{\text{aff}}}_{{}_{\text{ad}}}$ as a dense open ind-scheme and carrying an equivariant action of $L^{\ltimes}G\times L^{\ltimes}G$. Let $T_{{}_{\text{ad}}}:=T/Z(G)$, $T^{\ltimes}_{{}_{\text{ad}}}:=\mathbb{G}_{m}\times T_{{}_{\text{ad}}}\subset G^{{}^{\text{aff}}}_{{}_{\text{ad}}}$, where $\mathbb{G}_{m}$ is the rotational torus. In ${\bf X}^{{}^{\text{aff}}}$, the closure ${\bf Y}^{{}^{\text{aff}}}:=\overline{T^{\ltimes}_{{}_{\text{ad}}}}$ gives a torus- embedding. It is covered by the affine Weyl group $W^{{}^{\text{aff}}}$-translates of the affine torus embedding ${\bf Y}_{{}_{0}}^{{}^{\text{aff}}}:=\overline{T^{\ltimes}_{{}_{\text{ad},0}}}\simeq\mathbb{A}^{\ell+1}$ given by the negative Weyl alcove. #### 4.0.1. On the torus-embedding ${\bf Y}_{{}_{0}}^{{}^{\text{aff}}}$ Recall that $Z:={\bf Y}_{{}_{0}}^{{}^{\text{aff}}}\setminus T^{\ltimes}_{{}_{\text{ad}}}$ is a union $\cup_{{}_{\alpha\in\mathbb{S}}}H_{{}_{\alpha}}$ of $\ell+1$ many standard coordinate hyperplanes meeting at normal crossings. For $\alpha\in\mathbb{S}$, let the $\zeta_{{}_{\alpha}}$’s denote the generic points of the divisors $H_{{}_{\alpha}}$’s. Let $A_{{}_{\alpha}}=\mathcal{O}_{{}_{{\bf Y}_{{}_{0}}^{{}^{\text{aff}}},\zeta_{{}_{\alpha}}}}$ (4.0.1) be the dvr’s obtained by localizing at the height $1$-primes given by the $\zeta_{{}_{\alpha}}$’s. Let $K_{{}_{\alpha}}$ be the quotient field of $A_{{}_{\alpha}}$. Let $Y_{{}_{\alpha}}:={\rm Spec}\,(A_{{}_{\alpha}})$. Note that we can identify the open subset ${\rm Spec}\,(K_{{}_{\alpha}})$ with ${T^{\ltimes}_{{}_{\text{ad}}}}\cap Y_{{}_{\alpha}}$. Let $Y^{\prime}:={T^{\ltimes}_{{}_{\text{ad}}}}\cup_{\alpha}Y_{{}_{\alpha}}.$ (4.0.2) The complement ${\bf Y}_{{}_{0}}^{{}^{\text{aff}}}\setminus Y^{\prime}$ can again be realized as a colimit of open subsets of ${\bf Y}_{{}_{0}}^{{}^{\text{aff}}}$ whose codimension is at least $2$ in ${\bf Y}_{{}_{0}}^{{}^{\text{aff}}}$. #### 4.0.2. Construction of a finite-dimensional Lie algebra bundle $J$ on ${\bf Y}_{{}_{0}}^{{}^{\text{aff}}}$ together with parabolic structures This construction is exactly analogous to the construction of $\mathcal{R}$ on $\bf Y$ in §3.0.3. We let $T^{\ltimes}_{{}_{\text{ad}}}$ and $\mathbb{S}$ play the role of $T_{{}_{\text{ad}}}$ and $S$. More precisely, let $\lambda=\sum_{\alpha\in\mathbb{S}}k_{{}_{\alpha}}\omega_{{}_{\alpha}}^{{\vee}}$ be a non-zero dominant $1$-PS of $T_{{}_{\text{ad}}}^{\ltimes}$. We set $\eta_{\lambda}:=\sum_{\alpha\in\mathbb{S}}k_{{}_{\alpha}}\frac{(1,\theta_{{}_{\alpha}})}{\ell+1}\quad\text{and}\quad\theta_{\lambda}:=\sum_{\alpha\in\mathbb{S}}\frac{k_{{}_{\alpha}}}{\sum_{{}_{\alpha\in\mathbb{S}}k_{{}_{\alpha}}}}\theta_{{}_{\alpha}}$ (4.0.3) and $C_{{}_{\lambda}}$ the curve in ${\bf Y}^{{}^{\text{aff}}}$ defined by $\eta_{\lambda}$ and $U_{{}_{\lambda}}$ the formal neighbourhood of the closed point of $z_{{}_{\lambda}}$ of $C_{{}_{\lambda}}$. So if we substitute the rational number $a$ of §2.0.3 by the number $\frac{\sum_{\alpha\in\mathbb{S}}k_{{}_{\alpha}}}{\ell+1}$, we have the parahoric group identification $\mathfrak{P}_{{}_{\eta_{\lambda}}}=\mathfrak{P}_{{}_{\theta_{\lambda}}}.$ (4.0.4) We may prove the following theorem exactly like (3.3). There we viewed the standard alcove as a cone over the far wall. Note here that unlike the situation in (3.3), we do not expect to get standard parahoric structures down all the strata. However, for strata contained in the divisor associated to $\alpha_{{}_{0}}\in\mathbb{S}$, the parahorics which occur will be standard as before. ###### Theorem 4.1. There exists a canonical Lie-algebra bundle $J$ on ${\bf Y}_{{}_{0}}^{{}^{\text{aff}}}$ which extends the trivial bundle with fiber $\mathfrak{g}$ on ${T^{\ltimes}_{{}_{\text{ad}}}}\subset{\bf Y}^{{}^{\text{aff}}}$ and such that for $k_{{}_{\alpha}}\in\\{0,1\\}$ we have the identification of functors from the category of $k$-algebras to $k$-Lie- algebras: $L^{+}(\mathfrak{P}_{{}_{\eta_{{}_{\lambda}}}})=L^{+}(\mathcal{R}\mid_{{}_{U_{{}_{\lambda}}}}).$ (4.0.5) ###### Corollary 4.2. The Lie algebra bundle $J_{{}_{{\bf Y}^{{}^{\text{aff}}}}}$ gets canonical parabolic structures (§8) at the generic points $\xi_{{}_{\alpha}}$ of the $W^{{}^{\text{aff}}}$-translates of the divisors $H_{{}_{\alpha}}\subset{\bf Y}_{{}_{0}}^{{}^{\text{aff}}}$, $\alpha\in\mathbb{S}$. ###### Proof. We prescribe ramification indices $d_{\alpha}$ (2.0.3) on the divisor $H_{{}_{\alpha}}$. Then as in the proof of (3.6), the identification (3.0.22) of the Lie algebra structures of $J_{{}_{{\bf Y}^{{}^{\text{aff}}}}}$ and the parahoric Lie algebra structures on the localizations of the generic points of $H_{{}_{\alpha}}$ allows us to endow parabolic structures at the generic points of the divisors.∎ #### 4.0.3. The parahoric group scheme on the torus embedding ${{\bf Y}^{{}^{\text{aff}}}}$ ###### Theorem 4.3. There exists an affine “wonderful" Bruhat-Tits group scheme ${\mathcal{G}}_{{}_{{\bf Y}^{{}^{\text{aff}}}}}^{{}^{\varpi}}$ on $\bf Y$ together with a canonical isomorphism $\text{Lie}({\mathcal{G}}_{{}_{{\bf Y}^{{}^{\text{aff}}}}}^{{}^{\varpi}})\simeq J$. It further satisfies the following classifying property: For any point $h\in{{\bf Y}^{{}^{\text{aff}}}}\setminus T^{\ltimes}_{ad}$, let $\mathbb{I}\subset\mathbb{S}$ be a subset such that $h\in\cap_{\alpha\in\mathbb{I}}H_{{}_{\alpha}}$. Let $C_{{}_{\mathbb{I}}}\subset{{\bf Y}^{{}^{\text{aff}}}}$ be a smooth curve with generic point in $T^{\ltimes}_{{}_{\text{ad}}}$ and closed point $h$. Let $U_{{}_{h}}\subset C_{{}_{\mathbb{I}}}$ be a formal neighbourhood of the closed point $h$. Then, the restriction ${\mathcal{G}}_{{}_{{\bf Y}^{{}^{\text{aff}}}}}^{{}^{\varpi}}\|_{{}_{U_{{}_{h}}}}$ is isomorphic to the Bruhat-Tits group scheme $\mathcal{G}_{{}_{\mathbb{I}}}$ on $U_{{}_{h}}$. ###### Proof. Recall that ${{\bf Y}^{{}^{\text{aff}}}}$ is covered by affine spaces $Y_{{}_{w}}\simeq\mathbb{A}^{\ell+1}$ parametrized by the affine Weyl group $W^{{}^{\text{aff}}}$. Each $Y_{{}_{w}}$ is the translate of ${{\bf Y}_{{}_{0}}^{{}^{\text{aff}}}}$. The translates of the divisors $H_{{}_{\alpha}}$ meet each $Y_{{}_{w}}$ in the standard hyperplanes on $\mathbb{A}^{\ell+1}$ and thus we can prescribe the same ramification data at the hyperplanes on each of the $Y_{{}_{w}}$’s. On the other hand, although we have simple normal crossing singularities, we do not have an analogue of the Kawamata covering lemma for schemes such as ${{\bf Y}^{{}^{\text{aff}}}}$. The lemma is known only in the setting of quasi-projective schemes. So to construct the group scheme, we employ a different approach. We observe firstly that, the formalism of Kawamata coverings applies in the setting of the affine spaces $Y_{{}_{w}}\subset{{\bf Y}^{{}^{\text{aff}}}}$. Let $p_{w}:Z_{{}_{w}}\rightarrow Y_{{}_{w}}$ be the associated Kawamata cover (8.0.2) with Galois group $\Gamma_{w}$. In Corollary 4.2 we observed that the Lie-algebra bundle $J$ has a canonical parabolic structure like $\mathcal{R}$ in the proof of Theorem 3.10. Letting $Y_{{}_{w}}$ play the role of ${\bf Y_{{}_{0}}}$ and using all arguments in the proof of Theorem 3.10 we obtain $\mathcal{H}_{{}_{w}}\rightarrow Z_{{}_{w}}$ which is $\Gamma_{w}$-group scheme with fibers isomorphic to $G$ whose invariant direct image is a group scheme $\mathcal{G}_{w}$ such that $\text{Lie}(\mathcal{G}_{w})=J\|_{Y_{{}_{w}}}$. The induced parabolic structure on $J\|_{Y_{{}_{w}}}$ is the restriction of the one on $J$. Indeed, by (4.2) these parabolic structures are essentially given at the local rings at the generic points $\xi_{\alpha}$ of the divisors $H_{{}_{\alpha}}$ and hence these parabolic structures on $J$ agree on the intersections $Y_{{{}_{uv}}}:=Y_{{}_{u}}\cap Y_{{}_{v}}$. Let $Z_{{}_{uv}}:=p_{u}^{-1}(Y_{{}_{uv}})$. Let $\tilde{Z}_{{}_{uv}}$ be the normalization of a component of $Z_{{}_{uv}}\times_{Y_{{}_{uv}}}Z_{{}_{vu}}$. Then $\tilde{Z}_{{}_{uv}}$ serves as Kawamata cover (8.0.2) of $Y_{{}_{uv}}$ (see [18, Corollary 2.6, page 56]). We consider the morphisms $\tilde{Z}_{{}_{uv}}\to Z_{{}_{u}}$ (resp.$\tilde{Z}_{{}_{uv}}\to Z_{{}_{v}}$) and let $\mathcal{H}_{{}_{u,\tilde{Z}}}$ (resp. $\mathcal{H}_{{}_{v,\tilde{Z}}}$) denote the pull-backs of $\mathcal{H}_{{}_{u}}$ (resp. $\mathcal{H}_{{}_{v}}$) to $\tilde{Z}_{{}_{uv}}$. Let $\Gamma$ denote the Galois group for $\tilde{Z}_{{}_{uv}}\rightarrow Y_{{}_{uv}}$. Then by Lemma 3.7, the invariant direct images of the equivariant Lie algebra bundles $\text{Lie}(\mathcal{H}_{{}_{u,\tilde{Z}}})$ and $\text{Lie}(\mathcal{H}_{{}_{v,\tilde{Z}}})$ coincide with the Lie algebra structure on $J$ restricted to the ${Y_{{}_{uv}}}$ and also as isomorphic parabolic bundles. Therefore we have a natural isomorphism of equivariant Lie- algebra bundles $\text{Lie}(\mathcal{H}_{u,\tilde{Z}})\simeq\text{Lie}(\mathcal{H}_{v,\tilde{Z}}).$ (4.0.6) As in the proof of Theorem 3.10, this gives a canonical identification of the equivariant group schemes $\mathcal{H}_{{}_{u,\tilde{Z}}}$ and $\mathcal{H}_{{}_{v,\tilde{Z}}}$ on $\tilde{Z}_{{}_{uv}}$. Since the invariant direct image of both the group schemes $\mathcal{H}_{{}_{u,\tilde{Z}}}$ and $\mathcal{H}_{{}_{v,\tilde{Z}}}$ are the restrictions $\mathcal{G}_{{}_{u,{Y_{{}_{uv}}}}}$ and $\mathcal{G}_{{}_{v,{Y_{{}_{uv}}}}}$, it follows that on $Y_{{}_{uv}}=Y_{{}_{u}}\cap Y_{{}_{v}}$ we get a canonical identification of group schemes: $\displaystyle\mathcal{G}_{{}_{u,{Y_{{}_{uv}}}}}\simeq\mathcal{G}_{{}_{v,{Y_{{}_{uv}}}}}.$ (4.0.7) These identifications are canonically induced from the gluing data of the Lie algebra bundle $J$ for the cover $Y_{{}_{w}}$’s. Therefore the cocycle conditions are clearly satisfied and the identifications (4.0.7) glue to give the group scheme ${\mathcal{G}}_{{}_{{\bf Y}^{{}^{\text{aff}}}}}^{{}^{\varpi}}$ on ${\bf Y}^{{}^{\text{aff}}}$. The verification of the classifying property follows exactly as in the proof of Theorem 3.10. ∎ ## 5\. The Bruhat-Tits group scheme on ${\bf X}^{{}^{{aff}}}$ Let ${\bf X}^{{}^{{aff}}}$ as in §4. We begin with a generality. Let $\mathbb{X}$ be an ind-scheme. By an open-subscheme $i:\mathbb{U}\hookrightarrow\mathbb{X}$ we mean an ind-scheme such that for any $f:{\rm Spec}\,(A)\rightarrow\mathbb{X}$, the natural morphism $\mathbb{U}\times_{\mathbb{X}}{\rm Spec}\,(A)\rightarrow{\rm Spec}\,(A)$ is an open immersion. For a sheaf $\mathbb{F}$ on $\mathbb{U}$ by $i_{{}_{}}(\mathbb{F})$ we mean the sheaf associated to the pre-sheaf on the “big site" of $\mathbb{X}$, whose sections on $f:{\rm Spec}\,(A)\rightarrow\mathbb{X}$ are given by $\mathbb{F}(\mathbb{U}\times_{\mathbb{X}}{\rm Spec}\,(A))$. The ind-scheme ${\bf X}^{{}^{\text{aff}}}$ has a certain open subset ${\bf X}_{{}_{0}}$ whose precise definition is somewhat technical [16, Page 705]. Let us mention the properties relevant for us. Recall that ${\bf X}^{{}^{\text{aff}}}=(G_{{}_{\text{ad}}}^{{}^{\text{aff}}}\times G_{{}_{\text{ad}}}^{{}^{\text{aff}}})~{}{\bf X}_{{}_{0}}$ and in fact ${\bf X}^{{}^{\text{aff}}}=(G_{{}_{\text{ad}}}^{{}^{\text{aff}}}\times G_{{}_{\text{ad}}}^{{}^{\text{aff}}})~{}{{\bf Y}^{{}^{\text{aff}}}}$. Further, the torus-embedding ${\bf Y}^{{}^{\text{aff}}}$ is covered by ${\bf Y}^{{}^{\text{aff}}}_{{}_{w}}\simeq\mathbb{A}^{{}^{\ell+1}}$ which are $W^{{}^{\text{aff}}}$-translates of ${\bf Y}^{{}^{\text{aff}}}_{{}_{0}}$, where ${\bf Y}^{{}^{\text{aff}}}_{{}_{0}}={{\bf Y}^{{}^{\text{aff}}}}\cap{\bf X}_{{}_{0}}$. We remark that, analogous to the case of ${\bf Y}_{{}_{0}}\subset{\bf Y}\subset{\bf X}$, just as the toric variety ${\bf Y}_{{}_{0}}$ was associated to the negative Weyl chamber, the toric variety ${\bf Y}^{{}^{\text{aff}}}_{{}_{0}}$ is associated to the negative Weyl alcove. So let us denote by $0$ the neutral element of $W^{{}^{\text{aff}}}$ also. Let $U^{\pm}\subset B^{\pm}$ be the unipotent subgroups. Let $\mathcal{U}^{\pm}:=ev^{-1}(U^{\pm})$ where $ev:G(A)\rightarrow G(k)$ is the evalution map. Further by [16, Prop 5.3] ${\bf X}_{{}_{0}}=\mathcal{U}\times{\bf Y}^{{}^{\text{aff}}}_{{}_{0}}\times\mathcal{U}^{-}.$ (5.0.1) Let ${\bf X}_{{}_{w}}:=\mathcal{U}\times{\bf Y}^{{}^{\text{aff}}}_{{}_{w}}\times\mathcal{U}^{-}$. These cover $\mathcal{U}\times{{\bf Y}^{{}^{\text{aff}}}}\times\mathcal{U}^{-}$. For $g\in G_{{}_{\text{ad}}}^{{}^{\text{aff}}}\times G_{{}_{\text{ad}}}^{{}^{\text{aff}}}$, let ${\bf X}_{g}:=g{\bf X}_{0}\quad{\bf X}_{{}_{g,w}}:=g{\bf X}_{{}_{w}}\quad{\bf Y}^{{}^{\text{aff}}}_{{}_{g,w}}:=g{\bf Y}^{{}^{\text{aff}}}_{{}_{w}}.$ (5.0.2) Note that we have the projection ${\bf X}_{{}_{g,w}}\to{\bf Y}^{{}^{\text{aff}}}_{{}_{g,w}}$ which is a $\mathcal{U}\times\mathcal{U}^{-}$-bundle. By [16, Theorem 5.1] the ind-scheme ${\bf X}^{{}^{\text{aff}}}$ has divisors $D_{\alpha}$ for $\alpha\in\mathbb{S}$ such that the complement of their union is ${\bf X}^{{}^{\text{aff}}}\setminus G_{{}_{\text{ad}}}^{{}^{\text{aff}}}$. The next proposition shows the existence of a finite dimensional Lie algebra bundle on ${\bf X}^{{}^{\text{aff}}}$ which is analogous to the bundle $\mathcal{R}$ over ${\bf X}$. ###### Proposition 5.1. There is a finite dimensional Lie algebra bundle $\bf R$ on ${\bf X}^{{}^{\text{aff}}}$ which extends the trivial Lie algebra bundle $G_{{}_{\text{ad}}}^{{}^{\text{aff}}}\times\mathfrak{g}$ on the open dense subset $G_{{}_{\text{ad}}}^{{}^{\text{aff}}}\subset{\bf X}^{{}^{\text{aff}}}$ and whose restriction to ${\bf Y}^{{}^{\text{aff}}}$ is $J$. ###### Proof. Following the arguments for (3.6) and (3.9), let $J^{\prime}$ be the locally free sheaf obtained on the open subset ${\bf X^{\prime}}$ obtained by the union of $G^{{}^{\text{aff}}}$ and the height one prime ideals of ${\bf X}^{{}^{\text{aff}}}$. Let ${\bf R}$ denote its push-forward. To check that the push-forward is locally free, without loss of generality we may consider its restriction to ${\bf X}_{0}$. But on ${\bf X}_{0}$, the pushforward of $J^{\prime}$ restricts to the pull-back of a Lie-algebra bundle $J$ on ${\bf Y}^{{}^{\text{aff}}}$ constructed in (4.1), which completes the argument. The rest of the properties follows immediately. ∎ ###### Theorem 5.2. There exists an affine “wonderful" Bruhat-Tits group scheme ${\mathcal{G}}_{{}_{{\bf X}^{{}^{\text{aff}}}}}^{{}^{\varpi}}$ on ${\bf X}^{{}^{\text{aff}}}$ together with a canonical isomorphism $\text{Lie}({\mathcal{G}}_{{}_{\bf X^{aff}}}^{{}^{\varpi}})\simeq{\bf R}$. It further satisfies the following classifying property: For any point $h\in{\bf X}^{{}^{\text{aff}}}\setminus G_{{}_{\text{ad}}}^{{}^{\text{aff}}}$, let $\mathbb{I}\subset\mathbb{S}$ be defined by the condition $h\in\cap_{\alpha\in\mathbb{I}}D_{\alpha}$. Let $C_{{}_{\mathbb{I}}}\subset{\bf X}$ be a smooth curve with generic point in $G_{{}_{\text{ad}}}^{{}^{\text{aff}}}$ with closed point $h$. Let $U_{{}_{h}}\subset C_{{}_{\mathbb{I}}}$ be a formal neighbourhood of $h$. Then, the restriction ${\mathcal{G}}_{{}_{{\bf X}^{{}^{\text{aff}}}}}^{{}^{\varpi}}\|_{{}_{U_{{}_{h}}}}$ is isomorphic to the Bruhat-Tits group scheme $\mathcal{G}_{{}_{\mathbb{I}}}$ on ${\rm Spec}\,(A)$. ###### Proof. We begin by observing that ${\bf X}_{{}_{g,w}}$ and ${\bf R}\rightarrow{\bf X}^{{}^{\text{aff}}}$ play the role of ${{\bf Y}_{{}_{g,w}}^{{}^{\text{aff}}}}$ and $J\rightarrow{{\bf Y}^{{}^{\text{aff}}}}$ in the proof of Theorem 4.3. Therefore the group scheme $\mathcal{G}_{{}_{g,w}}$ glue together and we obtain the global group scheme ${\mathcal{G}}_{{}_{{\bf X}^{{}^{\text{aff}}}}}^{{}^{\varpi}}$. The verification of the classifying property follows exactly as in the proof of Theorem 3.10. ∎ ## 6\. The Bruhat-Tits group scheme on ${\bf X}^{{}^{{poly}}}$ In [16], apart from the loop group case Solis also discusses the polynomial loop group situation which in fact is relatively simpler. Consequently, the analysis as well as the results in the loop group situation go through with some simplifications. Let $L_{{}_{\text{poly}}}G$ be defined by $L_{{}_{\text{poly}}}G(R^{\prime})=G(R^{\prime}[z^{{}^{\pm}}])$ (6.0.1) and we define $L_{{}_{\text{poly}}}^{\ltimes}G$ similarly as $L_{{}_{\text{poly}}}^{\ltimes}G:=\mathbb{G}_{m}\ltimes L_{{}_{\text{poly}}}G$. Solis constructs a “wonderful" embedding of the polynomial loop group $L_{{}_{\text{poly}}}^{\ltimes}G/Z(G)$. We denote this space by ${\bf X}^{{}^{\text{poly}}}$. It has an open subset ${\bf X}^{{}^{\text{poly}}}_{{}_{0}}$ whose definition is somewhat technical [16, §5.3], but, which plays the role completely analogus to ${\bf X}_{{}_{0}}$ for ${\bf X}^{{}^{\text{aff}}}$. We will continue to denote by $\bf Y:=\overline{T^{\ltimes}_{{}_{\text{ad}}}}$ the toral embedding of $T^{\ltimes}_{{}_{\text{ad}}}$ obtained by taking its closure in ${\bf X}^{{}^{\text{poly}}}$. As in the loop group case, in the polynomial case too $\bf Y$ is covered by infinitely many affine spaces $\mathbb{A}^{\ell+1}$ parametrized by the affine Weyl group $W^{{}^{\text{aff}}}$. In fact, if $t^{-\alpha_{i}}$ are the regular functions on $T^{\ltimes}_{{}_{\text{ad}}}$ given by the character $-\alpha_{i}$, then $\overline{T^{\ltimes}_{{}_{ad,0}}}:=\overline{T^{\ltimes}_{{}_{\text{ad}}}}\cap{\bf X}_{{}_{0}}^{{}^{poly}}\simeq{\rm Spec}\,\mathbb{C}[t^{-\alpha_{0}},\ldots,t^{-\alpha_{\ell}}].$ We index these affine spaces by the affine Weyl group $W^{{}^{\text{aff}}}$ and denote them as before by $Y_{{}_{w}}$’s, $w\in W^{{}^{\text{aff}}}$. As before, we may construct a finite-dimensional Lie-algebra bundle $J$ on ${{\bf Y}^{{}^{\text{aff}}}}$, then ${\bf R}$ on ${\bf X}^{{}^{\text{poly}}}$ and then construct a group scheme ${\mathcal{G}}_{{}_{{\bf X}^{{}^{\text{poly}}}}}^{{}^{\varpi}}$. Since the inductive structure of ${\bf X}^{{}^{\text{poly}}}$ is identical to ${\bf X}^{{}^{\text{aff}}}$, the proofs are also identical. Therefore we omit them. Now we outline a construction in the polynomial loop situation which will be generalized in the next section. For any affine space $Y_{{}_{w}}$, we have an affine embeddings $\displaystyle i_{w}:T_{{}_{\text{ad},w}}\times\mathbb{G}_{m}=T^{\ltimes}_{{}_{\text{ad},w}}\hookrightarrow Y_{{}_{w}}$ (6.0.2) Furthermore, the projection $p:T_{{}_{\text{ad},w}}\times\mathbb{G}_{m}\to\mathbb{G}_{m}$ extends to a morphism $p_{w}:Y_{{}_{w}}\to\mathbb{A}^{1}$ such that $p^{{}^{-1}}(0)$ is the union of the hyperplanes. At the level of coordinates, this extension is simply the product of the coordinate functions. Further $i_{w}$ and $p_{w}$ glue to give (6.0.7) ## 7\. An analogue of a construction of Mumford In [13] towards the very end Mumford gives a beautiful construction of the geometric realization of the relative case of buildings via toroidal embeddings. He deals with the general situation of an arbitrary discrete valuation ring $R$ with algebraically closed residue field $k=R/\mathfrak{m}$. Our aim in this section is limited to the case when $k=\mathbb{C}$, $\mathfrak{m}=(z)\subset\mathbb{C}[z]$ and $R=\mathbb{C}[z]_{{}_{\mathfrak{m}}}$. Let $K=\mathbb{C}(z)$ and let $S={\rm Spec}\,R$ and $\eta$ be the generic point and $o\in S$ the closed point. For schemes $X$ over $S$, $X_{{}_{\eta}}$ will be the fibre over $\eta$ and $X_{{}_{o}}$ the fibre over $o$. We will briefly sketch an analogue of Mumford’s construction in our setting and outline of the relationship between this construction to Solis’s approach and as a consequence construct a “wonderful" Bruhat-Tits group scheme on the toroidal embedding of $G_{{}_{ad,}}\times S$. For the purposes of this section alone so as to remain consistent with the one in [13], we will have the following set of notations. $\displaystyle H:=G_{{}_{\text{ad}}}$ (7.0.1) $\displaystyle H_{{}_{S}}:=G\times S$ (7.0.2) $\displaystyle T_{{}_{S}}\subset H_{{}_{S}}$ (7.0.3) $\displaystyle T_{{}_{S}}:=T_{{}_{\text{ad}}}\times S$ (7.0.4) $\displaystyle T_{{}_{K}}:=T_{{}_{\text{ad}}}\times{\rm Spec}\,K$ (7.0.5) i.e. $H_{{}_{S}}$ is a split adjoint semisimple group scheme over $S$, $T_{{}_{S}}\subset H_{{}_{S}}$ a fixed split maximal torus and $G$ will as before stand for the simply connected group. We will denote $U^{\pm}_{S}:=U^{\pm}\times S$. Base-changing (6.0.7) by ${\rm Spec}\,R\rightarrow\mathbb{A}^{1}$, by the definition of $\bf Y$ we obtain: (7.0.10) We continue to denote by $Y_{{}_{w}}$ the base-change. The orbit space $H_{{}_{S}}/T_{{}_{S}}$ exists as an affine scheme over $S$ and $H_{{}_{S}}$ via the quotient map $\pi:H_{{}_{S}}\to H_{{}_{S}}/T_{{}_{S}}$ is a locally free principal $T_{{}_{S}}$-bundle over $H_{{}_{S}}/T_{{}_{S}}$. We now consider the associated fibre bundle: (7.0.15) Note that $T_{\eta}=Y_{w,\eta}$ and hence $H_{\eta}=(H_{S}\times^{{}^{T_{S}}}Y_{{}_{w}})_{{}_{\eta}}$. Let us denote by $\displaystyle\overline{Z}_{w}:=H_{S}\times^{{}^{T_{S}}}Y_{{}_{w}}$ (7.0.16) and observe that $U_{{}_{S}}^{-}\times Y_{{}_{w}}\times U_{{}_{S}}=Z_{{}_{w}}$ is an open subset of $\overline{Z}_{w}$. In fact, over $U_{{}_{S}}^{-}\times U_{{}_{S}}\subset H_{{}_{S}}/T_{{}_{S}}$, the quotient map $\pi:H_{S}\rightarrow H_{S}/T_{S}$ is trivial. Analogous to Mumford’s definition [13, page 206], we define $\displaystyle\overline{H_{{}_{S}}}:=\bigcup_{{}_{x\in H(K)}}(H_{S}\times^{{}^{T_{S}}}Y_{{}_{w}}).x$ (7.0.17) where the generic fibre $H_{\eta}$ is identified in each copy $(H_{S}\times^{{}^{T_{S}}}Y_{{}_{w}}).x$ by a translate of $H_{\eta}$ by right multiplication by $x$ in the Iwahori subgroup. It takes a bit to prove that there is such an action. Note that we have the identification: $\displaystyle H_{\eta}=\overline{H_{{}_{\eta}}}$ (7.0.18) Consider the inclusion $G_{{}_{\text{ad}}}\times\mathbb{G}_{m}\hookrightarrow L_{{}_{\text{poly}}}^{\ltimes}G/Z(G)$. Let ${\bf M}=\overline{G_{{}_{\text{ad}}}\times\mathbb{G}_{m}}$ in ${\bf X}^{{}^{poly}}$ as a scheme over $\mathbb{C}$. It is locally of finite type. ###### Proposition 7.1. The $S$-scheme $\overline{H_{{}_{S}}}$ as a $\mathbb{C}$-scheme is isomorphic to $\bf M$. In particular, $\overline{H_{{}_{S}}}$ is a regular scheme over $\mathbb{C}$ such that the closed fibre $\overline{H_{o}}$ is a complete scheme over $\mathbb{C}$ which is a union of smooth components meeting at normal crossings. The embedding $H_{{}_{S}}\subset\overline{H_{{}_{S}}}$ is a toroidal embedding without self-intersection. Finally, the strata of $\overline{H_{{}_{S}}}-H_{{}_{S}}$ are precisely the parahoric subgroups of $H(K)$. This bijection between the strata and the parahoric subgroups extends to an isomorphism of the graph of the embedding $H_{{}_{S}}\subset\overline{H_{{}_{S}}}$ with the Bruhat-Tits building of $H$ over $S$. By the methods in this note we immediately deduce the existence of a “wonderful" Bruhat-Tits group scheme $\mathcal{G}_{{}_{\bf M}}^{{}^{\varpi}}$ on $\bf M$ which behaves naturally with respect to the strata. ###### Remark 7.2. A difference between Mumford’s construction and ours is that in Mumford’s construction the fibre over the closed point in $S$ is a reducible divisor but not necessarily with simple normal crossing singularities. In fact, the components associated to the parahorics which are not hyperspecial come with multiplicity being the coefficient $c_{{}_{\alpha}}$ of the associated simple root in the expression of the highest root. So a Kawamata lemma is not immediately applicable in that situation. ###### Remark 7.3. Mumford’s construction works over algebraically closed fields of any characteristic while we assumed the characteristic to be $0$. We did this to make the construction in the loop group situation following [16]. On the other hand, it seems that with a bit more work the whole construction of $\mathcal{G}_{{}_{\bf M}}^{{}^{\varpi}}$ will go through for characteristics $p$ such that $p$ is coprime to the $d_{{}_{\alpha}}$’s in (2.0.3). ## 8\. Appendix on parabolic and equivariant bundles In this section we recall and summarize results on parabolic bundles and equivariant bundles on Kawamata covers. These play a central role in the constructions of the Bruhat-Tits group schemes made above. Consider a pair $(X,D)$, where $X$ is a smooth quasi-projective variety and $D=\sum_{{}_{j=0}}^{{}^{\ell}}D_{{}_{j}}$ is a reduced normal crossing divisor with non-singular components $D_{{}_{j}}$ intersecting each other transversely. The basic examples we have in mind in the note are discrete valuation rings with its closed point, the wonderful compactification $\bf X$ with its boundary divisors or affine toric varieties. Let $E$ be a locally free sheaf on $X$. Let $n_{{}_{j}},j=0\ldots,\ell$ be positive integers attached to the components $D_{{}_{j}}$. Let $\xi$ be a generic point of $D$. Let $E_{{}_{\xi}}:=E\otimes_{{}_{\mathcal{O}_{{}_{X}}}}\mathcal{O}_{{}_{X,\xi}}$ and $\bar{E}_{{}_{\xi}}:=E_{{}_{\xi}}/\mathfrak{m}_{{}_{\xi}}E_{{}_{\xi}}$, $\mathfrak{m}_{{}_{\xi}}$ the maximal ideal of $\mathcal{O}_{{}_{X,\xi}}$. ###### Definition 8.1. A parabolic structure on $E$ consists of the following data: * • A flag $\bar{E}_{{}_{\xi}}=F^{{}^{1}}\bar{E}_{{}_{\xi}}\supset\ldots\supset F^{r{{}_{{}_{j}}}}\bar{E}_{{}_{\xi}}$, at the generic point $\xi$ of each of the components $D_{{}_{j}}$ of $D$, * • Weights $d_{{}_{s}}/n_{{}_{j}}$, with $0\leq d_{{}_{s}}<n_{{}_{j}}$ attached to $F^{{}^{s}}\bar{E}_{{}_{\xi}}$ such that $d_{1}<\cdots<d_{r_{j}}$ By saturating the flag datum on each of the divisors we get for each component $D_{{}_{j}}$ a filtration: $\displaystyle E_{{}_{D_{{}_{j}}}}=F^{{}^{1}}_{{}_{j}}\supset\ldots\supset F^{r{{}_{{}_{j}}}}_{{}_{j}}$ (8.0.1) of sub-sheaves on $D_{{}_{j}}$. Define the coherent subsheaf $\mathcal{F}^{s}_{{}_{j}}$, where $0\leq j\leq\ell$ and $1\leq s\leq r_{{}_{j}}$, of $E$ by: $\displaystyle 0\to\mathcal{F}^{s}_{{}_{j}}\to E\to E_{{}_{D_{{}_{j}}}}/F^{s}_{{}_{j}}\to 0$ (8.0.2) the last map is by restriction to the divisors. In [2] it is shown that there is a Kawamata covering of $p:(Y,\tilde{D})\to(X,D)$ with suitable ramification data and Galois group $\Gamma$. It is also shown that there is a $\Gamma$-equivariant vector bundle $V$ on $Y$ such that the invariant direct image $p^{\Gamma}_{{}_{}}(V)=E$ which also recovers the filtrations $\mathcal{F}^{s}_{{}_{j}}$. For the sake of completeness, we give a self-contained ad hoc argument for this construction which is more in the spirit of the present note. The data given in (8.1) is as in [14] which deals with points on curves. Note that in our setting we have rational weights. Under these conditions, the aim is to set up an equivalence $\displaystyle p^{\Gamma}_{{}_{}}:\\{\Gamma-\text{equivariant~{}bundles~{}on}~{}Y\\}\to\\{\text{parabolic~{}bundles~{}on}~{}X\\}$ (8.0.3) and as the notation suggests, this is achieved by taking invariant direct images. Since $p:Y\to X$ is finite and flat, if $V$ is locally free on $Y$ then so is $p^{\Gamma}_{{}_{}}(V)$. An equivariant bundle $V$ on $Y$ is defined in an analytic neighbourhood of a generic point $\zeta$ of a component by a representation of the isotropy group $\Gamma_{{}_{\zeta}}$ and locally (in the analytic sense) the action of $\Gamma_{{}_{\zeta}}$ is the product action. This is called the local type in [15] or [14]. By appealing to the $1$-dimensional case, we get canonical parabolic structures on $E$ at the generic points $\xi$ of the components $D_{{}_{j}}$ in $X$. This construction is an equivalence. Given a parabolic bundle $E$ on $(X,D)$, to get the $\Gamma$-equivariant bundle $V$ on $Y$, such that $p^{\Gamma}_{{}_{}}(V)=E$, we proceed as follows. If we did know the existence of such a $V$ then we can consider the inclusion $p^{}(E)\subset V$. Taking its dual (and since $V/p^{}(E)$ is torsion, taking duals is an inclusion): $\displaystyle V^{}\hookrightarrow(p^{}(E))^{}=p^{}(E^{})$ (8.0.4) By the $1$-dimensional case (obtained by restricting to the height $1$-primes at the generic points) we see that the quotient $T_{{}_{\zeta}}:=p^{}(E^{})_{{}_{\zeta}}/V_{{}_{\zeta}}^{}$ is a torsion $\mathcal{O}_{{}_{Y,\zeta}}$-module and $T_{{}_{\zeta}}$ is completely determined by the parabolic structure on $E$. The parabolic structure on $E$ therefore determines canonical quotients: $\displaystyle p^{}(E^{})_{{}_{\zeta}}\to T_{{}_{\zeta}}$ (8.0.5) for each generic point $\zeta$ of components in $Y$ above the components $D_{{}_{j}}$ in $X$. The discussion above suggest how one would construct such a $V$; we begin with these quotients (8.0.5). Then we observe that there is a maximal coherent subsheaf $V^{\prime}\subset p^{}(E^{})$ such that $\displaystyle p^{}(E^{})_{{}_{\zeta}}/V_{{}_{\zeta}}^{\prime}=T_{{}_{\zeta}}.$ (8.0.6) Away from $p^{{}^{-1}}(D)$ in $Y$, the inclusion $V^{\prime}\hookrightarrow p^{}(E^{})$ is an isomorphism. Dualizing again, we get an inclusion $p^{}(E)\hookrightarrow(V^{\prime})^{{}^{}}$ which is an isomorphism away from $p^{{}^{-1}}(D)$. Set $W=(V^{\prime})^{{}^{}}$. Since $W$ is the dual of a coherent sheaf, it is reflexive. It is not hard to check that $W$ coincides with the vector bundle $V$ that Biswas constructs in codimension $2$ and hence everywhere by [12, Proposition 1.6, page 126]. This gives us the desired equivalence (8.0.3). #### 8.0.1. The group scheme situation Let $(X,D)$ be as above. Let $\xi$ be the generic point of a component $D_{{}_{j}}$ of $D$ and let $A:=\mathcal{O}_{{}_{X,\xi}}$ and $K$ be the quotient field of $A$. Let $\mathcal{G}_{{}_{\theta}}$ be a Bruhat-Tits group scheme on ${\rm Spec}\,(A)$ associated to a vertex $\theta$ of the Weyl alcove. We always assume that these group scheme are generically split. Let $B=\mathcal{O}_{{}_{Y,\zeta}}$ where $\zeta$ is the generic point of a component of $Y$ above $D_{{}_{j}}$ and let $L$ be the quotient field of $B$. We assume that the local ramification data for the Kawamata covering has numbers $d_{{}_{\alpha}}$ (2.0.3). Let $\Gamma_{{}_{\zeta}}$ be the stabilizer of $\Gamma$ at $\zeta\in Y$. The results of [1, Proposition 5.1.2 and Remark 2.3.3] show that there exists an equivariant group scheme $\mathcal{H}_{{}_{B}}$ on ${\rm Spec}\,(B)$ with fiber isomorphic to the simple connected group $G$ and such that $\text{Res}_{{}_{B/A}}(\mathcal{H}_{{}_{B}})^{{}^{\Gamma_{{}_{\zeta}}}}\simeq\mathcal{G}_{{}_{\theta}}$. Suppose that we have a group scheme $\mathcal{G}_{{}_{X^{\prime}}}$ on an open $X^{\prime}\subset X$ which includes all the height $1$ primes coming from the divisors $D_{{}_{j}}$’s with the following properties: • Away from the divisor $D\subset X$, $\mathcal{G}$ is the constant group scheme with fiber $G$. * • The restrictions $\mathcal{G}\mid_{{}_{{\rm Spec}\,(A)}}$ at the generic points $\xi$ are isomorphic to the Bruhat-Tits group scheme $\mathcal{G}_{{}_{\theta}}$ for varying $\xi$ and $\theta$. In other words, $\mathcal{G}_{{}_{X^{\prime}}}$ is obtained by a gluing of the constant group schemes with $\mathcal{G}_{{}_{\theta}}$’s along ${\rm Spec}\,(K)$ by an automorphism of constant group scheme $G_{{}_{K}}$. Now consider the inverse image of the constant group scheme $p^{{}^{}}(G_{{}_{X-D}})\simeq G\times p^{}(X-D)$. Then using the gluing on $X^{\prime}$, we can glue the constant group scheme $p^{{}^{}}(G_{{}_{X-D}})$ with the local group schemes $\mathcal{H}_{{}_{B}}$ for each generic point $\zeta$ to obtain a group scheme $\mathcal{H}_{{}_{Y^{\prime}}}$ on $Y^{\prime}=p^{{}^{-1}}(X^{\prime})$ such that: $\displaystyle\text{Res}_{{}_{Y^{\prime}/X^{\prime}}}(\mathcal{H}_{{}_{Y^{\prime}}})^{{}^{\Gamma}}\simeq\mathcal{G}_{{}_{X^{\prime}}}.$ (8.0.7) #### 8.0.2. Kawamata Coverings Let $X$ be a smooth quasi-projective variety and let $D=\sum_{{}_{i=0}}^{{}^{\ell}}D_{i}$ be the decomposition of the simple or reduced normal crossing divisor $D$ into its smooth components (intersecting transversally). The “Covering Lemma” of Y. Kawamata (see [18, Lemma 2.5, page 56]) says that, given positive integers $n_{{}_{0}},\ldots,n_{{}_{\ell}}$, there is a connected smooth quasi-projective variety $Z$ over $\mathbb{C}$ and a Galois covering morphism $\displaystyle\kappa:Z\to X$ (8.0.8) such that the reduced divisor $\kappa^{{}^{}}{D}:=\,({\kappa}^{{}^{}}D)_{{}_{\text{red}}}$ is a normal crossing divisor on $Z$ and furthermore, ${\kappa}^{{}^{}}D_{{}_{i}}=n_{{}_{i}}.({\kappa}^{{}^{}}D_{i})_{{}_{\text{red}}}$. Let $\Gamma$ denote the Galois group for the covering map $\kappa$. The isotropy group of any point $z\in Z$, for the action of $\Gamma$ on $Z$, will be denoted by ${\Gamma}_{{}_{z}}$. It is easy to see that the stabilizer at generic points of the irreducible components of $(\kappa^{{}^{}}D_{i})_{{}_{\text{red}}}$ are cyclic of order $n_{{}_{i}}$. ## References * [1] V.Balaji and C.S. Seshadri, Moduli of parahoric $\mathcal{G}$–torsors on a compact Riemann surface, J. Algebraic Geometry, 24, (2015),1-49. * [2] I. 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# Graphical Models for Financial Time Series and Portfolio Selection Ni Zhan<EMAIL_ADDRESS>, Yijia Sun<EMAIL_ADDRESS>, Aman Jakhar and He Liu Carnegie Mellon University (2020) ###### Abstract. We examine a variety of graphical models to construct optimal portfolios. Graphical models such as PCA-KMeans, autoencoders, dynamic clustering, and structural learning can capture the time varying patterns in the covariance matrix and allow the creation of an optimal and robust portfolio. We compared the resulting portfolios from the different models with baseline methods. In many cases our graphical strategies generated steadily increasing returns with low risk and outgrew the S&P 500 index. This work suggests that graphical models can effectively learn the temporal dependencies in time series data and are proved useful in asset management. dynamic clustering, portfolio selection, autoencoders ††journalyear: 2020††copyright: rightsretained††conference: ACM International Conference on AI in Finance; October 15–16, 2020; New York, NY, USA††booktitle: ACM International Conference on AI in Finance (ICAIF ’20), October 15–16, 2020, New York, NY, USA††doi: 10.1145/3383455.3422566††isbn: 978-1-4503-7584-9/20/10 ## 1\. Introduction Portfolio selection is a common problem in finance. In general, investors wish to maximize returns while minimizing risk. Markowitz theory suggests that portfolio diversification minimizes risk and the optimal portfolio takes into account correlated movements across assets. Existing works on porfolio selection use the historical covariance matrix of returns. However, factors such as market index, sector, industry and other stocks and commodities that may affect asset cash flows can result in a high degree of correlation among equities. This causes the historical covariance matrix to be ill conditioned and the optimal portfolio highly sensitive to small changes. Expanding the universe of assets also requires a greater amount of data to estimate the covariance matrix. Furthermore regime changes and the non-stationary nature of financial markets discourage the use of static covariance matrices. From a graph viewpoint, estimating the covariance using historic returns models a fully connected graph between all assets. The fully connected graph appears to be a poor model in reality, and substantially adds to the computational burden and instability of the problem. The goal of this work is to develop graphical models that can capture the time varying patterns in the covariance matrix and reflect the cross-series dynamics at multiple time indices. Using graph inference algorithms and thresholding, we plan to discover and incorporate the factor dependencies in a partially connected graph. Therefore, we leverage graphical models that are able to reflect temporal changes among stocks thus addressing the issue of correlations changing over time. Within a time period, a graph allows selection of independent assets for the portfolio, which should improve robustness of our solution. In order to learn the overall time series features, we use principle component analysis (PCA) and autoencoders to capture the latent space distribution. We employ variational autoencoders with Gaussian and Cauchy priors to model temporal dependencies and reflect multi-scale dynamics in the latent space. Additionally, we construct a sequence of graphical models using dynamic clustering techniques and structural learning. We associate a graphical model to each time interval and update the graph when moving to the next time point. We use price data of US equities from the S$\&$P 500 index to construct graphical models to create portfolios, simulate returns, and compare with benchmarks. ## Related Work Various graph methods have been used for the portfolio selection problem. The literature includes many examples of variance-covariance networks that analyze complicated interactions and market structure of financial assets (Ledoit and Wolf, 2003; Buser, 1977; Polson and Tew, 2000). Liu et. al. used elliptical- copula graphical models on stock returns, and used the graphs to select independent stocks (Liu et al., 2015). The paper chose elliptical-copula graph model over Gaussian graphical models because elliptical-copula models better model heavy tail distributions common in finance. The paper shows that graphs modeling asset “independence” can be learned from historical returns and used for effective portfolio strategies, however it did not consider dynamic graphs. Previously, time varying behavior was modeled as dynamic networks whose topology changes with time. Talih and Hengartner proposed a graphical model for sequences of Gaussian random vectors when changes in the underlying graph occur at random times, mimicking the time varying relationships among collection of portfolios (Talih and Hengartner, 2005). Time series data is separated into a pre-determined number of blocks. The sample precision matrix estimated within each block serves as the foundation to construct the time- varying sequences of graphs which arises in blocks as shown in Fig. 1. Figure 1. Blocks of Gaussian graph sequences Blocks of Gaussian graph sequences. The graphs $G_{b}$ and $G_{b+1}$ in successive blocks differ only by the addition or deletion of at most one edge. Markov Chain Monte Carlo is used to recover the segmented time varying Gaussian graphical sequences. Experiments using this model were done to estimate portfolio return in five U.S. industries. However, this paper assumed that the total number of distinct networks are known a priori and the network evolution is restricted to changing at most a single edge at a time. Additionally, dimensionality increases when temporal resolution is small, imposing significant computational burden. As an extension to dynamic dependence networks, Isogai (Isogai, 2017) proposed a novel approach to analyze a dynamic correlation network of highly volatile financial asset returns by using network clustering algorithms to mitigate high dimensionality. Two types of network clustering algorithms (Isogai, 2014, 2016) were employed to transform correlation network of individual portfolio returns into a correlation network of group based returns. Groups of correlation networks were further clustered into only three representative networks by clustering along the time axis to summarize information on the intertemporal differences in the correlation structure. A case study was conducted on Japanese stock dataset. Other studies used autoencoders to dimensionally reduce stock returns (Gu et al., 2019; Heaton et al., 2016). ## Methods The primary objective of this work is to exploit a collection of graphical models to analyze dynamic dependencies among stocks and aid trading strategies. We would like to have a better understanding of the stock features by learning the latent space of stock time series. With a good latent space representation, stocks with similar features will fall into the same cluster. To capture the time-varying correlation among stocks, we divide the overall time horizon into multiple time intervals, zoom in to each interval and construct a local graphical model, and update the local connectivities as we move on to the next temporal period. In each time interval, we aim to identify a suitable number of stock clusters based on the graphical model and develop a portfolio selection strategy. After selecting our portfolios, we test their performance using a backtest simulation and market data. We employ a few different clustering and structural learning techniques, including PCA, autoencoders, agglomerative clustering, affinity propagation clustering, and graphical LASSO, to create portfolios that maximize return and minimize risk. #### Dataset We used price data of US equities from tech, financials, and energy sectors of the S$\&$P 500 index. The data matrix has available stocks as data observations and daily returns as features, i.e. the data matrix has shape [number of stocks, number of daily returns]. Data was obtained on a closing basis at a daily frequency for January 1st, 2012 until January 1st, 2020. We excluded stocks which had missing data for this time period. We used adjusted close to account for splits, dividends etc. The daily closing price data for the stocks and other financial variables was obtained using yahoo finance for python. #### PCA We used empirical returns of one year prior as training data for the simulated year. We used PCA to dimensionally reduce the data matrix of the stocks to three components. We inverse transformed stock representations in reduced space to full data, and calculated the L-2 norm of difference between actual data and recovered data for each stock. The ten stocks with the largest difference in L-2 norm were selected for ”Max Difference” portfolio while ten stocks with the smallest difference were selected for ”Min Difference” portfolio. PCA extracts information about the stock returns, and stocks with large difference between recovered and actual data indicate unexpected or ”difficult to capture” behavior. #### Autoencoders To extend the dimension reduction method and capture more complex interactions, we tested two autoencoders. The observed variables $\mathbf{x}$ for the autoencoders are empirical returns of the selected stocks. The latent space $\mathbf{z}$ has two dimensions. The variational distribution $q_{\phi}(\mathbf{z}\|\mathbf{x})$ approximating $p(\mathbf{z}\|\mathbf{x})$ is assumed Normal for both autoencoders. The likelihood $p_{\theta}(\mathbf{x}\|\mathbf{z})$ is assumed Normal for one autoencoder, and Cauchy for the other. Specifically, for the Normal autoencoder, $p_{\theta}(\mathbf{x}\|\mathbf{z})=N(\mathbf{x};\mu_{\theta}(\mathbf{z}),\Sigma^{2}_{\theta}(\mathbf{z}))$, and for the Cauchy autoencoder, $p_{\theta}(\mathbf{x}\|\mathbf{z})=\text{Cauchy}(\mathbf{x};x_{0,\theta}(\mathbf{z}),\gamma_{\theta}(\mathbf{z}))$. For both autoencoders, the recognition and generative networks are parameterized by neural nets with one hidden layer with 100 reLU neurons and latent space of two dimensions. We chose a Cauchy distribution because stock return distributions usually have heavy tails, and therefore expect the Cauchy autoencoder to have more reliable results. To select portfolios, we found the latent space representation of each stock, and calculated the L-2 norm of difference between real data and generated data (from latent space) for each stock. The ten stocks with max and min L-2 norms were selected for Max and Min Difference portfolios, respectively, similar to the PCA strategy. #### Dynamic clustering and graphical sequences To better capture how stocks move in relation to one another throughout the time period, we utilize clustering and structural learning techniques to create dynamic graph structures. Even though the dataset contains stocks from three sectors, within the same sector, some stocks are more correlated than the others, and stocks from different sectors can also have non-negligible correlations. The goal here is to identify the most suitable number of clusters to assign stocks from three sectors into. Two clustering techniques are adopted here: agglomerative clustering and affinity propagation clustering. Agglomerative clustering divides stocks into a number of clusters according to pair-wise Euclidian distance. It requires the number of clusters to be pre- determined. We used hierarchical clustering with Ward’s minimum variance criterion to produce a dendrogram which in turn is used to determine the number of clusters by drawing a horizontal line and counting the number of vertical lines it intercepts. A total of 15 clusters were used for agglomerative clustering. Affinity propagation (Frey and Dueck, 2007) is a clustering technique that does not require an input number of clusters. It relies on similarity calculation between pairs of data points to determine a subset of representative examples in the dataset. The similarity between two points satisfies that $s(x_{i},x_{j})>s(x_{i},x_{k})$ if and only if $x_{i}$ is more similar to $x_{j}$ than to $x_{k}$. A responsibility matrix $\mathcal{R}$ and availability matrix $\mathcal{A}$ serve as message exchanging paths between data points. Clusters are updated by alternating between the responsibility matrix update and availability matrix update. $\displaystyle r(i,k)$ $\displaystyle=$ $\displaystyle s(i,k)-\max_{k^{\prime}\neq k}\\{a(i,k^{\prime})+s(i,k^{\prime})\\}$ $\displaystyle a(i,k)$ $\displaystyle=$ $\displaystyle\min\big{(}0,r(k,k)+\sum_{i^{\prime}\notin\\{i,k\\}}\max(0,r(i^{\prime},k))\big{)}\;i\neq k$ $\displaystyle a(k,k)$ $\displaystyle=$ $\displaystyle\sum_{i^{\prime}\neq k}\max(0,r(i^{\prime},k))$ This approach is able to identify a high quality set of exemplars and the corresponding clusters with much lower error and lower computational burdens compared to agglomerative clustering. The number of clusters is flexible and updated throughout the time horizon. It is suitable to identify clusters when the data size is large. The data matrix contains daily returns of each stock, and has shape [number of stocks, number of daily returns]. We used spectral embedding on the daily returns to transform the stocks to a 2D plane and reduce dimensionality. An example of the 2D embedding of stocks is shown in Fig. 2. Edge connectivities were created using graphical LASSO with thicker edge indicating stronger correlation. At the beginning of each quarter, we rely on daily returns from the previous quarter, construct a lower-dimensional embedding space, and generate clusters using the before-mentioned clustering techniques. The edge connectivities from graphical LASSO are connectivity input for agglomerative clustering. A new clustering is created at the beginning of each quarter and therefore updated throughout the time horizon quarterly. For agglomerative clustering and affinity propagation portfolios, the portfolio selection strategy is as follows. For each cluster, the top ten stocks with minimum Euclidean distance (in the embedding space) to cluster centers were added to the portfolio. If a cluster had fewer than ten stocks, no stocks were added to the portfolio from that cluster. Because the clusters were created quarterly, the portfolios were also selected quarterly. To benchmark the model performance, we also constructed portfolios using PCA followed by KMeans clustering. KMeans with static clusters as well as quarterly updated clusters were both used as benchmarks. In all cases with KMeans, we used a fixed number of clusters (ten clusters), and the stock with the minimum Euclidean distance to each cluster center was part of the portfolio. Therefore KMeans portfolios had ten total stocks. Figure 2. An example of connected graphs from dynamic clustering An example of connected graphs from dynamic clustering. ### Testing Frameworks We used portfolio rebalancing strategy to test the performance of stock selection. This method rebalances the portfolio at some frequency (we used monthly). The weights across the portfolio’s stocks are either equal weights or determined from mean-variance optimization (described below). The portfolio is initialized with weights of the selected stocks. At each rebalance time point, shares are bought or sold to renormalize the dollar amounts by weight across the stocks. We compared the PCA and autoencoder portfolio selection strategy with rebalancing and buying and holding the S&P 500. Metrics to evaluate strategies included total returns, daily return standard deviations, and Sharpe ratios across simulation time-periods. Note that high Sharpe is desirable and indicates high returns with lower risk. We followed the equal weight rebalancing strategy to test the performance of dynamic clustering, comparing affinity propagation, agglomerative clustering, PCA KMeans portfolios and the S&P 500. At the end of each quarter, stock clusters were updated by selling the existing portfolio and buying stocks from the new cluster with equal dollar amounts. #### Efficient frontier weights Efficient frontier weights were determined using PyPortfolioOpt, with expected means and covariance calculated from empirical returns of the year prior to simulation. The solver optimized for maximum Sharpe, and stock weights were unconstrained between 0 to 1. ## Results ### PCA, Autoencoder Portfolios The two autoencoders were trained with Auto-encoding Variational Bayes (AEVB) that optimizes a stochastic estimate of evidence lower bound objective (ELBO) (Kingma and Welling, 2013). The autoencoders were trained for over 200 epochs, and the lower bound of log-likelihood converged. The training for each year was repeated ten times as the training is stochastic. The max and min difference portfolios across the ten trainings were aggregated per year, and the ten stocks which appeared most frequently in a year were used for simulating the following year. The PCA and autoencoders portfolio selection and monthly rebalance strategy was implemented for five simulation years, 2014-2018 inclusive. We rebalanced using both equal weights and efficient frontier weights. To compare against simpler methods, we constructed additional portfolios: volatility (Vol) and ”average return over volatility” (AvgRet/Vol). The max and min Vol portfolios had ten stocks with highest or lowest standard deviation of returns, respectively, in year prior to simulation. A stock selected for Max PCA or autoencoder portfolio has a large error in its model representation, and may have high volatility. Therefore we wanted to compare volatility alone with PCA and autoencoder portfolios. The ”average return over volatility” represents a proxy for Sharpe ratio, and we wanted to test if individual stocks with high Sharpe combined would have good portfolio performance. Table 1 shows the simulation results: yearly returns (%) and standard deviation of daily returns (%). Daily return standard deviations are higher for max portfolios than min portfolios for PCA, autoencoder, and Vol, which is expected based on their construction. Table 1 also includes the average return and Sharpe ratio over the five simulation years. Risk free return in Sharpe ratio was one-year Treasury yield averaged for that year. Results for 2019 are reported as a forward-test (the data was completely withheld prior to reporting), and holding S&P 500 is shown for comparison. From Table 1, there are several observations which show the PCA and autoencoder strategies are useful in portfolio selection and able to create portfolios with higher return at lower risk, more so than volatility or AvgRet/Vol alone. The Max PCA and Max autoencoder portfolios perform better or on-par with Max Vol in terms of yearly return and Sharpe, from 2014-2018. For PCA and autoencoder, Max outperforms Min in both Sharpe and average returns, which is not the case with Vol. Specifically Min Vol has higher Sharpe but lower average returns than Max Vol. The Max AvgRet/Vol portfolio has higher Sharpe but lower average return than the other portfolios, and is not always able to obtain an optimal weights solution. Therefore, the model-based strategies aid in choosing portfolios which have better returns, Sharpe, and weights optimization capability, over Vol or AvgRet/Vol alone. Other observations are that PCA seems to have lower risk than autoencoder. This is likely because PCA is a simpler model and stochasticity was introduced in autoencoder training. Improvements to the autoencoder model can be considered for future work. For some returns such as 309% and 142%, the efficient frontier allocated all assets into one stock when given ten stocks. Picking one outperforming stock can give extraordinarily high returns compared with selecting multiple stocks for a portfolio. In the cases with 309% and 142% returns, the efficient frontier optimizer selected only one stock because portfolio weight allocations were unconstrained between 0 and 1. Future work can include constrained weight allocation for the efficient frontier portfolios. In 2018, the general market trended down, and Min Vol portfolio performed the best, while years 2014 through 2017 were bull markets. The best portfolio selection strategy is likely different depending on the overall market trend. It would be interesting to consider optimal strategies for different market trends and transitions. Table 1 included Normal autoencoder results only because the Cauchy autoencoder had similar results. We examined the latent space representations of the stocks. Between PCA and the two autoencoders, the Cauchy autoencoder had the best performance in separating stocks by sector, shown in Fig 3. This shows the model learned relevant information about the stocks. We find it quite remarkable that the autoencoder clustered the sectors considering the only training data was daily stock return for a year. It may show that stock returns are quite correlated within sectors. The strategies commonly selected some stocks while others were more specific to certain models. For example, for the Min portfolios, BRK-B, USB were common across Vol, PCA, and autoencoders. AFL was more often selected by Cauchy autoencoder, MMC by Vol, BLK by Normal autoencoder. For the Max portfolios, AMD, MU were common across Vol, PCA, and autoencoders, while AKAM was more specifically selected by PCA. The fact that some stocks were common across all models shows that models learned relevant information, while some stocks specifically selected by certain models shows that models learned distinct information. In addition, Max portfolio stocks exhibited ”outlier” behavior. Table 1. PCA, Autoencoder, Volatility Portfolio returns and standard deviations (%) by simulation year \| \| \| 2014 \| 2015 \| 2016 \| 2017 \| 2018 \| Avg Yr Ret \| Daily Ret Std \| Sharpe \| 2019 Test ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- Equal Weights \| PCA \| Max \| 47.7 \| 10.4 \| 65.4 \| 34.2 \| -3.1 \| 30.9 \| 1.39 \| 1.19 \| 36.5 \| Min \| 10.7 \| 1.69 \| 25.7 \| 19.6 \| -16.8 \| 8.2 \| 0.94 \| 0.48 \| 32.3 AutoEnc \| Max \| 29.8 \| 3.85 \| 75.8 \| 25.3 \| -5.02 \| 26.0 \| 1.5 \| 0.88 \| 28.2 \| Min \| 12.9 \| -1.66 \| 23.5 \| 16.0 \| -20.7 \| 6.0 \| 1.05 \| 0.31 \| 33.4 Vol \| Max \| 46.0 \| -0.86 \| 74.4 \| 18.9 \| -5.52 \| 26.6 \| 1.57 \| 0.84 \| 37.4 \| Min \| 5.54 \| 2.64 \| 21.7 \| 22.8 \| 3.54 \| 11.2 \| 0.79 \| 1.13 \| 31.2 AvgRet/ Vol \| Max \| 16.9 \| 6.29 \| 21.3 \| 24.3 \| 0.72 \| 13.9 \| 1.12 \| 1.39 \| 40.6 Min \| 12.5 \| -18.5 \| 33.5 \| 38.9 \| -22.7 \| 8.7 \| 1.27 \| 0.30 \| 2.0 Eff Frontier \| PCA \| Max \| 60.5 \| 19.5 \| 25.9 \| 65.5 \| -14.1 \| 31.5 \| 1.94 \| 1.03 \| 142 \| Min \| 22.3 \| -6.12 \| 27.6 \| 25.8 \| -9.66 \| 12.0 \| 1.05 \| 0.66 \| 44.2 AutoEnc \| Max \| 58.4 \| 19.5 \| 309 \| 65.5 \| -13.4 \| 87.8 \| 2.66 \| 0.76 \| 141 \| Min \| 15.9 \| 2.11 \| 35.4 \| 28.1 \| -10.2 \| 14.3 \| 1.18 \| 0.78 \| 11.6 Vol \| Max \| 57.7 \| 19.5 \| 25.9 \| 20.4 \| -13.3 \| 22.0 \| 2.0 \| 0.91 \| 142 \| Min \| 11.6 \| -1.93 \| 28.6 \| 15.2 \| 3.49 \| 11.4 \| 0.89 \| 0.98 \| 27.7 S&P \| 14.4 \| 1.29 \| 13.6 \| 20.7 \| -5.13 \| 9.0 \| 0.79 \| 0.82 \| 30.4 Figure 3. Latent space representation from a trained Cauchy autoencoder Latent space representation from a trained Cauchy autoencoder groups sectors together. ### Dynamic Clustering We used trading data from January 1st, 2012 to January 1st, 2019 to evaluate the performance of the model. On the first day of each quarter, clusters are formed following agglomerative and affinity propagation clustering techniques using daily closing price from the previous quarter, and new clusters are created each quarter. A portfolio is created by selecting stocks with minimum Euclidean distance to cluster centers in the lower-dimensional transformed space. We also generated two portfolios using PCA with KMeans clustering, with fixed clusters and quarterly updated clusters respectively, as our benchmark portfolios. Figure 4 compares the various portfolios using monthly rebalance strategy and holding the S&P 500. Both KMeans cluster portfolios lead to comparable performance with the S&P 500. KMeans with dynamic cluster update performs better than using the same clusters throughout the time horizon. Both agglomerative clustering and affinity propagation outperformed S&P 500 index, with affinity propagation generating the highest returns throughout seven years. Since affinity propagation identifies the most suitable number of clusters for the given data set, the resulting cluster size can differ from what is used in agglomerative and KMeans clustering. The likelihood of a stock being assigned to the correct cluster is higher for affinity propagation, therefore stocks that constantly beat the S&P 500 are likely to be assigned to the same cluster and correspond to minimum distance in the spectral embedding space. Creating portfolios using such stocks steadily outgrows the performance of S&P 500. We found that both cluster update and cluster size can influence the quality of the portfolio. Involving temporal changes to the clusters invariably boost the performance. Figure 4. Rebalancing with different clustering strategies vs. S&P 500 from 2012 to 2018 Affinity propogation dynamic clusters outperforms other strategies and benchmarks in simulation test from 2012 to 2018. ## Conclusion We showed that graphical models learn useful information and correlation between stocks only based on their returns. We developed a portfolio selection using PCA and autoencoder models and rebalance strategy that selects high return, low risk portfolios. We also explored the effect of dynamic clustering on overall portfolio returns. We observed that dynamic cluster update yields higher returns than using fixed clusters. A flexible cluster size also improves the performance than using a constant cluster size. When stocks are assigned to the correct clusters throughout the time horizon, with rebalancing strategies that minimizes risk, we are able to create a portfolio with steadily increasing returns. For future work, we can include more data into the analysis and model training, such as using trading volume, expanding number of years and stock sectors. There are other experimental factors which can be varied such as dimensions in latent space, stock selection strategy, rebalance frequency and timing. There are many questions to explore, and this work shows that graphical models have interesting and useful applications in asset management. ## References * (1) * Buser (1977) Stephen A. Buser. 1977\. Mean-Variance Portfolio Selection with Either a Singular or Nonsingular Variance-Covariance Matrix. _Journal of Financial and Quantitative Analysis_ 12, 3 (1977), 347–361. * Frey and Dueck (2007) Brendan J. Frey and Delbert Dueck. 2007. Clustering by Passing Messages Between Data Points. _Science_ 315, 5814 (2007), 972–976. * Gu et al. (2019) Shihao Gu, Bryan T. Kelly, and Dacheng Xiu. 2019\. 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11institutetext: Laboratory for Atmospheric and Space Physics, University of Colorado at Boulder, 3665 Discovery Drive, Boulder, CO, USA 11email<EMAIL_ADDRESS>22institutetext: NOAA/National Centers for Environmental Information, 325 Broadway, Boulder, CO, USA 33institutetext: High Altitude Observatory, National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO, USA 44institutetext: Naval Research Laboratory, Washington, DC, USA 55institutetext: Colorado Research Associates Division, NorthWest Research Associates, 3380 Mitchell Lane, Boulder, CO, USA 66institutetext: NASA Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD, USA 77institutetext: Institute of Physics & Kanzelhöhe Observatory for Solar and Environmental Research, University of Graz, A-8010 Graz, Austria 88institutetext: Royal Observatory of Belgium, Avenue Circulaire 3, 1180 Uccle, Belgium 99institutetext: Reflective X-ray Optics LLC, New York, NY, USA # SunCET: The Sun Coronal Ejection Tracker Concept James Paul Mason 11 Phillip C. Chamberlin 11 Daniel Seaton 22 Joan Burkepile 33 Robin Colaninno 44 Karin Dissauer 55 Francis G. Eparvier 11 Yuhong Fan 33 Sarah Gibson 33 Andrew R. Jones 11 Christina Kay 66 Michael Kirk 66 Richard Kohnert 11 W. Dean Pesnell 66 Barbara J. Thompson 66 Astrid M. Veronig 77 Matthew J West 88 David Windt 99 Thomas N. Woods 11 ###### Abstract The Sun Coronal Ejection Tracker (SunCET) is an extreme ultraviolet imager and spectrograph instrument concept for tracking coronal mass ejections through the region where they experience the majority of their acceleration: the difficult-to-observe middle corona. It contains a wide field of view (0–4 $R_{\odot}$) imager and a 1 Å spectral-resolution-irradiance spectrograph spanning 170–340 Å. It leverages new detector technology to read out different areas of the detector with different integration times, resulting in what we call “simultaneous high dynamic range”, as opposed to the traditional high dynamic range camera technique of subsequent full-frame images that are then combined in post-processing. This allows us to image the bright solar disk with short integration time, the middle corona with a long integration time, and the spectra with their own, independent integration time. Thus, SunCET does not require the use of an opaque or filtered occulter. SunCET is also compact — $\sim$15 $\times$ 15 $\times$ 10 cm in volume — making it an ideal instrument for a CubeSat or a small, complementary addition to a larger mission. Indeed, SunCET is presently in a NASA-funded, competitive Phase A as a CubeSat and has also been proposed to NASA as an instrument onboard a 184 kg Mission of Opportunity. ###### keywords: EUV instrument – Coronal Mass Ejections – high dynamic range – CubeSat ## 1 Introduction and Science Drivers The primary science question that the Sun Coronal Ejection Tracker (SunCET) instrument concept is designed to address is: What are the dominant physical mechanisms for coronal mass ejection acceleration as a function of altitude and time? In the standard model configuration of a coronal mass ejection (CME; Figure 1), a CME must overcome the constraint of overlying field in order to escape. Perhaps the simplest model of this defines a 1D, horizontal background magnetic field that declines in strength with height, characterized by the “decay index” (Bateman, 1978; Kliem and Török, 2006). If the background field decays too rapidly, the so-called torus instability of the embedded flux rope occurs, meaning the flux rope erupts. The decay index has a direct impact on the CME kinematics. The acceleration curves in the bottom of Figure 2, derived from magnetohydrodynamic (MHD) simulations by Török and Kliem (2007), correspond to decay index profiles, with each increase in acceleration corresponding to an increase in in the decay index profile and the final CME speed. Thus, the acceleration profile of a CME acts as a natural probe of the surrounding magnetic field. There are many complications layered on top of this simple model in reality, described later in this introduction. Figure 1: Standard cartoon CME model. The flux rope extends through the page. Overlying fields resist the flux rope’s elevation and expansion. Magnetic reconnection releases the energy stored in the field to accelerate the flux rope, producing a CME. Adapted from Forbes et al. (2018). The bulk of the CME acceleration profile in all cases occurs either in the observational gap or in the region where existing instruments are not optimized. This gap exists between extreme ultraviolet (EUV) imagers (widest outer field of view [FOV] of 1.7 $R_{\odot}$) and coronagraphs (typical inner FOV of 2.5 $R_{\odot}$ but effectively higher due to diffraction-degraded spatial resolution). Some instruments observe only part of the low-middle corona (Solar TErrestrial RElations Observatory [STEREO; Kaiser et al. 2007] / Coronagraph-1 [COR1; Howard et al. 2008], Geostationary Operational Environmental Satellite [GOES] / Solar Ultraviolet Imager [SUVI; Martínez- Galarce et al. 2010] , Project for On-Board Autonomy [PROBA2] / Sun Watcher with Active Pixels and Image Processing [SWAP; Seaton et al. 2013]). Some have low signal to noise in the middle corona (SUVI, SWAP). Some are ground-based with duty cycles $<20$% (K-Cor). Some have limitations on cadence (COR1). SunCET, however, avoids all of these issues because it is specifically optimized for this study of CMEs. Directly observing the CME height-time profile through the whole low and middle corona allows the derivation of complete speed-time and acceleration-time profiles, and thus strong model constraints, requiring accurate modeling of the magnetic environment to obtain the observed profiles. Such constraints do not presently exist, but SunCET can provide them. Figure 2: Top: Composite of SDO/AIA 171 Å image and SOHO/LASCO/C2 white-light coronagraph image. The longstanding observational gap is shown in dark grey. Bottom: Modeled acceleration profiles of torus instability CMEs, adapted from Török and Kliem (2007) Fig. 3. The different curves result from different background magnetic field decay index profile assumptions, with each higher acceleration peak corresponding to a larger decay index profile. Most of the acceleration occurs in the observational gap that SunCET fills. The torus instability is not the only mechanism involved in CME eruptions. Complicating factors are introduced by, e.g., the 3D structure of the erupting material and the surrounding magnetic field, by potential drainage of dense plasma, and by continued magnetic reconnection freeing more energy to drive the CME. The influence of these factors also evolve with altitude and time, as the CME dynamics play out. There have been at least 26 review papers on the topic over the last two decades (Green et al. 2018, and references therein) — a testament to the sustained, intense interest in this topic. Figure 3: Simulated CME kinematic profiles. Solid lines indicate the unperturbed torus instability. Dashed lines from right to left correspond to increasing durations (6 $\tau_{A}$ up to 10$\tau_{A}$) of an upward, linearly rising velocity perturbation, resulting in fundamentally different acceleration profiles. The SunCET FOV (0–4 $R_{\odot}$; indicated in light blue) covers and extends beyond this simulation. Adapted from Schrijver et al. (2008) Fig. 7. For example, a relatively modest complication to layer into the torus instability model is to add an upward velocity perturbation with finite duration. MHD simulations by Schrijver et al. (2008) showed that simply changing the duration of this perturbation results in fundamentally different acceleration profiles (Figure 3). With brief perturbations, the profile is single-peaked and occurs at later times. Increasing the duration of the perturbation does not simply result in an earlier peak, but in two peaks. Just as in Figure 2, the heights that these acceleration profiles differentiate themselves occurs across the Heliophysics System Observatory (HSO) measurement gap. SunCET observations can discriminate between single-peak versus double- peak CME acceleration profiles, which then determines the duration of a velocity perturbation in the torus instability model. Another CME initiation mechanism arises from the magnetic field topology of the flux rope. Hood and Priest (1981) showed that if the total twist in a flux rope exceeds a critical threshold (448°), a “helical kink” instability will occur, causing the flux rope to erupt. Such contortions lead to an impulsive acceleration and a large rotation of the flux rope (Fan 2016, Figure 4). Note the substantial differences in the simulated acceleration profiles between Figures 2, 3, and 4; and that they all occur in the under observed region. Figure 4: MFE simulation containing the helical kink instability, resulting in impulsive CME acceleration. The SunCET FOV (0–4 $R_{\odot}$; indicated in light blue) captures the impulse and small jerks. Adapted from Fan (2016). The other aspect of acceleration is direction: CMEs can be deflected away from “pure” radial propagation by as much as $\sim$30$\degree$, which is again determined primarily by Bex (Figure 5). This force has a non-radial component because the field is not perfectly symmetric about the flux rope, causing a magnetic gradient on the CME’s sides as the loops drape around the rising CME. The Forecasting a Coronal Mass Ejection’s Altered Trajectory (ForeCAT) analytical model accounts for these and other forces on a CME to determine its non-radial velocity (Kay et al., 2013, 2015, 2016; Kay and Gopalswamy, 2018). Furthermore, Kay and Opher (2015) modeled 200 CMEs in ForeCAT and found that deflection occurring in the middle corona accounts for nearly all of the deflection that occurs between initiation and 1 AU. The background magnetic field and radial CME speed are two free parameters in ForeCAT that are critical to get right; SunCET observations can strictly constrain them via forward modeling. Figure 5: ForeCAT simulations of a CME propagating through background magnetic fields (PFSS) of various strengths. R is radial distance. CMEs experience greater non-radial velocity in middle corona environments with stronger magnetic fields. The SunCET FOV (0–4 $R_{\odot}$; indicated in light blue) captures the majority of CME deflection. Adapted from Kay (2016). Additionally, coronal dimming often occurs as a result of CMEs. The faster a CME departs, the steeper the decline in coronal emission. The more mass the CME takes with it, the deeper the drop in coronal emission. A large number of studies have demonstrated this link with coronal imagers (e.g., Aschwanden 2009; Aschwanden et al. 2009; Dissauer et al. 2018, 2019; Thompson et al. 2000) and with spectral irradiance data (Woods et al., 2011; Mason et al., 2014, 2016, 2019). A major advantage of dimming measurements is that they are effective measures of CME kinematics even when they occur at disk center. Coronagraphs and imagers suffer from the problem of determining halo CME speed and/or mass. Dimming is an effective measure of CME kinematics both on and off-disk (Dissauer et al., 2019; Chikunova et al., 2020). Thus, instrument suites that can capture both the dimming and direct observations of limb CMEs are ideal for CME observation. This is precisely what SunCET does. SunCET will be the first mission that allows continuous measurements of CMEs during their initial acceleration phase using only a single instrument. This is advantageous compared to currently used instruments, where, e.g. EUV imagers in the low corona are combined with white-light coronagraphs higher up to track this phase. Artifacts can be introduced in the resulting CME kinematics using this combined data due to the tracking of different structures in the different instruments, since the observed emission is generated by different physical processes. SunCET is not dependent on other instruments to observe CME initiation and acceleration but does have a sufficiently wide field of view to overlap with coronagraphs for further expanded studies. The same challenges with different CME structures in EUV versus white light will be present, but SunCET’s broader temperature response should mitigate this somewhat. ## 2 Instrument Design SunCET is an instrument with a Ritchey-Chrétien, wide-field-of-view telescope (4 $R_{\odot}$), an off-rowland-circle EUV spectrograph, and a novel, simultaneous-high-dynamic-range detector. This new detector technology allows us to image the bright solar disk and CMEs through the dim middle corona simultaneously. It also allows us to measure solar irradiance spectra on the unused portion of the same detector with an integration time independent of the telescope image. The entire design is compact, fitting in a $\sim$15 $\times$ 15 $\times$ 10 cm volume; or about 2.5 CubeSat Units. This makes it ideal as a CubeSat or as a compact instrument suite to include on larger spacecraft that requires few physical resources. SunCET observes in the EUV rather than white light because 1) CMEs have already been demonstrated to be visible in the EUV and 2) it allows for major simplifications in the technical design of the instrument. While white-light observations are independent of temperature since they rely on light Thomson scattered from free electrons, SunCET observations do have the caveat that their temperature dependence (emission from ions at particular temperatures) means that CMEs whose plasma is not at ambient coronal temperatures will not be visible. The dynamic range between on and off disk in the EUV is already large ($\sim$105 by 2 $R_{\odot}$) but this is orders of magnitude larger in white light ($\sim$108), increasing the technical challenge. Moreover, the absolute brightness are vastly different; there are far more visible light photons. This presents a major challenge with scattered light: even small imperfections in optics would result in enough of the numerous disk photons to land on the part of the detector with the exceptionally faint middle corona, swamping out CME observations. This is further exacerbated by the fact that most surfaces scatter light more efficiently in visible light than in EUV light. Therefore, SunCET observes CMEs in the EUV. ## 3 Imager Design The SunCET imager was designed to provide high-dyamic range with moderate spatial resolution while providing a large field-of-view not heard of in historical on-disk EUV imagers out to 4 $R_{\odot}$. This section describes the technical design details that were traded in order to close on the science question. ### 3.1 Dynamic Range The SunCET imager requires a dynamic range of at least $7\text{\times}{10}^{4}$, based on GOES-16/SUVI observations of CMEs and SunCET’s design optimizations. The dimmest target of interest is a CME at the outer FOV, and the brightest is the coronal loops of an active region associated with a CME. SUVI-observed radiances are used to estimate brightness in SunCET (see Section 3.7). At 3.5 $R_{\odot}$, CMEs are $6.9\text{\times}{10}^{-4}$ W/m2/sr. A few of the brightest pixels in active regions reach $\sim$70 W/m2/sr, but are typically $\sim$4.8 W/m2/sr in SunCET. Another factor of 10 is included to distinguish the loops from the background solar disk. Thus, we have a required dynamic range of (4.8 / $6.9\text{\times}{10}^{-4}$) $\times$ 10 = $7\text{\times}{10}^{4}$. We allow solar flares and a small number of the brightest pixels inside active regions to saturate because 1) they are not our target of interest, 2) our entrance filter mesh mitigates diffraction (Section 3.4), and 3) the blooming in our detector is modest: only a few percent ranging across a few pixels (verified during the 33.336 NASA sounding rocket flight and in the lab). Projected performance: CME brightness at the outer SunCET FOV of 4 $R_{\odot}$ is $2.1\text{\times}{10}^{-4}$ W/m2/sr. That implies a dynamic range of $2.3\text{\times}{10}^{5}$. From 0–1.05 $R_{\odot}$, we run exposures of 0.025 seconds and from 1.05–4 $R_{\odot}$ the exposures will be 10 seconds — a factor of 400$\times$ dynamic range. Our detector has a native dynamic range of $\sim$$5\text{\times}{10}^{3}$. 2$\times$2 pixel binning provides an additional factor of 4. Combining these, we obtain SunCET’s high dynamic range of $8\text{\times}{10}^{6}$, well above the required range of $7\text{\times}{10}^{4}$. For comparison, the SDO/AIA dynamic range is $1\text{\times}{10}^{4}$ (Lemen et al., 2012). ### 3.2 Field of View Most CMEs accelerate through the low and middle corona (Bein et al., 2011; D’Huys et al., 2014). We set our required minimum field of view (FOV) at 0.5 $R_{\odot}$, corresponding to $\pm$30$\degree$ from disk-center. Lower than this and the events tend to be halo CMEs, which are difficult to obtain height-time profiles from. The outer FOV requirement is set to 3.5 $R_{\odot}$. SunCET covers the gap between existing instruments and includes enough overlap to ensure a smooth transition in any complementary height-time profiles. SOHO/LASCO’s inner FOV is 2.4 $R_{\odot}$ and its upcoming replacement, NOAA’s GOES-U/CCOR and SWFO/CCOR, will have an inner FOV of 3 $R_{\odot}$. The aforementioned traditional CME measurements, which are from white-light coronagraphs, use occulters that are mechanically restricted to be a limited distance away; therefore these observations have significantly degraded spatial resolution in their inner FOV that is much worse than their stated plate-scale resolution, sometimes upwards of 1 arc-min in the inner FOV. These effects are primarily due to vignetting (e.g. Koutchmy 1988; Aime et al. 2019). This is not the case with SunCET as it does not require an occulter to observe the CMEs in the low- and middle-corona, so its spatial resolution is not diffraction limited and is superior even in the FOV region that overlaps with the coronagraphs. Projected performance: The FOV of SunCET is 0–4 $R_{\odot}$ (5.6 $R_{\odot}$ in image corners). ### 3.3 Temporal Resolution: Exposure and Cadence SunCET is required to observe CMEs with speeds up to at least 1000 km/s, which accounts for 98% of all CMEs (Gopalswamy et al., 2009; Barlyaeva et al., 2018). Given the cadence described below and the field of view, SunCET’s projected performance is to observe CMEs with speeds up to 3900 km/s. The fastest CME in the CDAW catalog is $\sim$3400 km/s, meaning that SunCET will be able to track CMEs with any previously observed speed. SunCET requires an exposure time $\leq$23 seconds in order to avoid motion blur of the CME. Combining the fastest required CME to observe (1000 km/s), our required spatial resolution of 30/resolution-element, and the conversion of angular to spatial resolution at 1 AU ($\sim$750 km/arcsec), we obtain 750 $\times$ 30 / 1000 $\approx$ 23 seconds/resolution-element. Projected performance - exposure: SunCET’s exposure times are 0.025 seconds from 0–1.05 $R_{\odot}$ and 10 seconds beyond that. SunCET requires a cadence $\leq$3.2 minutes. SunCET must be able to track a 1000 km/s CME from the solar limb through its FOV, a range of 2.5 $R_{\odot}$, or $1.74\text{\times}{10}^{6}$ km. Therefore, the minimum time a CME would be in the FOV is 29 minutes. We require at least 9 height-time samples to distinguish acceleration profiles (Figure 3). Thus, our cadence must be less than 29 minutes / 9 samples = 3.2 minutes. Projected performance - cadence: The SunCET mission is designed to downlink 1 minute cadence data. The designed FOV actually extends to 4 $R_{\odot}$, meaning we will capture 38 height-time points for limb-CMEs traveling at a speed of 1000 km/s and more points for CMEs that start slightly on disk and/or with slower speeds. For example, the average CME speed is 490 km/s (Webb and Howard, 2012) and if it crosses from 0.7–4 $R_{\odot}$, we will obtain 78 height-time points. ### 3.4 Bandpass: Coatings and Filters Table 1: Strong emission lines in the SunCET bandpass. Irradiance measured by SDO/EVE (Woods et al., 2012) [HTML]1A73C9 Ion \| $\uplambda$ [Å] \| log10(T [K]) \| Quiet Sun Irradiance [$\upmu$W/m2/Å] ---\|---\|---\|--- Fe IX \| 171.1 \| 5.9 \| 67 Fe X \| 174.5 \| 6.1 \| 73 Fe X \| 177.2 \| 6.1 \| 48 Fe XI \| 180.4 \| 6.2 \| 77 \| Fe XI --- (doublet) 188.2 \| 6.2 \| 61 Fe XII \| 193.5 \| 6.2 \| 45 Fe XII \| 195.1 \| 6.2 \| 63 CMEs have been routinely identified in narrowband EUV imagers sensitive to temperatures between $\sim$0.6–1.6 MK (e.g., GOES/SUVI). Therefore, SunCET is required to observe at least one of the emission lines identified in Table 1. Projected performance: SunCET’s baseline bandpass is 170–200 Å — capturing all of the emission lines in Table 1, which boosts the signal (Section 3.7). The telescope mirrors employ reflective multilayer coatings designed to provide broad spectral response spanning the instrument bandpass. These coatings follow an aperiodic design, and comprise 15 repetitions of alternating layers of B4C, Mo, and Al, with individual layer thicknesses ranging from $\sim$5–100 Å. The aperiodic coating design provides an average reflectance of $\sim$33% from 170–200 Å, as shown in Figure 6. For reference, periodic multilayer coatings operating in this portion of the EUV are generally used for narrow- band response: for example, the periodic Si/Mo coatings used for the 195 Å channel of the GOES/SUVI instrument, also shown in Figure 6, achieve a peak reflectance of $\sim$34% with a spectral bandpass of $\sim$9.5 Å full-width- half-max (FWHM). Figure 6 also shows the periodic Al/Zr coatings used for the Hi-C rocket instrument (Kobayashi et al., 2014), which achieve a peak reflectance of $\sim$50% with a spectral bandpass of $\sim$8.5 Å FWHM. The aperiodic B4C/Mo/Al multilayer coatings are currently under development with funding from the NASA H-TIDeS program. Figure 6: Calculated reflectance near normal incidence (5°) of the broad-band, aperiodic B4C/Mo/Al multilayers used for the SunCET telescope mirrors (green), and for reference, the narrow-band, periodic Si/Mo multilayer coatings used for the GOES/SUVI instrument (red), and the Al/Zr multilayer coatings used for the Hi-C rocket instrument (blue). The C/Al/C entrance filter from Luxel Corporation prevents visible light from entering the chamber and has high heritage (24 of them in GOES/EXIS). It is supported by a 5 lines/inch mesh, which has heritage from the Hi-C sounding rocket flights and avoids the diffraction issues of the 70 lines/inch mesh used on SDO/AIA and TRACE (Lemen et al., 2012; Lin et al., 2001). A second C/Al/C filter directly in front of the detector eliminates visible light from possible pinholes in the primary filter or from stray light in the instrument. ### 3.5 Spatial Resolution SunCET requires spatial resolution better than 30. CME flux ropes often manifest observationally as a cavity which trails behind a bright front (Forsyth et al., 2006). The smallest cavities have a diameter of 0.2 $R_{\odot}$ (180) and are approximately circular, which corresponds to a circumference of $\sim$600 (Fuller and Gibson, 2009). To account for non- circularities, we require $\sim$20 points outlining the cavity, which results in our spatial resolution requirement of 600/20 = 30. Figure 7 shows a cavity observed in PROBA2/SWAP (3.16 resolution) binned down to demonstrate that cavities can be resolved at this resolution in practice. Projected performance: SunCET provides 20 resolution. Its plate scale is 4.8/pixel so 2$\times$2 binning can be applied, which meets the Nyquist sampling criterion. Figure 7: CME cavity observed in PROBA2/SWAP 174 Å binned down to SunCET required resolution of 30 (projected performance is 20). The cavity remains easily identifiable. SunCET’s SNR will be 9–30$\times$ higher off disk, making CME identification even easier. The 1.7 $R_{\odot}$ FOV shown here, the largest of any solar EUV imager to date, is SWAP’s; SunCET’s extends to 4 $R_{\odot}$. Adapted from Byrne et al. (2014). ### 3.6 Mirrors Figure 8: SunCET’s compact Ritchey-Chrétien telescope, which fits inside a 6U CubeSat with all typical bus components. SunCET contains a Ritchey-Chrétien (RC) telescope encased in a vacuum chamber with a one-time-release door (Figure 8). This type of telescope has good performance for wide fields of view (Figure 9) and has been used frequently for similar instruments (e.g., SOHO/EIT, STEREO/EUVI, GOES/SUVI). Despite its compact size, the telescope achieves nearly flat resolution across the wide FOV. The mount for the secondary mirror is designed with a coefficient of thermal expansion matching the mirror to account for focus sensitivity. Figure 9: Left: Ray trace of SunCET optics. Right: 80% encircled spot diameter over the FOV. This simple design yields excellent performance, with a mean resolution of 20 that is flat across nearly the entire FOV. ### 3.7 Signal to Noise Ratio (SNR) SunCET requires a signal to noise ratio (SNR) $\geq$10\. This is the international standard that defines digital image quality as “acceptable” (ISO 12232, 2019). The same standard defines SNR of 40 as “excellent”. These numbers are in line with the expectations of experts that have done CME image processing with coronagraph and EUV imager data. Table 2: SunCET SNRs for on-disk features and CME loops above the limb. Radiances are from GOES/SUVI 195 Å images of the 2017-09-10 CME (Seaton and Darnel, 2018) and are extrapolated beyond its FOV of 1.7 $R_{\odot}$. SNR at all heights is above the level that ISO 12232 defines as “excellent”. [HTML]1A73C9 \| Quiet Sun \| Active Region \| Flare \| 1.05 $R_{\odot}$ \| 1.5 $R_{\odot}$ \| 2 $R_{\odot}$ \| 3 $R_{\odot}$ \| 3.5 $R_{\odot}$ \| 4 $R_{\odot}$ ---\|---\|---\|---\|---\|---\|---\|---\|---\|--- \| Radiance --- [W/m2/sr] 0.1 \| 10 \| 40 \| 0.2 \| $1.5\text{\times}{10}^{-2}$ \| $3\text{\times}{10}^{-3}$ \| $3\text{\times}{10}^{-4}$ \| $1\text{\times}{10}^{-4}$ \| $3\text{\times}{10}^{-5}$ Effective exposure [s] \| 0.025 \| 0.025 \| 0.025 \| 0.025 \| 10 \| 10 \| 10 \| 10 \| 10 e-/res-element \| $1.48\text{\times}{10}^{4}$ \| $1.48\text{\times}{10}^{6}$ \| $5.94\text{\times}{10}^{6}$ \| $2.97\text{\times}{10}^{4}$ \| $8.9\text{\times}{10}^{5}$ \| $1.78\text{\times}{10}^{5}$ \| $1.78\text{\times}{10}^{4}$ \| $5.94\text{\times}{10}^{3}$ \| $1.78\text{\times}{10}^{3}$ \| Saturation limit --- [e-/res-element] $1.08\text{\times}{10}^{5}$ \| $1.08\text{\times}{10}^{5}$ \| $1.08\text{\times}{10}^{5}$ \| $1.08\text{\times}{10}^{5}$ \| $1.08\text{\times}{10}^{6}$ \| $1.08\text{\times}{10}^{6}$ \| $1.08\text{\times}{10}^{6}$ \| $1.08\text{\times}{10}^{6}$ \| $1.08\text{\times}{10}^{6}$ SNR \| 122 \| Saturated \| Saturated \| 172 \| 944 \| 422 \| 133 \| 77 \| 42 Projected performance: Table 2 shows the SunCET SNR as a function of distance from the sun, based on the parameters shown in Table 3. Conservative radiance estimates come from GOES/SUVI 195 Å images of a CME that was tracked all the way to the edge of the SUVI 1.7 $R_{\odot}$ FOV (Seaton and Darnel, 2018). For the solar disk, the effective exposure is the median of three 0.025-second images; for 1.05–4 $R_{\odot}$, it is the median of ten 1-second exposures. This removes energetic particle tracks and, for the long exposure, increases the full-well saturation limit of the detector by a factor of 10. These conservative estimates show that SunCET CME measurements would have an excellent SNR of 42 even out at 4 $R_{\odot}$. Table 3: SunCET instrument parameters needed to calculate SNR. [HTML]1A73C9 Instrument parameter \| Value \| Description ---\|---\|--- Wavelength \| 170–200 Å \| Broadband response defined by mirror coating \| Aperture --- size 44.9 cm2 \| \| 9.6 cm diameter truncated on two sides --- to a height of 7.62 cm and a 4.8 cm diameter secondary mirror obscuring its center \| Weighted factor --- for broadband 6.88 \| \| 7 emissions in the bandpass weighted by their --- quiet-Sun intensity to the 195 Å emission line (see Table 1) Pixel size \| 7 $\upmu$m $\times$ 7 $\upmu$m \| e2v CIS115 datasheet and confirmed in house Pixel array \| 1500 $\times$ 1500 \| Full array is 1504 $\times$ 2000; $\sim$5 rows dedicated to dark FOV \| 4 $R_{\odot}$ \| Design FOV (requirement is 3.5 $R_{\odot}$) Plate scale \| 4.8/pixel \| \| From pixel size, number of pixels, and FOV; --- Note that 2$\times$2 binning will be applied, resulting in 9.6/resolution-element \| Optics --- throughput 0.06 \| \| 2 mirrors with B4C/Mo/Al coatings (0.35 each), --- entrance Al/C filter (0.6) with 5 lpi filter mesh (0.98), Al secondary/pinhole filter (0.85) Quantum yield \| 18.3 e-/ph \| Average over 170–200 Å bandpass Dark noise \| $<$0.08 e-/pixel/sec \| At -10°C, from LASP lab tests Readout noise \| 5 e-/pixel \| From LASP lab tests Fano noise \| 1.3 e-/pixel \| Fano factor of 0.1 for Si Max read rate \| \| 0.1 sec (full frame) --- 0.025 (up to 500 rows) In SunCET, 500 rows corresponds to 0–1.33 $R_{\odot}$ Few observations of the extended corona above $\sim$2 $R_{\odot}$ have been made in the EUV, but among these there is clear evidence that the CME signal will be detectable (Tadikonda et al. 2019, Figure 10). At about 3 $R_{\odot}$, noise in SUVI becomes comparable to solar signal. SunCET, however, is optimized for this large FOV. SunCET has a larger primary mirror geometric area (3.5$\times$), broadband wavelength response (6.88$\times$), and larger pixel solid angle (16$\times$) for a total 385$\times$ boost in signal compared to SUVI. Furthermore, the SunCET mirrors will be polished to highest degree possible, up to 3 times the smoothness of SUVI’s, to minimize scattered light. Figure 10: Composite of GOES/SUVI 195 Å off-point images that shows solar structure out to 3 $R_{\odot}$ — even without a bright CME — before straylight in the instrument becomes comparable with the coronal signal. Adapted from Tadikonda et al. (2019). ## 4 Spectrograph Design The SunCET irradiance spectrograph channel is a high-heritage off-Rowland circle design based on the SDO/EVE Multiple EUV Grating Spectrographs A2 (MEGS-A2) channel (Crotser et al., 2007). It provides the full-Sun solar irradiance from 170-340 Å at 1 Å spectral resolution. This EUV range is important for overlapping with the SunCET imager EUV bands for calibration purposes and provides additional science capability. It observes Fe IX through Fe XVI emission lines that often experience coronal dimming during CMEs (Woods et al., 2011; Mason et al., 2014, 2016, 2019). This allows for halo CME kinematics to be tracked even if SunCET is not deployed on multiple platforms with stereoscopic viewing angles. It also enables study of the energetics powering the CME as a function of time. It shares the vacuum door and detector with the SunCET imager, but has its own optical path including the entrance slit, filters, and grating. These measurements are especially pressing because EVE/MEGS-A experienced a CCD electronics anomaly in 2014 May, preventing the continued solar observations by MEGS-A. While other EVE channels and new GOES EUV Sensor (EUVS) channels are continuing solar EUV observations in the 170-340 Å range, they are only broadband measurements that are not optimized for coronal dimming irradiance observations nor for detailed calibration of solar EUV imagers. ### 4.1 Spectrograph Dynamic Range The solar irradiance values, as measured from SDO/EVE (Woods et al., 2012), from 170-340 Å range from ${10}^{-6}$–${10}^{-2}$ W/m2/nm due to variations in the peaks of the emission line in this range, the reduced irradiance values between the strong emission lines, as well as solar activity including solar minimum times and during the largest solar flares; therefore, the required dynamic range of the spectrograph is $1\text{\times}{10}^{4}$. Projected performance: The $8\text{\times}{10}^{6}$ dynamic range discussed in Section 3.1 is more than two orders of magnitude better than needed for the spectrograph. ### 4.2 Spectrograph Spectral Range and Resolution The SunCET spectrograph requires a spectral range between 170-340 Å and 1 Å spectral resolution. The entrance to the spectrograph is a 3 $\times$ 0.028 mm in order to maximize the slit image height (cross-dispersion direction) on the allotted 500 pixel height of the detector to maximize the SNR, while the width is optimized to meet the 1 Å spectral resolution requirement — it is this slit width and the grating ruling that limits the spectral resolution. The grating ruling, distance and curvature are all optimized in order to meet the spectral range and resolution as well. The optical path after being dispersed from the grating goes through the hole in the secondary imager mirror and onto the common detector. The grating is a Type-I concave imaging grating in order to image the slit onto the detector. There is an Al/C entrance filter mounted to the entrance slit in order to limit the spectral bandpass close to the required range, and an additional Al filter prior for additional bandpass rejection at the entrance to the imager optical cavity as well as to reduce any stray light or pinholes that may develop in the first filter. Given the 1500 allotted pixels in the dispersion range, this gives a plate- scale resolution of approximately 0.11 Å per pixel; therefore the spectrograph will oversample the spectral resolution by about a factor of 9$\times$, or 4.5$\times$ with the 2$\times$2 pixel binning. This allows for fits to spectral lines to be performed and allow for Doppler shift measurements of emission lines and plasma velocity flows during flares to be calculated (Chamberlin, 2016; Hudson et al., 2011) Projected performance: SunCET provides 1 Å spectral resolution across the fully observed 170-340 Å spectral range. ### 4.3 Spectrograph Signal to Noise Ratio (SNR) The SunCET spectrograph also requires a SNR of 10 or better as discussed in 3.7. This is achieved by using a long-slit and minimal optical elements, along with the high QE detector. The slit was also sized, and filter thickness optimized, to maximize the SNR without while conservatively not saturating or even go beyond the linear full well capacity of the the CMOS sensor. Even with a very large factor of 10 increase (Chamberlin et al., 2008, 2018) during flares for these lines given in Table 4, they will still be almost another factor of 2 below the full-well of this sensor. Table 4: The SunCET spectrograph SNRs for various strong emission lines. Irradiances are from SDO/EVE (Woods et al., 2011). SNR at all heights is above the level that ISO 12232 defines as “excellent”. [HTML]1A73C9 Wavelength (Å) \| 171 \| 193.5 \| 195 \| 304 \| 335 ---\|---\|---\|---\|---\|--- \| Irradiance --- [W/m2/sr] $6.7\text{\times}{10}^{-4}$ \| $4.5\text{\times}{10}^{-4}$ \| $6.3\text{\times}{10}^{-4}$ \| $1.0\text{\times}{10}^{-3}$ \| $1.0\text{\times}{10}^{-4}$ Integration [s] \| 10 \| 10 \| 10 \| 10 \| 10 Counts/Pixel \| 737 \| 495 \| 693 \| 1100 \| 110 SNR \| 272 \| 237 \| 282 \| 444 \| 145 Projected performance: Table 4 shows the SunCET spectrograph SNR for five strong emission lines, based on the parameters shown in Table 5. These estimates show that SunCET solar spectral irradiance measurements would have an excellent SNR of better than 100. Table 5: SunCET spectrograph instrument parameters needed to calculate SNR. [HTML]1A73C9 Instrument parameter \| Value \| Description ---\|---\|--- Wavelength \| 170–340 Å \| \| Contains various strong emission lines, --- including some that show coronal dimming. Defined by grating equation. \| Aperture --- size 0.0098 cm2 \| 3.0 mm tall $\times$ 28 $\upmu$m wide \| Number of Pixels --- per emission line 2000 \| \| 500 pixels tall $\times$ 4 pixels wide --- (defined by slit) Pixel size \| 7 $\upmu$m $\times$ 7 $\upmu$m \| Teledyne e2v CIS115 datasheet and confirmed in house Pixel allocation \| 500 $\times$ 1500 \| Full array is 1504 $\times$ 2000; $\sim$5 rows dedicated to dark FOV \| Full Sun \| Solar Irradiance, image the slit Plate scale \| 0.011 nm \| \| From pixel size, number of pixels, wavelength range; --- Note: oversampling spectral resolution of 0.1nm \| Optics --- throughput 0.0122 \| \| Grating Efficiency (0.06), Pt Grating Coating (0.4), --- Al/C entrance filter (0.6), Al secondary/pinhole filter (0.85) Quantum yield \| 18.3 e-/ph \| Average over 170-200 Å bandpass Dark noise \| $<$0.08 e-/pixel/sec \| At -10°C, from LASP lab tests Readout noise \| 5 e-/pixel \| From LASP lab tests Fano noise \| 1.3 e-/pixel \| Fano factor of 0.1 for Si ## 5 Detector Figure 11: The Teledyne e2v CIS115 detector and LASP Compact Camera and Processor (CCAP) that flew successfully on a NASA sounding rocket in 2018; CCAP is now flying on the CSIM CubeSat launched in 2018. SunCET uses a Teledyne e2v CIS115 back-illuminated, back-thinned CMOS sensor (Table 3, Figure 11). This sensor is a 1504$\times$2000 pixel array, where a square area of 1500$\times$1500 pixels will be dedicated to the image while the remaining 500$\times$1500 pixels will record the spectrally dispersed slit image from the irradiance spectrograph. Using a single detector to record data from two technically different but scientifically complementary channels significantly reduces the technical resources needed while maximizing science potential. In 2017, LASP developed custom electronics for readout of this sensor that enables independent exposure control per row. A per-pixel readout is now being developed. LASP’s “Compact Camera and Processor” (CCAP; Figure 11) system with this detector was successfully flown in 2018 on the NASA 36.336 sounding rocket (PI: T. Woods, U. of Colorado/LASP) and more recently in January 2020 on the NASA 36.356 sounding rocket (PI: S. Bailey, Virginia Tech). CCAP includes a Xilinx Kintex-7 FPGA with an embedded 32-bit processor and dedicated image compression core. ## 6 Instrument Requirements on Spacecraft The instruments described above place requirements on the performance and capabilities of whatever spacecraft hosts them. They are primarily driven by the imager. Pointing accuracy must be better than 30with stability better than 30 RMS over 23 seconds and knowledge better than 10. This ensures that the center of the sun stays in the center of portion of the detector dedicated to the imager and does not drift significantly during or between integrations. This pointing performance is achievable even on CubeSat platforms as demonstrated by the Miniature X-ray Solar Spectrometer (MinXSS), Arcsecond Space Telescope Enabling Research in Astrophysics (ASTERIA), Compact Spectral Irradiance Monitor (CSIM), and others (Mason et al., 2017; Pong, 2018). Prime science data generation is heavily dependent on CME occurrence rates, but downlink schemes can easily be designed for flexibility and the “poorest” CMEs can be ignored if there are bandwidth limitations. For CME occurrence rates at the middle of the rising phase of the solar cycle, SunCET generates $\sim$28 MB/day for the imager, and $\sim$65 MB/day of data for the spectrograph. These data are compressed using a lossless JPEG-LS scheme. ## 7 Conclusions The SunCET instrument fills a crucial, historically under observed region of the Sun — the middle corona — precisely the region where CMEs experience the majority of their acceleration. This region is inherently very difficult to observe because of the extreme intensity dynamic range between the bright solar disk and the dim corona. SunCET introduces a new technology that avoids the limitations of previous instruments. By developing a detector that can vary exposure time across its surface, we can simultaneously observe the disk without saturating and the dim middle corona; allowing us to track CMEs from their initiation all the way through their primary acceleration phase. Moreover, we can image spectra on the same detector with their own, independent integration time. Figure 12: Tracking a very fast CME in GOES/SUVI 195 Å base-difference images. The CME quickly extended beyond the FOV of SUVI. SunCET’s FOV (light blue shading) is more than twice as large. Adapted from Veronig et al. (2018). There is a large body of knowledge for tracking CMEs in coronagraphs and EUV imagers (Sarkar et al., 2019; O’Hara et al., 2019; Veronig et al., 2018; Byrne et al., 2014; Mierla et al., 2013; Bein et al., 2011; Gopalswamy et al., 2009; Vršnak et al., 2007). SunCET data processing will employ the techniques already developed for other observatories but improve the results because of its wider FOV (e.g., Veronig et al. 2018; Figure 12) and that it does not require the serendipitous alignment between instrument off-point campaigns and CME occurrence (e.g., O’Hara et al. 2019). Below we summarize: 1. 1. The majority of CME acceleration occurs in a historical observational gap: the middle corona 2. 2. Observations of full CME acceleration profiles provide tight constraints on models and thus our physical understanding of how the magnetically-dominated corona influences CME kinematics 3. 3. SunCET provides these observations, overcoming the limits of traditional technologies with a novel simultaneous high-dynamic-range detector 4. 4. SunCET is compact and thus suitable for CubeSat missions or an instrument on a larger spacecraft SunCET is presently in a NASA-funded, competitive Phase A as a 6U CubeSat and has also been proposed to NASA as an instrument onboard a 184 kg Mission of Opportunity. ## 8 Acknowledgements J.P.M thanks the numerous people who contributed to the development of the SunCET concept design and the reviewers for their commments that made this paper stronger. A.M.V. and K.D. acknowledge the Austrian Space Applications Programme of the Austrian Research Promotion Agency FFG (ASAP-11 4900217 CORDIM and ASAP-14 865972 SSCME, BMVIT). ## References * Aime et al. (2019) Aime, C., C. Theys, R. Rougeot, and H. Lantéri, 2019. 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